Accurate representation of velocity space using truncated Hermite expansions. Joseph Parker Oxford Centre for Collaborative Applied Mathematics Mathematical Institute, University of Oxford Wolfgang Pauli Institute, Vienna 3 rd March 01 Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 1 / 34
Model for ITG instabilities (from Pueschel et al., 010) Test problem for GENE, simplified gyrokinetic equations: 1D slab, (z, v ) adiabatic electrons B = Bz homogeneous and static drift-kinetic ions electrostatic: E = Φ quasineutral uniform density and temperature gradients in background perturbation about Maxwellian F 0 (v ) = π 1/ exp( v ) nonlinearity dropped May also derive directly from Vlasov equation and quasineutrality. Electrostatic limit of Belli & Hammett (005) model Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 / 34
Model for ITG instabilities (from Pueschel et al., 010) Linearized kinetic equation F 1 t + α iv F 1 z + [ω n + ω Ti ( v 1 )] ik y ΦF 0 + τ e α i v Φ z F 0 = 0 Quasineutrality Φ = F 1 dv. Parameters ω n = 1, normalized density gradient ω Ti = 10, normalized ion temperature gradient k y = 0.3, perpendicular wavenumber τ e = 1, species temperature ratio (T i = T e ) α i = 0.34 nondimensional constant: velocity and length scales Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 3 / 34
Wave-like solutions Leads to eigenvalue problem, F 1 = f(v ) exp ( i ( k z ωt )) L 0 f = ωf ( where L 0 f = α i kv f + [ω n + ω Ti v 1 )] k y ΦF 0 + τ e α i kv ΦF 0 L real = eigenvalues in complex conjugate pairs = no damping So actually want to solve, lim L ɛf = ωf, for L ɛ = L 0 + iɛc ɛ 0 where C is collisions, e.g. Lénard Bernstein or BGK. Breaks symmetry, so decay rate may tend to nonzero limit as ɛ 0. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 4 / 34
0.4 Dispersion relation 0. growth rate, γ 0.0 0. 0.4 0.6 0.8 1.0 0 1 3 4 5 6 7 8 parallel wavenumber, k Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 5 / 34
Discretization Keep exp(ik z) behaviour, represent v on a uniform grid, Obtain linear ODE system f t = imf, Φ = N vf(v i ) i=1 M real N N matrix Matrix eigenvalue problem (quicker than running IVPs): iωf = f t = imf Growth rate is γ = max I(ω), for ω spectrum(m). Matrix M depends on: All parameters, particularly parallel wavenumber. Discretization and resolution of grid in v. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 6 / 34
0.3 0. Growth rate for simple discretized system N = 10 N = 100 dispersion relation growth rate, γ 0.1 0.0 0.1 0. 0 1 3 4 5 6 7 8 parallel wavenumber, k Capture growth, but convergence slow with N. No decay. ɛ 0 at fixed N resolution. Need to restore collisions. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 7 / 34
0.14 0.1 0.10 0.08 Distribution function in velocity space k =.75 k = 3.5 k = 3.75 f(v) 0.06 0.04 0.0 0.00 0.0 0.04 3 1 0 1 3 v Fine scales develop as I(ω) 0. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 8 / 34
Hermite expansion in velocity space Use expansion in asymmetric Hermite functions, f(v) = a m φ m (v), φ m (v) = H m(v) m m!, φ m(v) = F 0 (v)φ m (v) m=0 Bi-orthogonal polynomials with Maxwellian as weight function φ m φ n dv = δ mn Recurrence relation m + 1 vφ m = φ m+1 + m φ m 1 = particle streaming becomes mode coupling Represent velocity space scales, ( H m (v) cos v m mπ ), as m large m = fine scales Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 9 / 34
Relative Entropy and Free Energy Relative entropy is R[F F 0 ] = ( ) F F log F + F 0 dv F 0 F log F dv + U + function(density) Expansion F = F 0 + ɛf 1 gives (at leading order) R[F F 0 ] = ɛ F 1 F 0 dv = ɛ a m + O(ɛ 3 ) m=0 Define free energy of each Hermite mode, E m, E = E m, with E m = 1 a m m=0 cf. energy spectra in Fourier space for Navier Stokes turbulence. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 10 / 34
