Compressible Magnetohydrodynamic Equations Research Articles

High-order central ENO finite-volume scheme for ideal MHD

A high-order accurate finite-volume scheme for the compressible ideal magnetohydrodynamics (MHD) equations is proposed. The high-order MHD scheme is based on a central essentially non-oscillatory (CENO) method combined with the generalized Lagrange multiplier divergence cleaning method for MHD. The CENO method uses k-exact multidimensional reconstruction together with a monotonicity procedure that switches from a high-order reconstruction to a limited low-order reconstruction in regions of discontinuous or under-resolved solution content. Both reconstructions are performed on central stencils, and the switching procedure is based on a smoothness indicator. The proposed high-order accurate MHD scheme can be used on general polygonal grids. A highly sophisticated parallel implementation of the scheme is described that is fourth-order accurate on two-dimensional dynamically-adaptive body-fitted structured grids. The hierarchical multi-block body-fitted grid permits grid lines to conform to curved boundaries. High-order accuracy is maintained at curved domain boundaries by employing high-order spline representations and constraints at the Gauss quadrature points for flux integration. Detailed numerical results demonstrate high-order convergence for smooth flows and robustness against oscillations for problems with shocks. A new MHD extension of the well-known Shu–Osher test problem is proposed to test the ability of the high-order MHD scheme to resolve small-scale flow features in the presence of shocks. The dynamic mesh adaptation capabilities of the approach are demonstrated using adaptive time-dependent simulations of the Orszag–Tang vortex problem with high-order accuracy and unprecedented effective resolution.

A blow-up criterion of strong solutions to the 3D compressible MHD equations with vacuum

In this paper, we prove a blow-up criterion of strong solutions to 3D viscous isentropic compressible magnetohydrodynamic equations. It is shown that if ρ and H satisfy ‖ρ‖L∞(0,T;L∞)+‖H‖L∞(0,T;Lr)<∞, for any 24/5≤r≤∞ and 3μ>λ, then the strong solutions to the Cauchy problem of the compressible magnetohydrodynamic equations can exist globally on [0,T]. In addition, initial vacuum is allowed.

Blow-up of smooth solutions to the isentropic compressible MHD equations

This article studies the blow-up of smooth solutions to the isentropic compressible magnetohydrodynamic (MHD) equations (1.1) in , . We obtain that under physical conditions, if the initial density and magnetic have compact support, then any smooth radially symmetric solutions will blow up in finite time.

Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary

We study the compressible magnetohydrodynamic equations in a bounded smooth domain in \({{\mathbb{R}}^2}\) with perfectly conducting boundary, and prove the global existence and uniqueness of smooth solutions around a rest state. Moreover, the low Mach limit of the solutions is verified for all time, provided that the initial data are well prepared.

Global Classical Solutions to 3D Compressible Magnetohydrodynamic Equations with Large Oscillations and Vacuum

We consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution, where the flow density is allowed to contain vacuum states, and the large-time behavior of the solution is also shown.

Asymptotic Limits of the Full Compressible Magnetohydrodynamic Equations

The paper deals with the asymptotic limits of the full compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$ which are the coupling between the Navier--Stokes--Fourier system with the Maxwell equations governing the behavior of the magnetic field. It is rigorously shown that for the general initial data, the weak solutions of the full compressible MHD equations converge to the strong solution of the ideal incompressible MHD equations as the Mach number, the viscosity coefficients, the heat conductivity, and the magnetic diffusion coefficient go to zero simultaneously. Furthermore, the convergence rates are also obtained.

Blow-up criterion for 3-D compressible magnetohydrodynamics with vacuum and zero resistivity

We study strong solutions of the equations of compressible magnetohydrodynamics with zero resistivity in a domain Ω⊂R3. We establish a criterion for possible breakdown of such solutions at a finite time in terms of both ‖∇u‖L1(0,T;L∞) and ‖θ‖L∞(0,T;L∞). More precisely, if a solution of 3D compressible magnetohydrodynamics with zero resistivity is initially regular and loses its regularity at some later time, then the loss of regularity implies growth without bound of both ‖∇u‖L1(0,T;L∞) and ‖θ‖L∞(0,T;L∞) as the critical time approaches. In addition, initial vacuum states are allowed in our cases.

Decay Estimates for Isentropic Compressible Magnetohydrodynamic Equations in Bounded Domain

In this paper, under the hypothesis that ▪ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrodynamic equations and show that the weak solutions decay exponentially to the equilibrium state in L2 norm. Our result verifies that the method of Daoyuan Fang, Ruizhao Zi and Ting Zhang [1] can be adapted to magnetohydrodynamic equations.

Roles of super-Alfvenic shear flows on Kelvin–Helmholtz and tearing instability in compressible plasma

The Kelvin–Helmholtz (KH) instability and the tearing mode instability under super-Alfvenic shear flow have been studied using two-dimensional compressible magnetohydrodynamic equations. The shear flow is parallel to the magnetic field line. A broad range of the plasma and of the shear flow speed are investigated. In compressible plasma, it is found that there are two critical velocities and , which are -related. When , the KH instability is a pure growing mode. When , the KH instability grows with oscillation. The instability is fully stabilized when . When the KH instability is dominant, there is always an explosive phase triggered by the tearing mode instability after the linear growing phase. For a small shear flow thickness, the KH instability is dominant, while tearing instability becomes dominant when the shear flow thickness is large enough.

