i
Table of Contents Chapter 1. General introduction. 1.1. What is Thermonuclear Fusion? ..................................................................... 1-3 1.2. Lawson criterium............................................................................................. 1-5 1.3. Confinement schemes...................................................................................... 1-6 1.4. Why fusion?..................................................................................................... 1-7 1.5. Safety and environmental aspects. .................................................................. 1-8 1.6. Present state of fusion research. ...................................................................... 1-9
Chapter 2A. Context of the thesis. 2A.1. The Tokamak..................................................................................................2-1 2A.2. Flows in the plasma edge. ..............................................................................2-3 2A.2.1. Definition of the plasma edge............................................................2-3 2A.2.2. Importance of controlling the plasma edge. ......................................2-5 2A.2.3. Importance of flows in controlling the plasma edge.........................2-6 2A.2.4. Requirement of a new method for the poloidal flow measurement. 2-7
Chapter 2B. Plasma theory. 2B.1. Fluid equations..............................................................................................2-10 2B.2. System of reference. .....................................................................................2-12
Chapter 3. 1D fluid model for the determination of the parallel and perpendicular flow by an inclined Mach probe. 3.1. Introduction. ....................................................................................................3-1 3.2. Langmuir probes..............................................................................................3-3 3.2.1. Single unmagnetized probes.................................................................3-3 3.2.2. Double unmagnetized probe.................................................................3-7 3.2.3. The effects of a magnetic field. ..........................................................3-11 3.3. Current collected by a plate at an oblique angle to the magnetic field in the presence of a parallel and perpendicular flow. .............................................3-15 3.3.1. 1D fluid model. ...................................................................................3-15 3.3.2. Solutions..............................................................................................3-22 3.3.2.1. Spatial variation of the presheath.........................................3-23
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3.3.2.2. Application of the model. ....................................................3-28 3.3.3. Limitation of the model. .....................................................................3-34 3.3.4. Role of the cross-field transport. ........................................................3-36 3.4. Comparison of the 1D fluid model with existing models. ...........................3-40 3.4.1. The MacLatchy model........................................................................3-40 3.4.2. PIC-code. ............................................................................................3-44 3.4.3. “Rotating sandwich probe” model. ....................................................3-47 3.5. Comment on the validity of the fluid equations...........................................3-49 3.6. Conclusions. ..................................................................................................3-50
Chapter 4. Database of Mach probe edge parameters and validation of the 1D fluid model. 4.1. Introduction. ....................................................................................................4-1 4.2. Experimental set-up for the edge plasma. ......................................................4-2 4.2.1. Polarization set-up. ...............................................................................4-2 4.2.2. Mach probe. ..........................................................................................4-3 4.2.2.1. Probe design. ..........................................................................4-4 4.2.2.2. Measuring system. .................................................................4-6 4.2.2.3. Connection length. .................................................................4-7 4.2.2.4. Heat load. ...............................................................................4-8 4.2.3. Atomic beams. ......................................................................................4-9 4.3. Plasma conditions. ..........................................................................................4-9 4.4. Analysis of the probe data.............................................................................4-11 4.5. Database of Mach probe data........................................................................4-15 4.6. Derivation of the perpendicular Mach number and validation of the 1D fluid model. ............................................................................................................4-17 4.6.1. Electric field profiles. .........................................................................4-19 4.6.2. Derivation of the parallel and perpendicular Mach number..............4-20 4.6.3. Electron density and temperature.......................................................4-27 4.6.4. Validation of the 1D fluid model. ......................................................4-30 4.6.5. Time evolution of the parallel and perpendicular Mach number. .....4-34 4.7. Influence of the radial current on the parallel Mach number.......................4-35 4.8. Poloidal asymmetries. ...................................................................................4-36 4.9. Alternative derivation of the perpendicular velocity....................................4-39 4.10.Conclusions. ...................................................................................................4-43
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Chapter 5 . The physics of flows induced by electrode polarization in the edge plasma of TEXTOR-94. 5.1. Introduction. ....................................................................................................5-1 5.2. Flows for an incompressible plasma...............................................................5-2 5.2.1. Surface averaged velocities. .................................................................5-2 5.2.2. Toroidal momentum equation. .............................................................5-4 5.2.3. Parallel momentum equation. ...............................................................5-6 5.2.4. Link between the radial current and the radial electric field. ..............5-7 5.2.5. Method of solution................................................................................5-8 5.2.6. Dependency of the local toroidal flow on the electric field.................5-9 5.3. Flows for a compressible plasma..................................................................5-11 5.3.1. Notation...............................................................................................5-11 5.3.2. Poloidal variation of the velocities.....................................................5-12 5.3.3. Solution for n%n . ...............................................................................5-13 5.3.4. Results and discussion. .......................................................................5-15 5.4. Poloidal asymmetries. ...................................................................................5-21 5.5. Case IE=0.......................................................................................................5-23 5.6. Conclusions. ..................................................................................................5-25
Chapter 6. General conclusions. 6.1. This thesis........................................................................................................6-1 6.2. Future research. ...............................................................................................6-3
Nederlandse Samenvatting Bibliography
1-1
Chapter 1. General introduction.
Nuclear fusion holds the promise of a clean, safe and almost inexhaustible energy and is candidate for the energy production in the future. Fusion of light particles can only be obtained at extremely high temperatures, at which matter is in the plasma state. Magnetic confinement methods for such high temperature plasmas provide the possibility for thermonuclear fusion to be used as an energy reactor. The tokamak is one of the most promising devices in which plasma is magnetically confined. This confinement is however not perfect and a practical realisation of an economical reactor still awaits the solution of a number of physical and technological challenges. A major issue is that the particle and energy losses of the plasma are mainly set by turbulence. The reduction of this turbulence is therefore an important goal. Radial gradients, or shear, in the plasma flows can be shown to suppress turbulence. The shear in the perpendicular flow is thought to be a critical factor since it is believed to destroy the turbulence and to lead to the creation of a transport barrier (see H-mode [89]). Until now, the experimental as well as the theoretical tools to analyse these flows in the edge plasma were largely missing. The most appropriate diagnostic to measure the flows in the plasma edge with a sufficient spatial resolution is a Mach probe. However, the interpretation of such probes for flow measurements perpendicular to the magnetic field is not well established. In this thesis a model will be derived that describes the parallel as well as the perpendicular transport of particles towards a probe surface. The model delivers a practicable method to be applied on Mach probe data so that
1-2 it enables the determination of the parallel and the perpendicular velocities. In the edge of the TEXTOR-94 tokamak strong flow fields can be set up in electrode polarization or biasing experiments. In these, a radial current is drawn through the edge by means of an electrode and high radial electric fields and sheared flows are created, which have been shown to suppress turbulence [45] [91]. A large database of Mach probe data is obtained in such polarized discharges, the main aim of which is to experimentally verify the 1D model. These experimental findings are then confronted with a 1D fluid simulation of Van Schoor and Cornelis [21] [85], which predicts the velocities and the electric field in dependence of the radial current imposed by the biasing electrode. The originality consists in the completeness of the confrontation of theory and experiment. In earlier work a similar comparison was restricted to Er only. Here both the measurements and the computation of the local flows are added. In summary, the main aims of this thesis are: • To develop a model, which allows to determine when applied to Mach probe data, besides the parallel velocity, also the perpendicular velocity of the plasma flow. • To demonstrate the viability of the model by applying the model on Mach probe data obtained in plasma conditions, in which high poloidal rotation exist. • To identify the physics issues related to the setting up of the flows in the edge plasma in electrode polarised discharges, by comparing the measured flow velocities with the ones predicted by the 1D fluid model of Van Schoor and Cornelis. This first chapter is meant to introduce the outsider in the world of nuclear fusion. Also, the necessity of fusion research in the context of alternative energy sources is discussed. Chapter 2 describes the context of this thesis in more detail: Firstly, the tokamak TEXTOR-94 on which the experiments are performed. Then, the focus is set on the plasma edge, the importance of flows the plasma edge and how they will be studied. Chapter 3 is devoted to probe theory. In the beginning of this chapter the basis probe theory of a single and double probe is reviewed. The effects of a magnetic field on these theories will be outlined. This will be the starting point for the new 1D fluid model, which describes
1-3 the transport of particles parallel and perpendicular towards a probe surface. It will be shown how the model can be applied to Mach probe data. This model is then compared with the existing models. Chapter 4 discusses the experimental part of this thesis. The tools, which are used to control and to diagnose the plasma edge, the biasing set-up and the Mach probe, are described. The Mach probe data are organised in a large database. The structure and application of this database is outlined. The main aim is the application and experimental validation of the 1D fluid model presented in the previous chapter. This validation requires a detailed image of the edge plasma in terms of electric field, parallel and perpendicular flows and the electron density and –temperature. The chapter continues with a further description of the structure of the edge plasma, including the dependency of the flows on the radial current and observed poloidal asymmetries. At the end of the chapter, an alternative method for the determination of the perpendicular velocity is presented. Chapter 5 exploits the measured flow and electric field profiles. The theories of Van Schoor [85]and Cornelis [21] for the flows in an incompressible plasma will be summarized, followed by the inclusion of compressibility of the plasma in the models by Van Schoor [84] [86]. Then the confrontation between theory and experiment will be performed en de results discussed. The chapter continues with an explanation for the observed poloidal asymmetries. Finally the flows are studied in the case that the electrode is withdrawn from the plasma. Chapter 6 summarizes the conclusions of the thesis and gives suggestions for further research. 1.1. What is Thermonuclear Fusion? Apart from the three familiar states of matter (solid, liquid, gas), a fourth exists, which is less known, although it is the most common one in universe. This is the plasma, which can be defined as an ionized gas. The sun is a gigantic plasma, in which energy is produced by fusion reactions. In this process two light nuclei, like Hydrogen isotopes, are fused to heavier nuclei. It appears that the mass of the fusion products is less than the total mass before the reaction. According to Einsteins’s law E = mc 2 ( c = 3.108 m s is the speed of light in vacuum) a mass m may be converted into an amount of energy E and the so-called mass defect is released as energy. This amount of energy is called the
1-4 binding energy. Figure 2.1 represents the binding energy per nucleon as a function of atomic mass number. The decrease of this function for elements with a mass number greater than ≈ 54, corresponding to the element Iron, shows that fission of heavy atoms, like Uranium, in two similar parts leads to the release of energy. The fusion reactions of practical use in laboratory conditions involve the hydrogenic istotopes hydrogen (H or 11H ) deuterium (D or 2 3 4 3 1H ) and tritium (T or 1H ) as well as the helium isotopes 2 He and 2 He and neutrons n: D + T → 24He + n + 17.6 MeV ( = 3.5 MeV nucleon ) D + D → T + 11H + 4.0 MeV D + D → 23He + n + 3.3 MeV D + 23He → 24He + 11H + 18.3 MeV ,
Figure 1.1. Binding energy per nucleon as a function of mass. The energy is released as kinetic energy of the reaction products and shows very clearly the advantage of fusion with respect to fission. The fission of U235 for example, only releases 0.85 MeV/nucleon. The amount of released energy in a fusion reaction is enormous: one liter of water contains an amount of deuterium (about 33mg) that will produce as much energy as burning 260 liters of oil. The annual requirements of an electric power plant of 1000MW are compared for different types of
1-5 fuel in Table 1.1. The table shows that only small amounts of nuclear fuels are needed, relative to traditional fuels like coal and oil. Since deuterium can easily be extracted from water, the fuel for the D-D reaction is cheap and widely accessible. However, the first generation of fusion reactors will utilize the D-T reactions since the requirements for this reaction are less stringent. Tritium is not available in nature (it undergoes a radioactive decay with a half-life of 12.3 years) and has to be provided from Lithium by a breeding process from fusion neutrons: 7 3 Li
+ n → 24He + T + n − 2.5MeV
6 3 Li
+ n → 24He + T + 4.8MeV
Since there is enough retrievable Lithium to support global scale fusion energy for several hundred years at least, fusion energy is an almost inexhaustible enery source. fuel Requirements in tonnes 2100,000 Coal 1500,000 Oil 150 Fission 0.6 Fusion Table 1.1. Comparison of annual fuel requirements for a 1000MW electric power plant. To achieve fusion of the two positively charged nuclei, the repulsive Coulomb force between the particles has to be overcome. Therefore, the particles need a sufficiently high kinetic energy for the fusion process. In a star the kinetic energy is obtained due to gravitational compression of the stellar gas. In laboratory conditions, fusion can be achieved in very high temperature plasmas with a large number of particles. This is called thermonuclear fusion. The D-T reaction has the larger cross-section at energies which are lower than for the other reactions. This is the reason why the first fusion reactors will run on a D-T fuel. To maximise the reaction rate for this process, ion temperatures in excess of 10keV should be attained. This corresponds to a temperature of 100.106K. At such temperatures, matter is in the plasma state, the atoms are stripped from their electrons and the gas becomes a mixture of electrons and positively charged nuclei.
1-6 The handling of matter under these conditions is the challenge of the nuclear fusion research. 1.2. Lawson criterium. The achievement of high temperatures is unfortunately not enough. If the energy loss in the fusion plasma is larger than the energy gain due to fusion reactions the temperature will decrease and the process will die out. Therefore a fusion reaction can only be sustained if the energy is confined sufficiently long, which is usally indicated by the energy confinement time τE. Lawson derived a criterium for a self-sustained fusion reaction by balancing the energy losses and sources in a plasma with a density n. A self-sustained fusion reaction is achieved if the triple product of particle density, temperature and energy confinement time is large enough. The Lawson-criterion for a Deuterium-Tritium reaction is found to be: n.τ E .T > 5 x1021 keV .s.m −3 .
This criterion is known as the ignition criterion. Ignition is reached when the α-particles resulting from the D-T reactions can be kept in the plasma (in the case of magnetic confiment) until they thermalise, given their energy to the plasma. If the fusion producing state of the plasma can be maintained without external heating so that the only power source is the α-particle heating, the plasma is ignited. This can also expressed by the power gain factor Q, the ratio between fusion power and input power Q=Pfus/Pin. Q → ∞ is then the conditon for stationary burning, when no external heating is required. The condition Q = 1 is called break-even. 1.3. Confinement schemes. Two questions arise from the previous section: How can one reach such high temperatures and how does one confine such a hot plasma? Two major methods are followed to realize these goals: inertial and magnetic fusion. Inertial fusion is based on the heating of a frozen
1-7 pellet of D-T by very intense laser-beams to the relevant temperatures. The surface of the pellet will expand and compresses the interior mass to the required densities so that fusion can take place. The present-day technology is however not yet capable to produce the required intensity for the laser-beams. A second possibility is to confine a low-density plasma with a confinement time much longer than inertial fusion with the use of strong magnetic fields. A magnetic field has the property that it causes charged particles to gyrate around the field lines via the Lorentz force. A toroidal geometry is then choosen to prevent end-losses. However, the disadvantage of such a geometry is that the magnetic field has a gradient towards the centre of the configuration leading to charge separation of ions and electrons. This results in an electric field, which creates and additional drift outwards from the confined plasma. To eliminate the drift, a field in the poloidal direction, perpendicular to the first one, is added so that the total field line twists around the torus. Two concepts fulfill these requirements: the tokamak and the stellarator. In the stellarator (Figure 1.2) the toroidal as well as the poloidal field are generated by external magnetic field coils. This results in a very complex geometry, but it enables the machine to operate in steady state. The tokamak (see Figure 2.1 in chapter 2) on the other hand has so far proved to be the most promising solution, but has the disadvantage of a pulsed operation mode. The current is induced by a transformer of which the secondary coil is the plasma loop.
Figure 1.2. The stellarator concept
1-8 1.4. Why fusion? From the previous sections it is clear that energy provided by fusion is a costly long-term project. In this section it is argued that the investment is worthwhile since fusion is one of the few energy sources for the future. It is obvious that the demand of energy will increase in the next years. A rough estimate indicating how dramatic this increase will be, is given by the following considerations [57]: Presently 5.109 people consume on average 2.2kW, so that the total amount of energy consumed currently amounts to about 2.2kW x 5.109 people x 1 year = 11Twyr. The world population will rise to 10 109 billion people. However, the world energy consumption will increase more rapidly than just proportional with the increase of population due the growing economies of the developing countries. The average power consumption per capita will rise to 3kW. So the estimated future energy need will equal about 30Twyr. The question is then how will this huge energy be supplied in the future? Most of the energy is currently produced by burning fossil fuels. At the given consumption the amount, which is still available, would only cover a few decades. The continuous discoveries of new deposits and exploitation techniques might elongate this time considerably. However, other aspects will play an important role: the greenhouse effect, which is not well understood, can be the onset of non-reversible climate changes. Depletion of the world energy resources will inevitably lead to political instabilitities in the world. The Gulf war is perhaps only a small scale illustration. Moreover, fossil fuels are very valuable materials for chemical and pharmaceutical industries. It might be necessary to spare fossil fuel for these purposes. The only long-term alternatives to burning fossil fuels are renewables, fission and fusion. The renewables and their limitations are summarised in Table 1.2. Although renewable energy resources in the world are large and inexhaustible, they have only a limited potential but they will certainly play an important role for local applications. The second option if fission. The fuel in the form of U235 will last only for a few decades. Using breeder technology to transform non-fissile fuel into fissile elements could provide energy for several centuries under the condition that the nuclear-waste problem will be solved. The last option is nuclear fusion. It is the least developed of the three but it
1-9 holds the promise of being a safe, inexhaustible and rather clean energy production method. Method Photovoltaic panels Windmills
Biogas Bioalcohol
Bio-oil Biomass
Investment needed required for 1000MW. About 100km2 in Middle Europe (10% efficiency assumed) 6660 mills of 150kW (with rotor blades of 20m and at the average wind speed prevailing at the North Sea coast) 60 million pigs or 800 millions chickens 6200 km2 of sugar beet 7400 km2 of potatoes 16100 km2 of corn 27200 km2 of wheat 24000 km2 of rapeseed 30000 km2 of wood
Table 1.2. Limitations of renewable energy sources [57].
1.5. Safety and environmental aspects. Since nuclear fusion involves nuclear reactions, radioactivity and nuclear waste, safety and environmental aspects, compared to the fission reactor, are of considerable interest. They are briefly summarized [57]. Inherent and passive safety. First, the amount of fuel available at each instant is sufficient for only a few minutes, in sharp contrast with a fission reactor where fuel for several years of operation is stored in the reactor core. Second, fusion reactions take place at extremely high temperatures and the fusion process is not based on a chain reaction. With any malfunction or incorrect handling the reactions will stop. An uncontrolled burning (e.g. a Chernobyl-type accident) of the fusion fuel is therefore excluded.
1-10
Radioactivity. The basic fuels (D and Li) as well as the direct end product (He) of the fusion reaction are not radioactive. The radioactive Tritium is breeded in-situ and does not have to be transported as in a fission reactor. However, Tritium and the neutrons activate the reactor materials. Studies indicate that the intensity and the duration of the radioactivity of the reactor wall can be minimised by the adequate choice of the materials, such that the radioactive inventory reduces with a factor of 107 in thirty years. In contrast to a fission reactor, the factor for the same time equals 103. 1.6. Present state of fusion research. The early years of tokamak research until now show an enormous increase in performance. The progress in magnetic fusion research in the past three decades is clearly visible in Figure 2.3 where the evolution in the ‘fusion triple product’ is shown. This progress has been obtained by building successively larger devices, more powerful heating systems, and mastering the art of plasma positioning, shaping and profile control and of wall surface treatment and impurity control. Nowadays, the biggest tokamak is JET (Joint European Torus, major radius 3m and minor radius a=1m) in England. Recently, records were obtained with the production of 16.1 MW, resulting in a ratio between fusion power and input power Q=Pfus/Pin≈0.65 [50]. The apparent saturation of the ‘fusion triple product’ in Figure 1.3 reflects the fact that the large experiments are essentially at the limit of their capability and that a new device, such as ITER is required. The aim of this device is to reach plasma conditions for plasma burning over long pulses (Q~10) .
2-1
Chapter 2. Context of the thesis. In this chapter the context of this thesis is described. The first section describes the Tokamak device TEXTOR-94, on which the research is performed. The second section focuses on the region of interest, the edge plasma and more in particular flows in the plasma edge. 2.1. The tokamak. A schematic representation of the tokamak is sketched in Figure 2.1. It has a vacuum chamber in the form of a torus. The torus has a major radius Ro , from its centre to the axis of the vacuum chamber, which has a minor radius a . In this torus the plasma gas is injected and ionised to form a plasma. The plasma is confined by a magnetic field since charged particles gyrate around the field lines. The main component of the magnetic field Bφ is produced by external coils surrounding the vessel. The toroidal shape prevents end-losses. However, due to the curvature of the toroidal magnetic field lines ions and electrons will drift apart yielding charge separation. In order to prevent this, a poloidal magnetic field Bθ is required. This produces field lines, which are helices around a torus. In a tokamak Bθ is produced by a toroidal current I p in the plasma itself. This current is induced by using the plasma as the secondary winding of a transformer. External coils generate additional fields, such as the vertical field ( Bz ) which provides the JxB force (J being the current density) necessary to oppose the hoop force of the plasma and which provides control of the position of the plasma column. Apart from its stabilising effect, the current also heats the plasma due to Ohmic dissipation. Nevertheless, the efficiency of heating
2-2 the plasma decreases with increasing temperature. Hence the plasma can be heated ohmically to a maximum temperature of about 4keV only. Additional heating is required to achieve higher values. Auxiliary heating can be provided by neutral beams, i.e. the injection of very high energetic neutral hydrogen or deuterium atoms, or by electromagnetic waves. The combination of the poloidal and toroidal fields results in helical magnetic field lines. The helicity of the field lines is measured by the safety factor q , defined as the number of toroidal turns a field line must make to complete a full poloidal turn. To obtain macroscopic stability, the safety factor at the boundary must be larger than 1. Therefore, the magnetic field pitch Θ at the edge, which is the ratio of poloidal to toroidal magnetic field
Bθ a ≈ ( qe is the safety factor at Bφ qe R0
the radius r = a ) is always much less than one. A basic property of the magnetic configuration in a tokamak is that the magnetic field lines with the same helicity lie on closed nested surfaces, called magnetic or flux surfaces. The region where the flux surfaces never intersect material boundaries is the core region, whereas more outwards, in the edge region, they do cross other materials (see section 2.2). The transport of particles is mainly along the field lines due to the Lorentz force, and the particles are thus confined on magnetic flux surfaces. This confinement is however not perfect, and transport across the magnetic surfaces is expected to be governed by collisions and the effects of the toroidal geometry. However, the observed radial transport is considerably larger than theory predicts. This phenomenon is called anomalous transport. Turbulent fluctuations are held responsible for the enhanced transport. Understanding this anomalous transport leads to the understanding of the confinement of the plasma in a tokamak.
2-3
a
θ
Ro Figure 2.1 Schematic representation of a tokamak device. The current Ip is induced by the transformer and generates a poloidal magnetic field. A stronger toroidal field is produced by external coils surrounding the vacuum vessel. By adding a vertical magnetic field a stable configuration is established. The research in this thesis is performed on the tokamak TEXTOR-94 (Tokamak Experiment for Technology Oriented Research, last upgrade (pulse length) in 1994). TEXTOR-94 is a medium sized limiter-tokamak (see section 2.2.1) with a circular cross-section, dedicated primarily to the study of plasma-wall interaction. The main parameters are summarized in Table 2.1.
Major radius
Ro
1.75
m
Minor radius
a
0.46
m
Toroidal magnetic field Plasma current Standard operation loop voltage
Bφ
1.3-2.9
T
Ip
200-800 350
kA
Vloop
1
V
Pulse duration
τpulse
10
s
Auxiliary heating
PNBI
2x1.5
MW
2-4 PICRH
2x2.2
MW
Table 2.1. TEXTOR-94 parameters.
2.2. Flows in the plasma edge. This thesis focuses on the plasma flow in the edge region and more in particular on the measurement of the flow by means of a Mach probe. This will be outlined in the following subsections. 2.2.1. Definition of the plasma edge.
In the core region of the plasma, magnetic field surfaces close on themselves. Outside the Last Closed Flux Surface LCFS or separatrix, the plasma particles principally move along ‘open’ magnetic field lines and will mainly hit intersecting material boundaries (often referred to as ‘targets’ or target plates’), such as limiters or divertors (see end of this section). Ions that reach these boundaries are neutralized and then either absorbed into the material boundary or returned into the plasma, forming a cloud of neutral particles concentrated mainly in front of the material interface. As they are not influenced by the magnetic field they behave in a completely different way as the ions or electrons. They can even penetrate a limited distance inside off the separatrix before they are ionized again. This process between plasma and neutral particles is called ‘recycling’. The region where these border effects are important is defined as the plasma edge. An overview of the plasma edge is given in [76]. This thesis will focus on the region of the plasma edge inside the last closed flux surface. The region outside the separatrix is the Scrape-Off Layer (SOL). The SOL refers to the region where plasma particles are ‘scraped’ from the core plasma and directed towards the targets. Besides the phenomenon of recycling, the ion particle flux towards the material boundaries also induces atoms and molecules of the surrounding material in the plasma. Since they can reach a higher ion charge state, they will radiate more strongly. This phenomenon of radiation often occurs just inside the LCFS. Therefore this region is also referred to as the radiating layer.
2-5
Figure 2.2. Limiter and Divertor configuration. Basically two methods are distinguished in tokamaks in order to minimize damaging effects of plasma-wall interactions. A first method involves a material limiter, which is introduced in the plasma (Figure 2.2). The Last Closed Flux Surface is now defined by the position of the limiter tip. Three configurations of limiters can be distinguished: •
• •
toroidal limiter: the contact zone in between plasma and limiter is symmetric in the toroidal co-ordinate, i.e. the limiter extends the long way around the torus. This configuration retains the toroidal symmetry of the device. In Textor-94, such a limiter, called the ALT-II limiter is installed (Figure 2.3). poloidal limiter: one or more rings are introduced in poloidal cross-sections of the tokamak. local limiters: the plasma is limited by a localized material boundary. In this case there is neither toroidal nor poloidal symmetry.
Limiters are usually constructed so that the magnetic field lines intersect with grazing angle, in order to spread out the particle and energy heat fluxes over a larger area than with normal incidence of the field lines. As an alternative to the limiter configurations, the ‘magnetic divertor’ concept is used (Figure 2.2). This method consists of an arrangement of coils such that the magnetic field lines are diverted to special plates relatively far from the core plasma. The last closed flux surface can be seen to intersect itself in a point called the X-point.
2-6
Textor vacuum ALT-II Textor liner
Figure 2.3. The toroidal belt limiter ALT-II installed in TEXTOR-94.
2.2.2. Importance of controlling the plasma edge.
Since the plasma edge forms the boundary of the core plasma, it is clear that it will influence the conditions in the core. However, even more important, is its role in particle and energy exhaust. Indeed, both the produced α-particles and the energy they carry, have to be conveyed through the scrape-off layer. Also, the plasma interacting with the wall will free wall material, impurities which should be kept away from the main plasma, while the geometry and the magnetic configuration should be such that the particle and energy fluxes stay below the technical limits of the used materials of divertor plates or limiters. For these reasons it is desirable to be able to closely control the plasma in the edge region and the SOL. An active method in controlling the boundary plasma is polarization. This method will be used in this thesis and will be discussed in the next section. Passive methods in controlling the plasma boundary consist in wall-conditioning by carbonization and siliconization of the wall and are effective in reducing impurities and recycling. A recent method, developed at TEXTOR-94 is achieved by controlled seeding of the plasma boundary by well chosen impurities as Si and Ne. This results in an increased radiation of the plasma boundary
2-7 acting as a cold radiative mantle around the core plasma and a high increase of the confinement. The so-called Radiative Improved mode [56] is very promising for confinement regimes of future tokamaks. 2.2.3. Importance of flows in controlling the plasma edge.
The plasma edge can be actively controlled by biasing. The method consists in creating electric fields in the plasma edge by external means [90]. Recently, the importance of biasing activities increased due to the recognition of a link between the applied field and H-mode phenomena. The H-mode is a mode of operation of high confinement, observed in tokamaks. It is characterised by a sudden increase of the energy and the particle confinement times, together with an increase of the radial electric field Er and the poloidal velocity vθ in the edge [89]. The present favoured explanation for the confinement improvement is the stabilization of turbulence by the shear in the ExB flow induced by Er. The determination of flow velocities, and especially the poloidal flow, in edge tokamak plasmas has thus become of prime importance [24] [43] [68]. Different methods for creating electric fields have been proposed and investigated; one can distinguish mainly those where an electric field in the poloidal direction or in the radial direction is imposed. An overview is given in [90]. At TEXTOR-94, radial electric fields are created by the electrode polarization set-up (see Figure 2.4) inside of the separatrix. An electrode is therefore introduced in the plasma, beyond the separatrix. A voltage is then applied between the electrode and the ALT-II limiter, thus biasing an inner closed flux surface with respect to the separatrix. An insulating sleeve assures that the bias potential is only imposed on flux surfaces intersecting the electrode head. A radial current is hereby introduced in the edge plasma. High radial electric fields and sheared flows are created, which have been shown to suppress turbulence and thus improve the confinement [45]. The experiments require low densities and temperature. They are ideal to study plasma behaviour and properties such as the radial conductivity, H-mode transition, shear in flows and transport coefficients. They are not seen as a future control mechanism for a reactor. A theoretical model is developed by Van Schoor [85] and Cornelis [21]. In this model the radial current drives the radial electric field and the associated flows. The important damping mechanisms in the model are neutral friction
2-8 and parallel viscosity. The shape and magnitude of the field and the flows results from the balance of driving and damping. Later compressibility was added to the model by Van Schoor [84] [86]. The confrontation allows to check the sensitivity of the flows and electric field to the damping mechanisms and to demonstrate the importance of compressibility on the local flows.
Figure 2.4. Electrode polarization set-up at TEXTOR-94.
2.2.4. Requirement of a new method for the poloidal flow measurement.
From the previous section it is clear that the measurement of the poloidal rotation is thought to be a critical factor in H-mode physics. Several methods exist to measure the poloidal or perpendicular flow. Spectroscopic measurements of the Doppler shift of impurity lines are usually the most practical way of measuring the velocity [6]. The advantage is that this method is a passive one and does not perturb the plasma. However, a major assumption in these measurements is that the impurity ions drift with the same velocity as the bulk plasma and this may not be the case [49]. Even more of a problem in the edge plasma is the condition that the temperature and density are too low to generate impurity lines of sufficient intensity to be used in a Doppler measurements. A final drawback of this method is that the spatial resolution
2-9 depends on the number of lines of sight. Mostly the spatial resolution is therefore limited to a few centimetres, whereas the flow fields in the edge plasma are very narrow, in the order of a centimetre. Diagnostics such as ion beams can be used to measure the plasma potential, which is in turn used to calculate the electric field. These measurements provide an indirect method of calculating the flow velocity, assuming that ExB drifts are dominated by the poloidal flow. The different components of the flow can hereby not identified and a more direct method of measuring the flow velocities in the edge plasma is required. v∞ B Γsd
Γsu Side view
Top view
Figure 2.5. Schematic presentation of a Mach probe. The velocity far from the probe v∞ is derived from the ratio of the up- Γsu and downstream Γsd fluxes. The only appropriate method to measure flow velocities with sufficient spatial resolution in the edge plasma are Mach probes. In the simplest form they are formed by two graphite collectors shadowed from one another and when inserted in the plasma they sense directly the particle fluxes (see Figure 2.5). The radial resolution is only limited by the radial dimension of the collectors. They measure simultaneously the electron temperature, density and particle velocity. Moreover they are capable in measuring the floating potential and hence the radial electric field. The geometry of the Mach probe in Figure 2.5 shows that the collectors can only receive an ion flux from one direction. The velocity is then derived from the ratio of the up- and downstream ion fluxes. This approach is however limited to the velocity parallel to the magnetic field [16] [17] [38] [39] [40] [73]. There has been an
2-10 increasing effort to determine the perpendicular flow by means of an inclined Mach probe [1] [2] [19] [36] [44] [53] [58] [59] [93]. Due to the sometimes ‘intuitive’ nature of the underlying models or since such models are developed for low magnetized plasmas, the interpretation of Mach probes for flow measurements perpendicular to the magnetic field is not yet established. The 1D fluid model in the following chapter describes the parallel and perpendicular transport towards a probe surface and gives a practicable method to apply this model to Mach probe data, such that it enables the determination of both the parallel and perpendicular flow velocities.
2-11
Chapter 2B. Plasma theory. In this subchapter some basic elements of plasma theory are reviewed, which will be used throughout the thesis. The description of the plasma behaviour in terms of individual particles can for many purposes be cumbersome. In order to gain insight in the global behaviour of magnetized plasmas the plasma will be described as a two component fluid of electrons and ions. These equations can be solved in two co-ordinate systems, the ‘parallel’ and ‘toroidal’ co-ordinate system. 2B.1. Fluid equations. The starting point of the derivation of fluid equations is the kinetic equation [8]. By taking the appropriate moments of this equation in velocity space it is possible to derive fluid equations. Each moment equation of order n contains a term involving the moment of order (n+1) so that an infinite sequence of equations is obtained. The problem of each plasma transport theory therefore is to provide a closure relation, an expression for the moments of a certain order in function of basic plasma properties such as density and temperature, independent of the higher order moments, so that the hierarchy can be stopped. Braginskii provided a set of closure relations based on the assumption that the characteristic length of the plasma (the tokamak) L is much longer than the mean free path (λ) of the particles, resulting in a set of transport coefficients for the collisional regime, the momentum equations themselves however are universally valid. The oth order moment in velocity space leads to: ∂ ni ,e + ∇.(ni ,ev i ,e ) = Si ,e , ∂t
(2.1)
2-12 where the subscripts denotes ions and electrons, the fluid velocity and density is represented by v i ,e and ni ,e . Si,e is a particle source. This equation describes the conservation of particles and is known as the continuity equation. The first order moment leads to an equation describing the transport of momentum for ions: ∂ (mi ni v i ) + ∇.(m.ni v i v i ) = −∇pi − ∇.Π i + Z.eni (E + v i xB ) + R i + Smi v i , ∂t
(2.2)
and a simplified version for the transport of momentum of the electrons, based on the fact that the electron mass is much smaller than the ion mass ( me = mi ):
(
)
−∇pe − ene . E + v e xB + Re = 0 ,
(2.3)
The momentum transfer due to collisions between ions and electrons is represented by the friction term R = Re = −Ri and Sm v again is a source i i
term, which can be for example a neutral drag force Fneutrals . Z.e is the ion charge. The pressure term and the viscosity tensor are represented by p and Π . Sometimes the ion and electron momentum equations are summed leading to what is known as the Total Momentum Equation: ∂ ( mi .ni v i ) + ∇. ( mi .ni v i v i ) = −∇p − ∇.Π i + JxB + S mi v i ∂t
,
(2.4)
in which p = pi + pe is the total pressure and J the current density. In all of the equations quasi-neutrality ni = ne (with Z=1) is assumed. The transport coefficients which relate e.g. the viscosity and friction forces with quantities as the temperature and density are derived by Braginskii for the collisional regime. However this regime is not valid over the whole radial range of the tokamak. The collisional regime refers to the cold low-density region in the edge plasma. Once inside the separatrix, the collisionality drops and the mean free path of the particles increases. This regime is called the plateau regime. Further inside, in the hot core plasma the plasma is said to be in the long mean free path regime or banana regime. The distinction can be made in the following way [5] [33] [88]:
2-13 A dimensionless collision frequency is defined for ions and electrons: ν α ∗ = τ α .ν α
(2.5)
with 1 τα = v th,α qRο the ‘untrapped particle transit frequency’. It is the time required by a particle of species α, having the thermal velocity and travelling along the magnetic field lines, to go once around in the poloidal direction. vth,α the thermal speed, q the safety factor, Ro the major radius and ν α the species collision frequency computed as: 3
1
νi 1
νe
= 2.09 * 1013
3 2 T 11 e
= 3.44 * 10
1
Ti 2 2 µ (s ) ni Λ
ne Λ
(2.6)
(s )
All units are S.I. except the temperatures which are in eV, µ is the ion mass expressed in units of the proton mass µ = mi mp and Λ is the Coulomb logarithm. Based on these definitions, the collisionality regimes can be classified as:
• Pfirsch-Schlüter or collisional regime if να ∗ > 1 3
• Plateau regime 1 > ν α ∗ > ε 2 • Banana regime
3
ε 2 > να *
The factor ε = r Ro is the dimensionless minor radius (see section 2B.2). Based on classification, one can derive that the edge plasma of TEXTOR-94 is in the plateau-regime. Expressions for the transport coefficients in this regime will be given in the concerning sections. 2B.2. System of reference. Two systems of reference can be used to describe the motion of particles. One is linked to the local magnetic field line; the other is
2-14 linked to the toroidal axis of the machine. They will be called the parallel and the toroidal system of reference respectively. The radial direction is for both systems the same and is perpendicular to the magnetic surface pointing outwards. The pitch angle of the field, α, is the angle between the parallel and the toroidal direction. The parallel co-ordinate system simplifies the equations since all vector products with the magnetic field disappear.
B
e|| e⊥
er eφ eθ
er r
e⊥
eθ
θ
e|| α
Ro
eφ
Figure 2.6. The toroidal and parallel system of reference. With co-ordinates (r ,θ ,φ ) , for the radial, poloidal and toroidal direction, Figure 2.6. shows that the radius R reads: R = R0 + r cosθ
(2.7)
where r is the distance to the plasma centre and R the distance to the vertical torus axis. The Shafranov shift is hereby neglected since the region of interest is the edge region, where the displacement of the magnetic surfaces is rather unimportant. The parameter ε is the aspect ratio. It represents the dimensionless minor radius and is defined by
2-15 ε = r Ro . The transformation from toroidal (r,θ,φ) to parallel co-
ordinates (r,⊥,||) is given by the projection relations: v ⊥ = cos α .vθ − sinα .vφ v|| = sinα .vθ + cos α .vφ
(2.8)
and for the inverse transformation: vθ = cosα .v ⊥ + sinα .v|| vφ = − sinα .v ⊥ + cos α .v|| .
