Compressible Magnetohydrodynamic Flows Research Articles

On an inhomogeneous boundary value problem for steady compressible magnetohydrodynamics flow

In this paper, we are concerned with the three-dimensional stationary equations of compressible magnetohydrodynamic (MHD) isentropic flow in a bounded cylinder domain $$\Omega =(0,1)\times \Omega _0$$ with an inhomogeneous boundary condition. We obtain the existence and uniqueness of a strong solution provided that the data are around a given constant flow. From our result, it reveals that the magnetic Reynolds number plays an important role in the stability of steady compressible MHD equations.

Asymptotic Limit for Rotational Compressible Magnetohydrodynamic Flows

In this paper we consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We study the asymptotic limit for the compressible rotational magnetohydrodynamic flows with the well-prepared initial data such that we derive a rigorous quasi-geostrophic equation with diffusion term governed by the magnetic field from a compressible rotational magnetohydrodynamic flows. This paper covers two results: the existence of the unique global strong solution of quasi-geostrophic equation with good regularity on the velocity and magnetic field and the derivation of quasi-geostrophic equation with diffusion.

Low Mach limit to one‐dimensional nonisentropic planar compressible magnetohydrodynamic equations

This paper is concerned with a one‐dimensional nonisentropic compressible planar magnetohydrodynamic flow with general initial data, whose behaviors at far fields x→±∞ are different. The low Mach limit for the system is rigorously justified. The limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma.

Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions

In this paper, we prove the energy conservation for the weak solutions to the three-dimensional equations of compressible magnetohydrodynamic flows (MHD) under certain conditions only about density and velocity. This work is inspired by the seminal work by Yu [24] on the energy conservation of compressible Navier-Stokes equations. Our result indicates that even the magnetic field is taken into account, we only need some regularity conditions of the density and velocity as in [24] to ensure the energy conservation.

The relaxation limit for full compressible magnetohydrodynamic flows with the Maxwell law

In this paper, we study the full compressible magnetohydrodynamic (MHD) flows with the viscous stress tensor given by the Maxwell law in R3. First, local existence of solutions for general large initial data and global existence of solutions for small initial data are shown. Then we obtain the convergence of solutions for the system with the Maxwell law to the corresponding solution for the classical system with the Newtonian law when the relaxation time goes to 0.

As a Matter of Force—Systematic Biases in Idealized Turbulence Simulations

Abstract Many astrophysical systems encompass very large dynamical ranges in space and time, which are not accessible by direct numerical simulations. Thus, idealized subvolumes are often used to study small-scale effects including the dynamics of turbulence. These turbulent boxes require an artificial driving in order to mimic energy injection from large-scale processes. In this Letter, we show and quantify how the autocorrelation time of the driving and its normalization systematically change the properties of an isothermal compressible magnetohydrodynamic flow in the sub- and supersonic regime and affect astrophysical observations such as Faraday rotation. For example, we find that δ-in-time forcing with a constant energy injection leads to a steeper slope in kinetic energy spectrum and less-efficient small-scale dynamo action. In general, we show that shorter autocorrelation times require more power in the acceleration field, which results in more power in compressive modes that weaken the anticorrelation between density and magnetic field strength. Thus, derived observables, such as the line-of-sight (LOS) magnetic field from rotation measures, are systematically biased by the driving mechanism. We argue that δ-in-time forcing is unrealistic and numerically unresolved, and conclude that special care needs to be taken in interpreting observational results based on the use of idealized simulations.

Modeling Aurora Type Phenomena by Short Wave-Long Wave Interactions in Multidimensional Large Magnetohydrodynamic Flows

We establish the convergence of an approximation scheme to a model for aurora type phenomena. The latter, mathematically, means a system describing the short wave-long wave (SW-LW) interactions for compressible magnetohydrodynamic (MHD) flows, introduced in a previous work, which presents short waves, governed by a nonlinear Schr\"odinger (NLS) equation based on the Lagrangian coordinates of the fluid, and long waves, governed by the compressible MHD system. The NLS equation and the compressible MHD system are also explicitly coupled by an interaction potential in the NLS equation and an interaction surface force in the momentum equation of the MHD system, both multiplied by a small coefficient. Since the compressible MHD flow is assumed to have large amplitude data, possibly forming vacuum, the coefficient of the interaction terms may be taken as zero, due to the large difference in scale between the two types of waves. In this case, the whole coupling lies in the Lagrangian coordinates of the compressible MHD fluid upon which the NLS equation is formulated. However, due to the possible occurrence of vacuum, these Lagrangian coordinates are not well defined, and herein lies the importance of the approximation scheme. The latter consists of a system that formally approximates the SW-LW interaction system, including non-zero vanishing interaction coefficients, together with an artificial viscosity in the continuity equation, an artificial energy balance term, an artificial pressure in the momentum equation and approximate Lagrangian coordinates, which circumvent the possible occurrence of vacuum. We prove the convergence of the solutions of the approximation scheme to a solution of a system consisting of a NLS equation based on the coordinate system induced by the scheme, and a compressible MHD system.

Derivation of Inviscid Quasi-geostrophic Equation from Rotational Compressible Magnetohydrodynamic Flows

In this paper, we consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the rotational compressible magnetohydrodynamic flows with the well-prepared initial data. It is a first derivation of quasi-geostrophic equation governed by the magnetic field, and the tool is based on the relative entropy method. This paper covers two results: the existence of the unique local strong solution of quasi-geostrophic equation with the good regularity and the derivation of a quasi-geostrophic equation.

