Perhaps the most obvious difficulty in understanding the atmosphere is in dealing with its enormous range of scales. The single picture in Fig. 1 shows clouds with horizontal spatial variability ranging from millimetres to the size of the planet, a factor of 10 billion in scale. In the vertical direction the range is more modest but still huge: about 10 million. The range of temporal variability is extreme, spanning a range of 100 billion billion: from milliseconds to the planet's age (Fig. 2).

The earliest approach to atmospheric variability was phenomenological: weather as a juxtaposition of various processes with characteristic morphologies, air masses, fronts, and the like. Circumscribed by the poor quality and quantity of the then available data, these were naturally associated with narrow-scale-range, mechanistic processes.

At first, ice ages, “medieval warming”, and other evidence of low-frequency processes were only vaguely discerned. Weather processes were thought to occur with respect to a relatively constant (and unimportant) background: climate was conceived as simply long-term “average” weather. It was not until the 1930s that the International Meteorological Organisation defined “climate normals” in an attempt to quantify the background “climate state”. The duration of the normals – 30 years – was imposed essentially by fiat: it conveniently corresponded to the length of high-quality data then available: 1900–1930. This 30-year duration is still with us today with the implicit consequence that – purely by convention – “climate change” occurs at scales longer than 30 years.

Interestingly, yet another official timescale for defining “anomalies” has been developed. Again, for reasons of convenience (and partly – for temperatures – due to the difficulty in making absolute measurements), anomalies are defined with respect to monthly averages. Ironically, a month wavers between 28 and 31 d: it is not even a well-defined unit of time.

The overall consequence of adopting, by convenience, monthly and 30-year timescales is a poorly theorized, inadequately justified division of atmospheric processes into three regimes: scales less than a month, a month up to 30 years, and a lumping together of all slower processes with timescales longer than 30 years. While the high-frequency regime is clearly “weather” and the slow processes – at least up to ice age scales – are “climate”, until Lovejoy (2013) the intermediate regime lacked even a name. Using scaling – and with somewhat different transition scales – the three regimes were finally put on an objective quantitative basis, with the middle regime baptized “macroweather”. By using scaling to quantitatively define weather, macroweather, and climate, we can finally objectively answer the question: how long does weather last? A bonus, detailed in Sect. 2, is that scaling analyses showed that what had hitherto been considered simply climate is itself composed of three distinct dynamical regimes. Rather than lumping all low frequencies together, we must also distinguish between macroclimate and megaclimate.

To review how scaling defines dynamical regimes, let us define scaling using fluctuations – for example of the temperature or of a component of the wind. For the moment, consider only one dimension, i.e. time series or spatial transects. Temporal scaling means that the amplitudes of fluctuations are proportional to their timescale raised to a scale-invariant exponent. For appropriately nondimensionalized quantities, 1fluctuation=(scale)ζ. Every term in this equation needs an appropriate definition, but for now, consider the classical ones. First, the usual turbulence definition of a fluctuation is a difference (of temperature, wind components, etc.) taken over an interval in space or in time. This interval defines the timescale (Δt) or space scale (Δx) of the corresponding fluctuation. Also, classically, one considers the statistically averaged fluctuation (indicated by “〈〉”). If we decompose ζ into a random singularity γ and a non-random “fluctuation exponent” H, then the appropriately averaged fluctuation will also be scaling with 2〈fluctuation〉=(scale)H;〈(scale)γ〉=1;ζ=H+γ, where the symbol 〈〉 indicates statistical (ensemble) averaging.

Later, in Sect. 2.5, fluctuations as differences (sometimes called “poor man's wavelets”) are replaced by (nearly as simple) Haar fluctuations based on Haar wavelets (see also Appendix B) and in Sect. 3, and Eq. (1) is interpreted stochastically. Finally, in Sect. 4, the notion of scale itself is generalized by introducing a scale function that replaces the usual (Euclidean) distance function (metric). These anisotropic-scale functions are needed to handle scale in 2D or higher spaces, especially with regard to stratification.

In atmospheric regimes where Eq. (1) holds, average fluctuations over durations λΔt are λH times those at duration Δt; i.e. they differ only in their amplitudes, they are qualitatively of the same nature, and they are therefore part of the same dynamical regime. More generally (Eq. 1), appropriately rescaled probabilities of random fluctuations also have scale-invariant exponents (“codimensions”, Sect. 3), so that the entire statistical behaviour is scaling. Scaling therefore allows us to objectively identify the different atmospheric regimes.

Over the Phanerozoic eon (the last 540 Myr), the five scaling regimes are weather, macroweather, climate, macroclimate, and megaclimate (Lovejoy, 2015). Starting at around a millisecond (the dissipation time), this covers a total range of ≈ 1019 in scale (Sect. 2.5; Tuck (2022) argues that the true dissipation scale is much smaller: molecular scales). Scaling therefore gives an unequivocal answer to the question posed in the title: “How long does weather last?”. The answer is the lifetime of planetary structures, typically around 10 d (Sect. 2.6).

If the key statistical characteristics of the atmosphere at any given scale are determined by processes acting over wide ranges of scales – and not by a plethora of narrow-range ones – then we must conclude that the fundamental dynamical processes are in fact dynamical “regimes” – not uninteresting “backgrounds”. While there may also be narrow-range processes, they can only be properly understood in the context of the dynamical regime in which they operate, and in any event, spectral or other analysis shows that they generally contribute only marginally to the overall variability. The first task is therefore to define and understand the dynamical regimes and then – when necessary – the narrow-range processes occurring within them.

Before answering the quite different scaling question “How big is a cloud?”, it is first necessary to discuss a complication: that the scaling is different for every level of activity. It turns out that the wide range over which the variability occurs is only one of its aspects: even at fixed scales, the variability is much more extreme than is commonly believed. Interestingly, the extremeness of the variability at a fixed scale is a consequence of the wide range of scaling itself: it allows the variability to build up scale by scale in a multiplicative cascade manner. As a result, mathematically, the scaling of the average fluctuation (Eq. 2) gives only a partial view of the variability, and we need to consider Eq. (1) in its full stochastic sense. In particular, if the exponent ζ is random, then it is easy to imagine that the variability may be huge.

To graphically see this, it is sufficient to produce a “spike plot” (the right-hand-side columns of Fig. 2, time, and the corresponding spatial plot, Fig. 3). These spike plots are simply the absolute first differences in the values normalized by their overall means (in Fig. 3, the normalization is slightly different, by the standard deviation). In the right-hand-side column of Fig. 2 and the bottom of Fig. 3, we see – with a single but significant exception, macroweather in time (Fig. 2) – that they all have strong spikes signalling sharp transitions. In turbulence jargon, the series are highly “intermittent”.

