Magnetohydrodynamic Approximation Research Articles

Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations

The full compressible magnetohydrodynamic equations can be derived formally from the complete electromagnetic fluid system in some sense as the dielectric constant tends to zero. This process is usually referred as magnetohydrodynamic approximation in physical books. In this paper we justify this singular limit rigorously in the framework of smooth solutions for well-prepared initial data.

ON THE SPATIAL SCALES OF WAVE HEATING IN THE SOLAR CHROMOSPHERE

Dissipation of magnetohydrodynamic (MHD) wave energy has been proposed as a viable heating mechanism in the solar chromospheric plasma. Here, we use a simplified one-dimensional model of the chromosphere to theoretically investigate the physical processes and the spatial scales that are required for the efficient dissipation of Alfv\'en waves and slow magnetoacoustic waves. We consider the governing equations for a partially ionized hydrogen-helium plasma in the single-fluid MHD approximation and include realistic wave damping mechanisms that may operate in the chromosphere, namely Ohmic and ambipolar magnetic diffusion, viscosity, thermal conduction, and radiative losses. We perform an analytic local study in the limit of small amplitudes to approximately derive the lengthscales for critical damping and efficient dissipation of MHD wave energy. We find that the critical dissipation lengthscale for Alfv\'en waves depends strongly on the magnetic field strength and ranges from 10~m to 1~km for realistic field strengths. The damping of Alfv\'en waves is dominated by Ohmic diffusion for weak magnetic field and low heights in the chromosphere, and by ambipolar diffusion for strong magnetic field and medium/large heights in the chromosphere. Conversely, the damping of slow magnetoacoustic waves is less efficient, and spatial scales shorter than 10~m are required for critical damping. Thermal conduction and viscosity govern the damping of slow magnetoacoustic waves and play an equally important role at all heights. These results indicate that the spatial scales at which strong wave heating may work in the chromosphere are currently unresolved by observations.

Planet migration and magnetic torques

AbstractThe possibility that magnetic torques may participate in close-in planet migration has recently been postulated. We develop three dimensional global models of magnetic star-planet interaction under the ideal magnetohydrodynamic (MHD) approximation to explore the impact of magnetic topology on the development of magnetic torques. We conduct twin numerical experiments in which only the magnetic topology of the interaction is altered. We find that magnetic torques can vary by roughly an order of magnitude when varying the magnetic topology from an aligned case to an anti-aligned case. Provided that the stellar magnetic field is strong enough, we find that magnetic migration time scales can be as fast as ~100 Myr. Hence, our model supports the idea that magnetic torques may participate in planet migration for some close-in star-planet systems.

Magnetic energy fluxes in close-in star-planet systems

AbstractMagnetic interactions between a close-in planet and its host star have been postulated to be a source of enhanced chromospheric emissions. We develop three dimensional global models of star-planet systems under the ideal magnetohydrodynamic (MHD) approximation to explore the impact of magnetic topology on the energy fluxes induced by the magnetic interaction. We conduct twin numerical experiments in which only the magnetic topology of the interaction is altered. We find that the Poynting flux varies by more than an order of magnitude when varying the magnetic topology from an aligned case to an anti-aligned case. This provides a simple and robust physical explanation for on/off enhanced chromospheric emissions induced by a close-in planet on time-scales of the order of days to years.

Evolution of generalized two‐dimensional magnetotail equilibria in ideal and resistive MHD

AbstractWe present results of two‐dimensional (2‐D) magnetohydrodynamic (MHD) simulations of the terrestrial magnetotail. A regional adaptation of the Lyon‐Fedder‐Mobarry global MHD model is used. As initial conditions, we employ a class of asymptotic magnetotail equilibria with and without an accumulation of magnetic flux at the tailward end (a Bz hump). The former have been recently shown by full particle simulations to be unstable to a kinetic mode with formal properties of ion tearing. Thus, our goal here is to investigate the evolution of the same equilibria in the MHD approximation and assist in the physical interpretation of the kinetic simulations. This is additionally motivated by the energy principle considerations which suggest that if the system is unstable kinetically, it may also be unstable ideally. To seek dynamical MHD regimes similar to those observed in kinetic simulations, we implement two sets of boundary conditions (velocity balanced, VB, and momentum balanced, MB), one allowing plasma flows through the boundaries and the other inhibiting such flows. The use of more reflecting MB boundary conditions results in suppression of any significant dynamics, and we see no substantial changes beyond initial equilibrium relaxation. On the other hand, VB boundary conditions allow a more efficient relaxation of initial equilibrium and absorb subsequently generated plasma flows. With these boundary conditions we find the equilibrium without a flux accumulation (i.e., with constant magnetic field component normal to the current sheet) to develop an apparently resistive mode accompanied by tailward plasma flows. At the same time, the equilibria with a Bz hump of sufficiently large amplitude develop a different, ideal, mode characterized by spontaneous generation of earthward plasma flows and an exponential growth of the corresponding electric field. This growth is qualitatively similar to the corresponding fully kinetic simulations although no explosive growth of the earthward moving Bz peak is evident in the MHD calculations, just an earthward shift of a part of the initial flux accumulation. We discuss implications of our results for the possibility of existence and impact of such equilibria in the Earth's magnetotail and in global MHD simulations.

