The relaxation of vorticity fluctuations in approximately elliptical streamlines

Abstract

This paper generalizes previous asymptotic studies which describe the winding–up of vorticity fluctuations in axisymmetric streamlines. We consider a steady Euler flow in the plane that possesses a region of closed streamlines of general form. At the centre of the streamlines is assumed to be a stagnation point around which the streamlines are approximately elliptical. A long–time asymptotic solution is obtained that describes how superposed weak fine–scale vorticity fluctuations in the region of closed streamlines can be subject to spiral wind–up and fine scaling. At the elliptic point in the centre this process is less effective and an inner analysis yields scaling exponents that characterize the behaviour here. In particular the vorticity fluctuations increase as a power law of the distance from the elliptic point with the scaling exponent given in terms of the vorticity and angular velocity of the basic Euler flow. We also determine the contribution to the far field from the perturbation vorticity and show that it exhibits a power–law decay at large times.