Toroidal modelling of RMP response in ASDEX-Upgrade

Toroidal modelling of RMP response in ASDEX-Upgrade:coupling between field pitch aligned response and kinkamplification

D A Ryan1 2, Y Q Liu2, A Kirk2, W Suttrop3, B Dudson1, M Dunne3, RFischer3, J C Fuchs3, M Garcia-Munoz3 4, B Kurzan3, P Piovesan5, MReinke1, M Willensdorfer3 and the ASDEX-Upgrade team3

1 York Plasma Institute, Department of Physics, University of York, Heslington, York,YO105DQ, UK

2 CCFE, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB, UK

3 Max Planck Institute for Plasma Physics, Garching, Germany

4 FAMN Department, Faculty of Physics, University of Seville, 41012 Seville, Spain

5 Consorzio RFX, Euratom-ENEA Association, Padova, Italy

Abstract. Using the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681), the single fluidresistive MHD plasma response to applied n = 2 resonant magnetic perturbations is computed,for a plasma discharge in the ASDEX-Upgrade tokamak. The computation predicts strong kinkamplification, as previously predicted in DIII-D (Haskey et al 2014 Plasma Phys. Control.Fusion 56 035005), which is strongly dependent on the toroidal phase shift between theupper and lower coils, ∆φul. In particular, edge localised low n peeling modes with poloidalmode numbers just above pitch resonance - a subset of the kink response - are amplified.The robustness of the amplified peeling response with respect to truncation of the X point isinvestigated, by recomputing the plasma response for a range of edge geometries. It is foundthat the computed peeling response, when plotted against the safety factor, is not sensitive tothe numerical truncation near the X point. It is also predicted that near the plasma edge whereresistivity is large, the pitch resonant components are finite and also strongly dependent on∆φul. A previous proposal that the amplified peeling response may indirectly drive the pitchaligned components by spectral proximity (Lanctot et al 2013 Nuclear Fusion 53 083019), isinvestigated by applying magnetic perturbations of a single poloidal harmonic, as a boundarycondition at the plasma edge. It is found that poloidal harmonic coupling causes harmonicsto couple to and drive harmonics directly beneath them spectrally, and also that the pitchaligned components can be driven by this mechanism. As a refinement, the amplified peelingresponse is simulated in relative isolation by using a filtered boundary condition. In this model,the peeling response drives the pitch aligned components to be comparable to their values inresponse to the RMP coils. This suggests that it is quite possible that the amplified low n

peeling response can drive the pitch aligned components in some circumstances, which wouldalter the coil configuration for optimum plasma stochastization, with implications for ELMcontrol by RMPs.

Poloidal coupling on ASDEX-Upgrade 2

1. Introduction

Edge Localised Modes (ELMs) are bursty plasmainstabilities, which occur repetitively in HighConfinement mode tokamak plasmas [1]. Theyare common in modern tokamaks, but usuallydo not pose a problem at these machine scales.However, extrapolations to ITER suggest thatif ELMs were allowed to occur unmitigated,then machine components would be at risk ofdamage [?].

Since it was discovered that ELMs can bestrongly mitigated or completely suppressed bythe application of Resonant Magnetic Perturba-tion fields [2], intensive research has been under-way attempting to expand the parameter spaces ofELM suppression and mitigation, and to under-stand the underlying mechanisms. To date, ELMsuppression or mitigation by RMPs has beenachieved on DIII-D [2], KSTAR [3], JET [4],MAST [5] and ASDEX-Upgrade [6]. However,the interaction between tokamak plasmas, ELMsand RMPs is highly complex, and a robust pre-dictive theory of ELM suppression by RMPs iscurrently lacking [7]. Such a theory is urgently re-quired, in order to predict the efficacy of the ITERELM coils.

