Turbulence and Standing Waves in Oscillatory Chemical Reactions With Global Coupling
Turbulence and Standing Waves in Oscillatory Chemical Reactions With Global Coupling
Turbulence and Standing Waves in Oscillatory Chemical Reactions With Global Coupling
Chemical turbulence and standing waves in a surface reaction model: The influence of
global coupling and wave instabilities
Chaos: An Interdisciplinary Journal of Nonlinear Science 4, 499 (1994); https://
doi.org/10.1063/1.166028
Oscillatory surface reactions demonstrate a large variety The purpose of the present paper is to investigate how
of spatiotemporal patterns, both regular and chaotic. 1- 3 Spa- the effect of global coupling, which is common to all oscil-
tial coupling in these reactions under isothermal conditions is latory surface reactions, changes the properties of diffusion-
provided by two basic mechanisms: surface diffusion of mo- induced chemical turbulence. By varying the relative inten-
bile adsorbates on the catalyst surface and changes of the sity of the additional global coupling term in CGLE, a
educt pressures in the gas phase. The latter arise due to the transition from uniform synchronous oscillations (for a very
mass balance in the reaction and, since mixing in the gas strong global coupling) to periodic standing waves and fur-
phase is very fast, this coupling is global. Gas-phase cou- ther to a turbulent state (for a weaker coupling) is found as
pling is known to significantly influence the course of oscil- shown below. The analysis of the turbulent state, realized in
latory sUlface chemical reactions by tending to synchronize the presence of global coupling, shows that its properties are
the oscillations.4 - 5 Under certain conditions, periodically os- qualitatively different from those of diffusion-induced turbu-
cillating patterns of standing waves have been found on the lence realized in absence of global coupling. It retains a cer-
surfaces. 4 tain degree of long-range order and its temporal behavior can
To consider the effects of global coupling in oscillatory be characterized as intermittent, i.e., turbulent bursts appear
reaction-diffusion systems, we have proposed a simple on the background of almost synchronous oscillations. Based
mathematical model obtained by including an additional glo- on the results of our analysis we attempt a qualitative com-
bal coupling term into the complex Ginzburg-Landau equa- parison with experimental observations.
tion (CGLE).6,7 Although such a model can be fully justified Our mathematical model consists of a dynamical equa-
only in the vicinity of a Hopf bifurcation, its analysis also tion for the local complex oscillation amplitude 7](x,t) in a
reveals more general qualitative properties of the involved population of small-amplitude limit-cycle oscillators which
phenomena and thus provides a basis for the interpretation of are coupled both locally and globally. By choosing appropri-
experimental data. Using this model, we have investigated ate dimensionless units, it can be written in .the form 6
the breakdown of synchronization caused by strong super- 7]= (1- iw) 7]- (l + i,8) 17]1 2 7]+ (1 + i €) v2 7]- ,ue ix 1], (1)
critical inhomogeneities (surface defects), we studied propa-
gation of phase flips over the globally synchronized state and where
the spontaneous formation of large-scale oscillation
domains. 6 ,7 This analysis has been performed in a parameter
region where diffusional coupling between oscillators tends
1]= (1/S) f dX7](x,t) (2)
to synchronize the local oscillations. is the spatial (surface) average of the local oscillation ampli-
However, depending on the dynamical properties of os- tudes (S is the total surface area). It differs from the standard
cillations in the individual surface elements and on the ratio CGLE 14 by the last integral term in Eq. (1) which can be
of surface diffusion constants for different adsorbed reagents, interpreted asa driving force applied to each individual os-
local diffusional coupling may also destabilize uniform bulk cillator and collectively produced by all oscillators in the
oscillations and give rise to chaotic spatiotemporal regimes population. The intensity of global coupling is characterized
known as chemical turbulence. 8,9 Under proper choice of the by the coefficient f-L, the factor with X in the last term takes
parameters, these turbulent regimes can be described by the into account a possible phase shift between the driving force
CGLE. A detailed statistical analysis of turbulence in CGLE, and the averaged amplitude 1]. The Benjamin-Feir (BF) in-
based on its numerical simulations, has been carried out in stability leading to diffusion-induced turbulence occurs for
Refs. 10-13. Eq. (1) without global coupling (p,=O) if the condition
J. Chern. Phys. 101 (11), 1 December 1994 0021-9606/94/101 (11 )/9903/61$6.00 © 1994 American Institute of Physics 9903
9904 Mertens, Imbihl, and Mikhailov: Oscillatory chemical reactions
where O=w+ 13+ ,u(sin X- 13 cos X), Substituting Eq. (3) into
g.
