Equations of Motion

The primitive equations in Cartesian coordinates can be written:

${\frac {\partial u}{\partial t}}+{\vec {v}}\cdot \nabla u-fv=-{\frac {\partial \phi }{\partial x}}+{F}_{u}+{D}_{u}$

${\frac {\partial v}{\partial t}}+{\vec {v}}\cdot \nabla v+fu=-{\frac {\partial \phi }{\partial y}}+{F}_{v}+{D}_{v}$

${\frac {\partial T}{\partial t}}+{\vec {v}}\cdot \nabla T={F}_{T}+{D}_{T}$

${\frac {\partial S}{\partial t}}+{\vec {v}}\cdot \nabla S={F}_{S}+{D}_{S}$

$\rho =\rho (T,S,P)$

${\frac {\partial \phi }{\partial z}}={\frac {-\rho g}{\rho _{o}}}$

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$