Stochastic Transport Theory [8] is one ofseveral working theories of ELM suppressionby RMPs, though other working hypotheseshave also been proposed [9, 10, 11]. In thestochastic transport theory, RMP fields can drivethe formation of magnetic island chains at rationalsurfaces in the plasma, where the safety factorq is a rational number m/n. If these magneticislands are wide enough and close enough to otherisland chains, then a stochastic magnetic field iscreated in the overlapping regions. The stochasticfield enhances transport in these regions, loweringthe pressure gradient and reducing the drive

for ELMs. The extent of stochasticity ischaracterised by the Chirikov parameter, whichis dependent on the local geometry, and on thepitch aligned components of the perturbation,as described in the cylindrical approximationfor a given rational surface by the formulafollowing [12].

σChir = 4

√∣∣∣∣hq2R

rs

bm=nq

B

∣∣∣∣ (1)

In the above, q is the safety factor, R the majorradius, B the equilibrium magnetic field, h =

d(log q)/d(log r) is the normalised shear, rs isthe minor radius and bm=nq the pitch alignedcomponent of the magnetic perturbation. Allquantities evaluated at the given rational surface.

Since the Chirikov parameter is propor-tional to the square root of the pitch alignedcomponents, it is thought that maximising thepitch aligned components may lead to the largeststochastic region, and maximise the effect of theRMPs on the plasma. RMP experiments aretherefore motivated to chose parameters whichmaximise the pitch aligned components of theperturbation. The total amplitude of the pertur-bation can be controlled by varying the coil cur-rents, and the perturbation spectrum can also bemodified; the dominant toroidal mode numbern can be chosen (from a small range) and thepoloidal spectrum can be modified by changingthe toroidal phase offset between the upper andlower coil sets, ∆φul.

Numerical models are used to predict thepitch aligned RMP components for a givenplasma equilibrium and applied perturbation. Inmany studies [3, 13, 14], experimental results ofRMP experiments are interpreted assuming the’vacuum approximation’, that is, that the appliedfield perturbation is as it would be in the absenceof a plasma. However, it has been shown that the


plasma response to the applied perturbation canconstitute a significant correction to the vacuumfield [15, 16, 17, 18]. Applied perturbations drivecurrents at rational surfaces in the plasma, whichstrongly suppress the pitch aligned components,and in the limit of zero resistivity, screenthem completely. Furthermore, the appliedperturbation can couple to and drive marginallystable modes in the plasma, a phenomenon knownas resonant field amplification (RFA) [16]. Inorder to accurately calculate the stochasticityinduced by an applied perturbation, the plasmaresponse to the perturbation should be accountedfor. A review of approaches to modellingthe plasma response to applied perturbations isavailable in [19].

Previous investigations of the ideal singlefluid plasma response to applied n = 2 RMPson the DIII-D tokamak [15], show a strongamplification of marginally stable kink modes,and suggest the possibility that this amplificationmay indirectly drive the pitch aligned componentsby ’spectral proximity’ [20], ie, by having similarradial location and poloidal harmonic number,m. The specific mechanism could be poloidalharmonic coupling between the pitch alignedcomponents and the amplified peeling response.It may be helpful to explicitly distinguish atthis point between ’harmonic’, which refers toa single poloidal fourier harmonic m havingpoloidal dependence e(imχ) (where χ is thepoloidal angle coordinate in the straight fieldline coordinate system), and ’mode’, whichrefers to a particular type of plasma instabilityidentified by its radial displacement profile, and isgenerally comprised of many poloidal harmonics.The ’kink response’ refers to amplification ofharmonics just above pitch resonance (ie, m >

qn). We also further divide the kink response into’core kink’ and ’low n peeling’ response, whichrefer to the kink response in the plasma core and

Figure 1. Spectrogram of the total field in response toan even coil configuration (∆φul = 0◦). The spectralregions (ie, regions in m, s space) referred to as ’Kink’,’Core Kink’ and ’Peeling’ are labelled. More rigorously,the designations ’kink’ and ’peeling’ refer to specificradial profiles of the plasma displacement. The negativeharmonics (ie, m < 0) are of no interest to this study,since it is unlikely that these can influence the pitch alignedcomponents. Note that m is a discrete quantity, the m axisis smoothed to make structures more easily discernible.

near the edge respectively. This nomenclature isillustrated in figure 1.

