Numerical studies on the self-similar collapse of the alpha-patches problem

Details Numerical studies on the self-similar collapse of the alpha-patches problem Numerical studies on the self-similar collapse of the alpha-patches problem

2009-02-04

arxiv.org/abs/0902.0706v1

Mathematics

Analysis of PDEs

20 pages, 9 figures

Scientific paper

In this paper we study a class of contour dynamics equations depending on a parameter alpha for which has been provided numerical evidence of self-similar collapse. This family of equations connect the vortex patch problem of the 2D Euler equations (limit alpha to 0) with the surface quasi-geostrophic equation (alpha=1). We explore numerically the evolution of these equations, but in order to transform the finite time blowup in an asymptotic behaviour, the equations are expressed in self-similar variables. We discuss the role of exact self-similar solutions that act as geometrical separatrices of what numerically is distinguished as collapsing and non collapsing initial conditions.

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