8.7: Scale Height in an Isothermal Atmosphere

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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 ρgAdz. 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 = − ρgdz. But if we regard air as an ideal gas, it obeys the equation of state for an ideal gas, equation 6.1.7: P = ρRT/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% N₂ and 20% O₂, 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!