Hall-magnetohydrodynamic Equations Research Articles

Slow shock and rotational discontinuity in MHD and Hall MHD models with anisotropic pressure

AbstractPressure anisotropy may modify the characteristics of magnetohydrodynamic (MHD) waves, in particular, the slow mode wave and the corresponding shocks and discontinuities. In this study the formation of slow shocks (SSs) in anisotropic plasmas is examined by solving the gyrotropic MHD and Hall MHD equations numerically for one‐dimensional Riemann problem. The MHD shocks and discontinuities are generated by imposing a finite normal magnetic field on the Harris type current sheet with a guide magnetic By component. It is shown that anomalous SSs moving faster than the intermediate wave or with positive density‐magnetic field correlation may be generated in gyrotropic MHD and Hall MHD models. Moreover, for some parameter values SSs may exhibit upstream wave trains with right‐handed polarization in contrast with the earlier prediction that SSs shall possess downstream left‐hand polarized wave trains based on the isotropic Hall MHD theory. For the cases of By ≠ 0, SSs with increased density and decreased magnetic field followed by noncoplanar intermediate mode or rotational discontinuity (RD)‐like structures similar to the compound SS‐RD structures observed in space plasma environments may possibly form in symmetric and asymmetric current layers. The Walén relation of these anomalous RDs without the correction of pressure anisotropy may significantly be violated.

Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations

In this paper, we study the low Mach number limit of the compressible Hall-magnetohydrodynamic equations. It is justified rigorously that, for the well-prepared initial data, the classical solutions of the compressible Hall-magnetohydrodynamic equations converge to that of the incompressible Hall-magnetohydrodynamic equations as the Mach number tends to zero.

On Partial Regularity for the 3D Nonstationary Hall Magnetohydrodynamics Equations on the Plane

We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two.

The blow-up criterion via horizontal component of velocity for the Hall-MHD equations

In this paper, we study the blow-up criterion of the smooth solutions to the three-dimensional incompressible resistive viscous Hall-magnetohydrodynamic equations in terms of and or and in Lebesgue spaces or in Besov space and in space, where denotes . We also obtain a priori estimate of velocity and magnetic field in the space of on .

Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion

The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-\Delta)^\alpha$. Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $\alpha\ge 1$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $\alpha>\frac12$. The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.

On Partial Regularity for the Steady Hall Magnetohydrodynamics System

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $\Omega \subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $\Omega=\Bbb R^3$, we show that the set of possible singularities is compact.

On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics with horizontal dissipation

In this paper, we investigate the Cauchy problem for the incompressible resistive Hall-magnetohydrodynamic equations with horizontal dissipation. By using very delicate energy estimate, we establish a blow-up criterion for the classical solutions and show the global-in-time existence of the classical solution for small initial data.

Exact solutions of the incompressible dissipative Hall magnetohydrodynamics

By using analytical method, the exact solutions of the incompressible dissipative Hall magnetohydrodynamics (MHD) equations are derived. It is found that a phase difference may occur between the velocity and magnetic field fluctuations when the kinetic and magnetic Reynolds numbers are both very large. Since velocity and magnetic field fluctuations are both circular polarized, the phase difference makes them no longer parallel or anti-parallel like that in the incompressible ideal Hall MHD.

Regularity criteria for the incompressible Hall-magnetohydrodynamic equations

We establish some new regularity criteria for the three-dimensional incompressible Hall-magnetohydrodynamic equations.

A fast tearing mode instability driven by agyrotropic electron pressure

The collisionless plasma environment at the current sheet of the Earth’s magnetotail is subjected to fast dynamic evolutions such as tearing instability. By considering agyrotropic pressure for electron and ion components of a collisionless plasma, we analytically investigate the dynamics of tearing mode instability, in which, breaking the frozen-in condition can either be provided by the electron inertia or by agyrotropic electron pressure. A set of linearized Hall-Magnetohydrodynamic (MHD) equations describes the evolution of tearing mode in a sheared force-free field. The presented scaling analysis shows that if the plasma-β exceeds a specified value, then the main mechanism of magnetic reconnection process is the nongyrotropic electron pressure. In this regime, the role played by agyrotropic ion pressure inside the reconnection layer is out of significance. Therefore, the electron-MHD framework, adequately, describes the dynamics of tearing instability with a growth rate which is much faster compared to the cases with a dominated bulk inertia or a gyrotropic plasma pressure.

On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics

In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [4], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of B˙2,1s norm is essential for this result. Contrary to the usual MHD the global well-posedness in the 212 dimensional Hall-MHD is wide open.

On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations

In this paper we deal with weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall-Magneto-Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.

