  We present results for the equilibrium statistics and dynamic evolution of
moderately large ($n = {\mathcal{O}}(10^2 - 10^3)$) numbers of interacting
point vortices on the unit sphere under the constraint of zero mean angular
momentum. We consider a binary gas consisting of equal numbers of vortices with
positive and negative circulations. When the circulations are chosen to be
proportional to $1/\sqrt{n}$, the energy probability distribution function,
$p(E)$, converges rapidly with $n$ to a function that has a single maximum,
corresponding to a maximum in entropy. Ensemble-averaged wavenumber spectra of
the nonsingular velocity field induced by the vortices exhibit the expected
$k^{-1}$ behavior at small scales for all energies. The spectra at the largest
scales vary continuously with the inverse temperature $\beta$ of the system and
show a transition from positively sloped to negatively sloped as $\beta$
becomes negative. The dynamics are ergodic and, regardless of the initial
configuration of the vortices, statistical measures simply relax towards
microcanonical ensemble measures at all observed energies. As such, the
direction of any cascade process measured by the evolution of the kinetic
energy spectrum depends only on the relative differences between the initial
spectrum and the ensemble mean spectrum at that energy; not on the energy, or
temperature, of the system.
