Understanding and controlling plasma rotation in tokamaks de Bock, M.F.M.

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1 Understanding and controlling plasma rotation in tokamaks de Bock, M.F.M. DOI: /IR Published: 01/01/2007 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 22. Aug. 2018

2 Understanding and controlling plasma rotation in tokamaks Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 22 januari 2007 om uur door Maarten De Bock geboren te Sint-Niklaas, België

3 Dit proefschrift is goedgekeurd door de promotoren: prof.dr. N.J. Lopes Cardozo en prof.dr. F.C. Schüller Copromotor: dr. R. Jaspers CIP-data library Technische Universiteit Eindhoven De Bock, Maarten Understanding and controlling plasma rotation in tokamaks/ by Maarten De Bock. Eindhoven : Technische Universiteit Eindhoven, Proefschrift. ISBN-10: ISBN-13: NUR 926 Trefwoorden: plasmafysica / kernfusie / fusieplasma s / plasmarotatie / magnetische velden / magnetohydrodynamica / plasmadiagnostiek Subject headings: plasma / nuclear fusion / plasma rotation / magnetic fields / magnetohydrodynamics / plasma diagnostics The work described in this thesis was performed as part of a research programme of Stichting voor Fundamenteel Onderzoek der Materie (FOM) with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), the Forschungszentrum Jülich GmbH and EURATOM. It was carried out at the Institut für Plasmaphysik at the Forschungszentrum Jülich GmbH in collaboration with the FOM-Institute for Plasma Physics Rijnhuizen. Typeset in L A TEX 2ε. Printed by B-SG, Forschungszentrum Jülich GmbH, D Jülich, Germany. Cover design by Wim De Pagie, artist impression of an excited carbon ion that passes by.

4 Just remember that you re standing on a planet that s evolving, and revolving at 900 miles an hour. That s orbiting at 19 miles a second, so it s reckoned, a sun that is the source of all our power. The sun and you and me, and all the stars that we can see, are moving at a million miles a day. In an outer spiral arm, at 40,000 miles an hour, of the galaxy we call the Milky Way. Galaxy song, by Monty Python

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6 Understanding and controlling plasma rotation in tokamaks Contents Contents v 1 Introduction Thermonuclear Fusion Why do we need fusion? Fusion reactions Tokamaks Importance of plasma rotation This thesis List of publications Theory of plasma rotation Introduction From single particle motion to plasma rotation Magnetohydrodynamics Tokamak equilibrium Definition of plasma rotation Momentum transport Momentum confinement time Momentum sources and sinks Momentum balance Neoclassical transport and rotation Neoclassical theory Poloidal flow damping Spontaneous neoclassical rotation Impurity and MHD Rotation Impurity rotation MHD rotation Conclusion TEXTOR, DED and diagnostics Introduction The TEXTOR tokamak and its heating systems The Dynamic Ergodic Divertor Co- and counter directions in TEXTOR Plasma diagnostics Charge exchange recombination spectroscopy iii

7 3.5.2 Electron Cyclotron Emission Thomson Scattering Interferometer Charge Exchange Recombination Spectroscopy Introduction Principle Additional effects Energy dependence of the emission rate Non-thermal line broadening Non-CX emission CXRS at TEXTOR Summary Plasma rotation at TEXTOR Introduction Momentum and energy confinement time in TEXTOR Ohmic rotation in TEXTOR Impurity rotation in TEXTOR MHD rotation in TEXTOR Conclusion Measurements of plasma rotation during DED operation Introduction A general overview Change in rotation as a function of DED current Plasma rotation locked to the DED frequency Transition threshold Summary and conclusion Theory of perturbation fields Introduction Penetration of modes Tearing modes Rutherford equation, EM torque and Viscous torque Effect of an external helical perturbation A locked mode Mode penetration The effect of an external perturbation in the plasma edge Introduction The stochastic torque in the plasma edge Conclusion Conclusion Momentum input by a stochastic edge field Introduction /4 DED operation Analysis of discharge # iv

8 Understanding and controlling plasma rotation in tokamaks 8.4 Conclusion Mode penetration Introduction Force balance model Electromagnetic Torque Force balance at q = Conclusion Changing the plasma rotation by EM waves Introduction Changing the momentum by ion cyclotron radiation Influence of ECRH on rotation Conclusion Discussion, conclusions and outlook towards future devices Exciting tearing modes with the DED in TEXTOR Influence of ICRH and ECRH on rotation in TEXTOR Rotation in ITER Summary 151 Samenvatting 155 Curriculum Vitae 159 Acknowledgements 161 A List of the discharges used in this thesis 163 Bibliography 165 v

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10 Understanding and controlling plasma rotation in tokamaks Chapter 1 Introduction 1.1 Thermonuclear Fusion Why do we need fusion? The world s economy runs on fossil fuel. It heats our houses, it drives our cars and delivers our electricity. However, the supply of oil, coal and gas is exhaustible. If no action is taken an energy crisis is imminent in the next couple of centuries, maybe decades. Apart from the inevitable fact that we will run out of fuel, burning coal, oil and gas also have an impact on the environment. The signs of global warming due to CO 2 and the accompanying climate change become increasingly more alarming. Nevertheless, mankind will not give up the present standard of living, so the energy consumption is not expected to reduce. It will rather increase. Sources of energy, other than fossil ones, are therefore needed. Those that are presently available, however, have a too low energy density (solar, wind and bio-energy) or produce long term radioactive waste (nuclear fission). Thermonuclear fusion holds the promise of an abundant supply of energy, without affecting the climate and with a minimum of short term radioactive waste Fusion reactions Fusion is what keeps the sun and all the stars burning. It is a process where two light nuclei fuse to form a heavier nucleus. It is found that the sum of the masses of the two light nuclei is larger than the mass of the fusion products. The difference in mass is, according to Einstein s well-known formula E = mc 2, converted into energy. There are several fusion reactions possible. The reaction that is the best candidate for energy production on earth is the deuterium-tritium reaction, because its cross-section is reasonable large. In this reaction two hydrogen isotopes, deuterium and tritium, fuse resulting in a helium nucleus and a neutron: D + T He + n MeV (1.1) The produced energy is divided over the fusion products: the neutron receives a kinetic energy of 14.1 MeV, the helium nucleus, also called α-particle, has an energy of 3.5 MeV. Deuterium is largely available in the oceans on earth. Tritium is not, but can be produced by a nuclear reaction of lithium also widely available and an energetic neutron. The waste of this fusion reaction is helium: a non-toxic, non-radioactive gas. 1

11 Chapter 1 - Introduction T He D n Figure 1.1 : The two hydrogen isotopes, deuterium and tritium, fuse to form helium and a neutron. In this reaction 17.6 MeV of energy is released. The nuclei of deuterium and tritium do not fuse spontaneously. Because they both have a positive charge, the repelling Coulomb force prevents the reaction. A sufficiently high kinetic energy of the nuclei is needed to overcome the Coulomb force. This high kinetic energy is achieved in a gas that has a temperature T of about 100 million degrees centigrade. At this kind of temperature gasses are ionised. We do no longer call them a gas, but call them a plasma. It is common to express temperatures in a plasma with ev, where 1 ev C. Not only a high temperature is needed. In order to have enough collisions between the highly energetic nuclei the density n of the particles has to be high enough as well. A third important parameter is the energy loss. If a fusion plasma loses its energy faster to the outside world than it can gain energy, from fusion reactions and/or from external heating, then the process will die out. The rate at which a plasma loses its energy is given by 1/τ E, where τ E is called the energy confinement time. A fusion reaction will be self-sustained if the product of the above three parameters temperature T, density n and confinement time τ E is sufficiently high. For the deuteriumtritium reaction it is found to be: n T τ E > kev s/m 3 (1.2) The above criterion is the so-called Lawson criterion. The triple product n T τ E is a figure-of-merit for a fusion reactor: the higher it is, the better. In order to have a reasonable cross-section for the deuterium-tritium reaction, a temperature of 1 to 10 kev is needed. In order to keep a plasma at such high temperatures we should avoid contact between the plasma and any material wall. A possible way of doing this, is to use a magnetic trap that confines the plasma in the centre of a vacuum chamber. 2 Section Thermonuclear Fusion

12 1.2 Tokamaks Understanding and controlling plasma rotation in tokamaks The fact that a plasma mainly consists of charged particles can be used to confine it. Due to Lorentz forces the movement of charged particles perpendicular to magnetic field lines is restricted, causing the charge particle α to gyrate around the field line with cyclotron frequency ω c = e α B/m α and a gyroradius, also called Larmor radius, ρ L = u /ω c (see fig. 1.2), where B is the magnetic field, e α the charge of the particle, m α the mass of the particle and u the particle velocity perpendicular to the magnetic field. Figure 1.2 : Charged particles gyrate closely around the magnetic field lines. Hence, the geometry of the magnetic field determines the geometry of the plasma that consists of charged particles. The movement of a charged particle parallel to the magnetic field is not restricted. In order to confine a plasma effectively, the field lines should therefore close in themselves, hence form a toroidal geometry. However, just a torus shaped vacuum vessel with a toroidal magnetic field is insufficient to confine a plasma. The curvature of the magnetic field causes electrons and ions to drift to the bottom, respectively the top of the torus, resulting in an electric field. This electric field in its turn leads to an outward drift of all particles, and therefore to a loss of confinement. To neutralise this electric field, particles that drifted to the top of the machine should be brought to the bottom and vice versa. This can be achieved by adding a poloidal component to the magnetic field. In a tokamak configuration the poloidal magnetic field is generated by a toroidal plasma current (see fig. 1.3). This plasma current is induced by a transformer, using the plasma as the secondary winding 1. On top of these poloidal and toroidal field components radial and vertical components are added to the magnetic field by external positioning and shaping coils. Poloidal and toroidal components add up to a set of helical field lines. These field lines lie on magnetic surfaces. The magnetic structure in a tokamak can be seen as an infinite set of nested magnetic flux surfaces. An important parameter in tokamak physics is the safety factor q: q = m n r B φ R B θ (1.3) It describes the number of toroidal windings of a field line needed for one poloidal winding (e.g. m=3, n=2 means that, when you follow a field line on this q = 3/2 surface, you need 3 toroidal windings and 2 poloidal windings to reach your starting point again, or 1.5 toroidal windings for 1 poloidal winding). At places with a (low) rational q-number instabilities can easily arise. The effect of a minor fluctuation at a certain position is amplified because the particles pass every q toroidal revolutions at that same position. 1 Another method to generate a poloidal field is using additional or specially shaped coils. This is done e.g. in a stellarator. Those configurations will not be discussed in this thesis Section Tokamaks 3

13 Chapter 1 - Introduction Figure 1.3 : The principle of magnetic confinement in a tokamak. For stability reasons a helical magnetic field is needed. The toroidal component of this field comes from field coils, the poloidal component is induced by the plasma current. By making the plasma column the secondary winding of a transformer, high plasma currents can be reached easily by feeding a current ramp to the primary winding of the transformer. 1.3 Importance of plasma rotation A fusion relevant plasma must fulfill three main criteria: The first criterion is the Lawson criterion, that says that the energy and particle confinement must be high. This is achieved if the energy that is put into the plasma stays sufficiently long inside the plasma. In a fusion reactor we thus have to minimise the energy transport to the wall. The confinement must not only be high, it also has to be stable. Because the plasma in a tokamak is confined by a magnetic field, a good understanding of the magnetic stability is necessary. In other words: we need to know which magnetic instabilities can occur and how we can avoid them. The power that goes into a fusion reactor eventually ends up at the wall of the reactor. A good knowledge of the plasma-wall interaction is needed in order to (a) optimise the first wall materials, so that they can cope with the power load, and (b) keep impurities and dust out of the core plasma. Several experiments around the world have shown that a high plasma rotation can attribute to the first two criteria: the improvement of both stability and confinement. The confinement in a tokamak is governed by the radial transport of energy from the plasma centre to the plasma edge. A large part of this transport is driven by turbu- 4 Section Importance of plasma rotation

14 Understanding and controlling plasma rotation in tokamaks lence. Empirically it is observed that the confinement decreases when the power input is increased. It is generally agreed upon that this confinement degradation is a result of increased turbulent transport. An important breakthrough in fusion physics was the discovery of the H-mode H stands for high confinement in the early 80 s [87]. When the power input reached a certain threshold the confinement suddenly almost doubled. One of the observations made during this transition was a sudden change in plasma rotation in the edge. More precisely the gradient in poloidal rotation v θ and the radial electric field E r which is linked with rotation increased. This observation led to the question: Does a sheared rotation reduce turbulent transport? Figure 1.4 : A simulation of turbulence with (A) and without (B) sheared plasma flow. The contours of the fluctuation potential in a poloidal cross section are plotted. Large turbulent transport exists along the iso-potential contours. It is clearly seen that the turbulent cells in the case of sheared flow are much smaller than in the case without flow, which means that the overall radial transport is lower. [58] Now, twenty years later, the mechanism of turbulence suppression by rotational shear is widely accepted. The physical picture of turbulence suppression by flow shear can be looked upon as follows: turbulent transport is a result of fluctuations with a radial correlation length r c and a decorrelation time τ c. These fluctuations can be seen as turbulent cells with a radial extension of r c and a lifetime τ c. The transport induced by the turbulent cells is given by the diffusion coefficient D turb rc 2 /τ c. A sheared flow, i.e. a different velocity at each radial point, will shred these turbulent cells apart, leading to smaller cells r sheared < r c. Consequently the diffusion coefficient, hence the radial transport, is smaller [7]. In figure 1.4 the fluctuation potential is shown for a simulation with and without sheared flow [58]. It is clearly seen that the turbulent cells are much smaller in the case with flow (A) than in the case without flow (B). A sheared plasma flow will thus improve the confinement. A straightforward method for increasing the velocity gradient is increasing the velocity. Because the vacuum vessel of a tokamak does not move, a velocity gradient exist between the plasma and the wall. This gradient will be large if the plasma rotates fast. Apart from improving the confinement, a fast rotating plasma also increases the stability of the magnetic configuration. In an ideal world the plasma in a tokamak is confined in a set of perfectly nested flux surfaces. In the real world sources of free energy in the plasma can break up and reconnect a flux surface, hence changing the magnetic topology (see figure 1.5). These reconnected flux surfaces are called tearing modes or magnetic islands. Tearing modes have an unfavourable effect on plasma confinement and can even cause minor and major disruptions. We therefore usually try to avoid them. Section Importance of plasma rotation 5

15 Chapter 1 - Introduction Figure 1.5 : In (a) the ideal magnetic topology is shown: a set of nested flux surfaces. In (b) two of the flux surfaces the q=1 and q=2 surfaces are broken up and reconnected, resulting in m/n = 2/1 (dark grey) and m/n = 1/1 (black) islands. Not only an excess in free energy can drive tearing modes. They can also be formed by an externally applied perturbation field. This perturbation field can be applied on purpose, but usually it is the result of misalignment of coils, a non-axisymmetric wall of the vacuum vessel, et cetera. In this case we call it an error field. Because no machine is perfectly aligned or perfectly symmetric, every fusion device has an error field. And due to this error field tearing modes can develop. Error fields are static, and as a result the tearing modes they excite do not move as well. Tearing modes, however, have to rotate with the plasma velocity. In rotating plasmas, therefore, an error field does not excite tearing modes: the tearing modes are suppressed. The interaction between the suppressed tearing modes and the error field will, however, slow down the plasma. Once the suppressed tearing modes are at rest in the frame of the error field large tearing modes will develop. This is called mode excitation. Because the error field needs to slow down the plasma to great extent, we expect a high threshold for mode excitation in fast rotating plasmas. The threshold for mode excitation by a static perturbation field in TEXTOR, as a function of the plasma rotation is given in figure 1.6. One sees that for fast rotating plasmas the threshold is indeed higher than for slow rotating plasmas. Another type of magnetic instabilities are the resistive wall modes (RWM). These RWM s occur in plasmas with a high plasma energy (β). When they lock, i.e. when they do not move with respect to the vessel wall, RWM s cause a disruption. In high β devices, like ITER, the prevention of RWM s is therefore crucial. Although RWM s have a different topology than tearing modes, they can be treated with the same philosophy as tearing modes: (a) in a fast rotating plasma RWM s will be suppressed, (b) the interaction between the suppressed RWM and the conducting (i.e. resistive) tokamak wall will slow down the plasma and (c) once the RWM is at rest with respect to the wall the RWM grows and causes a disruption. A fast rotating plasma rotation has both good plasma confinement, through flow shear turbulence suppression, and stability, due to the high threshold for error field driven tearing modes and resistive wall modes. In order to optimise the performance of fusion devices, it is vital to know (a) how plasma rotation improves confinement and stability and (b) how a large plasma rotation can be driven. 6 Section Importance of plasma rotation

16 Understanding and controlling plasma rotation in tokamaks 2 Mode threshold (I DED / ka) Ω φ,0 (q=2) (rad/s) x 10 4 Figure 1.6 : The threshold at which an externally applied, static perturbation field excites a 2/1 tearing mode at the q=2 surface in TEXTOR, is plotted as a function of the plasma rotation frequency at the q=2 surface. At high rotation velocities, in both directions, the threshold is high. At low rotation velocities, the threshold is lower. 1.4 This thesis Above, two benign aspects of plasma rotation were mentioned: turbulence suppression by flow shear and increased stability against error field driven tearing modes and resistive wall modes. In this thesis will concentrate on the error field driven tearing modes. The reason for selecting this specific topic is the fact that the TEXTOR tokamak, where the work presented in this thesis was executed, is equipped with two tangential neutral beams, that allow us to control the plasma rotation, and a dynamic ergodic divertor (DED), that allows us to create and control the external perturbation field. For the next generation of fusion reactors, like ITER, the stability against resistive wall modes (RWM) will be more important than the stability against tearing modes. The investigation of RWM s is, however, difficult on TEXTOR due to the low β. Nevertheless we can gain insight in the process of RWM locking by investigating the excitation of tearing modes, because both processes are very similar. When we have a closer look at figure 1.6, we see that the simple statement the higher the rotation, the higher the threshold is not completely valid. The minimal threshold is not located at zero rotation. Furthermore the increase in threshold depends on the direction of the rotation; at the right side of the minimum the threshold increases faster with increasing rotation than at the left side of the minimum. The relation between error field modes and plasma rotation seems to be more complex than we have thought. It is advisable to have a closer look into this subject. A first question we will try to answer in this thesis is: How do we avoid the excitation of unfavourable modes by an external perturbation field? Section This thesis 7

17 Chapter 1 - Introduction To answer this question we can reverse it: How can we excite modes with an external perturbation field? Because it can be noticed on several tokamaks that the mode excitation is linked with the plasma rotation, we will narrow down the above question to: How does the threshold for mode excitation by an external perturbation field depend on the plasma rotation? with as specific sub-questions: - Why does the minimal threshold occur at a finite rotation velocity? - Why is the threshold plot asymmetric around the minimum? In the previous section it was said that the dependence of the threshold on the plasma rotation was a result of the fact that the error field had to slow down the plasma first. We should thus also ask ourselves how this slowing down mechanism works: How does the external perturbation field change the plasma rotation? Because the plasma rotation is driven by torques, the aim of this thesis is to identify the different torques that a perturbation field can exert onto a plasma. Once these torques are known the interplay between these perturbation field torques and the torques that are present in plasma without external perturbation field have to be investigated. As a result we hope to be able to reconstruct the change in plasma rotation as a function of the external perturbation, and consequently predict the threshold for mode excitation as a function of the plasma rotation. The thesis is structured as follows. In the second chapter the basic theoretical framework is introduced. Therefore a selection is made from what is known about plasma rotation in the literature. In chapter 3 the TEXTOR tokamak and the DED are introduced. The TEXTOR tokamak, with its two neutral beams, and the DED, that provides us with a fully adjustable perturbation field, are the tools that allow us to perform dedicated experiments on the interplay between a perturbation field and the plasma rotation. In order to investigate plasma rotation we have to be able to measure it. The technique used for rotation measurements is charge exchange recombination spectrometry and is introduced in chapter 4. Now that the theory, the machine and the measurement technique are introduced, chapter 5 briefly looks into what plasma rotation typically means at TEXTOR. Some theoretical predictions from chapter 2 are applied to TEXTOR and are compared with measurements. In chapter 6 we have a look into the measurements of plasma rotation during DED operation. These measurements do not fully match our expectations. In chapter 7 we therefore have a look into the literature. We find that our expectation is based on the assumption that there is only one type of force exerted on the plasma by the perturbation field: an electromagnetic force positioned at the rational q=surfaces comparable with the slip force in an electric induction motor. We also find in the literature that another 8 Section This thesis

18 Understanding and controlling plasma rotation in tokamaks force is possible. The perturbation field can create a stochastic region in the plasma edge. Parallel electron loss in the stochastic region causes subsequently a stochastic force in the plasma edge. To get a good description of the plasma rotation and the mode excitation both the electromagnetic force and the stochastic force have to be taken into account. In order to get a better insight into the stochastic force, an experiment was carried out under conditions in which the electromagnetic force could be neglected. This is presented in chapter 8. It allowed us to compare the theory of the stochastic force given in chapter 7 to the experimental observation. It also led to a straightforward expression for the stochastic force. Finally, in chapter 9 the interplay between perturbation field and plasma rotation is investigated in detail. Including both the electromagnetic and the stochastic force we were able to get a prediction for the change in plasma rotation and the threshold for mode excitation that is in agreement with the measurements. The work presented in chapters 6 to 9 confirms that a fast rotating plasma has a better resistance against the excitation of tearing modes than a slow rotating one. In present day devices high rotation velocities can be achieved using neutral beam injection. This is not the case for the next generation of fusion reactors. Extra sources of momentum input, apart from neutral beam injection, are therefore necessary. In the literature ion cyclotron heating (ICRH) is often mentioned as a good candidate for momentum input. And if ion cyclotron waves can change the plasma rotation, it is a small step to assume that also electron cyclotron waves (ECRH) will have an influence on plasma rotation. At the TEXTOR tokamak both ICRH as ECRH systems are available. In chapter 10 experiments are presented that investigate the influence of ICRH and ECRH on the plasma rotation. In a final chapter a prospect is given on what the results of this thesis mean for future devices like ITER. 1.5 List of publications In this section a list of publications and conference contributions is given. Those related to this thesis are marked with an asterisk. Journal publications [1] M. De Bock, K. Jakubowska, M. von Hellermann et al. Measuring one-dimensional and two-dimensional impurity density profiles on TEXTOR using combined charge exchange-beam emission spectroscopy and ultrasoft x-ray tomography. Review of Scientific Instruments, vol. 75 pp (2004). [2] M.F.M. de Bock, I.G.J. Classen et al. The influence of plasma rotation on tearing mode excitation in TEXTOR. submitted to Nuclear Fusion. [3] I.G.J. Classen, M.F.M. de Bock et al. Dynamics of tearing modes in the presence of a perturbation field. submitted to Nuclear Fusion. [4] K.H. Finken, S.S. Abdullaev, M.F.M. De Bock et al. Toroidal Plasma Rotation Induced by the Dynamic Ergodic Divertor in the TEXTOR Tokamak. Physical Review Letters, vol. 94 pp (2005). Section List of publications 9

