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Edited by: Fabrice Deluzet, UMR5219 Institut de Mathématiques de Toulouse (IMT), France

Reviewed by: Pablo S. Moya, Universidad de Chile, Chile; Jayr Amorim, Instituto Tecnológico de Aeronáutica, Brazil

This article was submitted to Plasma Physics, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Spectral (transform) methods for solution of Vlasov-Maxwell system have shown significant promise as numerical methods capable of efficiently treating fluid-kinetic coupling in magnetized plasmas. We discuss SpectralPlasmaSolver (SPS), an implementation of three-dimensional, fully electromagnetic algorithm based on a decomposition of the plasma distribution function in Hermite modes in velocity space and Fourier modes in physical space. A fully-implicit time discretization is adopted for numerical stability and to ensure exact conservation laws for total mass, momentum and energy. The SPS code is parallelized using Message Passing Interface for distributed memory architectures. Application of the method to analysis of kinetic range of scales in plasma turbulence under conditions typical of the solar wind is demonstrated. With only 4 Hermite modes per velocity dimension, the algorithm yields damping rates of kinetic Alfvén waves with accuracy of 50% or better, which is sufficient to obtain a model of kinetic scales capable of reproducing many of the expected statistical properties of turbulent fluctuations. With increasing number of Hermite modes, progressively more accurate values for collisionless damping rates are obtained. Fully nonlinear simulations of decaying turbulence are presented and successfully compared with similar simulations performed using Particle-In-Cell method.

Kinetic effects play a critical role in many problems in plasma physics. Correct description of kinetic processes typically requires solution of the Vlasov-Maxwell equations. Unfortunately, kinetic physics brings into the picture very small spatial (Debye length, electron gyroradius) and temporal (plasma frequency, electron gyrofrequency) scales, which could be several orders of magnitude smaller than system scales for most problems of interest. For example, in the Earth's magnetosphere the microscopic physics is key to magnetospheric dynamics during geomagnetic storms and substorms (Reeves et al., ^{9} m. Similar or even more extreme separation of scales is typical of the solar wind and the solar corona, where kinetic physics is tied for example to mechanisms of energy release by magnetic reconnection and turbulence, impacting such fundamental problems as as the origin of coronal heating or solar wind heating and acceleration (e.g., Aschwanden,

Recognizing the inability of fully-kinetic methods to simulate large-scale systems, alternative approaches are being sought that focus on the so-called ^{1}

A new method for the solution of the Vlasov-Maxwell equations that could resolve the issues outlined above was recently proposed by Delzanno (

In this paper, we describe a 3D parallel simulation code SpectralPlasmaSolver (SPS), which implements the method proposed in Delzanno (

We consider Vlasov-Maxwell equations for the evolution of the distribution function _{s}(

where _{x, (v)} represents the gradient operator in physical (velocity) space and _{pe} computed with a reference plasma density _{0}, _{s} is the mass of species _{0} is the permittivity of vacuum, _{pe}_{e} = _{pe}, _{0}, _{0}, _{s} is the charge of each plasma species

We assume periodic boundary conditions in physical space and consider domain [0, _{x}] × [0, _{y}] × [0, _{z}]. The distribution function in velocity space is assumed to go to zero at infinity.

The Vlasov-Maxwell Equations (1)–(3) are solved with a spectral discretization in both physical and velocity space. In particular, we expand the distribution function as

with _{x, y, z} (_{n, m, p}) the number of modes in physical (velocity) space along each spatial direction. Here index _{x}, _{y}, _{z}}.

The electric field is expanded as

and similarly for the magnetic field

where δ = _{n} is the Hermite polynomial of the n-th order (Holloway,

By substituting expansion (4) into Equations (1)–(3) and using the orthogonality relations for both Fourier and Hermite functions, we can rewrite the Vlasov-Maxwell equations as a system of non-linearly coupled ordinary differential equations (ODE) for the coefficients of the expansion. These are (Delzanno,

for the Vlasov equation, and

for Maxwell's equations (written in index notation with summation over repeated indices implied). Here ε_{βγδ} is the Levi-Civita tensor and (_{n} − 1], _{m} − 1] and _{p} − 1] and the convolution operator is defined as

where _{n}, _{m}, _{p}. In order to address potential problems arising from the filamentation typical of collisionless plasmas, we add a collisional term on the right hand side of Eq. (7)