In this work, the plasma response to ann = 2 RMP field applied to an ASDEX-Upgrade plasma equilibrium is computed. Thecode MARS-F [21], which is well benchmarkedagainst other codes [22] and validated againstexperiments [18, 23, 24], is used for thecomputation of the plasma response. MARS-F models the plasma response as a linearstatic 3D perturbation superimposed on a 2Dequilibrium, and computes it by numericallysolving the linearised perturbed equations ofsingle fluid resistive MHD. The linear theoryon which the MARS-F model built, is valid solong as the plasma displacement is sufficientlysmall compared with the equilibrium scale length.The limits of the validity are defined rigorouslyin [22]. The sensitivity of the plasma response tothe geometry near the X point, is also investigated


by varying plasma boundary shape. In orderto investigate the general behaviour of poloidalharmonic coupling, a magnetic perturbation of asingle poloidal harmonic is applied as a boundarycondition at the plasma edge. In order toinvestigate specifically the effect of the peelingresponse on the pitch resonant components,a filtered boundary condition was designed,which contained only harmonics constituting theamplified peeling response. The pitch alignedcomponents resulting from this applied boundarycondition were then computed.

This paper is organised as follows. Section2 describes the experimental equilibrium used forthis study, and also how the applied perturbationis modelled and benchmarked. Section 3introduces the nomenclature and components ofthe plasma response, and describes the computedplasma response to this applied perturbation, aswell as its dependence on X point geometry.Section 4 describes the plasma response tocustomised boundary conditions, designed toinvestigate poloidal harmonic coupling. Insection 5 some possible implications of theseresults are discussed.

2. Equilibrium and Applied Perturbation

2.1. Equilibrium

The 2D plasma equilibrium used in this study wasreconstructed from an ASDEX-Upgrade experi-ment, shot number 30835, which was designed tostudy ELM mitigation at low collisionality. Es-sential information on the plasma equilibrium islisted in table 1. The initial equilibrium recon-struction was done using the free boundary equi-librium code CLISTE [25]. Magnetic measure-ments, the q = 1 surface location from measure-ments of sawtooth instabilities, and the scrape-

off layer current were used as constraints. Theequilibrium was then refined and mapped to theMARS-F straight field line coordinate system us-ing the fixed boundary CHEASE [26] equilibriumsolver. Figure 2 shows curves fitted to experimen-tally measured radial profiles of parameters whichare also used as input for the MARS-F model: a)the electron temperature, b) ion temperature, c)plasma toroidal angular velocity, and d) the elec-tron density.

Shot num Time q0 q95

30835 3200ms 0.81 3.8

Ip (MA) B0 (T) β Te,0, Ti,0 (keV)0.77 1.7 1.9 % 6.96, 5.05

Table 1. Parameters of the plasma equilibrium of shot30835 at 3200ms; the equilibrium used for this numericalstudy. q0 and q95 are the safety factor at the core and theΨN = 0.95 flux surface respectively. Ip is the total plasmacurrent, B0 is the vacuum equilibrium field strength at themagnetic axis, β = (2µ0 < P >)/(B2

0) is the ratio ofvolume averaged plasma pressure to magnetic pressure onaxis, Te,0 and Ti,0 are the electron and ion temperatures atthe magnetic axis.

Figure 2. Ion and electron temperature, electron densityand rotational velocity, from ASDEX-Upgrade shot 30835,taken at 3200ms into the plasma shot.


2.2. Perturbation

In the experiment, two coil sets, the upper andlower set each consisting of 8 coils, were usedto apply a static magnetic perturbation to thisequilbrium, with dominant toroidal mode numbern = 2. Figure 3 is a sketch of the AUG coilset and plasma boundary. The RMP coils aremodelled as current perturbations, with delta-like functions of finite width at the poloidallocations of the coils, with a prescribed e(inφ)

toroidal dependence. In the toroidal direction, adiscrete set of 8 coils is modelled as a continuoussine wave with a single toroidal harmonic,n. The poloidal spectrum in contrast, containsmany poloidal harmonics, m. With this givenequilibrium and static perturbation, MARS-F canbe used to solve the single fluid resistive MHDequations for the plasma response, that is, thestate vector of perturbation quantities caused bythe applied magnetic perturbation.

