We derive rigorous estimates for finite-time averages of various mathematical objects, which involve the solutions of the 2D space-periodic Navier-Stokes equations, and correspond to relevant physical quantities in turbulence. The fundamental relations of the mathematical theory of turbulence are derived from the equations of motion. The main results of this first part concern the direct enstrophy cascade, lower bounds for finite-time averages of the energy and the rate of energy dissipation, Kolmogorov's dissipation law, upper and lower bounds for the dissipation cut-off. We allow time-dependent forcing in all length scales and examine several spatial and temporal regularity conditions on forces.

Next, we study the vertically averaged Rayleigh-Benard equations with periodic-Dirichlet mixed boundary conditions. We require global regularity, and show that the solutions have the necessary space regularity properties for an analysis of turbulence along the same lines as in the first part. The result of averaging is a 2D Navier-Stokes system. The body force is determined by the solutions of the 3D Rayleigh-Bénard equations, and consists of boundary terms and Reynolds' stresses. We analyze the averaged system for 2D turbulence, and derive similar and different aspects with the general 2D problem.

Finally, we study the horizontally averaged Rayleigh-Benard system. The goal is to analyze the role of convection in heat transport by exploring the relation between the adimensional heat flux (the Nusselt number), and another non-dimensional quantity (the Rayleigh number) representing the predetermined physical state. We prove some properties of the temperature profile and the velocity field, and deduce various power laws relating the Nusselt and the Rayleigh numbers under assumptions on velocity and temperature.