The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments

The point-vortex system on the surface of the sphere is examined by Monte Carlo methods. The statistical equilibria found in the system when it is constrained to keep circulation zero (but without other explicit constraints on site values) are found to be self-regulating in a sense. While site strengths will grow without bound as the number of sweeps increases, the Dirichlet quotient, the ratio of enstrophy to energy, is found to converge rapidly to a finite nonzero value. This unlimited growth in site values remains controlled. The dependences of this quotient on the temperature and on the mesh size are examined.