Structural Stochastic Stability of a Simple Auto-Oscillatory Climatic Feedback System

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  • 1 Department of Geology and Geophysics, Yale University, New Haven, CT 06520
  • | 2 The Center for the Environment and Man, Inc., Hartford, CT 06120
  • | 3 Department of Geology and Geophysics, Yale University, New Haven, CT 06520
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Abstract

Based on a more detailed model developed previously, governing possible feedbacks between sea ice extent (η), deep-ocean temperature (θ), and atmospheric carbon dioxide, we have constructed a simple, deterministic, dynamical system for η and θ that yields a stable limit cycle as a solution. To make the system more realistic we add random (white noise) forcing and explore the new response as a function of the amplitude of this stochastic forcing. Included in the analysis are examples of sample time series, sample phase trajectories, variance spectra, and “residence density” in the phase space. The results show that physically reasonable values of the stochastic amplitude do not completely obscure the basic limit cycle, but they do convert an intransitive, deterministic system to an almost-intransitive one that tends to reside longer in two distinct regions of the phase space, with relatively fast transitions between thorn. The solutions also show how stochastic forcing of one component of the climatic feed-back system (i.e., η) can lead to a damped response in another unforced component (θ).

Abstract

Based on a more detailed model developed previously, governing possible feedbacks between sea ice extent (η), deep-ocean temperature (θ), and atmospheric carbon dioxide, we have constructed a simple, deterministic, dynamical system for η and θ that yields a stable limit cycle as a solution. To make the system more realistic we add random (white noise) forcing and explore the new response as a function of the amplitude of this stochastic forcing. Included in the analysis are examples of sample time series, sample phase trajectories, variance spectra, and “residence density” in the phase space. The results show that physically reasonable values of the stochastic amplitude do not completely obscure the basic limit cycle, but they do convert an intransitive, deterministic system to an almost-intransitive one that tends to reside longer in two distinct regions of the phase space, with relatively fast transitions between thorn. The solutions also show how stochastic forcing of one component of the climatic feed-back system (i.e., η) can lead to a damped response in another unforced component (θ).

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