Integration of Fokker Planck calculation in full wave FEM simulation of LH waves O. Meneghini S. Shiraiwa R. Parker 51 st DPP APS, Atlanta November 4, 29 L H E A F * Work supported by USDOE awards DE-FC2-99ER54512 and DE-AC2-76CH373 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 1 / 23
Abstract A full wave simulation code based on Finite Element Method (FEM) was developed to solve LH waves in tokamaks 1. The FEM approach allows a seamless handling of the antenna, first wall, SOL, divertor and core regions. In the region of plasma inside of the separatrix, electron landau damping is modeled by means of an iterative procedure. The code has been recently coupled to a bounce averaged Fokker Planck solver, which self consistently calculates the electron 2D distribution function resulting from the balance between collisions and RF quasilinear diffusion (DQL) in a toroidal geometry. The evaluation of the DQL term from the full wave fields has been done by integration of the momentum equation of test particles. Results will be presented for the Alcator C-Mod tokamak. 1 S. Shiraiwa, TI3.3 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 2 / 23
Introduction Background Why full-wave based simulations? Modeling LH by ray tracing has several longstanding issues: WKB requires K K << 1 which for LH waves in Tokamak plasmas is questionable: at low densities (small K ) in fast changing density (big K) near cutoffs (P ) near caustics ( K r ) Ambiguity in the launched spectrum ray has to start inside the cutoff finite height of waveguide Impossible to get direct evaluation of the electric field GENRAY simulation by Greg Wallace O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 3 / 23
Introduction Maxwells equation for LH waves LH waves in a Maxwellian plasma Maxwell equation in the spectral domain k ( k E ) + ω2 c 2 ɛ( k) E( k) = LH waves are well described by cold plasma dielectric tensor for propagation and Electron Landau Damping (ELD) absorption term S id ɛ( k) = ɛ cold i χ ELD ( k) = id S i P χ ELD (k z ) Wave-number dependence of χ ELD (k z ) term results in an integro-differential equation in real space domain ( ) ( E( x)) + ω2 c 2 ɛ cold E( x) ẑ i χ ELD (z z )E z (z )dz = 2π O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 4 / 23
Introduction LH waves in a Maxwellian plasma Susceptibility χ ELD for a Maxwellian plasma ( ) χ ELD (k ) = 2σ exp ω2 πωpω 2 k 2 sgn(k ) w2 k 3 w2 w χ ELD (z) = 1 2π χ ELD (k )e i2πk z dk A.U. 1.8.6.4 A.U. 6 5 4 3 2 1.2 3 2 1 1 2 3 k z [m 1 ] 1 2.2.1.1.2 z [m] Plots of χ ELD in the wavenumber and real space domain, for 1 kev and 1 kev at 4.6 GHz O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 5 / 23
Introduction Solution in FEM by means of iterative procedure Solution with iterative procedure 2 the LHEAF code The original integro-differential equation is split into two parts: A conventional PDE which can be solved by FEM codes (COMSOL) ( E ) (N) ( x)) + ( ɛ ω2 c 2 cold + ɛ (N) Leff E (N) ( x) = A convolution integral (computed in MATLAB) ɛ (N+1) Leff = i E (N) and the solution is found iteratively ẑẑ 2π dz χ ELD (z z )E (N) z (z ) COMSOL Leff E Initial guess MATLAB Leff COMSOL L H E A F Converged E (N) =E (N 1) E N N th step Lower Hybrid wave Analysis based on FEM 2 O. Meneghini et al., PoP 29 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 6 / 23
Introduction Alcator C - Maxwellian plasma Alcator C - Maxwellian plasma 2.5 kev 3 kev Comparison of LHEAF with fullwave code Toric-LH a and raytracing shows very good agreement, especially in the single-pass absorption regime a P. Bonoli PP8.5, J. Wright PP8.6 5 kev 1 kev [W/cm 3 ] 11 1 9 8 7 6 5 4 3 2 1 Power density LHEAF Solid line Toric LH Dashed line Ray Tracing Dash Dotted line 2.5 kev 3 kev 5 kev 1 kev.2.4.6.8 1 Normalized minor radius * Normalized output of Toric-LH is scaled to match O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 7 / 23
Introduction Integration with Fokker Planck Integration with Fokker Planck solvers Development of quasi-linear plateau plays an important role in determining: Power absorption profile Current drive In order to achieve realistic modeling of LHCD in tokamak plasmas, coupling to Fokker-Planck solver is of critical importance 1D Fokker Planck) Uniform plasma Simple and efficient 2D Fokker Planck 3 Mirroring effect (bounce-averaged, -width banana width) Computationally more intensive 3 Computational Methods for Kinetic Models of Magnetically Confined Plasmas, Killeen et al., Springer-Verlag, 1986 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 8 / 23
