0-0 Applying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models B. Srinivasan, U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle, WA 98195-2600. IEEE Pulsed Power and Plasma Sciences Conference June 2007 Albuquerque, NM
Abstract Simulations are performed using the full two-fluid plasma system. The twofluid plasma model is investigated for its capabilities of capturing physics that is lost with simpler fluid models such as Magnetohydrodynamics (MHD). The regime in between the full two-fluid equation system and MHD is explored by the application of asymptotic approximations to the two-fluid system. These asymptotic approximations involve ignoring electron inertia, setting the speed of light to infinity and ensuring charge neutrality. Applying all three approximations together gives Hall-MHD. The results obtained by applying the asymptotic approximations are compared to the two-fluid plasma model to determine what physics is lost by the application of each approximation. Simulations of electromagnetic plasma shock and collisionless reconnection will be presented to demonstrate the various physical effects in comparison to the results of the two-fluid model.
Motivation To study the two-fluid plasma model To study the physical effects when each of the asymptotic approximations are applied independently to the two-fluid plasma model To implement Hall-MHD and compare it to the two-fluid model The equation sets used are balance laws of the form, Q t + F(Q) = S(Q)
Two-Fluid Plasma Model Euler equations for ion and electron fluids in 1D for simplicity n i n i u i 0 n i u i n i u 2 i + p n i q i i m i (E x + v i B z w i B y ) n i v i n i u i v i n i q i m i (E y + w i B x u i B z ) n i w i n i u i w i n i q i m i (E z + u i B y v i B x ) e i t + (e i + p i )u i n n e x = i q i (E x u i + E y v i + E z w i ) n e u e 0 Mn e u e Mn e u 2 e + p e n eq e m i (E x + v e B z w e B y ) Mn e v e Mn e u e v e n eq e m i (E y + w e B x u e B z ) Mn e w e Mn e u e w e n eq e m i (E z + u e B y v e B x ) (e e + p e )u e n e q e (E x u e + E y v e + E z w e ) e e
The electromagnetic fields are described by Maxwell equations: B t + E = 0, E t c2 B = 1 ɛ 0 E = 1 ɛ 0 (q i n i + q e n e ) B = 0. s q s m s ρ s v s Writing Maxwell s equations in balance law form in 1D: E x 0 1 ε 0 (n i q i u i n e q e u e ) E y c 2 B z 1 ε 0 (n i q i v i n e q e v e ) E z + c 2 B y 1 ε = 0 (n i q i w i n e q e w e ). t B x x 0 0 B y E z 0 0 B z E y
The ion and electron energies e i and e e are M: electron-to-ion mass ratio n: number density u, v, w: velocity in x-,y-,z-directions p: pressure E: electric field B: magnetic field q i, q e : ion and electron charges c: speed of light e i = p i γ 1 + 1 2 n i(u 2 i + vi 2 + wi 2 ) e e = p e γ 1 + 1 2 Mn e(u 2 e + ve 2 + we) 2
Asymptotic Approximations to be applied to the Two-Fluid Model Each of the following approximations are applied independently. Charge neutrality, n i = n e No displacement currents, ε 0 0, c Ignore electron inertia, M 0 Applying all 3 of the above together gives Hall-MHD
Approximation 1: Apply Quasineutrality only, n i = n e The Two-Fluid equations remain unchanged except that there is only one continuity equation for both fluids. The dispersion relation for parallel propagation is ωpi 2 + ωpe 2 + ω 2 ω2 c 2 k 2 ω 2 c 2 = 0. (1) sik2 The dispersion relation for perpendicular propagation is where c si = T i mi. ωpi 2 + ωpe 2 + ω 2 ω2 c 2 k 2 ω 2 c 2 + ω 2 c 2 k 2 si k2 ω2 ci ω 2 c 2 = 0 (2) sik2
w 100 80 60 40 20 20 40 60 80 100 k w 2 C 2 k 2 2 1.75 1.5 1.25 1 0.75 0.5 0.25 100 200 300 400 500 w Left: Dispersion relation for the quasineutrality condition, ω Vs k. Right: Plot of the phase velocity squared, v 2 φ = ω2 k 2 Black: Parallel propagation, v φ = c si c for ω = 0 Vs ω Red: Perpendicular propagation similar to parallel with an additional dependence on ω ci. v 2 φ = c2 ω 2 ci +c2 si ω2 p c 2 (ω 2 ci +ωp2 ) for ω = 0, where ω2 p = ω 2 pi + ω2 pe
Approximation 2: Apply ε 0 0 only, implies c and no displacement currents The Electric field time derivative drops out of Ampere s law to give: B = µ 0 (n i qu i n e qu e ) (3) Quasineutrality is automatically implied through the math The dispersion relation for parallel propagation is which is just the ion acoustic wave. ω 2 = c 2 sk 2 (4) The dispersion relation for perpendicular propagation is where c 2 s = T i+t e m i +m e. ω 2 = c 2 sk 2 + 2ω ci ω ce (5)
3.5 3 2.5 2 1.5 1 0.5 w 4 2 4 6 8 10 k 1.5 0.5-0.5-1 -1.5 w 2 2 1-2 k 2 0.5 1 1.5 2 w Left: Dispersion relation for the c condition, ω Vs k. Right: Plot of the phase velocity squared, v 2 φ = ω2 k 2 Black: Parallel propagation, ion acoustic wave Vs ω Red: Perpendicular propagation dispersion relation similar to parallel with an additional dependence on ω ci and a cutoff at ω = 2ω ce ω ci.
