Pressure work in a steady cross-isobaric wind ($-{\rho }^{-1}\mathbf{u}\cdot {\nabla }_{h}p=r|\mathbf{u}{|}^2$)

Dot the steady linear-drag momentum equation with the velocity. The Coriolis force does no work, so pressure-gradient work balances frictional dissipation. In a strictly geostrophic flow the wind is along isobars and pressure work vanishes. Cross-isobar motion supplies the positive pressure work needed to maintain a dissipative equilibrium.