Two highly singular intermittent structures: Rain and turbulence

Authors


Abstract

[1] Rainfall charges soil moisture and river basins among its many roles with respect to the hydrologic cycle. Research aimed at improved understanding and modeling of surface water processes includes attention to rainfall at a variety of space-time scales. Given the atmospheric environment in which rain events are observed, some similarities between certain rainfall data structures and fluid turbulence can be expected. So the space-time intermittency and large fluctuations observed in both rain rates and energy dissipation rates have provided an interest among hydrologists in developing physical theories, experiments, and mathematical models. In response to a request for insights into multiplicative cascade models, the main goal of this article is to single out a special mathematical transformation, namely, “size biasing” (or “tilting”), which has proven to be very powerful in the mathematical analysis of multiplicative cascades and which has also been successfully exploited within the context of turbulence from a physical perspective.

1. Introduction

[2] In view of the space-time scales at which modeling and understanding are sought, many problems of surface water hydrology are not amenable to the classic deterministic methods of physics and mathematics. A prominent example being the PUB problem, an acronym coined for Prediction from Ungauged Basins. Broadly speaking one seeks to predict the discharge from river basins using information gleaned from geographic (topographic) maps and local climate patterns, but in the absence of streamflow gauges and other such network measuring devices [see Gupta and Waymire, 1997, 1998, and references therein].

[3] The salient fact is that many of the problems for surface water hydrology fall within John von Neumann's “medium scale range” for fluid motions. More specifically, in contemplating the future impact of computational technology at a historical Conference on the Application of Numerical Integration Techniques to the Problem of the General Circulation held in the mid-1950s, von Neumann [1960] called attention to a “medium-range” scale which is too long from the point of view of an initial value problem, and too short for a steady state theory to be applicable. In particular, this is a scale range over which statistical fluctuations cannot be ignored.

[4] It is not surprising that stochastic models would find such widespread use as found in hydrology during the past fifty years. In the case of rainfall, many of the early attempts toward classes of models with a hierarchical structure fall within the broad framework introduced by LeCam [1961]. This broad class of random fields continues to provide a useful framework for modeling and data analysis in hydrology. However, there are problems such as the PUB problem for which such models cannot be adequately calibrated. These are problems for which one may attempt to identify and exploit certain statistical multiscaling structure, self-similarity, and fractal geometry; see Gupta [2004], Rodriquez-Iturbe et al. [1998b], Reggiani et al. [2001], and Burd et al. [2000] for recent general formulations and illustrative results in the context of river basin hydrology.

[5] Continued progress in connection with the PUB problem, for example, will depend on further research on both landform and atmospheric processes, and their physical and biological interactions. While outside the scope of this article, the development of the fractal geometry of river networks and flows illustrates a related facet of the overarching direction of such research; for example, see Rodriguez-Iturbe and Rinaldo [1997] and Veitzer et al. [2003] for general observations and specific results. For rainfall, one may cite the work by Lovejoy [1982] and Lovejoy and Mandelbrot [1985] as an early identification of scaling structure in rain fields that has continued to be explored and refined [see Lovejoy and Schertzer, 1985, 1990; Schertzer et al., 1991; Gupta and Waymire, 1993; Over and Gupta, 1994; Rodriguez-Iturbe et al., 1998a]. Also see Catrakis and Domotakis [1998] for new ideas in turbulence related to the main topic of this paper. In particular, it is here that rainfall and turbulence theory have found a common home as highly singular and intermittent data structures.

[6] This paper has an intentionally narrow technical focus and is far from comprehensive. Certainly exciting new ideas not mentioned here continue to be explored in the literature. The main objective of this paper is simply to highlight a mathematical notion which has proven to be powerful for analysis of singular structures and, with a broader aim, toward engaging the physical scientists in further developing its applications to data structures of the type found in rain and turbulence. To this end, in section 2 we very briefly recall the multiplicative cascade model. This is followed by section 3 in which we focus on the role of “size biasing” from a mathematical perspective. In section 4 its significance is discussed within statistical turbulence theory.

