Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wavefront slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including: (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three-dimensions, and spatial or/and time-dependent variation of excitability), and (iv) the dynamics of multi-armed spiral waves with the new prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHug-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semi-quantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models

Controlling domain patterns far from equilibrium

A high degree of control over the structure and dynamics of domain patterns in nonequilibrium systems can be achieved by applying nonuniform external fields near parity breaking front bifurcations. An external field with a linear spatial profile stabilizes a propagating front at a fixed position or induces oscillations with frequency that scales like the square root of the field gradient. Nonmonotonic profiles produce a variety of patterns with controllable wavelengths, domain sizes, and frequencies and phases of oscillations.Comment: Published version, 4 pages, RevTeX. More at http://t7.lanl.gov/People/Aric

Scroll waves in isotropic excitable media : linear instabilities, bifurcations and restabilized states

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.Comment: 30 pages 16 figures, submitted to Phys. Rev.

The Saffman-Taylor problem on a sphere

The Saffman-Taylor problem addresses the morphological instability of an interface separating two immiscible, viscous fluids when they move in a narrow gap between two flat parallel plates (Hele-Shaw cell). In this work, we extend the classic Saffman-Taylor situation, by considering the flow between two curved, closely spaced, concentric spheres (spherical Hele-Shaw cell). We derive the mode-coupling differential equation for the interface perturbation amplitudes and study both linear and nonlinear flow regimes. The effect of the spherical cell (positive) spatial curvature on the shape of the interfacial patterns is investigated. We show that stability properties of the fluid-fluid interface are sensitive to the curvature of the surface. In particular, it is found that positive spatial curvature inhibits finger tip-splitting. Hele-Shaw flow on weakly negative, curved surfaces is briefly discussed.Comment: 26 pages, 4 figures, RevTex, accepted for publication in Phys. Rev.

Disturbance of meromixis in saline Lake Shira (Siberia, Russia): possible reasons and ecosystem response

Saline Lake Shira (Southern Siberia, Russia) was meromictic through the observation period 2002-2015. During the under-ice periods of 2015 and 2016, complete mixing of the water column was recorded for the first time, and hydrogen sulphide temporarily disappeared from the water column of the lake; i.e. in those years the lake turned to holomixis. In the summer of 2015, a sharp increase in chlorophyll a, organic carbon, zooplankton, and phytoflagellates was observed in the lake, which was probably due to the release of nutrients from the monimolimnion. Purple sulfur bacteria completely disappeared from the lake after the first mixing in 2015, and did not reappear despite the restoration of meromixis in 2017. Thus, it was demonstrated that purple sulfur bacteria are sensitive to the weakening of the stratification of Lake Shira. Based on the data of the seasonal monitoring of temperature and salinity profiles over the period 2002-2017, it was presumed that the main cause of deep mixing in 2015 was the weakening of the salinity gradient due to strong wind impact and early ice retreat in the spring of 2014. In addition, it was shown that in previous years a significant contribution to the maintenance of meromixis was made by an additional influx of fresh water, which caused a rise in the lake level in the period 2002-2007. Thus, we identified a relationship between the stratification regime of the lake and the change in its level, which provides valuable information both for the forecast of water quality and for reconstruction of the Holocene climate humidity in this region of Southern Siberia from the sediment cores of Lake Shira