In practice, most measurements of the atmosphere are made of pressure not elevation due to the variation in gravity at different elevations and  latitudes. The pressure contours of the atmosphere are defined by a geopotential height (i.e. a “gravity-corrected elevation”).

To understand this mathematically, consider the equation for hydrostatic equilibrium:

\frac{dp}{dz} = -\rho g

We can replace \rho by invoking the Ideal Gas Law

\frac{dp}{dz} = -\frac{pg}{RT}

which can rearranged as

p\frac{dz}{dp} = -\frac{RT}{g} 

This is a solvable ordinary differential equation, which has the measured height as a function of pressure.

z(p_s) - z(p) = -R\int_{p}^{p_s} \frac{T}{gp}dp dx

Here, p_s defines the pressure at the surface (i.e. where the elevation is zero).

z(p) = R\int_{p}^{p_s} \frac{T}{gp}dp

From this, we can see the importance of the geopotential height. At the tropics, where the air is warm, the air column will expand and the geopotential height will be high. This means that where the air is warm, lower elevations are less dense because of expansion. Similarly, at the poles, where the air is cold, the geopotential height is lower.

The thickness of an atmospheric layer will be larger where the temperature is higher (i.e. at the tropics) and thinner where the temperature is colder (i.e. at the poles).

We can also approximate how the geopotential height will tilt from warmer regions to cooler regions.

\Delta z = \frac{R \Delta T}{g} \ln \frac{p_s}{p}

By examining the geopotential height of a region, we can say whether the region is colder than average (lower geopotential height) or warmer (higher geopotential height).

Figure 1: Predicted geopotential height (500 mb contours) 1/9/2020 – 1/13/2020

 

 

Image Credit: NOAA/National Weather Service, National Centers for Environmental Prediction, Weather Prediction Center (https://www.wpc.ncep.noaa.gov/mdltrend)