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Resonant magnetic perturbation effect on thetearing mode dynamics

Novel measurements and modeling of magnetic fluctuation induced momentumtransport in the reversed-field pinch

RICHARD FRIDSTROM

Doctoral ThesisStockholm, Sweden, 2017

TRITA-EE 2017:135ISSN 1653-5146ISBN 978-91-7729-549-5

KTH School of Electrical Engineering (EES)SE 1044 Stockholm

Sweden

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlagges till of-fentlig granskning for avlaggande av Teknologie doktorexamen i elektroteknik onsdagenden 13 december 2017 klockan 9.00 i Sal F3, Lindstedtsvagen26, Kungliga TekniskaHogskolan, Stockholm.

c© Richard Fridstrom, 13 December 2017

Tryck: Universitetsservice US AB

i

Abstract

The tearing mode (TM) is a resistive instability that can arise in magnetically confinedplasmas. The TM can be driven unstable by the gradient of the plasma current. When themode grows it destroys the magnetic field symmetry and reconnects the magnetic field inthe form of a so-called magnetic island. The TMs are inherentto a type of device called thereversed-field pinch (RFP), which is a device for toroidal magnetic confinement of fusionplasmas. In the RFP, TMs arise at several resonant surfaces,i.e. where the field linesand the perturbation have the same pitch angle. These surfaces are closely spaced in theRFP and the neighboring TM islands can overlap. Due to the island overlap, the magneticfield lines become tangled resulting in a stochastic magnetic field, i.e. the field lines fill avolume instead of lying on toroidal surfaces. Consequently, a stochastic field results in ananomalously fast transport in the radial direction. Stochastic fields can also arise in otherplasmas, for example, the tokamak edge when a resonant magnetic perturbation (RMP) isapplied by external coils. This stochastization is intentional to mitigate the edge-localizedmodes. The RMPs are also used for control of other instabilities. Due to the finite numberof RMP coils, however, the RMP fields can contain sidebands that decelerate and lock theTMs via electromagnetic torques. The locking causes an increased plasma-wall interaction.And in the tokamak, the TM locking can cause a plasma disruption which is disastrous forfuture high-energy devices like the ITER. In this thesis, the TM locking was studied in twoRFPs (EXTRAP T2R and Madison Symmetric Torus) by applying RMPs. The experimentswere compared with modern mode-locking theory. To determine the viscosity in differentmagnetic configurations where the field is stochastic, we perturbed the momentum via anRMP and an insertable biased electrode.

In the TM locking experiments, we found qualitative agreement with the mode-lockingtheory. In the model, the kinematic viscosity was chosen to match the experimental lock-ing instant. The model then predicts the braking curve, the short timescale dynamics, andthe mode unlocking. To unlock a mode, the RMP amplitude had todecrease by a factor tenfrom the locking amplitude. These results show that mode-locking theory, including therelevant electromagnetic torques and the viscous plasma response, can explain the experi-mental features. The model required viscosity agreed with another independent estimationof the viscosity. This showed that the RMP technique can be utilized for estimations of theviscosity.

In the momentum perturbation experiments, it was found thatthe viscosity increased100-fold when the magnetic fluctuation amplitude increased10-fold. Thus, the experimen-tal viscosity exhibits the same scaling as predicted by transport in a stochastic magneticfield. The magnitude of the viscosity agreed with a model thatassumes that transport oc-curs at the sound speed – the first detailed test of this model.The result can, for example,lead to a clearer comparison between experiment and visco-resistive magnetohydrodynam-ics (MHD) modeling of plasmas with a stochastic magnetic field. These comparisons hadbeen complicated due to the large uncertainty in the experimental viscosity. Now, the vis-cosity can be better constrained, improving the predictivecapability of fusion science.

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Sammanfattning

Magnetiskt inneslutna fusionsplasman (MCF) ar vanligtvis mottagliga for flera insta-bila moder. De med lang vaglangd beskrivs av magnetohydrodynamisk (MHD) teori. Medhansyn till plasmats andliga resistivitet, forutsparMHD en instabilitet som river upp ochaterkopplar magnetiska faltlinjer, en sa kallad ”tearing mod” (TM). Det nya tillstandet harlagre magnetisk energi jamfort med tillstandet innan bildandet av ”tearing moden”, vilkenalltsa minimerar den potentiella energin. Tearing moder uppkommer vid ytor dar mag-netiska faltlinjer sluter sig sjalva efterm toroidala ochn poloidala varv. Dessa ytor arresonanta i den meningen att magnetfaltet och stromstorningen har samma helicitet, vilketminimerar stabiliserande effekten fran bojning av magnetiska faltlinjen.

Externa resonanta magnetiska stortningar (RMP) har flera positiva effekter for fusion-splasman. Exempelvis, begransning av ”kant lokaliserademoder” (ELM) och styrning av”neoklassiska tearing modens” (NTM) position for stabilisering med elektron-cyklotronstromdrivning. Foljaktligen anses anvandandet av RMP som nodvandigt i framtida fusions-anlaggningar. Det finns tyvarr risk for negativa konsekvenser, till exempel kan en RMPleda till inbromsning av en TM och eventuellt vagglasning. Det ar darfor viktigt att forstainteraktionen mellan TM och RMP. En vagglast TM, det vill saga som inte roterar relativtmaskinvaggen, kan vaxa i storlek och darigenom forsamra fusionsplasmainneslutningen.I slutandan kan en vagglast TM leda till ”plasmaavbrott”, vilket kan orsaka vaggskador.Darfor bor vagglasta moder undvikas i fusionsmaskiner.

I den har avhandlingen studeras mekanismerna for TM-lasning och -upplasning pagrund av externa resonanta magnetiska storningar genom experiment. Studierna bedrivs itva MCF-maskiner av typen ”Reverserat-Falt Pinch” (RFP), vid namn EXTRAP T2R ochMadison Symmetric Torus (MST). De studerade maskinerna uppvisar flera roterande TMunder normal drift. TM-lasning och -upplasning har studerats i EXTRAP T2R genom attanvanda en RMP. Experimenten visar att efter en TM har blivit vagglast, sa maste RMPamplituden minskas avsevart for att lasa upp den igen. Liknande experimentella studierhar utforts i MST, men med en RMP bestaende av flera vaglangder. Samtidig inbromsningav samtliga TM observerades. I de fall TM blir vagglasta, observeras ingen upplasningefter RMP-avstangning. Foljaktligen karakteriseras l˚asning och efterfoljande upplasningav hysteresis, bade i EXTRAP T2R och i MST.

Resultaten visar kvalitativ overensstammelse med en teoretisk modell, som beskrivertearing modens tidsutveckling under inverkan av resonantamagnetiska storningar. Badeexperiment och teori visar att en RMP orsakar en reduktion avTM-rotation och plasmatsrotation vid resonansytan.Andringen av hastighet motverkas av ett vridmoment fran om-givande plasmat, som uppstar via dess viskositet. Men efter TM-lasning sa relaxeras helaplasmats rotation, pa grund av att hastighetsminskningensprids via viskositeten. Dettaresulterar i ett reducerat vridmoment fran omgivande plasma, vilket forklarar den ob-serverade hysteresen. Hysteresen fordjupas ytterligareav den okade amplituden hos etten vagglast tearing mod.

List of Papers

This thesis is based on the work presented in the following papers:

I The tearing mode locking–unlocking mechanism to an external resonant field inEXTRAP T2RL. Frassinetti,R. Fridstr om, S. Menmuir, P.R. BrunsellPlasma Phys. Control. Fusion, vol. 56, p. 104001, 2014

II Hysteresis in the tearing mode locking/unlocking due to resonant magnetic pertur-bations in EXTRAP T2RR. Fridstr om, L. Frassinetti, P.R. BrunsellPlasma Phys. Control. Fusion, vol. 57, p. 104008, 2015

III Tearing mode dynamics and locking in the presence of external magnetic perturba-tionsR. Fridstr om, S. Munaretto, L. Frassinetti, B.E. Chapman, P.R. Brunsell, J.S. SarffPhysics of Plasmas, vol. 23(6), p. 062504, 2016

IV Estimation of anomalous viscosity based on modeling of experimentally observedplasma rotation braking induced by applied resonant magnetic perturbationsR. Fridstr om, S. Munaretto, L. Frassinetti, B.E. Chapman, P.R. Brunsell, J.S. Sarff43th European Physical Society (EPS) Conference on Plasma Physics, Leuven, P2.049,July 4–July 8, 2016

V Multiple-harmonics RMP effect on tearing modes in EXTRAP T2RR. Fridstr om, P.R. Brunsell, L. Frassinetti, A.C. Setiadi44th European Physical Society (EPS) Conference on Plasma Physics, Belfast, P1.137,June 26–June 30, 2017

VI Dependence of perpendicular viscosity on magnetic fluctuations in a stochastictopologyR. Fridstr om, B. E. Chapman, A. F. Almagri, L. Frassinetti, P. R. Brunsell, T. Nishizawa,J. S. SarffTo be submitted

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iv LIST OF PAPERS

VII Modeled and measured magnetic fluctuation induced momentumtransport in thereversed-field pinchR. Fridstr om, B. E. Chapman, A. F. Almagri, L. Frassinetti, P. R. Brunsell, T. Nishizawa,J. S. SarffTo be submitted

Some other contributions by the author, which are not included in this thesis:

VIII Experimental study of tearing mode locking and unlocking inEXTRAP T2RR. Fridstr om, L. Frassinetti, P.R. Brunsell41st European Physical Society (EPS) Conference on Plasma Physics, Berlin, P5.083,June 23 – 27, 2014

IX A method for the estimate of the wall diffusion for non-axisymmetric fields usingrotating external fieldsL. Frassinetti, K.E.J. Olofsson,R. Fridstr om, A.C. Setiadi, P.R. Brunsell, F.A. Volpe,J. DrakePlasma Phys. Control. Fusion, vol. 55, p. 084001, 2013

X Braking due to non-resonant magnetic perturbations and comparison with neoclas-sical toroidal viscosity torque in EXTRAP T2RL. Frassinetti, Y. Sun,R. Fridstr om, S. Menmuir, K.E.J. Olofsson, P.R. Brunsell,M.W.M. Khan, Y. Liang, J.R. DrakeNucl. Fusion, vol. 55, p. 112003, 2015

XI Resistive Wall Mode Studies utilizing External Magnetic PerturbationsP.R. Brunsell, L. Frassinetti, F.A. Volpe, K.E.J. Olofsson, R. Fridstr om, A.C. SetiadiProceeding of the 25th IAEA Fusion Energy Conference, St. Petersburg, RussianFederation, Paper EX/P4-20, 2014

XII Local measurement of error field using naturally rotating tearing mode dynamicsin EXTRAP T2RR. M. Sweeney, L. Frassinetti, P.R. Brunsell,R. Fridstr om, F.A. VolpePlasma Phys. Control. Fusion, vol. 58(12), p. 124001, 2016

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The author’s contribution to the included papers

Paper II implemented the TM evolution model used in the simulations.

Paper III planned and performed the experiments, analyzed the experimental data, developed acode to model the TM time evolution in EXTRAP T2R, and wrote the paper. In all steps, Ihad guidance from my supervisors Per Brunsell and Lorenzo Frassinetti, who are also co-authors. In addition, technician Hakan Ferm was of great importance running EXTRAPT2R.

Papers III and IVI planned and performed the experiments in MST and analyzed the experimental data. Ialso developed a code modeling the TM time evolution, including the EM torque actingon several TMs due to the interaction with RMPs and the wall. Iwrote the papers. Inthe whole process I had help from my co-authors; Stefano Munaretto, Lorenzo Frassinetti,Brett Chapman, Per Brunsell and John Sarff. In addition, theMST-team was of utmostimportance in running the machine.

Paper VI planned and performed the experiments, analyzed the experimental data, developed acode to model the TM time evolution for multiple harmonics inEXTRAP T2R, and wrotethe paper. In performing the experiments, I had help from my supervisor Per Brunsell andthe magnetic feedback control expert Agung Chris Setiadi.

Paper VI and VIII planned and performed the experiments, with help from Brett Chapman and AbdulgaderAlmagri. Takashi Nishizawa was also important for the planning and performing of theexperiments. In addition, the MST-team was of great importance in running the machine.After the experiments, I implemented a numerical model, andcompared the simulationswith the experimental data. In the writing process, all the co-authors and especially BrettChapman gave important suggestions and help with the editing.

Acknowledgements

First and foremost, I would like to thank my supervisors Per Brunsell and Lorenzo Frassinetti.I am glad that I had you as my supervisors. Thanks for your guidance, support, and forproviding me the resources to perform the research. You always had good suggestionswhen I needed them. Thanks also go to technician Hakan Ferm and control expert AgungChris Setiadi for their help in running the EXTRAP T2R experiments.

I want to acknowledge the support from my collaborators in the MST-group (Universityof Wisconsin, Madison): Brett Chapman, Stefano Munaretto,Abdulgader Almagri, JohnSarff, Takashi Nishizawa, and the rest of the MST-group. Thanks for sharing your expertknowledge and giving me the opportunity to perform the experiments. Thanks also foryour hospitality, which made my visits to Madison most enjoyable.

Thanks to the people I met during my visits to ASDEX-Upgrade in IPP Garching. Spe-cial thanks to Sina Fietz for introducing me to the differentdiagnostics and computationaltools.

