Random Processes with Stationary Increments and Composite Spectra
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om Processes with Stationary Increments and Composite Spectra G. S. Golitsyna, * and M. I. Fortusa, ** a Obukhov
Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 119017 Russia *e-mail: [email protected] **e-mail: [email protected] Received October 8, 2019; revised December 13, 2019; accepted February 5, 2020
Abstract—A technique for deriving and interpreting fractal geophysical processes with power-law spectra of a different nature is described. Examples include the energy spectra of atmospheric processes and their role in the mixing of impurities, the frequency spectra of sea wind waves, and the spatial spectra of the surface relief of celestial bodies in the solar system. A.N. Kolmogorov’s works in the early 1930s, which were subsequently developed by his followers A.M. Obukhov, A.S. Monin, A.M. Yaglom, and others, are the most important for this. Kolmogorov’s probabilistic laws serve as a model for the analysis of the processes under consideration by methods of the similarity and dimensionality theory. Keywords: methods, random processes, structural functions, spectra, atmospheric turbulence, various natural processes DOI: 10.1134/S0001433820030081
1. INTRODUCTION A critical role in the development of the modern theory of random processes was played by the series of Kolmogorov’s studies under the general title Analytical Methods of Probability Theory, crowned by the work Zufällige Bewegungen (Random Motions), 1934 ([1], hereafter, ANK34). The latter study specified the accelerations as a sequence of uncorrelated random variables. In modern theoretical physics, they are called δ -correlated random variables in time. The analytical solution of a Fokker–Planck type equation proposed by Kolmogorov in [1] was discussed in detail by Monin and Yaglom [2] as applied to the theory of turbulence developed in 1941 by Kolmogorov and his graduate student Obukhov. In the mid-1950s, Obukhov began to study an equation for the probability distribution function of the form
∂p ∂p D ∂ 2 p + ui = , ∂t ∂xi 2 ∂ui2
(1)
which is called in our scientific literature the Fokker– Planck–Kolmogorov equation (FPKE34), where D is the diffusion coefficient. This equation operates in both two and three dimensions (it is only assumed that the probability distribution function (PDF) is homogeneous with respect to coordinates). A detailed description of this equation can be found in [2, section 24.4]. Obukhov indicated in [3] that the diffusion coefficient D in Eq. (1) is proportional to the kinetic energy dissipa-
tion rate ε 2, i.e., describes diffusion in the velocity space. This equation has three scales [3] given by its general solution for the probability distribution density 3
2 2 pt ( X ,U ) = 3 2 exp − 3X 3 + 3UX2 − U . (2) Dt 2πDt Dt Dt It can be seen from here that this is a normal distribution with time-dependent local extremes. The transform
( )
12 t = t τ, U = U ( ετ) , X = X ετ3
12
,
where τ is an arbitrary quantity, reduces Eq. (1) to the universal (i.e., self-similar) form
∂p
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