(2.9)
The magnetic field plays an important role in plasma theory. Some useful expressions for the toroidal and poloidal magnetic field can be obtained: Bφ =
Bo Ro = B cosα R
(2.10)
Bθ = B.sinα = Bφ .tgα = Bφ .Θ
(2.11) where Bo is the field on the magnetic axis and Θ the pitch of the magnetic field. The combination of the toroidal and poloidal fields results in helical magnetic field lines. The helicity of the magnetic field lines is measured by the safety factor q, defined as the number of toroidal turns a field line must make to complete a full poloidal turn: q=
1 1 Bφ r Bφ = ∫ 2π R Bθ Ro Bθ
(2.12) The integral is taken over a closed poloidal contour on the flux surface. The last equality is only valid for a large aspect ratio, i.e. ε = r Ro = 1 . The pitch then becomes:
2-16
Θ=
Bθ ε = . Bφ q
(2.13)
3-1
Chapter 3. 1D fluid model for the determination of the parallel and perpendicular flow by an inclined Mach probe.
3.1. Introduction. The aim of this chapter is to derive a practicable method for the determination of the flow vector, i.e. the components of the ion flow parallel and perpendicular to the magnetic field, using a probe. Flow velocities are usually measured with a Mach probe. In the simplest form a Mach probe consists of two Langmuir probes separated by an insulator. Such probes are referred to by several names including, ‘Janus’, ‘paddle’, ‘split Langmuir’ and ‘directional Langmuir’ probes, indicating a different design and mode of operation. The fundamental concept behind the operation of the Mach probe is that the drift speed of the ions can be inferred from the ratio R of the ion saturation current collected from the upstream Ii ,sat ,up and downstream Ii ,sat ,down directions, defined by the relation: R=
Ii ,sat ,up Ii ,sat ,down
= f (M ,...)
(3.1)
where f is a function, which depends on several quantities including the unperturbed Mach number. The Mach number is defined by
3-2
M=
v , cs
(3.2)
where cs is the ion sound speed and v is the velocity of the ions. The components of the Mach number parallel and perpendicular to the magnetic field are denoted as, M|| and M⊥. The major task is then to model the transport of particles towards the probe surface in order to derive an expression for the function f in equation (3.1). The method for the measurement of the parallel flow component is well established and commonly used [28] [34] [55] [61] [62]. The probe surfaces are then oriented perpendicular to the magnetic field. Equation (3.1) is then of the form: R = exp ( cM||,∞ )
(3.3)
with c a constant depending on the model used. Several models were developed for the determination of the parallel flow velocity in strongly magnetized plasmas; they can be classified as fluid [38] [73] or kinetic [16] models. These models assume that ion flow consist mainly of transport parallel to the magnetic field. To preserve continuity of particles, the parallel flow is balanced by a slow cross-field diffusive transport. Intuitively the simplest way to measure the perpendicular flow is to incline the Mach probe so that the probe surfaces would directly sense the presence of a perpendicular flow. However, cross-field transport consisting of a perpendicular flow is not incorporated in the foregoing models. Therefore, a satisfactory method that is able to determine the perpendicular flow is missing up to now. Usually, asymmetries in ion saturation current, collected by surfaces orientated parallel to the magnetic field are used as an indication of cross-field flow [30] [36] [44] [93]. Some approaches exist in the literature which provide an expression for the current collected by a Mach probe in the presence of perpendicular flow [27] [36] [53]. However, these expressions are based on 'intuition'. Also, a few models exist for weakly magnetised plasmas [1] [59], but they are not directly applicable to the considered case of strongly magnetized plasmas. This Chapter will start in section 3.2 with a review of the basic probe theory, not only for a clear understanding of the complete chapter but also because the Mach probe will be used as a diagnostic for the measurement of electron density and temperature. In section 3.3 a
3-3 model will be derived which deals with the cross-field transport consisting not only of diffusion but also out of a coherent perpendicular flow in the generalized case of an inclined probe surface. This model can be applied to Mach probe data to derive the perpendicular and parallel Mach. It will then be compared in section 3.4 with the existing methods to derive the perpendicular flow.
3.2. Langmuir probes. Irving Langmuir was the first to utilize a probe, which uses an electrical measurement technique (in contrast to surface collector probes). Generally, the probe is connected across a potential source to a reference electrode and the current flowing to the probe is measured as a function of the applied voltage. The resulting relation between the probe current and the probe voltage is called the ‘I-V characteristic’. This relation allows to deduce information concerning potential, electron density and temperature of the undisturbed plasma in the immediate vicinity of the probe. Electrical probes are therefore usually referred to as Langmuir probes. The theory of a single Langmuir probe is reviewed in section 3.2.1. The Mach probe, consisting of two collectors can be viewed as a set of two single Langmuir probe or as a double probe when they are interconnected. Since the latter configuration has advantages for the experiment, a double probe configuration will be used in Chapter 4 to measure flow velocities, electron density and electron temperature. Therefore the double probe theory is reviewed in section 3.2.2. The next two sections are mostly based on the work of Langmuir [51] [52] and Bohm [7]. Further information can be found in [37] [80] [69] [55] [12] [11]. The influence of a magnetic field on the basic probe theory is reviewed in section 3.2.3. This will be the starting point for the 1D fluid model. 3.2.1. Single unmagnetized probes.
A probe is called a single probe when it is so small in comparison with its reference electrode that the current density that the probe causes just in front of the reference-electrode, has an negligible influ-
3-4 ence on the plasma quantities at that place. In a Tokamak, a probe is called a single probe when the role of the reference electrode is taken over by the conducting wall or by the limiter (see Figure 3.1). Figure 3.2 shows the I-V characteristic of a single probe. For simplicity one assumes the plasma to be infinite, homogeneous and quasi-neutral, i.e. ni = ne , in the absence of the probe. The electrons and ions have an isotropic Maxwellian velocity distribution f (v ) with temperature T. Each charged particle hitting the probe is supposed to be absorbed and not to interact with the probe material. A plasma is highly conducting and it may be considered as an equipotential volume at the plasma potential Vp . In absence of an electric field between plasma and probe, the particle flux towards a surface is: Γ=
1 nv 4
with v =
8kT πm
(3.4) Plasma Vacuum vessel Probe
I V V Figure 3.1: Schematic drawing of a single probe. Positive current indicates current flowing out of the probe.
I Ie,sat
(A)
(B)
V (C)
Vfl
Vp
Ii,sat
3-5
Figure 3.2. I-V characteristic for a single probe. where v is the mean particle speed. Here the velocity distribution is given by a half Maxwellian distribution (this is because no particles are traveling away from the surface). Equation (3.4) holds for ions as well as electrons. The distributions stay Maxwellian and the plasma quasineutral almost up to the probe surface. However, if Te and Ti are comparable, the mean ion speed will be much smaller than the mean electron speed due their large difference in mass so that the total electric current of an unperturbed plasma towards a probe is dominated by electrons: 1 1 1 I = - eA ni v i - nev e ≈ eAnev e > 0 4 4 4
(3.5)
The sign convention is such that the probe would emit a net positive current. Since a plasma cannot support potential differences greater than :
1 kTe 2
without violating its quasi-neutrality, decreasing the potential on
the surface causes the plasma to form a sheath in front of the surface to shield the plasma from the potential on the solid surface. The plasma will split up in two distinct regions: a thin sheath in front of the probe surface and a presheath deeper in the plasma. In the sheath quasi-neutrality is violated and the electric field is strong. The thickness of the sheath is of order of a few Debye lengths, the Debye length being defined by:
3-6
λD = ε 0
kTe ne e 2
(3.6)
with εo the dielectric constant in vacuum and e the elementary charge. The sheath width will increase with decreasing probe voltage. Since λD = l i ,e , with li,e the mean free path of the ions and electrons, the sheath is collisionless. The potential drop in the presheath is small, :
1 kTe , and 2
acts to draw ions from the plasma into the sheath. Bohm [7] showed for the first time that the criterion for proper sheath formation demands that the ions have a minimum velocity at the plasma/sheath edge given by: v sh ≥ cs =
kTe mi
(3.7)
This inequality is the Bohm criterion. Since cs is the ion sound speed, equation (3.7) demands at least sonic speed at the sheath edge. The criterion can be understood such that only when the velocity of the ions is sufficiently high, the decrease of the ion density towards the surface caused by the acceleration in the electric field is smaller then the decrease of the electron density governed by the Boltzmann factor, thus the ion density in the entire sheath can indeed remain higher than the electron density. The function of the presheath is then to accelerate the ions to reach the sound speed at the sheath edge. When the probe is at a strong negative potential compared to Vp (region (C)), all ions (and no electrons) will be collected and the total ion current, in this case the ion saturation, current is then given by: Ii ,sat = - eni ,sh cs A = - ensh cs A = − ζ en∞ cs A ,
(3.8)
where ni,sh, nsh and n∞ denotes respectively the ion and electron density at the sheath edge and the electron density of the unperturbed plasma far away from the probe. Equation (3.8) relates thus the electron density at the sheath edge to the density of the unperturbed plasma. It will be shown in section 3.3.2 that the value of ζ depends on the flow of the plasma. For the case of a non-streaming plasma the value of ζ is approximated by ζ = 0.5 . The ion saturation current is independent of the applied probe voltage and gives as a constant contribution to the total
3-7 probe current. Strictly, the foregoing derivation is valid for Ti = 0 . However, the inclusion of a finite ion temperature changes the ion current only via the absolute value of cs. In the case of Ti ≠ 0 : k. (Te + γ Ti )
cs =
(3.9)
mi
with γ the specific heat ratio of the ions. Equation (3.7) remains valid and is now called the generalized Bohm-criterion. If the potential on the surface approaches the plasma potential, the potential barrier for the electrons is lowered (becoming 0 when V=Vp) and the amount of electrons collected increases exponentially (region (B)). From the foregoing consideration, it is concluded that the ideal I-V characteristic, for potentials smaller than the plasma potential, is given by: I
e (V − Vfl ) Isat 1 − exp kTe
=
for V < Vp
(3.10) where I=0 is obtained for V=Vfl. Another way of writing the I-V characteristic is as follows: I
=
AJi ,sat + AJe,sat exp
e (V − Vp ) kTe
(3.11) 1 4
1 4
with Je,sat = enshv = ensh
8kTe me
and
Ji ,sat = −ensh cs
(3.12) whereby the current is written as the product of the surface A and the current density J. For V>Vp, all ions will be reflected and the electron saturation current will flow. Equation (3.10) will thus be replaced by a constant current of the order given by equation (3.12). If one lowers the potential until the total current is zero, then this situation is equivalent with inserting a probe electrically insulated from other parts of the plasma device (“a floating probe”). The probe would rapidly charge up
3-8 negatively until the electrons are repelled and the net electrical current is brought to zero. An expression is derived for the potential difference between the floating and the plasma potential starting from equation (3.11) when I (V =Vfl ) = 0 : −
Ji ,sat Je,sat
=
exp
e (Vfl − Vp ) kTe
(3.13) Together with equation (3.12), this equation is transformed into: Vfl − Vp
=
kTe m T ln 2π e γ i + 1 2e mi Te
(3.14) As discussed above, the floating potential on the surface is the potential at which the surface will adjust itself so as to reduce the electron flux to the surface until it is equal to the ion flux. In the case of cold ions ( Ti = Te ) one has the well-know result Vfl − Vp ≈ − 3 kTe e , that the potential drop for a floating probe is approximately 3kTe. When an object is inserted in the plasma a sheath will thus always be created since the adopted potential is sufficiently lower than the plasma potential. The importance of this result will be revealed in 3.4.2. 3.2.2. Double unmagnetized probe.
There are situations when a single probe may be difficult to use (e.g. a well-defined counter-electrode may be absent). A system which floats at the plasma potential is then desirable. A double probe is such a system and is developed by Johnson and Malter [48]. The principal of a double probe is that the probe current flows between two probes of comparable dimensions and that both probes are inserted in the plasma.
Plasma Probe 2
Probe 1
Id
Vd
Vacuum vessel
3-9
Figure 3.3. Schematic drawing of a double probe In practice, the basic double probe consists of two Langmuir probes interconnected as shown in Figure 3.3. The I-V characteristic of each probe can be written as: I1 =
A1Ji 1,sat + A1Je1,sat exp
e (V1 - Vp1 )
(3.15) I2 =
A2 Ji 2,sat + A2 Je 2,sat exp
kTe
e (V2 - Vp 2 ) kTe
(3.16) V1, Vp1 ,V2 and Vp2 are respectively the voltage and plasma potential of probe 1 and 2. V1 and V2 are related by the equation: Vd = V1 − V2
(3.17) Since the two-probe system is floating the net current drawn from the plasma must be zero. Therefore, the following equation has always to be satisfied: I1 + I2 = 0
(3.18)
3-10
2 1.5
−I2 Vd
−2
Current [10 A]
1 Id
0.5 0 −0.5 −1 −1.5 −2 −150
I
1
−100
−50 Voltage [V]
0
50
Figure 3.4. The double probe viewed as a combination of two single probes. For the further discussion the current-voltage diagram of Figure 3.4 is used. There I1 and –I2 represent the two single probe characteristics immersed in the same plasma (equal density, temperature and plasma potential). If the two probes are connected together with no resistance between them (short circuit), they intersect (I1=-I2) on the I=0 axis at the floating potential. Now, if a potential Vd is introduced between the probes, a current Id will flow in the circuit, Id in probe 1 and –Id in probe 2 since the total current out of the plasma is strictly 0 for all values of Vd. The value of Vd is the voltage difference between the intercepts of the I=Id horizontal line and the two probe characteristics. First, this means that Vd is independent of Vp, i.e. the probe is floating. Secondly, since one curve is saturating (ion saturation) and the other increasing exponentially (electron current) it is the probe on the negative side that will take most of the voltage relative to ground. And finally, the current in the circuit will never exceed the ion saturation current of the single probes. Consider now the case where the plasma is slightly different at the two probes, characterized by a somewhat lower plasma potential in front of probe 2. In Figure 3.4, the current of probe 2 is calculated as the dashed line. The first striking difference is that the intercept of I1 and – I2 is not on the I=0 axis anymore. This means that in this case, when the two probes are shorted together, the current is not zero and that the volt-
3-11 age at which the probes will adjust themselves differs from the single probe floating potentials Vf1 and Vf2. Note that Vf1 and Vf2 are now different because the two probes are in different plasmas. Again, applying a non zero Vd between the probes will change the current in the circuit in the same fashion as above. The double probe characteristic in Figure 3.5 is derived from equations (3.15) and (3.16). It is assumed that Je,sat is the same for the two probes since the electron saturation current is much larger than the ion saturation current and that A1 = A2 = A :
Id
=
e.(Vd −(Vp1 −Vp 2 ) ) Ii 1,sat exp − Ii 2,sat kTe e.(Vd −(Vp1 −Vp 2 ) ) exp +1 kTe
(3.19) In the reasoning above, the only difference in front of the two probes was the plasma potential; the density and temperature were assumed to be the same. Considering the expression for the ion saturation current, equation (3.8) implies that the ion saturation currents of both probes should be the same. These currents can differ if one considers two probes hidden from one another immersed in a flow. This explains why the double probe characteristic in Figure 3.5 is an asymmetric tangent hyperbolic curve.
3-12
2 I
2i,sat
1.5 1
−2
Id [10 A]
0.5 0
−0.5 −1
I
1i,sat
−1.5 −2 −150
−100
−50
0 Vd [V]
50
100
150
Figure 3.5. I-V characteristic of a double probe. The electron temperature Te can now be calculated from the slope of the characteristic: Te =
Ii 1,sat + Ii 2,sat dI 4 d dVd V (I =0 )
(3.20) Once Te has been determined, the density is calculated using equation (3.8) for the ion saturation current. Strictly, this equation is valid for a non-streaming plasma. The derivation of the density from a double probe in a streaming plasma is treated in section 3.3.2. 3.2.3. The effects of a magnetic field.
In the foregoing analysis, it is assumed that no magnetic field is present, so that the particle transport is determined only by the electric field. In this section the effect of a strong magnetic field on the basic probe theory is discussed. Since the case of interest is the collection of the ions, a negative biased probe surface is assumed. A magnetic field is
3-13 defined as “strong” if the gyro-radii of the ions ρi and the electrons ρe are small in comparison with the smallest dimension of the probe projected on the plane perpendicular to the magnetic field. In case of e.g. a long cylindrical probe with its axis perpendicular on the magnetic field, the dimension mentioned is given by the radius a. The condition for a strong magnetic field is: ρi =
mi v th,i eB
(3.21) with v th,i the ion thermal velocity. Two major effects occur in the presence of a strong magnetic field in the case of an inclined surface: - A change of the geometry of the current density lines towards the probe. In the absence of collisions (a collisionless plasma), the gyro centers of the charge carriers are no longer able to move perpendicularly to the magnetic field, because the charges are now bounded by their gyro-motion to the magnetic field lines. As a result one can regard the presheath effectively as one-dimensional and the major task is to find solutions for the presheath. - A new region, the magnetic presheath, is set up in front of the sheath. The function of this region is to turn the trajectory of the particles being parallel in the presheath to being perpendicular to the surface in the sheath (see Figure 3.7). The particles have now to reach the ion sound speed at the Magnetic Presheath Entrance MPSE. The particles are assumed to be collected when entering the magnetic presheath. An overview of the function and the structure of the magnetic presheath will be given at the end of this section. As in the zero field case, a weak electric field exists in the presheath, which is needed to accelerate the ions up to the sound speed at the MPSE. More important is that a so-called flux tube is formed by all field lines intersecting the probe-surface. Theoretically, in the absence of collisions, the probe drains all charge carriers out of its flux tube. In practice however, the flux tube is connected to another object in the plasma, (e.g. limiter) or other plasma sources (e.g. ionization of neutrals) are present in the presheath. As a result of the anisotropy of the plasma in the presence of a magnetic field, the presheath region is thus
3-14 shaped by the flux tube, with a typical scale length denoted as L, and expands until plasma sources or collisional terms become significant and are able to balance the parallel collection flow. The cross-field transport can be due to diffusion and/or a perpendicular flow. The case of diffusion is well investigated and the method of solving the problem of the one-dimensional presheath is to seek solutions satisfying Poisson’s equation and the Boltzmann equation in the parallel direction, treating the perpendicular diffusion equation as a source term in the parallel equations. The schematic image of the presheath is shown in Figure 3.6. This kind of treatment has been applied from a kinetic theory viewpoint by Emmert et al. [23]. Stangeby [71] was the first in applying this kind of treatment to the theory of magnetized probes. However, the general problem in the foregoing models is that classical diffusion by itself cannot account for the observed diffusion [37]. This is a generally observed phenomenon in the present fusion experiments and denoted as anomalous transport. As Hutchinson [38] [39] [40] shows, it is necessary to introduce anomalous viscosity in the equations to model correctly the cross-field diffusive transport. In contrast to the classical viscosity, the anomalous viscosity links momentum transport to particle transport. This will be discussed in the following sections in more detail. Further refinements to the fluid approach are given by the kinetic derivation of Chung [16]. In contrast to fluid theory, a kinetic approach covers also the supersonic regime and gives information on other quantities such as heat flux within the presheath. Since these treatments do not give rise to analytical results, the fluid approach will be used in the following sections. The previous treatments of the presheath only consider crossfield transport due to diffusion. In a Tokamak, flow fields with components perpendicular to the magnetic field may also contribute to the anisotropy of the current collected by the probe. In the presence of high radial electric fields, the perpendicular flow component can even reach magnitudes, which are higher than the parallel one. Theoretical work that takes into account the anisotropy due to strong cross-field flow is lacking. Most of the existing models are trying to give an “intuitive” expression for the collected current rather than investigating how the parallel and the perpendicular flow combine at the MPSE. A new model has therefore to be derived. presheath
sheath
probe
Diffusive source
3-15
Figure 3.6 Schematic illustration of the geometry of the ion collection region. Ions accelerate towards the probe inside the presheath while the exchange of ions between the presheath and the outer plasma acts as a diffusive source in the one-dimensional equations.
Scale lengths caused by the presence of a magnetic field The rest of this section describes the different regions, which exist when an inclined surface is inserted in a plasma in which a strong magnetic field is present. L B E
ni = ne
presheath
v ≥ cs
v y ≥ cs
E
ni > ne
ρi : 4λD
magnetic sheath
surface
Figure 3.7. Schematic overview of the presheath, the magnetic presheath and the sheath. v y is the velocity component perpendicular to the surface. There are thus three spatial regions, differing in characteristics and in scale length: (a) the Debye sheath (scale λD); (b) upstream of it, the magnetic presheath (scale ρi) and (c) upstream of it, the presheath (scale L). This situation is shown in Figure 3.7. The scale lengths satisfy: λD = ρ i = L
3-16
The particles in the Debye sheath are not affected by the magnetic field due to the small dimension of the sheath and the fact that the existing electric field is so strong. The electrons are governed by the Boltzmann factor as before, when V is negative, so that the temperature can still be deduced from the slope of the characteristic. The ion saturation current is estimated by noting that the Bohm formula for the zero magnetic field case corresponds to ions flowing to the probe at approximately the ion sound speed. In a strong field this sonic flow can occur only along the field, not across it. Therefore, it is reasonable to suppose that the Bohm formula still applies except that the effective collection area is not the total probe surface but the projection of the surface in the direction of the magnetic field. So an estimate of the plasma density may be obtained from equation (3.8) using the projected area in place of A. In the sheath, the trajectory of the particles is perpendicular to the magnetic field. To turn the trajectory of being parallel in the presheath and perpendicular in the sheath, a magnetic presheath [13] [15] [22] [64] [65] [77] region arises upstream of the Debye sheath. In the magnetic presheath a significant electric field (perpendicular to the solid surface) is present. The scale size is the ion Larmor radius. The total potential drop in the magnetic presheath and the Debye sheath is about 3 kTe e , the same as for the normal incidence sheath where this drop completely occurs in the Debye sheath. The ion velocity parallel to B at the MPSE, must satisfy the condition: v||,MPSE ≥ cs
(3.22) This criterion, the so-called Bohm-Chodura plasma sheath criterion, for the ion velocity at the MPSE, and the generalized Bohm criterion at the sheath edge, equation (3.7), have to be simultaneously satisfied. The difference between the two criteria is that at the sheath edge, the ion velocity has reached the sound speed perpendicular to the surface. The function of the electric field within the magnetic presheath is to turn the ion flow from being (at least) sonic along B in the presheath to being (at least) sonic perpendicular to the solid surface. Now, the magnetic presheath, just like the presheath, but in contrast with the sheath is a region of quasi-neutrality, in which only ∆ne = ∆ni is permitted, and not ∆ne > ∆ni . One might therefore think that supersonic plasma flow into
3-17 the magnetic presheath is prohibited. Chodura [13], Riemann [64] and Stangeby [77] showed that supersonic flow parallel to the magnetic field at the MPSE is permitted. From the description of the Magnetic presheath, it is clear that the particles can assumed to be collected at the MPSE. 3.3. Current collected by a plate at an oblique angle to the magnetic field in the presence of a parallel and perpendicular flow. 3.3.1. 1D fluid model.
In this section the equations will be derived which describe parallel as well as perpendicular flow towards an inclined surface; and when applied to Mach probe data they will allow the determination of the parallel and the perpendicular flow velocity. The model is based on fluid theory and follows the model of Hutchinson for the determination of the parallel flow [38] [40] [41]. In extension to Hutchinson, the probe surface can be inclined with respect to the magnetic field defined by the rotation angle θ (The rotation angle is written as italic and should not be confused with the poloidal coordinate θ). Furthermore, the radial transport can be due to a diffusive influx and a perpendicular macroscopic flow. This model is reported in [83].
y
⊥
x
probe surface insulator
B
||
θ
r
Figure 3.8. Mach probe geometry showing the parallel (||), perpendicular (⊥) and radial (r) directions; the probe is inclined (angle θ) with respect to the magnetic field direction. The x and y direction are respectively parallel and perpendicular to the probe surface. De surface has a length l and a half radial width a.
3-18
The modeling starts from the full 3D ion continuity equation (2.1), and the parallel projection of the ion momentum equation (2.2). Anomalous shear viscosity η is taken into account; plasma sources (e.g. ionization) or collisional friction are absent. Furthermore parallel viscosity is neglected. The justification for this choice will be given later. The general fluid equations can be written as: Continuity equation ∂ ∂ ∂ ni v|| ) + ( ni v ⊥ ) + ( ni v r ) ( ∂ || ∂⊥ ∂r
=
0
(3.23) Momentum equation ∂ 2v|| ∂ ∂ ∂ mi ni v||2 + mi ni v||v ⊥ ) + mi ni v||v r ) − η 2 ( ( ∂ || ∂⊥ ∂r ∂r
(
)
=
− ∇|| pi + e ni E||
(3.24) mi, ni, pi, v , E|| are the ion mass, ion density, ion pressure, fluid velocity and the parallel electric field in the presheath. The following assumptions are made: The presheath is quasi-neutral. Thus Poisson’s equation is replaced by the quasi-neutrality equation, ni = ne .
(3.25) Since the case of interest is the collection of ions, the probe is assumed to be sufficiently negative so that the majority of the electrons is repelled. The electron density can be taken as given by a Boltzmann factor, ne = n∞ exp ( eV kTe )
(3.26) The subscript ∞ denotes quantities in the unperturbed plasma, far from the collection region, where the potential V = V∞ is taken to be zero. The
3-19 electron temperature Te is expressed in eV and is assumed to be constant. It follows, with (3.25)and (3.26), that the electric field in the presheath can be written as E|| = − ∇|| ( kTe e ) ln ( ni n∞ ) = − (Te eni ) ∇||ni .
(3.27) Equations (3.23) and (3.24) are completed with an ion energy equation [8] of the form: pi = cst ⋅ niγ
⇔ Ti = cst ⋅ niγ −1 ⇔
∇|| pi = γ kTi ∇||ni
.
(3.28) with γ the specific heat ratio of the ions. It is assumed that the ions are isothermal. The error hereby introduced can be neglected as Chung [17] showed via a kinetic approach. This assumption also implies that equation (3.28) is describing an isothermal ion fluid and γ = 1 . However, probe measurements do not generally give Ti, so Ti = cst is only a simplifying assumptions and the precise value of γ is left open. One can avoid this issue by writing the right-hand side of equation (3.28) by combining (3.27) and (3.28) as: −mi cs2∇||ni
(3.29) Since the probe causes a local disturbance of the plasma and acts as a sink for particles, the radial diffusive influx is modelled by: ni v r = − D
∂ ni ∂r
.
(3.30) The diffusion coefficient D is anomalous. In what follows it will become clear that D can be taken as a constant. The density and velocity are now non-dimensionalized in the following way:
3-20 n = ni ni ,∞ , (with ni ,∞ the unperturbed ion density);
(3.31) v|| cs = M|| , v ⊥ cs = M ⊥ ;
(3.32) and non-dimensionalizing co-ordinate transformation:
(
)
(
)
||' = D cs a 2 || , ⊥' = D cs a 2 ⊥ , r ' = r a .
(3.33) The primes on the new co-ordinates are dropped after transformation. It is now clear that D can be assumed to be constant since a change in D will only alter the length of the presheath. The anomalous shear viscosity η is assumed to be related the anomalous diffusion coefficient D by: η = α mi ni D
(3.34) with α an arbitrary constant. Taking the previous assumptions into account, the continuity and parallel momentum equation read as: ∂ ∂ ∂2 n nM|| ) + ( nM⊥ ) − 2 ( ∂ || ∂⊥ ∂r
=
0
(3.35) ∂ 2M|| ∂ ∂ ∂ ∂n α nM||2 + nM M − M − n ( || ⊥ ) ∂ r || ∂ r ∂ ⊥ ∂|| ∂r 2
(
)
=
−
∂n ∂||
(3.36) For the purpose of reducing these equations to one-dimensional forms, M|| times equation (3.35) is subtracted from equation (3.36). The momentum equation can thus be written as: ∂ M|| ∂M|| ∂ M|| ∂ n ∂ 2M|| ∂n + M||n + M⊥ n − − αn ∂ || ∂ || ∂⊥ ∂r ∂r ∂r2
(3.37)
=
0
3-21 Transformation to 1D Equations (3.35) and (3.37) will now be transformed into 1D equations. The cross-field gradients over the radial dimension of the probe are linearised by replacing ∇ rψ → (ψ ∞ − ψ )
and ∇2r ψ → (ψ ∞ − ψ ) ,
(3.38) where ψ is a general variable for density and flow velocity and ψ∞ is the unperturbed quantity. This linearisation is warranted as demonstrated by the 2D-calculation by Hutchinson in [40]. To incorporate the inclination of the probe surface, the equations are then transformed to the (x,y) co-ordinate system parallel and perpendicular to the probe surface (see Figure 3.8). ∂ ∂y ∂ ∂ ∂ ∂y ∂ ∂ sinθ and cosθ = = = =− ∂ || ∂ || ∂ y ∂ y ∂⊥ ∂⊥ ∂y ∂y
(3.39) The transformation equations, equations (3.39), imply the assumption that the quantities do not vary along the probe surface: ∂ =0. ∂x
(3.40) This assumption is identical to the condition of axi-symmetry in a Tokamak when treating the flow towards the limiter and divertor surface [3] [14] [87]. Hutchinson showed via a 2D calculation that the density at the surface is indeed constant [40]. Equation (3.35) and (3.37) are rewritten as: ∂ M|| M⊥ n ∂ M⊥ ∂n 1 ∂2n − − M|| − +n tgθ tgθ ∂y sinθ ∂ r 2 ∂y ∂y
(3.41)
=
0
3-22 ∂ M|| M⊥ ∂M|| ∂ 2M|| ∂n 1 ∂ M|| ∂ n + M||n − − + αn n ∂y ∂y tgθ ∂ y sinθ ∂r ∂r ∂ r 2
=
0
(3.42) Obtaining n
n
∂ M||
out of equations (3.41), i.e.:
∂y
∂ M||
=
∂y
M ∂n 1 ∂2n n ∂ M⊥ − M|| − ⊥ + tgθ ∂y tgθ ∂ y sinθ ∂ r 2
(3.43) and substituting in equation (3.42), gives
∂n ∂y
1 sinθ
=
∂ 2 M|| M ∂ 2 n ∂M|| ∂n n −α n M|| − ⊥ 2 − − 2 θ θ tg r r tg ∂ ∂ ∂ ∂ r r 2M⊥ M|| M ⊥2 M||2 − − 1+ 2 tgθ θ tg ( )
M⊥ ∂ M⊥ M|| − tgθ ∂ y
(3.44) Eliminating
∂ M|| ∂y
=
∂n ∂y
out of equations (3.41) and (3.42)gives:
∂ 2 M|| M ∂ M|| ∂ n 1 ∂ 2 n n ∂ M⊥ − 2 + M|| − ⊥ +α n − 2 θ θ ∂y sinθ ∂r ∂ ∂ tg r r tg r ∂ 2M ⊥ M|| M ⊥2 n M||2 − − 1+ tgθ ( tgθ )2
(3.45) The equations in the parallel direction are then calculated by making the inverse co-ordinate transformation, back to the parallel-perpendicular co-ordinate system:
3-23
∂n ∂ ||
∂ 2 M|| M⊥ ∂ 2 n ∂M|| ∂n M ∂M⊥ α − − − + n M|| − ⊥ M n || tgθ 2 ∂r ∂r tgθ ∂⊥ ∂r2 ∂r 2 M M|| − ⊥ − 1 tgθ
=
(3.46) −
∂M||
=
∂ ||
∂ 2 M|| ∂M⊥ M ⊥ ∂ M|| ∂ n ∂ 2 n α M n + − + + n || tgθ ∂ r ∂ r 2 2 ∂⊥ ∂r ∂ r 2 M n M|| − ⊥ − 1 tgθ
(3.47) The final assumption in the model is that the presence of the probe does not influence the perpendicular flow, thus that v⊥ is constant in the presheath, i.e. ∂ M⊥ = 0. ∂⊥
(3.48) The discussion of equation (3.48) is reserved for the end of this section. Applying equation (3.48) on equations (3.46) and (3.47), the resulting transport equations are: ∂n ∂ ||
=
M⊥ M|| − tgθ (1 − n ) − ( M||,∞ − M|| ) 1 − n (1 − α ) 2 M M|| − ⊥ − 1 tgθ
(3.49) ∂M|| ∂ ||
(3.50)
=
M − (1 − n ) + M|| − ⊥ ( M||,∞ − M|| ) 1 − n (1 − α ) tgθ 2 M⊥ n M|| − − 1 tgθ
3-24 Note that equations (14-15) of [40] are recovered for M⊥ = 0 or θ = 90o . The singularities in the denominators of equations (3.49) and (3.50) determine the parallel velocity at the MPSE. In this case M|| reaches M||,MPSE
=
M⊥ +1 tg θ
(3.51) This boundary condition implies that with the probe oriented at a certain angle and in the presence of a perpendicular flow, the parallel flow has to adapt itself such that equation (3.51) is fulfilled. The sign in equation (3.51) has to be positive so that M|| reaches 1 at the MPSE in the case of θ=90°. Stangeby and Chankin [10] anticipated equation (3.51) on the basis of "intuition". Hutchinson [42] derived it via a Galilean transformation. In contrast to these former treatments, equation (3.51) follows in a natural way from the transport equations. From equations (3.49) and (3.50), one achieves an equation that describes the presheath density as a function of parallel Mach number: ∂n ∂ M||
=
M⊥ M|| − tgθ (1 − n ) − ( M||,∞ − M|| ) n M − (1 − n ) + M|| − ⊥ ( M||,∞ − M|| ) tgθ
.
(3.52) whereby the value α = 1 is used. This choice will be justified in section 3.3.4. In the rest of this section the important assumption, equation (3.48) will be commented. The latter expresses that the perpendicular flow is constant in the presheath. So in the presheath, there’s no force acting on the particles in the perpendicular direction, which should alter the velocity in the perpendicular direction. However, if the perpendicular flow is caused by an Er xB drift, then the probe might have an influence on the perpendicular flow in the immediate vicinity of the probe. The radial electric field is the gradient of a radial potential profile. Inserting a probe would generate a short-circuit for the radial potential profile or, at least, would disturb the potential profile in the immediate vicinity of the probe surface and therefore also the perpendicular
3-25 flow. One has therefore to prove that the probe does not influence the potential profile. Consider therefore the situation, which is shown in Figure 3.9: the probe surface is inclined in the (||, ⊥)-plane and a cross-section AB-A’B’ is considered. The values at positions A, B and A’, B’ represent respectively the unperturbed quantities (potential, density and parallel velocity) at infinity and the perturbed ones at the MPSE. Projecting the crosssection AB-A’B’ in the (||,r)-plane results in the situation of Figure 3.9b. There the potential profile at infinity is defined by the potentials φA and φB whereby φA>φB is assumed. It has to be demonstrated that ∇φ AB = ∇φ A ' B ' or in other words, that the potential drop in the presheath for both paths AA' and BB' is the same, i.e. ∇φAA ' = ∇φBB ' . The sheath density, nsh, is related with n∞ via the electron momentum equation: −∇|| p + ( −e )E||n = 0
(3.53) assuming that Te is constant in the presheath results then in: −Te ∇||ne − eE||ne = 0
(3.54) Finally, equation (3.53) reads: ∇|| ( −Te ln ( n ) + eφ ) = 0 .
(3.55) Since Hutchinson's 2D calculation [40] has shown that the density at the sheath edge is constant along the probe-surface, one derives ∇φ AA ' = ∇φBB ' and also ∇φ AB = ∇φ A ' B ' . Therefore, it can indeed be assumed that the probe does not influence the perpendicular flow. This also means that only the parallel electric field, existing in the presheath, exerts a force on the particles in the direction of the probe surface. A few remarks should be taken into account. Despite the existence of strong radial density gradients in the presheath, the density seems to be constant along the probe surface. Further, it must be stressed that the situation addressed here is simplified. The density at infinity is assumed
3-26 to be constant. The case where strong density gradients exist in the radial direction is too complex. r
⊥ ||
Er
||
φA
AB
>
φB
E|| = −∇||φ
A'B'
probe
φA '
presheath
φB '
Figure 3.9a and b. Schematic view of an inclined probe surface respectively in the (||,⊥)- and (||,r)-plane.
3.3.2. Solutions.
In this section, the solutions of equations (3.49), (3.50) and (3.52) will be discussed. The numerical solutions, in section 3.3.2.1 give insight in the spatial variations of the presheath but do not offer practicable equations for the application of the model to experimental Mach probe data. Therefore, an analytical solution will be adopted in section 3.3.2.2. For a clear understanding, the following sign convention is defined. Since the geometry considered is periodic with 180°, the equations are solved for 180o < θ < 0o . The upstream collector is then defined as the collector, which faces the projection of the flow vector in the magnetic field direction. One can identify two cases, 90o ≥ θ > 0o and 180o > θ ≥ 90o which are shown in Figure 3.10a and b, and which will be referred to as case A and B. In case A the sign convention is straightforward: the upstream collector corresponds to positive values of M|| and negative values of M⊥ . In case B one expects the upstream side to correspond to positive values of M|| and positive values of M⊥ . Since the sign of tgθ in equation (3.51) is negative, negative values of M⊥
3-27 have to be taken to correctly account for the fact that in case B the perpendicular flow now is directed towards the downstream collector. In both cases the signs for M|| and M⊥ are reversed for the downstream side. B v⊥
B
⊥ upstream
v||
collector
θ
downstream collector ||
v||
v⊥
⊥
θ
upstream collector
Figure 3.10a and b. Definition of the upstream and downstream collector for respectively case A ( 90o ≥ θ > 0o ) and case B ( 180o > θ ≥ 90o ).
The solutions will be illustrated by examples, in which the value of the parallel flow and perpendicular are chosen as M||,∞ = 0.2 and M ⊥ = 0.4 . The
angles, which represent case A and B, are θ = 60o and
θ = 120o .