[7] As remarked by LeCam [1961, p. 165] in connection with rainfall models: “A true model should take into account the laws of fluid mechanics and thermodynamics.” The same can of course be said for turbulence models, where the basic physics is thought to be captured by the Navier-Stokes equations. However in neither case are the problems solved. In the final section we describe some recent mathematical efforts to analyze Navier-Stokes equations by means of certain natural multiplicative cascades recently uncovered in Fourier frequency space. For simplicity of the mathematical exposition, the main ideas behind this representation are explained in terms of Burgers equation. Interested readers are referred to Waymire [2005, and references therein] for the corresponding exposition in the context of three-dimensional incompressible Navier-Stokes equations. However it must be emphasized that neither the mathematical theory nor the physics of these ‘stochastic cascade representations’ to Navier-Stokes in Fourier frequency space is sufficiently well developed to claim any connection with the multiplicative cascade model in physical space; also see Ossiander [2005] for recent efforts to develop a corresponding multiplicative representation in physical space, but subject to the same caveat.

2. Multiplicative Cascade Models

[8] The random multiplicative cascades comprise a familiar class of models in both rainfall and turbulence research. These models are of special interest for their intermittency, extreme variability, and multiscaling structure. The origins of this class of models trace back to classic works of Richardson [1922], Kolmogorov [1941, 1962], and Yaglom [1966] in the statistical turbulence theory. A rich geometrical and scaling perspective was subsequently advanced by Mandelbrot [1974], and a complete mathematical treatment was initiated by Kahane and Peyrière [1976]. In particular, Mandelbrot [1974] stimulated consideration of alternatives to Kolmogorov's lognormal hypothesis based on consideration of scaling exponents.

[9] In the context of turbulence one imagines introducing energy into the fluid by a large-scale stirring motion, whereby smaller-scale eddies split off and dissipate energy in random proportions W to that available. These eddies in turn split off smaller-scale eddies, and so on. In the simplest mathematical formulation one considers random measures Mn(dx) on the one-dimensional unit interval [0,1] having a piecewise constant density ρn(x) over a dyadic subinterval

equation image

by a constant value

equation image

where the random factors Wv, v = 〈j1j2jk〉, are independent and identically distributed (IID) nonnegative mean one random variables indexed by the vertices v = 〈j1j2jk〉, ji ∈ {0, 1}, of the binary tree; one may view such v as the pixel address at the k level of resolution.Remark 1. As pointed out by Mandelbrot [1974], turbulence data is often in the form of one-dimensional projections. As a result, individual sample realizations will not be conservative. The mean one condition (EW = 1) provides conservation on average of Mn(dx) = ρn(x)dx where for any subregion Δ = (a, b) ⊆ [0, 1],

equation image

Thus the “conservation on average” is reasonable for the cases of models based on lower-dimensional traces since sample realizations of lower-dimensional projections of of flow will not conserve energy [see also Jouault et al., 2000].

[10] The multiplicative cascade M(dx) is the random measure obtained by passing to the fine-scale limit as n → ∞. The main point of emphasis for this paper is the following simple well-known observation: According to the Strong Law of Large Numbers and (strict) Jensen's inequality, one has for any product of IID mean one positive nondegenerate random variables

equation image

with probability one as n → ∞. Thus the limit of the ratio is negative and finite. This implies (for the numerator) that equation imageimage → 0 with probability one. Since there are uncountably many points in the spatial domain (i.e., paths through tree) events with probability zero can “add up.” Such deviations from the average behavior is the subject of section 3.

