Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes

Abstract

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 (...

From molecules to meteorology via turbulent scale invariance

“…The emphasis is on the physical picture and diagrams, rather than on mathematics and equations. There is a large literature on other atmospheric aspects, particularly in the rainfall and hydrological literature stemming from the publication of Schertzer and Lovejoy (1987). Surface temperature (Koscielny-Bunde et al, 1998;Syroka and Toumi, 2001) has been examined, as has ozone column density Varotsos, 2005).…”

Generalized hydraulic geometry: Derivation based on a multiscaling formalism

“…The multiscaling framework has been successfully employed to provide a theoretical basis for the empirically observed scale-dependent behavior of flood peaks in the context of regional quantile analysis [e.g., Gupta and Waymire, 1990;Gupta et al, 1994;Gupta and Dawdy, 1995]. Other applications of this concept have appeared in atmospheric turbulence, rain and clouds (e.g., see Schertzer and Lovejoy [1987] for an early reference), in river networks [e.g., Gupta and Waymire, 1989], and in solute transport [Sposito and Jury, 1988]. In this paper, it is shown that the multiscaling framework can provide a theoretical basis for interpreting and modeling the scale and frequency dependent relations between channel morphometry and discharge (known as HG).…”

Multi-fractal thermal characteristics of the southwestern GIN sea upper layer

“…over a large range of wavenumber k. The spectral exponent b contains information about the degree of stationarity of the data [5,6]. If b < 1, the field is stationary; if 1 < b < 3, the field contains nonstationary signal with stationary increments and in particular, the small-scale gradient field is stationary; if b > 3, the field is nonstationary with nonstationary increments.…”