معادلات آب کم‌عمق

${\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}+{\frac {\partial (\eta u)}{\partial x}}+{\frac {\partial (\eta v)}{\partial y}}&=0\\[3pt]{\frac {\partial (\eta u)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\eta u^{2}+{\frac {1}{2}}g\eta ^{2}\right)+{\frac {\partial (\eta uv)}{\partial y}}&=0\\[3pt]{\frac {\partial (\eta v)}{\partial t}}+{\frac {\partial (\eta uv)}{\partial x}}+{\frac {\partial }{\partial y}}\left(\eta v^{2}+{\frac {1}{2}}g\eta ^{2}\right)&=0.\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}-fv&=-g{\frac {\partial h}{\partial x}}-bu,\\[3pt]{\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}+fu&=-g{\frac {\partial h}{\partial y}}-bv,\\[3pt]{\frac {\partial h}{\partial t}}&=-{\frac {\partial }{\partial x}}{\Bigl (}u\left(H+h\right){\Bigr )}-{\frac {\partial }{\partial y}}{\Bigl (}v\left(H+h\right){\Bigr )},\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}-fv&=-g{\frac {\partial h}{\partial x}}-bu,\\[3pt]{\frac {\partial v}{\partial t}}+fu&=-g{\frac {\partial h}{\partial y}}-bv,\\[3pt]{\frac {\partial h}{\partial t}}&=-H{\Bigl (}{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}{\Bigr )}\end{aligned}}}$