  Turbulent flows are out-of-equilibrium because the energy supply at large
scales and its dissipation by viscosity at small scales create a net transfer
of energy among all scales. Here, the energy cascade is approximated by a
combined contribution of a forward drift and diffusion that recover accepted
phenomenological theories of turbulence. The fluctuation theorem (FT) is then
shown to describe the scale-wise statistics of forward and backward energy
transfer and their connection to irreversibility and entropy production. The
ensuing turbulence entropy may be used to formulate an extended turbulence
thermodynamics.