Figure 3. a) Sketch of the upper (blue) and lower (green)AUG RMP coil sets, in relation to the plasma boundary forshot 30835 (orange). b) A cross section showing the plasmaboundary (orange), and the locations of the upper and lowercoil sets.

2.3. Benchmark

In order to perform a benchmark of the vacuumfield, the vacuum perturbation was first computedusing MARS-F, and also separately using theBiot-Savart based ERGOS code [?]. The vacuum

pitch aligned field components computed withMARS-F and ERGOS differ by less than 2%.This vacuum benchmark gives confidence aboutthe representation of the applied perturbationin MARS-F. A more detailed explanation andvalidation of the coil representation used inMARS-F can be found in [15].

3. Plasma Response to Applied Perturbation

The single fluid MHD plasma response wascomputed including plasma toroidal rotation,realistic geometry (excluding X point), andresistivity, which was calculated with the Spitzermodel (ie, η ∝ T

−3/2e ). Since MARS-F is a

linear model, the principle of superposition canbe applied, and the response due to the upper andlower coil sets were computed separately. Thetotal plasma response to both coils can then becomputed in post-process for any ∆φul using therelation b∆φul = bupper + blower × e(i∆φul), asis done in previous studies of the linear plasmaresponse [15]. Figure 4 shows the vacuumperturbation, and the total magnetic perturbationincluding plasma response, ie, btotal = bvac +

bresponse. In this work, ’total’ is taken to mean’including the plasma response’. The quantityplotted is the absolute value of the normalcomponent of the perturbed magnetic field, |b1| =| b·∇ψBeq·∇φ

qR2

0B0|. Several features are of interest.

Firstly, at the inner rational surfaces the totalpitch aligned components (b1

m=qn) are almostzero. This is because in the bulk plasma theplasma response is close to ideal, and so thepitch aligned components are almost perfectlyscreened by internal currents. However, closer tothe plasma edge the temperature is lower, and soresistivity is higher, allowing the pitch resonantcomponents near the plasma edge to be finite.Secondly, there are some areas of the spectrogram


Figure 4. a) The poloidal spectrum of the applied field inthe vacuum approximation, with ∆φul = 180◦. The pitchaligned components are highlighted with white circles, andthe white dashed line follows the m = nq(s) contour.b) The poloidal spectrum of the applied field includingthe plasma response (ie, the total field). There is someamplification of the core kink, but the edge localisedpeeling response is far more prominent. The pitch alignedcomponents in the plasma bulk are completely screened bythe plasma response, but the components close to the edgecan remain finite.

which are amplified above their vacuum values byresonant field amplification. As figure 4 shows,the plasma response causes some amplification ofthe core kink mode, but the edge localised low n

peeling mode is the dominant response.

To measure the dependence of the peelingresponse on ∆φul, the field amplitude at arepresentative spectral point (ie, a point in m, s

space) is computed for a scan of ∆φul. Figure5 shows the magnetic field perturbation |b1| atspectral location (m, s) = (11, 0.99). s is theradial coordinate, defined as s =

√ψN where

ψN is the poloidal magnetic flux normalisedsuch that ψN = 1.0 at the plasma edge. Theplot also shows the vacuum field amplitude atthis spectral location. The figure shows thatthe plasma response amplifies the field in thisspectral region far above its vacuum value. Italso demonstrates the strong dependence of theresponse on the coil phase shift ∆φul. Thisfinding is qualitatively consistent with a similar

Figure 5. The magnitude of the perturbation at aspecific spectral location (m, s) = (11, 0.99), which isrepresentative of the amplified peeling response, with a scanof ∆φul. The solid line shows the vacuum perturbation, andthe dashed line is the total perturbation.

study of a DIII-D plasma, which also predictedamplification of the low n peeling response withstrong ∆φul dependence [15].

