# Lévy Walks as a Universal Mechanism of Turbulence Nonlocality

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## Abstract

**:**

^{3}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 t

^{3}scaling law to the combined regime of Lévy flights and Lévy walks in fluids and gases.

## 1. Introduction: Superdiffusion in the Biberman–Holstein Model

^{3}scaling law for turbulent relative dispersion, i.e., for the mean square of the mutual separation of a pair of particles, ${r}_{pair}\left(t\right)$, in a fluid or gaseous medium,

**r**, t) = δ(

**r**−

**r**

_{0})δ(t − t

_{0}). The commonly used concept of the average distance traveled by a photon in a given time turns out to be inapplicable in the case of superdiffusion, because the function f(r, t) falls too slowly with increasing distance from the primary source, and therefore the integral that determines the mean square of the displacement $\overline{{r}^{2}}$ diverges. The definition of $\overline{t}$(r) corresponding to the case of superdiffusion was given in [24] and takes the following form:

^{2}),

^{3}law for hydrodynamic turbulence). In Section 3, a scaling is proposed, which generalizes Richardson’s law to the combined mode of Lévy flights and Lévy walks for hydrodynamic turbulence. One of the arguments in favor of the Biberman–Holstein approach is that it is another way to derive Richardson’s t

^{3}law (1).

## 2. Superdiffusion in Plasma Turbulence

^{3}law (1) for the hydrodynamic turbulence of fluids and gases. Below, we will dwell on the physical model [12] in more detail, since the success of its application to plasma turbulence will allow us to propose a generalization of Richardson’s law for fluids and gases in Section 3.

- The spectrum of scattered radiation (Figure 4a in [54]).
- The phase and the modulus of the complex function $\widehat{C}\left(\omega ,\u2206r\right)$ at fixed values of the distance between two points along the minor radius of the plasma column (Figure 4b,c in [54]).
- The coordinate dependence of $\widehat{C}\left(\omega ,\u2206r\right)$ at different frequencies.

^{3}law (1), but also the similarity of the spectrum of the specific kinetic energy of density fluctuations in the tokamak plasma to the spectrum of the fluctuation velocity component in hydrodynamic turbulence make it possible to qualify the transport of density fluctuations in a tokamak plasma across a strong magnetic field as turbulence. Note that studies of the nonlocal properties of turbulence, including the deviation of statistics from the Gaussian one in various plasma turbulence phenomena, are reflected in the collective monograph [70].

## 3. Extension of Richardson’s t^{3} Law to the Combined Lévy Flight and Lévy Walk Regime for Turbulence in Fluids and Gases

^{3}law (1) given below to the combined regime of Lévy flights and Lévy walks. Such a generalization is suggested, as noted above, by the idea of Schlesinger and colleagues [10] and the success of the model [12] in interpreting experiments on cross-correlation reflectometry of tokamak plasma.

^{2}/s, ${\epsilon}_{0}$ characterizes the specific energy dissipation rate in units of m

^{2}/s

^{3}.

^{3}law (1) is supported by an extensive database, including, for example, recent experimental and theoretical studies in [76], the behavior of turbulent pair correlation (turbulent pair dispersion) at large times and the change of regimes (scalings) with time is of undoubted interest.

## 4. Conclusions

^{3}law for the mean square separation of a pair of particles in a fluid or gaseous medium. It is shown here that not only this fact, but also the similarity of the spectrum of the specific kinetic energy of density fluctuations in the tokamak plasma to the spectrum of the fluctuating velocity component in hydrodynamic turbulence, makes it possible to quantify the transfer of density fluctuations in the tokamak plasma across a strong magnetic field as turbulence. The key feature of the developed approach is that it was possible to establish the proximity of the kinetics of plasma density fluctuations to hydrodynamic turbulence within the general framework of interpreting the results of experiments using the phenomenological model of density fluctuations transfer in the Lévy walk mode, which does not require specifying the physical model of elementary excitations of the medium.

^{3}law for the combined regime of Lévy flights and Lévy walks in fluids and gases. Although Richardson’s t

^{3}law is supported by an extensive database, the behavior of turbulent pair correlation (turbulent pair dispersion) over long periods of time and the change in regimes (scalings) with time is of undoubted interest.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Correction Statement

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**Figure 2.**Dependence of pair dispersion on time for the nonlocality parameter γ = 2/3 and different values of the retardation parameter ${R}_{c}$: (

**a**) 10, (

**b**) 30, (

**c**) 100. The results are shown for Lévy flights (black line), Lévy walks (pink dashed line), combined Lévy flights and walks (34) (yellow curve); ballistic front (19) (red dotted line).

**Figure 3.**Dependence of pair dispersion on time for the nonlocality parameter γ = 1/2 and different values of the retardation parameter ${R}_{c}$: (

**a**) 10, (

**b**) 100. The results are shown for Lévy flights (black line), Lévy walks (pink dashed line), combined Lévy flights and walks (35) (yellow curve); ballistic front (19) (red dotted line).

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**MDPI and ACS Style**

Kukushkin, A.B.; Kulichenko, A.A.
Lévy Walks as a Universal Mechanism of Turbulence Nonlocality. *Foundations* **2023**, *3*, 602-620.
https://doi.org/10.3390/foundations3030036

**AMA Style**

Kukushkin AB, Kulichenko AA.
Lévy Walks as a Universal Mechanism of Turbulence Nonlocality. *Foundations*. 2023; 3(3):602-620.
https://doi.org/10.3390/foundations3030036

**Chicago/Turabian Style**

Kukushkin, Alexander B., and Andrei A. Kulichenko.
2023. "Lévy Walks as a Universal Mechanism of Turbulence Nonlocality" *Foundations* 3, no. 3: 602-620.
https://doi.org/10.3390/foundations3030036