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General stability criterion of inviscid parallel flow

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 $-μ_1<\frac{U&#39;&#39;}{U-U_s}<0$ or $0<\frac{U&#39;&#39;}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection point, $μ_1$ is the eigenvalue of Poincaré&#39;s problem. Second, this criterion is generalized to barotropic geophysical flows in $β$ 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&#39;d&#39;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.