The collisional operator given by Equation (10) is not meant to model any specific physical dissipation mechanism, such as Coulomb collisions. Rather, it is added to damp higher-order Hermite modes. This motivates the specific functional form of the operator, together with the requirement that it does not act on the first three modes of the Hermite expansion, which have connections to density, momentum and energy of the system (Delzanno,

Equations (7) and (8) can be cast in matrix form as

where _{1,2,3} represent the linear part of Equations (7) and (8) (advection in the case of Vlasov equations) while

System (11) is discretized in time with a fully-implicit Crank-Nicolson scheme (Crank and Nicolson,

where Δ^{θ}) = ^{θ}, and ^{θ+1/2} = (^{θ+1}+^{θ})/2 (and similarly for

The resulting numerical method formulated above and implemented in the SpectralPlasmaSolver (SPS) has several important properties. The main motivation for the choice of AW-Hermite spectral discretization of velocity space is the fact that the low order modes of the expansion correspond to the classic fluid description of the plasma (with a particular closure), while the kinetic physics is retained by adding more modes to the expansion. In other words, the fluid-kinetic coupling is an intrinsic feature of SPS. The Fourier discretization ensures that the constraints of Maxwell's equation (i.e., ∇·

The implementation of SPS described here extends 2D Fortran 90 code for shared-memory architectures that was described in Vencels et al. (

Computation of convolutions is based on a parallel Fast Fourier Transforms (FFTs). The code utilizes 1D and 2D domain decomposition of the solution space, such that each domain owns a portion of ^{2}

Figure ^{3} Fourier modes and _{h} = 4 Hermite modes in each direction. The weak scaling study was performed by proportionally scaling up the system size and the MPI rank count for a system with 63^{3} Fourier modes and _{h} = 4 Hermite modes per direction. These tests were performed on XE nodes of Blue Waters supercomputer and used 8 MPI ranks per node. GCC compilers version 4.9.3 and FFTW library version 3.3.4 were used to compile the code. Each Blue Waters XE node is equipped with 2 AMD Interlagos model 6276 CPU processors. Significant variability in timings for a given simulation size was observed in some tests, presumably due to how a particular job was placed with respect to the topology of the communication links. The results shown in Figure ^{3}^{3} spatial modes, 4^{3} Hermite functions, and initial conditions corresponding to Maxwellian plasma. It is apparent that saturation of the overall scaling is related to increasing relative cost of the communication with increasing number of processes.

Examples of strong

Example of strong scalings of SPS obtained on Electra supercomputer. Red circles represent computation time, blue squares mark the ratio of communication to computation time, and dashed lines denote ideal scalings.

These results point to a number of further enhancements in the code that would significantly increase parallelism and performance: (i) implementing decomposition in the Hermite coefficient space in addition to the

Solar wind is believed to represent one of the best accessible examples of astrophysical large-scale plasma turbulence (Bruno and Carbone, _{p} ~ 1 (where _{p} is the proton inertial length). The behavior at electron scales is less well characterized, but generally power law spectra are observed to extend to scales of the order of electron gyroradius, with subsequent steepening to either another power law or an exponential range reported in the literature (Sahraoui et al.,

In a classical paradigm of turbulence, the energy is transported by nonlinear interactions with no loss through the inertial range and is dissipated at small scales. In a weakly collisional plasma, such as solar wind, dissipation of the turbulent fluctuations is due to collective wave-particle interactions. These interactions can deposit the energy into various plasma components, such as thermal or suprathermal electrons, protons, and heavy ion species. Since turbulence has been proposed to play a significant role in the energy balance of solar wind, solar corona, and other space and astrophysical systems, understanding of the details of energy dissipation is a question of both significant theoretical interest and of great practical importance.

