The depth-averaged 2-D continuity and momentum equations are given by
∂ h ∂ t + ∂ ( h u j ) ∂ x j = S f o r j = 1 , 2 {\displaystyle {\frac {\partial h}{\partial t}}+{\frac {\partial (hu_{j})}{\partial x_{j}}}=Sforj=1,2}
∂ ( h u i ) ∂ t + ∂ ( h u i u j ) ∂ x j − ϵ i j 3 f c u j h = g ∂ η ∂ x j + 1 ρ 0 ∂ p a ∂ x j + ∂ ∂ x j [ ν t h ∂ u i ∂ x j ] + τ i ρ f o r i = 1 , 2 a n d j = 1 , 2 {\displaystyle {\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 }}fori=1,2andj=1,2}
where t {\displaystyle t} is time, u j {\displaystyle u_{j}} is the current velocity in the jth direction, h {\displaystyle h} is the total water depth, C s a {\displaystyle C_{sa}} is the salinity concentration, and K s a {\displaystyle K_{sa}} is the salinity mixing coefficient.
![{\displaystyle {\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 }}fori=1,2andj=1,2}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3e864f2087b56845f908c3e5de41e593f323ff20)
is time, 
is the current velocity in the jth direction, 
is the total water depth, 
is the salinity concentration, and 
is the salinity mixing coefficient.