PLASMA-NEUTRAL MODELING IN NIMROD Uri Shumlak*, Sina Taheri*, Jacob King** *University of Washington **Tech-X Corporation April 2016
Plasma-Neutral Model Physical Model is derived by E. Meier and U. Shumlak* The Model is a generalization of Braginskii Neutral (Ion + Electron) conversion is allowed. Ionization, recombination, and charge exchange collisions are included: e + n i + + 2e φ ion e + i + n + hν i + + n n + i + Further Assumptions: Only single ionization No bound excited states Effective ionization energy included Optically thin plasma / neutral fluid * E.T. Meier and U. Shumlak, Phys. Plasmas (2012)
Plasma-Neutral Governing Eqs. Continuity n t + (nv) = Γ ion rec i Γ n n n t + (n v ) = Γ rec ion n n n Γ i Momentum t (m nv)+ (m nvv + pι+ Π) = j B + R in en i i i + R e + Γ ion i m i v n Γ rec n m i v + Γ cx m i (v n v)+ R cx cx in R ni t (m in n v n )+ (m i n n v n v n + p n Ι+ Π n ) = R in en i R e Γ ion i m i v n + Γ rec n m i v Γ cx m i (v n v) R cx cx in + R ni Ionization/Recombination Elastic Neutral Collisions Charge Exchange Collisions C i i - Density production rate R c n - Frictional force - Collisional heat exchange Q
Plasma-Neutral Governing Eqs. cont. Energy ε t + εv + v pι+ Π Faraday s Law + h = j E + v R in i + v e R en e +Q in i +Q en ion 1 e + Γ i 2 m iv 2 n φ ion +Q ion rec 1 n Γ n 2 m iv 2 Q i + Γ cx 1 2 m i ( v 2 n v 2 ) + v n R cx in v R cx ni +Q cx cx in Q ni ε n t + ε nv n + v n ( p n Ι+ Π n ) + h n = +Q n in +Q n en Γ i ion 1 v n R i in + R e en Γ cx 1 2 m i v 2 n v 2 v n R in B t = v B 1 qn j B Ρ e R ie en i + R e 2 m v Q ion rec 1 i n n + Γ n 2 m i v2 +Q i cx + v R cx ni Q cx cx in +Q ni rec +Q e rec rec Q e rec
Ionization/Recombination/Charge Exchange Electron impact ionization, radiative recombination and charge exchange create source terms on the governing equations Production is quadratic with density Reaction rates are functions of T e Charge exchange depends on relative velocity Γ ion i = σ v ion nn n Γ rec n = σ v rec n 2 Γ cx = σ cx nn n V cx σ v ion = A 10 1+ P ( φ / T 6 ion e) 1/2 X +φ ion / T e σ v rec = 2.6 10 19 Z 2 T e φ ion T e K e φ ion /T e V cx = 4 π v 2 Ti+ 4 π v 2 2 Tn+ v in σ cx,h =1.12 10 18 7.15 10 20 ln V cx σ cx,d =1.09 10 18 7.15 10 20 ln V cx
Different Models of Cross-Section Constant Approximate the cross-section with a constant value Roughly good approximation for high temp. and low ion velocity Symmetric (n=0) Capturing the important dependence on temp and ion velocity Moderate computational cost Full Nonlinear Capture the nonlinear dependence High computational cost
Implementation in NIMROD Continuity Discretize equations using implicit leap-frog time advance Solve for Δ values Δn Δt + 1 j+1 ( Δn v ) 1 2 2 σ v n ion ( Δn + nδn n n) + 1 2 σ v rec Δn n Δt + 1 2 Δn v n n Use NIMROD solver for coupled equations ( 2nΔn) = ( nv) + σ v ion nn n σ v rec n 2 + 1 4 σ v ΔnΔn 1 ion n 4 σ v Δn rec 2 j+1 + 1 2 σ v n Δn + nδn ion n n 1 2 σ v rec ( 2nΔn) = ( n n v n ) σ v ion nn n + σ v rec n 2 1 4 σ v ΔnΔn + 1 ion n 4 σ v Δn rec 2 Currently nonlinear quadratic-in-δn term is neglected A 11 A 12 A 21 A 22 Δn Δn n = b + f Δn, Δn n
Implementation in NIMROD Momentum NIMROD is coded to advance plasma velocity as mn v t + v v = j B p Π Multiplying the continuity equation by m i v and subtracting from momentum mn v t + v v = j B p Π+ R in en i + R e v mn n n t Γ i ion m i ( v v n ) Γ cx m i ( v v n ) + R cx cx in R ni + v n v n = p Π R in en n n i R e + Γ n rec m i v v n + Γ cx m i ( v v n ) R in cx + R ni cx
Implementation in NIMROD Momentum cont. Discretize equations using implicit leap-frog time advance Solve for Δ values RHS is in time step j LHS is in Δ terms v j+1 v j Nonlinear terms, f Δv Δv, to be implemented with outer iteration Plasma and Neutral momentum equations are coupled Solving the coupled system simultaneously requires an expensive 6-vector solve Coupling terms are explicit Plan to try outer iteration to time-center coupling terms (as with implicit advection) Neutral momentum equation uses SI operator terms associated with the sound wave RHS is implemented LHS is being implemented Implementing the nonlinear terms are the next step
Output files in NIMROD Neutral output neutral.bin (drawneut.in): kinetic/internal energy and Fourier data at the probe location (as in history.bin) Neutral fields are available in hdf5 and binary dump files when neutral_mode input is /= none
What to do next? Add source terms due to ionization/recombination/charge exchange to plasma momentum equation Verify the validity of the code with benchmark simulations Density equilibrium (Saha problem) Gaussian flow relaxation Energy Conservation Add energy equation for neutral flow Add source terms due to ionization/recombination/charge exchange to plasma energy equation Verify the validity of the code as a whole plasma-neutral model