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A recent paper by Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743) on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Cartan–Poincaré differential form formulation of the system. We also correct some typographical errors (omissions) in Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743). We show that the vorticity–symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan–Poincaré form to obtain a class of differential forms that represent the system using Cartan’s geometric theory of partial differential equations
A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on the Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers that ensure mass conservation, entropy advection with the flow, the Lin constraint, and Faraday's equation (i.e. the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potential Ã where α=Ã⋅dx is Lie dragged with the flow, and B=∇×Ã. The multi-symplectic conservation laws give rise to the Eulerian momentum and energy conservation laws. The symplecticity or structural conservation laws for the multi-symplectic system corresponds to the conservation of phase space. It corresponds to taking derivatives of the momentum and energy conservation laws and combining them to produce n(n−1)/2 extra conservation laws, where n is the number of independent variables. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.
Small scale explosively driven fragmentation experiments have been performed on Aluminum (Al)-Tungsten (W) granular composite rings processed using cold isostatic compression of Al and W powders with a particle size of 4-30 microns. Fragments collected from the experiments had a maximum size of the order of a few hundred micrometers. This is a dramatic reduction in the fragment size when compared to the 1-10 mm typical for a homogeneous material such as solid aluminum under similar loading conditions. Numerical simulations of the experiment were performed to elucidate the mechanisms of fragmentation that were responsible for this shift in fragmentation size scales. Simulations were performed with a significantly stronger explosive driver to examine how the mechanisms of fragmentation change when the detonation pressure increases.
We generalize Silin's dispersion equation for electrostatic parametric resonances in an unmagnetized plasma to include interactions hitherto overlooked. Of particular interest to ionospheric physicists in this generalization is the interaction between an ion acoustic wave (shifted up in frequency by that of the pump) and an unshifted Langmuir wave. This yields growth rates equal to that of its counterpart in the Silin version, namely the ‘classical’ interaction between a Langmuir wave (shifted down by the pump frequency) and an unshifted ion acoustic wave. The more general (truncated) dispersion equation also displays resonances between ion acoustic sidebands, but this requires pump frequencies of the order of, or less than, the ion plasma frequency and therefore may be of little practical interest in the ionosphere. The system is analysed using Fourier techniques, which lead to two dispersion equations governing the Fourier transforms of the ion and electron perturbation densities. It is shown that, as far as the structure of the coupled recursion relations for the ion and electron Fourier components is concerned, there is an equivalence between the fluid and kinetic treatments. In the case of weak pump fields we calculate the growth rates associated with the various instabilities from a truncated three-wave interaction dispersion relation. Where it is appropriate, these growth rates are compared both with those from the Sum dispersion equation and the counterparts from the Zakharov model. We also discuss the case of very strong pump fields where the ‘natural’ interacting mode frequencies are not those associated with the usual Langmuir—ion acoustic modes of the unpumped plasma, but rather the oscillator frequencies corresponding to the coupled oscillator paradigm that is applicable to the system. Here again we find instabilities of a nature analogous to those arising in the weak pump case.
The hydromagnetic analogue of the Kelvin–Helmholtz problem is extended to include the effects of the Hall term. In contrast to other results in the literature it is shown that, in the case of incompressible fluids, the stability of a shear plane is unaffected by the introduction of the Hall term. The special case of a hot, uninagnetized fluid on one side of the interface and a cold, magnetized fluid on the other is studied in some detail. In this case it is shown that the presence of the Hall term can have either a stabilizing or a destabilizing effect, depending upon whether the sound speed in the hot fluid is very much greater than the Alfvén speed in the cold fluid or vice versa.
This paper investigates the stability of a cosmic ray shock to long-wavelength perturbations. The problem is formulated in terms of finding the transmission coefficient for compressive waves across a cosmic ray shock by solving the generalized, two-fluid Rankine-Hugoniot relations. For strong shocks, the transmission coefficient confirms that compressive waves can undergo considerable amplification on passage through such shocks. The resonances of the transmission coefficient provides us with the dispersion equation governing the stability of the shock to long-wavelength ripple-like distortions. By using the principle of the argument method, it is established that cosmic ray shocks are stable.
It is shown that the conservation law for total momentum of an ion-beam plasma system can be cast in the form of a classical energy integral of a particle in a potential well. By using boundary conditions appropriate to a solitary pulse, we derive conditions for the existence of finite-amplitude solitons propagating in the system. Under suitable conditions, as many as three forward-propagating solitary waves can exist. It is interesting to note that the criterion for their existence is intimately related to the absence of convective instabilities in an ion-beam plasma. Exact ‘sech2’ type solutions are available in the weakly nonlinear regime. Solitary-wave profiles for the general case are obtained numerically.