Multiscaling in Hall-Magnetohydrodynamic Turbulence: Insights from a Shell Model

We show that a shell-model version of the three-dimensional Hall-magnetohydrodynamic (3D Hall-MHD) equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-k and high-k power-law ranges of three-dimensional Hall-magnetohydrodynamic, and find that the extended-self-similarity procedure is helpful in extracting the multiscaling nature of structure functions in the high-k regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solar-wind measurements.

On the temporal decay for the Hall-magnetohydrodynamic equations

We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions.

A simulation study of Io-related Jovian decametric radiation: Control factor of occurrence probability

[1] Io-Jupiter interaction leads to auroral emissions and Jovian decametric radiations (Io-DAM). The longitudinal distribution of the Io-DAM occurrence probability has been considered to be controlled by the footprint magnetic intensity of each hemisphere. Recent observations revealed that the brightness of the main auroral spot is mainly modulated by Io's magnetic latitude. In the present study, we propose that the Io-DAM occurrence probability is controlled by the north-south asymmetry of the footprint magnetic intensity, in addition to Io's magnetic latitude. Hall magnetohydrodynamic equations are solved in the corotating meridional plane, which includes the Jovian ionosphere of finite thickness. We assume that the ionospheric Pedersen conductance is inversely proportional to the footprint magnetic intensity; the conductance is nearly the same for both ionospheres at a longitude of 290°, while twice higher for the south than for the north at a longitude of 110°. Above the northern ionosphere, the parallel current density further than 20° downstream of the main spot is estimated to be 1.5–2.0 times larger for 290° than for 110°. This indicates that if the Io-DAM lead angle is large, the suppressed Io-DAM occurrence probability for the northern hemisphere around 110° would be caused by the north-south asymmetry of the footprint magnetic intensity. A large current density conducted into the south around 110° would be the source of the Io-D component, radiated from the southern hemisphere. The observed brightness of the Io-related auroras is discussed in the context of the intensity of the parallel current density.

Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations

We establish the global well-posedness of the axisymmetric solutions to the incompressible viscous Hall-magnetohydrodynamic equations.

Spontaneous Chiral Symmetry Breaking of Hall Magnetohydrodynamic Turbulence

Hall magnetohydrodynamics (MHD) is investigated through three-dimensional direct numerical simulations. We show that the Hall effect induces a spontaneous chiral symmetry breaking of the turbulent dynamics. The normalized magnetic polarization is introduced to separate the right- (R) and left-handed (L) fluctuations. A classical k(-7/3) spectrum is found at small scales for R magnetic fluctuations which corresponds to the electron MHD prediction. A spectrum compatible with k(-11/3) is obtained at large-scales for the L magnetic fluctuations; we call this regime the ion MHD. These results are explained heuristically by rewriting the Hall MHD equations in a succinct vortex dynamical form. Applications to solar wind turbulence are discussed.

Hall magnetohydrodynamic reconnection in the plasmoid unstable regime

A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth dη≡SL-1/2LνA/γ, where SL is the Lundquist number, L, the length of the current sheet, νA, the Alfvén speed, and γ, the growth rate. Modifications to a recent resistive MHD analysis [N. F. Loureiro et al., Phys. Plasmas 14, 100703 (2007)] arise when collisions are sufficiently weak that dη is shorter than the ion skin depth di ≡ c/ωpi. Secondary islands grow faster in this Hall MHD regime: the maximum growth rate scales as (di/L)6/13SL7/13νA/L and the number of plasmoids as (di/L)1/13SL11/26, compared to SL1/4νA/L and S3/8, respectively, in resistive MHD.

Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.

Hall effect in a strong magnetic field: Direct comparisons of compressible magnetohydrodynamics and the reduced Hall magnetohydrodynamic equations

In this work we numerically test a model of Hall magnetohydrodynamics in the presence of a strong mean magnetic field: the reduced Hall magnetohydrodynamic model (RHMHD) derived by [Gomez et al., Phys. Plasmas 15, 102303 (2008)] with the addition of weak compressible effects. The main advantage of this model lies in the reduction of computational cost. Nevertheless, up until now the degree of agreement with the original Hall MHD system and the range of validity in a regime of turbulence were not established. In this work direct numerical simulations of three-dimensional Hall MHD turbulence in the presence of a strong mean magnetic field are compared with simulations of the weak compressible RHMHD model. The results show that the degree of agreement is very high (when the different assumptions of RHMHD, such as spectral anisotropy, are satisfied). Nevertheless, when the initial conditions are isotropic but the mean magnetic field is maintained strong, the results differ at the beginning but asymptotically reach a good agreement at relatively short times. We also found evidence that the compressibility still plays a role in the dynamics of these systems, and the weak compressible RHMHD model is able to capture these effects. In conclusion the weak compressible RHMHD model is a valid approximation of the Hall MHD turbulence in the relevant physical context.