Statistical mechanics of strong and weak point vortices in a cylinder

The motion of 100 point vortices in a circular cylinder is simulated numerically and compared with theoretical predictions based on statistical mechanics. The novel aspect considered here is that the vortices have greatly different circulation strengths. Specifically, there are 4 strong vortices and 96 weak vortices, the net circulation in either group is zero, and the strong circulations are five times larger than the weak circulations. As envisaged by Onsager [Nuovo Cimento, Suppl. 6, 279 (1949)], such an arrangement leads to a substantial amplification of statistical trends such as the preferred clustering of the strong vortices in either same-signed or oppositely signed pairs, depending on the overall energy level. To prepare the ground, this behavior is illustrated here first by a simple toy model with exactly solvable statistics. A microcanonical ensemble based on the conserved total energy E and angular momentum M for the whole vortex system is then used, in which the few strong vortices are treated as a subsystem in contact with a reservoir composed of the many weak vortices. It is shown that allowing for the finite size of this reservoir is essential in order to predict the statistics of the strong vortices accurately. Notably, this goes beyond the standard canonical ensemble with positive or negative temperature. A certain approximation is then shown to allow a single random sample of uniformly distributed vortex configurations to be used to predict the strong vortex statistics for all possible values of E and M. Detailed predictions for the energy, two-vortex, and radial distribution functions of the strong vortices are then made for comparison with three simulated cases of near-zero M and low, neutral, or high E. It is found that the statistical mechanics predictions compare remarkably well with the numerical results, including a prediction of vortex accumulation at the cylinder wall for low values of E. (C) 2002 American Institute of Physics.