Scale height

For planetary atmospheres, it is the vertical distance upwards, over which the pressure of the atmosphere decreases by a factor of 1/e. The scale height remains constant for a particular temperature. It can be calculated by

:$H\; =\; \backslash frac\{kT\}\{Mg\}$

where:

* k = Boltzmann constant = 1.38 x 10⁻²³ J·K⁻¹

The pressure at the Earth's surface (or at higher levels) is a result of the weight of the overlying atmosphere [force per unit area]. If at a height of z the atmosphere has density ρ and pressure P, then moving upwards at an infinitesimally small height dz will decrease the pressure by amount dP, equal to the weight of a layer of atmosphere of thickness dz.

Thus:

:$\backslash frac\{dP\}\{dz\}\; =\; -g\backslash rho$

where g is used to denote the acceleration due to gravity. For small dz it is possible to assume g to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore using the equation of state for an ideal gas of mean molecular mass M at temperature T, the density can be expressed as such:

:$\backslash rho\; =\; \backslash frac\{MP\}\{kT\}$

Therefore combining the equations gives

:$\backslash frac\{dP\}\{P\}\; =\; \backslash frac\{-dz\}\{\backslash frac\{kT\}\{Mg\}\}$

which can then be incorporated with the equation for H given above to give:

:$\backslash frac\{dP\}\{P\}\; =\; -\; \backslash frac\{dz\}\{H\}$

which will not change unless the temperature does. Integrating the above and assuming where P₀ is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:

:$P\; =\; P\_0\backslash exp(-\backslash frac\{z\}\{H\})$

This translates as the pressure decreasing exponentially with height.

In the Earth's atmosphere, the pressure at sea level P₀ averages about 1.01×10⁵Pa, the mean molecular mass of dry air is 28.964 u and hence 28.964 × 1.660×10⁻²⁷ = 4.808×10⁻²⁶ kg, and g = 9.81 m/s². As a function of temperature the scale height of the Earth's atmosphere is therefore 1.38/(4.808×9.81)×10³ = 29.26 m/deg. This yields the following scale heights for representative air temperatures.

:T = 290 K, H = 8500 m :T = 273 K, H = 8000 m :T = 260 K, H = 7610 m :T = 210 K, H = 6000 m

These figures should be compared with the temperature and density of the Earth's atmosphere plotted at NRLMSISE-00, which shows the air density dropping from 1200 g/m³ at sea level to 0.5³ = .125 g/m³ at 70 km, a factor of 9600, indicating an average scale height of 70/ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K.

Note: # Density is related to pressure by the ideal gas laws. Therefore with some departures caused by varying temperature—density will also decrease exponentially with height from a sea level value of ρ₀ roughly equal to 1.2 kg m⁻³ # At heights over 100 km, molecular diffusion means that each molecular atomic species has its own scale height.

See also

References

Category:Atmosphere Category:Atmospheric dynamics