Recent Advances in 2D and 2.5D Vortex statistics and dynamics

This review paper offers both an overview of the recent developments in vortex statistics, and a detailed critical review of some of the advances as well as open problems in the area. A fruitful new direction in vortex dynamics concerns vortical systems with very large numbers of vortices in which statistical methods play a pivotal role. After a review of some of the older equilibrium statistical mechanics approaches, such as Onsager’s Vortex gas theory, we survey the recent burst of interest and papers on the application of statistical mechanics to a wide range of problems. They include many interesting problems such as rotating stratified flows with natural applications to atmospheric and oceanic sciences. Special models that will be discussed include point vortex as well as vortex blob models in axisymmetric flows, in the plane, and in geophysical flows involving multi-layer baroclinic effects. A recent reverse application of a quantum path-integral Monte-Carlo algorithm to the Lions-Majda model for nearly parallel vortex filaments will be reviewed with numerical results. A connection between these statistical theories and the classical numerical method known as the vortex method will also be made. Many of these noteworthy advances are made possible by good Monte-Carlo algorithms, providing a cross-disciplinary aspect to the field. An important open problem in the field of the vortex gas is the exact partition function for any finite number of particles N and any finite temperature.