The stability of statically unstable layers

Abstract

We investigate the development of instability in a fluid with density locally of the form ρ₀[1 −(N² / g)z + A sin Kz], composed of an overall stable uniform gradient of buoyancy frequency, N, but with a superimposed sinusoidal variation of vertical wavenumber, K, and amplitude, A [Lt ] 1; g is the acceleration due to gravity and z is the upward vertical coordinate. Layers exist in which the fluid is statically unstable when the parameter r = N² / gKA, is less than unity.

When r is zero, the density is sinusoidal in z and the problem reduces to one studied by Batchelor & Nitsche (1991). Their solution, which finds a gravest mode of linear instability with terms having vertical motions independent of z and with horizontal scales large in comparison with K⁻¹, is extended to non-zero r. An effect of a small, but finite, r is to stabilize the fluid, increasing the critical Rayleigh number and the corresponding non-dimensional horizontal wavenumber. The vertical scale of the mode which first becomes unstable is reduced as r increases. A small sinusoidal shear destabilizes the fluid.

When r approaches unity, the density field contains regions of static instability which are of thickness small compared to K⁻¹. The problem then approximates to one studied by Matthews (1988). Consistent solutions for the growth of disturbances are obtained by truncated series and, following Matthews, by the solution of a Fourier-transformed equation. A small uniform shear, characterized by a flow Reynolds number, Re > O, is found to stabilize the fluid, in that it increases the critical Rayleigh number of the onset of instability. This suggests that convective Rayleigh–Taylor instability, with constant phase lines parallel to the flow, is then the favoured mode of onset of instability. At very large Rayleigh numbers and at a Prandtl number of 700, however, the growth rate of the most rapidly growing linear disturbances may increase as Re increases from zero, and the form of the evolving flow is then less certain.

The theory is used to estimate the scale and growth rates of instability in overturning internal gravity waves in the laboratory experiment described in a companion paper (Thorpe 1994).