Experimental studies of tearing mode and resistive wall mode dynamics in the reversed field pinch configuration

Transcription

1 Experimental studies of tearing mode and resistive wall mode dynamics in the reversed field pinch configuration Jenny-Ann Malmberg DOCTORAL THESIS Alfvén Laboratory Royal Institute of Technology Stockholm 2003

2 ISBN Experimental studies of tearing mode and resistive wall mode dynamics in the reversed field pinch configuration Jenny-Ann Malmberg, 6 June 2003 Norstedts tryckeri AB, Stockholm 2003

3 Malmberg, Jenny-Ann Experimental studies of tearing mode and resistive wall mode dynamics in the reversed field pinch configuration (in English), Alfvén Laboratory, Division of Fusion Plasma Physics, Royal Institute of Technology, Stockholm 2003 Abstract It is relatively straightforward to establish equilibrium in magnetically confined plasmas, but the plasma is frequently succeptible to a variety of instabilities that are driven by the free energy in the magnetic field or in the pressure gradient. These unstable modes exhibit effects that affect the particle, momentum and heat confinement properties of the configuration. Studies of the dynamics of several of the most important modes are the subject of this thesis. The studies are carried out on plasmas in the reversed field pinch (RFP) configuration. One phenomenon commonly observed in RFPs is mode wall locking. The localized nature of these phase- and wall locked structures results in localized power loads on the wall which are detrimental for confinement. A detailed study of the wall locked mode phenomenon is performed based on magnetic measurements from three RFP devices. The two possible mechanisms for wall locking are investigated. Locking as a result of tearing modes interacting with a static field error and locking due to the presence of a non-ideal boundary. The characteristics of the wall locked mode are qualitatively similar in a device with a conducting shell system (TPE-RX) compared to a device with a resistive shell (Extrap T2). A theoretical model is used for evaluating the threshold values for wall locking due to eddy currents in the vacuum vessel in these devices. A good correlation with experiment is observed for the conducting shell device. The possibility of succesfully sustaining discharges in a resistive shell RFP is introduced in the recently rebuilt device Extrap T2R. Fast spontaneous mode rotation is observed, resulting in low magnetic fluctuations, low loop voltage and improved confinement. Wall locking is rarely observed. The low tearing mode amplitudes allow for the theoretically predicted internal nonresonant on-axis resistive wall modes to be observed. These modes have not previously been distinguished due to the formation of wall locked modes. The internal and external nonresonant resistive wall modes grow on the time scale of the shell penetration time. These growth rates depend on the RFP equilibrium. The internal nonresonant resistive wall modes dominate in Extrap T2R, especially for shallow reversed discharges. The external nonresonant modes grow solely in deep reversal discharges. Descriptors Nuclear fusion, reversed field pinch, resistive instabilities, wall locked modes, tearing modes, resistive shell modes, field errors, EXTRAP-T2, EXTRAP-T2R, TPE-RX ISBN iii

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5 Preface What started out as a Master of science project at CIEMAT 1 in Madrid turned out to be more than just an adventure in Spain. This first practical encounter with Fusion ended up as an appetizer and I soon found myself applying for a PhD. position at the Alfvén laboratory. This was greatly due to my main supervisor for the Masters project, Prof. Elisabeth Rachlew. Many thanks for the inspiration and encouragement to go for it. One thing lead to another and I ended up within the field that intrigued me the most. Magnetics. I feel greatful to Prof. Jim Drake for making this possible. Since the EXTRAP T2 experiment was shut down within two weeks of my arrival at the Alfvén laboratory, I was soon engaged in a project at ETL 2 in Tsukuba, Japan. At ETL I got my first real introduction to reversed field pinch experimental work and I was inspired by the wonderful enthusiasm and curiosity that I experienced among my colleagues. My supervisor at ETL, Dr. Yasuyuki Yagi made me see many hidden and wonderful details in the data that I was analyzing. He also made me realize that saving time is usually a matter of doing something properly from the beginning. When I got back from Japan, I soon found myself digging in the database of EXTRAP T2. The more data I analyzed, the more questions appeared. I was lucky to have a supervisor like Dr. Per Brunsell, who always took the time to find the structure in my confusion. New ideas sprung from all these long discussions, ideas that would eventually lead to new confusion. To my mind, this is the process of research. The challenge lies in the constant flow of unexplored details. The small steps of progress are supported by momentary truth. I am greatful that I have had the opportunity to take these steps. Bromsten, April, Centro de Investigaciones Energéticas, Medioambientales y Technològicas 2 Electro Technical Laboratory v

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7 Contents 1 Introduction Toroidaldevices Thereversed-fieldpinch Mode dynamics Taylor relaxation and the dynamo MHD stability RFP stability Tearing mode dynamics Regimes for magnetic fluctuation reduction and improved confinement Resistivewallmodes Feedback stabilization of resistive wall modes Description of experiments ExtrapT ExtrapT2R TPE-RX Magnetic diagnostics ExtrapT ExtrapT2R TPE-RX Experimental data analysis Fourieranalysis Lastclosedfluxsurfacereconstruction Phasealignmentanalysis Mode rotation analysis Radial and tangential magnetic field measurements Results PaperI PaperII PaperIII PaperIV PaperV PaperVI PaperVII PaperVIII PaperIX vii

8 viii Contents 6 Conclusions 49 References 53

9 Contents ix This thesis is based on the following papers: I. Y. Yagi, H. Koguchi, H. Sakakita, S. Sekine, Y. Maejima, J.-A. Nilsson, T. Bolzonella and P. Zanca, "Mode-locking phenomena in the TPE-RX reversed-field pinch plasma", Physics of Plasmas 6, 3824 (1999). II. Y. Yagi, H. Koguchi, H. Sakakita, S. Sekine, P. R. Brunsell and J-A. Malmberg, "Evolution process of the mode wall-locking and phase-locking in a reversed-field pinch plasma", Physics of Plasmas 8, 1625 (2001). III. J.-A. Malmberg, P. R. Brunsell, Y. Yagi, H. Koguchi, "Locked modes in two reversed-field pinch devices of different size and shell system", Physics of Plasmas 7, 4184 (2000). IV. P. R. Brunsell, H. Bergsåker, M. Cecconello, J. R. Drake, R. M. Gravestijn, A. Hedqvist and J-A. Malmberg, "Initial results from the rebuilt Extrap T2R RFP device", Plasma Phys. and Control. Fusion 43, 1457 (2001). V. J.-A. Malmberg and P. R. Brunsell, "Resistive wall instabilities and tearing mode dynamics in the EXTRAP T2R thin shell reversed-field pinch", Physics of Plasmas 9, 212 (2002). VI. J.-A. Malmberg, M. Cecconello, P. R. Brunsell, Y. Yadikin and J. R. Drake, "Reversed-field pinch experiments in EXTRAP T2R with a resistive shell boundary", (in Proceedings of the 19th International Atomic Energy Agency Conference, Lyon, 2002). EX/P2-02. VII. P. R. Brunsell, J-A. Malmberg, Y. Yadikin, and M. Cecconello, "Resistive wall modes in the EXTRAP T2R reversed-field pinch", submitted to Phys. of Plasmas. VIII. J.-A. Malmberg, J. Brzozowski, P. R. Brunsell, and M. Cecconello, "Mode- and plasma rotation in a resistive shell reversed-field pinch", submitted to Phys. of Plasmas. IX. M. Cecconello, J.-A. Malmberg, P. Nielsen, R. Pasqualotto, and J. R. Drake, "Study of the confinement properties in a reversed-field pinch with mode rotation and gas fuelling", Plasma Phys. Control. Fusion 44, 1625 (2002).