I am lucky to have shared office with fellow (current and past)PhD students: Alvaro,Armin, Chris, Emmi, Estera, Igor, Kristoffer, Moon, Pablo,Petter, Waqas, and Yushan.Thanks for sharing your time with me during discussions at lunches, movie nights and otheroccasions. I am especially thankful to Armin and Estera for helping me in proofreadingthis thesis. I would also like to acknowledge the other colleagues for contributing to a goodwork environment.

Finally, I would like to thank my friends and family for theirencouragement and sup-port throughout the years. Thanks to my parents Birgitta andHakan, my sister Anna, mygrandmother Sonja, and the rest of our family. I also want to thank Lotta for her support.

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Acronyms

Here are the acronyms used in this thesis:ELM Edge Localized ModeEM ElectromagneticMCF Magnetic Confinement FusionMH Multiple HelicityMHD MagnetohydrodynamicsMRE Modified Rutherford EquationMST Madison Symmetric TorusNTM Neoclassical Tearing ModePPCD Pulsed Poloidal Current DriveQSH Quasi Single HelicityRFP Reversed-Field PinchRMP Resonant Magnetic PerturbationRWM Resistive Wall ModeTM Tearing Mode

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Contents

Li st of Papers iii

Acknowledgements vii

Acronyms ix

Contents 1

1 Introduction 31.1 Nuclear Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Magnetically confined fusion . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Magnetohydrodynamical model . . . . . . . . . . . . . . . . . . . . . . 61.4 Reversed-Field Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Tearing modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Stochastic magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Resonant magnetic perturbation effect on tearing mode dynamics . . . . . 111.8 Plasma viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 Focus of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Description of the experimental setup 172.1 EXTRAP T2R device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The Madison Symmetric Torus device . . . . . . . . . . . . . . . . . . . 21

3 Modeling the tearing mode dynamics 253.1 Geometry and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Magnetic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Electromagnetic torques . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Fluid equation of motion including EM-torque source terms . . . . . . . 293.5 Tearing mode island angular evolution . . . . . . . . . . . . . . . . . . . 293.6 Tearing mode amplitude evolution . . . . . . . . . . . . . . . . . . . . . 293.7 Example: Modeled plasma velocity evolution under influence of an RMP 30

1

2 CONTENTS

4.1 EXTRAP T2R results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 MST results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Discussion 455.1 RMP effect on the dynamics and locking of TMs . . . . . . . . . . . . . 455.2 The anomalous viscosity in the RFP stochastic magnetic field . . . . . . . 47

6 Summary and Conclusion 49

References 51

4 Results 33

Chapter 1

Introduction

The first part of this chapter serves as a brief introduction to nuclear fusion and magneticconfinement. The following sections describe the specific subjects studied in this thesisand the last section defines the focus of the thesis.

1.1 Nuclear Fusion

Nuclear fusion reactions provide the energy in the Sun and other stars. In the fusion processnuclei are joined (”fused”), forming a heavier nucleus. Since the reacting nuclei are ofequal charge sign, they must have high velocities (temperature) to overcome the repulsiveelectrical force. When they are sufficiently close, the attractive nuclear force dominatesover the electrical force, and fusion can occur. At requiredtemperatures, atoms are strippedof their electrons and enter a plasma state – a gas of unbound electrons and ions.

In the Sun, the fusion reactions take place inside the hot core. The main chain of fusionreactions in the Sun starts with Hydrogen nuclei (protons p)and ends with Helium (4He)[1]. The mass deficit determines the amount of released energy, per Einstein’s famousformula E = mc2 [2]. The main part of the energy is released in the form of gamma-ray photons. The photons interact with nuclei inside the Sunand leave the Sun’s outersurface about 10 million years (!) after the original fusionreaction [3]. After escaping theSun, a tiny fraction of the photons arrives at planet Earth after about 500 seconds. Sincethe free energy depends on the speed of light squared (c2 ≈ 9×1016 m2/s2), the energy-density is enormous compared to other energy sources, such as oil, coal, or wind. This is,unfortunately, most obvious in the devastating effects of nuclear weapons.

Nuclear power is already used for electricity production infission power plants usingneutron induced splitting of uranium isotopes. However, scientists are underway to de-velop fusion power plants, which would have several benefitsover the fission plants. Forexample, there is no risk of uncontrolled chain reactions with subsequent core melt-down,as in Harrisburg (1976), Chernobyl (1986) and Fukushima (2011). This safety-advantagepartly explains why controlled fusion is not a reality yet after 60 years [4] of research; it ischallenging to achieve the right conditions on Earth.

3

4 CHAPTER 1. INTRODUCTION

Fusion of hydrogen atoms is carbon dioxide (CO2) free and thus of interest for futureenergy production. Since the start of the industrial revolution CO2 emissions have contin-uously increased, and the increase in CO2 concentration in the atmosphere correlates witha global increase in the temperature on Earth. Based on temperature measurements, scien-tists estimate that the Earth’s surface temperature increased with a rate of 0.15–0.2oC perdecade from 1970 to 2009 [5]. A global temperature increase,in the order of a few degreesCelsius, can lead to many catastrophes. For example, it may lead to increased water levelsfrom the melting ice, increase of the deserts, and more extreme weather. In light of this, itis desirable to add fusion energy in our energy-mix to the reduce the CO2 emission. Herefollow some of the criteria required for future fusion reactors.

First, the reaction between Deuterium (D) and Tritium (T) isthe one with the highestprobability. It is, therefore, the reaction perused for thefirst generation of fusion reactors.The D-T fusion reaction yields a neutron and an alpha particle:

D+T → 4He+n+17.6 MeV,

where the neutron (n) carries 14.1 MeV of the kinetic energy and the alpha particle (4He)carries the remaining 3.5 MeV. The magnetic field confines the charged alpha particlefor a short time (∼ 1 s), allowing it to heat the plasma in collisions with the fuel ions.The neutron is unaffected by magnetic fields and directly escapes the plasma. Thereon,its energy heats the reactor wall and the water that is circulating the wall. The heat isconverted into electricity in the usual way (steam-turbine-generator).

The fusion fuels will be available for a long time compared with the fossil fuels. Deu-terium is available from seawater, but T that has a half-lifeof 12.3 years is not naturallyavailable on Earth. Currently, T is produced in fission powerplants. In future fusionreactors, however, the tritium may be produced in a lithium-blanket outside the machine-wall [6]. With the current energy demand, fuels are expectedto last more than 1000 years,being limited by the availability of lithium. In such a long time, fusion technology maydevelop to the point that a more difficult reaction could be utilized, such as the D-D fusion.

As already mentioned, to fuse two hydrogen atoms, the Coulomb barrier must be over-come. This requires a high relative velocity, i.e. a high-temperature plasma. For fusionreactions, another important factor is that the atoms have to meet each other. This is morelikely with a high particle density. However, even with a high kinetic energy and a highparticle density, the probability of a fusion reaction is relatively low. Therefore, the energyneeds to be confined for a long time. To summarize, we need a high temperature (T), highfuel number density (n), and long energy confinement time (τE). This is described by thetriple product (Lawson [7]) that has to fulfill the condition

nTτe ≥ 3×1021 keVsm−3

to reach ignition in a D-T fusion reactor. Here, ignition means that the alpha particles heatthe plasma and no auxiliary heating is needed.

In the Sun, gravity confines the plasma. On Earth, researchers pursue two other types ofconfinement schemes: magnetic confinement (MCF) and inertial confinement (ICF). In the

1.2. MAGNETICALLY CONFINED FUSION 5

latter scheme, a laser-induced implosion of a D-T pellet provides the (inertial) confinement.The mechanism resembles that of a (tiny) hydrogen bomb.

Since this thesis is performed within the realm of MCF in a type of device called thereversed-field pinch (RFP), the following sections introduce MCF and RFPs.

1.2 Magnetically confined fusion

In MCF, we use the fact that charged particles encircle magnetic field lines in helical paths– in other words we confine the particles by the Lorentz force.To overcome end-losses,scientists realized that the field lines need to close on themselves within the plasma con-tainer (vacuum vessel). The simplest geometry is a “doughnut” shape called a torus, whichis the geometry of most present-day MCF schemes. In these schemes, the field lines lie onnested toroidal surfaces on to which the particles are constrained, to the first approxima-tion. The two main MCF schemes are pinch machines and Stellarators. In terms of relativefusion output, the most successful configuration is a type ofpinch device called the toka-mak (Figure 1.1). The current record is 67% energy output (16MW from 24 MW inputpower) and was set in 1997 by the JET tokamak [8]. In France, 35nations collaborate tobuild the world’s largest tokamak ITER. One of ITER’s targets is to produce 500 MW offusion energy from 50 MW of input energy. Likely, ITER will bethe first MCF device toproduce a net energy gain.

In pinch machines (such as the tokamak and reversed-field pinch RFP), an important in-gredient to the confinement is provided by driving a toroidalcurrent (Jφ ) inside the plasma.The current creates a perpendicular (poloidal) magnetic field (Bθ ), and together they pro-duce a force that compresses the plasma (the compressionalJφ ×Bθ force is called a pincheffect). The current is induced by an external transformer.Because of the torus shape, theinside of the torus has a higher magnetic field than the outside. The difference in magneticfield pressure causes the plasma to drift outwards. This drift can be hindered by adding avertical magnetic field or a perfectly conducting wall to provide a restoring force. Unfor-tunately, the equilibrium is highly unstable. Scientists realized, however, that the stabilityproperties improve by adding a toroidal magnetic field. The toroidal field is created bydriving current in external coils (Figure 1.1).

The poloidal and toroidal magnetic fields are the two main knobs that are tuned to op-timize equilibrium and stability properties. This can be performed in many ways and itturns out that the plasma properties can vary to a great extent, which is shown in configu-rations such as tokamaks and RFPs. The tokamak has the best stability properties due toits high toroidal field (Btoroidal >>Bpoloidal), however for a reactor, this requires supercon-ducting magnetic field coils. The RFP which hasBtoroidal ≈ Bpoloidal is more unstable butcan be operated with normal copper coils. The main problems for the pinch devices, thatresearchers are trying to solve, are caused by the pulsed operation, plasma instabilities, andplasma wall-interaction.

In Stellarators, on the other hand, external coils generateall the 3D magnetic fieldsand no inductive current is being driven. The configuration requires optimal selection ofmagnetic field coils and a high precision in coil-alignment.Historically, the stellarator had


Toroidal

field coils

Toroidal

magnetic field

Poloidal

magnetic field

Inner poloidal field coils

(primary transformer circuit)

Outer poloidal field coils

(for plasma positioning

and shaping)

Plasma electric current

(secondary transformer circuit)

Resulting helical magnetic field

CPS14

.565

-1c

Figure 1.1: Typical layout of a tokamak device, whereBtoroidal >> Bpoloidal. The layoutresembles that of the RFP, but in the RFP the relative field strengths are similar (Btoroidal ∼Bpoloidal). Courtesy of EUROfusion [9].

been less successful than the tokamak due to magnetic field errors that destroy the toroidalfluxes surfaces and degrade the confinement. The field errors can occur due to the finitenumber of coils and/or misalignment of the coils. Stellarators have more recently started toshow tokamak-similar confinement. In the design of the newlybuilt Wendelstein 7-X [10],the coil positions were selected by supercomputers solvingan optimization problem formachine performance. If its performance can equal that of a similarly sized tokamak, itmight change the course for future fusion devices.

1.3 Magnetohydrodynamical model

Magnetohydrodynamical (MHD) theory describes the global behavior in MCF plasmas.In short, MHD couples the Navier-Stokes fluid equations withMaxwell’s electromagnetic

1.4. REVERSED-FIELD PINCH 7

equations. Thus, it describes the plasma as a conductive fluid. However, due to its com-plexity, there exist several simplifications/scalings of the MHD theory.

The MHD equations can be derived from either kinetic theory (Maxwell-Vlasov equa-tions) by taking moments of the ion and electron distribution functions, or from singleparticle guiding center theory [11]. The resistive form of the MHD equations are summa-rized below [11]:

mass conservation:dρdt

+ρ∇ ·v= 0 (1.1a)

momentum conservation:ρdvdt

= J×B−∇p (1.1b)

energy conservation:ddt

(

pργ

)

= 0 (1.1c)

Ohm’s law:E+ v×B= ηJ (1.1d)

Maxwell: ∇×E =−∂B∂ t

(1.1e)

∇×B = µ0J (1.1f)

∇ ·B = 0 (1.1g)

The conservation of momentum, Eq. (1.1b), describes the force balance of the plasma.Equation (1.1d) is the resistive Ohm’s Law. Even though the resistivity termηJ is smallcompared to the other terms, it is the only dissipative process in Ohm’s law. Hence, it isresponsible for the two main transport losses: particle diffusion and magnetic field diffu-sion. Resistivity also allows for a larger number of instabilities compared with the idealMHD which neglects the resistive term. One of these instabilities, ”the tearing mode”, isthe focus of the present thesis.

Quite often, the ideal MHD suffices to describe global stability in the plasma. IdealMHD neglects the resistivity (i.e. assumes infinite conductivity); hence the following rela-tion is used for Ohm’s law:

E+ v×B= 0 (1.2)

1.4 Reversed-Field Pinch

In an RFP, the toroidal magnetic field reverses its directionnear the plasma edge. Thelocation where the toroidal field changes sign is called the reversal-surface. Figure 1.2shows typical radial profiles of the magnetic field. These profiles result in a high magneticshear (change in magnetic field pitch angle with minor radius) that provides a stabilizingeffect on MHD modes. In contrast, the main stabilizing effect in the tokamak is due to fieldline bending of the strong toroidal field.