3.1. Pitch Aligned Components

The pitch aligned field components, marked aswhite circles in figure 1, refer to spectral points(in m, s space) where m = nq(s). The pitchaligned field components are of particular interestto RMP studies, because they determine the widthof any stochastic regions which may form inresponse to the RMPs. Figure 6 shows themagnitude of the vacuum (blue) and total (green)pitch aligned field components, computed witha coil phase difference of ∆φul = 120◦. InIdeal MHD, RMPs drive currents at rationalsurfaces which completely screen the pitchaligned components. This is why in thebulk plasma, where the resistivity is low, theplasma is approximately ideal and pitch alignedcomponents tend to be very small. However,close to the plasma edge (roughly s > 0.95)


Figure 6. The pitch aligned components of the appliedperturbation. In the plasma bulk, the pitch alignedcomponents are well screened, but can be finite in the edgeregion where resistivity is higher.

the resistivity is much higher, so the pitchaligned components can be finite. In the plasmabulk and most of the edge region where theelectron temperature is high, the Spitzer modelfor resistivity is quite acceptable. However, inthe limit approaching the plasma edge wherethe electron temperature can tend to zero, theSpitzer resistivity would tend towards infinity.In the current scheme a numerical singularityis avoided by fixing a maximum value of theresistivity, chosen for numerical stability. A morerefined model of resistivity at the plasma edgemay include kinetic effects [27].

Figure 7 shows the amplitude of the outer-most 3 pitch aligned components (at the m = 8,9, and 10 surfaces) as a function of coil phase dif-ference ∆φul. The dashed lines show the pitchaligned components of the vacuum field, whereasthe solid lines show the total field. The maxi-mum value of the vacuum pitch aligned compo-nents (ie, vacuum alignment) occurs at around∆φul = 30◦, whereas the maximum value of thetotal pitch aligned components occurs at around∆φul = 90◦; the coil phase which maximises the

Figure 7. Magnitudes of the outermost 3 pitch alignedcomponents, in the vacuum approximation (dashed lines)and the total field (solid lines). The maximum fieldincluding the plasma response is offset 60◦ from its vacuumvalue. This tells us that vacuum modelling alone isinsufficient to predict the coil phase for optimum pitchalignment.

total pitch aligned components, is offset from itsvacuum value by 60◦. This demonstrates thatwe may not expect a vacuum pitch aligned ap-plied field to maximise the total pitch alignedcomponents. Therefore in order to truly opti-mise the coil configuration to maximise total pitchaligned components, the plasma response mustbe accounted for. It should be noted that thisphase shift is not specific to the resonant compo-nents. Non-resonant components also experiencea shift in their dependence on ∆φul (see figure5), but this effect will not be examined here. The∆φul dependence of the total and vacuum fieldsis robust with respect to changes in qa caused bychanges in plasma shape, and hence so is the 60◦

offset of the total field from the vacuum. Howeverthe magnitudes of the outermost pitch alignedcomponents were found to be slightly sensitive toqa. The rotational and resistive radial profiles arefixed with respect to changes in qa, but alteringthe edge q profile changes the radial locations ofthe rational surfaces. Altering qa can therefore


change the values of resistivity and plasma rota-tion at the pitch aligned components, which mayexplain this sensitivity.

Another interesting feature of figure 7, is thatfor certain ranges of ∆φul the total pitch alignedcomponents are of the same order or largerthan their vacuum values, which is surprisingconsidering the strong screening effects whichact to reduce them. In the context of thislinear single fluid MHD model, the mechanismby which parts of the perturbation spectrumcould exceed their vacuum value, is resonantfield amplification (RFA). It has previouslybeen proposed [20], that the amplified edgelocalised peeling response may be indirectlydriving the pitch aligned components by spectralproximity. That is, poloidal harmonics of theamplified peeling response may couple to anddrive the pitch aligned components by poloidalharmonic coupling. This possibility motivated theinvestigation into poloidal harmonic coupling onASDEX-Upgrade, described in section 4.

