Surface flow formula (Water Overlay)

In the model, imbalances in the water surface elevation across the grid drive the flow of water until a state of equilibrium is reached in terms of w and flux. Behavior of the flow is described by a second-order semi-discrete central-upwind scheme produced by Kurganov and Petrova (2007), which is based on the 2-D Saint-Venant equations (a.k.a. shallow water equations):

$\begin{array}{rl}\frac{\partial h}{\partial t}& +\frac{\partial (hu)}{\partial x}+\frac{\partial (hv)}{\partial y}=0,\\ \frac{\partial (hu)}{\partial t}& +\frac{\partial }{\partial x}(h{u}^2+\frac{1}{2}g{h}^2)+\frac{\partial (huv)}{\partial y}=-gh\frac{\partial B}{\partial x}-gh{n}^{2}u\sqrt{u^2+v^2}{h}^{-\frac{4}{3}},\\ \frac{\partial (hv)}{\partial t}& +\frac{\partial (huv)}{\partial x}+\frac{\partial }{\partial y}(h{v}^2+\frac{1}{2}g{h}^2)=-gh\frac{\partial B}{\partial y}-gh{n}^{2}u\sqrt{u^2+v^2}{h}^{-\frac{4}{3}},\end{array}$

where

u	is the velocity in the x-direction
v	is the velocity in the y-direction
h	is the water depth
B	is the bottom elevation
g	is the acceleration due to gravity
n	is the Gauckler–Manning coefficient