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Title: Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows

Abstract

The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.

Authors:
;  [1]
  1. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef (Egypt)
Publication Date:
OSTI Identifier:
22599977
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Plasmas; Journal Volume: 23; Journal Issue: 8; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AXIAL SYMMETRY; COMPRESSIBLE FLOW; CONFINEMENT; DENSITY; EQUATIONS; EQUILIBRIUM; EXACT SOLUTIONS; INCOMPRESSIBLE FLOW; LIMITING VALUES; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; NONLINEAR PROBLEMS; PLASMA; STEADY-STATE CONDITIONS; THREE-DIMENSIONAL CALCULATIONS; VELOCITY

Citation Formats

Moawad, S. M., E-mail: smmoawad@hotmail.com, and Ibrahim, D. A.. Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows. United States: N. p., 2016. Web. doi:10.1063/1.4960043.
Moawad, S. M., E-mail: smmoawad@hotmail.com, & Ibrahim, D. A.. Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows. United States. doi:10.1063/1.4960043.
Moawad, S. M., E-mail: smmoawad@hotmail.com, and Ibrahim, D. A.. 2016. "Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows". United States. doi:10.1063/1.4960043.
@article{osti_22599977,
title = {Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows},
author = {Moawad, S. M., E-mail: smmoawad@hotmail.com and Ibrahim, D. A.},
abstractNote = {The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.},
doi = {10.1063/1.4960043},
journal = {Physics of Plasmas},
number = 8,
volume = 23,
place = {United States},
year = 2016,
month = 8
}
  • A sufficient condition for the linear stability of three dimensional equilibria with incompressible flows parallel to the magnetic field is derived. The condition refers to internal modes and involves physically interpretable terms related to the magnetic shear and the flow shear.
  • The equilibrium and Lyapunov stability properties for two-dimensional ideal magnetohydrodynamic (MHD) plasmas with incompressible and homogeneous (i.e., constant density) flows are investigated. In the unperturbed steady state, both the velocity and magnetic field are nonzero and have three components in a Cartesian coordinate system with translational symmetry (i.e., one ignorable spatial coordinate). It is proved that (a) the solutions of the ideal MHD steady state equations with incompressible and homogeneous flows in the plane are also valid for equilibria with the axial velocity component being a free flux function and the axial magnetic field component being a constant (b) themore » conditions of linearized Lyapunov stability for these MHD flows in the planar case (in which the fields have only two components) are also valid for symmetric equilibria that have a nonplanar velocity field component as well as a nonplanar magnetic field component. On using the method of convexity estimates, nonlinear stability conditions are established.« less
  • It is proved that (a) the solutions of the ideal magnetohydrodynamic equation, which describes the equilibrium states of a cylindrical plasma with purely poloidal flow and arbitrary cross-sectional shape [G. N. Throumoulopoulos and G. Pantis, Plasma Phys. Controlled Fusion {bold 38}, 1817 (1996)], are also valid for incompressible equilibrium flows with the axial velocity component being a free surface quantity and that (b) for the case of isothermal incompressible equilibria the magnetic surfaces necessarily have a circular cross-section. {copyright} {ital 1997 American Institute of Physics.}
  • A three-dimensional equilibrium theory (J. Geophys. Res. {bold 92}, 11 101 (1987)) for stretched plasma configurations, such as the Earth's magnetotail, is extended to include the effects of field-aligned flow. The magnetohydrodynamic (MHD) equations for this case can be solved in a general way by reduction to a set of ordinary differential equations and an ordinary integral. The solutions represent lowest-order solutions of an asymptotic expansion of the MHD equations for small electric field and weak time dependence. Simplified equations are presented for two-dimensional equilibria and for incompressible flow. Possible magnetospheric applications include the configuration of the geotail near andmore » beyond the termination of the closed field line region, the steady motion of a plasmoid (a plasma bubble severed from the Earth) through the distant geotail, and configurations at the magnetopause, the interface between the magnetosphere and the shocked solar wind plasma. For illustration, solutions for the steady motion of a two-dimensional plasmoid through the distant magnetotail are presented.« less
  • Using the magnetotail equilibrium theory and a solution method outlined by Birn (1987), we have constructed self-consistent three-dimensional models for the quiet average magnetotail beyond about 20 R/sub E/ distance but earthward of a potential distant neutral line, which take into account the decrease of the tail flaring with distance. We find that this effect is coupled with the presence of magnetic shear and thus with field-aligned electric currents. These currents have the signature of region 1 currents, toward the Earth on the dawnside and away on the duskside, and contribute about 5 x 10/sup 5/ A to the totalmore » Birkeland current. They are strongly concentrated near the plasma sheet-lobe boundary and increase toward the flanks of the tail. Associated with the field-aligned currents and the corresponding magnetic field shear there is a bulging effect that tends to deform a circular cross section of the tail near the Earth into one that has bulges in the low-latitude boundary region. We argue that this effect may be the cause for increased interaction with the solar wind in these regions, producing interconnected fields and tailward flowing plasma on magnetospheric-like fields in the low-latitude boundary layer, and deforming this boundary region into the observed dog bone shape of the plasma sheet cross section. copyright American Geophysical Union 1989« less