[11] The random cascade is defined by the branching number parameter b (equal to 2 in this exposition) and the random factors Wv, referred to as cascade generators. Kolmogorov's “lognormal hypothesis” refers to a choice of lognormal distribution for W. As suggested by the above, however, the structure of cascades is due to a sufficiently high rate of deviation from averages, rather than the universality of the Gaussian distribution furnished by the central limit theorem. Starting with the work of Mandelbrot [1974], tests of this hypothesis and physical arguments for alternative laws have been topics of lively discussion in modern statistical turbulence theory [e.g., Frisch, 1999; Molchan, 1997, Jouault et al., 2000; Vainshtein and Sreenivasan, 2004; Dubrulle and Graner, 1997; Padoan et al., 2003].

[12] We close this section noting that the multidimensional formulation of the multiplicative cascade model presented in this section is straightforward, and cascades over two-dimensional squares and/or three-dimensional boxes will be left to the reader to construct. Refinements for which the branching number b may be viewed as a random parameter are given by Peyrière [1977] and Burd and Waymire [2000] and as a “continuous parameter” by Barral and Mandelbrot [2002]. This latter class of models is a natural extension of the discrete parameter multiplicative cascade models. Interesting arguments for consideration of models in which the cascade generators are statistically correlated have recently been developed in the physics literature [see, e.g., Biferale et al., 1999; Benzi et al., 2003; Cleve et al., 2005]. Various extensions of the mathematical theory for correlated cascades were obtained by Waymire and Williams [1995, 1996]. However, such models, both physical and mathematical, are well outside the scope of this article.

3. Size-Biasing (or Tilting) Probabilities

[13] Size biasing, or tilting, is a change of measure transformation with mathematical roots in the theory of large deviations. Because of its proven importance for both the mathematical analysis of multiplicative cascades, as well as in more physical considerations for singular events in turbulence, we will sketch Cramer's main ideas from the point of view of large deviation theory in some detail. Ellis [1985] provides a comprehensive treatment with applications to statistical physics.

[14] Suppose X1, X2, … is an IID sequence of random variables with mean μ. Then according to the law of large numbers, one has the well-known stabilization of averages AN for large N:

equation image

In particular, given any δ > 0, the event that AN would fall outside the interval μ ± δ (i.e., “would deviate from μ by a positive amount δ”) is a rare event for large N. In fact, under suitable conditions on the distribution of the observations Xn, n ≥ 1, the probability is exponentially small for large N, e.g.,

equation image

where I(δ) = − c*(μ + δ) is the large deviation rate and is expressed in terms of the Legendre transform

equation image

of the cumulant generating function

equation image

assumed for simplicity of this exposition to exist and be finite for all h.

[15] To understand formula (4), first note the simple inequality

equation image

for all h ≥ 0; we use 1[F] to denote the 0–1 valued indicator of an event F. Since by independence, the moment generating function of NANX1 + … + XN may be expressed as eNc(h), one has for any h ≥ 0,

equation image

In particular, choose h = hδ to maximize the exponent and obtain an (upper bound) for the rate of decay of probability in (4).

[16] So let us see that this bound is the actual large deviation rate by computing the reverse inequality in the equality asserted by (4). For this it is useful to exponentially tilt or size bias the distribution of X in such a way that the deviant event is the rule, rather than the exception. For the given deviation μ + δ, suppose that the maximum defining the Legendre transform is achieved at h = hδ, with

equation image

and

equation image

at h = hδ. In particular μ + δ − equation imageimage = 0. So define a random variable equation image to have the size-biased distribution given by

equation image

where Zδ = image = image normalizes imagedy) to a probability distribution. This is referred to exponential size biasing of the distribution of X. Now observe that

equation image

That is, for the size-biased distribution, the deviation by δ is to be expected for the average behavior. In particular, the law of large numbers yields

equation image

From here one may obtain the reverse inequality by the law of large numbers under size biasing: Namely, let ε > 0, and consider deviations of size μ + δ (to within ±ε) defined by

equation image

Note that exp{−Nh(μ + δ + ε) + hequation imageXj} ≤ 1 upon the event [(X1, …, XN) ∈ DN]. Thus one has for h = hδ ≥ 0

equation image

Now, the law of large numbers under the size-biased distribution (having mean μ + δ) makes equation imageN → μ + δ and hence P(equation imageN ∈ (μ + δ − ε, μ + δ + ε)) → 1 as N → ∞. In particular, it follows from (5) that for any ε > 0,

equation image

Now let ε ↓ 0 to see the reverse inequality underlying (4).

