  The problem of low Reynolds number turbulence in active nematic fluids is
theoretically addressed. Using numerical simulations I demonstrate that an
incompressible turbulent flow, in two-dimensional active nematics, consists of
an ensemble of vortices whose areas are exponentially distributed within a
range of scales. Building on this evidence, I construct a mean-field theory of
active turbulence by which several measurable quantities, including the
spectral densities and the correlation functions, can be analytically
calculated. Because of the profound connection between the flow geometry and
the topological properties of the nematic director, the theory sheds light on
the mechanisms leading to the proliferation of topological defects in active
nematics and provides a number of testable predictions. A hypothesis, inspired
by Onsager's statistical hydrodynamics, is finally introduced to account for
the equilibrium probability distribution of the vortex sizes.
