Vortex Glass and Vortex Liquid in Oscillatory Media

Abstract

We study the disordered, multi-spiral solutions of two-dimensional homogeneous oscillatory media for parameter values at which the single spiral/vortex solution is fully stable. In the framework of the complex Ginzburg-Landau (CGLE) equation, we show that these states, heretofore believed to be static, actually evolve on ultra-slow timescales. This is achieved via a reduction of the CGLE to the evolution of the sole vortex position and phase coordinates. This true defect-mediated turbulence occurs in two distinct phases, a vortex liquid characterized by normal diffusion of individual spirals, and a slowly relaxing, intermittent, "vortex glass". 45.70.Ht, 45.70.Qj Spiral waves are ubiquitous in oscillatory and excitable two-dimensional active media [1]. Their cores are robust wave sources which determine the oscillating frequency of the entire system and may dominate the surrounding dynamics. Spirals (often called vortices) may spontaneously appear and annihilate in a typical manifestation of spatiotemporal chaos. When the single spiral solution is stable, one easily observes locked, quasi-frozen, multispiral disordered structures whose glassy character has been suggested [2][3][4]. Despite being found in most models of excitable or oscillatory media as well as in experiments (Belousov-Zhabotinskii reaction [5], surface growth [6], and others) surprisingly very little is known about the properties of these disordered states and the transitions leading to their formation.In this Letter we investigate, in the framework of the complex Ginzburg-Landau equation (CGLE), the general issue of these multi-spiral solutions and show that they actually evolve on ultra-slow timescales. This is achieved thanks both to long numerical simulations of the CGLE and to a quantitatively correct reduction of the dynamics to the evolution of the sole vortex position and phase coordinates. This true defect-mediated turbulence occurs in two distinct phases, a vortex liquid characterized by normal diffusion of individual spirals, and a slowly relaxing, intermittent, "vortex glass".The CGLE describes most properties of generic oscillatory media, at least at a qualitative level, even if one is not in the vicinity of a supercritical long-wavelength Hopf bifurcation, where it can be systematically derived (for reviews, see [7,8]). Under appropriate scaling of the physical variables, it takes the universal formwhere A is a complex amplitude, b and c are real parameters characterizing relative dispersion and nonlinear frequency shift, and ∆ is the Laplace operator. Intensive studies conducted over the last ten years revealed a wide variety of striking dynamical phenomena in one, two, and three space dimensions, many of which were also observed in various experimental contexts, sometimes up to a quantitative agreement with CGLE predictions. A distinctive feature of the two-dimensional CGLE is the existence of nontrivial sources of spiral waves (vortices) which determine the oscillating frequency of the entire system [9]. The single-spiral so...

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