Given a Geopotential $Φ$ , the geopotential height is defined as

Z (z) : = \frac{Φ ( z )}{g _{0}} = \int_{0}^{z} \frac{g}{g _{0}} d z

where $g_{0}$ is the gravitational field at $z = 0$ , the usual $9.81 \frac{m}{s}$ .

When $g \approx g_{0}$ , one has $Z (z) \approx z$ , differing from it when $g \neq = g_{0}$ , i.e. at higher altitudes.

For a gravitational field

Opening $g$ as¹

g (z) = \frac{GM}{( R + z ) ^{2}}

we have the same results for the respective geopotential, just also dividing by $g_{0} = \frac{GM}{R ^{2}}$ :

Z (z) = (\frac{R}{R + z}) z

When $z ≪ R$ , naturally we have

Z (z) \approx z

WALLACE, John Michael; HOBBS, Peter Victor. Atmospheric science: an introductory survey. 2nd ed ed. Amsterdam Paris: Academic press, 2006.

Naturally supposing that the Earth is spherical, i.e. integrating over $z$ is equivalent to integrating radially. ↩

Nicholas Funari Voltani