3.2. Robustness of peeling response with respectto X point truncation

In divertor experiments, the q profile is not de-fined at the X point where there is zero poloidalfield. In the MARS-F flux-based coordinate sys-tem, this would introduce a numerical singular-ity, and so a certain truncation scheme must beused to exclude the X point. Truncation effec-tively approximates the divertor configuration asa limiter configuration with an otherwise similarshape, imposing a finite q at the plasma edge, qa.It has previously been suggested [15] that the pre-dicted amplification of the low n edge localisedpeeling response may be sensitive to the trunca-tion around the X point, in particular to the valueof the edge safety factor qa.

In order to test the robustness of thepeeling response with respect to changes ingeometry around the X point, the plasmaresponse was recomputed as the plasma boundaryincrementally approached a separatrix. Figure 8shows the plasma boundaries and edge q profilesof 5 equilibria, identical except for differinglevels of truncation around the X point, and asa result slightly different edge q profiles. Figure9 shows the edge spectrograms computed fordifferent values of qa. The spectrograms showa distortion of the spectrum radially, but not achange in amplitude. Changing the edge q profilemoves the m = nq(s) curve, and the spectrogramis distorted radially following its movement. Thissuggests that while the response at a given radialposition s is sensitive to changes in geometry,the response at a given value of q is not. Figure10 shows the edge radial profile of the m = 12

harmonic of the total magnetic perturbation, for5 values of qa. The plots show that changingthe geometry has no effect on the amplifiedpeeling response when q(s) is used as the radialcoordinate, suggesting that altering the q profileonly radially distorts the structure of the peelingresponse, and does not affect its amplitude. Thissuggests that in the limit of including the Xpoint, the amplified peeling response would stillbe present. We point emphasise that this is astudy of the amplified plasma response of stableedge peeling modes, not the stability of thesemodes. Therefore this finding does not contradictprevious studies [28] which find that the stabilityof edge peeling modes can be sensitive to thepresence or absence of an X point.

The sensitivity of the plasma response toresistivity and plasma rotation, is as described inprevious studies [16], and will not be discussedhere.


Figure 8. Plasma boundaries and q profiles incrementallysharpening the X point, approaching a separatrix. A sharper(ie, less truncated) X point leads to higher values of qa inthe equilibrium construction.

4. Poloidal Harmonic Coupling on AUG

4.1. Generic poloidal coupling; single harmonicboundary condition

Analytically, there are 3 primary sources ofpoloidal harmonic coupling: toroidicity couplesharmonics with ∆m = |m − m′| = 1,ellipticity couples harmonics with ∆m = 2,and triangularity couples harmonics with ∆m =

3. Since the MARS-F code operates in StraightField Line (SFL) coordinates, geometrical andphysical quantities are inseparable, so poloidalcoupling can in principle occur between any pairof poloidal harmonics, and manifests both in thevacuum field, and the plasma response.

In order to investigate the generic behaviourof poloidal coupling in the MARS-F model, thecode was used in an unconventional manner, inwhich the perturbation was not applied by RMPcoils, but by a prescribed magnetic perturbationb1BC , applied at the plasma boundary. The

perturbation at any closed surface (eg, theplasma boundary) completely determines the

Figure 9. Edge regions of spectrograms from 5 differentgeometries, corresponding to 5 values of qa. Increasing qamoves them = qn line, ’compressing’ the peeling responsestructure radially, but not altering its global amplitude.

perturbation inside that surface, subject to thesame physics as a normal RMP coil calculation.A magnetic perturbation with a single poloidalharmonic mBC and unit amplitude (b1

BC = 1.0

for m = mBC , and zero for m 6= mBC)was applied at the plasma boundary, and theresulting magnetic field in the plasma bulk wascomputed. Figure 11 shows a spectrogram of avacuum field and total field, resulting from a unitamplitude mBC = 7 and mBC = 13 magneticperturbation applied at the plasma edge. At theplasma edge, the perturbation is purely single


