  In this article, we study the long time behavior of solutions of a variant of
the Boussinesq system in which the equation for the velocity is parabolic while
the equation for the temperature is hyperbolic. We prove that the system has a
global attractor which retains some of the properties of the global attractors
for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains
infinitely many invariant manifolds in which several universal properties of
the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.