19 Chapter 1 - Introduction [5] Y. Kikuchi, M.F.M. de Bock, K.H. Finken et al. Forced Magnetic Reconnection and Field Penetration of an Externally Applied Rotating Helical Magnetic Field in the TEXTOR Tokamak. Physical Review Letters, vol. 97 pp (2006). [6] H. Koslowski, E. Westerhof, M.F.M. de Bock et al. Tearing mode physics studies applying the Dynamic Ergodic Divertor on TEXTOR. Plasma Physics and Controlled Fusion, vol. 48 pp. B53 B61 (2006). [7] M. Lehnen, S. Abdullaev, W. Biel, M.F.M. de Bock et al. Transport and divertor properties of the dynamic ergodic divertor. Plasma Physics and Controlled Fusion, vol. 47 pp. B237 B248 (2005). [8] R. Wolf, W. Biel, M.F.M. de Bock et al. Effect of the Dynamic Ergodic Divertor in the TEXTOR Tokamak on MHD Stability, Plasma Rotation and Transport. Nuclear Fusion, vol. 45 pp (2005). Conference contributions [1] M. de Bock, R. Jaspers, M. von Hellermann et al. Plasma rotation during operation of the dynamic ergodic divertor in TEXTOR. 31st EPS Conference on Controlled Fusion and Plasma Physics, vol. 28G, pp. P (2004). [2] M. de Bock, C. Busch, K.H. Finken et al. Plasma rotation during operation of the dynamic ergodic divertor in TEXTOR. 32nd EPS Conference on Controlled Fusion and Plasma Physics, vol. 29C, pp. P (2005). [3] C. Busch, M. de Bock, K.H. Finken et al. Impact of the DED on ion transport and poloidal rotation at TEXTOR. 32nd EPS Conference on Controlled Fusion and Plasma Physics, vol. 29C, pp. P (2005). 10 Section List of publications

20 Understanding and controlling plasma rotation in tokamaks Chapter 2 Theory of plasma rotation 2.1 Introduction In order to tackle the questions raised in the previous chapter, a basic theoretical framework is needed. The basic theory of the dynamics of a plasma is very well described in the literature [88, 32, 2]. In this chapter we have selected all the information needed when discussing the rotation of a plasma. We start with the general description of a plasma as a collection of individual particles, with each their own velocity, in an electro-magnetic environment. This description can be transformed to a multiple fluids description, in which each species ions, electrons and neutrals have a collective fluid velocity and a thermal velocity distribution. A further simplification is made when only two plasma species are considered, electrons and ions. These two are combined to describe the plasma as a single, conducting fluid: the magnetohydrodynamic (MHD) equations of a plasma. At this point the magnetic configuration comes into discussion. It will be shown that the magnetic configuration in a tokamak consists of nested flux surfaces. The fluid velocity can now be split in two components: a fluid velocity perpendicular to the flux surfaces is referred to as convection, whereas fluid velocity on a flux surface is called rotation. Having defined rotation as being fluid velocity on a flux surface, we have a closer look into the direction of the rotation. We can split the rotation in a toroidal and poloidal component, compatible with the symmetry of a tokamak, or in a component parallel and a component perpendicular to the magnetic field, which has more physical relevance. We also discuss the sources, the sinks and the transport of rotation. More specifically the force or momentum balance equation is derived. This will turn out to be a powerful tool to interpret the experiments on plasma rotation. Up to this point, we have assumed the transport coefficients for particle, energy and momentum to be given. We have not looked into the underlying physical mechanisms. This means whether classical transport (collisional diffusion), neoclassical transport (taking into account the toroidal geometry of a tokamak) or turbulent transport is considered. When following neoclassical transport theory, we find out that poloidal plasma rotation is strongly damped, so that we can concentrate in this thesis on the toroidal rotation. It also follows from neoclassical theory that plasmas have a tendency 11

21 Chapter 2 - Theory of plasma rotation to rotate spontaneously. This intrinsic or spontaneous rotation could be very helpful in establishing a fast rotating, well-confined plasma. The influence that turbulent transport has on rotation e.g. zonal flows falls out of the scope of this thesis. Although the single fluid MHD picture provides good insight into the physics, plasma diagnostics measure the properties of the several plasma species rather than those of the single plasma fluid. In the last section we therefore return to the multiple fluid picture and try to link the electron fluid rotation and the impurity fluid rotation, to the single fluid plasma rotation. 2.2 From single particle motion to plasma rotation Magnetohydrodynamics A plasma is a large collection of several types of particles: ions, neutrals and electrons. The statistical behaviour of a large number of particles is governed by the Boltzmann equation (2.1). It describes the change of the distribution function f α (x, u, t) of the particles of the type α (α being ions or electrons or neutrals), where x describes the position and u the velocity of each particle. f α t + u f α + F m α u f α = ( ) fα t c (2.1) ( The ) distribution function f α (x, u, t) changes as a result of the forces F and collisions fα t. In a plasma the main forces are long-range Lorentz forces F = q α(e + u B). c Collisions between different plasma species result in friction forces (momentum sources and sinks), viscosity (momentum transport) and resistivity. In order to get a full description of the plasma the above Boltzmann equation has to be solved for each plasma species, together with the Maxwell equations for E and B and an appropriate description for the collisions. This set of Boltzmann, Maxwell and collision equations is called the kinetic model. Solving these equations is quite cumbersome and usually it is not necessary to describe plasma behaviour in terms of the distribution function and microscopic quantities. A simplified model is therefore derived: magnetohydrodynamics. A first step in simplifying the above kinetic model is evaluating the appropriate moments of equation (2.1). A moment A is defined by: A 1 n α Af α du (2.2) where n α = f α du is the density distribution. Multiplying equation (2.1) by A and integrating over the velocity space, we get the conservation equations for mass (A = 1) and momentum (A = mu): dn α dt + n α v α = 0 (2.3) n α m α dv α dt = n α q α (E + v α B) P α + R α (2.4) 12 Section From single particle motion to plasma rotation

22 Understanding and controlling plasma rotation in tokamaks The total time derivative in equations (2.3)-(2.4) is defined as d dt = t + v α. It is seen that by evaluating the moments of equation (2.1) we now have a set of equations that describes the plasma by macroscopic quantities: v α = u the fluid velocity P α = n α m α ww the pressure tensor 1 R α the momentum transfer due to collisions between different plasma species (i.e. friction) 1 w = u vα is the thermal velocity A next step is to consider only 2 particle species in the plasma: electrons and ions. When we combine equation (2.3) for electrons and ions, the 1 fluid conservation of mass relation is found: dρ dt + ρ v = 0, (2.5) where ρ = m i n i + m e n e is the mass density and v = m in i v i +m en ev e n i m i +n em e the centre of mass velocity. It is clear that m e m i and that, due to quasi-neutrality, electron and ion density are equal n = n e = n i. Hence it can be said that ρ nm, m m i and v v i. The combination of equation (2.4) for electrons and ions results in a new momentum conservation and a current conservation equation: ρ dv = j B p Π (2.6) dt E + v B = 1 ( m ) e j ne e t + j B p e Π e + R e (2.7) In short it can be said that the description of the momenta of the two fluids is replaced by the momentum equation of the centre of mass velocity v which is almost equal to the ion fluid velocity due to the large difference in electron and ion mass and a description of the electrical current density j which relates to the difference in ion and electron velocity. In equations (2.6)-(2.7) the pressure tensor P = P e + P i is split up in the scalar pressure p = p e + p i and the anisotropic part Π = Π e + Π i, so that P = p I + Π. The scalar pressure relates to temperature via p α = n α T α and p = nt, which implies T = T e + T i. Π α depends like R α on collisions, but in contrast to R α also on velocity gradients (see subsection and [80, 88, 9]). Π α describes the viscosity and is therefore also called the viscous stress tensor. Both Π e and R e in equation (2.7) give rise to resistivity. Equation (2.6) is also known as the force balance equation: it is indeed nothing more than Newton s law m dv dt = F. Equation (2.7) is the generalised Ohm s law, which links currents, flows, resistivity and electromagnetic fields. As seen in (2.3) and (2.4) in each conservation equation of a specific moment there appears a higher order moment. A term to close the fluid equations is therefore needed. Section From single particle motion to plasma rotation 13

23 Chapter 2 - Theory of plasma rotation Usually the assumption of an adiabatic fluid ( ) is made: no heat is transferred to or from the plasma fluid. This is expressed by d p dt ρ = 0. γ The adiabatic assumption, equations (2.5)-(2.7) and the Maxwell equations together form the MHD equations: a single fluid description of the plasma. The complete set of MHD equations is given below: Mass conservation: dρ + ρ v = 0 (2.8) dt Momentum conservation or force balance: ρ dv = j B p Π (2.9) dt Generalised Ohm s law: E + v B = 1 ( m ) e j ne e t + j B p e Π e + R e (2.10) Assumption of an adiabatic fluid: ( ) d p dt ρ γ = 0 (2.11) Faraday s law: E = B (2.12) t Ampère s law: B = µ 0 j + 1 E c 2 (2.13) t Gauss law: B = 0 (2.14) Charge conservation: j = 0 (2.15) Tokamak equilibrium The description of the equilibrium of a magnetically confined plasma in a tokamak allows a further reduction of the MHD equations. Obviously an equilibrium is a stationary situation, so all partial derivatives to time can be set to zero ( t = 0). Furthermore hot tokamak plasmas have low resistivity and viscosity. We can therefore in a first approximation neglect them. Finally we assume an incompressible plasma fluid. By definition incompressibility means dρ dt = 0. This is, according to mass conservation (2.8), equivalent with v = 0. The application of these reductions to the MHD equations leads to the ideal MHD equations. The most used equation is the ideal force balance equation, that expresses the balance between the kinetic pressure gradient and the Lorentz force: p = j B (2.16) From equation (2.16) it can easily be seen that in an equilibrium magnetic field lines and current lines lie on surfaces of constant pressure: 14 Section From single particle motion to plasma rotation

24 Understanding and controlling plasma rotation in tokamaks B p = 0 j p = 0 (2.17) These surfaces of constant pressure are also surfaces of constant poloidal magnetic flux (see below) and are therefore called flux surfaces. Most important plasma parameters, like pressure, density, temperature, field line helicity, are flux functions, i.e. they are constant on a flux surface. The exact geometry of these flux surfaces can be calculated with the Grad-Shafranov equation; this is the expression for the ideal force balance equation (2.16) in a toroidal geometry [32, 77, 88]. Figure 2.1 : Magnetic flux surfaces in a tokamak Grad-Shafranov equation To calculate the magnetic configuration in a tokamak it is useful to introduce the poloidal flux function Ψ = S B θ ds. It describes the poloidal flux per radian toroidal angle φ through a surface going from the magnetic axis to a point (Z, R) (see fig. 2.1). The definition of the flux function and the right-handed cylindrical coordinate system (Z, R, φ) given in figure 2.1 allows us to rewrite the magnetic field as: B = 1 R Ψ e φ + B φ e φ B R = 1 Ψ R Z, B Z = 1 Ψ R R (2.18) Substitution of (2.18) in (2.17) leads to Ψ p = 0, which implies Ψ = Ψ(p) or p = p(ψ). The latter form is more customary; it states that pressure is a flux function. Defining F RB φ and using Ampère s law (2.13), j can be rewritten as: j = 1 µ 0 R F e φ + j φ e φ j R = 1 F µ 0 R Z, j Z = 1 F µ 0 R R (2.19) Substituting (2.19) in (2.17) gives F p = 0. This proves F = F (p) and using p = p(ψ), it follows that F = F (Ψ) is a flux function as well. Including (2.18) and (2.19) in the force balance equation, we end up with an partial differential equation describing the equilibrium of in a tokamak: the Grad-Shafranov equation [32, 77, 88]. R ( ) 1 Ψ + 2 Ψ R R R Z 2 = µ 0R 2 dp dψ F df dψ (2.20) Section From single particle motion to plasma rotation 15

25 Chapter 2 - Theory of plasma rotation This equation has three variables Ψ, F and p, corresponding with B, j and p so that two out of the three variables have to be known in order to resolve the third one. In reality initial estimates are taken for p, j and B, based on measurements of density, temperature, the external B-fields and q, which is a measure for the field line helicity: q rb φ /RB θ. In an iterative process p, j and B subsequently converge to a self-consistent solution. As a result one gets the total magnetic equilibrium field within the plasma, the current density profile, and the pressure p as a flux function. The topology of the equilibrium field shown in figure 2.1. It shows that the magnetic flux surfaces are nested. Figure 2.2 gives typical values for the toroidal field B φ, the toroidal current density j φ and, corresponding to that, the q-profile Shafranov shift B φ (T), j φ (MA/m 2 ), q B φ q j φ R R 0 (m) Figure 2.2 : Magnetic field, current density and q-profile for a typical ohmic discharge in TEXTOR (B φ = 2.25 T, I p = 350 ka, T e = 1 kev, n e = m 3 ). One of the results of the Grad-Shafranov equation or more precisely: one of the consequences of the toroidal shape of a tokamak is that the magnetic axis lies not in the centre of the plasma. It is shifted to the outside of the tokamak. This shift is called the Shafranov shift. It is also indicated in figure 2.2. Influence of plasma velocity on the equilibrium The Grad-Shafranov equation expresses the balance between the force due to the pressure gradient and the Lorentz force. When a plasma rotates, there is another force we should take into account: the centrifugal force. Including the centrifugal force into the force balance will result in a different equilibrium. In other words: fluid velocity has an influence on the tokamak equilibrium. The Grad-Shafranov equation given above does not take plasma velocity into account. In [8] the Grad-Shafranov equation, including plasma velocity is derived. It is shown that the pressure p is no longer a flux function, it is a function of flux Ψ and R: p(ψ, R). Therefore a single modified Grad-Shafranov equation does not suffice to get a full description 16 Section From single particle motion to plasma rotation

26 Understanding and controlling plasma rotation in tokamaks of the equilibrium; a set of equations is needed. In [8] a pure toroidal plasma velocity is assumed; Ω φ = v φ /R. The set of equations describing the equilibrium then are: R ( ) 1 Ψ + 2 Ψ R R R Z 2 = µ 0 R 2 p Ψ F df dψ 1 p ρ R 2 Ψ = 1 2 Ω2 φ (Ψ) (2.21) Equation (2.21) shows that plasma rotation has an influence on the magnetic equilibrium in a tokamak. The equilibrium determines to great extent the plasma stability and behaviour. For large rotation velocities the plasma flow should therefore be taken into account when calculating the equilibrium configuration. It is however shown in [8] that for subsonic velocities solving the static Grad-Shafranov equation (2.20) suffices. In most tokamaks the plasma velocity is subsonic, so it is justified to neglect the influence of plasma velocity on the equilibrium Definition of plasma rotation Starting from the velocity of individual particles u, over the fluid velocity of each plasma species v α, we ended up with the fluid velocity v of the single plasma fluid described by the MHD equations. The last step is to define which part of this fluid velocity v is rotation and which part represents particle flux. The total plasma fluid velocity v consists of a toroidal component v φ and a component in the poloidal plane. The component in the poloidal plane can in its turn be split up in a part parallel to the flux surfaces v θ and a part perpendicular to the flux surfaces v r. For circular, unshifted flux surfaces θ and r are polar coordinates in the poloidal plane, for non-circular, shifted flux surfaces the situation is a bit more complicated. For simplicity we will use polar coordinates in the poloidal plane: v θ and v r will be called the poloidal and radial components of the plasma velocity respectively. Thanks to the symmetries of a tokamak configuration, the toroidal and poloidal velocities describe plasma rotation. Because both the poloidal and the toroidal component of the plasma velocity lie on a flux surface the definition of plasma rotation is: Plasma rotation is the part of the fluid velocity that lies on a flux surface. Plasma rotation is often expressed by the angular frequency rather than by the rotation velocity: Ω φ v φ /R, Ω θ v θ /r. The radial component of the plasma velocity perpendicular to the flux surfaces describes particle convection. It is usually expressed as a particle flux: Γ = nmv r. Perpendicular and parallel rotation The θ and φ coordinates correspond with the geometry of a tokamak. From a physics point of view it is also interesting to describe the plasma rotation by the velocity component v parallel to the magnetic field and the component v perpendicular to the field. Section From single particle motion to plasma rotation 17

27 Chapter 2 - Theory of plasma rotation B θ v θ v B v v v φ B φ Figure 2.3 : Plasma velocity tangential to a flux surface i.e. rotation is shown here. For this we cut open a flux surface. The rotation can be expressed in toroidal and poloidal components, or in components parallel or perpendicular to the magnetic field. Due to the fact that B φ /B θ 1, the parallel rotation is mainly in the toroidal direction, the perpendicular rotation is mainly in the poloidal direction. The velocity evolution is given by the force balance equation (2.9). For the parallel velocity v the only term at the right hand side of (2.9) is ( Π). This means that the parallel velocity depends on the viscosity and has no direct driving terms. The perpendicular rotation velocity i.e perpendicular to B, but not perpendicular to the flux surface also has a dependence on viscosity ( Π), but has a direct driving term j r B as well. Viscosity influences the rotation profiles, but it will never by itself make the plasma rotate. This means that we can only directly drive plasma rotation in the direction perpendicular to the magnetic field. This does not mean there is no parallel rotation in fact the parallel rotation is usually quite high but the parallel rotation is always a result of the redistribution of perpendicular velocity by the viscosity. E B-drift An example of the importance of perpendicular rotation is the E B-drift. In an ideal case i.e. steady state and no resistivity Ohm s law (2.10) reduces to: E + v B = 0 (2.22) Taking the cross product of equation (2.22) with the magnetic field one gets, v = E B B 2. (2.23) This velocity perpendicular to the magnetic field is called E B-drift. If we look at E B-rotation i.e the velocity component of v that is tangential to the flux surface and thus perpendicular to the radial direction it means that a radial electric field will contribute to the perpendicular rotation. Vice versa it can be said that the perpendicular part of plasma rotation induces a radial electric field. Because B φ /B θ 1, the perpendicular rotation is mainly in the poloidal direction (see figure 2.3). In the plasma centre this poloidal rotation is strongly damped (see later), which means that, in presence of an electric field, a strong parallel velocity exists so that the poloidal component of the parallel velocity compensates the poloidal component of the E B-rotation. One finds: v θ = 0, v φ = E r B θ and v v = B φ B θ (2.24) 18 Section From single particle motion to plasma rotation

28 Understanding and controlling plasma rotation in tokamaks 2.3 Momentum transport In this section we have a closer look into the force balance equation (2.9), that describes the evolution and distribution of rotation. One could do this by evaluating the parallel and perpendicular rotation velocities, or by looking into the toroidal and poloidal components. The latter option will be taken here. In fact we will just discuss the toroidal rotation. The reason for neglecting poloidal rotation will be given in section 2.4.2, where it will be shown that the poloidal rotation in a tokamak is strongly damped Momentum confinement time As said previously, equation (2.9) is Newton s law. Considering only the toroidal component and a constant force it can be rewritten as: dρv φ dt Integrated over the whole plasma this becomes: ( dρvφ ) dv = plasma dt ( a ) d (mnω φ R) Rdr 3 = 0 Lφ 0 = F φ (x). (2.25) plasma ( ) F φ (x) dv ( a ) F φ (r)rdr 3 dt τφ dl φ = T φ dt 0 0 τ φ = L φ T φ, (2.26) where L φ is the total angular momentum of the plasma, T φ is the total input torque and τ φ is the momentum confinement time [46]. Equation (2.26) basically states that the momentum confinement time τ φ is the time necessary to reach a certain angular momentum L φ, given a certain input torque T φ. Consequently it is also the time in which the total momentum is lost when the input torque is turned off. It is therefore a characteristic time for (radial) momentum transport. Momentum transport is described by the momentum diffusion coefficient D 1 φ. When the plasma radius is a, the average momentum diffusion coefficient is given by D φ a 2 /τ φ. The definition of τ φ is very similar to that of the energy confinement time τ E = 3 2 nt dr3 /P, P being the total input power [88]. Empirically it has been observed that the relationship between τ φ and τ E is actually stronger than just a similarity. Measurements of τ φ and τ E done in several tokamaks show that increasing τ E corresponds with increasing τ φ (see e.g. figure 2.4 and [46]). When comparing the ion energy confinement time τe ion i.e. the energy transport in the ion channel with the momentum confinement time τ φ the agreement is even better [19, 86]. Apart from the experimental evidence that τ φ and τe ion are related, several theoretical models, as e.g. described in [60], predict the ion thermal diffusivity and the momentum diffusion to be equal; χ i = D φ. 1 In the literature both D φ and χ φ are used for denoting the momentum diffusion coefficient Section Momentum transport 19

29 Chapter 2 - Theory of plasma rotation Figure 2.4 : Momentum confinement time τ φ plotted against energy confinement time τ E for different plasma conditions. Measurements were done in L-mode, neutral beam heated shots in ASDEX, during steady state phases.[46] Many experiments on machines throughout the world have shown that, both in L-mode and in H-mode regimes, the energy confinement time τ E reduces with increasing power input. Scaling laws of τ E describe the relation between τ E and the input power P as τ E P α, α = [67]. Given the fact that the momentum confinement τ φ is observed to be scaling with the ion energy confinement time τe ion, we can state that the momentum confinement τ φ also reduces with increasing power input [19, 18]. When investigating the plasma rotation in discharges with varied power input this dependence of τ φ on the power input has to be taken into account. Exceptions on the tight relation between τ φ and τ E are found at low densities, in transient phases (when e.g. extra momentum sources like neutral beams are suddenly switched on or off) and towards the plasma edge [46, 86, 91]. Also for plasmas rotating in the direction counter to the plasma current the energy confinement time is observed to be lower than the momentum confinement time [4, 90] Momentum sources and sinks The right hand side of equation (2.9) shows three forces that drive the plasma velocity v: j B, p and Π. The p term is a force perpendicular to the flux surfaces and hence has no contribution to the plasma rotation. The two other force terms do have an influence on plasma rotation. The divergence of the anisotropic part Π of the pressure tensor also known as the viscous stress tensor has components in the toroidal and poloidal direction. As said in subsection the different elements of Π depend among other things on the gradients of the velocity. This means that the plasma rotation at one position causes a driving force at an other position in the plasma. In other words: the term Π is responsible for momentum transport. In a steady state situation it determines the shape of the rotation profile. It will be discussed in more detail in subsection j B is a force perpendicular to the magnetic field. There are two mechanisms to apply a j B-force to the plasma. A first method is creating a radial current j r. This will result in both a toroidal force j r B θ and a poloidal force j r B φ. A second possibility is to perturb the magnetic field such that a radial component B r is created. If a radial field B r exists, the flux surface picture breaks down and we can no longer define plasma 20 Section Momentum transport

