<p>This paper is concerned with the explicit solutions for a class of geophysical gyre models. Under the relevant physical conditions, the current model can be regarded as a governing equation of the Antarctic Circumpolar Current (ACC). For the linear oceanic vorticity case, we derive some explicit two-dimensional solutions which depend on solving the associated Fuchs type equation, Hypergeometric equation, and Legendre’s differential equation, respectively. These explicit solutions provide a framework for analyzing the influence of various parameters on the flow field and offer valuable insights into the characteristics of the ACC. Our simulations under constant vorticity capture the trend of the ACC’s velocity increasing with latitude. Finally, for the nonlinear vorticity case, we establish the existence of nontrivial solutions to this equation by introducing energy functionals and imposing appropriate assumptions.</p>