  We inquire the statistical properties of the pair formed by the Navier-Stokes
equation for an incompressible velocity field and the advection-diffusion
equation for a scalar field transported in the same flow in two dimensions
(2d). The system is in a regime of fully developed turbulence stirred by
forcing fields with Gaussian statistics, white-noise in time and self-similar
in space. In this setting and if the stirring is concentrated at small spatial
scales as if due to thermal fluctuations, it is possible to carry out a
first-principle ultra-violet renormalization group analysis of the scaling
behavior of the model.
  Kraichnan's phenomenological theory of two dimensional turbulence upholds the
existence of an inertial range characterized by inverse energy transfer at
scales larger than the stirring one. For our model Kraichnan's theory, however,
implies scaling predictions radically discordant from the renormalization group
results. We perform accurate numerical experiments to assess the actual
statistical properties of 2d-turbulence with power-law stirring. Our results
clearly indicate that an adapted version of Kraichnan's theory is consistent
with the observed phenomenology. We also provide some theoretical scenarios to
account for the discrepancy between renormalization group analysis and the
observed phenomenology.
