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Nonequilibrium Oscillations, Probability Angular Momentum, and the Climate System

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Abstract

Though the Boltzmann–Gibbs framework of equilibrium statistical mechanics has been successful in many arenas, it is clearly inadequate for describing many interesting natural phenomena driven far from equilibrium. The simplest step towards that goal is a better understanding of nonequilibrium steady-states (NESS). Here we focus on one of the distinctive features of NESS—persistent probability currents—and their manifestations in our climate system. We consider the natural variability of the steady-state climate system, which can be approximated as a NESS. These currents must form closed loops, which are odd under time reversal, providing the crucial difference between systems in thermal equilibrium and NESS. Seeking manifestations of such current loops leads us naturally to the notion of “probability angular momentum” and oscillations in the space of observables. Specifically, we will relate this concept to the asymmetric part of certain time-dependent correlation functions. Applying this approach, we propose that these current loops give rise to preferred spatio-temporal patterns of natural climate variability that take the form of climate oscillations such as the El-Niño Southern Oscillation (ENSO) and the Madden–Julian Oscillation (MJO). In the space of climate indices, we observe persistent currents and define a new diagnostic for these currents: the probability angular momentum (\(\mathscr {L}\)). Using the observed climatic time series of ENSO and MJO, we compute both the averages and the distributions of \(\mathscr {L}\). These results are in good agreement with the analysis from a linear Gaussian model. We propose that, in addition to being a new quantification of climate oscillations across models and observations, the probability angular momentum provides a meaningful characterization for all statistical systems in NESS.

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Notes

  1. In much of the physics community, “phase space” is a term used for the space of x–p (coordinate and momentum). Significantly, these variables are even/odd under time reversal. In this paper, however, we use this term in the sense common in the dynamical systems and climate science communities. In the cases we consider here, there is no reason to regard the variables (e.g., temperature and volume, or two amplitudes of a principal component analysis) as having different symmetry under time reversal. For many in the community of statistical physics, the familiar term in this context is “configuration space.” We will use the two terms interchangeably and assume there is no confusion.

  2. Below, we will be considering time-dependent distributions, which we denote by \(P\left( C;t\right) \). The superscript (\(^{*}\)) signifies a stationary distribution.

  3. Though the form of our equation appears to be for continuous t and discrete C, it is simple to write equations for other types of variables, e.g., continuous t and C. Note that we have restricted ourselves to systems evolving with time-independent rates.

  4. In this form, the criterion for the W’s to satisfy DB appears to depend on \(P^{*}\). Kolmogorov [37] provided a criterion which involves only the W’s.

  5. Note that the discrete version of the \(\delta \) in the noise correlation is a Kronecker delta of the time steps divided by \(\varepsilon \). Note also that there is no correlation between \({\mathbf {x}}\left( t\right) \) and \(\varvec{\eta }\left( t\right) \), so that \( < {\mathbf {x}}\left( t\right) \varvec{\eta }\left( t\right) > \equiv 0\).

  6. The \(D_{\alpha \beta }\) here is the same as the one introduced above, the only difference being it is restricted to be x-independent in a LGM.

  7. To avoid confusion, we use different notation for quantities in a LGM from the general case, e.g., p and \({\mathbf {j}}\) instead of P and \({\mathbf {K}}\).

  8. Here, \( <\mathscr {O}> ^{*}\) refers to the average in the stationary state: \(\int \mathscr {O}\left( {\mathbf {x}}\right) p^{*}\left( {\mathbf {x}}\right) d{\mathbf {x}}\).

  9. Note that we use the same letter for both the density of a quantity and the total, the former having an additional argument, \({\mathbf {x}}\).

  10. Introduced in 1940 by Levy [43], this concept was subsequently developed in a series of articles. For a modern review and further generalizations, see, e.g., Ref. [44]. Most recently, it has been exploited in the context of noisy couple RC circuits [45, 46].

  11. Note that “diffusion” in this case also carries these units, as it is the noise covariance matrix.

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Acknowledgements

This work was supported in part by NSF INSPIRE Award #1245944 and NSF DMR #1507371. Work of JBW was partially carried out during a stay at the Institute for Marine and Atmospheric research Utrecht (Utrecht University, NL) which was supported by the Netherlands Centre for Earth System Science. Three of us (JBW, BF-K, RKPZ) are grateful for the hospitality of the MPIPKS, where some of this work was carried out during a workshop Climate Fluctuations and Non-Equilibrium Statistical Mechanics: an Interdisciplinary Dialogue in the summer of 2017. We would like to thank Kevin Bassler, Ronald Dickman and Beate Schmittmann for helpful discussions.

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Authors and Affiliations

  1. Department of Atmospheric and Oceanic Sciences, University of Colorado, Boulder, CO, 80309, USA

    Jeffrey B. Weiss

  2. Department of Earth, Environmental, and Planetary Sciences, Brown University, Providence, RI, 02912, USA

    Baylor Fox-Kemper

  3. Department of Physics, University of California, Berkeley, CA, 94720, USA

    Dibyendu Mandal

  4. Department of Earth and Environmental Sciences, University of Michigan, Ann Arbor, MI, 48109, USA

    Arin D. Nelson

  5. Department of Physics & Astronomy, University of North Carolina at Asheville, Asheville, NC, 28804, USA

    R. K. P. Zia

  6. Center for Soft Matter and Biological Physics, Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    R. K. P. Zia

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  1. Jeffrey B. Weiss
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Correspondence to Jeffrey B. Weiss.

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Communicated by Valerio Lucarini.

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Weiss, J.B., Fox-Kemper, B., Mandal, D. et al. Nonequilibrium Oscillations, Probability Angular Momentum, and the Climate System. J Stat Phys 179, 1010–1027 (2020). https://doi.org/10.1007/s10955-019-02394-1

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