Stability of a non-divergent Ekman layer

The stability of a non-divergent Ekman layer generated by the horizontal motion of a rigid lid lying on the surface of a rotating fluid is investigated for large αR where α is the wave number of the perturbation and R the Reynolds number. On the assumption that whenever instability occurs rolls are formed at an angle β to the direction of motion of the lid, it is shown that the solutions of the sixth order differential equations governing the perturbation fields can be expressed in terms of the four solutions of the Orr-Sommerfeld equation for a two-dimensional plane parallel flow. The velocity profile of this flow is obtained by projecting the velocity vector of the Ekman flow onto a vertical plane perpendicular to the rolls.

As the angle β is varied, the nature of the instability (viz. inviscid or viscous) is discussed by examining the upper branch of the neutral stability curve for infinite Reynolds number.

The Coriolis force which affects the structure of the perturbation fields does not influence the nature of the neutral stability curve which is similar to that of a plane parallel flow having the same velocity profile as the component of the Ekman layer flow perpendicular to the direction of the rolls.