Advection of a passive scalar by a vortex couple in the small-diffusion limit

Abstract

We study the advection of a passive scalar by a vortex couple in the small-diffusion (i.e. large Péclet number, Pe) limit. The presence of weak diffusion enhances mixing within the couple and allows the gradual escape of the scalar from the couple into the surrounding flow. An averaging technique is applied to obtain an averaged diffusion equation for the concentration inside the dipole which agrees with earlier results of Rhines & Young for large times. At the outer edge of the dipole, a diffusive boundary layer of width O(Pe−½) forms; asymptotic matching to the interior of the dipole yields effective boundary conditions for the averaged diffusion equation. The analysis predicts that first the scalar is homogenized along the streamlines on a timescale O(Pe−$\frac{1}{3}$). The scalar then diffuses across the streamlines on the diffusive timescale, O(Pe). Scalar that diffuses into the boundary layer is swept to the rear stagnation point, and a finite proportion is expelled into the exterior flow. Expulsion occurs on the diffusive timescale at a rate governed by the lowest eigenvalue of the averaged diffusion equation for large times. A split-step particle method is developed and used to verify the asymptotic results. Finally, some speculations are made on the viscous decay of the dipole in which the vorticity plays a role analogous to the passive scalar.