A systematic study of spirals and spiral turbulence in a reaction-diffusion system

Article abstract of DOI:10.1063/1.1554397

A study was conducted of the spiral behaviors in a reaction-diffusion system using the BZ reaction and observed certain new phenomena. Transition from the D type spiral turbulence to the T type turbulence on the D-T boundary can be readily explained by the change of the BZ reaction from excitable to oscillatory. A well established method, the hierarchical structure theory can be applied to identify this transition.

Full text of DOI:10.1063/1.1554397

A systematic study of spirals and spiral turbulence in a reaction-diffusion system
Hongyu Guo, Liang Li, Qi Ouyang, Jian Liu, and Zhensu She
Citation: The Journal of Chemical Physics 118, 5038 (2003); doi: 10.1063/1.1554397
View online: http://dx.doi.org/10.1063/1.1554397
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 118, NUMBER 11
15 MARCH 2003
A systematic study of spirals and spiral turbulence
in a reaction-diffusion system
Hongyu Guo, Liang Li, and Qi Ouyanga)
Department of Physics, State Key Laboratory for Mesoscopic Physics, Peking University,
Beijing, 100871, China
Jian Liu
Department of Mechanics and Engineering Science, State Key Laboratory for Studies of Turbulence
and Complex Systems, Peking University, 100871, China
Zhensu She
Department of Mechanics and Engineering Science, State Key Laboratory for Studies of Turbulence
and Complex Systems, Peking University, 100871, China and Department of Mathematics,
University of California, Los Angeles, Los Angeles, California 90095
͑
Received 11 September 2002; accepted 31 December 2002͒
We report our experimental study on chemical patterns in a spatial open reactor using the Belousov–
Zhabotinsky reaction. A phase diagram showing different regimes of spiral dynamics is presented.
The focus of the study is on transitions leading to defect-mediated turbulence ͑spiral turbulence͒.
Some new interesting phenomena are described, including a transition from one type of spiral
turbulence to another type, and the re-entry from spiral turbulence into ordered spiral waves. We
also try to characterize different states of turbulence using the hierarchical structure theory.
©
2003 American Institute of Physics. ͓DOI: 10.1063/1.1554397͔
I. INTRODUCTION
tion about spiral dynamics. For example, we know that a
spiral can be supported in excitable or oscillatory media,
During the past 30 years, spiral patterns are always con-
sidered as one of the most intriguing spatiotemporal struc-
tures in macroscopic systems driven far from thermodynamic
equilibrium, and the study of spiral dynamics has attracted a
great deal of attention from researchers in different fields,
such as nonlinear physics, mathematics, physical chemistry,
biology, and cardiology. There are several reasons for these
attractions. The first reason is that spiral patterns are ubiqui-
tous in different systems, including reaction-diffusion
both of which can be obtained in the BZ reaction.1
,8,22
Sec-
tions of phase diagrams using different control parameters
have been sketched to identify different regimes of spiral
dynamics.2
1,22
Transitions from regular spiral patterns to
defect-mediated turbulence have also been studied carefully
with the BZ reaction, and two important instabilities, the
Doppler instability1
6,22
and the long wavelength
instability,1
3,17,22
have been found. The Doppler instability
1
,2
aggregating slime mold cells,3 cardiac muscle
occurs in an excitable system, corresponding to regions
where the control parameter ⌬ϵ͓H SO ͔͓NaBrO ͔ is low.
media,
4
–6
7
tissue,
and intracellular calcium release. Second, as we
2
4
3
In such a parameter space, spiral patterns are trigger waves,
which are governed by a dispersion relation that relates the
speed to the period of traveling waves. There exists a mini-
mum period below which the system cannot recover to its
excitable state, and traveling waves cease to exist.23 Usually,
the period of regular spiral waves is larger than the minimum
period, thus spiral waves are stable. However, when a Hopf
bifurcation contributes to the spiral core and makes the spiral
tip meandering, the trajectory of the spiral tip changes from a
single circle to a flower. When the meandering speed is suf-
ficiently large, due to the Doppler effect, two adjacent wave
fronts near the core are so close that the local wavelength is
beyond the critical value, causing a spontaneous generation
of defects.16
have known, a spiral is self-organized by a topological de-
fect, which constructs the spiral core and plays the role of
1
,8
dominator. So the study of spiral behavior might give hints
about the defect dynamics. One important aspect in this re-
search field is studying transitions from ordered spiral pat-
terns to defect-mediated turbulence. Several spiral instabili-
ties leading to defect-mediated turbulence9–12or spiral
turbulence have been well studied in theory,
and ob-
served both in experiments1
3–17
and numerical
Study of such instabilities is one of the most
1
8–20
simulation.
