Geostrophic wind (${\mathbf{u}}_g=(\rho f{)}^{-1}\hat{\mathbf{z}}\times {\nabla }_{h}p$)

The geostrophic wind is the horizontal flow for which the pressure gradient force balances the Coriolis force. It runs along atmospheric isobars and has speed proportional to their pressure gradient, at fixed density and nonzero Coriolis parameter. Curvature, friction and small spatial scales can invalidate that approximation.

A steady horizontal parcel with drag acceleration $-r\mathbf{u}$ obeys $0=-{\nabla }_{h}p/\rho -f\hat{\mathbf{z}}\times \mathbf{u}-r\mathbf{u}$. Inverting the horizontal rotation-plus-drag matrix gives the title. For $r>0$, the velocity has a component toward lower pressure and a component along atmospheric isobars. For $p_x=0$, $f>0$, $p_y>0$, both $u,v$ are negative. The cross-isobar angle is $\arctan (r/f)$, and the speed is smaller than the corresponding frictionless wind. A vertically resolved turbulent stress is not generally equivalent to one constant $r$.

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.

With $x$ eastward and $y$ northward, a purely meridional pressure gradient gives the velocity in the title. For $f>0$, pressure increasing northward gives westward wind, leaving lower pressure to the wind's left. Reversing the pressure gradient reverses the velocity. At a fixed gradient and density the speed scales as $1/|\sin \varphi |$; the formal equatorial divergence signals breakdown of geostrophic balance, rather than a real infinite wind.