Hydrostatic Neutral Waves in a Parallel Shear Flow of a Stratified Fluid

William Blumen Dept. of Astro-Geophysics, University of Colorado, Boulder

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Abstract

The stability of a parallel flow, exhibiting both vertical and horizontal shear in a Boussinesq fluid, is investigated by a linear analysis. The effect of the earth's rotation is not considered and the disturbances are assumed to be hydrostatic, adiabatic and inviscid. A theorem, restricting the range of the complex phase speed c=cr+ici is exhibited. Then neutral wave solutions are found when the basic flow is represented by ū(y,z) = (a + z) tanhy and cr = ci = 0. It is concluded that these neutral waves are adjacent to unstable waves (cr = 0, ci > 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanhy. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.

Abstract

The stability of a parallel flow, exhibiting both vertical and horizontal shear in a Boussinesq fluid, is investigated by a linear analysis. The effect of the earth's rotation is not considered and the disturbances are assumed to be hydrostatic, adiabatic and inviscid. A theorem, restricting the range of the complex phase speed c=cr+ici is exhibited. Then neutral wave solutions are found when the basic flow is represented by ū(y,z) = (a + z) tanhy and cr = ci = 0. It is concluded that these neutral waves are adjacent to unstable waves (cr = 0, ci > 0), which owe their existence to inertial instability associated with the inflection point in the profile of tanhy. The effect of the vertical shear is passive, only modifying the characteristics of the instability but not being responsible for its occurrence.

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