Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

Nikolay V Antonov,Nikolay M Gulitskiy,German E Kochnev,Polina I Kakin

doi:10.3390/universe6090145

Abstract

We study a self-organized critical system under the influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [Phys. Rev. Lett.62: 1813 (1989)]. The motion of the environment is modelled by the isotropic Kazantsev–Kraichnan “rapid-change” ensemble for an incompressible fluid: it is Gaussian with vanishing correlation time and the pair correlation function of the form ∝δ(t−t′)/kd+ξ, where k is the wave number and ξ is an arbitrary exponent with the most realistic values ξ=4/3 (Kolmogorov turbulence) and ξ→2 (Batchelor’s limit). Using the field-theoretic renormalization group, we find infrared attractive fixed points of the renormalization group equation associated with universality classes, i.e., with regimes of critical behavior. The most realistic values of the spatial dimension d=2 and the exponent ξ=4/3 correspond to the universality class of pure turbulent advection where the nonlinearity of the Hwa–Kardar (HK) equation is irrelevant. Nevertheless, the universality class where both the (anisotropic) nonlinearity of the HK equation and the (isotropic) advecting velocity field are relevant also exists for some values of the parameters ε=4−d and ξ. Depending on what terms (anisotropic, isotropic, or both) are relevant in specific universality class, different types of scaling behavior (ordinary one or generalized) are established.

Highlights

The universality class where both the nonlinearity of the HK equation and the advecting velocity field are relevant exists for some values of the parameters ε = 4 − d and ξ
While an equilibrium system can become critical only if a certain control parameter is tuned to a precise critical value [1,2,3], systems with self-organized criticality (SOC) [4,5,6,7,8,9,10] evolve towards a critical state owing only to their intrinsic dynamics
We studied effects of isotropic turbulent advection described by Kazantsev–Kraichnan

Summary

Introduction

While an equilibrium system can become critical only if a certain control parameter is tuned to a precise critical value [1,2,3], systems with self-organized criticality (SOC) [4,5,6,7,8,9,10] evolve towards a critical state owing only to their intrinsic dynamics. The scaling behavior that corresponds to the regime where only the nonlinearity of the HK equation is relevant (the limit of the pure HK model) is realized through a kind of “dimensional transmutation”: the ratio u of the two diffusivity coefficients νk and ν⊥ acquires in this limit a nontrivial canonical dimension. The regime where both the turbulent advecting field (the isotropic one) and the nonlinearity (the anisotropic one) are relevant appears to be the most intriguing. This resembles modified types of scaling hypotheses (weak scaling in the spirit of Stell and generalized scaling in the spirit of Fisher) for systems which have more than one significantly different characteristic scales [67,68,69,70]

RG Analysis of the HK Model without Turbulent Advection

Renormalization of the Model with Turbulent Advection

Fixed Points of the Model with Turbulent Advection

Scaling Regimes and Critical Dimensions in the Model with Turbulent Advection

Conclusions