Shallow water equations

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For a one-layer fluid,

\frac{\partial \eta}{\partial t} + \nabla \cdot \left(h \mathbf{u}\right) =0 \, ,

\frac{\partial \mathbf{u}}{\partial t} + \left(\mathbf{u} \cdot \nabla \right) \mathbf{u} = -g \nabla \eta  \,,
where  h = H + \eta is the total depth, H is the mean depth, and \eta is the free-surface displacement.