Simulations Of Magnetohydrodynamic Turbulence Research Articles

Probing Three-dimensional Magnetic Fields. III. Synchrotron Emission and Machine Learning

Synchrotron observation serves as a tool for studying magnetic fields in the interstellar medium and intracluster medium, yet its ability to unveil three-dimensional (3D) magnetic fields, meaning probing the field’s plane-of-the-sky (POS) orientation, inclination angle relative to the line of sight, and magnetization from one observational data, remains largely underexplored. Inspired by the latest insights into anisotropic magnetohydrodynamic (MHD) turbulence, we found that synchrotron emission’s intensity structures inherently reflect this anisotropy, providing crucial information to aid in 3D magnetic field studies: (i) the structure’s elongation gives the magnetic field’s POS orientation and (ii) the structure’s anisotropy degree and topology reveal the inclination angle and magnetization. Capitalizing on this foundation, we integrate a machine learning approach—convolutional neural network (CNN)—to extract this latent information, thereby facilitating the exploration of 3D magnetic fields. The model is trained on synthetic synchrotron emission maps, derived from 3D MHD turbulence simulations encompassing a range of sub-Alfvénic to super-Alfvénic conditions. We show that the CNN is physically interpretable and the CNN is capable of obtaining the POS orientation, inclination angle, and magnetization. Additionally, we test the CNN against the noise effect and the missing low-spatial frequency. We show that this CNN-based approach maintains a high degree of robustness even when only high-spatial frequencies are maintained. This renders the method particularly suitable for application to interferometric data lacking single-dish measurements. We applied this trained CNN to the synchrotron observations of a diffuse region. The CNN-predicted POS magnetic field orientation shows a statistical agreement with that derived from synchrotron polarization.

A high-order finite-difference solver for direct numerical simulations of magnetohydrodynamic turbulence

This paper presents the development and validation of a Magnetohydrodynamics (MHD) module integrated into the Xcompact3d framework, an open-source high-order finite-difference suite of solvers designed to study turbulent flows on supercomputers. Leveraging the Fast Fourier Transform library already implemented in Xcompact3d, alongside sixth-order compact finite-difference schemes and a direct spectral Poisson solver, both the induction and potential-based MHD equations can be efficiently solved at scale on CPU-based supercomputers for fluids with strong and weak magnetic field, respectively. Validation of the MHD solver is conducted against established benchmarks, including Orszag-Tang vortex and MHD channel flows, demonstrating the module's capability to accurately capture complex MHD phenomena, providing a powerful tool for research in both engineering and astrophysics. The scalability of the Xcompact3d framework remains intact with the incorporation of the MHD module, ensuring efficient performance on modern high-performance clusters. This paper also presents new findings on the evolution of the Taylor-Green vortex under an external magnetic field for different flow regimes.

Magnetohydrodynamic Turbulence Simulations as a Testing Ground for PUNCH

The Polarimeter to UNify the Corona and Heliosphere (PUNCH) will image macroscopic features of the inner heliosphere and also admit sufficiently high spatial resolution to probe scales of turbulence within the upper end of the inertial range, close to the integral scale. As PUNCH is an imager, its measurements will relate differently to the underlying turbulent environment of the outer corona and inner heliosphere from more familiar in situ samples. We present a numerical study that combines magnetohydrodynamic simulations of turbulence together with FORWARD-modeling synthesis of white-light data via the FORWARD code. We show that (i) the “usual” turbulence scalings are modified by the integration along the LOS in an optically thin medium, and (ii) those scalings are still linked to the original properties of the turbulent field. This study is a first step in the process of analyzing and understanding the unprecedented information that PUNCH will provide.

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

Nonlocal contributions to the turbulent cascade in magnetohydrodynamic plasmas.

We present evidence for nonlocal contributions to the turbulent energy cascade in magnetohydrodynamic (MHD) plasmas. Therefore, we revisit a well-known result derived directly from the MHD equations, i.e., the Politano and Pouquet law for the transfer of kinetic and magnetic energy in scale. We propose adding a term that accounts for nonlocal transfer and represents the influence of fluctuations from large scales due to the Alfvén effect. Supported by direct numerical simulations of homogeneous and isotropic MHD turbulence, we verify that in some plasma configurations, neglecting the additional nonlocal term might consistently overestimate energy dissipation rates and, thus, the contributions of turbulent energy dissipation potentially affecting solar wind heating, i.e., a central puzzle in space plasma physics that motivates the present work.

Energy transfer and third-order law in forced anisotropic magneto-hydrodynamic turbulence with hyper-viscosity

The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ( $B_0$ ), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$ . Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$ . With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.

