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Register a new userViriato: a Fourier-Hermite spectral code for strongly magnetised fluid-kinetic plasma dynamics
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We report on the algorithms and numerical methods used in Viriato, a novel fluid-kinetic code that solves two distinct sets of equations: (i) the Kinetic Reduced Electron Heating Model (KREHM) equations [Zocco & Schekochihin, Phys. Plasmas 18, 102309 (2011)] (which reduce to the standard Reduced-MHD equations in the appropriate limit) and (ii) the kinetic reduced MHD (KRMHD) equations [Schekochihin et al., Astrophys. J. Suppl. 182:310 (2009)]. Two main applications of these equations are magnetised (Alfvenic) plasma turbulence and magnetic reconnection. Viriato uses operator splitting (Strang or Godunov) to separate the dynamics parallel and perpendicular to the ambient magnetic field (assumed strong). Along the magnetic field, Viriato allows for either a second-order accurate MacCormack method or, for higher accuracy, a spectral-like scheme composed of the combination of a total variation diminishing (TVD) third order Runge-Kutta method for the time derivative with a 7th order upwind scheme for the fluxes. Perpendicular to the field Viriato is pseudo-spectral, and the time integration is performed by means of an iterative predictor-corrector scheme. In addition, a distinctive feature of Viriato is its spectral representation of the parallel velocity-space dependence, achieved by means of a Hermite representation of the perturbed distribution function. A series of linear and nonlinear benchmarks and tests are presented, including a detailed analysis of 2D and 3D Orszag-Tang-type decaying turbulence, both in fluid and kinetic regimes.
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An overview of the algorithm and a sampling of plasma applications of the implicit, adaptive high order finite (spectral) element modeling framework, HiFi, is presented. The distinguishing capabilities of the HiFi code include adaptive spectral element spatial representation with flexible geometry, highly parallelizable implicit time advance, and general flux-source form of the partial differential equations and boundary conditions that can be implemented in its framework. Early algorithm development and extensive verification studies of the two-dimensional version of the code, known as SEL, have been previously described [A.H. Glasser & X.Z. Tang, Comp. Phys. Comm., 164 (2004); V.S. Lukin, Ph.D. thesis, Princeton University (2008)]. Here, substantial algorithmic improvements and extensions are presented together with examples of two- and three- dimensional applications of the HiFi framework. These include a Cartesian two-dimensional incompressible magnetohydrodynamic simulation of low dissipation magnetic reconnection in a large system, a two-dimensional axisymmetric simulation of self-similar compression of a magnetic plasma confinement configuration, and a three-dimensional Hall MHD simulation of spheromak tilting and relaxation. Some planned efforts to further improve and expand the capabilities of the HiFi modeling framework are discussed.
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A minimal model for magnetic reconnection and, generally, low-frequency dynamics in low-beta plasmas is proposed. The model combines analytical and computational simplicity with physical realizability: it is a rigorous limit of gyrokinetics for plasma beta of order the electron-ion mass ratio. The model contains collisions and can be used both in the collisional and collisionless reconnection regimes. It includes gyrokinetic ions (not assumed cold) and allows for the topological rearrangement of the magnetic field lines by either resistivity or electron inertia, whichever predominates. The two-fluid dynamics are coupled to electron kinetics --- electrons are not assumed isothermal and are described by a reduced drift-kinetic equation. The model therefore allows for irreversibility and conversion of magnetic energy into electron heat via parallel phase mixing in velocity space. An analysis of the exchanges between various forms of free energy and its conversion into electron heat is provided. It is shown how all relevant linear waves and regimes of the tearing instability (collisionless, semicollisional and fully resistive) are recovered in various limits of our model. An efficient way to simulate our equations numerically is proposed, via the Hermite representation of the velocity space. It is shown that small scales in velocity space will form, giving rise to a shallow Hermite-space spectrum, whence it is inferred that, for steady-state or sufficiently slow dynamics, the electron heating rate will remain finite in the limit of vanishing collisionality.
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.
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