Multifractals, Generalized Scale Invariance and Complexity in Geophysics

January 19, 2012 9:1 WSPC/S0218-1274 03064

International Journal of Bifurcation and Chaos, Vol. 21, No. 12 (2011) 3417–3456 c World Scientiﬁc Publishing Company DOI: 10.1142/S0218127411030647

MULTIFRACTALS, GENERALIZED SCALE INVARIANCE AND COMPLEXITY IN GEOPHYSICS

DANIEL SCHERTZER LEESU, Ecole´ des Ponts ParisTech, Universit´eParis-Est, 6-8 Av. B. Pascal, 77455 Marne-la-Vall´ee Cedex 2, France [email protected] SHAUN LOVEJOY Physics Department, McGill University, 3600, University St., Montreal, Que. H3A 2T8, Canada [email protected]

Received March 10, 2011; Revised May 6, 2011

The complexity of geophysics has been extremely stimulating for developing concepts and techniques to analyze, understand and simulate it. This is particularly true for multifractals and Generalized Scale Invariance. We review the fundamentals, introduced with the help of pedagogical examples, then their abstract generalization is considered. This includes the characterization of multifractals, cascade models, their universality classes, extremes, as well as the necessity to broadly generalize the notion of scale to deal with anisotropy, which is rather ubiquitous in geophysics.

Keywords: Multifractals; generalized scale invariance; scaling; geophysics.

1. Introduction is focused on the fundamentals of multifractals and Geophysics, especially atmospheric dynamics over generalized scale invariance, and shows that they various scale ranges (i.e. from micro-turbulence are rather at the crossroads of the aforementioned to climate dynamics), has been an inspiring com- concepts. This is illustrated with the help of exam- plex system for elaborating innovative concepts and ples to understand, analyze and simulate nonlin- techniques. For instance, everyone has in mind, the ear phenomena in geophysics over wide ranges of Lorenz’s paper of 1963 [Lorenz, 1963] and the cor- scales. responding “butterfly effect” paradigm. However, In a very general manner, multifractals are there have been numerous contributions, which space or space-time fields — the latter being often showed that the complexity of geophysical phenom- called processes — that have structures at all ena could be much more considerable and of a scales. They are, therefore, a broad generalization somewhat different nature. This includes a rather of the (geometrical) fractals. Indeed, the set indi- different approach to predictability by Lorenz him- cator function of a fractal set is a field that has self [Lorenz, 1969], fractals [Mandelbrot, 1983], only two values (= 0 or 1), whereas multifractal strange attractors [Nicolis & Nicolis, 1984], self- fields generally have multiple values, which are very organized criticality [Bak et al., 1987; Bak & Tang, often continuous. The property to have structures 1989], and stochastic resonance [Nicolis, 1981; Benzi at all scales is trivially scale invariant since it does et al., 1982; Nicolis, 1982; Benzi, 2010]. This paper not depend on the scale of observation. This shows

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3418 D. Schertzer & S. Lovejoy that a multifractal field can also be defined as being be understood as an infinite hierarchy of embed- invariant for a given scale transform, which is there- ded fractals, e.g. those supporting singularities fore a symmetry for this field. One says that it is a higher than a given order. This is the source of scale invariant field or a scaling field for short. We the terminology, but more importantly of a rather will give a very general, precise definition of this straightforward manner to understand intermit- invariance and the corresponding scale transform tency: higher and higher levels of “activity” of that could be either deterministic or stochastic the field are concentrated on smaller and smaller (e.g. involves only equality in probability distri- fractions of the space. This provides a rather bution or other statistical equivalences), isotropic straightforward way to quantify and analyze inter- or not. mittency. This easiness is in sharp contrast with Due to their definition, multifractals are the mathematical nature of a multifractal field: it therefore not only quite general, but also quite is a (mathematical) measure, which is furthermore fundamental. Indeed, not only are symmetry princi- multi-singular. The fact that it is a measure implies ples the building blocks of physics and many other that it is not a pointwise field with a given value disciplines [e.g. Weyl, 1952; Zee, 1986], but scale at any given point, but it only yields an average symmetry is an element of the extended Galilean value over any given (small) neighborhood of this invariance. Unfortunately, attention in mechanics, point or an integration of a suitable set of “test especially in point mechanics, has been initially functions”. The fact it is multisingular with respect focused on the space shifts between two (Galilean) to the Lebesgue measure, i.e. does not admit a frameworks that differ only by a constant rela- density with respect to the latter, means that its tive velocity or a given rotation that define the estimated density at larger and larger resolution pure Galilean group. But with extended bodies, diverges with various power-laws exponents. These therefore continuous mechanics, it broadened to singularity orders are called singularities for short. other transforms such as scale dilations. In par- This can be therefore seen as a very broad gen- ticular, [Sedov, 1972] pointed out in the wake of eralization of the singular Dirac’s measure (often the Π-theorem of Buckimgam [Buckingham, 1914, improperly called function), whose mass is concen- 1915; Sonin, 2004] the key role of the latter in fluid trated on a pointwise “atom” and therefore can mechanics, including for many applications (e.g. to be understood as a divergence of its density that estimate the blast wave of a nuclear explosion). becomes infinite at this point, whose dimension is More recently, Speziale [1985] demonstrated their zero. For multifractals, theirs mass is distributed on relevance in selecting and defining relevant sub- the embedded fractal sets mentioned above. grid models in turbulence. This has been widely As discussed below, these generalizations, as used under the denomination of self-similarity, but well as their necessity, step-by-step became patent with unnecessary limitations. Indeed, the main goal with the help of cascade models in turbulence and was to find a unique scale transform under which strange attractors. These models were first rather the nonlinear dynamical equations (in particular, crude (e.g. discrete in scales) and apparently sim- the Navier–Stokes equations associated with others, ple, but yielded nontrivial behaviors which were such as the advection-dissipation equation). This step-by-step recognized as escaping from the origi- was mainly achieved by adimensionalising these nal framework of analysis, i.e. fractal analysis. Their equations with the help of various so-called char- present refined offsprings (e.g. continuous in scales) acteristic quantities, including characteristic space are generic models providing multifractal fields and time scales. Multifractals are, in fact, invari- and related data analysis techniques. These models ant for a multiple scale symmetry and therefore and multifractals have been used in a wide range correspond once again to a broad generalization of of scientific disciplines, from ecology to financial properties that were previously perceived in a too physics, including high energy physics, geophysics, restrictive framework. We indeed have to go from astrophysics, etc. Apriori, all domains of non- scale analysis, a seminal example being the quasi- linear science and complex science are concerned, geostrophic approximation derivation by Charney but geophysics remains a preeminent domain of [1948], to scaling analysis [Schertzer et al., 2011] applications. that we will briefly discuss in Sec. 8.4. To cover various aspects of multifractals and A not-so-trivial consequence of this multiplic- generalized scale invariance, the paper is organized ity of scale symmetries is that these fields could as follows: January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3419

• Section 2 introduces the key notions of singular an adequate mathematical framework had been measures with the help of the pedagogical exam- elusive for a while and has been seriously elabo- ple of the rainrate at various resolutions, rated only during the last twenty five years. Indeed, • Section 3 introduces the cascade models in a the variability of precipitation — which occurs phenomenological manner and two pedagogical over a wide range of (space and time) scales and models, intensities — is beyond the scope of classical geo- • Section 4 introduces the general multifractal physics. A well-known symptom of this basic prob- framework with the codimension and the scaling lem is that the basic hydro-meteorological quantity, moment functions and their duality via the the rain rate r, has a strong scale dependency, since Legendre transform, as well as a comparison it depends on the duration on which it is measured. between multifractal formalisms, This is illustrated by Fig. 1(a) that displays (from • Section 5 goes a bit further in formalism by top to bottom) the Nˆımes time series of rainrates introducing cascade generators, characteristic rλ(t) from an hour resolution (λ = 365 × 24) down functions and multifractal scale symmetry in a to a year resolution (λ =1): general, yet precise manner, λ • Section 6 is devoted to the question of universal- rλ(t)= 1 τi,τi+1 (t)R([τi,τi+1[); T [ [ ity, in particular to drastically reduce the number i of parameters characterizing a multifractal, T τi − τi = (1) • Section 7 pursues, in a given way, this discussion +1 λ to address the question of models that are con- where R([τi,τi [) is the rain accumulation (i.e. tinuous in scales, +1 the rain received) during the time interval [τi,τi [ • Section 8 suggests how to broadly generalize +1 and 1A denotes the indicator function of any set the previous results with the help of general- A. The intensity scale strikingly decreases with the ized scales and Lie cascades in the framework resolution, e.g. the maximum decreases from 35 to of Generalized Scale Invariance, as well as their 0.1 mm/h, which gives some credence that the vari- application to differential systems. ability decreases with decreasing resolution. Nev- • Section 9 is focused on the question of extremes, ertheless, as emphasized in [Ladoy et al., 1993b] particularly on heavy tails that are easily gen- the variability still exists in the yearly resolution. erated by multifractals, the relationship with The classical assumption is that the rain accumu- self-organized criticality and related multifractal lation is a regular (mathematical) measure dR(t) transitions. with respect to the (Lebesgue) volume measure dt, Although this is already a long paper, it does i.e. dR(t) admits a pointwise density r(t), defined not cover all the aspects of multifractals in geo- almost everywhere: physics. For instance, the question of multifractal dR(t)=r(t)dt. (2) predictability [Schertzer & Lovejoy, 2004b] is not discussed, but only evoked, in spite of the fact This density r(t) is called the rainrate and that it corresponds to a broad generalization of the is considered as the basic quantity of interest results of [Lorenz, 1969]. More generally, and for the in hydrometeorology. However, this assumption same reason, although the concepts are illustrated implies that r(t) would correspond to a scale inde- with the help of geophysical examples, this paper pendent small scale limit of rλ(t)(λ →∞), which does not pretend to review in an exhaustive man- is in contradiction with the observed strong scale ner their applications to geophysics. dependency of rλ(t). On the contrary, as discussed at length in [Schertzer et al., 2010], as well as in 2. Singularities Everywhere in Sec. 4.1, the assumption that dR(t) is a (multiple) singular measure with respect to the Lebesgue mea- Geophysics? sure implies that rλ(t) exhibits power-law singular- To better appreciate the underlying physico- ities at larger and larger resolutions: mathematical nature of a multifractal, let us con- γ rλ(t) ≈ λ . (3) sider the precipitation intermittency with the help of common sense intuition: most of the time it does Figure 1(b) displays these singularities for the not rain, furthermore, when it does rain its intensity rainrate rλ(t) displayed in Fig. 1(a), resolution by can be extremely variable. In spite of this intuition, resolution. The intensity of these singularities γ’s is January 19, 2012 9:1 WSPC/S0218-1274 03064

3420 D. Schertzer & S. Lovejoy

Fig. 1. Rainfall time series at Nˆımes (1972–1975) from 1 h to 1 year duration: (a) the rainrate rλ(t)[Eq.(1)]exhibitsastrong scale dependency, since its maximal value decreases from 35 to 0.1 mm/h from 1 h to 1 year duration (the unit of the intensity scale corresponds to 0.1 mm/h); (b) the corresponding singularities γ ≈ logλ(rλ) show on the contrary a remarkable scale independency over the same range of durations. Reproduced from [Schertzer et al., 2010]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3421

