Segmented Waves from a Spatiotemporal Transverse Wave Instability

Spiral waves and, more generally, traveling waves have been thoroughly studied in excitable and oscillatory media. In most of these active media, the inhibitor species diffuses at a rate comparable to or less than the activator, or is immobilized, and the wave fronts are continuous and sta- ble. When waves in the Belousov-Zhabotinsky (BZ) reac- tion lose stability, they exhibit ripples or break up due to wave front instability (1). Spiral breakup was also reported in the BZ reaction due to convective instability (2). This form of breakup has been discussed in connection with the mechanism of cardiac fibrillation (3). In systems where the inhibitor diffuses more rapidly than the activator, breakup is associated with a negative eikonal-curvature relation (4,5). Spiral breakup typically leads to spiral turbulence or chaotic waves (6), except for a recent report of stable segmented spirals in the BZ-AOT (AOTsodium bis(2- ethylhexyl) sulfosuccinate) microemulsion system (7). In most cases of unstable waves, a faster-diffusing inhibitor plays a key role, a phenomenon which has drawn much attention recently. In fast inhibitor systems, fronts separat- ing bistable steady states can show interesting behavior as well, developing into labyrinthine patterns (8) or cellular structures (9-11), both of which show intrinsic wavelengths. Here, we examine waves in the chlorine dioxide-iodine- malonic acid (CDIMA) reaction, where the activator spe- cies is iodide, which diffuses slowly or is immobilized, due to the presence of a complexing agent in the gel medium. The chlorite inhibitor, in contrast, diffuses rapidly. This condition favors Turing instability, and it was in this sys- tem that Turing's idea (12) was first brought to experimen- tal reality (13). The coexistence between Turing patterns and waves has been observed previously as superposed (14) or juxtaposed (15) patterns, or as mixed turbulence (14) near a codimension-2 Turing-Hopf bifurcation point. Here, we report a different coexistence, far above the codimension-2 point, where the Turing structures serve as wave sources, and the continuous waves are unstable and break into segments; spirals separate into segmented spirals while preserving the striking spiral envelope. The segmentation does not settle down with a unique perfora- tion length. Instead, segments keep growing, splitting, shrinking, and vanishing in a ''staggering'' motion. We simulate these behaviors with a simple reaction-diffusion model, analyze the stability properties of this segmentation to find the relevant transverse spatiotemporal wave front instability, and calculate the dispersion relation. Our experiments are carried out in a one side fed un- stirred reactor. The working medium is an agarose gel (2%, Fluka), which is separated from the feeding chamber by an Anapore membrane (Whatman, pore size 0:2 � m) impreg- nated with agarose gel (4%) to avoid stirring effects and a cellulose nitrate membrane (Whatman, pore size 0.45 mm) for contrast. Three solutions are fed into the reactor, one containing I2 (Aldrich), another ClO2, and the third ma- lonic acid (MA, Aldrich) and poly (vinyl alcohol) (PVA, Aldrich, 80% hydrolyzed, average mol. wt. 9000-10 000), all prepared in a 10 mM solution of sulfuric acid (16). Figures 1(a)-1(c) show a sequence of snapshots of segmented waves emitted from Turing spots or outwardly propagating spirals. Initially the system shows continuous waves that move rapidly, but as the concentration of PVA