Measurement of the scaling slope of compressible magnetohydrodynamic turbulence by synchrotron radiation statistics

ABSTRACT Based on magnetohydrodynamic (MHD) turbulence simulations, we generate synthetic synchrotron observations to explore the scaling slope of the underlying MHD turbulence. We propose the new Q-U cross-intensity X and cross-correlation intensity Y to measure the spectral properties of magnetic turbulence, together with statistics of the traditional synchrotron I and polarization PI intensities. By exploring the statistical behaviour of these diagnostics, we find that the new statistics X and Y can extend the inertial range of turbulence to improve measurement reliability. When focusing on different Alfvénic and sonic turbulence regimes, our results show that the diagnostics proposed in this paper not only reveal the spectral properties of the magnetic turbulence but also gain insight into the individual plasma modes of compressible MHD turbulence. The synergy of multiple statistical methods can extract more reliable turbulence information from the huge amount of observation data from the Low-Frequency Array for radio astronomy and the Square Kilometer Array.

As a Matter of Dynamical Range – Scale Dependent Energy Dynamics in MHD Turbulence

Magnetized turbulence is ubiquitous in many astrophysical and terrestrial plasmas but no universal theory exists. Even the detailed energy dynamics in magnetohydrodynamic (MHD) turbulence are still not well understood. We present a suite of subsonic, super-Alfvénic, high plasma beta MHD turbulence simulations that only vary in their dynamical range, i.e., in their separation between the large-scale forcing and dissipation scales, and their dissipation mechanism (implicit large eddy simulation, ILES, and direct numerical simulation (DNS)). Using an energy transfer analysis framework we calculate the effective numerical viscosities and resistivities, and demonstrate that all ILES calculations of MHD turbulence are resolved and correspond to an equivalent visco-resistive MHD turbulence calculation. Increasing the number of grid points used in an ILES corresponds to lowering the dissipation coefficients, i.e., larger (kinetic and magnetic) Reynolds numbers for a constant forcing scale. Independently, we use this same framework to demonstrate that—contrary to hydrodynamic turbulence—the cross-scale energy fluxes are not constant in MHD turbulence. This applies both to different mediators (such as cascade processes or magnetic tension) for a given dynamical range as well as to a dependence on the dynamical range itself, which determines the physical properties of the flow. We do not observe any indication of convergence even at the highest resolution (largest Reynolds numbers) simulation at 20483 cells, calling into question whether an asymptotic regime in MHD turbulence exists, and, if so, what it looks like.

Probing 3D magnetic fields – I. Polarized dust emission

ABSTRACT Polarized dust emission is widely used to trace the plane-of-the-sky (POS) component of interstellar magnetic fields in two dimensions. Its potential to access 3D magnetic fields, including the inclination angle of the magnetic fields relative to the line of sight (LOS), is crucial for a variety of astrophysical problems. Based on the statistical features of observed polarization fraction and POS Alfvén Mach number $\overline{M_{\rm A}}_{,\bot }$ distribution, we present a new method for estimating the inclination angle. The magnetic field fluctuations raised by anisotropic magnetohydrodynamic (MHD) turbulence are taken into account in our method. By using synthetic dust emission generated from 3D compressible MHD turbulence simulations, we show that the fluctuations are preferentially perpendicular to the mean magnetic field. We find the inclination angle is the major agent for depolarization, while fluctuations of magnetic field strength and density have an insignificant contribution. We propose and demonstrate that the mean inclination angle over a region of interest can be calculated from the polarization fraction in a strongly magnetized reference position, where $\overline{M_{\rm A}}_{,\bot }^2\ll 1$. We test and show that the new method can trace the 3D magnetic fields in sub-Alfvénic, trans-Alfvénic, and moderately superAlfvénic conditions (0.4 ≲ MA ≲ 1.2). We numerically quantify that the difference of the estimated inclination angle and actual inclination angle ranges from 0° to 20° with a median value of ≤10°.

Dust dynamics in ramses – I. Methods and turbulent acceleration

ABSTRACT Supernova ejecta and stellar winds are believed to produce interstellar dust grains with relatively large sizes. Smaller grains can be produced via the shattering of large grains that have been stochastically accelerated. To understand this stochastic acceleration, we have implemented novel magnetohydrodynamic (MHD) particle-in-cell methods into the astrophysical fluid code ramses. We treat dust grains as a set of massive ‘superparticles’ that experience aerodynamic drag and Lorentz force. We subject our code to a range of numerical tests designed to validate our method in different physical conditions, as well as to illustrate possible mechanisms by which grains can be accelerated. As a final test as well as a foundation for future work, we present the results of decaying dusty MHD turbulence simulations with grain parameters chosen to resemble 1–2 μm grains in typical cold neutral medium conditions. We find that in these conditions, these grains can be effectively accelerated to well beyond their shattering velocities. This is true for both electrically charged and neutral grains. While the peak of the gas-grain relative drift velocity distribution is higher for neutral grains, the drift velocity distribution of charged grains exhibits an extended exponential tail out to much greater velocities. Even so, the shapes of the distributions are such that the extra gas-grain coupling provided by the Lorentz force offers grains relative protection from shattering. We also discuss the connection between our simulations and the relatively pristine ∼μm-sized pre-solar grains that do not appear to have undergone significant wear in their lifetimes.

