The role of a delay time on the spatial structure of chaotically advected reactive scalars

TL;DR

This paper investigates how delay times in reaction equations influence the spatial structure of chaotically advected reactive scalars, revealing new scaling regimes and transition scales in the presence of delays.

Contribution

It extends previous models by analyzing the effects of delay terms on scalar field structures and identifies new intermediate-scale regimes caused by delays.

Findings

Small-scale structures are shared across fields regardless of delay.

Delay introduces additional intermediate scaling regimes.

Numerical models confirm theoretical predictions with biological and chemical systems.

Abstract

The stationary-state spatial structure of reacting scalar fields, chaotically advected by a two-dimensional large-scale flow, is examined for the case for which the reaction equations contain delay terms. Previous theoretical investigations have shown that, in the absence of delay terms and in a regime where diffusion can be neglected (large P\'eclet number), the emergent spatial structures are filamental and characterized by a single scaling regime with a H\"older exponent that depends on the rate of convergence of the reactive processes and the strength of the stirring measured by the average stretching rate. In the presence of delay terms, we show that for sufficiently small scales all interacting fields should share the same spatial structure, as found in the absence of delay terms. Depending on the strength of the stirring and the magnitude of the delay time, two further scaling regimes that are unique to the delay system may appear at intermediate length scales. An expression for the transition length scale dividing small-scale and intermediate-scale regimes is obtained and the scaling behavior of the scalar field is explained. The theoretical results are illustrated by numerical calculations for two types of reaction models, both based on delay differential equations, coupled to a two-dimensional chaotic advection flow. The first corresponds to a single reactive scalar and the second to a nonlinear biological model that includes nutrients, phytoplankton and zooplankton. As in the no-delay case, the presence of asymmetrical couplings among the biological species results in a non-generic scaling behavior.