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):

${\begin{aligned}{\frac {\partial h}{\partial t}}&+{\frac {\partial (hu)}{\partial x}}+{\frac {\partial (hv)}{\partial y}}=0,\\[3pt]{\frac {\partial (hu)}{\partial t}}&+{\frac {\partial }{\partial x}}\left(hu^{2}+{\frac {1}{2}}gh^{2}\right)+{\frac {\partial (huv)}{\partial y}}=-gh{\frac {\partial B}{\partial x}}-ghn^{2}u{\sqrt {u^{2}+v^{2}}}h^{-{\frac {4}{3}}},\\[3pt]{\frac {\partial (hv)}{\partial t}}&+{\frac {\partial (huv)}{\partial x}}+{\frac {\partial }{\partial y}}\left(hv^{2}+{\frac {1}{2}}gh^{2}\right)=-gh{\frac {\partial B}{\partial y}}-ghn^{2}u{\sqrt {u^{2}+v^{2}}}h^{-{\frac {4}{3}}},\end{aligned}}$

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