Hypsometric Equation

The hypsometric equation, as discussed in Metr 201, expands
out to

(1)

where Ęz is the 1000-500 mb thickness. However, the equation is more general
than that (as you will see when you derive it) and the thickness between any
two pressure levels can be related to the mean temperature of that layer by
changing the upper and lower bounds.

Solving for the mean temperature of the 1000-500 mb layer
yields

where R is the gas constant of dry air. ln2=.693147; g=9.8 ms^{-2};
R=287.04 J kg^{-1} K^{-1}

or

Take the local derivative of both sides of the equation

which states that the changes in thickness in a column of
air that extends from 1000 to 500 mb at a fixed location will be proportional
to the local changes in the mean temperature of that layer (in this case, the
1000-500 mb layer). If these
changes are smoothly distributed through the whole layer, than this can be used
to forecast the surface temperature change.

A thickness change of 3 dm will result, according to this
equation, in a temperature change of 1.5^{o} K (or 1.5^{o}C or
about 3^{o}F)