10 x Contents Papers related to this work but not included in this thesis: Y. Yagi, H. Koguchi, J-A. Nilsson, T. Bolzonella, P. Zanca, S. Sekine, T. Osakabe and H. Sakakita, "Phase- and Wall-locked modes found in a large reversed-field pinch machine, TPE-RX", Jpn. J. Appl. Phys. 38, L 780 (1999) H. Koguchi, Y.Yagi, Y. Hirano, T. Shimada, H. Sakakita, S. Sekine, T. Osakabe, and J.-A. B. Nilsson, "Field error and its effect on the plasma performance in TPE-RX, reversed-field pinch device", Plasma Phys. Control. Fusion (2000). P. R. Brunsell, H. Bergsåker, J. H. Brzozowski, M. Cecconello, J. R. Drake, J-A. Malmberg, J. Scheffel and D. D. Schnack, Mode dynamics and confinement in the reversed-field pinch, (In Proc. of the 18th International Atomic Energy Agency conference, Sorrento, 2000). CN-77/EXP3/14. M. Cecconello, J-A. Malmberg, E. Sallander and J. R. Drake, Self-organisation and intermittent coherent oscillations in the Extrap-T2 reversed field pinch, Physica Scripta 65, 69 (2002). M. Rubel, M. Cecconello, J.-A. Malmberg, G. Sergienko, W. Biel, J. R. Drake, A. Hedqvist, A. Huber and V. Philipps, Dust particles in controlled fusion devices: morphology, observations in the plasma and influence on the plasma performance, Nuclear Fusion 41, p (2001). J. Linke, M. Rubel, J. A. Malmberg, J. R. Drake, R. Duwe, H. J. Penkalla, M. Rödig and E. Wessel, Carbon Particles Emission, Brittle Destruction and Co-deposit Formation: Experience from Electron Beam Experiments and Controlled Fusion Devices, Physica Scripta T91, p. 36 (2001). J-A. Malmberg and P. R. Brunsell, "Mode dynamics in the rebuilt EXTRAP T2R RFP device", (In Proc. of the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, 2001). J. R. Drake, P. R. Brunsell, J. Brzozowski, M. Cecconello, J-A. Malmberg, E. Sallander, "First results from the Extrap T2R RFP experiment", (In Proc. of the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, 2001). Y. Yagi, T. Bolzonella, A. Canton, K. Hayase, Y. Hirano, S. Kiyama, H. Koguchi, Y. Maejima, J- A. Malmberg, H. Sakakita, Y. Sato, S. Sekine, T. Shimada and K. Sugisaki, "Confinement in the TPE-RX reversed-field pinch" (In Proc. of the 18th International Atomic Energy Agency conference, Sorrento, 2000). EX4/06. P. Martin, et. al, "Overview of quasi single helicity experiments in reversed field pinches", (in Proceedings of the 19th International Atomic Energy Agency Conference, Lyon, 2002). P. R. Brunsell, J.-A. Malmberg, and D. Yadikin, "Resistive wall mode studies in the EXTRAP T2R RFP", (In Proc. of the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Montreux, 2002). M. Cecconello, J.-A. Malmberg, G. Spizzo, R. M. Gravestjin, P. Franz, P. Piovesan, P. Martin, B. E. Chapman, and J. R. Drake, "Current profile modification experiments in EXTRAP T2R", manuscript (included in PhD thesis by M. Cecconello, Experimental studies of confinement in the EXTRAP T2 and T2R reversed field pinches, January 2003).

11 Kalle...När man tänker på någon nästan jämt, då är det som att ligga i en myrstack, det är hemskt, man föreställer sig, man liksom fastnar på en tjärustick, det går bara runt runt, just för att det är så hopplöst, man kan inte tänka på annat... Jag menar: man blir som tokut. Fast det är ju bara att man tänker, man borde inte tänka, det blir som i en myrstack. Per Olov Enquist, Kapten nemos bibliotek

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13 Chapter 1 Introduction Nuclear fusion is the basic process in which the sun and the stars produce power to sustain their high temperatures. This process is based on the following fusion reaction between two hydrogen particles H + H D + e + (1.1) The probability of this reaction is very low unless very high temperatures and densities are reached. The high densities in the sun due to gravity allows for nuclear fusion to occur between two hydrogen particles. Since these high densities can t be achieved in a laboratory plasma, fusion reactions with lower temperature thresholds are found more suitable for a future power plant. The most promising reaction for this purpose is that between deuterium and tritium. D + T 4 He + n +17.6MeV (1.2) The cross-section for this reaction is a strong function of temperature. With the present designs, the temperature needed for such a reaction to occur is in the order of 10 8 C. The high temperatures required for fusion reactions to occur is related to the fact that the reacting forces must have enough energy to overcome the Coloumb barrier so that the short range nuclear forces dominate thus leading to the nuclear fusion reaction. The goal of fusion research is to confine a hot plasma for a sufficiently long time in order for fusion reactions to occur at such a rate that the energy output exceeds the energy required to sustain the fusion device. One of the main concepts that are being explored for practical realization of fusion is magnetic fusion. The plasma is in this case confined by magnetic fields that isolate the plasma from the inner walls of the device. In this way, the plasma can be heated to high temperatures without being cooled by the walls. Magnetic confinement of plasma in the shape of a torus is considered the most successful concept to date. The reversed field pinch (RFP) is one such concept. 1.1 Toroidal devices Toroidal configurations are conveniently described by the major radius R and the minor radius a and the coordinate system (e r, e θ, e φ ), where r is the minor radial coordinate (0 <r<a), θ is the poloidal angle and φ is the toroidal angle (see Fig. 1.1). Toroidal plasmas are axisymmetric if the equilibrium is independent of the angle φ. In an axisymmetric toroidal field configuration, the magnetic field lines wind around the torus forming a set of nested toroidal flux surfaces. Three basic equations describe an MHD equilibrium: 3

14 4 Chapter 1. Introduction Amperes law, B = µ 0 J, (1.3) Momentum equation, B =0, (1.4) J B = p. (1.5) A system is in equilibrium if no net forces are exerted on the plasma. This requires that the force due to plasma pressure is balanced by the magnetic force and is described by the momentum equation. If this condition is satisfied there is no pressure gradient along the magnetic field lines resulting in a constant pressure on the magnetic surfaces. Furthermore, the condition implies that the current lines lie on these surfaces. Figure 1.1. Coordinate system in a toroidal configuration. There are several basic plasma parameters that describe the properties of an equilibrium plasma. The plasma β measures the efficiency of plasma confinement by the magnetic field and is defined as the ratio of the average plasma energy to the average magnetic energy. In general, high values of β are desirable in fusion devices. The safety factor (q = rb φ /RB θ ) and the rotational transform (ι) are important parameters for both MHD equilibrium and stability since they give a measurement of the pitch of the magnetic field lines in a plasma. If the difference between the poloidal

