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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\ud 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\ud balances advection and the molecular diusion kappa. We study the two-dimensional velocity field u(x, y, t) = 2 cos2(omega*t)(0, sin x) + 2 sin2(omega*t)(sin y, 0). As omega 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. \ud \ud By time-averaging diagnostics measured from the scalar field we find that the time-averaged length of the scalar contours in the chaotic region grow like kappa^{-1/2} for small kappa, for all values of omega, while the dependence of the time-averaged maximum scalar value, C_max, depends strongly on omega as kappa varies. In the presence of islands C_max is proportional to kappa^{-alpha} for some alpha between 0 and 1 and with kappa small, and we demonstrate that there is a correlation between alpha and the area of the periodic islands, at least for large omega. The limit of small omega 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 omega limit for this flow is examined using the method of matched asymptotic expansions.\ud \ud 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

Topics:
G100 Mathematics

Publisher: American Institute of Physics

Year: 2009

DOI identifier: 10.1063/1.3159615

OAI identifier:
oai:eprints.brighton.ac.uk:6476

Provided by:
University of Brighton Repository

Downloaded from
http://eprints.brighton.ac.uk/6476/1/islands_steady_source_2.pdf

- (2004). A bound on mixing efficiency for the advection-diffusion equation.
- (1989). Analytical air pollution advection and diffusion models.
- (1986). Area of the stratospheric polar vortex as a diagnostic for tracer transport on an isentropic surface.
- (2003). Diagnosing transport and mixing using a tracer-based coordinate system.
- (1996). Diascalar flux and the rate of fluid mixing.
- (2000). Effective diffusion as a diagnostic of atmospheric transport,
- (2000). Effective diffusion as a diagnostic of atmospheric transport, 2. troposphere and lower stratosphere.
- (2008). Effective diffusion of scalar fields in a chaotic flow. Phys. Fluids,
- (2003). Enhanced diffusion regimes in bounded chaotic flows.
- (2004). Foundations of chaotic mixing.
- (1964). Handbook of Mathematical Functions.
- (2006). Multiscale mixing efficiencies for steady sources.
- (1979). Nonlinear Oscillations.
- (2006). Scalar decay in chaotic mixing. Transport and Mixing in Geophysical Flows,
- (1991). Semi–lagrangian schemes for atmospheric models – A review.
- (2004). Spectral properties and transport mechanisms of partially chaotic bounded flows in the presence of diffusion.
- (2005). Transient micromixing: examples of laminar and chaotic stirring.
- (1996). Two–dimensional mixing, edge formation and permeability diagnosed in area coordinate.

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