Andrei A. Kulichenko

The nonlocality (superdiffusion) of turbulence is expressed in the empiric Richardson t³ scaling law for the mean square of the mutual separation of a pair of particles in a fluid or gaseous medium. The development of the theory of nonlocality of various processes in physics and other sciences based on the concept of Lévy flights resulted in Shlesinger and colleagues’ about the possibility of describing the nonlocality of turbulence using a linear integro-differential equation with a slowly falling kernel. The approach developed by us made it possible to establish the closeness of the superdiffusion parameter of plasma density fluctuations moving across a strong magnetic field in a tokamak to the Richardson law. In this paper, we show the possibility of a universal description of the characteristics of nonlocality of transfer in a stochastic medium (including turbulence of gases and fluids) using the Biberman–Holstein approach to examine the transfer of excitation of a medium by photons, generalized in order to take into account the finiteness of the velocity of excitation carriers. This approach enables us to propose a scaling that generalizes Richardson’s t³ scaling law to the combined regime of Lévy flights and Lévy walks in fluids and gases.

The nonlocality (superdiffusion) of turbulence is expressed in the empiric Richardson t3 scaling law for the mean square of the mutual separation of a pair of particles in a fluid or gaseous medium. The development of the theory of nonlocality of various processes in physics and other sciences based on the concept of Lévy flights resulted in the idea of Shlesinger and colleagues about the possibility of describing the nonlocality of turbulence using a linear integro-differential equation with a slowly falling kernel. The close approach developed by us made it possible to establish the closeness of the superdiffusion parameter of plasma density fluctuations moving across a strong magnetic field in a tokamak to the Richardson law. In this paper, we show the possibility of a universal description of the characteristics of nonlocality of transfer in a stochastic medium (including turbulence of gases and fluids) using the Biberman-Holstein approach to the transfer of excitation of a medium by photons, generalized to take into account the finiteness of the velocity of excitation carriers. This approach enables us to propose a scaling that generalizes Richardson's t3 scaling law to the combined regime of Lévy flights and Lévy walks.

One of the most important properties of stochastic nonlinear processes, including the turbulence of the hydrodynamic motion of continuous media, is distant spatial correlations. To describe them, an approach was proposed by Shlesinger and colleagues based on a linear integro-differential equation with a slowly decaying kernel, which corresponds to superdiffusion (nonlocal) transfer in the regime of Lévy walks (Lévy flights when the finite velocity of the carriers is taken into account). In this paper, we formulate a similar approach that makes it possible to formulate the problem of determining these properties from the scattering spectra of electromagnetic (EM) waves and cross-correlation reflectometry. A universal description of the relationship between the observed symmetric quasi-coherent component in the spectrum of scattered EM waves in plasmas and a process of the Mandelstam–Brillouin scattering type is obtained. It is shown that the nonlocality of spatial correlations of density fluctuations in a turbulent medium is due to long-free-path carriers of the medium’s perturbations, for which the free path distribution function is described by the Lévy distribution. The effectiveness of the proposed method is shown by the example of the interpretation of the data of cross-correlation reflectometry of EM waves in the radio-frequency range for the diagnosis of turbulent plasma in magnetic confinement devices for axisymmetric toroidal thermonuclear plasma.

We obtain an analytic description of the excitation front propagation in a medium during nonstationary superdiffusive (nonlocal) transport in the case of a finite fixed velocity of perturbation carriers (so-called Lévy walks with stops). This problem embraces phenomena such as resonance radiation transport in astrophysical gases and plasma, biological migration, and energy transfer by waves in a plasma. In this approach, the result obtained by integrating the exact solution to the kinetic equation for the Green function is independent of the coordinate space dimensionality. The results are compared with data obtained using another more exact method of determining the front and with the results of numerical calculations of the statistics of trajectories using the Monte Carlo method. Comparison demonstrates the applicability of our results in a wide range of parameters of the problem. We propose a universal description of the perturbation front dynamics in the medium for an arbitrary (including infinitely high) fixed velocity of perturbation carriers. This corresponds to the combination of expressions for the front in the case of transport by Lévy flights and by Lévy walks. We consider the criteria of transition between these regimes of the superdiffusive transport, which corresponds, in particular, to the account for the finite velocity of light in the superdiffusive transport of resonance radiation in gases and plasmas. For Lévy walks, we have obtained a relation between the integral characteristic of perturbation of the medium and its carriers.

Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.