15 1.2. The reversed-field pinch 5 angles of a certain field line at a certain flux surface and toroidal position and the same field line after one transit around the torus is given by δθ, then the rotational transform for that flux surface is defined as the average difference of poloidal angle δθ for an infinite number of transits around the torus. The rotational transform is therefore a measure of the pitch of field lines on a specific flux surface and is related to the safety factor by q =2π/ι. The shear measures the difference of the pitch of the magnetic field lines from one flux surface to the next. It is an important factor for stabilization of MHD instabilities. Especially those driven by pressure gradients. The magnetic shear is related to the safety factor by χ = q(r) /q(r). In general, the nested flux surfaces of a toroidal pinch device do not share a common geometrical center. This is because the flux surfaces are shifted outwards. The shift depends on the inverse aspect ratio (ɛ = a/r), the current distribution and the β parameter. In such a configuration, any calculations would involve the dependence of two coordinates, which would greatly complicate the analysis. However, if the aspect ratio is large, or if the β parameter is small, then the outward shift is small. In this case the plasma can be approximated by a straight cylinder with the cylindrical coordinates (e r, e θ, e z ). One of the most successful devices, in terms of achieving parameters for controlled thermonuclear fusion, is the tokamak. Other fusion concepts that have made interesting progress during the past decade, is the stellarator, the spheromak and the RFP. Although the RFP is at a relatively early stage of development, it has several advantages. 1.2 The reversed-field pinch The RFP is an axisymmetric toroidal magnetic fusion device. The plasma is confined by a toroidal magnetic field partly generated by external coils, and a poloidal magnetic field due to a toroidal current flowing in the plasma. Unlike tokamaks, the ratio of the toroidal current to the toroidal magnetic field is relatively high in an RFP. Furthermore, the stability properties of the RFP are favourable due to the strong shear in the magnetic field, i.e. the toroidal magnetic field reverses sign at the edge. These stability properties can be maintained, even at a very high β value (plasma pressure/magnetic field energy density). The reversed field state is reached spontaneously and can be maintained against resistive diffusion given enough applied toroidal voltage. The reversed toroidal field is sustained by a combination of the paramagnetic [1] effect due to the helical form of the current flowing parallel to the magnetic field and a special phenomenon that appears in the RFP called the dynamo. However, the dynamo is essential for the RFP state, since the field reversal cannot be obtained solely by the paramagnetic effect.

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17 Chapter 2 Mode dynamics 2.1 Taylor relaxation and the dynamo Taylor relaxation is the dissipative process by which the plasma relaxes towards a minimum energy state [2, 3] by converting poloidal magnetic flux into toroidal magnetic flux and thereby forming a reversed field configuration. By formulating a theory for plasma self-organization, Taylor predicted the existence of force free relaxed states [2]. This formulation is based on the assumption that the plasma naturally reaches the lowest energy state under the constraint that the toroidal flux and the linkage of the toroidal and poloidal flux, the total magnetic helicity, remains constant. The profiles corresponding to this state can be calculated by solving B = µb (2.1) Where µ is a constant. µ is known as the normalized parallel current density and is given by µ = µ 0 J B/B 2. The solution to this equation in cylindrical geometry is given by Bessel functions where B z /B 0 = J 0 (µr), B θ /B 0 = J 1 (µr) and B r /B 0 =0. Here, B 0 is the toroidal field at the magnetic axis, and J 0 and J 1 are the zeroth and first order Bessel functions respectively. These solutions are the so called Taylor states and are generally referred to as the Bessel Function Model (BFM) profiles. Two basic global parameters which are useful for experimental studies of relaxation are the reversal parameter, F = B φ (a)/ <B φ > and the pinch parameter, Θ=B θ (a)/ <B φ >. In experiments, the F and Θ values deviate from those based on Taylor relaxation given by the BFM (see 2.1). This is because full relaxation in the strict sense of a closed system cannot be reached in the experiments. However, the macroscopic features of the RFP state show a close resemblance to the Taylor state. The main reasons why only partial relaxation is obtained in RFP experiments are the following: (i) experimental RFP states are not closed, i.e., they correspond to ohmic states with a relatively long term dissipation of energy; and (ii) the fully relaxed state requires a constant µ which is not physically possible considering that the edge of the plasma is cooler than that closer to the axis resulting in a higher resistivity at the edge. This will in turn lead to a peaking of the radial current profile, which is inconsistent with a constant value of µ. and (iii) the state of minimum energy is force free [4], i.e. it is pressureless and therefore requires β =0 whereas in reality, plasmas have a pressure gradient. Other models have been developed to include the effects of the pressure gradient and the edge effects. These are the Modified Bessel Function Model (MBFM) [5], the (α, Θ 0,β 0 ) model [6], the Polynomial Function Model (PFM) [7] and the Modified Polynomial Function Model (MPFM) [8]. One of the most commonly used models is the (α, Θ 0,β 0 ) model [6]. In this model, the current profile is no longer assumed to be constant. Instead it is approximated in terms of the normalized 7

18 8 Chapter 2. Mode dynamics Figure 2.1. Equilibrium parameters F and Θ based on the BFM model (full drawn line) and corresponding experimental values based on Extrap T2 data ( ). radius r/a as µ(r) =2θ 0 [1 (r/a) α ]/a, whereθ 0 = aµ(0)/2. Furthermore, the model assumes a finite pressure of the equilibrium. By specifying µ(r) and p(r), profiles for current densities and magnetic fields can be determined for a certain equilibrium from the equilibrium equations (Eqns. 1.3 to 1.5) and Eqn In Fig. 2.2, several profiles are plotted based on the (α, Θ 0,β 0 ) model as a function of minor radius (r). These include the toroidal and poloidal magnetic fields, the toroidal and poloidal current densities as well as the current densities parallel and perpendicular to the magnetic field. Plasma relaxation requires magnetic reconnection. The dynamo process requires fluctuating fluid velocity (ṽ) and magnetic fields ( b). These requirements are provided by the nonlinear evolution of unstable resistive magnetohydrodynamic (MHD) modes. The magnetic perturbations are conveniently expressed in terms of Fourier harmonics of the form b(r, t) =b(r)exp[γt + i(mθ + nφ)] whereγ is the growth rate, m is the poloidal mode number and n is the toroidal mode number. Figuratively speaking, the helical perturbation associated with these modes produces a flow necessary to tilt the poloidal magnetic field into the toroidal direction, thus converting poloidal flux into toroidal flux. This conversion is described in Ref. [9] in terms of the spatial average of Ohms law and distinguishing between the mean and fluctuating parts of the electric field E 0 = E f +ηj 0. E f =ṽ b is then the mean part of the electric field produced by the fluctuating velocity (ṽ) and magnetic fields ( b), which are associated with the tearing instabilities driving the dynamo. The net effect of E f is to suppress axial current on the axis and to drive poloidal current at the edge. In this way, the resistive diffusion of the negative toroidal field at the edge which gives rise to peaking of the radial current profile is counteracted, resulting in a flattening (relaxation) of the current profile. By reducing parallel current gradients, marginally stable current profiles are maintained. Experimentally, the dynamo and relaxation has been observed in many devices. The dynamo may be observed in terms of oscillations in several parameters such as F and Θ [10 14], corresponding to a cyclical peaking and flattening of the µ profile. An example of discrete dynamo oscillations in T2R are shown in Fig. 2.3.

