Navier-Stokes equations for a hydrostatic ocean on the rotating sphere

In a rotating frame of reference and in the absence of sources and sinks of mass or salt the Navier-Stokes equations are

\[ u_t + \vec{u} \cdot \nabla u - fv = -\frac{1}{\rho_0} p_x + k \nabla^2 u + F^x \] \[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - fv = -\frac{1}{\rho_0} \frac{\partial p}{\partial x} + k \nabla^2 u + F^x \]
\[ v_t + \vec{u} \cdot \nabla v + fu = -\frac{1}{\rho_0} p_y + k \nabla^2 v + F^y \] \[ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + fu = -\frac{1}{\rho_0} \frac{\partial p}{\partial y} + k \nabla^2 v + F^y \]
\[ 0 = -\frac{1}{\rho} p_z + g \] \[ 0 = -\frac{1}{\rho} \frac{\partial p}{\partial z} + g \quad \text{(hydrostatic approximation, } g \gg w_t\text{)} \]
\[ u_x + v_y + w_z = \nabla \cdot \vec{u} = 0 \] \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \quad \text{(incompressibility)} \]
\[ T_t + \vec{u} \cdot \nabla T = k \nabla^2 T + H \] \[ \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = k \nabla^2 T + H \quad \text{(heating/cooling)} \]
\[ S_t + \vec{u} \cdot \nabla S = k \nabla^2 S + S \] \[ \frac{\partial S}{\partial t} + u \frac{\partial S}{\partial x} + v \frac{\partial S}{\partial y} + w \frac{\partial S}{\partial z} = k \nabla^2 S + S \quad \text{(rain/evaporation)} \]
\[ \rho = f(T, S, p) \approx \rho_0 (1 - \alpha T + \beta S) \] \[ \rho = f(T, S, p) \approx \rho_0 \left( 1 - \alpha T + \beta S \right) \]
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