PRESENTER: This schematic shows the relationship between height surfaces and pressure surfaces. Typically, pressure surfaces slope downward in height from the equator, where it is warmer, to the pole, where it is colder. You were able to show this in Activity 2.2, and you saw it again in Lesson 2.4 on thickness.

Note that on the constant height surface, from the equator to the pole, the pressure surface is decreased with latitude. Now it also, on a constant pressure surface from the equator to the pole, the height surface is decreased on the constant pressure surface. Thus, changes in pressure are proportional to changes in height.

After a little math, we can show that 1/rho-- that is, one over the density-- times the change in pressure with respect to x or y on a height surface is equal to g times the change in z with respect to x or y on a constant pressure surface. Finally, we note that gdz is just a differential of the geopotential phi, which has units of meters squared per second squared, which are the same units as energy divided by mass. So changes in height on a constant pressure surface are the same as changes in geopotential on a constant pressure surface.