# What are the governing equations solved in coupled atmosphere-ocean models?

In (hydrostatic) atmospheric general circulation models, for example the so-calle Primitive Equations, consisting of the horizontal momentum equation, the hydrostatic balance, the continuity equation, and the energy balance

$$\begin{eqnarray} \nonumber \frac{\partial}{\partial t}\vec{v} & = & \vec{v}\times(f+\xi)\vec{e}_z - \omega\frac{\partial}{\partial p}\vec{v} -\nabla \left(\Phi+\frac{\vec{v}^2}{2}\right) + \vec{R} \\ \frac{\partial }{\partial p}\Phi & = & -\frac{1}{\rho} = -\frac{RT}{p}\\ \nonumber 0 & = & \nabla\cdot\vec{v} + \frac{\partial}{\partial p} \omega \\ \nonumber c_p\frac{d}{d t} T & = & \frac{\omega}{\rho} + Q_{rad} + Q_{lat} + \eta + \epsilon . \end{eqnarray}$$

are solved numerically. How are these equations modified and complemented, when the atmospheric model is to be coupled to a full ocean general circulation model, what additional effects and equations have to be included?

As the salinity plays an important role in driving the oceanic circulations among other things, for example a balance equation for the salinity, appropriately coupled to the other governing equations will certainly be needed.

• You will end up with boundary conditions at the interface (the sea surface) that couple the two models. There must certainly be a large amount of literature on how to formulate and solve such problems in the weather and climate modeling communities. – Wolfgang Bangerth Sep 28 '13 at 2:24
• I would start by looking at any of the aGCM models that have been successfully implemented - for example. – Isopycnal Oscillation Sep 28 '13 at 8:16