ELSEVIER Physica D 97 (1996) 509-516 PHYSICA Spirals in excitable media: the free-boundary limit with diffusion David A. Kessler a,., Raz Kupferman b a Department of Physics, Bar-llan University, Ramat-Gan 52900, Israel b AT&TBell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974, USA Received 18 September 1995; revised 4 December 1995; accepted 5 December 1995 Communicated by L. Kramer Abstract We solve numerically for the steady-state spiral in the thin-interface limit, including the effects of diffusion of the slow field. The calculation is performed using a generalization of the hybrid scheme of Keener. In this method, the diffusion equation is solved on a suitable mapped lattice while the eikonal equation relating the field on the interface to the interfacial velocity and curvature is solved independently. We present results for the selected frequency and tip radius as a function of the various parameters. We note that a stability analysis based on these results may be performed. The study of spiral patterns in excitable media has been the focus of considerable attention in recent years (for a review, see [1]). Much progress has been achieved in this area, through the use of simulations and the analysis of various limiting cases. Spirals ex- hibit a wide range of interesting dynamics, including transitions from simple rotation to meandering (com- pound rotation) to hypermeandering and yet more complex behaviors [2,3]. Spirals are relevant not only in the context of chemical reactions such as the famed Belousov-Zhabotinskii reaction [2,4] but also in vari- ous biological systems such as electrical conduction in heart tissue [5] and aggregation of the slime mold [6]. A key tool for analyzing spirals, and patterns in general, is the thin-interface, or free-boundary limit. This limit arises [7] from taking the ratio, 1/~, of the reaction rate of the bi-stable reaction to the other reac- tion to be large. In this limit, the dynamics reduces to that of the single "slow" field. The spiral can then be * Corresponding author. considered as a sharp interface between two different phases of the slow field, the so-called "excited" and "refractory" regions. In particular, there is much evi- dence to support the notion that in this limit, the spiral solution achieves a simple scaling form, known as the Fife ansatz [8], in which the parameter ~ only deter- mines the overall length and time scales of the pattern. In this limit, the spiral is characterized by only a few simple macroscopic "material parameters" derivable from the underlying dynamics. This Fife-scaling has been verified in two limits. One is the limit where the diffusion constant of the slow field is vanishingly small [9-11]. In this limit, the "tip radius," the distance of closest approach of the spiral to the center of rotation, goes to zero. Also, the curvature at the tip goes to infinity. One then has to treat the tip region separately, with a cutoff length scale given by the larger of the small diffusion length or the small interface width. When one analyzes the stability of the spiral in this limit [ 12], one finds that the tip always exhibits a single unstable mode. It is 0167-2789/96/$15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved PH S0167-2789(96)00135-2