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 (...

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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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Abstract. We present a selection of methodologies for using the palaeo-climate model component of the Coupled Model Intercomparison Project (Phase 5) (CMIP5) to attempt to constrain future climate projections using the same models. The constraints arise from measures of skill in hindcasting palaeo-climate changes from the present over three periods: the Last Glacial Maximum (LGM) (21 000 yr before present, ka), the mid-Holocene (MH) (6 ka) and the Last Millennium (LM) (850-1850 CE). The skill measures may be used to validate robust patterns of climate change across scenarios or to distinguish between models that have differing outcomes in future scenarios. We find that the multi-model ensemble of palaeo-simulations is adequate for addressing at least some of these issues. For example, selected benchmarks for the LGM and MH are correlated to the rank of future projections of precipitation/temperature or sea ice extent to indicate that models that produce the best agreement with palaeo-climate information give demonstrably different future results than the rest of the models. We also explore cases where comparisons are strongly dependent on uncertain forcing time series or show important non-stationarity, making direct inferences for the future problematic. Overall, we demonstrate that there is a strong potential for the palaeo-climate simulations to help inform the future projections and urge all the modelling groups to complete this subset of the CMIP5 runs.Published by Copernicus Publications on behalf of the European Geosciences Union.