On the Instability of Ekman Boundary Flow

Abstract

The stability of the two-dimensional boundary flow produced in a rotating tank with small inflow is investigated by means of perturbation analysis. The resulting differential eigenvalue problem is solved numerically both for the complete set of equations and for a truncated set (the Orr-Sommerfeld equation) shown by Stuart and Barcilon to be valid in the limit of large Reynolds number R. Both solutions exhibit instability, with a critical R of about 55 for the complete solution and 93 for the truncated set, where R is based on the boundary scale depth and the relative velocity of the fluid distant from the boundary. Stationary waves become unstable for R > 115. These values are to be compared with Faller's observed value of R ≈ 125 for nearly stationary waves. The difference between theory and observation is discussed and largely rationalized as being due to the difficulty of observing moving waves with Faller's method. The effect of the terms neglected in Stuart's analysis is shown to lead to a new mechanism of instability, dependent on the coriolis effect and viscosity. This instability mechanism is further investigated by means of a simplified analytic solution. These results are believed to reinforce Faller's suggested identification of Ekman instability with the ?large eddies? in the real turbulent atmosphere.