Laboratoire Eau Environnement et Systèmes Urbains (LEESU)

Résumé: Benoit B. Mandelbrot, who advanced the concept of power law scaling as a fundamental property of a broad range of natural processes and patterns in geophysics, economics, mathematics, and virtually all of science, died on 14 October 2010 in Cambridge, Mass., at the age of 85. Mandelbrot, known as the �father of fractal geometry,� was a mathematician who developed the scaling concepts of self-similarity and self-affinity and found examples in spatial, temporal, and size patterns across a broad spectrum of disciplines. He coined the term �fractal� (from the Latin noun �fractus,� meaning fragmented) for shapes and patterns that exhibit self-similarity, meaning that they are statistically scale independent. Such shapes are characterized by fractional power law exponents, between the integer (Euclidean) dimensions. He is best known through his books, including Les Objets Fractals: Forme, Hasard et Dimension; Fractals: Form, Chance and Dimension; The Fractal Geometry of Nature; and Multifractals and 1/f Noise: Wild Self-Affinity in Physics [Mandelbrot, 1975, Mandelbrot 1977, Mandelbrot 1982, Mandelbrot 1999].

Résumé: Lindborg et al. (2010) claim that the apparent spectrum power law E(k)  k−3 on scales  600 km obtained with the help of commercial jetliner trajectory deviations (GASP and Mozaic databases) could not be brought into question (Lovejoy et al., 2009a), because this spectrum corresponds to “a well known theory of quasi-geostrophic turbulence developed by Charney (1971)”. Lindborg et al. (2010) also claim that “limitations [of this theory] have been relaxed in many of the modern models of atmospheric turbulence”. We show that both claims are irrelevant and that generalized scale invariance (GSI) is indispensable to go beyond the quasi-geostrophic limitations, to go in fact from scale analysis to scaling analysis in order to derive better analytical models. In this direction, we derive vorticity equations in a space of (fractal) dimension D = 2+Hz (0  Hz  1), which corresponds to a first step in the derivation of a dynamical alternative to the quasi-geostrophic approximation and turbulence. The corresponding precise definition of fractional dimensional turbulence already demonstrates that the
classical 2-D and 3-D turbulence are not the main options to understand atmospheric dynamics. Although (2+Hz)-D turbulence with 0 < Hz < 1) has more common features with 3-D turbulence than with 2-D turbulence, it has nevertheless
very distinctive features: its scaling anisotropy is in agreement with the layered pancake structure, which is typical of rotating and stratified turbulence but not of the classical 3-D turbulence.

Résumé: Cet article utilise les propriétés multifractales d'un évènement pluvieux dans la région de Londres le 9 février 2009, pour mieux comprendre et quantifier 'incertitude associée à la variabilité spatio-temporelle des précipitations non mesurées par les radars météorologiques en bande C, dont la résolution estimée est de 1 km*1 km*5min, et comment elle se transfère aux prévisions des débits en réseaux d'assainissement. Le cas d'étude hydrologique est celui du bassin versant urbain de 900 hectares de Cranbrook (Londres). Les propriétés multifractales sont en accord avec le modèle spatio-temporel le plus simple, reposant sur un exposant d'anisotropie entre l'espace et le temps. Ceci permet de désagréger le champ de précipitation à l'aide de cascades multifractales spatio-temporelles. Un ensemble de champs de précipitations désagrégés réalistes est alors généré à l'aide des multifractals universels, puis l'ensemble des hydrogrammes correspondants est obtenu par un modèle urbain pluie-débit semi-distribué. Il apparait que les queues de probabilité issues de l'analyse de 100 échantillons de précipitation présentent un comportement en loi de puissance, qui est retrouvé sur les débits de pointe mais avec des exposants différents

Résumé: Dans le cadre des multifractals universels, il est possible de caracteriser la variabilite spatio-temporelle de la pluie sur une grande gamme d'echelle a l'aide de trois parametres invariants d'echelles. Dans cette etude, nous avons estime ces parametres multifractals sur des simulations numeriques effectuees avec le modele meso-echelle Meso-NH, developpe par Meteo-France et le Laboratoire d'Aerologie (Univ. P. Sabatier, Toulouse, France), et des images radar composites, couvrant le meme evenement pluvieux, a savoir un orage particulierement violent, dit de type Cevenol, ayant eu lieu sur la partie sud de la France du 5 au 9 Septembre 2005. La comparaison des resultats montre que les deux types de donnees presentent des domaines d'invariance d'echelle relativement similaires, et dont les proprietes sont en accord avec les modeles de precipitation spatio-temporels unifies et scalants les plus simples. Neanmoins l'evaluation de leurs exposants conduit a des valeurs parfois fortement differentes. Citation Gires, A., Tchiguirinskaia, I., Schertzer, D. Lovejoy, S. (2011) Analyses multifractales et spatio-temporelles des precipitations du modele Meso-NH et des donnees radar. Hydrol. Sci. J. 56(3), 380-396.

