Compressible Magnetohydrodynamics Research Articles

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with Perfectly Conducting Boundary

The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with Perfectly Conducting Boundary

Well-posedness and blow-up criterion for strong solutions of a class of compressible MHD equations

In this paper, we consider the initial boundary value problem of a class of one-dimensional magnetohydrodynamics (MHD) with viscous, compressible, and non-Newtonian flow. We prove the existence of the solution in the presence of the resistivity coefficient, obtain the convergence rate in L2-norm, and establish the blow-up criterion of the local strong solution in the presence of the resistivity coefficient. Moreover, the initial vacuum is allowed.

Stability and exponential decay for the compressible viscous non-resistive MHD system

How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R3 or T3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

Optimal temporal decay rates for 3D compressible magnetohydrodynamics system with nonlinear damping

Recently, Li–Fu–Wang (Li et al., 2022) established the optimal temporal decay rates of solutions near the equilibrium state to the 3D compressible magnetohydrodynamic system with nonlinear damping αuβ−1u for β⩾3. In this paper, we further extend Li–Fu–Wang’s result to the case β>1 by finer energy estimates.

Global well‐posedness of strong solutions to the compressible magnetohydrodynamic equations with Coulomb force and non‐flat doping profile

This paper establishes the global well‐posedness of strong solutions for the 3D compressible magnetohydrodynamics (MHD) with non‐flat doping profile and Coulomb force, which is of hyperbolic–parabolic–elliptic mixed type. It is essentially shown that for the initial value problem with initial density allowed to contain vacuum states, the strong solution exists globally under small energy but possible large oscillations.

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

A theoretical study on the bulk viscosity in a compressible flow of an electrically conducting gas in the presence of magnetoacoustic waves

Only a few studies in the current literature are dedicated to the analysis of the effect of a bulk viscosity (i.e., second viscosity coefficient), and little is known about how to determine this quantity for high frequency compressible flows involving magnetic effect. The dissipation phenomenon associated with rapid expansions in the flow and the consequent attenuation of magnetoacoustic waves by the bulk viscosity has been neglected by several works on gas dynamics at high frequency. In this paper, we present a theoretical study on a compressible flow of an electrically conducting barotropic gas in the presence of a bulk viscosity. The governing equations represent a coupling between hydrodynamics and Maxwell's equations in the context of magnetohydrodynamics (MHD), and the relevant physical parameters of the flow are presented after an appropriate dimensional analysis of these equations. First, an analysis of small perturbations around an equilibrium state of the electrically conducting gas results in a system of linearized equations in the frequency-wavenumber space. The dispersion relation for magnetic waves is determined in terms of magnetic and the bulk viscosity effects. Second, based on the dispersion relation for magnetoacoustic waves, we propose a new expression to estimate the bulk viscosity in terms of the gas barotropic coefficient, the wavenumber, the wave frequency, the orientation of the applied field, and the Euler magnetic parameter, which measures the relative importance between magnetic and thermodynamic pressures. The behavior of the bulk viscosity is examined as a function of the wavenumber for different magnetic Euler numbers and field orientation, revealing how the rate of energy dissipation associated with the bulk viscosity can be controlled by varying the intensity of an applied magnetic field and its orientation. We show that understanding magnetoacoustic waves not only provides a tool for estimating the bulk viscosity in plasma flows in terms of these wave parameters, but also offers a potential pathway to manipulating both the magnitude and orientation of a magnetic field in order to reduce the rate of energy dissipation in most of plasma flows. This rate of dissipation produced by the rapid expansion of a recombining gas can be orders of magnitude larger than the one produced by the standard shear viscosity. These insights into bulk viscosity in compressible MHD flow at high frequency have potential applications in reacting plasmas where magnetoacoustic waves are damped by bulk viscosity, in production and dissipation of turbulence and in phenomenon of sonoluminescence as occurs in liquid containing gas bubble oscillating at fairly high frequency.

Small-amplitude Compressible Magnetohydrodynamic Turbulence Modulated by Collisionless Damping in Earth’s Magnetosheath: Observation Matches Theory

Plasma turbulence is a ubiquitous dynamical process that transfers energy across many spatial and temporal scales and affects energetic particle transport. Recent advances in the understanding of compressible magnetohydrodynamic (MHD) turbulence demonstrate the important role of damping in shaping energy distributions on small scales, yet its observational evidence is still lacking. This study provides the first observational evidence of substantial collisionless damping (CD) modulation on the small-amplitude compressible MHD turbulence cascade in Earth’s magnetosheath using four Cluster spacecraft. Based on an improved compressible MHD decomposition algorithm, turbulence is decomposed into three eigenmodes: incompressible Alfvén modes and compressible slow and fast (magnetosonic) modes. Our observations demonstrate that CD enhances the anisotropy of compressible MHD modes because CD has a strong dependence on wave propagation angle. The wavenumber distributions of slow modes are mainly stretched perpendicular to the background magnetic field ( B 0) and weakly modulated by CD. In contrast, fast modes are subjected to a more significant CD modulation. Fast modes exhibit a weak, scale-independent anisotropy above the CD truncation scale. Below the CD truncation scale, the anisotropy of fast modes enhances as wavenumbers increase. As a result, fast-mode fractions in the total energy of compressible modes decrease with the increase of perpendicular wavenumber (to B 0) or wave propagation angle. Our findings reveal how the turbulence cascade is shaped by CD and its consequences for anisotropies in the space environment.