favorable routes to investigate spatiotemporal chaos, which
is poorly understood at present. From an application point of
view, the result of this study has a promising application
potential in cardiology, as transitions from spiral to spiral
turbulence occur in heart tissue and play an essential role in
cardiac arrhythmia and fibrillation.4
If the control parameter ⌬ϵ͓H SO ͔͓NaBrO ͔ is higher
2
4
3
,6
2
2
ϩ
than 0.2 M , the ferroin (Fe(phen)3 ) catalyzed BZ reaction
can be considered as an oscillatory system and the existing
spiral is phase wave. Near the onset, the system’s local vari-
able ͑c͒, such as the concentration of ferroin, can be written
as cϭc ϩA exp(i␼ t)ϩc.c., where ␼ is the Hopf fre-
Experimental investigations using the Belousov–
Zhabotinsky ͑BZ͒ reaction have provided plentiful informa-
a͒Electronic mail: qi@phy.pku.edu.cn
0
c
c
0021-9606/2003/118(11)/5038/7/$20.00
5038
© 2003 American Institute of Physics
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6

J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Spiral turbulence in a reaction-diffusion system
5039
quency. Suppose the system is near a supercritical Hopf bi-
furcation, the complex amplitude A obeys the complex
Ginzburg–Laudau equation ͑CGLE͒,
II. EXPERIMENTAL SETUP
Our experiments are conducted in a spatial open reactor
2
ϩ
using the ferroin (Fe(phen)3 ) catalyzed BZ reaction. The
heart of our reactor is a thin porous glass disk ͑0.4 mm thick
and 20 mm in diameter͒. The glass disk has 25% void space
and 100 Å average pore size, so that it can be considered as
a homogeneous medium for a macroscopic pattern with a
character length scale of a few hundred micrometers. The
porous glass prevents convection when used as the reaction
medium. Each side of the disk is contacted by a reservoir
where the reactants are continuously refreshed by highly pre-
cise chemical pumps ͑Pharmacia P-500͒ and kept homoge-
neous by stirring. The chemicals are fed to both reservoirs
separately so that we can keep one ͑A͒ in the reduced state of
the reaction system, and the other ͑B͒ in the oxidized state.
When the reactants diffuse into the porous glass and meet
there, pattern-forming reactions take place and spatiotempo-
ral patterns form. The chemical patterns are monitored in
transmitted light ͑halogen light filtrated to wavelength less
than 550 nm͒ with a charge coupled device ͑CCD͒ camera.
The signal received by the CCD camera is sent to a computer
where images are digitized and saved for further quantitative
analysis.
