# 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).