Cascades and statistical equilibrium in shell models of turbulence

Abstract

We study the GOY shell model simulating the cascade processes of turbulent flow. The model has two inviscid invariants governing the dynamical behavior. Depending on the choice of interaction coefficients, or coupling parameters, the two invariants are either both positive definite, analogous to energy and enstrophy of 2D flow, or only one is positive definite and the other not, analogous to energy and helicity of 3D flow. In the 2D like model the dynamics depend on the spectral ratio of enstrophy to energy. That ratio depends on wave-number as $k^{\alpha }$. The enstrophy transfer through the inertial sub-range can be described as a forward cascade for $\alpha <2$ and a diffusion in a statistical equilibrium for $\alpha >2$. The $\alpha =2$ case, corresponding to 2D turbulence, is a borderline between the two descriptions. The difference can be understood in terms of the ratio of typical timescales in the inertial sub-range and in the viscous sub-range. The multi fractality of the enstrophy dissipation also depends on the parameter $\alpha$, and seems to be related to the ratio of typical timescales of the different shell velocities.

29 pages, 11 figures

Cited by in corpus (19)

• Cascades and transitions in turbulent flows
• A note on shell models for MHD Turbulence
• Long short-term memory embedded nudging schemes for nonlinear data assimilation of geophysical flows
• A Note on the Regularity of Inviscid Shell Model of Turbulence
• Inverse Cascade Regime in Shell Models of 2-Dimensional Turbulence
• Fibonacci turbulence
• Cascades of energy and helicity in the GOY shell model of turbulence
• A Critical "Dimension" in a Shell Model for Turbulence
• The role of helicity in triad interactions in 3D turbulence investigated in a new shell model
• Revisiting the SABRA Model: Statics and Dynamics
• Bridging Inertial and Dissipation Range Statistics in Rotating Turbulence
• Chaotic and regular instantons in helical shell models of turbulence
• Quasi-solitons and asymptotic multiscaling in shell models of turbulence
• Temporal intermittency and cascades in shell models of turbulence
• Non-equilibrium statistical mechanics of the turbulent energy cascade: irreversibility and response functions
• A stochastic model of cascades in 2D turbulence
• Shell Models on Recurrent Sequences: Fibonacci, Padovan and Other Series
• Nonlinear phase synchronization and the role of spacing in shell models
• About the maximum entropy principle in non equilibrium statistical mechanics