Fig. 1. (Continued) strikingly independent of the scale, contrary to the by singularities. Figure 2 displays the statistical rainrate intensity. One may also not note that there moments of the rain surrogate (radar reflectivity) is not a unique value of γ, i.e. the measure dR(t) over nearly four orders of magnitude in scale but is a multiple singular measure, with respect to the could continue over another two orders of magni- Lebesgue measure dt (which would yield γ =0). tude to smaller scales as observed with the help of Therefore — contrary to the usual hypothesis — ground radar data [Schertzer & Lovejoy, 1987]. there is no self-consistent definition of an instanta- We will show that the stochastic multifractal neous pointwise rain rate defined as a function r(t). fields offer a very convenient and operational We can proceed to a similar analysis with framework to handle such stochastic (multi-) respect to space. This can be achieved with the singular measures. They indeed allow us to help of statistical moments that should, as discussed systematically characterize, model and under- in Sec. 4.2, display multiple power-laws generated stand extremely variable fields while avoiding the January 19, 2012 9:1 WSPC/S0218-1274 03064

3422 D. Schertzer & S. Lovejoy

q Fig. 2. Statistical moments rλ of order q of statistical moments (q =0, 0.1,...,2.9, bottom to top) with respect to their resolution λ in a log–log plot of the rainrate rλ estimated by the radar of the TRMM satellite (≈ 1100 orbits) during a duration ∆t and across a horizontal section ∆x∆y (x along the satellite path, y transverse to it), with ∆y =4.3km,∆x = Learth/λ (Learth = 20 000 km). Straight lines correspond to power law fits. Reproduced from [Lovejoy et al., 2008a]. restrictive homogeneity assumptions implicit in of nonlinear PDE’s: their scale invariance (see the conventional approaches. While traditional Fig. 4). Even in spite of these (over) simplify- numerical approaches are forced to use drastic ing assumptions, the consequences of such choices scale truncations, to transform partial differen- are ultimately complex and unwieldy numeri- tial equations (PDE’s) into ordinary differential cal codes. Often the relevance of such codes, is equations (ODE’s), to make arbitrary regularity questionable and furthermore as their scales are assumptions, and to perform ad-hoc and unjusti- quite different from those of observations, they fied parameterizations (if only for the nonexplicit are often only intercompared with other (simi- “subgrid” scales), these classical manipulations lar) models. Multifractal analyses of these models (and mutilations) violate a fundamental symmetry are therefore rather indispensable to evaluate their

v1,1

v2,1 v2,2 v 3,1 v3,2 v3,4 v3,3

vi,j

vi+1,2j-1 vi+1,2j

Fig. 3. A rather general scheme of a (deterministic) cascade process, which actually corresponds to a simpliﬁcation of the Navier–Stokes equations, where only direct interactions between eddies and their oﬀsprings are preserved. Reproduced from [Chigirinskaya & Schertzer, 1996]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3423

models, which we will review, are superficially quite simple phenomenological “toy models”. Neverthe- less, they yield exotic phenomena (exotic compared to conventional smooth mathematical descriptions of the real world...) and have highly nontrivial consequences. For example, as we will see later, simple cascade models already give rise to a fundamental difference between observables and truncated processes, and such a difference is a general property of the wide class of “hard” multifractal processes (which distinguish between “dressed” and “bare” properties respectively) as discussed in Sec. 9. These models produce hierarchies of self-organized random structures.

3. Phenomenology of Cascades Fig. 4. A polemical illustration of the drastic reduction of Partial Differential Equations to Ordinary Differential Starting with [Richardson, 1922], the phenomenol- Equations in General Circulation Models. Reproduced from ogy of (scalar) turbulent cascades was first dis- [Schertzer & Lovejoy, 1993]. cussed in the context of hydrodynamic turbulence where the structures are “eddies”. Since we simply follow how the “activity”, measured by the tur- performance [Stolle et al., 2009; Lovejoy & bulent energy flux to smaller scales in turbulence, Schertzer, 2011; Gires et al., 2011]. Further- becomes more and more inhomogeneous as large more, they could be as well indispensable to structures break up into smaller and smaller scales, develop stochastic parametrizations, whose neces- cascades are a very general paradigm. sity is more and more recognized [Palmer & The key assumption in phenomenological mod- Williams, 2010], in order to reach the intrinsic pre- els of turbulence (which became explicit with the dictability limits, which are themselve multifractal pioneering work of [Novikov & Stewart, 1964; [Schertzer & Lovejoy, 2004a]. Yaglom, 1966; Mandelbrot, 1974] is that succes- Concerning multifractal modeling, cascade sive steps (independently) define the fraction of models (see Fig. 3 for an illustration) are generic the flux of energy distributed over smaller scales. multifractal processes. Indeed, they are built up It should be clear that the small scales cannot be with the help of an elementary scale invariant pro- regarded as adding energy; they only modulate the cess that is hierarchically repeated scale after scale energy passed down from larger scales. The explicit along a hierarchical scaling tree of smaller and hypothesis is that the fraction of the energy flux smaller structures. They therefore generate scal- (more generally the “activity”) from a parent struc- ing fields and yield a rather straightforward way ture to an offspring will be determined in a scale to understand that extreme variability over a very invariant manner. large range of scales may merely result from the In the (pedagogical) case of “discrete (in scale) repetition scale after scale of a given elementary cascade models” “eddies” are defined by the hier- process or interactions between neighboring scales. archical and iterative division of a D-dimensional This phenomenological idea can be traced back cube into smaller subcubes, with a constant ratio of to Richardson’s celebrated poem on self-similar cas- scales λ (greater than 1, very often equal to 2, see cades [Richardson, 1922] describing atmospheric 1 Fig. 3). More precisely, for each n ∈ N, the initial dynamics as a cascade process. However, it required 0 D-dimensional cube ∆0 of size L is divided step by some time and various developments before provid- n i ing well-defined models [Yaglom, 1966; Mandelbrot, step into λ1 smaller disjoint subcubes ∆n (ij =0, n− n 1974; Frisch et al., 1978] and to become rigorously 1,...(λ1 1); j =1, 2,...,D)ofsizen = L/λ1 that 0 conceptualized with the help of fractals [Mandel- fully cover ∆0. In other words, the λ1-base coordi- −n i brot, 1977, 1983]. nates of the corner ci = iλ1 of the subcube ∆n Then it evolved (after 1983), into a multifractal has only n digits. The density of the flux energy εn approach. The earliest scale invariant multifractal at the step n is taken to be strictly homogeneous January 19, 2012 9:1 WSPC/S0218-1274 03064

3424 D. Schertzer & S. Lovejoy on each “subeddies” of scale n, i.e. εn is a step 3.1. Unifractal insights and the function: simplest cascade model n (β ) λ−1 -model i εn(x)= εn1 i (x)(4) ∆n The simplest cascade model, often called the β- i=0 model, takes the intermittency of turbulence into account by assuming [Novikov & Stewart, 1964; where 1 i is the indicator function of the subcube ∆n Mandelbrot, 1974; Frisch et al., 1978] that eddies i − ∆n. The energy density εn−1 at step n 1 will be are either dead (inactive) or alive (active). This multiplicatively distributed to subeddies: corresponds1 to the fact that the random variable µε, which deﬁnes the multiplicative increments, has εn(x)=µεn(x)εn− (x)(5) 1 only two states (see Fig. 5): with the help of the following multiplicative Pr(µε = λc )=λ−c (alive) increment decomposed on the same basis: 1 1 (9) − −c Pr(µε =0)=1 λ1 (dead). λn−1 i The boost µε = λc > 1 is chosen so that the µεn(x)= µεn1 i (x)(6) 1 ∆n i=0 ensemble averaged ε is conserved [Eq. (8)]. At each step in the cascade, the fraction of the alive eddies i where the multiplicative increment components µεn are usually assumed to be identically and independently distributed (i.i.d.), as well as indepen- ISOTROPIC i i d dent of the variables εn, i.e. µεn = µε. Ensemble = self similarity conservation, called “canonical conservation, of the ﬂux corresponds to (· denotes the ensemble average):

εn = ε0 (7) L -D -D N(L) ∝ L N(L) ∝ L s obviously it requires that the random variable µε defining the multiplicative increments should L satisfy: L/2 L/2 µε =1. (8) L/2 L/2 This is also a sufficient condition, except if the cascade is degenerate, i.e. dies away after a finite number of steps (εn → 0 almost surely). This arises when a too strong, unrealistic mean intermittency D = Ln 3 =1.58 is chosen and we will see that this can be easily s Ln 2 quantified. In spite of their apparently simple yet some- Ln 4 D = = 2 what awkward discretizations, these models are Ln 2 already able to give key insights into the fundamentals of cascade processes. This is confirmed Fig. 5. A schematic of an isotropic discrete in scale cascade. using more realistic (continuous in scale) cascades, The left-hand side shows a nonintermittent (“homogeneous”) cascade, the right-hand side shows how intermittency can be which are necessary to take into account other modeled by assuming that not all sub-eddies are “alive”. This (statistical) symmetries (e.g. translation invariance, is an implementation of the “β-model”. From [Schertzer & see Sec. 7). Lovejoy, 1993].

1 The β-model is often deﬁned more vaguely than this. We follow the more precise stochastic presentation in [Schertzer & Lovejoy, 1984]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3425

−c activity almost surely vanishes after a finite number decreases by the factor β = λ1 (hence the name “β-model”) and conversely their energy flux den- of steps. sity is increased by the factor 1/β to assure (average) conservation. After n steps, this drastic and 3.2. The simplest multifractal simple dichotomy is merely amplified by the total n variant (the α-model) scale ratio λ1 : We already pointed out the mere occurrence/ nc −nc Pr(εn = λ1 )=λ1 (alive) nonoccurence of rain is not too informative, e.g. −nc (10) in the Nimes time series [Fig. 1(a)] the daily aver- Pr(εn =0)=1− λ (dead). 1 age ≈ 2.1 mm mm/day is negligible compared to a Hence either the density goes on to diverge 228 mm in a few hours — the October 1988 catas- with an (algebraic) order of singularity c,orisat trophe in Nˆımes! Furthermore, we already pointed once “killed” (set to zero)! Following the usual def- out that this time series does not display a unique inition of a geometric codimension C(A)ofaset singularity [see Fig. 1(b) and the corresponding dis- A of dimension D(A)embeddedinaspaceE of cussion in Sec. 2]. The variability of time series in dimension D(E): Nimes is so significant that [Ladoy et al., 1993a] and [Bendjoudi et al., 1997] found evidence of diver- C(A)=D(E) − D(A) (11) gence of high order statistical moments (a subject we will discuss more in Sec. 9). Similarly, Fig. 6 c is the codimension of the alive eddies and shows the qualitatively similar spatial distribution their corresponding (geometrical) dimension Ds is of the radar reflectivity of rain. This variability (if c