The problem of correctly describing energy dissipation in weakly collisional turbulence can be viewed as a problem of understanding fluid-kinetic coupling. Indeed, the fluctuations in the inertial range are widely thought to be well-described by MHD approximation. At the same time, adequate description of the dynamics at sub-proton scales and especially of the energy dissipation mechanisms requires electron and ion kinetic effects. The large-scale fluctuations drive dynamics at sub-proton scales, while the small-scale fluctuations may influence the large scales e.g., by supplying dissipation of energy and momentum, breaking topological constraints through magnetic reconnection, or placing constraints on values of temperature anisotropy enforced by instabilities. Significant progress has been achieved in numerical modeling of turbulence in weakly collisional plasmas (e.g., Howes,

Since the linear dispersion and damping of plasma modes are thought to be an important factor in determining the turbulence dynamics at sub-proton scales, we begin by discussing the ability of the Fourier-Hermite (FH) expansion approach to capture these properties for kinetic Alfvén wave (KAW). Observations in the solar wind and existing theoretical results suggest that KAWs play a significant, if not dominant role in the dynamics at sub-proton scales (e.g., Howes et al.,

It is important to note that even when the number of Hermite basis functions is low (4–6 in each direction), the FH method can reproduce the linear properties of the waves with sufficient accuracy to yield an approximate model for sub-proton scale dynamics that compares well with fully kinetic PIC simulations (see section 4.2). Development of advanced fluid models targeting the sub-proton range of scales and capable of reproducing the damping rates of kinetic Alfvén and whistler modes has been the subject of intense research (e.g., Passot et al., ^{r} can be expressed

were _{n} are constant coefficients, Ψ^{n} is the dual Hermite basis function such that ^{3}

For example, for _{m} = _{n} = _{p} = 4, the system of evolution equations for coefficients _{m, n, p} is equivalent to a fluid system that retains evolution equations for density, mean velocity, pressure, and heat flux tensors. The closure approximation relates the 4-th order moment to the lower ones using Equation (13). Remarkably, such a purely numerical closure can yield similar or better results than purposefully constructed models.

Here we focus on the (weakly damped) modes that are thought to be relevant to astrophysical turbulence. The ability of the method to describe some other modes and instabilities has been demonstrated in prior publications (e.g., Delzanno, _{0}. The mass ratio is _{i}/_{e} = 100, ω_{pe}/Ω_{ce} = 2, and β_{e} = β_{i} = 0.5. Here _{pe}/Ω_{ce} is significantly lower than the one in many systems of interest, such as the solar wind, but is chosen to enable direct comparison with explicit PIC simulations (see section 4.2). The latter could not be conducted at realistic value of ω_{pe}/Ω_{ce} due to the requirement to resolve spatial scales of the order of Debye length

We consider angle of propagation θ = 88° with respect to the background magnetic field, as is characteristic of the kinetic Alfvén waves (KAW). For each value of

_{h} = 4 and _{h} = 6 respectively, while green square correspond to eigenvalues computed for _{h} = 6. The black curve shows numerical solution from a linear Vlasov solver. The angle of propagation is θ = 89°.

To provide more information on the linear properties of the method, Figures _{h}, the number of Hermite modes per direction. For each case, we show eigenvalues that are the closest to the Vlasov solution. Figure _{h} = 6, angle of propagation θ = 88°, _{e} = 0.5, and several values of the collisionality parameter ν. The latter plot illustrates the role played by finite collisionality: when ν is zero, all the eigenvalues are on the real axis and the solution corresponding to a damping mode in the linearized Vlasov equation is obtained as the result of phase-mixing between eigenmodes. With finite collisionality, the spectrum of eigenvalues becomes progressively sparser, but certain eigenvalues appear in the vicinity of the solution of the linearized Vlasov equation. It is those eigenvalues that give the solutions plotted in Figure

Eigenvalues of the linearized algorithm for the angle of propagation θ = 80.5°_{h} = 4, 6, 8 respectively. _{e} = 0.5, _{h} = 6, θ = 88° and several values of the collisionality parameter ν.

Having established that the FH method presented here is capable of adequately describing linear properties of the wave modes of interest, we present an example of a fully nonlinear simulation of turbulence targeting the physics at electron scales. We consider decaying turbulence, where no external forcing is applied and the turbulence develops as a result of the decay of an initial perturbation. The simulations are performed in an anisotropic rectangular simulation domain of size _{∥} = 400_{e} and _{⊥} = 50_{e}, where parallel and perpendicular refer to the orientation with respect to the background magnetic field. The simulation is initialized with Maxwellian uniform plasma and a perturbation of the magnetic field of the form _{x}), _{y}), _{z})}, with _{k} = 0, _{0}·δ_{k} = 0, _{k} = 0, and |δ_{k}| = |δ_{k}|. The mean energy _{e} = 0.5 and _{h} = 6 and _{h} = 5. The time step is ^{4} particles per cell per species at

Figure _{ci} = 10 (when these properties become quasi-stationary) for the two SPS simulations and the PIC case. The top panel shows the spectrum of magnetic fluctuations _{e} ~ 1, where _{e} = _{th, e}/Ω_{ce} and the Debye length λ_{e}, which are not well separated for the considered parameters since _{e} < 1. The saturation of the spectra in PIC simulations at high values of _{av}Ω_{ce}≈3.6.

