Full-Wave Maxwell Simulations for ECRH H. Hojo Plasma Research Center, University of Tsukuba in collaboration with A. Fukuchi, N. Uchida, A. Shimamura, T. Saito and Y. Tatematsu JIFT Workshop in Kyoto, Dec. 15-17, 003
Old version of Maxwell Equation Simulator ( Maxim ) This code is used mainly for plasma diagnostic problems. t t 1 ε B = c 1 E = B [ J + σ ( r) E] ε() r ε ε() r E pe 0 0 t ε0me 0 e J = ω [ n( r)] E J B ( r) Artificial terms introduced to set wall boundary Input: n () r = Plasma density ( arbitrary ) B () r = External magnetic field ( arbitrary ) 0 0 1 σ() r = ε() r = σ* ( > 0) ε* ( > 1) plasma otherwise Incident wave : arbitrary ( cw, pulse, etc )
A set of equations( m i = ) for simulation can describe R- and L-modes in parallel propagation, and O and X-modes in perpendicular propagation in high frequency range ( GHz or more ).
Two -d simulation models on EM wave propagation Sectional View Top View
Radial Profiles of Characteristic Frequencies in Tokamak Model f 0 =3 α=3
EM Wave Propagation (Vertical Injection) at ω/ω 0 = and (ω pe0 /ω 0 ) =3 Ex Ey Ez Bx By Bz
EM Wave Propagation (Tangential Injection) at ω/ω 0 = and (ω pe0 /ω 0 ) =3 Ex Ey Ez Bx By Bz
Radial Profiles of Characteristic Frequencies in Tokamak Model f 0 =6 α=3
EM Wave Propagation (Vertical Injection) at ω/ω 0 = and (ω pe0 /ω 0 ) =6 Ey // Bt
EM Wave Propagation (Tangential Injection) at ω/ω 0 = and (ω pe0 /ω 0 ) =6
Toroidal propagation of electromagnetic wave ( Movie )
Ultrashort-pulse propagation in LHD plasma EM Pulse of FWHM = 30ps 0 0 0 1-3 nxy (, ) = n[1 ψ( XY, )/ ψ ], n = 5 10 cm B( XY,, φ = 0) with p=5 [T. Watanabe and H. Akao: J. Plasma Fusion Res. 73 (1997) 186.] Red ellipse denotes plasma-vacuum boundary.
Ultrashort-Pulse Reflectometry for LHD Plasma ( FDTD Simulation ) Reflected waves from plasma ( incident pulse width = 30ps ) Signal at detector point Time delay for reflected waves Reconstructed density profile ( ) Abel inversion
Objective Development of a new type of computational tool for wave heating in place of ray-tracing method Ray-tracing method based on geometrical optics is simple, but cannot treat 1. wave diffraction,. wave tunneling, 3. mode conversion. Maxwell equation simulator based on electromagnetic optics can clear the above difficulty in ray-tracing method.
