From directed percolation to patterned turbulence

TL;DR

This paper demonstrates that ordered stripe patterns in turbulent flows can self-assemble from stochastic processes near a directed percolation critical point, challenging traditional views of pattern formation.

Contribution

It reveals that macroscopic patterns in turbulence can emerge from local stochastic rules without wavelength selection, linking pattern formation to directed percolation criticality.

Findings

Patterns self-assemble from stochastic processes at the DP critical point

Pattern expansion rates follow DP critical exponents

Order arises from local stochastic rules, not wavelength selection

Abstract

The transition to turbulence is characterized by an abrupt loss of order and predictability, featuring the intermittent proliferation and decay of localized turbulent structures. En-route to becoming fully turbulent, surprisingly order reappears when alternating laminar and turbulent regions arrange in regular stripe patterns. This macroscopic organization is believed to arise top down from a classic pattern forming instability of turbulence, imprinting a wavelength onto the disordered flow field. We here demonstrate that patterns instead self-assemble with increasing velocity. Starting from the intermittent stripe regime, specifically from the corresponding directed percolation (DP) critical point, regular patterns are established within the scaling range of the DP transition. Likewise the patterns' expansion rates are set by the DP critical exponents, attesting that all underlying processes are stochastic. This apparent contradiction between the inherent stochasticity and the displayed order is resolved by abandoning the common perception of laminar and turbulence as opposing states. More generally our study exemplifies that macroscopic patterns can arise solely from local stochastic rules, in the absence of wavelength selection typically associated with pattern formation.