30 Understanding and controlling plasma rotation in tokamaks rotation as the velocity tangential to flux surface. We therefore define the plasma rotation as being the toroidal and poloidal components of the plasma velocity. The cross-product of this B r with the main plasma current (j φ, j θ ) results again in poloidal (j φ B r ) and toroidal ( j θ B r ) forces. In order to come to the force balance equation the assumption was made that the plasma only consists of electrons and ions. This assumption is not valid in the edge of the plasma where the temperature is not high enough to ionise all atoms, hence neutral particles exist. Also neutral beam injectors add neutral particles to the plasma. These neutrals cause a need for two extra force terms in equation (2.9). The neutrals at the plasma edge are essentially a momentum sink: charge-exchange reactions replace the fast plasma ions by slow neutral particles, which of course slows down the plasma. The fast neutralised plasma ions lose their momentum due to collisions with other neutrals and the wall. Without the neutral particles in the edge, the plasma would never lose its momentum. The neutral particles introduce a force term into the force balance equation that slows down the plasma. This force depends on the rate coefficient for charge-exchange reactions σv cx, the neutral density n 0, the plasma density n and the plasma velocity v: F cx edge = nm σv cx n 0 v. A second source of neutrals is provided by the neutral beams. In the previous section it was said that the only way to drive plasma rotation was through the j B term: a force term perpendicular to the magnetic field. This is no longer true when tangential neutral beams are present. The neutral particles injected by a tangential beam have a velocity component in the toroidal direction. Ultimately these neutral particles are all ionised, transferring their toroidal momentum to the plasma. In [91] the process of the momentum transfer is discussed in detail. It is shown that a part of the neutral beam force is a perpendicular j B-force, but partly the NBI also directly drives momentum parallel to the magnetic field. In present day devices tangential neutral beams are the main sources of rotation. We will denote this beam force as F NBI (r) Momentum balance In a steady state situation and with the force terms due to neutral beam injection and friction with edge neutrals included, the force balance equation (2.9) becomes: nmv v = j B p Π nm σv cx n 0 v + F NBI (2.27) The steady state form of the conservation of mass equation (2.8): nmv = 0, allows us to rewrite the left hand side of equation (2.27) as nmv v + v nmv = (nmvv). This leads to the general momentum balance equation: (nmvv) + Π = j B p nm σv cx n 0 v + F NBI (2.28) The right hand side of the momentum balance equation (2.28) contains momentum sources and sinks. The left hand side contains the divergence of two tensors that describe inertia (nmvv) and viscosity (Π); they account for momentum transport. The viscous stress tensor Π depends on viscosity coefficients η and on the rate-of-strain tensor W [80, 88, 9]. The rate-of-strain tensor W comes from conventional fluid theory and has following definition: Section Momentum transport 21

31 Chapter 2 - Theory of plasma rotation W αβ = v α x β + v β x α 2 3 δ αβ v (2.29) The earlier mentioned dependence of Π on the velocity gradients comes from the contribution of W to Π. The viscosity coefficients η provide the link between the stresses in Π and velocity gradients in W. The viscosity coefficients η rely on the underlying physical transport mechanisms. Both the neoclassical transport model (see section 2.4) and anomalous, turbulent transport models can be used to derive these viscosity coefficients [15, 60, 79]. For an orthogonal coordinate system (x, y, z) with the z-coordinate along the magnetic field the relation between Π, η and W is as follows [88]: Π xx = 1 2 η 0(W xx + W yy ) 1 2 η 1(W xx W yy ) η 3 W xy Π yy = 1 2 η 0(W xx + W yy ) 1 2 η 1(W yy W xx ) + η 3 W xy Π zz = η 0 W zz Π xy = Π yx = η 1 W xy η 3(W xx W yy ) Π xz = Π zx = η 2 W xz η 4 W yz Π yz = Π zy = η 2 W yz + η 4 W xz, (2.30) where η 0 is a parallel viscosity term, η 1 and η 2 describe perpendicular viscosity and η 3 and η 4 represent gyro-viscosity. We now consider the toroidal component of equation (2.28) and take the average over the flux surfaces [74]: ( nmvv) φ + ( Π) φ = (j B) φ nm σv cx n 0 v φ + (F NBI ) φ, (2.31) where the definition of the flux-surface average on circular surfaces is A = 1 2π 2π 0 Adθ. The expression for the toroidal component of the divergence of a tensor in large aspect ratio, circular flux-coordinates is given by [80]: ( T ) φ = 1 r r (rt rφ) + 1 T θφ r θ + 1 T φφ R 0 φ (2.32) Due to axi-symmetry T φφ φ = 0. If the assumption T θφ θ = 0 is made, and flux surface averaging is applied, the terms on the left hand side of equation (2.31) become: ( Π) φ = 1 r r (rπ rφ) (2.33) ( nmvv) φ = 1 r r (rnmv rv φ ) (2.34) We first have a look at equation (2.33). The rφ-component of Π is [80]: Π rφ = η 2 R r ( vφ R 1) η 4R r θ ( vφ R 1) = η 2 R 0 Ω φ r, (2.35) 22 Section Momentum transport

32 Understanding and controlling plasma rotation in tokamaks where large aspect ratio and v φr 1 θ = Ω φ θ = 0 were assumed. Neglecting convective momentum transport, the perpendicular viscosity η 2 can be defined as η 2 nmd φ, D φ being the momentum diffusion coefficient. The first term on the left hand side of equation (2.31) is given by (2.34). The nmv r in (2.34) is a particle flux Γ = nmv r. Assuming only anomalous, diffusive transport the n following expression is valid: Γ = md p r, D p being an anomalous particle diffusion coefficient. We now assume, as was done in [74], that D φ = D p = D. When we use the above assumptions and the equations (2.33) - (2.34), the left hand side of equation (2.31) becomes: ( nmvv) φ + ( Π) φ = 1 r = 1 r ( r r = R 0 r = R 0 r ) Ω φ rγv φ rη 2 R 0 r ( r( md n This leads to the following angular momentum balance equation: R 0 r r r ) r )R Ω φ 0Ω φ r(nmd)r 0 r ( ( n rmd r Ω φ + n Ω )) φ r ( rd ) r (nmω φ) (2.36) ( rd ) r r (nmω φ) = j r B θ j θ B r nm σv cx n 0 v φ + F NBI, (2.37) where we assumed all terms on the right hand side to be independent of θ. When all forces are known, this angular momentum balance equation allows us to calculate the toroidal rotation profile. 2.4 Neoclassical transport and rotation The viscosity coefficients η and the momentum diffusion coefficient D φ depend on the physical mechanism of momentum transport. There are several ways to describe the transport of momentum, energy and particles: classical, neoclassical and turbulent. The classical model describes the transport caused by collisions in a cylindrical plasma. The plasma in a tokamak has however a toroidal shape. When we take the torus shape into account, we find that the collision driven transport is higher than in the classical, cylindrical case. The collision driven transport in a torus shaped, magnetically confined plasma is called the neoclassical transport. Apart from collisions, also turbulence can drive transport. The neoclassical model thus gives us the lower limit of particle, energy and momentum transport: the transport in absence of turbulence. In this chapter we will discuss the neoclassical transport and especially focus on the neoclassical momentum transport. From neoclassical theory it follows that the poloidal velocity is strongly damped. The mechanism of poloidal flow damping is often referred to as parallel viscosity. In section 2.3 only the toroidal momentum transport was discussed. The reason for neglecting poloidal momentum transport is this poloidal flow damping: due to poloidal flow damping the effect of a poloidal force on the poloidal rotation is small and Section Neoclassical transport and rotation 23

33 Chapter 2 - Theory of plasma rotation strongly localised to the position where the force is applied. There is virtually no poloidal momentum transport. Another result from neoclassical transport is the existence of a spontaneous rotation in a tokamak plasma, that will be discussed in section As said in the chapter 1 a fast rotating plasma has a better confinement and stability. In devices where the momentum input by e.g. neutral beams is limited, spontaneous rotation is therefore very important. The mechanisms of poloidal flow damping and the neoclassical expressions for spontaneous plasma rotation will be given in the following sections Neoclassical theory Classical transport is governed solely by collisions. It is a diffusion process with D α ρ 2 L,α /τ c,α the diffusion coefficient for particles α. Here τ c,α is the collision time and the step size ρ L,α is the Larmor radius. Classical transport is only valid when the magnetic field inside the plasma is straight, homogeneous and stationary. Due to the toroidal shape of a tokamak, the magnetic field is not homogeneous: it is stronger at the inner side of the torus than at the outer side. The curvature and gradient of the field leads to extra forces and drifts that are not present in e.g. cylindrical configurations. The classical model can therefore not describe the transport of particles, energy and momentum in a tokamak. Neoclassical theory takes into account this inhomogeneous, curved field, which results in the correct description of the collisional transport in a tokamak [2]. Figure 2.5 : Particles with a total velocity u α bounce back at a position R min = 2µB0R0 m αu. The 2 α orbit they follow is banana shaped due to drifts; here the poloidal projection is shown. Particles with a small pitch angle (u /u 1) follow their trajectory through the high-field side and are called passing. Due to drifts the orbits do not lie exactly on the magnetic flux surface, but are slightly shifted. One of the most visible aspects of neoclassical theory are so-called trapped particles. Charged particles that travel, parallel to the magnetic field, from the outer side of the torus to the inner side, go from a region with lower magnetic field to a region with higher magnetic field. In other words: these particles see a B and experience a force F = µ B, where µ = m α u 2 /2B is the magnetic moment which is a constant of motion. This force will slow down the parallel velocity of the particles when they are moving 24 Section Neoclassical transport and rotation

34 Understanding and controlling plasma rotation in tokamaks towards the high-field side of the tokamak. Particles with a low enough u will be stopped and reflected before reaching the high-field side of the tokamak: they oscillate in a socalled banana orbit (see fig. 2.5) and are called trapped particles. Particles that are fast enough do not bounce back at the high-field side, but continue their trajectory parallel to the field line; they are called passing particles. Due to drifts the orbits of passing and trapped particles are shifted with respect to the magnetic field lines they are connected to. For trapped particles this means they have a certain banana width w B. Below three important parameters for trapped particles are given: the width of a banana orbit w B = ε 1/2 ρ L q the density of trapped particles n trapped = n total ε the bounce frequency ω b = v th ε/qr, where ɛ = a/r 0 is the inverse aspect-ratio of the plasma. For large aspect-ratio tokamaks like TEXTOR ɛ < 1. The existence of trapped particles and the banana orbits have a significant influence on the transport. Let us, as an example, look at collisional transport of particles. If a trapped particle undergoes collisions in the time it needs to complete its banana orbit defined by ω b it does not know it is trapped and the transport will be close to classical, with a step size ρ L. When the collision frequency is lower than the bounce frequency, the trapped particle can at least complete one banana orbit without colliding. The step that trapped particles then make in a collision is no longer the Larmor radius ρ L, but the much larger banana width w B. This results in a completely different diffusion coefficient. In a tokamak one defines three different transport regimes: the low collisionality banana regime, the high collisionality Pfirsch-Schlüter regime and the intermediate plateau regime. All three have different transport coefficients. In figure 2.6 the diffusion coefficient is drawn as a function of the collisionality. The three transport regimes are easily recognised. D Plateau Pfirsch-Schlüter Banana Classical ε 3/2 1 ν* Figure 2.6 : The variation of the diffusion coefficient with the collisionality throughout the three neoclassical transport regimes. As a comparison also the classical diffusion coefficient is drawn. One sees that neoclassical theory predicts a larger transport than classical theory. Also for investigating rotation trapped particles are important. Trapped particles bounce back and forward, so they never make a poloidal turn. Their poloidal momentum Section Neoclassical transport and rotation 25

35 Chapter 2 - Theory of plasma rotation is therefore zero. Toroidally trapped particles do have a rotation, because the banana orbits can have a precession around the torus. The poloidal flow damping that follows from neoclassical theory is discussed in the next section Poloidal flow damping The reason for poloidal rotation damping is the toroidal shape of a tokamak. This causes an inhomogeneous magnetic field, stronger at the inside of the torus, weaker at the outside. A toroidally rotating plasma in its rest frame does not see a change in the magnetic field. However a poloidally rotating plasma does see a time-varying field when moving from the outside to the inside of the torus. Through the mechanism of magnetic pumping the energy of a time-varying field will be transferred to the plasma [6, 81]. In case of poloidal movement through a spatial periodic field, the energy of the time-varying field in the rest frame of the plasma is the kinetic energy of the rotating plasma. In other words: the poloidal rotation decays, because the kinetic energy of the poloidal rotation is used to heat the plasma. The magnetic pumping can be split in a collisional and a collision-free part. The collisional scheme is applied when the collision time is smaller than the period of the perturbation. The magnetic moment of the particle is µ = m α u 2 /2B. As long as particle does not collide this is a constant of motion. Hence, when the particle moves to the high-field side, B increases and the particle will gain perpendicular energy m α u 2 /2. Due to the conservation of energy, the parallel velocity u will reduce. When returning to the low-field side the particle will again lose its extra perpendicular energy and the net change in perpendicular energy over one period is zero. Also u will increase again when moving to the low field side, so also the net change in u over one period is zero. However, when a collision occurs when the particle is at the high-field side, it will redistribute its extra energy, effectively heating the plasma. This means that, after the collision, µ is lower than before the collision. The force F = µ B, that has to accelerate u when the particle moves to the low field side, will therefore also be lower and there will be a net reduction of u over one period. [6] Also in a collision-free situation, where the collision time is larger than the perturbation period, the poloidal rotation is damped. This is due to the trapped particles: those particles for which u gets zero before the high-field side is reached. Because trapped particles do not make a full poloidal turn they do not contribute to the poloidal momentum. This form of poloidal flow damping is usually called transit-time magnetic pumping. Seen from within the frame of a single particle, this process is identical to the Landau damping of an electromagnetic wave. One could think that the toroidal momentum of trapped particles is also zero, but this is not the case because the banana orbits of the trapped particles have a toroidal precession around the torus. Collisions between trapped and passing particles will further reduce the poloidal flow (the collisional part of poloidal flow damping). A more rigourous derivation of the decay of poloidal rotation is given in [81]. Using a drift-kinetic equation for ions it describes the cross-b currents due to the magnetic perturbation and viscous drag. These currents are then responsible for j B-forces that brake the poloidal rotation. The above damping occurs over the whole plasma volume. Depending on local density and temperature more transit-time or more collisional damping occurs. 26 Section Neoclassical transport and rotation

36 Understanding and controlling plasma rotation in tokamaks In the edge of the plasma at low temperature high electron-ion collisionality leads to strong electric fields and a non-zero poloidal (and toroidal) velocity. This spontaneous rotation in an ohmic plasma 2 will be discussed in subsection In general it can be said that the poloidal rotation is strongly damped. This means that any externally induced poloidal rotation decays towards the neoclassical value 3 within a decay time which is in the order of the ion-ion collision time. The effect of poloidal flow damping is included in the viscous term ( Π) θ of the poloidal force balance. This viscous term, taking into account poloidal damping, has following form [11, 71]: ( Π) θ = α(v θ vθ neo ), (2.38) where vθ neo is the neoclassical rotation (see section 2.4.3) and α is a factor that depends on the parallel viscosity in contrast to the momentum diffusion coefficient D in the toroidal balance equation (2.37) that depended on perpendicular viscosity. Poloidal flow damping is therefore often referred to as being the result of strong parallel viscosity. The exact values of vθ neo and α depend on the transport regime: banana, plateau or Pfirsch-Schlüter regime. The poloidal force balance equation, where inertia has been neglected, becomes [11, 71]: α(v θ v neo θ ) = j r, B φ (1 + q 2 )nmν cx v θ. (2.39) In contrast to the the toroidal balance equation (2.37) this equation has no radial derivative of the poloidal velocity. This means that a poloidal velocity at one radial position does not affect the poloidal rotation at another position. The effect of a poloidal force on the poloidal rotation is strongly localised to the position where the force is applied. Because no poloidal force terms are present in the plasma core and because the neoclassical poloidal velocity is low, the poloidal velocity can be neglected in the plasma core Spontaneous neoclassical rotation If no external momentum is provided (e.g. by neutral beams), plasma rotation is still possible due to density gradients, temperature gradients and local, radial electric fields. One could assume that the parallel electric field, that exerts a parallel force F = ee on the ions, is also a possible source of momentum. However, due to the higher mass of the ions compared with the electrons, E will predominantly accelerate the electrons, giving rise to current rather than rotation. For the derivation of this ohmic rotation we must depart from MHD one fluid model and return to the description of the ion fluid. The ion fluid velocity that follows will be approximately equal to the fluid velocity in the MHD model. First, in order to get a physical picture of the mechanisms at work, we will consider the mass and momentum conservation equations (2.3) and (2.4) for the ion fluid. In a steady state situation, with subsonic velocities, no momentum exchange with other species (R ie 0) and a isotropic pressure (P i p i I), these can be reduced to [13]: n i v i = 0 (2.40) p i = en i (E + v i B) (2.41) 2 in contrast to beam or RF heated plasmas, as beams and RF waves also supply extra momentum input 3 see section Section Neoclassical transport and rotation 27

37 Chapter 2 - Theory of plasma rotation Taking the cross product of equation (2.41) with B a perpendicular velocity is found: v i = E B B 2 1 en i p i B B 2 (2.42) Through equation (2.40) the parallel velocity v i is also partly determined. v i is the drift velocity vi = 1 F B en i, due to the forces en B 2 i E and p i. The first term on the right hand side of (2.42) is the E B drift that was discussed earlier. The second term in (2.42) is, for the one fluid MHD picture, hidden in the diamagnetic current: n i ev i n e ev e = p i B B 2 p e B B 2 j = p B B 2 (2.43) The above derivations indicate that electrical fields and pressure gradients can cause both poloidal and toroidal rotation. In [14, 37] neoclassical theory is used to derive a natural neoclassical poloidal velocity vθ neo and the derivative of the neoclassical toroidal velocity vφ neo: vθ neo = K 1 T i eb φ r v neo φ r = K 1T i q 2 eb θ (2.44) ( ) ln 2 Ti, (2.45) where K 1 depends on the collisionality: K 1 = 1.17 in the banana regime, K 1 = 0.5 in the plateau regime and K 1 = 1.83 in the Pfirsch-Schlüter regime according to [14, 37]. Also in [49] neoclassical expressions for the poloidal and toroidal velocities are derived. The expression for vθ neo is the same as the one derived in [14, 37], but the K 1 parameter given in [49] goes from 0.5 in the banana regime down to approximately 2 in the Pfirsch- Schlüter regime, where the exact values depend on the amount of impurities in the plasma. In [14] only the derivative of the toroidal plasma velocity is given (see (2.45)), whereas in [49] a expression for the local toroidal velocity is given. The expression for the toroidal velocity in [49] needs, apart from density and temperature measurements, an extra input: the radial electric field E r. This E r can e.g. be the result of momentum transport from spontaneous rotation at other positions in the plasma. The expression for vφ neo in the banana regime is: v neo φ r = E r T ( i 1 dn i B θ eb θ n i dr + 1 K 1 T i ) dt i. (2.46) dr Also in [49] the effect of the parallel electric field E is considered. As expected, the parallel electric field has little influence on the rotation of the bulk ions, but, surprisingly, impurity ions will experience a rotation in the counter-current direction due to the parallel electric field. Because plasma rotation improves the confinement and stability of a plasma, having spontaneous rotation is very beneficial. The above shows that a spontaneous rotation is expected in tokamaks. However, even in a well-established model like the neoclassical transport model, the actual values of spontaneous rotation can differ depending on the 28 Section Neoclassical transport and rotation

38 Understanding and controlling plasma rotation in tokamaks assumptions made when evaluating the model. Moreover the models used to derive the above expressions are far from complete; the friction forces due to neutrals were for example not included. Also turbulent transport was not taken into account. Due to the complexity of the subject there exists a wide range of models that yield an equally wide range of predictions for the spontaneous rotation. 2.5 Impurity and MHD Rotation The MHD equations allow us to describe the dynamics of a plasma by regarding it as a single conducting fluid. Although this picture gives us physical insight, the physical truth is that a plasma consists of many different species, that each have a fluid like behaviour. When a plasma is diagnosed, the properties of one of the plasma species are measured, instead of the properties of the single plasma fluid. It is therefore necessary to know the relationship between the plasma velocity v used in the MHD equations, and the fluid velocities of the main ions v i, the electrons v e and the impurities v I. As said in section the plasma fluid velocity is the centre of mass velocity of all fluids: v = m in i v i + m e n e v e + I m In I v I n i m i + n e m e + I m In I. (2.47) If we assume that the impurity density is low n I n i n e and take into account the mass difference between ions and electrons m e m i, we can approximate the plasma fluid velocity by the main ion velocity: v v i. Measurements of plasma rotation are mostly done through spectroscopic analysis of the emission of impurity ions. This means the v I is measured. Other diagnostics like e.g. electron cyclotron emission (ECE) or reflectometry measure fluctuations in electron temperature or density. These fluctuations are usually related to the velocity v e of the electron fluid. In the following sections we will discuss how measured impurity velocities v I can be translated into the plasma velocity v and how the frequency of electron temperature or density fluctuations can be linked to v Impurity rotation The collision frequency between impurities and the main ions is usually high enough to assume that the temperature of the impurities and the main ions is equal, within an energy confinement time: T i = T I. However the fluid velocity of the ions is not necessarily the same as that of the main ions. When there is a difference in pressure gradient p/z I, Z I being the nuclear charge of the impurity ions, then a differential rotation between the impurities and main ions exists. This occurs typically in a situation with continuous momentum input, e.g. by neutral beams. The neoclassical moment approach of Hirshman and Sigmar [40, 49] provides an expression for the differential rotation between the main ions and one impurity. In [49, 82] this difference in rotation v = v i v I is given by: v θ 1 dt 2eB φ dr [ 3K L ( T 1 Z )] il p,i L p,i Z I L p,i v φ 3 2eB θ dt dr K 2, (2.48) Section Impurity and MHD Rotation 29

39 Chapter 2 - Theory of plasma rotation where we have neglected the θ-dependence. L T is the temperature gradient length and L p,i and L p,i are the pressure gradient lengths of the main ions and impurity ions respectively. K 2 is a function depending on the collisionality. One sees that the difference in poloidal rotation depends on the impurity pressure and main ion pressure, whereas the difference in toroidal rotation only depends on the temperature gradient. One also notices that the difference in toroidal rotation will be significant in discharges with a strongly peaked temperature profile (dt/dr 0) and a low plasma current (1/B θ 0). The validity of the above equations (2.48) has been tested on several machines [5, 82] MHD rotation The difference between electron and ion velocities, where we neglect impurities, is expressed by the current in a plasma: j = ne(v i v e ). If we measure the current density and the electron fluid velocity, then also the main ion velocity is known. The main part of the current is parallel to the magnetic field, but equation (2.43) shows that a pressure gradient results in a diamagnetic current, and thus the difference between the electron and ion fluid velocity also has a component perpendicular to the magnetic field. It is quite difficult to directly measure the electron fluid velocity, although it can be done with tangential Thomson scattering [47]. Diagnostics like electron cyclotron emission (ECE) or reflectometry are, however, capable of measuring fluctuations in electron density are temperature. The frequency of these fluctuations can be translated into plasma velocity. These fluctuations in temperature and density occur when so-called MHD modes are present within the plasma. As said in subsection and shown in figure 2.1, the magnetic topology in a confined plasma consist of a set of nested flux surfaces. Under certain conditions flux surfaces can break up and reconnect, thus changing the magnetic topology. This can be seen in figure 2.7. These reconnected flux surfaces are a specific class of MHD modes, called tearing modes or magnetic islands. Magnetic islands occur at flux surfaces with a rational q = m/n, and have therefore a periodicity given by the poloidal and toroidal mode numbers m and n. Figure 2.7 : At rational q-surfaces the nested flux surfaces can break up. Here m/n = 2/1 (dark grey) and m/n = 1/1 (black) islands are shown at the q=2 and q=1 surfaces. 30 Section Impurity and MHD Rotation

