The pressure change with altitude $h$ due to hydrostatic effects is given by $d p=-\rho g d h$.
a) For an idealised atmosphere the temperature falls with altitude at a constant rate so that $\partial T \partial h=-k$, where $k$ is a constant with units $\mathrm{K} / \mathrm{m}$. Show that the pressure $p$ at altitude $H$ can be written
$$
p=p_{S l}\left\{1-k H T_{s l}\right\}^g R k=p_{S l}\left(T T_{s l}\right) g^{g / R k}
$$
where $p_{s l}$ and $T_{s l}$ are the static pressure and temperature at sea level, 101.3 kPa and 288.15 K .
For the International Standard Atmosphere the rate of change in temperature with altitude is taken to be 6.5 K per 1000 m up to the tropopause at 11 km . Show that when $g=9.81 \mathrm{~m} / \mathrm{s}^2$ and $R=287 \mathrm{J} / \mathrm{kgK}$, the pressure at altitude $H$, in metres, is given by
$$
p=p_{s l}\left(\pi T_{s l}\right)^{5.26}=p_{s l}\left\{1-2.26 \times 10^{-5} H\right\}^{5.26}
$$
up to the tropopause, above which the pressure is given by
$$
p=p T \exp \left\{-1.58 \times 10^{-4}\left(H-11.10^3\right)\right\}
$$
where $p_T$ is the pressure at the tropopause.
b) If the relationship between pressure and density were that for isentropic changes (i.e. reversible and adiabatic) $\rho / \rho \gamma=$ constant, show that the pressure at altitude $H$ can then be written as
$$
p=p_{S I}\left[1-\frac{\gamma-1}{\gamma} \frac{g H}{R T_{S I}}\right]^{\gamma /(\gamma-1)} .
$$
Plot a few values of pressure, density and temperature on Fig.1.1, the International Standard Atmosphere.
Notes: Atmospheric air is not dry. For saturated air the rate of temperature drop is given as 4.9 K per km, compared with 6.5 K per km in the International Standard Atmosphere. The isentropic calculation assumed dry air.
Different 'standard' atmospheres are sometimes used to model situations more closely: for example over Bombay in the monsoon season the atmosphere is very different from over Saudi Arabia in summer or northern Russia or America in winter.