Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called “Oseen’s vortex”. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.

/pdf/global-stability-of-vortex-solutions-of-the-two-dimensional-2qvxzmvibd.pdf

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

https://hal.archives-ouvertes.fr/hal-00081677/document

Physics of vortex merging

In Sect. 1.1 of this chapter we explain the subject of Geophysical Fluid Dynamics, and give the description of main vortex structures that have become objects of the present book: (a) Heton – a two-layer vortex with opposite rotations in different layers, and (b) Intrathermocline lens, which is studied in this work as a vortex patch in the intermediate layer of a three-layer ocean model. In this section, we propose also a short review of works in the related topics. Section 1.2 is a mathematical introduction. It contains all main formulae which are used further in Chaps. 2 and 3.

The Introductory Chapter

Columnar vortices are known to support a family of waves initially discovered by Lord Kelvin (1880) in the case of the Rankine vortex model. This paper presents an exhaustive cartography of the eigenmodes of a more realistic vortex model, the Lamb-Oseen vortex. Some modes are Kelvin waves related to those existing in the Rankine vortex, while some others are singular damped modes with a completely different nature. Several families are identified and are successively described. For each family, the underlying physical mechanism is explained, and the effect of viscosity is detailed. In the axisymmetric case (with azimuthal wavenumber m = 0), all modes are Kelvin waves and weakly affected by viscosity. For helical modes (m = 1), four families are identified. The first family, denoted D, corresponds to a particular wave called the displacement wave. The modes of the second family, denoted C, are cograde waves, except in the long-wave range where they become centre modes and are strongly affected by viscosity. The modes of the third family, denoted V, are retrograde, singular modes which are always strongly damped and do not exist in the inviscid limit. The modes of the last family, denoted L, are regular, counter-rotating waves for short wavelengths, but they become singular damped modes for long wavelengths. In an intermediate range of wavelengths between these two limits, they display a particular structure, with both a wave-like profile within the vortex core and a spiral structure at its periphery. This kind of mode is called a critical layer wave, and its significance is explained from both a physical and a mathematical point of view. Double-helix modes (m = 2) can similarly be classified into the C, V and L families. One additional mode, called F, plays a particular role. For short wavelenghs, this mode corresponds to a helical flattening wave, and has a clear physical significance. However, for long wavelenghts, this mode completely changes its structure, and becomes a critical layer wave. Modes with larger azimuthal wavenumbers m are all found to be substantially damped.

/pdf/kelvin-waves-and-the-singular-modes-of-the-lamb-oseen-vortex-3m5dggo1gv.pdf

Kelvin waves and the singular modes of the Lamb–Oseen vortex

We propose a quantitative criterion for the merging of a pair of equal two-dimensional co-rotating vortices. A cross-validation between experimental and theoretical analyses is performed. Experimental vortices are generated by the roll-up of a vortex sheet originating from the identical and impulsive rotation of two plates. The phenomenon is then followed up in time until a rapid pairing transition occurs for which critical parameters are measured. In the theoretical approach, the nonlinear Euler solution representing a pair of equal vortices is computed for various nonuniform vorticity distributions. The stability analysis of such a configuration then provides critical values for the onset of merging. From this data set, a criterion depending on global impulse quantities is extracted for different shapes of the vorticity distribution. This theoretical statement agrees well with our experimentally based criterion.

/pdf/a-merging-criterion-for-two-dimensional-co-rotating-vortices-1z9o4juo9o.pdf

A merging criterion for two-dimensional co-rotating vortices

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.

/pdf/rapid-relaxation-of-an-axisymmetric-vortex-4uoei261ub.pdf

Rapid relaxation of an axisymmetric vortex

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.

/pdf/quasi-steady-monopole-and-tripole-attractors-for-relaxing-2r6o5q3dg8.pdf

Quasi-steady monopole and tripole attractors for relaxing vortices

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.

/pdf/distortion-and-evolution-of-a-localized-vortex-in-an-2krao69fq1.pdf

Joseph F. Lingevitch

Papers

Rapid relaxation of an axisymmetric vortex

Quasi-steady monopole and tripole attractors for relaxing vortices

Distortion and Evolution of a Localized Vortex in an Irrotational Flow