# Vorticity of spirals in the rock-paper-scissors equation on the sphere

film

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Solution of a reaction-diffusion equation involving three chemicals, each of them dominating one of the others, and dominated by the other one.

At each point in space and time, there are three concentrations u, v, and w of chemicals, that we may call Red, Blue and Green. Denoting by rho = u + v + w the total concentration, the system of equations is given by

d_t u = D*Delta(u) + u*(1 - rho - a*v)

d_t v = D*Delta(v) + v*(1 - rho - a*w)

d_t w = D*Delta(w) + w*(1 - rho - a*u)

where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75 while D is equal to 0.2. The terms proportional to a*v, a*w and a*u denote reaction terms, in which Red is beaten by Blue, Blue is beaten be Green, and Green is beaten by Red. The situation is thus similar to the Rock-Paper-Scissors game, and there exist simpler cellular automata with similar properties,The equation is solved by finite differences, where the Laplacian is computed in spherical coordinates. Some smoothing has been used at the poles, where the Laplacian becomes singular in these coordinates.

To obtain the vorticity, the values of u, v and w are first converted into an angle with the following steps:

- divide u and v by the total density rho = u + v + w

- set x = (u - 1/3) + (v - 1/3)/2 and y = (v - 1/3)*sqrt(3)/2

- convert (x,y) to polar coordinates

The resulting transformation from (u,v,w) to (x,y) is similar to a discrete Fourier transform with 3 modes. The vorticity is then obtained by computing the curl (or rotational) of the gradient of the polar angle. For a smooth field, the curl of the gradient is zero. However, the curl has singularities where the gradient is not defined, as in the center of spirals. This allows to make the center of the spirals visible as peaks (I’m not sure where the other spirals come from, they may be due to round-off errors). In the course of the simulation, some centers of vortices and anti-vortices (rotating in opposing directions) annihilate.