Recent Progress in Self-consistent full wave simulations of lower hybrid waves John C. Wright P. T. Bonoli - MIT E .J. Valeo, C. K. Phillips - PPPL R. W. Harvey - Comp-X US Japan RF Workshop San Diego, March 8-10, 2010

Recent Progress in Self-consistent full wave simulations of lower hybrid waves E. J. Valeo1, P. T. Bonoli2, R. W. Harvey3, C. K. Phillips1, J. C. Wright2 1 PPPL, 2 MIT, 3 Comp-X US Japan RF Workshop San Diego, March 8-10, 2010

Participants in the Center for Simulation of Wave-Plasma Interactions L.A. Berry, D.B. Batchelor, E.F. Jaeger, E. D`Azevedo D. Green D. Smithe P.T. Bonoli, J.C. Wright H. Kohno, A.S. Richardson C.K. Phillips, E. Valeo N. Gorelenkov, H. Qin R.W. Harvey, A.P. Smirnov N.M. Ershov M. Brambilla R. Bilato M. Choi R. Maggiora D. Milanesio D. D’Ippolito, J. Myra

Lower Hybrid heating/current drive overview Modeling LH waves Outline Lower Hybrid heating/current drive overview Modeling LH waves Ray tracing Full Wave Comparisons between approaches Full wave effects + non-thermal electrons + relativistic effects Summary and Conclusions

LHRF Regime Compare with ICRF FW perp wavelength of 10cm LHCD Regime: ci<<<<ce and  ≥ 2LH Unmagnetized ions Strongly magnetized electrons [(ke)2  1] Wave equation is sixth order with two propagating modes, one damped: Mode converted ion plasma wave is not propagative, so drop sixth order term Electrostatic LH “slow wave” branch Electromagnetic LH “fast wave” branch Wave lengths are very short: Predicts an accessibility criterion: Index of refraction Compare with ICRF FW perp wavelength of 10cm wallace PP06.0076 parametric studies of LH coupling Bonoli PF 1982 Density

Tradeoffs and constraints Compare with ICRF FW perp wavelength of 10cm wallace PP06.0076 parametric studies of LH coupling

Approaches to solution WKB expansions: ray tracing (GENRAY, ACCOME) and beam tracing (LHBEAM) are asymptotic approximations to the wave equation, k a>>1 required. => can be problems with boundaries Full Wave(TORIC-LH): solves Maxwell's equations directly, yields the electric field. Hyperbolic Maxwell-Boltzmann Integral wave equation Time Harmonic -> Elliptic wave equation WKB -> raytracing terminated at 90%

TORIC Full Wave Code TORIC [Brambilla 1999] was developed with an FLR model for the plasma current response, JP, for ion cyclotron waves and recently extended with an asymptotic form for lower hybrid waves. The antenna is modeled as a current sheet, JA for ICRF and as the mouth of a wave guide with imposed E|| for LH. It solves Maxwell's equations for a fixed frequency (Helmholtz problem) assuming toroidal symmetry in a mixed spectral-finite element basis. This is the physical optics solution. Large Block Tri-diagonal matrix structure [ (8M)2 X NPSI ] Parallelized Serial

Simulation model is a coupled full wave and Fokker-Planck system TORIC-LH (Wright 2004CPC,2009PoP)‏ CQL3D (Harvey 1992IAEA)‏ Bounce averaged Fokker-Planck code that solves for new distribution function from RF quasilinear flux, provides f(v)‏ v0/vte v||0/vte fe v|| Dielectric for LH uses zero FLR electrons and unmagnetized ions. Calculates the RF fields and the quasilinear flux. => fe(v) and E(x) are self-consistent Electron distribution functions from iteration with a Fokker-Planck code are used for dielectric. Pre-tabulation of Im Chizz results in no speed loss compared to the evaluation of the Z function for the Maxwellian case. Simulations use EFIT reconstructed magnetic equilibria.

Full wave code algorithms TORIC-LH AORSA Jaeger Alcator C. 106 mesh pts. n||=-2.5, B0=8T, ne0=5x1019 m-3,Te0=5keV. COMSOL-LH Meneghini, Shiraiwa, Parker COMSOL with 2D FE TORIC-LH with 1D FE/1D Spectral AORSA with 2D spectral Each approach has advantages and difficulties: CPU-hrs: COMSOL-LH 13, TORICLH 80, AORSA 32k (ray tracing ~ minutes) but wall clock times for all three are comparable given number of processors used. (2hrs)‏ COMSOL: 2D elements can model wall and separatrix region; sparse matrix scales well The plasma dielectric: COMSOL-LH requires real space dielectric formulation, doesn't have algebraic k|| TORIC-LH needs care in treating FLR truncation AORSA – All Orders treatment most general, can handle fast ion interactions Alcator C-Mod. 512x512. n||=-4.0, B0=2T, ne0=6x1019 m-3,Te0=4keV.

Single pass is well converged with linear damping (Maxwellian)‏ Single pass damping on a Maxwellian plasma is converged. Little interference is evident. Power is very localized in single pass. <- Spectrum is converged at 1023 modes Use this slide to explain common measures that are used. Spectrum, power and poynting, 2D fields. Expect to spend at least 2minutes on this slide. r/a n|| = -2.55, ne0=7e19 m-3 Te0 = 10 keV, B0=4T

Comparison with ray tracing in weak damping limit Alcator C-Mod. n||=-1.55, B0=5.3T, ne0=7x1019m-3,Te0=2.3keV. Maxwellian damping in both cases. Good agreement in power deposition locations. Field amplitudes are magnified by “cavity effect” [Q~/γ~200]

Weak damping requires quasilinear fe(v) for self-consistent treatment Bulk damping Accessibility cutoff Width indicates poloidal broadening of Δn||/n||0 ~ Δm/q nφ r/a The poloidal power spectrum shows lack of convergence at outer flux surfaces, Also magnification of unit amplitude applied field.

<Dql>bounce for FP coupling Calculated following full wave treatment in Wright NF1997. Bounce averaged, Larmor cutoff [ J0(kρi) ] at high perpendicular energies. Non-relativistic. Note the symmetry in the trapped region. v0/vte v||0/vte <Dql> v0/vte v||0/vte

Effect of non Maxwellian damping Step0 Step1 Step2 Convergence achieved in two iterations. Note the reduction in peak amplitude. Fields are less space filling now.

Effect of non Maxwellian damping r/a Energy in poloidal modes plotted for selected flux surfaces. Power density and Poynting flux. Power is normalized to the total power. Spectrum is converged. Power is a bit broader.

Iteration between TORICLH and CQL3D with relativistic effects converges rapidly From TORIC_RUN 600. Damping first over damping then settles down to marginally multipass. Poynting flux units are A.U. (for unit applied field) Power is normalized to total coupled power. Iterations Converges after some oscillations in damping strength. Shows increased absorption Power used = 350 kW generating 150 kA, n||=-2.5, Te=2.5 keV Resolution, TORIC 400Nrx255Nm, CQL3D 60Nr x 88 Npitch x 160 Nu

Step 5

SUMMARY TORIC + CQL3D produce self-consistent, relativistically correct, solutions for full-wave lower hybrid fields and electron distribution functions in toroidal plasma with realistic magnetic geometry and profiles. X-ray spectral intensities computed from synthetic diagnostics have been validated against data from Alcator C-MOD. Inclusion of an improved radial transport model in the evolution of Fe may resolve the difference between the computed and the broader experimental radial profile. Near term plans: Import a newly developed (Lee, Wright) parallel algorithm with 3D decomposition (M1, M2, NPSI), shown to perform well (5X) in ICRF TORIC.

Additional material

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