Hall Magnetohydrodynamic Turbulence Research Articles

Fundamental units of triadic interactions in Hall magnetohydrodynamic turbulence: How far can we go?

A systematic study has been carried out to obtain the fundamental units of triad interaction in Hall magnetohydrodynamic turbulence. Instead of finding the elementary building blocks of nonunique mode-to-mode (M2M) transfer rates, we have investigated the fundamental units for uniquely defined combined transfers and convincingly showed that the mode-to-mode transfers can act as a practical base element for the same. In addition to the conventional field-specific mode-to-mode transfers, here we have introduced the idea of mode-specific transfers, which is found to be important for the turbulent cascade and the turbulent relaxed states. Whereas the Hall transfer is found to associate mode-to-mode transfers for mode-specific interactions (with a three-member basis), it presents a mixture of typical mode-to-mode (also with a three-member basis) and non-mode-to-mode (with a five-member basis) transfers for the field-specific interactions. The non-mode-to-mode transfers are shown to satisfy the triad conservation differently from the mode-to-mode transfers. However, they also possess an inherent non-uniqueness and hence cannot be determined unambiguously unlike the combined transfer rates.

Contribution of the Hall term in small-scale magnetohydrodynamic dynamos

The Hall effect plays an interesting role in the context of astrophysical dynamos. Using direct numerical simulation, the dynamo action in Hall magnetohydrodynamic turbulence has been systematically studied. Unlike previous studies, the total Hall contribution has been decomposed, separating the contribution of magnetic and current fields in feeding the large-scale magnetic field. Using scale-specific flux relations, it is shown that the small-scale magnetic and current fields contribute to the large-scale magnetic field growth. Unlike small-scale magnetic fields and large-scale current fields, the small-scale current fields appear to be the leading contributors to small-scale dynamo action.

Local energy transfer and dissipation in incompressible Hall magnetohydrodynamic turbulence: The coarse-graining approach.

We derive the coarse-graining (CG) equationsof incompressible Hall magnetohydrodynamic (HMHD) turbulence to investigate the local (in space) energy transfer rate as a function of the filtering scale ℓ. First, the CG equationsare space averaged to obtain the analytical expression of the mean cascade rate. Its application to three-dimensional simulations of (weakly compressible) HMHD shows a cascade rate consistent with the value of the mean dissipation rate in the simulations and with the classical estimates based on the "third-order" law. Furthermore, we developed an anisotropic version of CG that allows us to study the magnitude of the cascade rate along different directions with respect to the mean magnetic field. Its implementation on the numerical data with moderate background magnetic field shows a weaker cascade along the magnetic field than in the perpendicular plane, while an isotropic cascade is recovered in the absence of a background field. The strength of the CG approach is further revealed when considering the local-in-space energy transfer, which is shown theoretically and numerically to match at a given position x, when locally averaged over a neighboring region, the (quasi-)local dissipation. Prospects of exploiting this model to investigate local dissipation in spacecraft data are discussed.

Statistical properties of three-dimensional Hall magnetohydrodynamics turbulence

The three-dimensional (3D) Hall magnetohydrodynamics (HMHD) equations are often used to study turbulence in the solar wind. Some earlier studies have investigated the statistical properties of 3D HMHD turbulence by using simple shell models or pseudospectral direct numerical simulations (DNSs) of the 3D HMHD equations; these DNSs have been restricted to modest spatial resolutions and have covered a limited parameter range. To explore the dependence of 3D HMHD turbulence on the Reynolds number Re and the ion-inertial scale di, we have carried out detailed pseudospectral DNSs of the 3D HMHD equations and their counterparts for 3D MHD (di = 0). We present several statistical properties of 3D HMHD turbulence, which we compare with 3D MHD turbulence by calculating (a) the temporal evolution of the energy-dissipation rates and the energy; (b) the wave-number dependence of fluid and magnetic spectra; (c) the probability distribution functions of the cosines of the angles between various pairs of vectors, such as the velocity and the magnetic field; and (d) various measures of the intermittency in 3D HMHD and 3D MHD turbulence.

Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis

We present results of a multiscale study of Hall-magnetohydrodynamic (MHD) turbulence, carried out on a dataset of compressible nonlinear 2D Hall-MHD numerical simulations of decaying Alfvénic turbulence. For the first time, we identify two distinct regimes of fully developed turbulence. In the first one, the power spectrum of the turbulent magnetic fluctuations at sub-ion scales exhibits a power law with a slope of ∼−2.9, typically observed both in solar wind and in magnetosheath turbulence. The second regime, instead, shows a slope of −7/3, in agreement with classical theoretical models of Hall-MHD turbulence. A spectral-transfer analysis reveals that the latter regime occurs when the energy transfer rate at sub-ion scales is dominated by the Hall term, whereas in the former regime, the governing process is the dissipation (and the system exhibits large intermittency). Results of this work are relevant to the space plasma community, as they may potentially reconcile predictions from theoretical models with results from numerical simulations and spacecraft observations.

Fluid Energy Cascade Rate and Kinetic Damping: New Insight from 3D Landau-fluid Simulations

Abstract Using an exact law for incompressible Hall magnetohydrodynamics (HMHD) turbulence, the energy cascade rate is computed from three-dimensional HMHD-CGL (biadiabatic ions and isothermal electrons) and Landau-fluid numerical simulations that feature different intensities of Landau damping over a broad range of wavenumbers, typically 0.05 ≲ k ⊥ d i ≲ 100. Using three sets of cross-scale simulations where turbulence is initiated at large, medium, and small scales, the ability of the fluid energy cascade to “sense” the kinetic Landau damping at different scales is tested. The cascade rate estimated from the exact law and the dissipation calculated directly from the simulation are shown to reflect the role of Landau damping in dissipating energy at all scales, with an emphasis on the kinetic ones. This result provides new prospects on using exact laws for simplified fluid models to analyze dissipation in kinetic simulations and spacecraft observations, and new insights into theoretical description of collisionless magnetized plasmas.

Sub-grid-scale model for studying Hall effects on macroscopic aspects of magnetohydrodynamic turbulence

A new sub-grid-scale model is developed for studying influences of the Hall term on macroscopic aspects of magnetohydrodynamic turbulence. Although the Hall term makes numerical simulations extremely expensive by exciting high-wave-number coefficients and makes magnetohydrodynamic equations stiff, studying macroscopic aspects of magnetohydrodynamic turbulence together with the Hall term is meaningful since this term often influences not only sub-ion-scales but also macroscopic scales. Our new sub-ion-scale sub-grid-scale model for large eddy simulations of Hall magnetohydrodynamic turbulence is developed in order to overcome the difficulties. Large eddy simulations by the use of the new model successfully reproduce statistical signatures such as the energies and probability density functions of the vorticity and current density, keeping some signatures intrinsic to Hall magnetohydrodynamic turbulence. Our new sub-grid-scale model enables numerical simulations of homogeneous and isotropic Hall magnetohydrodynamic turbulence with a small computational cost, improving the effective resolution of an LES from that carried out with earlier models, and retaining the ion-electron separation effects by the Hall term in the grid scales.

Spectral Transfer and Kármán–Howarth–Monin Equations for Compressible Hall Magnetohydrodynamics

Abstract We derive two new forms of the Kármán–Howarth–Monin (KHM) equation for decaying compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results of a weakly compressible, 2D, moderate-Reynolds-number Hall MHD simulation and compare them with an isotropic spectral transfer (ST) equation. The KHM and ST equations are automatically satisfied during the whole simulation owing to the periodic boundary conditions and have complementary cumulative behavior. They are used here to analyze the onset of turbulence and its properties when it is fully developed. These approaches give equivalent results characterizing the decay of the kinetic + magnetic energy at large scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. The Hall cascade appears when the MHD one brings the energy close to the ion inertial range and is related to the formation of reconnecting current sheets. At later times, the pressure dilatation energy exchange rate oscillates around zero, with no net effect on the cross-scale energy transfer when averaged over a period of its oscillations. A reduced 1D analysis suggests that all three methods may be useful to estimate the energy cascade rate from in situ observations.

Spacetime Hall-MHD Turbulence at Sub-ion Scales: Structures or Waves?

