Focusing of waves in stratified and/or rotating fluids

A (modern) view of energy transfers in homogeneous and isotropic turbulence (lecture, 2 x 1h30) These two lectures are intended to recall basic phenomena that are taking place in fully developed three-dimensional turbulence, governed by the forced Navier-Stokes equations of incompressible fluids. All along the presentation, as a guide, we will repeatedly refer to measurements of the streamwise component of the velocity field as it is observed in wind tunnels and jets. In the first lecture, we begin with the precise derivation of the kinetic energy budget in a local fashion using a conservation equation formulation. This will lead us to state the peculiar asymptotic behaviors of viscous dissipation and energy injection as the Reynolds number goes to infinity, as they are extrapolated from experimental observations and as assumed in the axiomatic approach of Kolmogorov. To illustrate these non-intuitive irregular behaviors, we will then present the predictions (and failures) of the stochastic heat equation, and build up a Gaussian representation of the expected singular behaviors of velocity gradients using fractional random fields. In the second lecture, we go beyond kinetic energy budget, and derive a local version of the two-points velocity product budget. Doing so, we exhibit a novel term, that originates from the nonlinearity of the underlying equations, that governs the energy transfer across scales, also known as the cascade mechanism. We interpret this term in light of recent approaches related to weak formulations of the Euler equations, and relate its averaged behavior to the third moment of longitudinal velocity increments (i.e. the 4/5th-law). In the latter, we, furthermore, assume statistical stationarity and isotropy.