Ion Beam Instability in Hall Thrusters

Stability of an ion flux, bounded with an anode and cathode, in Hall thrusters is investigated by theoretical and numerical modeling. Two-fluid magnetohydrodynamic approximation with cold magnetized electrons and cold nonmagnetized ions is used. The inertia of the electrons is taken into consideration. The perturbations are assumed to be quasi-neutral, potential, and dependent on a single spatial coordinate only. For simplicity, the magnetic field is assumed to be uniform. It is shown that the presence of the boundaries, where the potential of the ion beam is fixed, can cause instability of the beam. The growth rate of instability and frequency excited oscillation is of the order of the reciprocal of the ion crossing time between the anode and cathode. In the limit of a uniform unperturbed state of plasma, the instability is analogous to the Pierce instability of an electron beam with the fundamental distinction that here, the perturbations are quasi-neutral. It is shown that the instability is alternately a periodical and oscillating depending on the range of the α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lh</sub> parameter, which includes induction of the magnetic field (through lower hybrid frequency), thickness of the acceleration layer, and velocity of the ion flux.

Three-dimensional magnetohydrodynamic model of the interaction between the solar wind and cometary atmospheres

A magnetohydrodynamic (MHD) model of the interaction between the solar wind and an expanding gas flow issuing from a comet nucleus is presented. The processes of photoionization of the neutral component of the comet matter and the resonance charge exchange between charged and neutral particles are taken into account, together with the interplanetary magnetic field effect. The adequacy of the single-fluid MHD approximation in describing the plasma flow with different temperatures of the components is shown. The model is numerically realized on the basis of the higher-order Godunov method using an approximate solution of the arbitrary MHD discontinuity breakdown problem. The results of a numerical investigation of the plasma and magnetic field parameters in the vicinity of a comet are presented for the conditions corresponding to the flyby of several spacecraft near comet Halley in March 1986 and the expected fly-round of comet Churyumov-Gerasimenko by the Rosetta spacecraft in 2015.

Full kinetic simulations of plasma flow interactions with meso- and microscale magnetic dipoles

We examined the plasma flow response to meso- and microscale magnetic dipoles by performing three-dimensional full particle-in-cell simulations. We particularly focused on the formation of a magnetosphere and its dependence on the intensity of the magnetic moment. The size of a magnetic dipole immersed in a plasma flow can be characterized by a distance L from the dipole center to the position where the pressure of the local magnetic field becomes equal to the dynamic pressure of the plasma flow under the magnetohydrodynamics (MHD) approximation. In this study, we are interested in a magnetic dipole whose L is smaller than the Larmor radius of ions riL calculated with the unperturbed dipole field at the distance L from the center. In the simulation results, we confirmed the clear formation of a magnetosphere consisting of a magnetopause and a tail region in the density profile, although the spatial scale is much smaller than the MHD scale. One of the important findings in this study is that the spatial profiles of the plasma density as well as the current flows are remarkably affected by the finite Larmor radius effect of the plasma flow, which is different from the Earth's magnetosphere. The magnetopause found in the upstream region is located at a position much closer to the dipole center than L. In the equatorial plane, we also found an asymmetric density profile with respect to the plasma flow direction, which is caused by plasma gyration in the dipole field region. The ion current layers are created in the inner region of the dipole field, and the electron current also flows in the region beyond the ion current layer because ions with a large inertia can closely approach the dipole center. Unlike the ring current structure of the Earth's magnetosphere, the current layers in the microscale dipole fields are not circularly closed around the dipole center. Since the major current is caused by the particle gyrations, the current is independently determined to be in the direction of the electron and ion gyrations, which are the same in both the upstream and downstream regions. The present analysis on the formation of a magnetosphere in the regime of a microscale magnetic dipole is significant for understanding the solar wind response to the crustal magnetic anomalies on the Moon surface, such as were recently observed by spacecraft.

