The statistical physics of turbulence

We give a broad overview of the statistical theory of turbulence, carefully basing it on important and consolidated notions of fluid dynamics, before going deeper into the discussion of specific models, prone to contemporary debates. In the so-called structural approach, the complexity of turbulence is perceived as the challenge to understand, from the point of view of vortex tubes dynamics, energy transport from large to small scales in the vanishing viscosity limit. The statistical properties of the energy cascade, such as the intermittency phenomenon, are modelled through seemingly diverse narratives associated to multiplicative stochastic processes or, alternatively, to the multifractal formulation of the singularity spectrum of the turbulent velocity field. The synthesis, the foundation from first principles and the integration of these two modelling strategies with the structural approach build up the essence of current theoretical challenges in turbulence.