We argue that the basic properties of rain and cloud fields (particularly their scaling and intermittency) are best understood in terms of coupled (anisotropic and scaling) cascade processes. We show how such cascades provide a framework not only for theoretically and empirically investigating these fields, but also for constructing physically based stochastic models. This physical basis is provided by cascade scaling and intermittency, which is of broadly the same sort as that specified by the dynamical (nonlinear, partial differential) equations. Theoretically, we clarify the links between the divergence of high-order statistical moments, the multiple scaling and dimensions of the fields, and the multiplicative and anisotropic nature of the cascade processes themselves. We show how such fields can be modeled by fractional integration of the product of appropriate powers of conserved but highly intermittent fluxes. We also empirically test these ideas by exploiting high-resolution radar rain reflectivities. The divergence of moments is established by direct use of probability distributions, whereas the multiple scaling and dimensions required the development of new empirical techniques. The first of these estimates the "trace moments" of rain reflectivities, which are used to determine a moment-dependent exponent governing the variation of the various statistical moments with scale. This exponent function in turn is used to estimate the dimension function of the moments. A second technique called "functional box counting," is a generalization of a method first developed for investigating strange sets and permits the direct evaluation of another dimension function, this time associated with the increasingly intense regions. We further show how the different intensities are related to singularities of different orders in the field. This technique provides the basis for another new technique, called "elliptical dimensional sampling," which permits the elliptical dimension rain (describing its stratification) to be directly estimated: it yields del =2.22+0.07, which is less than that of an isotropic rain field (del =3), but significantly greater than that of a completely flat (stratified) two-dimensional field (de1-2). 1. INTRODUCTION In theoretical terms the rain field can be considered to be the solution of a complex set of coupled nonlinear partial differential equations. These equations must clearly include the effect of the dynamical interactions of water vapor and liquid, latent heat release, radiation, wind fields, etc.. Structures in these fields are nonlinearly coupled over a range of over roughly 9 orders of magnitude in scale along the horizontal (-1 mm to-1000 km), and they are therefore way beyond the scope of direct deterministic numerical modeling. In order to function at all, global models of either climate or weather rely extensively on ad hoc "subgrid scale parameterizations." These parameterizations are unsatisfactory, not only because of their unphysical nature, but also because the theoretical (...

Drainage basins in many parts of the world are ungauged or poorly gauged, and in some cases existing measurement networks are declining. The problem is compounded by the impacts of human-induced changes to the land surface and climate, occurring at the local, regional and global scales. Predictions of ungauged or poorly gauged basins under these conditions are highly uncertain. The IAHS Decade on Predictions in Ungauged Basins, or PUB, is a new initiative launched by the International Association of Hydrological Sciences (IAHS), aimed at formulating and implementing appropriate science programmes to engage and energize the scientific community, in a coordinated manner, towards achieving major advances in the capacity to make predictions in ungauged basins. The PUB scientific programme focuses on the estimation of predictive uncertainty, and its subsequent reduction, as its central theme. A general hydrological prediction system contains three components: (a) a model that describes the key processes of interest, (b) a set of parameters that represent those landscape properties that govern critical processes, and (c) appropriate M. Sivapalan et al. 858 meteorological inputs (where needed) that drive the basin response. Each of these three components of the prediction system, is either not known at all, or at best known imperfectly, due to the inherent multi-scale space-time heterogeneity of the hydrological system, especially in ungauged basins. PUB will therefore include a set of targeted scientific programmes that attempt to make inferences about climatic inputs, parameters and model structures from available but inadequate data and process knowledge, at the basin of interest and/or from other similar basins, with robust measures of the uncertainties involved, and their impacts on predictive uncertainty. Through generation of improved understanding, and methods for the efficient quantification of the underlying multi-scale heterogeneity of the basin and its response, PUB will inexorably lead to new, innovative methods for hydrological predictions in ungauged basins in different parts of the world, combined with significant reductions of predictive uncertainty. In this way, PUB will demonstrate the value of data, as well as provide the information needed to make predictions in ungauged basins, and assist in capacity building in the use of new technologies. This paper presents a summary of the science and implementation plan of PUB, with a call to the hydrological community to participate actively in the realization of these goals.Key words drainage basins; predictions; uncertainty; heterogeneity; gauging; hydrological models; hydrological theory; field experiments La décennie de l'AISH sur les prévisions en bassins non jaugés (PBNJ), 2003-2012: émergence d'un futur passionnant pour les sciences hydrologiquesRésumé Les bassins versants de drainage de nombreuses régions du monde sont peu ou pas du tout jaugés, et dans certains cas les réseaux de mesures existants sont en déclin. Le problème est compliq...

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River flows have been known to be scaling for over 40 years and scaling notions have developed rapidly since the 1980s. Using the framework of universal multifractals and time series of rainfall and river runoff for 30 French catchments (basin sizes of 40 km2 to 200 km2) from 1 day to 30 years, we quantify types and extent of the scaling regimes. For both flow and rain series, we observed a scale break at roughly 16 days, which we associate with the “synoptic maximum”; the time scale of structures of planetary spatial extent. For the two scaling regimes in both series, we estimate the universal multifractal parameters as well as the critical exponents associated with multifractal phase transitions. Using these exponents, we perform (causal) multifractal time series simulations and show how a simple (linear) scaling transfer function can be used to relate the low‐frequency rainfall series to the corresponding river flow series. The high‐frequency regime requires nonlinear transforms.

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International audienceUrban catchments are typically characterised by high spatial variability and fast runoff processes resulting in short response times. Hydrological analysis of such catchments requires high resolution precipitation and catchment information to properly represent catchment response. This study investigated the impact of rainfall input resolution on the outputs of detailed hydrodynamic models of seven urban catchments in North-West Europe. The aim was to identify critical rainfall resolutions for urban catchments to properly characterise catchment response. Nine storm events measured by a dual-polarimetric X-band weather radar, located in the Cabauw Experimental Site for Atmospheric Research (CESAR) of the Netherlands, were selected for analysis. Based on the original radar estimates, at 100m and 1min resolutions, 15 different combinations of coarser spatial and temporal resolutions, up to 3000m and 10min, were generated. These estimates were then applied to the operational semi-distributed hydrodynamic models of the urban catchments, all of which have similar size (between 3 and 8km2), but different morphological, hydrological and hydraulic characteristics. When doing so, methodologies for standardising model outputs and making results comparable were implemented. Results were analysed in the light of storm and catchment characteristics. Three main features were observed in the results: (1) the impact of rainfall input resolution decreases rapidly as catchment drainage area increases; (2) in general, variations in temporal resolution of rainfall inputs affect hydrodynamic modelling results more strongly than variations in spatial resolution; (3) there is a strong interaction between the spatial and temporal resolution of rainfall input estimates. Based upon these results, methods to quantify the impact of rainfall input resolution as a function of catchment size and spatial-temporal characteristics of storms are proposed and discussed. © 2015 The Authors

Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we then show how to produce a causal stochastic model of intermittent fields and use it to study the predictability of these fields. Finally, by returning to the origins of the turbulent "shell models" and restoring a large number of degrees of freedom (the Scaling Gyroscope Cascade, SGC models) we partially close the gap between the cascades and the dynamical Navier–Stokes equations. Furthermore, we point out that beyond a close agreement between universal parameters of the different modeling approaches and the empirical estimates in turbulence, there is a rather common structure involving both a "renormalized viscosity" and a "renormalized forcing". We conclude that this gives credence to the possibility of deriving analytical/renormalized models of intermittency built on this structure.

The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian 1 noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. We therefore derive in this paper a Fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Levy-stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the Fractional Fokker-Planck equation.