3.3.2.1. Spatial variation of the presheath.
The solutions of equation (3.52) describe the normalized density as a function of the parallel Mach number in the presheath. These solutions can be obtained by solving the latter equation or by solving both the spatial equations (3.49) and (3.50). The following procedure is applied: A parallel Mach number M||,∞ and a normalized density value n = 1 characterize the unperturbed plasma, i.e. the plasma at infinite distance from the probe. They are the starting values for M|| and n in the coupled differential equations (3.49) and (3.50). In the presheath the particles are then accelerated towards the sound speed, which is reached at the MPSE. The parallel Mach number will then adopt the value M||,MPSE, which is imposed by the boundary condition, equation (3.51).
downstream
collector
||
3-28 The denominators of equations (3.49) and (3.50) show that indeed the solution of both equations end, when M|| reaches M||,MPSE. Note that the value of M||,MPSE is determined by the choice of M⊥ and θ and that for θ = 90o , M||,MPSE = 1 irrespective of the value of M⊥. Due to the conservation of particles in the presheath, the density will decrease in the presheath. The value, which is obtained in equation (3.50) associated with M||,MPSE is the density at the MPSE nsh. The solution of equation (3.52) is now obtained by taking the evolution of the density n as a function of the parallel Mach number M|| in the presheath. The evolution of M|| and n in the presheath for case A, B and θ = 90o are plotted in Figure 3.11 and Figure 3.12. The non-dimensional parallel distance is chosen such that ||= −∞ defines the unperturbed plasma, ||= 0 the MPSE. It is clear that at infinity n = 1 (Figure 3.11) and M||,∞ = 0.2 (Figure 3.12). The solution of equation (3.52) is shown in Figure 3.13. The parallel Mach number at infinity is herein the value at n = 1 . The curves describe the evolution of M|| as a function of n in the presheath and end at the MPSE, thereby defining the values M||,MPSE and nsh. The interpretation of the upstream curves of these Figures is as follows. If there’s a perpendicular flow present and if the probe is inclined, the parallel flow does not accelerate to the sound speed but to a value given by equation (3.51). This value is the lowest for case A and increases in the cases θ = 90o and B. The reason is that in case A the perpendicular flow is directed towards the upstream collector, in case B towards the downstream collector. Note that in case B, M||,MPSE >1. Since the acceleration is shorter in case A, the presheath length is shorter and the value of nsh is the lowest compared to the other cases. At the downstream side, this order is reversed.
3-29
1
0.8
A; downstream θ = 90°; downstream B; downstream
0.6
nsh
A; upstream θ = 90°; upstream B; upstream
0.4
0.2
0 −5
−4 −3 −2 −1 non−dimensional parallel distance (||)
0
Figure 3.11. Spatial variation of the density in the presheath for the upand downstream side in the cases A,B and θ = 90o . The parallel distance equal to zero corresponds with the MPSE. 1.4 1.2 1
M||
0.8 B; upstream
0.6
θ = 90°; upstream
0.4
A; upstream
0.2
A; downstream θ = 90°; downstream B; downstream
0 −0.2 −5
−4
−3 −2 −1 non−dimensional distance (||)
0
Figure 3.12. Spatial variation of the Mach number in the presheath for the up- and downstream side. The parallel distance equal to zero corresponds with the MPSE.
3-30
1 0.9 0.8
A; upstream
n
0.7
θ = 90°; upstream
0.6 B; upstream
0.5 0.4
θ = 90°; downstream A; downstream
0.3 0.2 −0.2
B; downstream
0
0.2
0.4
0.6 M||
0.8
1
1.2
1.4
Figure 3.13. Solutions for the normalized density as a function of Mach number in the presheath. The value of M|| at n=1 defines the parallel Mach number of the unperturbed plasma. The value for n at the point where the solutions break down defines n at the MPSE nsh.
Applying this procedure now to starting values for M||,∞ given by for a given M⊥ and θ and retaining the associated MPSE density, then a direct relation between nsh and M||,∞ is obtained. Figure 3.14 shows this relation in the cases A, B and θ = 90o . To cover the whole parameter range in addition M⊥ and θ have to be varied. However, not the whole 3D parameter region, −1 < M||,∞ < 1 , −1 < M⊥ < 1 and −1 < M||,∞ < 1
0o < θ < 180o , can hereby
covered due to the limitations of the model. These will be discussed in section 3.3.3. These limitations are the reason why in Figure 3.14, the curves are not shown for the whole region −1 < M||,∞ < 1 . Since equation (3.51) imposes the condition for the ions to reach the sound speed at the MPSE, the ion saturation current, given by equation (3.8), is proportional to the MPSE density and the curves in Figure 3.14 are therefore a relation between the ion saturation current collected by the probe and the parallel Mach number of the plasma for a given M⊥ and θ.
3-31
1
A, upstream side
0.8
θ = 90°
nsh
0.6
0.4
B, downstream side B, upstream side
0.2 A, downstream side
0 −1
−0.5
0 M||,∞
0.5
1
Figure 3.14. Density at the MPSE nsh vs. the unperturbed Mach number M||,∞ for cases A, B and θ = 90o .
For the experiment, the ratio R = nsh,up . nsh,down
nsh,up nsh,down
and the quantity
are important. The curves for the ratio R relate directly
the external Mach number and the current collected by the surface for a given M⊥ and θ. From the quantity nsh,up . nsh,down , the density measured by the probe is calculated. Both quantities are calculated from the values shown in Figure 3.14. The results are plotted in Figure 3.15. The curves for the quantity nsh,up . nsh,down show to be weakly dependent on only M||,∞ . This proves that density measurements using the Bohm formula for the ion saturation current when M||,∞ = 0 , i.e. equation (3.8), will give reasonable results even in the presence of a parallel and perpendicular flow. However, the value for ζ in equation (3.8) is usually approximated by ζ = 0.5 , whereas the model gives a value ζ = 0.35 for M||,∞ = 0 (see Figure 3.14). This discrepancy is explained by the fact that the Bohm criterion is defined for Ti = 0 , while in the case considered Ti ≠ 0 . So, the ion temperature has indirectly an influence via the expression for the ion sound speed. When Ti = Te is chosen, the ion saturation current is given by:
3-32
Ii ,sat = 0.35 x
Te T en∞ A = 0.49 e en∞ A , mi mi
2
(3.56) hereby recovering the factor ζ = 0.5 . 12 10 R (A)
R(θ = 90°)
R
8 6 (n
4
.n
sh,up
)0.5 × 10 (A)
(n
sh,down
)0.5 × 10 (B)
.n
sh,up
sh,down
0.5
2
(nsh,up.nsh,down)
× 10 (θ = 90°)
R (B)
0 0
0.2
0.4
M
0.6
0.8
1
||,∞
Figure 3.15. The ratio R =
nsh,up nsh,down
and nsh,up . nsh,down for cases A, B
and θ = 90o .
3.3.2.2. Application of the model.
The numerical solution in the previous section does not offer a practicable expression to relate the current collected by the probe surface with M||,∞ , M⊥ and θ. Therefore, an analytical solution of equation (3.52) is proposed of the form: n ≈ exp ( M||,∞ −M|| )
(3.57) At the MPSE, equation (3.51) imposes the value for M||:
3-33
M⊥ M⊥ nsh,up,down ≈ exp ± M||,∞ m c − 1 = exp cup,down ( M||,∞ ) ± M||,∞ m − o tgθ tgθ
(3.58) whereby the sign convention is used for the cases A and B. The parameters cup, cdown and co are introduced, so that equation (3.58) can be brought in agreement with the exact solutions in Figure 3.14. In general cup and cdown are functions of M||,∞, M⊥ and θ. However, it is found that cup and cdown are almost independent of M⊥ and θ and weakly dependent on M||,∞. To demonstrate this, first, the dependency of cup and cdown on M||,∞ is calculated for the case θ = 90o . Equation (3.58) reduces to:
(
(
)
)
nsh,up,down = exp cup,down ( M||,∞ ) ± M||,∞ − c0 ,
(3.59) where c0 is the fitting parameter for M||,∞=0:
( (
c0 = ln nsh M||,∞ =0
)) = ln(0.35) = 1.05
cup and cdown are then computed as: cup,down =
ln ( nsh,up,down ) + co ± M||,∞
(3.60) The values for cup and cdown are plotted in Figure 3.16, in which the values of cup and cdown are given by respectively M||,∞ < 0 and M||,∞ > 0 . These values are then used in equation (3.58) to see if the numerical solutions can be reproduced for θ ≠ 90o . The comparison is shown in Figure 3.17 where nsh is plotted as a function of M||,∞ for the cases A, B and θ = 90o . The solid and dashed curves represent the results of the numerical solution (Figure 3.14) and the markers are calculated by equation (3.58) in which for the up- and downstream side respectively cup and cdown is used. It is clear that indeed cup and cdown are almost independent of M⊥ and θ and weakly dependent on M||,∞.
3-34
1.45 1.4 1.35
cup,down
1.3 1.25 1.2
1.15 1.1 1.05 1 0
0.2
0.4
|M|||
0.6
0.8
1
Figure 3.16. cup (solid line) and cdown (dashed line) vs. M||,∞ . 1
0.8
A, upstream side θ = 90°
n
sh
0.6
0.4
B, downstream side B, upstream side
0.2 A, downstream side
0 −1
−0.5
0 M||,∞
0.5
1
Figure 3.17. Density at the MPSE nsh vs. the unperturbed Mach number M||,∞ for cases A, B and θ = 90o .
3-35 Since the ion saturation current is directly related to the sheath edge density, an expression for the ion saturation current collected by the probe surface on the up- and downstream side is obtained from equation (3.58): M⊥ Ii ,sat ,up,down = nsh,up,down n∞ cs A sinθ = exp cup,down ( M||∞ ) ± M||∞ m c e n c A sinθ − o ∞ s tgθ
(3.61) The ratio R of the upstream and downstream saturation currents, which will be used in the experiment, reads: R=
Ii ,sat ,up Ii ,sat ,down
M = exp c(M||,∞ ). M||,∞ − ⊥ . tg θ
(3.62) The parameter c is now related to cup and cdown by the expression: c = ( cup + cdown )
(3.63) The value of c as a function of M||,∞ is plotted in Figure 3.18 and allows to fit equation (3.62) to the numerical solutions for the ratio R in Figure 3.15. The comparison of the ratio R obtained by the numerical solution and the approximated solution is plotted in Figure 3.19. The small discrepancy is due to the fact that the parameter c is assumed to be only function of M||,∞. Note that equation (3.62) is independent of cs and thus of the assumption on the value of Ti. In the experiment, not the ratio R but the value of ln(R) is the interesting parameter since the latter is directly proportional to M||,∞, M⊥ and θ. In Figure 3.20, ln(R) is plotted versus θ. Note that these curves represent directly the boundary condition. The markers indicate the numerical solutions of ln(R). The natural logarithm of equation (3.62) is represented as the dashed curves. The analytical solution can now be used to obtain M⊥ and M||,∞ from experimental Mach probe data. Such a probe measures the upstream and downstream saturation currents. The value of ln(R) measured with the probe surface oriented at θ = 90o , gives directly c.M||,∞ . Since c is a function of M||,∞ , only an unique com-
3-36 bination can provide the value of ln(R). Therefore the value of c and M||,∞ are directly obtained from the value of ln(R) at θ = 90o . At least one further measurement with the probe surfaces oriented at a different angle is required to deduce M⊥. 2.5
c(M||,∞)
2.45
2.4
2.35
2.3
2.25 0
0.2
0.4
0.6
M
0.8
1
||,∞
Figure 3.18. c(M||,∞) as a function of the unperturbed Mach number.
12 10
R (A) R(θ = 90°)
R
8 6 4
0.5 (nsh,up.nsh,down) × 10 (A)
(nsh,up.nsh,down)0.5 × 10 (B)
0.5 (nsh,up.nsh,down) × 10 (θ = 90°)
2 R (B)
0 0
0.2
0.4
M
||,∞
0.6
0.8
1
3-37 Figure 3.19. Ratio R and Ii ,sat ,up . Ii ,sat ,down e n∞ cs A sinθ given by the numerical solution (solid and dashed line) and equations (3.62) and (3.64) (markers) for cases A, B and θ = 90o .
4 3
M =.4 ⊥
ln(nsh,up/nsh,down)
2
M⊥=.8
M⊥=.3
1 M⊥=.1
0
−1 −2 −3 −4 0
30
60
90 θ [°]
120
150
180
Figure 3.20. ln(nsh,up/nsh,down) vs. θ for various values of M||,∞ and M⊥: The markers indicate numerical solutions of equation (6) with ● : M||,∞=0.1; M⊥=0.1 and M⊥=0.3; ✧: M||,∞=0.5; M⊥=0.4 and M⊥=0.8. The dashed curves are curves obtained by equation (3.62).
The unperturbed density at infinity can be obtained from the square root of the product of the up- and down stream ion saturation currents, since this quantity shows only a slight dependence on the flow of the plasma. Using equation (3.61) for the up- and downstream saturation current, this quantity can be expressed as: Ii ,sat ,up . Ii ,sat ,down = nsh,up . nsh,down e n∞ cs A sinθ M⊥ = exp ( cup − cdown ) . M||,∞ − − 2 co e n∞ cs A sinθ tgθ
(3.64)
3-38 The quantity
Ii ,sat ,up . Ii ,sat ,down e n∞ cs A sinθ
is plotted in Figure 3.19 and
is in good agreement with the numerical solution. This demonstrates that in first and good approximation, the density at infinity can be calculated from the quantity Ii ,sat ,up . Ii ,sat ,down in the case of a non-streaming plasma ( M||,∞ = M⊥ = 0 ) . From equation (3.64) then follows: n∞ ≈
Ii ,sat ,up . Ii ,sat ,down exp ( −2 co ) e cs A sinθ
≈
Ii ,sat ,up . Ii ,sat ,down 0.35 e cs A sinθ
(3.65) Note that the density is a function of the sound speed and thus depends on the assumption of the value of Ti. 3.3.3. Limitation of the model.
The limitations of the model will be discussed in this section. They can be classified as model dependent and physics related limitations. Boundary condition of the 1D fluid model. The main restriction of the 1D fluid model is that due to the fluid approach, the velocity at the MPSE is limited to sonic speed. As a result the values of M||,∞, M⊥ and θ , which can be treated by the model are imposed by the boundary condition, equation (3.51). The unperturbed Mach number M||,∞ is accelerated towards the value at the MPSE given by the latter equation. The maximum parallel Mach number of the unperturbed plasma for a given M⊥ and θ is thus: M||,∞ = 1 +
M⊥ tgθ
(3.66) If M||,∞ is higher than imposed by equation (3.66), one would expect the parallel flow to adopt a reversed sign and there would be an outflux of particles from the presheath. In the present model, this is not possible
3-39 due to the assumed modeling of the density gradient in the presheath, equation (3.38): ∂n = 1− n , ∂r
(3.67) which imposes the maximum density in the presheath n=1. This density value corresponds with M||,∞ calculated from equation (3.66). If the unperturbed plasma is characterized by M||,∞ and M⊥ , then from equation (3.66) one can deduce the angle up to which the probe can be rotated such that equation (3.66) is not violated: m M⊥ 1 − M||,∞
θ = arctg
(3.68) where the appropriate sign has to be taken to treat case A and B of the previous section. For example, if the unperturbed plasma has a parallel Mach number M||,∞ = 0.3 and a perpendicular Mach number M⊥ = 0.2 then in case A ( − M⊥ ) the limiting angle attains the value θ = 16o and in case B ( + M⊥ ) θ = 164o . To cover the supersonic region, kinetic modeling needs to be applied. In the case M||,∞ −
M⊥ > 1 , a presheath does not exist tgθ
anymore
since the particles do not accelerate towards the MPSE to reach the ion sound speed. One can compare this situation with a satellite moving at supersonic speed in the ionosphere. The plasma does not experience the presence of the probe since no communication exists between the probe and the plasma via a perturbed region (presheath) [17] [73]. The probe is directly collecting particles at supersonic speed. Therefore, the collected current is expected to be expressed by the product of the unperturbed density and the supersonic speed: Isat
≈
e n∞ v A
with v > cs .
Sheath at grazing incidence angles of the magnetic field.
3-40 For small values of M||,∞ and M⊥ a different limitation can be reached. When the probe surface is at grazing incidence with respect to the magnetic field, i.e. θ → 0 , the sheath physics changes drastically. The sheath can have a different structure compared to the normal Debye sheath. For example, electrons can now be attracted instead of repelled. This modified structure makes an analysis of the collected current very complicated. The hypothesis is that the sheath extends into the plasma such that the probe will collect particles from the parallel and perpendicular directions (see Figure 3.21). The flush mounted probe theories try to model this extended sheath [4] [22] [26]. Although a correct modeling of the sheath and the parallel transport at small angles seems to be very difficult, the situation could be interesting from the point of the collection of the perpendicular flow. In the limit of θ → 0 and in the presence of a high perpendicular flow, one would intuitively expect that the perpendicular flow is directly collected by the probe surface. Under the condition that one has an estimate of the parallel flow contribution, the perpendicular velocity would be directly related to the probe current: I ~ en∞v ⊥ A
(3.69) In the following chapter, this relation will be experimentally investigated. B v⊥
Aeff
θ
Sheath Sheath
Sheath probe
Figure 3.21. Schematic view of the sheath when the probe surface is at grazing incidence angle of the magnetic field.
3-41 A further constraint at small angles is that the fluid approach, which is used in the previous section, requires that the region considered is collisional. In general, the presheath and the magnetic presheath are assumed to be collisional. Only if the presheath becomes of the dimension of the sheath, fluid theory is no longer valid. Since the dimension of the sheath can be estimated by λsh : 4λD , the criterion which defines the smallest angle possible up to which fluid theory is valid is given by: 4λD ≈ sinθ l
(3.70) with l being the probe length. 3.3.4. Role of the cross-field transport.
The cross-field transport in the model is carried by a perpendicular flow and diffusive influx. In this section the role of the different processes is discussed. Cross-field transport due to perpendicular flow. Considering equation (3.61), it appears that for θ = 0o , the ion saturation current is not affected by a perpendicular flow. This may look strange at first sight. Indeed, if the presheath is thought to be a longelongated fluxtube along the field line, then the trajectory of the perpendicular particles is bent towards the probe surface due to the parallel electric field in the presheath (see Figure 3.22). Intuitively, the expectation would be that also a perpendicular flow should affect the saturation currents even when the collectors are perpendicular to the magnetic field [36]. Hutchinson demonstrated that this is not the case. In [41] he calculated the current to a probe surface perpendicular to the magnetic field in the presence of a parallel flow balanced by only considering a cross-field macroscopic flow. The geometry is shown in Figure 3.23. The magnetic field is along the z-axis, x is the cross-field direction. A semi-infinite probe surface is located at x≥0, z=0. A flow is set up with a parallel velocity v z = v z,∞ at infinity where the density reaches n = n∞ and a perpendicular velocity v x , which is assumed constant everywhere. v|| v⊥
E||
presheath Probe surface
3-42
Figure 3.22. Intuitive picture of the presheath. The particles trajectories are bent towards the probe surface due to the parallel electric field in the presheath.
The 2D flow is described by means of the continuity equation and the parallel momentum equation. The solutions are: n = n∞ , M = M∞ ,
for
z M x x ≥ M∞ + 1
(3.71) n = n∞ exp( z M x x − M∞ − 1), M = z M x x - 1 for
z M x x < M∞ + 1.
(3.72) Figure 3.23 illustrates the solutions: The unperturbed plasma at infinity is described by equation (3.71). The flow lines are straight until they reach the perturbed region, presented by the dashed line. From then on, the solution is given by (3.72) and the trajectories of the particles are bent towards the probe surface. The current collected by the plate is thus given by: Ii ,sat = nsh en∞ cs A = exp (-1- M||,∞ ) en∞ cs A .
(3.73) The ratio of the up- to downstream currents reads: R = exp ( cM||,∞ )
(3.74)
3-43 with c = 2 . The current and the ratio R are thus independent of M⊥. Moreover, when M⊥ = 0 , the only solution compatible with the boundary condition is n = n∞ and M|| = M||,∞ = −1 . The cross-field transport is therefore essential to achieve a physically acceptable solution, but does not appear in the final expression of the collected current. z (1 + M∞ ) = x M⊥
Perturbed
x
region.
Probe surface
n=n∞ M=M∞
z vz vx
Flow lines
Figure 3.23. Geometry in the case that the cross-field transport is only due to a macroscopic perpendicular flow. Cross-field transport due to diffusive influx. Consider now the case where the cross-field transport is only due to a diffusive influx, i.e. equation (3.61) for M⊥ = 0 :
(
(
))
Ii ,sat ,up,down = exp cup,down ( M||∞ ) ± M||∞ − 1 en∞ cs A sinθ
,
(3.75) with co ≈ 1 . The up- to downstream current ratio is given by equation (3.62) for M⊥ = 0 : R = exp ( c M||,∞ ) .
(3.76) The value of c lies between 2.3 and 2.45 and is thus close to the value c=2 predicted by the v⊥=cst case of Hutchinson. As discussed in section 3.3.1, a change in cross-field diffusion alters the effective length of the presheath but does not appear in the final expression of the collected
3-44 current or the ratio R. However, indirectly the diffusion coefficient has an influence on the current via the ratio η mi n D = α where α = 1 is chosen and which give rise to the mentioned values of cup,down . Putting α = 0 , which is in an inviscid model, gives c = 0.95 [73]. Thus the v⊥=cst case gives results quite close to those of the viscous diffusive model α = 1 but substantially different from the inviscid α = 0 model. The underlying physical explanation is the following: By assuming α = 1 , the particles transfer also their momentum while diffusing into the presheath (in contrast to classical diffusion). The result for v⊥=cst is now the best described by a viscosity-diffusion ratio α = 1 . In the v⊥=cst case the influx is purely contributable to the local gradient of the density since the local derivative of the velocity is assumed to be zero. The good agreement with the viscid model indicates that there’s sufficient momentum transfer to fulfil the requirements of the boundary condition at the MPSE. The two cases lead to the important conclusion that the current collected by the probe is independent of the nature of the cross-field process. Either in the case of diffusion or perpendicular flow the cross-field transport acts as a source term in the 1D equations. Also, to measure the perpendicular flow the probe surface needs to be inclined since only then the boundary condition at the MPSE reveals the influence of a perpendicular flow. Consider now the cross-field influx to consist of a perpendicular flow and cross-field diffusion. For negligible perpendicular flow, the density required at the MPSE is assured by the radial diffusion and the presheath will extend in the parallel direction. When the perpendicular flow increases, the particle and the related momentum transport are taken over by the flow resulting in an influx governed by the perpendicular flow. Turning the probe alters the geometrical projection of the surface in the parallel direction and thus the collected current but also the velocity requirements for the parallel flow and thus the density required at the MPSE. In contrast to the case where the collectors are perpendicular to the magnetic field, the perpendicular flow does appear in the boundary condition and allows to relate the magnetic presheath density to the magnitude of the perpendicular Mach number. The question remains whether the value α = 1 is the correct one since the value of the viscosity depends strongly on the plasma conditions. Several reports in the literature indicate that α = 1 is indeed the correct value [25] [40] [55] [60]. Note that the so-called ‘visco-Mach’ probe [18] [19] tries to measure simultaneously the value of α and M|| .
3-45 At first sight this would exclude the uncertainty of the value however the applied method is not well established. At the end of this section, two comments are given with respect to the v⊥=cst case, the importance of which will become clear in the following sections. It is obvious that the trajectory of the particles is such that they are bent towards the surface. However, changing the value of Mx changes the detailed flow field, such as the effective flowcollecting surface, but not the current collected by the probe as demonstrated by equation (3.73). Further, the solutions, equations (3.71) and (3.72), are derived for a semi-infinite probe surface. One can consider such a plate as a series of finite probe surfaces separated by infinite small gaps (see Figure 3.24). Each of the surfaces will generate its own flow tube. To reach probe surfaces, which lie in the higher x direction, the flow has to cross more and more flux tubes, but without affecting the current collected by the probe surfaces. This means that the flow towards a probe surface is not influenced by the different flux tubes. A concatenation of probe surface oriented at different angles will not alter this conclusion because it is the boundary condition at the MPSE which determines the current collected by the surface.
x
Infinite small Probe sur-
z vz
flow lines
vx
leff
Figure 3.24. Geometry in the case that the cross-field transport is only due to a macroscopic perpendicular flow whereby the semi-infinite probe surface is considered as a series of finite surfaces separated by infinite small gaps.
3-46
3.4. Comparison of the 1D fluid model with existing models.
Some reports provide an alternative approach in order to account for the perpendicular flow. Although they are not based on first principles a quantitative comparison is possible. In this section, the fluid model is compared with these models. The development of the models goes hand in hand with a specific probe geometry. So the comparison allows also conclusions concerning an alternative probe design. In particular, the viability of the multi-collector geometry will be discussed. 3.4.1. The MacLatchy model.
The so-called ‘intuitive’ MacLatchy algorithm is developed by MacLatchy et al. [53] to derive the perpendicular and parallel flow components from the saturation currents collected with the Gundestrup probe. Such a probe consists of a circle of outward-facing planar electrodes (see Figure 3.25). The MacLatchy algorithm assumes a model for the collection of ions by the individual pins based on the Hutchinson model and fits then the polar array of modelled currents to the experimental values. One of these collecting surfaces is shown in Figure 3.26.
collector
Figure 3.25. The Gundestrup probe.
v //
J⊥
B J //
θ
x
y
3-47
Figure 3.26. A collecting surface of the Gundestrup probe
The current collected by the single pin is given by
[α , β ] AE ( J // sinθ ± J⊥ cosθ ) +K (1 − [α , β ]) J // AE sinθ
Ip =
(3.77) The current is thus thought to be a sum of three different contributions: a parallel, a perpendicular and a diffusion related term. The coefficients α and β in combination with the + and the - sign account for the upstream and downstream side. The constant is given by K = a L , with L the length of the presheath and a the dimension of the probe. As in the Hutchinson theory, the parallel current density at the probe surface can be written as: J|| = [α , β ] en∞ cs
.
(3.78) The expression for the perpendicular current density is given by: J ⊥ = en∞v ⊥ .
(3.79) Equation (3.77) and Figure 3.26 show that the parallel and perpendicular current contributions are thus determined by their geometrical projection. For each of the pins, the current is calculated using equation (3.77). Both J// and J⊥ are varied to minimize the value of chi-squared defined by χ 2 = ∑ ( Im -I p ) . 2
coll
(3.80)
3-48
where Im the measured saturation current and Ip is the calculated current. The sum is made of all collectors excluding those for θ = 0o . A comparison between the fluid model and the MacLatchy algorithm is possible by considering the Gundestrup as a collection of Mach probes oriented at different angles with respect to the magnetic field (see Figure 3.27). The current ratio of the pairs of surfaces is then compared with the value given by equation (3.62). Figure 3.28 shows such a comparison between the fluid model and the MacLatchy algorithm for the case M|| = 0.6 and M ⊥ = 0.4 . For θ = 90o , the two models describe the collection of the parallel flow. Since for this angle, both models have comparable expressions for the parallel Mach number, the current ratio gives the same result. The small discrepancy comes from the diffusion term in equation (3.77). For θ ≠ 90o , the behaviour of the up- to downstream ratio is different. The ratio of the currents given by equation (3.77) predicts a value, which is lower than given by the fluid model. This is because the magnetic presheath edge is not considered as a Mach-surface in the case of the MacLatchy algorithm. The model does not cope with the fact that the perpendicular and the parallel flow combine at the MPSE. The current density due to the perpendicular flow component is just geometrically added to the parallel current density. Since the parallel velocity always has to reach cs at the MPSE, the contribution of the perpendicular current to the total current is smaller, leading one to underestimate the perpendicular velocity in comparison with the fluid model.
θ=30°
v⊥
B
θ=60° ⊥
|| θ
θ=90° B
v||
θ=120° θ=150°
Figure 3.27. The Gundestrup probe can be viewed as an array of double probes oriented at different angles with respect to the magnetic field.
3-49 The collectors interconnected with a dashed line represent schematically a Mach probe. 2.5 M||=0.6; M⊥=0.4 o PIC code ∆ MacLatchy algorithm − Fluid model
ln(Isat,up/Isat,down)
2 1.5 1 0.5 0 −0.5 0
30
60
90
120
150
180
θα[[°]° Figure 3.28. Comparison between the PIC-code, the Maclatchy algorithm and the fluid model for M||=0.6 and M⊥=-0.4 .
The argumentation in the previous paragraph was based on the assumption that the Gundestrup probe was a collection of stand-alone Mach probes. In reality this is not the case and all collectors are simultaneously at a negative, ion attracting potential. Intuitively one expects that each collector creates its own flux tube (see Figure 3.29). The effect which the different flux tubes will have on the perpendicular flow is then not clear. Intuitively the trajectory of the perpendicular flow would deviate, leading one to overestimate the perpendicular flow for the downstream pins. Taking into account the comment at the end of section 3.4.1, the image of interfering flux tubes is incorrect and the Gundestrup probe can indeed be viewed as a collection of double probes oriented at different angles with respect to the magnetic field. v⊥
B
v||
3-50
Figure 3.29. Schematic view of the Gundestrup probe (only the upstream side is shown) whereby each collector creates its own flux tube. Intuitively the trajectory of the perpendicular flow is influenced by the different flux tubes.
3.4.2. PIC-code.
In order to interpret the Gundestrup polar diagrams, J.P. Gunn developed a Particle in Cell (PIC) code [27]. The simulation region (shown in Figure 3.30) is a plane in which lie the magnetic field lines. The two dimensions represent the parallel and perpendicular directions on a magnetic flux surface. The code takes only the guiding centres into account (i.e. Larmor radius small compared to the probe). Quasi-neutrality is guaranteed by replacing ne by ni in the equations. This means that electrons are not included in the simulation. Furthermore Te is assumed as a constant. The parallel electric field is given by a Boltzmann relation, i.e. determined from the ion density gradient at each time step. The probe is positioned at the center of the simulation region. Ions whose trajectories intercept the probe surface are absorbed (For a clear understanding, note that in the PIC simulation the function of the surface is similar to the magnetic presheath. In the following, the term ‘surface’ will denote the point where the particles are absorbed). This is the sink action that leads to the formation of the density gradients and the parallel electric field. Two types of cross-field motion are implemented. When M⊥ = 0 , diffusion, governed as a random walk process, is responsible to provide the radial influx. The second type is a constant drift speed M⊥ = cst imposed on all the ions.
3-51
B
⊥ collector
||
Figure 3.30. PIC code grid populated with ions (grey dots). The geometry of the probe consists of an array of collecting surfaces.
The comparison between the fluid model and the PIC code can be done in a similar way as in the previous subsection, i.e. considering the PIC geometry as a collection of individual double probes located at different angles with respect to the magnetic field. The comparison, plotted in Figure 3.28, shows that the PIC code and the fluid give the same results. This agreement is not surprising since several essential elements of the fluid model are also incorporated into the PIC code. Besides the condition of sound speed at the surface, in the code, each ion is pushed a small random step in the perpendicular direction at every time step. Its velocity before and after the step is kept identical. This is what allows perpendicular transfer of parallel momentum. The average step size dx determines the diffusion and viscosity coefficients. D = dx
2
2 dt
and η = ni mi D . Therefore the condition
∂ M⊥ =0 ∂⊥
and α = 1
are build in the PIC-code and the same results as given by the fluid model could be expected if the PIC geometry was consisting of one single surface. Further, the PIC geometry is ‘Gundestrup’-like in the sense that it consists of an array of collectors. The agreement in Figure 3.28 confirms that the region in front of a multifaceted probe geometry should not be seen as consisting of separated flux tubes but as one perturbed region, i.e. a presheath, shaped by the dimension of the probe (see Figure 3.31). Thus the mechanisms outlined in section 3.3.4 is applicable to the PIC geometry, namely that cross-field diffusion or a perpendicular flow is responsible for particle and momentum transport into the flux tube, balancing hereby the parallel flow. The influence of a per-
3-52 pendicular flow is reflected in a changed boundary condition at the surface and thus in a different density present at the surface. In the fluid model, the amount of particles present at the surface is dictated by the boundary condition, whereas in the PIC code this condition follows from the simulation. The advantage of the fluid model is then that it could be applied to a multifaceted geometry, since the PIC-code does not provide a practicable expression for the collected current. v⊥
B
v||
Figure 3.31. The presheath region in front of a multi-pin geometry according to the PIC-code simulation.
One can argue that the simulated geometry consists of an array of connected conductors, whereas a real probe would have isolators as spacers. To justify the conclusions above one has to answer the more general question what is the difference between a collecting and an insulating surface immersed in the plasma. Even in the case of one single collector, a real probe design consists of a highly conducting collector implemented in an insulating body. So also in this case the question arises: what is the influence of the insulating body on the collection of particles. As discussed in Section 3.2.1 a sheath will be established in front of each surface, which is at a potential a few kTe lower than the plasma potential, regardless of the fact whether the surface is an insulator or a collector. Usually the collector is biased negatively while the insulator is floating but in both cases the condition of sonic speed at the MPSE remains. Therefore only one electric field and associated presheath region exists, shaped by the dimensions of the probe. The sheath in front of the different materials will have a different width. For a collector, insulator or conductor, the amount of charged particles, which is
3-53 located directly on the surface, will vary. Therefore the electron repelling potential drop in the sheath and thus the width of the sheath will be different. A conductor will have a smaller sheath width because the charges are directly removed from the surface. Since the sheath dimension is negligible in comparison with the dimension of the presheath, the sheath will not alter the perpendicular flow upstream of the material. Therefore, the comparison of the fluid model and the PIC simulation and the accompanying conclusions still hold in a realistic situation. 3.4.3. “Rotating sandwich probe” model.
A last approach has been developed by K. Höthker [36] and is related to the sandwich probe design. The so-called sandwich probe (Figure 3.32) is a Mach probe, which can rotate continuously relative to the magnetic field. The procedure behind the operation of the sandwich probe is that the rotation of the probe allows to take probe signals for as many angles as the problem has parameters to be determined. Although the argumentation in [36] does not offer a practicable method to derive the perpendicular flow, it gives an onset of how the perpendicular transport towards a probe surface should be interpreted.
B collector
θ insulator
insulating shaft
Figure 3.32. Rotating sandwich probe.
The approach in [36] is to model the particle trajectories in the presheath. Figure 3.33 shows such a trajectory. The flow is bent towards the surface and results therefore in an effective collecting area, which is enlarged compared to the geometrical one. From Figure 3.33 an expression is derived for the effective collecting area.
3-54
e E|| l sin2 θ Aeff = a l sin (θ + α ) + sinα 2 2 mv ⊥
(3.81) With A = a ⋅ l the geometric area of the collecting plate, θ the angle of rotation relative to the magnetic field direction, α = arctan (v ⊥ v|| ) , v⊥ and v|| the perpendicular and parallel flow speed,
E|| the mean electric field in the presheath and m the mass of the ions. The effective collecting area is thus the sum of two terms. The first term describes the collecting area if the effect of the presheath on the ion orbits is not accounted for. Then Aeff is equal to the geometrical projected one. It becomes maximum for α + θ = 90° . The second term accounts for the effect of the acceleration of the ions in the electric field and it vanishes for α = 0 or θ = 0 . The coefficient of the latter term comprises the parameters to be determined, v⊥ and E||. These parameters can be obtained via an in situ calibration of equation (3.81) on the measured probe currents. However, it is stated in [36] that the simple determination of the effective collecting area in order to use the probe for the determination of the cross field ion flow speed is only a first step. A more realistic treatment of the transport close to the surface, in particular the magnetic presheath, has to be included. This is exactly what the 1D fluid model says: Although the detailed flow field might be such that the effective collecting surface is influenced by the perpendicular flow, it is indeed the boundary condition at the MPSE, which determines the current collected by the probe surface.
3-55
α
α
θ
Figure 3.33. Geometry of the sandwich probe: l’’ defines the geometrical collecting surface, the effective collecting surface is defined by l’+l’’.
3.5. Comment on the validity of the fluid equations.
In this chapter, fluid equations are used. A fluid approach, in contrast to a kinetic approach, gives the benefit to be analytically integrable, leading to compact formulas for ion density and current. The treatment of the perpendicular dynamics by a fluid approach will be justified, provided that the ion gyro radius is much smaller than the probe radius so that the particles can be described by their guiding centres. The necessary condition for the treatment of the parallel dynamics is the self-collisionality of the ions since in section 3.3.1 the parallel viscosity η|| is neglected. The ion gyro radius is calculated with equation (3.21): ρi =
mi v th,i eB
Inserting typical experimental parameters vth,i≈ 6*104m/s, B=2.33T, the ion gyro radius has a value ρi≈5.3*10-4m. Compared to the probe radius a=4*10-3m, one can conclude that the condition for a strong magnetic field is satisfied. The self-collisionality of the presheath requires the ion-ion mean free path to be shorter than the length of the
3-56 flux tube to the probe. In the case of pure diffusive influx, the length of the flux-tube is estimated by: L|| =
cs a 2 D
(3.82) Providing typical parameters D=1m/s2, cs=6*104m/s, equation (3.82) gives a collection length L||~ 3m. The ion mean free path is about 25% of this length. To justify the application of fluid theory the importance of collisions in the presheath has to be investigated. The importance of ion-ion collisions can be derived by comparing the viscid fluid approach with collisionless kinetic treatment [17] [23] [71]. Rather close agreement has been demonstrated between a completely collisionless kinetic model and a fluid model. For example, the particle flux density to a sur1 n∞ cs , where the colli2 sionless theory can be shown to give a value of f (Ti Te ) n∞ cs ,
face is computed by the fluid model to be :
where f (Ti Te ) varies monotically from 0.487 at Ti Te = 0 to 0.798 as Ti Te → ∞ . Thus, the discrepancies appear to be explicable in terms of the ion temperature variation omitted in the fluid approach and the different ion contribution to the acoustic speed. The specific heat ratio γ is set to one, where the kinetic treatment gives a value of 2-2.5. The key issue in both approaches, kinetic and fluidum, is the exchange of parallel momentum between presheath and the outer plasma, not the redistribution in the presheath. Thus ion-ion collisions (or the lack of them) constitute a minor effect. As a result, the viscous terms arising from parallel gradients can be dropped out of the momentum equation. In order for the quasineutral approximation to be satisfied over the presheath requires the sheath thickness to be small. Since the sheath has a thickness typically a few times the Debye length λD, this requires λD a = 1 , a condition which is easily satisfied. 3.6. Conclusions.
