Resistive Magnetohydrodynamic Equations Research Articles

Exact solutions for reconnective magnetic annihilation

A family of exact solutions of the steady resistive nonlinear magnetohydrodynamic equations in two dimensions (x, y) is presented for reconnective annihilation, in which the magnetic field is advec...

Effective gravity due to the Hall effect and quasi-acoustic–gravity waves in accelerated plasma flows

The equations of resistive magnetohydrodynamics taking into account the Hall effect are solved analytically for the case of a planar plasma flow across the magnetic field. Waves of purely acoustic nature are revealed that propagate in the plasma medium on the background of a steady magnetically accelerated plasma flow of a rather arbitrary form. The magnetic field manifests itself in this process only in that it produces an effective gravity force, the ‘gravitational’ acceleration being proportional to ωeτe. Like acoustic–gravity waves in the atmosphere, such quasi- acoustic–gravity (QAG) waves in a plasma can increase greatly in amplitude as they propagate ‘upwards’, i.e., in this case, counter to the direction of the electric field. In a plasma channel (where the plasma flows between the electrodes), a system of standing QAG waves arises, which has a stationary component. These waves are responsible for a number of distinctive features in the behaviour of Hall plasma flows, in particular for very large gradients and unexpected small-scale fluctuations of physical quantities near the anode, revealed earlier by numerical simulation. In the cases where the plasma-flow parameters provide a high intensity of these fluctuations, the flow appears to be unstable. In another range of parameter values, the QAG waves are strongly coupled with the electric currents that they induce in the electrodes, their amplitudes increase with time, and, again, the flow cannot be steady. The existence of a rather general dimensionless similarity criterion for resistive Hall plasma planar flows is also shown. This criterion can be found directly from the structure of the equations without any restrictions on the plasma parameters or on the form of the flow.

Nonlinear hydromagnetic waves in a nonideal magnetohydrodynamic plasma: Exact translationally symmetric solutions

The time evolution of translationally symmetric hydromagnetic waves in a nonideal magnetohydrodynamic (MHD) plasma is investigated. Exact time-dependent wave solutions of the nonlinear resistive and dissipative MHD equations for the plasma within a cylindrical annulus are presented. It is shown that the magnetic field and the velocity have different damping time and that in the asymptotic limit t→∞ the plasma relaxes toward the equilibrium state involving uniform rotation and no magnetic field.

Numerical simulations and simplified models of nonlinear electron inertial Alfvén waves

We describe nonlinear resonance absorption of compressional Alfvén waves in a model magnetosphere. It is shown that the ponderomotive force of excited standing shear Alfvén waves can lead to nonlinear saturation and spatial structuring of field line resonances and that in low‐beta plasmas ponderomotive saturation may occur before other nonlinear saturation processes such as the Kelvin‐Helmholtz instability. The effects of finite electron inertia are also considered, and it is shown that spatial structuring of field line resonances may occur with scale sizes compatible with those found in discrete auroral arcs. Results are discussed in the context of three‐dimensional numerical solutions to the resistive magnetohydrodynamic equations and based on a simplified analytical model of nonlinear resonance absorption.

Pfirsch–Schlüter transport for toroidal elliptic magnetic surfaces in tokamak

Transport equations in the collisional regime have been found for very general geometries in tokamaks, using resistive magnetohydrodynamic equations, neglecting the toroidal field Eφ and assuming toroidal axisymmetry. Detailed analytic results have been obtained for families of elliptic magnetic surfaces. In the limit case of equal semiaxis and large aspect ratio, the well known Pfirsch–Schlüter result is obtained.

Self-consistent models of incompressible stationary reconnection by weakly two dimensional extension of magnetic annihilation

Weakly two-dimensional self-consistent solutions of the stationary resistive magnetohydrodynamic equations are presented. The solutions are found by applying an asymptotic expansion procedure to the stationary resistive MHD equations with the annihilation solution as lowest order solution. Two different approximation schemes are investigated. Whereas the first scheme allows the introduction of a normal magnetic field component in the lowest order, the second scheme also introduces a weak variation of the current density along the current sheet in the lowest order. This is closely related to the fact that in the first case the inflow velocity does not vary along the current sheet, whereas in the second case the inflow velocity may vary weakly along the current sheet.