First-Principles Fermi Acceleration in Magnetized Turbulence.

This Letter provides a concrete implementation of Fermi's model of particle acceleration in magnetohydrodynamic (MHD) turbulence, connecting the rate of energization to the gradients of the velocity of magnetic field lines, which it characterizes within a multifractal picture of turbulence intermittency. It then derives a transport equation in momentum space for the distribution function. This description is shown to be substantiated by a large-scale numerical simulation of strong MHD turbulence. The present general framework can be used to model particle acceleration in a variety of environments.

Measurement of MHD Turbulence Properties by Synchrotron Radiation Techniques

It is well known that magnetohydrodynamic (MHD) turbulence is ubiquitous in astrophysical environments. The correct understanding of the fundamental properties of MHD turbulence is a pre-requisite for revealing many key astrophysical processes. The development of observation-based measurement techniques has significantly promoted MHD turbulence theory and its implications in astrophysics. After describing the modern understanding of MHD turbulence based on theoretical analysis and direct numerical simulations, we review recent developments related to synchrotron fluctuation techniques. Specifically, we comment on the validation of synchrotron fluctuation techniques and the measurement performance of several properties of magnetic turbulence based on data cubes from MHD turbulence simulations and observations. Furthermore, we propose to strengthen the studies of the magnetization and 3D magnetic field structure’s measurements of interstellar turbulence. At the same time, we also discuss the prospects of new techniques for measuring magnetic field properties and understanding astrophysical processes, using a large number of data cubes from the Low-Frequency Array (LOFAR) and the Square Kilometre Array (SKA).

Fundamental scales in the kinematic phase of the turbulent dynamo

ABSTRACT The turbulent dynamo is a powerful mechanism that converts turbulent kinetic energy to magnetic energy. A key question regarding the magnetic field amplification by turbulence, is, on what scale, kp, do magnetic fields become most concentrated? There has been some disagreement about whether kp is controlled by the viscous scale, kν (where turbulent kinetic energy dissipates), or the resistive scale, kη (where magnetic fields dissipate). Here, we use direct numerical simulations of magnetohydrodynamic turbulence to measure characteristic scales in the kinematic phase of the turbulent dynamo. We run 104-simulations with hydrodynamic Reynolds numbers of 10 ≤ Re ≤ 3600, and magnetic Reynolds numbers of 270 ≤ Rm ≤ 4000, to explore the dependence of kp on kν and kη. Using physically motivated models for the kinetic and magnetic energy spectra, we measure kν, kη, and kp, making sure that the obtained scales are numerically converged. We determine the overall dissipation scale relations $k_\nu = (0.025^{+0.005}_{-0.006})\, k_\text{turb}\, \mbox{Re}^{3/4}$ and $k_\eta = (0.88^{+0.21}_{-0.23})\, k_\nu \, \mbox{Pm}^{1/2}$, where kturb is the turbulence driving wavenumber and Pm = Rm/Re is the magnetic Prandtl number. We demonstrate that the principle dependence of kp is on kη. For plasmas, where Re ≳ 100, we find that $k_p= (1.2_{-0.2}^{+0.2})\, k_\eta$, with the proportionality constant related to the power-law ‘Kazantsev’ exponent of the magnetic power spectrum. Throughout this study, we find a dichotomy in the fundamental properties of the dynamo where Re > 100, compared to Re < 100. We report a minimum critical hydrodynamic Reynolds number, Recrit = 100 for bonafide turbulent dynamo action.

