TY - JOUR

T1 - The role of a delay time on the spatial structure of chaotically advected reactive scalars

AU - Tzella, A.

AU - Haynes, P.H.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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 Péclet number), the emergent spatial structures are filamental and characterized by a single scaling regime with a Hölder 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 nongeneric scaling behavior.

AB - 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 Péclet number), the emergent spatial structures are filamental and characterized by a single scaling regime with a Hölder 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 nongeneric scaling behavior.

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-70049112684&partnerID=8YFLogxK

U2 - 10.1063/1.3210778

DO - 10.1063/1.3210778

M3 - Article

AN - SCOPUS:70049112684

SN - 1070-6631

VL - 21

JO - Physics of Fluids

JF - Physics of Fluids

IS - 8

M1 - 087101

ER -