# 1.3.2. Atmosphere¶

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

where

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)

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):

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

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:

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).