From the Earth simulation, the generation and evolution of flux transfer events FTEs are studied. Kinetic phenomena, such as the crescent electron phase space distribution, the Larmor electric field, and the lower hybrid drift instability are identified from the global simulation. The reconnection dawn-dusk asymmetry also arises from the simulations; the reconnection jets are stronger on the dawn side, which agrees with the MESSENGER observations. Subject s : Fifth-order accurate scheme for hyperbolic equation, Magnetohydrodynamics with embedded particle-in-cell model. Size: Format: PDF.
About Deep Blue Data. 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 1 represent the lowest value. The break point observed in both strong and weak scalings is most likely related to the fact that simulation time is limited by communication time at high rank count. Each Skylake node is equipped with two Xeon Gold processors.
Intel compilers and MKL library version were utilized in these tests. Furthermore, mpiprof tool 3 was used to collect separate timings for computation and MPI communication. The scaling shown here was obtained for a system with 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.
Figure 1. Dashed black lines denote ideal scalings. The efficiency of the weak scaling is defined as the ratio between the number of MPI ranks and simulation time normalized to that value for the reference case with 4 MPI ranks. Figure 2. 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 k -space; ii overlapping communication and computation by using non-blocking collective MPI operations; iii performing transforms for several variables at a time. Solar wind is believed to represent one of the best accessible examples of astrophysical large-scale plasma turbulence Bruno and Carbone, The spectrum of magnetic fluctuations in the solar wind, as measured in-situ by spacecraft, exhibits power law behavior over many orders of magnitude in scales.
A large range of scales, where fluctuations have predominantly Alfvenic character, exhibits many features expected of the inertial range of turbulence. 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. Significant progress has been achieved in numerical modeling of turbulence in weakly collisional plasmas e.
In this section we describe an example of applying SPS to this problem. As the first step, we focus on the ability of the method to adequately describe kinetic dissipation and the small-scale dynamics.see
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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. We note that linear properties of the Fourier-Hermite method, especially in the electrostatic limit, as well as its application to several other wave modes and instabilities are well described in the literature e. The time-stepping formulation of the method is generally capable of following expected linear phase for a certain time, limited by recurrence. The accuracy of linear damping or growth rates increases with the number of Hermite basis functions.
Furthermore, recurrence can be suppressed by including physical or artificial collision operator in the method formulation. 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. Such models typically rely on advanced closures, often incorporating expressions for the heat flux consistent with the linear theory Hammett and Perkins, To make contact with the traditional fluid hierarchy, we first note that in asymmetrically-weighted formulation of the FH method an exact correspondence exists between the fluid moment equations and the evolution equations for coefficients of the expansion.
Indeed, for any power r , v r can be expressed. Then velocity moments of the kinetic equation can be related to the Hermite moments as. The closure approximation relates the 4-th order moment to the lower ones using Equation 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. We present results from two complementary methods of analyzing linear properties of the method: decay of the initial perturbation in time-stepping context and the eigenvalues of the linearized equations. We consider uniform maxwellian plasma in a background magnetic field of strength B 0. For each value of k , Figure 3 shows the eigenvalue that is the closest to the KAW solution, as given by a linear Vlasov solver that implements equations of Stix It is important to emphasize that damping of a generic perturbation is the result of phase mixing between different eigenmodes and is thus better described in time-stepping formulation of the FH method e.
Figure 3 also includes real frequency and the damping rate obtained by initializing fully nonlinear SPS simulations with small-amplitude perturbation corresponding to eigenvector from the linear Vlasov solution. Overall, it is apparent that even when small number of Hermite modes is used, the method yields a good approximation to the collisionless damping rates and frequency for these nearly perpendicularly propagating modes.
Such modes are the most relevant in the solar wind, since the turbulent cascade at both large and at the kinetic scales is highly anisotropic and supplies energy predominantly in the perpendicular direction e. Figure 3. The black curve shows numerical solution from a linear Vlasov solver. To provide more information on the linear properties of the method, Figures 4A—C show eigenvalues of the linearized algorithm for three different angles of propagation and several values of parameter N h , the number of Hermite modes per direction.
For each case, we show eigenvalues that are the closest to the Vlasov solution. 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 3. As the number of Hermite modes is increased for finite collisionality, the number of eigenvalues in the vicinity of the Vlasov solution increases not shown.
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Figure 4. 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 resolution in the Fourier space is modes in each perpendicular direction and 31 modes in the parallel direction. In order to compare the SPS results with another well established technique for solving Vlasov-Maxwell system, we have conducted a Particle-In-Cell simulation of the same problem. However, due to very short range of scales considered, the power law behavior cannot be established reliably. This transition is followed by yet steeper spectrum at scales below Debye length, where fluctuations are essentially electrostatic.
Overall, the spectra of magnetic fluctuations in SPS and PIC simulations are in good agreement, taking into account the lower energy of the initial perturbation in PIC. The only significant differences are observed at sub-Debye scales, similar to the results reported in Vencels et al.
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For example, grid effects strongly affect PIC results, while the presence of an explicit collisional operator may have a strong influence on the SPS results. The saturation of the spectra in PIC simulations at high values of k is due to the numerical noise. Various techniques exist for mitigating effects of the noise in PIC simulations, such as spatial filtering or temporal averages e.
Figure 5. The black curves represent PIC simulation. The properties of electron U e and ion U i velocity fluctuations are summarized in the top and bottom panels of Figure 6 , respectively. At the sub-proton scales, the magnitude of electron velocity fluctuations significantly exceeds that of ions since ions are demagnetized. Overall, the behavior of U e spectra is very similar to those of the magnetic field, with U e spectra shallower by approximately k 2.
Figure 6. PIC spectra flatten out due to effects of numerical noise. Finally, Figure 7 shows the evolution of the total energy in the simulations E t , normalized to the energy of the initial perturbation E 0. The SPS simulations, as expected, demonstrate excellent conservation properties Delzanno, In fact, no change in the total energy is present within the accuracy of the diagnostic output.
While this is likely acceptable for the present case, we had to use an unconventionally large number of particles to achieve such level of energy conservation. Figure 7. Energy conservation in the simulation of decaying turbulence.
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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.
Fully nonlinear simulations of decaying plasma turbulence were performed and successfully compared against corresponding PIC simulations. 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 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.
Jost analyzed our implementation of SPS and made a number of very useful recommendations for improving its performance.
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