Autor: David Dritschel (University of St Andrews, Reino Unido)
Resumo/ementa: This course will discuss both the mathematics and the numerical methods enabling the study of inviscid incompressible fluid flow on two-dimensional surfaces, with an emphasis on closed surfaces. Examples of closed surfaces include the sphere and the ellipsoid, both practically relevant to planetary atmospheric dynamics. Despite these applications, relatively little is known about fluid flows on such surfaces, compared to flow in the two-dimensional plane. We begin with the simplest fluid model, namely the motion of singular point vortices, a model first introduced by Kirchhoff in the 1800s. We shall see that surface geometry has a profound effect on the dynamics of such vortices. In particular, surface curvature tends to be destabilising, especially when the curvature varies across the surface. We then discuss an accurate numerical method for studying vortex motion on general surfaces of revolution. Students will be given the opportunity to use this method (together with associated visualisation tools) to explore various aspects of vortex motion. We then generalise the analysis to finite distributions of vorticity on closed surfaces - vortex patches. We discuss the `inversion problem', namely how to obtain the velocity field from the instantaneous distribution of vorticity. We show how this simplifies for surfaces of revolution and then discuss a hybrid Lagrangian-Eulerian numerical method for accurate flow simulation. The method, called the Contour-Advective Semi-Lagrangian (CASL) algorithm, blends the most accurate and efficient parts of a grid-free contour dynamics method with a grid-based conventional method. Again, students will have the opportunity to use this method to explore the complex vortex dynamics which may occur on closed surfaces. The schedule for the course is as follows: Lecture 1: Mathematical aspects of fluid flow on closed surfaces. Point vortex dynamics. Lecture 2: Numerical methods for point vortices on closed surfaces. Hands-on exploration of the dynamics. Lecture 3: Generalisation to vortex patches. The inversion problem and simplifications for surfaces of revolution. The CASL algorithm. Hands-on exploration of the dynamics.
Cronograma: O curso será dividido em 3 aulas, de 20 a 27 de janeiro.