In this chapter for the first time a 1D model is derived, which allows to determine the parallel and perpendicular flow velocity by using a Mach probe. For the modelling, the fluid continuity and momentum
3-57 equations are used to describe the transport of particles towards an inclined Mach probe. The cross-field transport is due to diffusive influx and/or a macroscopic flow. The equations are transformed to 1D. From the resulting equations the modified boundary condition for the velocity at the magnetic presheath edge, which was in the past derived on intuition, appears now in a natural way and determines the current collected by the probe surface. The boundary condition defines the MPSE as a Mach surface. Therefore the parallel and perpendicular flows have to combine in an intrinsic way at the MPSE to fulfil the boundary condition. The influence of the perpendicular flow is only reflected on the boundary condition when the probe is inclined. For the case where the surfaces are parallel to the magnetic field, the contribution vanishes. In the presheath, the particles are accelerated towards the magnetic presheath to fulfil the boundary condition. The cross-field transport, due to diffusion and a perpendicular flow acts as source terms in the 1D equations to balance the parallel influx. Therefore a necessary condition to measure the perpendicular flow is the inclination of the probe surfaces, since the perpendicular flow is only then reflected in the boundary at the MPSE. In this process the knowledge of the detailed flow field in the presheath is therefore not important. The MacLatchy algorithm supposes an intuitive expression for the collected current. It is assumed that the current density contributions in the parallel and perpendicular directions are geometrically addable. Therefore the MPSE is not considered as a Mach surface, which leads to an underestimate of the perpendicular flow. The comparison with the PIC code confirms that the fluid model is also applicable to a multi-pin geometry and that the latter does not suffer the problem of interfering flux tubes.
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Chapter 4 . Database of Mach probe edge parameters and validation of the 1D fluid model.
4.1. Introduction. In this chapter the experimental part of this thesis is discussed. From the Mach probe data a large database of edge parameters, including the parallel and perpendicular flows, the radial electric field and the electron density and temperature, is obtained. To our best knowledge, such a complete database of these parameters measured under the same plasma conditions is not found in the literature. The first aim is to validate the 1D fluid model presented in the previous chapter. This validation is reported in [81]. The database further allows to build a consistent image of the flows and radial electric fields induced by biasing in the edge. To demonstrate the viability of the 1D fluid model, a substantial perpendicular flow has to be present in the edge plasma. The Mach probe measurements are therefore performed in electrode biasing discharges since the polarization set-up at TEXTOR-94 offers the unique possibility to induce a perpendicular flow in the edge plasma in a controlled way. In these discharges a radial current is imposed which sets up a radial electric field Er and ensuing high toroidal and poloidal flows. The measurement of this radial electric field can then be compared with the electric field obtained from the radial momentum balance resulting in a qualitative analysis of the computed perpendicular Mach number. This analysis will be performed in two cases: One where the electric field is low and a second where a high electric field exists in the edge plasma. The outline of this chapter is as follows. In section 4.2 the experimental tools consisting of the Mach probe and the polarization set-up are described. How the Mach probe data are analysed to determine the Mach number, the electron density and tem-
4-2
perature and floating potential is explained in section 4.4. Section 4.5 shows how the Mach probe data are organised in the database and in which way the information will be used. Section 4.6 discusses the derivation of the parallel and perpendicular Mach number and the validation of the 1D fluid model. The polarization experiment gives the possibility to draw a radial current in the edge plasma. The change of the parallel Mach number with varying radial current will be investigated in section 4.7. Since the Mach probe can be installed at two poloidal locations, poloidal asymmetries of the measured parameters can be studied. Poloidal asymmetries of the parallel Mach number and the electron density are observed. These findings will be discussed in section 4.8. At the end of this chapter, in section 4.9, an alternative method for the derivation of the perpendicular velocity is presented. The ion saturation current is therefore measured with the probe surfaces parallel to the magnetic field. The hypothesis is that under these conditions the perpendicular flow is directly related to the collected probe current. Although this method has an intuitive basis, it offers a different view in the determination of the perpendicular velocity.
4.2. Experimental set-up for the edge plasma. In this section the tools are described which are used to influence and to diagnose the edge plasma. These tools are the polarization set-up and the Mach probe. Atomic beams are used for comparison of the probe results concerning the electron temperature and density.
4.2.1. Polarization set-up. A schematic view of the edge polarization set-up is shown in Figure 4.1. An electrode is located at the bottom of the vacuum vessel and is inserted with the leading edge 5 cm beyond the ALT-II limiter. The height of the electrode head is 1.5cm. The head is mushroom shaped with a diameter of 10cm. The voltage is applied between the electrode and all eight blades of the ALT-II limiter, which is grounded to the potential of the liner. The current therefore follows the path: power supply, electrode, plasma, ALT-II, power supply. It is experimentally found that the electrode current IE is equally distributed over all blades. It is important to note that no other limiter (e.g. poloidal limiter) but the ALT-II is inserted in the plasma. This to assure that the radial
4-3
current flows mainly towards ALT-II. An insulating boron-nitride sleeve assures that the bias potential is only imposed on flux surfaces intersecting the electrode head.
Figure 4.1. Schematic presentation of the polarization set-up showing a poloidal cross-section of the vacuumvessel. A mushroom shaped electrode is inserted from the bottom beyond the ALT-II limiter. The Mach probe is located in the equatorial outboard plane or in the top plane.
4.2.2. Mach probe. The Mach probe can be installed at two different poloidal locations, i.e. in the equatorial outboard plane and at the top of the machine (see Figure 4.1). In both locations, the probe is mounted on a steel shaft, which is then inserted in a slow manipulator. “Slow” indicates that the probe can only be moved in the radial direction in between discharges. A gate valve separates the vacuum vessel and the manipulator so that the probe can be mounted without disturbing the vacuum in the vessel. An oven is present in the manipulator to bake the probe. To obtain the required vacuum a turbo pump is used. The rotation angle θ is strictly the angle of the probe surfaces with respect to the magnetic field. But due to practical considerations the surfaces are aligned with respect to the toroidal axis rather than the magnetic field axis. This is warranted since the pitch angle at the considered radii is very small (α=2.44°). To change the ro-
4-4
tation angle, the vacuum in the manipulator has to broken and the probe together with the shaft can then be turned to the desired angle. As a result, the rotation angle can only be changed between successive experimental days.
4.2.2.1. Probe design. From the previous chapter it is clear that in order to measure the flow vector, one needs a probe surface which has a well-defined geometrical collecting area, i.e. a flat surface, located at different angles with respect to the magnetic field. A rotating double probe, e.g. a sandwich type probe, seems a good solution. The benefit is that the probe surfaces are continuously rotated with respect to the magnetic field; hereby giving a clear mark at precise the moment when the probe-surfaces are parallel to the magnetic field so that the error introduced by mounting the probe with respect to the toroidal axis could be eliminated. However, the technique required for the rotation implies serious technical complications for the shaft on which the probe is mounted. The design requires sliding contacts for the current wires inside the vacuum vessel, which causes serious limitations on the practicability and maintenance. Moreover, the time resolution of the measurement is limited to the rotation frequency, which is of the order of a few Hertz. The time constants in the target plasma under consideration are of the order of milliseconds. The low rotation frequency would result in a serious limitation of the obtained results. For these reasons, the Mach probe is chosen to be nonrotative. The Mach probe geometry is shown in Figure 4.2 and Figure 4.3. The probe consists of two graphite collectors which are square shaped to have a well defined collecting area and are incorporated in an insulating boron-nitride body. When a probe is exposed to hot plasma, the surface of the boron-nitride can be covered by a thin layer of graphite due to sublimation of the collector’s surface. The complete surface of the insulating body can therefore become electrically conducting. To avoid electrical contact between the collector and the insulating body, some space is left between the two materials. This implies that the total collecting surface is not the surface area of the collector, but the total circular area in which the probe is imbedded as can be seen in Figure 4.2 and Figure 4.3. The radius of this area is 0.4cm, giving a total collecting surface of A=0.5cm2. Considering the ion larmor radius, which has the value ρi≈0.05cm, the condition of a strongly magnetized probe is satisfied. The radial position of the probe will be referred to as the position of the centre of the collectors. Note that the leading edge of the probe is 1.2cm further in the plasma than the probe tip.
4-5
boron-nitride
•
0.8 cm
graphite
•
0.55 cm
3.4 cm
current wires 3.4 cm
~r Figure 4.2 Cross-section of the probe. Two graphite collectors are imbedded in an insulating body of boron-nitride.
9 cm boron-nitride
1.2 cm
0.55 cm
graphite
0.8 cm r
Figure 4.3. Side-view of the Mach probe. The collecting surface is defined by the total gray area.
4-6
4.2.2.2. Measuring system. The Mach probe can be used either to measure the ion saturation current or the floating potential. The measuring system for the Mach probe in the ion saturation current mode is schematically shown in Figure 4.4. The steel shaft and the manipulator are not displayed. The two collectors of the Mach probe are used in a floating double probe circuit. Therefore the current wires of the two collectors of the Mach probe are interconnected via a high power amplifier (Röhrer type). The maximum output voltage and current are V = ±200V and I = ±28 A . This power supply amplifies a triangular signal generated by a function generator. The sweeping frequency of this signal is 200Hz. Both the amplifier and the function generator are outside the bunker. Inside the bunker a box containing the measuring electronics is placed close to the manipulator to prevent noise pickup, which could be induced due to the length of the cables. The electronic circuit consists of two isolation amplifiers, which measure the voltage and the current signal of the probe. The current is obtained by measuring the potential drop over a small resistance. At the same time, the isolation amplifiers serve as a DC-break between the probe potential and the potential of the data-acquisition (CAMAC) modules. The sampling frequency of the acquisition is 25kHz. At the start-up phase of a discharge, high currents can exist in the plasma. To protect the isolation amplifiers from these currents, a trigger pulse is used to switch on the function generator after the start-up phase of the discharge. Note that since all components are separated from the ground, the total system is floating. The Mach probe can also be used to measure the floating potential. Therefore both collectors are connected to an isolation amplifier. Function generator
Trigger puls measuring electronics
probe V
High Power Amplifier (isolated) ~ 220V
I
(liner)
(bunker) Data-acquisition Textor-mass
Figure 4.4. Block diagram of the double probe measuring system.
4-7
4.2.2.3. Connection length. To apply the probe theory described in the previous chapter, the connection length of the probe, i.e. the distance between the probe and the nearest object in the plasma following the field lines, must be larger than the presheath length. When the probe is inside the LCFS, the field lines are closed and the connection length is infinite. In contrast the field lines are open in the scrape-off layer and the connection length must be investigated.
θ [°]
Field lines
Probe location
Lc
α
ALT-II limiter blades
φ [°] Figure 4.5. Magnetic field lines at TEXTOR-94 plotted in the toroidal and poloidal (φ, θ)-plane at r=46cm. In the region considered the ALT-II limiter is the nearest object. This can be seen in Figure 4.5 where the field lines are plotted as a function of the toroidal and poloidal coordinate. The probe is located at φ = 270o, θ = 90o . The upper edge of the ALT-II is located at θ = 60o . From Figure 4.5 it can then be derived that the shortest connection length Lc is given by: Lc =
2π r 12 sinα
(4.1)
Providing the values r = 0.46m and α = 2.44o , the connection length is Lc=5.7m. This length has to be compared with the presheath length, which can typically be esti2 -5 2 mated by equation (3.82). Providing the parameters D=1m/s , A=5*10 m ,
4-8
cs=6*104m/s, the presheath length is L|| = 3m . The probe theory described in the previous chapter is thereby applicable.
4.2.2.4. Heat load. An important aspect in the application of a probe is the heat load on the probe surface. The probe may not be overheated in order to allow a correct analysis of the data. In this section the accessibility of the plasma for the probe in terms of density and temperature is investigated. Therefore, the derivation in [54] is followed. The temperature rise of a surface is determined mainly by heat conduction into the bulk. In [54], an expression is given for the temperature rise ∆T of the surface at a time t after the start of a constant incident power Q: ∆T = 2
Q t A π .k.C.ρ
(4.2)
where k is the thermal conductivity, C the specific heat and ρ the density of the solid. A is the area of the probe surface. The incident power on the surface is given by:
Q
=
δ .Ii ,sat .Te ,
(4.3)
with Ii ,sat the ion saturation current approximated by Ii ,sat = 0.35ene cs A , δ is the sheath energy transmission factor, T m T eV + 2 i + 2 1 + i 2π e δ (V ) = − kTe Te T mi e
−
1 2
eV exp kTe
(4.4)
This expression shows that the heat flux to a surface strongly depends on the potential V of the surface. Using equation (4.3) and (4.4), one can define a critical density nc which should not be exceeded: nc <
Q 0.35.δ .e.cs .AT . e
(4.5)
To estimate the maximal power which the probe surface can sustain without sublimation of the surface, Q is calculated with the assumption that a temperature rise
4-9
equal to the sublimation temperature will occur during the time of the discharge. The sublimation temperature of the graphite (type EK98) has the value Ts=1873K. The duration of a typical discharge is t=4s. The other properties of graphite are: ρ = 1.85 g cm3 , C = 0.709 J gK and k = 0.9 W cmK . The maximal power the surface can sustain then follows from equation (4.2): Q = 382W cm 2 . From equation (4.3), the critical density can be calculated: nc = 8.7 ∗ 1012 cm −3 12
Te=Ti=30eV) and nc = 6 ∗ 10 cm
−3
(for
(for Ti=3Te ). One can conclude that in the edge
plasma, where ne typically attains the value ne ≈ 0.5 ∗ 1012 cm −3 (see also section 4.6.3), the probe is not expected to overheat. To avoid overheating after successive discharges, the probe is never operated in successive discharges at a position far inside the separatrix.
4.2.3. Atomic beams. By means of emission spectroscopy, the local electron density and temperature can be inferred from the line-integrated intensity [70] of well-chosen impurities. In the case of Li-atoms, the rate coefficient for ionisation and excitation are independent of the electron temperature, and the electron density can be determined from the emission profile. The operational limits of the Li-beam are: ne<4*1012m-3. The electron temperature can be deduced from the intensity of specific line ratios of Helium, which show a specific dependence on Te in the range between 10eV and 100eV but nearly no variation with the density. The He-beam temperature is only accurate if the density is above the value ne > 2*1012cm-3. In the low-density discharges here, the He-beam diagnostic is at the lower (or even below) limit of its applicability. The Li- and He-beam diagnostic is located in the equatorial outboard plane, close to the probe manipulator.
4.3. Plasma conditions. The global plasma parameters like the central line averaged density n e,o , plasma current I p , and toroidal magnetic field Bt are the same for all discharges reported in this thesis. Their respective values are: ne,o = 1019 m −3 , I p = 210kA , Bt = 2.33T .
Further, the discharges in which the Mach probe measurements are performed are pure Ohmic or polarized discharges. In both cases the target plasma is the same in
4-10
terms of global plasma parameters described above but in the Ohmic case the electrode is not inserted in the plasma. In the polarized discharges the electrode is always at the same radial position with its leading edge 5 cm beyond the limiter. The electrode voltage VE can be pre-programmed with a plateau-phase of high voltage or can be kept zero. In the zero voltage case, the electrode is grounded to the ALT-II limiter and a small radial current is drawn through the edge plasma. In the high plateau voltage case (VE=600-700V), this radial current is increased and will set up a high radial electric field.
−3
ne,o [10 cm ]
15
5
600 400
E
V [V]
12
10
200
150 100
E
I [A]
200
50
I [A]
0.5 0 −0.5 0
0.5
1
1.5
2 t [s]
2.5
3
3.5
4
Figure 4.6. Line averaged density, electrode voltage and current of a typical high voltage polarization discharge. The lower graph shows an example of a Mach probe current signal. The dotted vertical lines represent the time slices at which the parameters will be studied in detail. They represent the L- and H-mode case. Typical high voltage polarization discharge. A typical polarization discharge with a high voltage plateau phase will now be described in more detail. An exemplary time trace of the electrode voltage and current signal during such a discharge is plotted in Figure 4.6. The electrode voltage VE in the experiments reported is linearly ramped up between t=1.0s and 1.3s, kept at a constant value up to 2.6s and ramped down in 0.4s. The voltage of the plateau phase has
4-11
been varied between VE=600-700V. It will be demonstrated that this difference will have a negligible effect on the electric field. Note that before and after the biasing phase the electrode voltage is set to VE=0V and hence a small electrode current remains. In the biasing phase one can distinguish two major phases: a phase of low confinement (L-mode) and high confinement (H-mode). The L-mode is characterized by a low electric field value and high radial current, the H-mode by a high electric field value and lower radial current. The transition between the two phases is characterised by the so-called bifurcation phenomenon. This occurs when the applied voltage reaches a value of approximately VE≅350V. The bifurcation is attributed to a dramatic decrease of the parallel viscosity [89], resulting in a sudden drop of the radial current (see Figure 4.6) and, as will be demonstrated, a sudden increase of the radial electric field. In the following sections the typical behaviour of the L- and H-mode will be investigated by evaluating the plasma parameters in detail at two time slices, t=1.12s and t=2s (see Figure 4.6). Note that in the H-mode a steady state is reached. The Lmode is taken during the ramping up of the electrode voltage, which complicates its study. The two time slices agree with respectively VE=300V and VE=600V-700V. No conclusions for transient phenomena will be drawn, more in particular not at the time of the transition. The study for transient phenomena at the time of the bifurcation is restricted due to the low frequency of the applied probe voltage.
4.4. Analysis of the probe data. In this section the interpretation of the raw probe data will be discussed. The determination of the ion saturation currents gives the ratio R and the electron density ne. The slope of the double probe I-V characteristic gives the electron temperature Te. Finally, the floating potential data will be used to calculate the radial electric field Er. Analysis of the double probe I-V characteristic. Figure 4.7 shows a sample of the applied probe voltage between the two probe collectors and the resulting double probe current. To facilitate the data-evaluation a program was developed to obtain automatically the electron density and temperature evaluation of each shot. This program is based on a least-mean square fit of equation (3.19) to the I-V characteristic, which can be obtained from the probe current and voltage. An example of such an I-V characteristic and the result of the fitting is plotted in Figure 4.8. The voltage and current data are taken from Figure 4.7, marked by the open circles. The fit reveals the up- and down-stream ion saturation currents. The elec-
4-12
tron density is then calculated with equation (3.65) and the electron temperature with equation (3.20). 250
0.5
200
0.4 0.3
100
0.2 I [A]
50 0 −50
0.1 0
−100
−0.1 −150
−0.2
−200 −250 0.5
0.51
0.52
0.53
0.54
0.55
t [s]
0.5
0.51
0.52
t [s]
0.53
0.54
Figure 4.7. Applied probe voltage and collected current vs. time. The sweeping frequency of the applied voltage is 200Hz. The marked voltage and current traces (o) are used to plot the I-V characteristic in Figure 4.8.
0.4 0.3 probe current [A]
probe voltage [V]
150
0.2 0.1 0 −0.1 −0.2 −200
−100
0 100 probe voltage [V]
200
Figure 4.8 Measured I-V characteristic (+). The sample is taken from Figure 4.7. The solid line represents a fit using equation 1.24. The arrows indicate the error bar on the ion saturation current. An error bar on the value of the ion saturation current can be defined as shown in Figure 4.8. There the variation of the saturation is taken as an estimate of the ion saturation current. The origin of this variation is coming from variations of electron temperature and density in the plasma. Automation of such a procedure by a com-
0.55
4-13
I [A]
0.1 0 −0.1 −0.2 −0.3
I [A]
puter program is very difficult. Therefore, the estimate was done on a number of samples showing that the error bar was of the order of only a few percent of the total current signal. Another reason not to apply such a procedure is that such an estimate does not give an indication for the quality of the saturation of the current.
0.1 0 −0.1 −0.2 −0.3
ln(R)
0.5 0 −0.5 −1 −1.5 0.5
1
1.5
2 2.5 3 3.5 t [s] Figure 4.9. Probe current (upper signal), envelope (middle signal) and natural logarithm ln(R) vs. time. The up- and downstream side of the probe is given by respectively the positive and negative values of the probe current. The ion saturation current is the envelope of the probe current signal.
The envelope of the probe current delivers the up- and downstream ion saturation currents (see Figure 4.9). The up-stream ion saturation current is hereby defined as the saturation current of the collector, which faces the projection of the flow vector in the magnetic field direction. A positive sign will be given to the currents measured by that collector. Consequently, the current of the downstream collector will then have a negative sign. The geometry considered is shown in Figure 4.12. The up- and downstream currents are interpolated on an equally spaced time axis to ensure that the ratio is taken at the same time step. For the derivation of the Mach numbers, the interesting parameter is not the ratio R but the natural logarithm of R, i.e. ln(R ) , since this value is directly proportional to the parallel and perpendicular Mach number. Therefore, in
4-14
the following, the value of ln(R ) is plotted. An example of the probe current, the envelope and the value of ln(R ) are plotted in Figure 4.9. Following the sign convention for the up- and downstream current, a positive value of ln(R ) indicates that the projection of the flow vector is parallel to the magnetic field. Analysis of the floating potential. When the electronic circuit is switched to floating potential measurement, both collectors measure the floating potential separately. As a result of the plasma flow, a different floating potential will be measured at the up- and downstream side [47]. To compensate for this effect the average from the up- and downstream floating potential is taken. The involved error is small as can be seen from Figure 4.10. Since the plasma potential is related to the floating potential as discussed in section 3.2.1, Er is derived from the floating potential profiles by the following equation:
Er = − ∇Vpl ≈ − ∇Vfl − 3.2
k.∇T e e
(4.6)
In the polarization experiments [45], ∇Vfl is of the order of 100Vcm-1. Since the contribution from the temperature gradient to Er is typically less than 10Vcm-1 and remains small in the high confinement regime, the last term in equation (4.6) can be neglected. The electric field is thus directly calculated from the gradient of the floating potential. 500
400
Vfl [V]
300
200
100
0 0.5
1
1.5
2 t [s]
2.5
3
3.5
Figure 4.10. Floating potential for the up- (solid line) and downstream (dashed line) side vs. time. The average of the two traces is taken as the floating potential.
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4.5. Database of Mach probe data. This section discusses how the probe data are organised in the database and in which way the information will be applied. The database can be viewed as a collection of radial profiles. Since the Mach probe can only be moved in the radial direction in between discharges, a radial profile is build of a series of nearly reproducible discharges in terms of polarization parameters and probe settings. The target plasma can be Ohmic or polarized as described in section 4.3. As discussed in the foregoing sections, the schemes in which the probe can be operated are the ion saturation current or the floating potential measurement. A second probe setting is the rotation angle of the probe surfaces with respect to the magnetic field. A last free parameter is the poloidal location of the probe. The probe can be mounted in the equatorial outboard plane or on the top of the machine. The combination of these settings and plasma conditions, resulting in different profiles, are summarized in Table 1.
Probe angle [°]
Probe position
Equatorial plane
Top plane
Polarised discharges Applied electrode voltage VE [V] 600-700
0.0
X
X
22.5
X
X*
45.0
X
X*
90.0
X
X
X*
Ohmic discharges
0
X
X
135.0 X X* Table 1. Summary of the Mach probe database for the ion saturation current measurement at a certain probe angle, probe location and target plasma (marked by X). Floating potential profiles are marked with (*). The different sets of profiles are grouped. The database can be divided in four series of profiles. The first series of profiles are those obtained at the angles θ = 22.5o, 45o, 90o and 135o with the probe located in the equatorial outboard plane and taken in high plateau voltage discharges (grouped by the dashed line in Table 1). The profiles obtained from ion saturation cur-
4-16
rent measurements at these angles can be used to derive the perpendicular and parallel Mach number by applying the 1D fluid model, described in the previous chapter. It will be demonstrated that in these high voltage polarized discharges the perpendicular flow is proportional to the electric field. Therefore, it is important that the radial electric field for each shot series is comparable. The electric field that results from the floating potential profiles will be used as a criterion for the reproducibility of the electric field. Since these floating potential profiles are taken on the same experimental day as the ion saturation current profiles for the considered angles, the resulting electric field profiles can be considered as representative for the electric field in which the ion saturation current measurements are performed. A second set of profiles consists of those for which the ion saturation current is measured with the probe located in the equatorial plane at a probe angle θ = 90o (grouped by the dashed-dotted line in Table 1). At this angle the Mach number of the parallel flow can be obtained. The comparison of the Mach number profile in Ohmic and polarized conditions allows to investigate the influence of a radial current on the parallel Mach number in the edge plasma. The profile of the ion saturation current and the floating potential measured at the top plane at a probe angle θ = 90o (grouped by the dotted line in Table 1) will be used to investigate poloidal asymmetries of a number of parameters derived from the previous two sets of profiles. These parameters concern the electric field, the electron density and temperature and the parallel Mach number. In this chapter only the data is presented. A detailed discussion of the underlying physics will be given in the next chapter. A last series of discharges gives an ion saturation profile (indicated by the solid line in Table 1) for the probe surfaces parallel with the magnetic field, i.e. θ = 0o . This profile is taken in high plateau voltage discharges and will be used for an alternative derivation of the perpendicular Mach number Criteria for the reproducibility of the discharges. It is clear that the various profiles in the database consist of a large series of discharges, which have to be nearly identical in order to have a consistent set of profiles. Criteria for the reproducibility of the shots are therefore of considerable importance. A first selection of the discharges for inclusion in the database is done on the basis of the analysis of the global plasma parameters, i.e. plasma current, line averaged density, magnetic field, etc. The discharges included in the database can be considered as reproducible and their analysis will not further be discussed. A second check, more important for the region considered, can be done based on the edge parameters. Besides a comparison of the radial electric field profiles, discussed above, each discharge in the series can be compared with each other by comparing the electron density and
4-17
temperature with the values delivered by the atomic He- and Li-beam. Further, in the polarized discharges, the electrode current will be used as a criterion since it is this current that will set up the electric field in the edge. Last Close Fluxed Surface (LCFS). In principle, the LCFS, the radial position of which is denoted as a, is defined by the ALT-II limiter which is at r = 46cm . However, it is experimentally found that to assure that the electrode current flows mainly towards the limiter, the plasma needs to be shifted towards the limiter. The absolute value of the LCFS at the equatorial and top plane is thus not given by the radial position of the ALT-II limiter. The position of the LCFS is determined by floating potential measurements by S. Jachmich [46]. The radial axis will henceforth be labelled as r-a rather than absolute values of the radial position r.
4.6. Derivation of the perpendicular Mach number and validation of the 1D fluid model. In this section the first set of profiles in the database will be discussed in detail. As outlined in the previous section, this set consists of profiles of the ion saturation current measured in high plateau voltage discharges with the probe located in the equatorial outboard plane. The probe surfaces are oriented at the probe angles θ = 22.5o, 45o, 90o and 135o . The position of the probe surfaces at these angles is schematically shown in Figure 4.11. Application of the 1D fluid model, described in the previous chapter, on the up- and downstream saturation currents results in a quantitative derivation of the perpendicular and parallel Mach number. In the polarized discharges high radial electric fields and ensuing flows are created in a controlled way. Therefore, the database not only allows to obtain the magnitude and direction of the flow vector by applying the 1D fluid model but offers also the possibility to obtain a qualitative analysis of the related Mach numbers. This will be done via the radial momentum balance, which is given by: 1 E = ∇ p −v B +v B θ φ φ θ r en r i i
(4.7)
The first term on the right hand side is the diamagnetic term with ni and pi the ion density and pressure, the second and third term represent respectively the poloidal and toroidal velocity contributions. In the latter two terms vθ , vφ , Bθ , Bφ are respectively the poloidal and toroidal components of the flow vector and the magnetic field.
4-18
This equation allows to calculate the electric field that is set up during the polarization phase from the parallel and perpendicular Mach numbers and the diamagnetic contribution. The comparison with the measured electric field will then demonstrate the viability of the 1D model and the related Mach numbers. In the following sections the different contributions to this equation will be discussed and the measured and calculated electric field will be compared.
θ = 22.5o
θ = 45o
B
Ip
θ = 90o
⊥ B
||
θ
θ = 135o Figure 4.11. The Mach probe is positioned at different angles with respect to the magnetic field. The thick solid lines interconnected with a dashed line represent schematically the Mach probe.
⊥
θ
upstream collector
v⊥
v||
Ip
B
θ
isolator
r~ ||
downstream collector
φ
Figure 4.12. Mach probe geometry showing the parallel, perpendicular and the toroidal, poloidal coordinate system whereby the pitch angle is neglected. Also shown are the plasma current and magnetic field direction. For the probe, the up- and downstream collector is indicated.
4-19
The velocities in equation (4.7) are expressed in poloidal and toroidal coordinates. To obtain the parallel and perpendicular velocities, the pitch angle will be neglected as discussed in section 4.2.2.2. As a result, the toroidal and poloidal directions are assumed to match the parallel and perpendicular direction but with reversed sign. In Figure 4.12 the considered geometry is shown. Also there the convention of the upstream- and downstream collector is indicated.
4.6.1. Electric field profiles. The database contains floating potential profiles taken at the same probe angles as the ion saturation current. Since the floating potential is independent of the probe angle, they can be used as a criterion for the reproducibility of the electric field at each angle. Figure 4.13a and b show the measured floating potential profiles in the L- and H-mode. 300
700 600
250
500 400
fl
V [V]
150
fl
V [V]
200
300
100
200 100
50
0
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
−5
−4
−3
−2
−1 r−a [cm]
0
Figure 4.13a and b. Floating potential vs. radius in L- and H-mode. The various profiles are taken at a different probe angle. The markers indicate the radial position at which the Mach probe is positioned at the various angles: θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). In both cases it can be concluded that the various floating potential profiles are reproducible. The variation of the profiles in the L-mode is larger than for the profiles in the H-mode. This is due to the fast ramping of the applied electrode voltage. However, considering Figure 4.13, it is warranted to adopt a typical radial electric field resulting from the average of the various floating potential profiles. The electric field profile is then calculated with equation (4.6). The resulting electric field profiles in the L- and H-mode are plotted in Figure 4.14a and b.
1
2
4-20
200
700 600
E [V/cm]
500
r
100
r
E [V/cm]
150
400 300 200
50
100
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
Figure 4.14a and b. Electric field Er vs. radius r-a in L- and H-mode. The open circles represent the electric field values calculated from the average floating potential profile. The magnitude for the electric field in the L-mode is considerably lower compared with the one in the H-mode. This observation is as expected since the applied electrode voltage in the L-mode is much smaller compared to the value in the H-mode and is below the threshold value for transition. Further, the L-mode electric field is broader than the H-mode field. The electric field profiles in Figure 4.14 will henceforth be referred to as the low and high electric field. Their values are considered to be typical for the L- and H-mode case.
4.6.2. Derivation of the parallel and perpendicular Mach number. In this section the typical perpendicular and parallel Mach number in the Land H-mode will be derived by applying the 1D fluid model on the up- and downstream ion saturation currents of the considered set of profiles. Besides the comparison of the electric field profiles obtained at each angle, it is desirable to have some criterion for the reproducibility of the electric field of each discharge in the various profiles. Since the radial current imposes the electric field, the value of the electrode current for each discharge can be used as such a criterion. The electrode current rather than the applied electrode voltage is hereby important since for the same voltage the same electrode current is not always obtained because the conductivity of the plasma can differ from discharge to discharge. In Figure 4.15a and b the electrode current value for each discharge as a function of the radial probe position in the considered set of profiles is plotted. The two figures represent respectively the L- and H-mode.
1
2
200
150
150
IE [A]
200
100
100
E
I [A]
4-21
50
0 −5
50
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
Figure 4.15a and b. Electrode current IE vs. radial probe position in L- and H-mode. The various profiles are taken at a different probe angle. The markers indicate the radial position at which the Mach probe is positioned at the various angles; θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). In both cases the electrode current remains approximately constant for radial positions r − a > −3 cm and has a mean value of IE = 163 A and IE = 110 A in the case of respectively a high and low Er. When the probe is deeper in the plasma, the measured electrode current decreases systematically. A possible explanation is that at these radii the probe enters into the flux tube of the electrode and short-circuits the potential on the flux surface, which results in a lower electrode current. Remember that the leading edge of the probe head is located 1.2cm deeper in the plasma than the centre of the collecting area. One can expect that the electric field generated in these discharges has a lower value. The results of the probe data for radial positions r − a < −4 cm must be questioned. The first step in the application of the 1D fluid model is the calculation of the natural logarithm of the ratio R as discussed in section 4.4. The quantity ln(R) at each angle is displayed in Figure 4.16a – d as 2D contour plots in which the value of ln(R) is shown as a function of radius and time. Some discharges in the profiles disrupted after 2.7s due to technical errors. This is translated in errors for the value of ln(R), which can be seen in the plots after 2.7s.
2
4-22
θ = 22.5°
θ = 45.0° 2.5
2.5
2
2
1.5
1
2
1.5
1
1
1
0
0
ln(R)
0.5
−1
−0.5
−2
r−a [cm]
0.5
−1
0
−0.5
−2
−1
−1
−3
−3 −1.5
−4
−1.5
−4
−2
−2
(a) (b)(c)
−5 0.5
(a) (b)(c)
−5 0.5
−2.5
1
1.5
2 t [s]
2.5
3
3.5
θ = 90°
−2.5
1
1.5
2 t [s]
2.5
3
3.5
θ = 135.0° 2.5
2.5
2
2
2
1.5
1
2
1.5
1
1
1
0
−1
0
−0.5
−2
ln(R)
0.5
r−a [cm]
r−a [cm]
0
0.5
−1
0
−0.5
−2
−1
−3
−1
−3 −1.5
−4
−2
−1.5
−4
(a) (b)(c)
−5 0.5
−2.5
1
−2 (a)
1.5
2 t [s]
2.5
3
3.5
−5 0.5
(b) (c) −2.5
1
1.5
2 t [s]
2.5
3
3.5
Figure 4.16a, b, c and d. 2D contour plot of the natural logarithm of the ratio R vs. radius and time. The plots represent the value of ln(R) at different probe angles, respectively at 22.5°, 45°, 90° and 135°. The position of the LCFS refers here to the H-mode case. The dotted lines mark the time slices t=1.12s and t=2s. The dashed dotted lines are respectively at the time slices t=1s ((a): start of the voltage ramp), t=1.17s ((b): bifurcation) and t=1.3s ((c): voltage plateau phase). The interpretation of these contour plots is as follows. The high voltage phase is clearly distinguishable at each angle, starting at 1s and ending at 3s. Also the time of the transition is clearly marked. Before transition no significant variation of the value of ln(R) can be observed. The transition is followed by the constant voltage plateau. During this constant phase, two radial regions can be identified. One narrow region is situated around r − a = −0.5cm and shows a clear variation of the value of ln(R) with probe angle. In the light of the discussion in the previous chapter this indicates the presence of a substantial perpendicular flow. The location of this region agrees with the
ln(R)
r−a [cm]
0
ln(R)
2
4-23
region where the electric field becomes important. A second broader region is situated around r − a ≈ −2cm and does not show a variation of ln(R) with probe angle. The flow in this region is therefore mainly parallel to the magnetic field. Figure 4.17 shows the various profiles of the value of ln(R) as a function of radius, respectively for a time step taken at 1.12s and 2s, representing therefore the L- and H-mode case. As could be expected from the 2D contour plots only the profiles in the H-mode show a clear signature of a perpendicular flow. Besides the two regions, which are described above, the plots in Figure 4.17 show also a variation of ln(R) with probe angle for radii r − a < −4cm . This variation will not be considered as physical, considering the discussion concerning the electrode current. To apply the 1D fluid model, the various radial profiles are linearly interpolated to an equally spaced radial axis. For each radial position, the value of ln(R) as a function of the probe angle is then fitted to equation (3.62), which reads: M ln ( R ) = c . M|| − ⊥ (4.8) tg θ The parameter c in this equation is directly obtained from the value of ln(R) for θ = 90o . Strictly, it is the value of c ∗ M|| which is directly related to the sheath edge
density. However, to reduce the fitting error, both parallel and perpendicular Mach number are assumed to be free parameters in equation (4.8). The fitting is based on a least-mean square method where equation (4.8) is of the form: Y = A + B. X ,
(4.9)
in which the parameters A and B are the parallel and perpendicular Mach number. An example of this fitting procedure for the low and high field case is shown in Figure 4.18a and b. The radial position at which the two examples are taken is indicated in Figure 4.17 and is the position for which a maximum variation of ln(R) as a function of angle is observed. The respective radii are r-a=-0.2cm and r-a=-0.5cm. In the Hmode the radial position corresponds with the position of the Er maximum. The values of the perpendicular and parallel Mach number obtained from the fitting in the two examples are respectively M ⊥ = −0.09 , M|| = −0.43 and M ⊥ = −0.53 , M|| = −0.18 . The dotted vertical lines in Figure 4.18 indicate the angles up to which the fluid model is valid. Equation (3.68) is used to calculate these angles. As can be seen, in the L-mode all experimental values of ln(R) are well within this window. In the H-mode, the data point for θ = 22.5o is clearly outside the boundary. As discussed in section 3.5.3.3, the limiting angle is model dependent and there’s no physical reason why equation (4.8) should not apply above the sound speed. The good agreement of the
4-24
least mean square fit with the data clearly proves equation (4.8) to be a correct expression for the variation of ln(R) as a function of angle θ. 3 2
ln(R)
1 0 −1 −2 −3 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
−4
−3
−2
−1 r−a [cm]
0
1
2
3 2
ln(R)
1 0 −1 −2 −3 −5
Figure 4.17a and b. Natural logarithm of the ratio R vs. radius r-a in L- (upper Figure) and H-mode (lower Figure). The various profiles are taken at a different probe angle. The markers indicate the radial position at which the Mach probe is positioned at the various angles; θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). The dotted vertical lines
4-25
indicate the radial position for which the value of ln(R ) as a function of probe angle is plotted in Figure 4.18. The above-described fitting procedure is applied to all possible radial positions and time steps. The radial axis is hereby limited to radial positions in the range from r − a = − 3.6cm to r − a = 1.2cm , since only at these positions values of ln(R) are available at all angles. The time range goes from 0.5s to 3.5s. The resulting parallel and perpendicular Mach number are shown as 2D contour plots in Figure 4.19 as a function of radius and time. As already expected from the interpretation of Figure 4.16 and Figure 4.17, the perpendicular Mach number peaks in the region around r − a ≈ − 0.5cm and the parallel Mach number in the region around r − a ≈ −2cm .
8
8
6
6
4
4
2
2
ln(R)
ln(R)
Further, the perpendicular flow is strongly increased after the transition and remains constant during the whole plateau phase. The magnitude of the parallel flow increases in the L-mode whereas a small decrease is observed in the H-mode. Note that the spikes in the values of the Mach-numbers after t=2.7s are due to mathematical errors and do not have a physical meaning.