Moment system Replace F 0 φ 0 and v with Hermite functions [ F 1 t + iα ikvf 1 + ik y Φ ω n φ 0 + ω ] T i φ + i kτ eα i Φ φ 1 = 0 Put F 1 = a m φ m (implicit sum over repeated m) [ a m t φ m + iα i kva m φ m + ik y Φ ω n φ 0 + ω ] T i φ + i kτ eα i Φ φ 1 = 0 Use recurrence relation on particle streaming ( ) a m m + 1 m t φ m + iα i ka m φ m+1 + φ m 1 +... = 0 Gives infinite set of coupled algebraic equations for {a n }. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 11 / 34
System of equations for expansion coefficients ( ) m + 1 m ωa m = a m+1 + a m 1 [ ] + k y ω n a 0 δ m0 + ω T i a 0 τ e a δ m + 0 δ m1 α i k }{{}}{{} driving Boltzmann response We have also used φ 0 1, so that, Φ = F 1 φ 0 dv = m=0 a m φ m φ 0 dv = a 0 Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 1 / 34
Matrix equation ω nk y 1 α i k τ e + 1 1 k yω Ti αi k 1 3 3...... N N...... a = ωa Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 13 / 34
Simple Truncation ω nk y 1 α i k τ e + 1 1 k yω Ti αi k 1 3 3 a = ωa Assume a N+1 = 0 at point of truncation. Last row of matrix: N + 1 N a N+1 + a N 1 = ωa N Equivalent to discretization on Gauss Hermite points {v j } Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 14 / 34
10 0 Hermite spectra of eigenfunctions free energy, am 10 10 10 0 10 30 k =, γ > 0 k = 6, γ = 0 10 40 10 0 10 1 10 Hermite mode, m + 1 Growing mode: a m decay as m increases = a N+1 0 is okay (Not) decaying mode not resolved Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 15 / 34
Collisions F 1 t + Free energy should cascade to large m (fine scales in v). Free energy has nowhere to go when a N+1 = 0. Restore collisions: [ω n + ω Ti ( v 1 Desirable properties for C[F 1 ] )] ik y ΦF 0 + iα i kv F 1 + iτ e α i kv ΦF 0 = C[F 1 ] conserves mass, momentum and energy C[F 1 ] dv = vc[f 1 ] dv = v C[F 1 ] dv = 0 satisfies a linearized H theorem: dr dt = ɛ F 1 C[F 1 ] F 0 dv 0 represents small-angle collisions (contains v-derivatives) Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 16 / 34
Lénard Bernstein (1958) collisions Simple member of linearized Landau/Fokker Planck class: C [F 1 ] = ν [ vf 1 + 1 ] F 1 v v collision frequency ν Hermite functions are eigenfunctions: C [a m φ m ] = νma m φ m = easy to implement in Hermite space Conserves mass, but not momentum or energy Satisfies a linearized R theorem: dr dt = ɛ ν m m a m 0 Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 17 / 34
Lénard Bernstein in Hermite space ω nk y 1 α i k τ e + 1 k yω Ti iν 1 αi k 1 iν 3 3 3iν...... N N inν a = ωa matrix now complex roots not in complex-conjugate pairs = can find negative growth rates (we can manually conserve momentum and energy) Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 18 / 34
Lénard Bernstein in Hermite space ω nk y 1 α i k τ e + 1 iν 1 k yω Ti αi 1 iν k 3 3 3iν...... N N inν a = ωa matrix now complex roots not in complex-conjugate pairs = can find negative growth rates (we can manually conserve momentum and energy) Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 18 / 34
Growth rate, Lenard Berstein collisions, N = 100 0.4 0. growth rate, γ 0.0 0. 0.4 0.6 0.8 ν = 10 1 ν = 10 ν = 10 3 dispersion relation 1.0 0 1 3 4 5 6 7 8 parallel wavenumber, k If collision frequency ν large enough to get k > 4 correct, then shape for k < 4 distorted. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 19 / 34
Growth rate, Lenard Berstein collisions, ν = 10 0.4 0. growth rate, γ 0.0 0. 0.4 0.6 0.8 N = 100 N = 00 N = 300 dispersion relation 1.0 0 1 3 4 5 6 7 8 parallel wavenumber, k resolves dissipative scales expensive: 300 modes Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 0 / 34
