  We study the relaxation of a 2D ultracold Bose-gas from a nonequilibrium
initial state containing vortex excitations in experimentally realizable square
and rectangular traps. We show that the subsystem of vortex gas excitations
results in the spontaneous emergence of a coherent superfluid flow with a
non-zero coarse-grained vorticity field. The streamfunction of this emergent
quasi-classical 2D flow is governed by a Boltzmann-Poisson equation. This
equation reveals that maximum entropy states of a neutral vortex gas that
describe the spectral condensation of energy can be classified into types of
flow depending on whether or not the flow spontaneously acquires angular
momentum. Numerical simulations of a neutral point vortex model and a Bose gas
governed by the 2D Gross-Pitaevskii equation in a square reveal that a large
scale monopole flow field with net angular momentum emerges that is consistent
with predictions of the Boltzmann-Poisson equation. The results allow us to
characterise the spectral energy condensate in a 2D quantum fluid that bears
striking similarity with similar flows observed in experiments of 2D classical
turbulence. By deforming the square into a rectangular region, the resulting
maximum entropy state switches to a dipolar flow field with zero net angular
momentum.
