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Register a new userFourier-Hermite decomposition of the collisional Vlasov-Maxwell system: implications for the velocity space cascade
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Turbulence at kinetic scales is an unresolved and ubiquitous phenomenon that characterizes both space and laboratory plasmas. Recently, new theories, {it in-situ} spacecraft observations and numerical simulations suggest a novel scenario for turbulence, characterized by a so-called phase space cascade -- the formation of fine structures, both in physical and velocity space. This new concept is here extended by directly taking into account the role of inter-particle collisions, modeled through the nonlinear Landau operator or the simplified Dougherty operator. The characteristic times, associated with inter-particle correlations, are derived in the above cases. The implications of introducing collisions on the phase space cascade are finally discussed.
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We study Landau damping in the 1+1D Vlasov-Poisson system using a Fourier-Hermite spectral representation. We describe the propagation of free energy in phase space using forwards and backwards propagating Hermite modes recently developed for gyrokinetics [Schekochihin et al. (2014)]. The change in the electric field corresponds to the net Hermite flux via a free energy evolution equation. In linear Landau damping, decay in the electric field corresponds to forward propagating Hermite modes; in nonlinear damping, the initial decay is followed by a growth phase characterised by the generation of backwards propagating Hermite modes by the nonlinear term. The free energy content of the backwards propagating modes increases exponentially until balancing that of the forward propagating modes. Thereafter there is no systematic net Hermite flux, so the electric field cannot decay and the nonlinearity effectively suppresses Landau damping. These simulations are performed using the fully-spectral 5D gyrokinetics code SpectroGK [Parker et al. 2014], modified to solve the 1+1D Vlasov-Poisson system. This captures Landau damping via an iterated Lenard-Bernstein collision operator or via Hou-Li filtering in velocity space. Therefore the code is applicable even in regimes where phase-mixing and filamentation are dominant.
Plasma turbulence is investigated using high-resolution ion velocity distributions measured by the Magnetospheric Multiscale Mission (MMS) in the Earths magnetosheath. The particle distribution is highly structured, suggesting a cascade-like process in velocity space. This complex velocity space structure is investigated using a three-dimensional Hermite transform that reveals a power law distribution of moments. In analogy to hydrodynamics, a Kolmogorov approach leads directly to a range of predictions for this phase-space cascade. The scaling theory is in agreement with observations, suggesting a new path for the study of plasma turbulence in weakly collisional space and astrophysical plasmas.
We study the stability of spatially periodic, nonlinear Vlasov-Poisson equilibria as an eigenproblem in a Fourier-Hermite basis (in the space and velocity variables, respectively) of finite dimension, $N$. When the advection term in Vlasov equation is dominant, the convergence with $N$ of the eigenvalues is rather slow, limiting the applicability of the method. We use the method of spectral deformation introduced in [J. D. Crawford and P. D. Hislop, Ann. Phys. 189, 265 (1989)] to selectively damp the continuum of neutral modes associated with the advection term, thus accelerating convergence. We validate and benchmark the performance of our method by reproducing the kinetic dispersion relation results for linear (spatially homogeneous) equilibria. Finally, we study the stability of a periodic Bernstein-Greene-Kruskal mode with multiple phase space vortices, compare our results with numerical simulations of the Vlasov-Poisson system and show that the initial unstable equilibrium may evolve to different asymptotic states depending on the way it was perturbed.
Kinetic simulations based on the Eulerian Hybrid Vlasov-Maxwell (HVM) formalism permit the examination of plasma turbulence with useful resolution of the proton velocity distribution function (VDF). The HVM model is employed here to study the balance of energy, focusing on channels of conversion that lead to proton kinetic effects, including growth of internal energy and temperature anisotropies. We show that this Eulerian simulation approach, which is almost noise-free, is able to provide an accurate energy balance for protons. The results demonstrate explicitly that the recovered temperature growth is directly related to the role of the pressure-strain interaction. Furthermore, analysis of local spatial correlations indicates that the pressure-strain interaction is qualitatively associated with strong-current, high-vorticity structures, although other local terms -- such as the heat flux -- weaken the correlation. These numerical capabilities based on the Eulerian approach will enable deeper study of transfer and conversion channels in weakly collisional Vlasov plasmas.
To explain energy dissipation via turbulence in collisionless, magnetized plasmas, the existence of a dual real- and velocity-space cascade of ion-entropy fluctuations below the ion gyroradius has been proposed. Such a dual cascade, predicted by the gyrokinetic theory, has previously been observed in gyrokinetic simulations of two-dimensional, electrostatic turbulence. For the first time we show evidence for a dual phase-space cascade of ion-entropy fluctuations in a three-dimensional simulation of hybrid-kinetic, electromagnetic turbulence. Some of the scalings observed in the energy spectra are consistent with a generalized theory for the cascade that accounts for the spectral anisotropy of critically balanced, intermittent, sub-ion-Larmor-scale fluctuations. The observed velocity-space cascade is also anisotropic with respect to the magnetic-field direction, with linear phase mixing along magnetic-field lines proceeding mainly at spatial scales above the ion gyroradius and nonlinear phase mixing across magnetic-field lines proceeding at perpendicular scales below the ion gyroradius. Such phase-space anisotropy could be sought in heliospheric and magnetospheric data of solar-wind turbulence and has far-reaching implications for the dissipation of turbulence in weakly collisional astrophysical plasmas.
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