Surface Semi-Geostrophic Dynamics: mixing from local to nonlocal dynamics
Final Report Abstract
The surface semi-geostrophic model has been implemented with forcing and dissipation as well as with a passive tracer to study the turbulence and the tracer statistics in the passage from local to nonlocal dynamics. Several numerical problems, such as the formation of singularities that did not allow for the transformation of coordinates from the geostrophic to the physical coordinates and the suppression of local dynamics in the tracer by the required smoothing filter made us decide to change the original plan of the project. We then studied the effects of the passage from local to nonlocal dynamics making use of a family of models, called generalized Euler equations or α models. In this set of models, the parameter α controls the passage from local to nonlocal dynamics. Making use of this family of models we performed four studies: 1) we studied the stability of SQG vortices in the asymptotic limits of small and large scale instabilities and their interactions; 2) we wrote the point vortex approximation of the α models and, for the case α = 1, corresponding to SQG, we studied the collapse of three point vortices. In particular, we found that in some limiting cases the collapse can either be self-similar or non self-similar. This is particularly important as the results can shed a light on the formation of singularities in the SQG equations. The formation of such singularities is still an open problem. If these singularities exist, it has been suggested that their formation should happen in a self-similar way. We instead show a route for a non-self similar formation of them; 3) we studied the statistics of velocity field for the full α models. In particular we showed that the PDFs follow power laws for which we offered analytical forms dependent on α. We verified our results analytically; 4) using analytical tools from statistical mechanics we wrote a mean field equation for the full α models. We then used the selective decay to study the formation of singularities for the α = 1 model.
Publications
- 2018,"Collapse of generalized Euler and surface quasigeostrophic point vortices", Physical Review E, 98, 023110
G. Badin and A.M. Barry
(See online at https://doi.org/10.1103/physreve.98.023110) - 2019, "Asymptotic scale-dependent stability of surface quasi-geostrophic vortices: semi-analytic results", Geophysical and Astrophysical Fluid Dynamics, 113, 574-593 (special issue "Mathematical Developments in Geophysical Fluid Dynamics: Structure, Vortices, and Waves")
G. Badin and F.J. Poulin
(See online at https://doi.org/10.1080/03091929.2018.1453930) - 2019: "Velocity statistics for point vortices in the local α-models of turbulence", Geophysical and Astrophysical Fluid Dynamics, 113, 527-552 (special issue "Mathematical Developments in Geophysical Fluid Dynamics: Structure, Vortices, and Waves")
G. Conti and G. Badin
(See online at https://doi.org/10.1080/03091929.2019.1572750) - 2020: "Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts", Journal of Statistical Physics (special issue "Statistical Mechanics of the Climate System")
G. Conti and G. Badin
(See online at https://doi.org/10.1007/s10955-019-02472-4)