Damping of nonlinear hydromagnetic waves: Exact three-dimensional solutions of the nonideal magnetohydrodynamic equations

The time evolution of three-dimensional hydromagnetic waves in a nonideal magnetohydrodynamic (MHD) plasma is investigated. Two types of exact time-dependent wave solutions of the nonlinear resistive and dissipative MHD equations are found. One solution shows a damped Alfvén wave and the other one shows a magnetically compressional wave with the magnetic and velocity fields having different damping times. It is shown that in the asymptotic limit, t → ∞, the plasma relaxes toward the equilibrium state with no magnetic field and constant (for the former solution) or sheared (for the latter solution) velocity.

Electron diamagnetic drift effect on the trapped hot particle-driven resistive interchange mode

The linearized flux-surface-averaged resistive magnetohydrodynamic equations, which include the electron diamagnetic drift and hot particle effects, are derived and studied numerically. The effect of electron diamagnetic drift on the trapped hot particle-driven resistive interchange mode is then investigated. It is shown that the effect can have a strong stabilization effect on the mode, similar to the conventional resistive interchange mode case.

Problems and Progress in Computing Three-Dimensional Coronal Active Region Magnetic Fields from Boundary Data

An overview of the whole process of reconstructing the coronal magnetic field from boundary data measured at the photosphere is presented. We discuss the errors and uncertainties in the data and in the data reduction process. The problems include noise in the magnetograph measurements, uncertainties in the interpretation of polarization signals, the 180° ambiguity in the transverse field, and the fact that the photosphere is not force-free. Methods for computing the three-dimensional structure of coronal active region magnetic fields, under the force-free assumption, from these boundary data, are then discussed. The methods fall into three classes: the ‘extrapolation’ technique, which seeks to integrate upwards from the photosphere using only local values at the boundary; the ‘current-field iteration’ technique, which propagates currents measured at the boundary along field lines, then iteratively recomputes the magnetic field due to this current distribution; and the ‘evolutionary’ technique, which simulates the evolution of the coronal field, under quasi-physical resistive magnetohydrodynamic equations, as currents injected at the boundary are driven towards the observed values. The extrapolation method is mathematically ill-posed, and must be heavily smoothed to avoid exponential divergence. It may be useful for tracing low-lying field lines, but appears incapable of reconstructing the magnetic field higher in the corona. The original formulation of the current-field iteration method had problems achieving convergence, but a recent reformulation appears promising. Evolutionary methods have been applied to several real datasets, with apparent success.

Numerical Simulation of Solar Coronal X-Ray Jets Based on the Magnetic Reconnection Model

We performed two-dimensional numerical simulations of solar coronal X-ray jets by solving the resistive magnetohydrodynamic (MHD) equations. The simulations were based on the magnetic reconnection model, in which the plasma of an X-ray jet is accelerated and heated by reconnection between the emerging flux and a pre-existing coronal field. Many observed characteristics of X-ray jets could be successfully reproduced. Morphologically, the two observed types of jets, two-sided-loop type and anemone-jet type, were well reproduced. Here, the two-sided-loop type is a pair of horizontal jets (or loops), which occurs when an emerging flux appears in a quiet region where the coronal field is approximately horizontal. In contrast, the anemone-jet type is a vertical jet, which takes place when an emerging flux appears in a coronal hole where the coronal field is vertical or oblique. Quantitatively, the velocity, temperature, thermal energy, kinetic energy, and other parameters obtained in the simulation are in good agreement with the observations. Furthermore, the simulations reveal new features which might be associated with X-ray jets: (1) A fast-mode MHD shock is produced at the collision site of each reconnection jet with the ambient magnetic field. (2) Reconnection produces a cool jet as well as a hot jet (X-ray jet). The hot and cool jets are adjacent to each other, which is consistent with the observed simultaneous coexistence of X-ray jets and Hα surges in the sun.

