The Relaxation of Vorticity Fuctuations in Approximately Elliptical Streamlines

This paper generalizes previous asymptotic studies which describe the winding-up of vorticity ®uctuations in axisymmetric streamlines. We consider a steady Euler ®ow 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 ­ ne-scale vorticity ®uctuations in the region of closed streamlines can be subject to spiral wind-up and ­ ne scaling. At the elliptic point in the centre this process is less e¬ective and an inner analysis yields scaling exponents that characterize the behaviour here. In particular the vorticity ®uctuations 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 ®ow. We also determine the contribution to the far ­ eld from the perturbation vorticity and show that it exhibits a power-law decay at large times.