Alexander Lifschitz

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.

Three-dimensional stability of elliptical vortex columns in external strain flows

The Kirchhoff-Kida family of elliptical vortex columns in flows with uniform strain and rotation displays a rich variety of dynamical behaviours, even in a purely two-dimensional setting. In this paper, we address the stability of these columns with respect to three-dimensional perturbations via the geometrical optics method. In the case when the external strain is equal to zero, the analysis reduces to the stability of a steady elliptical vortex in a rotating frame. When the external strain is non-zero, the stability analysis reduces to the theory of a Schrödinger equation with quasi-periodic potential. We present stability results for a variety of different Kirchhoff-Kida flows. The vortex columns are typically unstable except when the interior vorticity is approximately the negative of the background vorticity, so that the flow in the inertial frame is nearly a potential flow.

Short-Wavelength Instabilities of Riemann Ellipsoids

Perturbations of incompressible S-type Riemann ellipsoids are considered in the limit of short wavelength. This complements the classical consideration of perturbations of long wavelength. It is shown that there is very little of the parameter space in which the laminar, steady-state flow can exist in a stable state. This confirms, in a physically consistent framework, hydrodynamic-stability results previously obtained in the context of unbounded, two-dimensional flows. The configurations stable to perturbations of arbitrarily short wavelength include the rigidly rotating configurations of Maclaurin and Jacobi, as well as a narrow continuum centred on the family of irrotational ellipsoids and including the non-rotating sphere. However, most configurations departing even slightly from axial symmetry are unstable. Some of the implications of these results for the complexity of astrophysical flows are discussed.

Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity

Abstract We consider three-dimensional flows of an inviscid (or asymptotically inviscid), incompressible fluid and present a method for describing the propagation of nonlinear small amplitude, short wavelength perturbations of these flows. Analyzing the evolution of these perturbations we obtain a local sufficient condition for the nonlinear instability of an arbitrary three-dimensional flow.

A new class of instabilities of rotating fluids

Standing waves in a rotating ideal fluid are considered. It is shown that all of them are unstable with respect to short‐wavelength perturbations. Moreover, it is demonstrated that the growth rate of the corresponding instabilities tends to infinity when either the amplitude of the standing wave increases or its spatial scale decreases without bound. It is suggested that the observed instabilities are akin to the Hadamard instabilities.

Exact description of the spectrum of elliptical vortices in hydrodynamics and magnetohydrodynamics

A method for studying natural oscillations of fluids and plasmas in the neighborhood of two‐dimensional elliptical flows is presented. The method uses scaling combined with the Fourier transformation to reduce the spectral stability problem for such flows to a spectral problem for an ordinary differential operator. This reduction is used to obtain a complete description of the spectrum for fluid flows and a qualitative description of the spectrum (including bounds for the complex part of the spectrum) for plasma flows. It is shown that a steady planar fluid flow with elliptical streamlines is spectrally unstable. It is also shown that all planar magnetized plasma flows with elliptical streamlines are spectrally unstable, except for the case when the magnitudes of the fluid velocity and the Alfven velocity are exactly equal to each other.

Short wavelength instabilities of rotating, compressible fluid masses

The equations describing the linear stability of a rotating, axisymmetric, selfgravitating fluid mass are considered in a short wavelength limit which takes proper account of boundary conditions. The well-known Høiland criterion for axisymmetric perturbations is recovered and found to be a sufficient condition for instability even if non-axisymmetric perturbations are allowed. The spectrum of the linear stability problem is not only continuous but also, in the unstable case, occupies a non-zero area in the complex spectral plane.

Local hydrodynamic instability of rotating stars

We reconsider the result, due to Goldreich & Schubert, that a differentially rotating star with an angular velocity depending on the vertical coordinate or violating Rayleighs criterion is unstable in the presence of a dissipative mechanism. The outcome confirms the result in a manner free of some of the objections that have been raised against it. It further shows that not only Rayleighs criterion but also Schwarzschilds criterion must be satisfied for stability to axisymmetric perturbations. The choice of small parameter singles out a wavelength band for which the linearized instability develops on the dynamical time scale

On the instability of three-dimensional flows of an ideal incompressible fluid

Abstract In a recent paper Lifschitz and Hameiri [Phys. Fluids A 34 (1991) 2644] demonstrated how the stability of general three-dimensional steady flows of an ideal incompressible fluid with respect to short wavelength perturbations can be studied via the geometrical optics method. The flow is unstable if for some stream line the transport equation associated with the wave which is advected by the fluid has a sufficiently rapidly growing solution. In the present paper we show that under certain conditions this equation can be solved explicitly. Using this fact we demonstrate that all axisymmetric flows such that the corresponding poloidal velocity has a point of stagnation, particularly all axisymmetric vortex rings, are unstable. We also describe explicitly exponentially growing solutions of the transport equation for three-dimensional flows with stretching.

On the spectrum of hyperbolic flows

By using a previously developed technique [A. Lifschitz, Phys. Fluids A 7, 1626 (1995)], it is shown that the spectrum of linear hyperbolic flows in the plane occupies a strip in the complex plane along the real axis which is independent of the geometry of the basic flow and that all the points in this strip are simple eigenvalues. The relevance of this result to the stability theory is explained.