 Pressure-coordinate thermal wind

Combining geostrophic balance, $fu=-{\Phi }_y$, $fv={\Phi }_x$, with hydrostatic balance, ${\Phi }_p=-RT/p$, and commuting the horizontal and vertical derivatives proves the displayed equations. Horizontal derivatives hold pressure fixed. This relates vertical shear to horizontal temperature gradients away from the equator; the derivation requires nonzero Coriolis parameter and the validity of geostrophic balance.