Hallatschek, K.

McKee, G.

English

MPG.PuRe

Theory of external geodesic acoustic mode excitationK. Hallatschek1, G.R. McKee21 Max-Planck-Institute for Plasma Physics, Garching, Germany2 University of Wisconsin-Madison, Madison, Wisconsin, USAIntroductionFrom the cornucopia of literature on flow shear suppression of turbulence it appears the fu-sion problem could be facilitated greatly if only the globalflows could be made strong enough tosuppress the convective turbulence, preferably by external me ns and not in reliance on the tur-bulence itself. It can be shown that injection of momentum byneutral particle beams or plasmawaves tends to be rather inefficient as the ratio of the applied ower to the momentum transferrate is given by the rather large phase- or particle velocity, respectively. In contrast, the resonantgeneration of geodesic acoustic modes – essentially oscillating zonal flows – by magnetic per-turbations by external coils is a potentially far more efficient method [1]. This could already bedone presently, e.g., in the DIII-D tokamak with the in-vessel ELM stabilization coils [I-coils].In contrast, inducing GAM plasma pressure oscillations by pulsed external heating has very lowefficiency independent of the spatial distribution of the heating, as the power deposited in theGAM motion is a factor of orderO(δT/T) (δT being the temperature oscillation associatedwith the GAM andT the background temperature) smaller than the heating powerused, as thisis thefree energydeposited by the heating [2].An elegant way to compute the action of external coil currents on the interior flux surfaceshas been discovered, which allows analytical estimates andthe coupling of a novel dynamicequilibrium code with a turbulence code to study the resonance layer in detail.Since GAMs are axisymmetric modes they can only interact with axisymmetric external cur-rents. Of these only toroidal currents need be considered, since any poloidal “ring coil” current(not encircling the plasma column) would just produce a fieldinternal to the coil and not af-fect the plasma. Heuristically, an external quadrupole (m= 2) toroidal current (fig. 1a) wouldattract/repulse the toroidal currents within an elliptic plasma column in such a way as to causea rotation (tilting force) of the column, i.e., act as a source for the GAMs. The current distribu-tion should be up-down and left-rightanti-symmetric, since otherwise the plasma column is justdeformed, not tilted. The corresponding perturbativemagnetic fieldhas left-right and up-downsymmetry, since it is a pseudo vector. (A perturbation withm= 1 corresponds to a pure shift ofthe plasma column, assuming high aspect ratio, and is ineffective.)39th EPS Conference & 16th Int. Congress on Plasma Physics O4.116(a) (b) (c)Figure 1:(a) Linear displacement and flux surface perturbation visualized for hypot etical quadrupoleperturbation currentI = 90kA for DIII-D plasma configuration from discharge #119527, including theeffect of screening currents in the limiter structures (not shown). Gray: unperturbed separatrix. Red/bluetoroidal loops: position of positive/negative perturbation current relative to the plasma current. Red/blueshells: volume traversed by positive/negative radial displacement of fluxsurfaces. Yellow arrows in crosssection: poloidal displacement component. White arrows: total displacementtangential to flux surfaces.GAM excitation in a turbulent nonlocal resistive ballooning scenario in a thin sell at the edge of a cir-cular tokamak discharge; (b) flux surface averaged poloidal flow velocity with resonance, (c) shearing ofturbulence fluctuations, top: cut from outboard midplane, bottom: steepening of on temperature profile.Dynamic equilibrium perturbationThe action of external coil currents on the interior flux surfaces of the plasma has been studiedby means of a novel dynamic equilibrium code as well as analytic lly, deriving a surprisinglysimple transfer function. Discounting very low aspect ratios R/a, the radial and poloidal dis-placement amplitude of the quasistatic equilibrium perturbation, augmented by the resonancequality factor, can serve as an estimate for the GAM amplitude. As shown exemplary for thecode output for a DIII-D configuration in fig. 1a, the flux surfaces are indeed repulsed/attractedby negative/positive currents.39th EPS Conference & 16th Int. Congress on Plasma Physics O4.116At this point the potentially greatly facilitating effectsof the presence of a depression ora null (X-point, saddle point ofψ) in the magnetic field become apparent. First, the radialdisplacement amplitude increases inversely proportionalto the poloidal fieldBpol = |∇ψ|/R,as visible close to the “X” on top in the displacement field shown in fig. 1a, since a given fluxperturbation(δψ) causes a larger shift at a smaller background flux gradient.Furthermore, the poloidal displacement resulting from theincompressibility of the toroidalmagnetic field is amplified by the nozzle effect of close flux surface spacing at the midplane.Specifically, for