Eliezer Hameiri

Local stability conditions in fluid dynamics

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.

The equilibrium and stability of rotating plasmas

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 Alfven 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.

Variational principles for equilibrium states with plasma flow

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.

Spectral estimates, stability conditions, and the rotating screw-pinch

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.

Turbulent magnetic diffusion and magnetic field reversal

The behavior of compressible plasmas caused by small fluctuations superimposed on slowly varying profiles is investigated self‐consistently within the framework of resistive magnetohydrodynamics. The effect of the fluctuations is shown to be the generation of an additional ‘‘eddy’’ resistivity, and not of a dynamo. This effect, nevertheless, suffices to explain magnetic field reversal in pinches. The functional form of the turbulent resistivity is derived from the dynamics of the fluctuations, taken as a bath of local and global modes that are believed to be responsible for field reversal. It is shown that the fluctuations tend to flatten the pressure and parallel current profiles and thus produce plasmas near a ‘‘Taylor state.’’

The ballooning spectrum of rotating plasmas

Ballooning modes are shown to be part of the spectrum by using a ‘‘singular sequence’’ of localized modes. We show that the modes arise from Alfven and slow magnetosonic waves propagating along rays confined inside the plasma. Different ballooning modes are seen, depending on the particular rotating frame of observation, indicating that there are accumulation points of eigenvalues. The effect of rigidly rotating flow is seen to be destabilizing due to an analog of the Rayleigh–Taylor instability associated with density gradients in the presence of a centrifugal force. Flow shear also modifies the stability criterion. A certain component of the flow shear will eliminate the ballooning modes.

Unstable continuous spectrum in magnetohydrodynamics

The continuous spectrum need not be stable for an equilibrium with flow. The case of a noncircular, rotating ϑ pinch is shown to have an unstable continuum with the instability interpreted as a parametric instability arising from coupling between the rotation frequency and the plasma wave frequencies. On the other hand, equilibria with sub‐Alfvenic flow parallel to the magnetic field in incompressible plasmas, as well as static equilibria, have a stable continuum. A formulation to determine the continuous spectrum is given in terms of characteristic surfaces for the equations and does not involve the use of any particular system of coordinates.

Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow

Lyapunov stability conditions for ideal magnetohydrodynamic (MHD) plasmas with mass flow in axisymmetric toroidal geometry are determined in the Eulerian representation. Axisymmetric equilibrium solutions of ideal MHD are associated to critical points of a nonlinearly conserved Lyapunov functional consisting of the sum of the total energy and the following flux‐weighted quantities: the circulation along field lines, the angular momentum, the toroidal flux, and the mass content within each flux tube. Conditions sufficient for Lyapunov stability of these equilibria against axisymmetric perturbations are found by taking advantage of the Hamiltonian formalism for ideal MHD. In particular [see Eq. (60)], it is sufficient for Lyapunov stability under linearized dynamics that an axisymmetric equilibrium be subsonic in the appropriate rotating frame, lie in the first elliptic regime of the Bernoulli–Grad–Shafranov (BGS) system of equations, and satisfy one additional, more complicated, condition. Effects of bound...

Shear stabilization of the Rayleigh-Taylor modes

The effect of flow shear on the stability of rotating ideal magnetohydrodynamic equilibria is investigated. It is found that a large enough shear can sometimes completely stabilize the flute(k=0) modes. The most notable result is a theorem which provides for an analytical determination of the exact number of unstable modes. A weaker sufficient condition for stability is also derived.

Canonical Hamiltonian mechanics of Hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics

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.