The primitive equations in Cartesian coordinates can be written:
∂ u ∂ t + v → ⋅ ∇ u − f v = − ∂ ϕ ∂ x + F u + D u ∂ v ∂ t + v → ⋅ ∇ v + f u = − ∂ ϕ ∂ y + F v + D v ∂ T ∂ t + v → ⋅ ∇ T = F T + D T ∂ S ∂ t + v → ⋅ ∇ S = F S + D S ρ = ρ ( T , S , P ) ∂ ϕ ∂ z = − ρ g ρ o ∂ 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.}
