### Ekman Layers

Near the upper and lower boundaries of the ocean (and lower boundary of the atmosphere), boundary layers form. These boundary layers, called Ekman layers, are typically modeled as geostrophic balance in the presence of an eddy viscosity. The model equation is

$$f\hat{k} \times \vec{u} = -\frac{\nabla p}{\rho_0} + \nu \frac{\partial^2 \vec{u}}{\partial z^2}$$

where $$\nu$$ is the eddy viscosity, and we take $$\rho_0$$ and $$f$$ to be constant (and gravity is absorbed into the pressure term). We further assume that $$\vec{u}=0$$ at the boundaries and that $$\vec{u}=\vec{u_g}$$ in the interior, where

$$\vec{u_g} \equiv \frac{1}{f\rho_0}\hat{k} \times \nabla p$$

is the geostrophic velocity.

Now consider specifically a bottom boundary with a surface stress:

$$\frac{\vec{\tau}}{\rho_0} = \nu \frac{\partial \vec{u}}{\partial z}$$

1. Show that the transport in the Ekman layer is$$\vec{U_e} \equiv \int\limits_{-H}^{-H+\delta} \vec{u_e}dz = \hat{k} \times \left(\frac{\vec{\tau}}{f\rho_0}\right)$$where $$\vec{u_e} = \vec{u}-\vec{u_g}$$, $$H$$ is the depth of the ocean, and $$\delta$$ is the thickness of the layer. Remember that the ageostrophic velocity induced by the surface stress vanishes at the edge of the Ekman layer.
2. Using the continuity equation and remembering that vertical velocity is identically zero at the bottom surface, show that the divergence of the total transport in the Ekman layer is$$\nabla \cdot \vec{U} = -w_e$$where $$w_e$$ is the vertical velocity at the upper edge of the Ekman layer.
3. Relate the vertical velocity to the divergence of the geostrophic transport in the Ekman layer and the applied stress.
4. Give a scaling estimate for the depth $$\delta$$ of the Ekman layer.
5. It is well known that the sediment in the bottom of a cup of fluid congregates toward the center of the cup when the fluid is stirred, regardless of the direction of stirring. Qualitatively modify the above Ekman-layer theory to explain this phenomenon. From Wikipedia.

From Wikipedia.