Compressible Magnetohydrodynamic Flows Research Articles

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R 3 and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.

Numerical Study of Compressible Magnetohydrodynamic Turbulence in Two Dimensions

We have studied forced turbulence of compressible magnetohydrodynamic (MHD) flows through two-dimensional simulations with different numerical resolutions. First, hydrodynamic turbulence with Mach number Msinit ≡ vrms/cs = 1 and density compression δρ/ρrms 0.45 was generated by enforcing a random force. Then, initial, uniform magnetic fields of various strengths were added with Alfvenic Mach number MAinit ≡ vrms/cA,init 1. An isothermal equation of state was employed, and no explicit dissipation was included. In our simulations, the maximum amplification factor of magnetic energy depends on resolution and is proportional to n, where nx is the number of grid cells spanned by the computational box size. After the MHD turbulence is saturated, the resulting flows are categorized as very weak field (VWF), weak field (WF), and strong field (SF) classes, which have MA ≡ vrms/cArms 1, MA > 1, and MA ~ 1, respectively. The flow character in the VWF cases is similar to that of hydrodynamic turbulence. In the WF cases, the magnetic energy is still smaller than the kinetic energy in the global sense, but the magnetic field can become locally important. Hence, not only in the SF regime but also in the WF regime, turbulent transport is suppressed by the magnetic field. In the SF cases, the energy power spectra in the inertial range, although no longer power-law, exhibit a range with slopes close to ~1.5, hinting at the Iroshnikov-Kraichnan spectrum. These characteristics of the VWF, WF, and SF classes are consistent with their incompressible turbulence counterparts, indicating that a modest compressibility of δρ/ρrms 0.45 or so does not play a significant role in turbulence. Our simulations were able to produce the SF-class behaviors only with a high resolution of at least 10242 grid cells. With lower resolutions, we observed the formation of a dominant flux tube, which accompanies the separation of the magnetic field from the background flow. The specific requirements for the simulation of the SF class should depend on the code (and the numerical scheme) as well as the initial setup, but our results do indicate that very high resolution would be required for converged results in simulation studies of MHD turbulence.

Variational principles and Lyapunov stability analysis for compressible cylindrical plasma flows with arbitrary cross-section

The equilibrium and stability properties of ideal compressible magnetohydrodynamic (MHD) flows are investigated. The domain is taken to be cylindrical with arbitrary cross-section. Variational principles for plasma equilibria with mass flow are formulated, where we associate the cylindrical MHD equilibrium states to critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross-helicity), the momentum and the magnetic helicity. Lyapunov stability conditions for compressible cylindrical ideal MHD flows with arbitrary cross-section are determined.

A conservative pressure‐correction method for the Euler and ideal MHD equations at all speeds

AbstractTo efficiently compute weakly compressible magnetohydrodynamic flows in astrophysical applications, approximate low Mach number reduced forms of the compressible MHD equations are frequently used. This is because standard characteristic‐based schemes for the full compressible MHD equations are inefficient and inaccurate for computing low Mach number magnetohydrodynamic flow, as a result of the increasing stiffness and weakening pressure/density coupling of the equations when M ↓ 0. Furthermore, these schemes are colocated, so that additional and artificial measures have to be taken to ensure solenoidality of the magnetic field. We present a new method with a staggered spatial discretization and a pressure‐correction solution algorithm that is more suitable for computing weakly compressible MHD flow, because of its Mach‐uniformity and efficient and accurate handling of the solenoidality constraint on the magnetic field. Copyright © 2002 John Wiley & Sons, Ltd.

On the inadmissibility of non-evolutionary shocks

In recent years, numerical solutions of the equations of compressible magnetohydrodynamic (MHD) flows have been found to contain intermediate shocks for certain kinds of problems. Since these results would seem to be in conflict with the classical theory of MHD shocks, they have stimulated attempts to reexamine various aspects of this theory, in particular the role of dissipation. In this paper, we study the general relationship between the evolutionary conditions for discontinuous solutions of the dissipation-free system and the existence and uniqueness of steady dissipative shock structures for systems of quasilinear conservation laws with a concave entropy function. Our results confirm the classical theory. We also show that the appearance of intermediate shocks in numerical simulations can be understood in terms of the properties of the equations of planar MHD, for which some of these shocks turn out to be evolutionary. Finally, we discuss ways in which numerical schemes can be modified in order to avoid the appearance of intermediate shocks in simulations with such symmetry.

