Comparison of Kinetic and Extended MHD Models for the Ion Temperature Gradient Instability in Slab Geometry D. D. Schnack University of Wisconsin Madison Jianhua Cheng, S. E. Parker University of Colorado Boulder D. C. Barnes TriAlpha Energy, Inc.
Un-indicted Co-Conspirators Ping Zhu Chris Hegna Eric Held Jake King Scott Kruger Carl Sovinec
Goals Verify the NIMROD code for the ITG instability Are the extended MHD equations being solved correctly? Validate the extended MHD model for the ITG When can extended MHD be used as a physical model for the ITG? Quantify the differences between extended MHD and fully kinetic model
Ion Temperature Gradient Instability Parallel sound wave destabilized by interaction with a perpendicular drift wave in the presence of an ion temperature gradient L is gradient scale length Perturbed perp. drift motions convect heat via V x dt i0 /dx Can amplify temperature perturbation in sound wave if phase and frequency are right Requires FLR/two-fluid effects for instability Stable in ideal and resistive MHD Threshold in ρ i /L or k y ρ i for instability Differs from g-mode, which is MHD unstable and is stabilized by FLR effects Good test for extended MHD model How far can the model be pushed into the kinetic regime?
Approach Solve local kinetic and fluid dispersion relations for complex eigenvalue Solve extended MHD model with NIMROD code for complex eigenvalue and global eigenfunction Solve Vlasov + field equations with hybrid kinetic δf code (Cheng, et. al.) for complex eigenvalue and global eigenfunction Compare all results for a range of k y ρ i and ρ i /L
Equilibrium Slab (x, y, z) geometry Quasi-neutral n i0 = n e0 = n 0 T i0 (x), B(x) = B 0 (x) e z, n 0 = const, T e0 = const. z is parallel, y is perpendicular, no shear E 0 + V s0 B 0 + 1 P s = 0 n 0 q s Specify P, determine B from MHD force balance Species force balance: E 0x determines frame of reference E 0x = 0 for all calculations here Ion drift velocity explicitly included in equilibrium
Local Kinetic Dispersion Relation No external forces or field line curvature; electrostatic Perturbations: Ignore x-dependence: local approximation Low frequency: ω << Ω e,i 1+ s 1 kλ Ds ξ s = ( k y ρ ) 2 i Fluid limit: ω se ω ds 1+ ω ( ) 2 ω f = fe i ( k y y+k z z ωt ) ω ds = T sk y q s B W dt s dx ω k z V ths 1 T s I 0 ( ω 2 2 ω s )ω + ω 2 se ω * Ti = 0 ( ξ s )e ξ s = 0 W (ζ ) = ( ζ / 2 )Z ( ζ / 2 ) +1 2 = k 2 T / M ω * = k y ω 2 z e s = ω 2 se + k 2 z (5 / 3)T i / M Ti eb dt i dx
Local Extended MHD Dispersion Relation XMHD mathematically equivalent to two-fluid model has same dispersion relation FLR effects captured through Braginskii closures (k y ρ i << 1): W i, j = V j x i Π i gv = η 3 2 ( ) + transpose ˆb W I + 3ˆbˆb Assume complete GV cancellations+electrostatic, k z /k y << 1: ( ω 2 2 ω s )ω + ω 2 se ω * Ti = 0 Same as fluid limit of kinetic equation!, Similar equation if GV cancellations are incomplete η 3 = P i 2B + V j x i 2 3 δ i, j V q igv = κ igv ˆb Ti, κ igv = 5 2 P i B
Cubic Dispersion Relation ω 3 ω 2 s ω + ω 2 * seω = 0 Ti Cubic Linear High frequency, or dt i /dx small: Cubic ~ Linear => Parallel sound waves: Low frequency, small dt i /dx : Linear ~ Constant => Drift wave: High frequency, Large dt i /dx : Constant Cubic ~ Constant => Instability: ω 2 = ω s 2 ω = ω 2 * seω Ti 2 ω s ~ dt i dx ( ) 1/3 e 2πil/3, l = 0,1,2 γ ~ L 1/3 ω = ω 2 * se ω Ti Interaction between sound and drift waves lead to instability Electromagnetic dispersion relation is quintic 2 new shear Alfven waves, same low frequency behavior
Behavior of Low Frequency Roots in Fluid Limit
Fluid Solution Depends on Single Non-dimensional Parameter g = 27 4 β e 2 β i β e + 5 3 β i 3 k k 2 ρ i L 2, g > 1 for instability Growth Rate: γ ω s = 1 2 Real Frequency: ω r = 1 ω s 2 3 ( g + g 1) 1/3 ( g + g 1) 1/3 + 1 ( g + g 1) 1/3 1 ( g + g 1) 1/3
Wave-Particle Interaction Effects Kinetic model includes wave-particle interaction effects (e.g., Landau damping) Not captured by extended MHD model Effects minimized when ω r /(k z V thi ) >> 1 (few particles resonant with wave) Also need k y ρ i << 1 For k ρ i ~ 0.2, resonant fraction ~ e ( ω r /(k V thi ) ) 2 ~ e (1.7)2 = 0.055