19 2.1. Taylor relaxation and the dynamo 9 Figure 2.2. Typical parameter profiles based on the (α, Θ 0,β 0 ) model. The underlying mechanism of the dynamo effect has been discussed in several papers based on code simulation studies. Sykes and Wesson first demonstrated that the dynamo mechanism could be obtained in a three-dimensional simulation of the nonlinear evolution of the m=1 resistive kink instability in the presence of externally applied toroidal voltage [15]. Nebel et al. [16] state that a few linearly unstable m =1modes are responsible for the reversal mechanism whereas the m =0modes play a minor role. However, three dimensional MHD simulations performed by Kusano and Sato [17, 18] showed that field reversal is essentially due to a combined effect of the m =0and m =1modes. The relaxation process occurs as a result of nonlinear reconnection [19] of the m = 0 mode which is driven by nonlinear coupling between unstable m = 1 tearing modes. Holmes et al. [20] identified two steps that were involved in the relaxation. The first process was an m =1spectral broadening towards high n resulting in a radial expansion of the zone of turbulence from the plasma core towards the field reversal surface, whereas in the second process, the unstable m =1modes were stabilized by a cascade of energy to small scales. Ho and Craddock [21] later gave a detailed description involving the quasilinear, nonlinear and diffusion effects on the dynamo.

20 10 Chapter 2. Mode dynamics Figure 2.3. Example of discrete dynamo oscillations in the average toroidal field, F, Θ and the soft x-ray signals for a T2R discharge. 2.2 MHD stability If a plasma is perturbed from its equilibrium state, the resulting perturbed forces will either act to restore the plasma to the original state (stability) or to enhance the perturbation further (instability). The basic destabilizing forces in an MHD plasma arise from current gradients and pressure gradients combined with unfavourable magnetic field curvature. A magnetically confined plasma inevitably has nonuniformity of current density and pressure leading to destabilization of MHD modes. Current driven modes originate from the property that parallel currents attract each other. These are often known as kink modes and exist even in a zero-pressure force-free plasma. The destabilizing effects of the current driven and pressure driven modes may be counteracted by several magnetic field properties. One of these properties is magnetic field line bending. A perturbed magnetic field acting in the direction perpendicular to the equilibrium magnetic field causes the field lines to bend, giving rise to a stabilizing force. Furthermore, compression of magnetic field lines as well as good magnetic field curvature also have stabilizing effects. The curvature is good if the center of curvature is in the opposite direction of the plasma pressure gradient. In a magnetically confined plasma, the stabilizing properties of magnetic field line bending are minimized in locations where the pitch of the helical mode pertubations is the same as that of the equilibrium field B = B θ e θ + B φ e φ. On these so called resonant surfaces, the wave vector of the perturbed field k = k θ e θ + k φ e φ = m r e θ + n R 0 e φ is perpendicular to the equilibrium magnetic field, where m and n are the poloidal and toroidal mode numbers respectively. This means that

21 2.2. MHD stability 11 Figure 2.4. A typical q profile based on Extrap T2 parameters. the perturbation wave fronts are in the direction along the magnetic field line and therefore do not give rise to magnetic field line bending. As a consequence, unstable displacements are localized near surfaces r = r s satisfying the condition Eqn. 2.2 then gives the expression k B 0 = m r B θ + n R 0 B φ =0 (2.2) rb φ R 0 B θ = m n = q(r) (2.3) where q is the safety factor. In an RFP, the safety factor is a decreasing function of the minor radius and changes sign at the edge of the plasma. This is due to the reversal of the toroidal field at the edge. The on-axis q value is around a/2r 0. A typical q profile for an RFP is shown in Fig Field reversal at the edge is crucial for maintaining stable RFP configurations since the q profile would otherwise have a minimum. A minimum in q would result in zero magnetic shear in this location which would lead to current-driven and pressure-driven internal MHD instabilities. Resonant surfaces arise for rational values of q. Therefore, these surfaces are also called rational surfaces. A mode that is resonant on the rational surface q = m/n traverses m toroidal turns for every n poloidal turns. The modes that are most likely to become unstable are those with long wavelength (low m and low n) where the m =0and m =1modes dominate the dynamics in the RFP. A mode that has a resonant surface in the plasma, i.e. that has a helical pitch that corresponds to that of the equilibrium field somewhere in the plasma, is called a resonant mode. Thus, a nonresonant mode refers to a mode that does not have a resonant surface in the plasma. Another distinction is made between internal and external modes. Internal modes have the same helical pitch as the equilibrium field inside the reversal surface whereas the pitch of the external modes corresponds to the equilibrium field outside the reversal surface. According to the definition of the coordinate system used here (see Fig. 1.1), the m =1internal modes have toroidal mode

22 12 Chapter 2. Mode dynamics Figure 2.5. Stability diagram for m =1modes in the α Θ 0 plane. Experimental values calculated from T2R data is also shown ( ). numbers n = 1/q(r) < 0 whereas the external modes correspond to toroidal mode numbers n =1/q(r) > 0. The m =0modes are resonant at the reversal surface. Instabilities are referred to as ideal or resistive. Ideal instabilities apply to perfectly conducting plasmas. This distinction is important since plasmas with even a small amount of resisitvity can alter the magnetic topology of the plasma and thereby change the stability properties. The ideal instabilities remain, but additional, resistive modes, are then introduced into the plasma. A stability diagram is plotted in Fig. 2.5 showing the stability regions of the various modes in the α Θ 0 plane. In a perfectly conducting plasma, the plasma fluid moves precisely with the magnetic field. In this case the magnetic field lines are said to be frozen in the plasma, keeping the magnetic helicity invariant. However, if a resistive term is introduced, the magnetic field lines will be allowed to reconnect across the rational surfaces and thereby form so called magnetic islands. The dominant resistive instability in the RFP is the tearing mode, which is the resistive form of the kink mode. The name tearing refers to the tearing of the magnetic field line at the rational surface. Magnetic islands may grow and overlap with magnetic islands formed by other resistive modes. In this case, traces of the magnetic field lines can fill the resulting volume formed by the overlapping magnetic islands, and are not restricted to a single rational surface. The magnetic field lines are then stochastic in this volume. Magnetic island growth implies growth of the perturbed radial magnetic field and is therefore limited by the resistive diffusion of this field according to B t = η µ 0 2 B (2.4) which is the resistive diffusion equation derived from Maxwells equations and Ohms law. The growth rate of the island depends on the resistivity as well as the width of the diffusion region. The growth rate of a tearing mode can be expressed in terms of two basic time scales. The first is the Alfvén time, τ A = a µ 0 ρ/b 0 where ρ is the plasma density, a is the minor radius and B 0 is the magnetic field. This time corresponds to the time it takes for an Alfvén wave to travel the