Figure 10. The edge radial profile of the m = 12 harmonicof the total magnetic perturbation, using a) q as radialcoordinate, b) s as radial coordinate. The plot shows thatwhen s is used as a radial coordinate, the peeling responseappears to be sensitive to the X point truncation. Howeverwhen q is used as the radial coordinate, the edge profilescoincide up to each value of qa.

m, but moving radially inwards, the spectrumbroadens by poloidal harmonic coupling, and alsomoves to lower |m|. This behaviour appearsto be general for any m. Figure 12 shows thetotal pitch aligned components, resulting fromboundary perturbations with mBC = 11, 12

and 13. For these computations, mBC werechosen such that they had no rational surface inthe plasma. This means that the pitch alignedcomponents can be non-zero only by poloidalcoupling. The results show that a unit amplitudesingle m perturbation applied at the plasmaboundary, can drive the pitch aligned components

Figure 11. Spectrograms showing the perturbationresulting from applying a unit amplitude single mBC = 7

and mBC = 13 perturbation as a boundary condition atthe plasma edge. Vacuum field and total field are shown.The spectrum broadens and shifts towards lower |m| as itpenetrates into the plasma bulk. Note that for clarity thecolormap in these plots is reversed with respect to previousspectrograms.

Figure 12. The total pitch aligned components of fieldsresulting from mBC = 11, 12, and 13 unit amplitudeboundary perturbations. The magnitude of the pitch alignedcomponents is up to 20% of the applied unit field. Sincethere are no m = 11, 12 or 13 surfaces in the plasma, thesepitch aligned components are non-zero only by poloidalharmonic coupling.

to be quite large relative to the size of the appliedperturbation (in this work, up to |b1

m=nq|/|b1BC | =

0.2). This result suggests that it is quite possiblefor the pitch aligned components to be driven bypoloidal harmonic coupling.


4.2. Amplified Peeling Response BoundaryCondition

By careful choice of boundary condition appliedat the plasma edge, the question of whether theamplified peeling response could be driving thepitch aligned components on ASDEX-Upgrade30835, can be directly studied. The value ofthe magnetic perturbation due to the RMP coilsand plasma response was computed in section2 (figure 4b). This total perturbation was thenfiltered such that only harmonics which constitutethe peeling response remain (ie, m > 11 in thiscase), and then this filtered perturbation was usedas the boundary condition applied at the plasmaedge. Applying this boundary condition b1

BC =

b1peeling isolates the peeling response, allowing us

to investigate its effects.

A simple check of the validity of thistechnique is to remove the filtering step beforeapplying the boundary condition, which shouldrecover the amplified peeling response predictedby the RMP computation. It was foundthat the unfiltered boundary condition exactlyrecovered the result from the full spectrum RMPcomputation, as expected.

Figure 13 shows the spectrograms of thevacuum and total fields, resulting from applyingthe peeling boundary condition, computed for∆φul = 0◦ and ∆φul = 180◦. Allspectrograms demonstrate the same shifts andbroadening in the poloidal spectrum as seen inthe single m boundary condition computations(figure 11). Figure 14 shows the vacuumand total pitch aligned components resultingfrom the applied peeling response boundarycondition, and compares them to the full spectrumcomputation. The figure shows that the pitchaligned components due to poloidal couplingwith the amplified peeling response, can be

Figure 13. Spectrograms showing the perturbation in theplasma bulk, resulting from applying a boundary conditionconsisting of the amplified peeling response at ∆φul = 0◦

and ∆φul = 180◦. a) vacuum field, b) total field.

comparable to or larger than the pitch alignedcomponents from the full spectrum. Thisresult further suggests that the amplified peelingresponse can strongly affect the pitch alignedcomponents, by poloidal harmonic coupling.

