Local stability conditions in fluid dynamics

Abstract

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.

On recent developments in the spectral problem for the linearized Euler equation

“…However, the identities (27)-(30) remain valid even if λ max = 0 provided the underlying fluid flow has arbitrarily long trajectories, the condition usually needed for translational and rotational invariance of spectra. In this case we simply have (31) σ ess (e…”

“…[19] and references therein). This approach was refined in the work of Friedlander and Vishik [25,26,52], and Lifschitz and Hameri [32,31] who developed geometric optics techniques to produce effective criteria for detecting instabilities in the essential spectrum of the Euler equation. Such instabilities are associated with stretching along the streamlines of the flow u(x).…”

On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids

“…These results are validated experimentally using an extended version of the set-up of Lacaze et al [16], with stronger imposed magnetic fields. These results are then extended to the large magnetic Reynolds number, large Reynolds number limit relevant to planetary applications, using a short-wavelength Lagrangian theory [18]. An analytical expression of the growth rate of the elliptical instability is determined and results are finally applied to the case of Io, highlighting the importance of the elliptical instability at the planetary scale.…”

Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

“…Recently, an important breakthrough has been made in understanding the essential spectrum of L and e tL , see [FV,FV2,FSV,FSV2,LH1,LH2,V,VF] and the bibliography therein. In particular, using asymptotic expansions for integral Fourier operators, the boundary of the essential spectrum of e tL (in dimensions two and three) was related to the maximal Lyapunov exponent of a so-called bicharacteristic amplitude system, see [V,VF,S2] and also (8) below.…”