10 0 10 1 Lenard Bernstein collisions, ν = 10 k = k = 6 free energy, am 10 10 3 10 4 10 5 10 6 10 7 10 8 10 0 10 1 10 Hermite mode, m + 1 Appreciable damping along whole spectrum. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 1 / 34
Hypercollisions Would prefer: low modes undamped high modes strongly damped... whatever the truncation point N Iterate Lénard Bernstein collisions: L[F 1 ] = ( vf 1 + 1 ) F 1, C[F 1 ] = ν( N) n L n [F 1 ] v v (like hyperdiffusion: ( ) n in physical space) Hermite functions are still eigenfunctions: ( m ) n C [a m φ m ] = ν am φ m N ν sets the decay rate of the highest mode linearized R-theorem: dr ( m ) n dt = ɛ ν am 0 N m=0 n = 1 corresponds to Lénard Bernstein collisions. two parameters: n 6, ν 10 (robust to variation) Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 / 34
0. Growth rate with hypercollisions, ν = 10, n = 6 growth rate, γ 0.0 0. 0.4 0.6 0.8 simple truncation, N = 100 hypercollisions, N = 10 exact dispersion relation 1.0 0 1 3 4 5 6 7 8 parallel wavenumber, k Captures decaying parts of spectrum. Excellent fit, fast convergence. Appropriate for nonlinear problems. Two parameters, n, ν, robust to variation. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 3 / 34
10 0 Hermite spectra with hypercollisions, k = 10 1 free energy, am / 10 10 3 10 4 10 5 10 6 10 7 N = 10 N = 100 10 8 10 0 10 1 10 Hermite mode, m + 1 Low moments largely undamped Damping at high m, for any N. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 4 / 34
10 0 Hermite spectra with hypercollisions, k = 6 free energy, am / 10 1 10 10 3 10 4 10 5 10 6 10 7 10 8 10 9 N = 10 N = 100 10 10 10 0 10 1 10 Hermite mode, m + 1 Low moments largely undamped Damping at high m, for any N. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 5 / 34
Energy equations and theoretical spectra Equation for coefficients, valid for m 3, ( ) m + 1 m ωa m = a m+1 + a m 1 + driving + Boltzmann response Treat as a finite difference approximation in continuous m. Energy equation for E m = a m / (Zocco & Schekochihin, 011) E m + ( ) ( m ) n mem = ν Em. t m N For a mode with growth rate γ, ( E m = C exp γ ( m m γ m γ ) 1/ ( ) ) m n+1/, m c with the cutoffs, m γ = 1 8γ, m(n+1/) c = [ N n ] (n + 1/) ν Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 6 / 34
Energy equations and theoretical spectra Equation for coefficients, valid for m 3, ( ) m + 1 m ωa m = a m+1 + a m 1 + hypercollisions Treat as a finite difference approximation in continuous m. Energy equation for E m = a m / (Zocco & Schekochihin, 011) E m + ( ) ( m ) n mem = ν Em. t m N For a mode with growth rate γ, ( E m = C exp γ ( m m γ m γ ) 1/ ( ) ) m n+1/, m c with the cutoffs, m γ = 1 8γ, m(n+1/) c = [ N n ] (n + 1/) ν Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 6 / 34
10 0 10 1 10 Growing mode, k = collision dominated free energy, Em 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 m 1/ behaviour theoretical spectrum numerical spectrum growth rate cutoff collisional cutoff growth rate dominated 10 0 10 1 10 E m = Hermite mode, m + 1 ( C exp γ ( ) m 1/ ( ) ) m n+1/ m γ m γ m c Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 7 / 34
10 1 Decaying mode, k = 6 free energy, Em 10 10 3 10 4 10 5 m 1/ growth rate dominated theoretical spectrum numerical spectrum growth rate cutoff collisional cutoff collision dominated 10 0 10 1 10 E m = Hermite mode, m + 1 ( C exp γ ( ) m 1/ ( ) ) m n+1/ m γ m γ m c Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 8 / 34
How strong should collisions be? We can find the collision strength required for a given resolution. Write hypercollisions as, C[F 1 ] = ν( L) n [F 1 ] (i.e. remove N n, damping strength expressed just by ν.) Need to resolve collisional cutoff, ( ) (n + 1/) 1/(n+1/) N > m c = ν as ν 0 Need infinite resolution to resolve collisionless case. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 9 / 34