Blow‐up criterion for 3D viscous‐resistive compressible magnetohydrodynamic equations

In this paper, we establish a blow‐up criterion of strong solutions for 3D viscous‐resistive compressible magnetohydrodynamic equations, which depends only on and . Our result improves the previous criterion in Lu's paper (Journal of Mathematical Analysis and Applications 2011; 379: 425–438.) for compressible magnetohydrodynamic equations by removing a stringent condition on the viscous coefficients μ &gt; 4λ. In addition, initial vacuum states are also allowed in our cases. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in ℝ3 for any specific heat ratio γ > 1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence. According to the author’s knowledge, it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with Γ > 1.

A regularizing effect of radiation in one-dimensional compressible MHD equations

This paper is concerned with the global existence of classical solutions to an initial–boundary value problem of the one-dimensional (1D) equations of compressible radiative magnetohydrodynamics (MHD). The key point here is that there is no growth restriction imposed on the heat conductivity, in particular, the heat-conducting coefficient is allowed to be a positive constant. This in particular implies that the radiation is indeed a mathematically “regularizing” effect on the fluid dynamics, since the global existence of the classical solution to the one-dimensional full perfect MHD equations without radiation is still unknown when all the viscosity, magnetic diffusivity and heat conductivity coefficients are constant.

Time periodic solutions of the compressible magnetohydrodynamic equations

The compressible magnetohydrodynamic equations with a time periodic external force in Rn are studied in this paper. We establish the existence, uniqueness and time-asymptotic stability of time periodic solutions when n≥5. Our proof is essentially based on a combination of the energy method and the time decay estimates of solutions to the linearized system.

Spine-fan reconnection

Context. From observations, the atmosphere of the Sun has been shown to be highly dynamic with perturbations of the magnetic field often lacking temporal or spatial symmetry. Despite this, studies of the spine-fan reconnection mode at 3D nulls have so far focused on the very idealised case with symmetric driving of a fixed spatial extent. Aims. We investigate the spine-fan reconnection process for less idealised cases, focusing on asymmetric driving and drivers with different length scales. We look at the initial current sheet formation and whether the scalings developed in the idealised models are robust in more realistic situations. Methods. The investigation was carried out by numerically solving the resistive compressible 3D magnetohydrodynamic equations in a Cartesian box containing a linear null point. The spine-fan collapse was driven at the null through tangential boundary driving of the spine foot points. Results. We find significant differences in the initial current sheet formation with asymmetric driving. Notable is the displacement of the null point position as a function of driving velocity and resistivity (η). However, the scaling relations developed in the idealised case are found to be robust (albeit at reduced amplitudes) despite this extra complexity. Lastly, the spatial variation is also shown to play an important role in the initial current sheet formation through controlling the displacement of the spine foot points. Conclusions. We conclude that during the early stages of spine-fan reconnection both the temporal and spatial nature of the driving play important roles, with the idealised symmetrically driven case giving a “best case” for the rate of current development and connectivity change. As the most interesting eruptive events occur in relatively short time frames this work clearly shows the need for high temporal and spatial knowledge of the flows for accurate interpretation of the reconnection scenario. Lastly, since the scalings developed in the idealised case remain robust with more complex driving we can be more confident of their use in interpreting reconnection in complex magnetic field structures.

Modelling magnetic flux emergence in the solar convection zone

[Abridged] Bipolar magnetic regions are formed when loops of magnetic flux emerge at the solar photosphere. Our aim is to investigate the flux emergence process in a simulation of granular convection. In particular we aim to determine the circumstances under which magnetic buoyancy enhances the flux emergence rate (which is otherwise driven solely by the convective upflows). We use three-dimensional numerical simulations, solving the equations of compressible magnetohydrodynamics in a horizontally-periodic Cartesian domain. A horizontal magnetic flux tube is inserted into fully developed hydrodynamic convection. We systematically vary the initial field strength, the tube thickness, the initial entropy distribution along the tube axis and the magnetic Reynolds number. Focusing upon the low magnetic Prandtl number regime (Pm<1) at moderate magnetic Reynolds number, we find that the flux tube is always susceptible to convective disruption to some extent. However, stronger flux tubes tend to maintain their structure more effectively than weaker ones. Magnetic buoyancy does enhance the flux emergence rates in the strongest initial field cases, and this enhancement becomes more pronounced when we increase the width of the flux tube. This is also the case at higher magnetic Reynolds numbers, although the flux emergence rates are generally lower in these less dissipative simulations because the convective disruption of the flux tube is much more effective in these cases. These simulations seem to be relatively insensitive to the precise choice of initial conditions: for a given flow, the evolution of the flux tube is determined primarily by the initial magnetic field distribution and the magnetic Reynolds number.

Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity

In this paper we investigate three-dimensional compressiblemagnetohydrodynamic equations with partial viscosity. Local strongsolutions to the compressible magnetohydrodynamic equations withlarge data are established.

A blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.

Global existence and convergence rates of smooth solutions for the full compressible MHD equations

In this paper, we consider the global smooth solutions and their decay for the full compressible magnetohydrodynamic equations in R 3. We prove the global existence of smooth solutions near the constant state in Sobolev norms by energy method and show the convergence rates of the L p -norm of these solutions to the constant state when the L q -norm of the perturbation is bounded.

Optimal decay rates of the compressible magnetohydrodynamic equations

We use a pure energy method recently developed by Guo and Wang to prove the optimal time decay rates of the solutions to the compressible magnetohydrodynamic equations in the whole space. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained.

Global existence and exponential stability for a 1D compressible and radiative MHD flow

In this paper, we prove the global existence and exponential stability of solutions in H+i (i=1,2,4) for one-dimensional compressible and radiative magnetohydrodynamics (MHD) equations in a bounded domain Ω:=(0,1).