Tipping mechanisms in a carbon cycle model

Rate-induced tipping (R-tipping) occurs when a ramp parameter changes rapidly enough to cause the system to tip between co-existing, attracting states, while noise-induced tipping (N-tipping) occurs when there are random transitions between two attractors of the underlying deterministic system. This work investigates R-tipping and N-tipping events in a carbonate system in the upper ocean, in which the key objective is understanding how the system undergoes tipping away from a stable fixed point in a bistable regime. While R-tipping away from the fixed point fits the framework of an established scenario, N-tipping poses challenges due to a periodic orbit forming the basin boundary for the attracting fixed point of the underlying deterministic system. Furthermore, for N-tipping, we are interested in the situation where noise is away from the small noise limit as it is more appropriate for the application. We postulate that two key points on the basin boundary are critical to understanding the noisy behavior: the exit point of what we find to be the most probable escape path (MPEP), which is determined by the Onsager-Machlup functional, and the pivot point, a point identified through the Maslov index, which appears as an obstacle to the movement of the escape region of noisy trajectories through the periodic orbit as noise increases.