Compressible Plasma Research Articles

Stability of a Rotating Compressible Stratified Plasma in the Presence of Hall Currents and Magnetic Resistivity

AbstractThe dynamic instability in a horizontal layer of a rotaing compressible plasma of variable density has been investigated to examine the influence of the simultaneous presence of the effects of Hall currents and finite magnetic resistivity. The linearized stability analysis has been carried out through the normal mode technique. By making use of the existence of a variational principle which is shown to characterize the problem, proper solutions have been obtained for a semiinfinite plasma in which there is an exponential density gradient along the vertical. The dispersion relation obtained has been solved numerically and it is found that both the resistivity and the Hall currents have a destabilizing influence as the growth rate of the unstable disturbances increases with increasing values of the parameters characterizing these effects. On the other hand, the Coriolis forces are found to have a stabilizing influence for in this case the growth rate decreases with increasing rotation.

Turbulent relaxation of compressible plasmas with flow

Relaxation of compressible plasmas to an equilibrium with flows is studied. The magnetohydrodynamic (MHD) equations with large parallel thermal conductivity and ergodic field lines show that ∫v⋅B is an invariant even with compression. Also, S=∫ρ ln( p/ργ) is the only entropy-like invariant of the MHD equations with infinite parallel thermal conductivity; S increases in time with finite thermal conductivity. The other invariants are energy, helicity ∫A⋅B, mass ∫ρ, and possibly angular momentum. Equilibria are found by extremizing energy while conserving these invariants or by maximizing entropy with the energy and other invariants as constraints. These invariants are complete in the sense of generating all equilibria that form after relaxation with ergodic field lines. For parallel flows, there are three classes of solutions characterized by the sign of dρ/dB2 and the mirror mode parameter. A sufficient condition for stability is derived. This condition is never satisfied by the class with dρ/dB2>0, indicating the possibility of unstable resistive interchanges. The class with dρ/dB2<0 is stable if generalizations of the local firehose and mirror criteria are satisfied, and if generalized Taylor helicity eigenvalues exceed those of the equilibrium. A comparison of these conditions with those of Frieman and Rotenberg is discussed.

Canonical maps between poisson brackets in eulerian and Lagrangian descriptions of continuum mechanics

Noncanonical, Lie algebraic, hamiltonian structures in the eulerian description of ideal continuum mechanics are shown to be compatible with nearly canonical structures in the lagrangrian description. Examples are given for compressible ideal fluid dynamics, magnetohydrodynamics, nonlinear elasticity, multifluid plasmas, superfluids 4He and 3He-A, and chromohydrodynamics.

Thermal instability of a compressible plasma with finite Larmor radius

The effects of compressibility and finite ion Larmor radius on the thermal instability of a plasma in the presence of a uniform horizontal magnetic field are investigated. When the instability sets in as stationary convection, the compressibility and finite Larmor radius are found to have stabilizing effects. The case of overstability is also studied leading to the establishment of conditions sufficient for avoiding overstability.

Kelvin-Helmholtz instabilities in a sheared compressible plasma

The Kelvin–Helmholtz dispersion equation for a sheared homogeneous compressible plasma containing a magnetic field is derived. It is shown that, in the incompressible limit, K–H modes, irrespective of wavelength, do not grow if |$U_0\enspace \text {cos}\enspace\theta\leqslant 2\enspace V_\text A\enspace \text {cos}\enspace\beta$|⁠, where U0, VA, θ and β are the total velocity jump across the shear layer, the Alfven speed and the angles the mode makes with the flow and magnetic field, respectively. Sufficiently low parallel magnetic fields are found to have a slightly destabilizing effect. For the compressible case, and assuming modes and magnetic fields parallel to the flow, it is shown that one has stability if the sound speed |$a\leqslant V_\text A$|⁠, providing |$M\geqslant 1$|⁠.

On dual spaces of differential Lie algebras

We present a mathematical scheme which serves as an infinite-dimensional generalization of Poisson structures on dual spaces of finite-dimensional Lie algebras, which are well known and widely used in classical mechanics. These structures have recently appeared in the theory of Lax equations, long waves in hydrodynamics, and various other physical models: compressible hydrodynamics, magnetohydrodynamics, multifluid plasmas, elasticity, superfluid 4He and 3He-A, Ginzburg-Landau theory of superconductors, and classical chromohydrodynamics (the generalization of plasma physics to Yang-Mills interactions).

