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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?
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).