ץ2A
ץr2
ץA
ץt
2
0
2A,
␶0
ϭ␮Aϩ͑g ϩig ͒
Ϫ␰ ͑d ϩid ͒
͉A
͉
͑1͒
1
2
1
2
where ␶ and ␰ are, respectively, the characteristic time and
the correlation length of the system, g , g , d , and d are
related to the diffusion coefficients and ensemble of reaction
kinetics. After rescaling, using ␶ and ␰ as time and space
0
0
1
2
1
2
0
0
units, we have
ץA
ץ2A
ץr2
ϭAϩ͑1ϩic1͒
Ϫ͑1ϩic3͒
͉
A
͉
2A,
͑2͒
ץt
where c ϭd /d and c ϭg /g are control parameters. In
1
2
1
3
2
1
the one-dimensional case, Eq. ͑2͒ has traveling wave solu-
tions A ϭF exp͓i(kxϪ␼t)͔, where F ϭ1Ϫk and ␼
ϭc k ϩc (1Ϫk ). Stability analyses show that, under
2
2
0
2
2
24
1
3
certain conditions, the system will be unstable against long
wavelength perturbations. After the instability occurs, sus-
tained long-wavelength modulated waves emerge. The long
wavelength instability has a convective nature, which ex-
plains the existence of stable modulated waves in a finite
space while the system has been unstable. In the two-
dimensional case, the instability usually induces a meadering
movement of spiral tips; the frequency of meadering move-
ment determines the wavelength of the modulation.22 Due to
the convective nature, this modulative perturbation travels
downstream as it amplifies, leaving ordered modulated spiral
waves. The fundamental difference between long-
wavelength modulated spiral waves and meandering spirals
is that the former has a nonzero convective velocity,1
while the latter is zero. When the control parameter is in-
creased to across a threshold, spiral waves break and defects
generate far away from the spiral center, as a result of two
neighboring wave fronts being too close. At last, when the
control parameter is further increased, absolute instability
occurs and defect-mediated turbulence invades the whole
We choose the concentration of malonic acid in reservoir
A
A
A (͓CH (COOH) ͔ or ͓MA͔ ) and sulfuric acid in reser-
2
2
B
voir B (͓H SO ͔ ) as the control parameters, which are ad-
2
4
justable in a range of 0.3–1.0 M and 0.4–1.6 M, respec-
tively. Other conditions are kept fixed during the whole
A,B
A
experiment:
͓
NaBrO
3͔
ϭ0.4 M;
͓ ͔
KBr ϭ0.03 M;
B
͓
ferroin͔ ϭ1.0 mM. The resident time in each reservoir is
0 s; the ambient temperature is 25Ϯ0.5 °C. Since there
3
1
are multiple concentration gradients across the reaction me-
dium, the observed pattern is quasi-three-dimensional, and
an instability in the gradient direction may happen. In this
study, we focus on spiral patterns which are stable in the
gradient direction, so that patterns are entrained in this
direction.26 As a result, the observed patterns can be consid-
ered as quasi-two-dimensional most of the time. Under cer-
tain conditions, the three-dimensional effect does exist and
might be the cause of some new phenomena. We will give
discussions later.
7,22
1
3,17,22
space.
The initial condition of the experiment is that there is
only one spiral tip residing in the middle of the reaction
medium. This condition can be achieved by using a helium–
neon laser ͑3 mW, ␭ϭ633 nm) to generate and guide spiral
tips. In a regime of stable spiral ͑see phase diagram of Fig.
1͒, after suitable reactant solutions are pumped into reser-
voirs, a train of traveling waves automatically appears from
the boundary of the reaction medium. We use a beam of laser
light to break a chemical wave front and create a couple of
defects, which develop into a pair of counter-rotating spiral
waves. Then we use laser light to lead one spiral tip to the
edge of the reactor and delete it, and drive the other one to
the center of the reaction medium. Once a spiral is ready, we
study it by changing one reactant concentration stepwisely
while fixing all other conditions. A record is taken after the
pattern relaxes into its asymptotic state and no laser is ap-
plied for a sufficient amount of time ͑around an hour͒.
Despite fruitful results of previous researches, some
problems have not been clarified and certain parameter space
deep into turbulence regimes has not been explored. In this
paper, after describing the experimental setup in the follow-
ing section, we present our systematic studies of dynamical
behaviors of spiral waves and spiral turbulence. A phase dia-
gram with the feeding concentrations of sulfuric acid and
malonic acid as control parameters is built, and the lines of
onset for each transition are identified. Several new phenom-
ena are described, including a transition between two types
of spiral turbulence and the re-entry of spirals from spiral
turbulence. In Sec. IV, we introduce a method, the hierarchi-
cal structure theory ͑HST͒,25 to characterize the states of
spiral turbulence. We show that a phase transition between
two types of spiral turbulence can be identified using this
method. In Sec. V, we give a discussion on the new discov-
ered phenomena.
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5040
J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Guo et al.
