An Exact, Steady, Purely Azimuthal Flow as a Model for the Antarctic Circumpolar Current

The problem of flow moving purely in the azimuthal direction on a sphere is considered. An exact solution for an incompressible (constant density), inviscid fluid, which admits a velocity profile below the surface and along the surface, is constructed; this can be regarded as a model for the Antarctic Circumpolar Current (ACC). The new approach adopted here is to model the processes that produce the observed structure of the ACC by the introduction of a nonconservative body force. It is shown that if the body force is conservative, then the governing equations necessarily lead to profiles that are quite unrealistic. However, with a suitable choice of body force, which reverts to conservative outside the ACC, any velocity profile of any width can be constructed as an exact solution of the system. A fairly simple choice is made in this note in order to present some specific results: a profile on the surface that is zero outside the arc of the ACC, with a maximum at its center and decaying with depth. It is shown that the methods developed here can be used to produce ever more complicated profiles to correspond to different data. Indeed, the basic example that this study introduces can be regarded as one of the jets that compose the ACC, and the results allow for any number of such jets. Although only one velocity profile is described, it is emphasized that many different choices, motivated by direct velocity observations in specific regions, are possible within the model. In conclusion, a few comments are made outlining the way in which this exact solution can be embedded within more general and complete discussions of the ACC and its properties.