Cascades and transitions in turbulent flows

Turbulent flows are characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal three and two dimensional homogeneous and isotropic cases characterized by the presence of a strictly direct and inverse energy cascade, respectively. New phenomena appear that alter the global and local transfer properties. In this review, we provide a critical summary of historical and recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse energy cascades, the different scenarios include: split cascades for which an invariant flows both to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a large scale condensate is formed. We classify all possible transitions from one scenario to another as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, magnetohydrodynamics (MHD) turbulence, and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as, e.g., the Reynolds, Rossby, Froude, Peclet, or Alfv ´ en numbers. We briefly discuss the presence ´ of anomalous scaling laws in 3D hydrodynamics and in other configurations, in connection with the intermittent nature of the energy dissipation in configuration space. A quick overview is also provided concerning the importance of cascades in other applications such as bounded flows, quantum fluids, relativistic and compressible turbulence, and active matter, together with a discussion of the implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address