[17] In applications to multiplicative cascades one considers the behavior of (cascade) products W1WN of mean one positive random variables in place of sums. However, in view of the transformation

equation image

one effectively considers large deviations in the sum X1 + … + XN where X = log W. In particular, the exponential size bias of the distribution of X coincides with the power function size bias of the distribution of W in the sense that

equation image

For a simple explicit example illustrating the size bias transform within the present framework note that if W has the (mean one) uniform distribution on [0, 2], then the hth power function size bias is given by the pdf 2−(h+1)(h + 1)yhdy, y ∈ [0, 2], with mean 2(h + 1)/(h + 2) → 1 and variance 4(h + 1)/{(h + 3)(h + 2)2} → 0 as h → ∞.

[18] On the purely mathematical side, a spatial extension of the size bias change of measure has been demonstrated to be an extremely powerful mathematical transformation for the analysis of multiplicative cascades. The general idea for the spatial extension is to first fix an infinite path t of vertices 〈j1〉, 〈j1j2〉, …, 〈j1j2jn〉, …, ji ∈ {0, 1}, through the cascade binary tree and change the joint probability distribution P, say, of the cascade generators Wv: ∣v∣ ≤ n down to scale n by

equation image

Then the joint probability of a sample of generators dω and randomly selected path dt is defined by

equation image

Now observe that the marginal distribution of cascade generators Wv: ∣v∣ ≤ n is obtained by summing over the paths to get

equation image

where Zn = Mn([0, 1]) is the total cascade measure. If, for example, the distribution P of generators is that of IID positive mean one random variables with density f(x) then the Pt distribution of the generators are IID with density f(x) off the t path, but generators along the t path will have size-biased density xf(x). Similarly one may more generally size bias along paths by equation imageP. The reader is referred to Kahane and Peyriére [1976], Burd and Waymire [2000], Holley and Waymire [1992], Waymire and Williams [1994, 1995, 1996], and Ossiander and Waymire [2000] for a variety of mathematical illustrations of the utility of this spatial size bias change of measure. From an intuitive perspective, since as explained earlier, any product along a cascade path of IID mean one positive random variables a.s. tends to zero as the number of factors increases in the fine-scale limit, the nondegenerate, albeit highly singular, structure of the multiplicative cascade is due to large deviations from this average behavior. The continuous nature of space, [0,1] for this illustration, means that uncountably many such products occur in the limit. The spatial extension of the size bias (change of measure) transform is precisely the correct mathematical tool to capture the frequency of these fluctuations relative to the scale resolution of the cascade.

4. Size-Biased Moments and Turbulence

[19] There is not a universally accepted definition of turbulence. Physically, one might describe it as: A fluid motion involving extreme fluctuations and high degrees of intermittency in space and time. From a mathematical perspective it may be defined by a model. A purely analytic formulation as a lower-dimensional (“intermittent”) manifold of singularities (“extreme fluctuations”) in 3-D incompressible Navier-Stokes equations was spawned by pioneering work of Jean Leray in the early 1930s. However, that such a definition is nonvacuous ranks among the outstanding unsolved mathematical problems of modern applied mathematics. Alternative deterministic models have been given in the form of chaotic dynamical systems and various modifications of the Navier-Stokes equations. Statistical considerations can be traced to early work of G. I. Taylor and others during this same period. The conceptualizations introduced and refined by Richardson [1922], Kolmogorov [1941, 1962], Yaglom [1966], and Mandelbrot [1974] comprise the statistical cascade theory explored in this paper.