Fokker Planck equation Introduction Integration with Fokker Planck f( r, v, t) has to satisfy the Boltzmann equation: f + v f + F m f v = ( ) f c Fokker Planck equation expresses all processes as diffusion f = (D f): ( ) ( ) ( ) f f f f = + + +... c DC RF Validity of RF as (quasilinear) diffusion, relies on the assumption: The spectrum of modes as seen by the particle, is sufficiently dense and broad that phase sensitive effects such as particles trapping disappear through desctructive interference and phase mixing The change of particle velocities during the wave-particle interaction is small and the interaction takes place over many wavelengths O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 9 / 23
1D Fokker Planck 1D Fokker Planck equation 1D Fokker Planck equation for electrons 4 f e = ( ) fe + c ( ) fe ql1d Collisional term for a Maxwellian background plasma: ( ) ( ) fe = v c D c (v ) v + ν c (v )v fe v Plasma-wave interaction in terms of Fokker-Planck diffusion ) term: ( fe = ql1d v D ql1d (v ) fe v ( ) Steady state solution fe = : f e (v ) =» ν c(v ) = βz Γ v 2 3/2 v Te 3 1 + v Te D c(v ) = βz Γ v Te»1 + Γ = ln(λ)nee4 4πɛ 2 m 2 e β z = (1 + Z)/5 ( ) 1 v ν c (v exp )v 2πv 2 T e D c (v ) + D ql1d(v )dv If D ql1d (v ) then f e (v ) Maxwellian 4 Lower Hybrid Simulation Code Manual,D. W. Ignat et al., 2 v v Te 2 3/2 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 1 / 23
1D Fokker Planck Evaluation of D ql1d Single particle D v in a running wave One dimensional motion (along the field lines) of a single particle in a running wave. Considering th and 1 st order motions: z = z + (v + v)t m d v dt Results in v = qe m = qe cos(kz + kvt ωt) sin(k(tv+z ) tω) sin(kz ) kv ω Diffusion as the mean square spread in velocity: Averaging over initial position z : D v = ( v)2 2t = qe m for t, D v π 2 2 sinc(t(ω kv)/2) 2 ( qe 1.9.8.7.6.5.4.3.2.1 Variation of <(Δ v) 2 > with (ω kv) ) 2 m δ(ω kv) 2 1 1 2 (ω kv) t=.1 t=.2 t=.3 t=.4 t=.5 t=.6 t=.7 t=.8 t=.9 t= 1 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 11 / 23
1D Fokker Planck Evaluation of D ql1d Evaluation of D ql1d from wave fields 5 Starting from the single particle picture, the wave electric field from full-wave code is Fourier decomposed: D ql1d (v ) = πe2 2πn/L 2 E k (k ) 2m 2 e L 2 δ(ω k v ), n =, ±1, ±2,... k = 2πn/L lim D ql1d(v ) = πe2 2π L 2m 2 E k (k ) 2 δ(ω k v )dk = πe2 e L 2m 2 e For the Alcator-C (5 kev) example at r/a =.5: 15 1 5 E along field line 2 4 6 8 L 2 15 1 5 E spectrum 4 2 2 4 n 8 6 4 2 x 1 18 D ql1d ( ) 2π ω 2 Lv E k v 1 1 v x 1 8 5 J. Lee PP8.4 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 12 / 23
1D Fokker Planck Evaluation of D ql1d Evaluation of f e from D ql1d By substituting into 1DFP equation f e = ( fe one finds the parallel distribution function. For the Alcator-C (5 kev) example at r/a =.5: ) c + ( ) fe ql1d 3 x 18 log 1 (f e ) 2.5 2 v 1.5 1.5 3 2 1 1 2 3 v x 1 8 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 13 / 23
1D Fokker Planck Susceptibility χ ELD Evaluation of χ ELD from f e (v, v ) Susceptibility for a hot non-relativistic plasma, considering only ELD: ω2 p χ(k ) = ω 2 m 3 e ( +ê ê f p p p 2 2πp dp [ dp 1 p f 2 p f p + p k v ω k v f p p 2 p 2 The imaginary part of the susceptibility arises from the residual of the term having v = ω k and is equal to: I[χ(k )] = ω2 p2π ω 2 ω2 k 2 f v v = ω k )] v dv = χ ELD (k ) O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 14 / 23
1D Fokker Planck Alcator-C (5 kev) example at r/a =.5 E D ql1d f e χ ELD D ɛ Leff 15 E along field line 2 E spectrum 1 15 5 1 5 1 2 3 4 5 6 7 8 9 L 4 3 2 1 1 2 3 4 n x 1 18 D ql1d f e 8 6 4 1 1 2 1.5 1.5.5 1 1.5 v x 1 8 χ ELD 2 15 1 5 1.5 1.5.5 1 1.5 v x 1 8 8 6 4 2 x 1 5 D along field line 1 8 6 4 2 2 4 6 8 1 n 1 2 3 4 5 6 7 8 9 L ε Leff 3 25 2 15 1 5 1 2 3 4 5 6 7 8 9 L O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 15 / 23
1D Fokker Planck Alcator C - 1D Fokker Planck plasma Alcator C - 1D Fokker Planck plasma (preliminary) 2.5 kev 3 kev By shooting 4 kw, f e is significantly modified and waves penetrate more deeply into the plasma. Consistent with the picture of D ql spoiling the absorption at high power. 11 Power density 5 kev 1 kev [W/cm 3 ] 1 9 8 7 6 5 4 3 2 1 2.5 kev 3 kev 5 kev 1 kev.1.2.3.4.5.6.7.8.9 Normalized minor radius O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 16 / 23
