Three-dimensional flows of an inviscid incompressible fluid and an inviscid subsonic compressible gas are considered and it is demonstrated how the WKB method can be used for investigating their stability. The evolution of rapidly oscillating initial data is considered and it is shown that in both cases the corresponding flows are unstable if the transport equations associated with the wave which is advected by the flow have unbounded solutions. Analyzing the corresponding transport equations, a number of classical stability conditions are rederived and some new ones are obtained. In particular, it is demonstrated that steady flows of an incompressible fluid and an inviscid subsonic compressible gas are unstable if they have points of stagnation.

In a rotating equilibrium state, the velocity and magnetic fields are shown to share the same flux surfaces. A simplified derivation is given of a second-order (not necessarily elliptic) partial differential equation which determines axisymmetric equilibrium states. For general configurations, equations on flux surfaces which determine the Alfvén and cusp continuous spectrum are derived and the stability investigated. These equations are written without the use of any particular coordinate system. Similar equations yield a sufficient condition for global stability of axisymmetric equilibria if the flow is parallel to the magnetic field up to a rigid rotation of the plasma. This condition is also necessary for stability in a mirror configuration with no toroidal field and a pure rigid rotation.

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The theory of ballooning modes, which are modes localized to a particular magnetic field line, is extended to corLfiguratlons relevant to space plasmas. Included are the effects of gravity and rotation and, in particular, boundary effects on magnetic field lines which intersect the plasma boundary. Three types of boundary conditions are considered, corresponding to perfectly conducting, conducting, and insulating boundaries. The interchange instability is also examined and is shown to be a special case of the ballooning instability. bution of mass in planetary magnetospheres, the subject of the interchange instability was extensively discussed, even recently [Cheng, 1985; Rogers and Sonnerup, 1986; Southwood and Kivelson, 1987, 1989]. As we later show in this article, the interchange instability is but a special case of the ballooning instability where the mode does not perturb the equilibrium magnetic field. It is, in fact, possible to derive the interchange stability criterion directly from the ballooning stability criterion. It is therefore to be expected that ballooning modes are as relevant and useful for the understanding of plasma circulation processes as are the interchange modes. The extension of ballooning modes theory to space plasmas presents us with new tasks on two different levels. On the technical level, it is desirable (and not too difficult) to incorporate additional effects which are significant in the space environment, such as the effects of gravity and of plasma rotation [Lakhina et al., 1990a, b]. Indeed, for the cold Iogenic plasma torus in the rapidly cotorating Jovian magnetosphere, these two effects may be more important than the pressure. A different level of work, requiring a more fundamental thinking, is called for in order to take 1513 E. Hameiri,

A new constant of the motion is utilized to formulate a variational principle for plasma equilibria with general flow fields. Two additional variational principles are derived from the original one. None of these formulations leads to a stability criterion if the velocity is not parallel to the magnetic field since the functionals used, the first of which being the energy, do not possess in this case a minimum but only stationary points. It is shown that other stability criteria already reported in the literature also suffer from the same deficiency. It is suggested that the lack of a minimum is due to the presence of ballooning modes.

While a microscopic system is usually governed by canonical Hamiltonian mechanics, that of a macroscopic system is often noncanonical, reflecting a degenerate Poisson structure underlying the coarse-grained phase space. Probing into symplectic leaves (local structures in a foliated phase space), we may be able to elucidate the order of transition from micro to macro. The Lagrangian guides our analysis. We formulate canonized Hamiltonian systems of Hall magnetohydrodynamics (HMHD) which have a hierarchized set of canonical variables; the simplest system is the subclass in which the ion vorticity and magnetic field have integral surfaces. Renormalizing the singularity scaled by the reciprocal Hall parameter (as the ion vorticity surfaces and the magnetic surfaces are set to merge), we delineate the singular limit to ideal magnetohydrodynamics (MHD). The formulated canonical equations will be useful in the study of ordered structures and dynamics (with integrable vortex lines) in HMHD and their singular limit to MHD, such as magnetic confinement systems, shocks or vortical dynamics.

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This article presents two sufficient conditions for the linear stability of rotating ideal plasmas, the first based on conservation of circulation and the second based on circle theorems applicable to linear Hamiltonian systems. The circle theorems also provide bounds on eigenmodes in the complex plane. All results are applied to the rotating screw-pinch which can be described by a single second-order ordinary differential equation.