5. Summary and Discussion

Using the MARS-F code, the resistive singlefluid plasma response to applied static RMPfields on AUG was investigated. Numericalcomputations predict a strongly amplified low n

peeling response, localised near the edge in thes > 0.7, m > qn region of the spectrum. Theamplified peeling response showed no tendencyto be reduced as the plasma boundary approachedX point geometry (ie, for increasing qa), ratherthe peeling response was distorted in the radialdirection in response to the changing edge q

profile. That is, the peeling response is notsensitive to the numerical truncation near the X-point when plotted as a function of q. It is alsopredicted that with finite resistivity, the outermost


Figure 14. The pitch aligned field components resultingfrom applying the filtered peeling boundary condition, andalso from the unfiltered RMP coil computation, for a)∆φul = 0◦ and b) ∆φul = 180◦. The dashed linesdenote the vacuum field, whereas the solid lines includethe total field. The plots, in particular plot b), show thatthe applying the filtered peeling boundary condition drovethe pitch aligned components to be large compared to theirvalues in a full spectrum RMP coil computation.

pitch aligned components can be finite andcomparable to the vacuum values. Both the pitchaligned components and the peeling responsehave a strong dependence on upper/lower coilphase difference ∆φul. Also, the value of ∆φulat which the total pitch aligned components aremaximised, is offset from optimum vacuum pitchalignment by 60◦. This result highlights theimportance of including the plasma response

when calculating optimum coil phase for futureRMP experiments. A vacuum calculation alonewould not predict the correct coil configurationfor maximum edge stochasticity. A comparisonbetween the MARS-F modelling results and theexperimental scans of ∆φul in ASDEX-Upgradehas been partially performed in [29], and asystematic comparison will be carried out in thefuture.

To investigate the proposal that the ampli-fied peeling response could drive the pitch alignedcomponents by poloidal harmonic coupling, astudy of poloidal coupling on AUG was un-dertaken. To demonstrate general poloidal har-monic coupling in realistic tokamak geometry,single m perturbations were applied as bound-ary conditions at the plasma edge, and the re-sulting bulk perturbation computed. The resultsshowed that even when perturbations have only asingle poloidal mode number at the plasma sur-face, in the plasma bulk the spectrum broadensby poloidal harmonic coupling, and also shifts to-wards lower |m|. Computations of the plasma re-sponse also showed that poloidal harmonic cou-pling could drive the total pitch aligned compo-nents. To isolate the effect of the amplified peel-ing response on the pitch aligned components,the amplified peeling response was prescribed asa boundary condition with other harmonics fil-tered out, and the resulting pitch aligned compo-nents computed. The computations predict thatthe amplified peeling response can drive the pitchaligned components by poloidal harmonic cou-pling in some circumstances.

In experiments it is possible to measure thepeeling response using external magnetic pickupcoils [30], and it is often possible to measuremagnetic island widths using Thomson Scatter-ing [31] or Electron Cyclotron Emission [32],from which the pitch aligned field components


can be calculated. If the pitch aligned compo-nents are being driven primarily by the ampli-fied peeling response, then we may expect to finda correlation between the measured peeling re-sponse and the measured island widths. However,if the peeling response were varied via an experi-mental scan of ∆φul, then it is possible that a cor-relation may not be expected. This is because thetotal field is the sum of the plasma response andvacuum field, which have dependencies on ∆φuloffset from each other. However, a scan of q95 forfixed vacuum field may change the amplitude ofthe peeling response independently of the vacuumfield.

If a large stochastic edge region is an impor-tant component of RMP effects on tokamak plas-mas, then it follows that the RMP configurationwhich should have the largest effect on the plasmais the one which maximises the pitch alignedcomponents, and hence the stochastic region.These results imply that the optimal coil config-uration for stochasticity, may depend closely onthe amplified peeling response, rather than solelythe vacuum field. This may be a useful consid-eration when designing future RMP experiments,and further motivates the creation of plasma re-sponse ’parameter maps’, such as was calculatedfor DIII-D [15].

Acknowledgements

This work is part-funded by the EPSRCthrough the Fusion Doctoral Training Network(grant number EP/K504178/1), and part-fundedby the RCUK Energy Programme (under grantEP/I501045) and the European Communities.To obtain further information on the data andmodels underlying this paper please [email protected]. The views andopinions expressed herein do not necessarilyreflect those of the European Commission.

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