RG approach to the inviscid limit for shell models of turbulence
arXiv:2408.04659 · doi:10.1088/1361-6544/adf407
Abstract
We consider an initial value problem for shell models that mimic turbulent velocity fluctuations over a geometric sequence of scales. Our goal is to study the convergence of solutions in the inviscid (more generally, vanishing regularization) limit and explain the universality of both the limiting solutions and the convergence process. We develop a renormalization group (RG) formalism representing this limit as dynamics in a space of flow maps. For the dyadic shell model, the RG dynamics has a fixed-point attractor, which determines universal limiting solutions. Deviations from the limiting solutions are also universal and given by a leading eigenmode (eigenvalue and eigenvector) of the linearized RG operator. Application to the Gledzer shell model reveals the RG attractor in the form of a closed invariant curve, while the Sabra shell model yields chaotic RG dynamics. An important consequence of the RG formalism is the understanding of the different roles of symmetry-preserving (canonical) and symmetry-breaking (e.g. viscous) regularizations.
arXiv comment35 pages, 8 figures
References in corpus (23)
- An optimal shell model of turbulence
- Potentially Singular Solutions of the 3D Incompressible Euler Equations
- Analytic Study of Shell Models of Turbulence
- Analogies between Scaling in Turbulence, Field Theory and Critical Phenomena
- A Note on the Regularity of Inviscid Shell Model of Turbulence
- From the butterfly effect to intrinsic randomness: the spontaneous growth of singular shear flows
- The vanishing viscosity limit for a dyadic model
- Spontaneously stochastic solutions in one-dimensional inviscid systems
- Functional renormalisation group for turbulence
- Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection
- Rayleigh-Taylor turbulence with singular nonuniform initial conditions
- Spontaneous stochasticity of velocity in turbulence models
- Hidden scale invariance of intermittent turbulence in a shell model
- Renormalization and universality of blowup in hydrodynamic flows
- A Renormalization Group Approach to Spontaneous Stochasticity
- Hidden spatiotemporal symmetries and intermittency in turbulence
- Toward analytic theory of the Rayleigh-Taylor instability: lessons from a toy model
- Hidden scale invariance of turbulence in a shell model: from forcing to dissipation scales
- Continuous representation for shell models of turbulence
- Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics
- Statistical determinism in non-Lipschitz dynamical systems
- Dyadic models for fluid equations: a survey
- RG analysis of spontaneous stochasticity on a fractal lattice: stability and bifurcations