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Assume $f>0$; replacing it by $|f|$ gives the thickness for either hemisphere. With $W=(u-U)+iv$, the steady anomaly equations reduce toThus the Bottom Ekman layer hasIts integrated anomalous transport isIt is the transport required by the vertically integrated momentum balance between Coriolis acceleration and bottom stress. For slowly varying geostrophic flow, $\nabla \cdot {\mathbf{U}}_g=0$ and $\nabla \cdot (\hat{z}\times {\mathbf{U}}_g)=-{\zeta }_g$, soMass conservation therefore gives the interior Ekman pumping condition

Assume a steady, small-Rossby number surface boundary layer, neglect horizontal viscosity and nonlinear acceleration, take the pressure gradient to be independent of depth, impose no normal flow at the surface, and let the viscous stress vanish at $z=-h$. Subtract the depth-independent geostrophic balance from horizontal momentum. For the ageostrophic velocity,Integrating from $-h$ to $0$ and usinggives the Ekman transport

Depth-integrated mass conservation, with $w(0)=0$, givesHence the vertical velocity entering the ocean interior is the Ekman pumping velocityFor constant $f$ this reduces to

Put $D(x,y)=H(x,y)-h(x,y)$ and let $\mathbf{u}=(u,v)$ be the depth-independent interior velocity. The kinematic boundary conditions on the sloping upper and lower surfaces areIntegrating mass conservation through the layer givesThe inviscid vertical-vorticity equation isConsequently the shallow-water potential vorticityobeys the forced evolution equationPositive upward Ekman pumping removes layer thickness and raises the potential vorticity of the remaining column.

For a steady small-Rossby number flow, $q\simeq f/D$, soEquivalently,The same result follows from the integrated vortex-stretching balance

When $\nabla D=(c,0)$ with $c>0$,If the upper pumping is absent or weak and the upper surface is locally level, the impermeable-bottom condition is $w_b=-cu$. The stipulated $w_b<0$ then gives $u>0$, and in the Northern HemisphereThe steady interior flow is therefore directed northeastward, along contours of $f/D$ in the unforced limit. As a parcel moves eastward into deeper water, its vortex column stretches; a poleward displacement increases $f$ and preserves potential vorticity. Nonzero Ekman pumping drives motion across the $f/D$ contours. This is topographic potential-vorticity steering and the associated topographic Sverdrup balance.

Bottom Ekman pumping damps a horizontal quasi-geostrophic Fourier mode while its influence penetrates vertically only over $|f|/(Nk)$. For a half-space with a vertically uniform initial mode, the boundary amplitude decays exponentially while the remote interior remains unchanged.