[20] Although we do little more than point to the Navier-Stokes equations in this section, it will be useful to have them for reference. The 3-D incompressible Navier-Stokes equations can be expressed in nondimensional form as

equation image

The region G may be thought of as a box of fluid, and the velocity u is assumed to satisfy some boundary condition, e.g., no slip or periodic. The left hand side provides the Eulerian acceleration, while the terms on the right hand side represent viscous dispersion, a pressure gradient, and external forcing, respectively. The nonlinear convective term u · ∇u is intrinsic to the definition of acceleration and not an artifact of modeling.

[21] The nondimensional form of the Navier-Stokes equations arises by setting a reference length scale l*, say the diameter of the region G, a timescale t*, say the time required for a large scale eddy to cycle. This gives rise to a characteristic velocity U* = equation image. By comparing the relative strengths of the inertial term ∣u · ∇u∣ = equation image to that of the dispersive term ∣νΔu∣ = νequation image one arrives at the celebrated Reynolds number dimensionless ratio given by

equation image

Turbulent motion is reflected in the predominance of the inertial term, i.e., Re ≫ 1.

[22] As energy is introduced at the macroscale l*, e.g., by a stirring of the fluid, it will gradually dissipate in the form of heat at sufficiently small scales. This (micro) dissipation length scale l* was identified by Kolmogorov by simply comparing the inertial term to the dissipative term. In particular one sees a transition from the relative sizes occurs at the length scale

equation image

Note that l* → 0 in the limit of large Reynolds number. The range of length scales l* < l < l* is referred to as the Kolmogorov inertial range. Further considerations of the redistribution of energy over this scale range involve first calculating the time rate of change of total kinetic energy. In particular, multiplying (dot product) the Navier-Stokes equations with u, applying integration by parts and some standard vector identities leads to

equation image

The first integral term, referred to as enstrophy, is a rate of decrease of energy. The localized rate (integrand)

equation image

is referred to as the energy dissipation rate, per unit volume per unit time.

[23] Over the inertial range Kolmogorov proposed the multiplicative statistical cascade as a spatially homogeneous and stationary statistical model of turbulence. Of course the essential open question for this model is determination of the distribution of the cascade generators W.

[24] By dimension considerations Kolmogorov indicated that one may expect the following fundamental scaling relations in the limit as ll*:

equation image

where

equation image

for a small region of integration Δl of diameter l, and a specified unit vector displacement direction e. Also, here 〈q〉 denotes the average of the quantity q. Kolmogorov observed by dimensional considerations that, denoting units of a quantity q by [q], since [ε] = l3/t2 and [u] = l/t, one has [u3] = [ε] · l and therefore

equation image

In response to the issue of intermittency of turbulence raised early on by Landau, Kolmogorov assumed the cascade generators to be lognormally distributed. However this leads to a quadratic polynomial form of the exponent ζh which has been argued to be inconsistent with incompressibility [see Frisch, 1999, pp. 133–135, 172].

[25] On the analytic side there have been a number of attempts to obtain the behaviors of u and ε over the inertial scale range directly from analysis of the Navier-Stokes equations by estimates of Hölder exponents, Hausdorff (fractal) dimensions etc. These have been somewhat successful in showing consistency with the general statistical cascade framework. On the numerical side there is also a large body of computational work based directly on Navier-Stokes equations. In any case one may consult Foias et al. [2001, and references therein] and Frisch [1999, and references therein] for many of the basic references on both purely analytic and numerical results. In this paper we focus on a hypothesis formulated by She and Levesque [1994] on the basis of numerical simulations of the Navier-Stokes equations. For this, first define

equation image

SL hypothesis

equation image

where Ah does not depend on the length scale l and β ∈ (0, 1).

[26] From this, upon substituting (12), one obtains the second-order nonhomogeneous linear difference equation

equation image

with initial conditions τ0 = 0, τ1 = 0. In this way, from the SL Hypothesis one obtains the exact solution

equation image

Remark 2. It may be noted that the appearance of equation image is not result of fitting Kolmogorov's −equation image (wave number) exponent. Here one obtains ζ2 = .696.