FIG. 1. The phase diagram with ͓MA͔A and ͓H SO ͔ as the control pa-
B
2
4
rameters. The solid lines indicate the onsets of different instabilities. The
dashed lines are the extrapolation of the solid lines.
III. EXPERIMENTAL RESULTS
Some previous studies27 point out that the dynamics of
spiral patterns is most sensitive to three parameters: the con-
centration of malonic acid ͓͑MA͔͒, the concentration of sul-
furic acid (͓H SO ͔), and the concentration of sodium bro-
2
4
mate (͓NaBrO ͔). Others provide the phase diagrams which
3
use ͓H SO ͔ and ͓MA͔ as control parameters and divide
2
4
different domains according to different observed
2
1,22
patterns.
However, there is still a certain parameter space
that has not been reached. This work explores a part of the
unexplored parameter space, and especially pays attention to
the phenomena occurring at high ͓H SO ͔. Some new phe-
2
4
nomena are found, including a transition between two types
of spiral turbulence and the re-entry of stable spirals from a
state of spatiotemporal chaos.
FIG. 2. Examples of different patterns observed in the experiment. ͑a͒
Simple rotating spiral ͑S͒; ͑b͒ meandering spiral ͑M͒; ͑c͒ a state of spiral
turbulence due to the Doppler instability ͑D͒; ͑d͒ modulated spiral due to the
B
Figure 1 shows the phase diagram using ͓H SO ͔ and
2
4
MA͔A as the control parameters. More than 200 points in
͓
long wavelength instability (C ); ͑e͒ coexistence of a modulated spiral and
1
the phase diagram are studied, and eight different dynamics
regimes are categorized. They are marked using different
capital letters: ͑1͒ Simple spiral ͑S͒, where the system is
either in an excitable or an oscillating medium, the period,
wavelength, and speed of the spiral waves are constant, as
shown in Fig. 2͑a͒. ͑2͒ Meandering spiral ͑M͒, where the
system is in an excitable medium and the movement of spiral
tip follows a hypocycloid or epicycloid trajectory instead of
a simple cycle. Due to the Doppler effect, both the local
period and the local wavelength change as a function of time.
The modulation forms a superspiral, as shown in Fig. 2͑b͒.
spiral turbulence; ͑f͒ a state of chemical turbulence due to the long wave-
length instability ͑T͒; ͑g͒ a state of spiral turbulence in region A; ͑h͒ stable
spiral in region R.
modulated spirals and spiral turbulence (C ). Because of the
2
convective nature of the long wavelength instability, pertur-
bations generated in the spiral core area travel out as they
grow, this makes a finite region around the spiral core where
traveling waves are asymptotically stable.1
3,17
Figure 2͑e͒
gives an example of such state. ͑6͒ Defect-mediated turbu-
lence due to the long wavelength instability ͑T͒, as shown in
Fig. 2͑f͒. ͑7͒ A newly observed chemical turbulence ͑A͒, and
͑8͒ stable spiral due to the reentry of spiral ͑R͒. The details
of the last two states will be described later in this section,
and examples of such states are presented in Figs. 2͑g͒ and
2͑h͒.
͑3͒ Spiral turbulence due to the Doppler instability ͑D͒,
where the minimum of local period goes beyond the critical
value, making traveling waves unstable. An example of such
a state is shown in Fig. 2͑c͒. ͑4͒ Modulated spirals due to the
long wavelength instability, (C ). Figure 2͑d͒ gives such an
1
example. Here, the system is in an oscillatory medium in-
stead of an excitable medium. ͑5͒ A state of coexistence of
Each of the boundaries that separate different regimes
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J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Spiral turbulence in a reaction-diffusion system
5041
FIG. 3. Images illustrating transitions from meandering spiral to D type
A
spiral turbulence and to T type spiral turbulence with ͓MA͔ ϭ0.3 M. The
B
values of ͓H SO ͔ is in ͑a͒ 0.733 M, ͑M͒; in ͑b͒ 1.100 M, ͑D͒; in ͑c͒ 1.133
2
4
M, near the critical point; and in ͑d͒, 1.200 M, ͑T͒.
defines a transition. Some transitions have been carefully
studied in former works. For example, S–M boundary de-
fines the onset where the spiral tip movement undergoes a
Hopf bifurcation. As a result, the trajectory of the tip changes
from a simple circle to an epicycloid or a hypocycloid, lead-
2
1
ing the spiral into the meandering state; M–D line gives the
onset of the Doppler instability;16 and S–C , C –C , and
1
1
2
C –T lines represent, respectively, the onset of the long
2
wavelength instability, the beginning of spiral wave break
up, and transition to full spiral turbulence ͑absolute
1
3,17,22
instability͒.