Kelvin:Helmholtz Instability at the magnetopause: Solution for compressible plasmas

The Kelvin‐Helmholtz (K‐H) instability of a tangential discontinuity in a compressible plasma is reexamined in the linear magnetohydrodynamic (MHD) approximation. For fixed plasma conditions, two different kinds of surface waves (labeled F for fast and S for slow) may exist simultaneously with different tangential wave vectors kt. The surface waves can be excited only for a limited range of U, the relative flow speed of the plasmas on the two sides of the interface. Thus the instability requires Ucm < U < Uum, where Ucm and Uum are the lower and upper critical velocities, respectively, and the subscript m distinguishes the fast and slow surface waves, with UcS < UcF. In the incompressible limit, there is only one surface wave with lower critical velocity, Uci, and no upper critical velocity. It has frequently been remarked that compressibility reduces the lower critical velocity. We show that the effect is small for the F wave but that UcS is considerably smaller than either UcF or Uci. However, for a given kt, the S wave is unstable only for a very small range of U. Growth rates, εm, of the surface waves are calculated and εF exceeds slightly the growth rate for the incompressible plasma (εi) when U ≳ Ui, but the slow wave growth rate is small in comparison with both εF and εi. Consequently, plasma compressibility is relatively ineffective in reducing the critical velocity for surface wave growth below that for an incompressible plasma. In particular, for the nominal conditions of the dayside equatorial magnetopause, S mode waves do not occur near the minimum UcS (i.e., with kt, perpendicular to the magnetospheric magnetic field) because εS → 0 and UuS = UcS. On the terrestrial magnetopause the F(S) surface waves can couple a quasi‐fast (quasi‐slow) MHD mode in the magnetospheric plasma to either a quasi‐fast or quasi‐slow MHD mode in the magnetosheath. These waves decay as they propagate into the bounding plasmas. Consideration of the limit U → UuF reveals the significance of the upper critical velocity. In this limit, the phase velocity of the fast surface wave approaches the MHD fast mode speed in both bounding plasmas. The imaginary part of normal component of k vanishes and the unstable surface waves change to stable MHD waves that propagate away from the boundary without damping. The above points are discussed in relation to previous work on the K‐H instability. Possible applications to observations in the terrestrial magnetosphere are mentioned.

Finite Amplitude Effects of Torsional Alfvén Waves in a Flux Tube

Consider a cylindrical, homogeneous magnetic flux tube embedded in a field-free compressible plasma. We employ the method of successive approximations to solve the magnetohydrodynamic equations analytically for an axisymmetrical (m=0) torsional Alfvén mode. The m=0 mode inside the tube may excite surface-, stationary- or progressive waves in the field-free plasma. We conclude that, in contrast to the linear theory, the flux tube may lose energy by wave radiation to the surroundings.

The mean electromotive force generated by random hydromagnetic waves in a collisionless plasma

The mean electromotive force generated by random hydromagnetic waves is calculated by using Fourier analysis methods for the cases of an incompressible Hall plasma and a compressible plasma with finiteβ. It is shown that in the case of a Hall plasma theαeffect can exist in the interaction between two waves which propagate in opposite directions and have different phase velocities and several newβ–terms are produced by the Hall effect, while in the case of a compressible plasma theα–effect can exist only in the interaction between the Alfvén mode and the fast or slow magneto-acoustic mode, and not between the fast and slow magneto-acoustic modes. The results are discussed in the context of the substorms in the earth's magnetosphere.

Nonlinear Evolution of the Magnetohydrodynamic Kelvin-Helmholtz Instability

Simulation of the magnetohydrodynamic Kelvin-Helmholtz instability in a compressible plasma shows that a super-Alfv\'enic shear flow parallel to the magnetic field develops into small eddies, which strongly compress, twist, and hence amplify the magnetic field, and that a sub-fast (magnetosonic) shear flow transverse to the magnetic field evolves into a fast shock. The anomalous viscosity due to the instability is important in understanding a viscous drag at the solar-wind-magnetosphere interface.

