<p>Potential temperature is conserved considering adiabatic processes and thus: <span class="math-container">$\text{d} \theta/\text{d}t = 0$</span>. Otherwise the term needs to be retained.</p>
<p>On page 84 (just below figure 4.3) they say that given an adiabatic process, parcel flow is constrained to a surface of potential temperature, so I think they just assume adiabatic flow.</p>

CC BY-SA 4.0
2023-06-22T14:26:46.980Z
but then the last term should also vanish.
MichaelW
CC BY-SA 4.0
2023-06-22T15:02:45.767Z
I agree. Maybe someone else spots something?
Joscha Fregin
CC BY-SA 4.0
2023-06-22T22:03:07.667Z
could be an error in the book... (?)
MichaelW
CC BY-SA 4.0
2023-06-23T11:09:58.043Z
Tend to agree again :)
Joscha Fregin
CC BY-SA 4.0
2023-06-22T09:30:29.333Z
2023-06-22T09:30:29.333Z
Joscha Fregin
<p>In  the book by Jonathan E. Martin &quot;Mid-Latitude Atmospheric Dynamics - A First Course&quot;</p>
<p>the continuity equation for isentropic coordinates is derived by applying conservation  of mass:</p>
<p><a href="https://i.sstatic.net/fJt6a.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fJt6a.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$\frac{d}{dt}(\frac{\partial p}{\partial \theta})  + \frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$</span></p>
<p>Now, following the book, by expanding the total derivative on the left I would get:</p>
<p><span class="math-container">$\underbrace{\frac{\partial}{\partial t}(\frac{\partial p}{\partial \theta})  + u \frac{\partial }{\partial x}(\frac{\partial p}{\partial \theta}) + v \frac{\partial }{\partial y}(\frac{\partial p}{\partial \theta}) + \frac{d \theta}{dt}\cdot \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})}_{\frac{d}{dt}(\frac{\partial p}{\partial \theta})} + \underbrace{\frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}}_{\frac{\partial p}{\partial \theta} \vec \nabla \cdot \vec V}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$</span></p>
<p>Using divergence operator, this has 6 terms, however, in the final result given by the book</p>
<p><a href="https://i.sstatic.net/SC1AP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SC1AP.png" alt="enter image description here" /></a></p>
<p>I would miss the term</p>
<p><span class="math-container">$\frac{d \theta }{d t} \cdot  \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})$</span></p>
<p>Why? I see no reason why this term is skipped.</p>
<p><a href="https://i.sstatic.net/iMZCR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iMZCR.png" alt="enter image description here" /></a></p>

<meteorology><atmosphere><atmosphere-modelling><fluid-dynamics>
Why is one term missing in the derivation of continuity equation in isentropic coordinates?
MichaelW
# Why is one term missing in the derivation of continuity equation in isentropic coordinates?

(Asked by: username_0 on 2023-06-21)

In the book by Jonathan E. Martin "Mid-Latitude Atmospheric Dynamics - A First Course"

the continuity equation for isentropic coordinates is derived by applying conservation of mass:

[![enter image description here](https://i.sstatic.net/fJt6a.png)](https://i.sstatic.net/fJt6a.png)

$\frac{d}{dt}(\frac{\partial p}{\partial \theta}) + \frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$

Now, following the book, by expanding the total derivative on the left I would get:

$\underbrace{\frac{\partial}{\partial t}(\frac{\partial p}{\partial \theta}) + u \frac{\partial }{\partial x}(\frac{\partial p}{\partial \theta}) + v \frac{\partial }{\partial y}(\frac{\partial p}{\partial \theta}) + \frac{d \theta}{dt}\cdot \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})}\_{\frac{d}{dt}(\frac{\partial p}{\partial \theta})} + \underbrace{\frac{\partial p}{\partial \theta} \frac{\partial u}{\partial x}+ \frac{\partial p}{\partial \theta} \frac{\partial v}{\partial y}}\_{\frac{\partial p}{\partial \theta} \vec \nabla \cdot \vec V}+\frac{\partial p}{\partial \theta} \frac{\partial}{\partial \theta}(\dot \theta)=0$

Using divergence operator, this has 6 terms, however, in the final result given by the book

[![enter image description here](https://i.sstatic.net/SC1AP.png)](https://i.sstatic.net/SC1AP.png)

I would miss the term

$\frac{d \theta }{d t} \cdot \frac{\partial }{\partial \theta}(\frac{\partial p}{\partial \theta})$

Why? I see no reason why this term is skipped.

[![enter image description here](https://i.sstatic.net/iMZCR.png)](https://i.sstatic.net/iMZCR.png)

**Tags:** meteorology, atmosphere, atmosphere-modelling, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2023-06-22. Score: 3 | Accepted answer

Potential temperature is conserved considering adiabatic processes and thus: $\text{d} \theta/\text{d}t = 0$. Otherwise the term needs to be retained.

On page 84 (just below figure 4.3) they say that given an adiabatic process, parcel flow is constrained to a surface of potential temperature, so I think they just assume adiabatic flow.

> **Comments:**
> **username_0**: but then the last term should also vanish.  
> **username_1**: I agree. Maybe someone else spots something?  
> **username_0**: could be an error in the book... (?)  
> **username_1**: Tend to agree again :)  

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