6 3 Fig. 6. A 1-D subsatellite section of satellite radar reﬂectivity (Z; units mm /m ) of rain at a 4.3 km horizontal resolution, 6 3 one full orbit showing ﬂux-like spikes; the mean Z is 53 mm /m .From[Lovejoyet al., 2008a]. January 19, 2012 9:1 WSPC/S0218-1274 03064

3426 D. Schertzer & S. Lovejoy

CASCADE LEVELS ε y l x 0 -- 0

multiplication by 4 independent random λ1 (multiplicative) l0 / increments 1 --

λ2 l 0 /

2 -- multiplication by 16 . . independent random . (multiplicative) increments λn l0 / n --

Fig. 7. A schematic of few steps of a discrete multiplicative cascade process, here the “α-model” with two “pure” singularities γ γ γ+ > 0,γ− < 0 (corresponding to the two values taken by the independent random increments λ + > 1,λ − < 1) leading to the appearance of mixed singularities γ (γ− ≤ γ ≤ γ+). From [Schertzer & Lovejoy, 1989b]. most of the peculiar properties of the β-model n where nr is the number of nr-combinations of n are lost. Indeed, let us consider the more realistic objects and the Stirling formula is used to derive α-instability allowing subeddies to be either “more asymptotics for large n and for any given (and active” or “less active”: ﬁxed) ratio r = n+/n.The“p-model” [Meneveau & γ+ c+ Pr(µε = λ1 )=λ1 Sreenivasan, 1987] and the “binomial multifractal γ− c+ c− (13) measure” correspond to microcanonical versions of Pr(µε = λ )=1− λ = λ 1 1 1 the α-model, which means that the ﬂux of energy with γ+ > 0,γ− < 0. The β-model is recovered is strictly conserved, not only on the ensemble with γ+ = c = c+; γ− = −∞ and the “canonical” average. This constraint fundamentally changes the conservation [Eq. (7)] implies that there are really properties of the processes, as we shall see below. only two free parameters out of c+, c−,γ+,γ−,since it corresponds to: γ −c γ −c 4. The General Multifractal λ + + + λ − − =1. (14) Framework At step n, we have “mixed” singularities 4.1. The codimension function c(γ) γr, 0 ≤ r ≤ 1 resulting from linear combinations of the “pure” singularities γ0 = γ−,γ1 = γ+,where The pedagogical example of the α-model provides r = n+/n is the fraction of outcomes of γ+ along helpful insights into the general formalism necessary the cascade branch leading to γr: for more general cascade processes. For instance, in the α-model as the number of cascade steps n γr = rγ +(1− r)γ− + becomes large, one obtains asymptotic expressions n γr n n rc++(1−r)c− n c(γr) [from Eq. (11)] which depend only on the total ratio Pr(ε =(λ1 ) )= (λ1 ) (λ1 ) nr of scale (denoted λ = L/ ≥ 1): − − c(γr)=r(Logλ1(r)+c+)+(1 r)(Logλ1 (1 r)+c−) γ −c(γ) (15) Pr(ελ ≥ λ ε1) ∼ λ (16) January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3427

This is a basic multifractal relation for mul- the codimension function c(γ). This graphical rep- tifractal processes, which merely states that the resentation helps also to estimate the limitations measure of the fraction of the probability space due to the finite size of a sample with the help of corresponding to the events the “sampling dimension” Ds [Schertzer & Lovejoy, γ 1989b; Lavallée et al., 1991] defined as being the Aλ(γ)= (x, ω) ∈ E × Ω | ελ(x, ω) ≥ λ ε1 (17) scaling exponent of the number Ns of independent has a (statistical) codimension c(γ), the precise samples of resolution λ: meaning of the asymptotic equivalence (λ →∞) Ds denoted by ∼ will be discussed below. As already Ns,λ ≈ λ (18) emphasized, there is generally no upper bound on as well as the corresponding “sampling singular- c(γ). On the other hand, due to the nested hierarchy ity” γs that is the almost sure maximal singularity of these events (∀ λ, γ ≤ γ : Aλ(γ) ⊂ Aλ(γ )) c(γ) present in a sample of sampling dimension Ds.This is necessarily an increasing function of γ. singularity γs has therefore a codimension equal Other fundamental properties are that c(γ) to the overall effective dimension of sampling [see must be convex and that if the process is con- Fig. 8(b)]: servative [Eq. (7)], then c(γ) has the fixed point: c(C1)=C1,whereC1 is at the same time a singu- c(γs)=D + Ds =∆s; (19) larity corresponding to the mean of the process and its codimension: at this point c(γ) is tangent to the and it will be discussed further in Sec. 9.4. bisectrix. Figure 8(a) illustrates these properties of 4.2. The multiscaling of moments γ K(q) and the Legendre rare c( ) events transformation Under fairly general conditions, a random variable can be specified by either its probability distribution or by (all) its statistical moments. For a

C1 non-negative random variable x,thesetworepre- sentations are linked by a Mellin transformation M, which is: γ q−1 C1 x = M(p) extreme events ∞ q−1 (a) = x p(x)dx (20) 0 p(x)=M −1(xq−1) rare c(γ) N ≈ λDs events s 1 c+i∞ = xq−1x−qdq (21) 2πi c−i∞ D+Ds (essentially these are simply the Laplace and inverse D Laplace transforms for the logs). In fact, if the moments increase slowly with q (when they satisfy the “Carleman criterion” — see [Feller, 1971]), γ only the knowledge of the integer order moments is γ D s suﬃcient. The relevance of the Carleman criterion extreme events for turbulence has been discussed in [Orszag, 1970], (b) but in any case the full Mellin duality [Schertzer & Lovejoy, 1993; Schertzer et al., 2002] will hold. Fig. 8. A schematic illustration of a conserved multifrac- This is somewhat more general than the tal, showing (a) the relations c(C1)=C1 and c (C1)=1, where C1 is the singularity of the mean, (b) how the sam- Legendre duality pointed out in [Parisi & Frisch, pling dimension imposes a maximum order of singularities 1985], but we can check that the Legendre trans- γ’s. From [Schertzer & Lovejoy, 1993]. form is the asymptotic (λ →∞) result linking the January 19, 2012 9:1 WSPC/S0218-1274 03064

3428 D. Schertzer & S. Lovejoy corresponding exponents. Since the codimension nonlinearity of the scaling exponents of the velocity c(γ) is the scaling exponent of the probabilities, structure functions (the statistical moments of the let us introduce the corresponding scaling moment velocity increments) empirically observed in [Ansel- function K(q): met et al., 1984]. Parisi and Frisch [1985] considered that the singularities of the velocity increments q ≈ K(q) ελ λ . (22) deﬁned as local Holder exponents should be geo- For large log λ, we can use the saddle point metrically and rather deterministically distributed approximation (Laplace’s method, see for example over embedded fractals. The f(α) formalism [Halsey [Bender & Orszag, 1978]) which yields asymptotic et al., 1986], which dealt with multifractal strange approximations to integrals of exponential form. attractors, in many respects further emphasized this One obtains that K(q) is related to c(γ)by: implicit nonrandom and geometric framework. At the notation level, a dimension formalism such as q q qγ f(α) formalism is formally related to codimensions ε = dPr(ελ)ε ∼ dPr(ελ)λ λ λ according to: ∞ − − = Log(λ)dc(γ)λqγλ−c(γ) (23) αD = D γ; fD(αD)=D c(γ). (26) −∞ This is without fundamental formal problems which yields the asymptotic behavior (λ →∞): as far as c(γ)

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3429

which corresponds to singularity γ for infinitely often resolutions λ’s. Let us illustrate this question by pointing out that stochastic multifractal singularities are nonlocal [Schertzer & Lovejoy, 1992]. Indeed, when we add more and more cascade steps in a stochastic cascade, the singularity γλ(x)ata given location x and increasing resolution λ under- goes a random walk (see e.g. Fig. 10), whereas a complete localization of the singularity would correspond to a pointwise limit: γ(x) = limλ→∞ γλ(x). Therefore, the relevant limit notion is the upper limit [Eq. (30)] rather than the much more stringent lower limit [Eq. (29)]. For applications, this means that the multifractal field is nonlocal, and one cannot always track a given singularity value by locally Fig. 9. Schematic diagram of the “pushforward” transform refining the analysis of the field, e.g. with the help Tλ,∗ of a (mathematical) measure µ due to a time contraction of wavelet analysis. This may yield spurious results. Tλ with a scale ratio λ (reproduced from [Schertzer et al., 2010]). Parisi and Frisch [1985], Frisch [1995] acknowledged that within their formalism they could get only a therefore the limit inferior: bounded range of singularities (in fact c(γ) λ On the overall, we reviewed the fact that whereas, the asymptotic scaling of the probability beyond their strong communality, various multi- [Eq. (16)] is rather related to the limit superior fractal formalisms have important differences due [Schertzer et al., 2002]: to basic assumptions on the nature of the process (e.g. stochastic or deterministic), the hypothesis to A(γ)= lim AΛ(γ) ≡ AΛ(γ) (30) be done is more or less stringent, but the resulting Λ→∞ λ Λ>λ hierarchy of fractals can be also rather different: it

Fig. 10. Nonlocal development of singularities γλ(x) (at location x and resolution L/λ) in a perspective view (along the number of steps n =Log2[λ]) in a discrete cascade of elementary scale ratio λ1 = 2 and of universal parameters α =1.5,C1 = 0.5 (see Sec. 6.3) over an overall ratio of scale Λ (≥ λ). It is rather obvious that for any given x, γλ(x) ﬂuctuates for increasing λ, contrary to the hypothesis of localness of singularity. January 19, 2012 9:1 WSPC/S0218-1274 03064

