Perturbations of Plane Couette Flow in Stratified Fluid and Origin of Cloud Streets

Abstract

Small perturbations of plane Couette flow in stably and unstably stratified fluid are considered. It is found that the system is more unstable when it is bounded both above and below than when its depth is infinite, but a finite negative Richardson number J is required to maintain the perturbation for both cases. For the former case, this limiting Richardson number is −3k 1 2 /4(k 1 2 + k 2 2 ), while for the latter it is −2k 1 2 /(k 1 2 + k 1 2 ), where k 1 and k 2 are wavenumbers in the mean flow and the transversal direction. These results show that in an unstably stratified layer of Couette flow, the preferred mode of motion is roll‐type convection, with the dimension in the wind direction much larger than in the transversal direction. The amplification factor σ for the perturbations has been determined as a function of the modified Richardson number J̄ = gS z U z −2 (1 + k 1 −2 k 2 2 ), and the dimensionless wavenumber α = h(k 1 2 + k 1 2 ) 1/2 . Four different regimes have been found, each corresponding to a different type of perturbation. An application of the theory is made to the formation of longitudinal cloud rolls observed in the earth's atmosphere and in certain laboratory experiments.