Résumé: In large urban areas such as the Paris and London one, storm water management is a challenge. Indeed because there is a significant proportion of impervious surface on large areas, great amounts of effective rainfall need to be handled. In this study, we use multifractal characterization of rainfall to quantify the uncertainty on sewer discharge forecasts associated to unmeasured small scale rainfall variability, i.e at a higher resolution than 1 km * 1 km * 5 min which is usually available with C-band radar networks. Two urban areas are used as cases study and compared: a catchment in the county of Seine-Saint-Denis in the North of Paris, and the Cranbrook catchment in the North of London. Several types of rainfall events (frontal or convective) are analysed. Concerning the rainfall data, Nimrod mosaics of the Met Office is used for the London catchment. For Paris' we use the data from the C-band radar of Trappes, located in the East of Paris. First an ensemble of realistic rainfall fields downscaled to a higher resolution is generated with the help of multifractal space-time cascades. The characteristic exponents used are the one estimated on the radar data. Second the corresponding ensemble of hydrographs is simulated by inputting each rainfall realization into a semi-distributed urban hydrological model. It appears that the uncertainty on the simulated peak flow is significant, reaching 40% for some rainfall events. Moreover the probability distribution of the extremes of both the rainfall and the peak flow exhibit a power-law falloff, indicting a high dispersion of the results. These results found on two independent models suggest that rainfall extremes play a key role in conditioning discharge extremes. The relationship between the characteristic exponents is discussed. In conclusion, we highlight the need to develop the use X-band radars in urban areas. Indeed such radars provide higher resolution data that would enable a better management of storm water. The results were obtained within the framework of the project GARP-3C (program R2DS, région Iles-de-France), and as part of the Flood Risk Management Research Consortium (FRMRC2, SWP3).

Résumé: We consider the space-time scaling properties of the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis products for the wind (u, v, w), humidity (h(s)), temperature (T), and geopotentials (z) and their corresponding turbulent fluxes using the daily 700 mbar products for the year 2006. Following previous studies on T, h(s), and u, we show that that the basic predictions of multiplicative cascade models are well respected over space-time scales below similar to 5000 km, shorter than similar to 5-10 days providing precise scale by scale determination of the reanalysis statistical properties (needed for example for stochastic parameterizations in ensemble forecasting systems). We innovate by including the meridional and vertical wind components (v, w) and geopotential (z), and by considering their horizontal anisotropies, their latitudinal variations and, perhaps most importantly, by directly analyzing the fields (not just fluxes). Whereas the fluxes have nearly isotropic exponents in space-time with little latitudinal variation (displaying only scale independent “trivial” anisotropy), the fields have significant scaling horizontal anisotropies. These complicate the interpretation of standard isotropic spectra and are likely to be artifacts. Many of the new (nonconservation) exponents (H) are nonstandard and currently have no adequate theoretical explanation although the key horizontal wind and temperature H exponents may be consequences of horizontal Kolmogorov scaling, combined with sloping isobaric surfaces. In time the scaling is broken at around 5-10 days, i.e., roughly the lifetime of planetary structures; lower frequencies are spectrally flatter: the “spectral plateau,” weather-low-frequency weather regime.

Résumé: 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.

Résumé: One of the key thematic areas for the development of future research in wind energy is turbulence. To better understand this topic requires the development of new numerical methods and measuring techniques capable of reaching micro scale effects. My PhD thesis will focus on advanced characterisation of micro-scale wind turbulence with respect to non-Gaussian heavy tailed statistics and short term extreme events (gusts) on the scales of 1 to 1000m and/or 1 to 100sec. Based on experimental data, multifractal wind field models with high frequency turbulent dynamics will be developed. Such models are promising candidates for providing initial flow conditions for turbulent dynamic computational fluid dynamics calculations. A combination of multifractals with other more classical models can open new perspectives for many industrial applications.

