The Antarctic Circumpolar Current (ACC) is the only ocean current that flows continuously around the globe, and it plays a crucial role in global climate. It encircles Antarctica and keeps warm waters away, helping to maintain the continent’s cold environment. This report studies a simplified mathematical model to capture the ACC’s steady-state behaviour, incorporating the effects of density stratification. The model studied is originally developed by Ding, who reformulated the ocean dynamics of Arctic gyres as a second-order non-linear differential equation for a stream function u(t) that describes the flow field. This includes the effect of Earth’s rotation (Coriolis effect), represented by ω, a prescribed vorticity distribution F (u), and a stratified density function ρ(u).  

In this report, the model is adapted to describe the ACC. This is achieved by imposing Dirichlet boundary conditions representing the two latitudinal limits. This leads to a BVP defined on a finite domain, providing an appropriate mathematical framework for capturing the flow of the ACC. After simplifying the differential equation to an integral equation, our analytical analysis of the model showed that under reasonable assumptions, an unique solution exists. Before computing a numerical solution, the assumptions introduced in the analytic analysis were verified for the specific choices of F and ρ, confirming their regularity and positivity. The equation was then solved numerically using a collocation method, yielding a stream function profile, consistent with observed ACC behaviour. The numerical solution was validated against an exact solution in a special case (constant F and ρ), showing excellent agreement (error < 10−6). Stability tests confirmed that the solution changes only slightly under small perturbations in boundary conditions.  

Finally, the influence of different density stratification profiles was explored. Gentle stratification (constant or linear ρ(u)) led to a stable, symmetric jet, whereas strongly non-linear stratification (quadratic or exponential ρ(u)) resulted in oscillatory or unstable solutions. This highlights the model’s sensitivity to stratification.