3430 D. Schertzer & S. Lovejoy could be either defined for each realization or with The name pullback evokes the fact that this trans- looser probabilistic constraints. form acts in the opposite direction (“contravari- antly”) to that of the original transform Tλ.This general notion is particularly useful when dealing 5. Generators, Characteristic with differential equations [Schertzer et al., 2012]. Functions and Multifactal Scale It is dual to the “pushforward” transform Tλ,∗ act- Symmetry ing (“covariantly”) on measures according to the ∈ N { Discrete cascades, i.e. with scale ratio λ λ1 = 1, following duality equation: 2 3 } λ1,λ1,λ1 ... , point out that the ελ form a multi- ∗ plicative group, a property that will be extended to fTλ,∗(dµ)= Tλ (f)dµ. (34) continuous cascades. It is defined by the fact that a cascade from resolution 1 to Λ, can be obtained by The pushforward transform Tλ,∗ is particularly multiplying a cascade from resolution 1 to λ ≤ Λ useful to mathematically deal with singular mea- by a cascade from λ to Λ. Because, the latter corre- sures such as rain accumulation [Schertzer et al., spond to a rescaled cascade from resolution 1 to Λ/λ 2010]. It is rather easy to check that the (trivial) (see Fig. 11), this group property corresponds to: group property of the original transform Tλ extends to both the pullback and pushforward transforms ∀ ≥ ≥ ∗ Λ λ 1: εΛ = ελTλ (εΛ/λ) (31) and that both are linear respectively on vector ∗ spaces of functions and their dual spaces of mea- where T is the “pullback” transform for any λ sures. function f: Like for all one-parameter groups, we are inter- ∗ ∀ x ∈ E: Tλ (f)(x)=f(Tλ(x)) (32) ested to characterize the infinitesimal generator of a cascade, which can be stochastic. Loosely speaking defined by a geometric point transform Tλ on the this allows us to return to an additive group. Let space E,onwhichελ is defined (i.e. time and/or us consider the generator Γλ of the cascade over a space). For the moment, up to Sec. 8 devoted (noninfinitesimal) scale ratio λ defined by: to Generalized Scale Invariance, Tλ is taken to be trivial scale transformation, i.e. the isometric ελ =exp(Γλ). (35) contraction: The additive group property corresponding to the x Tλ(x)= (33) multiplicative property displayed by Eq. (31) is: λ ∗ ∀ Λ ≥ λ ≥ 1: Γ =ΓλT (Γ ). (36) although it is already clear that the generality of the Λ λ Λ/λ pullback transform will allow broad generalizations The generator is well defined for any finite res- of the results obtained for the isometric contraction. olution λ, but its limit for asymptotically large

bare cascade 1 1 1

λ λ = X λ Λ / λ

Λ * Tλ Λ dressed cascade hidden cascade

Fig. 11. A schematic diagram (horizontal segments schematically represent involved scales) showing a (“dressed”) cascade from resolution 1 to a (“bare”) cascade from resolution 1 to λ, multiplied by a (“hidden”) cascade from resolution Λ/λ to Λ, ∗ which is the pushback transformed with the help of Tλ of a cascade constructed from resolution 1 to Λ/λ. The terminology of “dressed, bare and hidden” cascades will be discussed in Sec. 9.2. Reproduced from [Schertzer et al., 2002]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3431 resolutions is not trivial. This can be seen with dimension to an infinite hierarchy of dimensions. the help of the moment scaling function K(q), This is emphasized by the fact that there is only which gains a new and convenient meaning. Indeed, a convexity constraint on the nonlinear functions Eq. (22) can be rewritten under the form: K(q)andc(γ), therefore apriorian infinity of parameters are required to determine a multifractal qΓλ Kλ(q) e = Zλ(q)=e (37) process. For obvious theoretical and empirical rea- where Zλ(q)andKλ(q) are none other than respec- sons, physics abhors infinity. This is the reason why tively the (Laplace) first and second characteristic in many different fields of physics the theme of uni- functions — or respectively the moment and cumu- versality appears: among the infinity of parameters, lant generating functions — of the generator Γλ. it may be possible that only a few are relevant. For This further shows that the characteristic function instance, in critical phenomena most of the expo- of Γλ, should be logarithmically divergent (with the nents describing phase transitions depend only on scale ratio λ): the dimensionality of the system. This is especially true as soon as we go beyond ideal systems to more Kλ(q)=K(q)Log λ (38) realistic ones that should be robust to perturbations hence a nontrivial small scale limit of the genera- or self-interactions. Such perturbations or interac- tor Γλ. tions may wash out many of the peculiarities of the We may now define fractal and multifractal theoretical model, retaining only some essential fea- symmetries in a very general and precise man- tures. Loosely speaking, a theoretician may concoct ner. Following [Lamperti, 1962], scale invariance for a model for an isolated system depending on a very stochastic fractals is usually defined by: large number of parameters, but most natural sys- d H tems are open and the resulting interactions can ε(Tλx) = λ ε(x) (39) wash out most of the details, just leaving the (few) where H is a given unique scaling exponent and ε essentials. This was the basic idea of the Renormal- can be an increment of random walk or field, as izing Group approach [Wilson & Kogut, 1974]. well as a measure. Considering the latter case, this The system can therefore be expected to con- can be rewritten with the help of the pushforward verge to some universal attractor, in the sense that transform into a standard form of a symmetry Sλ: a whole class of models/processes with rather dis- tinct parameters will be nevertheless attracted to d Sλε = ε; Sλ = T˜λ o Tλ,∗ (40) the same process defined by only a small number of relevant parameters: the larger the basin of which corresponds to the product of a pushforward attraction, the more universal the attractor. transform Tλ,∗ by a codomain contraction, i.e. a contraction T˜λ on the space where ε is valued (up until now the real space ): 6.2. Universality in multiplicative −H T˜λ = λ . (41) processes? The study of multiplicative random processes has a A multifractal symmetry is obtained for long history [Aitchison & Brown, 1957] going back Eq. (40) as soon as T˜λ is no longer defined by a to at least [McAlsister, 1879], who argued that mul- unique scaling exponent, but by a full set of singu- tiplicative combinations of elementary errors would larities γ’s: lead to lognormal distributions. Kapteyn [1903] gen- d −γ −Γλ T˜λ = λ ≡ e . (42) eralized this somewhat and stated what came to be known as the “law of proportional effect”, which has Generalized scale invariance (Sec. 8) will fur- been frequently invoked since, particularly in biol- ther generalize this symmetry due to nonscalar ogy and economics (see also [Lopez, 1979] for this ˜ anisotropic Tλ and/or Tλ,∗. law in the context of rain). This law was almost invariably used to justify the use of lognormal dis- 6. Universality tributions i.e. it was tacitly assumed that the lognormal was a universal attractor for multiplicative 6.1. The concept of universality processes. Although Kolmogorov [1962], Obukhov The jump from fractals to multifractals is huge, [1962] did not explicitly give the law of propor- since it corresponds to a jump from a unique tional effect as motivation, it was almost certainly January 19, 2012 9:1 WSPC/S0218-1274 03064

3432 D. Schertzer & S. Lovejoy the reason why they suggested a lognormal distribution for the energy dissipation in turbulence. This claim seemed to be secured by the explicit “lognormal” cascade model developed in [Yaglom, 1966]. The claim of universality of the lognormal model was first criticized in [Orszag, 1970] and then in [Mandelbrot, 1974]. Whereas Orszag’s criticism was on the grounds that the (infinite) hierarchy of integer order moments would not determine a lognormal process, Mandelbrot’s criticism was based on the fact that even if the cascade process were lognormal at each finite step, that in the small scale limit, the spatial averages would not be lognormal. This limit already Fig. 12. Scheme of densification of scales: each horizontal line schematizes to a step of the cascade from large to small poses a nontrivial mathematical problem, since it scales (up to bottom). Scale densification of a cascade (left, corresponds to a weak limit of random measures with a rather large elementary scale ratio) corresponds to [Kahane, 1985], as discussed in Sec. 9.1. Further- more and more intermediate steps (right, with a smaller ele- more, since the particularities of the discrete models mentary ratio). From [Schertzer & Lovejoy, 1997]. (e.g. the α-model) remain at each scale, therefore in its small-scale limit, this lead to the opposite claim: where aN ,bN (with the nonrestrictive choice b1 =1) that multiplicative cascades do not admit any uni- (i) are renormalizing constants and ελ are N indepen- versal behavior [Mandelbrot, 1989, 1991; Gupta & dently identically distributed cascade processes on Waymire, 1993]. the same range of scales with an overall resolution (i) λ for nonlinear mixing or ελ are the successive ele- 6.3. Universal multifractals and the mentary cascade steps (of elementary resolution λ) multiplicative central limit for a scale densification over the same scale ratio λ. theorem Secondly, one has to check that these stable points As long as we keep the total range of scale fixed are attractive over a given domain of processes. and finite mixing (by multiplying them) indepen- A multiplicative central limit theorem [Sch- dent processes of the same type (identical distribu- ertzer & Lovejoy, 1997] is obtained with the help tion, therefore the same codimension function c(γ) of the cascade generator. Indeed, Eq. (43) cor- and scaling moment function K(q)), and then take responds to the generalized (additive) stability the limit Λ →∞[Schertzer & Lovejoy, 1987, 1988], problem [Lévy, 1925, 1937; Gnedenko, 1943; Gne- we avoid the difficulties that we mentioned above denko & Kolmogorov, 1954], i.e. the following fixed and a totally different limiting problem is obtained. point: For instance, this may correspond to a (renor- N (i) Γλ − aN Log λ malized) densification of the excited scales by Γλ − a1Log λ = . (44) bN introducing more and more intermediate scales i=1 (see Fig. 12), e.g. to obtain a continuous scale For instance, iteration of Eq. (43) or (44) shows cascade model as discussed in Sec. 7. Alterna- that the bN ’s form a multiplicative group with tively, we may also consider the (renormalized) aN = a : nonlinear mixing of identically independently dis- 1 tributed (i.i.d.) cascade models which correspond to bNM = bN bM (45) their multiplication. In both cases, we first look for the cascade processes that are stable under either and one then may introduce the exponent α to nonlinear mixing or densification, then to their generate this group: attractivity and their domain of attraction. Stable 1/α d (1) bN = N . (46) cascades are the fixed points ελ = ελ of: On the other hand, taking ensemble averages −a1 −aN (i) 1/bN λ ελ = (λ ελ ) (43) of both sides of Eq. (43) one obtains that the scal- i=1,N ing moment function K(q)ofελ is the fixed point January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3433

K(q)=K(1)(q)of: where C1 is the singularity of the mean field (q =1), which with the help of the Legendre transform is q aN q defined by: − (1) − K(q) a1q = N K (47) bN bN d C1 = K(q) . (53) dq which yields the solutions: q=1 The constraint on α mentioned in Eq. (52) α K(q)=cq + a1q. (48) results from the requirement that the (first) characteristic function Zλ(q) [Eq. (37)] of the genera- Attractivity is rather immediate. Indeed, let us tor should be positive definite so as to be a Mellin (i) consider the following K (q)(withβ>α): transform of a probability (according to the Schoen- berg’s theorem [Schoenberg, 1938], which comple- (i) β K (q)=K(q)+O(q ) ments the Bochner’s theorem [Bochner, 1955] for α β the Laplace transform). The two extreme cases = cq + a q + O(q ) (49) 1 correspond respectively to the (uni/mono) fractal we have therefore: β-model (α = 0) and the lognormal model (α =2), whereas the case α = 1 is derived from Eq. (52) → (1) q aN q with α 1: K(q) − a1q = lim N K − (50) N→∞ bN bN K(q)=C1qLog q; α =1. (54) and therefore the corresponding ελ is attractive: Equation (52) shows that for 0 <α<2the generator Γλ follows an “extremely asymmetric” −a1 −aN (i) 1/bN λ ελ = lim (λ ελ ) . (51) or “skewed” Levy distribution (i.e. its skewness N→∞ − i=1,N parameter β = 1) so that the fat tail (power- law tail) is present only for negative fluctuations, Finally, with the help of the constraint K(1) = otherwise the field ελ will have divergent statistical 0 of a conservative field, Eq. (48) yields the following moments (K(q)=∞) for all positive order q.Nev- scaling moment function of a conservative universal ertheless, all the negative statistical moments of ελ multifractal [Schertzer & Lovejoy, 1987] (see Fig. 13 diverge, which is neither a problem, nor surprising for an illustration): due to its frequent extremely low or even zero values (especially for the β-model). C α α K(q)= 1 (qα − q); 0 ≤ α ≤ 2 (52) Due to the fact that q /α and γ /α are α − 1 Legendre duals for (1/α)+(1/α) = 1, one obtains