Résumé: La qualité des statistiques des précipitations, notamment les courbes Intensité-Durée-Fréquence, dépend étroitement de la fiabilité des données disponibles. Or, il a été montré que la plupart des séries temporelles provenant de pluviomètres à augets basculants ont une fréquence d'enregistrement inférieure à celle assumée. Cette question est particulièrement importante pour l'hydrologie urbaine, car les estimations sont sensibles aux fluctuations hautes fréquences des précipitations. Le déficit en données à haute fréquence peut conduire à d'apparentes ruptures des lois d'échelle, ce qui complique inutilement et notoirement la modélisation des précipitations. Il est donc indispensable de quantifier la qualité des données avant de les utiliser. Nous avons présenté une procédure SERQUAL qui permet de répondre à cette question et d'extraire d'une base de données des sous-séries ayant les qualités requises. Dans cette présentation, nous utilisons cette procédure SERQUAL pour sélectionner sous-séries ayant les qualités requises pour des analyses à haute résolution. Nous présentons les caractéristiques multifractales des données sélectionnées, qui peuvent être utilisées pour calibrer ou valider des modèles statistiques ou stochastiques. Nous montrons aussi que l'évolution de ces caractéristiques peut être aussi utilisée pour évaluer des conséquences hydrologiques du changement climatique en région Ile-de-France

Résumé: In Part I we considered continuous in scale cascade processes which were spatially continuous these are needed for modeling many geofields In this second part we consider the effects of spatial discretization this allows us to make numerical simulations We show how to correct the simulations for the leading finite size effects for both space and (causal) space-time processes The resulting processes have significantly improved small scale statistical properties in practice it can lead to great savings in computer time and memory usage In an appendix we give a Mathematica code for the corresponding space-time simulations (C) 2010 Elsevier Ltd All rights reserved

Résumé: In spite of the unprecedented quantity and quality of meteorological data and numerical models, there is still no consensus about the atmosphere's elementary statistical properties as functions of scale in either time or in space. This review paper proposes a new synthesis based on a) advances in the last 25 years in nonlinear dynamics, b) a critical re-analysis of empirical aircraft and vertical sonde data, c) the systematic scale by scale, space-time exploitation of high resolution remotely sensed data and d) the systematic re-analysis of the outputs of numerical models of the atmosphere including reanalyses, e) a new turbulent model for the emergence of the climate from “weather” and climate variability. We conclude that Richardson's old idea of scale by scale simplicity today embodied in multiplicative cascades can accurately explain the statistical properties of the atmosphere and its models over most of the meteorologically significant range of scales, as well as at least some of the climate range. The resulting space-time cascade model combines these nonlinear developments with modern statistical analyses, it is based on strongly anisotropic and intermittent generalizations of the classical turbulence laws of Kolmogorov, Corrsin, Obukhov, and Bolgiano. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.

Résumé: Aircraft measurements of the power spectra of the horizontal wind field typically find a transition from approximate to k(-5/3) to approximate to k(-2.4) at scales somewhere around 40 km (k is a wave number). In the usual interpretation this represents a transition between an isotropic three-dimensional (3-D) (k(-5/3)) and an isotropic 2-D(k(-3)) turbulence; we have recently argued that the turbulence is so highly anisotropic that it has different exponents in the horizontal and vertical. When coupled with gently sloping isobaric aircraft trajectories this predicts the break as a transition from a roughly horizontal spectrum at small scales to the spurious appearance of the vertical spectrum at large scales. If the atmosphere indeed has wide-range horizontal scaling, then it is important to test out the multiplicative cascade models that predict its statistical behavior. In this paper, we do this by analyzing wind, temperature, pressure, and humidity data from the Winter Storm 2004 experiment using 24 aircraft legs, each 1120 km long and at 280 m resolution. We analyze both the turbulent fluxes and the fluctuations showing that in spite of the nonflat trajectories, there is good evidence of roughly planetary-scale multiplicative cascades. By carefully determining the scale-by-scale effects of intermittency on the aircraft altitude and measurements, we estimate the corresponding scaling exponents. We argue that our results should finally permit the emergence of a long-needed consensus about the basic scale-by-scale statistical properties of the atmosphere.

