Fokker‐Planck Kinetic Description of Small‐scale Fluid Turbulence for Classical Incompressible Fluids§

Extending the statistical approach proposed in a parallel paper [1], purpose of this work is to propose a stochastic inverse kinetic theory for small‐scale hydrodynamic turbulence based on the introduction of a suitable local phase‐space probability density function (pdf). In particular, we pose the problem of the construction of Fokker‐Planck kinetic models of hydrodynamic turbulence. The approach here adopted is based on the so‐called IKT approach (inverse kinetic theory), developed by Tessarotto et al. (2004–2008) which permits an exact phase‐space description of incompressible fluids based on the adoption of a local pdf. We intend to show that for prescribed models of stochasticity the present approach permits to determine uniquely the time evolution of the stochastic fluid fields. The stochastic‐averaged local pdf is shown to obey a kinetic equation which, although generally non‐Markovian, locally in velocity‐space can be approximated by means of a suitable Fokker‐planck kinetic equation. As a side r...