the outermost shown flux surface displacement in the figureδBpol ∼ 25mT,|ξrad,max| ∼ 4.8cm|ξpol| ∼ 13cm, and the nozzle effect amounts to max(|∇ψ|)/min(|∇ψ|)∼2.9– the displacement diverges at the separatrix.In addition it becomes clear that the plasma is not screening(as one might expect) but ratheramplifying the perturbation current. This is due to the attraction/repulsion of the toroidal plasmacurrents by the external current in the same/opposite direction.Interaction with turbulenceTo describe the induced resonance layer within a turbulencesimulation it is essential to retainthe radial variation of the GAM frequency throughout the computational domain, i.e. use a“nonlocal” code and not rely on the flux tube/local approximation.Fig. 1b,c show the possible characteristic features in an electrostatic turbulence run withthe NLET code [3] for the following reference parameters at the middle of domain (r = 0):α = 0.1,εn = 0.05,ηi = 1,τ = 1,q = 3,s = 1. The externally induced poloidal displacementwasL0/2, (L0 is the ballooning scale length; typical ballooning scale lengths in a tokamak edgeare 0.3cm). The resonance layer is obvious in the time series of theflow profiles (fig. 1b) fromthe amplitude peaking and the characteristic radial phase vriation due to the radial variation ofthe local oscillator frequency. The quality factor at the resonance is found to beQ∼ 25, i.e.,the externally induced displacement is amplified by a factorQ at the resonance. The peakingand phase variation lead to a strong flow shear at the resonance, d a modulation and partialsuppression of the turbulence by the GAMs. For high enough external forcing, this leads tosomething akin to a transport barrier (fig. 1c).DiscussionAccording to the definition of the resonance quality factorQ the power lost by GAM dampingis PGAM = ωGAMEGAM/Q. The energy stored in the GAM oscillation isEGAM = mω2GAMd2/2,whered ∼ ξpolQ is the resonance-amplified displacement amplitude,m= 4π2aRwnmi the os-cillating plasma mass,w the width of the GAM resonance layer,n the ion density andmi the39th EPS Conference & 16th Int. Congress on Plasma Physics O4.116ion mass. For circular slender plasma, the GAM frequency isωGAM =√2γcs/R, cs =√2T/mi .Inserting typical tokamak edge valuesT ∼ 200eVn∼ 1019m−3, R∼ 1.5m, a∼ 0.5m, and as-suming the width and displacement of the GAMs to be relevant to the turbulence,w,d∼ 2cm,and deuterium as medium results inωGAM ∼ 175krad/s, EGAM = md2ω2GAM/2 ∼ 0.09J andPGAM = EGAMωGAM/Q ∼ 0.7kW for Q ∼ 20. For realistic tokamak experiments the powerrequirements are about an order of magnitude smaller, sincethe observedωGAM is reducedcompared to the value for a circular plasma by about a factor of two [4, 5].To achieve GAM amplitudes sufficient to significantly affectthe turbulent diffusivity certainlyrequires a dedicated setup, which to our knowledge presently is nowhere installed. Howeverfor a first demonstration or even diagnostic purposes (to measur plasma acoustic resonancespectra) in many machines existing positioning coils or perturbation coils for edge instabilitycontrol (e.g. resistive wall, edge localized or quasi coherent modes) could be used, providedthey can be driven at the GAM frequency. For example in DIII-Dthe ELM suppression coils(I-coils) can produce a perturbation field of 0.02mT at 7kHz yielding a displacement of∼ 6mmunder good conditions with the above estimates, while a displacement amplitude of∼ 2mmmagnitude is routinely detected by spectroscopic imaging and Doppler measurements [6, 7].The parameters could even be chosen more favorable to the GAMexcitation, e.g. by loweringthe equilibrium magnetic field and the GAM frequency. Moreovr, even while not sufficient tosuppress the turbulence completely, raising the GAM amplitude somewhat may lower the LHtransition threshold [4].The optimal conditions for GAM generation can be found by computing the induced dis-placement for varying plasma shapes, coil and passive condutor positions. In addition, the per-turbation itself may also be maximized. For example elongati the plasma column increasesthe sensitivity to changes of the vertical magnetic forces,up to the well known vertical dis-placement instability. Higher order shaping, such as indentation, can convert the thus amplifiedvertical shift into the desired poloidal displacement.References[1] K. Hallatschek, G.R. McKee, Phys. Rev. Lett.submitted[2] K. Hallatschek, Phys. Rev. Lett.93, 125001 (2004)[3] K. Hallatschek, et al., Phys. Plasmas7, (2000) 2554[4] G.D. Conway, et al., Plasma Phys. Control. Fusion50, 085005 (2008)[5] K. Hallatschek, Plasma Phys. Control. Fusion49, B137 (2007)[6] G. McKee, et al., Phys. Plasmas5, 1712 (2003)[7] G.D. Conway, et al., Plasma Phys. Control. Fusion50, 085005 (2008)39th EPS Conference & 16th Int. Congress on Plasma Physics O4.116

Theory of external geodesic acoustic mode generation

http://hdl.handle.net/21.11116/0000-0008-8C96-1

Theory of external geodesic acoustic mode generation

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