Stochastic attractors for shell phenomenological models of turbulence

TL;DR

This paper investigates stochastic shell models of turbulence, demonstrating the continuous dependence of solutions on a parameter and the existence and upper semicontinuity of finite-dimensional random attractors, advancing understanding of turbulence modeling.

Contribution

It extends previous deterministic results to stochastic models, proving the existence of random attractors and their upper semicontinuity with respect to a bridging parameter.

Findings

Solutions depend continuously on the parameter λ.

Finite-dimensional random attractors exist for each λ.

Random attractors vary upper semicontinuously with λ.

Abstract

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $\lambda$ which bridges continuously between the two systems. That is, for $\lambda=1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $\lambda=0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $\lambda$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $\lambda$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $\lambda$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models.