Secondary instability of the stably stratified Ekman layer

Abstract

AbstractThe Ekman flow, an exact solution of the Boussinesq equations with rotation, is a prototype flow for both atmospheric and oceanic boundary layers. The effect of stratification on the finite-amplitude longitudinal rolls developing in the Ekman flow and their three-dimensional stability is studied by means of linearized and nonlinear numerical simulations. Similarities and differences with respect to billows developing in the Kelvin–Helmholtz (KH) unidirectional stratified shear flow are discussed. Prandtl number effects are investigated as well as the role played by the buoyant-convective instability. For low Prandtl number, the amplitude of the saturated rolls vanishes at the critical bulk Richardson number, while at high Prandtl number, finite-amplitude rolls are found. The Prandtl number also affects how the growth rate of the secondary instability evolves as the Richardson number is increased. For low Prandtl number, the growth rate decreases as the Richardson number increases while it remains significant for large Prandtl number over the range of stratification studied. This behaviour is likely a result of the differing amplitudes of the roll vortices. Furthermore, the most unstable wave vector is much lower than for the secondary instability of KH billows. Examination of the energetics of the secondary instability shows that buoyant-convective instability is present locally at high Reynolds and Prandtl numbers but plays an overall minor role despite the presence in the base flow of statically unstable regions characterized by a high Richardson number.