Summary
We study a conceptual model that describes the vertical mixing process in the North Atlantic Ocean between warmer surface water and cold, deep water. This mechanism, which is crucial for the transport of warm water to Northern latitudes, is subject to seasonal forcing due to variations in the amount of fresh glacial meltwater that enters the North Atlantic and seasonal atmospheric variations. The model takes the form of a periodically forced planar vector field for (appropriately rescaled) temperature and salinity, and it has two parameters: the density threshold $\eta$ for a change in mixing strength, and the virtual salinity flux $\mu$ that is subject to periodic variation of strength $c$. In the absence of forcing ($c=0$), the system has a stable periodic orbit in a bounded region of the $(\mu,\eta)$-plane. For $c \neq 0$ there is an interaction between this intrinsic oscillation and the periodic forcing. For fixed $\eta$, one finds dynamics on invariant tori in a bounded region of the $(c,\mu)$-plane. We determine the associated overall resonance structure by computing the rotation number $\rho$ over this region, as well as individual curves of saddle-node bifurcations that bound selected (lower-order) resonance tongues. This allows us to study how the resonance tongues in the $(c,\mu)$-plane change with the parameter $\eta$. More specifically, we present generic bifurcations of resonance tongues that explain the observed changes.