These transitions have been comprehen-
sively understood. Besides these transitions, there are several
notable new phenomena shown in the phase diagram: the
D–T boundary, the C –A boundary, and an isolated R re-
gion. In the following, we will describe these transitions in
detail.
1
FIG. 4. Images illustrating different patterns observed with ͓MA͔A
B
ϭ0.4 M. The value ͓H SO ͔ is in ͑a͒ 0.400 M, ͑M͒; in ͑b͒ 0.633 M, ͑S͒; in
2
4
͑c͒ 0.767 M, (C ); in ͑d͒ 0.967 M, ͑A͒; in ͑e͒ 1.067 M, (C ); in ͑f͒ 1.133 M,
1
2
Notice that the D and T regions are defect-mediated tur-
bulence due to different instability mechanisms; one is due to
͑R͒; in ͑g͒ 1.200 M, (C2); and in ͑h͒ 1.267 M, ͑T͒.
1
6,22
the Doppler instability
while the other experiences the
long wavelength instability.1
3,17,22
A new transition from D to
fixed ͓MA͔A at 0.4 M and increased ͓H SO ͔ stepwisely.
B
2 4
A
B
T is discovered along the line ͓MA͔ ϭ0.3 M. Figure 3 gives
examples of patterns in different states ͓͑a͒, ͑b͒, ͑d͔͒ and
patterns near the critical point ͓͑c͔͒. We notice in the experi-
ments that the turbulence states in D and T regimes are dif-
ferent in time scale and length scale, indicating that there is a
transition between the two states. In the following section,
we will use the HST to identify this transition.
The system behaved normally in the low ͓H SO ͔ region:
2 4
As the increase of the control parameter, we observed mean-
B
dering spirals at ͓H SO ͔ ϭ0.4 M, ͓Fig. 4͑a͔͒, a transition
2
4
B
to simple spirals at ͓H SO ͔ ϭ0.633 M ͓Fig. 4͑b͔͒, another
2
4
transition to a state of long wavelength modulated spirals at
B
͓H SO ͔ ϭ0.767 M ͓Fig. 4͑c͔͒. Then a new scenario took
2
4
place: As the control parameter was further increased, the
spiral broke up rapidly, and the entire system plunged into a
regime of spiral turbulence without experiencing the C2
state, as shown in Fig. 4͑d͒. Without fully understanding this
behavior, we name this regime A to distinguish it from other
Another phenomenon that we find is a new regime of
spiral turbulence ͑A͒ located in a region between C and C ,
1
2
as shown in Fig. 1. Besides, an isolated island of stable spiral
inside the domain of C appears. These phenomena are very
2
B
robust; we repeated the experiments several times and found
the same behaviors. We also used another piece of glass disk,
which has little difference in diffusion coefficient, and simi-
lar phenomena were observed, too. In the experiments, We
states of turbulence, such as D and T. When ͓H SO ͔ was
increased again, a local region of stable spiral grew up, and
2
4
the system entered into the C regime ͓see Fig. 4͑e͔͒. The
2
spiral gradually took up the whole domain of the reaction
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5042
J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Guo et al.
2
2 1/2
medium, driving the defects sea out of the boundary ͓Fig.
