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Fluids

Isothermal-barotropic approximation

Previously we found for the pressure gradient of a fluid the expression

dP/dy = -ρ_fluid g

For liquids, which are approximately incompressible, we found that the pressure in a liquid is proportional to the depth under the surface of the liquid. However, for a gas we need a relation between the density ρ_fluid and the height y.
In the isothermal approximation one assumes that the temperature T is the same anywhere in the gas. As a consequence one finds that the ideal gas law for the volume V yields

V = NkT/P

where N represents the number of molecules. Hence, when m_molecule is the mass of one molecule we obtain for the mass M of N molecules

M = N×m_molecule

For the local density of the gas results

ρ_gas = M/V = (N×m_molecule)/(NkT/P) = (m_molecule /kT) P

The pressure gradient of a gas takes the form

dP/dy = -(m_molecule g/kT) P = -P/H

where the constant H (remember that the temperature T is supposed to be constant in an isothermal approximation) is called the scale height and is defined by

H = kT/m_molecule g

This model, when applied to the Earth's atmosphere is, furthermore, barotropic, since it assumes that the molecular composition is the same at any altitude, whereas also the gravitational acelleration g is taken constant.

atmosphere