0
0
−2
−2
−4
−4
−6
−6
−8 0
45
90 θ [°]
135
180
−8 0
45
90 θ [°]
135
Figure 4.18a and b. Natural logarithm of the ratio R vs. probe angle θ in L- and Hmode. The respective radii are those for which a maximum variation of ln(R) is observed in Figure 4.17 (respectively r-a=-0.2cm and r-a=-0.5cm). The round dots (•) represent the experimental values, the dashed curves are fitted curves following equation (4.8) The maximum angle up to which the 1D fluid is valid is shown by the vertical dotted lines. In Figure 4.20 and Figure 4.21, the perpendicular and parallel Mach number for the L- and H-mode are plotted. The error bars are obtained by calculating the standard deviation on the parameters A and B in equation (4.9). Since the errors on the ion saturation currents are small, they are not included. The magnitude of the error bars is
180
4-26
in all cases acceptable considering the fact that the profiles are obtained from a large number of discharges. 0.3
2
0.2
0.1
1
0
−0.1
−1
−0.2
M⊥
r−a [cm]
0
−0.3
−2
−0.4
−3 −0.5
−4 (a)
−5 0.5
−0.6
(b)(c)
1
1.5
2 t [s]
2.5
3
3.5
0.3
2
0.2
0.1
1
0
−0.1
−1
−0.2
M||
r−a [cm]
0
−0.3
−2
−0.4
−3 −0.5
−4
−0.6 (a)
−5 0.5
1
(b)(c)
1.5
2 t [s]
2.5
3
3.5
Figure 4.19a and b. 2D contour plots of the perpendicular and parallel Mach number as a function of radius and time. The position of the LCFS refers here to the H-mode case. The dotted lines mark the time slices t=1.12s and t=2s. The dashed dotted lines are respectively at the time slices t=1s ((a): start of the voltage ramp), t=1.17s ((b): bifurcation) and t=1.3s ((c): voltage plateau phase).
4-27
0.3
0.3
0.2
0.2
0.1
0.1 0
−0.1
−0.1 M⊥
M||
0
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−5
−4
−3
−2
−1 r−a [cm]
0
1
2
−5
−4
−3
−2
−1 r−a [cm]
0
1
2
1
2
Figure 4.20. Parallel and Perpendicular Mach number vs. radius in L-mode. 0.3
0.3
0.2
0.2
0.1
0.1 0
−0.1
−0.1
M⊥
M
||
0
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−5
−4
−3
−2
−1 r−a [cm]
0
1
2
−5
−4
−3
−2
−1 r−a [cm]
0
Figure 4.21. Parallel and Perpendicular Mach number vs. radius in H-mode.
4.6.3. Electron density and temperature. The results for the electron density and temperature obtained from the Mach probe data for the profiles considered are the topic of this section. The electron density and temperature are plotted at the time slices t=1.12s and t=2s, representing the L- and H-mode case, respectively in Figure 4.22 and Figure 4.23. In both cases a typical Libeam density profile and He-beam temperature is added, providing an independent comparison. In Figure 4.22a and Figure 4.23a, the density and electron temperature
4-28
profile is also plotted for θ = 135o at t=0.9s, i.e. before the start of the high voltage ramp. 12
12
3.5
3.5 3
2.5
2.5 ne [cm−3]
3
ne [cm ]
−3
x 10
2
2
1.5
1.5 1
1
0.5
0.5
0 −5
x 10
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
Figure 4.22a and b. Electron density ne vs. radius in L- and H-mode. The various profiles are taken at a different probe angle. The markers indicate the radial position at which the Mach probe is positioned at the various angles; θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). For comparison a typical Li-beam density profile is added (x, solid line). The dashed line in the L-mode represents the electron density at t=0.9s and θ =135°. The density profiles of the probe at the different angles all show the same trend and they are comparable in magnitude except the profile for θ = 22.5o which has a lower value for both cases. A possible explanation is that the projected surface along the magnetic field becomes too small so that the criterion for a magnetized probe is not satisfied. The general observation is that the density profiles do not show significant changes unless the voltage is above the value for transition. In contrast to the profiles before transition, all density profiles in Figure 4.22b show the characteristic feature of the steepening of the density gradient in the H-mode [45] [89]. The position of the steep gradient of the density agrees with the inner gradient of the electric field. A second feature observed at all angles in the H-mode is that the profiles for radii smaller than r − a < −3cm remain flat whereas a steeper increase of the density is expected. This observation seems unphysical and one could question the interpretation of the probe data. At these radii the probe could be overheated. In section 4.2.2.4, it is shown that at these radii the probe is not expected to overheat. A further remark is that the profiles before transition show a normal increase towards the inner plasma and therefore a physically correct behaviour. A proper interpretation of the probe data is therefore justified and one has to search an explanation for this behaviour in the plasma itself. This could be a cooling of the plasma due to the presence of neutral par-
2
4-29
ticles. However, the recycling at the electrode is very local effect and cannot be expected to influence the neutral density at the considered radii [89]. Despite this unclear feature, it is the aim to work with the probe results as they are measured. Therefore a typical density profile is adopted in L- and H-mode, represented by the solid line. Despite the fact that the Li-beam profile is more smoothened, the profile reveals in the H-mode the same two features as the probe density, namely a flattening deeper inside the plasma and an increase of the gradient at the same position as the inner electric field gradient. In contrast to the probe, the Li-beam shows an additional fine structure as a local flattening at the position r − a = −0.5cm , i.e. at the position where the electric field peaks. The resolution, at which the probe data are taken, is too low to resolve this feature.
80
80
60
60
Te [eV]
100
Te [eV]
100
40
20
0 −5
40
20
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
Figure 4.23a and b. Electron temperature Te vs. radius in L- and H-mode. The various profiles are taken at a different probe angle The markers indicate the radial position at which the Mach probe is positioned at the various angles; θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). For comparison a typical He-beam temperature profile is added (∗, solid line). As for the density profiles, the temperature profiles do not undergo significant changes up to the transition. After the transition (Figure 4.23b), the profiles look quite different. In the region of high electric field, clearly a peakening of the temperature is observed: In the H-mode the electric field and hence the drop in the floating potential occurs in a narrow region. This means that the power of the electrode current is dissipated in a narrow region, leading to a local increase of the temperature. In the Lmode the dissipation occurs over a broader region and therefore such a local heating of the plasma is not observed. In a region deeper inside the plasma (around r − a ≈ −2cm ) a local cooling of the plasma compared to the low electric field case can be observed. This observation is in line with the increased radiation, which is ob-
2
4-30
served in the edge during the H-mode [89]. A typical electron temperature profile in the L- and H-mode is adopted as represented by the solid lines. Comparing these profiles with the He-beam profiles should be done with great caution since the density in the considered region is below the limit for the application of the He-beam diagnostic. This can explain why the temperature provided by the He-beam is systematically higher than the temperature given by the probe data. However, the trend of the probe profiles is also found in the He-beam profiles: an increase at the position of high electric field and a cooling deeper in the plasma.
4.6.4. Validation of the 1D fluid model. The results of the previous sections will now be used to calculate the different contributions in the radial momentum balance, equation (4.7). The diamagnetic term is calculated with the values for the density and temperature which have been adopted in the previous section. On account of the quasi-neutrality the ion density equals the electron density. The velocities in equation (4.7) are related to the Mach numbers, obtained in section 4.6.2, via equation: v = cs .M ,
(4.10)
whereby the sound speed cs is defined by the equation: cs =
Te + Ti , mi
(4.11)
in which Te = Ti is assumed. As mentioned before the poloidal and toroidal velocities are assumed to equal the perpendicular and parallel velocities but with reversed sign. Note that the sign convention also implies that the magnetic field in the (φ,θ)-coordinate system considered has a negative value. The electric field obtained in this way is compared with the electric field presented in section 4.6.1. Figure 4.24 shows these electric fields in the L- and H-mode. For the derivation of the errorbars on the calculated electric field only the errors on the Mach numbers are taken into account.
4-31
200
Er [V/cm]
150
100
50
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
−4
−3
−2
−1 r−a [cm]
0
1
2
1000
600
r
E [V/cm]
800
400
200
0
−5
Figure 4.24a and b. Comparison of the electric field calculated with the radial momentum equation and the measured electric field (o) in L- and H-mode. The 1 calculated electric field (solid line) consists of ∇ pi (dotted line), − v B θ φ eni r 1
− ∇ p φ θ (dashed-dotted line). Note that eni r i is plotted. In the H-
(dashed line), v B
mode the calculated Er is added whereby cs is calculated from equation (4.11) with values for Ti obtained from equation (4.13) (solid line, +) and from equation (4.11) with Ti=3Te (solid line, ◊).
4-32
The magnitude of the diamagnetic term is comparable in L- and H-mode. In the H-mode its contribution to Er is negligible. Since the poloidal magnetic field is one order of magnitude smaller than the toroidal magnetic field, the poloidal velocity term is much larger than the toroidal one, although the parallel and perpendicular Mach numbers are comparable. In the H-mode the contribution of the toroidal term is even negligible. It is clear that in the H-mode a qualitative comparison is possible since there large poloidal Mach numbers are measured. In the H-mode the measured electric field is reconstructed in magnitude and width by the values of the parallel and perpendicular Mach number provided by the 1D fluid model. The main conclusion is that in the 1D fluid model the transport of the perpendicular flow is correctly implemented. The application of the model on Mach probe data proves to be a reliable method to determine the perpendicular Mach number. The method not only delivers quantitative, but also precise values of the Mach numbers. A final conclusion is that the electric field in the H-mode is mainly created by the poloidal velocity term in equation (4.7). High electric fields go thereby hand in hand with high poloidal rotations. One should note that Te = Ti is assumed in the calculation of cs. In reality Ti can be larger then Te. In Figure 4.24b the calculated Er is added with the assumption Ti = 3Te , showing that the uncertainty of Ti in cs has a large influence on the flow velocities. An independent measurement of Ti is however not available. An estimate for the value of Ti and cs is obtained by using equation (3.65), derived in the previous chapter. This equation enables the calculation of the electron density from the up- and downstream saturation currents. Since the Li-beam density is almost independent of Te, the substitution of this density in equation (3.65) allows the calculation of a realistic value of cs:
cs =
Ii ,sat ,up .Ii ,sat ,down 0.35 e nLi A sinθ
.
(4.12)
In Figure 4.25 the ratio of the value of cs calculated with equation (4.12) (cs,calculated) and cs calculated with equation (4.11) with the assumption Te =Ti (cs,assumed) is shown for the H-mode. The ratio Ti Te is then obtained from the expression: Ti cs,calculated = 2. Te cs,assumed
2 − 1
(4.13)
The value of Ti Te is plotted in Figure 4.26 and shows that the value of Ti is mostly higher than the value of Te except in the region of high electric field, where the value of Te peaks. In Figure 4.24b, the calculated Er is plotted whereby cs is calculated from
4-33
equation (4.11) with values for Ti obtained from equation (4.13). It is clear that the conclusion remains that the fluid model reconstructs the measured Er. 2.5
Cs,calculated/Cs,assumed
2
1.5
1
0.5
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.25. cs,calculated cs,assumed vs. radius in H-mode. The value of cs,calculated is computed with equation (4.12), the value of cs,assumed with equation (4.11) with the assumption Te =Ti . The various profiles are taken at a different probe angle. The markers indicate the radial position at which the Mach probe is positioned at the various angles; θ =22.5° (dashed-dotted line, ), θ =45° (dashed line,ο), θ =90° (solid line, +), θ =135° (dotted line, ∆). The solid line represents the average value of the different profiles.
5
Ti / Te
4
3
2
1
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
Figure 4.26. Ti Te vs. radius in H-mode.
2
4-34
4.6.5. Time evolution of the parallel and perpendicular Mach number.
I [A] and V [V]
Finally, the time evolution of the Mach number at the electric field maximum is shown in Figure 4.27. The Mach numbers are only plotted from 0.5s to 2.7s since no meaningful data are available after this time. As mentioned earlier, one should be careful in interpreting the time evolution, especially at the time of the transition. Nevertheless, some important features can be clearly recognised. To account for the pitch angle, the Mach numbers will be described in the toroidal and poloidal coordinates.
VE
E
600 400 200
600 400
r
E [V/cm]
E
IE*3
200
M
φ
0.4 0.2
Mθ
0.4 0.2 0 −0.2 0.5
1
1.5
2 t [s]
2.5
3
3.5
Figure 4.27. Time evolution of the Mach numbers and the electric field at the radial position of the electric field maximum. The upper graph shows a typical electrode voltage signal (solid line) and electrode current (dashed line). The electrode current value is multiplied by a factor 3. The vertical dotted lines indicate the time slices t=1.12s and t=2s. The interpretation of Figure 4.27 is as follows. At the start of the electrode voltage ramp, the plasma rotates mainly in the toroidal direction. This is due the fact that the electrode is grounded (VE = 0V ) . A small radial current will therefore be drawn in the edge plasma and will force the plasma to rotate in the toroidal direction.
4-35
The magnitude of this toroidal rotation increases during the voltage ramp with increasing radial current. Since in this phase a small radial electric field is created, a moderate poloidal rotation exists. At the bifurcation, the radial electric field increases strongly together with the poloidal rotation. Both values remain constant during the plateau phase of the electrode voltage. The toroidal rotation does not increase with the electric field and shows even a small decrease. When the electrode voltage ramps down, the above described scenario is reversed.
4.7. Influence of the radial current on the parallel Mach number. The database contains several profiles consisting of ion saturation current measurements with the probe surfaces perpendicular to the magnetic field and for the probe located in the equatorial plane. Under these conditions, the parallel Mach number can directly be derived from the up- and downstream ion saturation currents. The available profiles (see Table 1) are plotted in Figure 4.28. They represent the Mach number for different values of the radial current: negligible radial current, i.e. the electrode is withdrawn from the plasma, IE=40A, i.e. electrode voltage VE=0V, IE=163A, and IE=110A. The latter two values are the profiles obtained in section 4.6.2 and are the values for the L- and H-mode. The difference of the radial current drawn in the edge plasma characterises the profiles of the parallel Mach number. It is the aim to report here only the experimental observations. A detailed discussion of the underlying physics of Figure 4.28 will be given in the following chapter. The parallel Mach number is clearly influenced by the presence of a radial current in the plasma. When the plasma is pure Ohmic, high flow, parallel to the magnetic field exists in the scrape off layer and decreases inside the LCFS. Inside the SOL, this flow is due to the streaming of the flow towards the limiter [85] [87]. The latter is indeed the nearest object in the plasma following the field lines. Inside the separatrix, no substantial driving force is present in the case IE = 0 and the plasma rotation decreases towards the inner plasma. When a radial current is induced in the edge plasma, a JxB -force acts on the particles, resulting in a rotation of the plasma antiparallel to the magnetic field. This flow increases non-linearly with increasing radial current. The toroidal rotation attains the same value although the current is decreased from the L- to H-mode. Both profiles show a maximum of the same magnitude in a region, which differs from the region where the electric field peaks. A minimum in the parallel Mach number is observed at the position of maximum electric field. This minimum decreases slightly in the H-mode.
4-36
0.6 0.4
M
||
0.2 0
−0.2 −0.4 −0.6 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.28. Radial profiles of the parallel Mach number for different values of the radial current; electrode out (x, dashed-dotted line), IE=40A (∇, dashed line), IE=160A (dotted line), IE=110A (solid line).
4.8. Poloidal asymmetries. Information concerning poloidal asymmetries of the electric field, electron density and temperature, and the parallel Mach number can be obtained by comparing the profiles with the probe surfaces perpendicular to the magnetic field, measured at the top plane and the equatorial plane. The asymmetries will be investigated in the Hmode case where the plasma conditions are stable and the effect on the profiles is more pronounced. The electrode current as a function of radial probe position in the discharge series considered has the same value and radial behaviour no matter at which poloidal probe position the probe has been operated. Therefore a qualitative comparison is justified. In Figure 4.29 - Figure 4.32 respectively the floating potential profiles, electron temperature, electron density and parallel Mach number in the H-mode at the two different probe locations are shown. The equatorial profiles are represented by the typical profiles obtained in section 4.6. For comparison, also the profiles in the equatorial plane at θ = 90o are shown.
4-37
700 600 500 Vfl [V]
400 300 200 100 0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.29. Floating potential profile at the top plane (o, dashed line) and the equatorial plane (+, solid line; typical profile, dashed-dotted line). 100
60
e
T [eV]
80
40
20
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.30. Electron temperature profile at the top plane (o, dashed line) and the equatorial plane (+, solid line; global profile, dashed-dotted line). From Figure 4.29 and Figure 4.30, it is clear that the profiles for the floating potential and the electron temperature are similar in shape and magnitude for both poloidal probe locations. There’s thus no evidence for a poloidal asymmetry in the electric field and electron temperature.
4-38
3
x 10
12
2.5
−3
Ne [cm ]
2 1.5 1 0.5 0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.31. Electron density profile at the top plane (o, dashed line) and the equatorial plane (+, solid line; typical profile, dashed-dotted line).
0.6 0.4
M||
0.2 0
−0.2 −0.4 −0.6 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figure 4.32. Parallel Mach number profile at the top plane (o, dashed line) and the equatorial plane (+, solid line; fitted profile, dashed-dotted line). For the electron density (Figure 4.31) and parallel Mach number (Figure 4.32) indications exist for poloidal asymmetries. The magnitude and shape of the electron density profile differs in the region of high Er. In that region the density gradient at the top plane is not as large as in the equatorial plane. Also, the magnitude of the density at the position of the electric field maximum is higher at the top plane than at
4-39
the equatorial position. The parallel Mach numbers have lower values in the top plane than in the equatorial plane. Yet, the profiles show the same trend at both poloidal probe locations: the parallel Mach number profile has a maximum in a region which differs from the position of the electric field maximum and a decrease in the region where the electric field peaks.
4.9. Alternative derivation of the perpendicular velocity. In this section an alternative derivation of the perpendicular velocity is presented. Instead of measuring the ion saturation current at different rotation angles, one could align the probe surfaces parallel to the magnetic field. In this way one might expect the perpendicular flow to be directly proportional to the current measured by the probe surfaces. Usually, this kind of measurement is used as an indication of the presence of a perpendicular flow [30] [36] [44] [93]. Here the unique situation exist that in the experimental conditions considered the perpendicular velocity is known from section 4.6.2. The database contains one profile where the probe surfaces are almost parallel to the magnetic field. “Almost” indicates that the probe surfaces are aligned with respect to the toroidal direction, meaning that the inclination angle is the pitch angle α≈2.44°. The profile is limited to the region of high electric field, which is the region of interest. Due to experimental constraints, no data is available after the bifurcation. 1
0.8
0
0.6
0.4
r−a [cm]
0
ln(R)
0.2
−0.5
−0.2
−1
−0.4
−0.6
−1.5
−0.8
−1
0.75 0.8 0.85 0.9 0.95 t [s]
1
1.05 1.1
Figure 4.33. 2D contour plot of the upstream to downstream current ratio vs. time for the probe surface nearly parallel to the magnetic field.
4-40
The analysis of the profile by applying the 1D fluid model is not possible since the equations are not valid at such a small rotation angle. Nevertheless, it is instructive to calculate the ratio of the envelope of the up- and downstream currents. The result of this calculation is shown in Figure 4.33. There, one can clearly identify that the value of ln(R ) is strongly increased at the start of the high voltage, at 1.05s, over a narrow region. Since the probe surfaces are nearly parallel to the magnetic field this increase can only be due to the presence of a substantial perpendicular flow. To obtain the perpendicular velocity the following intuitive method is proposed. The concept is that the perpendicular velocity can be derived directly from the current of the upstream surface since the perpendicular flow should be directly collected by this collector. In Figure 4.34, a 2D contour plot is shown which represents the envelope of the upstream current as a function of radius and time. However, as outlined in the previous chapter, the ion saturation current is not defined for this angle since no conventional sheath is formed. Furthermore, the parallel flow can still have a substantial contribution to the collected current. In the polarization experiment this contribution can be estimated from the current collected before the application of the voltage ramp. As shown in section 4.6.5, Figure 4.27, the plasma rotates then mainly in the parallel direction. The following expression is therefore proposed: ∆Ii ,up = ( Ii ,up )
VE = 300V
− ( Ii ,up )
VE = 0V
= e ne v ⊥ A
(4.14)
The part of the upstream current, which is associated with the perpendicular velocity, is defined as the difference of the upstream current before and after the start of the electrode voltage ramp. The two chosen time steps are 0.98s and 1.12s, corresponding with respectively VE=0V and VE=300V. The two time slices together with their difference are shown in Figure 4.35.
4-41
0.09
0.08
0 0.07
0.06
0.05
−1
Iup [A]
r−a [cm]
−0.5
0.04
0.03
−1.5 0.02
0.75 0.8 0.85 0.9 0.95 t [s]
1
1.05 1.1
Figure 4.34. 2D contourplot of the upstream current vs. radius and time. To calculate the perpendicular velocity, the electron density and the probe surface are required. For the latter, a first approximation is to use the geometrical probe surface. The electron density is of more concern since it cannot be calculated from the probe current. Therefore the electron densities which are derived in section 4.6.3 in the L-mode obtained by the probe and the Li-beam are used. The resulting perpendicular velocities are plotted in Figure 4.36. The velocity has a negative sign in agreement with the convention that the flow is directed towards the surface. For comparison the perpendicular velocity obtained in section 4.6.2 is added. The two velocities show a remarkable agreement in width and magnitude. However, the interpretation of this result should be handled with caution. The model, which is used to calculate the perpendicular flow, is very simplified. Nevertheless, the fact that the perpendicular flow can be obtained by using the unperturbed density at infinity indicates that the flow is indeed directly collected by the probe surface. Although the underlying model is based on ‘intuition’, this method can be used to determine the perpendicular velocity. The benefit of this method is that the probe does not have to rotate with respect to the magnetic field. Moreover, the uncertainty concerning the value of Ti in the expression for the sound speed is not present.
4-42
0.1
0.08
I [A]
0.06
0.04
0.02
0
−1.5
−1
−0.5 r−a [cm]
0
Figure 4.35. Current collected by the upstream surface at the time step 0.98s (dashed line) and 1.12s (dashed-dotted line). The difference of the two currents is plotted as the solid line. The markers indicate the radial positions of the Mach probe.
−5
⊥
v [106cm/s]
0
−10
−15
−1.5
−1
−0.5 r−a [cm]
0
Figure 4.36. Perpendicular velocity for the probe surfaces nearly parallel to the magnetic field calculated by equation (4.14). In there, the density is taken obtained from the probe data (o, solid line) and the density from the Li-beam diagnostic (o, dashed-dotted line). For comparison, the perpendicular velocity obtained in section 4.6.2 is added (+, dashed line).
4-43
4.10. Conclusions. In this Chapter a large database for probe measurements has been presented. For the first time the electric field, parallel and perpendicular Mach number, electron density and temperature have been derived under the same plasma conditions. Hereby a global image is obtained of all the important parameters in the edge plasma. The electrode polarization experiment is shown to be an useful tool since it can create in a controlled way high radial electric fields in the edge plasma. The ensuing high flows in the H-mode enable to clearly demonstrate that the influence of a perpendicular flow is reflected on the ratio of the up- and downstream current when rotating the probe surfaces. The comparison of the calculated and the experimental radial electric field via the radial momentum equation shows the viability of the 1D fluid model. The experimental electric field profile is reconstructed by the parallel and perpendicular Mach number, computed by the application of the 1D fluid model on the ratio of the saturation currents. Further, the radial momentum equation gives insight in the structure of the electric field. The magnitude of the parallel and perpendicular flow contributions proves that in the H-mode the radial electric field is created mainly by the perpendicular flow. The shape of the edge plasma parameters changes drastically in the presence of a high radial electric field. The profiles in the H-mode, compared to the L-mode, show that the density gradient increases substantially. The electron temperature increases in the region of the high electric field due to the dissipated power of the electrode current in a narrow region. In the presence of a high electric field (H-mode), the density and the parallel Mach number profiles show significant poloidal asymmetries. This could be seen from measurements taken at the equatorial and at the top plane. The parallel flow is very sensitive to a radial current in the edge plasma. When the electrode is withdrawn or floating, a negligible radial current is present in the plasma and only a small rotation exist, vanishing towards the inner plasma. When a substantial radial current is imposed in the edge plasma, the parallel Mach numbers increases non- linear with increasing current. Large parallel Mach numbers have been measured in the L- and H-mode. In both cases, the profiles show a maximum at a position which differs from the electric field maximum and a minimum at the position where the electric field is high. In the H-mode, this minimum decreases. A last result of this Chapter concerns the measurement of the perpendicular flow when the probe surfaces are parallel to the magnetic field. Although the underlying method is of an ‘intuitive’ nature, the perpendicular velocity profile has been recovered. This method delivers an alternative way to determine the perpendicular velocity.
4-44
In this Chapter, the importance and the reliability of probes as a diagnostic tool in the edge plasma has been clearly demonstrated, despite the difficulty in interpreting the probe data.
5-1
Chapter 5 . The physics of flows induced by electrode polarization in the edge plasma of TEXTOR-94.
5.1. Introduction. The aim in this chapter is to discuss the physics of the flow profiles presented in the previous chapter. A theoretical fluid model has been developed by Cornelis and Van Schoor [21][85] to predict the velocities and the electric field in dependence of the radial current imposed by the biasing electrode. Later, compressibility was added in the model by Van Schoor [84] [86]. A summary of these theories will be given here. The database consisting of flow and electric field profiles offers then the unique possibility to benchmark these models and to identify the underlying physics of the flows. The originality of this Chapter consists in the completeness of the confrontation between theory and experiment. The results are reported in [82] and [84]. The models use a fluid approximation, for which the steady state continuity and total momentum equation are given by equations (2.1) and (2.4): ∇. ( nv ) = 0
(5.1)
(
)
∇. ( m.nvv ) = − ∇p − ∇.Π + JxB + Fneutrals ,
(5.2)
where v is the flow velocity, n the ion density, p the total pressure, Π the viscosity tensor, J the current density, m the ion mass and B the magnetic induction. As discussed in Chapter 2B, the collisionality regime of the edge plasma of TEXTOR-94 is the Plateau-regime. The last term on the right hand side of equation (5.2) is the neutral drag force which is given by:
5-2
Fneutrals = − ν .v
(5.3)
where ν is a drag coefficient yet to be defined. The toroidal and parallel projections of the total momentum equation (5.2) will be used to calculate vθ, vφ and Er . The 1D fluid model without compressibility in [21] and [85] will be reviewed in section 5.2. In addition to [85] the local velocities at the equatorial outboard plane, i.e. at a poloidal angle θ = 0o , will be calculated here in order to make the comparison with the measured flow profiles. It will be shown that the model cannot reproduce the measured toroidal flow and that compressibility of the plasma might have a large impact on the local velocities [82]. The influence of the variation of density on the local velocities has already been adressed by Hazeltine [31] [32] applying a fluid model. Neutrals and viscosity were not included in the model, leading to solutions interpreted as shocks in plasma density. Roshansky and Tendler [66] [67] estimated the poloidal variation of the density, caused by the presence of a strong radial electric field using a kinetic approach. As their model does not contain neutral friction, it cannot be applied directly. The effects of compressibility will be incorporated in the model in section 5.3. In section 4.8 of the previous chapter, poloidal asymmetries were reported for the toroidal Mach number and the density. In section 5.4, it is investigated whether the magnitude of the poloidal density variation can explain the observed asymmetries. In the final section 5.5 the case is treated where the electrode is retracted from the plasma. The measured profiles also in this case are confronted with theory. Recently large velocities parallel to the magnetic field were measured by Höthker [35] in Ohmic discharges. These results are in contradiction with the observations presented in the previous chapter. A possible explanation will be given for the large velocities in [35].
5.2. Flows for an incompressible plasma. In this section the theory of Van Schoor and Cornelis is reviewed. Since compressibility is not taken into account here, the equations can be simplified as all quantities can be considered as surface functions.
5.2.1. Surface averaged velocities. The starting point is the ion radial momentum and the continuity equation. They provide an expression for the perpendicular and poloidal velocities. The velocities will then be expressed in terms of two flux functions V(r) and F(r).
5-3
As shown in section 4.6 of the previous chapter, the perpendicular velocity is directly related with the radial electric field via the radial momentum equation which can be written as: v⊥ =
R cosα 1 ∂ pi −Er + Ro Bo en ∂ r
(5.4)
Since the magnetic surfaces are equi-potential, equi-pressure and equi-density surfaces, equation (5.4) can be written as: v⊥ =
R V (r ) cosα Ro Bo
(5.5)
with V(r) a surface function describing the driving force due to the electric potential and the pressure. The poloidal velocity is introduced via the continuity equation in curvi-linear coordinates and for an axi-symmetric configuration: n ∂ ∂ ( R.n.vθ ) = ( R.vθ ) = 0 θ ∂ rR rR ∂θ 1
(5.6)
Note that the derivative in the radial direction is neglected. This is warranted since the radial velocity is modelled by an anomalous diffusion. The anomalous diffusion is decreased by the electric field [45] and the radial velocity will become small. The poloidal velocity must then be of the form: vθ =
Ro F (r ) R Ro
(5.7)
with F(r) a flux function which can be understood by computing the flux surface average of the poloidal velocity: vθ =
F (r ) Ro
whereby the flux surface average value of a quantity X is defined as:
(5.8)
5-4
X =
1 R X ( r ,θ )dθ 2π ∫ Ro
(5.9)
F(r) and V(r) are the two unknown flux functions in the theory through which the other quantities will be expressed. Combining equations (5.5) and (5.7), the following expressions can be derived for the parallel and toroidal velocities using the projection relations (2.6) and (2.7): v|| = vφ =
R V (r ) 1 Ro F ( r ) − cos2 α Ro Bo sinα R Ro cosα Ro F ( r ) R V ( r ) − sinα R Ro Ro Bo
(5.10) (5.11)
Note that v|| and vφ are functions of V(r) and thus of Er. This pure geometric effect will have a large impact on the obtained parallel and toroidal velocities.
5.2.2. Toroidal momentum equation. The toroidal momentum equation provides an expression for the radial current density: Jr =
1 ν vφ Bθ
(5.12)
Herein the friction coefficient ν is approximated by [29]:
(
ν = n.m.nn . σ v = n.m.nn . 10−8 T 0.318 i
)
(5.13)
where nn is the neutral density and Ti the ion temperature expressed in eV. The friction coefficient is supposed to be constant on a magnetic surface. Integrated over a magnetic surface, equation (5.12) gives the current flowing through the surface: Ir =
1 1 R ν ∫ R 2vφ 2π rdθ = ν vφ .S , ΘBo Ro θ ΘBo Ro
(5.14)
5-5 where Θ = tgα is the pitch of the magnetic field, S the surface of a magnetic flux surface, i.e. S = 2π r .2π Ro . Defining now the surface function G(r): G( r ) =
R vφ , Ro
(5.15)
then, with equation (5.14) an expression for the radial current is obtained: Ir =
S ν G(r ) . ΘBo
(5.16)
G(r) is thus related to the surface averaged toroidal velocity. One sees that the effect of a radial current induced in the plasma is that it acts as a driving force setting up a plasma rotation in the toroidal direction. A link between G(r), F(r) and V(r) can be obtained by computing equation (5.15) by taking the surface average of equation (5.11) multiplied by R Ro : ΘG( r ) =
F (r ) V (r ) 3 2 1+ ε − Ro Bo 2
(5.17)
The velocities, equations (5.10) and (5.11) can now be expressed as functions of G(r) and V(r). Together with equation (5.5) this results in the following set equations for the velocities: R V (r ) 1 Ro vφ = G(r ). o + R Bo Θ R
3 2 R 1 + 2 ε − R o
Ro V (r ) 3 2 1+ ε ΘG(r ) + R Bo 2 R Θ V ( r ) Ro 3 2 R cos2 α v|| = G(r ). o 1 + ε − + 2 Ro sinα R sinα Bo R vθ =
(5.18) (5.19) (5.20)
To compute G(r) and V(r) a supplementary equation is needed. As will be shown in the next section, the parallel momentum equation is the most convenient choice.
5-6
5.2.3. Parallel momentum equation. Consider the expression for the total momentum equation (5.2) in which only parallel viscosity and neutral friction is retained since they are considered as main contributors to the radial current. The contribution of convection is negligible since the convection is function of the radial velocity, which is small as explained in the discussion of the continuity equation, equation (5.6). After multiplication with B and averaging this equation reads:
( )
B. ∇.Π
+ ν B.v = 0
(5.21)
The viscosity term can be written as [5]:
( )
B . ∇.Π
F (r ) = η. − v NEO R o
(5.22)
in which η is the parallel viscosity. v neo is called the neo-classical velocity [33] and is given by:
v neo = β
1 ∂ Ti , ZeBo ∂ r
(5.23)
β is a parameter which depend on the collisionality regime; β = −0.5 in the plateau regime. This velocity plays an important role in neo-classical theory. Indeed in the core region where no neutrals are present, the parallel momentum equation reduces to
( )
B . ∇.Π
= 0 . The average poloidal rotation attains now the value given by the neo-
classical velocity. For the viscosity η, a phenomenologic expression is adopted as the viscosity in the plateau regime [79]: π q.v th,i −U 2 1 ν i q2 e pm + 2 2 Ro 2 1 + U pm
η = Bo Θmi n
(5.24)
5-7
with U pm =
−Er the undimensional electric field, v th,i the ion thermal velocity, Bθ v th,i
and νi the ion collision frequency defined by equation (2.6) From equation (5.24), it is seen that the viscosity can be destroyed by the electric field. The second term in the momentum equation is calculated by using equation (5.20), resulting in:
ν v||B =
ν B0 ε2 1 V (r ) 1 cos2 α 2 − G( r ) (1 + ) + 1 + 2ε cos α 2 cos α Bo sinα sinα
(
)
(5.25) Combination of equations (5.22) and (5.25) results in an expression for the parallel momentum equation: F (r )
η
Ro
ν B0 1 V (r ) 1 cos2 α ε2 2 − VNEO + − G(r ) (1 + ) + 1 + 2ε = 0 2 cos α Bo sinα sinα cos α
(
)
(5.26)
5.2.4. Link between the radial current and the radial electric field. With the ingredients described in the previous sections, the link between a radial current and an electric field can be understood. When a plasma is being charged, the plasma reacts by setting up an electric field which goes hand in hand with perpendicular flows by virtue of the radial momentum balance. This field would grow indefinitely, unless the plasma reacts by setting up a compensating return current, Ir. In the case of biasing, the electrons extracted from the plasma by the electrode constitute an experimentally measured electrode current IE flow in the external circuit. The plasma must drive a reaction current Ir, between the outer radius of the electrode and the limiter radius a satisfying the following condition:
J r S ( r ) = IE
(5.27)
where S(r) is the area of a magnetic surface. The mechanisms by which the return current is generated is quite complicated. The radial ion momentum equation imposes that the combination of the electric potential and ion pressure gradients are balanced by a vxB force so that a rotation in the perpendicular direction emerges. The parallel momentum equation contains only passive terms and redistributes via the viscosity the velocity between the parallel and
5-8
the toroidal direction. This toroidal rotation balances, via the neutral drag, the JxB force in the toroidal momentum equation, thus provoking a radial current which has to compensate the external imposed current. It is important to note that even when no biasing is applied, particle losses take place, generating a net current Iloss. This loss current is made up of the loss of banana ions, hereby charging the plasma negatively. This loss current is small compared to IE and is therefore neglected in equation (5.27). The case where only a loss current is present will be studied in more detail in section 5.5.
5.2.5. Method of solution. It is the aim to reconstruct the measured electric field and flow profiles by solving the toroidal and parallel momentum equations, equations (5.16) and (5.26), i.e. to find solutions for F(r) and V(r). Therefore density, temperature and radial current values are assumed to be provided by the experiment. The neutral density will be used as a fitting parameter for the damping of the electric field. The following strategy will therefore be applied: With a given neutral density profile, G(r) can be directly derived from the toroidal momentum equation, i.e. equation (5.16). An expression for V(r) can then be obtained by eliminating G(r) between the latter equation, and the parallel momentum equation, i.e. equation (5.26). V(r) results in:
V (r ) = Bo
−
Ir ΘBo Sν
Boν ε 2 η Θ + 1 + + ηv neo 2 cos2 α . ε2 B ν η 1 + + o κ 2 cos α
(5.28)
with
κ =
1 cos2 α 1 + 2ε 2 − sinα sinα
(
)
F(r) is then computed by equation (5.17). Once the neutral density and the two flux functions are known, the local velocities at the position where the probe is located
(θ = 0 ) can be computed with equations (5.18)-(5.20). Note that the viscosity coeffio
cient η in equation (5.28) is a function of the electric field. Therefore, a solution for V(r) can only be obtained in an iterative way. The measured electric field is taken as the starting value.
5-9
5.2.6. Dependency of the local toroidal flow on the electric field. A detailed comparison between theory and experiment will be given in section 5.3.4. At this stage only the comparison for the local toroidal velocity profiles in the L- and H-mode are important. The considered profiles are plotted in Figure 5.1a and b. From Figure 5.1 it can be concluded that a large discrepancy in magnitude exists between the calculated and measured local vφ-values. Moreover, both of the calculated profiles show a maximum at the position of maximal Er, which is not found experimentally. Indeed, the value of vφ in the experimental profiles is suppressed at the position of maximal Er and shows a maximum at a position deeper in the plasma. In the model, the maxima in the calculated vφ profiles result from a pure geometrical effect as can be seen by analysing equation (5.18). Writing this expression to 2 O(ε ) in terms of the aspect ratio gives:
vφ (r ,θ ) = (1- ε cosθ ).G(r ) −
2ε V (r ) cosθ Θ Bo
(5.29)
the value of which at the outboard equator ( θ = 0o ) becomes:
vφ ( r ,θ = 0°) = (1 − ε ).G(r ) −
2ε V ( r ) . Θ Bo
(5.30)
The latter equation shows that at small pitch angles (in the case of TEXTOR-94 α=2.44°), V(r) and hence Er has a strong influence on the local toroidal velocity. The higher Er grows, the more the theoretical local velocity will differ from the surface average value. According to equation (5.15), G ( r ) ; vφ : G(r) should not at all be af-
fected by Er and is in that respect in line with the experimental observation. What is at stake here is the insistence on sticking to the magnetic surface as a constant density surface. As shown in [66] [67], at high poloidal rotation velocities, a significant poloidal modulation of the plasma density can occur which in first approximation results in a cancelling of the second term on the right in equation (5.30). The inclusion of the poloidal density variation is the subject of the next section.