K(q)/C1 versus q 2.0

1.5 α 1.0 0.0000 1 0.2000 0.4000 0.5 0.6000 α = 2.0 0.8000 1.0000 K(q)/C 1.2000 0.0 1.4000 1.6000 1.8000 2.0000 -0.5 α = 0.0 -1.0 0.0 0.5 1.0 1.5 2.0 q

Fig. 13. The universal multifractal moment scaling exponent K(q) normalized by C1. They lie between the parabolic (α =2) “log-normal” model and the linear (α =0)β model. January 19, 2012 9:1 WSPC/S0218-1274 03064

3434 D. Schertzer & S. Lovejoy

γ γ c( )/C versus /C1 2.0

α 0.0000 1.5 0.2000 α = 0 0.4000 0.6000 1 0.8000 1.0000 1.0 1.2000 1.4000 1.6000 c( γ ) / C 1.8000 2.0000 0.5

α = 2.0

0.0 -1.0 2.01.51.00.50.0-0.5 γ / C1

Fig. 14. The universal multifractal probability exponent c(γ/C1) normalized by C1. They lie between the parabolic (α =2) “log-normal” model and the bilinear (α =0)β model. from Eq. (52): not uniformly distributed. Since all the statistical α interrelations between different structures depend γ 1 1 1 on this ultra-metric, not on the usual metric, this c(γ)=C1 + ; + = 1 (55) C1α α α α has drastic consequences. In particular, there is no see Figs. 13 and 14 for illustration of the parameter hope of obtaining a (statistically) translation invari- sensitivity of the scaling moment and codimension ant cascade, since such invariance depends on the functions. metric, not the ultra-metric. In summary, discrete cascades are useful for 7. Continuous Scale or “Infinitely grasping many of the fundamentals, but one has Divisible” Cascades to avoid being blocked by their artifacts. As a final note on discrete scale cascades, let us empha- 7.1. The limitations of discrete size that almost all rigorous mathematical results scale cascades on cascade processes have been derived in this An important consequence of universality is the restricted framework; this is not only because possibility of starting with a discrete cascade model it is convenient, but also for complex historical reasons — including the debate on universality (e.g. one with an elementary scale ratio λ1 =2)and with the help of a scale densification to obtain a con- discussed above. As a consequence, the issue of continuous scale or infinitely divisible cascades has not tinuous in-scale process λ1 → 1. Such cascades are necessary since physically we expect real systems to been discussed enough. involve a continuum of scales and there is usually no physically based quantization rule that would 7.2. Continuous scale cascades and restrict scale ratios to integer values. Furthermore the discrete models involve a hierarchical splitting their generators rule of structures into substructures which is based The general idea of cascades as one-parameter on a notion of distance which is not a metric, but multiplicative groups discussed in Sec. 5, especially rather an “ultra-metric”. More precisely, it cor- Eq. (31) becomes essential for continuous scale responds to the λ-adic ultrametric: the distance cascades. For instance, the corresponding addi- between two structures at a given level of a discrete tive property for generators [Eq. (36)] show that cascade process is defined by the level of the cascade they should be infinitely divisible, i.e. they can be where they first share a common ancestor, i.e. not divided into smaller and smaller additive compo- by the usual distance. This implies that the distance nents. On the other hand, we also saw that scaling between the centers of two contiguous eddies is requires a logarithmic divergence of the generator. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3435

For a concrete and generic example, let us con- (β =0,K(q)=q2) and has therefore no divergence sider the case of universal multifractals (Sec. 6.3), for q<0. Figure 15 compares the subgenerators in whose generators are not only infinitely divisible, the α<2, α = 2 cases. but also stable and attractive. They are therefore This can be seen under a (fractional) differen- robust under nonlinear interactions. To satisfy the tial form of order D/α using a fractional power of logarithmic divergence their generators [Eq. (33)] the Laplacian ∆x (with a scale resolution L/λ): should be (colored) stable Lévy noises [Schertzer & Lovejoy, 1987] obtained by a fractional integration − − D/2α ≈ (α) ( ∆x,λ) Γλ(x) γ0 (x) (57) over a given “subgenerator”, which is white Lévy (α) or under the corresponding and more explicit (frac- noise γ0 with a Lévy stability index α and furthermore “unitary” in the sense that its (Laplace) tional) integration form: second characteristic function is: 1/α D (α) α var(α) d γ0 (x ) − · ∞· Γλ(x)= K0(q)=sign(α 1)q 1q≥0 + 1q<0 (56) | − |D/α mD−1(∂BL) BL\BL/λ(x) x x (α) which means that γ0 is extremely asymmetric − var(α)Log λ; [Schertzer et al., 1988] for α<2, i.e. with a − C skewness β = 1, to have only strong negative var(α)= 1 extremes. This is no longer relevant for the Gaus- α − 1 sian case (α = 2), which is necessarily symmetric (58)

Fig. 15. A schematic showing a gaussian and extremal Levy subgenerator (with α =1.6); note that there are extremes only for the negative ﬂuctuations. From [Wilson, 1991]. January 19, 2012 9:1 WSPC/S0218-1274 03064

3436 D. Schertzer & S. Lovejoy where BL(x) is the ball of center x and size L (which 1/α D (α) var(α) d γ0 (x ) could be arbitrarily chosen), mD−1(∂BL)isthe dγλ = D/α mD− (∂BL) ∂B x |x − x | (D − 1)-dimensional measure of its (hyper-)surface 1 L/λ( ) ∂BL. For isotropic cases, BL(x)={x ||x − x |≤ (60) L/2}, BL\B (x)={x | L/2λ<|x − x |≤ L/λ where var(α) corresponds to a generalization of the } D−1 D/2 L/2 and mD−1(∂BL)=2(L/2) π /ΓE(D/2) (quadratic) variation of a (semi-) martingale (e.g. (where ΓE is the Euler Gamma function). In [Metivier, 1982]) and this explains why: order to get some convergent moments (finite (α) var(α)dλ K(q)), the subgenerator γ0 must be extremely dγλ − dΓλ = . (61) assymmetrical. It is also interesting to note that λ ελ due to Eq. (58) is the solution of the following differential equation: 7.3. Nonconservative multifractals and the Fractionally Integrated dελ = ελdγλ (59) Flux model (FIF ) where dγλ is the infinitesimal generator of the Integration of Eq. (59) [or equivalent use of the gen- multiplicative group of ελ: erator defined by Eq. (58)] yields a conservative field

Fig. 16. Isotropic (i.e. self-similar) multifractal simulations showing the eﬀect of varying the parameters α and H (C1 =0.1 in all cases). (From left to right) H =0.2, 0.5 and 0.8. (From top to bottom) α =1.1, 1.5 and 1.8. As H increases, the ﬁelds become smoother and as α decreases, one notices more and more prominent “holes” (i.e. low smooth regions). The realistic values for topography (α =1.79, C1 =0.12, H =0.7) correspond to the two lower right-hand simulations. All the simulations have the same random seed. Reproduced from [Gagnon et al., 2006]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3437

ελ, which is not always a desirable property. The moments of the increments ∆ν(x,x)=|ν(x) − classic example of such nonconsevative processes is ν(x)|: the turbulent velocity field; in Kolmogorov three- | − |q∝| − |qH−K(q) dimensional isotropic turbulence, whose structure ν(x) ν(x ) x x . (63) 2 ≈ function is not scale invariant: ∆v (L/λ) These increments have therefore a scale depen- −2H 2 λ ∆v (L) ,with:H =1/3. This constraint is dent mean, with H = 0 as scaling exponent (see easily satisfied with the help of a fractional integra- Figs. 16 and 17). Figures 15 and 18 display the tion of order H: effect of changing the order of this scale invariant D smoothing of order H. The convolution involved in ∝ d ε(x ) νλ(x) | − |D−H . (62) Eq. (63) corresponds to a fractional integration of BL\B (x) x x L/λ (fractional) order H over ε. However, there is not In the framework of this Fractionally Integrated a unique definition of such a noninteger integra- Flux model [Schertzer et al., 1997], the singularities tion. For instance, to respect causality for processes are redistributed with an average shift of −H as in time, the corresponding kernel or Green func- seen on the structure functions, i.e. the statistical tion should be totally asymmetric in order not to

Fig. 17. All simulations have H =0.35 (this is a scale invariant smoothing, see Sec. 7.3). The parameter C1 increases from 0.05 to 0.8 in steps of 0.15 from left to right and α increases from 0.4 from top to bottom in units of 0.4 to 2.0 (bottom). January 19, 2012 9:1 WSPC/S0218-1274 03064

3438 D. Schertzer & S. Lovejoy

Fig. 18. All simulations have C1 =0.05. The parameter H increases from 0.05 to 0.8 in steps of 0.15 from left to right and α increases from 0.4 from top to bottom in units of 0.4 to 2.0 (bottom). integrate over the future [Marsan et al., 1996]. This 8. Generalized Scale Invariance (GSI) is particularly important for predictability studies 8.1. Generalized scales [Schertzer & Lovejoy, 2004a] and multifractal fore- casts [Macor et al., 2007]. In fact, it suffices to intro- Up until now we have considered “self-similar” frac- duce a Heaviside function (1t≥0)asaprefactorto tals and multifractals i.e. geometric sets and fields the noncausal kernel to obtain a causal Green func- that are not only scaling, but also statistically rotation g(t, t), e.g.: tionally invariant. However real world systems are not isotropic so that it is important to generalize 1t−t≥ 0 this to strongly anisotropic systems. The resulting g(t, t )=| − |D−H . (64) t t “Generalized Scale Invariance” (GSI) was actually Furthermore, “delocalized” kernels can be motivated by the need to account for the strong defined for space-time processes to have a given scaling stratification of the atmosphere [Schertzer & scaling exponent D−H and at the same time singu- Lovejoy, 1985b, 1985a]: Fig. 19 gives a (scaling) larities (e.g. poles) in their Fourier transforms that anisotropic version of the isotropic cascade scheme generate waves. This will be discussed further in the (Fig. 5) originally developed to model atmospheric next section that introduces the necessary concept stratification, which has gained increasing empiri- and tools of Generalized Scale Invariance. cal confirmation (Fig. 20) with the help of various January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3439