Abstract Spatiotemporal properties of two-dimensional (2D) Hall-magnetohydrodynamic (MHD) turbulence at intermediate plasma β = 2 are studied by means of Fast Iterative Filtering (FIF), a new technique for the decomposition of nonstationary nonlinear signals. Results show that the magnetic energy at sub-ion scales is concentrated in perturbations with frequencies smaller than the ion-cyclotron (IC) frequency and with polarization properties that are incompatible with both kinetic Alfvén waves (KAWs) and IC waves. At higher frequencies, we clearly identify signatures of both whistler waves and KAWs; however, their energetic contribution to the magnetic power spectrum is negligible. We conclude that the dynamics of 2D Hall-MHD turbulence at sub-ion scales is mainly driven by localized intermittent structures, with no significant contribution of wavelike fluctuations.

A compact exact law for compressible isothermal Hall magnetohydrodynamic turbulence

Using mixed second-order structure functions, a compact exact law is derived for isothermal compressible Hall magnetohydrodynamic turbulence with the assumptions of statistical homogeneity, time stationarity and infinite kinetic/magnetic Reynolds numbers. The resulting law is written as the sum of a Yaglom-like flux term, with an overall expression strongly reminiscent of the incompressible law, and a pure compressible source. Being mainly a function of the increments, the compact law is Galilean invariant but is dependent on the background magnetic field if one is present. Only the magnetohydrodynamic source term requires multi-spacecraft data to be estimated whereas the other components, which include those introduced by the Hall term, can be fully computed with single-spacecraft data using the Taylor hypothesis. These properties make this compact law more appropriate for analysing both numerical simulations and in situ data gathered in space plasmas, in particular when only single-spacecraft data are available.

Asymmetry of Quadratic Energy Transfer Between Ion Cyclotron and Whistler Modes in Fully Developed Hall Magnetohydrodynamic Turbulence

Asymmetry of Quadratic Energy Transfer Between Ion Cyclotron and Whistler Modes in Fully Developed Hall Magnetohydrodynamic 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.

Energy Cascade Rate Measured in a Collisionless Space Plasma with MMS Data and Compressible Hall Magnetohydrodynamic Turbulence Theory.

The first complete estimation of the compressible energy cascade rate |ϵ_{C}| at magnetohydrodynamic (MHD) and subion scales is obtained in Earth's magnetosheath using Magnetospheric MultiScale spacecraft data and an exact law derived recently for compressible Hall MHD turbulence. A multispacecraft technique is used to compute the velocity and magnetic gradients, and then all the correlation functions involved in the exact relation. It is shown that when the density fluctuations are relatively small, |ϵ_{C}| identifies well with its incompressible analog |ϵ_{I}| at MHD scales but becomes much larger than |ϵ_{I}| at subion scales. For larger density fluctuations, |ϵ_{C}| is larger than |ϵ_{I}| at every scale with a value significantly higher than for smaller density fluctuations. Our study reveals also that for both small and large density fluctuations, the nonflux terms remain always negligible with respect to the flux terms and that the major contribution to |ϵ_{C}| at subion scales comes from the compressible Hall flux.

On Exact Laws in Incompressible Hall Magnetohydrodynamic Turbulence

Abstract A comparison is made between several existing exact laws in incompressible Hall magnetohydrodynamic turbulence in order to show their equivalence, despite stemming from different mathematical derivations. Using statistical homogeneity, we revisit the law proposed by Hellinger et al. and show that it can be written, after being corrected by a multiplicative factor, in a more compact form implying only flux terms expressed as increments of the turbulent fields. The Hall contribution of this law is tested and compared to other exact laws derived by Galtier and Banerjee & Galtier using direct numerical simulations of three-dimensional electron MHD turbulence with a moderate mean magnetic field. We show that the studied laws are equivalent in the inertial range, thereby offering several choices on the formulation to use depending on the needs. The expressions that depend explicitly on a mean (guide) field may lead to residual errors in estimating the energy cascade rate; however, we demonstrate that this guide field can be removed from these laws after mathematical manipulation. Therefore, it is recommended to use an expression independent of the mean guide field to analyze numerical or in situ spacecraft data.