5-10
9 8 7
vφ [10 cm/s]
6
6
5 4 3 2 1 0 −5
−4
−3
−2 −1 r−a [cm]
0
1
9 8
6
vφ [10 cm/s]
7 6 5 4 3 2 1 0 −5
−4
−3
−2 −1 0 1 r−a [cm] Figure 5.1a. and b. Toroidal velocity vφ at θ = 0o vs. radius r-a in L- and H-mode. Comparison between measured (o) and calculated vφ (∗). The values of the calculated vφ in the H-mode are divided by a factor 4.
5-11
5.3. Flows for a compressible plasma. In this section, the addition of the compressibility by Van Schoor [84] [86] in the model is summarised. Expressions for the velocities will be derived, consisting of a surface averaged part and a poloidally varying part. The surface averaged toroidal and parallel momentum equation of the previous section can still be used to compute the surface averaged part of the velocities. An expression for the poloidal varying part is obtained via the variable part of the parallel momentum equation.
5.3.1. Notation. It will be assumed that all quantities X are composed of a poloidal constant part (only function of the minor radius r) and a poloidally dependant part (function of the poloidal angle θ as well as of r): X ( r ,θ ) = X (r ) + X%( r ,θ ) .
(5.31)
For the poloidally constant part the flux surface average will be taken defined by equation: X (r ) = X =
1 R X (r ,θ )dθ , 2π ∫ Ro
(5.32)
so that the variable part can be computed as: X%(r ,θ ) = X (r ,θ ) − X (r ) .
(5.33)
The assumption is now that X% X is at most of the same order as the inverse aspect ratio ε = r Ro , which will be used as a small parameter. All corrections of O(ε2) will be neglected. The neutral friction coefficient and the viscosity are proportional to the ion density (see equations (5.13) and (5.24)) and can be written in the form:
ψ = nψ ∗ = ( n + n%)ψ ∗ ,
(5.34)
where ψ represents the neutral friction coefficient or the viscosity, whereas ψ* is the density independent part of these parameters.
5-12
5.3.2. Poloidal variation of the velocities. The continuity equation is given by equation (5.6). Here the density is considered to be of the form of equation (5.33) so that the continuity equation now can be written as:
( n + n%) . (vθ
+ v% θ ) =
f (r ) R
(5.35)
or, to O(ε2), as: % nvθ + nv% θ + nvθ =
f (r ) R
(5.36)
It is clear that nvθ =
f (r ) , Ro
(5.37)
so that one can pose: vθ =
1 f (r ) F (r ) = , n Ro Ro
(5.38)
hereby defining the flux function F(r) in analogy with equation (5.8). The variable part of the poloidal velocity then follows from equation (5.33): v% θ =
F (r ) Ro n%F (r ) − 1 − Ro R n Ro
(5.39)
The perpendicular velocity is defined via the radial momentum equation and is given by equation (5.4). To compute the surface averaged value, the radial electric field Er will be taken poloidally constant. To simplify the calculations the pressure gradient term will be dropped. For the H-mode this simplification can be justified by noting that the latter term is always much smaller than the electric field. To O(ε2) one then finds:
5-13
v ⊥ = cosα
V (r ) Bo
v%⊥ = cosα
V (r ) R − 1 . Bo Ro
(5.40)
Application of the projection relations allows to compute the other velocity components: vφ =
1 F (r ) Ro V ( r ) R n%F ( r ) − − Θ Ro R Bo Ro n Ro
(5.41)
v // =
1 F ( r ) Ro n%F (r ) cos α V (r ) R − − Θ Bo Ro sinα Ro R n Ro
(5.42)
The poloidal and perpendicular velocities are given by: vθ =
F (r ) Ro n%F (r ) − Ro R n Ro
V (r ) R cos α . v⊥ = Bo Ro
(5.43)
Note that equations (5.41)-(5.43) are expressed as functions of F(r) and V(r). With equation (5.17) they can be written in terms of G(r) and V(r).
5.3.3. Solution for n%n . An approximated analytical expression for the density perturbation n%n is computed from the variable part of the parallel momentum equation. n%n is found to be of the form: ∆ F (r ) 2ε 1 + 2 2 q Ro Ro n% cosθ =− 4 ∆ F (r ) V (r ) n + 3 q 2Ro2 Ro Bo
The viscosity and neutral friction coefficients are introduced via the parameter
(5.44)
5-14
∆=
η∗ ν∗
(5.45)
At high electric field values the viscosity is destroyed by the field (see equation (5.24)). This results in a small value for ∆. It follows that for ∆ → 0, equation (5.44) results in the same limit for the density variation as Rozhansky and Tendler [66] [67] calculated, namely: n% = −2ε .cosθ . n
(5.46)
In Figure 5.2 the comparison between the approximated and high field solution for n%n at θ = 0o in the H-mode is shown. The two solutions show that the correction due to the poloidal density variation will become significant in the region where the electric field peaks. The approximated solution shows also an unexpected significant density variation in regions where the electric field is small. This is due to the fact that in the derivation of equation (5.44) the diamagnetic term was not taken into account. The contribution of the diamagnetic term becomes important for low electric field values. 0 −0.1
∼ − n/n
−0.2 −0.3 −0.4 −0.5
−5
−4
−3
−2 −1 r−a [cm]
0
1
Figure 5.2. Poloidal variation n%n vs. radius r-a in the H-mode at θ = 0o : approximated solution (dashed line) and high Er limit value (solid line).
5-15
To demonstrate that indeed the dependency of the local toroidal velocity on V(r) disappears at the high electric field limit by the poloidal density variation, equation (5.46) is introduced in equation (5.41). This results to O(ε2) in: vφ ( r ,θ ) = (1 + ε cosθ ).G(r )
(5.47)
where equation (5.41) is written as a function of G(r) and V(r) to make the direct comparison with equation (5.30). The toroidal velocity is now only function of G(r) as one would expect.
5.3.4. Results and discussion. In this section the results provided by the experiment and the theoretical model are discussed. The comparison is performed for the L-mode and H-mode. Further, the discussion holds for the local situation at the equatorial plane, i.e. θ = 0o . The outline for this section is such that first the relevant experimental profiles are presented. Then the strategy of the fitting procedure is described, followed by a discussion of the obtained results. Experimental parameters. The relevant experimental parameters are the local electron density, electron temperature, toroidal and poloidal velocity, described in the previous chapter for θ = 0o . They are plotted in Figure 5.4-Figure 5.8. The left (a) and right (b) column represent respectively the L- and H-mode case. The velocities are calculated from the Mach numbers by using v|| = M|| .cs and v ⊥ = M⊥ .cs , whereby the sound speed is de-
fined by cs =
(Te + Ti )
mi with Ti = Te .
The Ir -profile (Figure 5.3) is thought to have the following structure. The electrode’s conducting tip is situated between −5cm < r − a < −3.5cm , so that between these two radii the current increases from zero to its maximal value, Ir=IE=163A and 110A respectively in the L- and H-mode. The value of Ir remains then constant between the outer most tip of the electrode and the separatrix, i.e. for radial positions −3.5cm < r − a < 0cm . Once outside of the separatrix, the radial current is converted into a poloidal current flowing to the limiter. To account for this effect, it is assumed that the radial current in this region decreases exponentially with a decay length of about λ=0.25cm. Justification of this length follows in the fitting procedure. The other parameters of importance for the modeling are:
Ro = 1.75m
5-16
Bo = 2.33T q(0) = 0.88 q( r − a ) = 6.7
Fitting procedure. As outlined in section 5.2.5, the aim is to reconstruct the measured electric field and the toroidal and poloidal flow profiles by retrieving solutions for the flux functions F(r) and V(r). Once these functions are known, the radial profiles for the poloidal density variation, the toroidal and poloidal velocities are obtained by equations (5.44), (5.41) and (5.43). The key equation to be solved is the expression for the flux function V(r), equation (5.28). The input parameters in this equation are the radial electric field Er, the ion temperature Ti, the neutral density nn and the ion density n: Er and Ti enter the equation via the viscosity η, equation (5.24), nn and n via the neutral friction coefficient ν, equation (5.13). Several considerations have to be taken into acount concerning these input parameters: • V(r) represents the surface averaged electric field. The measured Er is therefore a starting value for the viscosity in equation (5.28) and V(r) is solved in an iteratif way. • Since the expression for V(r) is derived from the surface averaged toroidal and parallel momentum equation, the input ion density is the surface average value. On account of the quasi-neutrality, the ion density equals the electron density. The electron density then used for the modeling is the measured local density at θ = 0o which is enhanced such that the calculated local density equals the Libeam density. The Li-beam density rather than the probe density is chosen as the reference for the absolute value of the density since this density is almost independent of the temperature. In the calculation of the density from the probe data, it is assumed that Te=Ti. Also, the collecting area A is taken as the total area in which the collectors are embedded. These assumptions could cause an error in the absolute value of the probe density. The local density is calculated from the surface average density by using equation (5.31): n% nlocal θ = 0o = n. 1 + n
(
•
)
(5.48)
The neutral density will be used as a fitting parameter for the damping of the electric field. Inside the separatrix, the neutral density is thought to have an exponential profile of the form:
5-17
r − rEr ,max nn = nn,Er ,max .exp , λn
(5.49)
where rEr ,max is the radial position of maximum Er and the value nn,Er ,max is chosen such as to reach fair agreement between the measured and computed electric field. The proper choice of the decay length of this profile is explained later. In the SOL, the neutral density attains a constant value given by equation (5.49) at the radius r − a = 0 .
160
160
140
140
120
120
100
100
Ir [A]
180
Ir [A]
180
80
80
60
60
40
40
20
20
0 −5
−4
−3
−2 −1 r−a [cm]
0
0 −5
1
−4
−3
−2 −1 r−a [cm]
0
1
0
1
Figure 5.3a and b. Radial current Ir vs. radius r-a in L- and H-mode.
80
80
60
60
Te [eV]
100
Te [eV]
100
40
40
20
20
0 −5
−4
−3
−2 −1 r−a [cm]
0
1
0 −5
−4
−3
−2 −1 r−a [cm]
Figure 5.4a and b. Electron temperature Te vs. radius r-a in L- and H-mode.
5-18
11
x 10
7
2
n
1.5
3 2
2
0.5
1 0 −5
−4
−3
−2 −1 r−a [cm]
0
1
1.5
4
nLi
3
1
nLi
nprobe
−3
−3
nn
4
2
5 n [cm ]
n
2.5
n
nlocal
6
nlocal
x 10 3
x 10
7
2.5
5 n [cm ]
8
n [cm−3]
6
11
12
x 10 3
1
nprobe
nn
1 0 −5
0
−4
−3
−2 −1 r−a [cm]
0
1
Figure 5.5a and b. Surface averaged density n (dashed-dotted line), calculated local density nlocal (dotted line), Li-beam density (+), measured probe density (solid line) and neutral density (dashed line) nn vs. radius r-a in L- and H-mode.
700
200
600
Er [V/cm]
500
100
r
E [V/cm]
150
400 300 200
50
100 0 −5
−4
−3
−2 −1 r−a [cm]
0
1
0 −5
−4
−3
−2 −1 r−a [cm]
0
Figure 5.6a and b. Electric field Er vs. radius r-a in L- and H-mode. Comparison between measured (o) and calculated Er (solid line).
1
0.5 0
nn [cm−3]
12
8
5-19
5
2
4
v [106cm/s]
1
3
θ
6
vθ [10 cm/s]
1.5
0.5
0 −5
2
1
−4
−3
−2 −1 r−a [cm]
0
0 −5
1
−4
−3
−2
−1
0
1
r−a [cm]
9
9
8
8
7
7
6
6
vφ [10 cm/s]
φ
5
6
6
v [10 cm/s]
Figure 5.7a and b. Local poloidal velocity vθ at θ = 0o vs. radius r-a in L- and Hmode. Comparison between measured (o) and calculated vθ ,with n%n (∗) and without n%n (dashed line).
4
5 4
3
3
2
2
1
1
0 −5
−4
−3
−2 −1 r−a [cm]
0
1
0 −5
−4
−3
−2 −1 r−a [cm]
0
Figure 5.8a and b. Local toroidal velocity vφ at θ = 0o vs. radius in L- and H-mode. Comparison between measured (o) and calculated vφ, with n%n (∗) and without n%n (dashed line). The calculated profile in the H-mode without n%n is divided by a factor 4.
1
5-20
Discussion. Er -, vφ - and vθ - profiles. Figure 5.6-Figure 5.8 allow to compare theory and experiment. From Figure 5.6 is it clear that the experimental electric field is rather well reproduced by the model. For the flow profiles the attention goes to the influence of the poloidal density variation on the flow profiles. The most important effect of the inclusion of the poloidal density variation is seen on the toroidal velocity profiles (Figure 5.8). In contrast to the situation without inclusion of n%n , the calculated toroidal velocity profiles in both L-
and H-mode show now a maximum at a radial position r − a = −2.5cm . One should conclude that a fair agreement is reached between the calculated and experimental toroidal velocities and that the discrepancies which remain are secundary order effects. The correction due the poloidal density variation is also reflected on the poloidal velocity profiles (Figure 5.7). Whereas the correct shape and order of magnitude is reached without n%n , the inclusion of n%n improves the agreement of the calculated and experimental poloidal velocity profiles in both L- and H-mode. Ir- profiles. The structure of the Ir-profile in the region of the electrode and in the SOL layer has a specific influence on the calculated profiles: At the position of the electrode, i.e. −5 < r − a ( cm ) < −3.5 , the current is assumed to increase linearly from 0A to its ma-
ximum value. This leads to the linear increase of the velocities in this region. In the SOL, for radii r − a > 0cm , the current is assumed to decrease exponentially. The fall-off length is chosen such that the part of Er and the velocities, which lie in the SOL is shaped by the fall-off length of the radial current. Since this is only an assumption, one should be careful in interpreting the results of the model in that region.
nn - profiles. In the neutral density profiles, two parameters have to be adjusted: The value of the neutral density at position of maximal electric field nn,Er ,max and the decaylength λn. The value of nn,Er ,max is adjusted until a fair agreement between the experimental and calculated electric field is reached. The electric field in the L-mode is lower than in the H-mode, resulting in a higher neutral density value at rEr ,max in the L-mode compared to the H-mode. Deeper in the plasma, a region exist where the toroidal velocity shows a maximum. The toroidal momentum equation imposes there the value of the neutral density. The product of the ion and neutral density has to be high in order to reach the measured toroidal velocity. The required high neutral density results in a slow decay of the assumed exponential profile.
5-21
The assumed neutral density value at the LCFS decreases from nn=1.3x1011cm-3 in the L-mode to nn=0.3x1011cm-3 in the H-mode. These values and their relative difference are in good agreement with the experimental observations of the neutral density in the ALT-II limiter [21].
5.4. Poloidal asymmetries. Knowing the magnitude of the poloidal density variation from section 5.3.3, a study of the poloidal asymmetries in the H-mode observed for the density and toroidal Mach number profiles (see section 4.5 of the previous chapter) is now possible. Considering expression (5.44) for the poloidal density, the density variation should cancel for a poloidal angle θ = 90o . Equation (5.31) expresses then that the quantities measured at the top plane represent the surface averaged quantities. 6
Vφ [106 cm/s]
5 4 3 2 1 0 −5
−4
−3
−2 −1 0 1 r−a [cm] Figure 5.9. Measured local toroidal velocity vφ at θ = 0o (o) and θ = 90o (+, solid line). The dashed line represents the approximated local toroidal velocity calculated at θ = 90o from θ = 0o with equation (5.50).
Since G(r) is related to the surface averaged toroidal velocity, the toroidal velocity at θ = 90o can be directly calculated from the local vφ at θ = 0o by using equation (5.47) for the high field limit:
5-22
(
vφ r ,θ = 0o
(1 + ε )
) = G (r ) ; v
φ
( r ,θ = 90 ) o
(5.50)
From Figure 5.9, where the different profiles of vφ are plotted, it can be concluded that vφ calculated with equation (5.50) approximates vφ measured at the top plane. The density at the equatorial outboard plane can be calculated from the averaged profile using equation (5.31): n(r ,θ =0o ) = n(r ,θ =90o ). (1 + n%n )
(5.51)
The comparison of the different density profiles is shown in Figure 5.10.
3
x 10
12
2.5
ne [cm−3]
2 1.5 1 0.5 0 −5
−4
−3
−2
−1 0 1 r−a [cm] Figure 5.10. Measured density n at θ = 0o (o, dashed line), θ = 90o (dashed-dotted
line). and n calculated at θ = 0o from θ = 90o with equation (5.51) (o, solid line).
In the region where Er is high, the calculated density equals the measured density at the equatorial plane, showing that in this region the correct density correction is obtained by the analytical expression for n%n . The calculated density profile shows also a steepening of the density gradient although not so steep as the measured profile. This is due to the moderate accuracy of the model for low Er values since the pressure gradient is not included in the derivation of n%n .
5-23
5.5. Case IE=0. When the electrode is retracted, it is true that IE=0, but a small loss current still exists in the plasma, which is neglected in the previous sections. This current originates from ion orbit losses. The toroidal flow which it generates is the subject of this section. In the core region, far away from the separatrix, where the plasma is in the banana regime, the ions follow closed orbits. The projection of these orbits resembles a so-called banana (see Figure 5.11 trajectory (a)). The trajectories however have a certain width, which is narrow in comparison with the poloidal extent of the banana. If the particle is close to the separatrix, due to this width it can cross the separatrix thus coming in the region where the field lines are open and so being lost at the limiters or divertor plates (see Figure 5.11 trajectory (b)). Analytical expressions for the magnitude of the so emerging current can be found in the literature [68]:
Jloss = −e
( a − r )2 ∂n π . .r1 exp − ∂t 2 r12
(5.52)
where ∂n ∂t is the rate at which ions are lost by the plasma: 1 * 4 G ∂n 2 U exp ν α = n.ν ii − + ( ) i S pm 1 ∂t ν * + (α U )4 2 i S pm
(5.53)
with
ν i* = ν i
−1
Rq Vthi ε
3
2
1 , αS = 2 ε 2 (1 + ε ) 2 .
G is a geometrical factor close to one and r1 is the maximum radius from which ions are lost. The separatrix is situated at r = a . Due the difference in Larmor radius between ions and electrons the plasma will get a negative charge so that a radial electric field will be produced resulting in an inward current balancing the ion orbit losses. This is necessary as the plasma cannot loose charge indefinitely. From equation (5.52) and (5.53) it is clear that in the case of biasing, when Er and hence Upm reaches high values, the loss current becomes indeed negligible.
5-24
Figure 5.11. Different banana orbits in a tokamak.
0.6 0.4
M||
0.2 0
−0.2 −0.4 −0.6 −5
−4
−3
−2 −1 0 1 r−a [cm] Figure 5.12. Parallel Mach number for IE=0: Experimental values (+, dashed dotted line), measurements of Höthker (o, solid line). For comparison the measured Mach number in biasing conditions where the electrode was grounded (IE=40A) is added (∇, dashed line).
In [85], the ion loss current is estimated to be Iloss=11A. The toroidal and parallel momentum equations are solved together with equation (5.52) and setting -Ir=Iloss. As one could expect, a narrow zone of toroidal rotation in the magnetic field direction is set up close to the separatrix, due to the presence of the loss current. The
5-25
perpendicular and poloidal rotation stay fairly small since the electric field is also small. The magnitude of computed Mach number is of the order of Mφ=0.03. Now the comparison of theory and experiment can be performed. In section 4.7 of the previous chapter, the parallel Mach number was investigated as a function of the radial current in the edge plasma. The profile for the case where the electrode was retracted is reproduced in Figure 5.12. It is assumed that the parallel Mach number equals the toroidal one, since the pitch angle at the considered radii is small. Although the magnitude is much higher then the predicted theoretical value, the measured toroidal flow is in the direction of the magnetic field and it decreases towards the inner plasma. This experimental result is in sharp contrast with the values reported by Höthker in [35] (see Figure 5.12). Although a direct comparison may not be straightforward since the plasma current and the density were substiantially higher, the Mach number profile in [35] disagrees in magnitude and direction. However, a fair agreement is reached with the profile obtained under biasing conditions for which the electrode was grounded (see Figure 5.12). In this case an electrode current IE=40A was measured. As explained in the previous sections, the toroidal rotation under biasing conditions results from a JxB -force. Although the Mach number profile in [35] is not obtained under these biasing conditions, it is stated that a poloidal limiter, floating with respect to plasma potential, was inserted in the plasma at a radius rlim,pol = rALT-II –2.7cm. If one surmises that the limiter was in reality grounded, then the results become consistent with the results presented here. The ascertainment in [35] that the poloidal limiter does not affect the Mach number is based on one measurement in which the limiter was retracted. However, this measurement point is taken at a radius r = rALT-II + 3.4cm where the JxB -force, resulting from the induced radial current, is not effective.
5.6. Conclusions. In this chapter the measured flow profiles were confronted with a theoretical fluid model of Van Schoor and Cornelis. When a radial current is induced by biasing in the edge plasma a radial electric field is created. To prevent this electric field from growing infinitely, damping mechanisms as neutral interaction and viscosity set up a compensating return current. The neutral interaction enters via the toroidal momentum equation, the viscosity via the parallel momentum equation. The flow profiles are then calculated from the surface averaged toroidal and parallel momentum equation. It is shown that with these ingredients the model is not capable of reproducing the measured toroidal flow. In contrast to the calculated profile, the measured rotation shows a minimum in the region where the electric field becomes important.
5-26
It is demonstrated that the poloidal density variation has a large influence on the flows. Together with the neutral friction and the viscosity a good agreement is reached between theoretical and experimental profiles. In the toroidal flow profile, two distinct regions can be identified. One region is situated in the region of high electric field. There the poloidal density variation has a large impact on the rotation and suppresses the rotation. Deeper in the plasma the toroidal rotation shows a maximum. These high velocities are explained by the toroidal momentum equation, which demands a relatively high amount of neutrals. The poloidal flow profile on the other hand is shown to be directly proportional to the electric field. The agreement is improved by the inclusion of the compressibility term. The observed poloidal asymmetries for the density and the toroidal rotation are explained by the poloidal density variation. The correct magnitude of the asymmetries is reproduced in the region of high electric field. In the case where no biasing is applied, a small loss current is present in the plasma. The theory predicts Mach numbers which are lower, but the direction of the measured rotation is correct. The measurements of Höthker are in sharp contrast with these predictions and may be caused by the poloidal limiter acting as an electrode, thereby inducing an radial current in the edge plasma.
6-1
Chapter 6. General conclusions.
6.1. This thesis.
The main realisations of this thesis are: • The development of a new 1D fluid model, which allows to determine besides the parallel velocity, also the perpendicular velocity of the plasma flow by application on Mach probe data. • The measurement of a large database of edge parameters under biasing conditions. For the first time, the parallel, perpendicular velocities and the electric field are determined under the same conditions. The database allows to demonstrate the validity of the 1D fluid model. • The identification of the mechanisms, by which the flows in the edge plasma are set up during biasing discharges. The first contribution consists in the modelling of the parallel and perpendicular transport of particles towards an inclined probe surface. The new model transforms the 3D fluid continuity and momentum equations into 1D equations, which describe the transport of particles parallel to the magnetic field. The cross-field transport, due to diffusion and a perpendicular flow, acts as source terms in the 1D equations to balance the parallel influx. The boundary condition, which appears in the transport equations, defines the MPSE as a Mach surface. Therefore the parallel and perpendicular flow has to combine in an intrinsic way at the MPSE to fulfil the boundary condition. The influence of the perpendicular flow is only reflected on the boundary condition when the probe is inclined. For the case that the surfaces are parallel to the magnetic field, the contribution vanishes. Therefore a necessary condition to measure the perpendicular flow is the inclination of the probe surfaces.
6-2
The model provides an analytical expression for the current collected by an inclined probe surface. This expression can be used to derive the perpendicular and parallel Mach number from the ratio of the up-and downstream ion saturation currents collected by a Mach probe, orientated at different angles with respect to the magnetic field. The second contribution follows from the experimental part of this thesis. A large database of edge parameters was obtained in electrode biasing discharges. For the first time profiles of the parallel and the perpendicular flow and the electric field were obtained under the same conditions. The flow velocities are obtained from the application of the 1D fluid model on the ratio of the up- and downstream saturation currents, measured by the Mach probe. The polarization set-up served as a useful tool in the validation of the obtained Mach numbers. It offers the unique possibility to create high poloidal flows in the edge plasma. These flows are generated by electric fields which can be measured by the Mach probe. This enables one to validate the Mach numbers by comparing the measured electric field with the field calculated via the radial momentum equation. In the H-mode, where the electric field and the associated flows are significant, it is shown that with the obtained Mach numbers the measured electric field could indeed be constructed. It is then concluded that mainly the poloidal rotation sustains the electric field. From the analysis of the I-V characteristic of the raw Mach probe data, profiles for the electron density were obtained. These results reveal the typical features in the H-mode of the steepening of the density gradient at the radial position where the electric field increases and the increase of the electron temperature in the region where the electric field is high. Poloidal asymmetries for the density and the parallel Mach number could be observed by comparing the results from Mach probe measurement at two poloidal positions. A final result is derived from the current measurement with the collectors of the Mach probe orientated at grazing incidence to the magnetic field. It is demonstrated that the current, collected by the probe at this position, is directly related to the perpendicular flow. This method is an alternative way to determine the perpendicular velocity. The third contribution concerns the mechanisms, which are responsible for setting up the flows in the edge plasma in polarized discharges. The experimental profiles were therefore confronted with a theoretical fluid model of Van Schoor en Cornelis. The model links the radial current to the electric field. The toroidal and parallel projection of the total momentum equation, in which neutral friction and parallel viscosity are retained, are used to predict the flow and the electric field profiles in L- and H-mode.
6-3
The first result of this confrontation is that the model could not reproduce the toroidal flow. The calculated local toroidal flow profile has a strong dependency on the electric field, which is not observed experimentally. It is demonstrated that the compressibility of the plasma has a large influence on the flow profiles, especially the toroidal flow profile. Indeed, when the poloidal density variation is included in the model, a fair agreement is reached between the experimental and calculated profiles. Both the calculated and measured local toroidal flow profiles show now a maximum at a position, which differs from the electric field, maximum. This high toroidal flow is explained by the interaction of the neutral density via the toroidal momentum equation. The poloidal density variation further explains the magnitude of the observed poloidal asymmetries for the toroidal flow and the density.
6.2. Future research. The 1D fluid model for the determination of the parallel and perpendicular flow should be extended such that it becomes valid for smaller rotation angles. This requires the incorporation of parallel viscosity and neutral friction in the model such that higher parallel Mach numbers are allowed to fulfil the boundary condition at the MPSE than in the present case. Additional measurements at smaller angles can give insight in the behaviour of the boundary condition at these angles. Also a more detailed cross-check with the PIC-code can provide additional information in this study. For the application of the 1D fluid model, the ion saturation current needs to be measured with the Mach probe positioned at different angles with respect to the magnetic field. These measurements could only be obtained by a series of experimental shot days in which the probe was mounted at a different angle with respect to the magnetic field. This procedure limits the practicability of the method to determine the perpendicular flow. A Gundestrup type probe, consisting of an array of collecting surfaces at different angles with respect to the magnetic field would satisfy the requirements of high practicability. The question of the interfering flux tubes can be solved experimentally, since a multi-pin probe can be viewed as a collection of double probes, orientated at different angles with respect to the magnetic field. One way of obtaining the perpendicular Mach number is by applying the 1D fluid model on the ion saturation currents of each pair of surfaces in a series of reproducible discharges, as is done in this thesis. This result should then be compared with the Mach number obtained from the measurement in which all collectors are biased at the same time.
6-4
The Mach numbers in the L-mode are determined during the ramping up of the electrode voltage. Their accuracy can be improved by obtaining them from discharges in which the electrode voltage in the plateau phase has a value VE=300V. In imitation of the parallel Mach number, also the perpendicular Mach number should be measured at a different poloidal location. Throughout the thesis the assumption Ti=Te is made. In chapter 4 it is shown that the uncertainty in the value of Ti has a large influence on the value of cs and the magnitude of the velocities. From the ion saturation currents and the Li-beam density an estimate for the value of Ti was obtained. Nevertheless, an independent measurement of Ti should reveal a more exact value of cs. This value should be taken into account in the calculation of the flow velocities and in the theoretical modelling of chapter 5.
Nederlandse samenvatting 1. Algemene inleiding. ......................................................................................................I 2. Situering van de dissertatie. ....................................................................................... III 3. 1D vloeistof-model voor de bepaling van de parallelle en loodrechte stroming door een schuine Mach-sonde. ................................................................................................ V 3.1. Langmuirsondes. .................................................................................................VI 3.2. Stroom gecollecteerd door een sonde-oppervlak, schuin geplaatst t.o.v. het magnetische veld, in de aanwezigheid van een parallelle en loodrechte stroming. .XI 3.3. Vergelijking van het 1D model met bestaande modellen. ..............................XVI 4. Databank van Mach-sonde randplasma parameters en validatie van het 1D vloeistofmodel. ..........................................................................................................................XIX 4.1. Experimentele opstelling................................................................................... XX 4.2. Analyse van de Mach-sonde gegevens. ........................................................ XXIII 4.3. Validatie van het 1D model. ........................................................................XXVII 4.4. Tijdsevolutie van het parallel en loodrecht Mach-getal. ..............................XXIX 4.5. Invloed van de radiale stroom op het parallelle Mach-getal. ........................ XXX 4.6. Poloïdale asymmetrieën. ...............................................................................XXXI 4.7. Alternatieve bepaling van de loodrechte stromingssnelheid........................XXXI 5. De fysica van de stromingen geïnduceerd door elektrode polarisatie in het randplasma van TEXTOR-94. .............................................................................. XXXIII 5.1. Stromingen voor een onsamendrukbaar plasma........................................ XXXIII 5.2. Stromingen voor een samendrukbaar plasma............................................ XXXVI 6. Besluiten en verder onderzoek................................................................................ XLI
I
1. Algemene inleiding. Het doel van het onderzoek naar gecontroleerde kernfusie is een technologie te ontwikkelen die kan bijdragen tot de wereldenergievoorziening op lange termijn. Dit onderzoek wordt uitgevoerd in laboratoria verspreid over de geïndustialiseerde wereld, met een grote mate van internationale samenwerking. De fusiereactie die het meest in aanmerking komt om fusie te realiseren in laboratorium omstandigheden is [88]: D + T → 24He + n + 17.6 MeV ( = 3.5 MeV nucleon )
Hierbij zijn de waterstof-isotopen (Deuterium (D of 21H ) en Tritium (T of 31H )), de Helium-isotopen ( 24He ) en neutronen n betrokken. Het Tritium wordt aangemaakt door middel van een reactie tussen Lithium (Li) en de neutronen die vrijkomen uit de fusiereactie: 7 3 Li
+ n → 24He + T + n − 2.5MeV
6 3 Li
+ n → 24He + T + 4.8MeV
Een eerste vereiste voor rendabiliteit van een fusiereactor is dat de reactiviteit van de reactie voldoende hoog moet zijn. Voor de eenvoudigst realiseerbare reactie, de D-T reactie, wordt in een mengsel van 50% deuterium en 50% tritium de reactiewaarschijnlijkheid pas groot genoeg bij een temperatuur van tenminste 10keV (ongeveer 108K). Bij een dergelijke temperatuur is het D-T-gas volledig geïoniseerd. Men noemt dit geïoniseerde medium een plasma. Men spreekt van thermonucleaire fusie. Een tweede vereiste van economische rendabiliteit wordt bepaald door een voldoende hoge waarde van nτE, waarin n de plasmadichtheid is en τE de energie opsluitingstijd. Voor een plasma met een dichtheid van n≈1020m-3 betekent dit dat τE ≥1s moet zijn. Éen van de bestaande ontwerpen van een fusiereactor is de tokamak (zie Figuur 1.1). Dit reactor-type is gebaseerd op het principe van magnetische opsluiting: geïoniseerde deeltjes zullen in eerste benadering magnetische veldlijnen volgen en kunnen zodoende opgesloten worden in een magnetische configuratie. Om een goede
II
opsluiting te realiseren worden torusvormige magnetische flux-oppervlakken gecreëerd waarin de magnetische veldlijnen helicoïdaal geschroefd zijn. transformatorkern spoelen voor het vertikaal veld
r spoelen voor het toroïdaal veld
helische veldlijnen
θ
Ro
Figuur 1.1. Schematische voorstelling van een tokamak: de plasmastroom, Ip, wordt geïnduceerd door een transformator en genereert een poloïdaal magnetisch veld Bθ. Een sterker toroïdaal veld Bφ is geproduceerd door externe spoelen die het vacuumvat omringen. Door het toevoegen van een vertikaal magnetisch veld wordt een stabiele configuratie bekomen.
III
2. Situering van de dissertatie. De experimenten in deze dissertatie zijn uitgevoerd op de tokamak TEXTOR94 (“tokamak EXperiment for Technology Oriented Research”). TEXTOR-94 is een middelgrote tokamak met een circulaire doorsnede. De grote en kleine straal, Ro en r (zie Figuur 1.1) bedragen respectievelijk 1.75m en 0.46m. Het onderzoek op TEXTOR-94 spitst zich toe op de studie van de plasma-wand interactie. Doordat het hete plasma in de rand van de tokamak in contact komt met de wand en hierdoor neutrale deeltjes in het plasma terechtkomen, heeft het randplasma een grote invloed op het plasma in de kern. Verschillende methodes, passieve en actieve, worden onderzocht om het randplasma te beïnvloeden. Een passieve methode bestaat in de installatie van een limiter zoals op TEXTOR-94 (zie Figuur 4.2). Dit is een structuur die in het plasma binnendringt en de buitenlaag afschraapt. Een actieve methode die in deze dissertatie aangewend wordt, is het beïnvloeden van het randplasma door polarisatie. Deze methode bestaat erin om elektrische velden in het randplasma te creëren door middel van externe middelen. Recentelijk werd aangetoond dat er een nauw verband is tussen het aangelegde veld en de H-mode [89]. De H-mode is een plasmaregime van hoge opsluiting, gekarakteriseerd door een plotse stijging van de energie- en deeltjesopsluitingstijd, tezamen met een stijging van het elektrische veld en de poloïdale snelheid. De verbetering van de opsluiting wordt toegeschreven aan de stabilisatie van de turbulentie door de afschuiving in de ExB stroming geïnduceerd door het elektrische veld. De stromingen en in het bijzonder de poloïdale stroming spelen dus een belangrijke rol in het randplasma. In het geval van TEXTOR-94 worden hoge elektrische velden en geassocieerde stromingen in het randplasma opgewekt door het aanleggen van een spanning tussen een in het plasma ingebrachte elektrode en de limiter (zie Figuur 4.2). Hierbij wordt een radiale stroom in het randplasma geïnduceerd. Van Schoor [85] en Cornelis [21] ontwikkelden een theoretisch model dat een verband legt tussen de radiale stroom en het radiale elektrische veld. Experimenteel onderzoek naar de stromingen in deze plasma condities wordt echter belemmerd door een gebrek aan geschikte diagnostieken voor de meting van de poloïdale stromingssnelheid. De enige methode die deze meestal zeer nauwe stromingsvelden kan meten met een voldoende hoge spatiale resolutie is een Mach-sonde
IV
(zie Figuur 2.1). In zijn meest eenvoudige vorm bestaat een Mach-sonde uit twee grafiet collectoren gescheiden door een isolator. Een Mach-sonde heeft het grote voordeel dat het gelijktijdig de elektronentemperatuur, elektronendichtheid en de snelheid van de plasmastroming kan meten. De bepaling van de snelheid is echter beperkt in de bestaande modellen tot de snelheid parallel met het magnetische veld. Er zijn recentelijk pogingen ondernomen om ook de loodrechte snelheid te bepalen met een Mach-sonde. De ontwikkelde modellen of methodes zijn echter gebaseerd op ‘intuïtie’ ofwel zijn ze bedoeld voor laag gemagnetiseerde plasmas. Er is dus noodzaak aan een gefundeerde methode om de loodrechte snelheid te bepalen. De belangrijkste doelstellingen van deze dissertatie zijn: • De ontwikkeling van een model dat toelaat om naast de parallelle, ook de loodrechte snelheid van de plasmastroming te bepalen met een Mach-sonde •
•
(Hoofdstuk 3) [83]. De validatie van het model door het toe te passen op Mach-sonde gegevens die bekomen zijn in plasma condities waarin een grote poloïdale plasma rotatie bestaat. (Hoofdstuk 4) [81]. Het identificeren van de onderliggende fysische processen waarmee de stromingen gecreërd worden in elektrode gepolariseerde ontladingen door de gemeten stromingen te vergelijken met een theoretisch model. (Hoofdstuk 5) [82][84]. isolator
collector
Figuur 2.1. Schematische voorstelling van een Mach-sonde. Twee collectoren zijn verborgen van elkaar door een isolator
V
3. 1D vloeistof-model voor de bepaling van de parallelle en loodrechte stroming door een schuine Mach-sonde. Het hoofddoel van dit hoofdstuk is een methode te ontwikkelen die bij toepassing op Mach-sondegegevens toelaat om de parallelle en loodrechte stroming te bepalen. Dit model is gerapporteerd in [83]. De driftsnelheid van de ionen wordt meestal gemeten door middel van een zogenaamde Mach-sonde. Deze bestaat uit twee sterk geleidende collectoren gescheiden door een isolator (zie Figuur 3.5). Het fundamentele concept achter de werking van een Mach-sonde is dat de stroming van de ionen kan afgeleid worden uit de verhouding R van de saturatiestromen stroomop- en stroomneerwaarts, Ii ,sat ,op en
Ii ,sat ,neer , gedefinieerd door de relatie: R=
Ii ,sat ,op Ii ,sat ,neer
= f (M,...)
(3.1)
Hierin is f een functie die afhankelijk is van verschillende parameters, het ongestoorde Mach-getal inbegrepen. Dit Mach-getal is gedefinieerd door: M=
v , cs
(3.2)
waarbij cs de geluidssnelheid voorstelt en v de stromingssnelheidsvector van de ionen. De componenten M|| en M⊥ zijn het parallelle en loodrechte Mach-getal. De opdracht bestaat er dan in om het transport van deeltjes naar het sonde-oppervlak te modelleren om zo een uitdrukking te vinden voor de functie f in vergelijking (3.1).
