 Solution

Here $s$ is downstream distance, $A$ the wetted cross-sectional area, $u$ the section-averaged downstream velocity, and $M$ the lateral water input per channel length and time. Thus $A_t$ is local water storage and $(Au{)}_s$ the divergence of discharge. Their balance is mass conservation.

The momentum terms $u_t$ and $u{u}_s$ are local and advective acceleration. The bed slope $S$ gives the downstream gravitational acceleration $gS$. Bed shear stress $\tau$ acting over wetted perimeter $l$ produces resisting force per unit mass $\tau l/(\rho A)$; the hydraulic radius is $R=A/l$. Finally $-g{\overline{h}}_s$ is the hydrostatic free-surface pressure-gradient term in the chosen bed-following description. The Saint-Venant approximation depth-averages the flow, assumes a long shallow geometry and hydrostatic pressure, and closes the boundary shear through a roughness law.