[27] Interestingly, the SL hypotheses may be precisely viewed in terms of the (first order) moment of size biased energy dissipation rates. Within the framework of the statistical multiplicative cascade model this was independently shown by Dubrulle [1994] and by She and Waymire [1994, 1995] to imply the log Poisson distribution of cascade generators W. Specifically, in view of the localized definition of ε one has

equation image

In particular

equation image

Thus, taking logs and dividing by log length scale, it follows that

equation image

So the determination of W can be made by simply noting that for W = adY+c, with Y having a Poisson distribution with parameter λ > 0, one precisely has

equation image

Thus the SL hypothesis has been shown to be equivalent to a log Poisson distribution distribution of cascade generators with parameters satisfying λ = 2, ad = 2/3, ac = e2/3 and, as such, provides an interesting alternative to Kolmogorov's lognormal hypothesis for cascade generators.

[28] Approaches to precise statistical error bars for scientifically sound empirical estimations and tests of hypothesis for multiplicative cascades are given by Troutman and Vecchia [1999] and Ossiander and Waymire [2000]. However, technical difficulties remain which rely on explicit measurements of energy dissipation rates for rigorous statistical inference; see Ossiander and Waymire [2002] for a discussion of the current status of theoretical tests and data requirements within the context of statistical turbulence theory.

5. Navier-Stokes Cascades

[29] The physics of fluid motion is embodied in the Navier-Stokes equations as the result of basic conservation principles. As a result, one would expect turbulent motions to somehow be captured within the framework of these equations, at least in the form of weak (distributional) solutions. As noted in the previous section, this is far from being realized mathematically and represents one of the greatest challenges to modern applied mathematics; see Foias et al. [2001] for a relatively recent account from both physical and mathematical points of view.

[30] An interesting development occurred for three-dimensional incompressible Navier-Stokes equations which will be described in this section. Namely, an explicit representation of the Fourier transform of classes of solutions to 3-D incompressible Navier-Stokes equations in the form of an expected value of a certain product of initial and forcing data evaluated at the nodes of a branching random walk was recently uncovered by LeJan and Sznitman [1997]. While probability models have long enjoyed important connections to partial differential equations, most notable being Brownian motion and the heat equation, this is an exciting mathematical connection between multiplicative cascades and nonlinear equations of fluid motion.

[31] For a simple mathematical illustration of ideas, it is instructive to consider the one-dimensional Burgers equation

equation image

Remark 3. The original representation obtained by LeJan and Sznitman [1997] was not valid for dimensions less than 3 due to serious integrabiltiy issues required. This was overcome by Bhattacharya et al. [2003] through the introduction of the notion of “majorizing kernels.” As a consequence it is possible to illustrate the main mathematical ideas in the simpler context of one-dimensional Burgers equation provided in this paper.

[32] Spatial Fourier transform will be denoted by equation image. For simplicity of exposition, assume equation image0(ξ) = 0 for ξ ≤ 0. Taking spatial Fourier transforms one obtains, with the aid of an exponential integrating factor and ξ > 0 and writing

equation image

that

equation image

Expressed in this form, the equation (16) takes on a probabilistic meaning. Namely, this is a recursive equation for the expected values of a multiplicative stochastic process initiated at ξ = ξ. The first term on the right hand side e−λ(ξ)tequation image0(ξ) is the product of the initial data equation image0(ξ) times the probability e−λ(ξ)t that an “exponentially distributed clock” with parameter λ(ξ) rings after time t; in this context we refer to a sample realization of an exponentially distributed random variable as a “ring of an exponential clock.” The second integral term is an expected value in the complimentary event of probability density λ(ξ)e−λ(ξ)s, that the clock rings at time s prior to t. Given that the clock rings at a time s prior to t, a product is formed with the factor m(ξ) and a random selection of a pair of new “offspring” wave numbers (or Fourier frequencies) η, ξ − η from the interval [0, ξ] with (uniform) probability density 1/ξ to complete the recursion over the remaining time ts. That is to say, the unique solution (in the appropriate function space) is furnished by the expected value

