ASYMPTOTICS FOR PARTICULAR SOLUTIONS OF ORR-SOMMERFELD'S EQUATION

Abstract

We study the evolution of hydrodynamic perturbation initially localized in a thin layer and having initial vorticity harmonic along the flow. We introduce two small parameters, namely, the velocity profile curvature and the viscosity. We treat the solution of the Cauchy problem as the evolution of a continuum spectrum wave in a viscous flow. We find that solutions for viscous and inviscid fluids do not differ significantly outside of a critical layer during periods comparable with viscous time $t\nu^{-1/3}\sim\nu^{-1/3}(kU')^{-2/3}$, where $\nu$ is the viscosity, $k$ is the wavenumber, and $U'$ is the transverse gradient of the flow. The critical layer grows with time as $(\nu t)^{1/2}$. In case of a viscous fluid, we use a detailed asymptotics for minors of the fundamental matrix of Orr-Sommerfeld's equation. In some cases the leading terms of the asymptotics for second-order minors involve higher-order terms from asymptotics of particular solutions.

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Computational Seismology, Vol. 2.