Critical noise for advanced dynamic B-tipping in nearly non-smooth Stommel-type models

We study critical relationships between the smoothness parameter for the underlying fold bifurcation and the noise level in the context of B-tipping near smooth and non-smooth dynamic fold bifurcations. The motivation is the Stommel 2-box model, a piecewise-smooth continuous dynamical system modeling thermohaline circulation in the North Atlantic, and related climate models. These contain non-smooth fold bifurcations which arise when a saddle-point and a stable focus meet at a border collision bifurcation. An asymptotic analysis of the corresponding Fokker-Planck Equation (FPE) for the stochastic system provides insight into critical noise levels, depending on the relative rate of parameter variation and a measure of smoothness of the underlying bifurcation. Critical scales are obtained from different reductions of the FPE, identifying cases where noise may advance tipping relative to deterministic behavior. Applying this approach for B-tipping near both smooth and non-smooth folds shows that the non-smooth case has greater sensitivity to smaller noise levels, with a smaller critical scale for noise-advanced tipping in the non-smooth case. Since these results do not depend on obtaining a solution of the FPE, the approach can be adapted to multi-degree-of-freedom models and in other applications.