Emergence and Secondary Instability of Ekman Layer Rolls

Abstract

The authors revisit the idealized scenario by which long-lived rolls are believed to emerge in the neutral planetary boundary layer, that is, through the saturation of the shear instability of the neutrally stratified Ekman flow. First, the nonlinear stages of the primary instability are studied, using a constant turbulent viscosity with Reynolds numbers up to 1000. Two-dimensional equilibrated rolls are found to exist, as predicted earlier based on a weakly nonlinear expansion. However, the flow may not saturate into those equilibrated rolls if the turbulent Reynolds number is too high. Second, a linear stability analysis of these equilibrated rolls is performed, which finds that they are subject to a three-dimensional instability. The growth rate of the most unstable mode is comparable to the growth rate of the primary instability; the selected horizontal length scale is about 4 times shorter. The unstable mode draws its energy by interacting with both across-roll and along-roll shear, the latter interaction being stronger. The latitude and the direction of the geostrophic wind affect the dynamics through the horizontal component of the Coriolis vector; their influence is investigated in both studies. At Reynolds numbers sufficiently higher than the threshold of the primary instability, the saturated rolls depend negligibly on latitude and wind direction. However, the growth rate of the secondary instability depends substantially on latitude and wind direction over the range of Reynolds numbers considered.