Viscous evolution of 2D dipolar vortices

J.H.G.M. van Geffen and G.J.F. van Heijst
Fluid Dynamics Research 22, 191-213 (1998)

1. Introduction

Two-dimensional turbulence shows, unlike three-dimensional turbulence, the emergence of large-scale structures from small(er)-scale structures; see e.g. McWilliams (1984). This process of self-organisation in 2D turbulence finds its origin in the inverse energy cascade: energy initially distributed over all scales eventually ends up in the large scales and thus coherent vortex structures are formed. The most common type of such structures that have been found to form is the monopolar vortex. Dipolar vortices are common too, though they are less frequently observed than monopoles. A tripolar vortex structure has also been seen to form (see e.g. Legras et al., 1988). In the present study the attention is restricted to dipolar vortices.

If, for instance, a turbulent blob of dye is injected horizontally at the appropriate density level in a stratified fluid (the stratification suppresses vertical motions and thus tends to make the flow quasi-2D), this blob collapses to a flat pancake-like structure which consists of two closely packed patches of oppositely signed vorticity (Van Heijst and Flór, 1989; Flór and Van Heijst, 1994). The vertical dimension of such a dipole is much smaller than its horizontal size, hence it can be treated as a planar, quasi-2D structure. Similar dipolar vortices can be created by towing a cylinder through a rotating fluid (in which the rotation of the fluid provides the two-dimensionalizing mechanism of the motions; Velasco Fuentes and Van Heijst, 1994) or through a thin soap film (Couder and Basdevant, 1986): the flow in the wake of the cylinder consists of one or more dipolar vortices. Electric pulses in a layer of mercury, which is subjected to a magnetic field to make the motions 2D, can also lead to the formation of dipolar vortices (Nguyen Duc and Sommeria, 1988).

The dipole formed in these 2D flows is fairly stable: the vortex retains its dipolar structure during its motion (caused by the dipole's net linear momentum), even though the dipole gradually decays due to viscous effects, as discussed e.g. by Flór and Van Heijst (1994) and Flór et al. (1995). Under geophysical circumstances dipolar vortices are often disturbed or even torn apart by external forces, such as strain and shear. These effects are not discussed here: the present study focusses on the evolution of dipolar vortices themselfs. A numerical and experimental study of the effects of a strain on dipoles has been done by Trieling et al. (1997), and the effects of a shear are the subject of future work.

From the analysis of the dipoles formed in laboratory experiments it appears that its characteristics are well described by the so-called Lamb dipole, a dipolar vortex with a circular form, which is a solution of the inviscid vorticity equation in an infinite domain constructed by Lamb (1932). [In fact, this solution was already formulated in a more general form by Chaplygin (1903); see also Meleshko and Van Heijst (1994).] In an inviscid fluid the Lamb dipole moves along a straight line with constant velocity and without change of form. This paper presents the results of a numerical study of the effects of viscosity and the finiteness of the domain on the evolution of a Lamb dipole. The numerical method used, a finite difference method, is outlined in the next section. Section 3 presents the Lamb dipole and the results of the numerical simulations of the evolution are discussed in Sect. 4. Since a dipolar structure with Lamb-like characteristics appears to be a stable structure in 2D turbulence, Sect. 5 focusses on the (numerical) evolution of some other initial dipolar structures (such as two monopolar vortices of opposite sign next to each other) to study whether a Lamb-like dipolar structure emerges. Some conclusions are formulated in Sect. 6.

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