Nonlocal stability analysis of the MHD Kelvin‐Helmholtz instability in a compressible plasma

A general stability analysis is performed for the Kelvin‐Helmholtz instability in sheared magnetohydrodynamic flow of finite thickness in a compressible plasma. The analysis allows for arbitrary orientation of the magnetic field B0, velocity flow v0, and wave vector k in the plane perpendicular to the velocity gradient, and no restrictions are imposed on the sound or Alfvén Mach numbers. The stability problem is reduced to the solution of a single second‐order differential equation, which includes a gravitational term to represent coupling between the Kelvin‐Helmholtz mode and the interchange mode. In the incompressible limit it is shown that the Kelvin‐Helmholtz mode is completely stabilized for any velocity profile as long as the condition is satisfied, where V0 is the total velocity jump across the shear layer. Numerical results are obtained for a hyperbolic tangent velocity profile for the transverse (B0 ⊥ v0) and parallel (B0∥v0) flow configurations. Only modes with kΔ < 2 are unstable, where Δ is the scale length of the shear layer. The fastest growing modes occur for kΔ ∼ 0.5‐1.0. Compressibility and a magnetic field component parallel to the flow are found to be stabilizing effects. For the transverse case, only the fast magnetosonic mode is destabilized, but if k · B0 ≠ 0, the instability contains Alfvén‐mode and slow‐mode components as well. The Alfvén component gives rise to a field‐aligned current inside the shear layer. In the parallel case, both Alfvén and slow magnetosonic components are present, with the Alfvén mode confined inside the shear layer. The results of the analysis are used to discuss the stability of sheared plasma flow at the magnetopause boundary and in the solar wind. At the magnetopause boundary, the fastest growing Kelvin‐Helmholtz mode has a frequency of 0 (V0/2Δ), which overlaps with the frequency range of geomagnetic pulsations (Pc 3‐5). It is suggested that the MHD Kelvin‐Helmholtz instability could serve as a dynamo process driving small‐scale field‐aligned currents in the presence of the sheared plasma flow in the magnetosphere.

Stationary waves at the interface between a plasma stream and magnetic field

A uniformly-streaming compressible and infinitely-conducting plasma is confined by a magnetic field aligned with the stream. The system is disturbed by introducing magnetic dipoles into the field. A Fourier-transform method is used to determine the displacement of the interface between the streaming plasma and the magnetic field within the framework of a ‘shallow-water’ approximation. For the case of a subsonic plasma stream, stationary waves appear on the interface upstream of the dipoles, and it is found that (i) these stationary waves are possible only if the gravity effects on the plasma are weak enough; (ii) the effect of surface tension at the interface is to reduce the amplitude and increase the wavelength of these waves. For the case of a supersonic plasma stream, however, stationary waves at the interface are not possible.

A generalised theory of stability of superposed fluids in hydromagnetics

This paper gives a generalised theory of hydromagnetic stability of the interface between two infinitely-conducting, compressible plasmas with the latter accelerated perpendicular to the interface and streaming parallel to the interface and subjected to a constant magnetic field parallel to the streaming direction. The method used is adapted from the one given by Plesset and Hsieh (1964) for the hydrodynamical case and the general dispersion relation is found to involve Whittaker's functions and their first derivatives. The familiar Rayleigh-Taylor and Kelvin-Helmholtz instability problems are recovered from the general dispersion relation in the appropriate special cases.

Parametric excitation of alfvén wave by magnetosonic wave with oblique propagation

A parametric excitation of the Alfvén wave ( k A, ω a ) by the magnetosonic wave ( K 1 fs , ω 1 fs ), which propagates obliquely to the static magnetic field, has been analyzed. The theoretical model for a one-fluid with uniform, unbounded, ideally conducting and compressible plasma is employed. The resonance conditions are chosen such as, k 1 fs = k 1 fs + k A and ω 1 fs − ω A = δ ∼ ω 2 fs . The p wave is assumed to be strong enough, so that the pump wave is given as a constant. In both the case of the standing and the propagating pump the growth rates of the excited waves depend on not only the pump power but also the β-ratio. In the standing pump the threshold pump intensity of the oscillating instability is zero at the perfect matching. It is found that we can obtain a larger growth rate of the parametric excitation of Alfvén wave by the fast magnetosonic pump wave for θ 1f ∼ 70–80° and the occurrence regions of parametric excitations are localized at the resonance point in the magnetosphere ( β ∼ m e / m i ). It is concluded that the parametric instability of Pc3 range HM-waves is the more possible theory than the linear resonance theory.