The RFP is like a little sibling to the tokamak but has some interesting aspects of itsown. First, the lower toroidal field enables the use of coppercoils instead of superconduct-ing coils. Furthermore, it might be possible to heat the plasma with only ohmic heating,


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Minor Radius r/a

Mag

netic

Fie

ld (

T)

B

φB

θ

Figure 1.2: Typical radial profiles of the RFP equilibrium magnetic field in the EXTRAPT2R device (KTH, Stockholm).Bθ and Bφ are the poloidal and toroidal components,respectively.

which implies a simpler and more economical reactor construction. Another advantageis a higher ratio of plasma pressure to magnetic field pressure, which is quantified by theso-called beta parameter. For a fixed magnetic field strength, the advantage of high beta isthat the device sustains a higher plasma pressure, i.e. higher temperature×density. This isfavorable for fusion reactions according to the Lawson criterion (Eq. 1.1). Unfortunatelyfor the RFP, the energy confinement time (τE) is short compared with the tokamak. Onereason for this is the lower safety factor at the edge which leads to resistive MHD turbu-lence [11]. A second reason is the usually stochastic magnetic field in the core region. Thestochastic magnetic field is caused by overlapping tearing mode (TM) islands, as describedin the two following sections.

1.5 Tearing modes

The tearing mode is an instability driven by the current gradient. The origin of the name”tearing” is that the instability tears and reconnects magnetic field lines, allowed by thefinite plasma resistivity. In the reconnection event, a magnetic island forms inside theplasma. A magnetic island can form its own magnetic axis in the shape of a screw, i.e.helical. The field lines on the islands magnetic axis, the so-called O-point, closes uponitself afterm toroidal andn poloidal turns. Figure 1.3 illustrates two field lines lyingon thetwo different magnetic islands: (m= 1,n= 6) and (m= 1,n= 7).

Tearing modes appear at magnetic surfaces where the magnetic field has the same he-licity as the mode. Here, the mode wave vectork is perpendicular to the equilibrium

1.5. TEARING MODES 9

magnetic fieldB. This implies that the stabilizing field line bending is minimized. Thelocations of the resonant surfaces are found by solving

k ·B = 0. (1.3)

To solve this equation we first assume that the equilibrium magnetic field only has twocomponents, the poloidalBθ and the toroidalBφ . Second, we assume that the torus can beapproximated by a periodic cylinder with length 2πR0, whereR0 is the torus major radius.We adopt cylindrical coordinates(r,θ ,z), wherez is related to toroidal angle byφ = z/R0.Furthermore,m andn are the poloidal and toroidal mode number, respectively. Hence, thecorresponding mode wavelengths areλθ = 2πr/mandλφ = 2πR/n. In Eq. 1.3, the aboveassumptions result in

mr

Bθ +nR

Bφ = 0

⇔ q(r)≡ rR

Bφ

Bθ=−m

n, (1.4)

whereq(r) is called the safety factor, and the resonant modes can occurwhereq(r) equalsa rational number−m/n.

At the resonant surfaces, the ideal magnetohydrodynamics (ideal MHD) approximationcannot describe the TMs, and the resistivity (resistive MHD) has to be considered in asurrounding layer. Furth, Killeen and Rosenbluth [12] introduced the so-called∆′-methodfor TM stability analysis, where the tearing stability index ∆′ is a measure of the jump in thespatial derivative of the flux function at the resonant surface. For a classical tearing modethat is not perturbed by external fields, the criteria for tearing stability is∆′ < 0, and viceversa∆′ > 0 for instability [12]. The stability is actually describedby the magnetic fieldoutside the resistive-layer and here the ideal MHD applies (e.g. Newcomb’s equation [13]which is used in Chapter 3).

In standard operation, the RFP exhibits several tearing modes that are resonant insidethe plasma. This state is calledmultiple helicity(MH). The tearing modes are usuallylinearly unstable, but non-linearly stable. Tearing modescan have both negative and pos-itive effect on RFP plasmas. On the positive side, the tearing modes sustain the toroidalmagnetic field throughout the discharge in the so-called dynamo process. In the dynamoprocess, the central TMs slowly increase their amplitude, followed by a fast relaxation inwhich the central modes exchange energy with the outer modes. The slow phase is charac-terized by resistive diffusion and causes a decrease in the toroidal field. The fast relaxationprocess is typically characterized by non-linear interaction betweenm= 1 TMs resonantin the core-region andm= 0 TMs resonant at the reversal-surface. This process restoresthe toroidal field. Lastly, if one island grows large it can dominate the magnetic topologyinside the plasma, as observed in thequasi-single helicityQSH-regime [14]. Increasedconfinement is observed and numerical calculation indicates this is connected with closedflux surfaces [15]. On the negative side, TMs can lead to locally increased plasma-wallinteraction, following a process of mode growth and subsequent locking relative to the vac-uum vessel. Furthermore, the RFP typically exhibits several overlapping TM islands. Theoverlap leads to a stochastic magnetic field followed by an increased radial transport [16].


Figure 1.3: The two central tearing modes [(m= 1,n= 6) and (m= 1,n= 7)] in a MadisonSymmetric Torus RFP plasma. Typically, they are embedded ina stochastic magnetic fieldas shown by the poloidal cross section in Figure 1.4.

1.6 Stochastic magnetic fields

A stochastic field line fills a 3D volume, as opposed to the confined field lines that lie on2D surfaces. Since the charged particles travel faster parallel to a field line, the stochasticfield leads to a quicker radial transport in a torus.

The stochastic magnetic fields are created if magnetic islands have overlapping clas-sical widths. The classical width of an island with the poloidal mode numberm and thetoroidal mode numbern is wmn= 4

√

rmn|br,mn|/(nBθ |q′mn|), wherermn is the minor radiusat the resonant surface,br,mn is the radial component of the magnetic fluctuation that formsthe island, andq′mn is the derivative of the safety factor. All quantities are evaluated atr = rmn. The level of island-overlap between two neighboring modes, (m,n) and (m′,n′),can be quantified by the Chirikov parameter [17]

s=12

wmn+wm′,n′

|rmn− rm′,n′ |. (1.5)

The overlap of the islands (s> 1) causes the field lines to become tangled. In this case, theradial excursion∆r after a distanceL along the field line can be described by a stochasticprocess. Averaging over several steps of lengthL, the diffusion coefficient of the magneticfield in space is

Dm =< ∆r2 >

2L. (1.6)

1.7. RESONANT MAGNETIC PERTURBATION EFFECT ON TEARING MODEDYNAMICS 11

In collisional plasmas, the wandering of the field lines in a stochastic field allows for aquicker interaction between the different radial positions, where collisions can occur. Inthe collisionless case, the transport can occur directly along a single field line over thewhole stochastic region.

The diffusion of electrons in a stochastic magnetic field wasfirst described by Rech-ester and Rosenbluth (R-R) [18], who stated that the heat diffusivity in the collisionlesslimit is χe = veDm,RR, whereve is the electron thermal velocity. The magnetic diffusioncoefficient in the quasilinear approximation [19] is given by

Dm,RR= Lc ∑m,n

(

br,mn

B

)2

, (1.7)

whereLc is the autocorrelation length [20],br,mn is the perpendicular component of themagnetic perturbation that creates the stochastic field, and B is the equilibrium magneticfield atr = rmn. The (m,n) modes that overlap are included in the sum, which we from hereon denote(br/B)2. In the model, R-R assume that the Chirikov island-overlap parameter[Eq. (1.5)] is much larger than one.

To illustrate the stochastic magnetic field, we solve Eq. (1.8) that traces a field line adistancedl along the torus

dxBx

=dyBy

=dzBz

=dlB. (1.8)

The equation was solved in cylindrical coordinates including both the equilibrium mag-netic field and perturbed radial magnetic field (see Chapter 3for calculation of the per-turbed field). When only a single tearing mode is included, the field lines form a singleisland structure with only closed flux surfaces [Figure 1.4a]. However, when the six cen-tral tearing modes are included in Eq. (1.8), the width of theinner island is reduced, andit is surrounded by a stochastic region [Figure 1.4b]. Note that only two of the six islandstructures are visible.

1.7 Resonant magnetic perturbation effect on tearing modedynamics

Except for the TMs, the RFP also exhibits other modes, of which the (non-resonant) re-sistive wall mode (RWM) is the most unstable. However, experiments show that RWMscan be suppressed by magnetic coils outside a resistive shell, operating in a feedback sys-tem [21–23]. Thanks to this advance, the RWM no longer limitsthe attainable dischargeduration in RFP experiments, such as EXTRAP T2R [21] (Stockholm, Sweden) and RFX-mod [24] (Padua, Italy). The magnetic feedback systems can also be used to perturb theplasma to study other aspects of the TMs and RFP physics. In the present work, we appliedresonant magnetic perturbations (RMPs) and studied its effect on the TMs.

Here follows a brief introduction to the RMP effect on the velocity and amplitude ofa tearing mode, which is general to the RFP and the tokamak. Let us first assume that the


−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

R−R0 [m]

Z [m

]

(a) One tearing mode

−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

R−R0 [m]

Z [m

](b) Six tearing modes

Figure 1.4: Poincare maps showing the poloidal cross section of the magnetic field in thecore-region. The plot is based on data from an MST RFP discharge.1

TMs rotate, which is the case in many tokamaks and RFPs. The coils that induce the RMPare typically placed outside the machine-wall to protect them from damage. However, thewall hinders penetration of any fast rotating fields via the induced eddy (image) currents.This means that the phase of the RMP has to be slowly rotating or static relative to the TMphase (the experiments in this thesis use a static phase for the RMP). The relative phasebetween the RMP and the TM island, [∆αm,n(t)], then changes with the TM rotation. TheRMP-field interaction with the plasma causes an induced current (j ind) at the correspondingresonant surface [25], which acts to shield the plasma from the RMP-field. The inducedcurrent interacts with the tearing mode (bmode) at the same resonant surface in the twofollowing ways:

(i) The Lorentz force (j ind ×bmode) results in an electromagnetic (EM) torque that changesthe velocity of the TM. The EM torque,TEM, is proportional to the amplitude of theinduced current, the TM amplitude, and sine of the relative phase difference, i.e.TEM ∝ |j ind ||bmode|sin(∆αm,n(t)). Moreover, the induced current is proportional tothe RMP amplitude|bRMP |.

(ii) The cos(∆αm,n(t)) component of the RMP-TM interaction modulates the amplitudeof the TM, which is described by the modified Rutherford equation [26]. The modu-lation is proportional to the RMP amplitude, the TM amplitude, and cos∆(αm,n(t)).

At first, the TM rotates relative to the RMP. This causes oscillation in TM amplitudeand velocity. Theory predicts that the RMP on average reduces both the amplitude andvelocity of a rotating TM [25,27]. If the RMP amplitude increases above a threshold value,

1.7. RESONANT MAGNETIC PERTURBATION EFFECT ON TEARING MODEDYNAMICS 13

Figure 1.5: An overview of the torques acting on the TM island. The plasma typicallyrotates in the poloidal and toroidal direction, but the source of its rotation is not shownhere. The TM and the plasma accelerate or decelerate each other via the viscous torque(Tvisc). The TM is decelerated by its interaction with induced eddycurrents in the wall(TEM,W). Additionally, the TM can be accelerated or decelerated byresonant magneticperturbations (TEM,RMP), which are imposed by RMP coils or by field errors.

the TM locks relative to the RMP phase. In the case of a static RMP, the TM becomeswall-locked. Wall-locking causes the TM amplitude to increase in amplitude and finallysaturate. Experimentally, the TM velocity reduction and subsequent wall-locking has beenobserved in RFPs [28–31] and in tokamaks [32, 33]. The increase in TM amplitude afterwall-locking has been observed in EXTRAP T2R [Paper I].

In addition to the RMP torque, the tearing modes are decelerated by their interactionwith the wall. As the TMs rotate, the wall observes a changingmagnetic field, and eddycurrents are induced in the wall. The phase-lag between these currents and the TM leadsto a mutualj ×b torque. The torque acts to equate the rotation of the mode andthe wall,i.e. decelerate the mode. The process resembles that of the induction motor. In our experi-ments, the wall torque is typically smaller than the appliedRMP torque.

To model the TM dynamics, we must consider both the EM torquesand the viscoustorque from the surrounding plasma (Figure 1.5). It is oftenassumed that the TM andplasma co-rotate at the resonant surface, which is called the no-slip condition [25]. Thus,the change in TM rotation caused by the EM torque will change the local plasma rotation inthe same way. The plasma at the resonant surface is connectedto the bulk plasma throughthe viscosity. Consequently, a differential rotation between plasma at the resonant surfaceand the bulk plasma results in a viscous torque that acts to equate the rotation.

The TM locking can have severe consequences on fusion machines, for example, in-creased plasma-wall interaction and, in tokamaks, a plasmadisruption. Therefore, it would


be necessary to unlock a locked TM, but the unlocking of a modecan be difficult due to twomechanisms [26]. First, after TM locking, the velocity reduction spreads via the viscosityuntil reaching a new steady-state plasma flow profile. This causes a substantial reduction inthe viscous torque, and to unlock a mode the RMP amplitude (EMtorque) must decreasesignificantly below the locking threshold. Second, the EM torque increases after lockingdue to an increase in the TM amplitude. The effects describedabove make it important toavoid locked TMs in fusion devices.

