<p>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?</p>

<p>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 (<a href="http://doi.org/10.1002/qj.49710644905" rel="nofollow">http://doi.org/10.1002/qj.49710644905</a>): $\frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=-{Q}$, where $u$ and $v$ are horizontal ( in $x,\ y$ directions) perturbation velocities and $p$ is proportional to the perturbation pressure.</p>

CC BY-SA 3.0
2016-02-18T06:37:31.047Z
Could you please clarify your question? It is not clear to me what is being asked.
Isopycnal Oscillation
CC BY-SA 3.0
2016-02-18T06:46:45.877Z
Is it possible to derive $\nabla.\mathbf{u}=−Q$
from the thermodynamic equation? $Q$ can be something proportional to the diabatic heating rate.
vijay
CC BY-SA 3.0
2016-02-18T07:28:57.863Z
If we are talking about water then $-Q$ must be zero as water is almost perfectly incompressible.
Isopycnal Oscillation
CC BY-SA 3.0
2016-02-18T13:36:16.970Z
@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
<atmosphere-modelling><tropical-cyclone><atmospheric-circulation><synoptic>
Forced shallow-water equation with diabatic heating as the source term
vijay
# Forced shallow-water equation with diabatic heating as the source term

(Asked by: username_0 on 2016-02-18)

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.

**Tags:** atmosphere-modelling, tropical-cyclone, atmospheric-circulation, synoptic

> **Comments:**
> **username_1**: Could you please clarify your question? It is not clear to me what is being asked.  
> **username_0**: Is it possible to derive $\nabla.\mathbf{u}=−Q$ from the thermodynamic equation? $Q$ can be something proportional to the diabatic heating rate.  
> **username_1**: If we are talking about water then $-Q$ must be zero as water is almost perfectly incompressible.  
> **username_2**: @username_0: 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.  

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## 0 answers