Résumé: The end users of hydrological models may be justified for being tired of the excessive uncertainty of these models, not to mention their simplistic approximations and crude modelling. The ever-increasing sophistication of model parameter fitting is simply a smoke-screen that hides the models' lack of physical basis, their scale dependence, and their inability to fit widely diverse behaviours. More generally, we have to admit a lack of qualitative improvement in hydrological modelling in recent times. In fact, operational hydrology may have suffered for some time from ignoring the advances in theoretical hydrology, which have, in contrast, greatly stimulated the nonlinear sciences. For instance, more than a century ago fractals were considered as geometrical monsters, whereas decades ago river networks became classical fractal objects, and rainfall and discharges are now classical examples of multifractal fields. These hydrological characteristics are still often ignored by operational hydrology, whereas they explain not only its current limitations, but also how to overcome them. To illustrate these problems, this paper focuses on the fact that hydrological fields are most likely singular with respect to measures of time and volume. This would not only explain the ubiquitous scale dependence of hydrological observations, but would also give the possibility to transform them into scale-independent quantities. The upscaling of a rainfall time series from an hour to a year is therefore discussed in detail, and enables us to quickly introduce other examples.

Résumé: Is the numerical integration of nonlinear partial differential equations the only way to tackle atmospheric complexity? Or do cascade dynamics repeating scale after scale lead to simplicity? Using 1000 orbits of TRMM satellite radiances from 11 bands in the short wave ( visible, infra red) long wave ( passive microwave) and radar regions and 8.8 to 20,000 km in scale, we find that the radiance gradients follow the predictions of cascade theories to within about +/- 0.5%, +/- 1.25%, +/- 5.9% for the short waves, long waves and reflectivities respectively and with outer scales varying between approximate to 5,000 to approximate to 32,000 km. Since the radiances and dynamics are strongly coupled, we conclude that weather can be accurately modeled as a cascade process. Citation: Lovejoy, S., D. Schertzer, V. Allaire, T. Bourgeois, S. King, J. Pinel, and J. Stolle ( 2009), Atmospheric complexity or scale by scale simplicity?, Geophys. Res. Lett., 36, L01801, doi: 10.1029/2008GL035863.

Résumé: Due to both systematic and turbulent induced vertical fluctuations, the interpretation of atmospheric aircraft measurements requires a theory of turbulence. Until now virtually all the relevant theories have been isotropic or 'quasi isotropic' in the sense that their exponents are the same in all directions. However almost all the available data on the vertical structure shows that it is scaling but with exponents different from the horizontal: the turbulence is scaling but anisotropic. In this paper, we show how such turbulence can lead to spurious breaks in the scaling and to the spurious appearance of the vertical scaling exponent at large horizontal lags. We demonstrate this using 16 legs of Gulfstream 4 aircraft near the top of the troposphere following isobars each between 500 and 3200 km in length. First we show that over wide ranges of scale, the horizontal spectra of the aircraft altitude are nearly k(-5/3). In addition, we show that the altitude and pressure fluctuations along these fractal trajectories have a high degree of coherence with the measured wind (especially with its longitudinal component). There is also a strong phase relation between the altitude, pressure and wind fluctuations; for scales less than 40 km (on average) the wind fluctuations lead the pressure and altitude, whereas for larger scales, the pressure fluctuations leads the wind. At the same transition scale, there is a break in the wind spectrum which we argue is caused by the aircraft starting to accurately follow isobars at the larger scales. In comparison, the temperature and humidity have low coherencies and phases and there are no apparent scale breaks, reinforcing the hypothesis that it is the aircraft trajectory that is causally linked to the scale breaks in the wind measurements. Using spectra and structure functions for the wind, we then estimate their exponents (beta, H) at small (5/3, 1/3) and large scales (2.4, 0.73). The latter being very close to those estimated by drop sondes (2.4, 0.75) in the vertical direction. In addition, for each leg we estimate the energy flux, the sphero-scale and the critical transition scale. The latter varies quite widely from scales of kilometers to greater than several hundred kilometers. The overall conclusion is that up to the critical scale, the aircraft follows a fractal trajectory which may increase the intermittency of the measurements, but doesn't strongly affect the scaling exponents whereas for scales larger than the critical scale, the aircraft follows isobars whose exponents are different from those along isoheights (and equal to the vertical exponent perpendicular to the isoheights). We bolster this interpretation by considering the absolute slopes (vertical bar delta z/delta x vertical bar) of the aircraft as a function of lag delta x and of scale invariant lag delta x/delta z(z)(1/H). We then revisit four earlier aircraft campaigns including GASP and MOZAIC showing that they all have nearly identical transitions and can thus be easily explained by the proposed combination of altitude/wind in an anisotropic but scaling turbulence. Finally, we argue that this reinterpretation in terms of wide range anisotropic scaling is compatible with atmospheric phenomenology including convection.