1.8 Plasma viscosity

The dynamic viscosity of a fluid is a measure of its resistanceto a shear or a tensile stress.The dynamic viscosity divided by the density is called the kinematic viscosity, or mo-mentum diffusivity. The viscosity can play an important role in both the stability and themomentum transport of a fluid.

In a liquid consisting of neutral particles, the viscosity is caused by the interactionbetween particles at adjacent surface layers. In plasmas, however, the particle interactionsare usually dominated by the Coulomb collisions that act on alonger scale length. In two-fluid plasmas, the momentum is mainly carried by the ions, andthe ion dynamic viscosityis a factor

√

(mi/me) larger than the electron one. Furthermore, in a magnetic field, theviscosity of the plasma is anisotropic since the particles follow the field lines (at least to thefirst approximation). This means that the parallel viscosity equals that of an unmagnetizedplasma, but the perpendicular viscosity is reduced becausethe free path of the particles isreduced from the standard mean free path (λ ) to the Larmor radius (ρL).

In the classical two-fluid MHD equations, the viscosity coefficients were first derivedby S.I. Braginskii [34]. The Braginskii viscosity is used inboth astrophysical and labo-ratory plasmas, for example, the flaring solar corona [35], clusters of galaxies [36], andthe tokamak fusion plasma [37]. Direct experimental measurements that confirm the Bra-ginskii viscosity are scarce, but in one experiment Dorf et al [38] measured the viscosityin a screw pinch plasma column and found agreement with the Braginskii viscosity. Onthe other hand, there exist plasmas where the perpendicularviscosity and other transportcoefficients are one or several orders of magnitude larger than the classical value. For ex-ample, in the core of a standard RFP plasma, the viscosity is anomalous [39, 40]. One ofthe results [39] indicated that the anomaly was due to the magnetic field being stochastic,and the order of magnitude agreed with a model [41].

The model [41], originally derived for the tokamak, assumesparticles and momentumare propagated along the stochastic field by sound waves. Accordingly, the kinematicviscosity is

ν⊥ = csDm,RR, (1.9)

wherecs is the plasma sound speed, and the magnetic diffusion coefficient is given byEq. (1.7). The model resembles the R-R model that describes the heat diffusivity, but theelectron thermal velocity is replaced bycs. The R-R model is widely used and tested in bothastrophysical plasmas [42, 43] and laboratory plasmas [16,44]. Until recently, however,only one experimental measurement of the viscosity [39] hadbeen compared with the

1.9. FOCUS OF THIS THESIS 15

model by Finn et al. The experiment was performed in Madison Symmetric Torus (MST)RFP [45]. The experimentalν⊥ was about 100 times larger than the classical Braginskiivalue. Instead, the experimentalν⊥ agreed with Finn’s model [Eq. (1.9)]. However, sinceit was only a single-point measurement, at a single magneticfluctuation amplitude(br/B),it was not enough evidence for a firm conclusion.

The first detailed test of Eq. (1.9) was performed in Paper VI,where both the intrinsicmagnetic fluctuation amplitude(br/B) and thecs were varied in the MST experiment.

1.9 Focus of this thesis

Interaction between RMP and tearing modes

Due to the different instabilities arising in the plasma, different control techniques are re-quired to sustain a plasma discharge. Many of these techniques use resonant magneticperturbations, which are applied by external coils that arefeedback controlled. For ex-ample, in tokamaks the RMPs are used to steer the TM island O-point in position forstabilization by electron cyclotron current drive [46] andfor mitigation of edge-localizedmodes [47]. The tailoring of such controlled perturbationscan be limited by the numberof coils. A limited number of RMP coils can lead to a perturbation field consisting of bothresonant and non-resonant modes. The non-resonant component leads to a global torqueon the plasma, which is called the neoclassical toroidal viscosity torque. The present the-sis, however, focuses on the TM’s interaction with the resonant component of a magneticperturbation.

The interaction between resonant magnetic fields and tearing modes can lead to modelocking. The locking further enhances the mode amplitude and increases the plasma-wallinteraction. In tokamaks, locked modes can cause plasma disruptions that cause strongforces on the machine structures. Therefore, locked modes should be avoided in the currenttokamaks and in ITER.

In this thesis the following questions are addressed regarding the interaction betweenan RMP and tearing modes:

• What are the mechanism’s behind TM locking and unlocking in the experiment?

• Can theoretical models predict, simultaneously, the experimental RMP amplitudethresholds for locking and unlocking of a TM?

• How is the locking threshold affected by plasma density and viscosity?

• Can the current models describe all the experimental features?

• What is the role of the torque exerted on the TMs due to inducededdy currents inthe conducting shell?

We answer these questions by performing both experiments and modeling. The ex-periments are based on RFP plasmas with rotating tearing modes and externally appliedRMPs. The modeling is based on well-known theoretical models, as described in Chapter3.


Measurement and modeling of the viscosity in stochastic magnetic fields

Stochastic magnetic fields can be present in space plasmas and laboratory plasmas. Thetokamak can become stochastic, for example, (I) during a disruption, (II) when TMs over-lap and (III) with an applied RMP. The RFP usually has a stochastic region, which extentis largest during the MH-regime. Visco-resistive MHD is usually employed to simulatethe plasmas described above. In general, the viscosity is not well-known in present experi-ments where the field is stochastic, which complicates the comparison with simulations. Inthis thesis work, we determine the viscosity in a 10-fold range of the magnetic fluctuationamplitude and compare our result with a theoretical model [41], the first in-depth test ofthis model.

Chapter 2

Description of the experimental setup

Experiments presented in this thesis were performed in the two RFPs: EXTRAP T2R andMadison Symmetric Torus (MST). For both machines, we investigated the effect of res-onant magnetic perturbations on the TM dynamics. The studied plasmas exhibit severalrotating tearing modes that are resonant inside the plasma,i.e. the so-calledmultiple helic-ity regime. The inner TM rotates the fastest, and the TM velocity falls off for larger minorradii [29, 48][Paper II]. The core TMs rotates mainly in the toroidal direction, and fromhere on only the toroidal component will be considered.

The main difference between the two experiments is that in EXTRAP T2R the appliedRMP spectrum is single harmonic, i.e. a specific(m= 1,n) was applied, whereas in MSTthe applied spectrum contains the poloidalm= 1 and a broad toroidaln spectrum. Belowfollows a summary of the two experiments.

2.1 EXTRAP T2R device

EXTRAP T2R [49] is located at KTH (Royal Institute of Technology) Stockholm, Sweden.It has a major radiusR0 = 1.24 m and a minor radiusa= 0.18 m. Hence, the aspect ratio islarge (R0/a≈ 6.8) and the torus can be approximated by a cylinder. The first wall is madeof stainless steel. Outside of the first wall sits a (double-layer) resistive copper shell.

One essential role of the shell is to reduce the growth rate ofunstable modes, throughthe induced eddy currents. For example, the shell diffusiontime is longer than the currentrise time, which helps to avoid wall-locking of TMs during the start-up phase. In general,the shell has a stabilizing effect on a TM if its rotation frequency is much higher than theinverse shell diffusion time [25]. At most times, this condition is satisfied in EXTRAPT2R, where typical rotation frequencies of the central TMs are f n,m ≈ 200−600×103 s-1

and the inverse shell diffusion time is 1/τw ≈ 100 s-1. In addition, the shell permits controlof the wall-locked RWMs by an external feedback system [21,50].

RFP equilibria are usually described with the reversal parameter (F) and the pinchparameter (Θ). The reversal parameter is defined as F= Bφ (a)/ < Bφ >, whereBφ (a) isthe toroidal magnetic field at the plasma edge, and< Bφ > is the poloidal-cross-section

17

18 CHAPTER 2. DESCRIPTION OF THE EXPERIMENTAL SETUP

average ofBφ . The pinch parameter is defined asΘ = Bθ (a)/ < Bφ >, whereBθ (a) is thepoloidal magnetic field at the plasma edge.

Typical plasma parameters in the EXTRAP T2R study are presented in Table 2.1.

Table 2.1: Typical plasma parameters in the EXTRAP T2R device for this study.

Plasma current,Ip 70-85 kAPinch parameter,Θ 1.6Reversal parameter, F -0.2Core electron temperature,Te 200-400 eVCore electron density,ne ≈ 1×1019 m−3

Equilibrium profiles were reconstructed by applying theα −Θ0 model [51], and usingthe reversal parameterF = −0.2 and pinch parameterΘ = 1.6. The corresponding safetyfactor radial profile,q(r), is shown in Figure 2.1. Equation 1.4 provides the locationsof the resonant modes. Them= 0 modes are at the reversal surface whereq(r) = 0.Inside the reversal surface are modes with rational numbers−m/n= q(r) > 0, of whichthose withm= 1 are the most unstable. Outside reversal surface are modes that fulfill−m/n= q(r)< 0. In Figure 2.1, the positions ofm= 1 resonant surfaces are indicated byblack dots, and(m= 1,n=−12) is the most central tearing mode.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.02

0

0.02

0.04

0.06

0.08

Saf

tey

fact

or, q

Minor radius, r/a

n=−12n=−13

m=0

n=−15n=−14

Figure 2.1: Typical safety factor profile in EXTRAP T2R (F = −0.2 andΘ = 1.6). Theresonant positions are indicated by black dots. The equilibrium magnetic fields responsiblefor theq-profile are plotted in Figure 1.2.

2.1. EXTRAP T2R DEVICE 19

The feedback system and RMP application

One of EXTRAP T2Rs characteristics is the feedback system for controlling the radialmagnetic field at the edge,br(a). It suppresses the RWMs, which otherwise limit theduration of the discharge. The system comprises 128 sensor coils, 128 actuator coils,and a controller. Both the sensor and actuator coils are atM = 4 poloidal andN = 32toroidal positions covering the whole shell-surface, as depicted in Figure 2.2. The sensorarray is positioned on the inner surface of the shell and the actuator array is on the outsideof the shell. To measure the unstablem= 1 modes, the coils at a toroidal position arepair-connected (up/down and inboard/outboard). The toroidal mode number resolution is−16≤ n ≤ 15. Several types of feedback controllers have been developed, such as theintelligent shell (IS) [21], revised intelligent shell (RIS) [52, 53], and model predictivecontrol (MPC) [54].

Figure 2.2: EXTRAP T2R saddle coil arrays consisting of 128 actuator-coils (red) and 128sensor-coils (blue). The coils are located at 4 poloidal and32 toroidal positions coveringthe whole shell-surface. The figure is created by K. Erik J. Olofsson [55]

In this work, we applied a single harmonic(m= 1,n = −15) RMP using the RIS-controller. The controller suppressed the other modes at the wall, as shown in Figure 2.3.Then=−15 RMP amplitude isbm=1,n=−15

r ≈ 0.8 mT. In comparison, the other harmonicsare negligible. Consequently, the RMP interaction with theplasma is localized to theresonant(m= 1,n=−15) TM.

The tearing mode measurements

The TMs were measured with an array of poloidal magnetic pick-up coils. The coil arrayis situated in-between the vacuum vessel and shell at minor radiusr = 0.191 m. The arraycomprises 4 poloidal×32 toroidal positions. Similar to the radial sensor coils, the pick-up coils are pair-connected to measure them= 1 component. Each coil measures the timederivative in the poloidal magnetic field (bθ ). The signals are time integrated and Fourierdecomposed to get the amplitude|bm,n

θ | and helical phase angleαm,n, of each harmonic


−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8

n

b r1,n (

mT

)

Figure 2.3: Mode spectrum in EXTRAP T2R when a perturbation with harmonics(m=1,n=−15) is applied. The feedback system suppressed the other modes.

(m= 1, −16≤ n≤ 15). To measure only the rotating TMs, the slowly changing field wasremoved by high-pass filtering. The time derivative of the phaseαm,n equals the helicalangular velocity of the (m,n) TM. Figure 2.4 shows the angular velocity of each TM whenusing full feedback suppression ofbr(a). This unperturbed velocity is sometimes calledthe natural velocity. The center has the highest rotation, dαm,n/dt ≈ 500 krad/s, and therotation falls off to zero at minor radiusr/a≈ 0.6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

100

200

300

400

500

600

r/a

d/dt

(αm

,n)

(kra

d/s)

Figure 2.4: Helical rotation frequency of the TMs plotted atcorresponding resonant radius.

In the experiment, the radial profile of the plasma flow can be approximated via therotation of the TMs, which is motivated as follows. The perturbed magnetic field of a sin-

2.2. THE MADISON SYMMETRIC TORUS DEVICE 21

gle (m,n) mode can be expressed asb(r , t) = bm,n(r, t)exp[i(mθ −nφ +αm,n(t))], whereαm,n(t) =

∫ t0 ωm,n(t)dt describes the time variation of the phase. The angular phaseve-

locity can be divided into the poloidal and the toroidal component:ωm,n(t) = mωm,nθ (t)−

nωm,nφ (t). In the EXTRAP T2R core, the modes rotate mainly in the toroidal direction

[48]. By neglecting the poloidal rotation, TM angular velocity is approximatelyωm,n(t)≡dαm,n/dt ≈ −nωm,n

φ = −nvm,nφ /R, wherevm,n

φ is the toroidal phase velocity of the(m,n)TM andR is the major radius. To a first order approximation, the TMs co-rotate with theplasma flow at each resonant surface [48]. Therefore, the measured TM velocities approx-imately represent the plasma flow radial profile (see Paper IIfor detailed discussion).

