Non-equilibrium statistical mechanics for a vortex gas

Kinetic equations governing the evolution of the reduced probability distribution functions associated with low- order vortex interactions making up an inviscid two- dimensional turbulent. fluid on the surface of a sphere are derived. The vortex gas is viewed as a coupling, via the Liouville equation, between monopoles, dipoles, and tripoles, and as such, is based on ' integrable' ballistic elements whose complex interactions generate turbulent. fluctuations. Assuming the dilute regime of a neutral vortex gas, pairwise interactions between monopoles and dipoles, dipoles and tripoles, monopoles and tripoles dominate over triple interactions such as monopole - dipole - tripole ones. Our general closure assumption used to arrive at a lower- order system is based on the idea of decomposing quadrupolar interactions into all possible lower- order interactions among monopoles, dipoles, and tripoles using a weighted average of these interactions with coefficients based on combinatorial likelihood. This is a far less restrictive assumption than the vortex - dipole - chaos assumption of Marmanis ( Marmanis H 1998 Proc. R. Soc. A 454 587 - 606) and leads to a fully coupled system of nine equations for the interaction of ballistic elements where each of the three ' building blocks' making up the turbulent gas is treated on an equal footing.