Hypsometric equation


The hypsometric equation, also known as the thickness equation, relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer considering the layer mean of virtual temperature, gravity, and occasionally wind. It is derived from the hydrostatic equation and the ideal gas law.

Formulation

The hypsometric equation is expressed as:
where:
In meteorology, and are surfaces. In radiosonde observation, the hypsometric equation can be used to compute the height of a pressure level given the height of a reference pressure level and the mean virtual temperature in between. Then, the newly computed height can be used as a new reference level to compute the height of the next level given the mean virtual temperature in between, and so on.

Derivation

The hydrostatic equation:
where is the density , is used to generate the equation for hydrostatic equilibrium, written in differential form:
This is combined with the ideal gas law:
to eliminate :
This is integrated from to :
R and g are constant with z, so they can be brought outside the integral.
If temperature varies linearly with z,
it can also be brought outside the integral when replaced with, the average virtual temperature between and.
Integration gives
simplifying to
Rearranging:
or, eliminating the natural log:

Correction

The Eötvös effect can be taken into account as a correction to the hypsometric equation. Physically, using a frame of reference that rotates with Earth, an air mass moving eastward effectively weighs less, which corresponds to an increase in thickness between pressure levels, and vice versa. The corrected hypsometric equation follows:
where the correction due to the Eötvös effect, A, can be expressed as follows:
where
This correction is considerable in tropical large-scale atmospheric motion.