Depth-integrated shallow-water transport ($(U,V)=H(u,v)$)

For a depth-independent horizontal velocity in a layer of reference depth $H$, the linear transports are $U=Hu$, $V=Hv$. They have dimensions of area per time and satisfy the surface continuity equation ${\eta }_t+U_x+V_y=0$. Their momentum equations are obtained by multiplying the depth-independent velocity equations by $H$.