Large-Scale Pattern Formation in the Presence of Small-Scale Random Advection

TL;DR

This paper investigates how small-scale random advection influences large-scale pattern formation in turbulent systems, revealing shifts in pattern onset and wave number through theoretical and simulation approaches.

Contribution

It introduces a conceptual framework combining pattern formation theory with stochastic analysis to understand the impact of random advection on large-scale patterns.

Findings

Random advection shifts pattern onset

Wave number of patterns is altered

Implications for turbulent convection flows

Abstract

Despite the presence of strong fluctuations, many turbulent systems such as Rayleigh-B\'{e}nard convection and Taylor-Couette flow display self-organized large-scale flow patterns. How do small-scale turbulent fluctuations impact the emergence and stability of such large-scale flow patterns? Here, we approach this question conceptually by investigating a class of pattern forming systems in the presence of random advection by a Kraichnan-Kazantsev velocity field. Combining tools from pattern formation with statistical theory and simulations, we show that random advection shifts the onset and the wave number of emergent patterns. As a simple model for pattern formation in convection, the effects are demonstrated with a generalized Swift-Hohenberg equation including random advection. We also discuss the implications of our results for the large-scale flow of turbulent Rayleigh-B\'{e}nard convection.