# Pattern Formation Induced by Time-Dependent Advection

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 6, Issue: 1, page 138-148
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topStraube, A. V., and Pikovsky, A.. "Pattern Formation Induced by Time-Dependent Advection." Mathematical Modelling of Natural Phenomena 6.1 (2010): 138-148. <http://eudml.org/doc/197708>.

@article{Straube2010,

abstract = {We study pattern-forming instabilities in reaction-advection-diffusion systems. We
develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a
spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal
with the flows periodic in space that may have arbitrary time dependence. We propose a
discrete in time model, where reaction, advection, and diffusion act as successive
operators, and show that a mixing advection can lead to a pattern-forming instability in a
two-component system where only one of the species is advected. Physically, this can be
explained as crossing a threshold of Turing instability due to effective increase of one
of the diffusion constants.},

author = {Straube, A. V., Pikovsky, A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {pattern formation; reaction-advection-diffusion equation; Lyapunov-Bloch exponents; spatially periodic mixing},

language = {eng},

month = {6},

number = {1},

pages = {138-148},

publisher = {EDP Sciences},

title = {Pattern Formation Induced by Time-Dependent Advection},

url = {http://eudml.org/doc/197708},

volume = {6},

year = {2010},

}

TY - JOUR

AU - Straube, A. V.

AU - Pikovsky, A.

TI - Pattern Formation Induced by Time-Dependent Advection

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/6//

PB - EDP Sciences

VL - 6

IS - 1

SP - 138

EP - 148

AB - We study pattern-forming instabilities in reaction-advection-diffusion systems. We
develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a
spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal
with the flows periodic in space that may have arbitrary time dependence. We propose a
discrete in time model, where reaction, advection, and diffusion act as successive
operators, and show that a mixing advection can lead to a pattern-forming instability in a
two-component system where only one of the species is advected. Physically, this can be
explained as crossing a threshold of Turing instability due to effective increase of one
of the diffusion constants.

LA - eng

KW - pattern formation; reaction-advection-diffusion equation; Lyapunov-Bloch exponents; spatially periodic mixing

UR - http://eudml.org/doc/197708

ER -

## References

top- T. M. Antonsen, Z. Fan, E. Ott, E. Garcia-Lopes. The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids, 8 (1996), 3094–3104. Zbl1027.76662
- L. M. Pismen. Differential flow induced chemical instability and Turing instability for Couette flow. Phys. Rev. E, 58 (1998), 4524–4531.
- J. Huisman, N. N. P. Thi, D. M. Karl, B. Sommeijer. Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum. Nature, 439 (2002), 322–325.
- Y. Khazan, L. M. Pismen. Differential flow induced chemical instability on a rotating disk. Phys. Rev. Lett., 75 (1995), 4318–4321.
- M. Leconte, J. Martin, N. Rakotomalala, D. Salin. Pattern of reaction diffusion fronts in laminar flows. Phys. Rev. Lett., 90 (2002), 128302.
- G. Nicolis, G. Prigogine. Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. Wiley & Sons, New York, 1977. Zbl0363.93005
- A. S. Pikovsky. Spatial development of chaos in nonlinear media. Phys. Lett. A, 137 (1989), 121–127.
- A. Pikovsky, O. Popovych. Persistent patterns in deterministic mixing flows. Europhys. Lett., 61 (2003), 625–631.
- L. Pismen. Patterns and interfaces in dissipative dynamics. Springer, Berlin, 2006. Zbl1098.37001
- D. Rothstein, E. Henry, J. P. Gollub. Persistent patterns in transient chaotic fluid mixing. Nature, 401 (1999), 770–772.
- A. B. Rovinsky, M. Menzinger. Differential flow instability in dynamical systems without an unstable (activator) subsystem. Phys. Rev. Lett., 72 (1994), 2017–2020.
- A. Straube, M. Abel, A. Pikovsky. Temporal chaos versus spatial mixing in reaction-advection-diffusion systems. Phys. Rev. Lett., 93 (2004), 174501.
- T. Tél, A. de Moura, C. Grebogi, G. Károlyi. Chemical and biological activity in open flows: a dynamical system approach. Physics Reports, 413 (2005), 91–196.
- A. M. Turing. The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London, Ser. B237 (1952), 37–72.
- D. A. VasquezChemical instability induced by a shear flow. Phys. Rev. Lett., 93 (2004), 104501.
- V. Z. Yakhnin, A. B. Rovinsky, M. Menzinger. Convective instability induced by differential transport in the tubular packed-bed reactor. Chemical Engineering Science, 50 (1995), 2853–2859.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.