23 2.4. Tearing mode dynamics 13 distance a, from the center of the plasma to the edge. The Alfvén time determines the growth rate of ideal modes. The second time scale is very long in comparison to τ A, and is called the resistive diffusion time τ R = µ 0 a 2 /η, whereη is the resistivity of the plasma. The tearing mode grows on a combined time scale τ tear = τ 3/5 R τ 2/5 A which is slower than the Alfvén transit time scale but faster than the resistive diffusion time scale. The ratio between the resistive diffusion time and the Alfvén time gives the Lundquist number, S = τ R /τ A [22]. 2.3 RFP stability Conventional RFP plasmas are surrounded by a close-fitting perfectly conducting shell, with a shell penetration time that is longer than the pulse duration time. RFPs depend on the conducting shell for field reversal since the flux conserving properties of the shell are needed for the relaxation process to occur. Furthermore, conducting shells can stabilize a wide range of MHD modes. On time scales longer than the magnetic penetration time of the shell these modes become resistive wall modes. These are the m=0 modes, internal and external nonresonant modes, as well as the external resonant modes [23 26]. However, the m =1internal resonant tearing modes are marginally stable with a conducting shell and are further destabilized when introducing resistive wall boundary conditions. These modes are responsible for driving the dynamo (described in section 2.1) which sustains the RFP discharge against ohmic decay [27, 28]. Both plasma rotation and the location of the shell with regard to the plasma edge are important factors for MHD stability [24 26]. If a mode is rotating with a rotation frequency that exceeds the inverse shell penetration time, the resistive shell will behave effectively as an ideal shell as far as this mode is concerned. Therefore, as long as the mode is rotating with sufficient rotation velocity, the stability of this mode will be essentially unaffected by the shell resistivity [26,29,30]. Resonant modes often rotate since the rotating plasma exerts a viscous force on the rational surfaces of these modes. Thus, both the m =0and m =1, resonant modes may be stabilized by mode rotation in combination with a relatively close fitting liner. The nonresonant resistive wall modes however, are not affected unless the rotation is some fraction of the Alfvén velocity [23]. These m =1 internal nonresonant and m =1,low-n external nonresonant resistive wall modes pose a potential problem for future long pulse operation. The shell vertical field penetration time also determines the time scale for equilibrium control. Therefore, in the trade-off between the requirements of equilibrium control and MHD stability an optimal range for the shell time scale is needed. This time scale should however be long enough to allow for the RFP to be established. This means that the shell time should be longer than the dynamo relaxation cycle time as well as the start up time of the RFP. An added factor in this tradeoff is the possibility of compensating for the degraded MHD stability introduced when the shell is resistive by using an extended coil feedback stabilization system. 2.4 Tearing mode dynamics The dominant tearing modes in RFP plasmas are the m =1core resonant modes with resonant surfaces inside the reversal surface. These modes are responsible for the dynamo sustainment of the RFP [27, 28]. Furthermore, the internal resonant modes reconnect magnetic field lines and generate overlapping chains of magnetic islands. These magnetic islands stochasticize the magnetic field which leads to degraded plasma confinement [32]. The innermost resonant tearing mode has a toroidal mode number corresponding to about n 2R 0 /a. The number of unstable tearing modes in an RFP plasma depends to a great extent on the aspect ratio (R 0 /a) of the device. If the aspect ratio is high, more modes will tend to be unstable [33]. Resonant modes often co-rotate with the plasma fluid [34 36]. The flowing plasma exerts a

24 14 Chapter 2. Mode dynamics Figure 2.6. The time evolution of a locked mode in T2. The calculations of the last closed flux surface shift are made using a model described in Ref. [31]. viscous torque on the rational surfaces of the modes, thereby causing them to rotate. The plasma flow is maintained by the diamagnetic and E B drifts. Mode rotation is generally not observed in resistive shell experiments. Instead, the tearing modes are wall locked. This was the case in the thin shell device T2 [37]. In T2R however, fast spontaneous rotation of the internal resonant tearing modes characterizes the mode dynamics in this device (Paper V). In practically every RFP experiment the m =0modes and the m =1core tearing modes phase lock together to form a toroidally localized structure in the perturbed magnetic field typically known as a "slinky" mode [38]. The phase locking occurs as a result of three wave nonlinear coupling between m = 1 modes and the m = 0 modes [39 42]. The modes are thought to phase lock in such a way that the magnitudes of the electromagnetic torques exerted on the various resonant surfaces of the modes are minimized [43, 44]. The slinky mode was first observed in the OHTE [45] experiment and was named after the spring toy with the same name. Observations of slinky modes have been made in several experiments since then [14, 46 48]. The slinky may be either stationary in the laboratory frame (wall locked) or rotating. Wall locked slinkys are generally referred to as wall locked modes. The localized nature and the amplitudes of these perturbations may result in problems associated with plasma wall interaction and wall loading problems leading to influx of impurities in the plasma. This was observed in both T2 [37] and RFX [47]. The time evolution of a typical wall locked mode in T2 is shown in Fig In papers I and II, detailed descriptions of the dynamics of the locked mode in TPE-RX are given. Paper III gives a further analysis of locked modes and provides a comparison of the locked mode phenomena in TPE-RX and T2. Wall locking may occur as a result of locking of tearing modes to a static external field error [34,49]. The field errors interact with the rotating helical perturbations in the plasma producing an electromagnetic force on these which opposes the viscous torque produced by the plasma flow and causes the mode rotation to arrest. The role of field errors in the mode locking process has

25 2.4. Tearing mode dynamics 15 Figure 2.7. An example of a rotating slinky mode in T2R observed in the inverse dispersion of phases. been analyzed in several studies [35, 46, 50, 51]. In conducting shell RFPs, locking at the gap is normally observed due to field errors. For instance, in the conducting shell RFP TPE-RX, locking occurs at the shell gap with a probability of 18 26% (Paper I). In addition to field errors, wall locking may occur in the presence of a resistive vacuum vessel. Eddy currents in the vessel may produce a retarding torque on the plasma which causes the mode rotation to slow down [29,34,52]. Furthermore, a resistive vessel also has the effect to lower the field error locking threshold [49]. Therefore, the combined effect of eddy currents induced in the resistive vessel and field errors due to imperfections in the shell often results in wall locking. The importance of the retarding torque from eddy currents in the vacuum vessel and its relation to the threshold values of the mode amplitude for wall locking was studied in Ref. [52]. The critical threshold values depend on the size and details of the shell configuration of the experiments as well as on plasma parameters, which in turn exerts limits on RFP operation if mode locking is to be avoided. These issues are discussed further in papers II and III. Phase locking among modes implies that the differences of the phases of adjacent modes

