Compressible Magnetohydrodynamic Equations Research Articles

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops

Context. The evolution of Alfvén waves in cylindrical magnetic flux tubes, which represent a basic model for loops observed in the solar corona, can be affected by phase mixing and turbulent cascade. Phase mixing results from transverse inhomogeneities in the Alfvén speed, causing different shells of the flux tube to oscillate at different frequencies, thus forming increasingly smaller spatial scales in the direction perpendicular to the guide field. Turbulent cascade also contributes to the dissipation of the bulk energy of the waves through the generation of smaller spatial scales. Both processes present characteristic timescales. Different regimes can be envisaged according to how those timescales are related and to the typical timescale at which dissipation is at work. Aims. We investigate the interplay of phase mixing and the nonlinear turbulent cascade in the evolution and dissipation of Alfvén waves using compressible magnetohydrodynamic numerical simulations. We consider perturbations in the form of torsional waves, both propagating and standing, or turbulent fluctuations, or a combination of the two. The main purpose is to study how phase mixing and nonlinear couplings jointly work to produce small scales in different regimes. Methods. We conducted a numerical campaign to explore the typical parameters, such as the loop length, the amplitude and spatial profile of the perturbations, and the dissipative coefficients. A pseudo-spectral code was employed to solve the three-dimensional compressible magnetohydrodynamic equations, modeling the evolution of perturbations propagating in a flux tube corresponding to an equilibrium configuration with cylindrical symmetry. Results. We find that phase mixing takes place for moderate amplitudes of the turbulent component even in a distorted, nonaxisymmetric configuration, building small scales that are locally transverse to the density gradient. The dissipative time decreases with increasing the percentage of the turbulent component. This behavior is verified both for propagating and standing waves. Even in the fully turbulent case, a mechanism qualitatively similar to phase mixing occurs: it actively generates small scales together with the nonlinear cascade, thus providing the shortest dissipative time. General considerations are given to identify this regime in the parameter space. The turbulent perturbation also distorts the background density, locally increasing the Alfvén velocity gradient and further contributing to accelerating the formation of small scales. Conclusions. Our campaign of simulations is relevant for the coronal plasma where Reynolds and Lundquist numbers are extremely high. For sufficiently low perturbation amplitudes, phase mixing and turbulence work synergically, speeding up the dissipation of the perturbation energy: phase mixing dominates at early times and nonlinear effects at later times. We find that the dissipative time is shorter than those of phase mixing and the nonlinear cascade when individually considered.

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.

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

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 Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum

In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy (∥ρ∥L∞(0,T;L∞)+∥u∥C([0,T];L3)<∞), then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over [0,T]. Notably, our analysis accommodates the presence of an initial vacuum.

On local well-posedness of problems with characteristic free boundary for hyperbolic systems of conservation laws

The proof of the local-in-time existence and uniqueness of a smooth solution to a free boundary problem for a hyperbolic system of conservation laws has some additional difficulties if the free boundary is a characteristic of this system. They are connected with the loss of control of the normal derivatives and the possible non-ellipticity of the symbol of the free boundary. Another peculiarity of problems with characteristic free boundary is that usually a loss of derivatives of the coefficients and source terms occurs in a priori estimates for the corresponding linearized problems. Moreover, the boundary conditions in the linearized problem can be non-dissipative, which makes it difficult to use the energy method. We describe methods for overcoming these difficulties. Our main examples are free boundary problems for Euler's equations and the equations of ideal compressible magnetohydrodynamics, for which we review the results on their local well-posedness. Bibliography: 61 titles.

Global weak solutions to the compressible magnetohydrodynamic and Vlasov equations

Global weak solutions to the compressible magnetohydrodynamic and Vlasov equations

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.

Global strong solutions to the compressible magnetohydrodynamic equations with slip boundary conditions in a 3D exterior domain

Global strong solutions to the compressible magnetohydrodynamic equations with slip boundary conditions in a 3D exterior domain

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.

On the incompressible and non-resistive limit of 3D compressible magnetohydrodynamic equations in bounded domains

In this paper, we investigate the incompressible and non-resistive limit for the initial boundary value problem of isentropic compressible resistive magnetohydrodynamic equations with ill-prepared initial data in three-dimensional bounded domains. We establish the higher-order uniform estimates with respect to both the Mach number and the resistivity coefficient in the framework of new type of weighted Sobolev spaces. Then we obtain the strong convergence of the magnetic field and the divergence-free component of the velocity field, as both the Mach number and the resistivity coefficient tend to zero.

Global classical solutions for three-dimensional compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in bounded domains

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier–Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier–Stokes–Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468–8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356–1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity.

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.

Formation of Singularity for Full Compressible Magnetohydrodynamic Equations with Zero Resistivity in Two Dimensional Bounded Domains

Formation of Singularity for Full Compressible Magnetohydrodynamic Equations with Zero Resistivity in Two Dimensional Bounded Domains

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

AbstractThe question of whether the two‐dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite‐time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well‐posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy‐type inequality.