ANISOTROPIE = COMPRESSION et REDUCTION

L - D y - D N(L) - L el N(L) - L s

L x L x /4 L x /4

L y /2 L y /2

Ln 6 ___ D s = = 1.29 Ln 4

Ln 8 D = ___ = 1.5 el Ln 4

Fig. 19. Anisotropic cascade scheme: compare with Fig. 5. From [Schertzer et al., 2002]. Fig. 20. The symbols show the mean absolute fluctuations (the first order structure functions) of the lidar backscatter ratio ρ (a surrogate for the passive scalar aerosol density). The points near the shallow slope (slope 1/3, Kolmogorov atmospheric measurements [Chigirinskaya et al., value) are the horizontal fluctuations, the points near the steeper line (slope 3/5, Bolgiano–Obukhov value) are for the 1994; Lazarev et al., 1994; Lilley et al., 2004; Love- vertical fluctuations. The averages are over an ensemble of joy & Schertzer, 2007; Lovejoy et al., 2007]. nine vertical airborne lidar cross-sections spanning the range The usual approach to scaling is to first posit of ≈ 100 m–100 km in the horizontal and 3 m–4 km in the ver- (statistical) isotropy and only then to consider scal- tical. ∆r is either the vertical or horizontal distance measured ing. This approach is so prevalent that the terms in meters. The lines have the theoretical slopes 3/5, 1/3, they intersect at the sphero-scale here graphically estimated from “scaling” and “self-similarity” are frequently used the intersection point as ≈ 10 cm. From [Lilley et al., 2004]. interchangeably! The most famous example of this is Kolmogorov’s hypothesis of “local isotropy” from which he derived the k−5/3 spectrum for the wind the positive real scale ratio λ (λ ≥ 1forasemi- fluctuations (k is a wavenumber). The GSI approach group), i.e. is the converse: it first posits scale invariance (scal- ∀ ∈ + ◦ ing), as the main symmetry and then considers λ, λ R : Tλ Tλ = Tλ λ (65) the remaining nontrivial symmetries. One may eas- and admits a generalized scale denoted x (to ily check that the type of anisotropic construction distinguish it from the usual Euclidean metric shown in Fig. 19 reproduces itself from scale to scale |x|), which satisfies the following three properties without introducing any characteristic scale. Since [Schertzer et al., 1999] further to that of being non- it involves scaling anisotropy in fixed directions, it negative: is called “self-affinity”. This anisotropic scheme is apparently the first explicit model of a physical (i) nondegeneracy, i.e. system involving a fundamental self-affine fractal mechanism [Schertzer & Lovejoy, 1984, 1985a]. x =0⇔ x =0 (66) GSI corresponds to the fact that the contraction operators Tλ [Eq. (33)] or T˜λ [Eq. (34)], which (ii) linearity with the contraction parameter 1/λ, define the scaling symmetry [Eq. (40)], are no longer i.e. isotropic [Eq. (33)] nor merely scalar [Eq. (41)]. For + −1 simplicity sake we first focus on the anisotropy of ∀ x ∈ E, ∀ λ ∈ R : Tλx≡Tλ · x = λ x Tλ, keeping T˜λ scalar, but mutadis muntandis the (67) same apply to an anisotropic T˜λ and a scalar Tλ, and finally both being anisotropic. In an abstract (iii) balls defined by this scale, i.e. manner, Tλ is a generalized contraction on a vector space E, if it is a one-parameter (semi-) group for B = {x |x≤} (68) January 19, 2012 9:1 WSPC/S0218-1274 03064

3440 D. Schertzer & S. Lovejoy

must be strictly decreasing with the contrac- It is straightforward to check that for tion Tλ: Spec(G) > 0, where Spec(·) denotes the spectrum, the scale can be deﬁned for any given positive + ∀ L ∈ R , ∀ λ>1: BL/λ ≡ Tλ(BL) ⊂ BL 0 <α≤∞as: (69) 1/α i iα xie = xie (73) and therefore: i i

+ ∀ L ∈ R , ∀ λ ≥ λ ≥ 1: BL/λ ⊂ BL/λ. (70) with: i 1/µi i ∀ i, xi : xie = ||xi| e | (74) The usual Euclidean norm |x| of a metric space is the scale associated to the isotropic contraction where |·|is a given norm on the vector space. These Tλx = x/λ. The two first properties are rather equations are very convenient to define anisotropic identical to those of a norm, whereas the last one processes, including space-time processes. Indeed, is weaker than the triangular inequality, which is the Green functions defining either a generator or required for a norm. a FIF model could be defined either in the physical space or the Fourier space with the help of these scales, e.g. for a (causal) space-time process: 8.2. Linear GSI D i k x ωt This corresponds to the simplest, but already gˆ(k,x) ≡ d xdte ( + )g(x, t) startling generalization of scale invariance: Tλ is no longer a scalar transform, but remains a linear =(iω + kHt )−H/Ht (75) transform. In this case, the group property Eq. (65) corresponds to the fact that this linear trans- where Ht is the scaling anisotropy exponent form is power law of a given linear (infinitesimal) between time and space. Furthermore, to obtain a generator G: pole, therefore waves, still keeping the same scaling behavior, it suffices to consider [Lovejoy et al., −G Tλ = λ ≡ exp(−G · Log(λ)). (71) 2008b]:

Ht −H/Ht gˆw(k,x)=(iω −k ) (76) This can be easily understood with the help of the corresponding matrices. This “integral” form is as well any weighted products of g(k,x)and equivalent to the following differential one: gw(k,x), i.e. weighted convolutions of g(k,x)and gw(k,x). This already provides a wide, although d λ Tλ = −G ◦ Tλ. (72) presumably not exhaustive framework to study tur- dλ bulence and waves. Equation (74) also confirms that |·| This differential equation also governs the evo- the transformation of the norm to the generalized scale |·| corresponds to a nonlinear transform lution (λ ≥ 1) of the point xλ = Tλx ,whosetra- 1 (except for the scalar case: G =1),whichmaybe jectory starts from an initial ball BL and crosses defined as: the balls BL/λ. The classical isotropic contraction 1/µi corresponds to the scalar case: G =1,where1 xi → sign(xi)|xi| . (77) denotes the identity application. The next simplest case corresponds to the “self-affine” case [Mandel- Nevertheless, the most interesting cases cor- brot, 1985], where G is diagonalizable, with real respond to complex eigenvalues, which obviously i eigenvalues µi and eigenvectors e . Figure 21 shows introduce rotations, and nondiagonalizable genera- that “zooming” into such a multifractal leads to sys- tors (Jordan matrices). This is illustrated in Figs. 23 tematic changes in shapes of the basic structures. and 24 for a two-dimensional generator G.Thereal- This graphically demonstrates the “phenomenolog- ity of the eigenvalues corresponds to: ical fallacy” [Lovejoy & Schertzer, 2007] in which Tr(G) 2 differences in morphology are confounded with dif- a2 = − Det(G) > 0 (78) 2 ferences in mechanism. Figure 22 shows the effect of varying the “sphero-scale”; the scale where struc- and a dominant stratification effect, whereas a2 < tures are roughly “roundish”. 0 corresponds to a dominant rotation effect. For January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3441

Fig. 21. A sequence from a zoom of a stratified universal multifactal cloud model with α =1.8,C1 =0.1,H =1/3, Hz =5/9. (From top left to bottom right) Each represents a blow-up by a factor 1.31 (total blow-up is a factor ≈ 24 000 from beginning to end). If the top left simulation is an atmospheric cross-section 8 km left to right, 4 km thick, then the final (lower left) image is about 32 cm wide by 16 cm high; the sphero-scale is 1 m as can be roughly visually confirmed since the left-right extent of the simulation second from bottom on the right is 1.02 m where structures can be seen to be roughly roundish. From [Lovejoy & Schertzer, 2010]. reasons discussed below, G is parametrized in the with: following way: Del =Tr(G) (81) d + cf− e which corresponds to an effective dimension called G = ; a2 = c2 + f 2 − e2. (79) “elliptical dimension” in reference to the elliptic- − f + ed c like shape of the balls under a GSI contraction. It is merely the topological dimension d =Tr(1) All the simulations have α =1.8,C =0.1,H = 1 of the vector space for the isotropic contraction. 0.33 (the empirical parameters for clouds), and are On the other hand, differentiation of the “integral” simulated on 256 × 256 grids with the same start- property of linearity with the contraction parameter ing seed so that the differences are only due to the [Eq. (67)] yields: anisotropy (the colors go from blue to white indi- cating values low to high). However, there are much d λ xλ = ∇xxλ·G · xλ (82) more general properties. For instance, the Jacobian dλ of any contraction is Tλ: which rather corresponds to a generalization of Euler’s theorem for homogeneous function −D det(Tλ)=λ el (80) (obtained for a self-affine G). This differential form January 19, 2012 9:1 WSPC/S0218-1274 03064

3442 D. Schertzer & S. Lovejoy

Fig. 22. The eﬀect of the sphero-scale, Cloud parameters: α =1.8,C1 =0.1,H = 1/3, r = vertical/horizontal aspect ratio top to bottom: r = 1/4, 1, 4, left to right, sphero-scale = 1, 8. From [Lovejoy & Schertzer, 2010]. can be used to assure that balls are strictly decreas- decreasing with the contraction group Tλ: ing with the contraction Tλ. This can be achieved for: Spec(sym(AG)) > 0 (84) 1/2 · BL = {x | (x,Ax) ≤ L} (83) where sym( ) denotes the symmetric part of a linear application. When A is furthermore positive and where A is a given bilinear application, Eq. (82) symmetric, i.e. the ball BL is an ellipsoid, this con- indeed yields [Schertzer & Lovejoy, 1985a; Schertzer dition [Eq. (84)] reduces to: et al., 1999] the following necessary and suﬃcient condition that the balls Tλ(BL) ≡ BL/λ are strictly Spec(sym(G)) > 0 ⇔ Re(Spec(G)) > 0. (85) January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3443

Fig. 23. (Top row) k =0,wevaryc (denoted i)from−0.3, −0.15,...,0.45 left to right and e (denoted j)from −0.5, −0.25,...,0.75 top to bottom. On the right we show the contours of the corresponding scale functions. (Middle row) Same except that k = 10. (Bottom row) e =0thec is increased from −0.3, −0.15,...,0.45 left to right, from top to bottom, k is increased from 0, 2, 4,...,10. See text for more details. From [Lovejoy & Schertzer, 2007]. January 19, 2012 9:1 WSPC/S0218-1274 03064

3444 D. Schertzer & S. Lovejoy

Fig. 24. (Top row) The same as the bottom row of Fig. 23 except that e =0.75. (Middle row) c =0ande left to right is: −0.5, −0.25,...,0.75. (Bottom row): Same as the middle row except that c =0.15. In all rows, from top to bottom, k is increased (0, 2, 4,...,10), the right hand shows the corresponding scale functions. From [Lovejoy & Schertzer, 2007]. January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3445