26 16 Chapter 2. Mode dynamics should have a constant value [43, 44]. This constant value should correspond to the position of the locking of the slinky according to α m,n α m,n+j = jφ lock m (2.5) where α m,n is the phase of the Fourier mode amplitude with poloidal mode number m and toroidal mode number n. φ lock m corresponds to the locking position of the slinky. This condition should be valid even if the modes are rotating. If this is the case, the slinky has a toroidal rotation velocity corresponding to ω lock m = Ω m,n Ω m,n+j j (2.6) where Ω m,n is the angular phase velocity of the mode with poloidal mode number m and toroidal mode number n. The angular phase velocity of the slinky is ωm lock. It is clear from Eqn. 2.6 that if the modes that form the slinky rotate with the same angular phase velocities, the slinky mode will be stationary [44]. Thus, mode rotation does not necessarily imply slinky mode rotation. The slinky mode formation and the three wave relation between m =0and m =1 modes are further discussed in paper VIII. If the modes are rotating, problems associated with plasma wall interaction can to a great extent be avoided. This is due to two reasons. First of all, with the rotation, the effect of the perturbation is not localized in one spot. Secondly, the rotating modes do not reach the same radial field amplitudes as in the locked case since they are stabilized by the mode rotation [26, 30]. A typical example of a rotating phase locked slinky mode is reflected in the analysis of the inverse dispersion of phases (1/σ) in Fig The inverse dispersion of phases is a measurement of the degree of phase alignment between the modes. A high value therefore reflects a high degree of phase alignment Regimes for magnetic fluctuation reduction and improved confinement Magnetic fluctuations account for most of the energy losses in the plasma core. Growth of the internally resonant modes leads to overlapping of magnetic islands which results in a stochastic core. Parallel particle and energy transport along stochastic magnetic field lines is a significant source of power loss and limitation to the confinement properties in an RFP. Many studies have focused on the issues of fluctuation reduction. One method that has been developed for this purpose is called pulsed poloidal current drive (PPCD). The aim of PPCD [53] is to modify the parallel current gradients in order to reduce the free energy for internally resonant tearing instabilities. The fluctuation reduction thereby leads to improved confinement. The PPCD technique was originally developed in the MST RFP [53]. Since then the method has been successfully implemented in a number of devices [54, 55] and progress has been made on the development of this technique [56 59]. The current profile control is obtained by driving an externally applied poloidal electric field at the edge. The dynamo electric field acts in such a way that it flattens the current profile by supressing the parallel current at the center and driving current at the edge. By inducing a poloidal current at the edge, the dynamo electric field required to flatten the profile is reduced. In this way the tearing mode fluctuations are reduced resulting in improved confinement. Recently, PPCD experiments have been performed in Extrap T2R [60]. A reduction of the m = 1 internally resonant tearing modes is observed during the PPCD phase. An increase in angular phase velocities of these modes is also observed and can be attributed to the increased central

27 2.4. Tearing mode dynamics 17 electron temperatures in the core that result from the PPCD operation. An increase in poloidal β as well as doubled energy confinement times are obtained. In Fig. 2.8, the soft x-ray signals, root mean square (rms) amplitudes of the internally resonant tearing modes and the corresponding average angular phase velocities are shown for discharges in Extrap T2R with PPCD ( ) and without PPCD (- - -). A reduction in tearing mode amplitudes is observed during the PPCD phase. The strongly increased soft x-ray signal is typical during PPCD. Figure 2.8. Soft x-ray signals (top), rms amplitudes of the internal resonant m =1modes (center) and corresponding angular phase velocities for typical discharges with ( ) and without (- - -) PPCD in Extrap T2R. Another approach to reducing magnetic fluctuation levels is based on the theoretical prediction that the RFP plasma can spontaneously reach a less chaotic magnetic configuration by means of a self-organization process called single helicity [61 63]. The transition of the plasma from multiple helicity states to single helicity states has been shown in 3D simulations [64,65]. Ideally, the selforganization is a result of a transition to a state in which the mode spectrum of the magnetic configuration is dominated solely by one helical mode. In experimental observations made so

28 18 Chapter 2. Mode dynamics far [58, 66 69] however, perfect single helicity states are not observed, instead, so-called quasi single helicity states are reached. Soft x-ray tomographic imaging revealed the formation of a helical coherent structure in the core during the quasi single helicity state [70]. The helical structure had closed helical flux surfaces and the same pitch as the dominant mode. The mappings of soft x-ray emissivity taken in multiple helicity and single helicity plasmas turned out rather different. In the multiple helicity case, a poloidally symmetric emissivity was observed in the plasma core, consistent with a region of strong heat transport. In the single helicity case however, a bean shaped hot structure in the core was evident. Experimental measurements have shown that this helical coherent structure in the core is a region of reduced transport [71]. Quasi single helicity states are often observed in combination with PPCD. An example of quasi single helicity during PPCD in Extrap T2R is shown in Fig. 2.9 Figure 2.9. The time evolution of F (top), soft x-ray (center) and tearing mode amplitudes normalized to the poloidal magnetic field at the edge (bottom) for a PPCD discharge in Extrap T2R. Quasi single helicity is observed during PPCD (marked with vertical dashed lines) with the m = 1, n = 14 mode dominating the mode spectrum. During the current ramp down phase, another mode (m =1, n = 13) becomes dominant.

29 2.5. Resistive wall modes Resistive wall modes As mentioned in section 2.3, RFPs operating with a perfectly conducting shell sufficiently close to the plasma can be stable to ideal and resistive current driven MHD modes [24,72]. In the presence of a finite conductivity (resistive) wall, the stability boundaries are changed. Modes that would be stable if the wall were perfectly conducting are destabilized. These modes are called resistive wall modes [24 26]. Several MHD modes can be stabilized by a resistive shell and moderate mode rotation. However, the nonresonant resistive wall modes cannot be stabilized by plasma rotation unless the rotation is some fraction of the Alfvén velocity [23]. Therefore, other means of stabilizing these internal and external nonresonant resistive wall modes have to be implemented. Linear MHD calculations have shown that internal and external nonresonant resistive wall modes grow on the time scale of the shell penetration time τ w [23, 25, 29]. These growth rates depend on the plasma equilibrium. At higher values of pinch parameter Θ, corresponding to deeper reversal of the toroidal magnetic field at the edge, the external nonresonant modes are expected to be more unstable whereas for lower values of equilibrium parameters, the growth rates of the internal nonresonant ideal modes dominate [29, 73]. Experimental results on RFP operation with a resistive shell are somewhat limited. In the HBTX-1C resistive shell RFP, wall locked perturbations were observed to grow to large amplitudes on the wall time scale, resulting in fast discharge termination [73]. These observations were thought to be due to a combination of internal resonant tearing modes and the ideal resistive wall modes predicted by linear stability theory [23]. In the Reversatron RFP, observations of resistive wall modes were also reported [74]. However, in the Extrap T2 [37] thin shell RFP, formerly known as the ohmic heating toroidal experiment (OHTE) [45], resistive wall mode activity was not directly observed. Instead, the devices displayed pulse lengths greatly exceeding the wall time [37, 38, 75]. Supposed explanations for this was the formation of wall locked slinky modes. It is feasible to believe that the equilibrium was such in these devices, that the internal nonresonant resistive wall modes dominated with only low growth rates of the external nonresonant resistive wall modes. Nonlinear interaction between the internal nonresonant modes and the wall locked internal resonant modes was suggested as a possible explanation for the differing observations in Extrap T2 and OHTE [38] compared to other thin shell devices. Furthermore, the similar time evolution of the internal resistive wall modes and the stationary, unstable internal resonant modes rendered the distinction between these modes difficult. This implies that the possible role of the internal resistive wall modes in T2 cannot be entirely discarded. These issues are addressed further in papers III and V. In the rebuilt configuration of Extrap T2, Extrap T2R, the nonresonant ideal resistive wall modes have been identified (Papers V, VI and VII). The possibility of identifying the on-axis resistive wall mode in T2R was due to fast spontaneous rotation of the internal resonant tearing modes in contrast to the typical wall locked mode scenario in T2. Rotation suppresses the radial magnetic field amplitudes of these modes at the wall. These low amplitude, rotating modes are therefore distinguished from the stationary internal nonresonant modes. A demonstration of this distinction is shown in Fig where the radial magnetic field amplitudes and angular phase velocities of an internal resonant (n = 12) and nonresonant (n = 10) mode for a typical T2R discharge are plotted. The stationary nonresonant mode grows exponentially throughout the discharge. However, the radial amplitude of the resonant mode is suppressed as long as the rotation frequency is sufficiently high. In agreement with linear theory [29, 73], the growth rates of the external nonresonant resistive wall modes are low in T2R. In paper VII, the experimental growth rates of the resistive wall modes in T2R are studied and compared with growth rates obtained from linear MHD theory. A reasonable agreement is found. The sensitivity of the growth rates of the nonresonant resistive wall modes can be demonstrated

