Bilato, R.

Bonoli, P.

Brambilla, M.

Wright, J.

English

MPG.PuRe

Numerical simulation of Ion Cyclotron waves in toroidal plasmasM. Brambilla1, R. Bilato1, J.C. Wright2, P. Bonoli21 Max-Planck Institut für Plasmaphysik - Euratom Association, Garching, Germany2 Massachusetts Institute of Technology, Cambridge, Massachusetts, USAIntroduction. The code TORIC [1] solves Maxwell equations in the Ion Cyclotron (IC) rangeof frequencies in toroidal axisymmetric plasmas. To make possible selfconsistent simulationsof IC experiments in tokamaks, TORIC has been coupled with SSFPQL [2], which solves thequasilinear Fokker-Planck (QLFP) equation for ions. The TORIC-SSFPQL package has beendescribed in some details in [3]. Here we will briefly recall the physics included in the codes, anddiscuss some features of the model related to specifically toroidal effects. Use of this packagefor parametric studies of IC heating scenarios in ASDEX Upgrade is illustrated in a companioncontribution to this Conference [4].TORIC. The model implemented in TORIC includes propagation and damping of the ex-ternally excited fast wave (FW), and of Ion Bernstein (IBW) and Ion Cyclotron (ICW) waveswhich can be excited by linear mode conversion (LMC) near ion-ion resonances. The absorptionchannels are fundamental and first harmonic IC heating of ions, and Landau and Transit Timedamping by electrons. Optionally, damping of the FW at higher IC harmonics can be simulated,although not simultaneously with mode conversion [5]. The wave equations are derived fromVlasov equation by expanding the e.m. fields in Fourier modes in the toroidal and poloidal an-gles [6]. While for the FW an expansion to second order in the Larmor radius is adequate, largeLarmor radius effects are taken into account to describe IBWs [7], in such a way that the localdispersion relation of the wave equations solved by TORIC is everywhere a good approxima-tion of the full hot-plasma dispersion relation. As a stand-alone, TORIC assumes local thermalequilibrium; for iteration with SSFPQL the coefficients of the wave equations are evaluated(to all orders in the Larmor radius) using the distributions predicted by the quasilinear kineticequation as explained in [3].The spectral representation of the hf field allows a rigorous definition of the local wavevec-tor~kn,m for each mode of toroidal and poloidal numbers n and m. The physical componentsof ~kn,m depend on position only on the scale of the MHD equilibrium, and, therefore, playmuch the usual role in determining the plasma response, in particular in defining particle-wavesresonances. There are, nevertheless, specifically toroidal effects which cannot be neglected.The first is toroidal broadening of IC resonances (in addition to the usual Doppler width) dueto the final length of the stationary-phase region of individual ions. In [8] it has been shown33rd EPS Conference on Plasma Phys. Rome, 19 - 23 June 2006 ECA Vol.30I, O-2.013  (2006)-31 <= m <= +31-63 <= m <= +63Figure 1: 1 – 6% H in D, ASDEX Upgrade: logarithmic plot of mod E+(m) versus ω/k‖vtheon 5 equidistant magnetic surfaces, toroidal mode n = 12.that this can be taken into account by “correcting” the local parallel wavevector with a factorwhich depends on the parallel gradient of the static magnetic field. This approximation is quiteaccurate, and guarantees that TORIC predicts that in a stable plasma power is everywhere (andnot only on average) transferred from the waves to the resonant ions.Secondly, the parallel component of~kn,m, and thus the parallel phase velocity, is different fordifferent Fourier modes. Because of the strong coupling between modes in toroidal geometry,the resulting “toroidally induced broadening” of Cherenkov resonances is by no means a smalleffect. In fig. (1) of [3] it is shown that even for the FW, whose poloidal number spectrum islimited by the relatively large wavelength, the range of parallel phase velocities for a singletoroidal mode extends from infinity to well below the electron thermal speed. It follows that thespectral representation is essential for an accurate estimate of absorption by the electrons: theapproximation k‖ ∼ n/R, for example, can be at best qualitatively correct.Numerical resolution of IBWs and ICWs excited by LMC requires a quite broad range ofpoloidal Fourier modes. We can distinguish two very different situations, roughly correspondingto the minority and the mode-conversion regimes, respectively.Fig. 1 compares the amplitude of modes with different parallel phase velocities in two runswith different resolution of a typical minority scenario [9] on 5 equidistant magnetic surfaces,from r/a = 0.1 (deep blue) to r/a = 0.9 (red). In this case, lack of convergence only affectsthe representation of IBWs in the plasma periphery. Remarkably, the low-order poloidal modesare essentially unaffected when new modes are taken into account: the global power balanceand the radial deposition profiles are already accurate to better that 0.5% in a run with only 31modes (−15≤m ≤+15), which strictly speaking should be sufficient only to resolve the