Dynamic tipping in the non-smooth Stommel-box model, with fast oscillatory forcing

Chris Budd, Cody Griffith, Rachel Kuske

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We study the behavior at tipping points close to non-smooth fold bifurcations in non-autonomous systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modeling thermohaline circulation. We obtain explicit asymptotic expressions for the behavior at tipping points in the settings of both slowly varying freshwater forcing and rapidly oscillatory fluctuations. The results, based on combined multiple scale and local analyses, provide conditions for the sudden transitions between temperature-dominated and salinity-dominated states. In the context of high frequency oscillations, a multiple scale averaging approach can be used instead of the usual geometric approach normally required for piecewise-smooth continuous systems. The explicit parametric dependencies of advances and lags in the tipping show a competition between dynamic features of the model. We make a contrast between the behavior of tipping points close to both smooth Saddle–Node Bifurcations and the non-smooth systems studied on this paper. In particular we show that the non-smooth case has earlier and more abrupt transitions. This result has clear implications for the design of early warning signals for tipping in the case of the non-smooth dynamical systems which often arise in climate models.

Original languageEnglish
Article number132948
JournalPhysica D: Nonlinear Phenomena
Early online date12 May 2021
Publication statusPublished - 30 Apr 2022


  • Border collision
  • Conceptual climate models
  • Dynamic bifurcation
  • Non-autonomous systems
  • Non-smooth dynamics
  • Tipping

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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