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Ekman transport is the integrated horizontal flow in the boundary layer resulting from the balance between frictional forces (at the surface or bottom) and Coriolis. Meanwhile, Ekman pumping is a vertical motion at the base of the boundary layer resulting from the curl of the wind.

What is the relationship between Ekman transport and Ekman pumping?

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The link is the conservation of mass equation: $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0. $$ Briefly, Ekman pumping is the horizontal divergence of the Ekman transport.

If we integrate this expression over the Ekman layer, $$ \int_{-D}^0 \frac{\partial u}{\partial x} dz + \int_{-D}^0 \frac{\partial v}{\partial y} dz + \int_{-D}^0 \frac{\partial w}{\partial z} dz = 0$$ where $D$ is the height of the Ekman layer we take $z=0$ to be the surface. This gives $$ \frac{\partial}{\partial x} \int_{-D}^0 u dz + \frac{\partial }{\partial y} \int_{-D}^0 v dz + w_{z=0} - w_{z=-D} = 0$$ or, $$ \frac{\partial}{\partial x} U + \frac{\partial }{\partial y} V + w_{z=0} - w_{z=-D} = 0$$ where $U$ and $V$ are the horizontal volume (Ekman) transports. As a boundary condition we take $w_{z=0} = 0$ leaving $$ w_{z=-D} = \frac{\partial}{\partial x} U + \frac{\partial }{\partial y} V. $$ So, the Ekman pumping velocity is the horizontal divergence of the Ekman transport (and thus also proportional to the curl of the wind stress).

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