equation image

for a multiplicative cascade X(t, ξ) defined by the following stochastic recursion in Fourier wave number space (see Figure 1): A particle of type ξ = ξ waits for an exponentially distributed time S with mean 1/λ(ξ) = equation image. If S > t then a value equation image0(ξ) is assigned to the initial vertex ∅ and the process terminates. On the other hand, if St then an independent coin flip is made. If the outcome is a tail then the particle dies and a value 0 is assigned, but if head occurs then the particle branches into two particles 〈1〉, 〈2〉 of respective types ξ〈1〉 = η and ξ〈2〉 = ξ − η selected according to the uniform distribution on [0, ξ]. Two independent exponential clocks S〈1〉, S〈2〉 with respective parameters λ(ξ〈1〉), λ(ξ〈2〉), are set, and the process is repeated independently from each of these two given types for the termination time t reduced to tS.

Figure 1.

A sample tree graph for Burgers' equation.

[33] The multiplicative cascade X(t, ξ) is, up to a (random) power of m ≡ −equation image/ν, a product of the assigned values of equation image0 at the selected frequencies; e.g., for the sample realization depicted in Figure 1 one has

equation image

In particular the premature death of 〈2〉 means that this particular sample realization will not contribute a positive value to the mathematical expectation in (17). A numerical Monte Carlo implementation based on this representation is given by Ramirez [2006]. The representation for three-dimensional incompressible Navier-Stokes equation is along precisely the same lines after one exploits incompressibility to project out the pressure term in Fourier space; see Waymire [2005] for an overview of the current status of this theory and open mathematical problems.

6. Conclusions

[34] The significance of size biasing for analyzing the structure of multiplicative cascades is well documented in the mathematics literature. That said, the analysis of both energy dissipation rates and/or rain rates in terms of size-biased averages continues to provide an intriguing area of inquiry for physical scientists. She and Leveque [1994] initiated physical considerations based on size-biased averages which have been shown to have unexpected and interesting consequences. At the very least, mathematical research suggests that consideration of size-biased averages for the analysis of highly singular and intermittent data sets may well capture interesting and useful structure. This may be summarized in terms of two basic principles: (1) Mathematical cascades inherit their structure from large deviations from average behavior, and (2) size-biased moments are intimately related to large deviation rates. To the author's knowledge this has not yet been considered in the analysis of rainfall data as an approach to identify and formulate possible underlying structural relations.

[35] The connection to multiplicative cascades in the Fourier domain has proven to be an interesting mathematical development. As illustrated by the one-dimensional Burgers equation, the basic mathematical ideas exploit a natural branching recursion in the equations, a point being the role of the nonlinear term in branching in Fourier space. The extension to three-dimensional incompressible Navier-Stokes follows precisely the same scheme after one projects out the pressure. As a result of the projection, the “multiplication” is not a simple pointwise product as it is in the case of the scalar Burgers equation, but the mathematical development is along precisely the same lines; see Bhattacharya et al. [2003] for details. In any case, it must be emphasized that the precise connection between Kolmogorov's multiplicative cascades in physical space to Navier-Stokes equations remains completely open as a substantial challenge to research on all fronts.

[36] Since the central theme of this paper grew out of a focus on mathematical notions, it seems fitting to end as we began, by quoting from LeCam [1961, p. 165]: “Also, we hope that the mathematical technique used here will remain applicable in some realistic studies.”

Acknowledgments

[37] This work was partially supported by a grant from the National Science Foundation (CMG 0327705). The author would also like to thank colleagues Vijay Gupta and Enrique Thomann for comments and suggestions based on an earlier draft of this paper.

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