30 20 Chapter 2. Mode dynamics Figure (a) Plasma current of a typical T2R discharge. (b) Amplitude of the internal nonresonant on-axis RWM, m = 1,n = 10 ( -) and an internal resonant tearing mode m = 1,n = 12 (- - -). (c) Helical angular phase velocity (Ω 1,n )ofthem =1internal nonresonant on-axis RWM, m =1,n= 10 ( -) and an internal resonant tearing mode m =1,n= 12 (- - -). in individual discharges. In Fig. 2.11, the time evolution of the radial amplitude of the m =1, n = 10 mode is shown for a discharge with PPCD (- - -) and without PPCD ( ). The growth rate of the internal resistive wall mode is clearly affected by the strong equilibrium change produced by the PPCD pulse. It should be noted however, that such an equilibrium change acts to increase the growth rate of the external nonresonant modes Feedback stabilization of resistive wall modes Nonresonant resistive wall modes are largely unaffected by plasma rotation unless the rotation velocities are some fraction of the Alfvén wave velocity [23]. The unstable nature of these modes implies that some kind of method to control the resistive wall modes is necessary in order to achieve long pulse operation. Active feedback stabilization of MHD modes is one potential method to improve the global plasma performance in toroidal devices. An elegant idea for stabilizing the perturbations was discussed by Bishop [76] who suggested the idea of an intelligent shell. The intelligent shell concept is a system where the boundary surface of the torus is fully covered by an array of active magnetic saddle coils, each with a dedicated sensor that measures the local magnetic field perturbation. There is a closed-loop feedback system between the sensor and the saddle coil that freezes the flux through the coil thus acting as an element of an intelligent (ideal) shell. In this way, resistive wall modes can be stabilized as well as

31 2.5. Resistive wall modes 21 Figure The time evolution of the pinch parameter Θ (top) and the n = 10 radial mode amplitude for a discharge with PPCD (- - -) and without PPCD ( ). The vertical line shows the time when PPCD begins. A clear difference in growth rate of the internal nonresonant mode can be observed due to the strong equilibrium change due to PPCD. other types of instabilities. Since then, practically realizable systems based on this concept have been explored both theoretically [77 80] and experimentally [81, 82]. In RFP configurations, active feedback schemes are complicated by the relatively large number of potentially unstable MHD modes. Two main concepts for feedback stabilization have been studied. The first concept [77] is based on especially designated helical feedback coils for each potentially unstable mode. This concept was successfully tested in the HBTX-1C resistive shell device [81, 82], concentrating solely on one single mode. However, the requirement of full stabilization of the revesed field configuration would require several helically winded coils (two for each mode). Given the amount of modes to feedback stabilize, as well as the inflexibility in using especially designated coils, this idea would be rather impractical. The second concept is a simplified version of the intelligent shell concept using a discrete set of equally distributed feedback controlled coils located outside the plasma vessel (the intelligent shell idea would require an infinite amount of such coils). In Ref. [78], successful feedback stabilization of multiple resistive wall modes in various RFP equilibria was demonstrated. In this work, a finite number of radial coil sensors in the toroidal direction were used. However, the problem was simplified by assuming a large number of coils in the poloidal direction so that coupling of different poloidal harmonics by the feedback currents could be neglected. The coupling occurs as a result of using a discrete set of coils. The current in the control coils will in this case give rise to an infinite number of harmonics corresponding to multiples of the number of active coils in poloidal or toroidal directions. This is the so-called sideband effect. In Ref. [78] it was also shown that the minimum amount of coils N, in the toroidal direction has to be at least greater than the range of the unstable resistive wall mode toroidal numbers n (N >n max n min ) in order to avoid

32 22 Chapter 2. Mode dynamics Figure The radial magnetic field signals measured on the side of the torus at 180 toroidal angle. Plasma current (top), radial magnetic field signals (center), and the corresponding current in the active feedback coil (bottom) for pulses with feedback ( ) and without feedback (- - -). coupling of unstable sidebands. In Ref. [80], this work was extended to include discrete coils also in the poloidal direction. The effect of coupling of different Fourier components from the discrete coils were discussed. Issues regarding feedback stabilization using sensors for the three different magnetic components as well as suitable geometrical dimensions for these coils were also studied. The best performances in terms of critical gain were found for toroidal and poloidal field sensors. The performance of the radial sensors was however found to improve if the feedback coils were slightly overlapping in toroidal direction [78, 80]. Experimental feedback stabilization of the m =1, n =2external nonresonant kink mode was demonstrated in the HBTX-1C thin shell RFP [81,82]. The mode was successfully stabilized, however, improved global plasma performance was not observed. It was suggested that this was due to the growth of the internal resonant modes. The Extrap T2R resistive shell RFP has many advantages for studying active feedback of resistive wall modes. Most importantly, the dominating radial field perturbations at the wall in T2R are due to resistive wall modes. This is partly due to the fact that the resistive shell penetration time is long compared to the time scale of the dynamo and relaxation events so that mode activity associated with these events is practically unaffected by the finite resistivity of the shell. Instead, the internal resonant modes exhibit fast spontaneous rotation. Other important advantages include the comprehensive set of magnetic diagnostics in T2R as well as a relatively good access to the shell surface. Active feedback coils can therefore be fitted onto the torus structure relatively easily. The favourable conditions for studying resistive wall mode dynamics and active feedback in T2R underline the importance of performing active feedback experiments on this device. Active feedback experiments have been initiated in T2R. The feedback concept which is being

33 2.5. Resistive wall modes 23 Figure The thermo couple signals measured for pulses with feedback ( ) and without feedback (- - -). The feedback coil was in this case placed on the side of the torus at 180 toroidal angle. This is a position where field errors are always present, leading to local heating of the vessel. implemented is based on arrays of radial magnetic field sensor coils that provide the signal to the feedback controller. These sensor coils are located on the outer surface of the vacuum vessel inside the thin shell (see Chapter 3.4). Active saddle coils placed outside the resistive shell produce the externally generated feedback field. In Fig. 2.12, magnetic field sensor coil signals are shown for pulses with ( ) and without (- - -) feedback at the toroidal position 180. In this location, field errors are typically present, giving rise to local heating of the vessel. These temperatures are measured by thermocouples placed on the outer surface of the liner. In Fig. 2.13, the temperature measurements are shown for the discharges in Fig with ( ) and without (- - -) feedback at 180 toroidal angle. The temperatures are lower at this position when the feedback is activated. Since the aim is to feedback stabilize the m =1modes, these sensor coils may be connected in series, thus providing an m =1signal to the controller. In this way, the amount of required feedback controllers and amplifiers are reduced. An example of first trials with feedback coils connected in series is shown in Fig The m =1signals are shown for pulses with ( ) and without feedback (- - -). The corresponding current in the active feedback coil shows the response of the controller to the input signal measured by the sensor coils. The signals are significantly lowered. The radial signals without feedback are largest towards the end of the discharges. This may be attributed to the growth of resistive wall modes. Feedback stabilization of the external and internal nonresonant modes in T2R will require a large set of active coils evenly distributed around the torus. A full coverage would correspond to active coils located at 4 poloidal x 32 toroidal positions. This type of configuration would enable feedback on m = 1 modes with different helicity, thereby avoiding the problem of coupling of internal modes to external modes. However, one of the interesting issues in this study is to determine whether partial coverage is sufficient for stabilization of these modes. Further ideas involve the investigation of different coil geometries in order to find the most efficient feedback system.

