Cascading tipping points: on the dynamics of coupled critical transitions
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We introduce the new concept of cascading tipping. It is defined as the event of a critical transition in a first, leading system, altering background conditions such that another critical transition in a second, following system occurs. A mathematical framework is created, where systems with saddle-node and Hopf bifurcations are used to investigate the behavior of systems involving abrupt transitions. Four deterministic cascading event types are defined, including the (1) double-fold, (2) fold-Hopf, (3) Hopf-fold and (4) double-Hopf cascade. Considering stochastic systems, we discussed the effect on the probability density function (PDF) and flickering effects, allowing for more subtle critical transitions. Statistical indicators and analysis tools for critical transitions are discussed, including the general theory of critical slowing down, degenerate fingerprinting and detrended fluctuation analysis (DFA). These are applied to the concept of cascading tipping in the form of detrended cross correlation analysis (DCCA) and a special case of extrapolation using the DFA of the following system. Using ensemble simulation runs, these statistical indicators are analyzed for the double-fold cascade and the fold-Hopf cascade. The concept of cascading tipping is applied to two climatological cases: (1) the overturning circulation coupled to El-Niño Southern Oscillation (ENSO) and (2) the overturning circulation coupled to southern hemispheric land ice formation. For the first case, we couple two conceptual models to show a case where a collapse of the overturning leads to an intensification of ENSO. In the second case, an existing conceptual box model is perturbed such that a transition from a southern sinking towards a thermohaline overturning state is followed by southern hemispheric land ice formation. These examples suggest the theoretical possibility of such events in the climate system.