Analysis of energy transfer through direct numerical simulations of magnetohydrodynamic Orszag–Tang vortex

Direct numerical simulations of magnetohydrodynamic turbulence with Orszag–Tang vortex initial conditions are presented here for two different Reynolds numbers. Simulations of this transition-to-turbulence problem are carried out using a pseudo-spectral code with a grid resolution fine enough to resolve the Kolmogorov scale. The power-law behavior of bulk quantities like kinetic energy, magnetic energy, integral length scales, and enstrophy is analyzed. For a low Reynolds number, weak turbulence spectra are observed with a slope of −2.0 in the inertial subrange, while for a higher Reynolds number, the spectra exhibit a slope very close to the Kolmogorov spectra of −5/3. The Q-criteria isosurfaces at the time instant of maximum enstrophy are used for the visualization of turbulence developed within the flow. Furthermore, a new approach is presented to understand the dynamics of the transfer between kinetic and magnetic energies in the early period of development using the energy flux spectra, rate of change of kinetic and magnetic energies, and current density structures, which employ a three-dimensional representation of the alignment between velocity and magnetic field across the structures.

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence

ABSTRACT The rate of magnetic field diffusion plays an essential role in several astrophysical plasma processes. It has been demonstrated that the omnipresent turbulence in astrophysical media induces fast magnetic reconnection, which consequently leads to large-scale magnetic flux diffusion at a rate independent of the plasma microphysics. This process is called 'reconnection diffusion' (RD) and allows for the diffusion of fields, which are dynamically important. The current theory describing RD is based on incompressible magnetohydrodynamic (MHD) turbulence. In this work, we have tested quantitatively the predictions of the RD theory when magnetic forces are dominant in the turbulence dynamics (Alfvénic Mach number MA < 1). We employed the Pencil Code to perform numerical simulations of forced MHD turbulence, extracting the values of the diffusion coefficient ηRD using the test-field method. Our results are consistent with the RD theory ($\eta _{\rm RD} \sim M_{\rm A}^{3}$ for MA < 1) when turbulence approaches the incompressible limit (sonic Mach number MS ≲ 0.02), while for larger MS the diffusion is faster ($\eta _{\rm RD} \sim M_{\rm A}^{2}$). This work shows for the first time simulations of compressible MHD turbulence with the suppression of the cascade in the direction parallel to the mean magnetic field, which is consistent with incompressible weak turbulence theory. We also verified that in our simulations the energy cascading time does not follow the scaling with MA predicted for the weak regime, in contradiction with the RD theory assumption. Our results generally support and expand the RD theory predictions.

Numerical simulations of gravitational waves from early-universe turbulence

We perform direct numerical simulations of magnetohydrodynamic turbulence in the early universe and numerically compute the resulting stochastic background of gravitational waves and relic magnetic fields. These simulations do not make the simplifying assumptions of earlier analytic work. If the turbulence is assumed to have an energy-carrying scale that is about a hundredth of the Hubble radius at the time of generation, as expected in a first-order phase transition, the peak of gravitational wave power will be in the mHz frequency range for a signal produced at the electroweak scale. The efficiency of gravitational wave (GW) production varies significantly with how the turbulence is driven. Detectability of turbulence at the electroweak scale by the planned Laser Interferometer Space Antenna (LISA) requires anywhere from 0.1% to 10% of the thermal plasma energy density to be in plasma motions or magnetic fields, depending on the model of the driving process. Our results predict a new universal form below the spectral peak frequency that is shallower than previously thought. This implies larger values of the GW energy spectra in the low-frequency range. This extends the range where turbulence is detectable with LISA to lower frequencies, corresponding to higher energy scales than the assumed energy-carrying scale.