In this paper we develop similarity solutions for the problem of nonlinear Landau damping of Alfvén waves. These solutions which are applicable to power-law wave spectra illustrate not only the basic feature of the damping process, namely that short-wavelength waves decay more rapidly than long-wavelength waves, but also how the damping depends on the initial strength of the power spectrum and its distribution in wavenumber.
It is shown that a plasma in which the background magnetic field varies in a direction perpendicular to its line of action can support ‘Rossby-type’ electrostatic waves at frequencies very much less than the ion gyrofrequency. The intrinsic wave propagation mechanism at work is structurally similar to that in the atmospheric Rossby wave, which comes about from fluid perturbations being in quasi-geostrophic equilibrium (i.e. the Coriolis force nearly balances the pressure gradient) and the latitudinal variation of the vertical component of rotational frequency vector (the β-effect) so that the time rate of change of the vertical component of the fluid vorticity is equal to the northward transport of the planetary vorticity. In a plasma this ‘geostrophic balance’ arises from the near-vanishing of the Lorentz force on the ion motion while the β-effect is provided by the transverse spatial variation of the ambient magnetic field. Unlike the atmosphere, however, such a magnetized plasma is capable of supporting two distinct types of Rossby wave. The interesting dispersive and anisotropic features of these waves are revealed by the properties of their wave operators and described in terms of the geometry of their wavenumber surfaces. Since these surfaces intersect, inhomogeneity or nonlinearity will give rise to strong mode-mode coupling in regions where the phases of both modes nearly match.
In this paper a multi-fluid approach is used to describe electrostatic interactions in an ion-beam plasma system. The structure of the wave equation governing the system exhibits the anisotropic and dispersive nature of the waves, whose properties are analysed in terms of the dispersion relation. The main purpose of this paper is to classify the different waves that can arise in an ion-beam plasma system in a systematic fashion. The classification is facilitated by introducing a three-parameter CMA diagram that illustrates the topological changes in not only the wavenumber, or refractive-index, surface but also the ray-velocity surface. Furthermore, an analytic expression governing wave amplification in an ion-beam plasma is incorporated within the framework of a generalized CMA diagram. Such a description provides a simple interpretation for the onset of wave amplification in terms of a topological change in the refractive-index surface. It is hoped that by collating the wave properties in a unified form, many of the complicated wave features observed in an experiment may be interpreted more easily.
Hydrodynamical equations describing the mutual interaction of cosmic rays, thermal plasma, magnetic field and Alfvén waves scattering the cosmic rays used in cosmic ray shock acceleration theory (e.g. McKenzie & Völk 1982; Drury 1983; Webb 1983) are analysed for long-wavelength linear compressive instabilities. The Alfvén wave field may contain a pre-existing component as well as a component excited by the cosmic ray streaming instability. In the case of no Alfvén wave damping, adiabatic wave growth and Alfvén wave generation by the cosmic ray streaming instability, it is found that the backward propagating slow magneto-acoustic mode is driven convectively unstable by the pressure of the self-excited Alfvén waves, provided the thermal plasma β is sufficiently large. The equations are also analysed for the case where the Alfvén wave growth is balanced by some nonlinear damping mechanisms. In the latter case both the forward and backward propagating slow magneto-acoustic modes may be driven unstable if the plasma β is sufficiently small. The conditions under which the instabilities occur are delineated, and sample calculations of growth rates given. Possible applications of the instabilities to astrophysical situations are briefly discussed.
In this paper the characteristics for a single- and a bi-ion plasma in the presence of Alfvén waves are given. In the single-ion case, the analysis is extended to the situation where Alfvén waves saturate and dissipatively heat the plasma. When there is no dissipation, there are three sound waves and one entropy wave in the single-ion plasma. Each sound wave is associated with two Riemann invariants relating the changes in density and wave pressure to changes in the flow. In the case when the Alfvén waves saturate and heat the plasma, there are two sound waves and one modified entropy sound wave. Each wave is associated with two Riemann invariants relating changes in density and entropy to changes in the flow. The analysis for the bi-ion plasma is simplified to very sub-Alfvénic flows. In this case the Alfvén waves behave like another plasma component, and both the electric and Alfvén wave forces have the same structure. The system possesses two entropy waves and four sound waves. Each sound wave is associated with two Riemann invariants relating changes in density and flow velocity along the characteristic curves.