1D Fokker Planck Alcator C-Mod - 1D Fokker Planck plasma Alcator C-Mod - 1D Fokker Planck plasma (preliminary) Log 1( E + 1) Sample simulation for experimentally relevant plasma parameters (n e =.8 1 2 m 3, T e = 4keV, P in = 6kW, n = 1.9) Radial power profile is broad Power density 3 Intermediate steps (1...9) Last step (1) 2.5 [W/cm 3 ] 2 1.5 1.5.2.4.6.8 Normalized minor radius O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 17 / 23
2D bounce-averaged Fokker Planck 2D bounce-averaged Fokker Planck equation 2D BAed linearized relativistic Fokker Planck equation Cylindrical coordinates (u, θ) in phase space, where u = γv ( ) 1 f = 1 G a 1 H a Γ a c u 2 + u u 2 sin(θ + λs a (t) ) θ G a = A a f + B a f u + C a f θ H a = D a f + E a f u + F a f θ Here A a G a are the bounced averaged diffusion coefficient, which are expressed as a function of g and h, the Rosenburg potentials (integrals over the background Maxwellian distribution function). S a (t) is the particle source term which is used to conserve the momentum which is lost onto the background Maxwellian plasma. Hence we need to solve a time dependent integro partial-differential equation in the 2D velocity space. O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 18 / 23
2D bounce-averaged Fokker Planck 2D bounce-averaged Fokker Planck equation 2D BAed linearized relativistic Fokker Planck equation Bouncing motion of a particle in a magnetic well Distribution function f e at arbitrary poloidal location s is obtained through mapping of the midplane distribution function f e(u, θ) by considering magnetic mirror effect: B(s) B = sin2 (θ) sin 2 (θ ) Diffusion terms are averaged over the bounce orbit τ B = H dl u Zero banana width approximation Linearization of the Fokker Planck equation: ) = F (f a (t), f b (t)) = F (M a + f a(t), M b + f b (t)) ( f c F (f a(t), M b ) + F (M a, f b (t)) F (f a(t), M b ) describes the collision of the test particle distribution function onto the background Maxwellian plasma. This interation is described by the A a G a terms, which are time independent terms F (M a, f b(t)) describes the collision of the background Maxwellian plasma onto the test particle distribution function. This interaction is described by the time dependent source term S a(t) O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 19 / 23
2D bounce-averaged Fokker Planck Evaluation of D ql2d Evaluation of D ql2d from wave fields Single particle picture suggests we can evaluate D ql2d by: Considering a mesh of particles in the 2D distribution function (u i, θ j ) Solve the momentum equation for the particles, when moving in E as evaluated from the full wave code Evaluate ( u ij ) 2 by averaging over the initial phase of the fields Consequently D ql2d (u i, θ j ) = ( uij)2 2t For the Alcator-Cmod example at r/a =.5: O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 2 / 23
2D bounce-averaged Fokker Planck Evaluation of D ql2d Evaluation of f e (u, θ) from D ql2d For t, the 2D BAed FP equation converges to a steady state solution. Shown is the 2D distribution function when in the presence of LHCD and DC electric field. Worth of notice are: Strong u asymmetry RF induced plateau Trapped region For the Alcator-Cmod example at r/a =.5: Log 1 (f e (γ v,θ)) x 1 8 24 26 5 28 4 3 1 24 1 26 Log 1 (f e (γ v,θ)) θ : [deg] θ : 45 [deg] θ : 9 [deg] θ : 135 [deg] θ : 18 [deg] γ v 3 32 1 28 2 34 1 36 38 1 3 5 4 3 2 1 1 2 3 4 5 γ v x 1 8 4 1 2 3 4 5 γ v x 1 8 O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 21 / 23
2D bounce-averaged Fokker Planck Susceptibility χ ELD Susceptibility χ ELD Susceptibility χ ELD can be computed for the 2D FP as for 1D FP. 2D distribution function depends on poloidal location via magnetic mirror mapping, consequently also χ ELD is now a function of space. Integration of LHEAF with 2D BAed FP is still under way... O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 22 / 23
Future work 2D bounce-averaged Fokker Planck Future work Systematic verification of 1D FP is under way... Comparison with Ray-tracing with 1D FP and GENRAY with CQL3D Comparison with TORIC-LH with CQL3D As an intermediate step we could assume that the deposition profile of 1D FP and 2D BAed FP are the same. Hence, proceed as follows: Compute E field using iterative routine coupled to 1D FP Assume 2D effects do not significantly affect power deposition profile Take E from 1D FP iteration and evaluate current profile using 2D BAed FP First comparison with experimental current profile In the long term: Complete integration with 2D FP Development of hard-xray synthetic diagnostic for direct comparison with experiment O. Meneghini (51 st DPP APS, Atlanta) Integration 2D BAed FP in LH full wave FEM November 4, 29 23 / 23