2.2 The Madison Symmetric Torus device

The MST [45] is located in the University of Wisconsin, Madison, USA. It has a majorradiusR0 = 1.5 m and a minor radiusa= 0.51 m. Hence, the aspect ratio (R0/a≈ 3) isless than half the one of EXTRAP T2R. In standard discharges the poloidal beta is aboutβθ ≈ 7 % [56]. Considering the fairly high aspect ratio and the lowbeta, the system canbe well approximated by a periodic cylinder [56,57]. Table 2.2 shows typical equilibriumparameters used in the MST study, Paper III.

Table 2.2: Typical plasma parameters for the MST study. Here, < ne > is the centralline-averaged electron density.

Plasma current,Ip 300 kAPinch parameter,Θ 1.8Reversal parameter, F -0.3Central electron temperature,Te 300-400 eVElectron density,< ne > 0.3−1.3×1019 m−3

The plasma current,Ip, is about 3–4 times the one in EXTRAP T2R. The plasmashad relatively high field-reversal (F = −0.3). The temperature was measured using theThomson scattering diagnostic. The central line-averageddensity< ne > was measuredwith a far-infrared interferometer [58]. In total, there are 11 lines-of-sight and they are usedfor reconstruction of the density profiles [59]. The densitywas varied from shot-to-shot.This information is used to examine dependence on density inthe TM braking due to anRMP.

Figure 2.5 shows a typical safety factor radial profile,q(r), calculated by the MST-Fit [59] (a non-linear, fixed boundary Grad-Shafranov solver) using experimental mea-surements as input. Historically, the toroidal mode numbers n in MST are defined withthe opposite sign to EXTRAP T2R, i.e. the resonant conditionis instead:q(r) = m/n.Thus, the modes inside the reversal surface have positiven and the most central mode isharmonics(m= 1,n= 6). MST has a lower absolute value of the central mode number(|n|= 6) compared to EXTRAP T2R (|n|= 12), which is mainly due to the different torusdimensions (aspect ratioRa/a).


0 0.2 0.4 0.6 0.8 1

−0.05

0

0.05

0.1

0.15

0.2

Saf

tey

fact

or, q

Minor radius, r/a

m=0

n=7n=6

Figure 2.5: Typical safety factor profile in MST. The resonant TMs are indicated by blackdots.

A special feature of MST is the highly conducting aluminum shell, which also actsas a vacuum vessel. With a shell time constant about 10 times longer than the dischargeduration, the shell can be considered thick and the mode amplitudes are zero at somedistance inside the shell [56]. However, to let the flux into the vessel, the shell has apoloidal and a toroidal gap. These gaps are also a source of magnetic field errors. At thepoloidal gap, the error-field is corrected by radial coils ina feedback system.

Feedback system and RMP application

The magnetic feedback system consists of 32 sensor coils, 38actuator coils, and a con-troller [60]. The coils are located at the poloidal gap at onetoroidal location, see Figure 2.6.The extent of the gap is less than 0.5 degree in toroidal angle. Therefore, then spectrumis broad and uncontrollable. Both vacuum measurements and calculations considering thegeometry show that the applied amplitude is similar for all then modes of interest [PaperII]. Thanks to numerous coils in poloidal direction, the poloidal mode numbers 0≤m≤ 16are controllable.

In this study, the feedback system is used to produce anm= 1 perturbation. Thecontribution to eachn is approximately 1/100 of the total appliedm= 1 amplitude, i.e.bm=1,n

r ≈ bm=1r /100. The RMP has constant phase (αRMP= 0) and the maximum amplitude

is varied from shot-to-shot.

2.2. THE MADISON SYMMETRIC TORUS DEVICE 23

Figure 2.6: Lower part of MST’s toroidal shell, with actuator coils (green) and sensorcoils (white) located around the poloidal gap. Courtesy of MST-group at University ofWisconsin, Madison, USA.

Momentum injection via a biased probe

As an alternative method to determine the viscosity, momentum was injected in the plasmaby using a biased insertable probe (Figure 2.7). The probe [39] was biased to +400 Vrelative to the plasma potential. The biasing leads to a charge-up of the flux surface near theprobe-tip. As a consequence, a radial electric field and a return current is created betweenthe flux surface and MST’s conductive shell. In interaction with the magnetic equilibriumfield, the return current produces a torque that spins up the edge plasma. Thereafter, thecore follows via the viscous momentum transfer. Here, we determined the edge flow bymeasuring the Doppler shift of the CIII ions which are located in the edge-region. Thecore flow was inferred from the measured phase velocities of the (core) resonant tearingmodes. By matching the momentum equation to the measured flowchange we estimatedthe experimental viscosity.

The tearing mode measurements

The core-resonantm= 1 tearing modes (TMs) are measured with a toroidal array of 32equally spaced poloidal magnetic field pick-up coils, whichare located at the inner surfaceof the shell (r = 0.52 m). The measurement and analysis are essentially the sameas forEXTRAP T2R [Paper II].

Similar to EXTRAP T2R, the tearing modes in MST co-rotate with the plasma [56].Therefore, we applied the no-slip condition in the modeling.


Figure 2.7: Schematic sketch of the biased probe. Courtesy of MST-group at University ofWisconsin, Madison, USA.

Chapter 3

Modeling the tearing mode dynamics

Tearing mode time evolution under the influence of an RMP has been theoretically de-scribed by several works [25,27,61,62]. Typically the TM-island evolution is modeled viathree coupled equations; (I) the plasma fluid equation of motion, (II) the no-slip conditionand (III) the modified Rutherford equation. Here, (I) describes the fluid motion consideringthe EM-torque due to the RMP, which is acting locally at the resonant surface. The no-slip condition (II) ensures the co-rotation of the plasma and TM at the resonant surfaces.The co-rotation is a reasonable assumption for the investigated experimental cases [Pa-per I-VII]. The modified Rutherford equation (III) describes the TM amplitude evolutionconsidering the effect of the RMP.

The following sections describe the model used for MST (thick resistive shell). Themodel includes the time evolution of multiple (m= 1) TMs with toroidal harmonicsn,which are affected by a multi-harmonic RMP. In the text below, it will be mentioned whenEXTRAP T2R (thin resistive shell) differs from the presented MST model.

3.1 Geometry and equilibrium

The model assumes a large aspect ratio and zero pressure. These are reasonable first-orderapproximations for EXTRAP T2R and for MST in standard discharges, as mentioned inChapter 2. With these assumptions, the equilibrium is well approximated by a periodiccylinder and cylindrical coordinates are adopted(r,θ ,z), where the toroidal angle is definedby φ = z/R0.

The force balance for the equilibrium magnetic field (B = [0,Bθ ,Bφ ]) is ∇×B =σ(r)B, whereσ(r) is the parallel current profile. MST equilibrium profiles arecalcu-lated by MSTFit [59], using the experimental values as input. For EXTRAP T2R, theequilibrium profiles are calculated with theα −θ0 model [51] using experimental values.

25

26 CHAPTER 3. MODELING THE TEARING MODE DYNAMICS

3.2 Magnetic perturbations

For a single mode, the perturbed magnetic field can be expressed as

b(r , t) = bm,n(r, t)exp[i(mθ −nφ +αm,n(t))], (3.1)

where,

bm,nr =

iψm,n

r, (3.2)

bm,nθ =− m

m2+n2ε2

dψm,n

dr+

nεσm2+n2ε2 ψm,n, (3.3)

bm,nφ =

nεm2+n2ε2

dψm,n

dr+

mσm2+n2ε2 ψm,n, (3.4)

andε = r/R0. ψm,n is the linearized magnetic flux function that satisfies Newcomb’sequation [13]

ddr

[ f m,n dψm,n

dr]−gm,nψm,n = 0, (3.5)

where

f m,n(r) =r

m2+n2ε2 , (3.6)

and

gm,n(r) =1r+

r(nεBθ +mBφ )

(m2+n2ε2)(mBθ −nεBφ )

dσdr

+ (3.7)

+2mnεσ

(m2+n2ε2)2 − rσ2

m2+n2ε2 .

Equation (3.5) is singular at the TM resonant surface,r = rm,ns , which satisfies

q(rm,ns ) =

rnBφ (rm,ns )

R0Bθ (rm,ns )

=mn. (3.8)

Here, the safety factor radial profile,q(r), is calculated using experimental measure-ments. (As mentioned in Chapter 2, the definition of the resonant condition for EXTRAPT2R includes a minus sign, i.e.q(r) =−m/n.)

The model assumes that the resonant torques arise from the interaction between thehelical current that flows inside the tearing mode and currents flowing in the external RMP-coils or/and in the resistive shell. Therefore, we divide the flux function into three parts,corresponding to the three different currents [57]

ψm,n(r, t) = Ψm,ns (t)ψm,n

s (r)+Ψm,nb (t)ψm,n

b (r)+Ψm,nc (t)ψm,n

c (r), (3.9)

3.2. MAGNETIC PERTURBATIONS 27

whereΨm,ns (t), Ψm,n

b (t) andΨm,nc (t) contain the phase and the amplitude of the pertur-

bation at the resonant radiusr = rm,ns , the shellr = rb, and RMP coilsr = rc, respectively.

The eigenfunctionsψm,ns (r), ψm,n

b (r), andψm,nc (r) are solutions to Eq. (3.5).

There is a difference in flux function for EXTRAP T2R and MST caused by the dif-ferent time scales of the shell diffusion. Considering EXTRAP T2R, the field at the innershell-surface approximately equals that on the outer surface. The radial variation inside theshell is assumed to be zero. This is described by the thin-shell approximation [63], whichleads to the following thin-shell dispersion relation [57]

∆Ψm,nb = inωm,n(t)τbΨm,n

b , (3.10)

whereωm,n(t) is the angular velocity of the TM,τb is the shell diffusion time. Contraryto EXTRAP T2R, MST has a thick conductive shell that hinders the field penetration andthe thick-shell approximation is reasonable. By assuming that the field outside the shell iszero, the following dispersion relation is obtained for theshell [57]:

∆Ψm,nb =

(

inωm,n(t)τbrb

δb

)1/2

Ψm,nb , (3.11)

whererb is the shell minor radius andδb is the thickness of the shell.For MST, the radial profiles of the eigenfunctions are drawn in Figure 3.1 and boundary

conditions are:

• ψm,ns (0) = 0, ψm,n

s (rs) = 1, andψm,ns (rb) = 0

• ψm,nb (rs) = 0, andψm,n

b (rb) = 1

• ψm,nc (rs) = 0, ψm,n

c (rc) = 1

where it is assumed that the RMP coils are positioned at minorradiusr = rc = 0.53 m,which means 1 cm outside the inner shell surfacer = rb. This assumption was made sincethe perturbation enters through the poloidal gap.

In EXTRAP T2R, the function describing the interaction withthe wall is non-zerooutside the wall and instead zero at the coils, i.e.ψm,n

b (rc) = 0. Other boundary conditionsequal those of MST.

The discontinuities in flux function are represented by

∆Ψm,ni = r

dψm,n

dr

∣

∣

∣

∣

r+i

r−i

, (3.12)

and

Em,ni j = r

dψm,nj

dr

∣

∣

∣

∣

r+i

r−i

, (3.13)

wherei and j can bes (TM island),b (shell) orc (RMP coils). For more details seereferences [56,57].


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

r/a

ψim

,n ψsm,n

ψcm,n

ψbm,n

rc

rb

rsm,n

Figure 3.1: Typical shape of the three tearing mode eigenfunctionsψm.ns , ψm.n

b andψm.nc ,

which are representing the interaction at the resonant surface (s), the wall (b) and the RMPcoils (c), respectively.

3.3 Electromagnetic torques

Here follows a brief derivation of the EM-torque terms for MST. The toroidal componentof the EM-torque acting on the TM is given by [25]

Tm,nEM (t) =Cm,nIm∆Ψm,n

s (Ψm,ns )∗= Tm,n

wall(t)+Tm,nRMP(t), (3.14)

where

Cm,n =2π2R0

µ0

nm2+(nrm,n

s /R0)2 . (3.15)

Using Eq. (3.9) and Eq. (3.11-3.13), we get:

Tm,nwall(t) =−Cm,n 1√

2

√

Km,n(t)Em,nbs Em,n

sb

Km,n(t)−√

2Km,n(t)Em,nbb +(Em,n

bb )2|Ψm,n

s (t)|2, (3.16)

and

Tm,nRMP(t) =−Cm,nEm,n

sc |Ψm,nc (t)||Ψm,n

s (t)|sin[∆αm,n(t)] , (3.17)

where

Km,n(t) = nωm,n(t)τbb/δb, (3.18)

3.4. FLUID EQUATION OF MOTION INCLUDING EM-TORQUE SOURCE TERMS29

Theαm,n(t) is the phase of the TM and∆αm,n(t) its phase difference to the RMP. Thesine function inTRMP means that the RMP will accelerate or decelerate the TM dependingof the relative phase∆αm,n(t). However, the average effect is a reduction of TM rotation.The electromagnetic torqueTm,n

EM depends on both the RMP and TM amplitude through|Ψm,n

c | and |Ψm,ns | respectively. TheEm,n

i j constants are calculated according to Section3.2. The exception isEm,n

bb . Instead,Em,nbb = −Em,n

sb Em,nbs /(Em,n

ss (rp)−Em,nss (rb)) where the

tearing stability index (Ess) is calculated assuming a perfectly conducting shell at thewallr = rb and atr = rp → ∞, respectively [57].