Coexistence of Weak and Strong Wave Turbulence in Incompressible Hall Magnetohydrodynamics

We report a numerical investigation of three dimensional, incompressible, Hall magnetohydrodynamic turbulence with a relatively strong mean magnetic field. Using helicity decomposition and cross-bicoherence analysis, we observe that the resonant three--wave coupling is substantial among ion cyclotron and whistler waves. A detailed study of the degree of non-linearity of these two populations shows that the ion cyclotron component experiences a transition from weak to strong wave turbulence going from large to small scales, while the whistler fluctuations display a weak wave turbulence character for all scales. This non-trivial coexistence of the two regimes with the two populations of waves gives rise to anomalous anisotropy and scaling properties. The weak and strong wave turbulence components can be distinguished rather efficiently using spatio-temporal Fourier transforms. The analysis shows that while resonant triadic interactions survive the highly non-linear bath of ion cyclotron fluctuations at large scales for which the degree of non-linearity is low for both populations of waves, whistler waves tend to be killed by the non-linear cross-coupling at smaller scales where the ion cyclotron component is in the strong wave turbulent regime. Such situation may have far-reaching implications for the physics of magnetized turbulence in many astrophysical and space plasmas where different waves coexist and compete to transfer non-linearly energy across scales.

Von Kármán–Howarth Equation for Hall Magnetohydrodynamics: Hybrid Simulations

Abstract A dynamical vectorial equation for homogeneous incompressible Hall-magnetohydrodynamic (MHD) turbulence together with the exact scaling law for third-order correlation tensors, analogous to that for the incompressible MHD, is rederived and applied to the results of two-dimensional hybrid simulations of plasma turbulence. At large (MHD) scales the simulations exhibit a clear inertial range where the MHD dynamic law is valid. In the sub-ion range the cascade continues via the Hall term, but the dynamic law derived in the framework of incompressible Hall-MHD equations is obtained only in a low plasma beta simulation. For a higher beta plasma the cascade rate decreases in the sub-ion range and the change becomes more pronounced as the plasma beta increases. This break in the cascade flux can be ascribed to nonthermal (kinetic) features or to others terms in the dynamical equation that are not included in the Hall-MHD incompressible approximation.

On the origin of the energy dissipation anomaly in (Hall) magnetohydrodynamics

Incompressible Hall magnetohydrodynamics (MHD) may be the subject of energy dissipation anomaly which stems from the lack of smoothness of the velocity and magnetic fields. I derive the exact expression of which appears to be closely connected with the well-known 4/3 exact law of Hall MHD turbulence theory. This remarkable similitude suggests a deeper mathematical property of the fluid equations. In the MHD limit, the expression of differs from the one derived by Gao et al (2013 Acta Math. Sci. 33 865–71) which presents miscalculations. The energy dissipation anomaly can be used to better estimate the local heating in space plasmas where in situ measurements are accessible.

Exact law for homogeneous compressible Hall magnetohydrodynamics turbulence.

We derive an exact law for three-dimensional (3D) homogeneous compressible isothermal Hall magnetohydrodynamic turbulence, without the assumption of isotropy. The Hall current is shown to introduce new flux and source terms that act at the small scales (comparable or smaller than the ion skin depth) to significantly impact the turbulence dynamics. The law provides an accurate means to estimate the energy cascade rate over a broad range of scales covering the magnetohydrodynamic inertial range and the sub-ion dispersive range in 3D numerical simulations and in in situ spacecraft observations of compressible turbulence. This work is particularly relevant to astrophysical flows in which small-scale density fluctuations cannot be ignored such as the solar wind, planetary magnetospheres, and the interstellar medium.

Helical Mode Absolute Statistical Equilibrium of Ideal Three-Dimensional Hall Magnetohydrodynamics**Supported by the National Natural Science Foundation of China under Grant Nos 11375190 and 11547137.

Using the Fourier helical decomposition, we obtain the absolute statistical equilibrium spectra of left- and right-handed helical modes in the incompressible ideal Hall magnetohydrodynamics (MHD). It is shown that the left-handed helical modes play a major role on the spectral transfer properties of turbulence when the generalized helicity and magnetic helicity are both positive. In contrast, the right-handed helical modes will play a major role when both are negative. Furthermore, we also find that if the generalized helicity and magnetic helicity have opposite signs, the tendency of equilibrium spectra to condense at the large or small wave numbers will be presented in different helical sectors. This indicates that the generalized helicity dominates the forward cascade and the magnetic helicity dominates the inverse cascade properties of the Hall MHD turbulence.

An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence

We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity , vorticity , magnetic field and the current , the vectors , and are also found to play a key role in the turbulent cascades. The current methodology offers a simple algebraic form which is specially interesting to study anisotropic space plasmas like the solar wind, with, a faster statistical convergence than the classical laws written in terms of third-order correlators.