Smooth and filamental structures in chaotically advected chemical fields

Alexandra Tzella, Peter H. Haynes

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Abstract

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large Peclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, gamma q, of the qth-order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, alpha, and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), (h) over bar. Under a homogeneous stretching approximation, gamma(q)/q= max{alpha/(h) over bar ,1} which implies that a well-defined filamental-smooth transition occurs at alpha = (h) over bar. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as q varies and less on the effects of alpha and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, (gamma) over tilde (q), depend nonlinearly on alpha with (gamma) over tilde (q) (gamma) over tilde (q) for (gamma) over tilde (q) < q. The magnitude of the correction to the homogeneous stretching approximation, <(gamma)over tilde>(q)-(gamma) over tilde (q), grows as alpha increases, reaching a maximum when the leading-order transition is reached (alpha=(h) over bar ). The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for alpha>h(max), where h(max) denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for alpha < <(h)over bar>, with an intermediate character for alpha between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Cramer function.

Original languageEnglish
Article number016322
Pages (from-to)-
Number of pages15
JournalPhysical Review E - Statistical, Nonlinear and Soft Matter Physics
Volume81
Issue number1
DOIs
Publication statusPublished - Jan 2010

Keywords

  • SMALL-SCALE STRUCTURE
  • PASSIVE SCALARS
  • SPATIAL STRUCTURE
  • FINITE LIFETIME
  • TURBULENCE
  • DIFFUSION
  • EXPONENTS
  • SPECTRA
  • REGIME
  • FLOWS

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