# Shallow water height as vertical coordinate

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.

• How would this be different than using a 3-D barotropic hydrostatic model? – casey Jan 30 '15 at 15:37
• 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 Jan 30 '15 at 16:35
• 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 Jan 30 '15 at 17:24
• Yes, there is exactly what you are looking for, the multi-layer shallow water equations. Geoff Vallis' GFD book is a good start. – milancurcic Jan 30 '15 at 17:36
• Thanx, the knowledge that this exists is already precious to me ;) – AtmosphericPrisonEscape Feb 1 '15 at 8:16