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Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid

AXEL DELONCLE, JEAN-MARC CHOMAZ, PAUL BILLANT

2007
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Journal of Fluid Mechanics
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This paper investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, F h ,isvariedfrom∞
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... to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for F h ≪ 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally. Introduction Because of its importance in industrial and geophysical applications, the evolution of shear flows has been much studied (Ho & Huerre 1984) . The basic case of a flow in a homogeneous fluid is well known. Squire's theorem (Squire 1933) states that any threedimensional unstable mode is less unstable than two-dimensional modes, implying that shear flows are dominated by two-dimensional instabilities. Necessary conditions for their existence are given by Rayleigh's inflection point criterion (Rayleigh 1887) and Fjørtoft's criterion (Fjørtoft 1950) on the basic velocity profile in the inviscid case. Howard (1961) also showed that the temporal growth rate of two-dimensional perturbations are confined inside a semi-circle in the complex plane. Much attention has also been devoted to the situation of a vertical shear in a stably stratified fluid. Miles (1961) and Howard (1961) established a sufficient condition for stability as Ri > 1/4 with Ri = N 2 /S 2 , the Richardson number, where N is the Brunt-Väisälä frequency and S the shear rate. In contrast, the case of a non-vertical shear in a vertically stratified fluid has been little addressed. Blumen (1970) extended the application of Howard's semicircle theorem for hydrostatic perturbations in a stratified fluid for an arbitrary orientation of the shear. However, the Squire theorem does not hold for such flows, meaning that it is not at present known whether two-dimensional perturbations dominate the dynamics of a stratified sheared flow. In order to address this question, we investigate the

doi:10.1017/s0022112006003454
fatcat:5kvjeeskcjghfoqfl2qx5zgdpu