# 8.7: Scale Height in an Isothermal Atmosphere

- Page ID
- 8602

The material in this chapter doubtless has countless applications, most of which I am unaware of, in meteorology. Two simple topics are easy to mention, namely the *scale height in an isothermal atmosphere*, dealt with in this section, and the adiabatic lapse rate dealt with in the next section.

Let us imagine a column of air of cross-sectional area *A* in an isothermal atmosphere – that is to say the temperature *T* is uniform throughout. Consider the equilibrium of the portion of the air between heights *z* and *z + dz*. The weight of this portion is ρ*gAdz*. Let *P* be the pressure at height *z* and *P + dP* be the pressure at height *z + dz*. (Note that *dP* is negative.) The net upward force on the portion dz of the air is −*AdP*. Therefore *dP* = − ρ*gdz*. But if we regard air as an ideal gas, it obeys the equation of state for an ideal gas, equation 6.1.7: *P* = ρ*RT**/*µ where ρ and µ are respectively the density and the “molecular weight” (molar mass) of the gas. Therefore \( \frac{R T}{\mu} d \rho=-\rho g d z\), or \( \frac{d \rho}{\rho}=-\frac{\mu g}{R T} d z\). Integrate to obtain

\[ \rho=\rho e^{-z / H}\]

where \( H=\frac{R T}{\mu g}\) is the *scale height*. It is large if the temperature is high, the gas light and the planet’s gravity feeble. It is the height at which the density is reduced to a fraction 1/*e*, or 36.8%. of its ground value. What would it be, in kilometres, for an atmosphere consisting of 80% N_{2} and 20% O_{2}, at a temperature of 20 ºC, where the gravitational acceleration is 9.8 m s^{−2}? What fraction is this of the radius of Earth? If you made a model of Earth one metre in diameter (radius = 50 cm), how thick would be the atmosphere? You’d better look after it - our atmosphere is a very thin skin clinging to the surface!