Plasma Instability Modeling With Mixed Wave/Diffusion Field Behavior

Recent efforts have extended fluid dynamic modeling to include low-electrical-conductivity (wavelike) electromagnetic field effects in plasmas. Here, we present results of a new study to explore and illustrate the capability of resolving such effects in the development of plasma instabilities. Recent work[1, 2] has explored the development of an efficient numerical procedure for simulating the coupled fluid dynamic and full electromagnetic (Maxwell) equations. This system of equations is more general than the resistive magnetohydrodynamic (MHD) model. The physical advantage to this system is that the inclusion of the displacement current and charge separation effects extends fluid dynamic modeling of plasmas to include regions of extremely low (or zero) electrical conductivity, where the electromagnetic fields possess a wave nature. Capturing the correct physical behavior of plasma instabilities in the limit of extremely low electrical conductivity is of significance for modern aerospace applications, including energy bypass arrangements for scramjet propulsion using magnetohydrodynamic augmentation, due to the extremely low electrical conductivity of the aerospace environment. The results depicted in this paper illustrate the potential for the numerical method realized in Ref. 1 to effectively resolve this behavior in cases where the MHD approximation may not be valid due to an exceedingly low electrical conductivity. A new study was initiated to explore more fully the capabilities to resolve regions of both low and high electrical conductivity using this numerical scheme. The problem selected for this study was the classic MHD KelvinHelmholtz instability. The initial and boundary conditions are provided in Ref. 2. A periodic box is initialized with two fluids of different density moving in opposite directions. An initial small perturbation is applied to the interfaces between the fluids, seeding an instability. A 256x256 grid size using a Roe solver for the coupled system of Navier-Stokes and full Maxwell equations was used, as detailed in [1, 2]. Three cases were explored: Case A was assigned a zero electrical conductivity, σ = 0 (purely hydrodynamic), Case B was given a small electrical conductivity of σ = 1, and Case C was given a very high conductivity of σ = 10 6 . Case A provides a hydrodynamic benchmark. Case B permits an investigation of a simulation that includes the effects of electromagnetic wave propagation in the plasma evolution. Case C effectively realizes the magnetohydrodynamic approximation. Results for the simulations are presented in a unique fashion in Fig. 1. For each subfigure, three spacetime isosurfaces of the density (blue, green and orange corresponding to the color range) have been depicted. Isosurfaces are the same for all three presented subfigures. Time flows from right (initial conditions) to left (final solution). For Case A, the fluid achieves a highly chaotic solution. The propagation of the initial disturbance at the interfaces between the two fluids is clear, as well as their development into a highly turbulent flow. Case B possesses an extremely low electrical conductivity, and therefore is not subject to the magnetic diffusion approximation of MHD. In fact, electromagnetic wave propagation was visible in this solution. The presence of a finite electrical conductivity provides some diffusion, but it does not match very well to the MHD prediction. However, the conductivity is large enough to abate the low-wavelength instabilities somewhat, and provide stability to the flow. Case C realizes the magnetohydrodynamic limit, where the magnetic field tends to stabilize the flow and strongly suppresses low-wavelength instabilities. A much more organized series of flow structures evolve through time to the final state. The magnetic field is effectively “frozen” to the fluid with such a large electrical conductivity, and the field strongly diffuses with little wave propagation.

Bifurcation phenomena and control for magnetohydrodynamic flows in a smooth expanded channel

This work reports the effects of magnetic field on an electrically conducting fluid with low electrical conductivity flowing in a smooth expanded channel. The governing nonlinear magnetohydrodynamic (MHD) equations in induction-free situations are derived in the framework of MHD approximations and solved numerically using the finite-difference technique. The critical values of Reynolds number (based on upstream mean velocity and channel height) for symmetry breaking bifurcation for a sudden expansion channel (1:4) is about 36, whereas the value in the case of the smooth expansion geometry used in this work is obtained as 298, approximately (non-magnetic case). The flow of an electrically conducting fluid in the presence of an externally applied constant magnetic field perpendicular to the plane of the flow is reduced significantly depending on the magnetic parameter (M). It is found that the critical value of Reynolds number for smooth expansion (1:4) is about 475 for the magnetic parameter M = 2. The separating regions developed behind the smooth symmetric expansion are decreased in length for increasing values of the magnetic parameter. The bifurcation diagram is shown for a symmetric smoothly expanding channel. It is noted that the critical values of Reynolds number increase with increasing magnetic field strength.

