Steady state: \( \delta(Anything)_t = 0 \)
Adiabatic: no diffusion, heating or adding of mass, \( k = S = T = 0 \)
Salinity and temperature are constant

\[ \begin{aligned} & \vec{u} \cdot \nabla u - fv = -\frac{1}{\rho_0} p_x + F^x \quad \text{(top/bottom)} \\ & \vec{u} \cdot \nabla v + fu = -\frac{1}{\rho_0} p_y + F^y \quad \text{(top/bottom)} \\ & 0 = -\frac{1}{\rho} p_z + g \\ & u_x + v_y + w_z = \nabla \cdot \vec{u} = 0 \end{aligned} \] The force of the wind can be expressed as the vertical divergence of the horizontal stress \( \vec{F} = \vec{\tau}_z \)

Can we ignore any of the other terms? Let’s compare the magnitudes of momentum advection and Coriolis force:

\( \begin{aligned} \mathcal{O}\left( \frac{u \, \frac{\partial u}{\partial x}}{fu} \right) &= \mathcal{O}\left( \frac{u}{fu} \cdot \frac{\partial u}{\partial x} \right) = \mathcal{O}\left( \frac{1}{f} \cdot \frac{\partial u}{\partial x} \right) \approx \mathcal{O}\left( \frac{1}{f} \cdot \frac{\Delta u}{\Delta x} \right) = \mathcal{O}\left( \frac{\Delta u / \Delta x}{f} \right) \\[8pt] &\approx \mathcal{O}\left( \frac{U / L}{f} \right) = \mathcal{O}\left( \frac{U}{fL} \right) \end{aligned} \)

with \( U \) and \( L \) being magnitudes of the variables