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).