Ideal MHD Equations Research Articles

General two-dimensional magnetohydrodynamic equilibria with mass flow

A set of reduced ideal MHD equations is derived to investigate equilibria of plasmas with mass flow in general two-dimensional geometry. These equations provide a means of investigating the effects of flow on self-consistent equilibria in a number of new two-dimensional configurations, such as helically symmetric configurations with helical axis, which are relevant to stellarators, as well as axisymmetric configurations. It is found that, as in the axisymmetric case, general two-dimensional flow equilibria are governed by a second-order quasilinear partial differential equation for a magnetic flux function, which is coupled to a Bernoulli-type equation for the density. The equation for the magnetic flux function becomes hyperbolic at certain critical flow speeds which follow from its characteristic equation. When the equation is hyperbolic, shock phenomena may exist. As a particular example, unidirectional flow along the lines of symmetry is considered. In this case, the equation mentioned above is always elliptic. An exact solution for the case of helically symmetric unidirectional flow is found and studied to determine flow effects on the magnetic topology.

High- n ideal and resistive shear Alfvén waves in tokamaks

Ideal and resistive MHD equations for the shear Alfvén waves are studied in a low-β toroidal model by employing the high- n ballooning formalism. The ion sound effects are neglected. For an infinite shear slab, the ideal MHD model gives rise to acontinuous spectrum of real frequencies and discrete eigenmodes (Alfvén-Landau modes) with complex frequencies. With toroidal coupling effects due to nonuniform toroidal magnetic field, the continuum is broken up into small continuum bands and new discrete toroidal eigenmodes can exist inside the continuum gaps. Unstable ballooning eigenmodes are also introduced by the bad curvature when β > β c . The resistivity (η) can be considered perturbatively for the ideal modes. In addition, four branches of resistive modes are induced by the resistivity: (1) resistive entropy modes which are stable with frequencies going to zero with resistivity as η 1 3 ; (2) tearing modes which are stable (Δ′ < 0) with frequencies approaching zero as η 3 5 ; (3) resistive periodic shear Alfvén waves which approach the finite frequency end points of the continuum bands as η 1 2 ; and (4) resistive ballooning modes which are purely growing with growth rate proportional to η 1 3 β 2 3 as η → 0 and β → 0.

A 2-D Equilibrium model of toroidal belt pinches

An analytical method is described to extend an existing equilibrium model of a straight cylindrical plasma column with elliptic cross-section, traversed by a uniform axial current, and surrounded by a vacuum. This model is the zero order in a procedure in which the ideal MHD equations are linearized after the introduction of small perturbations which represent toroidicity, deviation from ellipticity and external force-free currents. The solution of the resulting equations is used to analyse the effects of the perturbations, major effects being the shift of stagnation points and reshaping of the outer flux surfaces. Numerical examples are given and a comparison is made with other existing theories and with belt screw-pinch experiments.

Wave-wave interaction in magneto-gravitating plasma

view Abstract Citations (1) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Wave-wave interaction in magneto-gravitating plasma Aggarwal, S. S. ; Kalra, G. L. Abstract Resonant interaction of coherent magnetoacoustic waves in a hot, tenuous, self-gravitating plasma in a uniform magnetic field is discussed. The plasma is described by the single-fluid ideal MHD equations. It is found that the resonant interaction between the wave-modes triggers instability at wavelengths shorter than the Jeans critical wavelength of instability as found in the linear theory. Instability initiated in the presence of a magnetic field grows at a faster rate than the corresponding instability in the absence of a magnetic field. Publication: The Astrophysical Journal Pub Date: May 1984 DOI: 10.1086/162052 Bibcode: 1984ApJ...280..792A Keywords: High Temperature Plasmas; Jeans Theory; Magnetoacoustic Waves; Plasma Resonance; Plasma-Electromagnetic Interaction; Wave Interaction; Magnetic Field Configurations; Magnetohydrodynamic Stability; Nebulae; Astrophysics full text sources ADS |