free energy, am / Hermite spectra with different ν, k = 6, n =4, N =100 10 0 10 10 4 10 6 10 8 10 10 10 1 10 14 10 16 10 18 ν = 0.8 10 7 ν = 1.0 10 7 ν = 1. 10 7 10 0 0 0 40 60 80 100 Hermite mode, m + 1 Weaker collisions = finer scales in spectra Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 30 / 34
growth rate, γ Growth rate against collision strength, k = 6, n =4 0.0 0.1 0. 0.3 N = 0 N = 50 N = 100 correct γ 0.4 0.5 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 hypercollision strength, ν a range of ν give the correct growth rate range extends to smaller ν as N increases Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 31 / 34
Hypercollisions on other grids Easiest to implement in Hermite space: C[a m ] = ν(m/n) n a m C[a] = Da But may be used on any grid. Map function values f to Hermite space by a = Mf, collide and map back, C[f] = M 1 DMf Only need to calculate M 1 DM once. growth rate, γ 0.4 0. 0.0 0. 0.4 0.6 0.8 Growth rate calculated on uniform grid, N = 16 simple truncation hypercollisions exact dispersion relation 1.0 0 1 3 4 5 6 7 8 parallel wavenumber, k Example: hypercollisions implemented on a uniform grid. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 3 / 34
Summary 1D model for ITG instability velocity space discretization = eigenvalue problem finite composition of normal modes = no Landau damping Hermite representation in decaying modes, have energy pile-up at small scales Damping with collisions Lénard Bernstein collisions finds damping, but requires O(100) terms hypercollisions excellent agreement with dispersion relation theoretical expression for eigenfunctions only 10 terms robust parameters easy to implement in Hermite space can use on any grid Vanishing collisions is different from collisionless. Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 33 / 34
References BELLI, E. A. & HAMMETT, G. W. (005) A numerical instability in an ADI algorithm for gyrokinetics. Computer Physics Communications 17 (), 119 13. LANDAU, L. D. (1946) On the Vibrations of the Electronic Plasma. Journal of Physics-U.S.S.R. 10 (5). LÉNARD, A. & BERNSTEIN, I. B. (1958) Plasma oscillations with diffusion in velocity space. Physical Review 11 (5), 1456 1459. PUESCHEL, M. J., DANNERT, T. & JENKO, F. (010) On the role of numerical dissipation in gyrokinetic Vlasov simulations of plasma microturbulence. Computer Physics Communications 181, 148 1437. VAN KAMPEN, N. G. (1955) On the Theory of Stationary Waves in Plasmas. Physica 1, 949 963. ZOCCO, A. & SCHEKOCHIHIN, A. A. (011) Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Physics of Plasmas 18 (10), 10309. Supported by Award No KUK-C1-013-04 from King Abdullah University of Science and Technology (KAUST). Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 34 / 34
Drift approximation in the Vlasov equation E + v B c Vlasov equation becomes = 0, v = v ẑ c Φ B, B = Bẑ, E = Φ. B f ( t + v ẑ c ) B Φ B f q Φ m z f = 0. v Perturb about stationary equilibrium, f(x, v, t) = F 0 (x, v) + ɛf 1 (x, v, t) Impose density and temperature gradients in x, [ ( F 0 x = 1 v + v )] ω n + ω Ti L vth 3 F 0 Assume a Fourier mode in y: y ik y. Integrate out perpendicular directions in v. ( F 1 [ω t + n + ω Ti v 1 )] F 1 ik y ΦF 0 + α i v z + τ Φ eα i v z F 0 = 0 Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01 1
Quasineutrality Poisson s equation Boltzmann electrons Φ = 4πq(n i n e ), ( ) qφ n e = n e exp Nondimensionalize (with ɛ = ρ s /L) ( ɛt e 4π n i q ρ s L Φ = 1 + ɛ ( ɛ Φ = 1 n ) e + ɛ n i ( Φ, potential for electrostatic perturbation Φ = T e ) F 1 dv n e exp (ɛφ). n i F 1 dv n ) e Φ n i F 1 dv. + O(ɛ ). Joseph Parker (University of Oxford) Velocity space using Hermite expansions 3 rd March 01