Steady state: \( \delta(Anything)_t = 0 \)
Adiabatic: no diffusion, heating or adding of mass, \( k = S = T = 0 \) \[ \begin{aligned} & \cancel{u_t} + \vec{u} \cdot \nabla u - fv = -\frac{1}{\rho_0} p_x + \cancel{k \nabla^2 u} + F^x \quad \text{(top/bottom)} \\ & \cancel{v_t} + \vec{u} \cdot \nabla v + fu = -\frac{1}{\rho_0} p_y + \cancel{k \nabla^2 v} + F^y \quad \text{(top/bottom)} \\ & \cancel{T_t} + \vec{u} \cdot \nabla T = \cancel{k \nabla^2 T} + \cancel{H} \quad \text{(heating/cooling)} \\ & \cancel{S_t} + \vec{u} \cdot \nabla S = \cancel{k \nabla^2 S} + \cancel{S} \quad \text{(rain/evaporation)} \\ \end{aligned} \]
Salinity and temperature are constant \[ \begin{aligned} & \vec{u} \cdot \nabla u - fv = -\frac{1}{\rho_0} p_x + F^x \quad \text{(top/bottom)} \\ & \vec{u} \cdot \nabla v + fu = -\frac{1}{\rho_0} p_y + F^y \quad \text{(top/bottom)} \\ & 0 = -\frac{1}{\rho} p_z + g \\ & u_x + v_y + w_z = \nabla \cdot \vec{u} = 0 \end{aligned} \]