From zero-mode intermittency to hidden symmetry in random scalar advection

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier-Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It ultimately identifies the scaling exponents as the eigenvalues of a Perron-Frobenius operator acting on invariant measures of the rescaled equations. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode scenario represents only one facet of intermittency, here prescribing the scaling exponents of even-order correlators. Besides, we argue that hidden symmetry provides a more generic mechanism, fully prescribing intermittency in terms of scaling anomalies, but also in terms of its multiplicative random nature and fusion rules required to explicitly compute zero-modes from first principles.