40 Understanding and controlling plasma rotation in tokamaks Inside an island the electron density and temperature is usually flat or slightly peaked. The velocity of the electrons perpendicular to the magnetic field is restricted: they gyrate around the field. This means that, when a magnetic island rotates the electrons rotate along. As a result the flat electron temperature and density periodically appears in the measuring volume of the ECE or reflectometry, which is seen as a fluctuation. The frequency of the fluctuation depends on the island rotation velocity and the poloidal and toroidal mode numbers m and n. In order to get an expression for the rotation of these magnetic perturbations, an inverse approach is used: looking a the change of magnetic flux through a surface S, with contour S, moving through the plasma with a velocity v MHD. dφ dt = B S t ds (v MHD B) dl (2.49) S In an ideal flux conserving plasma dφ B dt is zero. Using Faraday s law t = E and the generalised Ohm s law (2.10) E +v B = 1 ne (j B p e) where the terms me j e t, Π e and R e were neglected equation (2.49) can be rewritten as [85]: dφ dt = (v + v e v MHD ) B dl = 0, (2.50) S where v e = j e ne = 1 p e B en is the electron diamagnetic drift. Equation (2.50) implies B 2 that dφ dt = 0 if the MHD mode moves with a perpendicular velocity v MHD = v + v e. Measuring the MHD velocity is rather complicated, one usually measures the MHD frequency, i.e. the frequency of the fluctuations on the ECE or reflectometry measurements. When the poloidal and toroidal mode numbers m and n are known, however, one finds the following relation between plasma velocity, electron diamagnetic drift and MHD frequency: f MHD = nv φ,mhd 2πR 0 + mv θ,mhd 2πr f MHD nv φ 2πR 0 + mv θ 2πr + mv θ,e 2πr, because: nvφ,e mv θ,e 2πR 0 2πr f MHD nv φ 2πR 0 + mv θ 2πr + f e (2.51) The electron diamagnetic frequency is given by: f e = m 2πr B φ n e eb 2 dp e dr m 2πr 1 n e eb φ dp e dr. (2.52) Because the pressure gradient and local density are often not very well diagnosed, the so-called natural profile shape of density and pressure can be used [34, 76]. The natural density and pressure profiles are given by n e (r) = n e (0).f(r) and p e = n e (0).T e (0).f 3 (r), with: [ ( ) ] 2/3 qa f(r) = r2 q 0 a 2, (2.53) where a is the plasma radius, q 0 the value of q at the magnetic axis, q a the value of q at the plasma edge and the q-profile is assumed to be quadratic. The resulting diamagnetic Section Impurity and MHD Rotation 31

41 Chapter 2 - Theory of plasma rotation frequency at a specific q-surface is then given by: f e = 2m T e(0)[kev] π B φ [T] a 2 [m] ( ) ( ) qa q 7/3 1 [khz] (2.54) q 0 q 0 When q 0 is also unknown, it can be approximated by q 0 = q a /(q a + 1) for a sawtoothing plasma [34, 76]. Discussions on the influence of v θ on equation (2.51) indicated that the poloidal term can be neglected due to strong poloidal flow damping and the difficulty of relating nonuniform poloidal velocity to a poloidal frequency [92]. So equation (2.51) can be simplified to: f MHD = n v φ + fe. (2.55) 2πR 0 When a magnetic island is present in the plasma, and its toroidal mode number n is known, then the toroidal plasma velocity v φ can be derived from the diamagnetic frequency f e and the measured MHD frequency f MHD. 2.6 Conclusion In this chapter we followed the derivation of the single fluid MHD equations as it is described in the literature. The toroidal and poloidal components of the fluid velocity of the plasma were defined as plasma rotation. The toroidal momentum balance in steady state equation (2.37) allows us to calculate the toroidal rotation profile when the toroidal forces are known. The radial derivatives in (2.37) indicate that a toroidal force at one position will have an influence over the whole plasma. How strong that influence is and how far it goes depends on the level of toroidal momentum transport. The momentum transport is governed by the diffusion coefficient D φ. The averaged momentum diffusion coefficient is given by D φ a 2 /τ φ, where a is the plasma radius and τ φ the momentum confinement time. In this thesis it is assumed that the momentum diffusion coefficient D φ equals the particle diffusion coefficient D. It is also often observed that the ion energy confinement time τe ion equals the momentum confinement time τ φ. Poloidally the rotation is strongly damped. This is a neoclassical effect; trapped particles can not make a poloidal turn and therefore do not contribute to the poloidal momentum. Collisions of passing with trapped particles will further slow down the poloidal rotation. The result of this poloidal flow damping is the poloidal momentum balance given in equation (2.39). This equation shows that the poloidal rotation will only change at the position where a poloidal force is applied. There is virtually no radial transport of poloidal momentum. Because there are no strong poloidal forces present in the core of the plasma, the poloidal rotation velocity can assumed to be zero in the plasma core. Also without external momentum input a plasma will rotate. The neoclassical expressions for spontaneous poloidal and toroidal rotation are given in section It is generally believed that for a good description of the spontaneous rotation, the neoclassical theory does not suffice. More advanced transport models including turbulence, however, give a wide range of predictions on spontaneous rotation. Diagnostics do not see the plasma as the single fluid that MHD equations describe; they measure the properties of the different species in the plasma. If we want to compare the measurements with the MHD theory, it is necessary to convert the fluid velocity of the 32 Section Conclusion

42 Understanding and controlling plasma rotation in tokamaks measured species to the MHD fluid velocity. In section the link between impurity rotation and main ion rotation is given. In section the relationship between MHD frequency and toroidal plasma rotation is derived. In this chapter we have summarised the theoretical tools to deal with plasma rotation. Section Conclusion 33

43

44 Understanding and controlling plasma rotation in tokamaks Chapter 3 TEXTOR, DED and diagnostics 3.1 Introduction In order to investigate the influence that plasma rotation has on the excitation of modes by an external perturbation field in a tokamak, we need a tokamak, a way to control the rotation and a way to control the perturbation field. The TEXTOR 1 tokamak provides us with all three. TEXTOR is a circular, medium-sized tokamak (R 0 = 1.75 m, a = 0.47 m). It has two tangential neutral beam injectors (NBI), injecting neutral particles in two opposite directions. By balancing the two NBI s, toroidal rotation profiles can be established going from full rotation in the direction of the plasma current, via a plasma with no rotation, to full counter-rotation. Furthermore, TEXTOR is equipped with a set of helical perturbation coils at the high-field side, called the Dynamic Ergodic Divertor (DED). These coils can be used to set up a static or dynamic perturbation field. In a first section we will discuss the general setup of the TEXTOR tokamak and its heating systems. Section 3.3 will introduce the properties of the DED. We do not only wish to control the plasma rotation and the perturbation field, we also need to measure the plasma response to the perturbation field. The most important measurement is that of the plasma rotation. For this, charge exchange recombination spectroscopy (CXRS) is used. Due to its importance for this thesis it is discussed in a separate chapter (chapter 4). The diagnostics that measure plasma parameters like temperature and density are introduced in section 3.5. This chapter gives a short introduction to the TEXTOR tokamak, at which the work presented in this thesis was carried out. For a more detailed overview we refer to [26] and [73]. 3.2 The TEXTOR tokamak and its heating systems TEXTOR is a limiter tokamak. Three types of limiters can be used: A toroidal bumper limiter at the high-field side. This limiter that also protects about one third of the inner wall in case of a disruption. A pumped, toroidal belt limiter, located near the bottom of the vacuum vessel at the low-field side. 1 Tokamak EXperiment for Technology Oriented Research 35

45 Chapter 3 - TEXTOR, DED and diagnostics The poloidal limiters that are located at one toroidal position. These poloidal limiters can be remotely moved, hence changing the plasma radius a. During a plasma discharge of course only one of these three limiters is really limiting the plasma. [64] A set of 16 coils around the vacuum vessel induce the toroidal magnetic field B φ of TEXTOR. This field can go up to 2.9 T. The iron core transformer provides the flux swing needed for plasma breakdown and plasma current I p. In TEXTOR the maximum plasma current is 800 ka, but typically the machine is operated with a plasma current of 400 ka. The coil system of TEXTOR is completed by a set of vertical and horizontal plasma positioning and shaping coils. A TEXTOR discharge can last up to 10 s, however most TEXTOR discharges last about 6 s. [64] The heating of the plasma in TEXTOR is provided by several sources. First of all, the plasma current supplies Ohmic heating, which is typically 0.3 M W. Additional heating is provided by ion cyclotron heating (ICRH), electron cyclotron heating (ECRH) and neutral beam injection (NBI). Ion Cyclotron Resonance Heating When a wave is launched into the plasma, of which the electric field rotates with an angular frequency ω IRCH, the power of the wave will be absorbed by ions gyrating around the magnetic field lines with a cyclotron frequency ω i = ω IRCH. This process is called Ion Cyclotron Resonance Heating (ICRH). In TEXTOR, two independent antenna systems are capable of each coupling 2 MW of ICRH power into the plasma. The ICRH antennae generate waves with frequencies in the range of MHz. The power can be injected continuously for up to 3 s. In TEXTOR usually minority heating is used. In that case the ICRH power is transferred to a minority of H ions (about 10 %) in a D plasma. Through collisions the H ions subsequently release their energy to the plasma bulk. [50] Electron Cyclotron Resonance Heating The principle of electron cyclotron resonance heating (ECRH) is comparable to that of ICRH, where the power is in the ECRH case absorbed by electrons. The frequencies are therefore higher at TEXTOR there is e.g a 140 GHz ECRH system and the wavelength is small 2 mm for the ECRH system at TEXTOR. Thanks to the low wavelength the heat deposition of the ECRH power is very well localised. When the electron cyclotron waves are injected under a specific toroidal angle, Doppler shift causes ECRH to preferentially heat the electrons that move in the direction of the ECRH antenna. The heated electrons have less collisions and thus lose their momentum slower than other electrons. As a result a net current is driven: electron cyclotron current drive (ECCD). The possibility of current drive, and its localisation, make ECRH/ECCD a very powerful tool for tailoring the current profile in a tokamak. In TEXTOR the ECRH power is produced by a 800 kw gyrotron. This gyrotron generates 140 GHz waves during a pulse of up to 10 s. The waves are in the extra-ordinary polarisation mode (X-mode) and are absorbed in the plasma at the second harmonic of the electron cyclotron frequency ω e. A specially designed steerable launcher allows to deposit the ECRH power at various radial positions, by changing the injection angle in the vertical direction. 36 Section The TEXTOR tokamak and its heating systems

46 Understanding and controlling plasma rotation in tokamaks The launcher also allows to change the toroidal injection angle, thus enabling ECCD. At full gyrotron power and a toroidal injection angle of ±10, depending on the temperature and density up to 50 ka of current can be driven, either co or counter to the inductive plasma current I p. [89] Neutral Beam Injection The two Neutral Beam Injectors (NBI) form a third heating source at the TEXTOR tokamak. A NBI generates a beam of highly energetic neutral particles, that are injected into the plasma. In the plasma the neutral beam particles are ionised through charge exchange reactions with the plasma ions. The fast ionised beam particles transfer their energy and their momentum to the bulk ions and electrons through collisions. Whether mostly the ions or the electrons are heated depends on the ratio of the beam energy and the electron temperature. When a NBI is directed tangentially to the magnetic axis of a tokamak, it will not only heat the plasma, but it will also supply an net toroidal momentum input. Tangential NBI s are commonly used to create fast rotating plasmas. Because the momentum transfer to ions and electrons is usually not equal, tangential neutral beams drive current as well. NBI 2 (counter-direction) B φ I p NBI 1 (co-direction) Figure 3.1 : Top view of TEXTOR tokamak. The two neutral beams and the usual direction of the plasma current I p and the toroidal magnetic field B φ are indicated. At TEXTOR two tangential heating beams are installed. One (NBI1) injects its neutral particles in the direction of the plasma current I p, the other (NBI2) injects its particles Section The TEXTOR tokamak and its heating systems 37

47 Chapter 3 - TEXTOR, DED and diagnostics counter to the plasma current. Figure 3.1 gives a top view of TEXTOR with the two NBI s and the typical direction of plasma current and toroidal magnetic field indicated. The type of particles that can be injected are H, D and He the usual injection species is H. Each of the NBI s launch a maximum of 1.5 MW into the plasma at a maximum energy of 55 kev/amu. The duration of the beam pulses can be as long as 10 s. To regulate the power of the neutral beams, V-targets are used. This V-target consists of two plates in a V-shaped configuration at the end of the injector. By bringing the plates closer together the aperture is reduced and the beam profile is partially scraped off. This means the power of both neutral beams can be tuned by changing the V-target opening. [83] In table 3.1 the TEXTOR machine parameters are summarised, together with some typical plasma parameters. Table 3.1 : TEXTOR machine and typical plasma parameters Major radius (R 0 ) 1.75 m Plasma radius (a) 0.47 m Plasma volume 7.0 m 3 Magnetic field (B) T (typical 2.25 T ) Plasma current (I p ) ka (typical 400 ka) Pulse length < 10 s (typical 5 s) Ohmic power MW Neutral beams 2 tangential beams NBI1 1.5 M W, co-current direction NBI2 1.5 M W, counter-current direction ICRH 4.0 MW, MHz ECRH 0.8 MW, 140 GHz ECCD < 50 ka Electron temperature (T e ) 1 kev Ion temperature (T i ) 1 kev Electron density (n e ) m 3 Toroidal rotation frequency (Ω φ ) rad/s 38 Section The TEXTOR tokamak and its heating systems

48 Understanding and controlling plasma rotation in tokamaks 3.3 The Dynamic Ergodic Divertor A poloidal divertor called the Dynamic Ergodic Divertor or DED is installed at TEX- TOR [26]. This DED consist of a set of helical perturbation coils located at the high-field side of the torus, behind the bumper limiter that takes up the task of divertor target. The main objectives of the DED are: Study and control of transport in the plasma edge. One important aim is the distribution of the heat load over the large area of the divertor target. Another is the creation of a region with a stochastic magnetic field and large transport near the plasma edge, which could be beneficial for e.g. impurity screening. The study of the effect that external perturbations have on the plasma; how external fields influence the plasma stability, the plasma rotation, the confinement et cetera. 18 magnetic coils are installed at the high-field side of TEXTOR: 16 perturbation coils, grouped in four sets of four coils, and two compensation coils, above and below the perturbation coils, that reduce the stray field. The perturbation coils enter the torus at the bottom, make one toroidal turn around the torus in clockwise direction when looking from the top, and leave at the top. Each set of four coils enters, and leaves, at a different toroidal position, separated by 90. The numbering of the coils goes from top to bottom. The pitch angle of the perturbation coils matches that of the field lines at the q=3 surface. In figure 3.2 the coils of the DED are schematically drawn. Figure 3.2 : The DED coils are located inside the TEXTOR vacuum vessel at the high-field side. There are 16 perturbation coils (yellow, black, red and gray), in 4 groups of 4 coils, and 2 compensation coils (green). Coils with the same colour carry identically phased currents. Figure (a) shows the coil configuration in 3/1 mode, figure (b) represents the coil configuration in 12/4 mode. When we apply current to these divertor coils we form a magnetic perturbation field. Four differently phased currents (0, 90, 180 and 270 ) can be fed to the coils. By feeding identically phased currents to several coils simultaneously, the principal mode numbers of the perturbation field can be changed. There are three modes of operation: 3/1 mode All coils in one group have the same phase. The next group has a 90 phase shift. This is sketched in figure 3.2 (a). Section The Dynamic Ergodic Divertor 39

49 Chapter 3 - TEXTOR, DED and diagnostics 12/4 mode Each coil in the group has a different phasing; the first 0, the second 90, the third 180 and the fourth 270. The first coil of next group then has 0 phasing again. This is shown in figure 3.2 (b). 6/2 mode The operational mode in between 3/1 and 12/4. The first two coils of the first group have 0 phasing, the last two have 90 phasing. The first two coils of the second group have a 180 phase, the last two 270. For the third and fourth group the situation is the same. The work presented in this thesis will focus on the 3/1 mode of DED operation, because the effects on plasma rotation and mode stability are found to be influenced strongest by this mode of operation. The 12/4 DED mode only affects the very edge of the plasma and does not create any detectable island activity. This operational mode is used in chapter 8 to look at the influence of the DED on the plasma rotation, in the situation where tearing modes are not excited. The 6/2 DED mode will not be discussed in this work. x x B r (T) n number m number B r (T) n number m number Figure 3.3 : The amplitudes of the different (m, n) components of the DED vacuum field in 12/4 mode at the q = 3 surface. The calculation was done for a DED coil current of 7.5 ka, a plasma current of 400 ka and a toroidal magnetic field of 1.9 T. Important Fourier components are: Br 8,4 = T and Br 12,4 = T. [27] Figure 3.4 : The amplitudes of the different (m, n) components of the DED vacuum field in 3/1 mode at the q = 3 surface. The DED coil current is now 1.5 ka, plasma current and toroidal field are 400 ka and 1.9 T, respectively. Important Fourier components are: Br 1,1 = T, Br 2,1 = T and Br 3,1 = T. [27] For each of these operational modes the vacuum field of the DED can be calculated [1]. The vacuum field is defined as the superposition of the field induced by the DED coils and the TEXTOR equilibrium field. The amplitudes of the toroidal (n) and poloidal (m) Fourier components of the radial field B r at the q = 3 surface are given in figure 3.3 for 12/4 operation and in figure 3.4 for 3/1 operation. The Fourier components are as expected centred around the desired mode numbers (m = 12 and n = 4 for 12/4 operation, m = 3 and n = 1 for 12/1 operation). For the toroidal mode number n, the band is very narrow. For the poloidal mode number m the band is wider and the components with mode numbers different from m = 12 or m = 3 cannot be ignored. In figure 3.5 (a) a Poincaré plot of the vacuum field in 3/1 mode is superimposed on a complete poloidal cross section. Each dot represents the crossing of a field line through 40 Section The Dynamic Ergodic Divertor

50 Understanding and controlling plasma rotation in tokamaks the plotted poloidal cross section. One can observe a strong m/n = 2/1 structure, as could be expected from the wide m-band in the Fourier spectrum shown in figure 3.4. Figure 3.5 (b) zooms in on the edge of the plasma and unfolds the poloidal angle. One can distinguish three areas in this plot: Very close to the edge there is a zone where the field lines connect to the wall after a limited number of toroidal turns. This is the laminar zone. A bit further into the plasma the field lines do no longer have a short connection length to the wall, but they do have a radial component. This causes them to fill a volume rather than laying on a (flux) surface. This is the stochastic (ergodic) zone. Closer to the plasma core the field lines become regular; they form circular, flux surfaces. Figure 3.5 : A Poincaré plot of the vacuum field in 3/1 DED operation. In (a) a poloidal cross section of the vacuum field in the TEXTOR vessel is drawn; also the DED coils are indicated. In (b) the same vacuum field is drawn, but now with the poloidal direction unfolded. On this plot the three regions laminar, stochastic and regular can easily be recognised. The parameters for which this vacuum field was calculated are: I p = 300 ka, B φ = 2.25 T and I DED = 3.75 ka. [27] In 3/1 operation the current supplied to the DED coils can go up to 3.75 ka, in 12/4 operation a coil current up to 15 ka is allowed. Both the current amplitude and the mode of operation determine the strength of the perturbation field. For high mode numbers e.g. in 12/4 operation the field rapidly decays with increasing distance from the DED coils. Therefore the 12/4 mode will only affect the very edge of the plasma, whereas in 3/1 operation the influence of the DED penetrates deeply into the plasma. The current fed to the DED coils can be either DC or AC. For DC DED the resulting perturbation field is static. When we apply AC current the phase in each coil constantly changes, resulting in a dynamic perturbation field. The phase can increase (AC + ) or decrease (AC ). Consider the 3/1 coil configuration sketched in figure 3.2 (a), with as a starting point the first group of coils (yellow) at 0, the second (black) at 90, the third Section The Dynamic Ergodic Divertor 41

51 Chapter 3 - TEXTOR, DED and diagnostics (red) at 180 and the fourth (gray) at 270. When we apply AC +, the phase in each group will increase, such that after a quarter period the phases are as follows: yellow = 90, black = 180, red = 270 and gray = 0. This means that the field created by the DED has rotated. This rotation is mainly in the poloidal direction and directed from top to bottom at the high-field side. In a poloidal cross section, with the high-field side at the left (like is the case in figure 3.5 (a)), the poloidal rotation of the AC + field is counterclockwise. The rotation of the field also has a small component in the toroidal direction. The toroidal rotation of the AC + field is clockwise when looking on top of the torus. With AC DED the field also rotates, but the direction is reversed. For 12/4 DED operation the result is of course the same. To summarise: an AC + field rotates counterclockwise in the poloidal direction and clockwise in toroidal direction. An AC field rotates poloidally clockwise and toroidally counterclockwise. The rotation frequency of the AC fields can go up to 10 khz, but usually 1 khz and 3.75 khz are used. To conclude this section, the main DED parameters are summarised in table 3.2. Table 3.2 : DED parameters Set of 16 perturbation coils (and 2 compensation coils) Modes of operation: 4 consecutive coils with same phasing - 3/1 operation 2 consecutive coils with same phasing - 6/2 operation Phase shift for each coil - 12/4 operation DED coil currents: DC - Static perturbation field f DED = 0 khz AC + - Rotating perturbation field toroidal component clockwise poloidal component from top to bottom at HFS f DED = 1 khz or 3.75 khz AC - Co-rotating perturbation field toroidal component counterclockwise poloidal component from bottom to top at HFS f DED = 1 khz or 3.75 khz 3.4 Co- and counter directions in TEXTOR In this thesis directions play an important role. We therefore give some attention to the definition of positive, co- and negative, counter-directions in the TEXTOR tokamak. Throughout this thesis, a righthanded coordinate system (r, θ, φ) is used, where r represents the radial direction and acts as the flux surface coordinate; θ and φ represent 42 Section Co- and counter directions in TEXTOR

52 Understanding and controlling plasma rotation in tokamaks the poloidal and toroidal direction, respectively. The radial coordinate r starts at 0 in the plasma centre and increases towards the plasma edge. This means a vector with a positive radial component points outwards, while a vector with a negative r-component points in the direction of the plasma centre. Toroidally the counterclockwise direction, when looking on top of the TEXTOR tokamak, is defined as the positive or co-direction; consequently the toroidal, negative or counter-direction is clockwise. With the positive direction defined for r and φ, and taking into account the righthanded (r, θ, φ) coordinate system, the positive direction in the poloidal plane is now fixed. When we look at a poloidal cross section, with the high-field side at the left, the poloidal, positive or co-direction is clockwise; the negative or counter-direction is counterclockwise in a poloidal plane. Co, I p, DED AC Counter, B φ, DED AC + Counter, f e *, DED AC + e φ Co, B θ, f i *, DED AC e r e θ Figure 3.6 : Definition of co- (positive) and counter-directions (negative) in TEX- TOR, both toroidally and poloidally. Although it is possible to reverse the direction of the plasma current I p and the toroidal magnetic field B φ, all data presented in this thesis were obtained from discharges using the common TEXTOR settings. Toroidally these settings imply that the plasma current I p is in the co-direction, hence positive. The toroidal magnetic field B φ is directed in the counter-direction and negative. In figure 3.1 it is seen that NBI1 injects momentum in the co-direction, inducing positive, toroidal plasma rotation, while NBI2 causes negative or counter-rotation. In the poloidal plane the poloidal magnetic field B θ is directed in the co-direction. Diamagnetic drifts of ions and electrons are perpendicular to the magnetic field: v α = 1 p α B e α n B 2, (3.1) with α = i or e. Due to the large toroidal component of the B-field, compared to the poloidal component, the diamagnetic drift of ions and electrons is almost completely poloidal. When we take into account the charge of the particle in question, the fact that p α is always directed towards the plasma centre and that B φ is in the counter- Section Co- and counter directions in TEXTOR 43

