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Pattern Formation Induced by Time-DependentAdvection

Published online by Cambridge University Press:  09 June 2010

A. V. Straube  and
A. Pikovsky
A. V. Straube*
Affiliation:
Department of Physics, Humboldt University of Berlin, Newtonstr. 15, D-12489, Berlin, Germany Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
A. Pikovsky
Affiliation:
Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
*
* Corresponding author. E-mail:[email protected]
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Abstract

We study pattern-forming instabilities in reaction-advection-diffusion systems. Wedevelop an approach based on Lyapunov-Bloch exponents to figure out the impact of aspatially periodic mixing flow on the stability of a spatially homogeneous state. We dealwith the flows periodic in space that may have arbitrary time dependence. We propose adiscrete in time model, where reaction, advection, and diffusion act as successiveoperators, and show that a mixing advection can lead to a pattern-forming instability in atwo-component system where only one of the species is advected. Physically, this can beexplained as crossing a threshold of Turing instability due to effective increase of oneof the diffusion constants.

Keywords

pattern formationreaction-advection-diffusion equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 1: Instability and patterns. Issue dedicated to the memory of A. Golovin , 2011 , pp. 138 - 148
Copyright
© EDP Sciences, 2010

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References

Antonsen, T. M., Fan, Z., Ott, E., Garcia-Lopes, E.. The role of chaotic orbits in the determination of power spectra of passive scalars . Phys. Fluids, 8 (1996), 3094–3104.CrossRefGoogle Scholar
Pismen, L. M.. Differential flow induced chemical instability and Turing instability for Couette flow . Phys. Rev. E, 58 (1998), 4524–4531.Google Scholar
Huisman, J., Thi, N. N. P., Karl, D. M., Sommeijer, B.. Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum . Nature, 439 (2002), 322–325.CrossRefGoogle ScholarPubMed
Khazan, Y., Pismen, L. M.. Differential flow induced chemical instability on a rotating disk . Phys. Rev. Lett., 75 (1995), 4318–4321.CrossRefGoogle ScholarPubMed
Leconte, M., Martin, J., Rakotomalala, N., Salin, D.. Pattern of reaction diffusion fronts in laminar flows . Phys. Rev. Lett., 90 (2002), 128302.Google Scholar
G. Nicolis, G. Prigogine. Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. Wiley & Sons, New York, 1977.
Pikovsky, A. S.. Spatial development of chaos in nonlinear media . Phys. Lett. A, 137 (1989), 121–127.CrossRefGoogle Scholar
Pikovsky, A., Popovych, O.. Persistent patterns in deterministic mixing flows . Europhys. Lett., 61 (2003), 625–631.CrossRefGoogle Scholar
L. Pismen. Patterns and interfaces in dissipative dynamics. Springer, Berlin, 2006.
Rothstein, D., Henry, E., Gollub, J. P.. Persistent patterns in transient chaotic fluid mixing . Nature, 401 (1999), 770–772.Google Scholar
Rovinsky, A. B., Menzinger, M.. Differential flow instability in dynamical systems without an unstable (activator) subsystem . Phys. Rev. Lett., 72 (1994), 2017–2020.CrossRefGoogle ScholarPubMed
Straube, A., Abel, M., Pikovsky, A.. Temporal chaos versus spatial mixing in reaction-advection-diffusion systems . Phys. Rev. Lett., 93 (2004), 174501.CrossRefGoogle ScholarPubMed
Tél, T., de Moura, A., Grebogi, C., Károlyi, G.. Chemical and biological activity in open flows: a dynamical system approach . Physics Reports, 413 (2005), 91–196.CrossRefGoogle Scholar
Turing, A. M.. The chemical basis of morphogenesis , Philos. Trans. Roy. Soc. London, Ser. B 237 (1952), 37–72. CrossRefGoogle Scholar
Vasquez, D. A. Chemical instability induced by a shear flow . Phys. Rev. Lett., 93 (2004), 104501.CrossRefGoogle ScholarPubMed
Yakhnin, V. Z., Rovinsky, A. B., Menzinger, M.. Convective instability induced by differential transport in the tubular packed-bed reactor . Chemical Engineering Science, 50 (1995), 2853–2859.CrossRefGoogle Scholar