Out-of-time-ordered correlators for turbulent fields: a quantum-classical correspondence

An extended formulation of out-of-time-ordered correlators (OTOCs), which quantify noncommutative operator growth and information scrambling in quantum many-body systems, is developed for turbulence dynamics as a representative of non-canonical Hamiltonian systems.

Based on the Wigner-Weyl transform and the Moyal bracket formalism, the semiclassical limit of OTOC for turbulent plasmas governed by the Hasegawa-Mima equation is derived as an ensemble-averaged squared Lie-Poisson bracket between two chosen functionals of the turbulent fields.

The classical-limit OTOC provides a quantitative measure of how a variational perturbation applied to one functional propagates across scales in the turbulent dynamics and how it affects another functional at a later time, thereby capturing scale-dependent or field-dependent transfer processes.

In a quasilinear approximation with a strong zonal flow, we provide a closed analytic expression of the classical-limit OTOC to characterize the interaction between zonal and non-zonal modes.

An asymptotic analysis shows that the OTOC grows quadratically at early time, while in the long-time strong-shear regime it approaches a finite saturated value with an inverse-square algebraic dependence.

This behavior is attributed to zonal-flow shearing, which rapidly scrambles the non-zonal perturbation toward higher wavenumbers, thereby reducing the low-wavenumber non-zonal content that can feed back onto large-scale zonal modes.