2013 US-Japan JIFT workshop on New Aspects of Plasmas Kinetic Simulation NIFS, November 22-23, 2013 Kinetic damping in gyro-kinetic simulation and the role in multi-scale turbulence cf. Revisit for Landau damping Y. Kishimoto, P. Hilscher, K. Imadera, J.Q. Li Graduate School of Energy Science, Kyoto University Cheng and Knorr, JCP 22, 330 (1976) Hilsher, Imadera, Li and Kishimoto, The effect of weak collisionality on damped modes and its contribution to linear coupling in gyro-kinetic simulation Phys, Plasmas 20, 082127 (2013) C.S. Ng et al., Kinetic eigen-modes and discrete spectrum of plasma oscillations in a weakly collisional plasma, PRL 83, 1974 (1999) V. Bratanov, F. jenko et al., Aspects of linear Landau damping in discretized systems, PoP 20, 022108 (2013)
Motivation Outline Various types of interaction among different scale fluctuations in fusion plasmas and the importance of stable/damped modes Example of multi-scale interaction 1/1 kink mode and high m.n ballooning modes Turbulent spectrum dominated by zonal flows Ion turbulence intermittency in the presence of ETG driven zonal flows MHD driven vortex flow/island and ITG turbulence Characteristics of stable/damped mode in gyro-kinetic simulation in slab geometry Recurrence phenomenon in collision less GK simulation and Case va Kampen (CvK) continuum mode Landau damping and phase mixing, and the role of collision in discretized Vlasov-Maxwell equation system Triad mode coupling and the role of damped mode Summary
Scattering of turbulence to high k x modes by zonal flows and the importance of high (k x,k y ) damping mode Turbulent spectrum dominated by zonal flows Spectral shape from GF ITG simulation Li and Kishimoto, PoP 17, 072304 (2010) Gurcan, et al., PRL 102, 255002 (2009) Theoretical spectral scaling (shell model to HW turbulence) n ~ k 2 ~ ~ k 2 3 k ~ ( 1 k 2 ) 2 e k 3 k x 2 2 x ) 2 0.002 13.0k ( k x x ) ~ e (1 k Coupled Kolmogorov-exponential law Damping to high-kx modes
k y 1 e 1 i Effect of ETG driven ZF on ITG turbulence Nonlocal energy transfer of ITG turbulence due to TEG driven micro-scale ZF. Li and Kishimoto, PRL 89, 115002 (2002) Scattering of ITG energy to high-k Reducing ITG zonal flows Ion scale Electron scale (k x ) 2 /2 10-1 10-5 10-9 (a) low-k (b) φ k q A 0 Γ 1 2 0 (q)cos(qx) 1 a 1 a 1 i 1 ρ e k x 10-13 high-k φ 0.1 1 10 k k q x 0.3 0.25 0.2 0.15 0.1 0.05 0 i (b) (a) < d zf /dx 2 >/2 0 500 1000 1500 2000 2500 3000 Upper state No flow case Lower state
Wang, Li, Dong, Kishimoto, PRL 89, 115002 (2002)
Importance of damped mode in mode coupling system Difference between fluid turbulence and plasma turbulence Hatch et al., PRL 106, 115003 (2011) Dissipation occurs at all spatial scales including those where the instability drives the turbulence Makwana et al., PoP 19, 062310 (2012) Not only unstable modes, but also stable/damping mode is important in the cross-scale coupling as energy sink and effective dissipation. Many of multi-scale interaction have been studied using fluid model, where damping and dissipation are modeled. Whether kinetic stable/damped modes are properly reproduced in gyro-kinetic simulations specifically in long time scale dynamics is an important issue. ITG Short-w ITG
Study of slab damped ITG mode based GK simulation 2D gyro-kinetic equation system in Fourier space with LB collision Numerical simulation (initial value problem : IVP) f 1 is equidistantly discretized in velocity space Shear-less slab ITG mode : Initial perturbation in velocity space (A) Random distribution : (B) Maxwellian distribution : (the largest growth rate)
Collision less Linear GK simulation of slab ITG mode (A)Random : (B) Maxwellian : Cheng and Knorr, JCP 22, 330 (1976) discretization in velocity space recurrence No damping solution Instantaneous growth rate ~ Landau No effective damping in long time scale simulation Whether the role of stable/damped modes works in collision-less nonlinear simulation
Collisional linear GK simulation of slab ITG mode (A)Random : (B) Maxwellian : w/o collision Inclusion of collisionality : with collision level of round-off error Collisionality plays a role in diminishing recurrence
Landau and CvK treatments for plasma normal mode L. D. Landau, J. Phys. USSR 10, 25 (1946) Eliminate f 1 and solve IV problem using Laplace transform, looking for the normal mode of E. Only one normal mode for each k, and obtain eigenvalue as the time asymptotic solution N. G. van. Kampen, Physica 21, 949 (1955) Eliminate E, looking for normal mode for f 1 using Fourier transport. The linearized Vlasov-Poisson equations have a continuous spectrum of singular normal modes, now known as Case and van Kampen (CvK) modes; A. Lenard and B. Bernstein, Physical Review 1012 1456 (1958) They derived dispersion relation with collision and showed that the solution converges to the Landau s solution. K. M. Case, Annals of Physics 7, 349 (1959) Prove the equivalence of the Landau and Van Kampen treatments of the initial value problem as the phase mixing of complete set of continuum spectrum
Case van Kampen (CvK) mode in the ITG system Eigen-value analysis with non-hermitian operator due to temperature gradient Descretized with N v v elements ω, ˆ n f n : linear gyr-kinetic operator Eigen-value corresponds to the Landau mode and modes with continuum singular structure corresponds to the CvK modes real real Landau solution Damping can not be reproduced by the eigen-mode
