text: The first derivation of the potential vorticity equation under a relaxed form of this condition was performed by Jule Charney using Ertel's theorem.
full_question: The first derivation of the potential vorticity equation under a relaxed form of this condition was performed by Jule Charney using Ertel's theorem. That potential vorticity equation under a loosened form of this condition reduces synoptic-scale flow to a non-linear PDE in terms of the stream function. That loosened condition is used because real systems tend to be baroclinic rather than barotropic and prefixes this condition with "quasi-". An incompressible fluid's flow under this condition forms a Taylor column and is independent of the coordinate parallel to the rotation axis according to the Taylor-Proudman theorem. This condition occurs when the ratio of horizontal flow velocity to twice the angular velocity of Earth's rotation times the horizontal scale of the flow, is much less than unity. Like hydrostatic equilibrium, it requires that there be no acceleration, and that ratio is the Rossby number. For 10 points, name this condition in which the pressure gradient force for a flow is balanced by the Coriolis effect.
first_sentence: The first derivation of the potential vorticity equation under a relaxed form of this condition was performed by Jule Charney using Ertel's theorem.
raw_answer: geostrophic [or word forms, such as geostrophy; or quasi-geostrophic until "incompressible" is read]