Propagation in Transversely Magnetized Compressible Plasma Between Two Parallel Planes

The propagation of waves in compressible single fluid macroscopic plasma between two parallel, perfectly conducting planes, with a transverse static magnetic field parallel to the boundaries is investigated. It is shown that the TE waves are not affected by either the static magnetic field or the compressibility of the plasma, while the TM wave will be affected by both.

5116. Transition from diffusion-convection to shealth-convection of a cold Langmuir probe in a moving compressible plasma: R M Clements and P R Smy,J Phys D: Appl Phys,14 (6), 1981, 1001–1008

5116. Transition from diffusion-convection to shealth-convection of a cold Langmuir probe in a moving compressible plasma: R M Clements and P R Smy,J Phys D: Appl Phys,14 (6), 1981, 1001–1008

Radiation characteristics of ring antennas for electron plasma waves

AbstractA theoretical analysis and experiments on one‐ and two‐element ring antennas in an isotropic compressible plasma are repeated. First, a two‐element ring antenna connected to the inner and outer conductors of a coaxial feeder is analyzed by decomposing the system into a zero phase type and a positive phase type. The analysis uses Fourier series expansion. Numerical results reveal that in both one‐ and two‐element systems the current is approximated by a triangular distribution, i.e., a charge with a uniform distribution, if the ring diameter is much smaller than the free space wavelength. In the case of two‐element systems, the charge distributions of elements #1 and #2 are out of phase. The input impedance of the two‐element ring is much smaller than that of the one‐element system. The measured and calculated radiation patterns of a one‐element ring for the electron plasma wave agree well except near θ = 90°. The side‐lobe level of the two‐element ring is much lower than that of the one‐element ring and the radiation characteristics are much improved. It is theoretically and experimentally confirmed that the radiation efficiency of the two‐element ring is much better than that of the one‐element ring for electron plasma waves.

Reflection of a transient plane wave obliquely incident on compressible plasma half-space

The expression for the reflected transient pulses from the sharp surface of a compressible plasma half-space are obtained in a series form with the first perturbation order explicitely evaluated and valid for small to moderate compressibility (α = u/C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ≤ 0.3, u is the electron acoustic velocity and C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is the velocity of light in free space). The reflected waveform are close to the waveform of cold plasma (α = 0) with noticeable change of amplitude and time delay of the first maxima.

Parametric instability of hydromagnetic waves with oblique propagation

The parametric excitation of Alfvén waves by two magnetosonic waves, which propagate obliquely to the static magnetic field, is analysed. The theoretical model used is the uniform, unbounded, ideally conducting and compressible one-fluid plasma, with suitable resonance conditions. Our consideration is restricted to determining the conditions under which some initially small perturbation grows, so that the magnetosonic pump wave is regarded as constant. It is found that, both in the case of the standing and the propagating pump, the growth rates of the excited waves depend not only on the pump power but also on β, and that the threshold pump intensities are proportional to β. In the case of the standing pump, the threshold pump intensity of the oscillating instability is zero at perfect matching. The parametric excitation of Alfvén waves by two magnetosonic waves, which propagate obliquely to the static magnetic field, is analysed. The theoretical model used is the uniform, unbounded, ideally conducting and compressible one-fluid plasma, with suitable resonance conditions. Our consideration is restricted to determining the conditions under which some initially small perturbation grows, so that the magnetosonic pump wave is regarded as constant. It is found that, both in the case of the standing and the propagating pump, the growth rates of the excited waves depend not only on the pump power but also on β, and that the threshold pump intensities are proportional to β. In the case of the standing pump, the threshold pump intensity of the oscillating instability is zero at perfect matching.

Rayleigh instability of compressible plasma in a vertical magnetic field

A study of the Rayleigh instability of a compressible plasma of density stratified in horizontal planes and subjected to a vertical magnetic field is made. The special case of a plane interface separating two superposed uniform plasmas of different densities and speeds of sound is treated as an example to illustrate the compressibility effects on the hydromagnetic Rayleigh instability. It is found that the hydromagneticcompressibility effects act toward reducing the growth rate in a hydrodynamically unstable situation.