"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

/pdf/the-phenomenology-of-small-scale-turbulence-3a5uzrupts.pdf

The phenomenology of small-scale turbulence

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).

/pdf/physical-modeling-and-analysis-of-rain-and-clouds-by-5aswiv2xlo.pdf

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

The alignment between vorticity and eigenvectors of the strain‐rate tensor in numerical solutions of Navier–Stokes turbulence is studied. Solutions for isotropic flow and homogeneous shear flow from pseudospectral calculations using 1283 grid points have been examined. The Taylor Reynolds number is 83 or greater. In both flows there is an increased probability for the vorticity to point in the intermediate strain direction and at three‐fourths of the sample points this strain is positive (extensive). This propensity for vorticity alignment with a positive intermediate strain is a consequence of angular momentum conservation, as shown by a restricted Euler model of the coupling between strain and vorticity. Probability distributions for intermediate strain, conditioned on total strain, change from a symmetric triangular form at small strain to an asymmetric one for large strain. The most probable value of the asymmetric distribution gives strains in the ratios of 3:1: −4. The evolution of the distribution from a symmetric to an asymmetric form as the strain magnitude increases is essentially the same in both flows, indicating a generic structure of intense turbulence. The alignment between the gradient of a passive scalar and eigenvectors of the strain‐rate tensor for Prandtl numbers of 0.1, 0.2, 0.5, and 1.0 has also been studied. There is an increased probability for the scalar gradient to align in the most compressive strain direction, and the average gradient is larger when it is pointing in that direction. Estimates for the scalar dissipation from the turbulent kinetic energy, its dissipation, and the root‐mean‐square scalar value are in reasonable agreement with calculated scalar dissipation if no explicit Prandtl number dependence is used in the estimate. Statistical analysis of scalar dissipation conditioned on energy dissipation yields a power‐law relation between conditioned mean values. Both simulated flows are found to obey the qualitative predictions of the Gurvich–Yaglom (lognormal) intermittency model. Energy and scalar intermittency exponents are estimated and compared to measured values.

Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence

The intermittency of the rate of turbulent energy dissipation e is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of e are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of e is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001830

The multifractal nature of turbulent energy dissipation

A rigorous mathematical description is given for the physical concept of the cascade process of sequential breakdown of turbulent eddies. It is assumed that the probability distribution for the ratio of typical values of turbulent fluctuations averaged over a small volume v to those averaged over a surrounding larger volume V is invariant under the group of space similarity transformations, provided that the length scales of both volumes are much smaller than the external scale of turbulence and much larger than the Kolmogoroff scale. This assumption is in general agreement with the conclusions of Kolmogoroff and Obukhov and with the existing measurements of the power spectrum for the square of the velocity derivative. Some measurements of probability distributions for smallscale turbulent fluctuations are also discussed.

A. S. Gurvich

Papers

Breakdown of Eddies and Probability Distributions for Small‐Scale Turbulence