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A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

Author

Listed:
  • James Walsh

    (Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA)

  • Esther Widiasih

    (Department of Mathematics, University of Hawaii–West Oahu, Kapolei, HI 96707, USA)

Abstract

We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary.

Suggested Citation

  • James Walsh & Esther Widiasih, 2020. "A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport," Mathematics, MDPI, vol. 8(3), pages 1-24, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:316-:d:326740
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    References listed on IDEAS

    as
    1. Peter Huybers, 2011. "Combined obliquity and precession pacing of late Pleistocene deglaciations," Nature, Nature, vol. 480(7376), pages 229-232, December.
    2. Didier Paillard, 1998. "The timing of Pleistocene glaciations from a simple multiple-state climate model," Nature, Nature, vol. 391(6665), pages 378-381, January.
    3. Edward J. Brook & Christo Buizert, 2018. "Antarctic and global climate history viewed from ice cores," Nature, Nature, vol. 558(7709), pages 200-208, June.
    4. Ayako Abe-Ouchi & Fuyuki Saito & Kenji Kawamura & Maureen E. Raymo & Jun’ichi Okuno & Kunio Takahashi & Heinz Blatter, 2013. "Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume," Nature, Nature, vol. 500(7461), pages 190-193, August.
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