FW.The situation is less favourable in simulations of mode conversion scenarios, particularly33rd EPS 2006; M.Brambilla et al. : Numerical Simulation of Ion Cyclotron Heating in Toroidal Plasmas 2 of 4Figure 2: 2 – 6% H in D, ASDEX Upgrade: Effective minority tail temperatures, and quasilinearpower deposition profiles in the minority (per MW coupled), at 4 and 8 MW.when the minority fundamental IC resonance is ‘isolated’, i.e. does not coincide with the firstharmonic of the majority ions. Convergence requirements are then particularly severe, becauseof the rapid spatial variations of the plasma dielectric properties near the MC layer. Reproducingnumerically the sharp minimum of E+ due to screening by resonant ions requires the destructiveinterference of a sufficiently large number of Fourier modes. Lack of convergence can result inoverestimating |E+|2, and therefore power absorption by the minority ions, by quite a largefactor. Indeed, satisfactory simulations in this regime, such as those reported in [4], have beenmade possible only by the parallelization of the code [10].TORIC has successfully reproduced IC heating results in several tokamaks, and shows goodagreement with the AORSA code [11], which solves for the same physics (to all orders in theLarmor radius) with a different numerical approach. Perhaps the best benchmark has been theaccurate simulation of mode conversion to ICW, a toroidal effect first observed in Alcator C-Mod [12].SSFPQL. The code SSFPQL [2] evaluates the steady-state distribution function of ionsheated at the fundamental and first harmonic IC resonance by balancing the Kennel-Engelmannquasilinear operator (QLO) [13] with the linearized collision operator. The form of the bounce-averaged QLO is simplified by neglecting the acceleration of ions in the resonance region. Thesolution is represented as a series in even Legendre polynomials including typically 5 to 8 terms;a special Bessel function identity is used to guarantee that the truncated expansion of the QLOis positive definite up to a sufficiently high energy, about 7 times the unperturbed temperature.Due to this approach, SSFPQL is very fast (less than 20 sec for two ion species on 100 magneticsurfaces on a laptop).33rd EPS 2006; M.Brambilla et al. : Numerical Simulation of Ion Cyclotron Heating in Toroidal Plasmas 3 of 4Fig. 2 shows the tail effective temperatures, estimated from the logarithmic slopes of thetails in the distribution function calculated by SSFPQL, and the power absorption profile of theminority for the same ASDEX Upgrade shot as in fig. 1, at two power levels, after 3 iterationswith TORIC. Of 4 MW, the power effectively coupled in the experiment, about 30% is predictedto heat the electrons, 10% directly from the wave, and 20% by collisional transfer, in goodagreement with the experimental observations [9]. 8 MW is the power level which would havebeen necessary for half of the power to end up in the electrons. As expected, increase in Teff‖broadens the minority absorption profile, but increase in Teff⊥ decreases the peak heating rate.Even at the relatively high collisionality of this experiment the anisotropy of the suprathermaltail is large, and increases with power, so that the second effect dominates, and the fraction ofpower absorbed by the minority ions decreases with increasing total power.SSFPQL is not adequate to determine the fate of the most energetic ions. For the exponen-tially more numerous ions at intermediate energies, which absorb the bulk of the IC power,it nevertheless provides sufficiently accurate distribution functions for coupling with TORIC.Important for this purpose are a good estimate of the anisotropy of the suprathermal tails, andthe inclusion of large Larmor radius corrections in the coefficients of the wave equations. Themajor source of inaccuracy is the fact that SSFPQL, like all otherwise more sophisticated QLFPsolvers based on surface averaging, neglects the finite width of the banana orbits of fast ions.References.1 M. Brambilla, Plasma Phys. Contr. Fusion 41, 1, (1999).2 M. Brambilla, Nucl. Fusion 34, 1121, (1994).3 M. Brambilla, R. Bilato, Nucl. Fusion 46 (2006), in press.4 R. Bilato, M. Brambilla, J.R. Wright, P.T. Bonoli, this Conference.5 M. Brambilla, Plasma Phys. Contr. Fusion 44 2423 (2002).6 M. Brambilla, Plasma Phys. Contr. Fusion 41 775 (1999).7 M. Brambilla, Nucl. Fusion 38, 1805, (1998).8 M. Brambilla, Phys. Letters A 188, 376, (1994).9 M. Brambilla, R. Bilato, C. Maggi, et al 16th Topical Conf. on Radio Frequency Power inPlasmas, AIP Conf. Proc. 787 (2005), p. 38.10 J.C. Wright, P. Bonoli, M. Brambilla, et al, Phys. Plasmas 11, (2004) 2473.11 E.F. Jaeger, L.A. Berry, E.D’Azevedo, et al Phys. Plasmas 8 1573 (2001).12 E. Nelson-Melby, M. Porkolab, P.T. Bonoli, et al, Phys. Rev. Letters 90 155004 (2003).13 C.F. Kennel, F. Engelmann, Phys. Fluids 9, 2377, (1966).33rd EPS 2006; M.Brambilla et al. : Numerical Simulation of Ion Cyclotron Heating in Toroidal Plasmas 4 of 4

Numerical simulation of Ion Cyclotron waves in toroidal plasmas

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