͑f͔͒, and we thus experienced a re-entry of the spiral ͑R
ץf
ץf
G͑ f͑x,y͒͒ϭ
ͫ
ͩ ͪ
ϩ
ͩ ͪ
ͬ
,
͑3͒
4
ץx
ץy
B
regime͒. When the ͓H SO ͔ reached a high enough value,
2
4
where f(x,y) is the field of image intensity. Then we inves-
tigate the behavior of contrast changes at a scale ᐉ by means
of a bidimensional integral,
defects appeared far away from the spiral center, and the core
of stable waves was invaded by the sea of defects little by
little again. Finally, the system got across the C –T bound-
2
ary and spiral turbulence dominated the whole domain. The
newly observed phenomena appear in a relatively narrow
range of control parameters. As shown in the phase diagram
xϩᐉ yϩᐉ
1
ᐉ2
␧͑ᐉ͒ϭ
͵ ͵
G͑ f͑x,y͒͒dxdy.
͑4͒
x
y
A
of Fig. 1, the system behaves differently for ͓MA͔
We study the static properties of ␧͑ᐉ͒ of images taken in
A
ϭ0.5 M and ͓MA͔ ϭ0.3 M.
2
56ϫ256 pixels, and characterize the field ␧͑ᐉ͒ by structure
p
Our experimental observations also indicate some typi-
cal critical phenomena near the onset, such as large fluctua-
tions and critical slowing down. In the experiment, once we
change the control parameter, the system needs a certain pe-
riod of time to relax into its newly established asymptotic
state. Generally 1 h is enough. But if the system approaches
the critical points, one needs to wait 2 or more hours in order
for the system to relax to its final state, otherwise one will
observe pseudohysteresis when changing back and forth the
control parameter. Large fluctuations are also observed near
the onset. This phenomenon is especially pronounced near
function S (ᐉ)ϭ
͗
͉
␧(ᐉ)
͉
͘
. The hierarchical structure model
(ᐉ)/S (ᐉ),
p
introduces a hierarchy of functions ␧ (ᐉ)ϭS
each describing more closely the intensity of fluctuations.
␧p(ᐉ) is associated with higher intensity fluctuations with
increasing p. She and Leveque used the hypotheses that an
invariant relation exists between ␧ and ␧ , which is re-
p
pϩ1
p
2
5
p
pϩ1
ferred to as a hierarchical symmetry relation ͑a symmetry
with respect to a translation in p͒,
␧pϩ1͑ᐉ͒ϭA ␧ ͑ᐉ͒ ␧ ͑ᐉ͒1Ϫ␤, 0Ͻ␤Ͻ1,
␤
͑5͒
p
p
ϱ
where Ap is independent on ᐉ and ␧ (ᐉ)ϭlim
which characterize the most intermitted structures. The term
can be eliminated and the hierarchical symmetry
relation becomes
␧p(ᐉ),
ϱ
p→ϱ
the A–C boundary in the A region. In this regime, large
2
␧ϱ(ᐉ)1
Ϫ␤
spiral patterns may spontaneously self-organize in the sea of
turbulence, and will disappear very slowly as we hold on the
same experimental conditions for a long time ͑a few hours͒.
We also notice a new feature of spiral patterns in the R
region. As shown in Fig. 4͑f͒, the amplitude of spiral waves
is not homogeneous in the R region; we observe a large
bright filament pattern, which self-organizes into a large spi-
ral, stretching out from the original spiral center and rotating
around it. Moreover, the brightness of this pattern oscillates
as a function of time: it is very bright at one time, then
slowly gets dark along with its rotation, then becomes bright
again in a cycle.
␧pϩ1͑ᐉ͒ Ap ␧ ͑ᐉ͒ ␤
p
ϭ
ͩ
ͪ
.
͑6͒
␧2͑ᐉ͒
A1 ␧1͑ᐉ͒
From a log–log plot of ␧
can examine the linearity. The value of ␤ is the slope of the
(ᐉ)/␧ (ᐉ) vs ␧ (ᐉ)/␧ (ᐉ), we
pϩ1
2
p
1
line. In Fig. 5 we show the calculation result of the values of
␤
for patterns of spiral turbulence at four separate control
B
parameters ͓H SO ͔ .
2
4
The parameter ␤ measures the degree of intermittency of
a turbulent flow. In the limit ␤→1, there is no intermittency.