How strong are the spikes? Using classical (Gaussian) statistics, we may use probability levels to quantify them. For example, Fig. 2 (right) shows solid horizontal lines that indicate the maximum spike that would be expected from a Gaussian process with the given number of spikes. For the 1000 points in each series in Fig. 2, this line thus corresponds to a Gaussian probability p=10-3. In addition, horizontal dashed lines show spikes at levels p=10-6 and p=10-9. Again, with the exception of macroweather, we see that the p=10-6 level is exceeded in every series and that the megaclimate, climate, and weather regimes are particularly intermittent, with spikes exceeding the p=10-9 levels. In Sect. 3 we show how the spikes can be tamed by multifractal theory and the maxima predicted reasonably accurately (Appendix A) by simply characterizing the statistics of the process near the mean (i.e. using their non-extreme behaviour).

The spikes visually underline the fact that variability is not simply a question of the range of scales that are involved: at any given scale, variability can be strong or weak. In addition, events can be highly clustered, with strong ones embedded inside weak ones and even stronger ones inside strong ones in a fractal pattern repeating to smaller and smaller scales. This fractal sparseness itself can itself become more and more accentuated for the more and more extreme events/regions: the series will generally be multifractal.

Scaling is also needed to answer the question “how big is a cloud?” (here “cloud” is taken as a catch-all term meaning an atmospheric structure or eddy). Now the problem is what we mean by “scale”. The series and transects in Figs. 2 and 3 are 1D, so that it is sufficient to define the scale of a fluctuation by the duration (time) or length (space) over which it occurs (actually, time involves causality, so that the sign of Δt is important; see Marsan et al. (1996): we ignore this issue here). However, the state of the atmosphere is mathematically represented by fields in 3D space evolving in time.

Consider Fig. 4, which displays a cloud vertical cross section from the CloudSat radar. In the figure, the gravitationally induced stratification is striking, and since each pixel in the figure has a horizontal resolution of 1 km but a vertical resolution of 250 m, the actual stratification is 4 times stronger than it appears. What is this cloud's scale? If we use the usual Euclidean distance to determine the scale, should we measure it in the horizontal or vertical direction? In this case, is the cloud scale its width (200 km) or its height (only ≈ 10 km)?

If the horizontal–vertical aspect ratio were the same for all clouds, the two choices would be identical to within a constant factor, and the anisotropy would be “trivial”. The trouble is that the aspect ratio itself turns out to be a strong power-law function of (either) horizontal or vertical scale, so that, for any cloud, 3vertical scale=(horizontal scale)Hz. In Sect. 4, we will see that, theoretically, the stratification exponent is Hz=5/9, a value that we confirm empirically on various atmospheric fields, including clouds.

To further appreciate the issue, consider the simulation in Fig. 5 that shows a vertical cross section of a multifractal cloud liquid water density field. The left-hand-side column (top to bottom) shows a series of blow-ups in an isotropic (“self-similar”) cloud. Moving from top to bottom, blow-ups of the central regions by successive factors of 2.9 are displayed. In order for the cross sections to maintain a constant 50 % “cloud cover”, the density threshold distinguishing the cloud (white or grey) from the non-cloud (black) must be systematically adjusted to account for this change in resolution. This systematic readjustment of the threshold is required due to the multifractality, and with this adjustment, we see that the cross sections are self-similar; i.e. they look the same at all scales.

The effect of differential (scale-dependent) stratification is revealed in the right-hand-side column that shows the analogous zoom through an anisotropic multifractal simulation with a stratification exponent Hz=5/9. The low-resolution (top) view of the simulation is highly stratified in the horizontal. Now, the blow-ups reveal progressively more and more roundish structures. Eventually – with the bottom cross section (a blow-up of a total factor of ≈ 5000) – we can start to see vertically oriented structures “dangling” below more roundish ones.

In the isotropic simulations (left-hand side), the only difficulty in defining the size of the cloud is the multifractal problem of deciding, for each resolution, which threshold should be used to distinguish cloud from no cloud. However, in the more realistic anisotropic simulation on the right, there is an additional difficulty in answering the question of “how big is a cloud?” Should we use the horizontal or vertical cloud extent? It turns out (in Sect. 4) that, to ensure that the answer is well defined, we need a new notion of scale itself: generalized scale invariance (GSI).

This review primarily covers scaling research over the last 4 decades, especially multifractals, generalized scale invariance, and their now extensive empirical validations. This work involved theoretical and technical advances, revolutions in computing power, the development of new data analysis techniques, and the systematic exploitation of mushrooming quantities of geodata. The basic work has already been the subject of several reviews (Lovejoy and Schertzer, 2010c, 2012b), but especially a monograph (Lovejoy and Schertzer, 2013). Although a book covering some of the subsequent developments was published more recently (Lovejoy, 2019), it was nontechnical, so that this new review brings its first four chapters up to date and includes some of the theory and mathematics that were deliberately omitted so as to render the material more accessible. The last three chapters of Lovejoy (2019) focused on developments in the climate-scale (and lower-frequency) regimes that will be reviewed elsewhere. The present review is thus limited to the (turbulent) weather regime and its transition to macroweather at scales of ≈ 10 d.

In order to maintain focus on the fundamental physical scaling issues and implications, the mathematical formalism is introduced progressively – as needed – so that it will not be an obstacle to accessing the core scientific ideas.

This review also brings to the fore several advances that have occurred in the last 10 years, especially Haar fluctuation analysis (developed in detail in Appendix B), and a more comprehensive criticism of scalebound approaches made possible by combining Haar analysis with new high-resolution instrumental and paleodata sources (Lovejoy, 2015). On the other hand, it leaves out an emerging body of work on macroweather modelling based on the fractional energy balance equation for both prediction and climate projections (Del Rio Amador and Lovejoy, 2019, 2021a, b; Procyk et al., 2022) as well as their implications for the future of climate modelling (Lovejoy, 2022a).

The presentation is divided into three main sections. Keeping the technical and mathematical aspects to a minimum, Sect. 2 focuses on a foundational atmospheric science issue: what is the appropriate conceptual and theoretical framework for handling the atmosphere's variability over huge ranges of scales? It discusses how the classical scalebound approach is increasingly divorced from real-world data and numerical models. Scaling is discussed but with an emphasis on its role as a symmetry principle. It introduces fluctuation analysis based on Haar fluctuations that allow for a clear quantitative empirical overview of the variability over 17 orders of magnitude in time. Scaling is essential for defining the basic dynamical regimes, underlining the fact that between the weather and the climate sits a new macroweather regime.

Section 3 discusses the general scaling process: multifractals. Multifractals naturally explain and quantify the ubiquitous intermittency of atmospheric processes. The section also discusses an underappreciated consequence, the divergence of high-order statistical moments – equivalently power-law probability tails – and relates this to “tipping points” and “black swans”. The now large body of evidence for the divergence of moments is discussed and special attention paid to the velocity field where the divergence of moments was first empirically shown 40 years ago in the atmosphere, then in wind tunnels, and most recently in large direct numerical simulations of hydrodynamic turbulence.