53 Chapter 3 - TEXTOR, DED and diagnostics direction, equation (3.1) results in an ion diamagnetic drift in the poloidal, co-direction and an electron diamagnetic drift in de counter-direction. In section 3.3 it was seen that the AC + DED field rotates poloidally in the counterclockwise direction, and toroidally it rotates clockwise. With the definition of co- and counter-direction given above, this means that the rotation of the AC + field is, both poloidally and toroidally, in the counter-direction and thus negative. Consequently, the AC DED field is rotating in the positive, co-direction. In figure 3.6 the co- and counter-directions for TEXTOR are shown. In table 3.3 the typical directions of several plasma parameters are summarised. Table 3.3 : Directions of plasma parameters in TEXTOR Co-direction, Positive Counter-direction, Negative Toroidal Plasma current I p Toroidal magnetic field B φ Momentum input NBI1 Momentum input NBI2 Toroidal rotation AC DED field Toroidal rotation AC + DED field Poloidal Poloidal magnetic field B θ Ion diamagnetic drift vi f i Poloidal rotation AC DED field Electron diamagnetic drift ve fe Poloidal rotation AC + DED field 3.5 Plasma diagnostics The plasma parameters that are of importance for this thesis are the plasma rotation velocity, the ion and electron temperature and the density. In the following paragraphs several diagnostics capable of measuring these parameters will be introduced. A description of all the diagnostics available at TEXTOR can be found in [10] and [20] Charge exchange recombination spectroscopy Charge exchange recombination spectroscopy (CXRS) is the principle diagnostic used in this thesis. It is a technique that analyses the light emitted by impurity ions in the plasma core when they undergo a charge exchange reaction. In such a charge exchange reaction, a fully stripped impurity ion receives an electron from a neutral particle, e.g. provided by a neutral beam. The impurity ion receives this electron in an excited state and will return to the ground state, losing its excess energy through line-emission. This line spectrum can be analysed with a spectrometer, yielding following information: Ion temperature The width of the emission line is caused by Doppler broadening. It is therefore a measure of the thermal velocity distribution, and hence of the ion temperature. Plasma rotation The Doppler shift of the emission line gives the velocity in the direction of the line-of-sight. Taking into account the geometry of the CXRS system this velocity component along the line-of-sight can be transformed into a toroidal or poloidal rotation. 44 Section Plasma diagnostics

54 Understanding and controlling plasma rotation in tokamaks Impurity content The intensity of the emission line is proportional to the density of the impurity ions. At TEXTOR there are two main CXRS systems. The NBI-CXRS system uses the NBI1 heating beam as a source of neutrals. The system has lines-of-sight with a toroidal geometry, which means that the measured Doppler shifts can be translated in toroidal rotation frequencies Ω φ. The time resolution of the NBI-CXRS system is 50 ms. The necessity of NBI1 prohibits CXRS measurements with this system during Ohmic discharges. The RUDI-CXRS system uses a low power, radially injecting, diagnostic neutral beam. The influence of this neutral beam on the global plasma behaviour is negligible, such that this diagnostic can be used to investigate Ohmic discharges. However, due to the lower beam power, the CXRS signal is low and a longer exposure time is needed to collect enough signal. This results in a time resolution of over 1 s. The lines-of-sight of the RUDI-CXRS system are located in a poloidal plane, allowing for poloidal rotation measurements. In chapter 4 CXRS is discussed in detail Electron Cyclotron Emission The electrons that gyrate around the magnetic field lines emit electron cyclotron radiation. At the electron cyclotron frequency the plasma is optically thick. This means that the electron cyclotron emission (ECE) is a black body radiation. According to Raleigh- Jeans law, the long wavelength approximation of Planck s law, the intensity of black body radiation is related to the temperature. The ECE intensity is therefore proportional to the electron temperature T e. The frequency of the ECE radiation ω e = eb/m e is proportional to the magnetic field B that in turn depends on the major radius R. This means that the ECE intensity as a function of frequency can be transformed to the electron temperature as a function of major radius: T e (R). Microwave detectors with a high sampling rate allow for measurements of T e (R, t) profiles with a fast time resolution. An absolute measurement of the intensity is needed, which means that an intensity calibration of the detectors is needed. This can be done by using a hot source or by cross calibration with another T e -diagnostic like e.g. Thomson scattering. Several ECE systems are installed at TEXTOR. The two commonly used systems are: EC11 and ECE-imaging. The EC11 system is a 11-channel, fixed frequency system. The frequencies of the 11 channels lie in the range of GHz. For TEXTOR operation with a central, toroidal magnetic field of B φ = 2.25 T this means the whole plasma region R = m is covered by the EC11 system. The time resolution of the EC11 system goes up to 10 khz. The ECE-imaging system provides 2D T e profiles. The radial position R comes from 8 frequency channels, separated by 0.5 GHz. The vertical Z-coordinate is resolved by imaging a plasma column of 16 cm in height onto an array with 16 detectors. The result is a matrix of 8 16 channels that images the T e in an area of 8 16 cm 2. The measuring area can be radially positioned by changing the frequency of the central channel over GHz in steps of 0.5 GHz. For a standard TEXTOR B φ field of 2.25 T this means from R = 1.40 m to R = 1.84 m with a step of approximately 1 cm. For measurements with a high time resolution, sampling at 500 khz is possible for a duration of 2 s. At 200 khz sampling the whole plasma discharge typically 6 s is covered. Section Plasma diagnostics 45

55 Chapter 3 - TEXTOR, DED and diagnostics Thomson Scattering The Thomson scattering diagnostic is based on the scattering of laser light at free electrons. The velocity of the electrons causes the scattered light to have a Doppler shift with respect to the wavelength of the incoming laser light. For a Maxwellian velocity distribution of the electrons, the spectrum of the scattered light has a Gaussian shape. The width of the spectrum gives the electron temperature T e. The intensity of the scattered light is proportional to the electron density n e. So apart from being a T e measurement, Thomson scattering provides a n e measurement as well. For the density measurement an absolute calibration is needed. The scattered light originates where the laser passes through the plasma. Imaging the scattered light therefore gives the local T e and n e along the path of the laser light. For Thomson scattering measurements a powerful laser is needed, so that the intensity of the scattered light is high enough. It also has to be a pulsed laser, so that the scattered light rises significantly above the plasma background light. At TEXTOR a multi-pulse Thomson scattering system is installed. It uses a highpower, intra-cavity ruby laser. The advantage of an intra-cavity laser is the fact that all laser light is used, whereas in a normal laser only a few percent of the laser light is coupled out of the cavity. The technical challenge of an intra-cavity laser is the enormous length of the cavity, that has to include the plasma. In TEXTOR the laser cavity is 18 m long. This unique system allows the measurement of T e and n e with a very high spatial and time resolution. The intra-cavity laser currently allows for up to 40 laser pulses over a period of 4 ms with each an energy of 15 J. The laser passes vertically through the plasma and is shifted 9 cm from the centre of the tokamak, towards the low-field side. The core Thomson scattering system images the full plasma diameter 90 cm on 120 channels. This means that during a 4 ms laser burst, T e (z, t) and n e (z, t) profiles are obtained with a spatial resolution of 7.5 mm and a 10 khz time resolution. When the plasma position and equilibrium are known, the z-coordinate can be transformed in a radial r-coordinate. Apart from the core system, also a high-resolution edge system is present at TEXTOR. It images 16 cm in the plasma edge onto 98 channels, yielding a spatial resolution of 1.7 mm. Because the core and edge system use the same detection equipment, they cannot be used simultaneously Interferometer Another technique for measuring the electron density is interferometry. In interferometry the light of a laser beam that has gone through the plasma is compared with that of a reference beam that travelled over the same length, but in vacuum. As the phase velocity of laser light depends on the refractive index of the medium it is going through plasma for one beam, vacuum for the reference beam there is a phase difference between the light of the plasma beam and the reference beam. From the interference pattern of the two beams the phase difference and thus the line-averaged refractive index of the plasma can be determined. The refractive index of a plasma is proportional to the electron density, so that interferometry yields the electron density n e averaged over the line along which the laser passes through the plasma. In TEXTOR the light of a far-infrared HCN laser goes vertically through the plasma at 9 different radial positions, yielding the line-integrated density at 9 different positions. 46 Section Plasma diagnostics

56 Understanding and controlling plasma rotation in tokamaks The time resolution is quite high 20 khz and the line-integrated density signals are used for real time density and positioning control. Using Able-inversion the line-integrated densities can be transformed into density profiles n e (r, t). The estimated error in the local density due to the Able inversion is about 5%. Section Plasma diagnostics 47

57

58 Understanding and controlling plasma rotation in tokamaks Chapter 4 Charge Exchange Recombination Spectroscopy 4.1 Introduction For the study of plasma rotation it is necessary to have a reliable method to measure that rotation. Charge exchange recombination spectroscopy (CXRS) is a powerful technique that allows local measurements of the ion velocity distribution, from which the random thermal velocity or ion temperature and the fluid velocity are derived. In CXRS the line-emission by ions in the plasma core is analysed. The Doppler shift and broadening of this line-emission are a measure for the fluid velocity and the ion temperature. Line-emission from ions in the plasma core does not occur spontaneously. Due to the high temperature the ions in the centre of the plasma are except for heavy impurities fully stripped. In order to emit light, they must at least receive one electron. This happens when they undergo a charge exchange reaction. Neutral atoms, usually provided by neutral beam injection, will in such a charge exchange reaction deliver one of their electrons to the plasma ion. The ion receives this electron in a high quantum level. It will return from its excited state to the ground state, losing its excess energy through line-emission. Charge exchange recombination spectroscopy is well described in the literature [42, 38]. In this chapter we bring together the elements that are essential in the interpretation of the measurements at TEXTOR. The CXRS setup used at TEXTOR will be discussed in the last section of this chapter. 4.2 Principle In the charge exchange process a neutral atom loses an electron to a plasma ion [42]. Depending on the type of the plasma ion and the neutral particle, the electron is most likely to be transferred to a specific, preferential quantum level of the plasma ion. The probability for electron transfer to both higher and lower levels decreases monotonically. In case of a charge exchange reaction H + C 6+ H + + C 5+ the C 5+ level most likely to be populated is n = 4, but also for the n = 7 and n = 8 levels the chances of population are significant [38]. These quantum levels differ from the ground state, which means that a charge exchange reaction will yield a plasma ion in an excited state. It will return to 49

59 Chapter 4 - Charge Exchange Recombination Spectroscopy H + A + H + + A H + + A + hν 1 + hν (4.1) the ground state through a cascade of transitions. The radiation emitted by the transitions varies over a wide wavelength range, depending on the populated states. For diagnostic use of charge exchange, especially transitions that emit visible light are of interest. The advantage of visible light is that it can be easily collected by common optical elements and also plenty spectroscopical methods are available for the analysis of visible light. In the TEXTOR tokamak the n : 8 7 transition of C 5+ is used. It has a wavelength of 529 nm, which is in the visible range. The reason for analysing the charge exchange light of an impurity carbon instead of the charge exchange light of the bulk ions deuterium or hydrogen is the high complexity of the hydrogen-deuterium spectrum. Usually the bulk plasma is a mixture of deuterium and hydrogen, where the charge exchange (CX) emission lines of deuterium and hydrogen overlap. Apart from CX emission, neutral deuterium and hydrogen in the plasma edge have a strong line-emission due to excitation by electrons. This emission lies in the same wavelength range as the charge exchange emission. Finally the neutral beam species is commonly hydrogen as well, resulting in a beam spectrum on top of the charge exchange spectrum. As a result one gets a spectrum that is the sum of two charge exchange lines, electron excited emission and beam emission. Decomposing this spectrum into its components in order the regain the charge exchange line results in large error bars on the derived quantities. Therefore, as said above, CXRS is usually done at impurity ions. Commonly used are helium and carbon. Helium is investigated because of its relevance: it is a reaction product of the fusion reaction. Carbon is looked into because its spectrum is compared to the spectrum of hydrogen or helium relatively easy to analyse. Also, carbon is always present in tokamaks because it is commonly used as a wall material. Of course, when analysing the CX spectrum of carbon, the temperature, rotation velocity and density that result from this analysis are those of carbon, not those of the bulk ions. The energy transfer time between carbon and hydrogen is, however, quite low. More specifically: it is shorter than the energy confinement time τ E. This means that the assumption that the carbon temperature equals the bulk ion temperature is valid, within the energy confinement time. The carbon rotation velocity does differ from the bulk rotation velocity. In section the expression for this difference is given, so that from the carbon fluid rotation the plasma bulk rotation can be calculated. The fact that 50 Section Principle

60 Understanding and controlling plasma rotation in tokamaks carbon CXRS gives us a measure for the carbon density, allows us to use CXRS data for impurity transport studies. In spectroscopy the light emitted by C 5+ ions is called C VI emission. The C VI (n : 8 7) emission has a natural wavelength of λ n = 529 nm and a natural line width of λ n = nm [63]. If an ion moves with a velocity u z in the direction of the observer, the light emitted by this ion is Doppler shifted by λ d = u z /cλ n. For an ion temperature of 10 ev the Doppler shift due to the thermal velocity is λ d = nm. Compared to this Doppler shift the natural line width λ n can be neglected, so that we can use a delta-function f(λ) = δ (λ λ n (1 + u z /c)) to describe the emission line. Let us consider a Cartesian observation system, with the z-axis in the direction of observation. The intensity of the emission coming from a volume (dx, dy, dz) around the point (x, y, z), depends on the density of neutral hydrogen n H, the density of the carbon ions n C and the effective emission rate σv. This effective emission rate depends on the cross section σ for charge exchange reactions and expresses the likelihood of interactions between carbon and hydrogen in which the n = 8 state is populated. The velocity of the carbon ions is given by a Maxwellian distribution: ( ) 3 mc g(u x, u y, u z ) = exp ( (u x v x ) 2 + (u y v y ) 2 + (u z v z ) 2 ), (4.2) 2πT C 2T C /m C where v x, v y and v z are the components of the carbon fluid velocity. The spectrum of the light coming from the volume (dx, dy, dz) emitted in the direction of the observer then is: where f local (λ) = 1 4π n Hn C ( = I exp (λ λ 0) λ 2 ( σv δ (λ λ n 1 + u z c )) g(u x, u y, u z )du x du y du z, ), (4.3) ( λ 0 = λ n 1 + v ) z c 2TC λ n λ = m C c I = n H n C σv λ n 4π 3/2 c λ. (4.4) The above learns us that we can derive v z the carbon velocity in the direction of the observer from the peak position of the Gaussian line-emission, that the carbon temperature T C can be derived from the width of the Gaussian and that the intensity of the Gaussian emission line gives us the carbon density n C. The total charge exchange spectrum that is observed is the sum of all the light locally emitted along the line-of-sight: ( f(λ) = I exp (λ λ ) 0) λ 2 dz,. (4.5) In the largest part of the plasma there is no neutral hydrogen present, thus n H = 0. There are however two regions in the plasma where there is a considerable amount of Section Principle 51

61 Chapter 4 - Charge Exchange Recombination Spectroscopy neutral particles: neutral particles that come from the wall will penetrate over a certain distance into the plasma before being ionised, and the neutral beam will inject highly energetic neutral particles all the way to the plasma centre. This means that at two positions on the line-of-sight the neutral density is non-zero: where the line-of-sight goes through the plasma edge and at the position where the line-of-sight crosses the neutral beam. The CX spectrum can therefore be split in two components: a passive charge exchange component coming from the plasma edge and an active charge exchange component caused by the neutral beam. This is shown in figure 4.1. The passive part of the spectrum contains of course only data on the edge properties. The active part of the charge exchange light originates from the crossing of neutral beam and line of sight, ensuring a local measurement of n C, T C and v z. Because the neutral beam supplies neutral particles up to the plasma centre, we can measure n C, T C and v z over the whole plasma. With multiple lines-of-sight, profiles of n C, T C and v C can be derived from the active part of the CX spectrum. f(λ) = edge ( I exp (λ λ 0) λ 2 ) dz } {{ } passive ( I exp (λ λ ) 0) beam λ 2 dz. (4.6) }{{} active + passive CX emission active CX emission passive CX emission y z x Neutral beam Figure 4.1 : A top view of a tokamak with a CX line-of-sight. The passive component of the CX spectrum is caused by neutral particles in the edge, the active CX signal is emitted where the line-of-sight crosses the neutral beam. In an optimised viewing geometry the line-of-sight is tangential to a flux surface where it crosses the neutral beam, like it is drawn in figure 4.1. Because n C, T and v z are flux functions, they can be taken out of the integral and the active part of the CX spectrum becomes: ( f active (λ) = I exp (λ λ ) 0) λ 2, (4.7) 52 Section Principle

62 Understanding and controlling plasma rotation in tokamaks with ( λ 0 = λ n 1 + v ) z c 2TC λ n λ = m C c σv λ n I = n C 4π 3/2 c λ beam n H dz. (4.8) When we use CXRS as a diagnostic tool, we are mainly interested in the active CX emission. The active part of the CX spectrum gives us the local values of T C, n C and v z. If we assume that the radial part of the carbon fluid velocity v C is a lot smaller than the toroidal and poloidal component, v z can be written as: v z = v φ cos α + v θ cos β, where α is the angle between the line-of-sight and the toroidal direction and β is the angle between the line-of-sight and the poloidal direction. When we choose a line-of-sight that is perpendicular to the poloidal direction, v z is proportional to the toroidal rotation velocity and vice versa Intensity (a.u.) active CX emission passive CX emission Background radiation CIII impurity line Wavelength (nm) Figure 4.2 : A typical CX spectrum. The black dots represent the measured spectrum, the solid gray line is the result of a fitting routine. The different components of the CX spectrum are indicated: the active CX in blue, the passive CX in red, a CIII impurity line in green and the background radiation is yellow. The undesirable passive component can be dealt with in two ways. The first is using a modulated neutral beam. During the period that the neutral beam is switched off, the only emission left is the passive emission. When the passive spectrum is subtracted from the total spectrum when the neutral beam is switched on, only the active CX spectrum remains. This method is widely used. There are, however, some disadvantages to this method. First of all there is a loss in time resolution, because the neutral beam is switched off half of the time. Another weak point is the fact that it assumes that the passive emission during neutral beam is the same as without neutral beam. Because a neutral beam can change the conditions in the edge of the plasma, this is not necessarily the case. Modulation Section Principle 53

63 Chapter 4 - Charge Exchange Recombination Spectroscopy also increases the errorbars: with A the total spectrum and B the passive spectrum the error on the subtracted spectrum A B becomes: (A B) = ( A) 2 + ( B) 2, which is larger than the error on both A and B. A second method for subtracting the passive emission is modelling of the passive CX spectrum. For this modelling measurements of temperature and density in the plasma edge can be used, as well as e.g. the spectrum measured by a line-of-sight that does not cross the neutral beam. When the passive spectrum is modelled a continuous neutral beam can be used. One needs, however, a very accurate model of the passive emission. In the wavelength range of passive and active CX emission, also light from other sources is present; e.g. a continuum background of Bremsstrahlung and line radiation from partly ionised impurity ions. Figure 4.2 a measured charge exchange spectrum is given. When this spectrum is fitted, all different components active CX, passive CX, impurity lines and background radiation have to be taken into account. 4.3 Additional effects The simple picture given above, where the width of the active CX line only depends on the temperature, the peak position is a simple function of the plasma rotation and the impurity density solely determines the intensity of the line-emission, is not fully correct. Several other mechanisms cause a broadening and shift of the CX emission line. Also, as was already mentioned, passive CX and active CX are not the only emissions in the observed wavelength range: line-emission from impurities at the edge, secondary emission by the CX ions, background emission, et cetera, they all complicate the charge exchange spectrum. These complicating effects will be discussed in this section Energy dependence of the emission rate In the previous section the effective emission coefficient σv was, for simplicity, considered to be constant. In reality σv depends on the collision velocity between the neutral particle and the plasma ion [38]: v col = vb 2 + u2 x + u 2 y + u 2 z 2v b (cos δu y sin δu z ), (4.9) where the same Cartesian system is used as above, v b is the velocity of the neutral beam particle and δ is the angle between the neutral beam and the y-axis. In most situations v b is constant and v b (u x, u y, u z ); in TEXTOR e.g. the beam energy is 50 kev whereas the thermal energy is in the order of 1 kev. In a first approximation it is acceptable to say σv (v col ) σv (v b ), thus equation (4.3), where σv was taken out of the velocity integral, is correct. Equation (4.9) shows however that v col has a distribution around v b. If the value of σv changes significantly around v b or it the distribution of v col around v b is wide i.e. at high temperatures the effective emission coefficient σv can not be taken out of the integral of equation (4.3). In figure 4.3 the effective emission rate σv for C VI (n : 8 7) emission is plotted as a function of collision velocity v col. In the same figure also the distribution of v col is given for a temperature of 1 kev in a TEXTOR CXRS geometry with the TEXTOR beam energy of 50 kev and for a temperature of 15 kev in ITER with the ITER beam energy of 100 ev. It is seen that σv changes significantly over the FWHM of the v vol distributions. Especially in the ITER case. 54 Section Additional effects

64 Understanding and controlling plasma rotation in tokamaks 1.4 x x σ v (m 3 /s) f collision TEXTOR ITER 1 kev 15 kev v x 10 6 collision (m/s) Figure 4.3 : The effective emission coefficient σv is as a function of the collision velocity v col (dashed line). The distribution of v col is also plotted (solid lines), for a temperature of 1 kev in a TEXTOR geometry and for a temperature of 15 kev in ITER. The neutral beam in TEXTOR has a beam energy of 50 kev, whereas the beam energy in ITER is 100 kev. This explains the different positions of both v col distributions. The gray regions indicate the FWHM of both distributions. [38] The result of the v col distribution and dependence of σv on v col is a change in Doppler broadening and Doppler shift of the emission line. This has an effect on the measurement of temperature, rotation velocity and intensity. Usually the width of the emission line is smaller than the width that one would expect in the case of a constant σv, hence the observed temperature is lower than the true temperature. In figures 4.4 and 4.5 the effect of the energy dependence of σv on temperature and rotation is given for the TEXTOR core CXRS system and for the ITER core CXRS system. Due to the low temperatures in TEXTOR below 5 kev we can neglect this effect in TEXTOR. Despite the higher temperatures that can be achieved in ITER the effect of the energy dependence of σv is less significant than in the TEXTOR case. The reason for this is the geometry of the core CXRS system in ITER. The lines-of-sight in the ITER core CXRS system are almost perpendicular to the neutral beam. The shape of the CX spectrum comes from the Doppler shift due to movement in the direction of the observer u z. The collision velocity v col however depends strongly on the velocity towards and away from the neutral beam, thus on u y in the ITER geometry. Therefore the shape of the CX spectrum depending on u z and the intensity of the spectrum depending on the collision velocity, thus u y are independent in the ITER geometry. As a result the effect on rotation and temperature measurement is negligible. Gyration of plasma ions The energy dependence of σv is also responsible for a gyration-dependent shift of the poloidal rotation. The gyromotion of ions is predominantly in the poloidal direction, due to the large toroidal magnetic field, so the effect is only seen in the poloidal rotation, not in the toroidal rotation. During one part of its gyro-obit a C 6+ ion will move towards the neutral beam, having a larger collision velocity, in the other Section Additional effects 55