Approximation 3: Ignore electron inertia only, M 0 The time derivative and convective derivative terms drop out of the electron momentum equation to give Ohm s law: P e n e q = (E + u e B) (6) The massless electrons respond instantaneously to the ions in the direction of the magnetic field, so E =0.
The dispersion relation for parallel propagation for M 0 is ω 2 = c 2 sik 2. (7) The dispersion relation for perpendicular propagation is 1 = k2 c 2 ω 2 + ω2 pi ω 2 + ω2 pi ωci 2 k 2 T e 1 m i ω 2 ω4 pi ω 2 ci ω2 + 2k2 Te m i + ωci 2 + ω2 pi ω 2 k 2 c 2 si + ω2 ci + ω2 pi ( ω 4 pi ωci 2 ω2 pi ω2 ω 2 (8) ) where T e is electron temperature.
w 30 25 20 15 10 5 w 2 C 2 k 2 1 0.8 0.6 0.4 0.2 5 10 15 20 25 30 k 5 10 15 20 25 30 w Left: Dispersion relation for the me mi 0 condition, ω Vs k Right: Plot of the phase velocity squared, v 2 φ = ω2 k 2 Black: Parallel propagation, ion sound wave Vs ω Red: Perpendicular propagation, asymptotes at the ion sound wave Blue: Perpendicular propagation, asymptotes at the speed of light
Applying all 3 asymptotic approximations gives Hall-MHD The dispersion relation for Hall-MHD is ( (ω 2 vak 2 [ω 2 ) 4 c 2 si + T ) e + va 2 k 2 ω 2 + m i = v2 A k2 k2 ω 2 n 0 eµ 0 ( c 2 si + T e m i ) v 2Ak 2 k2 ] ( ω 2 c 2 sik 2 T ) e k 2 m i (9) (10) where v A = B 0 n0 m i µ 0 is the Alfven speed.
Waves propagating parallel to the magnetic field are ω = ± 1 2 ω ciδ 2 i k 2 + where δ i is the ion skin depth. w 10 v 2 A k2 + 1 4 ω2 ci δ4 i k4, ω = c si k (11) 8 6 4 2 1 2 3 4 5 k The Whistler wave (Green) grows quadratically without bound when δ i and ω ci are significant and Hall effects become important. If λ δ i >> 1 then the dispersion diagram sits near the origin and resembles Ideal MHD.
For perpendicular propagation, the dispersion relation gives ω = k va 2 + c2 s (12) where, c 2 s = T e+t i m i. This is the magnetosonic wave. The Whistler wave, the Alfven wave and the Magnetosonic wave need to be resolved in this system to capture the Hall effects.
Hall-MHD Equation system for cold electrons Written as Ideal MHD + source terms: ρ i t + (ρ iu i ) = 0 ρ i u i + (ρ i u i u i + P i I BB + B2 I) = 0 t µ 0 2µ 0 ε i t + ) [(ε i + P i + B2 u i (B u ] i) B = 1 2µ 0 µ 0 [ ] ( B) B B t + (u ib Bu i ) = n i eµ 0 eµ 2 0 ( ) B 2 n i ( B) Induction equation source term is tough to handle with shock problems such as the BrioWu shock. Need to either add artificial viscosity, or could use the electron energy equation with the P e source term in the induction equation.
BrioWu shock problem using Ideal MHD and the Two-Fluid Plasma Model 1 n Vs x 1.1 1 n Vs x Two Fluid model MHD model 0.9 0.8 0.8 number density 0.6 0.4 number density 0.7 0.6 0.5 0.4 0.2 0.3 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Left: An MHD shock is initialized with a jump in number density, pressure and magnetic field Right: The Two-Fluid solution compared to the ideal MHD solution. Want to see how Hall-MHD and the other reduced fluid models compare to these results.