Nonlinear interaction of tearing modes in an infinite-aspect-ratio tokamak

The nonlinear interaction of tearing modes centered on different resonant surfaces in a low-β tokamak is studied. Center manifold theory is applied to the reduced equations of resistive magnetohydrodynamics, including the electron energy equation. For the equilibrium current profile a polynomial parametrized by the values of shear and safety factor at several given radii is chosen. Amplitude equations describing the asymptotic time behavior of interacting modes are obtained. The coefficients of these equations depend on the mode numbers of the modes involved and on the parameters of the problem. They were calculated for the interaction of the (2,1)- and the (3,2)-tearing mode. The dependence of the corresponding solutions on equilibrium parameters and transport coefficients is investigated.

Damping of nonlinear hydromagnetic waves: Exact solutions of the nonideal magnetohydrodynamic equations

The time evolution of hydromagnetic waves in a nonideal magnetohydrodynamic (MHD) plasma is investigated. Exact time-dependent wave solutions of the nonlinear resistive and dissipative MHD equations for a plane geometry are presented to show that the magnetic field and the velocity have different damping times and that in the asymptotic limit t → ∞ the plasma relaxes toward the equilibrium state with no magnetic and velocity fields.

Dynamics of current sheet formation and reconnection in two-dimensional coronal loops

Current sheet formation and magnetic reconnection in a two-dimensional coronal loop with an X-type neutral line are simulated numerically using compressible, resistive magnetohydrodynamic equations. Numerical results in the linear and nonlinear regimes are shown to be in good agreement with a recent analytical theory [X. Wang and A. Bhattacharjee, Astrophys. J. 420, 415 (1994)]. The topological constraint imposed by helicity-conserving reconnection is discussed. It is found numerically that helicity-conserving reconnection causes the initial X-point structure of the loop to change to Y points, with current sheets at the separatrices encompassing the Y points. Implications for observations are discussed.

Fast reconnection in high temperature plasmas

Pressure forces acting on electrons are shown to dramatically alter magnetic field line reconnection in high temperature plasmas. The electron pressure introduces a new physical scale length ρs, the ion gyroradius based on the electron temperature, into the resistive magnetohydrodynamic (MHD) equations. The single dissipation layer of resistive MHD is split into two distinct layers by this effect: a very small inner current layer and a larger flow layer. Unlike resistive MHD, the current layer is microscopic in the outflow, as well as the inflow, direction. As a consequence, the current layer is not unstable to the formation of secondary magnetic islands at low values of resistivity and patchy reconnection does not occur. The absence of a strong current sheet in the outflow region enables the magnetic nozzle controlling the outflow to open up. The magnetic reconnection rate therefore remains large as the resistivity η and ρs become small.

Relaxation of a nonideal magnetohydrodynamic plasma in a cylindrical column

The time evolution of a nonideal magnetohydrodynamic (MHD) plasma is investigated. An approach for the reduction of the nonlinear vector MHD equations for a current carrying plasma with a strong longitudinal external magnetic field to a set of scalar partial differential equations is supposed. Analytical time-dependent solutions of the nonlinear resistive and dissipative MHD equations for a cylindrical plasma column are presented. It is shown that the internal magnetic field and the velocity have different damping times and that in the asymptotic limit t→∞ the plasma slowly relaxes toward the static equilibrium state. Analytical reduced MHD solutions describing slowly evolving states in tokamaks are found.