65 Chapter 4 - Charge Exchange Recombination Spectroscopy T observed (kev) TEXTOR ITER Ω observed (rad/s) x TEXTOR, T = 1 kev e TEXTOR, T = 2 kev e TEXTOR, T e = 5 kev ITER, T e = 2 kev ITER, T e = 15 kev T (kev) true Ω x 10 4 true (rad/s) Figure 4.4 : The observed ion temperature, due to the energy dependence of σv plotted against the true temperature for the TEXTOR (blue) and the ITER (red) case. The dotted line indicates T true = T observed. Figure 4.5 : The observed toroidal rotation, due to the energy dependence of σv plotted against the true rotation for the TEXTOR (blue) and the ITER (red) case. For ITER we assumed that no poloidal rotation is present. The dotted line indicates Ω true = Ω observed. part of the gyro-obit the collision velocity is lower. This means that in one part of the gyro-orbit more excited C 5+ ions are produced due to the higher effective emission rate σv than in the other part. If the lifetime τ of the excited state is larger than the gyroperiod this does not matter. However, with τ being of the order of 10 9 s and the cyclotron frequency ω c being of the order of 10 8 Hz for carbon impurities and typical TEXTOR magnetic fields of 2.25 T, the excited impurity ions only perform about a tenth of their gyration before emitting the charge exchange photon. This means that predominantly ions with a gyromotion that has a high σv will emit CX light, leading to a shift in the measured poloidal rotation Non-thermal line broadening In this section we discuss two broadening effects of the CX emission line that are independent of the ion temperature. These effects are the Zeeman effect, due to the magnetic field, and so-called l-state mixing. They are both discussed in more detail in [42] and [75]. Non-degeneracy of the l-states In the previous section we assumed that the transition between the n = 8 and the n = 7 level was only one transition between two levels with each only one corresponding energy. As a result the emission consisted of one single line with a natural wavelength of λ n = 529 nm and a natural line width of λ n = nm [63]. In reality the n = 8 and n = 7 levels of the C 5+ ion have a substructure. As a result there is not just one transition between the n = 8 and n = 7 level, but there is a number of possible transitions between the sublevels of n = 8 and the sublevels of n = Section Additional effects

66 Understanding and controlling plasma rotation in tokamaks In figure 4.6 the energy level diagram of the C 5+ ion is given. It shows that each principle quantum shell defined by the quantum number n has n subshells defined by the orbital quantum number l = 0,..., n 1 and also referred to as s, p, d, f, g,.... When electron spin is taken into account each l-level except l = 0 splits in two levels, defined by J = l + 1/2 and J = l 1/2. The allowed radiative transitions between the n = 8 and n = 7 level are those for which l = ±1 and J = 0, ±1. (l) s 0 p 1 d 2 f 3 g 4 h 5 i 6 k 7 J=1/2 (0.00) n = ev J=3/2 (0.25) J=1/2 (0.00) J=5/2 (0.32) J=3/2 (0.25) J=7/2 (0.36) J=5/2 (0.32) J=9/2 (0.40) J=7/2 (0.36) J=11/2 (0.40) J=9/2 (0.40) J=13/2 (0.43) J=11/2 (0.40) J=15/2 (0.43) J=13/2 (0.43) λ n = 529 nm J=3/2 (0.36) J=5/2 (0.47) J=3/2 (0.36) J=7/2 (0.54) J=5/2 (0.47) J=9/2 (0.58) J=7/2 (0.54) J=11/2 (0.61) J=9/2 (0.58) J=13/2 (0.65) J=11/2 (0.61) J=1/2 (0.00) n = ev J=1/2 ( 0.04) Figure 4.6 : Energy level diagram for the principle (n = 8)- and (n = 7)-shells of C 5+. Each principle n-shell is divided in l-subshells and J-levels. The energy within the principle n-shell differs slightly. That difference is exaggerated in this plot. The energy difference with the (l = 0)-level, within the same n-shell, is given in parentheses (10 4 ev ). The energy level of the (l = 0)-level, for both n = 8 and n = 7 is given at the left (ev ). [63] If all these different sublevels within the same n shell would have the same energy, i.e. if they would be degenerated, the sum of the emission by all the allowed transitions between n = 8 and n = 7 would still be one single line with a wavelength of λ n = 529 nm. However, the energy of the different sublevels within one principle n-shell is not exactly the same; there is a fine-structure. Due to the fine-structure each allowed transition has a slightly different wavelength, and the total emission spectrum consists of a set of lines instead of one line. Which of the l-levels of the n = 8 shell of C 5+ are populated in a charge exchange reaction depends on the beam energy. In TEXTOR the l-levels are approximately statistically populated due to the 50 kev neutral beam. Furthermore, if the lifetime τ of the excited states is considerably larger than the ion-ion collision time, collisions will cause a transfer between the different l-states of n = 8 shell before the charge exchange electron drops to a lower level and emits a photon. This phenomenon is called collisional l-mixing. It means that, even if the population of the l-levels would not be statistic, collisional l-mixing would make sure that the l-levels are statistically populated. The statistic population of all l-levels results in 37 allowed transitions, instead of one. The spectrum of these 37 lines is given by the black lines in figure 4.7. These lines cover a wavelength range of about Section Additional effects 57

67 Chapter 4 - Charge Exchange Recombination Spectroscopy nm. This corresponds with the Doppler broadening caused by a ion temperature of 4 ev. Zeeman effect In a tokamak a strong magnetic field in the order of B = 2 T is present. Without a magnetic field the energy levels within the same n shell differ slightly due to the fine-structure. The presence of a magnetic field will cause Zeeman splitting of one J-level into 2J + 1 energy levels separated by E z = µ b gbm j, m j = J,..., +J, where µ b is the Bohr magneton ( ev/t ) and g is called the Landé factor (of the order 1, depends on l, J and the spin). The allowed transitions are now given by l = ±1, J = 0, ±1 and m j = 0, ±1. When, for simplicity, we set g = 1, then every line in the fine-structure spectrum is split into three lines: λ λ λ Z, λ, λ + λ Z, with λ Z = µ b B. For B = 2 T the Zeeman split is λ Z = 0.03 nm. This is twice as large as the wavelength range of the fine-structure spectrum, which implies that in the presence of a strong magnetic field the total line spectrum of the C VI (n : 8 7) transition consists of three groups of fine-structured lines, where the distance between the groups is determined by Zeeman splitting. In figure 4.7 the total line spectrum in a magnetic field of 2 T is drawn. It covers a wavelength range of 0.07 nm, which corresponds with a Doppler broadening caused by an ion temperature of 90 ev λ (nm) Figure 4.7 : Line spectrum of the C VI (n : 8 7) transition. In case there is no magnetic field present the line spectrum is given by the black lines. It is determined solely by the fine-structure. When a strong magnetic field is present, Zeeman splitting causes every line to split in three lines. The resulting spectrum has three groups of fine-structure lines: the original in black and the groups shifted by ± λ Z in grey. The total CX spectrum, where non-thermal broadening due to l-mixing and Zeeman splitting is included, is the sum of the Doppler spectra for every emission line, where the relative intensity of every transition and the population of every sublevel is taken into account. When we treat every emission line separately the analysis of a CX spectrum gets quite complicated. Therefore the total profile of all transition lines is presented as a single, but broadened Gaussian, of which the width and the peak position depends on the l-mixing and Zeeman splitting. This Gaussian replaces the δ-function in equation (4.3). Apart from the introduction of this non-thermal Gaussian into the CX spectrum, nonthermal broadening also puts a lower limit to ion temperature measurements with CXRS. Temperatures below 100 ev are usually hard to resolve Non-CX emission In the same wavelength range as the charge exchange emission there are other sorts of emission that have nothing to do with the CX reaction or are a secondary effect of charge exchange. Especially when the active CX emission is low e.g. in the plasma centre where 58 Section Additional effects

68 Understanding and controlling plasma rotation in tokamaks the neutral beam density is low due to beam attenuation these non-cx emissions can swamp active CX signal and thus make the measurement of T C, n C and Ω φ,c virtually impossible. Below, we give a short overview of some of these non-cx emissions. Halo and plume In a charge exchange interaction a plasma ion receives an extra electron. After some the time it will of course lose this electron again. However, if the ionisation time is long enough, the plasma ion can be excited again by electron impact. This causes a delayed, secondary emission. This secondary emission is especially important for low Z ions, like hydrogen and helium, because they don t have to be excited to a high level in order to emit visible light. For C 5+ the electron impact should excite the ion from the ground state n = 1 to the n = 8 state in order to have a secondary emission at 529 nm. The chance for such an excitation by electron impact is negligible. If the plasma ion remains charged after the CX interaction, it still moves along the magnetic field and the delayed emission is seen as a plume. If the plasma ion is neutralised in the CX interaction, e.g. when H + or He + undergoes a CX reaction, it is no longer confined to the magnetic field and it can escape in all possible directions. The delayed emission is therefore called halo. Impurity lines In the edge of the plasma the temperature is too low to fully ionise all particles. The non-fully stripped ions usually impurity ions can be excited by electron impact and thus emit line radiation. If the wavelength of these impurity lines overlaps with the wavelength of the CX emission, the analysis of the CX spectrum is difficult. Getting rid of these unfavourable impurity lines can be done in the same way as the passive CX emission is treated: by beam modulation or modelling of the impurity emission. Impurity emission is however not always an annoyance. If the wavelength of the impurity line does not overlap with the CX emission but lies within the vicinity, the impurity line can be used for real-time wavelength calibration. Line-emission by impurity ions comes from the very edge of the plasma. There the temperature is very low, resulting is a narrow emission line who s peak position can be easily determined. Also the plasma rotation in the very edge of the plasma is negligible, such that the wavelength of the impurity emission is the natural wavelength λ n. The measured pixel position of this impurity line on the camera, together with the known λ n, is then used to derive the relation between the pixels on the camera and the wavelengths. Continuum emission The whole spectrum of active and passive CX emission and impurity line-emission is superposed onto a background of continuum emission. This continuum emission is Bremsstrahlung due to the relative movement of the charged particles in the plasma. The formula for the intensity of Bremsstrahlung in the visible region is [88]: di dλ = ḡn2 ez eff λ, (4.10) T e where T e is in kev, n e in m 3 and λ in nm. ḡ is called the Gaunt factor and can be approximated by 2 3/π. It shows that the intensity of the Bremsstrahlung can be used to determine Z eff, i.e. the purity of the plasma. If carbon is the major impurity in the plasma, then Z eff 1 + Z C (Z C 1)n C /n e. Combining the Z eff from Bremsstrahlung and Z eff from the intensity of the carbon CX emission, an intensity calibration of the system can be performed. Section Additional effects 59

69 Chapter 4 - Charge Exchange Recombination Spectroscopy The continuum radiation also depends on the square of the density. This can pose a diagnostic problem in fusion reactors with a high density like ITER. For a TEXTOR plasma typical values for T e, n e are 1 kev and m 3. The resulting continuum intensity is quite small, as can be seen in figure 4.2. For ITER both the density and the temperature are much higher: T e 10 kev and n e m 3. This means that the local intensity of Bremsstrahlung is eight times higher in ITER. Also the length of the line-of-sight through the plasma is about four times larger in ITER than in TEXTOR. As a result the line-integrated continuum intensity is 30 times higher in ITER than it is in TEXTOR. Because Bremsstrahlung is a statistical process the noise level goes with the square root of the signal, resulting in a five times higher noise level in ITER compared to TEXTOR. This continuum noise contaminates the active CX signal. When we take into account that the beam attenuation in ITER is stronger than in TEXTOR and that the effective emission rate σv is lower for the 100 kv beams at ITER than it is for the 50 kv beam at TEXTOR (see figure 4.3), we expect the active CX signal to be much lower in ITER than in TEXTOR. Therefore in ITER the high noise due to the high Bremsstrahlung intensity could possibly drown the low active CX signal. 4.4 CXRS at TEXTOR There are two main CXRS systems at TEXTOR. One system uses the NBI1 heating beam as a source of neutrals. We will therefore refer to it as the NBI-CXRS system. The other system uses a low power, radially injecting, diagnostic neutral beam. This system is called RUDI 1. The NBI-CXRS system has lines-of-sight tangential to the toroidal direction and is mainly used for ion temperature, carbon density and toroidal rotation measurements. The RUDI system has poloidal lines-of-sight. Apart from ion temperature and carbon density, it measures poloidal rotation. In this thesis we are mainly interested in the toroidal plasma rotation. Therefore most plasma rotation data presented in this work comes from the NBI-CXRS system. NBI-CXRS system The NBI-CXRS system has three sets of lines-of-sight. They are plotted in figure 4.8. These lines-of-sight are all located in the equatorial plane and are directed tangentially to the flux surfaces, which means that the measured Doppler shifts can be translated in toroidal rotation frequencies Ω φ. Two sets of lines-of-sight the red core system and the blue the edge system observe the plasma in the counter-direction and one set the green lines in figure 4.8 looks in the co-direction. The NBI-CXRS system uses the heating beam NBI1 as a source of neutral particles. Because this is a high power beam, there is a large neutral density up to the plasma centre and the resulting CX signal is high. This allows us to reduce the exposure time of the camera and hence have a reasonably fast time resolution. In normal operating conditions a time resolution of 50 ms is used. Faster sampling is possible up to 10 ms but the quality of the CX spectra reduces significantly with shorter exposure times. A disadvantage of the heating beam is its large width. It was said previously that the CXRS measurement is a local measurement as long as the line-of-sight is tangential to a flux surface where it crosses the neutral beam. When a wide neutral beam is used, this condition is difficult to reach. The neutral particle density of NBI1 in TEXTOR has 1 this stands for RUssian DIagnostic, which immediately reveals its origin. 60 Section CXRS at TEXTOR

70 Understanding and controlling plasma rotation in tokamaks 0 (a) (b) 0.5 Y (m) X (m) R (m) Figure 4.8 : The NBI-CXRS setup at TEXTOR. The left figure shows a top view of TEXTOR with the three sets of lines-of-sight (core in red, edge in blue and ). The gray area represents the full width of NBI1 at half maximum. In the right figure an arbitrary CXRS profile is plotted, with the radial resolution. The horizontal lines indicate the FWHM of the neutral beam over the lines-of-sight. a Gaussian shape perpendicular to the beam axis. In the equatorial plane, where the lines-of-sight are located, the full width half maximum (FWHM) is in the order of 20 cm. The grey area in figure 4.8 indicates the FWHM of NBI. In an optimised geometry the major radius of the flux surface where the line-of-sight enters the beam i.e. where the neutral beam density is at half maximum does not differ from the major radius of the flux surface where the line-of-sight crosses the centre of the beam and the major radius of the flux surface where the line-of-sight leaves the neutral beam again. In that case the measurement is truly local. In reality the line-of-sight does cross several flux surfaces when it goes through the neutral beam. As a result the CX signal is the line integrated emission over the crossed flux surfaces. The range of flux surfaces that are crossed determines the radial resolution of the CXRS system. In figure 4.8 (b) an arbitrary Gaussian profile is shown to indicate the radial resolution of each of the sets of lines-of-sight. It is clear from this figure that the co-observing system and the edge system yield the best radial resolution. These two systems were however both in development when the experiments for this thesis were done. The CXRS data presented in this work therefore comes only from the core, counter-observing system. RUDI system Apart from the NBI-CXRS system, there is also a CXRS system installed at TEXTOR that uses a low power, radially injecting, diagnostic neutral beam: the RUDI- CXRS system. The lines-of-sight of this system are radially as well, which means that the RUDI system measures poloidal rotation. Of course also the ion temperature and the carbon density are diagnosed with this system. Because a low power beam is used, the influence on the plasma is minimal. The RUDI beam does not drive rotation and it hardly heats or fuels the plasma. This makes the RUDI system suitable for measurements of T C, n C and v θ,c in e.g. Ohmic discharges. The Section CXRS at TEXTOR 61

71 Chapter 4 - Charge Exchange Recombination Spectroscopy RUDI beam is also modulated. The passive CX component and the impurity lines are thus removed from the spectrum by subtracting the spectrum measured during the off-phase of the beam from the spectra obtained during the on-phase of the beam. The disadvantage of a low power, diagnostic beam is the low CXRS signal. As a result a long exposure time is needed to collect enough signal. For RUDI the time resolution is rather poor: in the order of 1 s. In this thesis the main part of the data comes from the NBI-CXRS system, that measures toroidal rotation. The poloidal rotation measurements from the RUDI system are only used on a few occasions. Spectrometers, cameras and calibration As mentioned above the NBI-CXRS system has three sets of lines-of-sight. The light of each of these sets goes through a bundle of optical fibres to a spectrometer. The spectrometers are located well away from TEXTOR. The spectrometers that are used are of the Littrow type. In figure 4.9 the spectrometer setup is drawn schematically. The light comes from TEXTOR through an optical fibre bundle and is focussed with two lenses on the entrance slit of the spectrometer. The entrance slit is either 50 µm, 100 µm or 300 µm. The width of the entrance slit is important because it greatly determines the width of instrument function. A narrow entrance slit gives a narrow instrument function, but also reduces the intensity of the light. Littrow spectrometer camera lens grating movable mirror wavelength calibration lamp Fibre bundle to TEXTOR Figure 4.9 : The spectrometer setup for the NBI-CXRS system at TEXTOR. The light comes from TEXTOR through the a fibre bundle. It is focussed on the entrance slit of the spectrometer. In the spectrometer a large lens assures a full illumination of the grating by a parallel bundle of light. The dispersed light is finally imaged, through the same lens, on the camera. Behind the entrance slit there is a small mirror that reflects the light approximately 90. Subsequently the light goes through a large lens that causes the grating at the back of the spectrometer to be fully illuminated by a parallel bundle of light. The grating of the 62 Section CXRS at TEXTOR

72 Understanding and controlling plasma rotation in tokamaks spectrometers that we used has 1200 lines per mm. The light reflected from the grating is imaged on the camera. The wavelength of the reflected light depends on the number of lines on the grating g, the angle of the grating γ and the order in which you want to measure m. To calculate the dispersion wavelength per meter at the position of the camera also the focal length f of the large lens is needed. For the used spectrometer this is f = 750 mm. The imaged wavelength and dispersion are then: λ = 2 m sin γ (4.11) g ( ) dλ dx = 1 m λ g 2 1 (4.12) m f g 2 For CXRS emission of carbon (λ = 529 nm) and a measurement in first order (m = 1), the dispersion is: dλ/dx = 1 nm/mm. Two types of cameras are used at TEXTOR: The Wright cameras have a CCD with pixels. Only half of the chip ( ) is illuminated. The pixels have an area of 22.5 µm 22.5 µm. The camera is positioned such that the narrow side (298 pixels) is in the direction of the wavelengths. Along the other direction (576 pixels) the different fibres are imaged. In this direction we can bin rows of pixels that are illuminated by the light of the same fibre. The output of the Wright camera than consists of n spectra n being the number of fibres. Each spectrum has 298 data points, representing the wavelengths nm to nm with a dispersion of nm/pixel (for carbon). The read-out time of the camera depends on the binning of the pixels. For 10 fibres the fastest read-out time is 30 ms. This limits the time resolution of the CXRS system. The other camera is a PixelVision camera. The active region of the CCD chip is pixels. The pixels are 11.8 µm 11.8 µm. Here the camera is positioned such that the wide side (655 pixels) is in the wavelength direction. Again binning is possible along the other direction the direction of the different fibres. The output of the PixelVision camera also consists of n spectra n being the number of fibres. Each spectrum now has 655 data points, representing the wavelengths nm to nm with a dispersion of nm/pixel. The PixelVision camera is faster than the Wright camera. It can read out the spectra of 20 fibres in 8 ms. The PixelVision camera can therefore reach a higher time resolution, if the intensity of the CX light allows it. The spectra that are the output of the cameras, give the intensity per pixel, not per wavelength. The properties of the spectrometer (γ, g, f and m) give us an idea about the pixel to wavelength conversion, but this is far from accurate. A good wavelength calibration is therefore needed. To do so, we use a wavelength calibration lamp that emits a spectrum of which the wavelengths are well documented. Ideally this calibration lamp should be put inside TEXTOR, such that the light of the calibration lamp follows the same path as the light from a plasma discharge would. This is however only possible during major service-shutdowns of TEXTOR, when the TEXTOR vessel is accessible. Section CXRS at TEXTOR 63

73 Chapter 4 - Charge Exchange Recombination Spectroscopy Instead of placing the calibration lamp inside TEXTOR, we put the calibration lamp close to the spectrometer. In between TEXTOR discharges a mirror can be placed in the optical beam path of the CXRS setup, as is shown in figure 4.9. The light of the calibration lamp is than focussed on the entrance slit of spectrometer and a calibration spectrum is recorded. Because the wavelengths of the lines of the calibration lamp are known, we can determine the pixel to wavelength conversion from this calibration spectrum. The conversion can then be applied to the spectra of the plasma discharges. This only works however if the optical axis of the calibration lamp spectrometer system exactly coincides with the optical axis of the TEXTOR spectrometer system. Because there is always some deviation between these optical axes, we empirically found that it is not possible to increase the accuracy above ±10 µm at the camera position. For carbon this corresponds with systematic wavelength error of ±0.01 nm or a systematic error in the velocity measured by Doppler shift of ±5.5 km/s. This error is the same for every position and every time frame, such that rotation gradients and the time evolution of the rotation are not influenced by this inaccuracy. For the intensity calibration a integrating sphere is used. The intensity of the light emitted in every direction by such an integrating sphere is known for every wavelength. The sphere is put inside TEXTOR. The intensity measured by the camera can then be compared to the intensity emitted by the integrating sphere. Because the sphere has to be placed inside TEXTOR, an intensity calibration can only be done during a serviceshutdown. 4.5 Summary We introduced the technique of charge exchange recombination spectroscopy. This technique analyses the light emitted when a fully stripped impurity ion receives an electron form a neutral particle. The Doppler shift of this line-emission is proportional to the plasma rotation velocity. Apart from plasma rotation, also the ion temperature and the impurity density can be determined via CXRS. A charge exchange spectrum consists of an active CX part, caused by the neutral beam, passive CX emission and impurity emission from the edge. The most interesting data comes from the active CX light, which means that we have to filter the passive and impurity light out of the spectrum. This can be done by beam modulation or by modelling the passive and impurity emission. The effective emission coefficient of the CX light depends on the collision velocity between the neutral particles and the plasma ions. This means that ions moving with a different velocity will emit light with a different intensity. Because the plasma ions have a thermal velocity distribution, the measured temperature and rotation may therefore differ from the true temperature and rotation. For low temperatures (order 1 kev ) the difference can be neglected. For high temperatures this effect can be important. One can however outsmart this effect by choosing the lines-of-sight of the CXRS system perpendicular to the neutral beam. This is e.g. the case for the core CXRS system at ITER. Zeeman splitting and l-state mixing are causes of non-thermal broadening of the CX spectrum. To take this non-thermal broadening into account the ideal CX spectrum has to be convoluted with a Gaussian of which the width is determined by the Zeeman splitting and l-state mixing. Non-thermal broadening makes it difficult to measure temperatures below 100 ev with CXRS. 64 Section Summary

74 Understanding and controlling plasma rotation in tokamaks The light emitted by impurities can be annoying, when it is located at the same wavelength position as the CX emission. But if the impurity emission is located in the same wavelength range but not at the same wavelength position it can be used for wavelength calibration. The continuum radiation is Bremsstrahlung. Combining the Bremsstrahlung with the intensity of the CX emission the impurity density can be determined without the need for an intensity calibration. The Bremsstrahlung can however also be malignant. For large machines with a high plasma density, like ITER, the Bremsstrahlung is very high. The noise on the continuum radiation goes with the square root of the intensity and will therefore be high as well. A high enough active CX signal is necessary in ITER, because it will otherwise drown in the continuum noise. In TEXTOR there are two CXRS systems. One uses the heating beam NBI1 as a source of neutrals, the other system RUDI has its own diagnostic neutral beam. The NBI- CXRS system is capable of measuring the toroidal rotation, the RUDI system measures poloidal rotation. Because of our interest in toroidal rotation, mainly the NBI-CXRS system was used for this work. There are three sets of lines-of-sights for the CXRS- NBI system, but only one of these sets the 9 channel core system is used for the measurements in this thesis. The time resolution of this system is 50 ms. The limited accuracy of the wavelength calibration for this system, results in a systematic error in the rotation velocity of ±5.5 km/s. Because this error is systematic the rotation gradients and the time evolution of the rotation are not influenced by this inaccuracy. Section Summary 65

