Inverse-square law between time and amplitude for crossing tipping thresholds

Abstract

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold (in a run-away positive feedback loop). We study how rapidly one needs to turn around once one has crossed the threshold. We derive a simple criterion that relates the peak and curvature of the parameter path in an inverse-square law to easily observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping for parameter paths that are turning around near the tipping point. We apply these approximations to investigate when dynamic changes of the albedo across a critical value cause tipping in a model for the Indian Summer Monsoon. The model, originally derived by Zickfeld et al, describes the positive moisture advection feedback between the Indian Ocean and the Indian subcontinent using two dynamic variables, the atmospheric temperature and specific humidity over land. The inverse-square law between time spent at elevated albedos and amplitude of increase beyond the tipping threshold is visible in the level curves of equal probability when the system is subject to random disturbances.