Effects of a poloidal divertor separatrix on resistive magnetohydrodynamic instabilities in a tokamak

A computer code package has been developed to simulate the linear and nonlinear evolution of long-wavelength resistive magnetohydrodynamic (MHD) instabilities in a four-node poloidal divertor tokamak (e.g., Wisconsin Tokapole II [Nucl. Fusion 19, 1509 (1979)]). Distinguishing features of this package include the use of a full set of three-dimensional (3-D) nonlinear resistive MHD equations and the inclusion of the divertor separatrix and the plasma outside the divertor separatrix in the computational domain. The present numerical results suggest that the plasma current outside the divertor separatrix tends to linearly stabilize the resistive MHD instability dominated by the m=2, n=1 mode, and, to a lesser extent, that dominated by the m=1, n=1 mode. (Here, m and n are poloidal and toroidal mode numbers, respectively.) However, the nonlinear evolution of the m=1, n=1 dominant instability is not significantly affected by the divertor configuration; the m=1, n=1 island is shown to reconnect totally by developing a large region of magnetic stochasticity. Hence, the cause of the partial reconnection observed in Tokapole II seems to lie beyond the scope of the classical resistive MHD model.

Nonlinear evolution of resistive tearing mode instability with shear flow and viscosity

The nonlinear evolution of the tearing mode instability with equilibrium shear flow is investigated via numerical solutions of the resistive magnetohydrodynamic (MHD) equations. The two-dimensional simulations are in slab geometry, are periodic in the x direction, and are initiated with solutions of the linearized MHD equations. The magnetic Reynolds number S was varied from 102 to 105, a parameter V that measures the strength of the flow in units of the average Alfvén speed was varied from 0 to 0.5, and the viscosity as measured by the Reynolds number Sν satisfied Sν≥103. When the shear flow is small (V≤0.3) the tearing mode saturates within one resistive time, while for larger flows the nonlinear saturation develops on a longer time scale. The two-dimensional spatial structure of both the flux function and the streamfunction distort in the direction of the equilibrium flow. The magnetic energy release decreases and the saturation time increases with V for both small and large resistivity. Shear flow decreases the saturated magnetic island width, and generates currents far from the tearing layer. The validity of the numerical solutions was tested by verifying that the total energy and the magnetic helicity are conserved. The results of the present study suggest that equilibrium shear flow may improve the confinment of tokamak plasma.

Statistical Analysis of Anomalous Transport in Resistive Interchange Turbulence

A new anomalous transport model for resisistive interchange turbulence is derived from statistical analysis applying two-scale direct-interaction approximation to resistive magnetohydrodynamic equations with a gravity term. Our model is similar to the K - e model for eddy viscosity of turbulent shear flows in that anomalous transport coefficients are expressed in terms of by the turbulent kinetic energy K and its dissipation rate e while K and e are determined by transport equations. This anomalous transport model can describe some nonlocal effects such as those from boundary conditions which cannot be treated by conventional models based on the transport coefficients represented by locally determined plasma parameters.

On the Lagrangian of the Linearized MHD Equations

The Lagrangian for the linearized, ideal and resistive magnetohydrodynamic (MHD) equations is discussed by introducing the perturbation of the total pressure. In the resistive MHD equations, the Lagrangian is expressed in terms of the electric displacement vector (time integral of the electric current) as well as the plasma displacement. The NOVA and NOVA-R formulations can be derived by using the obtained Lagrangian.

Nonlinear interactions of tearing modes in the presence of shear flow

The interaction of two near-marginal tearing modes in the presence of shear flow is studied. To find the time asymptotic states, the resistive magnetohydrodynamic (MHD) equations are reduced to four amplitude equations, using center manifold reduction. These amplitude equations are subject to the constraints due to the symmetries of the physical problem. For the case without flow, the model that is adopted has translation and reflection symmetries. Presence of flow breaks the reflection symmetry, while the translation symmetry is preserved, and hence flow allows the coefficients of the amplitude equations to be complex. Bifurcation analysis is employed to find various possible time asymptotic states. In particular, the oscillating magnetic island states discovered numerically by Persson and Bondeson [Phys. Fluids 29, 2997 (1986)] are discussed. It is found that the flow-introduced parameters (imaginary part of the coefficients) play an important role in driving these oscillating islands.