Hall-magnetohydrodynamic Equations Research Articles

The two-and-a-half-dimensional incompressible Hall-MHD system with partial dissipation

In this paper, we investigate the Cauchy problem for the two-and-a-half-dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. We prove the global regularity of a certain system of equations with horizontal or vertical dissipation and magnetic hyper-diffusion for large initial data.

Large time behavior for the Hall-MHD equations with horizontal dissipation

This paper examines the large time behavior of solutions to the 3D Hall-magnetohydrodynamic equations with horizontal dissipation. As preparations we establish the global well-posedness of solutions and their global explicitly uniform upper bounds for Hk (k ≥ 1) to this system with initial data small in H2. Furthermore, if the initial data also belongs to homogeneous negative Besov spaces, we prove the optimal decay rates of the aforementioned global solutions and their higher order derivatives.

New Liouville type theorems for the stationary Navier–Stokes, MHD, and Hall–MHD equations

We establish new Liouville-type theorems for weak solutions of the stationary Navier–Stokes equations, stationary magnetohydrodynamics (MHD) equations and stationary Hall–MHD equations under some conditions on the growth of certain Lebesgue norms of the velocity and the magnetic field.

Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations

We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.

Effects of magnetic field direction on [formula omitted] vertical transport in the Martian ionosphere

The directions of Martian magnetic field, which depends on the interaction between the crustal magnetic field and the time-various interplanetary magnetic field (IMF), is in a state of disorder and plays a significant role in ions vertical transport, such as ions upwelling and precipitation. In general, vertical diffusion transport of ionospheric plasma tends to be promoted around vertical magnetic field regions but be suppressed around horizontal magnetic field area. However, the mechanism behind this phenomenon is not clear to date. Based on three-dimensional multi-fluid Hall magneto-hydrodynamic (MHD) equations in conjunction with an equivalent source dipole (ESD) model, we investigated mechanism behind the control of vertical transport by different magnetic field directions in the Martian ionosphere. Numerical results showed that the same direction of interplanetary magnetic field with ESD results in the formation of mini-magnetosphere to protect the ionosphere, which reflected in higher density distributions comparing to the no-ESD case due to joint effects of thermal pressure increasing and plasma vertical transport. At low altitude of ionosphere, O+ ion inward transport (precipitation) tends to be inhibited particularly around horizontal magnetic field, with high contribution from the motional electric force. In addition, at high altitude, O+ outward transport (upwelling) is likely to be facilitated over vertical-field areas in significant part controlled by the ambipolar electric force, whereas it is likely to be inhibited in horizontal-field region, contributed mainly from the Hall electric force. These results will enrich our understanding of mechanisms behind the control of ions vertical transport by magnetic field directions.

Four-thirds law of energy and magnetic helicity in electron and Hall magnetohydrodynamic fluids

In this paper, by exploiting the feature of the Hall term, we establish some local version four-thirds laws for the dissipation rates of energy and magnetic helicity in both the inviscid electron and Hall magnetohydrodynamic equations in the sense of Duchon–Robert type. Four-thirds law involving the dissipation rates of energy for the Hall magnetohydrodynamic equations generalizes corresponding the work of Galtier and new 4/3 laws for the dissipation rates of magnetic helicity in these systems are first observed. To this end, we derive local Kármán–Howarth–Monin type equations of the energy and magnetic helicity in these equations.

Extending the Spectral Difference Method with Divergence Cleaning (SDDC) to the Hall MHD Equations

The Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral difference scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral difference method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature.

Martingale solutions of the stochastic Hall-magnetohydrodynamics equations on [formula omitted

We prove the existence of a global martingale solution of stochastic Hall-magnetohydrodynamics equations on R3 with multiplicative noise. Using the Fourier analysis we construct a sequence of approximate solutions. The existence of a solution is proved via the stochastic compactness method and the Jakubowski generalization of the Skorokhod theorem for nonmetric spaces, in particular, the spaces with weak topologies. The main difficulty is caused by the Hall term which makes the equations strongly nonlinear.

Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain

<abstract><p>This paper focuses on the energy equality for weak solutions of the nonhomogeneous incompressible Hall-magnetohydrodynamics equations in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n\geqslant2) $. By exploiting the special structure of the nonlinear terms and using the coarea formula, we obtain some sufficient conditions for the regularity of weak solutions to ensure that the energy equality is valid. For the special case $ n = 3 $, $ p = q = 2 $, our results are consistent with the corresponding results obtained by Kang-Deng-Zhou in [Results Appl. Math. 12:100178, 2021]. Additionally, we establish the sufficient conditions concerning $ \nabla u $ and $ \nabla b $, instead of $ u $ and $ b $.</p></abstract>

Temporal decay for the highest-order derivatives of solutions of the compressible Hall-magnetohydrodynamic equations

Recently, Gao and Yao established the global existence and temporal decay rates of solutions for a system of compressible Hall-magnetohydrodynamic fluids (Gao and Yao in Discrete Contin. Dyn. Syst. 36: 3077–3106, 2016). However, because of the difficulty of derivative loss in the nonlinear terms, Gao and Yao could not provide the temporal decay for the highest-order derivatives of classical solutions. In this paper, motivated by the decomposition technique of both low and high frequencies of solutions in (Wang and Wen in Sci. China Math. 65: 1199–1228 2022), we further derive the temporal decay for the highest-order derivatives of the strong solutions. Moreover, the decay rate is optimal, since it agrees with the solutions of the linearized system.

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.

On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space $$H^{s}$$ and even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $$L^{2}$$ norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear $$H^{s}$$ -illposedness (for s arbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

Well-posedness results for the 3D incompressible Hall-MHD equations

In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range.

On three-dimensional Hall-magnetohydrodynamic equations with partial dissipation

In this paper, we address the Hall-MHD equations with partial dissipation. Applying some important inequalities (such as the logarithmic Sobolev inequality using BMO space, bilinear estimates in BMO space, Young’s inequality, cancellation property, interpolation inequality) and delicate energy estimates, we establish an improved blow-up criterion for the strong solution. Moreover, we also obtain the existence of the strong solution for small initial data, the smallness conditions of which are given by the suitable Sobolev norms.

Structure-Preserving and Helicity-Conserving Finite Element Approximations and Preconditioning for the Hall Mhd Equations

We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.

Energy conservation for the nonhomogeneous incompressible Hall-MHD equations in a bounded domain

In this paper, we study the energy conservation of the nonhomogeneous incompressible Hall-magnetohydrodynamic equations in a bounded domain Ω⊂Rd with d=2,3. By exploring the special structure of the nonlinear terms in the Hall-magnetohydrodynamic equations, we obtain the sufficient conditions for the regularity of the weak solutions for energy conservation. The treatment of the boundary terms mainly relies on the coarea formula.

Global smooth solutions to the 3D generalized Hall-magnetohydrodynamic equations with large data

In this paper, based on a perturbation argument and fully exploring the nonlinear structure of the considered system, we establish the global existence to the three-dimensional generalized incompressible Hall-MHD equations for a class of large initial data, whose norms can be arbitrarily large. Our obtained result improves considerably the recent result in Li et al. [A class large solution of the 3D Hall-magnetohydrodynamic equations. J Differ Equ. 2020;268(10):5811–5822] to the generalized Hall-MHD system with equipping with different viscosity and magnetic diffusion coefficients.

Low Mach number limit of the full compressible Hall-magnetohydrodynamic equation with general initial data

In this paper, we are concerned with the low Mach number limit for the full compressible Hall-magnetohydrodynamic equations within the frame work of local smooth solution in R3. Under the assumption of large temperature variations, we first obtain the uniform estimates of the solutions in a ε-weighted Sobolev space, which establishes the local existence theorem of the full compressible Hall-magnetohydrodynamic equations on a finite time interval independent of the Mach number. Then, the low Mach limit is proved by combining the uniform estimates and the dispersive estimates on the wave equation due to Métivier and Schochet (2001).

Global well-posedness of the 3D incompressible Hall-MHD equations for small initial data in certain Besov spaces

We prove global well-posedness of solution to the 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) equations, where the initial data is in critical Besov space. Since the initial vertical velocity may be large enough, our work improves some recent results in WanZ2,WZ19.

Global Wellposedness and Large Time Behavior of Solutions to the Hall-Magnetohydrodynamics Equations

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in \mathbb R^n (n \geq 3) . Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all the previous wellposedness results in [6, 7, 21], the initial energy we considered here may not be finite.