Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid

Abstract

In a previous paper we considered the nonlinear stability of a cylindrical mixing layer in an incompressible fluid at large Reynolds numbers. Nonlinear evolution results in the formation of vortex structures in the vicinity of the corotation radius rc. This paper considers the same model but in a compressible fluid. A fundamental difference implied by the presence of compressibility is the possibility of the generation of disturbances which are no longer localized near the shear layer but embrace the entire region. These are acoustic waves generated in the region of corotation resonance and emitted into the periphery. In the r > rc region lines of equal density are trailing spirals. The nonlinear evolution of such disturbances is determined by redistribution of the mean flow inside the critical layer (CL). It is shown that only two possible types of CL, viscous and unsteady, can be realized here. For both types of these regimes, evolution equations describing the dynamics of a spiral density wave amplitude are obtained and their solutions analysed. It appears that at any values (provided that they are small enough) of initial supercriticality of the flow, an explosive growth of amplitude occurs which continues as long as values comparable with background ones are reached.

Conditions That Inhibit the Spontaneous Radiation of Spiral Inertia–Gravity Waves from an Intense Mesoscale Cyclone

“…It is not our intention to present a rigorous theory for the nonlinear interaction of a VR wave and its critical layer (cf. Balmforth et al 2001;Shukhman 1991). Our simplified analysis seems adequate to explain the principal simulation results of section 6.…”

“…In particular, the results were not explained as the outcome of a competition between critical layer damping and radiative pumping, as they are here. This paper also overlaps astrophysical studies of modal stability in compressible, differentially rotating discs, with gravitational centers of attraction (e.g., Papaloizou and Pringle 1987;Shukhman 1991;Papaloizou and Lin 1995;Li et al 2001). The usual example is a stellar accretion disc.…”

Gravitational loss-cone instability in stellar systems with retrograde orbit precession

“…It is widely used in problems related to the so‐called critical layer, that is, a narrow domain near the resonance of the wave and the shear flow of fluid (see e.g. Hickernell 1984; Shukhman 1991). ] We obtain The imaginary part of can be simplified using relation (4.24), reflecting the balance between growth and damping for a neutral mode.…”

Damping and pumping of a vortex Rossby wave in a monotonic cyclone: Critical layer stirring versus inertia–buoyancy wave emission

“…These papers analyze waves in straight shear flow [47][48][49] and in stellar accretion disks. [50][51][52] In general, they discuss how PV stirring in the critical layer affects wave stability. The accretion disk papers further discuss the positive feedback of sound-wave emission ͑as opposed to IB wave emission͒.…”