Equilibrium for Global T i (x) = T i0 1+ 0.9 tanh x L Walls are far away Calculations Used for both XMHD and kinetic calculations η i peaks at x < 0
Local Fluid Growth Rate vs. x Maximum local growth rate biased toward x < 0
Comparison of NIMROD and Local Fluid Growth Rates
Growth Rate is a Function of T e /T i When T e = 0 the drift wave does not propagate
Comparison of Local Kinetic and Fluid Growth Rates with NIMROD ρ i / L = 4 10 4 1/ L = 3 / m k = 0.1 m Results Ω i = 1.9 10 8 / sec β 0 = β e + β i = 0.05 T e / T i = 4 NIMROD and local fluid in fair agreement for k y ρ i < 0.2 NIMROD, local fluid, and local kinetic agree on marginal point Local kinetic stabilizes at k y ρ i ~ 1 Global hybrid kinetic calculation impractical for this value of ρ i /L
Comparison of Local and Global Kinetic and Fluid Results Larger values of ρ i /L allow global kinetic calculation k y ρ i = 0.2 for all results
Comparison of Kinetic and Fluid Eigenfunctions x = 0 Max. η i Max. local fluid growth rate Biased toward x < 0 Shape consistent with ion diamagnetic drift
Verification of NIMROD When ρ i /L << 1, NIMROD growth rate in good agreement with local fluid theory as a function of 1/L for fixed k y ρ i = 0.14 Difference at marginal point For fixed ρ i /L = 4 10-4, NIMROD growth rate in good agreement with local fluid theory as a function of k y ρ i Agreement on marginal point, k y ρ i = 0.025 Excellent agreement for k y ρ i < 0.1 Good agreement for k y ρ i < 0.2 Divergence due to spatial dependence of equilibrium Accurate and correct solutions of extended MHD equations for this parameter range NIMROD is verified for the ITG Hybrid Kinetic Model also Verified
Validation of Extended MHD Model in NIMROD Direct comparison with more physically accurate kinetic models (both local and global) For ρ i /L < 10-3, extended MHD has same marginal point in k y ρ i as local kinetic solution Good agreement for k y ρ i < 0.05 Begin significant divergence for k y ρ i > 0.2 Wave particle interactions For k y ρ i = 0.2, agreement on marginal point in ρ i /L (= 0.013), but significant disagreement for larger ρ i /L Wave particle interactions Global extended MHD and hybrid kinetic eigenfunctions have similar character for L/ρ i = 30 and 20 Extended MHD is reliable physical model for ρ i /L < 10-3 and k y ρ i < 0.2, and is validated in this parameter range
Implications for Nonlinear Extended MHD Computations Integrated model of MHD-scale dynamics in presence of ITG turbulence? ITG growth rate increases as (k y ρ i ) 1/3 g-mode (interchange) stabilized by large k y ρ i Increasing resolution for nonlinear computations introduces modes with larger growth rates Impossible to converge nonlinear spectrum? Kinetic model stabilizes for k y ρ i ~ 1 Suggests adding hyper-dissipation ~ (k y ρ i ) 4 Control unphysical large k y ρ i modes with little effect for k y ρ i < 0.2
For NIMRODs only..
Full Disclosure Late time numerical instability Growth rate and real frequency are functions of x Growth rate and real frequency are different for different components of eigenvector Global kinetic and XMHD eigenfunctions differ significantly at L/ρ i = 10 Is it real, or is it..????
Numerical Instability ITG 2Δy mode
Growth Rate and Real Frequency are Functions of x Growth Rate Real Frequency
Different Components of Eigenvector Have Different Eigenvalues B x, B z, V x, V z : B y, V y : γ = 3.125 10 4 ± 0.5 / sec γ = 3.195 10 4 ± 0.9 / sec Difference is 7 10 2 / sec, larger than statistical uncertainty in calculation of growth rate?????????????
Global Kinetic and XMHD Eigenfunctions Differ at L/ρ i = 10
Future Directions Can we improve the closures in extended MHD? Particle ions as part of bulk species? Eric s kinetic closures (on grid in phase space)? Can we go further into kinetic regime? Nonlinear ITG Slab geometry 2 Δy instability? Hyper-dissipation Thermal conductivity? ITG turbulence? Effective transport? Annulus calculations? Global toroidal simulations Sawtooth + core ITG turbulence? Can all this be captured in a single ` integrated fluid calculation? Return to Giant Sawtooth??
The Latest..
Nonlinear ITG n = 0 2 modes k_perp rho_i = 0.1 Run eventually dies (???) n = 1
Effect on Profiles Flattens Ion Temperature Suppresses Equilibrium Flow