VI
3.1. Langmuirsondes. Voor een goed begrip van het gehele hoofdstuk wordt een overzicht gegeven van de theorie van Langmuir [51] [69] [80] [48] betreffende de enkelvoudige en dubbelsonde zonder magneetveld. Gewoonlijk is de sonde verbonden met een referentie-elektrode via een spanningsbron. De stroom gecollecteerd door de sonde wordt gemeten als functie van de aangelegde spanning. De relatie tussen stroom en spanning wordt de ‘I-V karakteristiek’ genoemd. Deze relatie laat toe om informatie te bekomen omtrent potentiaal, elektronendichtheid en -temperatuur van het ongestoorde plasma in de onmiddellijke omgeving van de sonde. I Ie,sat
(A)
(B)
V
(C) Vvl
Vp
Ii,sat
Figuur 3.1. I-V karakteristiek van een enkelvoudige sonde.
In een tokamak wordt een sonde enkelvoudig genoemd als de rol van de referentie-elektrode is overgenomen door de gehele geleidende wand of limiter (zie principeschets in Figuur 3.2). De I-V karakteristiek is schematisch weergegeven in Figuur 3.1: Een plasma is gewoonlijk in hoge mate geleidend en het kan beschouwd worden als een equipotentiaalvolume op de plasmapotentiaal Vp. Een sonde die in het plasma geplaatst wordt zal deze plasmapotentiaal niet aannemen. Door het verschil in
VII
snelheid tussen ionen en elektronen zal het oppervlak negatief opladen en een spanningsverschil zal zich spontaan ontwikkelen om de elektronen-flux naar het oppervlak toe te reduceren zodat deze gelijk wordt aan de ionen-flux naar het oppervlak. De stroom naar het oppervlak is nul en de sonde bevindt zich nu op de vlottende potentiaal, Vvl, welke negatief is t.o.v. de plasmapotentiaal. Het spanningsverschil tussen plasma- en vlottende potentiaal wordt gewoonlijk benaderd door:
Vfl − Vp ≈ 3
kTe e
(3.3)
De potentiaalval vindt plaats in een dunne grenslaag die is opgebouwd tussen het plasma en het sonde-oppervlak. De dikte van deze laag is van de orde van enkele Debye lengtes. Het plasma zelf is in hoge mate quasi-neutraal: ni = ne . In de grenslaag wordt de quasi-neutraliteit verbroken. De functie van deze laag is om het plasma af te schermen van de negatieve potentiaal op het oppervlak. De opbouw van een dergelijke ruimteladingslaag rond de sonde vereist dat de ionen bij het verlaten van het quasineutrale plasma een minimum stromingssnelheid, gelijk of groter dan de geluidssnelheid cs, in de richting van de sonde toe hebben:
v sh ≥ cs =
k (Te + γ Ti ) mi
(3.4)
Hierbij is γ de isentrope toestandsverandering van het ionengas. Deze voorwaarde wordt het veralgemeende Bohm-Chodura criterium genoemd. Daar waar de grenslaag zelf in het Engels ‘sheath’ wordt genoemd, heet de zone waarin deze versnelling plaatsvindt, ‘presheath’ of ‘voorgrenslaag’. Indien een sterk negatieve potentiaal aangelegd wordt op de sonde, zullen alle elektronen worden gereflecteerd en alleen ionen worden gecollecteerd die de zogenaamde ionensaturatiestroom vormen (zone (C) in Figuur 3.1):
Ii ,sat = - eni ,sh cs A = - ensh cs A = − ζ en∞ cs A
(3.5)
Hierbij verwijst ni,sh, nsh en n∞ respectievelijk naar de ionen- en elektronendichtheid aan de grenslaag en de elektronendichtheid van het ongestoorde plasma ver weg van de sonde. Vergelijking (3.5) legt dus een verband tussen de elektronendichtheid aan de
VIII
grenslaag en de dichtheid in het ongestoorde plasma. In de volgende paragrafen zal duidelijk worden dat de waarde van ζ afhankelijk is van de stroming is van het plasma. In het geval van een niet-stromend plasma wordt de waarde van ζ benaderd door de waarde ζ = 0.5 . Wanneer het oppervlak positief gepolariseerd wordt t.o.v. de vlottende potentiaal, zal de afstotende barriëre voor de elektronen afnemen (en wordt 0 voor V=Vp). De elektronenstroom neemt dan exponentieel toe (zone (B) in Figuur 3.1). Uit het voorgaande kan nu begrepen worden dat de I-V karakteristiek in Figuur 3.1 uitgedrukt word door de vergelijking: =
I
e (V − Vvl ) Isat 1 − exp kTe
voor V < Vp
(3.6)
In geval V>Vp zal de elektronensaturatiestroom Ie,sat gecollecteerd worden (zone (A) in Figuur 3.1). Vacuüm vat Plasma Sonde
Plasma Sonde 2
Sonde 1
I
Id V
V
Vd
Figuur 3.2. Principeschets van een enkelvoudige (links) en dubbelsonde (rechts). In het experiment worden de twee collectoren in een zogenaamde dubbelsonde configuratie gebruikt. Hierbij zijn de twee collectoren met elkaar verbonden via een spanningsbron (zie principe schets in Figuur 3.2; Vd is de opgelegde spanning tussen de twee sondes en Id de dubbelsonde stroom). Deze techniek heeft het voordeel dat het gehele systeem geïsoleerd is van de aarde en dat de maximale stroom in de keten be-
IX
paald wordt door de ionensaturatiestroom die veel lager is dan de hoge elektronensaturatiestroom. De dubbelsondekarakteristiek kan afgeleid worden uit de I-V karakteristiek van twee enkelvoudige sondes in combinatie met de voorwaarde dat de totale stroom naar de dubbelsonde altijd 0 moet zijn. De I-V karakteristiek voor een dubbelsonde is afgebeeld in Figuur 3.3 en wordt beschreven door de vergelijking: e.(Vd −(Vp1 −Vp 2 ) ) Ii 1,sat exp − Ii 2,sat kTe e.(Vd −(Vp1 −Vp 2 ) ) exp +1 kTe
=
Id
(3.7)
Bij de afleiding is aangenomen dat Ie,sat dezelfde is voor de twee sondes daar de elektronensaturatiestroom veel groter is dan de ionensaturatiestroom en dat de oppervlakken van beide sondes gelijk is: A1 = A2 = A . Tevens is verondersteld dat beide sondes in een plasma geplaatst zijn met gelijke dichtheid en temperatuur, maar met verschillende plasmapotentiaal. Vergelijking (3.5) impliceert dan dat de ionensaturatiestromen eveneens gelijk zijn. Deze kunnen echter verschillen als men twee sondes beschouwt die verborgen zijn t.o.v. elkaar in een stromend plasma (een Mach-sonde). 2 I2i,sat
1.5
0.5 0
d
I [10−2A]
1
−0.5 −1
I
1i,sat
−1.5 −2 −150
−100
−50
0 V [V]
50
100
d
Figuur 3.3. I-V karakteristiek van een dubbelsonde.
150
X
De elektronentemperatuur Te kan berekend worden uit de helling van de I-V karakteristiek: Te =
Ii 1,sat + Ii 2,sat
(3.8)
dI 4 d dVd V (I =0 )
Eens Te bepaald is, kan de dichtheid worden berekend met behulp van vergelijking (3.5) voor de ionensaturatiestroom. Deze vergelijking is echter slechts geldig voor een niet-stromend plasma. Een vergelijking die toelaat om de dichtheid uit een dubbelsonde te bepalen in het geval van een stromend plasma zal later worden afgeleid.
L B E ni = ne
E
voorgrenslaag
v ≥ cs
v y ≥ cs
ni > ne
ρi : 4λD
magnetische grenslaag grenslaag
sonde-oppervlak Figuur 3.4. Schematische weergave van de grenslaag, the magnetische grenslaag en de voorgrenslaag. v y is de snelheidscomponent loodrecht op het sonde-oppervlak. De aanwezigheid van een sterk magneetveld legt een aantal ingrijpende implicaties op de voorgaande theorie. Een magneetveld wordt als sterk beschouwd als zowel ρi en ρe, respectievelijk de gyratiestralen van ionen en elektronen, klein zijn t.o.v. de kleinste dimensie van de sonde geprojecteerd op het vlak loodrecht op het magneetveld. De twee voornaamste effekten van een sterk magneetveld op een sterk negatief gepolariseerde sonde die schuin wordt geplaatst t.o.v. het magneetveld zijn de formatie van een nieuwe zone, de magnetische grenslaag [63], en een verandering van de geometrie van de stroomdichtheidslijnen naar de sonde toe [37]. De nieuwe zone, de magnetische grenslaag, wordt gevormd tussen de grenslaag en de voorgrenslaag. De dikte van deze zone is van de orde van de ionen
XI
larmorstraal. De functie van deze quasi-neutrale zone bestaat erin om de deeltjesbanen die parallel zijn met het magnetische veld in de voorgrenslaag, af te buigen zodat ze loodrecht zijn op het oppervlak aan de grenslaag (zie Figuur 3.4). De ionen moeten nu de geluidssnelheid bereiken aan de magnetische grenslaag en kunnen als gecollecteerd beschouwd worden als ze de magnetische grenslaag binnen gaan. Een tweede effekt van de aanwezigheid van een sterk magnetish veld is de verandering van de geometrie van de stroomdichtheidslijnen naar de sonde toe. De beweging van de gyratiecentra van de deeltjes is sterk gereduceerd omdat de deeltjes nu gebonden zijn aan het magnetische veld. Dit impliceert dat de voorgrenslaag effectief als één-dimensionaal kan beschouwd worden en de taak bestaat er nu in om het transport van deeltjes in de voorgrenslaag te modelleren. Het basis idee achter de 1D modellering is dat het deeltjestransport bestaat uit snelle parallelle stroming gebalanceerd door een trage diffusie van de deeltjes haaks op de magnetische veldlijnen. De voorgrenslaag zal zich nu langgerekt uitstrekken langsheen het magnetische veld in een zogenaamde fluxbuis met lengte L. De deeltjes worden in de quasi-neutrale fluxbuis versneld naar het sonde-oppervlak waar ze aan de magnetische grenslaag als gecollecteerd kunnen beschouwd worden. De oplossing van de transportvergelijkingen in de voorgrenslaag laten toe om de snelheid van het ongestoorde plasma te relateren met de dichtheid aan de grenslaag. De saturatiestroom gecollecteerd door de sonde is proportioneel met deze dichtheid (zie vergelijking (3.5)) en een relatie tussen gemeten stroom en ongestoorde dichtheid kan aldus bekomen worden.
3.2. Stroom gecollecteerd door een sonde-oppervlak, schuin geplaatst t.o.v. het magnetische veld, in de aanwezigheid van een parallelle en loodrechte stroming. De methode beschreven op het einde van vorige paragraaf is tot nu toe beperkt tot de bepaling van de parallelle stromingssnelheid omdat transport van deeltjes haaks op het magneetveld beschouwd werd als diffusief. De voornaamste auteurs van deze modellen zijn Chung [16], Hutchinson [38] en Stangeby [73]. Het nieuwe model opgesteld in deze paragraaf is gebaseerd op de theorie van Hutchinson maar houdt rekening met het transport haaks op het magnetische veld veroorzaakt door diffusie en een loodrechte rotatie van het plasma hetgeen moet toelaten ook de loodrechte stroming van het plasma te bepalen. In het model wordt tevens rekening gehouden met een sonde-oppervlak dat schuin t.o.v. het magnetische veld geplaatst is. De gebruikte geometrie is weergegeven in Figuur 3.5.
XII
⊥
x
y
sonde-oppervlak isolator
B
||
r
θ
Figuur 3.5. Mach-sonde geometrie. De afleiding van het nieuwe model vertrekt van de 3D ionencontinuïteitsvergelijking en de parallelle projectie van de ionenmomentumvergelijking: anomale afschuifviscositeit is in rekening gebracht, plasma bronnen en wrijvingstermen tussen het plasma en neutrale deeltjes zijn afwezig: ∂ ∂ ∂ ( ni v ⊥ ) + ( ni v r ) ( ni v|| ) + ∂⊥ ∂ || ∂r
=
0
∂ 2v|| ∂ ∂ ∂ mi ni v||2 + mi ni v||v ⊥ ) + mi ni v||v r ) − η 2 ( ( ∂ || ∂⊥ ∂r ∂r
(
)
(3.9) =
− ∇|| pi + e ni E||
(3.10) waarbij mi, ni, pi, v , E|| respectievelijk de ionen massa, - dichtheid, - druk, - snelheid en het parallelle elektrische veld in de voorgrenslaag voorstellen. De Mach-getallen worden bekomen met vergelijking (3.2) waarin de geluidssnelheid cs gedefinieerd wordt door
cs =
k. (Te + T )i mi
(3.11)
waarbij γ = 1 wordt gesteld. Vergelijkingen (3.9) en (3.10) beschrijven dus een isothermisch vloeistof. De afschuif-viscositeit η is anomaal en gegeven door
η = α mi ni D
(3.12)
met α een constante waarvan de waarde α = 1 aangenomen wordt, en D een constante anomale diffusiecoëfficient haaks op het magneetveld.
XIII
De volgende dimensieloze variabelen worden geïntroduceerd:
n = ni ni ,∞ , (met ni ,∞ de ongestoorde ionendichtheid);
(3.13)
v|| cs = M|| , v ⊥ cs = M ⊥ ;
(3.14)
en volgende coördinatentransformatie:
(
)
(
)
||' = D cs a 2 || , ⊥' = D cs a 2 ⊥ , r ' = r a .
(3.15)
Na transformatie worden de nieuwe coördinaten geschreven zonder accenten. Het is nu duidelijk dat D als constant kan verondersteld worden omdat een verandering in D alleen de lengte van de voorgrenslaag veranderd. In hetgeen volgt worden de vergelijkingen (3.9) en (3.10) tot 1D herleid. Daar de sonde een locale storing in het plasma veroorzaakt en fungeert als een deeltjesput, kan de radiale diffusieve flux beschreven worden door: ni v r = − D
∂ ni ∂r
(3.16)
De gradienten over de radiale dimensie van de sonde worden gelineariseerd door de vervanging: ∇ rψ → (ψ ∞ − ψ ) a and ∇ 2r ψ → (ψ ∞ − ψ ) a 2 ,
Hierbij is ψ een algemene variabele voor dichtheid en stromingssnelheid en ψ∞ is de ongestoorde grootheid. Deze linearisatie is gerechtvaardigd door de 2D simulatie van Hutchinson [40]. De vergelijkingen worden dan getransformeerd naar het (x,y)-coördinatensysteem parallel en loodrecht op het sonde-oppervlak (zie Figuur 3.5). Hierbij wordt aangenomen dat de grootheden niet veranderen langsheen het sonde-oppervlak: ∂ =0. ∂x
(3.17)
Deze aanname is identisch aan de voorwaarde van axi-symmetrie in een tokamak bij het modelleren van de stromingen naar een limiter of divertor oppervlak [3] [14] [87].
XIV
Op deze manier worden transport vergelijkingen in de y-richting bekomen. Vervolgens wordt de omgekeerde coördinatentransformatie doorgevoerd, terug naar het parallelloodrechte coördinatensysteem. Tenslotte wordt aangenomen dat in de loodrechte richting v⊥ constant wordt verondersteld in de voorgrenslaag, i.e.:
∂M⊥ = 0. ∂⊥
(3.18)
De volgende transportvergelijkingen voor de dichtheid en het parallelle Mach-getal in de voorgrenslaag worden dan bekomen:
∂n ∂ ||
∂M|| ∂ ||
=
=
M⊥ M|| − tgθ (1 − n ) − ( M||,∞ − M|| ) [1 − n ] 2 M M|| − ⊥ − 1 tgθ M − (1 − n ) + M|| − ⊥ ( M||,∞ − M|| ) (1 − n ) θ tg 2 M n M|| − ⊥ − 1 tgθ
(3.19)
(3.20)
Deze vergelijkingen drukken uit dat deeltjes met een initiële snelheid M||,∞ zullen versnellen in de voorgrenslaag naar de magnetische grenslaag toe waarbij tezelfdertijd de dichtheid zal dalen. De randvoorwaarde die de waarde van het parallelle Mach-getal aan de magnetische grenslaag M||,MPSE oplegt, wordt gegeven door de singulariteit in de noemer van vergelijkingen (3.19) en (3.20): M||,MPSE
=
M⊥ +1 tg θ
(3.21)
Deze randvoorwaarde werd door andere auteurs op een intuïtieve manier afgeleid maar wordt nu voor de eerste maal afgeleid uit de transportvergelijkingen. Ze drukt uit dat de ingang van de magnetische grenslaag een Mach-oppervlak is. Dit houdt in dat, in het geval er een macroscopische loodrechte stroming bestaat in het plasma, de snelheid van de parallelle stroming zich zo moet aanpassen dat aan vergelijking (3.21) wordt voldaan. Merk op dat voor θ = 90o , de invloed van de loodrechte stroming wegvalt en
XV
dat dus M||,MPSE = 1 . Om de loodrechte stroming te bepalen moet de sonde dus schuin geplaatst worden t.o.v. het magnetische veld. De transportvergelijkingen kunnen numeriek opgelost worden. Deze oplossingen leveren echter geen praktisch bruikbare uitdrukking voor de bepaling van de snelheden. Daarom is een benaderende analytische oplossing voorgesteld die dan in overeenkomst wordt gebracht met de numerieke oplossingen. De benaderende analytische uitdrukking voor de saturatiestromen is van de vorm: M Ii ,sat ≈ exp M||,∞ − ⊥ − 1 e n∞ cs A sinθ tg θ
(3.22)
Merk op dat voor θ = 90o geen van beide processen, diffusie of macroscopische stroming expliciet verschijnt in de uitdrukking voor de saturatiestroom. De enige functie van deze processen is dat ze toelaten dat de dichtheid en de snelheid van de deeltjes ver weg van de sonde zich aanpassen aan de dichtheid en snelheid die opgelegd wordt door de randvoorwaarde aan de magnetische grenslaag. Alleen als de rotatiehoek verschilt van 90° verandert de opgelegde randvoorwaarde voor het parallelle Mach-getal zodat de invloed van de loodrechte stroming zichtbaar wordt. De asymmetrie van deze stroom aan de stroomop- en neerwaartse zijde van de sonde kan nu uitgedrukt worden door de verhouding van de stromen aan beide zijden: R=
Ii ,sat ,op Ii ,sat ,neer
M = exp c(M||,∞ ). M||,∞ − ⊥ θ tg
(3.23)
De functie c brengt de benaderende oplossing in overeenstemming met de numerieke. De waarde van c is hoofdzakelijk afhankelijk van M||,∞ en ligt in het bereik tussen 2.3 en 2.45. In Figuur 3.6 is de overeenkomst tussen de benaderende analytische en numerieke oplossing weergegeven. Vergelijking (3.23) maakt het mogelijk om de parallelle en loodrechte stroming van het plasma te bepalen indien het toegepast wordt op Mach-sonde gegevens. De verhouding van de saturatiestromen bij θ=90°, geeft onmiddellijk M||,∞. Tenminste één bijkomende meting met de sonde gepositioneerd onder een andere hoek is nodig om het loodrechte Mach-getal te bepalen.
XVI
4 3
M =.4 ⊥
ln(R) ln(nsh,up /nsh,down)
2
M⊥=.8
M⊥=.3
1 M⊥=.1
0
−1 −2 −3 −4 0
30
60
90 θ [°]
120
150
180
Figuur 3.6. ln(R) vs. θ voor verschillende waarden van M||,∞ and M⊥: De markeringen stellen de numerieke oplossingen voor van de transportvergelijkingen: ● : M||,∞=0.1; M⊥=0.1 en M⊥=0.3 ; ✧: M||.∞=0.5; M⊥=0.4 en M⊥=0.8. De onderbroken lijnen zijn de oplossingen volgens vergelijking (3.23).
Het 1D model geeft verder een uitdrukking voor de elektronendichtheid ne in een stromend plasma. In eerste en goede benadering wordt ne gegeven door:
n∞ ≈
Ii ,sat ,up . Ii ,sat ,down exp ( −2 co )cs A sinθ
≈
Ii ,sat ,up . Ii ,sat ,down 0.35 cs A sinθ
(3.24)
3.3. Vergelijking van het 1D model met bestaande modellen. Enkele rapporten in de literatuur verschaffen een alternatieve benadering om de loodrechte stroming in rekening te brengen. Daar deze modellen meestal gepaard gaan met een specifieke sonde-geometrie, laat de vergelijking met het 1D model toe om ook conclusies te trekken aangaande de toepasbaarheid van het model op alternatieve sonde-geometrieën.
XVII
θ=30°
v⊥
B
θ=60° θ=90°
⊥ B
||
θ
v||
θ=120° θ=150°
Figuur 3.7. De Gundestrup geometrie kan beschouwd worden als een rij van dubbelsondes geörienteerd onder een verschillende hoek t.o.v. het magnetische veld. De collectoren verbonden door de stippellijn representeren schematisch een Machsonde. Het zogenaamde intuïtieve MacLatchy-algoritme [53] is ontwikkeld om de loodrechte en parallelle stromingscomponenten te bepalen van de saturatiestromen gecollecteerd door de Gundestrup-sonde (zie Figuur 3.7). Zulk een sonde bestaat uit een reeks van collectoren die onder een vaste hoek t.o.v. het magneetveld ingeplant zijn in een isolator. De stroom gecollecteerd door elk oppervlak wordt gegeven door een vooropgestelde uitdrukking voor de saturatiestroom. De magnetische grenslaag wordt hierbij niet als een Mach-oppervlak beschouwd. Alleen de parallelle component van de stromingsvector moet de geluidssnelheid bereiken aan de magnetische grenslaag. De stroomdichtheidscomponenten parallel en loodrecht op het magneetveld worden geometrisch opgeteld en combineren niet op een intrinsieke manier zoals de randvoorwaarde in het 1D model oplegt. Voor een gegeven saturatiestroom bij een bepaalde hoek volgt hieruit dat de bijdrage van de loodrechte stroming altijd onderschat zal woren. De vergelijking tussen het 1D model en het MacLatchy-algoritme is mogelijk door de Gundestrup-geometrie te beschouwen als een verzameling van Mach-sondes (zie Figuur 3.7). In Figuur 3.8 is de vergelijking gemaakt tussen het MacLatchy algoritme en het 1D model aan de hand van de waarde van ln(R) (zie vergelijking (3.23)). Voor θ = 90o hebben de beide modellen bij benadering eenzelfde uitdrukking voor de bepaling van het parallelle Mach-getal en geven dus ook ongeveer eenzelfde resultaat. Voor θ ≠ 90o wordt inderdaad een beduidend lagere waarde van ln(R) voorspeld door het MacLatchy-algoritme.
XVIII
Figuur 3.8. Vergelijking tussen de deeltjes-in-cel-simulatie, het MacLatcy-algoritme en het 1D-model. Door J.P. Gunn werd een deeltjes-in-cel-simulatie ontwikkeld om het transport van deeltjes parallel en loodrecht naar het oppervlak te modelleren [27]. Uit Figuur 3.8 blijkt dat deze simulatie een goede overeenkomst geeft met het 1D model. Dit is niet verwonderlijk daar de simulatie dezelfde onderliggende fysische aannames maakt als het 1D-model. De deeltjes-in-cel-geometrie gebaseerd op de Gundestrupgeometrie, dus op een verzameling van Mach-sondes. Intuïtief verwacht men dat elke collector zijn eigen voorgrenslaag creëert en dat deze verschillende voorgrenslagen een storende invloed kan hebben op de collectie van de loodrechte stroming door de verschillende collectoren. Uit de simulatie blijkt dat dit niet het geval is en dat de Gundestrup-geometrie inderdaad als een verzameling van Mach-sondes kan beschouwd worden. Hiermee bevestigt de deeltjes-in-cel simulatie het 1D-model. Het 1D-model heeft het voordeel t.o.v. de deeltjes-in-cel-simulatie van een praktisch bruikbare uitdrukking te verschaffen voor de bepaling van de parallelle en loodrechte stroming. Het ‘roterende “sandwich” sonde’ model is ontwikkeld door K. Höthker [36] en is verbonden met het ontwerp van de zogenaamde “sandwich” sonde. Dit is een Mach-sonde die continu kan roteren t.o.v. het magneetveld. De benadering in [36] bestaat in het modelleren van de deeltjesbanen in de voorgrenslaag. Op die manier wordt een uitdrukking bekomen voor het effectief collecterend oppervlak dat de invloed reflecteert van de loodrechte stroming. Deze uitdrukking is slechts een eerste stap in de bepaling van de loodrechte stroming.
XIX
4. Databank van Mach-sonde randplasma parameters en validatie van het 1D vloeistofmodel. In dit hoofdstuk wordt het experimentele deel van deze dissertatie beschreven. Uitgaande van de Mach-sonde gegevens is een grote databank bestaande uit randplasma parameters opgesteld. De databank omvat de parallelle en loodrechte stromingssnelheden, het radiale elektrische veld en de elektronendichtheid en -temperatuur. Zulk een volledige databank bestaande uit deze parameters gemeten onder dezelfde condities is, naar het beste weten van de auteur, niet beschikbaar in de literatuur. De voornaamste toepassing van de databank bestaat erin om het 1D-vloeistofmodel, beschreven in het voorgaande hoofdstuk te valideren. Deze validatie is gerapporteerd in [81]. De databank laat verder toe om een zeer gedetailleerd beeld te vormen van de stromingen en de elektrische velden, geïnduceerd door plasmarand polarisatie. v⊥
⊥
θ v||
Ip
B
θ r~ ||
φ Figuur 4.1. Mach-sonde geometrie waarbij het aangenomen parallel-loodrecht en het toroidaal-poloidaal coördinatensysteem wordt getoond. Hierbij is de pitch angle (de hoek tussen de parallelle en toroïdale richting, α = 2.44o ) verwaarloosd.
Om het theoretisch 1D model uit het voorgaande hoofdstuk experimenteel te valideren moet er een substantiële poloïdale rotatie van het plasma aanwezig zijn. De polarisatie opstelling op TEXTOR-94 geeft de unieke mogelijkheid om op een gecon-
XX
trolleerde manier hoge radiale elektrische velden op te wekken. Dat deze velden onafscheidelijk verbonden zijn met plasmastromingen wordt weergegeven door de radiale momentum vergelijking: 1 E = ∇ p −v B +v B θ φ φ θ r en r i i
(4.1)
De eerste term aan de rechterzijde is de diamagnetische term waarbij ni en pi de ionendichtheid en -druk voorstellen. De tweede en derde term representeren respectievelijk de poloïdale en toroïdale snelheidscontributies; vθ , vφ , Bθ , Bφ stellen respectievelijk de poloïdale en toroïdale componenten van de stromingsvector en het magnetische veld voor. De gebruikte geometrie is afgebeeld in Figuur 4.1. Door meting van de vlottende potentiaal kan het radiaal elektrische veld bepaald worden. Uit de analyse van de ionensaturatiestromen kunnen de snelheden, elektronentemperatuur en –dichtheid bepaald worden. Een kwalitatieve studie van het model kan dus bekomen worden door het gemeten veld te vergelijken met het veld berekend met behulp van vergelijking (4.1).
4.1. Experimentele opstelling. De globale plasma parameters zijn voor alle ontladingen dezelfde; de centrale lijn gemiddelde dichtheid, plasmastroom en magnetisch veld bedragen respectievelijk: ne,o = 1012 cm −3 , Ip=210kA, Bφ=2.33T. De polarisatie opstelling wordt schematisch voorgesteld in Figuur 4.2. Een paddestoelvormige elektrode wordt 5cm binnen de separatrix gebracht en een spanning wordt aangelegd tussen de elektrode en de limiter. Een boor-nitride huls isoleert de steel van de elektrode. De aangelegde spanning en gemeten elektrodestroom voor een typische gepolariseerde ontlading is weergegeven in Figuur 4.3. De voorgeprogrammeerde elektrodespanning heeft een plateau-vorm die aanvangt bij 1s en eindigt bij 3s. Hierin kunnen twee fasen onderscheiden worden: een fase van lage opsluiting (L-mode) en een van hoge opsluiting (H-mode). De L-mode is gekarakteriseerd door een klein elektrisch veld waarde en grote radiale stroom. De H-mode door een groot elektrisch veld en een kleinere radiale stroom. De overgang tussen de twee fasen is gekenmerkt door het zogenaamd bifurcatie fenomeen dat plaatsgrijpt bij een zekere waarde van de elektrodespanning. De bifurcatie is gekenmerkt door een plotse daling
XXI
van de radiale stroom en een sterke stijging van het elektrische veld. Het typische gedrag van de L- en H-mode zal worden onderzocht door de sonde gegevens in detail te evalueren voor twee tijdstippen: bij t=1.12s en t=2s. Op deze tijdstippen heeft de aangelegde spanning respectievelijk de waarde VE=300V en VE≅650V. Merk op dat voor en na de aanbrenging van de hoge elektrodespanning de elektrode geaard is (VE=0V) waarbij een kleine radiale stroom gemeten wordt.
Figuur 4.2. Polarisatie opstelling. De Mach-sonde kan op twee poloïdale posities op de machine gemonteerd worden zoals schematisch weergegeven in Figuur 4.2. Dit geeft de mogelijkheid om poloïdale asymmetrieën waar te nemen. De Mach-sonde is op deze twee posities geplaatst in een manipulator waarmee de radiale positie van de sonde tussen twee opeenvolgende ontladingen kan veranderd worden. Op een experimentele dag wordt de sonde onder een vaste hoek ten opzichte van het magneetveld gemonteerd op de machine. Een radiaal profiel van de saturatiestromen en de vlottende potentiaal wordt bekomen door een aantal opeenvolgende reproduceerbare ontladingen waarbij de sonde zich telkens op een verschillende radiale positie bevindt. Deze procedure wordt herhaald met de sonde gemonteerd onder een andere hoek. Op deze manier wordt de databank opgebouwd uit een verzameling van radiale profielen.
15 10 5
n
e,o
[1012cm−3]
XXII
VE [V]
600 400 200 200
IE [A]
150 100 50 0
I [A]
−0.5 0.5 0 −0.5 0
0.5
1
1.5
2 t [s]
2.5
3
3.5
4
Figuur 4.3.Tijdsevolutie van de lijn gemiddelde centrale dichtheid, aangelegde spanning VE en gemeten elektrodestroom IE. Het onderste signaal is een voorbeeld van de gemeten dubbelsonde stroom.
Boor-nitride
•
Grafiet
•
0.55 cm
3.4 cm
stroom draden 3.4 cm
~r Figuur 4.4. Dwarsdoorsnede van de Mach-sonde.
XXIII
9 cm boor-nitride
1.2 cm
0.55 cm
grafiet
0.8 cm r Figuur 4.5. Langsdoorsnede van de Mach-sonde.
De bouw van de Mach-sonde is schematisch weergegeven in Figuur 4.4 en Figuur 4.5: twee grafiet collectoren zijn ingeplant in een boor-nitride materiaal dat als isolator functioneert. Voor de meting van de ionensaturatiestromen is de Mach-sonde opgenomen in een zogenaamd vlottend dubbelsonde meetcircuit dat geïsoleerd is van de aarde. De radiale positie van de sonde wordt genomen als de positie van het midden van de collectoren.
4.2. Analyse van de Mach-sonde gegevens. Uit de analyse van de ruwe sonde gegevens kunnen nu de parameters bepaald worden die toelaten om met vergelijking (4.1) het 1D model te valideren. De radiale as in de figuren die volgen wordt uitgedrukt relatief t.o.v. de positie, a, van het laatst gesloten flux-oppervlak. Het radiaal elektrische veld wordt bekomen als de gradient van het gemeten vlottende potentiaalprofiel. De elektrische velden op de tijdstippen t=1.12s en t=2s die als typisch worden beschouwd voor de L- en H-mode zijn weergegeven in Figuur 4.6. Het elektrische veld in de H-mode is aanzienlijk groter en is ook nauwer dan in de Lmode.
XXIV
200
700 600
Er [V/cm]
500
100
r
E [V/cm]
150
400 300 200
50
100
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
1
2
Figuur 4.6a en b. Radiaal elektrisch veld Er in L- en H-mode. De open cirkels representeren de waarden van het elektrische veld berekend uit het vlottende potentiaalprofiel.
12
12
3.5
3.5 3
2.5
2.5 ne [cm−3]
3
ne [cm ]
−3
x 10
2
2
1.5
1.5 1
1
0.5
0.5
0 −5
x 10
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
Figuur 4.7a and b. Elektronendichtheid ne vs. radius in L- and H-mode. De profielen zijn bekomen bij verschillende rotatiehoeken van de sonde-oppervlakken. De markeringen duiden de radiale positie van de Mach-sonde aan bij de verschillende hoeken; θ=22.5° (onderbroken-gestipte lijn, ), θ =45° (onderbroken lijn,ο), θ =90° (volle lijn, +), θ =135° (gestipte lijn, ∆). Ter vergelijking is een typisch elektronendichtheidsprofiel gemeten door de Li-straal diagnostiek toegevoegd (x, volle lijn). De onderbroken lijn in de L-mode representeert de elektronendichtheid bij t=0.9s and θ=135°.
XXV
80
80
60
60
Te [eV]
100
Te [eV]
100
40
20
20
0 −5
40
−4
−3
−2
−1 r−a [cm]
0
1
2
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
Figuur 4.8a and b. Elektronentemperatuur Te vs. radius in L- and H-mode. De profielen zijn bekomen bij verschillende rotatiehoeken van de sonde-oppervlakken. De markeringen duiden de radiale positie van de Mach-sonde aan bij de verschillende hoeken: θ =22.5° (onderbroken-gestipte lijn, ), θ =45° (onderbroken lijn,ο), θ =90° (volle lijn, +), θ =135° (gestipte lijn, ∆). Ter vergelijking is een typisch elektronentemperatuursprofiel gemeten door de He-straal diagnostiek toegevoegd (∗, volle lijn). Uit de analyse van de I-V karakteristiek van de ruwe dubbelsonde karakteristiek worden de elektronendichtheid (vergelijking (3.24)) en elektronentemperatuur (vergelijking (3.8)) berekend. De resultaten zijn respectievelijk weergegeven in Figuur 4.7 en Figuur 4.8. Uit deze resultaten kan besloten worden dat zowel de temperatuur als de dichtheid nagenoeg niet veranderen tot het tijdstip van de bifurcatie. Na de bifurcatie (H-mode) vertonen de dichtheidsprofielen de typische verhoging van de dichtheidsgradient op een radiale positie die overeenkomt met de binnenste gradient van het elektrische veld. Tevens is er in de H-mode een verhoging van de elektronentemperatuur waargenomen in de omgeving waar het elektrische veld hoog is. Door fitting van vergelijking (3.23) op de saturatiestromen, opgemeten onder verschillende hoeken van de sonde t.o.v. het magneetveld, wordt het parallelle en loodrechte Mach-getal bepaald. In deze procedure zijn het parallelle en loodrechte Mach-getal als vrije parameters beschouwd. De resulterende Mach-getallen zijn weergegeven in Figuur 4.9 en Figuur 4.10, respectievelijk voor de L-mode en H-mode. De foutenvlaggen geven de statistische fout weer, bekomen bij het toepassen van het 1D-model. De conclusies zijn dat in de H-mode een substantiële loodrechte rotatie is waargenomen waarbij het maximum overeenkomt met de positie van het elektrische
2
XXVI
veld maximum. De grootte van de parallelle rotatie blijft nagenoeg dezelfde in zowel L- als H-mode en vertoont een maximum op een positie die verschilt van het elektrische veld maximum. Op de positie van het elektrische veld maximum vertoont de parallelle rotatie een minimum.
Figuur 4.9a en b. Parallel en loodrecht Mach-getal in de L-mode.
Figuur 4.10a en b. Parallel en loodrecht Mach-getal in de H-mode.
XXVII
4.3. Validatie van het 1D model. Met de resultaten beschreven in voorgaande paragraaf is het mogelijk om met uitdrukking (4.1) de vergelijking te maken tussen het berekende en gemeten elektrische veld. Deze vergelijking is weergegeven in Figuur 4.11. In het bepalen van de foutenvlaggen op het berekende veld is alleen de fout op de Mach-getallen in rekening gebracht. De andere parameters in de berekening zijn constant verondersteld. In zowel L- als H-mode is een goede overeenkomst gevonden tussen het berekende en gemeten elektrische veld. In beide gevallen is het duidelijk dat vooral de poloïdale snelheidsterm verantwoordelijk is voor het elektrische veld. Uit deze overeenkomst volgt het belangrijke besluit dat het transport van deeltjes in de loodrechte richting op een correcte manier wordt beschreven door het 1D-model. De toepassing van het model op Mach-sonde gegevens bewijst een betrouwbare methode te zijn voor de bepaling van het loodrechte en parallelle Mach-getal. Merk op dat de stromingssnelheden berekend zijn uit de gemeten Machgetallen met vergelijking (3.2). De geluidssnelheid cs is gedefinieerd door vergelijking (3.11) waarbij Te = Ti is aangenomen. In werkelijkheid kan Ti beduidend groter zijn dan Te waardoor de waarde van de stromingssnelheden sterk kan toenemen. (zie Figuur 4.11b waar Er weergegeven is waarbij cs berekend is met de aanname Ti=3Te) Gebruik makend van vergelijking (3.24) en de Li-straal dichtheid kan een schatting bekomen worden van de waarde van Ti en de reële waarde van cs. Op basis van deze waarde kan opnieuw cs en Er berekend worden (zie Figuur 4.11b). Hieruit blijkt duidelijk dat het 1D vloeistof-model het gemeten Er reconstrueert.
XXVIII
200
100
r
E [V/cm]
150
50
0 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
−4
−3
−2
−1 r−a [cm]
0
1
2
1000
600
r
E [V/cm]
800
400
200
0
−5
Figuur 4.11a en b. Vergelijking tussen het gemeten (o) en berekende elektrische veld () in L- (bovenste figuur) en H-mode (onderste figuur). Het berekende elektrische veld 1 heeft de volgende bijdragen: ∇ pi (gestipte lijn), − v B (onderbroken lijn), θ φ eni r 1
− ∇ p φ θ (onderbroken gestipte lijn). Merk op dat eni r i is weergegeven. In de H-
v B
mode is tevens Er weergegeven waarbij de waarde van cs is berekend met vergelijking (3.11) waarbij Ti=3Te (volle lijn, +). en cs berekend met vergelijking (3.11) waarbij de waarde van Ti bekomen is met behulp van vergelijking (3.24) en de Li-straal dichtheid (volle lijn, ◊).