Spectrum of magnetic fluctuations _{ci} = 10. The vertical lines mark scales corresponding to _{e} = 1 and _{e} = 1. The horizontal lines in _{h} = 6 and _{h} = 5 respectively. The black curves represent PIC simulation. The top panel also shows a characteristic local slope of the magnetic spectrum in the vicinity of _{e} ~ 1.

The middle panel of Figure _{∥} = (1/2)β_{i}(1+τ)/[1+β_{i}(1+τ)] in the range _{e}/_{i} (Boldyrev et al., _{∥} is about 15% higher in PIC simulations relative to both SPS cases in the range _{e} ≲ 1. In all of the simulations, the compressibility increases after _{e} ~ 1. The increase is somewhat sharper in PIC simulations and this behavior is better tracked by the _{h} = 6 case. Finally, the bottom panel in Figure _{e} ≲ 1 the properties of the fluctuations are consistent with KAWs. The values of electron compressibility in this range are very close between PIC and SPS simulations, but deviate at scales _{e}≳1, with a sharper increase in the PIC case.

The properties of electron _{e} and ion _{i} velocity fluctuations are summarized in the top and bottom panels of Figure _{e}| spectra is very similar to those of the magnetic field, with _{e} spectra shallower by approximately ^{2}. A good agreement between PIC and SPS _{e} spectra is seen at scales _{e} ≲ 1. In contrast, the (weak) fluctuations of ion velocity exhibit more substantial differences between PIC and SPS cases. In particular, the PIC spectrum of _{i} is shallower at small scales, leading to about a factor of 3 difference _{e} ~ 1, even though the overall energy of ion velocity fluctuations is lower in PIC.

Spectra of electron _{ci} = 10. The vertical lines mark scales corresponding to _{e} = 1 and _{e} = 1. In all panels, the blue and red curves correspond to SPS simulations with _{h} = 6 and _{h} = 5 respectively. The black curves represent PIC simulation. PIC spectra flatten out due to effects of numerical noise.

Finally, Figure

Energy conservation in the simulation of decaying turbulence. The curves show evolution of total energy

In this paper we summarized recent development of a highly parallelized version of the code SPS. SPS is a spectral code that implements dual Fourier-Hermite transform to obtain numerical solution of the Vlasov-Maxwell system. The method is particularly suitable for describing fluid-kinetic coupling since a direct analogy could be drawn between traditional fluid hierarchy and the equations for the evolution of the coefficients of expansion in the case of asymmetrically-weighted Hermite basis. In contrast to the traditional multi-moment models, the number of coefficients in the expansion is limited only by available computational resources, which opens up intriguing possibilities for building models capable of adaptively increasing the number of coefficients in selected regions of physical or wavenumber space.

As an example, we have discussed application of SPS to the problem of turbulence in weakly collisional plasmas. A model of plasma dynamics capable of capturing essential features of turbulence at electron scales could be obtained with only a few Hermite modes, as was demonstrated by considering linear properties of Kinetic Alfvén waves. Fully nonlinear simulations of decaying plasma turbulence were performed and successfully compared against corresponding PIC simulations. While some differences in the decay of the imposed perturbation of ion flow were observed between PIC and SPS, SPS simulations successfully reproduce most of the statistical properties of magnetic and electron fluctuations observed in PIC simulations at _{e} ≲ 1. Significant differences in the spectra are observed at scales approaching Debye scale and below. In practice, this range of scales might not be significant for turbulence investigations, since most of cascading energy is expected to dissipate before it reaches Debye scales.

The datasets for this study can be obtained by sending request to VR.

VR and GLD closely collaborated on development and validation of the version of SPS discussed here and preparation of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We greatly acknowledge fruitful discussions on the physics of kinetic turbulence with S. Boldyrev. G. Jost analyzed our implementation of SPS and made a number of very useful recommendations for improving its performance.

^{1}Ho, A., Datta, I., and Shumlak, U.

^{2}Intel® Math Kernel Library. Available online at:

^{3}Jin, H.