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Earth System Dynamics An interactive open-access journal of the European Geosciences Union
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Discussion papers
https://doi.org/10.5194/esd-2019-65
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/esd-2019-65
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: research article 23 Oct 2019

Submitted as: research article | 23 Oct 2019

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Earth System Dynamics (ESD).

Π-theorem generalization of the ice-age theory

Mikhail Y. Verbitsky1 and Michel Crucifix2 Mikhail Y. Verbitsky and Michel Crucifix
  • 1Gen5 Group, LLC, Newton, MA, USA
  • 2UCLouvain, Earth and Life Institute, Louvain-la-Neuve, Belgium

Abstract. Analyzing a dynamical system describing the global climate variations requires, in principle, exploring a large space spanned by the numerous parameters involved in this model. Dimensional analysis is traditionally employed to deal with equations governing physical phenomena to reduce the number of parameters to be explored, but it does not work well with dynamical ice-age models, because, as a rule, the number of parameters in such systems is much larger than the number of independent dimensions. Physical reasoning may however allow us to reduce the number of effective parameters and apply dimensional analysis in a way that is insightful. We show this with a specific ice-age model (Verbitsky et al., 2018) which is a low-order dynamical system based on ice-flow physics coupled with a linear climate feedback. In this model, the ratio of positive-to-negative feedback is effectively captured by an adimensional number called the V-number, which aggregates several parameters and, hence, reduces the number of governing parameters. This allows us to apply the central theorem of the dimensional analysis, the π-theorem, efficiently. Specifically, we show that the relationship between the amplitude and duration of glacial cycles is governed by a property of scale-invariance that does not depend on the physical nature of the underlying positive and negative feedbacks incorporated by the system. This specific example suggests a broader idea, that is, the scale invariance can be deduced as a general property of ice age dynamics, if the latter are effectively governed by a single ratio between positive and negative feedbacks.

Mikhail Y. Verbitsky and Michel Crucifix
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Mikhail Y. Verbitsky and Michel Crucifix
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Supplementary code and data to ESD paper "Π-theorem generalization of the ice-age theory" by Verbitsky, M.Y. and Crucifix, M. (Version 1.0) M. Y. Verbitsky, M. Crucifix, and D. M. Volobuev https://doi.org/10.5281/zenodo.3473957

Mikhail Y. Verbitsky and Michel Crucifix
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Latest update: 16 Nov 2019
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Short summary
Using the central theorem of the dimensional analysis, the π-theorem, we show that the relationship between the amplitude and duration of glacial cycles is governed by a property of scale-invariance that does not depend on the physical nature of the underlying positive and negative feedbacks incorporated by the system.
Using the central theorem of the dimensional analysis, the π-theorem, we show that the...
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