For the two-dimensional case [and parame- skewed product called the Lie bracket [·, ·] that fur- trization given by Eq. (79)], this reduces to the thermore satisfies the Jacobi identity. For matrices, following: the Lie bracket is defined to be the commutator: Tr(G) > 0and X, Y = XY − YX (88) 2 2 (86) det(G) > 0 ⇔ d>0andd >a . In the example of the two-dimensional generator G we have: 8.3. Nonlinear GSI and Lie 2I = J, K , 2J = I,K , 2K = J, I (89) cascades whereas 1 obviously commutes with any element We will use the example of the two-dimensional gen- of this Lie algebra l(2,R) of the two-dimensional erator G with its above parametrization [Eq. (79)] real matrices. Recall that any Lie algebra g is said that corresponds to a “pseudo-quaternions” repre- to be abelian if its bracket, whatever is its expres- sentation: sion, vanishes (i.e. ∀ X, Y ∈ g :[X, Y ]=0,in G = d1 + eI + fJ + cK; short: [g, g]=0),aLiesubalgebra s of g is a subspace of g that is closed under the Lie bracket (i.e. 10 0 −1 1 = , I = , [s, s] ⊂ s), a subspace l of g is an ideal of g if s is 01 10 (87) not only closed with itself but with g (i.e. [g, l] ⊂ l), 01 10 the largest abelian ideal of g is called its radical J = , K = and if it is zero g is called semi-simple. The cru- 10 0 −1 cial importance of abelian (sub-)algebra is due to to point out that in a rather general manner linear the fact that the corresponding Lie (sub-)groups are GSI results can be extended, albeit given technical indeed commutative, i.e. the symmetries commute. difficulties briefly mentioned below, to the nonlinear With the help of these definitions, it is rather GSI and lead to the general notion of Lie cascades. straightforward to check that the one-dimensional Let us first point out that there are at least two rea- subalgebra R generated by 1 is the radical of l(2,R), sons to look for nonlinear GSI. The first one is that whereas s spanned by {I, J, K} is semi-simple. The one has often to work with manifolds rather than latter is classically known as sl(2,R), the special with vector spaces, the second one [Schertzer & two-dimensional real linear algebra of matrices with Lovejoy, 1991], not necessarily independent of the zero trace. We have: previous one, is that one has often to deal with l(2,R)=R ⊕ sl(2,R) (90) local symmetries, e.g. the original (Weyl’s) local gauge invariance, rather than with global ones. The which is merely a particular example of the Levi main point is that the generators of the (local) decomposition of any Lie algebra into its radical symmetries define Lie algebra whose structure is and a semi-simple subalgebra. It is also important essential to understand the interrelations between to note that the three two-dimensional subalgebra various symmetries. si (i =1, 3) spanned respectively by {1, I}, {1, J}, Indeed, let us consider that the symmetries {1, K} are all abelian (s1 is merely equivalent to the Tλ and T˜λ, therefore the whole scale symmetry set of complex numbers), but they are not ideals of Sλ [Eq. (40)], together with all other potential l(2,R). It means that the corresponding subgroups symmetries (e.g. more classical symmetries such are commutative, but do not commute with any ele- as rotations) smoothly vary with respect to their ment of the full group generated by l(2,R). A simple parameters. These symmetries not only form a consequence is that if the generator G of Tλ (resp. group with respect to their composition, but also G˜ of T˜λ) belongs to s3, then it will commute with a smooth manifold and therefore a Lie group G any symmetry generated by K, but not with those (e.g. [Sattinger & Weaver, 1986]). In a rather gen- generated by I, i.e. rotations. eral manner, this group is generated from the Whereas the mapping from Lie groups to Lie symmetries that are infinitesimally close to the algebra is rather one-to-one, the inverse is often identity (for infinitesimally small parameter vari- more complex: different Lie groups may have the ation), which spans the tangent space to the iden- same Lie algebra. Nevertheless, the exponential tity transformation. In fact, these generators form map allows to fully capture the local structure of the a Lie algebra g, i.e. a vector space with a (bilinear) group from its algebra. Here, the exponential map January 19, 2012 9:1 WSPC/S0218-1274 03064

3446 D. Schertzer & S. Lovejoy

20 10

10 5

−5 5 10 −10 −5 5 10

− −10 5

−10 −20

(a) (b) Fig. 25. GSI balls for λ =1to10bystepsδλ =0.5: (a) linear case for g(x)=Gx [Eq. (91)] where the matrix is defined according to Eq. (79) with the set of parameters (d, c, f, e)=(1, 0.5, 0.5, 1), i.e. stratification being dominant, (b) nonlinear 3/4 case: the parameter e now depends on the location x : e(x)=|x| . This introduces an extreme differential rotation. from a tangent space to its manifold correspond to systems. Indeed, the classical manner to deal with the generalization of the exponential function, par- such problems was to first proceed to a scale anal- ticularly seen under its differential form as: ysis, i.e. to define so-called characteristic quanti- dxλ g(xλ) ties that help to select a few corresponding rele- = − . (91) dλ λ vant terms that will be retained to build up approx- Loosely speaking, one has only to carefully imations of the original equations. However, as dis- distinguish the tangent space from its manifold, cussed by analysis [Schertzer et al., 2012], on the whereas both are confused in the (linear) case of particular example of the quasi-geostrophic approx- vector spaces. The above mentioned limitation to imation [Charney, 1948], there are many reasons to the local structure is that the integration can intro- expect that the scaling of the approximation will be different from the original equations. GSI allows to duce an external outer scale, i.e. the inverse of Tλ = proceed in a different manner: an anisotropic analy- T1/λ, i.e. a dilation, for large λ is not always defined. As simple examples, we can consider the inte- sis enables us to select the relevant interactions that gration of the nonlinear differential equation (91), occur on a wide range of scales (contrary to the scale analysis, which is local in scale) and obtain a new with the initial condition x1 belonging to a given set of equations, which corresponds to a statistical ball BL, still on a vector space E, but where g breaking of an isotropic statistical symmetry, which is nonlinear. Its gradient ∇xg is equivalent to a local (infinitesimal) generator G. Therefore, the can be seen as a broad generalization of the spon- condition of (local) growth of the balls for linear taneous symmetry breaking. Here, the anisotropic scaling splits a vector equation that admits isotropic GSI applies now to ∇xg. For the two-dimensional case, the same parametrization as before for G scaling into separated component equations. In the [Eq. (87)] corresponds to spatially variable coef- case of the vorticity equation this can be obtained by first decomposing the fields and operators into ficients (d, c, e, f)andifBL is a symmetric ellip- horizontal and vertical components (with respective soid, ∇xg needs only to satisfy the same condition [Eq. (86)] as G. indices h and v), e.g. for the velocity field u and gra- ∇ Figure 25(b) is one among many figures that dient operator : can be obtained in this nonlinear manner. Note due ∇ ∇ ∇ to nonlinearity the balls Bλ, are no longer convex u = uh + uv; = h + v (92) sets, contrary to the linear case. and with the help of the following pushback trans- 8.4. GSI, scaling analysis and forms [Schertzer et al., 2012]: differential systems T ∗(u)=λγ(u + λhu ); GSI has a huge potential of applications when λ h v ∗ γ −h (93) dealing with anisotropy and classical differential Tλ (∇)=λ (∇h + λ ∇v) January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3447 where h is the scaling anisotropy exponent of the 9. The Extremes: Self-Organized vertical vs. the horizontal, as well as the correspond- Criticality and Multifractal ing decomposition of the vorticity ω into: Phase Transitions

ω = ωv + σ + τ; ωv = ∇h × uh, 9.1. The singular small-scale limit (94) of a cascade process σ = ∇h × uv,τ= ∇v × uh The small-scale limit λ →∞of a cascade process is where ωh = σ+τ is the horizontal vorticity, which is very singular since for any positive singularity γ, γ assumed to be negligible at large scales where the the density ελ ≈ λ diverges. These divergences (almost vertical) Earth rotation Ω is assumed to are statistically significant for γ>C1, since it be dominant. A careful examination of the scaling corresponds to q>1andK(q) > 0, therefore K(q) exponents resulting from Eq. (93) involved in the to ελ = λ →∞. This singular small-scale vorticity equation (D./Dt denotes the Lagrangian divergence means that if a limit exists, it is not in time derivative): the sense of functions. This is similar to the Dirac δ-“function”, in fact the Dirac δ-measure, which is Dω = ω ·∇u (95) not a function at all but a “generalized function” Dt defined as a limit of functions. The Dirac δ-measure points out that the latter splits into the three fol- is indeed only meaningful if we integrate over it. It lowing equations: is rather obvious that the β-model corresponds to a (random) generalization of the Dirac δ-measure Dσ for points belonging to a fractal set of codimen- = σ ·∇huh Dt sion c = D − Ds. Conversely, the Dirac δ-measure can be understood as the particular (deterministic) Dτ =(τ ·∇h + ωv ·∇v)uh (96) case corresponding to a codimension c = D and Dt dimension Ds = 0, i.e. to isolated points. Dωv As a consequence, one has to consider the limit =(τ ·∇h + ωv ·∇v)uv → Dt of the corresponding measures Πλ(A) Π∞(A) over compact sets A of dimension D, i.e. the D- Conversely, these three equations, together dimensional integration of the density ελ over A: with the large scale condition ωv ≈ Ω, allow an D anisotropic scaling like that displayed by Eq. (93), Π∞(A) = lim Πλ(A) = lim ελd x. (98) as well as a nonlinear growth of the horizontal λ→∞ λ→∞ A vorticity. The latter mechanism is absent from In agreement with turbulent nomenclature, the the quasi-geostrophic approximation, which corre- integral Πλ can be termed a “flux” (of energy) sponds to σ ≡ τ ≡ 0 and the approximation: through the scale l = L/λ,whereasελ can be called (energy) flux density at this scale. Therefore, we Dωv ≈ Ωv ·∇vuv. (97) anticipate that the fluxes (but not the densities) Dt converge as λ →∞. Still due to the singular nature Contrary to the quasi-geostrophic equations, of the limit, we may expect that convergence can the fractional vorticity equations (96) are not only occur over a limited range of order q of the an approximation of the vorticity equations (95), statistical moments: because solutions of the former are also solutions of q ∃ qD > 1, ∀ q ≥ qD : Π∞(A) = ∞ (99) the latter. They rather correspond to select relevant interactions to provide solutions having a scaling as whereas: prescribed by Eq. (93). This explains the extreme q ∀ λ<∞ : Πλ(A) < ∞. (100) contrast between Eq. (96) with a 3D nonlinear vorticity stretching and Eq. (96) with only a weak, lin- The subscript D of the critical order q under- ear stretching. We believe that this scaling analysis scores its dependence on the dimension of the space is rather illustrative of the potential of GSI to bet- over which it is integrated. This dependence can ter investigate the nonlinear generating equations be used [Schertzer & Lovejoy, 1984] to demonstrate of anisotropic scaling fields, well beyond the scale that cascade processes are generically multifractals: analysis and the resulting classical approximations. the dependence of qD on D defines a hierarchy of January 19, 2012 9:1 WSPC/S0218-1274 03064

3448 D. Schertzer & S. Lovejoy

∗ fractal sets. An important consequence of the diver- the pushback operator Tλ λ. For small enough order gence of moments [Eq. (99)] is also the exponent of moments, the flux prefactor remains finite in the the power-law fall-off of the probability distribution: limit Λ →∞, the bare and dressed scaling proper- −q ties are the same. However for q ≥ qD this prefactor ∃ qD > 1,π 1:Pr(Π∞(A) >π) ≈ π D . begins to scale with Λ/λ, so that a drastic change (101) occurs; it diverges with Λ. Equations (99)–(101) are equivalent and the In a real system, the scaling is cutoff by dissi- divergence of moments has many practical impli- pation so that while the moments qqD will depend on the small scale considered the hallmark of self-organized critical- details. Since it is this small scale activity that ity (SOC; [Bak et al., 1987, 1988]) so that cascade causes the divergence, in analogy with the classical processes can be considered a nonclassical route to (chaotic) metaphor for sensitive dependence, this SOC. In this respect, we could note an important has been called the “multifractal butterfly effect” difference: classical SOC is a cellular-automaton [Lovejoy & Schertzer, 1998]. model (the prototype being the “sandpile” and its avalanches) that requires a “zero-flux” limit: in the 9.3. Scale dependence and sandpile each grain must be added one at a time divergence of the flux : The only after the activity induced by the previous grain heuristic argument is over. In contrast, the cascades require a quasi- constant flux, and this is generally more physical. Let us first consider some simple heuristics whose To graphically appreciate the propensity for main interest is that they are model independent. multifractal processes to generate extremes at all They are based on the fact that a D-dimensional scales, see Fig. 26. In order to make the point that integration of a singularity γ just corresponds to − occasionally very rare events occur, a single value shifting the latter by D, which corresponds to (the maximum) of the subgenerator was boosted by the scaling exponent of the elementary volume of afactorN (placed at the centre of the simulation). integration. As a consequence, all singularities of order γ