Magnetic field suppression characteristics in interaction process between shock wave and light gas cylinder

Based on ideal compressible magnetohydrodynamics (MHD) equations, the interface instabilities induced by the interaction between planar shock wave and the light gas (Helium) cylinder under the influence of the magnetic fields with different directions are investigated numerically by using the CTU(corner transport upwind)+CT (constrained transport) algorithm. The numerical results elucidate the evolution of flow field characteristics and wave structures with and without magnetic field. Moreover, we examine the influence of the magnetic field direction on a characteristic scales (including the length, height and width of the central axis of gas cylinder), as well as the volume compressibility. Then, the mechanism of the magnetic field direction affecting the interface instability is studied in depth by integrating the analyses of the circulation, energy, velocity and magnetic force distribution within the flow field. The core of this study, is to explore the suppression mechanism of interface instability by magnetic field force. The results show that the magnetic pressure plays a crucial role in driving vorticity away from the interface, thereby reducing its deposition on the density interface. Simultaneously, it adheres to the divided vortex layer, thereby effectively isolating the influence of Richtmyer-Meshkov instability on the interface. On the other hand, the magnetic tension adheres to the separated vortex layer, and its direction is opposite to that of the vorticity generated by the shear of interface velocity. This action effectively suppresses the Kelvin-Helmholtz instability and the rolling-up of vortices on the density interface. Additionally, under the control of a longitudinal magnetic field, the direction of magnetic tension is opposite to the direction of the central jet, effectively suppressing the development of Rayleigh-Taylor instability.

The Diffusion and Scattering of Accelerating Particles in Compressible MHD Turbulence

We numerically study the diffusion and scattering of cosmic rays (CRs) together with their acceleration processes in the framework of the modern understanding of magnetohydrodynamic (MHD) turbulence. Based on the properties of compressible MHD turbulence obtained from observations and numerical experiments, we investigate the interaction of CRs with plasma modes. We find that (1) the gyroradius of particles exponentially increases with the acceleration timescale; (2) the momentum diffusion presents the power-law relationship with the gyroradius in the strong turbulence regime, and shows a plateau in the weak turbulence regime implying a stochastic acceleration process; (3) the spatial diffusion is dominated by the parallel diffusion in the sub-Alfvénic regime, while it is dominated by the perpendicular diffusion in the super-Alfvénic one; (4) as for the interaction of CRs with plasma modes, the particle acceleration is dominated by the fast mode in the high β case, while in the low β case, it is dominated by the fast and slow modes; and (5) in the presence of acceleration, magnetosonic modes still play a critical role in the diffusion and scattering processes of CRs, which is in good agreement with earlier theoretical predictions.

Efficient Implementation of Modern Entropy Stable and Kinetic Energy Preserving Discontinuous Galerkin Methods for Conservation Laws

Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D, respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems, including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.

Global small solutions to the 3D compressible viscous non-resistive MHD system

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space [Formula: see text] or the periodic box [Formula: see text] even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.

Mathematical Theory of Compressible Magnetohydrodynamics Driven by Non-conservative Boundary Conditions

We propose a new concept of weak solution to the equations of compressible magnetohydrodynamics driven by ihomogeneous boundary data. The system of the underlying field equations is solvable globally in time in the out of equilibrium regime characteristic for turbulence. The weak solutions comply with the weak–strong uniqueness principle; they coincide with the classical solution of the problem as long as the latter exists. The choice of constitutive relations is motivated by applications in stellar magnetoconvection.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

AbstractWe establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. The formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

Nonlinear interaction between double tearing mode and Kelvin–Helmholtz instability with different shear flows

The nonlinear interaction between the double tearing mode (DTM) and Kelvin–Helmholtz (KH) instabilities with different shear flow profiles has been numerically investigated via the use of a compressible magnetohydrodynamics (MHD) model. We focus on KH instabilities in weak and reversed magnetic shear plasmas with strong stabilizing effect of field line bending. Results show that KH instabilities coupled with DTMs occur in these plasmas and the KH mode dominates the instability dynamics, suggesting the crucial role of weak magnetic shear in the formation of high-mode harmonics. For symmetric flows, an asymmetric forced magnetic reconnection configuration is maintained during the growth phase, leading to interlocking of the modes. Additionally, this investigation of the DTM-KH instability interaction contributes to our understanding of the nonlinear reconnection mechanism in the regime of weak and reversed magnetic shear plasmas, which is relevant for astrophysical and fusion studies.

On vacuum free boundary problems in ideal compressible magnetohydrodynamics

AbstractWe survey some recent results on vacuum free boundary problems in three‐dimensional ideal compressible magnetohydrodynamics, restate the main theorems in our works (Secchi and Trakhinin, Nonlinearity 27(1) (2014) 105–169; Trakhinin and Wang, Arch. Ration. Mech. Anal. 239(2) (2021) 1131–1176; Trakhinin and Wang, Math. Ann. 383(1–2) (2022) 761–808; Trakhinin and Wang, SIAM J. Math. Anal. 54(6) (2022) 5888–5921), and provide an alternative proof for the linear well‐posedness of the full plasma–vacuum interface problem under the noncollinearity condition.

Nonlinear Stability and Existence of Two-Dimensional Compressible Current-Vortex Sheets

We are concerned with the nonlinear stability and existence of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with a characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash–Moser iteration, provided that the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin–Helmholtz instabilities.