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

\[ \begin{aligned} & u_t + \vec{u} \cdot \nabla u - fv = -\frac{1}{\rho_0} p_x + k \nabla^2 u + F^x \quad \text{(top/bottom)} \\ & v_t + \vec{u} \cdot \nabla v + fu = -\frac{1}{\rho_0} p_y + k \nabla^2 v + F^y \quad \text{(top/bottom)} \\ & 0 = -\frac{1}{\rho} p_z + g \quad \text{(hydrostatic approximation, } g \gg w_t\text{)} \\ & u_x + v_y + w_z = \nabla \cdot \vec{u} = 0 \quad \text{(water is almost incompressible)} \\ & T_t + \vec{u} \cdot \nabla T = k \nabla^2 T + H \quad \text{(heating/cooling)} \\ & S_t + \vec{u} \cdot \nabla S = k \nabla^2 S + S \quad \text{(rain/evaporation)} \\ & \rho = f(T, S, p) \approx \rho_0 (1 - \alpha T + \beta S) \end{aligned} \]
  • Divergence
  • Reynolds averaging has been done already, and k is turbulent diffusivity
  • \(f = 2\Omega \sin\Theta, \text{ where } \Omega = \frac{2\pi}{86400 \, \text{s}} \text{ and } \Theta \text{ is latitude}\)
  • For now, we avoid using the spherical coordinate system, so we use Taylor \( f(x) \approx f(x_0) + f'(x_0)\Delta x \) to approximate f: \(f = 2\Omega \sin\theta \approx 2\Omega \left( \sin\Theta_0 + \frac{1}{R} \cos\Theta_0 \, \Delta \Theta \right) \Delta R = f_0 + \beta y\)
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