Special relativistic magnetohydrodynamic shock waves in a perfect gas of infinite conductivity and constant adiabatic index are analysed. It is shown that the Rankine-Hugoniot equations for such shocks may be reduced to a seventh degree polynomial for the downstream dynamical volume ω, with the polynomial coefficients depending on the upstream state (ω equals the specific volume times the ratio of the energy density of the fluid (omitting electromagnetic terms) to the fluid rest mass energy density). In the non-relativistic limit, the polynomial equation reduces to a relation between the upstream and downstream Alfvénic Mach numbers, previously obtained by Cabannes. The equation for ω classifies in a natural way both shocks in which the electric field may be eliminated by transforming to the de Hoffman–Teller frame, and shocks for which this is not possible. The equation is used to determine the downstream state of relativistic shocks for a given upstream state as specified by the plasma beta, magnetic field obliquity, and flow speed.
This paper develops a theoretical framework for the description and classification of small-amplitude waves with frequencies much less than the ion gyrofrequency, propagating in an ion-beam plasma system. In this respect, the results extend to the strongly magnetized regime the results obtained previously by Zank and McKenzie and applied by Greaves et al. to study wave propagation in such a system for frequencies in excess of the ion gyrofrequency but less than the electron plasma frequency. For completeness, the full wave equation governing an ion-beam plasma system for any strength of applied magnetic field is derived. In specializing to the strong-magnetic-field limit, we find that the class of refractive-index topologies (which characterize the kinematic properties of wave propagation) is less rich than in the un-magnetized case. After investigating the topology of the refractive-index surface and the phase-, ray- and group-velocity surfaces, we construct a CMA diagram appropriate to the strongly magnetized ion-beam plasma system. The temporal stability and spatial amplification of the slow ion-acoustic mode for frequencies less than the stationary ion plasma frequency is investigated. We show that a strong magnetic field normal to the drift direction of the ion beam stabilizes long-wavelength modes that would be unstable in the unmagnetized case.
The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a small-amplitude, long-wavelength, but nonlinear wave equation. The present paper derives Burgers' equation for three different hydrodynamical models of cosmic ray shocks. The main development of the paper concerns the hydrodynamical model in which Alfvénic effects are neglected. For this model it is shown that the steady-state solution of Burgers' equation is equivalent to the weak, but smooth, transition solutions of the shock structure equation obtained previously. It is shown that Burgers' equation may be regarded as a wave energy equation, in which the interaction of the wave with the background medium is taken into account. Burgers' equation is also derived for Alfvénic models in which the Alfvén waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfvén velocity relative to the background fluid.
This paper explores the properties of Rossby-type electrostatic electron plasma waves at frequencies very much less than the electron gyrofrequency but very much greater than the ion gyrofrequency. Such waves represent the electron counterpart of ion Rossby waves, which propagate at frequencies very much less than the ion gyrofrequency in a plasma in which the ambient magnetic field possesses a spatial gradient perpendicular to its line of action. This feature simulates the ‘β-effect’ that operates in the classical atmospheric Rossby wave: the wave dynamics associated with both ion and electron Rossby waves are structurally similar to those associated with wave perturbations in a rotating fluid, where the β-effect arises from a spatial gradient in the Coriolis acceleration. It is shown that this plasma β-effect gives rise to a ‘new’ mode of the Rossby type, and in addition considerably modifies the conical wave propagation properties characteristic of the electron cyclotron mode. The highly dispersive and anisotropic nature of these waves is described in terms of the topology of the wavenumber surfaces concomitant with plane-wave solutions of the wave equation for the system as a whole.
In this paper we discuss the stability of three genetically similar non-uniform flows to compressive disturbances whose wavelengths are much shorter than the length scales characterizing the background flow. The results are relevant to theoretical models of cosmic ray shocks and solar wind type flows involving heat conduction. A JWKB expansion solution yields an equation which determines how the amplitudes of the perturbations may grow (or decay) as they propagate within such structures. It is shown that, in all three of the models considered, the perturbations exhibit spatial growth if the background flow is sufficiently supersonic and decelerating. The associated equations describing the evolution of the wave action are also studied with a view to deciding whether or not the behaviour of this attractive variable can provide an unambiguous answer to the question of stability. In the case of a shock transition dominated by heat conduction, it is shown that the effects of dissipative heating within the transition more than offset those of wave growth, with the result that wave amplification is accompanied by wave action decay. Therefore in general it would appear that the wave action equation alone cannot unambiguously settle stability questions.
In this paper it is shown that a stationary plasma can be accelerated by a moving neutral gas only if the velocity of the neutral gas exceeds Alfvén's critical velocity. An expression for the terminal velocity of the interaction is given which shows that, in the limit of high incoming neutral gas speeds, the composite plasma is accelerated up to one quarter of the gas speed. We also discuss terminal velocities associated with the inverse problem, namely the deceleration of a magnetized plasma as a result of its motion through, and interaction with, a stationary neutral gas.