COLLAPSE AND FRAGMENTATION OF MAGNETIC MOLECULAR CLOUD CORES WITH THE Enzo AMR MHD CODE. II. PROLATE AND OBLATE CORES

We present the results of a large suite of three-dimensional (3D) models of the collapse of magnetic molecular cloud cores using the adaptive mesh refinement (AMR) code Enzo2.2 in the ideal magnetohydrodynamics (MHD) approximation. The cloud cores are initially either prolate or oblate, centrally condensed clouds with masses of 1.73 or 2.73 $M_\odot$, respectively. The radial density profiles are Gaussian, with central densities 20 times higher than boundary densities. A barotropic equation of state is used to represent the transition from low density, isothermal phases, to high density, optically thick phases. The initial magnetic field strength ranges from 6.3 to 100 $\mu$G, corresponding to clouds that are strongly to marginally supercritical, respectively, in terms of the mass to magnetic flux ratio. The magnetic field is initially uniform and aligned with the clouds' rotation axes, with initial ratios of rotational to gravitational energy ranging from $10^{-4}$ to 0.1. Two significantly different outcomes for collapse result: (1) formation of single protostars with spiral arms, and (2) fragmentation into multiple protostar systems. The transition between these two outcomes depends primarily on the initial magnetic field strength, with fragmentation occurring for mass to flux ratios greater than about 14 times the critical ratio for prolate clouds. Oblate clouds typically fragment into several times more clumps than prolate clouds. Multiple, rather than binary, system formation is the general rule in either case, suggesting that binary stars are primarily the result of the orbital dissolution of multiple protostar systems.

Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid

In this paper, we prove the uniform estimates with respect to the dielectric constant and the global-in-time existence of the 1-D full equations for a thermo-radiative electromagnetic fluid in a bounded interval without vacuum. We establish the uniform estimates in the dielectric constant, which gives that, as the dielectric constant tends to zero, the solutions to the 1-D full thermo-radiative electromagnetic equations will converge to the one for the 1-D full magnetohydrodynamic equations with thermo-radiations and magnetic diffusions. This approximation is usually referred to as the magnetohydrodynamic approximation, where the displacement current is not taken into account.

Uniform existence of the 1-d complete equations for an electromagnetic fluid

In this paper, we prove the global-in-time and uniform-in-ϵ existence of the 1-d complete equations for an electromagnetic fluid in a bounded interval without vacuum, where ϵ is the dielectric constant. Consequently, the limit as ϵ→0 can be established. This approximation is usually referred to as the magnetohydrodynamic approximation, and is equivalent to the neglect of the displacement current.

EVOLUTION OF PLASMOID-CHAIN IN POYNTING-DOMINATED PLASMA

We present our recent results of the evolution of the plasmoid-chain in a Poynting dominated plasma. We model the relativistic current sheet with cold background plasma using the relativistic resistive magnetohydrodynamic approximation, and solve its temporal evolution numerically. Numerical results show that the initially induced plasmoid triggers a secondary tearing instability. We find the plasmoid-chain greatly enhances the reconnection rate, which becomes independent of the Lundquist number, when this exceeds a critical value. Since magnetic reconnection is expected to play an important role in various high energy astrophysical phenomena, our results can be used for explaining the physical mechanism of them.

Magnetic reconnection under anisotropic magnetohydrodynamic approximation

We study the formation of slow-mode shocks in collisionless magnetic reconnection by using one- and two-dimensional collisionless MHD codes based on the double adiabatic approximation and the Landau closure model. We bridge the gap between the Petschek-type MHD reconnection model accompanied by a pair of slow shocks and the observational evidence of the rare occasion of in-situ slow shock observations. Our results showed that once magnetic reconnection takes place, a firehose-sense (p∥&amp;gt;p⊥) pressure anisotropy arises in the downstream region, and the generated slow shocks are quite weak comparing with those in an isotropic MHD. In spite of the weakness of the shocks, however, the resultant reconnection rate is 10%–30% higher than that in an isotropic case. This result implies that the slow shock does not necessarily play an important role in the energy conversion in the reconnection system and is consistent with the satellite observation in the Earth's magnetosphere.

Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model

We use the expansion-normalized variables approach to study the dynamics of a non-tilted Bianchi Type I cosmological model with both a homogeneous magnetic field and a viscous fluid. In our model the perfect magnetohydrodynamic approximation is made, and both bulk and shear viscous effects are retained. The dynamical system is studied in detail through a fixed-point analysis which determines the local sink and source behavior of the system. We show that the fixed points may be associated with Kasner-type solutions, a flat universe FLRW solution, and interestingly, a new solution to the Einstein Field equations involving non-zero magnetic fields, and non-zero viscous coefficients. It is further shown that for certain values of the bulk and shear viscosity and equation of state parameters, the model isotropizes at late times.

EVOLUTION OF RELATIVISTIC PLASMOID CHAINS IN A POYNTING-DOMINATED PLASMA

In this paper, we investigate the evolution of the plasmoid-chain in a Poynting-dominated plasma. We model the relativistic current sheet with cold background plasma using the relativistic resistive magnetohydrodynamic approximation, and solve its temporal evolution numerically. We perform various calculations using different magnetization parameters of the background plasma and different Lundquist numbers. Numerical results show that the initially induced plasmoid triggers a secondary tearing instability, which gradually fills the current sheet with plasmoids, as has also been observed in the non-relativistic case. We find the plasmoid-chain greatly enhances the reconnection rate, which becomes independent of the Lundquist number, when this exceeds a critical value. In addition, we show the distribution of plasmoid size becomes a power law. Since magnetic reconnection is expected to play an important role in various high energy astrophysical phenomena, our results can be used for explaining the physical mechanism of them.

Counter effects of meridional flows and magnetic fields in stationary axisymmetric self-gravitating barotropes under the ideal MHD approximation: clear examples – toroidal configurations

We obtain the general forms for the current density and the vorticity from the integrability conditions of the basic equations which govern the stationary states of axisymmetric magnetized self-gravitating barotropic objects with meridional flows under the ideal magnetohydrodynamics (MHD) approximation. As seen from the stationary condition equations for such bodies, the presence of the meridional flows and that of the poloidal magnetic fields act oppositely on the internal structures. The different actions of these two physical quantities, the meridional flows and the poloidal magnetic fields, could be clearly seen through stationary structures of the toroidal gaseous configurations around central point masses in the framework of Newtonian gravity because the effects of the two physical quantities can be seen in an amplified way for toroidal systems compared to those for spheroidal stars. The meridional flows make the structures more compact, i.e. the widths of toroids thinner, while the poloidal magnetic fields are apt to elongate the density contours in a certain direction depending on the situation. Therefore, the simultaneous presence of the internal flows and the magnetic fields would work as if there were no such different actions within and around the stationary gaseous objects such as axisymmetric magnetized toroids with internal motions around central compact objects under the ideal MHD approximation, although these two quantities might exist in real systems.

Allowance for the interaction between the underwater electric discharge channel plasma and the shock wave reflected from the chamber’s wall

The numerical simulation of a spark discharge formed along the axis of a cylindrical chamber filled with water has been carried out in the magnetohydrodynamic approximation. The simulation results are compared with the data known from the literature. The analysis of the spatiotemporal distribution of the pressure and temperature in the discharge chamber has been performed with due account for the interaction between the shock waves excited by the spark discharge and reflected from the chamber’s wall and the plasma channel.

Lattice Boltzmann magnetohydrodynamics with current-dependent resistivity

Lattice Boltzmann magnetohydrodynamics is extended to allow the resistivity to be a prescribed function of the local current density. Current-dependent resistivities are used to model the so-called anomalous resistivity caused by unresolved small-scale processes, such as current-driven plasma microturbulence, that are excluded by the magnetohydrodynamics approximation. These models closely resemble the Smagorinsky eddy viscosity model used in large eddy simulations of hydrodynamic turbulence. Lattice Boltzmann implementations of the Smagorinsky model adjust the collision time in proportion to the local rate of strain, as obtained from the non-equilibrium parts of the hydrodynamic distribution functions. This works successfully even with a single relaxation time collision operator. However, the existing lattice Boltzmann magnetohydrodynamic implementation contains a spurious term in the evolution equation for the magnetic field that violates the divergence-free condition when the relaxation time varies in space. A correct implementation requires a matrix collision operator for the magnetic distribution functions. The relaxation time imposed on the antisymmetric component of the electric field tensor is calculated locally from the non-equilibrium part of the magnetic distribution functions, which determine the current, while the symmetric component remains subject to a uniform relaxation time to suppress the spurious term. The resulting numerical solutions are shown to converge to independent spectral solutions of the magnetohydrodynamic equations, and to preserve the divergence-free condition up to floating point round-off error.