On the direct and inverse energy transfer in 2-dimensional and 3-dimensional turbulent flows and in turbulent models*

In this seminar, I will discuss a few important open problems in ``Fully Developed Turbulence'' concerning its most idealized realization, i.e. the case of statistically homogeneous and isotropic flows. I will discuss the importance of inviscid conserved quantities in relation to the most striking statistical properties shown by all turbulent flows: the growth of small-scales, strongly non-Gaussian fluctuations, including the presence of anomalous scaling laws. By using unconventional numerical methodology, based on a Galerkin decimation of helical Fourier modes [1-3], I will argue that some phenomena characterizing homogeneous and isotropic flows might be important also for a much larger spectrum of applications, including flows with geophysical and astrophysical relevance as for the case of rotating turbulence and/or conducting fluids. Results about both real 3D decimated Navier-Stokes equations and dynamical models of it will be presented. \\[4pt] [1] L. Biferale, S. Musacchio and F. Toschi, Phys. Rev. Lett. 108 164501(2012).\\[0pt] [2] L. Biferale, E.S. Titi, J. Stat. Phys. 151, 1089(2013).\\[0pt] [3] L. Biferale, S. Musacchio and F. Toschi, J. Fluid Mech. 730, 309(2013).