0
0.8 0.8 Q)
I/)
0
Eq. (l), two coupled dynamical equations for the variables 0.6 0.6 s:::
a.
p(x,t) and cp(x,t) are obtained:
0.4 0.4
p=(l- p2)p_ \1 2p- p(\1 cp)2+ €p\12cp+2€\1 p'Y cp
FIG. 3. Breathing standing waves at ,u=0.02 for f3= -1.0, ,,= 1.5, X=7T,
L '=256, and T=800; local values of the real oscillation amplitude are shown
in grayscale.
1.0 0.10
"v
0::
0.5 0.05
o~---~-----~------~~~~~-~o
o 0.1 0.2 0.3
11
[Fig. 5(b)] the bursts become more frequent and the state of 0.2 0.2
'0. ,
the medium between them approaches that of the phase tur- o...... ~
bulence. However, it is still significantly different from the o -'0_ 0
0.1 0.1
developed amplitude turbulence without global coupling [cf. ------...2 ___ _ _0 _ _
Fig. 2(b) for the same values of the system parameters but
JL=O]. 0 0
0 500 1000 1500 2000 2500
The intermittency which persists in the presence of glo- L
bal coupling is qualitatively different from the one recently
found in Ref. 18 in a narrow parameter region in two- FIG. 1. Dependence of the time-averaged global amplitude (black circles)
dimensional simulations of COLE without global coupling. and its standard deviation (white circles) on the system size L at p.=0.2,
In the latter case the defects multiplied until they filled the /3=-1.5, X=1T, and E=2.0; the averaging interval is At= 1600.
4S. Jakubith, H. H. Rotermund, W. Engel, A. von Oertzen, and G. Ertl, 13 A. Weber, L. Kramer, L S. Aranson, and L. Aranson, Physica D 61, 279
Phys. Rev. Lett. 65, 3013 (1990). (1992).
'G:Veser and R. ImblhI, J. Chern. Phys. 96, 7155 (1992). 14 A. Newell, Lect. Appl. Math. 15, 157 (1974).
6G. Veser, F. Mertens, A. S. Mikhailov, and R. Imbihl, Phys. Rev. Lett. 77, IS P. Coullet and K. Emilsson, Physica A 88, 190 (1992); Physic a D 61, 119
975 (1993). (1992).
7F. Mertens, R. Imbihl, and A. Mikhailov, J. Chern. Phys. 99, 8668 (1993). 16H. Levine and X. Zou, Phys. Rev. Lett. 69, 3013 (1990).
~ Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer,
17N. Bekki and K. Nozaki, Phy,s. Lett. A 110, 133 (1985).
Berlin, 1984).
ISr. Aranson, A. Weber, and L. Kramer, Phys. Rev. Lett. 72, 2316 (1994).
9 A. S. Mikhallov and A. Yu. Loskutov, Foundations ofSynergetics II. Com-
plex Patterns (Springer, Berlin, 1991). 19G. Veser, F. Esch, and R. Imbihl, CataI.Lett. 13,371 (1992).
lOp. Coullet, L. Gill, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989). 2oH. Hakim and W-J. Rappel, Phys. Rev. A 46, 7347 (1992).
11 Th. Bohr, A. W. Petersen, and M. H. Jensen, Phys. Rev. A 42, 3226 (1990). 21 M. Tsodyks, I. Mitkov, and H. Sompolinsky, Phys. Rev. Lett. 71, 1280
12B. I. Shraiman, A. Pumir, W. van Saarlos, P. C. Hohenberg, H. Chate, and (1993).
M. Holen, Physica D 57, 241 (1992). 22U. Middya, D. Luss, and M. Sheintuch, J. Chern. Phys. 100, 3568 (1994).