General stability criterion of inviscid parallel flow

Abstract

A more restrictively general stability criterion of two-dimensional inviscid parallel flow is obtained analytically. First, a sufficient criterion for stability is found as either $-{\mu }_1<\frac{U^{\prime \prime }}{U-U_s}<0$ or $0<\frac{U^{\prime \prime }}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection point, ${\mu }_1$ is the eigenvalue of Poincaré's problem. Second, this criterion is generalized to barotropic geophysical flows in $\beta$ plane. Based on the criteria, the flows are are divided into different categories of stable flows, which may simplify the further investigations. And the connections between present criteria and Arnol'd's nonlinear criteria are discussed. These results extend the former criteria obtained by Rayleigh, Tollmien and Fjørtoft and would intrigue future research on the mechanism of hydrodynamic instability.

Revtex4, 4 pages, 2 figures, extends the first part of physics/0512208, Accepted, to be continued

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