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

In this paper we study a class of contour dynamics equations depending on a parameter α for which has been provided numerical evidence of selfsimilar collapse [2]. This family of equations connect the vortex patch problem of the 2D Euler equations (limit α → 0) with the surface quasigeostrophic equation (α = 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. Mathematics Subject Classification: 65M99, 35Q35