Review Article:  Scaling, dynamical regimes and stratification: How long does weather last? How big is a cloud? 

Abstract

Abstract. Until the 1980’s, scaling notions were restricted to self-similar homogeneous special cases. I review developments over the last decades, especially in multifractals and Generalized Scale Invariance (GSI). The former is necessary for characterizing and modelling strongly intermittent scaling processes while the GSI formalism extends scaling to strongly anisotropic (especially stratified) systems. Both of these generalizations are necessary for atmospheric applications. The theory and (some) of the now burgeoning empirical evidence in its favour is reviewed. Scaling can now be understood as a very general symmetry principle. It is needed to clarify and quantify the notion of dynamical regimes. In addition to the weather and climate, there is an intermediate “macroweather regime” and at time scales beyond the climate regime, (up to Milankovitch scales) there is a macroclimate and megaclimate regime. By objectively distinguishing weather from macroweather it answers the question “how long does weather last?”. Dealing with anisotropic scaling systems – notably atmospheric stratification – requires new (non-Euclidean) definitions of the notion of scale itself. These are needed to answer the question “how big is a cloud?”. In anisotropic scaling systems morphologies of structures change systematically with scale even though there is no characteristic size. GSI shows that it is unwarranted to infer dynamical processes or mechanisms from morphology. Two “sticking points” preventing the more widespread acceptance of the scaling paradigm are also discussed. The first is an often implicit phenomenological “scalebounded” thinking that postulates a priori the existence of new mechanisms, processes every factor of two or so in scale. The second obstacle is the reluctance to abandon isotropic theories of turbulence and accept that the atmosphere’s scaling is anisotropic. Indeed there is currently appears to be no empirical evidence that the turbulence in any atmospheric field is isotropic. Most atmospheric scientists rely on General Circulation Models, and these are scaling – they inherited the symmetry from the (scaling) primitive equations upon which they are built. Therefore, the real consequence of ignoring wide range scaling is that it blinds us to alternative scaling approaches to macroweather and climate – especially to new models for long range forecasts and to new scaling approaches to climate projections. Such stochastic alternatives are increasingly needed notably to reduce uncertainties in climate projections to the year 2100.

The Fractional MacroEvolution Model: a simple quantitative scaling macroevolution model

“…If Mitchell's spectrum had been accurate, two consecutive million year average Earth temperatures would only have differed by about 10 microKelvins (μK)-yet this patently false implication was not noticed because the units of the spectrum (K 2 yr) were not intuitive, whereas an RMS Haar fluctuation of 10 μK would have been obviously problematic. Even spectral updates as recent as 2020 are in error by a factor 10 11 (see the review by Lovejoy [2023]). The comparison of the fluctuation analyses in this paper with those of the spectra (Appendix 3) highlights the power of fluctuation analysis when applied to irregularly sampled data.…”

“…Before attempting to understand processes at specific timescales, it is important to understand their context, that is, the dynamical regime in which they operate. Dynamical regimes are objectively defined by scaling; they are regimes over which fluctuations are scaling (see the review by Lovejoy [2023]). By definition, a scaling regime is one in which fluctuations ΔT (in some quantity such as temperature) are of the form ΔT(Δt) ∝ Δt H , where Δt is duration, or "lag," scale, and H is an exponent.…”

Geographic variability of dust and temperature in climate scaling regimes over the Last Glacial Cycle

“…where the canonical factor 2 multiplying the difference is a calibration parameter necessary to expand the working range from anomaly (H < 0) and differences (H > 0) to the Haar range −1 ≤ H ≤ 1 (Lovejoy, 2023).…”

“…Research that delves into climate variability across various scales has revealed that dynamical atmospheric regimes are characterised by scaling. Notably, over the past 540 million years (the Phanerozoic eon), five distinct scaling regimes have been identified (Lovejoy, 2023). These are the weather regime (spanning from 6 hours up to 20 days), the macroweather regime (covering 20 days to 50 years), the climate regime (encompassing 50 years to 80 kyr), the macroclimate regime (ranging from 80 to 500 kyr), and the megaclimate regime (denoting time scales larger than 500 kyr).…”

Review article: Interdisciplinary Perspectives on Climate Sciences – Highlighting Past and Current Scientific Achievements

“…These concepts are strictly related to those relying on the description of turbulent flows, as the existence of a hierarchy of exponents to fully characterise the nature of the system (i.e., the so-called multifractal view, Frisch, 1995), the observation of a global vs. a local scale invariance (Kuzzay et al, 2017), and the role of extreme events as singularities (Alberti et al, 2023). Shaun Lovejoy applied the above concepts to describe different dynamical regimes in the climate system: the weather, the macroweather, and the climate (Lovejoy, 2019). Indeed, weather and climate models are based on thermodynamics and continuum mechanics and are successful because they retain only some relevant "macroscopic" variables.…”