<p>In atmospheric physics we use often a variety of vertical coordinates, $z$ the height above some reference surface, or $r$ as distance from the planetary center.</p>

<p>However sometimes it simplifies the equations we want to solve by using other coordinates, Pressure $P$, LogPressure, Potential temperature $\theta$,...</p>

<p>One vertical coordinate I want to specifically ask about is the shallow water height. It is defined by looking at the hydrostatic stability approximation
$$ \frac{\partial P}{\partial z} = \rho_0 g$$
then integrating over a variable height $h(x,y)$ so that we arrive at constant pressure $P(h)$ we get easily
$$P(z)=\rho_0 g (h-z)+P(h) $$</p>

<p>So $h(x,y)$ is now the variable height between two Iso-Pressure surfaces in some planetary fluid.<br>
Now we can express the navier-stokes-equation, mass equation, thermodynamic equation in terms of $h$. I guess this will be very familiar to those who could answer me.</p>

<p>My question is now:</p>

<ul>
<li>Assuming no vertical mixing (=geostrophic movement on isobaric surfaces), can I now build a vertical model of the planet consisting of a multitude of shallow fluid-layers? Namely I'd take $h1(x,y)$ as between $p1$ and $p2$, $h2(x,y)$ between $p2$ and $p3$,... and thus build up and atmosphere of a gas giant between $\sim 10^{-1} bar$ and several $10^2$ bars.</li>
<li>And more important: when could this approach be totally wrong?</li>
</ul>

<p>Naively I'd expect this to be OK in my case, as I'm still effectively integrating over the whole planet, but I hoped for some experts opinion on this. </p>

CC BY-SA 3.0
2015-01-30T15:37:29.360Z
How would this be different than using a 3-D barotropic hydrostatic model?
casey
CC BY-SA 3.0
2015-01-30T16:35:40.677Z
There are analytic solutions for the type of motions I'm interested in. So I'd just evaluate the formulas for each level as function of 2 parameters and let the parameters evolve, instead of the hole 3D-sphere. Effectively I have then a 1-D modell, which is what I want. I'm not interested in turbulent transport etc. just large-scale motion.
AtmosphericPrisonEscape
CC BY-SA 3.0
2015-01-30T17:24:38.407Z
Ah sorry, so the point of the whole thing is: I want to know the differences between my modell and a full 3D modell. If my approximations are fine, then there would be no questions.
AtmosphericPrisonEscape
CC BY-SA 3.0
2015-01-30T17:36:06.220Z
Yes, there is exactly what you are looking for, the multi-layer shallow water equations. Geoff Vallis' GFD book is a good start.
milancurcic
CC BY-SA 3.0
2015-02-01T08:16:14.090Z
Thanx, the knowledge that this exists is already precious to me ;)
AtmosphericPrisonEscape
CC BY-SA 3.0
2015-02-02T09:21:37.013Z
Unless I'm missing something (which I may well be), using pressure as a vertical coordinate is equivalent to using depth, under a hydrostatic assumption. There are many hydrodynamic modelling systems that do this, including (as identified in comments) many that are thought of as "shallow water models".
Semidiurnal Simon
<atmosphere-modelling><fluid-dynamics>
Shallow water height as vertical coordinate
AtmosphericPrisonEscape
# Shallow water height as vertical coordinate

(Asked by: username_0 on 2015-01-30)

In atmospheric physics we use often a variety of vertical coordinates, $z$ the height above some reference surface, or $r$ as distance from the planetary center.

However sometimes it simplifies the equations we want to solve by using other coordinates, Pressure $P$, LogPressure, Potential temperature $\theta$,...

One vertical coordinate I want to specifically ask about is the shallow water height. It is defined by looking at the hydrostatic stability approximation
$$ \frac{\partial P}{\partial z} = \rho\_0 g$$
then integrating over a variable height $h(x,y)$ so that we arrive at constant pressure $P(h)$ we get easily
$$P(z)=\rho\_0 g (h-z)+P(h) $$

So $h(x,y)$ is now the variable height between two Iso-Pressure surfaces in some planetary fluid.  
Now we can express the navier-stokes-equation, mass equation, thermodynamic equation in terms of $h$. I guess this will be very familiar to those who could answer me.

My question is now:

* Assuming no vertical mixing (=geostrophic movement on isobaric surfaces), can I now build a vertical model of the planet consisting of a multitude of shallow fluid-layers? Namely I'd take $h1(x,y)$ as between $p1$ and $p2$, $h2(x,y)$ between $p2$ and $p3$,... and thus build up and atmosphere of a gas giant between $\sim 10^{-1} bar$ and several $10^2$ bars.
* And more important: when could this approach be totally wrong?

Naively I'd expect this to be OK in my case, as I'm still effectively integrating over the whole planet, but I hoped for some experts opinion on this.

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

> **Comments:**
> **username_1**: How would this be different than using a 3-D barotropic hydrostatic model?  
> **username_0**: There are analytic solutions for the type of motions I'm interested in. So I'd just evaluate the formulas for each level as function of 2 parameters and let the parameters evolve, instead of the hole 3D-sphere. Effectively I have then a 1-D modell, which is what I want. I'm not interested in turbulent transport etc. just large-scale motion.  
> **username_0**: Ah sorry, so the point of the whole thing is: I want to know the differences between my modell and a full 3D modell. If my approximations are fine, then there would be no questions.  
> **username_2**: Yes, there is exactly what you are looking for, the multi-layer shallow water equations. Geoff Vallis' GFD book is a good start.  
> **username_0**: Thanx, the knowledge that this exists is already precious to me ;)  
> **username_3**: Unless I'm missing something (which I may well be), using pressure as a vertical coordinate is equivalent to using depth, under a hydrostatic assumption. There are many hydrodynamic modelling systems that do this, including (as identified in comments) many that are thought of as "shallow water models".  

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## 0 answers