Résumé: In Part I of this paper, we developed asymptotic approximations for single photon scattering in thick, highly heterogeneous, “Log-Levy” multifractal clouds. In Part II, theoretical multiple scattering predictions are numerically tested using Monte Carlo techniques, which show that, due to long range correlations, the photon paths are “subdiffusive” with the corresponding fractal dimensions tending to increase slowly with mean optical thickness. We develop reasonably accurate statistical relations between N scatter statistics in thick clouds and single scatter statistics in thin clouds. This is explored further using discrete angle radiative transfer (DART) approach in which the radiances decouple into non-interacting families with only four (for 2-D clouds) radiance directions each. Sparse matrix techniques allow for rapid and extremely accurate solutions for the transfer: the accuracy is only limited by the spatial discretization. By “renormalizing” the cloud density, we relate the mean transmission statistics to those of an equivalent homogeneous cloud. This simple idea is remarkably effective because two complicating effects act in contrary directions: the “holes” which lead to increased single scatter transmission and the tendency for multiply scattered photons to become “trapped” in optically dense regions, thus decreasing the overall transmission. (C) 2009 Elsevier B.V. All rights reserved.

Résumé: By direct statistical analysis we show that over almost all their range of scales and to within typically better than +/- 1%, atmospheric fields obtained from analyses and numerical integrations of atmospheric models have the multifractal structure predicted by multiplicative cascade models. We quantify this for the horizontal wind, temperature, and humidity fields at 5 different pressure levels for the ERA40 reanalysis, the Canadian Meteorological Centre Global Environmental Multiscale (CMC, GEM) model, as well as the National Oceanographic and Atmospheric Administration Global Forecasting System (NOAA, GFS). We investigate the additional prediction that the cascade belongs to a universal multifractal basin of attraction. By demonstrating a 'Levy collapse' of the statistical moments to within +/- 2 to +/- 5% over most of the range of scales, we conclude that there is good evidence for this. Finally, we discuss how this stochastic multiplicative cascade structure can be exploited in improving ensemble forecasts.

Résumé: We critically re-examine existing empirical studies of vertical and horizontal statistics of the horizontal wind and find that the balance of evidence is in favour of the Kolmogorov k(x)(-5/3) scaling in the horizontal, Bolgiano-Obukov scaling k(z)(-11/5) in the vertical corresponding to a D-s = 23/9 stratified atmosphere in (x, y, z) space. This interpretation is particularly compelling once one recognizes that the 23/9-D turbulence can lead to long-range biases in aircraft trajectories and hence to spurious statistical exponents in wind, temperature and other statistics reported in the literature. Indeed, we show quantitatively that one is easily able to reinterpret the major aircraft-based campaigns (GASP, MOZAIC) in terms of the model. In part I, we have seen that this model is compatible with 'turbulence waves' which can be close to classical linear gravity waves in spite of their very different nonlinear mechanism. We then use state-of-the-art lidar data of atmospheric aerosols (considered as passive tracers) in order to obtain direct estimates of the effective ('elliptical') dimension of the spatial part: D-s = 23/9 = 2.55 +/- 0.02. This result essentially rules out the standard 3-D or 2-D isotropic theories or the anisotropic quasi-linear gravity wave theories which have D-s = 3, 2, 7/3 respectively. In this paper we focus on the multifractal (intermittency) statistics showing that there is a very small but apparently real variation in the value of D-s, ranging for the weak and intense structures so that Ds ranges from roughly 2.53 to 2.57. We also show that the passive scalars are well approximated by universal multifractals; we estimate the exponents to be alpha(h) = 1.82 +/- 0.05, alpha(v) = 1.83 +/- 0.04, C-1h = 0.037 +/- 0.0061 and C-1v = 0.059 +/- 0.007 (h for horizontal, v for vertical). Copyright (c) 2008 Royal Meteorological Society.