GEM challenge problem - Magnetic Reconnection Cannot simulate this in Ideal MHD because the fluid is frozen to the magnetic field and there are no non-ideal terms that allow the field lines to connect to each other The Hall term implies that the electrons are frozen to the magnetic field instead of the ions, so to get reconnection we need non-ideal terms like artificial viscosity, resistivity or an electron pressure gradient with the Cold Electron Hall-MHD model discussed. Two-Fluid GEM challenge is simulated, and the reduced fluid models will be compared to these results in the future
Two-Fluid Magnetic Reconnection initial conditions ( 1 ( n e = n i = n 0 5 + y ) ) sech2 λ (13) P e = 1 B 12µ 0n 2 e 0 (14) P i = 5 B 12µ 0n 2 i (15) 0 ( y B x = B 0 tanh + λ) ( ) ( ) 0 π 2πx πy cos sin (16) 10 L y B y = B 0 10 2π L x sin J ez = µ 0B 0 λ where λ = 0.5d i, d i = 128 64 ( ) ( ) 2πx πy cos L x sech2 ( y λ ) c ω pi, L x = 25.6, L y L y L x L y (17) (18) = 12.8, with a grid resolution of
y y y 0 0 0 x x x Two-Fluid Magnetic Reconnection: 6 0 5 0 4 0 3 0 2 0 1 0 6 0 5 0 4 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 3 0 2 0 1 0 6 0 5 0 4 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 3 0 2 0 1 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 Left: Electron density is shown over characteristic transit times, t=0, 25, 50 Right: The blue dots represent the two-fluid reconnected flux, plot from [2]
Numerical Algorithms Used System to be solved: Q t + F x = S, where Q = Conserved variables, F = Flux, S = Source terms Finite Volume Method: High Resolution Wave Propagation method Finite Element Method: Runge-Kutta Discontinuous Galerkin method Both methods involve solving the Riemann problem across each cell interface
Finite Volume Method: High Resolution Wave Propagation Method First solve the hyperbolic, homogeneous equation using the 2 nd order update formula [3] Q n+1 i = Q n i t x t x Q t + F x = 0 [ A + Q i 1/2 + A Q i+1/2 ] ([ F] i+1/2 [ F] i 1/2 ) where the eigenvalues, eigenvectors and flux differences give: A ± Q i±1/2 : right- and left-going fluctuations at each cell interface, F] i±1/2 : correction fluxes at each cell interface to get to 2 nd order accuracy Limiters are applied to reduce the oscillations that form due to this higher order (Lax-Wendroff type scheme).
Source Term Handling Strang splitting is used. First solve ODE dq/dt = S over t/2 Solve homogeneous equation over t Again solve ODE over t/2 To advance the ODE dq/dt = S, a higher order Runge-Kutta solver is used with Strang splitting.
Finite Element Method: Discontinuous Galerkin Method Basic Scheme Apply polynomial basis functions to conserved quantities within each cell: Q(x) = Q r v r (x) r=0 where v r (x): basis functions locally defined within each cell Q r : expansion coefficients r: spatial order Balance law: t I i v r (x)qdx + I i v r (x) F x dx = I i v r (x)s.
Legendre polynomials chosen as basis functions (orthogonality property). The scheme becomes [4]: Q r t = F i+1/2 ( 1) r F i 1/2 x + 1 x 1 1 dp r (η) dη Fdη + 1 2 1 1 P r Sdη where F(η) F(Q(x(η), t)) and S(η) S(Q(x(η), t)). Interface fluxes computed using Riemann problem solver like with Wave Propagation method Integrals computed using Gaussian quadrature ODE dq r dt = RHS solved using Runge-Kutta time integration (RKDG scheme). Temporal order spatial order.
Limiters for RKDG scheme Coefficient of linear term modified using a min-mod limiter. Limiter limits order of algorithm in regions where it is applied. 2 types of limiters used: Characteristics-based limiters: Conserved variables transformed to characteristic variables and then apply limiter. Better accuracy. Component-based limiters: Directly apply limiters to conserved variables without transforming them. Faster, no Jacobian required For stability, CFL = 1/(2r 1), r: spatial order
Conclusions and Further Study The MHD shock problem is presented using Ideal MHD and the Two-Fluid Plasma Model The Magnetic Reconnection problem is presented using the Two-Fluid Model The dispersion relations are studied for each of the approximations that are independently applied to reduce the Two-Fluid Model. When all 3 approximations are applied together Hall-MHD is obtained. Hall-MHD is studied and will be implemented with either artificial viscosity or an electron pressure gradient and the electron energy equation. Hall-MHD and the other reduced fluid models will be used to simulate the MHD shock and Magnetic Reconnection problems and the results will be compared to Ideal MHD and the Two-Fluid Model.
References [1] D. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith. A wave-propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM Journal of Scientific Computing, 24:955 978, 2002. [2] M.A. Shay et al. Alfvenic collisionless magnetic reconnection and the hall term. Journal of Geophysical Research, 106(A3):3759 3772, 2001. [3] A. Hakim, J. Loverich, and U. Shumlak. A high resolution wave propagation scheme for ideal Two-Fluid plasma equations. Journal of Computational Physics, 2006. [4] J. Loverich, A. Hakim, and U. Shumlak. A discontinous Galerkin method for ideal two-fluid plasma equations. Journal of Computational Physics, 2006. [5] J. Loverich and U. Shumlak. Approximate Riemann solver for the two-fluid plasma model. Journal of Computational Physics, 187:620 638, 2003. [6] S. Ohsaki and S. Mahajan. Hall current and alfven wave. Physics of Plasmas, 11(3):898 902, 2003. [7] P. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357 372, 1981.