In Sect. 4 a totally different aspect of scaling is covered: anisotropic scaling, notably scaling stratification. The section outlines the formalism of GSI needed to define the notion of scale in anisotropic scaling systems. By considering buoyancy-driven turbulence, the 23/9D model is derived: it is a consequence of Kolmogorov scaling in the horizontal and Bolgiano–Obukhov scaling in the vertical. This model is “in between” flat 2D isotropic turbulence and “voluminous” isotropic 3D turbulence – it is strongly supported by now burgeoning quantities of atmospheric data. It not only allows us to answer the question “how big is a cloud?”, but also to understand and model differentially rotating structures needed to quantify cloud morphologies.

In the introduction, the conventional paradigm based on (typically deterministic) narrow-range explanations and mechanisms was contrasted with the alternative scaling paradigm that builds statistical models expressing the collective behaviour of high numbers of degrees of freedom and that provides explanations over huge ranges of scales.

Let us consider the narrow-range paradigm in more detail. It follows in the steps of van Leuwenhoek, who – peering through an early microscope – was famously said to have discovered a “new world in a drop of water” – microorganisms (circa 1675). Over time, this evolved into a “powers of ten” view (Boeke, 1957) in which every factor of 10 or so of zooming revealed qualitatively different processes and morphologies. Mandelbrot (1981) termed this view “scalebound” (written as one word), which is a useful shorthand for the idea that every factor of 10 or so involves something qualitatively new: a new world, new mechanisms, new morphologies, etc.

The first weather maps were at extremely low spatial resolution, so that only a rather narrow range of phenomena could be discerned. Unsurprisingly, the corresponding atmospheric explanations and theories were scalebound. Later, in the 1960s and 1970s under the impact of new data, especially in the mesoscale, the ambient scalebound paradigm was quantitatively made explicit in space–time Stommel diagrams (discussed at length in Sect. 2.6) in which various conventional mechanisms, morphologies, and phenomena were represented by the space scales and timescales over which they operate. For a recent inventory of scalebound mechanisms from seconds to decades, see Williams et al. (2017).

While Stommel diagrams reflected scalebound thinking, the goal was the modest one of organizing and classifying existing empirical phenomenology, and it did this in the light of the prevailing mechanistic analytic dynamical meteorology. It was Mitchell (1976), writing at the dawn of the paleoclimate revolution, who, more than anyone, ambitiously elevated the scalebound paradigm into a general framework spanning a range of scales from (at least) an hour to the age of the planet (a factor of tens of billions, upper left, Fig. 6). Mitchell's data were limited, and he admitted that his spectrum was only an “educated guess”. He imagined when the data would become available that their spectra would consist of an essentially uninteresting white-noise “background” interspersed with interesting quasi-periodic signals representing the important physical processes. Ironically, Mitchell's scalebound paradigm was proposed at the same time as the first GCMs (Manabe and Wetherald, 1975). Fortunately, the GCMs are scaling, inheriting the symmetry from the governing equations (Schertzer et al., 2012); see Chap. 2 of Lovejoy and Schertzer (2013).

Mitchell's schematic (upper-left panel) was so successful that, more than 4 decades later, his original figure is still faithfully reproduced (e.g. Dijkstra, 2013) or updated by very similar scalebound schematics with only minor updates. Even though the relevant geodata have since mushroomed, the updates notably have less quantification and weaker empirical support than the original. The 45-year evolution of the scalebound paradigm is shown in the other panels of Fig. 6. Moving to the right of the figure, there is a 25-year update, modestly termed an “artist's rendering” (Ghil, 2002). This figure differs from the original in the excision of the lowest frequencies and by the inclusion of several new multimillennial-scale “bumps”. In addition, whereas Mitchell's spectrum was quantitative, the artist's rendering retreated to using “arbitrary units”, making it more difficult to verify empirically. Nearly 20 years later, the same author approvingly reprinted it in a review (Ghil and Lucarini, 2020).

As time passed, the retreat from quantitative empirical assessments continued, so that the scalebound paradigm has become more and more abstract. The bottom left of Fig. 6 shows an update downloaded from the NOAA paleoclimate data site in 2015 claiming to be a “mental model”. Harkening back to Boeke (1957), the site went on to state that the figure is “intended … to provide a general “powers of ten” overview of climate variability”. Here, the vertical axis is simply “variability”, and the uninteresting background – presumably a white noise – is shown as a perfectly flat line.

At about the same time, Lovejoy (2015) pointed out that Mitchell's original figure was in error by an astronomical factor (Sect. 2.4), so that – in an effort to partially address the criticism – an update in the form of a “conceptual landscape” was proposed (Fig. 6, bottom right, von der Heydt et al., 2021). Rather than plotting the log of the spectrum E(ω) as a function of the log frequency ω, the “landscape's” main innovation was the use of the unitless “relative variance” ωE(ω) plotted linearly, indicated as a function of log ω (bottom right of the figure). Such plots have the property that areas under the curves are equal to the total variance contributed over the corresponding frequency range. Before returning to these schematics, let us discuss the scaling alternative.

The basic signature of scaling is a power-law relation of a statistical characteristic of a system as a function of space scale and/or timescale. In the empirical test of the Richardson 4/3 law (Fig. 7, left panel), it is the turbulent viscosity as a function of horizontal scale that is a power law. On the right-hand side, it is rather the complicated (fractal) perimeters of clouds and rain zones that are power-law functions of the corresponding areas. The Fig. 7 analysis methods lack generality, so let us instead consider spectra (Fourier space) and, then, fluctuations (real space, Sect. 2.5, Appendix B).

Following Mitchell, we may consider variability in the spectral domain: for example, the power spectrum of the temperature T(t) is 5Eω∝T̃ω2, where T̃ω is its Fourier transform and ω is the frequency. A scaling process E(ω) has the same form if we consider it at a timescale λ times smaller or, equivalently, at a frequency λ times larger: 6Eλω=λ-βEω, where β is the “spectral exponent”. The solution of this functional equation is a power law: 7Eω∝ω-β. Therefore, a log–log plot of the spectrum as a function of frequency will be a straight line; see Fig. 12 for early quantitative applications to climate series.

