Smooth and Nonsmooth Bifurcations in Welander's Ocean Convection Model

Abstract

Welander's model is a conceptual ocean convection model, that describes ocean convection with a few, simple dynamical equations. Welander's goal in formulating his model was to show that internally driven oscillations could be caused solely by switching between strongly convective and nonconvective states. Because of the conceptual importance of the switching mechanism, Welander created two versions of the model, one with a smooth transition between convective states, and one with an abrupt nonsmooth transition. He was able to numerically find a periodic orbit in both versions of the model. The climactic import of the model is in the idea that oscillations can be internally driven, but the model also has interesting mathematical import. Welander's implicit assumption that the nonsmooth model is easier to analyze is mathematically suspect, and a rigorous comparison between the smooth and nonsmooth models is not immediately clear. In this work, we introduce the model with scientific context, and complete a rigorous nonsmooth analysis of the model. We find one known nonsmooth bifurcation analogous to a supercritical Hopf bifurcation, but we also find a bifurcation that has not been previously described. Finally, we compare the smooth and nonsmooth models, paying close attention to the dangers inherent in such a comparison.