Geostrophic Wind from the Slope of a Pressure Surface

The pressure push, read as a slope. The geostrophic wind is the wind that would blow if the push of the pressure gradient and the sideways push of the Earth's spin were in exact balance. On a chart of a constant-pressure surface such as the 500 hPa level, the pressure push shows up as a slope in the height $Z$ of that surface, so $V_{\mathrm{geo}} = \frac{g}{f}\,\frac{\partial Z}{\partial n}$: where the surface is steeper the wind is stronger, and it runs along the height contours with the low height on its left in the northern hemisphere.

The terms. The slope $\partial Z/\partial n$ is the change of height per distance across the contours; 1 m per km is typical. The Coriolis parameter $f$, twice the Earth's rotation rate $\Omega$ times the sine of the latitude, is about 1.03e-4 per second at 45°, entered as a positive number. Balloons report the height of the 500 hPa level at each station, so the slope between stations gives the wind with no need for the density of the air. The sibling formula written for a pressure difference on a level surface does need the density; with $\Delta p = \rho g\,\Delta Z$ it gives the same wind for any $\rho$ and $\Delta n$.

A worked case. A slope of 0.4 m per km at 45° N, where f = 2 × 7.292115e-5 × sin 45° = 1.0312608e-4 per second, gives 9.80665 × 4e-4 / 1.0312608e-4 = 38.0375 m/s, or 38.04 m/s. Run backward, 38.04 m/s at that latitude implies a 0.4 m per km slope, and 38.04 m/s over a 0.4 m per km slope returns the $f$ of 45° N.

Limits. The balance is an assumption: it holds when the Rossby number is small, and the Rossby-number page tells you whether it does. The height field this page differentiates is what the hypsometric equation builds from layer temperatures, and the coriolis-parameter page gives $f$ from the latitude. The source is Holton and Hakim (2013), chapter 3, which treats the geostrophic wind in isobaric coordinates.