  We theoretically explore key concepts of two-dimensional turbulence in a
homogeneous compressible superfluid described by a dissipative two-dimensional
Gross-Pitaeveskii equation. Such a fluid supports quantized vortices that have
a size characterized by the healing length $\xi$. We show that for the
divergence-free portion of the superfluid velocity field, the kinetic energy
spectrum over wavenumber $k$ may be decomposed into an ultraviolet regime
($k\gg \xi^{-1}$) having a universal $k^{-3}$ scaling arising from the vortex
core structure, and an infrared regime ($k\ll\xi^{-1}$) with a spectrum that
arises purely from the configuration of the vortices. The Novikov power-law
distribution of intervortex distances with exponent -1/3 for vortices of the
same sign of circulation leads to an infrared kinetic energy spectrum with a
Kolmogorov $k^{-5/3}$ power law, consistent with the existence of an inertial
range. The presence of these $k^{-3}$ and $k^{-5/3}$ power laws, together with
the constraint of continuity at the smallest configurational scale
$k\approx\xi^{-1}$, allows us to derive a new analytical expression for the
Kolmogorov constant that we test against a numerical simulation of a forced
homogeneous compressible two-dimensional superfluid. The numerical simulation
corroborates our analysis of the spectral features of the kinetic energy
distribution, once we introduce the concept of a {\em clustered fraction}
consisting of the fraction of vortices that have the same sign of circulation
as their nearest neighboring vortices. Our analysis presents a new approach to
understanding two-dimensional quantum turbulence and interpreting similarities
and differences with classical two-dimensional turbulence, and suggests new
methods to characterize vortex turbulence in two-dimensional quantum fluids via
vortex position and circulation measurements.
