# Profiles of the atmosphere

The properties of the atmosphere that vary with altitude include the pressure, temperature and the composition of constituents such as water vapor. This chapter provides some insight into these dependencies with simple derivations that hold under idealized conditions. The stage will be set for the following chapter which provides methods for analyzing the conditions at the time of observation.

### 6.1 Pressure versus height

Atmospheric pressure drops off dramatically with height above the surface. This is indicated by the graph in Figure 6.1 which shows the dependence of pressure on altitude for the US Standard Atmosphere.1

By balancing the vertical components of force on a slab of air at an arbitrary height z it is possible to derive a formula for the average pressure as a function of height, p(z). Consider a column of air with cross-section 1 m2. In the column

4km) 14

Figure 6.1 Dependence of pressure p (hPa) on altitude z (km) for the US Standard Atmosphere.

1 The US Standard Atmosphere is a model of the atmospheric profile of various properties. It is used primarily in aviation and satellite drag calculations. It attempts to give a global average of conditions. More can be learned about it on the internet. Figure 6.1 Dependence of pressure p (hPa) on altitude z (km) for the US Standard Atmosphere.

dz t

Figure 6.2 Diagram of a column of air of cross-section area A with a slab of thickness dz at height z. The pressure and gravitational forces on the slab are indicated.

we picture a thin horizontal slab of air whose bottom surface is located at height z above sea level and whose thickness is dz (see Figure 6.2). The mass of material in the slab is (density times volume): dM = pAdz, where A = 1m2 is the horizontal cross-sectional area of the slab. The weight of the slab of gas is (dM)g = (pAdz)g.

Beneath the slab is a pressure force pushing upwards: p(z)A. Above the slab is a pressure force pressing downwards, p(z + dz)A « (p(z) + dp dz) A. (6.1)

Equating the net pressure force on the slab to the gravitational force, dP dzA =-p g dzA. (6.2)

After cancellations we obtain the hydrostatic equation:

dp dz

[hydrostatic equation].

We can use the Ideal Gas Law to write:

where we indicate explicitly that both temperature and pressure are functions of altitude z. In the last step we used the ideal gas equation of state. The hydrostatic equation has many uses, but it is particularly useful if the dependence of T on z is known. This may often be the case, at least approximately. If it is true we can write:

Then integrating each side from the surface up to level z leads to: