The primitive equations in Cartesian coordinates can be written:
∂u∂t+v→⋅∇u−fv=−∂ϕ∂x+Fu+Du{\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+fu=−∂ϕ∂y+Fv+Dv{\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=FT+DT{\displaystyle {\frac {\partial T}{\partial t}}+{\vec {v}}\cdot \nabla T={F}_{T}+{D}_{T}}
∂S∂t+v→⋅∇S=FS+DS{\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=−ρgρo{\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}