Alternatively, we can consider scaling in real space. Due to “Tauberian theorems” (e.g. Feller, 1971), power laws in real space are transformed into power laws in Fourier space (and vice versa). This result holds whenever the scaling range is wide enough – i.e. even if there are high- and/or low-frequency cutoffs (needed if only for the convergence of the transforms). If we consider fluctuations ΔT over time interval Δt, then if the system is scaling, we can introduce the (“generalized”, qth-order) structure function as 8SqΔt=ΔTΔtq∝Δtξq, where the “〈〉” sign indicates statistical (ensemble) averaging (assuming statistical stationarity, there is no t dependence). Once again, classical fluctuations are defined simply as differences, i.e. ΔT(Δt)=T(t)-T(t-Δt), although more general fluctuations are needed as discussed in Sect. 2.5. For stationary scaling processes, the Wiener–Khintchin theorem implies a simple relation between real space and Fourier scaling exponents: 9β=1+ξ2 (the “2” is because the variance is a second-order moment). If, in addition, the system is “quasi-Gaussian”, then S2 gives a full statistical characterization of the process. Therefore, often only the second-order structure function S2(Δt) is considered (e.g. Fig. 12, top). However, as discussed above, geo-processes are typically strongly intermittent and rarely quasi-Gaussian, and the full exponent function ξ(q) is needed (Sect. 3.1). In the next section, we discuss this figure in more detail, including its physical implications. For the moment, simply note the various linear (scaling) regimes in the log–log plots.

In spite of its growing disconnect with modern data, Mitchell's figure and its scalebound updates continue to be influential. However, within 15 years of Mitchell's famous paper, two scaling composites, over the ranges 1 h to 105 years and 103 to 108 years, already showed huge discrepancies (Lovejoy and Schertzer, 1986b; Fig. 12, top panel; Shackleton and Imbrie, 1990, bottom left; see also Pelletier, 1998, bottom right, and Huybers and Curry, 2006). Returning to Mitchell's original figure, Lovejoy (2015) superposed the spectra of several modern instrumental and paleo series; the differences are literally astronomical (Fig. 13). While over the range 1 h to 109 years, Mitchell's background varies by a factor ≈ 150, the spectra from real data imply that the true range is a factor greater than a quadrillion (1015).

Returning to the artist's rendering, Fig. 14 shows that, when compared to the data, it fares no better than Mitchell's educated guess. The next update – the NOAA's mental model – only specified that its vertical axis be proportional to “variability”. If we interpret variability as the root-mean-square (rms) fluctuation at a given scale and the flat “background” between the bumps as white noise, then we obtain the comparison in Fig. 15. Although the exact definition of these fluctuations is discussed in Sect. 2.5, they give a directly physically meaningful quantification of the variability at a given timescale. In Fig. 15, we see that the mental model predicts that successive average Earth temperatures of 1 million years would differ by only tens of micro-Kelvin. A closely similar conclusion would hold if we converted Mitchell's spectrum into rms real-space fluctuations.

The most recent scalebound update – the “conceptual landscape” – is compared with modern data in Fig. 16. Although the various scaling regimes proposed in Lovejoy (2013) (updated in Fig. 18 and discussed below) are discreetly indicated in the background, in many instances, there is no obvious relation between the regimes and the landscape. In particular, the word “macroweather” appears without any obvious connection to the figure, but even the landscape's highlighted scalebound features are not very close to the empirical curve (red). Although the vertical axis is only “relative”, this quantitative empirical comparison was made by exploiting the equal-area property mentioned above. The overlaid solid red curve was estimated by converting the disjoint spectral power laws shown in the updated Mitchell graph (Fig. 8). In addition, there is also an attempt to indicate the amplitudes of the narrow spectral spikes (the green spikes in Fig. 13) at diurnal, annual, and – for the epoch 2.5–0.8 Myr – obliquity spectral peaks at (41 kyr)-1. In conclusion, the conceptual landscape bears little relation to the real world.

The scalebound framework for atmospheric dynamics emphasized the importance of numerous processes occurring at well-defined timescales, the quasi-periodic “foreground” processes illustrated as bumps – the signals – on Mitchell's nearly flat background. The point here is not that these processes and mechanisms are wrong or non-existent: it is rather that they only explain a small fraction of the overall variability, and this implies that they cannot be understood without putting them in the context of their dynamical (scaling) regime. This was also demonstrated quantitatively and explicitly over at least a significant part of the climate range by Wunsch (2003).

One of the lessons to be drawn from the educated guesses, artists' renderings, and conceptual landscapes is that, although spectra can be calculated for any signal, the interpretations are often not obvious. The problem is that we have no intuition about the physical meaning of the units – K2 s, K2 yr, or even K2 Myr – so that often (as here) the units used in spectral plots are not even given. It then becomes impossible to take data from disparate sources and at different timescales to make the spectral composites needed to make a meaningful check of the scalebound paradigm.

The advantage of fluctuations such as in Fig. 12 (top) is that the numbers – e.g. the rms temperature fluctuations at some scale – have a straightforward physical interpretation. However, the differences used to define fluctuations (see Fig. 17, top) have a non-obvious problem: on average, differences cannot decrease with increasing time intervals (in Appendix B, this problem is discussed more precisely in the Fourier domain). This is true for any series that has correlations that decrease with Δt (as physically relevant series always do). A consequence is that whenever the value of ξ(2) is negative – implying that the mean fluctuations decrease with scale – the difference fluctuations will at best give a constant result (the flat parts of Fig. 12, top).

However, do regions of negative ξ(2) exist? One way to investigate this is to try to infer the exponent ξ(2) from the spectrum that does not suffer from an analogous restriction: its exponent β can take any value. In this case we can use the formula β=1+ξ(2) (Eq. 9). The latter implies that negative ξ(2) corresponds to β<1, and a check on the spectrum in Fig. 7 indicates that several regions (notably the macroweather regime) are indeed flat enough (β<1) to imply negative ξ(2). How do we fix the problem and estimate the correct ξ(2) when it is negative?

It took a surprisingly long time to clarify this issue. To start with, in classical turbulence, ξ(2)>0 (e.g. the Kolmogorov law), there was no motivation to look further than differences. Mathematically, the main advance came in the 1980s from wavelets. It turns out that, technically, fluctuations defined as differences are indeed wavelets, but mathematicians mock them, calling them the poor man's wavelet, and they generally promote more sophisticated wavelets (see Appendix B2): the simplicity of the physical interpretation is not their concern. This was the situation in the 1990s, when scaling started to be systematically applied to geophysical time series involving negative ξ(2) (i.e. to any macroweather series, although at the time this was not clear). A practical solution adopted by many was to use the detrended fluctuation analysis (DFA) method (Peng et al., 1994). One reason the DFA method is popular is that the raw DFA fluctuations are not too noisy. However, this is in fact an artefact since they are fluctuations of the running sum of the process, not of the process itself. When DFA fluctuations of the process are used, they are just as variable as the Haar fluctuations (Lovejoy and Schertzer, 2012a; Hébert et al., 2021a). Unfortunately, DFA fluctuations are difficult to interpret, so that typically only exponents are extracted: the important information contained in the fluctuation amplitudes is not exploited (see Appendix B).

