1.3.2. Atmosphere

In the atmosphere, (see Figure 1.18), we interpret:

(1.10)\[r=p\text{ is the pressure}\]

(1.11)\[\dot{r}=\frac{Dp}{Dt}=\omega \text{ is the vertical velocity in p coordinates}\]

(1.12)\[\phi =g\,z\text{ is the geopotential height}\]

(1.13)\[b=\frac{\partial \Pi }{\partial p}\theta \text{ is the buoyancy}\]

(1.14)\[\theta =T \left( \frac{p_{c}}{p} \right)^{\kappa} \text{ is potential temperature}\]

(1.15)\[S=q \text{ is the specific humidity}\]

where

\[T\text{ is absolute temperature}\]

\[p\text{ is the pressure}\]

\[\begin{split}\begin{aligned} &&z\text{ is the height of the pressure surface} \\ &&g\text{ is the acceleration due to gravity}\end{aligned}\end{split}\]

In the above the ideal gas law, \(p=\rho RT\), has been expressed in terms of the Exner function \(\Pi (p)\) given by (1.16) (see also Section 1.4.1)

(1.16)\[\Pi (p)=c_{p} \left( \frac{p}{p_{c}} \right)^{\kappa},\]

where \(p_{c}\) is a reference pressure and \(\kappa = R/c_{p}\) with \(R\) the gas constant and \(c_{p}\) the specific heat of air at constant pressure.

At the top of the atmosphere (which is ‘fixed’ in our \(r\) coordinate):

\[R_{\rm fixed}=p_{\rm top}=0.\]

In a resting atmosphere the elevation of the mountains at the bottom is given by

\[R_{\rm moving}=R_{o}(x,y)=p_{o}(x,y) ,\]

i.e. the (hydrostatic) pressure at the top of the mountains in a resting atmosphere.

The boundary conditions at top and bottom are given by:

(1.17)\[\omega =0~\text{at }r=R_{\rm fixed} \text{ (top of the atmosphere)}\]

(1.18)\[\omega =~\frac{Dp_{s}}{Dt}\text{ at }r=R_{\rm moving}\text{ (bottom of the atmosphere)}\]

Then the (hydrostatic form of) equations (1.1)-(1.6) yields a consistent set of atmospheric equations which, for convenience, are written out in \(p-\)coordinates in Section 1.4.1 - see eqs. (1.59)-(1.63).