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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. Thesignature of this blowup is represented by self-similar instantonic structurestraveling coherently through the inertial range. These solutions mightinfluence the energy transfer and the anomalous scaling properties empiricallyobserved for the forced and viscous models. In this paper we present a study ofthe instantonic solutions for a set of four shell models of turbulence based onthe exact decomposition of the Navier-Stokes equations in helical eigenstates.We find that depending on the helical structure of each model, instantons arechaotic or regular. Some instantonic solutions tend to recover mirror symmetryfor scales small enough. Models that have anomalous scaling develop regular nonchaotic instantons. Conversely, models that have non anomalous scaling in thestationary regime are those that have chaotic instantons. The direction of theenergy carried by each single instanton tends to coincide with the direction ofthe energy cascade in the stationary regime. Finally, we find that whenever thesmall-scale stationary statistics is intermittent, the instanton is less steepthan the dimensional Kolmogorov scaling, independently of whether or not it ischaotic. Our findings further support the idea that instantons might be crucialto describe some aspects of the multi-scale anomalous statistics of shellmodels.

Chaotic and regular instantons in helical shell models of turbulence