The influence of the horizontal component of Earth's angular velocity on the instability of the Ekman layer

Abstract

A systematic study is made of the effect of latitude on the linear, normal mode stability characteristics of the laminar barotropic Ekman layer. The outcome depends upon the direction of the geostrophic flow (in the case of flows modelling the atmospheric Ekman layer) or, alternatively, upon the direction of the applied stress (in the case of flows modelling the oceanic Ekman layer). The minimum critical Reynolds number Rc is a function of latitude. For the atmospheric Ekman layer Rc = 30.8 for all latitudes less than 26.2° and increases monotonically with latitude to 54.2. At a latitude of 45° N, Rc is 33.9 and arises for a geostrophic wind directed towards a compass heading 252° (clockwise from north), corresponding to rolls with axes pointing due west and having wavenumber k (with unit of length taken to be the Ekman layer depth) of 0.594. The minimum Rc for the oceanic boundary layer is 11.6 for latitudes less than 81.1°, and increases with latitude to 11.8. At 45° N latitude, the critical condition arises for a surface-current compass heading of 345.2°, roll axis of 351° and a wavenumber k = 0.33. The results for Rc are all symmetric about the equator, with roll axes and associated basic flow directions rotated by 180°. As the Reynolds number R increases, the effects of the perturbation Coriolis acceleration on the instability diminish, as has been previously shown, and the error caused by neglect of the horizontal component of angular velocity therefore decreases. The high Reynolds number limit is systematically explored. It is shown that the lower branch of the neutral curve is not inviscid as R → ∞; rather kR → constant. The upper branch is inviscid in the limit R → ∞, and corresponds to a regular or singular neutral mode depending on whether the angle ε between the outer geostrophic flow and the roll axis is greater or less than 15.93°. ‘Inflectional’ modes, thought to be relevant by some investigators, do not exist for ε < 15.93°. Lastly, the most unstable inviscid mode corresponding to zero phase speed, a condition to which certain well-known experiments are sensitive, occurs at ε = 11.8° with wavenumber k = 0.6. This is in good agreement with published experimental data.