The influence of periodic islands in the flow on a scalar tracer in the presence of a steady source
Turner, M. R.
Gilbert, Andrew D.
Physics of Fluids
American Institute of Physics
In this paper we examine the influence of periodic islands within a time periodic chaotic flow on the evolution of a scalar tracer. The passive scalar tracer is injected into the flow field by means of a steady source term. We examine the distribution of the tracer once a periodic state is reached, in which the rate of injected scalar balances advection and diffusion with the molecular diffusion К. We study the two-dimensional velocity field u(x,y,t)=2 cos2(ωt)(0,sin χ)+2 sin2(ωt)(sin y,0). As ω is reduced from an O(1) value the flow alternates through a sequence of states which are either globally chaotic, or contain islands embedded in a chaotic sea. The evolution of the scalar is examined numerically using a semi-Lagrangian advection scheme. By time-averaging diagnostics measured from the scalar field we find that the time-averaged lengths of the scalar contours in the chaotic region grow like К−1/2 for small К, for all values of ω, while the behavior of the time-averaged maximum scalar value, Cmax, for small К depends strongly on ω. In the presence of islands Cmax˜К−α for some α between 0 and 1 and with К small, and we demonstrate that there is a correlation between α and the area of the periodic islands, at least for large ω. The limit of small ω is studied by considering a flow field that switches from u=(0,2 sin x) to u=(2 sin y ,0) at periodic intervals. The small К limit for this flow is examined using the method of matched asymptotic expansions. Finally the role of islands in the flow is investigated by considering the time-averaged effective diffusion of the scalar field. This diagnostic can distinguish between regions where the scalar is well mixed and regions where the scalar builds up.
Copyright © 2009 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluids 21 (2009) and may be found at http://link.aip.org/link/?PHFLE6/21/067103/1
21 (6), article 067103