The vanishing obstacle limit and the vortex method for the surface quasi-geostrophic equation

Abstract

The aim of this BEPE proposal is to develop qualitative and practical results for recent problems concerning the surface quasi-geostrophic (SQG) equations, a versatile geophysical model of oceanic and atmospheric fluid dynamics which has been useful for various applications, including climate models and the interaction of ocean eddies. Moreover, the SQG is a prototype nonlocal reaction-diffusion model and such models have found many recent applications. The SQG equations have been the subject of numerous recent papers but for the most part are treated in the case of the whole space. Our aim, however, is to prove and extend these results in the presence of a boundary, in particular, outside an obstacle. At the same time, the problems we are focused on are connected to the computational implementation of the SQG and other models, and would serve to deepen the application of state of the art numerical methods to them. This would potentially provide new insights into the aspects of turbulence for which these models are advantageous. Our first problem is the vanishing obstacle limit for the SQG. In this problem one aims to ascertain the limiting properties of the solutions outside a vanishing obstacle whose size becomes smaller than a reference spatial scale. This is crucial to understand the long-range effects of the obstacle and the effects of small-scale motions, which is specifically relevant for the SQG. Next we propose to study a closely related problem, which is that of justifying the point vortex model for the SQG. This is connected to the first problem because the dynamics of point vortices can be seen as equivalent to the dynamics of vanishing obstacles under specific conditions. Furthermore, the practical motivation for the point vortex model is that it provides an efficient numerical approximation for the main dynamics, via the so-called vortex method. The justification for this model would thus provide a robust numerical scheme to apply to the SQG in various boundary value problems. Lastly, our third proposed problem concerns determining the rate of growth in time of the support of smooth temperature patches. This is central to understand the long range effects mentioned above and provides a qualitative measure to determine the interaction of point vortices. These problems are thus closely related to our main FAPESP project, where one of our main results deals with an ocean circulation model for the surface temperature. In this kind of models, the effect of boundaries, and the interaction of ocean eddies, which can be seen as point vortices, are crucial, so that we hope that our results will help incorporate such conditions and provide more physically realistic models. (AU)