CMS-Flow:Hydro Eqs

Governing Equation

The depth-averaged 2-D continuity and momentum equations are given by

        ${\frac {\partial h}{\partial t}}+{\frac {\partial (hu_{j})}{\partial x_{j}}}=S$

for $j=1,2$

        ${\frac {\partial (hu_{i})}{\partial t}}+{\frac {\partial (hu_{i}u_{j})}{\partial x_{j}}}-\epsilon _{ij3}f_{c}u_{j}h=g{\frac {\partial \eta }{\partial x_{j}}}+{\frac {1}{\rho _{0}}}{\frac {\partial p_{a}}{\partial x_{j}}}+{\frac {\partial }{\partial x_{j}}}{\biggl [}\nu _{t}h{\frac {\partial u_{i}}{\partial x_{j}}}{\biggr ]}+{\frac {\tau _{i}}{\rho }}$  

for $i=1,2$ and $j=1,2$

Symbol	Description	Dimensions
$t$	Time	T
$h$	Total water depth $h=\zeta +\eta$	L
Failed to parse (unknown function "\sigmaf"): {\displaystyle \sigmaf }	Still water depth	L
$\eta$	Water surface elevation with respect to the still water elevation	L
$u_{j}$	Current velocity in the jth direction	L / T
$S$	Sum of Precipitation and evaporation per unit area	L / T
$g$	Gravitational constant	L / T2
$\rho$	Water density	M / L3
$p_{a}$	Atmospheric pressure	M / (L T 2 )

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