The Non-Boussinesq Taylor–Caulfield Instability

Abstract

The study of the conditions under which a stratified shear flow becomes turbulent is important, as turbulence is the source of mixing and dissipation in the atmosphere and can significantly influence the momentum and temperature structure of the atmospheric circulation. Oftentimes, the density structure of atmospheric flows is organized in thick layers of constant density separated by thin layers of sharp density gradients. It has been shown by previous studies that such multilayered flows can become unstable under shear. In this work, we investigate Taylor–Caulfield Instability (TCI), which occurs in a three-layer fluid moving with a constant shear flow. Previous studies examined the instability under the Boussinesq approximation, which is not expected to hold in cases of sharp density gradients. The non-Boussinesq limit is therefore investigated in this work. TCI is studied using the classical perturbation theory, that is by examining the evolution of small perturbations to the base flow. The wavelength of the waves expected to dominate the flow as well as the time in which these waves will emerge are calculated. In addition, the characteristics of the unstable waves are studied under a variety of conditions for the shear and the stratification. It is found that under the Boussinesq approximation, the wavelength of the instability waves is underestimated and the time for the evolution of the waves is overestimated.

Keywords

• stratified shear instability
• turbulence
• Boussinesq approximation
• layering