Chaotic and regular instantons in helical shell models of turbulence

Abstract

Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations in helical eigenstates. We find that depending on the helical structure of each model, instantons are chaotic or regular. Some instantonic solutions tend to recover mirror symmetry for scales small enough. Models that have anomalous scaling develop regular non chaotic instantons. Conversely, models that have non anomalous scaling in the stationary regime are those that have chaotic instantons. The direction of the energy carried by each single instanton tends to coincide with the direction of the energy cascade in the stationary regime. Finally, we find that whenever the small-scale stationary statistics is intermittent, the instanton is less steep than the dimensional Kolmogorov scaling, independently of whether or not it is chaotic. Our findings further support the idea that instantons might be crucial to describe some aspects of the multi-scale anomalous statistics of shell models

Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

“…The model is formulated in a form identical to the original Euler equations, but with the algebraic structure defined on the 3D logarithmic lattice. We show that the blowup in this model is associated with a chaotic attractor of a renormalized system, in accordance with some earlier theoretical conjectures [37][38][39][40]; one can also make an interesting connection with the chaotic Belinskii-Khalatnikov-Lifshitz singularity in general relativity [8,41]. A distinctive property of the attractor is its anomalous multiscale structure, which explains the diversity of the existing DNS results, discloses fundamental limitations of current strategies, and provides new guidelines for the original blowup problem.…”

Cascades and transitions in turbulent flows

“…The possibility to have a critical discontinuous transition from forward to inverse cascade at changing the model's parameters has been discussed by [179,182]. Shell models are also characterized by the presence of quasi-coherent instantonic solutions that bring energy from large to small scales [572][573][574][575]. As a result, the energy transfer must be seen as the superposition and interaction of a statistical background plus quasi-coherent burst-like solutions travelling to small scales.…”

Fluid dynamics on logarithmic lattices

“…The chaotic behavior justifies the high sensitivity to perturbations, which is encountered in full DNS [56] and has theoretical foundation in developed turbulence [90]. Instability of blowup solutions is also observed in other simplified models [95,74,37] and was proved recently for the full incompressible 3D Euler equations [96]. The chaotic attractor restores the isotropy in the statistical sense, even though the solution at each particular moment is essentially anisotropic, in similarity to the recovery of isotropy in the Navier-Stokes turbulence [50,11].…”