XXIX
4.4. Tijdsevolutie van het parallel en loodrecht Mach-getal. De tijdsevolutie van het elektrische veld maximum, tezamen met deze van het poloïdale en toroïdale Mach-getal op deze radiale positie is weergegeven in Figuur 4.12. Hieruit kan afgeleid worden dat in de L-mode het plasma voornamelijk in de toroïdale richting roteert. Op het moment van de bifurcatie vertonen zowel het elektrische veld als het poloïdale Mach-getal een sterke stijging en blijven constant geduren-
IE [A] and VE [V]
de de plateau-fase. Het parallelle Mach-getal vertoont deze stijging niet en daalt enigszins in waarde.
VE
600 400 200
Er [V/cm]
IE
600 400 200
Mφ
0.4 0.2
Mθ
0.4 0.2 0 −0.2 0.5
1
1.5
2 t [s]
2.5
3
3.5
Figuur 4.12. Tijdsevolutie van het elektrische veld maximum en de Mach-getallen op deze positie. De bovenste grafiek toont een typisch elektrodespannings- (volle lijn) en elektrodestroom signaal (onderbroken lijn). De waarde van de elektrodestroom is vermenigvuldigd met een factor 3.
XXX
4.5. Invloed van de radiale stroom op het parallelle Mach-getal. De polarisatie opstelling heeft het grote voordeel dat de radiale stroom in het randplasma kan gevariëerd worden. De invloed van het parallelle Mach-getal werd bestudeerd door profielen van deze grootheid te vergelijken bij verschillende waarden van de radiale stroom. In Figuur 4.13 zijn deze profielen van het parallelle Mach-getal samengevat. Het is duidelijk dat de radiale stroom een grote invloed heeft op de grootte en de richting van de parallelle stroming in het randplasma. In het geval dat de elektrode niet aanwezig is in het plasma (IE=0), is een kleine ionenverliesstroom verantwoordelijk voor een stroming parallel met het magnetische veld. Deze stroming verkleint snel dieper in het plasma. In gepolariseerde ontladingen (IE≠0) is een stroming anti-parallel met het magnetische veld waargenomen, veroorzaakt door een JxB kracht. De stroming verandert niet-lineair met de elektrodestroom. In het geval de elektrode geaard is (IE=40A) is de stroming kleiner vergeleken met deze in de L- (IE=163A) en H-mode (IE=110A). In deze laatste twee gevallen blijft ze nagenoeg gelijk.
0.6 0.4
M
||
0.2 0
−0.2 −0.4 −0.6 −5
−4
−3
−2
−1 r−a [cm]
0
1
2
Figuur 4.13. Radiale profielen van het parallelle Mach-getal voor verschillende waarden van de radiale stroom; elektrode verwijderd (x, onderbroken-gestipte lijn), IE=40A (∇, onderbroken lijn), IE=160A (gestipte lijn), IE=110A (volle lijn).
XXXI
4.6. Poloïdale asymmetrieën. Poloïdale asymmetrieën kunnen bestudeerd worden door de resultaten van de Mach-sonde gegevens op de twee poloïdale locaties met elkaar te vergelijken. Deze vergelijking is uitgevoerd voor de resultaten in de H-mode betreffende de elektronendichtheid en –temperatuur, het elektrische veld en het parallelle Mach-getal. Uit deze studie blijkt dat er voor het parallelle Mach-getal en de elektronendichtheid poloïdale asymmetrieën waar te nemen zijn. Het dichtheidsprofiel op de top van de machine vertoont in de H-mode een kleinere gradient dan het profiel in het equatoriale vlak. Tevens is er een grotere dichtheidswaarde in de omgeving waar het elektrische veld groot is. De parallelle Mach-getallen hebben een kleinere waarde op de top van de machine dan in het equatoriale vlak. Beide profielen vertonen wel dezelfde kenmerken: een minimum op de positie waar het elektrische veld hoog is en een maximum dieper in het plasma.
4.7. Alternatieve bepaling van de loodrechte stromingssnelheid. Tot slot van dit hoofdstuk wordt een alternatieve manier voorgesteld om de loodrechte rotatie experimenteel te bepalen. Deze methode is gebaseerd op de stroommeting waarbij de sonde-oppervlakken parallel georiënteerd zijn met het magnetische veld. Gewoonlijk wordt deze meting gebruikt als een aanwijzing voor de aanwezigheid van een loodrechte stroming [30] [36] [44] [93]. Hier bestaat de unieke situatie dat de loodrechte stromingssnelheid gekend is. Het onderliggende ‘intuïtieve’ model is gebaseerd op de aanname dat de sondestroom gemeten met de sonde-oppervlakken parallel met het magneetveld proportioneel is met de ongestoorde dichtheid en de loodrechte stromingssnelheid: Ii : e nev ⊥ A
(4.2)
Hoewel het onderliggende model gebaseerd is op intuïtie, tonen de stromingssnelheden in Figuur 4.14, enerzijds gebaseerd op vergelijking (4.2) en anderzijds deze uit paragraaf 4.2, experimenteel aan dat uit deze sonde-stroom de loodrechte snelheid kan bepaald worden. Het voordeel van deze methode is dat de sondeoppervlakken niet moeten gedraaid worden t.o.v. het magnetische veld en dat geen veronderstellingen moeten gemaakt worden betreffende de bepaling van de geluidssnelheid.
XXXII
−5
⊥
6
v [10 cm/s]
0
−10
−15
−1.5
−1
−0.5 r−a [cm]
0
Figuur 4.14. Loodrechte snelheid gebaseerd op vergelijking (4.2) waarbij de sonde-oppervlakken parallel zijn met het magneetveld en waarbij de ongestoorde dichtheid is bekomen uit de sonde gegevens (o, volle lijn) en de dichtheid uit de Listraal diagnostiek (o, onderbroken-gestipte lijn). Ter vergelijking is de loodrechte snelheid uit paragraaf 4.2 toegevoegd (+, onderbroken lijn).
XXXIII
5. De fysica van de stromingen geïnduceerd door elektrode polarisatie in het randplasma van TEXTOR-94. In dit hoofdstuk worden de onderliggende mechanismen beschreven die verantwoordelijk zijn voor het opzetten van de stromingsprofielen gepresenteerd in het vorige hoofdstuk. Een theoretisch model is ontwikkeld door Van Schoor [85] en Cornelis [21] om deze stromingsprofielen en het elektrische veld te voorspellen als functie van de radiale stroom geïnduceerd door de polarisatie elektrode. Later voegde Van Schoor samendrukbaarheid van het plasma toe aan dit model [86] [89]. In dit hoofdstuk wordt een samenvatting gegeven van deze theoriën. De gemeten stromingsen elektrische veldprofielen bieden dan de unieke mogelijkheid om dit model te toetsen aan de werkelijkheid en om de onderliggende mechanismen die verantwoordelijk zijn voor het opzetten van deze stromingen te identificeren. De originaliteit van dit hoofdstuk bestaat in de volledigheid van de confrontatie tussen theorie en experiment. De resultaten zijn gerapporteerd in [82] and [84].
5.1. Stromingen voor een onsamendrukbaar plasma. Het model van Van Schoor en Cornelis legt een verband tussen de radiale stroom geïnduceerd door de polarisatie elektrode en het elektrische veld. Dit verband kan op volgende manier begrepen worden. Door de aangelegde elektrodespanning zal een elektrode stroom IE vloeien waardoor het plasma opgeladen wordt. Het plasma reageert hierop met de creatie van een elektrisch veld en geassociëerde stromingen. Dit veld zal onbegrensd groeien tenzij het plasma een compenserende tegenstroom Ir opzet. De belangrijkste bijdragen tot deze radiale stroom in het model zijn de wrijving van het plasma met neutralen en de parallelle viscositeit. Deze bijdragen vormen de dempingsmechanismen voor de stromingen en het elektrische veld. De uiteindelijke grootte en vorm van de stromingen en het elektrische veld resulteert dan uit de balans van creatie en demping. In het model worden de toroïdale en parallelle projecties van de totale momentum vergelijking gebruikt om vθ, vφ en Er te berekenen. In hetgeen volgt wordt de afleiding van de vergelijkingen voor deze grootheden samengevat.
XXXIV
De toroïdale momentum vergelijking verschaft een uitdrukking voor de radiale stroomdichtheid: 1 ν vφ Bθ
Jr =
(5.1)
Hierin is ν de neutrale wrijvingscoëfficient. Geïntegreerd over een magnetisch oppervlak geeft deze vergelijking de stroom doorheen dat oppervlak: Ir =
1 R 1 ν ν G ( r ) .S vφ .S = ΘBo ΘBo Ro
(5.2)
Hierin stelt <X> de integratie voor van de quantiteit X over een magnetisch oppervlak. Verder is Θ = tgα de ‘pitch’ van het magnetische veld, S de oppervlakte van een magnetisch flux-oppervlak, i.e. S = 2π r .2π Ro . De integratie over een magnetisch oppervlak van het product van de toroïdale snelheid en R Ro definieert de fluxfunctie
G(r). Het elektrische veld is geïntroduceerd in de vergelijkingen via de fluxfunctie
V(r): V ( r ) = −E r +
1 ∂pi . en ∂r
(5.3)
De loodrechte snelheid is dan bekomen via de radiale momentum vergelijking: v⊥ =
R V (r ) . Cosα Ro Bo
(5.4)
Via de continuïteitsvergelijking in een axi-symmetrische configuratie definieert de fluxfunctie F(r) de poloïdale snelheid: vθ =
Ro F (r ) R Ro
(5.5)
Een supplementaire vergelijking is vereist om G(r), F(r) en V(r) te berekenen. Daarvoor is de parallelle momentum vergelijking waarin parallelle viscositeit en
XXXV
wrijving tussen het plasma en neutralen zijn behouden, het beste geschikt. Hierin is voor de viscositeit η een fenomenologische uitdrukking ingevoerd die aantoont dat het elektrische veld de viscositeit kan vernietigen. Een vergelijking voor V(r) kan nu afgeleid worden:
V (r ) = Bo
−
Ir ΘBo Sν
Boν ε 2 ηΘ + 1+ + ηVneo 2 Cos 2α . ε2 B ν η 1 + + o κ 2 Cosα
(5.6)
Merk op dat η een niet-lineaire functie is van Er zodat een oplossing van vergelijking (5.6) iteratief moet bekomen worden. Het gemeten elektrische veld is de startwaarde in deze procedure. Eens de twee fluxfuncties G(r) en V(r) gekend zijn, kunnen de lokale snelheden in het equatoriale vlak ( θ = 0o ) en het elektrische veld berekend worden. De neutralendichtheid wordt in deze procedure gebruikt als een fitparameter. Met deze oplossingsmethode werd een redelijke overeenkomst gevonden tussen het experimentele en theoretische elektrische veldprofiel en het lokale poloïdale snelheidsprofiel (zie figuren Figuur 5.3 en Figuur 5.4). Het theoretisch lokale toroïdale snelheidsprofiel vertoont echter een maximum op de positie van maximaal elektrische veld dat experimenteel niet observeerd wordt. De oorzaak in het model volgt uit een puur geometrisch effect dat kan begrepen worden wanneer de uitdrukking voor de lokale ( θ = 0o ) toroïdale snelheid expliciet als een functie geschreven wordt van G(r) en V(r):
vφ (r ,θ ) = (1 − ε ).G( r ) −
2ε V (r ) cosθ Θ Bo
(5.7)
Bij een kleine hoek α (in het geval van TEXTOR-94 is α=2.44°) vertoont V(r) en dus ook Er een grote invloed op de lokale toroïdale snelheid. Merk op dat uit de definitie van G(r) volgt dat G ( r ) ; vφ : G(r) vertoont deze afhankelijkheid dus niet en is daarmee in lijn met de experimentele observatie. Door enkele auteurs is gesuggereerd dat samendrukbaarheid van het plasma een grote invloed kan hebben op de lokale toroïdale snelheid in het geval van sterke poloïdale stroming. Daar deze modellen niet onmiddellijk toepasbaar zijn, werd de compressibiliteit op de volgende manier door Van Schoor in het model ingebouwd.
XXXVI
5.2. Stromingen voor een samendrukbaar plasma. De aanname is dat alle quantiteiten samengesteld zijn door een poloïdaal constant deel (alleen functie van de kleine straal r) en een poloïdaal afhankelijk deel (functie van de poloïdale hoek θ als ook van r): X ( r ,θ ) = X (r ) + X%(r ,θ )
(5.8)
De aanname in het model is dat X% X ten hoogste van dezelfde orde is als de parameter ε = r Ro . Volgende uitdrukkingen voor de snelheden worden dan bekomen: vφ =
1 F ( r ) Ro V (r ) R n%F (r ) − − Bo Ro n Ro Θ Ro R
1 F (r ) Ro n%F (r ) Cosα V (r ) R − − Θ Bo Ro Sinα Ro R n Ro F ( r ) Ro n%F (r ) − vθ = Ro R n Ro
v // =
v⊥ =
V (r ) R Cosα . Bo Ro
(5.9) (5.10) (5.11) (5.12)
De integratie van de toroïdale en parallelle momentum vergelijking over een magnetisch oppervlak kan nog steeds gebruikt worden om de fluxfuncties F(r) en V(r) te berekenen. Het variabel deel van de parallelle momentum vergelijking verschaft een uitdrukking voor de poloïdale dichtheidsperturbatie n%n . Een benaderende analytische uitdrukking voor de dichtheidsvariatie is gegeven door: ∆ F (r ) 2ε 1 + 2 2 % n q Ro Ro Cosθ =− 4 ∆ F (r ) V (r ) n + Bo 3 q 2Ro2 Ro
(5.13)
waarbij de parameter ∆ = η ∗ ν ∗ de dichtheidsonafhankelijke delen van de viscositeit
η∗ en neutrale wrijvingscoëfficient ν∗ introduceert. Bij hoge elektrische veldwaarden wordt de viscositeit door het veld vernietigd. Dit resulteert in een kleine waarde voor ∆. Voor de limiet ∆ → 0 volgt uit verge-
XXXVII
lijking (5.13) dezelfde limietwaarde voor de dichtheidsvariatie als gevonden door Rozhansky en Tendler [66] [67], namelijk: n% = −2ε Cosθ . n
(5.14)
In Figuur 5.1 zijn de benaderende analytische en hoge-veld limietoplossing voor n%n bij θ = 0o in de H-mode weergegeven. De twee oplossingen tonen aan dat de correctie door de poloïdale dichtheidsvariatie op de stromingen beduidend is in het gebied waar het elektrische veld piekt.
0 −0.1
∼ − n/n
−0.2 −0.3 −0.4 −0.5
−5
−4
−3
−2 −1 r−a [cm]
0
1
Figuur 5.1. Poloïdale variatie n%n vs. radius in de H-mode bij θ = 0o : benaderende analytische oplossing (onderbroken lijn) en hoge-veld limietwaarde (volle lijn).
Het introduceren van de hoge-veld limietwaarde in de uitdrukking voor de toroïdale snelheid, vergelijking (5.9), levert: vφ ( r ,θ) = (1 + ε cosθ).G(r )
(5.15)
Hierdoor is aangetoond dat bij hoge velden inderdaad de invloed van Er op de lokale toroïdale snelheid verdwijnt door het in rekening brengen van de samendrukbaarheid van het plasma.
XXXVIII
200
700 600
150
E [V/cm]
Er [V/cm]
500
r
100
400 300 200
50
100 0 −5
−4
−3
−2 −1 r−a [cm]
0
0 −5
1
−4
−3
−2 −1 r−a [cm]
0
1
Figuur 5.2a and b. Gemeten (o) en berekend (volle lijn) elektrische veld Er vs. radius r-a in L- en H-mode.
5
2
4
vθ [106cm/s]
6
vθ [10 cm/s]
1.5
1
0.5
0 −5
3
2
1
−4
−3
−2 −1 r−a [cm]
0
1
0 −5
−4
−3
−2
−1
r−a [cm]
Figuur 5.3a and b. Poloïdale snelheid vθ bij θ=0° vs. radius r-a in L- en H-mode: gemeten (o) en berekende vθ , met n%n (∗) en zonder n%n (onderbroken lijn).
0
1
9
9
8
8
7
7
6
6
vφ [106cm/s]
6
vφ [10 cm/s]
XXXIX
5 4
5 4
3
3
2
2
1
1
0 −5
−4
−3
−2 −1 r−a [cm]
0
1
0 −5
−4
−3
−2 −1 r−a [cm]
0
Figuur 5.4a and b. Toroïdale snelheid vφ bij θ=0° vs. radius in L- en H-mode. Vergelijking tussen gemeten (o) en berekende vφ met n%n (∗) en zonder n%n (onderbroken lijn). Het berekende profiel in de H-mode zonder n%n is gedeeld door 4.
In Figuur 5.2 tot Figuur 5.4 wordt de confrontatie weergegeven tussen experiment en theorie. De linker- en rechterkolom stellen respectievelijk de L- en H-mode voor. Het is duidelijk dat het experimentele veld wordt gereproduceerd door het model. Voor de lokale poloïdale snelheid is de overeenkomst verbeterd in vergelijking met het profiel dat voorspeld wordt zonder de inbreng van de poloïdale dichtheidsvariatie. Het lokale toroïdale snelheidsprofiel heeft nu de correcte grootte en vertoont nu ook een maximum dieper in het plasma. De conclusie is dat een behoorlijke overeenkomst is bereikt tussen berekende en experimentele profielen en dat de overblijvende verschillen tweede orde effecten zijn, veroorzaakt door aannames in het model. In het vervolg van dit hoofdstuk is aangetoond dat de poloïdale asymmetrieën die geobserveerd zijn voor de dichtheid en de toroïdale snelheid, verklaard kunnen worden door de poloïdale dichtheidsvariatie in rekening te brengen. Tot slot van dit hoofdstuk wordt de situatie bestudeerd waarin de elektrode uit het plasma verwijderd is (IE=0). In dit geval zorgt een kleine ionen verliesstroom voor een rotatie parallel met het magnetische veld Uit de vergelijking tussen theorie en experiment blijkt dat de richting van de waargenomen stroming correct is, maar groter is dan theoretisch voorspeld. De resultaten gepresenteerd door Höthker [35] wijken echter sterk af van deze theoretische voorspelling. Ze vertonen echter een goede overeenkomst met de stromingsprofielen die bekomen werden wanneer de elektrode geaard is. Een mogelijke verklaring bestaat erin dat de poloïdale limiter die in het plasma
1
XL
aanwezig was, geaard is en hierbij een radiale stroom induceert in het randplasma. Deze poloïdale limiter oefent daarbij een zelfde functie uit als de elektrode.
XLI
6. Besluiten en verder onderzoek. Het onderzoek in deze dissertatie heeft een aantal belangrijke bijdragen gerealiseerd. Een eerste belangrijke realisatie bestaat in de ontwikkeling van een nieuw 1D model dat toelaat om door middel van een Mach-sonde de parallelle en loodrechte stroming van het plasma te bepalen. Dit model is bovendien ook toepasbaar op een multi-collector geometrie. Een tweede belangrijke realisatie is het opstellen van een zeer volledige databank van randplasma parameters met behulp van een Mach-sonde. Hierbij zijn voor de eerste maal de parallelle en loodrechte stromingssnelheden en het radiale elektrische veld onder dezelfde condities bekomen. De voornaamste toepassing van deze databank is de experimentele validatie van het 1D model. Hiermee is experimenteel aangetoond dat het transport van deeltjes op een correcte manier in het 1D model is beschreven en dat het model preciese Machgetallen levert. De richting en de grootte van het parallelle Mach-getal blijkt zeer sterk beïnvloed te zijn door de grootte van de radiale stroom. Poloïdale asymmetrieën voor de elektronendichtheid en het parallelle Machgetallen zijn geobserveerd. Een alternatieve methode is gepresenteerd voor de bepaling van de loodrechte stroming. Hierbij volgt de loodrechte stromingssnelheid uit de stroommeting waarbij de sonde-oppervlakken parallel met het magnetische veld geörienteerd zijn. Een laatste belangrijke resultaat van deze dissertatie is de confrontatie van de gemeten stromings- en elektrische veldprofielen met een theoretisch model voor elektrode gepolariseerde ontladingen. Hierbij is aangetoond dat samendrukbaarheid van het plasma grote invloed heeft op de stromingen in het randplasma en in het bijzonder op de toroïdale stroming. Tezamen met neutrale wrijving en parallelle viscositeit is het model in staat om de experimentele profielen te reconstrueren. Verder onderzoek zou het 1D model voor de bepaling van de parallelle en loodrechte stroming moeten uitbreiden voor supersone snelheden. Bijkomende metingen bij kleinere hoeken moeten verder inzicht geven in de randvoorwaarde aan
XLII
de magnetische grenslaag. Tevens zou de toepasbaarheid van het model op een multicollector geometrie experimenteel moeten geverifiëerd worden. De loodrechte stromingssnelheid zou, in navolging van de parallelle stromingssnelheid, eveneens op een andere poloïdale locatie moeten gemeten worden. Tenslotte zou de ionentemperatuur op een onafhankelijke manier bepaald moeten worden zodat de onzekerheid op de geluidssnelheid en daarmee op de stromingssnelheden weggenomen worden.
1
Bibliography [1]
Antoni V., Desideri D., Martines E., Serianni G., Trmontin L., ‘Plasma flow in the outer region of the RFX reversed field pinch experiment’, Nucl. Fus. 36(11) 1996 1561
[2]
Back R. and Bengtson R.D., ‘A Langmuir/Mach-sonde array for edge plasma turbulence and flow’, Rev. Sci. Instr. 68(1) 1997 377
[3]
Baelmans M 1993 Ph.D. Thesis, Institut für Plasmaphysik Jül-2891
[4]
Bergmann A., ‘Two-dimensional particle simulation of Langmuir sonde sheaths with oblique magnetic field’, Phys. Plasmas 1(11) 1994 3598
[5]
Balescu R., ‘Transport processes in plasmas’, Amsterdam: Elsevier 1988
[6]
Bitter M., Wong K.L., Scott S., Hsuan H., Grek B., Johnson D. and Tait G., ‘Measurements of the toroidal plasma rotation velocity in TFTR major-radius compression experiments with auxiliary neutral beam heating’, Phys. Fluids B2 1990 1503
[7]
Bohm D., ‘Minimum ion kinetic energy for a stable sheath’, In Guthrie A. and Wakerling R.K. eds., The characteristics of Electrical discharges in Magnetic Fields, vol. 5 of National Nuclear Energy Series, p77. McGraw-Hill Book Company, Inc., 1st edtion ,1949
[8]
Braginskii S., Reviews of plasma physics Vol I 1965 (New York: Plenum Publishing Corporation)
[9]
Campbell D. J., ‘Physics and goals of RTO/RC-ITER’, Plasma Phys. Control. Fus. 41 1999 381
[10]
Chankin A.V. and Stangeby P.C., ‘The effect of diamagnetic drift on the boundary conditions in tokamak scrape-off layers and the distribution of plasma fluxes near the target’, Plasma Phys. Control. Fus. 36 1994 1485
2
[11]
Chen, Francis F., Electrical Sondes, In Huddlestone R.H. and Leonard S. LO. eds., Plasma diagnostic Techniques, New York Academic Press, 1965. p. 113
[12]
Chen, Francis F., Introduction to plasma Physics, Plenum Press, 2nd edition, 1974.
[13]
Chodura R., ‘Plasma-wall transition in an oblique magnetic field’, Phys. Fluids 25(9), 1982 162
[14]
Chodura R., ‘Basic problems in edge plasma modelling’, Contrib. Plasma Phys. 28 303
[15]
Chodura R., ‘The Bohm-Chodura plasma sheath criterion’, Phys. Plasmas 2(3), 1995.
[16]
Chung K.-S. and Hutchinson I.H., ‘Kinetic theory of ion collection by probing objects in flowing strongly magnetized plasmas’, Phys. Rev. A38 1988 4721
[17]
Chung K.-S. and Hutchinson I.H., ‘Effects of a generalized presheath source in flowing magnetized plasmas.’, Phys. Fluids B3(11), 1991
[18]
Chung K.-S., ‘Development of a Visco-Mach-sonde for the simultaneous measurement of viscosity and flow velocity in tokamak edge plasma’, Nucl. Fus. 34(9) 1994 1213
[19]
Chung K.-S. and Bengtson R.D., ‘Simultaneous measurement of viscosity and flow velocity in Texas Experimental Tokamak-Upgrade (TEXT-U) edge plasmas by using a Visco-Mach-sonde’, Phys. Plasm. 4(8) 1997 2928
[20]
Cohen, ‘Electric drift, plasma current and the Bohm condition in the SOL of a Tokamak with a toroidal limiter’, Comm. Plasma. Phys. Control. Fusion 16(4) 1995 255
[21]
Cornelis J., Sporken R., Van Oost G., Weynants R. R., ‘Predicting the radial electric field imposed by externally driven radial currents in tokamaks.’, Nucl. Fus. 34(2) 1994 171
3
[22]
Daube Th., 'The ion current of a flush mounted langmuir sonde in an oblique magnetic field.', Contrib. plasma phys. 38S 1998 145
[23]
Emmert G.A., Wieland R.M., Mense A.T. and Davidson J.N., ‘Electric sheath and presheath in a collisionless, finite ion temperature plasma’, Phys. Fluids 23(4) 1980 803
[24]
Groebner R. J., Burell K. H. and Seraydarian R.P., ‘Role of Electric Field and Poloidal Rotation in the L-H Transition’, Phys. Rev. Lett. 64 1990 3015
[25]
Gulick S.L., Stansfield B.L., Abou-Assaleh Z., Boucher C., Matte J.P., Johnston T.W. and Marchand R., ‘Measurement of pre-sheath flow velocities by laser-induced fluorescence’, J. Nucl. Mat. 176&177 1990 1059
[26]
Gunn J.P., ‘The influence of magnetization strength on the sheath: implications for flush-mounted sondes’, Phys. Plasmas 4 (12) 1997 4435
[27]
Gunn J.P., ‘Two-dimensional quasineutral PIC simulation of a gundestrup sonde’, Czech. J. Phys. 48 1998 S2
[28]
Harbour P.J. and Proudfoot G., ‘Mach number deduced from sonde measurements in the divertor and boundary layer of DITE’, J. Nucl. Mater. 121 222 1984
[29]
Harrison M., 1986 Phys. of Plasma-Wall-Interactions in Controlled Fusion NATO ASI Series B 131 (New York: Plenum Press)
[30]
Hassam A.B., ‘Poloidal rotation of tokamak plasmas at super ploidal sonic speeds’, Nucl. Fus. 36 (6) 1996 707
[31]
Hazeltine R., Lee E., Rosenbluth M, ‘Rotation of tokamak equilibria’, Phys. Rev. Letter 25(7) 1970 427
[32]
Hazeltine R., Lee E., Rosenbluth M, ‘Resistive plasma rotation and shock formation in toroidal geometry’, Phys. Fluids 14(2) 1971 361
4
[33]
Hinton F.L. and Hazeltine R.D., ‘Theory of plasma transport in toroidal confinement systems’, Rev. Mod. Phys. 48(2) 1976
[34]
Höthker K., Bieger W. and Belitz H.-J., ‘A new sonde to determine the mach number of plasma flow in a magnetized plasma’, Proc. 17th Conf. Contr. Fus. Pl. Heating, Amsterdam, 14B Part IV 1990 1568
[35]
Höthker K., Belitz H.-J., Schorn R.P., Bieger W., Boedo J.A., ‘Experimental study of neoclassical plasma flow and bootstrap current in the Tokamak TEXTOR’, Nucl. Fus. 34 (11) 1994 1461
[36]
Höthker K., Bieger W. and Van Goubergen H., ‘On the determination of the drift perpendicular to the magnetic field by means of a sonde’, Contrib. Plasma Phys. 38S 1998 121
[37]
Hutchinson, I.H., ‘Plasma Particle flux’, In principles of Plasma diagnostics, p50-86, Cambridge, University Press, 1987
[38]
Hutchinson I.H., ‘A fluid theory of ion collection by sondes in strong magnetic fields with plasma flow.’, Phys. Fluids 30 1987 3777
[39]
Hutchinson I.H., ‘Ion collection by sondes in strong magnetic fields with plasma flow.’, Proc. 14th EPS Conf. Contr. Fus. and Plasma Phys., Madrid, 1987, Vol 11D, PtIII, 1987 1330
[40]
Hutchinson I.H., ‘Ion collection by sondes in strong magnetic fields with plasma flow’, Phys. Rev. 37 1988 4358
[41]
Hutchinson I.H., ‘Reply to the comments of Stangeby’, Phys. Fluids 31(9) 1998 2728
[42]
Hutchinson I.H., ‘The magnetic presheath boundary condition with ExB drifts’, Phys. Plasmas 3(1) 1996 6
[43]
Itoh S.-I. and Itoh K., ‘Model of L- to H-mode Transition in Tokamak’, Phys. Rev. Lett. 60 1988 2276
5
[44]
Jain K.K., ‘Poloidal plasma rotation and its effect on fluctuations in a toroidal plasma’, Nucl. Fus. 36(12) 1996 1661
[45]
Jachmich S., Van Oost G., Weynants R.R. and Boedo J. A., ‘Experimental investigations on the role of EXB flow shear in improved confinement’, Plasma Phys. Control. Fusion 40 1998 1105
[46]
Jachmich S., Ph.D. Thesis, to be published
[47]
Jachmich S., Van Goubergen H. and Weynants R. R., ‘Influence of plasma flow on the floating potential’, to be published in Proc. 27th EPS Conf. on Contr. Fus. and Plasma Phys. 2000 Budapest
[48]
Johnson E.O. and Malter L., ‘A floating sonde Method for measurements in Gas Discharges’, Phys. Rev. 80(1) 1950 58
[49]
Kim Y.B., Diamond P.H. and Groebner R.J., ‘Neoclassical poloidal and toroidal rotation in tokamaks’, Phys. Fluids B3(8) 1991 2050
[50]
Keilhacker et al., ‘Fusion physics progress on the Joint European Torus (JET)’, Plasma Phys. Control. Fusion 41 1999 B1
[51]
Langmuir I., ‘The effect of Space Charge and Initial Velocities on the Potential Distribution and Thermionic Current between Parallel Plane Elektrodes’, Phys. Rev. 21 1923 419
[52]
Langmuir I., ‘The interaction of electron and positive ion space charges in cathode sheath’, Phys. Rev. 33 1929 954
[53]
MacLatchy C., Boucher C., Poirier D. and Gunn J., ‘Gundestrup: A Langmuir/Mach-sonde array for measuring flows in the scrape-off layer of TdeV’, Rev. Sci. instrum. 63 (8) 1992 3923
[54]
Manos D.M. and McCracken G.M., ‘Sondes for edge diagnostics in magnetic confinement fusion devices’, Physics of Plasma-Wall Interactions in Controlled Fusion, edited by Post, D.E. and Behrisch, R., vol. 131 of NATO ASI series, Series B, Physics, 1986, p. 135, Plenum Press.
6
[55]
Matthews G.F., ‘Tokamak Plasma diagnosis by electrical sondes’, Plasma Phys. Control. Fus. 36 1994 1595
[56]
Ongena J., Messiaen A. M. et al., ‘Overview of experiments with radiation cooling at high confinement and high density in limited and divertred discharges.’, Plasma Phys. Contr. Fusion 41 1999 A379
[57]
Ongena J. and Van Oost G.,’ Energy for future centuries’, Trans. Fus. Techn. 37 (2T) 2000 3
[58]
Othsuka H., Kimura H., Shimomura S., Maeda H., Yamamoto S., et al., ‘Sonde measurements in the Scrape-off layer of a tokamak’, Pl. Phys., 20 1978 749
[59]
Peterson B.J., Talmadge J.N., Anderson D.T., Anderson F.S.B. and Shohet J.L., ‘Measurement of ion flows using an “unmagnetized” Mach-sonde in the interchangeable module stellarator’, Rev. Sci. Instrum. 65(8) 1994 2599
[60]
Poirier D., Boucher C., ‘Validation of plasma velocity measurements with Mach-sondes using Laser Induced Fluorescence’, Proc. 25th EPS Conf. On Contr. Fus. and Plasma Phys., Prague, Vol 22C 1998 1602
[61]
Proudfoot G. and Harbour P.J., ‘A new electrostatic sonde for instantaneous measurement of gradients in a plasma boundary layer’, J. Nucl. Mat. 111&112 1982 87
[62]
Proudfoot G., Harbour P.J., Allen J., and Lewis A J., ‘Poloidal and radial variations in plasma transport in a limiter scrape-off layer in DITE’, J. Nucl. Mat. 128-129 1984 180
[63]
Riemann K.-U., ‘The Bohm criterion and sheath formation’, J. Phys. D: Appl. Phys. 24 1991 493
[64]
Riemann K.-U., ‘Theory of the collisional presheath in an oblique magnetic field’, Phys. Plasmas 1(3) 1994 552
7
[65]
Riemann K.-U., ‘Bohm's Criterion and Plasma-Sheath transition’, Contr. Plasma Phys. 36S 1996 19
[66]
Rozhansky V., Samain A., Tendler M., ‘Fast poloidal rotation and improved confinement’, 18th EPS Conf. on Contr. Fus. and Plasma Phys., Berlin, 3-7 june, ECA 15C, 1991 IV-133
[67]
Rozhansky V. and Tendler M., ‘The effect of the radial electric field on the L-H transitions in tokamaks’, Phys. Fluids B4 1992 1877
[68]
Shaing K.C. and Crume E.C. Jr, ‘Bifurcation theory of Poloidal Rotation in Tokamaks: A Model for the L-H Transition’, Phys. Rev. Lett. 63, 1989 2369
[69]
Schott L., Electrical Sondes, Plasma diagnostics, edited by Lochte-Holtgreven, North Holland publishing company, Amsterdam 1968, 668-725
[70]
Schweer B. , ‘Atomic beam diagnostics’, Trans. Fus. Techn. 37 (2T) 2000 368
[71]
Stangeby P.C., ‘Effect of bias on trapping sondes and bolometers for tokamak edge diagnosis’, J.Phys. D: Appl. Phys. 15 1982 1007
[72]
Stangeby P.C., ‘Plasma sheath transmission factors for tokamak edge plasmas’, Phys. Fluids 27(3) 1984 682
[73]
Stangeby P.C., ‘Measuring drift velocities in tokamak edge plasmas using sondes.’, P. C. Stangeby, Phys. Fluids 27 1984 2699
[74]
Stangeby P.C., 'The plasma sheath', Physics of Plasma-Wall Interactions in Controlled Fusion, edited by Post, D.E. and Behrisch, R., volume 131 of NATO ASI series, Series B, Physics, 41-97, 1986, Plenum Press.
[75]
Stangeby P.C., ‘Comments on “A fluid theory of ion collection by sondes in strong magnetic fields with plasma flow”’, Phys. Fluids 31(9) 1988 2726
[76]
Stangeby P.C. and McCracken G.M., ‘Plasma boundary Phenomena in Tokamaks’, Nucl. Fus. 30 1990 1225
8
[77]
Stangeby P.C., ‘The Bohm-Chodura plasma sheath criterion’, Phys. Plasmas 2(3) 1995 702
[78]
Stangeby P.C. and Chankin A.V., 'The ion velocity (Bohm-Chodura) boundary condition at the entrance to the magnetic presheath in the presence of diamagnetic and ExB drifts in the scrape-off layer', Phys. Plasmas 2(3) 1995 707
[79]
Stix T.H., ‘Decay of poloidal rotation in a Tokamak plasma’, Phys. Fluids 16 1973 1260
[80]
Swift, J.D. and Schwar M.J.R., Electrical sondes for plasma diagnostics, Iliffe Books Ltd., 1970
[81]
Van Goubergen H., Jachmich S., Van Oost G., Van Schoor M., Weynants R.R. and Desoppere E., ‘The inclined Mach-sonde as a diagnostic for perpendicular flow measurements in a strongly magnetized plasma’, Proc. 26th EPS Conf. on Contr. Fus. and Plasma Phys., Maastricht, 14-18 june, ECA 23J, 1999 1049
[82]
Van Goubergen H., Van Schoor M., Jachmich S., Van Oost G., Weynants R.R. and Desoppere E., ‘Theoretical and experimental study of toroidal and poloidal flows in the edge plasma of TEXTOR-94 polarisation discharges.’, Proc. 26th EPS Conf. on Control. Fusion and Plasma Phys., Maastricht, 14-18 june, ECA 23J 1999 693
[83]
Van Goubergen H., Weynants R.R., Jachmich S., Van Schoor M., Van Oost G. and Desoppere E., ‘A 1D fluid model for the measurement of perpendicular flow in strongly magnetized plasmas’, Plasma Phys. Control. Fusion 41 1999 L17
[84]
Van Goubergen H., Van Schoor M. Jachmich S., Weynants R.R., Van Oost G. and Desoppere E., ‘Assessment of the important physics of flows induced by biasing in the edge of a Tokamak’, to be published in Phys. Rev. Lett.
[85]
Van Schoor M., Ph.D. Thesis 1998 University of Antwerp.
9
[86]
Van Schoor M, Van Goubergen H., Weynants R.R., ‘Importance of compressibility to explain the observed rotation velocities in edge biasing experiments’, to be published in Proc. 14th Conf. on Plasma Surf. Int., 2000 Rosenheim
[87]
Van Schoor M. and Weynants R. R., ‘Radial Current and flows in the scrapeoff Layer of a tokamak’, Plasma Phys. Control. Fusion 40 1998 403
[88]
Wesson J., ‘Tokamaks’ Clarendon Press, Oxford 1997
[89]
Weynants R. R., Van Oost G., Bertschinger G., Boedo J. et al., ‘Confinement and profile changes induced by the presence of positive or negative radial electric fields in the edge of the TEXTOR Tokamak’, Nucl. Fus. 32 (5) 1992 837
[90]
Weynants R.R. and G. Van Oost, ‘Edge biasing in Tokamaks’, Plasma Phys. Control. Fusion 35 B177 1993
[91]
Weynants R. R., Jachmich S. and Van Oost G., ‘Demonstration of the role of ExB flow shear in improved confinement.’, Plasma Phys. Control. Fusion 40 1998 635
[92]
Wolf G. and Unterberg B., ‘Thermonuclear Burn Criteria’, Trans. Fus. Techn. 29 1996 17
[93]
Xiao C., Jain K.K., Zhang W. and Hirose A., ‘Measurement of plasma rotation velocities with elektrode biasing in the Saskatchewan Torus-Modified (STORM) tokamak’, Phys. Pl. 1(7) 1994 2291