Résumé: Using a unique data set of three-dimensional drop positions and masses (the HYDROP experiment), we show that the distribution of liquid water in rain displays a sharp transition between large scales which follow a passive scalar-like Corrsin-Obukhov (k(-5/3)) spectrum and a small-scale statistically homogeneous white noise regime. We argue that the transition scale l(c) is the critical scale where the mean Stokes number (= drop inertial time/turbulent eddy time) St(l) is unity. For five storms, we found l(c) in the range 45-75 cm with the corresponding dissipation scale St(eta) in the range 200-300. Since the mean interdrop distance was significantly smaller (approximate to 10 cm) than l(c) we infer that rain consists of 'patches' whose mean liquid water content is determined by turbulence with each patch being statistically homogeneous. For l > l(c), we have St(l) < 1 and due to the observed statistical homogeneity for l < l(c), we argue that we can use Maxey's relations between drop and wind velocities at coarse grained resolution l(c). From this, we derive equations for the number and mass densities (n and rho) and their variance fluxes (psi and chi). By showing that chi is dissipated at small scales (with l(rho,diss) approximate to l(c)) and psi over a wide range, we conclude that rho should indeed follow Corrsin-Obukhov k(-5/3) spectra but that n should instead follow a k(-2) spectrum corresponding to fluctuations scaling as Delta rho proportional to l(1/3) and Delta n proportional to l(1/2). While the Corrsin-Obukhov law has never been observed in rain before, its discovery is perhaps not surprising; in contrast the Delta n approximate to l(1/2) number density law is quite new. The key difference between the Delta rho, Delta n laws is the fact that the microphysics (coalescence, breakup) conserves drop mass, but not numbers of particles. This implies that the timescale for the transfer of the density variance flux chi is determined by the strongly scale-dependent turbulent velocity whereas the timescale for the transfer of the number variance flux is determined by the weakly scale-dependent drop coalescence speed. We argue that the l(1/2) law may also hold (although in a slightly different form) for cloud drops. Because they are consequences of symmetries, we expect the l(1/3), l(1/2) laws to be robust. Since the large-scale turbulence determines the n and rho fields which are the 0th and 1st moments of the drop-size distribution, they constrain the microphysics: dimensional analysis shows that the cumulative probability distribution of nondimensional drop mass should be a universal function dependent only on scale; we confirm this empirically. The combination of number and mass density laws can be used to develop stochastic compound multifractal Poisson processes which are useful new tools for studying and modelling rain. We discuss the implications of this for the rain rate statistics including a simplified model, which can explain the observed rain rate spectra.

Résumé: In this first of a three-part series, we argue that the dynamics of turbulence in a stratified atmosphere should depend on the buoyancy over a wide range of vertical scales and on energy flux over a wide range of horizontal scales; it should be scaling, but anisotropic, not isotropic. We compare the leading statistical theories of atmospheric stratification which are conveniently distinguished by the elliptical dimension D-s which quantifies their degree of spatial stratification. This includes the mainstream isotropic 2-D (large scales), isotropic 3-D (small scales) theory but also the more recent linear gravity wave theories (D-s = 7/3) and the classical fractionally integrated flux (FIF) 23/9-D unified scaling model. In the latter, the horizontal wind has a k(-5/3) spectrum as a function of horizontal wavenumber determined by the energy flux and a k(-11/5) energy spectrum as a function of vertical wavenumber determined by the buoyancy force variance flux. In this model, the physically important notion of scale is determined by the turbulent dynamics, it is not given a priori (i.e. the by usual Euclidean distance). The 23/9-D FIF model is the most physically and empirically satisfying, being based on turbulent (spectral) fluxes. The FIF model as originally proposed by Schertzer and Lovejoy is actually a vast family of scaling models broadly compatible with turbulent phenomenology and with the classical turbulent laws of Kolmogorov, Corrsin and Obukov. However, until now it has mostly been developed on the basis of structures localized in space – time. In this paper, we show how to construct extreme FIF models with wave-like structures which are localized in space but unlocalized in space – time, as well as a continuous family of intermediate models which are akin to Lumley – Shur models in which some part of the localized turbulent energy 'leaks' into unlocalized waves. The key point is that the FIF requires two propagators (space – time Green's functions) which can be somewhat different. The first determines the space – time structure of the cascade of fluxes; this must be localized in space – time in order to satisfy the usual turbulence phenomenology. In contrast, the second propagator relates the turbulent fluxes to the observables; although the spatial part of the propagator is localized as before, in space – time it can be unlocalized. (It is still localized in space, now in wave packets.) We display numerical simulations which demonstrate the requisite (anisotropic, multifractal) statistical properties as well as wave-like phenomenologies. In parts II and III we will examine the empirical evidence for the spatial and temporal parts, respectively, of the model using state-of-the-art lidar data of aerosol backscatter ratios (which we use as a surrogate for passive scalar concentration). Copyright (c) 2008 Royal Meteorological Society.