75

76 Understanding and controlling plasma rotation in tokamaks Chapter 5 Plasma rotation at TEXTOR 5.1 Introduction In chapter 2 several statements on the rotation of a plasma were made: The momentum confinement time τ φ scales with the energy confinement time τ E. Plasmas rotate even when no external momentum input is applied. Impurities rotate at a different speed than the main plasma ions. The frequency of MHD modes is the toroidal rotation frequency plus the electron diamagnetic drift frequency. The experimental verification of this is given here for the TEXTOR case. 5.2 Momentum and energy confinement time in TEXTOR In several tokamaks it is experimentally found that very often the momentum confinement time τ φ is strongly related to the energy confinement time τ E (see section 2.3.1). Especially when the ion energy confinement time is compared to the momentum confinement time, it is observed that for most steady-state discharges, with densities above m 3, τ φ = τe ion. In transient phases and at low densities (n < m 3 ), the relation τ φ = τe ion does not hold [86]. Figure 5.1 shows that also in TEXTOR τ φ and τ E are coupled. The momentum confinement time τ φ was calculated for shots with a known momentum input by the neutral beams: T NBI = R NBI1P NBI1 v NBI1 R NBI2P NBI2 v NBI2, (5.1) with R NBI the impact radius of the neutral beam, P NBI the neutral beam power and v NBI the velocity of the neutral beam particles. The total angular momentum of the plasma, L φ, is calculated on the basis of rotation and density measurements during these shots. The resulting momentum confinement time τ φ then is: τ φ = L φ T NBI. (5.2) 67

77 Chapter 5 - Plasma rotation at TEXTOR The energy confinement time τ E = E dia /P total is calculated with the measured total power input and the stored plasma energy measured with the standard diamagnetic loop diagnostic. Due to poor diagnostic coverage it was not possible to determine the ion energy confinement time τe ion in TEXTOR Figure 5.1 : The measured momentum confinement time τ φ is plotted against the measured energy confinement time τ E. Just like in other tokamaks, we find that τ φ scales with τ E. Within the error of the measurement we can even say τ φ = τ E. This allows us to determine an unknown torque T φ if τ E and the rotation are measured: T φ = L φ τ E. measured τ φ (s) measured τ E (s) The fact that also in TEXTOR τ φ τ E allows us to determine the momentum input T φ from the measured L φ and the measured τ E. This is interesting when the total momentum input is not known. The dynamic ergodic divertor at TEXTOR, for example, is expected to exert a torque onto the plasma. The total momentum input of the plasma during the application of the DED is the sum of the known T NBI and the unknown T DED. This total momentum input is also L φ /τ E. Hence the total torque exerted by the DED that we find is: T DED = L φ /τ E T NBI. 5.3 Ohmic rotation in TEXTOR In chapter 2 we indicated that plasmas rotate even when no external momentum input is applied. In present day tokamaks the plasma rotation is mainly driven by neutral beams and the small fraction of spontaneous rotation is usually neglected. In ITER neutral beams will not be able to drive a large amount of toroidal momentum, due to large plasma mass, and the spontaneous rotation will become very important. It is therefore worthwhile to look into spontaneous rotation, even in a medium-size tokamak like TEXTOR. We will apply the neoclassical expressions given in section to the TEXTOR case in order to get an estimate for the spontaneous rotation in TEXTOR. According to neoclassical theory, the poloidal rotation is proportional to the gradient of the ion temperature. For the toroidal rotation several different expressions are available in the literature. The expressions for the toroidal velocity are not closed : they are given as a derivative of the toroidal velocity, or an assumption on the radial electric field has to be made. In figure 5.2 the density, temperature, magnetic field and the collisionality in an Ohmic TEXTOR discharge are given. It is seen that over a large part of the plasma the collisionality is low and the transport regime of interest is the banana regime. Only in the edge of the plasma, the expressions for transport in the plateau regime must be used. We use the collisionality factor K 1 (ν ) as it is given in [49]. 68 Section Ohmic rotation in TEXTOR

78 Understanding and controlling plasma rotation in tokamaks B φ (T) / B θ (0.1 T) / q T (kev) / n (10 19 m 3 ) Collisionality ν* T = T i = T e Plateau regime q B φ B θ n = n i = n e Pfirsch Schlüter regime Figure 5.2 : The magnetic fields, the density and temperature of a typical Ohmic discharge in TEXTOR are given. From Banana regime R (m) this the collisionality follows. This collisionality defines which transport regime has to be used when the spontaneous neoclassical rotation is calculated. The sign of K 1 determines the direction of the poloidal rotation. In the banana regime the poloidal rotation is in the ion diamagnetic drift direction and in the Pfirsch-Schlüter regime the rotation is in the electron diamagnetic drift direction. In the plateau regime the poloidal rotation goes from the ion to the electron diamagnetic drift direction. Because the largest part of the TEXTOR plasma is in the banana regime, the neoclassical poloidal rotation in TEXTOR is in the ion diamagnetic drift direction. The expected poloidal rotation in TEXTOR is also very low. At its maximum it reaches only 0.8 km/s. The poloidal rotation is given by the blue line in figure 5.3. For the toroidal neoclassical velocity there are several expressions. The expression given by equation (2.45) in chapter 2 was found in [14]. Here, not the toroidal velocity v φ is given, but the derivative dv φ /dr. We made the assumption that the v φ is zero at the plasma edge and that equation (2.45) is valid over the whole plasma. The resulting toroidal velocity is in the counter-current direction and has a maximum velocity of 6 km/s. The toroidal velocity profile according to [14] is drawn by the solid red line in figure 5.3. Although the assumption v φ = 0 at the edge is reasonable, one should be careful when stating that equation (2.45) is valid over the whole plasma. The equation considers a pure ion-electron plasma, while in the edge neutral and not fully ionised particles are present as well. It also assumes the plasma to be confined in nested flux surfaces, but this picture breaks down at the plasma edge. If we use the expression (2.46) found in [49] we have a local expression of v φ. The assumption we have to make here lies in the radial electric field E r. A possible assumption is E r = 1/en i dp i /dr, which basically states that the electric field, by balancing the force due to the ion pressure gradient, prevents the loss of ions. If we substitute E r into equation (2.41), we see that this assumption also means that we assume a parallel plasma flow. Section Ohmic rotation in TEXTOR 69

79 Chapter 5 - Plasma rotation at TEXTOR Equation (2.46) then returns: v φ = (K 1 /eb θ ) dt i /dr, which together with the poloidal rotation v θ = (K 1 /eb φ ) dt i /dr indeed yields a parallel rotation v = (K 1 B/eB θ B φ ) dt i /dr. In figure 5.3 v φ, according to [49] and with assumption of a parallel flow, is given by the dashed red line. Because v θ is in the co-direction and the total plasma flow is parallel, v φ is counter-rotating. The toroidal rotation velocity in the plasma centre is 8 km/s. v θ (m/s) / v φ (m/s) Plateau regime Poloidal velocity (v θ ) Toroidal velocity (v φ ) according to [14] Toroidal velocity (v φ ) according to [49] Toroidal velocity from MHD frequency Ohmic sawtooth precursor Banana regime Plateau regime R (m) Figure 5.3 : Poloidal and toroidal velocity profiles calculated for an Ohmic discharge at TEX- TOR (I p = 400 ka, B φ = 1.9 T, T and n as given in figure 5.2). The poloidal rotation is in the ion diamagnetic drift direction (i.e. co-rotation) due to the fact that almost the whole plasma is in the banana regime. For the toroidal rotation we used two different theoretical expressions. If we follow the approach of [14] and assume v φ = 0 in the edge, the toroidal rotation in the centre will be in the counter direction. Again due to the banana regime. If we use the expression given in [49] and use the assumption that E r = 1/en i dp i /dr, we find a parallel flow. The resulting v φ is in the counter-direction. An indirect measurement (red star), using the MHD frequency of the sawtooth precursor in an Ohmic discharge, shows that toroidal rotation in the plasma centre is indeed in the counter-direction. The NBI driven rotation is a factor 5 to 10 higher than the expected spontaneous rotation in TEXTOR. The NBI-CXRS system at TEXTOR can not be used to compare the theoretical expectation with a measurement of the rotation velocity, because this system uses a neutral beam. Neutral beam driven plasmas reach rotation velocities up to 100 km/s. This is a lot more than the expected spontaneous rotation (6 to 8 km/s) and therefore, whenever the neutral beam is used, the plasma rotation velocity is dominated by the beam driven rotation. We can however use MHD frequency measurements in TEXTOR to derive the toroidal rotation velocity at a specific q surface (see section 5.5). When we measure the MHD frequency of the sawtooth precursor, located at the q = 1 surface, we can measure the central toroidal rotation velocity in an the Ohmic discharge. This measured rotation velocity is 9 km/s in the counter direction, which is consistent with the expected spontaneous toroidal rotation. 70 Section Ohmic rotation in TEXTOR

80 Understanding and controlling plasma rotation in tokamaks The subject of spontaneous rotation is a very important issue in plasma physics, due to its relevance for the next generation of fusion devices like ITER. To get a neoclassical value for this rotation, quite large assumptions have to be made. On top of that the neoclassical theory does not take turbulence into account. It is however expected that turbulence is a key parameter in the description of spontaneous rotation. This section served as an illustration on the amount of spontaneous rotation we could expect in TEXTOR. For a detailed discussion on spontaneous rotation we would like to refer to the literature [17, 41, 62, 69]. 5.4 Impurity rotation in TEXTOR In TEXTOR the plasma rotation is measured using charge exchange recombination spectroscopy. The line-emission that is analysed is emitted by carbon ions. The measured velocity therefore is the bulk velocity v C of the carbon fluid. What we are actually interested in is the plasma velocity v; i.e. the centre of mass velocity of all plasma species. For low impurity densities, the plasma velocity v can be approximated by the fluid velocity of the main plasma ions. In the case of TEXTOR the main ions are deuterium ions. Z eff, which is a measure for the impurity content, is usually about 2. The plasma velocity is then given by v = 0.7v D + 0.3v C. In section the neoclassical expression for the difference between the velocity of the impurity ions and the velocity of the main ions is given. For TEXTOR we can calculate the difference v D,C = v D v C from equation (2.48). The relation between the measured v C and the plasma velocity v then is: v = v C v D,C. According to the neoclassical theory the difference between the toroidal rotation of the carbon ions and the plasma fluid is large for plasma discharges with a low plasma current and a peaked temperature profile. For TEXTOR the plasma current is usually around 400 ka, which is not particulary low, and the temperature is not extremely high or peaked (central temperatures in the order of 1 kev ). The difference between v φ,c and v φ is therefore expected to be low. In figure 5.4 the measured carbon toroidal rotation frequency and the resulting plasma rotation frequency when using (2.48) is plotted for a typical TEXTOR, beam driven discharge. The magnetic field and plasma current for the presented discharge are the same as those given in figure 5.2; the electron temperature (T central = 1.5 kev ) and density (n central = m 3 ) are higher due to the fact that this is a beam driven plasma, not an Ohmic discharge. The beam used was NBI1, which injected 600 kw of power in the co-direction. The calculated difference between the carbon rotation velocity and the plasma rotation velocity is approximately 2 km/s = 10 3 rad/s. It is directed in the co-direction. We could choose to neglect this correction, because it is very close to the statistical error that is made when the CXRS spectra are fitted. The correction is, however, systematic and therefore must be taken into account. In chapter 4 it was seen that the wavelength calibration of the CXRS system only allows for a measurement accuracy of about 5 km/s. This means that the plasma rotation velocity lies within the systematic error of the carbon rotation velocity measurement. Instead of trying to get a better calibration for the carbon rotation velocity and then apply the neoclassical correction to get the plasma rotation velocity, usually a different Section Impurity rotation in TEXTOR 71

81 Chapter 5 - Plasma rotation at TEXTOR 2.5 x Ω φ (rad/s) R (m) Figure 5.4 : The measured carbon rotation frequency plotted together with the statistical fitting error (dotted line and error bars). The solid line represents the profile of the plasma rotation, that was calculated with eq. (2.48). It shows that the plasma rotation is higher, or more precisely is more in the co-current direction, than the carbon rotation. The correction is low: Ω φ 10 3 rad/s, which is of the order of the statistical error and lower than the calibration error for the CXRS system (indicated by the shaded area). Nevertheless, the correction should be taken into account because it is systematic. approach is used at TEXTOR. As will be shown in the next section the frequency of MHD modes is directly linked with the toroidal plasma velocity. From the measurements of the MHD frequency the plasma velocity at one location the q surface of the mode can be determined. By calculating the difference, the carbon rotation velocity at this location is derived from the plasma rotation velocity. This gives us a accurate calibration of the carbon rotation velocity profile. Subsequently we use the now calibrated profile of the carbon rotation velocity and the neoclassical difference between carbon and plasma rotation velocity, to determine the profile of the plasma rotation velocity. 5.5 MHD rotation in TEXTOR In both previous sections we mentioned that the frequency of MHD modes, measured with e.g. ECE, can be used to determine the toroidal plasma rotation at the q surface where the mode is located. This allows us to figure out the direction of the rotation in Ohmic discharges and to calibrate the measured carbon rotation velocity. The relationship between the toroidal rotation velocity and the MHD frequency is given in section 2.5.2: f MHD = (nv φ )/(2πR) + f e. In TEXTOR the electron diamagnetic drift is in the counter-direction, which means that f e < 0. Because, in the literature, the diamagnetic frequency is always given as a positive number, we will do so as well in this thesis. With f e positive the above formula for TEXTOR becomes: f MHD = nv φ 2πR f e. (5.3) When both f e, calculated with equation (2.54), and f MHD, measured with ECE or a 72 Section MHD rotation in TEXTOR

82 Understanding and controlling plasma rotation in tokamaks magnetic diagnostic, are known, then the toroidal velocity is known as well. The validity of the above equation has already been checked in [85]. We have taken over the results from [85] here as an illustration. In figure 5.5, for a number of discharges, the MHD frequency of the sawtooth precursor (m/n = 1/1) is plotted against the toroidal rotation frequency (f φ = ) at the q = 1 surface. Positive frequencies correspond to co-rotation, negative frequencies to counterrotation. In these discharges the electron temperature (T e = 1.2 kev ), the magnetic field (B φ = 2.25 T ), q at the edge (q a = 4) and q at the magnetic axis (q 0 q a /(q a + 1) = 0.8) where kept constant, such that the diamagnetic frequency calculated with (2.54) is also constant: fe = 3.6 khz. v φ 2πR Figure 5.5 : The MHD frequency of the 1/1 sawtooth precursor plotted against the toroidal rotation frequency f φ = v φ 2πR. It shows a linear relationship: f MHD = f φ 2.0 khz. This plot proves that the expression f MHD = nf φ fe is valid. Thus that f MHD measurements can be used to determine v φ. The offset of the line gives us the local fe. The value of the local fe is observed to be lower than the fe that is calculated with equation (2.54). A lower local pressure gradient, due to the presence of modes itself, is probably responsible for this. With the local fe known, the measurement of f MHD during an Ohmic discharge gives us the toroidal rotation in an Ohmic discharge. This point is indicated by Ω. The corresponding toroidal rotation velocity is 10 4 m/s in the counter-direction. [85]. According to equation (5.3) we expect to see a straight line when we plot the precursor frequency against the toroidal rotation frequency. Figure 5.5 indeed shows a linear relationship between the measured f φ and the measured f MHD. We also expect the slope of this line to be 1, because n = 1 for the sawtooth precursor. Linear regression analysis of the data points learns that the slope is 0.998, which is consistent with this. For f φ = 0 we expect to see f MHD = fe. Figure 5.5 shows us then that fe = 2.0 khz. The calculation of fe with equation (2.54) resulted in fe = 3.6 khz. This is higher than the measured fe. For the calculation of fe the local pressure gradient is needed. In equation (2.54) this local pressure gradient was derived by assuming natural density and temperature profiles. The presence of modes can however locally change pressure and density profile. Over the O-point of a magnetic island the pressure gradient is zero, only through the X-point there is a pressure gradient. The averaged pressure gradient over the Section MHD rotation in TEXTOR 73

83 Chapter 5 - Plasma rotation at TEXTOR island region could therefore be lower than the one that follows from the natural pressure profile. A lower fe, typically a factor two, would result from that. From figure 5.5 also the toroidal rotation at the q = 1 surface during an Ohmic discharge can be determined. The precursor frequency during an Ohmic discharge was measured to be 3.0 khz. Because this frequency decreased when the co-beam (NBI1) was switched on, we find that the precursor rotated in the counter-direction: f MHD = 3.0 khz. With the local fe = 2.0 khz given, equation (5.3) yields a toroidal frequency of f φ = 1.0 khz, or a rotation velocity of v φ = 10 4 m/s at the q = 1 surface. This is a rotation in the counter-direction. It is indicated by the Ω-mark in figure 5.5 and by the red star in figure 5.3. The work presented in [85], and repeated here, shows that measurements of the MHD frequency of modes is a very valuable addition to the rotation measurements with CXRS. 5.6 Conclusion In this chapter we experimentally verified some of the topics presented in chapter 2 for the specific case of the TEXTOR tokamak. First of all we found that for steady state TEXTOR discharges τ φ τ E. This allows us to use τ E instead of τ φ for discharges where τ φ can not be determined from measurements; e.g. when we want to determine the momentum input by the DED. Secondly we tried to find out how much spontaneous rotation is expected in TEXTOR based upon the neoclassical approach of [14] and [49]. The expected spontaneous, toroidal rotation velocity is approximately 6 to 8 km/s in the counter-direction. In TEXTOR, CXRS can not be used to measure the toroidal rotation in an Ohmic discharge, but a measurement of the sawtooth precursor frequency allowed us to derive a central toroidal rotation velocity of 9 km/s in the counter-direction, which is consistent with the expectation. The velocity of the spontaneous rotation is an order of magnitude smaller than the beam driven velocities. Therefore, in TEXTOR spontaneous rotation is usually negligible. However, it is interesting to study the phenomenon of spontaneous rotation in a quantitative manner, as it will be a very important effect in ITER. The toroidal rotation velocity of the carbon ions, that is measured with CXRS, differs about 2 km/s with the toroidal plasma rotation. This difference is of the order of the statistical error when fitting a CX spectrum, but has to be taken into account because it is a systematic deviation: the plasma rotation is always more in the co-direction than the carbon rotation. Together with the measurement of the MHD frequency, the correction on the carbon velocity can be used to get an absolute calibration of the plasma velocity profile. A final topic was the relationship between the toroidal velocity and the frequency of MHD modes. The validity of f MHD = nf φ f e was tested. With the above expression the measurement of the MHD frequency at one position the rational q surface gives us the toroidal plasma rotation at that location. This value can then be used to calibrate the velocity profile measured with CXRS. The measurement of f MHD also showed that the plasma rotates in the counter-direction during an Ohmic discharge. 74 Section Conclusion

84 Understanding and controlling plasma rotation in tokamaks Chapter 6 Measurements of plasma rotation during DED operation 6.1 Introduction Whereas most tokamaks in the world have a fixed error field, the ergodic dynamic divertor (DED) on TEXTOR allows us to manipulate the error field. We will therefore not call it an error field anymore but a controlled perturbation field. In the introduction we mentioned that error or perturbation fields can excite MHD modes. Because these modes degrade the plasma confinement and stability they can even lead to a disruption we want to avoid their excitation. The best way to know how to avoid them, is to know how to excite them. The controlled perturbation field of the DED will be used to carefully study the threshold at which mode excitation occurs. In this chapter we try to find an answer to the questions raised in the introduction: How does the application of a perturbation field change the plasma rotation? How does the excitation of tearing modes depend on the plasma rotation velocity? In the literature the prediction is made that the plasma rotation velocity changes with increasing perturbation strength [28, 55]. This change in rotation velocity is expected to be monotonic towards the plasma rotation velocity for which the MHD modes are at rest in the frame of the perturbation field. Once this plasma velocity is reached, the MHD modes grow and form large magnetic islands: the so-called mode excitation. To test these hypotheses, we carried out a research programme in which we applied a systematic variation of DED perturbation level and the momentum input by the neutral beams. The rotation velocity during the application of the DED and the threshold for mode excitation were measured. The results of these measurements are then compared to the predictions that are found in the literature. 6.2 A general overview A linear increase of the current in the DED coils causes a linear increase of the perturbation field. And, as said in the introduction, we expect to see a change in the plasma rotation when the perturbation is increased. 75

85 Chapter 6 - Measurements of plasma rotation during DED operation In figure 6.1 time traces of the toroidal rotation frequency are given during a linear ramp up of the DED current in three different modes: AC (co), DC (static) and AC + (counter). In figures 6.2 to 6.4 the evolution of the toroidal rotation profile is given during the DED current ramp up, for all three modes of operation (AC, DC and AC + ). For the DC and AC + case, one can observe two phases: a phase where the rotation frequency increases slightly with increasing DED current and where the shape of the rotation profile is conserved; and a phase where the rotation profile is flattened and the frequency stays constant even when the DED current is further increased. When the DED is operated in AC, only the first phase of increasing rotation frequency with increasing DED strength is observed. x 10 4 Ω φ (rad/s) Co rotating DED field (AC ) Static DED field (DC) Counter rotating DED field (AC + ) 1 0 I DED (ka) time (s) Figure 6.1 : Time traces of the toroidal rotation frequency at R = 1.83 m, for 3 plasma discharges with a static (DC), co-rotating (AC ) and counter-rotating (AC + ) perturbation field induced by the DED. For the co-rotating field the rotation increases with the perturbation strength ( I DED ). For the static and counter-rotating fields the increase in plasma rotation is cut short at the transition threshold. The time traces of the toroidal rotation frequency in figure 6.1 are taken at the position R = 1.83 m (r/a = 0.1) for three TEXTOR discharges with DC, AC + and AC DED operation. All other plasma parameters where identical in these discharges. The plasma rotation before the DED is applied is in the co-direction (positive). One observes an initial increase in rotation frequency when the DED current is increased. For the co-rotating DED field (AC ) the increase in rotation frequency continues up to the maximum value of the DED current; the plasma rotation stays in the first phase. For a static (DC) or counter-rotating DED field (AC + ), after a slight initial increase, a sharp drop of the rotation frequency is observed. After this sharp drop the toroidal plasma rotation frequency stays constant, even though the DED current is further increased. The plasma rotation frequency during this constant phase depends on the DED frequency: it is 76 Section A general overview

