Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles

Conceptual climate models provide an approach to understanding climate processes through a mathematical analysis of an approximation to reality. Recently, these models have also provided interesting examples of nonsmooth dynamical systems. Here we discuss a conceptual model of glacial cycles consisting of a system of three ordinary differential equations defining a discontinuous vector field. We show that this system has a large periodic orbit crossing the discontinuity boundary. This orbit can be interpreted as an intrinsic cycling of the Earth's climate giving rise to alternating glaciations and deglaciations.

[1]  S. Peckham,et al.  The physical basis of glacier volume-area scaling , 1997 .

[2]  S. Marshall,et al.  Coupled energy‐balance/ice‐sheet model simulations of the glacial cycle: A possible connection between terminations and terrigenous dust , 1995 .

[3]  Michael Ghil,et al.  Free oscillations in a climate model with ice-sheet dynamics , 1979 .

[4]  G. North,et al.  New parameterizations and sensitivities for simple climate models , 1993 .

[5]  R. Pierrehumbert.  Climate dynamics of a hard snowball Earth , 2005 .

[6]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[7]  Peter John Huybers,et al.  Glacial variability over the last two million years: an extended depth-derived agemodel, continuous obliquity pacing, and the Pleistocene progression , 2007 .

[8]  Barry Saltzman,et al.  A low‐order dynamical model of global climatic variability over the full Pleistocene , 1990 .

[9]  J. Oerlemans.  Glacial cycles and ice-sheet modelling , 1982 .

[10]  J. D. Hays,et al.  Variations in the Earth ' s Orbit : Pacemaker of the Ice Ages Author ( s ) : , 2022 .

[11]  Eli Tziperman,et al.  On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times , 2003 .

[12]  P. Huybers.  Combined obliquity and precession pacing of late Pleistocene deglaciations , 2011, Nature.

[13]  Christopher Rackauckas,et al.  On the Budyko-Sellers energy balance climate model with ice line coupling , 2015 .

[14]  G. North.  Theory of Energy-Balance Climate Models. , 1975 .

[15]  Mario di Bernardo,et al.  Bifurcations in Nonsmooth Dynamical Systems , 2008, SIAM Rev..

[16]  M. J. Newman,et al.  Variations in the Earth ' s Orbit : Pacemaker of the Ice Ages , 2007 .

[17]  F. Parrenin,et al.  Amplitude and phase of glacial cycles from a conceptual model , 2003 .

[18]  J. Oerlemans.  Glacial cycles and ice-sheet modelling , 1982 .

[19]  Esther R. Widiasih,et al.  Dynamics of the Budyko Energy Balance Model , 2011, SIAM J. Appl. Dyn. Syst..

[20]  Didier Paillard,et al.  The Antarctic ice sheet and the triggering of deglaciations , 2004 .

[21]  Max J. Suarez,et al.  Simple albedo feedback models of the icecaps , 1974 .

[22]  P. Welander.  A simple heat-salt oscillator , 1982 .

[23]  S. Warren,et al.  The ablation zone in northeast Greenland: ice types, albedos and impurities , 2010, Journal of Glaciology.

[24]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[25]  Richard McGehee,et al.  A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth's Orbit , 2012, SIAM J. Appl. Dyn. Syst..

[26]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[27]  J. Jouzel,et al.  Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica , 1999, Nature.

[28]  P. Ashwin,et al.  The middle Pleistocene transition as a generic bifurcation on a slow manifold , 2015, Climate Dynamics.

[29]  Didier Paillard,et al.  The timing of Pleistocene glaciations from a simple multiple-state climate model , 1998, Nature.

[30]  Richard McGehee,et al.  A Quadratic Approximation to Budyko's Ice-Albedo Feedback Model with Ice Line Dynamics , 2014, SIAM J. Appl. Dyn. Syst..

[31]  M. Budyko.  The effect of solar radiation variations on the climate of the Earth , 1969 .

[32]  Milutin Milankovictch,et al.  Canon of insolation and the ice-age problem : (Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem) Belgrade, 1941. , 1998 .

[33]  Michael Ghil,et al.  A climate model with cryodynamics and geodynamics , 1981 .

[34]  A. Abe‐Ouchi,et al.  Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume , 2013, Nature.

[35]  H. E. Wright,et al.  Plant trash in the basal sediments of glacial lakes , 2004 .

[36]  W. Ruddiman.  Orbital changes and climate , 2006 .

[37]  William D. Sellers,et al.  A Global Climatic Model Based on the Energy Balance of the Earth-Atmosphere System. , 1969 .

[38]  Stephen John Hogan,et al.  On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in ℝ3 , 2014, SIAM J. Appl. Dyn. Syst..

[39]  Johannes Weertman,et al.  Milankovitch solar radiation variations and ice age ice sheet sizes , 1976, Nature.