The Kolmogorov’s picture of turbulence29 belongs to this
limit. In contrast, in the limit ␤→0, only the most intermit-
tent structures persist; this is the ordered limit. One example
in this category is the black and white ␤ model,30 where only
one type of structure ͑white͒ is responsible for the energy
dissipation and system disorder. In Fig. 5, one observes that
IV. QUANTITATIVE ANALYSIS OF SPIRAL
TURBULENCE
As discussed in the previous section and shown in Fig. 3,
we notice that the turbulence state from the Doppler instabil-
ity ͑D͒ and that from the long wavelength instability ͑T͒ are
different in time scale and length scale, and the patterns
change abruptly when the control parameter varies across the
D–T boundary, indicating that there is a new transition. In
this section, we report that the HST ͑Ref. 25͒ should be a
proper candidate to distinguish the two states of spiral turbu-
lence and to describe spatiotemporal chaos.
B
␤ at ͓H SO ͔ ϭ0.800 M and 1.100 M is larger than
2
4
B
͓H SO ͔ ϭ1.200 M and 1.600 M. Figure 6 gives a transi-
tion diagram, the change of ␤ as a function of ͓H SO ͔ . It
shows that the measured value of ␤ has a decreasing ten-
dency, and the values of ␤ in regime D are higher than in
regime T. At the onset (͓H SO ͔ ϭ1.133 M), the value of ␤
has a sudden decrease. We thus suggest that the HST can be
a suitable candidate in describing different states of spiral
turbulence obtained in a reaction-diffusion system.
Another measurement of defect-mediated turbulence is
the density of defects. In Fig. 3, we qualitatively realize that
the defect density of spiral turbulence in the T region is
higher than in the D region. At the critical point, it has a
sharp change. So the density of defects can also identify the
new transition and the result is consistent with the HST
analysis. However, getting the exact number of defects from
noisy experimental data in a highly turbulent regime is not
easy at least, so that in our case the HST has a clear advan-
2
4
B
2
4
B
2
4
In our case, the data that we can quantitatively analyze
are images gained from experiments. The magnitude of fluc-
tuations depends on the diversity of the image intensity in
the data series. In hydrodynamics, the basic field in turbu-
lence is the velocity from which the local energy dissipation
can be defined. A natural candidate for a variable analog to
the local energy dissipation is a quantity which takes its larg-
est contributions from the places where large changes in con-
2
8
trast occur. We thus choose the variable G(f(x,y)) defined
as below to study
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J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Spiral turbulence in a reaction-diffusion system
5043
process,32,33 so the model is also called the ‘‘log-Poisson’’
model. When we do multiscaling and hierarchical analysis,
some transformations or symmetry can be useful to clarify
the complexity of a system. We think that these quantitative
properties are useful and revelatory to characterize defect-
mediated turbulence and some other spatially extended sys-
tems, and further research should be done. The detailed and
deep discussion of the use of HST in spatially extended sys-
tems will be reported elsewhere.
V. DISCUSSIONS
This work revealed three new phenomena: transition
from the D type of spiral turbulence to the T type turbulence
on the D–T boundary; transition from spiral to spiral turbu-
lence on the S–A boundary; and the re-entry of spiral from a
state of defect-mediated turbulence on the A–C boundary
2
͑Fig. 1͒. Transition between two types of turbulent states
could be related to the change of the system between an
excitable medium and an oscillatory medium. As discussed
in the Introduction, although both types of spiral turbulence
belong to defect-mediated turbulence, one is due to the Dop-
pler instability ͑meandering͒ which takes place in an excit-
able medium, the other is related to the long wavelength
instability, which occurs in an oscillatory medium that is near
a supercritical Hopf bifurcation, and becomes dominating af-
ter absolute instability occurs. We also know that as the in-
FIG. 5. The hierarchical symmetry relations obtained using images with
B
four different values of ͓H SO ͔ . Each set of points is fitted by a straight
2
4
line, the slope is the value of ␤. ͑a͒ 0.800 M, ␤ϭ0.935 17; ͑b͒ 1.100 M,
ϭ0.8901; ͑c͒ 1.200 M, ␤ϭ0.772 46; and ͑d͒ 1.600 M, ␤ϭ0.738 04. The
␤
A
other control parameter is ͓MA͔ ϭ0.3 M.
tage in analyzing experimental data. In the HST, the higher
value of ␤ means lower intermittency or lower singularity, so
that the value of ␤ of a system describes the degree of order;
we say that a system with smaller ␤ has more order. From
regime D to regime T, the degree of singularity enhances so
that the density of defects increases. The saturation of defects
in regime T is remarkably larger than that in regime D. With
a sufficiently large saturation of defects, interaction of de-
fects makes the fluctuation weaker, so the system will be-
come more ‘‘order.’’
