<p>I was reading an article on fluid dynamics of tropical cyclones, where I came across a condition which states that,</p>

<blockquote>
  <p>For adiabatic friction less flow, equations (1) to (5) have a
  solution, v(r, z), for a steady freely spinning vortex in which u and
  w are identically zero and v(r, z) is an arbitrary function of r and
  z.</p>
</blockquote>

<p>Where these 5 equations are basic primitive equations (2 horizontal and 1 vertical equation of motion,conservation of mass and thermodynamic equation) written in cylindrical coordinates.</p>

<p>If I try to consider the horizontal(u component only) and vertical equations of motion when they are in gradient wind and hydrostatic balance, u and w will be conserved i.e du/dt=0 and dw/dt=0.
How can one conclude that they are identically zero?
For the article, please open the following link 
<a href="https://www.google.co.in/url?sa=t&amp;source=web&amp;rct=j&amp;url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&amp;ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&amp;usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00" rel="nofollow noreferrer">https://www.google.co.in/url?sa=t&amp;source=web&amp;rct=j&amp;url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&amp;ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&amp;usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00</a></p>

<p>The primitive equations are given in section 2.1 and the lines highlighted above are taken from the first paragraph of section 2.2</p>

CC BY-SA 3.0
2018-01-22T17:58:07.307Z
Please, cite your source. Context matters. For example, are you sure that those u,v,w are not formulated in a locally co-rotating frame?
AtmosphericPrisonEscape
CC BY-SA 3.0
2018-01-22T18:24:23.247Z
Please open  the following link https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00
Agni
CC BY-SA 3.0
2018-01-22T18:28:47.077Z
Please learn to cite properly. I'm not gonna read a 24 page paper for this. Cite page, section and edit your question accordingly.
AtmosphericPrisonEscape
CC BY-SA 3.0
2018-01-22T18:34:32.460Z
The authors are not very explicit about it, but the section title clearly says "The equations in cylindrical polar coordinates." Thus I'd assume $u,v,w$ have meaning in the cyclone-centered coordinate system, not in the planetary one. Therefore $\dot{u} = 0$, as per definition of a cyclone in this coordinate system.
AtmosphericPrisonEscape
CC BY-SA 3.0
2018-01-22T18:36:18.150Z
Yes, but that does not say that u is zero identically. Zero derivative doesn't imply that the function is also zero. Please make it clear.
Agni
CC BY-SA 3.0
2018-01-22T18:57:57.723Z
They say they're using a 'steady, freely spinning vortex' so they're already plugging in u and w as zero as initial conditions. Then by requirement u and w stay that way. Plugging that into the equations and checking, shows that there is no contradiction, so $u=w=0$ can be (part of) the solution.
AtmosphericPrisonEscape
CC BY-SA 3.0
2018-01-22T22:43:26.910Z
Ok, thanks. I am still learning so understanding everything at my own pace. By the way why did I get my question downvoted?
Agni
CC BY-SA 3.0
2018-01-22T23:14:48.650Z
No problem. I'm not 100% sure about this, but this is how i'd read it. Taking a look at the reference they give might help as well. The downvote's not me this time!
AtmosphericPrisonEscape
<atmosphere><tropical-cyclone><mesoscale-meteorology>
Why does the absence of diabatic heating rate and frictional terms make the zonal and vertical velocity identically zero?
Agni
# Why does the absence of diabatic heating rate and frictional terms make the zonal and vertical velocity identically zero?

(Asked by: username_0 on 2018-01-22)

I was reading an article on fluid dynamics of tropical cyclones, where I came across a condition which states that,

> For adiabatic friction less flow, equations (1) to (5) have a
> solution, v(r, z), for a steady freely spinning vortex in which u and
> w are identically zero and v(r, z) is an arbitrary function of r and
> z.

Where these 5 equations are basic primitive equations (2 horizontal and 1 vertical equation of motion,conservation of mass and thermodynamic equation) written in cylindrical coordinates.

If I try to consider the horizontal(u component only) and vertical equations of motion when they are in gradient wind and hydrostatic balance, u and w will be conserved i.e du/dt=0 and dw/dt=0.
How can one conclude that they are identically zero?
For the article, please open the following link
<https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00>

The primitive equations are given in section 2.1 and the lines highlighted above are taken from the first paragraph of section 2.2

**Tags:** atmosphere, tropical-cyclone, mesoscale-meteorology

> **Comments:**
> **username_1**: Please, cite your source. Context matters. For example, are you sure that those u,v,w are not formulated in a locally co-rotating frame?  
> **username_0**: Please open  the following link https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00  
> **username_1**: Please learn to cite properly. I'm not gonna read a 24 page paper for this. Cite page, section and edit your question accordingly.  
> **username_1**: The authors are not very explicit about it, but the section title clearly says "The equations in cylindrical polar coordinates." Thus I'd assume $u,v,w$ have meaning in the cyclone-centered coordinate system, not in the planetary one. Therefore $\dot{u} = 0$, as per definition of a cyclone in this coordinate system.  
> **username_0**: Yes, but that does not say that u is zero identically. Zero derivative doesn't imply that the function is also zero. Please make it clear.  
> **username_1**: They say they're using a 'steady, freely spinning vortex' so they're already plugging in u and w as zero as initial conditions. Then by requirement u and w stay that way. Plugging that into the equations and checking, shows that there is no contradiction, so $u=w=0$ can be (part of) the solution.  
> **username_0**: Ok, thanks. I am still learning so understanding everything at my own pace. By the way why did I get my question downvoted?  
> **username_1**: No problem. I'm not 100% sure about this, but this is how i'd read it. Taking a look at the reference they give might help as well. The downvote's not me this time!  

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## 0 answers

