Two dimensional singularity turbulence

Using the self-focusing non-linear Schrödinger (NLS) equation, we suggest a singularity-mediated turbulent scenario. This equation is taken as a simple (toy) model to investigate the role of potential singularities in fully developed turbulence and intermittencies. The self-focusing NLS equation has the advantage to exhibit finite-time singular solutions in any dimension D, which are described by a simple solution of a nonlinear self-similar universal equation. We investigate in this paper in particular the two dimensional (2D) dynamics which provides a more complex behavior than the 1D case studied formerly. The current scenario of turbulence offers an understanding of the role of singularities in dissipation in a turbulent flows and reproduces, among other aspects, the observed anomaly of dissipation in the limit of zero viscosity.