Geostrophic adjustment is an initial-value problem involving the evolution of an unbalanced initial state toward one of geostrophic balance. Here we consider the three-dimensional hydrostatic primitive equations, and an initial state that consists of a Gaussian sphere of low geopotential, and positive u and v wind components. Notice that this state is far from geostrophic balance, with the strongest winds located where the geopotential gradient is zero. The first loop shows the evolution of the geopotential field in an x--z cross section. Notice the inertial gravity waves radiating away from the initially localized field. Although the end state looks similar to the initial state, it has adjusted so that it is in geostrophic balance with the velocity field, as shown below.
The second loop shows the evolution of the meridional wind (v) in an x--z cross section. Notice that the wind rapidly adjusts to a dipole around the point of lowest geopotential, indicative of geostrophic balance.
Finally, here is an x--y plan view of the potential vorticity (PV) and wind at the end of the solution. The wind is nearly azimuthal around the region of low geopotential, and positive PV. The solution is not yet exactly steady, and there are numerical artifacts, but you should get the sense the wind has adjusted. The PV, of course, is frozen in the flow for all time; u, v, w, etc. all evolve and radiate energy from the localized initial condition, but the PV is stuck on fluid particles.