Nonlinear Evolution of External Ideal MHD Modes Near the Boundary of Marginal Stability

Abstract The nonlinear evolution of external ideal MHD-modes is determined from the equations of ideal MHD by employing a reductive perturbation method which uses a driving parameter for expansion. The reduction of the plasma equations is the same as for internal modes and was treated previously [1]. A main problem arising in addition for external modes is the reduction of the nonlinear boundary conditions. The set of reduced boundary conditions is obtained on the undisplaced boundary in the marginally stable equilibrium position. Another additional problem arises from the fact that the linear MHD operator is only selfadjoint for linear eigenmodes but not for the higher order mode corrections. This complicates the determination of nonlinear amplitude equations for the marginal mode which are obtained from solubility conditions. The amplitude equations are qualitatively the same as for internal modes. Quantitatively, the calculation of the coefficients in these is different. Explicit expressions for the coefficients are derived in full generality. The effect of higher order corrections to the nonlinear amplitude equations is discussed quantitatively for one of two possible cases and qualitatively for the other.

On the topological stability of magnetostatic equilibria

The topological stability of MHD equilibria is investigated by exploring the formal analogy, in the ideal MHD limit, between the topology of magnetic lines of force in coordinate space and the topology of integral surfaces of one- and two-dimensional Hamiltonian systems in phase space. It is demonstrated that in an astrophysical setting, symmetric magnetostatic equilibria satisfying the ideal MHD equations are exceptional. The principal result of the study is that previous infinitesimal perturbation theory calculations can be generalized to include finite-amplitude and symmetry-breaking effects. The effect of the ergodicity of perturbed symmetric equilibria on heat dispersal in magnetically dominated plasmas is discussed.

Magnetically Induced Plasma Rotation and Nuclear Fusion

Introductory views are presented on the physically different but remarkably good fusion reaction performance of the dense plasma focus compared to the mainline magnetic fusion research assemblies. Reference is given to our attempt in explaining a reputed fundamental but not understood phenomenon in the focus plasma by an ion acceleration mechanism of basically the betatron type. Paradoxically, this magnetically induced plasma rotation is both contained in and denied by standard plasma equations, briefly derived and discussed with respect to validity. The neglect of the Hall term in the usual "Ideal MHD Equations" is shown to be in error with regard to current magnetic fusion research plasmas. Magnetically induced plasma rotation is derived from the ion fluid motion equation, from basic MHD fluid equations and by an analysis of the electro-magnetic torque distribution. General validity is proved by invoking usually overlooked consequences of the noncentral character of the forces sustaining quasi-neutrality. Arguments are given in favor of new approaches to magnetic fusion

Nonlinear Evolution of Internal Ideal MHD Modes Near the Boundary of Marginal Stability

A family of ideal MHD equilibria is considered introducing the concept of a driving parameter λ the increase of which beyond a certain threshold λ0 drives the plasma from a linearly stable to an unstable state. Using reductive perturbation theory, the nonlinear ideal MHD equations of motion are expanded in the neighbourhood of λ0 with respect to a small parameter ε. An appropriate scaling for the expansions is derived from the linear eigenmode problem. Integrability conditions for the reduced nonlinear equations yield nonlinear amplitude equations for the marginal mode. Nonlinearly, the instabilities are either oscillations about bifurcating equilibria, or they are explosive. In the latter case, the stability limit depends on the amplitude of the perturbation and is shifted into the linearly stable regime. Generally bifurcation of dynamically connected equilibria is observed at λ0

Normals of non‐WKB Alfvén waves in the solar wind

A theoretical treatment of wave normals of monochromatic non‐WKB Alfvén waves in the solar wind is developed. The method is based on the one‐fluid ideal MHD equations governing small‐amplitude, undamped, poloidal Alfvén waves propagating in the solar equatorial plane of a Weber and Davis spiral field. Numerical results for waves propagating from the sun to 1 AU are presented. The purpose of the calculations is to determine whether the inclusion of non‐WKB effects can significantly alter the geometrical optics result of Völk and Alpers that the wave normals tend to align themselves with the radial direction as the waves propagate through the solar wind. The conclusion is that there are differences, depending on the wave period, but that they are relatively small: the wave normals at 1 AU of waves of any period which have evolved from a reasonable distribution of normals in the corona are confined to within about 20° from the radial direction. The implication is that any lack of radial alignment is unlikely to be explained by appeal to finite wavelength effects.