For MST,Tm,nEM is given by Eq. (3.14). While, the form ofTm,n

EM is different in EXTRAPT2R due to the thin shell (see derivation in Ref. [61]).

3.4 Fluid equation of motion including EM-torque source terms

Theoretically, the modification of the fluid velocity due to an RMP is determined by twocompeting torques; the viscous torque (Tvisc) that acts to oppose changes in the plasmarotation and the EM-torque (TEM) produced by the interaction of the RMP and a TM islandresonant at minor radiusrm,n

s [25, 27, 61]. The equation of fluid motion can be expressedas [26,27,61]:

ρ(r)∂∆Ω

∂ t=

1r

∂∂ r

(

rµ⊥(r)∂∆Ω∂ r

)

+∑n

Tm,nEM (t)

4π2rR30

δ (r − rm,ns ), (3.19)

where the first term on the right-hand side is the viscous torque density,ρ(r) is theplasma mass density and∆Ω(r, t) the plasma angular velocity reduction profile. The per-pendicular dynamic viscosityµ⊥(r) is related to the kinematic viscosityν⊥(r) throughµ⊥(r) = ρ(r)ν⊥(r). In the second term, the delta function implies that the EM torque actsonly at the resonant radius.

3.5 Tearing mode island angular evolution

The no-slip condition [26] equates the rotation of the TM island and the plasma flow at theresonant surface,

dαm,n(t)dt

≡ ωm,n(t) = nΩ0(rm,ns , t)+n∆Ω(rm,n

s , t) (3.20)

whereΩ0 is the unperturbed toroidal angular velocity profile andrm,ns the resonant radius.

3.6 Tearing mode amplitude evolution

The TM amplitude evolution is modeled by the modified Rutherford equation (MRE) [64,65]. Frassinetti et al derived the MRE in EXTRAP T2R thin-shell regime [28]. A similar


derivation, following the logic in Ref. [65] and using the thick-shell approximation, resultsin the MRE used for MST

dΨm,ns (t)dt

=Gm,n

Λτm,nr

(

[

Em,nss +Dm,n(t)

]√

|Ψm,ns | +

|Ψm,nc |

√

|Ψm,ns |

Em,nsc cos[∆αm,n(t)]

− ΛGm,n (λ

m,n0 )2 ln

[

Gm,n√

|Ψm,ns |

]

)

, (3.21)

where

Dm,n(t) = Em,nsb Em,n

bs

√

Km,n

2−Em,n

bb

Km,n−√

2Km,n+(Em,nbb )2

, (3.22)

Gm,n =

(

Bθ R0

n

[

rσ(m2+n2ε2)−2mnε]

)(1/2)

r=rm,ns

, (3.23)

λ m,n0 =

r2σ ′

rσ −2mnε/(m2+n2ε2)

∣

∣

∣

∣

r=rm,ns

, (3.24)

τm,nr =

µ0r2

η

∣

∣

∣

∣

r=rm,ns

, (3.25)

Λ ≈ 1.6 andη is the parallel resistivity. The time scale is in the order ofthe resistivetime scaleτr . In Eq. (3.21) on the right side, the first term is the linear instability drive(or stability if Em,n

ss < 0). The second term is the TM’s interaction with the shell. The thirdterm is the TM’s interaction with the RMP, which produce oscillations in the amplitudeaccording to cos[∆αm,n(t)]. The last term is the non-linear saturation.

3.7 Example: Modeled plasma velocity evolution under influence ofan RMP

This section shows an example of the change in the plasma flow,∆vφ (r, t), under the impactof an RMP.

The model equations are implemented in MATLAB1. The fluid equation of motion (Eq.3.19) is discretized in space using Method-of-Lines (MOL),which allows solving the fullsystem as a set of coupled ordinary differential equations (ODEs). The initial conditionis ∆vφ (r,0) = 0 and boundary conditions are∆vφ (a, t) = d∆vφ (0, t)/dt = 0, wherer =a = 0.51 m is the plasma minor radius in MST, and∆vφ = ∆ΩR is the toroidal velocityreduction profile of the plasma.

Throughout this thesis, we assumed a uniform viscosity in the whole plasma,µ⊥(r) =ρ(0)ν⊥(0), andν⊥ was changed to match the experimental deceleration-curve of the TM.Other input-parameters are measured in MST.

1MATLAB Release R2012b, The MathWorks, Inc., Natick, Massachusetts, United States

3.7. EXAMPLE: MODELED PLASMA VELOCITY EVOLUTION UNDERINFLUENCE OF AN RMP 31

The RMP amplitude is from experimental data (same as shown later in Chapter 4 Figure4.4). For clarity, only a single harmonic(m= 1,n= 6) is considered in the model. TheRMP is applied in the time intervalt ≈ 0.5−6.5 ms. Within this time, the RMP amplitudeis increased frombm=1,n=6

r ≈ 0 mT to maximumb1,6r ≈ 0.13 mT. After 6.5 ms, the RMP is

turned off and the amplitude falls tob1,6r ≈ 0 mT.

The modeled∆vφ (r, t) is shown in Figure 3.2. The RMP acts at the TM resonantradius,rs ≈ 0.15 m. Initially, phases of acceleration and deceleration are produced, dueto the EM torque acting on the TM island. However, on average the RMP decelerates theplasma flow. The(1,6) mode locks att ≈ 6.5 ms. After TM locking, the plasma velocityreduction spreads due to viscosity and the profile relaxes. The relaxation results in a lowerviscous restoring torque. Therefore, when the RMP is turnedoff, the mode remains locked(t ≥ 6.5 ms).

Figure 3.3 shows a top view of∆vφ (r, t) in a time interval of the braking and subse-quent locking. The first interaction is clearly at the resonant radius, whereby the velocityreduction travels radially via the viscosity. There is still a remaining viscous torque butit is too small to unlock the TM and the plasma. As described inthe next Chapter, theelectromagnetic interaction with the conducting wall can sustain the locked state.

Figure 3.2: Modeled velocity reduction profile∆vφ (r, t) under the influence of an RMPramp.


Figure 3.3: Modeled velocity reduction profile∆vφ (r, t); top-view at the time around lock-ing in figure 3.2. The RMP acts at the (1,6) resonant radius (r = 0.15 m).

Chapter 4

Results

The first part of this chapter presents experimental measurements and modeling of TMdynamics under the influence of external RMPs. The experimental results are comparedwith the model that is outlined in the previous chapter. Then, we give a short summary ofthe viscosity measurements performed in MST at varying TM amplitudes.

4.1 EXTRAP T2R results

Mechanisms behind the hysteresis in TM-locking and -unlocking

In the experiments, an RMP of the harmonic (m= 1,n = −15) was applied during theplasma current flat-top. We found that the RMP amplitude required to lock a (m= 1,n=−15) TM in EXTRAP T2R, i.e. the locking threshold, isbRMP

r ≈ 0.6 mT. To release thelocked TM, the RMP amplitude had to be reduced tobRMP

r ≈ 0.05 mT. Hence, there ishysteresis in the locking/unlocking process. In Paper II [66], the mechanism behind thehysteresis was experimentally studied in EXTRAP T2R and compared with theory.

Theoretically, the lower unlocking threshold is mainly ascribed to two mechanisms.The first is the relaxation of the plasma velocity in the core that occurs after the TM locks.The relaxation is caused by the viscous momentum transfer from the resonant surface.Naturally, the reduced plasma flow leads to a lower viscous restoring torque, as describedby the model in Chapter 3. Second, there is a deepening of the hysteresis because ofthe increase in TM amplitude, which occurs at the TM locking.We have observed bothmechanisms in EXTRAP T2R, in qualitative agreement with thetheory.

To illustrate the above mechanisms, Figure 4.1 shows the experimental TM evolutionin a shot where the (m= 1,n= −15) RMP amplitude is first ramped up and then rampeddown. Braking of then= −15 TM starts at≈15.2 ms (bRMP

r ≈ 0.4 mT) and the lockingoccurs at≈16.0 ms (bRMP

r ≈ 0.6 mT). The TM unlocked at≈18ms, when the RMP wasreduced tobRMP

r ≈ 0.05 mT. The reason for this behavior is examined by the radial ve-locity reduction profile (∆vφ ) during braking and after locking. Just before then = −15mode locks, the velocity reduction profile peaks at then = −15 resonant surface wherethe RMP is acting [blue dots in Figure 4.1(d)]. After lockingof then= −15, the velocity

33

34 CHAPTER 4. RESULTS

reduction spreads via the plasma viscosity. The relaxed profile [red diamonds in Figure4.1(d)] reduces the restoring viscous torque. This partly explains the hysteresis. Further-more, the increase in TM amplitude [Figure 4.1(c)] deepens the hysteresis by increasingthe EM torque.

t1t2

t1 t2

profile

relaxes

Figure 4.1: (a) Experimental (m= 1,n=−15) perturbation amplitude, and (b-c) the (m=1,n=−15) tearing mode time evolution. Data are time-averaged in a0.5 ms time window.The vertical lines in frames (a-c) indicate the time instants of the velocity reduction profilesin frame (d), where the velocity reduction of each TM (n< −12) is plotted at its resonantradius. The initial (pre-RMP) velocity is taken att = 14 ms.

To compare with the experiments, the model in Chapter 3 was adapted to EXTRAPT2R experimental parameters. The kinematic viscosity (ν⊥) was adjusted to match theexperimental TM braking. The value required by the model wasin the order ofν⊥ ≈ 1m2/s. This value is anomalous with about one order of magnitude. The source of theanomaly is discussed in Chapter 5.

Figure 4.2 (b-c) shows the modeled TM behavior under the influence of an RMP [Fig-ure 4.2 (a)]. The model reproduces, qualitatively, the samebehavior of TM braking, lock-ing and unlocking. The threshold amplitudes for locking andunlocking differs slightlycompared with the experiment. However, a perfect agreementmay not be expected con-sidering the uncertainties in model input, for example, plasma viscosity and density pro-files. Similar to the experiment, the modeled velocity reduction profiles [Figure 4.2(d)]

4.1. EXTRAP T2R RESULTS 35

relaxes after the TM locking. Furthermore, the increased width of the locked TM-islandFigure [4.2(c)] qualitatively agrees with the experimentally observed increase in TM am-plitude. Thus, both mechanisms that cause the hysteresis are in reasonable agreement withexperiment.

0 5 100

1

2

W/W

0v φT

M (

km/s

)

0

10

20

time (ms)

b rRM

P (

mT

)

0.00.20.40.60.8

t1 t

2

TM velocity

TM island width

(b)

(a)RMP

(c)

0 0.5 1−30

−25

−20

−15

−10

−5

0

5

r/a

∆vφ (

km/s

)

t1t2

(d)

profilerelaxes

Figure 4.2: (a) Modeled (m= 1,n = −15) perturbation amplitude, and (b-c) the (m=1,n= −15) tearing mode time evolution. The vertical lines indicate the time instants ofthe velocity reduction profiles plotted in frame (d).

To further investigate the RMP threshold amplitude required for mode locking and un-locking, we applied (1,-15) RMPs of various amplitudes in EXTRAP T2R. First, the max-imum amplitude of the RMP was changed from shot-to-shot to find the locking threshold.To find the unlocking threshold, we locked the (1,-15) mode via the RMP and then de-creased the RMP to a lower amplitude, which was also varied from shot-to-shot. In Figure4.3, the TM velocity is plotted during the first RMP step (black circles) and during/after theRMP step-down (white circles). There is an obvious hysteresis since the locking thresholdis ten times higher than that for unlocking.

Under experimental like conditions, the model (Chapter 3) was used to simulate TMlocking and unlocking to an RMP. To investigate the locking threshold, the simulationwas repeated several times applying an RMP square waveform,each time using differentmaximum amplitude. TM unlocking was explored by the introducing a second step inwhich the RMP amplitude is decreased after the TM locking. This amplitude varied ineach simulation to find the model estimate of the unlocking threshold.