On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics

We consider an initial-boundary value problem for the one-dimensional equations of compressible isentropic magnetohydrodynamic (MHD) flows. The non-resistive limit of the global solutions with large data is justified. As a by-product, the global well-posedness of the compressible non-resistive MHD equations is established. Moreover, the thickness of the magnetic boundary-layer of the value with is proved, where is the resistivity coefficient. The proofs of these results are based on a full use of the so-called ‘effective viscous flux’, the material derivative and the structure of the equations.

Global existence and large time behavior of solutions for compressible quantum magnetohydrodynamics flows in [formula omitted

The present paper concerns with compressible quantum magnetohydrodynamic flows in T3. This new model was derived by Haas [13] from a Wigner–Maxwell system. First, we reformulate this system as the quantum Euler–Maxwell system by using effective velocity. Then, the global existence of weak solutions is obtained by adding artificial viscosity terms and the Faedo–Galerkin method together with weak compactness techniques. Moreover, large time behavior of solutions is established.

Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows

We consider the compressible models of magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We derive a rigorous quasi-geostrophic equation governed by magnetic field from the stratified flows of the rotational compressible magnetohydrodynamic flows with the well-prepared initial data and the tool of proof is based on the relative entropy. Furthermore, the convergence rates are obtained.

The existence of stationary star solutions for compressible magnetohydrodynamic flows

In this paper, we are concerned with the compressible Euler-Poisson system coupled to a magnetic field in the three-dimensional space. Based on a variational method and the exact expression of the Green’s function for an elliptic equation in spherical coordinates, we prove the existence of stationary star solutions.

Existence of weak solutions to steady compressible magnetohydrodynamic equations

The three-dimensional equations of steady compressible magnetohydrodynamic non-isentropic flows are considered. A boundary value problem for the velocity and the temperature is studied in a bounded domain with arbitrarily large data. The existence of weak solutions to these equations is established under the assumption that the pressure P(ρ,θ)=c1ργ+c2ρθ for γ>73.

Blowup of smooth solution for non‐isentropic magnetohydrodynamic equations without heat conductivity

In this paper, we establish a blow‐up criterion for the three‐dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial‐boundary‐value problem with initial density allowed to vanish, the strong or smooth solution for the three‐dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.

Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows

The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.

A remark on blow-up criterion for 3D compressible magnetohydrodynamic flows with vacuum

In this paper, we concerned a blowup criterion of strong solutions for 3D viscous, compressible magnetohydrodynamic(MHD) flows with vacuum. We proved that any strong solutions of viscous, compressible MHD fluids will blow up in finite time if limT→T*(∥ρ∥L∞(0,T;Ls)+∥H∥L∞(0,T;Ll))=∞, for some 1 < s < ∞ large enough, and 245≤l≤∞, provided that the viscosity coefficients satisfy 3μ > λ. This work is motivated by the recent works of Wang and Li (2014) and Yu (2013) on blow up criterion for 3D MHD flows.

Discrete filters for large-eddy simulation of forced compressible magnetohydrodynamic turbulence

We discuss results of the applicability of discrete filters for the large-eddy simulation (LES) method of forced compressible magnetohydrodynamic (MHD) turbulent flows with the scale-similarity model. New results are obtained for cross-helicity and residual energy. Cross-helicity and residual energy are important quantities in magnetohydrodynamic turbulence and have no hydrodynamic counterpart. The influences and effects of discrete filter shapes on the scale-similarity model are examined in physical space using finite-difference numerical schemes. We restrict ourselves to the Gaussian filter and the top-hat filter. Representations of this subgrid-scale model, which correspond to various 3- and 5-point approximations of both Gaussian and top-hat filters for different values of parameter (the ratio of the cut-off length-scale of the filter to the mesh size), are investigated. Discrete filters produce more discrepancies for the magnetic field. It is shown that the Gaussian filter is more sensitive to the parameter ϵ than the top-hat filter in compressible forced MHD turbulence. The 3-point filters at and give the least accurate results whereas the 5-point Gaussian filter shows the best results at and . There are only very small differences deep into the dissipation region in favor of . For cross-helicity, the 5-point discrete filters are in good agreement with the results of direct numerical simulation (DNS), while the 3-point filter produces the largest discrepancies with DNS results. There is no strong dependence on the choice of the parameter and order approximation is a much more important factor for the cross-helicity. The difference between the filters is less for the residual energy compared with total energy. Thus, the total energy is more sensitive to the choice of discrete filter in the modeling of compressible MHD turbulence using the LES method.

Blow-up criterion for two-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows

This paper establishes a blow-up criterion for the two-dimensional (2D) viscous, compressible, and heat conducting magnetohydrodynamic(MHD) flows. It is essentially shown that for the initial boundary value problem of the 2D full compressible MHD flows with initial density allowed to vanish, the strong solution exists globally if the divergence of velocity satisfies ‖divu‖L1(0,T;L∞)<∞. In particular, the criterion is independent of both the temperature and the magnetic field and is just the same as that of the full compressible Navier–Stokes equations.

A regularity criterion of strong solutions to the 2D compressible magnetohydrodynamic equations

This paper establishes a regularity criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the L∞-norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed.

Local conservative regularizations of compressible magnetohydrodynamic and neutral flows

Ideal systems like magnetohydrodynamics (MHD) and Euler flow may develop singularities in vorticity (w=∇×v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper, we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length λ which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/ωpe. Our regularization preserves the symmetries of the original systems and, with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied to generalize the constitutive relation to bound higher moments of vorticity. A “swirl” velocity field is identified, and shown to transport w/ρ and B/ρ, generalizing the Kelvin-Helmholtz and Alfvén theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch, and a plane vortex sheet. The proposed regularization could facilitate numerical simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma dynamic systems arising from our work are briefly discussed.