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McDougall (1989) proved that neutral surfaces possess an exact geostrophicstreamfunction, but its form has remained unknown. The exact geostrophicstreamfunction for neutral surfaces is derived here. It involves a pathintegral of the specific volume along a neutral trajectory. On a neutralsurface, the specific volume is a multivalued function of the pressure on thesurface, $\underset{\sim}{p}$. By decomposing the neutral surface into regionswhere the specific volume is a single-valued function of $\underset{\sim}{p}$,the path integral is simply a sum of integrals of these single-valuedfunctions. The regions are determined by the Reeb graph of$\underset{\sim}{p}$, and the neutral trajectory is encoded by a walk on thisgraph. Islands, and other holes in the neutral surface, can create cycles inthe Reeb graph, causing the exact geostrophic streamfunction on a neutralsurface to be multivalued. Moreover, neutral surfaces are ill-defined in thereal ocean. Hence, the topobaric geostrophic streamfunction is presented: asingle-valued approximation of the exact geostrophic streamfunction for neutralsurfaces, for use on any well-defined, approximately neutral surface. Numericaltests on several approximately neutral surfaces reveal that the topobaricgeostrophic streamfunction estimates the geostrophic velocity with an errorthat is about an order of magnitude smaller than that for any previously knowngeostrophic streamfunction. Also, the Montgomery potential is generalized,leading to an alternate form of the exact geostrophic streamfunction forneutral surfaces. This form is used to construct the orthobaric Montgomerypotential, an easily computable geostrophic streamfunction that estimates thegeostrophic velocity more accurately than any previously known geostrophicstreamfunction, but less so than the topobaric geostrophic streamfunction.

The exact geostrophic streamfunction for neutral surfaces