<p>The first equation is presented when the vector identities are not used. That is when students may not have knowledge of advanced vector algebra, for example, curls, dot and cross products, eigenvalues and eigenvectors. The second uses vector identity concepts. While the first is given to undergraduate courses because it doesn't require the concept of advanced vector algebra, the second is given to senior level undergrads and graduate classes as it requires advanced concepts of vector algebra.</p>

CC BY-SA 4.0
2023-02-20T17:06:32.277Z
This makes sense. I was thinking in a completely wrong direction
MichaelW
CC BY-SA 4.0
2023-02-19T15:59:43.520Z
2023-02-19T16:07:41.527Z
2023-02-19T16:07:41.527Z
Gemechu Fanta Garuma
<p>I'm a student of meteorology in first year. This time my question is not about a particular problem I have, but rather about teaching and learning: In a recent lecture on atmospheric dynamics we learned among other things the &quot;vorticity equation&quot;: Its derivation was carried out starting from the horizontal momentum equations:</p>
<p><a href="https://i.sstatic.net/nTSJV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nTSJV.png" alt="enter image description here" /></a></p>
<p>After taking the y derivative of the u equation and the x derivative of the v equation and subtracting them, we get a lot of terms, which were grouped together and partially cancel each other. After a struggling time, we end up finally in the vertical component <span class="math-container">$\zeta $</span> of the vorticity <span class="math-container">$\vec \omega $</span>:</p>
<p><a href="https://i.sstatic.net/D8jaS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/D8jaS.png" alt="enter image description here" /></a></p>
<p>Note: the pictures of the equations above are taken from YouTube: <a href="https://www.youtube.com/watch?v=lqDx2uiKx_Q&amp;list=PL_cuIb7hx5lg_zHfUVsUrw6I66U4jq8Dq&amp;t=163s" rel="nofollow noreferrer">AtmosphericDynamics Chapter04 Part02 VorticityEquation</a>, but they were identical in the lecture (the derivation was skipped in the video).</p>
<p>Now I have found the following after my own search, whereby the derivation is a lot easier (see chapter 3.4.) of <a href="https://www2.cgd.ucar.edu/staff/islas/teaching/3_Circulation_Vorticity_PV.pdf" rel="nofollow noreferrer">Vorticity, Circulation and Potential Vorticity</a>.</p>
<p>(see 3.4)</p>
<p><a href="https://i.sstatic.net/HVMFq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HVMFq.png" alt="enter image description here" /></a></p>
<p>This approach, using well know vector identities is independent of the coordinate system and gives to my surprise really the same result for <span class="math-container">$D \zeta/Dt$</span> although it looks totally different at the first sight (I ignore the friction term).</p>
<p>In Holton, a book which I like very much, the derivation is essentially the same as in our lecture: quite log and non-motivating, because so many partial derivative terms had to be arranged and grouped with each other to get the final result after doing some pages of simple but totally boring algebra. Contrary, the other derivation is not only shorter, but more elegant in terms of vector analysis and gives in only a few steps the generic result for all components of <span class="math-container">$\vec \omega$</span>, not only for <span class="math-container">$\zeta$</span></p>
<p>Why this completely cumbersome derivation and not doing the same as in the other text, which is much more appealing and more general?</p>
<p>Is there a particular reason to write the final equation in a rather &quot;nasty&quot; way in local cartesian coordinates instead using the generic variant? It would have been more fun and more motivating to learn it in the other way around from the beginning.</p>

<meteorology><atmosphere-modelling><fluid-dynamics><vorticity>
Why are there different Notations of Vorticity Equation?
MichaelW
# Why are there different Notations of Vorticity Equation?

(Asked by: username_0 on 2023-02-18)

I'm a student of meteorology in first year. This time my question is not about a particular problem I have, but rather about teaching and learning: In a recent lecture on atmospheric dynamics we learned among other things the "vorticity equation": Its derivation was carried out starting from the horizontal momentum equations:

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

After taking the y derivative of the u equation and the x derivative of the v equation and subtracting them, we get a lot of terms, which were grouped together and partially cancel each other. After a struggling time, we end up finally in the vertical component $\zeta $ of the vorticity $\vec \omega $:

[![enter image description here](https://i.sstatic.net/D8jaS.png)](https://i.sstatic.net/D8jaS.png)

Note: the pictures of the equations above are taken from YouTube: [AtmosphericDynamics Chapter04 Part02 VorticityEquation](https://www.youtube.com/watch?v=lqDx2uiKx_Q&list=PL_cuIb7hx5lg_zHfUVsUrw6I66U4jq8Dq&t=163s), but they were identical in the lecture (the derivation was skipped in the video).

Now I have found the following after my own search, whereby the derivation is a lot easier (see chapter 3.4.) of [Vorticity, Circulation and Potential Vorticity](https://www2.cgd.ucar.edu/staff/islas/teaching/3_Circulation_Vorticity_PV.pdf).

(see 3.4)

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

This approach, using well know vector identities is independent of the coordinate system and gives to my surprise really the same result for $D \zeta/Dt$ although it looks totally different at the first sight (I ignore the friction term).

In Holton, a book which I like very much, the derivation is essentially the same as in our lecture: quite log and non-motivating, because so many partial derivative terms had to be arranged and grouped with each other to get the final result after doing some pages of simple but totally boring algebra. Contrary, the other derivation is not only shorter, but more elegant in terms of vector analysis and gives in only a few steps the generic result for all components of $\vec \omega$, not only for $\zeta$

Why this completely cumbersome derivation and not doing the same as in the other text, which is much more appealing and more general?

Is there a particular reason to write the final equation in a rather "nasty" way in local cartesian coordinates instead using the generic variant? It would have been more fun and more motivating to learn it in the other way around from the beginning.

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


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## 1 answers

### Answer by username_1 on 2023-02-19. Score: 4 | Accepted answer

The first equation is presented when the vector identities are not used. That is when students may not have knowledge of advanced vector algebra, for example, curls, dot and cross products, eigenvalues and eigenvectors. The second uses vector identity concepts. While the first is given to undergraduate courses because it doesn't require the concept of advanced vector algebra, the second is given to senior level undergrads and graduate classes as it requires advanced concepts of vector algebra.

> **Comments:**
> **username_0**: This makes sense. I was thinking in a completely wrong direction  

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