Applications of Integral Relations and Conservation Laws in the Theory of Hydrodynamic Stability

Abstract

In order to describe the spectrum (or eigenvalues) of the reduced operator of stability (14.1′), (14.2′) in the complex plane, it is usually useful to study the properties of its quadratic form, a well-known method in the theory of linear operators in a Hilbert space. In the framework of linear theory we begin with a rigorous proof of the Miles stability criterion for a flow of a stratified fluid.

Theorem 16.1 (Miles 1961) A plane-parallel flow of a stratified fluid with its Richardson number

$$\mathit{Ri}=-\frac{g d\bar{\rho}/dz}{\rho_{0} ( dU/dz )^{2}} \equiv \frac{g \beta}{ ( U^{\prime} )^{2}}\quad \mbox{\textit{everywhere} } >\frac{1}{4}, $$

is stable.