Noise perturbations of Point Vortex model

The investigation of stochastic perturbations in fluid dynamics has been a standard procedure in the physical and mathematical literature. In particular, we recall the use of transport-type noise (advection by a stochastic velocity field) in the development of the theory of turbulence, due to the work of Kraichnan on stochastic transport of passive scalars. A very classical particle system in the fluid dynamics context is the Point Vortex model. This system is a discretization of the Euler equation in the vorticity formulation, and it has been introduced and studied by Helmholtz; however, it is due to the investigation on two-dimensional turbulence carried on by Onsager that this model reached popularity and recognition in the context of fluid discretizations. The Point Vortex model is also an object of studies in the context of dynamical systems, due to the Hamiltonian aspects of vortex dynamics. In this thesis, we too focus on the study of two-dimensional models; this is not a limitation in our investigation, as many phenomena in nature can be modeled by almost-two-dimensional systems, examples are atmospheric or geostrophic turbulence; this modelization is possible if the vertical scale of the system is negligible with respect to the horizontal scale or vice-versa. By investigating both the Point Vortex model in its standard formulation and its stochastic modifications, we explore two long standing problems in fluid dynamics: turbulence modeling by stochastic perturbations of transport type and the long time behavior of an inviscid fluid. In the first case, we investigate both the Point Vortex model with stochastic modification of transport type, and the motion of a passive scalar transported by a stochastic velocity field. This stochastic velocity field is very similar to the one studied in the Point Vortex model context, and it is a sum of independent and compactly supported vector fields. This simplified setting allows us to study more in detail the noise. In the second case, we investigate the long time behavior of a two- dimensional fluid: we study the equilibrium dynamics’s temporal structure by computing correlation of local observables, function of the vorticity, of a large number $N$ of point vortices under the invariant measure $dx_1 dots dx_N$ and we exibit evidence of persistence in time correlations, in the form of power law decay of the latter