Atmospheric equation in meteorology
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:
[1]
where:
In
meteorology, and are
isobaric 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 [kg/m3], 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 (e.g., given a small change in 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:
[2]
where the correction due to the
Eötvös effect, A, can be expressed as follows:
where
- = Earth rotation rate,
- = latitude,
- = distance from Earth center to the air mass,
- = mean velocity in longitudinal direction (east-west), and
- = mean velocity in latitudinal direction (north-south).
This correction is considerable in tropical large-scale atmospheric motion.
See also
References