To use the geostrophic equation to analyze flow in a planetary atmosphere, we need to know the three-dimensional pressure field. While this can be measured for the Earth's atmosphere, it is very difficult to derive from remote-sensing observations of the giant planets. However, remote-sensing observations can determine the three-dimensional temperature field and, if we know the wind speeds at a certain pressure level (e.g., by tracking discrete cloud features at the cloud tops), then the wind speeds at all other levels can be deduced from the thermal wind equation.
To derive the thermal wind equation, the hydrostatic equation dp = —pg dz, must first be rearranged to give
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