Ray Tracing Method Dispersion Relation : D( ω, kr, ) = 0 d D/ k d D/ r r =, k = d t D/ ω d t D/ ω W 1 Re { E * σ E} = E = E 0 exp( Im( k)d s)
Maxwell Equation Simulator Good Points: resonance cutoff 1. Full wave analysis. Wave diffraction 3. Wave tunneling 4. Mode conversion incident evanescent transmitted resonance cutoff reflected Bad Points: transmitted evanescent incident 1. Applicable to fundamental heating ( Harmonic resonances can be included in the simulator equations, however, which cannot be solved in time domain as initial value problems.) To trace waves, it is of importance that wave equations are solved in time domain as initial value problems.. Approximated kinetic effect ( Ray tracing results could be covered. )
Maxwell Wave Equation: σ k ω ω ω σ E 1+ i E = 0 c εω 0 J = σ E Conductivity tensor is generally obtained from Vlasov equation. In the limit of 0 ( ce ) : ωpe ω pe ω ω ce S = 1 + i π exp ω ω ce ω k v te k vte S 1 id 0 σ i = id S 1 0 εω 0 0 0 P 1 D ω ω ce pe ω pe ω ω ce = + i π exp ω ω ωce ω k v te k vte P ωpe ωω pe ω = 1 + i π exp 3 ω ( k vte ) k vte Approximation : k in S, Dand Pis determined by local dispersion equation. Dk (, r, ω ) = 0
Equations for Simulations: Previous conductivity tensor is covered by the following equations. ( These time-evolution equations can be solved as initial value problems. ) B = E t 1 1 E = c B J σ E t ε ε 0 0 1 e ν J = E J B J ε ε ε ωpe 0 0 t 0me 0 Cyclotron and Landau damping Collisional damping σ xx σxy σxz σ1bz + σ3bx i σ1bz ( σ1 σ3) bxb z σ = σyx σyy σyz = iσ1bz σ1 iσ1bx σzx σzy σ zz ( σ1 σ3) bb x z iσ1bx σ1bx + σ3bz σ ω 1 pe ω ω ce = π exp εω 0 ω k v te k vte σ ωω 3 pe ω = π exp 3 εω 0 ( k vte ) k vte B 0 b = = bx xˆ + bz zˆ B 0
Absorbed power deposition in ECRH : W 1 = Re { E* σ E} 1 1 = σ1( bz Ex + Ey + bx Ez ) + σ3( bx Ex + bz Ez ) 1 i + ( σ σ ) ( + ) + σ ( ) ( ) 5 1 bb x z ExEz EE x z 1 be z x be x z Ey be z x be x z Ey When b z = 1, b = 0 x 1 1 W = σ3 Ez + σ1 Ex iey 1 1 = σ + + + ( ) Re( ) Im( ) Im( ) Re( ) 3 Ez σ1 Ex Ey Ex Ey Ex Ey
Cross Polarization Scattering ( O and X-modes) Time domain version of Fidone-Granata equation: mode conversion by magnetic shear ( + ω pe + ωce) Ex + ωpeωcee = 0 t t E ( c + ω pe + c θ ) E + ωce Ex = c [ θ + θ E ] t x x X-mode E ( c + ω pe + c θ ) E = c [ θ + θ E ] t x x O-mode B( x) = B ( x) zˆ + B ( x) yˆ z θ ( x) = arctan( B / B ) y y z θ = d θ /d x, θ = d θ /dx 1 Incident O-mode to X-mode at 48GHz in LHD Plasma t=000 O-mode X-mode n(x) 0-1 B p (x) - 0 500 1000 1500 x[mm]
-d simulation for fundamental ECRH ( Right-handed circularly polarized wave with ω = 8GHz ) E y E z W x cyclotron resonance
Tunneling of right-handed circularly polarized mode ( -d simulation ) 1.5 1 L-mode is not affected by resonance and cutoff layer. R-wave L-wave 0.5 0-0.5-1 -1.5 0 50 100 150 00 50 300 z Resonance and cutoff layer Launched beam is not a plane wave, and then wave diffraction occurs.
Tunneling of Right-handed circularly polarized mode 1 0.5 ( 1-d simulation ) E y 0-0.5 cyclotron resonance and cutoff layer -1 1 0.5 E y 0-0.5-1 1 0.5 E y 0-0.5-1 0 00 400 x 600 800 1000
Transmittance of right-handed circularly polarized mode T = ( E B) ( E B) x x=+ x x= resonance cutoff incident evanescent transmitted T = exp( π kl) in Budden s problem 1-d simulation -d simulation k=k(x=- )
Summary We developed a Maxwell equation simulator, which approximately takes into account wave-particle interactions such as cyclotron resonance. The code can treat wave diffraction, mode tunneling and also mode conversion of electromagnetic waves, in addition to estimate power deposition profile in ECRH.