New clarity was achieved with the help of the (first) Haar wavelet (Haar, 1910). There were two reasons for this: the simplicity of its definition and calculation and the simplicity of its interpretation (Lovejoy and Schertzer, 2012a). To determine the Haar fluctuation over a time interval Δt, one simply takes the average of the first half of the interval and, from this, subtracts the average of the second half (Fig. 17, bottom; see Appendix B2 for more details). As for the interpretation, when H is positive, then it is (nearly) the same as a difference, whereas whenever H is negative, the fluctuation can be interpreted as an “anomaly” (in this context, an anomaly is simply the average over a segment length Δt of the series with its long-term average removed, Appendix B2). In both cases we also recover the correct value of the exponent H. Although the Haar fluctuation is only useful for H in the range -1 to 1, this turns out to cover most of the series that are encountered in geoscience.

Figure 18 shows a modern composite using the rms Haar fluctuation, spanning a range of scales of ≈ 1017 (compare this with Fig. 12, top, for the earlier version using fluctuations as differences). The same five regimes as in Fig. 13 are shown, but now the typical variations in temperature over various timescales are very clear.

Also shown in Fig. 16 are reference lines indicating the typical scale dependencies. These correspond to typical temperature fluctuations ΔT∝Δtξ(2)/2≈Δtξ(1), where ξ(1)=H is the “fluctuation exponent” (the exponent of the mean absolute fluctuation, the relationship ξ(2)=2H, is valid if we ignore intermittency: it is the quasi-Gaussian relationship still often invoked; see Eq. 15). In the figure, we see that the character of the regimes alternates between regimes that grow (H>0) and ones that decrease (H<0) with timescale. The sign of H has a fundamental significance; to see this, we can return to typical series over the various regimes in Fig. 2 (left-hand-side column). In terms of their visual appearances, the H>0 regimes have signals that seem to “wander” or “drift”, whereas for H<0 regimes, fluctuations tend to cancel. In the former, waiting longer and longer typically leads to larger changes in temperature, whereas in the latter, longer and longer temporal scales lead to convergence to well-defined values.

With the help of the figure, we can now understand the problem with the usual definition of climate as “long-term” weather. As we average from 10 d to longer durations, temperature fluctuations do indeed tend to diminish – as expected if they converged to the climate. Consider for example the thick solid line in Fig. 18 (corresponding to data at 75∘ N), which shows that, at about 10 d, the temperature fluctuations are ≈± 3 K (S2(Δt)1/2≈6 K), diminishing at 20 years to ≈±0.3 K. Since H<0, the Haar fluctuations are nearly equivalent to the anomalies, i.e. to averages of the series with the long-time mean removed. Over this range, increasing the scale leads to smaller and smaller fluctuations about the point of the apparent point of convergence: the average climate temperature. Figure 18 also shows the longer scales deduced purely from paleodata (isotope ratios from either ice or ocean cores).

The interpretation of the apparent point of convergence as the climate state is supported by the analysis of global data compared with GCMs in “control runs” (i.e. with fixed external conditions, Fig. 19). When averaged over long enough times, the control runs do indeed converge, although the convergence is “ultra slow” (at a rate characterized by the exponent H≈-0.15 for the GCMs). Extrapolating from the figure shows that, even after averaging over 1 million simulated years, the GCMs would still typically be only within ±0.01 K of their respective climates.

Returning to Fig. 18, however, we see that, beyond a critical timescale τc, the convergence is reversed and fluctuations tend rather to increase with timescale. In the Anthropocene (roughly since 1900), the ≈ 15-year timescale where fluctuations stop decreasing and begin increasing with scale is roughly the time that it has taken for anthropogenic warming (over the last decades) to become comparable to the natural internal variability (about ±0.2 K for these globally averaged temperatures). However, for the preindustrial epoch (see the “S”-shaped paleotemperature curve from the EPICA (European Project for Ice Coring in Antarctica) ice core, Fig. 18), the transition time is closer to 300 years. The origin of this larger τc value is not clear; it is a focus of the PAGES–CVAS (Past Climate – Climate Variability Across Scales) working group (Lovejoy, 2017b).

Regarding the last 100 kyr, the key point about Fig. 18 is that we have three regimes – not two. Since the intermediate regime is well reproduced by control runs (Fig. 19), it is termed “macroweather”: it is essentially averaged weather.

If the macroweather regime is characterized by slow convergence of averages with scale, it is logical to define a climate state as an average over durations that are long enough so that the maximum convergence has occurred – i.e. over periods Δt>τc. In the Anthropocene, this gives some objective justification for the official World Meteorological Organization's otherwise arbitrary climate-averaging period of 30 years. Similarly, the roughly 10 d to 1-month weather–macroweather transition at τw gives some objective justification for the common practice of using monthly average anomalies: these define analogous macroweather states. The climate regime is therefore the regime beyond τc where the climate state itself starts to vary. In addition to the analyses presented here, there are numerous papers claiming evidence for power-law climate regimes: Lovejoy and Schertzer (1986b), Shackleton and Imbrie (1990), Schmitt et al. (1995), Ditlevsen et al. (1996), Pelletier (1998), Ashkenazy et al. (2003), Wunsch (2003), and Huybers and Curry (2006); for a more comprehensive review, see the discussion and Table 11.4 in Lovejoy and Schertzer (2013).

Again from Fig. 18, we see that the climate state itself starts to vary in a roughly scaling way up until Milankovitch timescales (at about 50 kyr, half the period of the main 100 kyr eccentricity frequency) over which fluctuations are typically of the order ±2 to ±4 K: the glacial–interglacial “window” (Lovejoy and Schertzer 1986) over typical variability that is quite clear in the figure (the most recent estimate is a total range of 6 K or ±3 K; Tierney et al., 2020). At even larger scales there is evidence (from ice core and benthic paleodata, notably updated with a much improved megaclimate series by Grossman and Joachimski, 2022, bold curve on the right) that there is a narrow macroclimate regime and then a wide-range megaclimate regime, but these are outside our present scope (see Lovejoy, 2015, for more discussion).

In the previous section, we discussed the space–time relations of structures of size l that maintained their identities over a lifetime τ – these space–time diagrams are Lagrangian (albeit deduced from Eulerian data and reanalyses). Unsurprisingly, it turns out to be much simpler to empirically check the corresponding fixed-frame Eulerian relations; we consider this in this subsection.

The key difference between the Eulerian and Lagrangian statistics is that the former involves an overall mean advection velocity V. When studying laboratory turbulence, Taylor (1938) proposed that the turbulence is “frozen”, such that the pattern of turbulence blows past the measuring point sufficiently quickly so that it does not have time to evolve. Frozen turbulence requires the existence of a scale separation between small and large scales, so that the large (nearly frozen) structures really do blow the small ones' structures without much evolution. When this approximation is valid, the spatial statistics may be obtained from time series by the deterministic transformation VΔt→Δx, where V is a constant.