86 Understanding and controlling plasma rotation in tokamaks x 10 4 Ω φ (rad/s) t = 1.65 s, I DED = 0.06 ka t = 1.90 s, I = 0.28 ka DED t = 2.15 s, I = 0.50 ka DED t = 2.40 s, I = 0.73 ka DED t = 2.65 s, I DED = 0.96 ka t = 2.90 s, I DED = 1.18 ka 1 q = 1 q = R (m) Figure 6.2 : The evolution of the toroidal rotation profiles during a ramp up of the current in the DED coils in AC operation. The rotation frequency increases when the DED current is increased. The shape of the rotation profile does not change: the rotation profile is just lifted. x 10 4 Ω φ (rad/s) t = 1.65 s, I = 0.06 ka DED t = 1.90 s, I DED = 0.28 ka t = 2.15 s, I DED = 0.50 ka t = 2.40 s, I DED = 0.73 ka t = 2.65 s, I = 0.96 ka DED t = 2.90 s, I = 1.18 ka DED 1 q = 1 q = R (m) Figure 6.3 : The evolution of the toroidal rotation profiles during a ramp up of the current in the DED coils in DC operation. black profiles: Initial phase, increasing rotation with increasing DED current. grey profiles: Second phase, plasma rotation locked with DED frequency. More towards the edge around the q = 2 the plasma rotation monotonically increases towards the final locked rotation. In the plasma centre the around the q = 1 the plasma rotation velocity first increases and then suddenly drops to the locked value. Section A general overview 77

87 Chapter 6 - Measurements of plasma rotation during DED operation x 10 4 Ω φ (rad/s) t = 1.65 s, I DED = 0.06 ka t = 1.90 s, I = 0.28 ka DED t = 2.15 s, I = 0.50 ka DED t = 2.40 s, I = 0.73 ka DED t = 2.65 s, I DED = 0.96 ka t = 2.90 s, I DED = 1.18 ka 1 q = 1 q = R (m) Figure 6.4 : The evolution of the toroidal rotation profiles during a ramp up of the current in the DED coils in AC + operation. black profiles: Initial phase, increasing rotation with increasing DED current. grey profiles: Second phase, plasma rotation locked with DED frequency. Here both around the q = 2 surface and around the q = 1 surface the plasma rotation velocity first increases and then decreases again. There is no location in the plasma where the change in plasma rotation is monotonic. lower for AC + DED operation than for DC operation. From figure 6.1 one can determine the difference between the plasma rotation frequency in AC + and DC operation. This difference is about rad/s or 1 khz. Taking into account that an AC + DED field rotates in the counter direction (i.e. negative) with a frequency of 1 khz and the DC field has of course zero frequency, this indicates that in the constant phase the plasma rotation frequency is directly related to the DED frequency. The transition between the increasing phase and the constant phase is reasonably fast; in the order of 100 ms. The threshold of the perturbation amplitude at which this transition occurs is different for different plasma conditions. In figures 6.2 to 6.4 give the evolution of the toroidal rotation frequency profile during the linear increase of the DED current. The data in these figures corresponds with the AC, DC and AC + time traces in figure 6.1. The black profiles are in the initial phase; the rotation frequency increases with increasing DED current. For the DC and AC + current ramp, figures 6.3 and 6.4, the increase is marginal. In case of a current ramp of the DED in AC operation, figure 6.2, the increase in rotation frequency is substantial. We also observe that the shape of the rotation profile does not change. The grey profiles in figures 6.3 and 6.4 are in the second phase where the rotation frequency does not change anymore with increasing DED current. The grey profiles are significantly flattened in comparison with the peaked profile before DED operation and during the initial phase. The rotation frequency in this flat region is related to the DED frequency. The transition from the peaked, black profiles to the flattened, grey profiles is different in DC and in AC + operation. For DC DED operation the plasma rotation in the core 78 Section A general overview

88 Understanding and controlling plasma rotation in tokamaks drops when the transition threshold is reached, while the rotation more towards the edge increases. In the case of AC + DED operation the rotation decreases over the whole measured region. Due to the flattened profile after the transition, the drop in rotation is of course larger in the centre than it is near the edge of the plasma. 6.3 Change in rotation as a function of DED current In the previous section we saw that, when increasing the DED current, the response of the plasma rotation can be separated in two clearly different phases. We now focus on the initial phase where the change in plasma rotation depends on the DED current. When increasing the current in the DED coils, in a first phase the rotation changes as a function of the DED current. The shape of the rotation profile does not change during this phase; the profile is just lifted as can be clearly seen in figure 6.2. This can be attributed to momentum transport. The toroidal momentum balance equation (2.37) shows that if the toroidal rotation changes at one position, this change will have an effect on the whole plasma. How strong this effect is depends on how large the momentum transport, defined by the momentum diffusion coefficient D, is. A toroidal torque localised near the plasma edge is able to lift the whole rotation profile. If we look at the change in rotation for a large number of discharges, we notice that corotating plasmas show an increase in rotation velocity, when the DED current is increased. Counter-rotating plasmas will slow down. This means the rotation does not spin up when the DED current is increased, but that there is a change of the plasma velocity in the codirection. In most cases this change in co-rotation is a monotonically increasing function of the DED current; it lasts until the threshold for the transition to the constant phase or the maximum DED current is reached. For plasmas that have a large rotation velocity in the co-direction, it is observed that they have an initial increase in co-rotation. This increase however slows down and reverses at higher DED currents, so finally the change of rotation is in the counter-direction. In figure 6.5 the change in toroidal rotation is plotted versus effective DED current for different plasma discharges, where the effective DED current is defined as: { Icoil (DC) I DED = 2 2+ I 2 coil (AC), (6.1) The data sets in figure 6.5 cover a wide range of plasma parameters: low and high power input, discharges rotating in co-direction or counter-direction before the DED was applied and discharges where DC, AC + and AC DED was used. The data can be divided in two groups: three sets of three discharges where in each set all plasma parameters where kept constant, only the frequency of the DED was changed form AC + over DC to AC. Between the sets of discharges the plasma parameters are different. The data points plotted in black come from discharges with low power and low momentum input by the neutral beams. The data points plotted in red come from discharges with strongly counterrotating plasmas and high power input. Section Change in rotation as a function of DED current 79

89 Chapter 6 - Measurements of plasma rotation during DED operation 2 x 104 3/1 DC, P NBI = 0.25 MW, T NBI = 0.15 Nm 3/1 AC +, P NBI = 0.25 MW, T NBI = 0.15 Nm Ω φ (rad/s) Effective I DED (ka) 3/1 AC, P NBI = 0.25 MW, T NBI = 0.15 Nm 3/1 DC, P NBI = 1.65 MW, T NBI = 0.65 Nm 3/1 AC +, P = 1.65 MW, T = 0.65 Nm NBI NBI 3/1 AC, P = 1.65 MW, T = 0.65 Nm NBI NBI 3/1 DC, P NBI = 1.30 MW, T NBI = 0.20 Nm 3/1 AC +, P NBI = 1.30 MW, T NBI = 0.20 Nm 3/1 AC, P NBI = 1.30 MW, T NBI = 0.15 Nm 3/1 DC, P NBI = 1.35 MW, T NBI = 0.85 Nm 3/1 DC, P NBI = 1.40 MW, T NBI = 0.75 Nm 3/1 DC, P NBI = 1.30 MW, T NBI = 0.35 Nm 3/1 DC, P = 1.30 MW, T = 0.20 Nm NBI NBI 3/1 DC, P NBI = 1.35 MW, T NBI = 0.15 Nm 3/1 DC, P NBI = 1.35 MW, T NBI = 0.35 Nm Figure 6.5 : The change in toroidal rotation versus effective DED current for several discharges. The change in rotation is mostly in the co-direction. Only for discharges with a high momentum input in the co-direction, the toroidal rotation changes in the counter-direction during the application of the DED. The change in rotation frequency also differs for discharges with different power input. DED torque (N m) Effective I (ka) DED 3/1 DC, P NBI = 0.25 MW, T NBI = 0.15 Nm 3/1 AC +, P = 0.25 MW, T = 0.15 Nm NBI NBI 3/1 AC, P NBI = 0.25 MW, T NBI = 0.15 Nm 3/1 DC, P = 1.65 MW, T = 0.65 Nm NBI NBI 3/1 AC +, P NBI = 1.65 MW, T NBI = 0.65 Nm 3/1 AC, P NBI = 1.65 MW, T NBI = 0.65 Nm 3/1 DC, P NBI = 1.30 MW, T NBI = 0.20 Nm 3/1 AC +, P NBI = 1.30 MW, T NBI = 0.20 Nm 3/1 AC, P NBI = 1.30 MW, T NBI = 0.15 Nm 3/1 DC, P NBI = 1.35 MW, T NBI = 0.85 Nm 3/1 DC, P = 1.40 MW, T = 0.75 Nm NBI NBI 3/1 DC, P = 1.30 MW, T = 0.35 Nm NBI NBI 3/1 DC, P NBI = 1.30 MW, T NBI = 0.20 Nm 3/1 DC, P = 1.35 MW, T = 0.15 Nm NBI NBI 3/1 DC, P NBI = 1.35 MW, T NBI = 0.35 Nm Figure 6.6 : The torque exerted by the DED versus effective DED current for several discharges. This torque was calculated by dividing the measured change in rotation by the momentum confinement time. One sees that the influence of the power input on the change in rotation was mainly due to the difference in momentum confinement time. 80 Section Change in rotation as a function of DED current

90 Understanding and controlling plasma rotation in tokamaks The data points plotted in green come from discharges with similar momentum input as the black data, but with higher power input. a rotation scan where the power input was kept constant and the momentum input was changed from discharge to discharge. The DED was operated in DC mode. These data are plotted in blue. Despite the variety in plasma parameters the change in co-rotation shows the same kind of behaviour for most discharges. The change is mostly in the co-direction and has a non-linear dependence on the DED current. Only for strongly co-rotating plasmas indicated by the blue + and signs the plasma slows down; i.e. the change in rotation is in the counter-direction. Figure 6.5 also shows that the in rotation depends on the power input as well. This is not unexpected; as mentioned in section and section 5.2 the toroidal momentum confinement time τ φ is related to the energy confinement time τ E, and hence decreases with increasing power input. This means that for plasmas with high power input the momentum input by the DED is lost faster than in low power plasmas. In other words, the change in plasma rotation will be different for discharges with a different power input, even if the torque exerted by the DED does not depend on the power input. So, in order to find out whether the torque exerted by the DED is influenced by the power input, we should plot the change in torque, instead of the change in rotation, versus the effective DED current. This change in torque is calculated from the rotation measurements using the definition of the momentum confinement time ( see also equation (2.26)): T DED = R2 0 ρ Ωφ dr, (6.2) τ φ where the momentum confinement time τ φ is determined before the DED is applied, assuming that it does not change during the initial DED phase. Figure 6.6 shows the torque exerted by the DED versus the effective DED current. It reveals that the influence of the power input on the torque caused by the DED is marginal. The non-linear dependence on the DED current and the sign reversal for highly co-rotating plasmas of course remains. The amount of change in torque due to the DED can go up to 1 Nm, which is of the same order of magnitude as the torque caused by the neutral beam. Usually the change in torque is lower because the threshold for transition to the locked phase is reached before the effective DED current is at maximum. Not only the toroidal rotation of the core plasma is affected during DED operation. The poloidal velocity at the edge of the plasma - measured with both CXRS and passive emission of CIII just inside the last closed flux surface - shows a change in the co-direction during DED as well, as is shown in figure 6.7 [11]. Also turbulence rotation measured with reflectometry shows an increase in the ion diamagnetic drift direction [53]. A fourth observation made during DED operation is a positive change E r of radial electric field at the very edge of plasma. This is seen in figure 6.8 where the floating potential profile in the outer few centimetres of the plasma gets less steep during DED operation [43]. Also sign reversal of the edge electric field has been observed in some cases. A positive E r is, through the E B drift, linked with an increase in co-rotation, both toroidally and poloidally. An important remark has to be made concerning the measurements of poloidal rotation, turbulence rotation and floating potential in the edge of the plasma. For the toroidal Section Change in rotation as a function of DED current 81

91 Chapter 6 - Measurements of plasma rotation during DED operation Figure 6.7 : The change of poloidal rotation as function of the DED current. For DC, AC + and AC these changes are in the co-direction, i.e. the ion diamagnetic direction or the direction of the poloidal magnetic field. [11] rotation in the core an increase in co-rotation was observed in most cases, but for some plasmas that were rotating fast in the co-direction, a decrease of the co-rotation was Figure 6.8 : The profile of the floating potential V fl near the plasma boundary in the case of no DED, AC + DED and AC DED. For both AC + and AC the profile of V fl gets less steep. This means that the radial electric field E r gets less negative and hence the change in electric field E r is positive.[43] 82 Section Change in rotation as a function of DED current

92 Understanding and controlling plasma rotation in tokamaks seen. This is not seen in the measurements poloidal rotation and floating potential near the plasma edge; they always show an increase in co-direction. Also in contrast to the measurements of the central toroidal rotation, these measurements do not show as clearly the two phases below and above the DED threshold. Where the core plasma rotation is locked to the rotation frequency of the perturbation field when the DED current exceeds the threshold, the edge measurements still see an increase of the rotation in co-direction. Moreover this increase in co-rotation with increasing DED current is even slightly enhanced when the threshold is exceeded [11]. Summary and discussion In summary, these experiments show that, when the level of the perturbation is increased, the plasma rotation changes. In other words, the perturbation field of the DED exerts a torque onto the plasma. We have seen that the total power input has little influence on this torque. We have also seen that the direction of this torque is mostly directed in the co-direction, resulting in a slowing down of counter-rotating plasmas and a spin-up of co-rotating plasmas. An exception to this behaviour was found for plasmas that rotated fast in the co-direction: after an initial increase in rotation velocity, these plasmas started to slow down with increasing perturbation strength. This exception was only found for the toroidal rotation in the plasma core: the poloidal rotation in the edge and the floating potential profile in the edge were for all discharges compatible with an increase in co-rotation. Also in the literature it is found that the plasma rotation is expected to change when the level of perturbation is increased [28, 55]. This change in rotation should monotonically go towards the rotation velocity for which the tearing modes are at rest in the frame of the perturbation field. As seen in section 5.5 the rotation frequency of tearing modes in TEXTOR is given by: ω t = nω φ 2πf e, (6.3) where f e is the local electron diamagnetic frequency. Hence the modes will be at rest when the slip frequency ω = 0, with: ω = ω t 2πf DED = nω φ 2π(f e + f DED ). (6.4) The change in ω is given in [55] as: [ ω = 1 ω ( ) ] 2 1/2 IDED 1 for I DED < I threshold, (6.5) 2 I threshold and hence ω > ω 0 /2, In the above formula ω 0 is the initial slip frequency before the perturbation field is applied, and I threshold is the DED current at which the slip frequency is half the initial slip frequency ω = ω 0 /2. At this point the theory predicts that ω jumps to zero. In [55] it is found that I threshold is proportional to ω 0. At ω = 0 the toroidal rotation frequency is given by Ω φ = (2π/n) (f e + f DED ). In TEXTOR f e is typically about 2 khz and f DED = 0, 1 or 1 khz for DC, AC or AC + operation, respectively. This means that in TEXTOR ω = 0 corresponds with a positive, i.e. co-rotating, toroidal rotation frequency Ω φ. The lowest Ω φ corresponding with ω = 0, occurs for AC + operation, where f DED = 1 khz, hence Ω φ rad/s for a toroidal mode number n = 1. Section Change in rotation as a function of DED current 83

93 Chapter 6 - Measurements of plasma rotation during DED operation In other words, when the perturbation level is increased the rotation is expected to change according to (6.5), which means towards a positive toroidal rotation frequency Ω φ. Counter-rotating plasmas and plasmas with a co-rotation frequency below Ω φ = (2π/n) (fe + f DED ) will experience a change in the co-direction. Plasmas that rotate in the co-direction with Ω φ larger than (2π/n) (fe + f DED ) are expected to slow down, hence experience a change in the counter-direction. Qualitatively this seems to agree with our experimental results. To make a quantitative comparison, we need to know at which radial position we need to measure the rotation frequency. Equation (6.5) predicts the rotation at the position of the MHD modes. In the next section we will see that 1/1 and 2/1 modes are excited at the q = 1 and q = 2 surfaces. We will therefore compare the measurements of the plasma rotation frequency at the location of the q = 1 and q = 2 surfaces to the ω-evolution given by (6.5). In figures 6.3 and 6.4 we immediately see that the rotation at the position of the q = 1 surface does not correspond with the rotation predicted in (6.5). Instead of monotonic change in rotation, the rotation first increases in the co-direction and than suddenly drops. We therefore conclude that influence of the interaction between the perturbation field and modes at the q = 1 surface will have little influence on the change in plasma rotation. The interaction between the perturbation field and modes at the q = 2 surface is a far more promising candidate to explain the change in plasma rotation. We see that at the q = 2 surface there is a monotonic change in rotation frequency in the DC case (fig. 6.3). For AC + operation (fig. 6.4) there still is an initial increase followed by a decrease, but this non-monotonic change is in the order of the measurement error. x f DED ) (rad/s) ω = Ω φ (q=2) 2 π (f e * I DED (ka) Figure 6.9 : A comparison between the measured slip frequency ω as a function of I DED and the ω-evolution as predicted in [55]. One immediately sees that the prediction does not agree with the measurement. 84 Section Change in rotation as a function of DED current

94 Understanding and controlling plasma rotation in tokamaks In figure 6.9 we compare the slip frequency ω calculated from the rotation frequency Ω φ at the q = 2 surface, with the expected ω that follows from equation (6.5). This is done for two situations: with an initial ω 0 < 0 and with a ω 0 > 0. It clearly shows that there is a discrepancy between the expected change in rotation and the measured change in rotation. So at first sight there seems to be an agreement between the measurements and the prediction: a change in co-rotation for counter-rotating and slowly co-rotating plasmas and a change in the counter-direction for plasmas that rotate fast in the co-direction. A quantitative comparison, however, shows that the prediction and measurement do not agree the change in not monotonic if ω 0 > 0 and is faster than expected if ω 0 < 0 (see figure 6.9). The discrepancy could possibly be explained if, apart from a torque trying to bring ω to zero, the perturbation field is responsible for a second torque acting on the plasma. This torque should be directed in the co-direction, such that for ω 0 < 0 it would bring ω faster to zero, while for ω 0 > 0 it would oppose the torque that tries to reduce ω. Because for ω 0, ω initially increases, for low DED currents this co-torque should be stronger than the torque that tries to bring ω to zero. On the other hand, because ω eventually becomes zero, for higher DED currents the torque reducing ω prevails. The measurements of the poloidal rotation and the floating potential in the plasma edge, suggest that this co-torque could be located in the edge region: the edge measurements always show a change of the rotation in the co-direction and this change continues even when the core toroidal rotation has already reached the constant phase. In the next section it will be shown that in this constant phase ω = 0 at the q = 2 surface. The fact that the poloidal rotation and the floating potential still change, although the rotation at the q = 2 surface is kept constant, suggests that a co-torque is present outside the q = 2 surface that increases with increasing DED current. 6.4 Plasma rotation locked to the DED frequency Once the DED perturbation reaches a high enough amplitude the rotational behaviour of the plasma changes very rapidly. In 50 to 100 ms the toroidal rotation profile flattens over practically the whole measured area. This can be seen in figures 6.1, 6.3 and 6.4. A flat area in a rotation profile means that the plasma rotates as a rigid body within this area. In figures 6.3 and 6.4 the positions of the q = 1 and q = 2 surfaces are indicated, where a quadratic q-profile was assumed. The rigid body rotation stretches out between these two q surfaces. After the transition the rotation frequency of stays constant, even with increasing DED current. Figure 6.1 shows that for discharges with a counter-rotation AC + DED field the rotation frequency is 6.3 rad/s or 1 khz lower than for discharges with a static DC field. This strongly suggests that, in this constant phase, the toroidal rotation frequency Ω φ is such that tearing modes within the plasma are locked to the DED field: ω t = ω DED = 2πf DED, where ω t is the tearing mode frequency. The relation between Ω φ and ω t is given by equation (6.3): ω t = nω φ 2πfe. In 3/1 DED operation the main Fourier components of the perturbation field have n = 1 as toroidal mode number. Figure 6.10 shows ω DED versus the MHD frequency ω t for n = 1 at the q = 2 surface. It is clearly seen that ω DED indeed equals ω t, as would be expected when a 2/1 mode is locked to the DED field. Section Plasma rotation locked to the DED frequency 85

95 Chapter 6 - Measurements of plasma rotation during DED operation 3 x AC DED 3.75 khz ω DED = 2 π f DED (rad/s) 1 AC DED 1.00 khz 0 DC DED 0.00 khz AC + DED 1.00 khz 1 2 AC + DED 3.75 khz ω t = Ω φ 2 π f * (rad/s) e x 10 4 Figure 6.10 : The data points, with errorbars, show the frequency of the DED field versus the tearing frequency in the locked phase. The data points coincide with the dashed line indicating ω DED = ω t. Of course a plasma rotation frequency that corresponds with the presence of locked modes is not a solid prove for the existence of these modes. The direct observation of tearing modes locked to the DED in the phase with a constant plasma rotation, is therefore needed. Because the electron temperature T e and density n e are uniform or slightly peaked in a magnetic island, diagnostics that measure electron temperature and/or density are well suited for mode detection. In figure 6.11 (a) the T e profile with Thomson scattering during DC DED operation is shown. A flat region, indicating the presence of an island, is seen at the position where q = 2 is expected. The fact that the island is only seen in the lower part of the profile (z < 0), and not, as the 2/1 island symmetry requires, in both lower and upper part, is due to the off-axis measurement of the Thomson scattering system. When the time evolution of the T e profile is measured during DC DED operation, no changes in the profile are observed as a function of time. This is expected, because the island is locked to the static DED field and hence does not move. During AC DED operation the island is locked to a rotating field, therefore the time evolution of the Thomson measurements should show the appearance and disappearance of the flat T e region with the frequency of the perturbation field. These fluctuations in T e, but also in n e, are shown in figure 6.11 (c). From this plot an island width in the order of W 10 cm can be determined. Islands can also be detected with ECE measurements. During DED in AC operation, fluctuations in T e are seen at specific locations in the plasma. Taking into account the rigid body rotation in the toroidal direction measured with charge exchange and assuming no poloidal rotation, the time information of the ECE signals corresponds with different toroidal positions. Through the q-profile these toroidal angles can be projected onto poloidal angles, thus reconstructing the 2 dimensional spatial electron temperature profile 86 Section Plasma rotation locked to the DED frequency

96 Understanding and controlling plasma rotation in tokamaks Figure 6.11 : Thomson Scattering measurements of the 2/1 islands during the locked DED phase: (a) T e profile during DC DED operation. The flattened region indicates a magnetic island. (b) The flattening in (a) is only seen at the lower part side of the profile because the measurement of the Thomson scattering system does not go through the plasma centre. (c) The time evolution of electron density and temperature during AC + DED operation. Both T e and n e show fluctuations with the DED frequency of 1 khz; n e is peaked in the island O-points, T e is flattened. The horizontal dotted lines indicate the position of the q = 2 surface, the vertical dashed lines indicate the time points of the measurements. [84] in a poloidal cross section. The result of such a rotational reconstruction is given in figure 6.12, where a quadratic q-profile is assumed. One can clearly see a 2/1 mode, but apart from that more towards the centre of the plasma a 1/1 internal kink mode is visible. Thomson scattering and ECE are not the only diagnostics that observe the excitation of islands when the rotation jumps into the constant phase. Measurements with soft X-ray cameras indicate the presence of a 2/1 island with an island width of about W = 8 cm and a coupled 1/1 internal kink mode [57]. And also magnetic probes and coils located just outside the plasma, pick up a change in magnetic flux when the transition is made to the locked phase. Apart form the Ω φ profile, that is constant and flat, and the observation of 2/1 and 1/1 modes with ECE, Thomson scattering and other diagnostics, a third observation is made in the phase with constant rotation frequency. For TEXTOR discharges with sawteeth, the sawteeth disappear quite abruptly, when the constant rotation phase is reached; this Section Plasma rotation locked to the DED frequency 87