The result is shown in Figure 4.3. The blue continuous line shows the modeled ve-locity reduction up to the TM locking (increasing RMP amplitude) and the red continu-ous line shows the TM unlocking (decreasing RMP amplitude).Qualitatively, the modelagrees with the experiment. Concerning the absolute value of the locking threshold, the


30

20

10

0

0.0 0.2 0.4 0.6 0.8 1.0

br

RMP(mT)

br

RMPincreasi

br

RMPdecreasi

LockingUn-locking

vTM( km/ s)

ng

ng

Model

Figure 4.3: Velocity of the (1,-15) TM with increasing RMP amplitude (black circles) anddecreasing RMP amplitude (white circles) for a series of shots. The shaded area aroundbRMP

r ≈ 0.6 mT indicates the locking threshold, and the shaded area aroundbRMPr ≈ 0.05

mT indicates the unlocking threshold. The blue line is the simulated TM velocity withincreasing RMP amplitude, and the error bars indicate the maximum and minimum ofthe velocity oscillations. The red line is the simulated TM velocity with decreasing RMPamplitude.

agreement was obtained by adjusting the kinematic viscosity (ν⊥ ≈ 1m2/s). Then, the un-locking threshold amplitude estimated by the model isbr ≈ 0.01 mT, which is lower thanthe experimental value (br ≈ 0.05−0.1 mT). There are some possibilities for the disagree-ment. First, in some of the experimental shots, the velocityreduction profile was not fullyrelaxed before the RMP ramp-down, e.g. Figure 4.1(d). In this case, the experimentalunlocking threshold should increase because the restoringviscous torque is larger than inthe fully relaxed case. Second, in the experiment, other MHDevents (that are not mod-eled) may produce unlocking by redistribution of momentum.This is sometimes observedin the so-called sawtooth events, which includes non-linear mode coupling. Third, thereare uncertainties in model input parameters that complicate a quantitative comparison. Forexample, the density was not measured in these EXTRAP T2R discharges, but instead as-sumed based on previous results. Possibly, a shot-to-shot deviation in the density couldexplain the spread in the experimental velocity (black circles in Figure 4.3). As shownlater, the plasma density affect the mode deceleration and the locking threshold in MST(Figure 4.8).

4.2. MST RESULTS 37

4.2 MST results

The RMP effect on TM locking and unlocking

Tearing mode dynamics was experimentally studied in MST under the influence of anRMP. The RMP was applied during the plasma current flat-top phase and its harmonicswere poloidalm= 1 and several toroidaln, as described in Chapter 2. As a consequence,all the tearing modes decelerated simultaneously. This is also predicted by the modelin Chapter 3 by considering the multiple-harmonics RMP spectrum, which produces anelectromagnetic torque on each TM.

For the modeled TM amplitudes, either the Rutherford equation or the experimentalTM amplitudes can be used as model input. In the first example,the Rutherford model wasused. The model included the eight inner TMs, i.e. the EM torques acting on the (m= 1)tearing modes with toroidal mode numbersn= 6 – 13.

Figure 4.4 shows experimental and modeledn= 6 TM velocity, amplitude, and phase.The RMP amplitude [Figure 4.4 (a)] increased from zero to about 13 mT. When the ampli-tude reached|bm=1

r | ≈ 10 mT, the tearing modes decelerated until they became arrested inthe lab frame (Figure 4.4 (b)). When the RMP was turned off, the TMs remained locked.Theoretically, this may be explained by the reduced rotation of the surrounding plasma andthus reduced viscous (restoring) torque (as shown in Chapter 3, Figures 3.2–3.3). Morespecifically, the viscous torque is proportional to the second derivative of the plasma ve-locity reduction profile, which is expressed as∆vφ (t, r) = vφ (0, r)−vφ (t, r) wherevφ (0, r)is the natural (unperturbed) velocity. Thus, to examine thereduction of viscous torque afterTM locking, the experimental velocity reduction of the tearing modes were compared withthe modeled plasma velocity reduction radial-profiles (Figure 4.5).

Figure 4.4 contains three time intervals (shaded areas) which were used for calculatingthe velocity reduction profiles. The initial velocity,vφ (t1), was subtracted from the time-averaged velocities att2 andt3, respectively. The resulting velocity reduction profiles areshown in Figure 4.5. The agreement with the model is good, within the 1σ error bars, bothduring the braking phase (t2), and the locked phase (t3).

The reduction of plasma flow in the vicinity of the tearing modes may partly explainwhy the modes remain locked after the RMP is turned off. Thereis, however, anotherimportant factor in the unlocking mechanism namely the wall-torque. Initial studies sug-gest that the wall-torque, which depends onb2

TM (see Chapter 3), can explain whether themodes in MST unlock or not. To get a more accurate estimation of the wall-torque, themeasured TM amplitudes were used as model input instead of using the Rutherford equa-tion. Figures 4.6-4.7 illustrates the role of the wall-torque. In Figure 4.6, the modes remainlocked after the RMP is turned off but the model can only replicate this behavior with thewall-torque included. In Figure 4.7 the mode amplitudes aresmaller than in the previouscase and the wall-torque is too low to sustain the locked state. Note that the wall-torque istypically negligible during the RMP application (Figure 4.6) but can still be large enoughto sustain the locked state after the RMP is turned off.

To summarize, the model including several modes can qualitatively explain the experi-mental behavior and quantitative agreement is obtained by adjusting the kinematic viscos-


10

5

0

(a)

|brm

=1 | (

mT

)

1

0.5

0

(b)

|bθ1,

6 | (m

T)

(c)

time (ms)

v φ1,6 (

km/s

)

17.5 19.5 21.5 23.5 25.5

0

20

40

exp.model

0

20

40

v φ1,6 (

km/s

) (d)

23.0 23.3 23.6 23.9 24.2 24.5

1 (e)

−1

0

time (ms)

∆α1,

6 /πt1

t2

t3

Figure 4.4: Time evolution of (a) the experimentalm= 1 RMP amplitude (also used asmodel input), and then = 6 mode’s (b) amplitude and (c) velocity. The data is time-averaged in a 0.5 ms time window. The filled areas indicate thethree time intervals (t1, t2andt3) that were used for calculating the velocity reduction profiles in Figure 4.5. In ashort time interval, frame (d-e) contains the non-smoothedand the modeledn= 6 velocityand phase.

4.2. MST RESULTS 39

0 0.2 0.4 0.6 0.8 1

−40

−30

−20

−10

0

r/a

∆vφ (

km/s

)

t2 exp.

t2 model

t3 exp.

t3 model

Figure 4.5: Experimental and modeled velocity reduction attime intervalst2 andt3 com-pared to the initial velocity (t1). The time intervals are indicated in Figure 4.4. The radiallocations are given by the resonant condition [Eq. ( 3.8)] wheren= 6 is the most centralandn= 15 is furthest out.

ity. The viscosity is further discussed at the end of this chapter.

The short timescale TM dynamics

In Figure 4.4 (a-c) the data were time-averaged in a 0.5 ms time window, to highlight thelong (millisecond) time-scale behavior. The short time-scale behavior is shown in Figure4.4 (d-e), where the authentic (non-smoothed)n = 6 TM signals are plotted in a timeinterval around the braking/locking. Qualitatively, the model is able to explain the shorttimescale behavior. Quantitatively, the TM locking att ≈ 24 ms is well described, butthe amplitude of the TM velocity oscillations is overestimated in the earlier braking phase(t ≈ 23.0−23.4 ms).

Both in experiment and model, the peaks of the velocity oscillations are narrower thanthe valleys, which is most apparent just before locking. Furthermore, the TM spends alonger time out-of-phase (|∆αm,n| ≥ π) than in-phase (|∆αm,n| ≤ π) with the RMP, asshown in Figure 4.4 (e). In theory this is explained by the phase dependence of the EMtorque, which isTEM ∝ −sin(∆αm,n). Thus, the TM is decelerated in the phase interval(0< ∆αm,n < π) and accelerated in the interval(−π < ∆αm,n < 0), which means that theminimum TM velocity should be reached at∆αm,n = π and the maximum at∆αm,n = 0.


|br| (

mT

)

05

10

0

10

20

v φ1,5 (

km/s

)

time (ms)

mod. w/o Twall

mod. w/ Twall

exp.

0

10

20

v φ1,6 (

km/s

)

mod. w/o T

wall

mod. w/ Twall

exp.

20 25 30 350

1

2

b θ(r=

b) (

mT

)

time (ms)

RMP on

n=5n=6

(d)

(c)

(b)

(a)

Figure 4.6: Time evolution in an experimental case with plasma current 200 kA. (a) RMPamplitude, (b-c) the (1,5) and (1,6) mode velocities, and (d) the mode amplitudes measuredat the edge. The modeled solution applies the RMP and the walltorque (red full lines) orthe RMP torque only (dashed lines).

The theoretical model agrees with the experimentally measured phase. Furthermore, theexperimental locking occurs in phase with the RMP, as predicted by the model.

The mode-locking dependence on plasma density

A set of plasma discharges were performed under similar equilibrium conditions (plasmacurrentIp, pinch-parameterΘ and reversal-parameterF) and similar initial TM velocities.However, the central line-averaged electron density varied between 0.35 and 1.25×1019

m−3. In each shot, the RMP amplitude was increased to a constant amplitude, whichvalue varied from shot-to-shot. The experimental data in Figure 4.8 shows that the lockingthreshold increase with increasing electron density. In the analysis, shots were divided intothree groups of different electron density; low density< ne>= 0.35−0.5×1019 m−3 (redcircles), medium density< ne >= 0.5− 0.8× 1019m−3 (blue squares) and high density

4.2. MST RESULTS 41

0

5

10|b

r| (m

T)

0

10

20

v φ1,5 (

km/s

)

time (ms)

mod. w/ T

wall

exp.

0

10

20

v φ1,6 (

km/s

)

time (ms)

mod. w/ Twall

exp.

24 26 28 30 32 34 36 38 400

1

2

b θ(r=

b) (

mT

)

time (ms)

RMP on

n=6n=5

(c)

(b)

(d)

(a)

Figure 4.7: Time evolution in a case with plasma current 125 kA. (a) RMP amplitude, (b-c)the (1,5) and (1,6) mode velocities, and (d) the mode amplitudes measured at the edge. Themodeled solution applies the RMP and the wall torque (red full lines).

< ne >= 0.8−1.25×1019 m−3(green diamonds).

The model was run for three different plasma densities, corresponding to the experi-mental densities. The RMP was increased to a constant amplitude that varied from simulation-to-simulation to find the locking threshold. In Figure 4.8, the model produced a similarincrease in TM locking threshold with the increasing density. The kinematic viscosity waskept constant in the three cases, which indicates that it is not strongly dependent on theplasma density. The Braginskii [34] classical kinematic viscosity is about two orders ofmagnitude lower than the experimental value. The result is,however, in-line with the onein Ref. [67], where it was found that the momentum transport in the MST core-region wasconsistent with magnetically driven transport [Eq. (1.9)]. The good agreement motivatedfurther exploration of the viscosity at varying levels of magnetic fluctuation amplitude [Pa-pers VI-VII].


0 5 10 15

0

10

20

30

40

br/B

θ(a) (%)

v φm=

1,n=

6 (km

/s)

Exp.

Mod.

<ne> 1019 m−3 0.35−0.5

0.5−0.80.8−1.25

0.410.640.90

Figure 4.8: The model compared with experimental data for then= 6 TM velocity versusapplied perturbation field (as percentage of the poloidal equilibrium field at the edge). Theelectron density is divided into three intervals. The velocities are time averaged in theinterval 2.0-2.25 ms into the perturbation phase. The poloidal equilibrium field at the edgeis similar for all shots,Bθ (a)≈ 0.13 T.

Momentum injected via a biased probe

In addition to the RMP-technique, we also determined the viscosity by modeling of thetransport of momentum injected via a biased probe, which wasbriefly described on page24. The probe-technique provided a valuable cross-check tothe viscosity inferred by theRMP-technique. Paper VII describes the results and conclusions of the method. Here,we provide an example of the probe’s effect on the plasma flow and compare with themodeling (Figure 4.9). The probe was biased to about +400 V and current drawn by theprobe was∼ 1 kA. In response to the biasing, the edge flow increased quickly [Figure 4.9(b)]. The core flow increased gradually and the accelerationstarted in the modes closest tothe edge, i.e. closest to the probe [Figure 4.9(d)]. In the end of the 10 ms probe pulse, theincrease in the core flow was similar to that in the edge which indicates a quick momentumtransport, i.e. a relatively large viscosity. The deceleration phase, after the probe wasturned off, resembles the acceleration phase when the probewas turned on. First, the edgeflow decreased quickly. Whereas, the deceleration of the core mode was most delayed.The time evolution of the flow could qualitatively be described by momentum diffusingfrom the edge to the core with a uniform and time-independentviscosity. The timescale ofmomentum transport is about hundred times shorter than the classical prediction. Instead,the anomalous transport agrees with momentum diffusion along the stochastic magneticfield [39,41].

4.2. MST RESULTS 43

(a)

(b)

(c)

t t t

t t t

(d)

t

1

2 3

32

0 1

voltage

current

Figure 4.9: (a) The bias voltage (Vmax≈ 0.4 kV) and the current (Imax≈ 1.0 kA) in theprobe. (b) The CIII toroidal flow velocity and the modeled plasma flow atr/a = 0.81.(c) The toroidal phase velocity of the central (1,5) tearingmode, and the modeled plasmaflow at the (1,5) resonant surface. (d) The perturbed velocity profile at the three instants oftime [vertical lines in (a-c)]. The experimental profiles are the change in the rotation of themodes (n= 5−10) and the modeled profiles are the change in plasma flow. The 1σ errorbars represents the change in each 2 ms time window.


Viscosity dependence on the amplitude of the tearing modes

To estimate the viscosity at varying level of intrinsic magnetic fluctuation, we separatelyapplied the probe-technique and the RMP-technique in various MST plasmas. In eachcase, the viscosity was determined by modeling the momentumtransport, as described inthe previous sections.

0 1 20

10

20

30

40

50

60

70

br/B (%)

ν ⊥ (

m2 /s

)

RMPProbe

(a)

br/B (%)

Dm

= ν

⊥/c

s (10

−4 m

)

0 1 20

1

2

3

4

5

6

(1.07±0.12)x2.13±0.19

(b)

Figure 4.10: (a) The experimental viscosity and (b) the magnetic diffusion coefficient, bothplotted versus the magnetic fluctuation amplitude (brms/B).

