Steady motion of hairpin-shaped vortex filaments in excitable systems
Sumana Dutta and Oliver Steinbock
Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA
Received 17 December 2009; published 28 May 2010
We demonstrate the existence of steadily translating filaments in the Belousov-Zhabotinsky reaction. The
filaments have self-reinforcing shapes tracing planar hairpins and constant velocities that are inversely propor-
tional to their width. These features are well described by an analytical solution of the mean curvature flow
problem. Using numerical simulations based on an excitable reaction-diffusion model, we also probe the
solution’s large basin of attraction and show that entangled hairpins reconnect during collisions.
DOI: 10.1103/PhysRevE.81.055202 PACS numbers: 05.45.-a, 82.40.Ck, 82.40.Qt
Curvature-dependent motion of curves and surfaces is at
the heart of many natural and man-made processes. Such
geometric flows are relevant to the evolution of soap films
and cell membranes, flame propagation, problems in quan-
tum field theory, as well as applications in image processing
1–3. The simplest case are planar one-dimensional curves s
that contract according to
ds
dt
= N
ˆ
, 1
where describes a system-specific constant line tension,
equals the local curvature, and N
ˆ
denotes the curve’s unit
normal vector. Mathematical analyses show that, under this
flow, all nonintersecting closed curves evolve toward a circle
and vanish in finite time. During the collapse, the enclosed
area decreases at a constant rate of -2 4,5. Furthermore
there are interesting shape-preserving solutions including
self-shrinkers such as circles and Abresch-Langer curves, ro-
tating “yin-yang” patterns, and translating curves 2,6.
To date only few of these solutions have been observed in
experiments. This situation is surprising because Eq. 1 ap-
plies to a wide range of phenomena including two-
dimensional grain boundaries and various one-dimensional
phase singularities 4. Among the latter examples,
curvature-dependent motion has been used to describe vortex
lines in superconductors 7 and scroll wave filaments in ex-
citable and oscillatory reaction-diffusion systems 8.
Scroll waves have been studied in numerous models in-
cluding the complex Ginzburg-Landau equation CGLE9.
They exist in experimental systems such as the chemical
Belousov-Zhabotinsky BZ reaction, the cellular slime mold
Dictyostelium discoideum, and cardiac tissue 10–12. Espe-
cially the latter system attracts considerable attention be-
cause scroll waves and their filament dynamics have been
linked to dangerous cardiac arrhythmias in humans. Further-
more, negative filament tension induces dynamically inter-
esting chaotic states 12–14.
In general, scroll wave filaments also move in binormal
direction at curvature-dependent velocities. This contribution
to the overall motion depends in a nontrivial way on reaction
kinetics and diffusion coefficients. However, it is zero in the
three-dimensional CGLE 9 and also vanishes in excitable
systems in which all relevant reaction species have equal
diffusion coefficients 15. Consequently, shape-preserving
solutions of Eq. 1 could exist in a broad spectrum of excit-
able and oscillatory systems. However, this prediction has
been tested so far only for circular filament loops 10,16.
In this Rapid Communication, we describe experiments
demonstrating the existence of steadily translating filaments
in three-dimensional excitable systems. These constant speed
constant shape structures are among the hallmark solutions
of Eq. 1 and have been referred to as “grim reapers” or
hairpins. The study is complemented by numerical simula-
tions based on a FitzHugh-Nagumo-like reaction-diffusion
model.
Our experiments employ disk-shaped systems of the
ferroin-catalyzed BZ reaction. The disks have a diameter of
10 cm and measure 8.0 mm in height. The lower 4.8 mm of
the medium are contained in agrose gel 0.8 % weight/
volume while the upper 3.2 mm are liquid solution. The
total height corresponds to approximately 1.7 wavelengths of
the excitation vortex. The initial reactant concentrations are
constant throughout the two layers: H
2
SO
4
= 0.16 mol / L,
NaBrO
3
= 0.04 mol / L, malonic acid = 0.04 mol / L, and
Fephen
3
SO
4
= 0.5 mmol / L. The corresponding solutions
are prepared in nanopure water 18 M cm and all experi-
ments are carried out at room temperature.
For the creation of scroll waves, we first initiate a nonro-
tating expanding wave front. This initiation step is carried
out by placing a silver wire on the gel-liquid interface for
about 20 s, which locally decreases the concentration of in-
hibitory bromide ions. The shape of the exposed wire con-
trols the shape of the triggered wave and subsequently the
form of the filament. Here, we use small localized nuclei
creating nearly spherical waves and long wire segments trig-
gering fronts resembling capped cylinders. Then we rapidly
swirl the reactant vessel in order to mix the solution phase,
which erases the upper portion of the wave. At this time, we
also place a glass plate onto the upper solution interface to
prevent undesired fluid flow during the main experiment. As
the fluid comes to rest, the rim of the gel-bound wave begins
to curl spontaneously and nucleates a scroll wave filament of
corresponding shape. Notice that the filament loop is formed
in very close vicinity of the gel-solution interface and there-
fore planar.
Detection of wave and filament dynamics is performed by
recording image sequences with a charge coupled device
CCD camera mounted over the system. This method uti-
lizes the color difference between the chemically reduced
rest state of the system red and its oxidized excitable state
PHYSICAL REVIEW E 81, 055202R2010
RAPID COMMUNICATIONS
1539-3755/2010/815/0552024 ©2010 The American Physical Society 055202-1