Exact solutions of the stationary MHD equations for a rotating toroidal plasma

For a toroidal plasma rotating about its axis of symmetry, the ideal MHD equations can be reduced to an equation which generalizes the static equilibrium equation if either the entropy or the temperature is constant on the magnetic surfaces. For both of these cases, exact analytic solutions are obtained and their properties are discussed.

The dissipation of hydromagnetic surface waves

view Abstract Citations (60) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The dissipation of hydromagnetic surface waves. Wentzel, D. G. Abstract When hydromagnetic surface waves travel along a surface with a thin but finite boundary layer, velocities within this layer become singular when computed according to the ideal MHD equations. The present paper computes the corresponding rate of wave damping. Sufficiently weak surface waves are dissipated either by a resonant conversion into kinetic Alfven waves or by viscosity. Astrophysically important surface waves may involve such large velocity amplitudes outside the narrow zone of linear dissipation that nonlinear phenomena limit the singularity and the method of dissipation. Even these, however, are confined to extremely narrow layers. Publication: The Astrophysical Journal Pub Date: October 1979 DOI: 10.1086/157437 Bibcode: 1979ApJ...233..756W Keywords: Magnetohydrodynamic Waves; Solar Corona; Surface Waves; Wave Attenuation; Differential Equations; Rates (Per Time); Viscosity; Wave Equations; Solar Physics; Hydrodynamics; Magnetohydrodynamics; Plasma full text sources ADS |

Collisionless damping of Alfvén waves in an inhomogeneous plasma: Solution of the initial value problem

The initial value problem of the linearized ideal MHD equations for a low‐β plasma in an inhomogeneous magnetic field is solved for a broad class of initial conditions. The validity of the solutions breaks down after a time, which is determined by the magnitude of the initial conditions. The energy is transported to the inhomogeneous region. The mechanism described here could therefore possibly be used for shaping of the plasma temperature profile.

Dynamical grid method for time-dependent simulations of axisymmetric instabilities in tokamaks

A natural nonorthogonal time-dependent coordinate transformation based on the magnetic field lines is utilized for the numerical integration of the two-dimensional axisymmetric time-dependent ideal MHD equations in tokamak geometry. The finite-difference grid is treated as a dynamical variable, and its equations of motion are integrated simultaneously with those for the fluid and magnetic field. The method is applicable to tokamak systems of arbitrary pressure and cross section. It is particularly useful for the nearly incompressible ideal MHD modes which are of interest in tokamak stability studies.

Axisymmetric ideal MHD stellar wind flow

The ideal MHD equations are reduced to a single equation under the assumption of axisymmetric flow. A variational principle from which the equation is derivable is given. The characteristics of the equation are briefly discussed. The equation is used to rederive the theorem of Gussenhoven and Carovillano.

Energy absorption in the continuous spectrum of ideal MHD

The relationship between energy absorption and the continuous frequency spectrum of the linearized equations of ideal MHD is investigated. We limit ourselves to incompressible fluid perturbations for which the continuum stems from resonant surfaces where the oscillation frequency equals the local frequency of an Alfvén wave. Details of this absorption process are illustrated by obtaining the response of the diffuse sheet pinch to an external current source. Expressions are derived that give the rate at which energy is transferred to the plasma, and show the spatial distribution of the absorbed energy. The results indicate that the absorption is enhanced if the plasma profile has regions of large spatial gradients. This enhanced absorption is a consequence of the presence of surface waves.