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3449

(a) (b) Fig. 26. This shows two realizations (a) of a random multifractal process with a single value of the maximum of the subgen- (1/2) erator (at the centre of a 512 × 512 grid) boosted by factors of N from 16 to 64 (increasing by 2 from top to bottom) in order to simulate very rare events. The scaling is anisotropic with complex eigenvalues of G, the scale function is shown in (b). duality (Sec. 4.2), we see that the divergence of the This heuristic argument can be made rigor- statistical moments for q ≥ qD is a consequence ous by introducing [Schertzer & Lovejoy, 1987] of the fact that a straight line is singular for the the Trace Moments of the ﬂux which are sim- Legendre transform. We therefore obtain: pler to handle than the statistical moments of the ﬂux. q

3450 D. Schertzer & S. Lovejoy moments: and larger sampling singularity γs [Eq. (19) and Fig. 8(b) for illustration]. q q X = x dPX (105) The Legendre transform of cs(γ)=c(γs)with γ ≥ γs leads to a spurious linear estimate Ks we only deal with estimates, the most usual ones instead of the nonlinear K for q>qs where qs = being an average over the Ns independent samples: c (γs) is the maximum moment that can accurately Ns be estimated: q 1 q {X }s = Xi . (106) q ≥ qs : Ks(q)=γs(q − qs)+K(qs); Ns i=1 (108) qphase transition and therefore is rather q q X = lim {X }s. (107) model-independent. Ns→∞ One may also consider space/time averages and 9.5. Multifractal phase transitions ergodicity assumptions. In our case, we must con- 9.5.1. Flux dynamics and thermodynamics sider a combination of statistical and space/time averaging. A first consequence of finite Ns is that As discussed by various authors [Schuster, 1988; only a limited range of q’scaninfactbesafely Tel, 1988; Schertzer & Lovejoy, 1992, 1994; explored: as we now show, estimates of moments Schertzer et al., 1993], there are formal analogies of higher order give no real information about the between multifractal exponents and standard ther- process and may lead to erroneous interpretations, modynamic variables. However, depending on the e.g. there had been speculations on the significance multifractal framework there are notable differences of the q →∞limit on the basis of finite empirical of appreciation. For the codimension multifractal samples of turbulence data. formalism, Table 1 displays the analogues between The limits imposed by finite sample sizes can what can be called (statistical) “fluxdynamics” best be understood with the help of the sampling [Schertzer & Lovejoy, 1991] and classical thermody- dimension Ds [Eq. (19) and Sec. 4.1]. As we increase namics. The expression “flux dynamics” comes from Ds (i.e. the number Ns of samples), we gradually the fact that the main quantity of interest is the explore the entire probability space encountering energy flux for systems out of equilibrium instead extreme but rare events that would almost surely of the energy for systems in thermodynamic equi- be missed on any finite sample (Fig. 27), i.e. larger librium. Since the analogies are based on the exponents of the probability and number densities these A analogues are more easily and naturally obtained Probability in the codimension framework, which define respec- Space tively the codimension c(γ) of a singularity γ and the D −S(E) entropy of a state energy E.TheLeg- Physical endre conjugate variables of the singularity and the space energy are respectively the moment of order q and the (reciprocal) temperature β =1/T . Similarly, the scaling moment function K(q) is the analogue of a (Massieu) potential (= the free energy/T ). D Discontinuities in the analogues of the free N ~ λ s s energy (the dual codimension function C(q)= − Physical Independent K(q)/(q 1)) and the thermodynamic potential space Realizations (K(q)) correspond to multifractal phase transitions (see Fig. 28 for illustrations). However, thermody- Fig. 27. Illustration showing how in random processes the effective dimension of space (D) can be augmented by con- namics applies to systems in equilibrium and with- sidering many independent realizations Ns.AsNs →∞,the out dissipation, whereas flux dynamics applies to entire (infinite dimensional) probability space is more and systems far from thermodynamic equilibrium which more explored. From [Schertzer & Lovejoy, 1993]. are strongly dissipative. A practical consequence is January 19, 2012 9:1 WSPC/S0218-1274 03064

Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3451

Table 1. Correspondence between ﬂuxdynamics and thermodynamics (setting for notation simplicity kB =1for the Boltzman’s constant kB): Σ(β) being the Massieu potential, F (β) the Helmhotz free energy. From [Schertzer & Lovejoy, 1993].

Flux Dynamics Thermodynamics

Probability space Phase space −1 Moment order: q (Reciprocal) Temperature: β = T Singularity order: γ (Negative) Energy: −E Generator (Negative) Hamiltonian Singularity codimension: c(γ) Codimension of entropy: D − S(E) Scaling moment function: K(q)=max(qγ − c(γ)) (Negative) Massieu potential: −Σ(β)=− min(βE − S(E)) γ E Dual codimension function: C(q)=K(q)/(q − 1) (Negative) Free energy −F (β)=−Σ(β)/β Dimension of integration: D External field: h Ratio of scales: λ Correlation length: ξ that in order to define its statistics, a multifractal to discrete cascades and furthermore to Ds =1. process fundamentally requires an infinite hierarchy On the other hand, the notion of second order of temperatures, rather than a single temperature. phase transition is interesting, because it is rather Therefore observing (i.e. with a finite Ns,Ds)amul- model-independent as it is based on the analogies of tifractal process at a given temperature (i.e. q)only the statistical exponents of the cascade. Indeed, it gives very limited information about the process. should occur as soon as there are no bounds on the Similarly, a multifractal phase transition is associ- singularities or their range exceeds the critical γs. ated with a qualitative change of observation of the same system when one changes the observation tem- 9.5.3. First order phase transition perature (i.e. changes q), whereas a thermodynamic phase transition corresponds to a qualitative change We can now revisit the question of the divergence of the behavior of the system under observation. of moments (Sec. 9.3) taking care of the sample

9.5.2. Second order phase transition 17 Finite sample size eﬀects (Sec. 9.4) can be now 15 understood as corresponding to a phase transition 13 of second order and in fact a “frozen free energy” 11 transition which has been discussed in various con- texts [Derrida & Gardner, 1986; Mesard et al., 1987; 9 K(q) Brax & Pechanski, 1991]. Indeed, we saw that the 7 almost sure highest order singularity (γs)whichcan 5 s be observed on N realizations, yields with the help 3 of the Legendre transform a linear behavior of the 1 observed Ks [Eq. (108)] for q>qs, whereas it is nonlinear as K(q)forq

3452 D. Schertzer & S. Lovejoy size finite effects, in the heuristic and very general K(q) framework we discussed in Sec. 4. We pointed out 3.0 that above a critical singularity γD, the dressed codimension cd(γ) becomes linear [Eq. (102)]. Due 2.5 to the definition of the codimension [Eq. (16)], this corresponds to a power-law for the probabil- 2.0 ity distribution [Eq. (101)], and by consequence to a divergence of statistical moments [Eq. (99)]. 1.5 However, due to the finite size of the samples, one 1.0 obviously cannot observe with the help of the statistical moment estimates this divergence, but in 0.5 fact only a first order transition, instead of the sec- q ond order transition discussed above (Sec. 9.5.2). 0.0 D Indeed, following the argument for Eq. (19), the maximum observable dressed singularity γd,s is the -0.5 543210 solution of: q

cd(γd,s)=∆s. (110) Fig. 29. The empirical scaling exponent moment function Kd,s(q) of the atmospheric turbulence energy flux using By taking the Legendre transform of cd with respectively 4 (crosses) and 704 (dots) 10 Hz time series, the restriction γd ≤ γd,s, we no longer obtain the compared to the (bare) K(q) (continuous line) with universal multifractal parameters α =1.45 and C1 =0.24. All theoretical Kd(q)=∞ for q>qD [Eq. (103)], but these curves collapse together until qD ≈ 2.4, whereas for then obtain the finite sample dressed Kd,s(q): larger moment orders q’s the empirical estimates become linear. The curve Kd,s(q) corresponding to an average of 704 q ≤ qD : Kd,s(q)=K(q); realizations is clearly above Kd,s(q), which is only possible (111) ≥ − for a first order multifractal transition, whereas a second q qD : Kd,s(q)=γd,s(q qD)+K(qD). order transition curve would correspond to a tangent to K(q). Reproduced from [Schmitt et al., 1994]. As expected, Eq. (103) is recovered for Ns → ∞, due to the fact that γd,s →∞.ForNs large but finite, there will be a high q (low tempera- universal multifractals (see Fig. 28), as well on ture) first order phase transition, whereas the scale atmospheric data (see Fig. 29). breaking mechanism proposed for phase transitions in strange attractors [Szépfalusy et al., 1987; Csor- das & Szépfalusy, 1989; Barkley & Cumming, 1990] 10. Future Directions is fundamentally limited to high or negative tem- The goal of this paper was to present the main peratures (small or negative q). This transition cor- theoretical developments of multifractals and gen- responds to a jump in the first derivative K (q)of eralized scale invariance during the two and a half the potential analogue [Schertzer et al., 1993]: decades of their explicit existence, including numerical simulations and data analysis. We hope that it ∆K (qD) ≡ K (qD) − K (qD) d,s reveals the awesome simplification that multifrac- ∆s − c(γD) tals can bring to complex systems by reducing their = γd,s − γD = . (112) qD complexity with the help of a quite diverse and widely applicable scale symmetries. On small sample size (∆s ≈ c(γD)), this transi- Although multifractals are increasingly under- tion will be missed, the free energy simply becomes stood as a basic framework for analyzing and frozen and we obtain: Kd,s(q) ≈ (q−1)D,whichwas simulating extremely variable processes, in many already discussed with the help of some empirical scientific disciplines and in particular in geophysics, data [Schertzer & Lovejoy, 1984], whereas Eq. (111) there is still a wide gap between their potential corresponds to an improvement of earlier works and their actual use. Beyond the familiar slow on “pseudo scaling” [Schertzer & Lovejoy, 1984, and inhomogeneous pace of the diffusion of scien- 1987]. The above relations, especially Eq. (112), tific knowledge, an additional reason for this could were tested numerically with the help of lognormal be obstacles inherited from older, preexisting and January 19, 2012 9:1 WSPC/S0218-1274 03064

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