The primitive equations in Cartesian coordinates can be written:
∂u∂t+v→⋅∇u−fv=−∂ϕ∂x+Fu+Du∂v∂t+v→⋅∇v+fu=−∂ϕ∂y+Fv+Dv∂T∂t+v→⋅∇T=FT+DT∂S∂t+v→⋅∇S=FS+DSρ=ρ(T,S,P)∂ϕ∂z=−ρgρo∂u∂x+∂v∂y+∂w∂z=0.{\displaystyle {\frac {\partial u}{\partial t}}+{\vec {v}}\cdot \nabla u-fv=-{\frac {\partial \phi }{\partial x}}+{\cal {F}}_{u}+{\cal {D}}_{u}{\frac {\partial v}{\partial t}}+{\vec {v}}\cdot \nabla v+fu=-{\frac {\partial \phi }{\partial y}}+{\cal {F}}_{v}+{\cal {D}}_{v}{\frac {\partial T}{\partial t}}+{\vec {v}}\cdot \nabla T={\cal {F}}_{T}+{\cal {D}}_{T}{\frac {\partial S}{\partial t}}+{\vec {v}}\cdot \nabla S={\cal {F}}_{S}+{\cal {D}}_{S}\rho =\rho (T,S,P){\frac {\partial \phi }{\partial z}}={\frac {-\rho g}{\rho _{o}}}{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0.}