In Taylor's laboratory turbulence, V is determined by the fan and by the wind-tunnel geometry, and within limits, the approximation is valid. However, although the transformation has been frequently used to interpret meteorological data, due to the horizontal scaling, there is no scale separation between large and small scales, so that atmospheric turbulence is not frozen. However, we are only interested in the statistical relations between time and space, and if the system is scaling, then advection can be taken into account using the Galilean transformation r→r-Vt, t→t. Since V is now a random (turbulent) velocity, its effects must then be averaged; this is discussed in Sect. 4.1.5. The full theory of space–time scaling requires the consideration of anisotropic space–time and was developed in Schertzer et al. (1997a), Lovejoy et al. (2008b), and Pinel et al. (2014) and reviewed in Lovejoy and Schertzer (2013).

In order to test the space–time scaling on real-world data, the best sources are remotely sensed data such as the space–time lidar data discussed in Radkevitch et al. (2008) or the global-scale data from geostationary satellites in the infrared (IR), whose spectra are shown in Fig. 25 (Pinel et al., 2014). The figure uses 1440 consecutive hourly images at 5 km resolution over the region 30∘ N to 30∘ S and 80 to 200∘ E. A full analysis based on the 3D (x,y,t) space data is given in Pinel et al. (2014); the figure shows only 1D subspaces (EW, NS, and time).

There are two remarkable aspects of the figure. The first is that, in spite of an apparently slight curvature (normally a symptom of deviations from perfect scaling), it is in reality largely a “finite-size effect” on otherwise excellent scaling. This can be seen by comparison with the black curve that shows the consequences of the averaging over the (roughly rectangular) geometry of the observing region combined with the “trivial” anisotropy of the spectrum (implied by the matrix C in Eq. 11). This is clearly visible in the various subspaces (x,y),(x,t),(y,t) analysed in Pinel et al. (2014), where the full theory and analysis are given. Comparing the spectra to the theoretical black curve, we see that there are only small deviations from scaling, and this holds over the range in space from 60 km to ≈ 5000 km (space) and from ≈ 2 h to ≈ 7 d.

The second remarkable aspect of Fig. 25 is the near-perfect superposition of the 1D spectra Eω,Ekx,Eky over the same range (obtained by successively integrating out the conjugate variables (e.g. Eω=∫Pkx,ky,ωdkxdky). Writing K¯=kx,ky,ω, the overall result is that the full 3D, horizontal space–time spectrum PK¯ respects the symmetry: Pλ-1K¯=λsPK¯ (see Sect. 4.1.5 for the theory). The full relationship between the Lagrangian and Eulerian statistics is derived in full in Pinel et al. (2014) and Chap. 8 of Lovejoy and Schertzer (2013). By averaging over the turbulent advection, the final theoretical result is 11PK∝C-1K¯-s,C=10-μx01-μy001-μx2+a2μy2, where the nondimensional μx and μy are related to the average zonal and meridional advection velocities and their variances, and a is the average “trivial” zonal to meridional aspect ratio. The vertical bars indicate the usual vector norm. Empirically, s=3.4, a≈1.6. Equation (11) implies that the 1D spectra in Fig. 25 respect Eω∝Ekx∝Eky as shown.

Given the space–time scaling, one can use the real space statistics to define Eulerian space–time diagrams. Using the same data, this is shown in Fig. 26, where we see that the relationship is nearly linear in a linear–linear plot (i.e. with a constant velocity) up to about 10 d, corresponding to near-planetary scales as indicated in the figure. Note some minor differences between the EW and NS directions.

Up until now, we have discussed scaling at a fairly general level as an invariance under scale changes, contrasting it with scaleboundedness and emphasizing its indispensable role in understanding the atmosphere, the ocean, and, more generally, the geosphere. There are two basic elements that must be considered: (a) the definition of the notion of scale and scale change and (b) the aspect of the system or process that is invariant under the corresponding change (the invariant).

We have seen that, in general terms, a system is scaling if there exists a power-law relationship (possibly deterministic, but usually statistical) between fast and slow (time) or small and large (space, or both, space–time). If the system is a geometric set of points – such as the set of meteorological measuring stations (Lovejoy et al., 1986), then the set is a fractal set and the density of its points is scaling – it is a power law whose exponent is its fractal codimension. Geophysically interesting systems are typically not sets of points but rather scaling fields such as the temperature T(r¯,t) at the space–time point (r¯,t). Although we will generally use the term scaling “fields”, for multifractals, the more precise notion is of singular densities of multifractal measures.

In such a system, some aspect – most often a suitably defined fluctuation ΔT – has statistics whose small and large scales are related by a scale-changing operation that involves only the scale ratios: the system has no characteristic size. In one dimension the scaling of temporal series or spatial transects can be expressed as 12ΔTΔt=dφΔtΔtH, where Δt is the time interval (scale) over which the fluctuations are defined (for transects, replace Δt by the spatial interval Δx), H is the fluctuation exponent, and φΔt is a random variable at resolution ϕΔt. Dimensionally, the units of φ determine H, and physically φ is a turbulent flux that drives the process (or a power of a flux; see Eq. 2). In turbulence theory, (in the relevant “direct” cascades) the fluxes are in Fourier space from small to large wavenumbers. The subscript Δt indicates that φ generally depends on the scale (resolution). The equality sign =d is in the sense of random variables; this means that the random fluctuation ΔTΔt has the same probability distribution as the random variable φΔtΔtH. We suppressed the t dependence since we consider the case where the statistics are independent of time or space – statistical stationarity or statistical homogeneity (see Appendix B4). Physically, this is the assumption that the underlying physical processes are the same at all times and everywhere in space. Equation (12) is the more formal expression of Eq. (1) or of the classical turbulence laws Eq. (4). For example, if we consider space rather than time, the Kolmogorov law has H=1/3 with φ=ε1/3.

The simplest case is where the fluctuations in a temporal series T(t) follow Eq. (12) but with φ a random variable independent of resolution: φΔt=dφ, with φ the same as in Eq. (12): a Gaussian random variable. This is the classical special case of non-intermittent, quasi-Gaussian turbulence. Examples are fractional Gaussian noise (fGn, -1<H<0) and fractional Brownian motion (fBm, 0<H<1), with special cases Gaussian white noise (H=-1/2) and standard Brownian motion (H=1/2).

Equation (12) relates the probabilities of small and large fluctuations; it is usually easier to deal with the deterministic equalities that follow by taking qth-order statistical moments of Eq. (12) and then averaging over a statistical ensemble: 13ΔTΔtq=φΔtqΔtqH, where “〈〉” indicates statistical (ensemble) averaging.

Equation (13) is the general case where the resolution Δt is important for the statistics of φ; indeed, quite generally, φΔt is a random function (1D) or field averaged at resolution Δt. If φΔt is scaling, its statistics will follow: 14φλq=λKq;λ=τ/Δt≥1, where τ is the largest, “outer” scale of the scaling regime satisfied by the equation, and the Δt resolution is λ times smaller. K(q) is a convex (K′′ (q)≥0) exponent function; since the mean fluctuation is independent of the scale (φλ=λK1=const), we have K(1)=0 (see Eq. 2). This is the generic statistical behaviour of cascade processes: it displays general “multiscaling” behaviour – a different scaling exponent K(q) for each statistical moment q. Since large q moments are dominated by large, extreme values and small q moments by common, typical values, K(q)≠0 implies that fluctuations of various amplitudes change scale with different exponents – multiscaling – and each realization of such a process is a multifractal. In general, K(q)≠0 is associated with intermittency, a topic we treat in more detail in Sect. 3.2 and 3.3.

