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