Mandelbrot Cascade Measures Independent of Branching Parameter (original) (raw)
Related papers
Scaling exponents and multifractal dimensions for independent random cascades
Communications in Mathematical Physics, 1996
This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(#), and (ii) multifractal dimensions are considered for cascades with a generator vector (wι w c ) of the general type. These problems were previously studied for independent strongly bounded variables w l : 0 < a < w/ ^ c. Consequently, a broad class of models used in applications, including Kolmogorov's log-normal model in turbulence, log-stable "universal" cascades in atmospheric dynamics, has not been covered. Roughly speaking, problems (i), (ii) are here solved under the condition that the scaling exists; the τ-function is calculated for all arguments (previously this was done for positive q) and a new effect emerges: the τ-function can generally involve discontinuities in the first derivative as well as in the second.
Stochastic selfsimilar branching and turbulence
Physica A: Statistical Mechanics and its Applications, 1996
A new stochastic multifractal cascade model for inertial-subrange turbulence is introduced which can be fitted to a recent experimental determination of the spectrum of generalized dimensions. The local scaling behaviour (set of scale-invariant multiplier distributions)
On intermittency in a cascade model for turbulence
1993
Physica D 65 (1993) 163-171 North-Holland SDI: 0167-2789(92)00018-J On intermittency in a cascade model for turbulence R. Benzi, L. Biferale and G. Parisi Dipartimento di Fisica, Universit"Tor Vergata", Via della Ricerca Scientifica, 1-00133 Rome, Italy Received 28 April 1992 Revised manuscript received 8 September 1992 Accepted 16 October 1992 Communicated by U. Frisch MCA CI In this note we study the possibility of performing analytic computations of the exponents characterizing the multifractal behaviour of turbulence.
Statistical properties of a turbulent cascade
Physica D: Nonlinear Phenomena, 1997
Statistical properties of a turbulent cascade are evaluated by considering the joint probability distribution p(v[, L i; v2, L2) for two velocity increments Vl, v2 of different length scales Ll, L2. We present experimental evidence that the conditional probability distribution p(v2, L21vl, L l) obeys a Chapman-Kolmogorov equation. We evaluate the Kramers-Moyal coefficients and show evidence that higher-order coefficients vanish except for the drift and diffusion coefficient. As a result the joint probability distributions obeys a Fokker-Planck equation. We calculate drift and diffusion coefficients and discuss their relationship to universal behaviour in the scaling region and to intermittency of the turbulent cascade.
Cumulant ratios in discrete cascade models of turbulence
2000
In the context of random multiplicative cascade processes, we derive analytical expressions for translationally invariant oneand two-point cumulants in logarithmic field amplitudes. On taking ratios of cumulants, geometrical effects due to spatial averaging cancel out. Unlike multifractal scaling exponents and multiplier distributions, these ratios can successfully further distinguish between cascade generators. Although the underlying hydrodynamic equations are deterministic, the statistical description of fully developed turbulence has by now a long tradition [1]. Random multiplicative cascade models on ultrametric trees form a particularly simple and robust class of such statistical models, reproducing important observed features such as multiplier distributions and their correlations [2,3] and related Kramers-Moyal coefficients reflecting Markovian properties [4]. The models have worked almost too well in the sense that different cascade generators for the multiplicative weights...
Multifractal Cascade Dynamics and Turbulent Intermittency
Fractals, 1997
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 aniso...
Finite-size scaling of two-point statistics and the turbulent energy cascade generators
Physical Review E, 2005
Within the framework of random multiplicative energy cascade models of fully developed turbulence, finite-size-scaling expressions for two-point correlators and cumulants are derived, taking into account the observationally unavoidable conversion from an ultrametric to an Euclidean two-point distance. The comparison with two-point statistics of the surrogate energy dissipation, extracted from various wind tunnel and atmospheric boundary layer records, allows an accurate deduction of multiscaling exponents and cumulants, even at moderate Reynolds numbers for which simple power-law fits are not feasible. The extracted exponents serve as input for parametric estimates of the probabilistic cascade generator. Various cascade generators are evaluated.
EPL (Europhysics Letters), 2008
In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe "negative dimensions" in random multifractals. For that purpose, we define a new way to study scaling where the observation scale τ and the total sample length L are respectively going to zero and to infinity. This "mixed" asymptotic regime is parametrized by an exponent χ that corresponds to Mandelbrot "supersampling exponent". In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter χ can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the "hidden" negative part of the singularity spectrum, corresponding to "latent" singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics.