On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

Watkins, N. W.; Chapman, S. C.; Chechkin, Aleksei V.; Ford, Ian; Klages, Rainer; Stainforth, David A.

doi:10.48550/arxiv.2007.06464

Cited by 2 publications

(2 citation statements)

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“…Whereas the highest-order term in the fractional relaxation equation is fractional, in the FLE the fractional term is of lowest order. Watkins et al (2020) provides a useful review that compares and contrasts various stochastic equations that appear in climate models and points out the relationship between the FLE and the FEBE.…”

Section: 3mentioning

confidence: 99%

“…Mathematically this implies that both variabilities have the same Green's function, opening up the possibility of estimating climate parameters directly from the internal variability. Basing himself on the weather-macroweather scale separation and assuming that the internal forcing from the weather regime was a Gaussian white noise, Hasselmann (1976) proposed a stochastic climate model that reduces to the stochastic box-model (h = 1); in the zero-dimensional case, the latter is therefore sometimes called the "Hasselmann equation" (see Watkins et al (2020) for a useful discussion, review as well as various fractional extensions).…”

Section: Internal Variability and Macroweather Forecastsmentioning

confidence: 99%

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The fractional energy balance equation

Lovejoy

Procyk

Hébert

et al. 2021

Quart J Royal Meteoro Soc

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Classical Energy Balance Equations (EBEs) are differential equations of integer order (h = 1), here we generalize this to fractional orders: the Fractional EBE (FEBE, 0 < h ≤ 1). In the FEBE, when the Earth is perturbed by a forcing, the temperature relaxes to equilibrium via a slow power-law process: h = 1 is the exceptional (but standard) exponential case. Our FEBE derivation is phenomenological, it complements derivations based on the classical continuum mechanics heat equation (that imply h = 1/2 for the surface temperature) and of the more general Fractional Heat Equation which allows for 0 < h < 2. Unlike some of the earlier "scale free" models based purely on scaling, the FEBE has an extra blackbody radiation term that allows for energy balance. It therefore has two scaling regimes (not one), it has the advantage of being stable to infinitesimal step-function perturbations and it has a finite Equilibrium Climate Sensitivity. We solve the FEBE using Green's functions, whose high-and low-frequency limits are power laws with a relaxation scale transition (several years). When stochastically forced, the high-frequency parts of the internal variability are fractional Gaussian noises that can be used for monthly and seasonal forecasts; when deterministically forced, the low-frequency response describes the consequences of anthropogenic forcing, it has been used for climate projections. The FEBE introduces complex climate sensitivities that are convenient for handling periodic (especially annual) forcing. The FEBE obeys Newton's law of cooling, but the heat flux crossing a surface nonetheless depends on the fractional time derivative of the temperature. The FEBE's ratio of transient to equilibrium climate sensitivity is compatible with GCM estimates. A simple ramp forcing model of the industrial-epoch warming combining deterministic (external) with stochastic (internal) forcing is statistically validated against centennial-scale temperature series.

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Section: 3mentioning

confidence: 99%

Section: Internal Variability and Macroweather Forecastsmentioning

confidence: 99%

The fractional energy balance equation

Lovejoy

Procyk

Hébert

et al. 2021

Quart J Royal Meteoro Soc

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The challenge of non-Markovian energy balance models in climate

Watkins,

Calel,

Chapman

et al. 2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We first review the way in which Hasselmann’s paradigm, introduced in 1976 and recently honored with the Nobel Prize, can, like many key innovations in complexity science, be understood on several different levels. It can be seen as a way to add variability into the pioneering energy balance models (EBMs) of Budyko and Sellers. On a more abstract level, however, it used the original stochastic mathematical model of Brownian motion to provide a conceptual superstructure to link slow climate variability to fast weather fluctuations, in a context broader than EBMs, and led Hasselmann to posit a need for negative feedback in climate modeling. Hasselmann’s paradigm has still much to offer us, but naturally, since the 1970s, a number of newer developments have built on his pioneering ideas. One important one has been the development of a rigorous mathematical hierarchy that embeds Hasselmann-type models in the more comprehensive Mori–Zwanzig generalized Langevin equation (GLE) framework. Another has been the interest in stochastic EBMs with a memory that has slower decay and, thus, longer range than the exponential form seen in his EBMs. In this paper, we argue that the Mori–Kubo overdamped GLE, as widely used in statistical mechanics, suggests the form of a relatively simple stochastic EBM with memory for the global temperature anomaly. We also explore how this EBM relates to Lovejoy et al.’s fractional energy balance equation.

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On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

Cited by 2 publications

References 30 publications

The fractional energy balance equation

The fractional energy balance equation

The challenge of non-Markovian energy balance models in climate

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