## 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

0

)/k

d+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.

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

0

)/k

d+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.

Original language | English |
---|---|

Number of pages | 17 |

Journal | Universe |

Volume | 6 |

Issue number | 145 |

Publication status | Published - 6 Sep 2020 |

## Scopus subject areas

- Statistical and Nonlinear Physics