34 24 Chapter 2. Mode dynamics Figure The plasma current (top), m =1radial magnetic field signals (center) and the corresponding current in the active feedback coil for pulses with feedback ( ) and without feedback (- - -).

35 Chapter 3 Description of experiments This thesis covers work performed on three different RFP experiments, Extrap T2 [37], Extrap T2R (paper IV) and the Toroidal Pinch Experiment (TPE-RX) [83]. Extrap T2 is a medium-sized resistive shell experiment operated at the Royal Institute of Technology in Stockholm. Extrap T2R is a rebuilt configuration of the Extrap T2 experiment. The dimensions of the experiment are the same as Extrap T2, however, the device is now operating in different parameter regimes and several improvements have been observed in the operation of the new device (paper IV). TPE-RX is a relatively large RFP operated at the Electrotechnical Laboratory (AIST, Tsukuba) in Japan since The characteristics of the three devices are summarized in table Extrap T2 Extrap T2 [37] is a medium sized RFP with major radius R=1.24 m and minor radius a= T2 was previously operated as the Ohmically Heated Toroidal Experiment (OHTE) [45] at the General Atomic company in San Diego but was later disassembled and moved to the Royal Institute of Technology in Stockholm. The device is provided with a thin brass shell (0.8 mm thick) with a field penetration time of about 1.5 ms (about 1/10 of the pulse duration time). The shell has four poloidal and two toroidal gaps, all electrically shorted except for one of the poloidal gaps. The vacuum vessel is made of stainless steel. The plasma-facing inside wall is fully covered with graphite tiles and is not provided with any discrete limiters. The vertical field required for radial equilibrium is obtained by ohmic heating primary coils. Two interlaced helical coils with 16 poloidal and 3 toroidal turns each provide the toroidal field. This helical copper conducter is 10 cm wide and 4 cm thick covering about 50% of the torus surface. The handedness of the helical coil corresponds to the internal magnetic field, but the pitch is nonresonant corresponding to q = 3/16 = There are toroidal coils for the return current which also cancel the vertical field produced by the helical coil. 3.2 Extrap T2R Extrap T2 was shut down in the spring of 1999 for a complete change of the front-end system. In September 2000, operation resumed on the device which was thereafter called Extrap T2R (paper IV). One of the main concerns in thin shell RFP operation is that modes lock to the wall forming localized slinky structures. Detailed studies on the resistive shell device Extrap T2 of the wall locked modes showed that the mode caused localized plasma-wall interaction, intense heating of the graphite tiles and finally, influx of gas and impurities from the tiles, leading to discharge degradation [37, 48]. However, if favourable conditions for mode rotation can be achieved, it 25

36 26 Chapter 3. Description of experiments Extrap T2 Extrap T2R TPE RX a(m) R 0 (m) R/a b/a I φ kA kA kA n e m m m 3 T e eV eV eV T i eV eV eV τ E 20 60µs µs 0.5 1ms β θ < 10% 5 15% 5 10% V loop V 20 30V 15 30V Θ F τ pulse < 16ms < 22ms < 100ms τ w 1.5ms 6.3ms 330ms Shell Brass Copper Copper First wall material Graphite tiles Stainless steel (316L) Stainless steel (316L) Table 3.1. Typical parameters in TPE-RX, Extrap T2 and Extrap T2R. would be reasonable to assume that discharges may be successfully sustained. This was one of the main aims for the new configuration of the device. The main changes on Extrap T2 included the replacement of the graphite armour first wall with a metal first wall, the replacement of the helical coil with a conventional solenoid-type coil and, the replacement of the single layer thin resistive shell with a double layer shell corresponding to a four-fold increase of the shell time. The vacuum vessel of Extrap T2R still has the same global dimensions as the previous device. The vessel is made of stainless steel Type 316L which is protected by a total of 180 molybdenum limiters. These cover about 8% of the inner wall. The plasma boundary is thereby limited to a minor radius of m giving a shell proximity b/a of The thin resistive shell of Extrap T2 has been replaced by a double layered copper shell. Each layer is 0.5 mm thick giving a shell vertical field diffusion time of the double layer of about 6.3 ms. The thin shell layers are overlapped in such a way that the poloidal gaps are in opposite toroidal positions in order to reduce magnetic field errors. Each layer has one toroidal gap. These are placed on different sides to provide shielding of the toroidal gap error field. Since the vacuum vessel is only partly protected it has been necessary to avoid the risk of damaging the vessel. This may occur as a result of localized plasma heat flux due to locked modes or inaccurate vertical field control. Therefore, at present, operation at low plasma currents ( 100 ka) has been chosen on Extrap T2R. The densities are low, around n e = m 3. The loop voltage has decreased to below 30 V (Paper IV).

37 3.3. TPE-RX 27 Figure 3.1. The Extrap T2 reversed field pinch. 3.3 TPE-RX The Toroidal Pinch Experiment, TPE-RX [83] is a relatively large RFP device with an aspect ratio of 1.72m/0.45m=3.8. The maximum designed plasma current and the discharge duration time is 1 MA and 100 ms respectively. However, the present capacity of the device is limited to a maximum current of 0.5 MA and a maximum discharge duration time of 100 ms due to the available energy of the power supply. The device operates with line averaged electron densities in the range m 3 [84]. TPE-RX has a conducting shell system which consists of a 50 mm thick wall conducting aluminium shell with a penetration time τ 330 ms and a double layered resistive shell consisting of two 0.5 mm thick copper shells (with τ 10 ms for the double layer). The two thin shells and the thick shell overlap at the positions of the poloidal gaps to minimize the poloidal gap field error. The shell proximity, b/a, for the inner double layered shell is 1.08, where b is the inner minor radius of the innermost conducting shell and a is the minor radius of the plasma boundary. The proximity of the thick shell is on average b/a =1.16. The plasma boundary is limited by a total of 244 mushroom shaped molybdenum limiters which are welded on the inside of the vacuum vessel. The vacuum vessel is made of stainless steel. A feedback controlled saddle coil (response time of 200 Hz) compensates for field errors at the thick shell gap. Furthermore, the center of the thick shell is shifted 17.5 mm from the center of the vacuum vessel and the thin shell so that the last closed flux surface is almost centered with respect to the vacuum vessel without the application of an external dc vertical field [85, 86]. In the discharges analyzed in this thesis, the dc equilibrium field was not applied. The toroidal angle is defined to increase in the direction of the plasma current, counter clockwise seen from above. The position of the thick shell gap is φ = 180