Abstract
Abstract. Universal multifractals (UMs) have been widely used to simulate and characterize, with the help of only two physically meaningful parameters, geophysical fields that are extremely variable across a wide range of scales. Such a framework relies on the assumption that the underlying field is generated through a multiplicative cascade process. Derived analysis techniques have been extended to study correlations between two fields not only at a single scale and for a single statistical moment as with the covariance, but across scales and for all moments. Such a framework of joint multifractal analysis is used here as a starting point to develop and test an approach enabling correlations between UM fields to be analysed and approximately simulated. First, the behaviour of two fields consisting of renormalized multiplicative power law combinations of two UM fields is studied. It appears that in the general case the resulting fields can be well approximated by UM fields with known parameters. Limits of this approximation will be quantified and discussed. Techniques to retrieve the UM parameters of the underlying fields as well as the exponents of the combination have been developed and successfully tested on numerical simulations. In a second step tentative correlation indicators are suggested. Finally the suggested approach is implemented to study correlation across scales of detailed rainfall data collected with the help of disdrometers of the Fresnel platform of Ecole des Ponts ParisTech (see available data at https://hmco.enpc.fr/portfolio-archive/taranis-observatory/, last access: 12 March 2020). More precisely, four quantities are used: the rain rate (R), the liquid water content (LWC) and the total drop concentration (Nt) along with the mass weighed diameter (Dm), which are commonly used to characterize the drop size distribution. Correlations across scales are quantified. Their relative strength (very strong between R and LWC, strong between DSD features and R or LWC, almost null between Nt and Dm) is discussed.
Highlights
Numerous geophysical fields exhibit intermittent features with sharp fluctuations across all scales, skewed probability distribution and long-range correlations
The behaviour of two fields consisting of renormalized multiplicative power law combinations of two universal multifractals (UMs) fields is studied. It appears that in the general case the resulting fields can be well approximated by UM fields with known parameters
We used the framework of joint multifractal analysis to characterize the correlation across scales between two multifractal fields
Summary
Numerous geophysical fields exhibit intermittent features with sharp fluctuations across all scales, skewed probability distribution and long-range correlations. Meneveau et al (1990), who suggested studying the properties of joint moments of two multifractal fields (i.e. the product of the two fields raised to two different powers) across scales. The behaviour of the scaling exponent as a function of the two moments provides information on the correlations between the two fields They tested their framework on velocity and temperature as well as velocity and vorticity. Seuront and Schmitt (2005a, 2005b) suggested a refinement of this framework and introduced a re-normalization of these joint moments to define an exponent called “generalized correlation function” and used the properties of this function to better understand the coupling between fluorescence (which is related to phytoplankton concentration) and temperature for various levels of turbulence. The framework is implemented on rainfall data to study the correlation between rain rate, liquid water content and quantities characterizing the drop size distribution
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