Our results showed that the viscosity increased with the level of magnetic fluctuations[Figure 4.10 (a)]. In a model that describes momentum transport in a stochastic magneticfield [Eq. (1.9)], the viscosity is expected to be dependent on the magnetic fluctuation tothe power of two(br/B)2 and the sound speedcs. In the experiment, both thecs and(br/B)varies (see Papers VI–VII). To investigate the(br/B) dependence, Figure 4.10 (b) showsthe viscosity divided by the calculated plasma sound speed,i.e. the magnetic diffusion co-efficientDm = ν⊥/cs. The experimental scaling ofDm (Figure 4.10) is in close agreementwith the(br/B)2 scaling predicted by the model in Eq. (1.7).

Chapter 5

Discussion

5.1 RMP effect on the dynamics and locking of TMs

The effect of an applied RMP on initially rotating tearing modes has been investigatedin the two RFPs: EXTRAP T2R (Papers I-II and V) and MST (Paper III). We observedthat the RMP reduced the tearing mode velocity. Locking of a TM relative to the staticRMP phase was observed if the RMP amplitude increased above athreshold. The RMPamplitude had to be significantly reduced to unlock the TM. The mechanisms behind thelower unlocking threshold agreed with those of a theoretical model described by Fitzpatricket al [27]. On a shorter timescale, the RMP produced oscillations in the TM velocity thatgrew in amplitude until the mode locking. The short timescale oscillations have previouslybeen observed in, for example, EXTRAP T2R [40] and in ASDEX-upgrade tokamak [33].In MST, we simultaneously matched the model with the (long timescale) mode lockingand the (short timescale) oscillations. The match was obtained by varying the kinematicviscosity, which was the only free parameter. The model-required viscosity was anomalousby two orders of magnitude. However, the agreement with a previous viscosity estimationusing a different technique increased confidence to the modeling and motivated furtherstudies of the viscosity.

It is interesting to compare the RMP threshold amplitude required for mode lockingin MST with that in EXTRAP T2R. In MST, them= 1 RMP contributes to each toroidalharmonicn by about 1/100 [Paper III]. At the time of locking when the totalm= 1 am-plitude is 13 mT, the RMP amplitude of eachn is |bm=1,n

r | ≈ 0.13 mT. In EXTRAP T2R,the locking-threshold was|bm=1,n

r | ≈ 0.6 mT for a single harmonics RMP. Considering themulti-harmonics RMP in MST, it seems reasonable that the RMPthreshold amplitude permode (|bm=1,n

r |) is lower in the MST.In MST, we observed that the locking-threshold increased with increasing plasma den-

sity. When modeling the mode locking at different plasma densities, we found agreementwith modeled momentum transport assuming a constant kinematic viscosity. This indicatesthat kinematic viscosity is independent of the density.

To release a locked mode, the RMP amplitude had to be significantly reduced from the

45

46 CHAPTER 5. DISCUSSION

locking threshold amplitude, i.e. there is hysteresis in the locking and unlocking process.In EXTRAP T2R, we found two mechanisms responsible for the hysteresis (Paper II).First, the viscous torque on the TM location decreases significantly after mode locking.The reduced viscous torque is explained by that, after mode locking, the whole velocityprofile relaxes to a lower rotation state. Second, the maximum of the RMP torque increasesby a factor two after mode locking, which is directly relatedto that TM amplitude increasesby a factor two after the locking. The EM wall torque increases by a factor four butstarting from a low magnitude. In EXTRAP T2R, the wall torqueis not sufficient to sustainthe locked state when the RMP amplitude is ramped down. This is in contrast to MST,where we found that the TMs stay locked if they are above a certain TM amplitude, evenwhen the RMP was reduced to zero. The modeled momentum transport, including the EMwall torque, agreed well with the MST discharges that the unlocked and those that stayedlocked. Since the RMP torque is zero during the unlocking process, the model agreementwith the experimental spin-up of the mode (Figure 4.7) increases the confidence in themodeled viscous torque and the viscosity.

Previous measurements [56] show that the CV impurity ions, located in MST’s core,rotate atvφ ≈ 5 km/s after the modes lock. This indicates some viscous restoring force. Theresidual plasma flow after TM locking might imply that the no-slip condition is violatedafter TM locking. However, it is uncertain since flow measurement is not local to the reso-nant surface. Before TM locking, both in EXTRAP T2R and MST, plasma flow reductionsimilar to TM velocity reduction is observed and the no-slipcondition is assumed valid asa first order approximation. The qualitative agreement withthe model also supports thisview.

In some discharges where the TMs stayed locked after the RMP phase, it is observedthat TMs can spin-up in connection to a dynamo reconnection event. In the dynamo event,the amplitudes of the core TMs are significantly reduced which implies a reduction of theEM torques. This reduction in EM torque amplitude might be sufficient to unlock the TMs.However, the reconnection event is quite complex. For example, it contains non-linearmode coupling with them= 0 modes near the edge and momentum is quickly (≤ 100µs)redistributed between radial locations in the plasma [68].Since the model we tested doesnot contain the physics of the reconnection event, RMP phases that contained reconnectionevents were not compared with the model.

We will now discuss possible problems in directly comparingthe model with the ex-periments. For example, the theoretical model does not contain (I) the intrinsic torque thatis responsible for the initial rotation, (II) the non-linear EM torque that exists between themodes, and (III) the neoclassical toroidal viscosity (NTV).

(I) The intrinsic torque in the RFP is expected to be the fluctuation-based kinetic stress[69]. We find two reasons that suggest that the momentum source is not largely affected bythe RMP: (1) neither the RMP or the probe has much effect on thefluctuation amplitudessuggesting the kinetic stress is unaffected, and (2) while the probe and the RMP act on theplasma differently the estimated viscosity is about the same for the two methods.

(II) While the non-linear torque between modes is importantwithout the RMP [70], itis negligible during the RMP application (see Paper III).

(III) Lastly, the NTV torque [71] might be expected to affectthe rotation in the MST

5.2. THE ANOMALOUS VISCOSITY IN THE RFP STOCHASTIC MAGNETICFIELD 47

experiment since the magnetic perturbation contains non-resonant components. We pro-pose three arguments why the NTV is small relative to the EM-torques. (1) Since onlythe locking time is optimized in e.g. Figure 4.6, the reasonable fit with the whole experi-mental time evolution indicates that the experimental braking torque scales as the modeledEM-torque. (2) The agreement with the experimental anharmonic oscillations prior to thelocking [Figure 4.4 (d-e)] suggests that the EM torque is dominant. (3) The third argumentis based on an EXTRAP T2R study [72] that compared the largestmagnitude non-resonantNTV torque with an intermediate magnitude resonant EM torque. While the volume aver-aged NTV torque was larger, the EM torque was four times larger at the resonant surface.In MST, there are EM torques on all the TMs and, if the TV scalessimilar to EXTRAPT2R, the accumulative effect of the EM torques should dominate the NTV. This does notmean that the NTV-torque is negligible, but indicates that it is not large relative to theEM-torque.

The results in Papers I-III and V show that most experimentalobservations can beexplained by mode locking theory, which is based on the RMP-TM electromagnetic inter-action and the viscous plasma response. The result bolstersthe confidence in the predictivecapability of the mode-locking theory. Further investigations related with the RMP may in-clude, e.g., (I) a study of the plasma response to the RMP field, (II) modeling of the NTV ina similar manner as done for non-resonant magnetic perturbations in EXTRAP T2R [72],and (III) examination of the core plasma flow after mode locking by using Charge Ex-change Recombination Spectroscopy (CHERS).

5.2 The anomalous viscosity in the RFP stochastic magnetic field

In Papers IV and VI-VII we estimated the viscosity in MST. We found that the kinematicviscosity increased with the magnetic fluctuation level (br/B). Theν⊥ increased almost100-fold when the (br/B) increased 10-fold. The scaling is in close agreement with the(br/B)2 expected from magnetic field line diffusion [19]. While the magnitude of viscos-ity agreed with the momentum being propagated at the sound speed along the stochasticfield [Eq. (1.9)]. The viscosity is anomalously large compared with the classical Braginskiiprediction, but the anomaly varied by a factor 5 to 300 depending on the type of plasmaequilibrium. The smallest deviation from the classical model is in the lowest(br/B), whereconfinement is tokamak-like [73]. The largest deviation is at intermediate(br/B). Inter-estingly, at the highest(br/B) the experimental viscosity is only a factor∼ 20 higher thanthe classical value. This plasma has a lowB equilibrium magnetic field which increasesthe Larmor radius and the classical viscosity.

In the calculation of the quasilinear magnetic diffusion coefficient [Eq. (1.7)], we cal-culated the autocorrelation length in accordance with Krommes model [20]. The model haspreviously been shown to agree with numerical simulations of the RFP [74]. In Eq. (1.7)we included the number of TMs that had overlapping island widths. The lowest fluctuationplasma, however, has a central TM that is relatively large and the Chirikov overlap criterionis not valid (it assumes similar widths of the overlapping islands). Simulations [Chapter1.6] that traced the magnetic field lines showed a central island structure with closed flux

48 CHAPTER 5. DISCUSSION

surfaces. Therefore, the most inner TM was neglected in Eq. (1.7 for the lowest (br/B) (asa side note, the inclusion of the central mode resulted in a factor ten overestimation of theexperimentalν⊥). In three types of plasmas, we calculated the magnetic fieldline diffusioncoefficient via the field line simulation [Eq. (1.6)]. The resulting Dm agreed within a factortwo to the quasilinear formula. The calculatedDm showed a flat profile in the core. This,combined with the relatively flat temperature profile in MST,suggests that the viscosityprofile is relatively flat too.

In EXTRAP T2R, the viscosity was estimated toν⊥ ≈ 1−2 m2/s. The error in theestimate is relatively large due to the lack of plasma density measurement which causeslarge uncertainty in the momentum equation. A more exact estimation was made in anotherEXTRAP T2R discharge [29] in which the density was measured.The estimated kinematicviscosity in the core-region wasν⊥ ≈ 4−40 m2/s. This value was anomalously large bybetween one to two orders of magnitude, which is similar to the observation in MST.The source of the anomaly in EXTRAP T2R was not determined butis likely caused bymagnetic fluctuations that destroy the magnetic flux surfaces, as shown in MST [Paper VI].

To summarize, we performed a first-time detailed test of a theory modeling the mo-mentum transport in a stochastic magnetic field. We found good agreement with the theoryover a wide parameter space where the magnetic field is stochastic. The results implythat the experimental viscosity can be estimated from the magnetic fluctuations. This willclarify the comparison between the experiments and the visco-resistive MHD [75], whichpreviously has been complicated by the fact that the experimental viscosity was not wellknown. Possible future studies may be: (1) study the limits of the theoretical model by,e.g., going to lowerB (lower plasma current) where the increase of the Larmor radiusshould increase the Braginskii viscosity, (2) study how collisions would modify the theoryand compare with experiments, (3) perform detailed numerical simulations ofDm for alltypes of plasmas, and (4) determine the experimentalν⊥ in the tokamak configuration byusing the RMP and the probe.

Chapter 6

Summary and Conclusion

The experimental response of a rotating tearing mode to a static RMP exhibits the samecharacteristics as a theoretical model that is based on the electromagnetic interaction be-tween the RMP and the TMs. The agreement with experiment is good both in the shorttimescale fluctuations in velocity and in the longer timescale locking process. The viscos-ity was chosen to match the experimental locking instant. Assuming the same viscosity,the model predicts the whole braking curve, the short timescale velocity oscillations, andthat the RMP amplitude has to be significantly reduced to unlock the mode. In EXTRAPT2R, the unlocking threshold amplitude is about ten times lower than the locking one. Inthe MST experiment, the discharges with high mode amplitudes tend to stay locked af-ter RMP was turned off. The modeling showed that the EM torque, which depends onthe mode amplitude squared, could predict the unlocking process in MST. In general, itcan be difficult to reduce the error field and/or the mode amplitude to reach the unlockingthreshold. Therefore locked modes should be avoided in fusion machines.

In MST, we found that the locking threshold increased with the plasma density, allother parameters being similar in the experiment. The modelreproduces the experimen-tal locking assuming a constant kinematic viscosity. The magnitude of the viscosity wasanomalous by two orders of magnitude. Instead, a model describing the momentum trans-port in a stochastic magnetic field showed agreement with theexperimental viscosity.

The reasonable agreement with magnetically driven transport motivated a broader studyof the viscosity. In the study, both the RMP and the biased probe were used to perturb themomentum. A valuable result is an agreement between the RMP and probe predicted vis-cosities in the same plasma condition. The study showed thatthe viscosity increased as thesquare of the relative magnetic fluctuations and both parameters span 100-fold. This wasin agreement with transport in a stochastic magnetic field and provided the first detailedtest of a model originally derived for the tokamak. The result can simplify comparisonbetween experiment and visco-resistive MHD simulations. These comparisons have beencomplicated due to the large uncertainty in the experimental viscosity in stochastic mag-netic fields. The visco-resistive MHD is, however, used to model plasmas with a stochasticmagnetic field, such as the RFP plasma, and the tokamak duringa disruption.

49

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