Planck2018 results

Observations of the submillimetre emission from Galactic dust, in both total intensityIand polarization, have received tremendous interest thanks to thePlanckfull-sky maps. In this paper we make use of such full-sky maps of dust polarized emission produced from the third public release ofPlanckdata. As the basis for expanding on astrophysical studies of the polarized thermal emission from Galactic dust, we present full-sky maps of the dust polarization fractionp, polarization angleψ, and dispersion function of polarization angles 𝒮. The joint distribution (one-point statistics) ofpandNHconfirms that the mean and maximum polarization fractions decrease with increasingNH. The uncertainty on the maximum observed polarization fraction,pmax= 22.0−1.4+3.5% at 353 GHz and 80′ resolution, is dominated by the uncertainty on the Galactic emission zero level in total intensity, in particular towards diffuse lines of sight at high Galactic latitudes. Furthermore, the inverse behaviour betweenpand 𝒮 found earlier is seen to be present at high latitudes. This follows the 𝒮 ∝ p−1relationship expected from models of the polarized sky (including numerical simulations of magnetohydrodynamical turbulence) that include effects from only the topology of the turbulent magnetic field, but otherwise have uniform alignment and dust properties. Thus, the statistical properties ofp,ψ, and 𝒮 for the most part reflect the structure of the Galactic magnetic field. Nevertheless, we search for potential signatures of varying grain alignment and dust properties. First, we analyse the product map 𝒮 × p, looking for residual trends. While the polarization fractionpdecreases by a factor of 3−4 betweenNH = 1020 cm−2andNH = 2 × 1022 cm−2, out of the Galactic plane, this product 𝒮 × ponly decreases by about 25%. Because 𝒮 is independent of the grain alignment efficiency, this demonstrates that the systematic decrease inpwithNHis determined mostly by the magnetic-field structure and not by a drop in grain alignment. This systematic trend is observed both in the diffuse interstellar medium (ISM) and in molecular clouds of the Gould Belt. Second, we look for a dependence of polarization properties on the dust temperature, as we would expect from the radiative alignment torque (RAT) theory. We find no systematic trend of 𝒮 × pwith the dust temperatureTd, whether in the diffuse ISM or in the molecular clouds of the Gould Belt. In the diffuse ISM, lines of sight with high polarization fractionpand low polarization angle dispersion 𝒮 tend, on the contrary, to have colder dust than lines of sight with lowpand high 𝒮. We also compare thePlanckthermal dust polarization with starlight polarization data in the visible at high Galactic latitudes. The agreement in polarization angles is remarkable, and is consistent with what we expect from the noise and the observed dispersion of polarization angles in the visible on the scale of thePlanckbeam. The two polarization emission-to-extinction ratios,RP/pandRS/V, which primarily characterize dust optical properties, have only a weak dependence on the column density, and converge towards the values previously determined for translucent lines of sight. We also determine an upper limit for the polarization fraction in extinction,pV/E(B − V), of 13% at high Galactic latitude, compatible with the polarization fractionp ≈ 20% observed at 353 GHz. Taken together, these results provide strong constraints for models of Galactic dust in diffuse gas.

Turbulent Rayleigh–Bénard convection in a strong vertical magnetic field

Direct numerical simulations are carried out to study flow structure and transport properties in turbulent Rayleigh-B\'{e}nard convection in a cylindrical cell of aspect ratio one with an imposed axial magnetic field. Flows at the Prandtl number 0.025 and the Rayleigh and Hartmann numbers up to $10^9$ and 1400 are considered. The results are consistent with those of earlier experimental and numerical data. As anticipated, the heat transfer rate and kinetic energy are suppressed by strong magnetic field. At the same time, their growth with the Rayleigh number is found to be faster in flows at high Hartmann numbers. This behaviour is attributed to the newly discovered flow regime characterized by prominent quasi two-dimensional structures reminiscent of vortex sheets observed earlier in simulations of magnetohydrodynamic turbulence. Rotating wall modes similar to those in the Rayleigh-B\'{e}nard convection with rotation are found in flows near the Chandrasekhar linear stability limit. Detailed analysis of the spatial structure of the flows and its effect of global transport properties is reported.

Turbulent drag reduction in magnetohydrodynamic and quasi-static magnetohydrodynamic turbulence

In hydrodynamic turbulence, the kinetic energy injected at large scales cascades to the inertial range, leading to a constant kinetic energy flux. In contrast, in magnetohydrodynamic (MHD) turbulence, a fraction of kinetic energy is transferred to the magnetic energy. Consequently, for the same kinetic energy injection rate, the kinetic energy flux in MHD turbulence is reduced compared to its hydrodynamic counterpart. This leads to relative weakening of the nonlinear term ⟨|(u·∇)u|⟩, (where u is the velocity field) and turbulent drag, but strengthening of the velocity field in MHD turbulence. We verify the above using shell model simulations of hydrodynamic and MHD turbulence. Quasi-static MHD turbulence too exhibits turbulent drag reduction similar to MHD turbulence.

Hall magnetohydrodynamic turbulence with a magnetic Prandtl number larger than unity

Turbulence structures with the magnetic Prandtl number larger than unity are studied by means of direct numerical simulations of homogeneous, isotropic, and incompressible Hall magnetohydrodynamic (MHD) turbulence driven by a random force. Spectral and spatial structures on the scales smaller than the ion skin depth are focused upon in this numerical paper. The numerical simulations reveal the emergence of a new power law in the velocity field whereas it is not observed in MHD turbulence simulation without the Hall term. An order estimate of the energy budget in the spectral space shows that this new power law appears in association with the Hall effect and that a balance between the Lorentz force and the viscous dissipation is crucial for formation of the power law. A resemblance to an elastic turbulence is found in the power-law-formation mechanism. Frequent eruptions of strong current ribbons accompanying strong palinstrophy density are observed, showing generation of the palinstrophy density by the Lorentz force at the scales below the ion skin depth. These properties in spectral and spatial structures characterize a high magnetic Prandtl number Hall MHD turbulence at the scales smaller than the ion skin depth.