Combining Eqs. (13) and (14), we obtain 15SqΔt=ΔTΔtq=φΔtqΔtqH∝Δtξq;ξ(q)=qH-Kq, where Sq is the qth-order (“generalized”) structure function and ξ(q) is its exponent defined in Eq. (15). Since K(q) is convex, Eq. (15) shows that, in general, ξ(q) will be concave (ξ′′≤0). The structure functions are scaling since the small and large scales are related by a power law: 16SqλΔt=λξqSqΔt. As discussed in Sect. 2.3, the spectra are power laws Eω≈ω-β with the exponents related as β=1+ξ2=1+2H-K2 (Eq. 9).

In the case of “simple scaling” where φ has no scale dependence (e.g. fGn, fBm), we find φΔtq=Bq, where Bq is a constant independent of scale Δt, and hence we have K(q)=0 and SqΔt∝ΔtqH, so that 17ξ(q)=qH; i.e. ξ(q) is a linear function of q. Simple scaling is therefore sometimes called “linear scaling”, and it respects the simpler relation β=1+2H. Linear scaling arises from scaling linear transformations of noises; the general linear scaling transformation is a power-law filter (multiplication of the Fourier transform by ω-H) or, equivalently, fractional integrals (H>0) or fractional derivatives (H<0). Appropriate fractional integrals of Gaussian white noises (of order H+1/2) yield fBm (1>H>0) and fractional derivatives yield fGn (-1<H<0). The analogous Lévy motions and noises are obtained by the filtering of independent Lévy noises (in this case, ξ(q) is only linear for q<α<2; for q>α, the moments diverge, so that both ξ(q) and Sq→∞).

The more general “nonlinear scaling” case, where K(q) is nonzero and convex, is associated with fractional integrals or derivatives of scaling multiplicative (not additive) random processes (cascades, multifractals). These pure multiplicative processes (φ in Eq. 12) have H=0, and they are sometimes called “conservative multifractals” since their exponent of the mean is ξ(1)=H=0 (i.e. the mean is independent of scale). The exponent H in Eq. (15) still refers to the order of fractional integration (H>0) or differentiation (H<0) of φ that adds the extra term qH in the structure function exponent: ξ(q)=qH-K(q). Note that while the symbol H is in honour of Edwin Hurst, the interpretation of H as the “Hurst exponent” is only valid for Gaussian processes: more generally, it is a fluctuation exponent describing the behaviour of the mean (q=1) fluctuations.

Note that, in the literature, the notation “H” is not used consistently. It was introduced in honour of Edwin Hurst, a pioneer of long-memory processes sometimes called “Hurst phenomena” (Hurst, 1951). Hurst introduced the rescaled range exponent, notably in the study of Nile River streamflow records. To explain Hurst's findings, Mandelbrot and Van Ness (1968) developed Gaussian scaling models (fGn, fBm) and introduced the symbol H. At first this represented a Hurst exponent, and they showed that, for fGn processes, it was equal to Hurst's exponent. However, by the time of the landmark Fractal Geometry of Nature (Mandelbrot, 1982), the usage was shifting from a scaling exponent to a model specification: the “Hurst parameter”. In this new usage, the symbol H was used for both fGn and its integral fBm, even though the fBm scaling exponent is larger by 1. To avoid confusion, we will call it HM. Subsequently, a mathematical literature has developed using HM with 0<HM<1 to parameterize both the process (fractional Gaussian motion – fBm) and its increments (fGn). However, also in the early 1980s (Grassberger and Procaccia, 1983; Hentschel and Procaccia, 1983; Schertzer and Lovejoy, 1983b), much more general scaling processes with an infinite hierarchy of exponents – multifractals – were discovered, clearly showing that a single exponent was not enough. Schertzer and Lovejoy (1987) showed that it was nevertheless possible to keep H in the role of a mean fluctuation exponent (originally termed a cascade “conservation exponent”). This is the sense of the H exponent discussed here. As described above, using appropriate definitions of fluctuations (i.e. by the use of an appropriate wavelet), H can take on any real value. When the definition is applied to fBm, it yields the standard fBm value H=HM, yet when applied to fGn, it yields H=HM-1.

We could mention that, here and in Sect. 3.3, where we discuss the corresponding multiscaling probability distributions, we use the c(γ), K(q) codimension multifractal formalism that is appropriate for stochastic multifractals (Schertzer and Lovejoy, 1987). Another commonly used multifractal formalism is the α, f(α), τ(q) dimension formalism of Halsey et al. (1986) that was developed for deterministic chaos applications. The relationship between the two formalisms is fα=d-cγ, where d is the dimension of the space in which the multifractal process is defined and α=d-γ is the singularity of the measure of the multifractal – not its density, whose singularity is γ. For the moment exponent functions, we have τq=dq-1-Kq.

The α, f(α), and τ(q) “dimension” formalism was developed to characterize deterministic chaos in low-dimensional (i.e. small d) spaces. Here we are interested in stochastic multifractal processes that are defined on probability spaces with d=∞. Therefore, a codimension formalism that is independent of d is required. Another difference between the formalisms is that the singularity of the multifractal measure α is assumed to be defined at a mathematical point (it is a “Holder exponent”), whereas in the codimension formalism, γ is the singularity of the density of the multifractal measure, and only a looser convergence in the neighbourhood of a point is assumed. This lack of pointwise convergence of the singularities is a general feature of (stochastic) cascade processes and hence the codimension formalism is more relevant in the atmosphere.

Using spike plots, we can simply demonstrate the unique character of the macroweather regime: low intermittency in time but high intermittency in space. We introduced the c(γ) function that for each spike level λγ characterizes the probability (fraction) of a transect or series (or more generally space) whose spikes exceed the threshold. In this section we discuss a particularly simple way to analyse the intermittency.

