Three-dimensional spiral waves in an excitable reaction system:Initiation and dynamics of scroll rings and scroll ring pairsTamás Bánsági, Jr. and Oliver Steinbocka兲Department of Chemistry and Biochemistry, Florida State University, Tallahassee,Florida 32306-4390, USA共Received 2 January 2008; accepted 19 February 2008; published online 27 June 2008兲We report experimental results on spiral and scroll waves in the 1,4-cyclohexanedione Belousov–Zhabotinsky reaction. The propagating concentration waves are detected by two-dimensional pho-tometry and optical tomography. Wave pulses can disappear in front-to-front and front-to-backcollisions. This anomaly causes the nucleation of vortices from collisions of three nonrotatingwaves. In three-dimensional systems, these vortices are scroll rings that rotate around initiallycircular filaments. Depending on reactant concentrations, the filaments shrink or expand indicatingpositive and negative filament tensions, respectively. Shrinkage results in vortex annihilation. Ex-pansion is accompanied by filament buckling and bending, which is interpreted as developingWinfree turbulence. We also describe the initiation of scroll ring pairs in four-wave collisions. Thetwo filaments are stacked on top of each other and their motion suggests filament repulsion. © 2008American Institute of Physics. 关DOI: 10.1063/1.2896100兴Spiral waves of excitation exist in a variety of experimen-tal systems including chemical reaction-diffusion media,living cells, and cellular tissues. In two space dimensions,their shape is close to that of an Archimedean spiral.Their tip orbits along a simple, often circular, trajectory.The spiral analog in three dimensions is the scroll wavewhich rotates around dynamic, one-dimensional spacecurves. A specific case is the scroll ring for which thefilament is a closed loop. The latter curve shrinks or ex-pands according to the system’s filament tension. We re-port an experimental study of scroll rings in a chemicalreaction solution which can generate positive as well asnegative filament tension. We also discuss the creation ofspiral and scroll waves during the collision of nonrotatingwave fronts. All experiments on three-dimensional mediaemploy optical tomography for the reconstruction of thewave patterns.I. INTRODUCTIONFar from thermodynamic equilibrium, pattern formationis a common process.1A frequently studied case is the propa-gation of excitation waves in reaction-diffusion media.2These nonlinear waves exist in a multitude of physical,chemical, and biological systems. In two-dimensional media,excitation waves can form rotating spiral waves.3Experi-mental examples for these spirals are found in variouschemical reactions,4–6yeast extracts,7aggregating cellcolonies,7as well as cardiac8and neuronal tissue.9Typically,these rotors have the shape of Archimedean spirals, and spi-ral tips describe simple trajectories such as circles. In three-dimensional systems, spiral waves are called scroll wavesand rotate around one-dimensional space curves often re-ferred to as “filaments.” These filaments are not stationarybut move according to their local curvature and gradient inrotation phase.10,11Moreover, filaments must terminate at thesystem boundary, form closed loops, or be pinned to thewake of another wave pulse. The latter option, however, re-quires that the excitable medium shows a particular type ofanomalous dispersion.12The experimental study of three-dimensional excitationwaves is hindered by at least two major factors: wave controland wave detection. These technical challenges are particu-larly problematic in systems such as the mammalian heart,which is nontransparent and in its entirety difficult to control.Nonetheless, a major motivation for the study of three-dimensional excitation waves is related to the heart, whererotating vortex structures are believed to cause cardiac ar-rhythmia and ventricular fibrillation. This dilemma makes ituseful to investigate nonbiological models such as thechemical Belousov–Zhabotinsky 共BZ兲 reaction. The exis-tence of three-dimensional scroll waves in the BZ reaction iswell documented.13,14However, only a few authors have per-formed tomographic experiments yielding detailed data onthe spatiotemporal evolution of the wave patterns 共see, e.g.,Refs. 15–17兲.In this article, we employ optical tomography for thestudy of three-dimensional scroll rings in a modified BZ re-action. The critical chemical modification involves the sub-stitution of the classic organic substrate malonic acid with1,4-cyclohexanedione 共CHD兲. The resulting CHD-BZ sys-tem shows a particular feature that is essential for the controlof scroll ring nucleation. It relates to the dispersion relationof waves in this system, which describes the velocity of aninfinite wave train c as a function of the interpulse spacing ␭.In many experimental system, the dispersion relation c共␭兲 isa monotonically increasing function that saturates at thewave speed of the solitary pulse c0. Moreover, there is al-a兲Author to whom correspondence should be addressed. Electronic mail:[email protected] 18, 026102 共2008兲1054-1500/2008/18共2兲/026102/8/$23.00 © 2008 American Institute of Physics18, 026102-1Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jspways a minimal spacing ␭minbelow which no wave trainsexist. In the CHD-BZ system, one finds anomalies for whichdc / d␭⬍0 and c共␭min兲⬎c0. Such systems show transientwave packets in which the trailing pulses propagate fasterthan the leading front of the packet. Consequently, thesepulses approach the back of the frontier pulse. There theyundergo a front-to-back collision that results in the annihila-tion of the trailing pulses. These dynamics have the slightlymisleading name “wave merging” and have been studied indetail elsewhere.18,19Notice that large-scale wave patternsform despite continuing destruction of pulses because theannihilation zone moves outwards with the leading pulse.II. EXPERIMENTALPropagation of three-dimensional excitation waves isstudied in highly viscous Belousov–Zhabotinsky 共BZ兲 sys-tems. The classic BZ reaction involves malonic acid as itsorganic substrate, which is brominated in the presence oftypical redox catalysts such as Ce共III兲 / Ce共IV兲 and ferroin/ferriin 共关Fe共phen兲3兴2+/3+兲. A problem arising from the oxida-tion