B
crease of ͓H SO ͔ , the chemical system changes from ex-
2
4
citable to oscillatory. Since the correlation length scale for a
pattern in an excitable medium is determined by diffusion of
chemical species, while the correlation length scale for a
pattern in an oscillatory medium is governed by phase diffu-
sion, when the system changes from excitable to oscillatory,
the correlation length of the turbulent pattern changes sud-
denly. This is verified both by experimental observations ͑see
Fig. 3͒ and quantitative analysis ͑see Fig. 6͒.
It should be stressed that HST has an invariance ͑sym-
metry͒ which defines a transformation group;31 this symme-
The chaotic state in region A ͑see Fig. 1͒ belongs to
defect mediated turbulence. It is similar to the spiral turbu-
lence in region D. However, the route to the state of spiral
turbulence is different from either the Doppler instability or
the long wavelength instability. It is mix of both. In this
scenario, the defects generate far from the spiral core, just
like the long wavelength instability, but the transition from
modulated spiral to turbulence is fast, similar to the Doppler
instability. In the phase diagram, it is clear that region A
try can be exactly simulated by a log-Poisson cascade
located between region M and region C , where both mean-
2
dering and long-wavelength instability cannot be ignored.
So, region A should be the result of the combination of both
instabilities. Besides, in this region the system tends to form
ordered spirals from the sea of defects. Away from the A–C2
boundary, this tendency is suppressed by the defects. With
B
the increase of ͓H SO ͔ , a stable spiral coexists with the
2
4
defects sea; finally, the newly formed stable spiral drives all
the defects out of the boundary and takes up the whole reac-
tion medium. We thus witness the re-entry of spirals. How-
ever, due to the finite reactor size, we do not know whether
the reentry spirals is stable in extended space or just stable
for a finite space and as a long-lived transient. The same
B
FIG. 6. The change of mean value of ␤ as a function of ͓H SO ͔ in the
2
4
range from 0.800 to 1.600 M. The value of ␤ in regime D is larger than that
in regime T. There exists a phase transition point at ͓H SO ͔ ϭ1.133 M.
A
B
trend is also observed in the C2 region with ͓MA͔
2
4
ϭ0.5 M; the difference is that no stable spiral can win. At
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5044
J. Chem. Phys., Vol. 118, No. 11, 15 March 2003
Guo et al.
present, we do not have a satisfactory explanation of this
experimental result. The most possible explanation is a codi-
mension effect. One notices in the phase diagram of Fig. 1
that region A is near the point where all the transition lines
join together. In this codimensional area the mechanisms of
spiral instability may mix together, a new route to spiral
turbulence may appear, and new patterns may form. Due to
the limit precision of control in the experiments, at present
we cannot study the dynamical behavior in this area in detail.
Another phenomenon worth discussing is the bright pat-
terns in Fig. 2͑h͒, which have a markedly long wavelength
and an oscillation in brightness. In fact such bright waves
exist in a wider parameter space. We think the three-
dimensional effect is the most possible reason for such a
super wave, but we have no reasonable explanation for the
oscillation now. As mentioned in Sec. II, despite the reaction
medium can be considered as quasi-two-dimensional most of
time, the three-dimensional effect exists and might play an
important role in spiral dynamics. The bright patterns in Fig.
fruitful discussions, and H. L. Swinney for providing us with
the porous glass. This work is supported by the National
Natural Science Foundation of China.
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ACKNOWLEDGMENTS
The authors thank S. F. Bai and Y. Li for computer pro-
gramming, W. D. Su, L. We, H. L. Wang, and L. Q. Zhou for
32
3
3
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