Consider the data shown in Fig. 31 (macroweather time series and spatial transects, top and bottom respectively). Figure 32 compares the rms fluctuations (with exponent ξ(2)/2) and the mean fluctuations (with exponent H=ξ(1)) from macroweather temperature time series (bottom) and for the spatial transects (top). When the system is Gaussian (so that K(q)=0), we obtain ξ(2)/2=ξ(1), so that moments of the mean and rms fluctuations are in a constant ratio. In this case, log⁡<ΔT(Δt)> is parallel to log 〈ΔT(Δt)2〉1/2. Figure 32 (bottom) shows that to a good approximation this is indeed true of the non-spiky temporal series (Fig. 31, top). However, the corresponding statistics of the spatial transect (the top lines in Fig. 31) tend to converge at large Δx corresponding to the highly spiky transect (Fig. 32, bottom). To a first approximation, it turns out that ξ(2)/2-ξ(1)≈K′(1)=C1, which characterizes the intermittency near the mean. However, there is a slightly better way to estimate C1, using the intermittency function (see Fig. 33 and caption) whose theoretical slope (for ensemble-averaged statistics) is exactly K′(1)=C1. As a point of comparison, we could note that fully developed turbulence in the weather regime typically has C1≈0.09. The temporal macroweather intermittency (C1≈0.01) is indeed small, whereas the spatial intermittency is large (C1≈0.12).

For many applications, the exceptional smallness of macroweather intermittency makes the “monoscaling” approximation (i.e. ξ(q)≈Hq) acceptably accurate, so that macroweather processes are relatively easy to statistically characterize. In this case, the fluctuation exponent H, the spectral exponent β, and the DFA exponent a (Appendix A5) are equivalent and sufficient (the general relations are H=β-1+K2/2=a-1+K2/2, and here, with no intermittency, K(2)=0, and these simplify to H=β-1/2=a-1). For examples of macroweather scaling, see Tessier et al. (1996), Pandey et al. (1998), Koscielny-Bunde et al. (1998), Bunde et al. (2004), Eichner et al. (2003), Blender et al. (2006), Huybers and Curry (2006), Rybski et al. (2006), Lennartz and Bunde (2009), Lanfredi et al. (2009), Fraedrich et al. (2009), Franzke (2010, 2012), Varotsos et al. (2013, 2009), and de Lima and Lovejoy (2015).

In the preceding sections, we gave evidence that diverse atmospheric fields are scaling up to planetary scales. In addition, we argued that they generally were multifractal, with each statistical moment q having a different exponent (K(q), Eq. 14). In Sect. 3.2, we saw that this nonlinear, convex part of the structure function exponent ξ(q) arises due to variability building up scale by scale from a large external scale L to smaller scales Δx (ratio λ=L/Δx or, in time, λ=τ/Δt). By analysing the statistics of the fluxes φλq, this gives us the possibility of directly determining the effective outer scale of the process, i.e. the scale at which the variability starts to grow. As a bonus, our method is based on isolating the flux: in the exponent, it yields K(q) rather than ξ(q) (fluctuations). By effectively removing the qH term, i.e. ξ(q)-K(q), it is only sensitive to K(q), which for small q is often a small correction to ξ(q).

Before proceeding to empirical analyses of the fluxes, a few comments are required. The flux in Eq. (14) is assumed to be normalized, i.e. φλ=1. For empirical estimates, one starts with unnormalized fluxes, and one does not know a priori the effective outer scale of the variability L that is needed to estimate the ratio λ. The normalization problem is easy to solve (see Eq. 18). For empirical estimates, one therefore starts with these normalized fluxes at the smallest available resolution (i.e. Δx=1 pixel); using this, lower-resolution estimates (i.e. larger Δx) are obtained simply by averaging. However, to verify Eq. (14), we need the scale ratio λ, which is the ratio of the (a priori unknown) outer scale L to the resolution Δx. The simplest procedure is to use the largest planetary-scale Lref (half the Earth's circumference) as an initial reference scale, a guess hopefully not far from the true outer scale L, a kind of “bootstrap”. When this is done, the statistics of the various moments as functions of the reference-scale ratio (i.e. with λref=Lref/Δx in place of λ:φLref/Δxq) are plotted in a log–log plot. For each moment order of q, the regressions with slopes K(q) all converge to the true outer scale; this is because at that scale λ=1, and (Eq. 14) shows that φλ=1q=1 for all q.

Figure 34 shows the first empirical trace-moment estimate (Schertzer and Lovejoy, 1987). It was applied to data from a land-based radar whose 3 km altitude reflectivity maps were 128 km wide with a 1 km resolution. The vertical axis is Log10Mq, where Mq=φλq and λ=Lref/Δx, with Lref=128 km. Although this gives a tantalizing hint that atmospheric cascades start at planetary scales, it was not until 20 years later when a decade of satellite radar data was released over the Internet that this was confirmed directly (Fig. 35, Lovejoy et al., 2009e). The poor scalings (curvature) for the low q values (bottom) were quantitatively explained as artefacts of the fairly high minimum detectable signal. Figure 36 shows similar results, but this time using the same geostationary satellite data whose spectra were analysed in Fig. 25. An interesting comparison of horizontal and vertical cascade structures from vertical sections of lidar aerosol backscatter is shown in Fig. 37. Although this is discussed in Sect. 4.1, we can already note that the outer scales are roughly the largest available (≈ 20 000 km in the horizontal and ≈ 10 km in the vertical), but analysis also shows that the slopes K(q) are the theoretically predicted ratio Khor/Kvert=Hz=5/9 (for all q, Sect. 4).

The trace moments characterize a fundamental aspect of the atmosphere's nonlinear dynamics – its intermittency – in fully developed turbulence, which is expected to be a “universal” feature, i.e. found in all high Reynolds number flows. In our case, the closest universality test is to compare Earth with Mars (using the same reanalyses as in Fig. 9). Figure 38 shows the result when this technique is applied to both terrestrial and Martian reanalyses for pressure, wind, and temperature (for both planets, the reanalyses were at altitudes corresponding to about 70 % of surface pressure). One can note that, (a) as predicted, the turbulence is universal, i.e. not sensitive to the forcing mechanisms and boundaries, so that the behaviour is nearly identical on the two planets, (b) there is clear multiscaling (the logarithmic slopes K(q)≠0), and (c) the effective outer scales (where the lines converge) are indeed nearly the size of the planet. For more detailed discussion and analyses (including spectra and horizontal anisotropy), see Chen et al. (2016).

Table 1 shows typical values of multifractal parameters estimated from trace moments (Sect. 3.4) of various atmospheric fields. Over the decades, many multifractal analyses of geofields have been performed, including of atmospheric boundary conditions, notably the topography on Earth (Lavallée et al., 1993; Gagnon et al., 2006), Mars (Landais et al., 2015), and the sea surface temperature (Lovejoy and Schertzer, 2013). We can see that the universal multifractal index (α) is typically fairly close to the log-normal value (α=2), although due to divergence of moments even when α=2, the statistics of the extremes are power laws (not log-normal; see the next section on divergence of moments). In the table we also see that with the notable exception of the highly intermittent precipitation field, the parameter for the intermittency near the mean (C1) might seem small. However, it should be noted that since the cascades operate over huge ranges of scales, the resulting fields are nevertheless highly intermittent. In addition, it should be recalled that, since α≈2, the intermittency increases very rapidly for the higher-order moments; so, for example, the kurtosis (q=4) has an “effective” intermittency 12 times larger (K(4)=C1 (42-4) = 12C1).