Mathematical model of the Antarctic Circumpolar Current with density variations
Analysing a boundary value problem modelling the Antarctic Circumpolar Current
T.A.B. Slavenburg (TU Delft - Electrical Engineering, Mathematics and Computer Science)
K. Marynets – Mentor (TU Delft - Mathematical Physics)
Dion Gijswijt – Graduation committee member (TU Delft - Discrete Mathematics and Optimization)
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Abstract
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.