# Forced shallow-water equation with diabatic heating as the source term

How to obtain the divergence of velocity field $\left(\nabla.\mathbf{u}\right)$ with ${Q}$ (something proportional to the diabatic heating rate) as the source term from the thermodynamic equation?

Such an equation (non-dimensionalized and in isobaric co-ordinates) is presented in A. E. Gill's 1980 paper on tropical circulation in the atmosphere (http://doi.org/10.1002/qj.49710644905): $\frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=-{Q}$, where $u$ and $v$ are horizontal ( in $x,\ y$ directions) perturbation velocities and $p$ is proportional to the perturbation pressure.

• Could you please clarify your question? It is not clear to me what is being asked. – Isopycnal Oscillation Feb 18 '16 at 6:37
• Is it possible to derive $\nabla.\mathbf{u}=−Q$ from the thermodynamic equation? $Q$ can be something proportional to the diabatic heating rate. – vijay Feb 18 '16 at 6:46
• If we are talking about water then $-Q$ must be zero as water is almost perfectly incompressible. – Isopycnal Oscillation Feb 18 '16 at 7:28
• @vijay - Looking at Isopycnal Oscillation's comments you may want to expand your question some more. Maybe give the context from the paper from which you extracted that equation which you want to be derived. – gansub Feb 18 '16 at 9:41
• @vijay: One can derive this from combining the mass-conservation, energy equation and equation of state into one single equation. In the logic of the shallow water approximation this makes sense, as there is only the shallow-water height characterizing mass, temperature and pressure. If this hint is not enough, I can re-derive it for you. I however don't know if Matsuno and Gill did more than just add a source term for the sake of it. – AtmosphericPrisonEscape Feb 18 '16 at 13:36