The Magnetohydrodynamic Kelvin‐Helmholtz Instability: A Three‐dimensional Study of Nonlinear Evolution

We investigate through high-resolution three-dimensional simulations the nonlinear evolution of compressible magnetohydrodynamic flows subject to the Kelvin-Helmholtz instability. As in our earlier work, we have considered periodic sections of flows that contain a thin, transonic shear layer but are otherwise uniform. The initially uniform magnetic field is parallel to the shear plane but oblique to the flow itself. We confirm in three-dimensional flows the conclusion from our two-dimensional work that even apparently weak magnetic fields embedded in Kelvin-Helmholtz unstable plasma flows can be fundamentally important to nonlinear evolution of the instability. In fact, that statement is strengthened in three dimensions by this work because it shows how field-line bundles can be stretched and twisted in three dimensions as the quasi-two-dimensional Cat's Eye vortex forms out of the hydrodynamical motions. In our simulations twisting of the field may increase the maximum field strength by more than a factor of 2 over the two-dimensional effect. If, by these developments, the Alfvén Mach number of flows around the Cat's Eye drops to unity or less, our simulations suggest that magnetic stresses will eventually destroy the Cat's Eye and cause the plasma flow to self-organize into a relatively smooth and apparently stable flow that retains memory of the original shear. For our flow configurations, the regime in three dimensions for such reorganization is 4 ≲ MAx ≲ 50, expressed in terms of the Alfvén Mach number of the original velocity transition and the initial Alfvén speed projected to the flow plan. When the initial field is stronger than this, the flow either is linearly stable (if MAx ≲ 2) or becomes stabilized by enhanced magnetic tension as a result of the corrugated field along the shear layer before the Cat's Eye forms (if MAx ≳ 2). For weaker fields the instability remains essentially hydrodynamic in early stages, and the Cat's Eye is destroyed by the hydrodynamic secondary instabilities of a three-dimensional nature. Then, the flows evolve into chaotic structures that approach decaying isotropic turbulence. In this stage, there is considerable enhancement to the magnetic energy due to stretching, twisting, and turbulent amplification, which is retained long afterward. The magnetic energy eventually catches up to the kinetic energy, and the nature of flows becomes magnetohydrodynamic. Decay of the magnetohydrodynamic turbulence is enhanced by dissipation accompanying magnetic reconnection. Hence, in three dimensions as in two dimensions, very weak fields do not modify substantially the character of the flow evolution but do increase global dissipation rates.

The formation of large-scale structures in supersonic magnetohydrodynamic flows

In this paper we investigate the existence of an inverse cascade of magnetic helicity for statistically steady compressible magnetohydrodynamic flows with rms Mach numbers that go up to unity. This has been done by performing a set of three-dimensional numerical simulations that were carried out using the RIEMANN code for astrophysical fluid and magnetofluid dynamics [D. S. Balsara, Astrophys. J. Suppl. 116, 133 (1988)]. The random forcing is made up of ABC flows corresponding to helical wave patterns at an intermediate scale. Both time histories of integral quantities and Fourier spectra are shown, illustrating the inverse cascade process. The observed build-up of large-scale magnetic helical structures for a wide range of Mach numbers indicates that, in the context of the generation of the magnetic field of the Galaxy where supersonic flows are common, a dynamo mechanism based on the nonlinear dynamics of magnetic helicity may be at work.

The effect of hall currents on the magneto hydro dynamic (MHD) boundary layer flow over a flat plate

Studies are made on laminar compressible magneto hydro dynamic (MHD) boundary layer flow over a flat-plate with Hall and ionslip currents. Viscous drag, magnetic drag and heat transfer coefficients are determined for the values of interaction parameter, electromagnetic load parameter, Hall parameter and ionslip current parameter. It is found that the overall drag (magnetic plus viscous) coefficient increases and the heat transfer coefficient decreases with interaction parameter.

Self-organization in three-dimensional compressible magnetohydrodynamic flow

A three-dimensional self-organization process of a compressible dissipative plasma with a velocity-magnetic field correlation is investigated in detail by means of a variational method and a magnetohydrodynamic simulation. There are two types of relaxation, i.e., fast relaxation in which the cross helicity is not conserved and slow relaxation in which the cross helicity is conserved approximately. In the slow relaxation case the cross helicity consists of two components with opposite sign that have almost the same amplitude in the large wavenumber region. In both cases the system approaches a high correlation state, dependent on the initial condition. These results are consistent with observational data of the solar wind. Selective dissipation of magnetic energy, normal cascade of magnetic energy spectrum, and inverse cascade of magnetic helicity spectrum are observed for the sub-Alfvénic flow case, as was observed previously for the zero-flow case. When the flow velocity is super-Alfvénic, the relaxation process is altered significantly from the zero-flow case.

Boundary layer flow with variable tensor electrical conductivity

Studies are made on laminar compressible magnetohydrodynamic (MHD) boundary layer flow over an insulated wall with variable tensor electrical conductivity. The effects of the electromagnetic loading parameter, the magnetic interaction parameter and the ratio of the wall temperature to the free stream temperature on the boundary layer physical parameters have been discussed.

The limit behavior of relaxation time for full compressible magnetohydrodynamic flows with Cattaneo’s law

The limit behavior of relaxation time for full compressible magnetohydrodynamic flows with Cattaneo’s law