Relaxation oscillations in an idealized ocean circulation model

TL;DR

This paper analyzes a simplified ocean circulation model with multiple time scales, demonstrating the existence of relaxation and canard oscillations, and showing how obliquity forcing can produce climate-like oscillatory patterns.

Contribution

It introduces a piecewise-smooth, three-time-scale model of thermohaline circulation and applies geometric singular perturbation theory to reveal complex oscillatory behaviors including canard cycles.

Findings

The model exhibits classical relaxation oscillations.

It also shows small amplitude canard cycles under certain parameters.

Obliquity forcing modulates oscillation amplitude and frequency, resembling climate proxy data.

Abstract

This work is motivated by a desire to understand transitions between stable equilibria observed in Stommel's 1961 thermohaline circulation model. We adapt the model, including a forcing parameter as a dynamic slow variable. The resulting model is a piecewise-smooth, three time-scale system. The model is analyzed using geometric singular perturbation theory to demonstrate the existence of attracting periodic orbits. The system is capable of producing classical relaxation oscillations as expected, but there is also a parameter regime in which the model exhibits small amplitude oscillations known as canard cycles. Forcing the model with obliquity variations from the last 100,000 years produces oscillations that are modulated in amplitude and frequency. The output shows similarities with important features of the climate proxy data of the same period.