Rapid Relaxation of an Axisymmetric Vortex

In this paper it is argued that a two‐dimensional axisymmetric large Reynolds number (Re) monopole when perturbed will return to an axisymmetric state on a time scale (Re1/3) that is much faster than the viscous evolution time scale (Re). It is shown that an arbitrary perturbation can be broken into three pieces; first, an axisymmetric piece corresponding to a slight radial redistribution of vorticity; second, a translational piece which corresponds to a small displacement of the center of the original vortex; and finally, a nonaxisymmetric perturbation which decays on the Re1/3 time scale due to a shear/diffusion averaging mechanism studied by Rhines and Young [J. Fluid Mech. 133, 133 (1983)] for a passive scalar and Lundgren [Phys. Fluids 25, 2193 (1982)] for vorticity. This mechanism is verified numerically for the canonical example of a Lamb monopole. This result suggests a physical explanation for the persistence of monopole structures in large Reynolds flows, such as decaying turbulence

Distortion and Evolution of a Localized Vortex in an Irrotational Flow

This paper examines the interaction of an axisymmetric vortex monopole, such as a Lamb vortex, with a background irrotational flow. At leading order, the monopole is advected with the background flow velocity at the center of vorticity. However, inhomogeneities of the flow will cause the monopole to distort. It is shown that a shear‐diffusion mechanism, familiar from the study of mixing of passive scalars, plays an important role in the evolution of the vorticity distribution. Through this mechanism, nonaxisymmetric vorticity perturbations which do not shift the center of vorticity are homogenized along streamlines on a Re1/3 time scale, much faster than the Re decay time scale of an axisymmetric monopole. This separation of time scales leads to the quasisteady evolution of a monopole in a slowly varying flow. The asymptotic theory is verified by numerically computing the linear response of a Lamb monopole to a time‐periodic straining flow and it is shown that a large amplitude, O(Re1/3), distortion results when the monopole is forced at its resonant frequency

Quasi-steady Monopole and Tripole Attractors in Relaxing Vortices

Using fully nonlinear simulations of the two-dimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasi-steady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasi-steady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing

Advection of a Passive Scalar by a Vortex Couple in the Small-diffusion Limit

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