The effect of ekman suction on a flow over a shallow topography in the beta plane

The effect of bottom Ekman layer suction on a homogeneous, constant depth, eastwards, low Rossby number flow over a shallow bottom topography in the beta plane is studied. The governing vorticity equation is obtained by expanding the velocities in the continuity and momentum equations in powers of the Rossby number, ?, and matching the vertical velocity with the vertical velocity at the outer edge of the bottom Ekman layer obtained from the Ekman layer solution. The suction effect is then linearized using an Oseen approxiamation and the resulting linear model is solved using Fourier transforms with the requirement that the solution behave like a vortex near the origin which is equivalent to the effect of an isolated bump, i.e., a Green's function solution is obtained. An analytical solution is thus, obtained in integral form and then numerically integrated. The effect of Ekman suction is found to be a damping of the downstream Rossby waves in a distance of order $\text{2√2U/f}{}_0\text{E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, an increased upstream influence, and a counterclockwise rotation of the closed streamline region about the origin. It is pointed out that the vortex solutions can be superimposed in order to obtain the solution for flow over topographies of finite horizontal text. This technique was used to compute the flow over a right circular cylinder. The results agree favorably with the experimental results of McCartney (1975).