Damping due to the phase mixing of CvK modese IVP simulation using the CvK eigen-mode leads no grow and no damping both for unstable and stable poloidal modes Decompose the Maxwellian perturbation in terms of the CvK eigenfunctions fˆ n, n Random initial perturbation c n ~ R Equivalent to the Landau damping ~ Landau Maxwellian initial perturbation Recurrence reproduced Damping with recurrence in collsion-less IVP simulation not results from the Landau eigen-mode, but from the phase mixing of the CvK eigen-functions in the discretized Vlasov-Poisson system
Phase mixing and energy transfer in velocity space (Ⅰ) Expansion of the distribution function of GK equation using Hermite basis f 1 2 u a H u exp u n n n a 0 is directly related to the potential the first term : Eigen-value equation : a : a vector whose nth component is the coefficient for the Hermite function Tridiagnal structure describing the energy transfer among neighboring Hermite modes
Phase mixing and energy transfer in velocity space (Ⅱ) Maxwellian F.C. Grant and M.R. Feix, Phys. Fluids 10, 696 (1967) Transfer from a0 to higher order moments Random noise All modes are simultaneously excited with no directionality 32th order Hermite function
Role of collision on the damping and recurrence IVP of ITG mode with collision Instantaneous growth rate during recurrence rec ~ LD Envelope of recurrence damping * c LD LD Damping rate averaged over log time scale exceeding the recurrence time is influenced by the collision, while less influenced when it becomes larger than the critical value, * i.e.. c c * c c Collision only has a physical meaning for. c * c c LD ~ * * c c LD ~ * * c ~ 2.110 3
Eigen value analysis of the ITG mode with collision (I) if ˆ ˆ 1 L C f Eigen-value analysis including collision effect ky LB 1 c 0.810 3 3 3 c 210 c 510 Majority of CvK modes When the majority of CvK modes exceeds the damping rate predicted by the Landau theory, one of the CvK modes changes its structure to the Landau mode Ng, Bhattacharjee, and Skiff, PRL 83, 1974 (1999) Theoretical approach using an expansion based on Hermite polynomials The CvK continuous spectrum is eliminated in the limit of zero collision frequency and replaced by a discrete spectrum, which are the Landau solutions. For small but nonzero collision frequency, the spectra and eigenmodes are qualitatively different from their counterparts in the collisionless theory
Critical collisionality for getting Landau solution in numerically discretized system Damping rate with respect to collisionality for different velocity space resolutions Majority (continuum) line of CvK mode Landau damping solution The CvK modes less affected by the collisionality c, while quickly affected as be come to c *. Around c c the Landau eigenmode emerges. As the velocity space resolution increases, the collisionality which damps the CvK mode decreases. As N v, c v Bratanov, Jenko et al., PoP 20, 022108 (2013) Comparison of different discretization scheme in capturing the CvK modes in gyrokinetic modeling.
Damping role of stable/damped mode in collisionless discretized system Whether and how the triad mode coupling relation and associated energy transfer works in collisionless discretized system where the expected damped mode suffers from recurrence and/or undamped nature ITG mode : k y =1.6 unstable mode k, k ex, ex Vortex flow : k ex =0.3 e.g. zonal flow, GAM, vortex flows Shearing, the process that enhances the transfer of energy to shorter wavelength mode k, Stable/damped mode ITG mode : k y =1.9
Coupling of stable/damped ITG mode via vortex flow Vortex flow : k ex 0.5 Unstable ITG mode : f1, k, k y 1.3 y Stable ITG mode : f1, k, k y 1.8 y 0 (No coupling) 2N v eigen-value Each N v eigen-vectors to and
Coupling of stable/damped ITG mode via vortex flow No coupling case : 0 No Landau solution f1, k y f 1, k y f1, k y f 1, k y stable unstable stable unstable
Coupling of stable/damped ITG mode via vortex flow with coupling case : 0.15 f1, k y f 1, k y f1, k y f 1, k y stable unstable
Coupling of stable/damped ITG mode via vortex flow Damping characteristics with respect to collisionality for different coupling strengths Unstable global mode Unstable global mode CvK global mode Stable global mode No stable eigen-mode, but continuum CvK spectrum Stable eigen-mode appears Reduction of the growth rate for the unstable mode always happens without depending on whether stable/damped modes exists or not as an eigen-mode
Collisionless linear GK simulation in coupling system External flow : k ex =0.3 k y =1.6 ITG mode : k y =1.9 CvK mode : Global ITG mode k y =1.6 k y =1.9 k y =1.6 ITG mode : k y =1.9 ITG mode : k y =1.6 k y =1.9
Summary Not only unstable modes which drive instability and turbulence but also stable/ damped modes which play an role of energy sink/dissipation is important in GK turbulent simulation specifically aiming at multi scale interaction. Such dissipations place at wider scales, leading to characteristic turbulent spectrums different from those in neutral fluid turbulence.. In the collisionless GK system, the potential is subject to recurrence beyond which the simulation becomes ambiguous, which results from the CvK continuum. The finite collision leads to damping of the CvK modes, so that the Landau mode as the eigen mode is recovered and then the recurrence disappear. The required critical collisionaality depends on the resolution in velocity space. Two approaches In theory Landau theory CvK theory and phase mixing Collsion less case Collsional case Vlasov simulation Even in the collisionless system with marginal CvK modes, the mode coupling between unstable and stable components through a tertiary mode and the resultant energy transfer can be properly calculated such that the stable/damped mode has been persisted as an eigenstate.