Abstract:
We review the role played by long-lived coherent vortices in freely evolving and forced two dimensional turbulence, and discuss a new way of conceptualising scaling theories for the distributions of vortex areas and intensities, and their time evolutions. Vortex populations in two-dimensional turbulence are many-body systems far from equilibrium, and such systems generically exhibit multiple regimes with distinct power law behaviour, each associated with transport of a conserved quantity across scales. Canonical examples in two-dimensional turbulence are the enstrophy cascade, in which classical theory predicts a k-independent enstrophy ux to small scales, and the inverse energy cascade, in which a k-independent energy ux to large scales is expected [1, 2]. Here we extendthese inertial range arguments to the vortex subfield in forced two-dimensional turbulence, with transport across scales in vortex area space mediated by vortex interactions taking the place of transport through wavenumber space [3]. We construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, n(A) _ t_iA��ri ; i 2 1; 2; 3, conserving the first three moments of !2vn(A) in three distinct scaling ranges. Here !2v is the `vortex intensity’, or mean square vorticity evaluated over vortices, and n(A) is the number density distribution of vortices with area A, where areas are defined as intense regions of vorticity bounded by vorticity isolines. Conservation is enforced in `comoving intervals’, whose endpoints evolve at the vortex growth rate; this amounts to assuming invariance under the dilatation of ow features associated with the inverse cascade, and in addition assuming that the growth in vortex area is the appropriate measure of dilatation in all scaling ranges. We discuss how the same conceptual framework can be applied to model vortices in decaying two-dimensional turbulence, signing the way toward a unied description of vortex populations in the two systems [4, 5]. The predictions are well-veried by high resolution numerical simulations, and insensitive to the vorticity threshold used to isolate the regions. We further discuss modications to the theory taking into account time- and A-dependent vortex intensity !2v.
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