Interacting Vortices and Spin-Up in Two-Dimensional Turbulence

Abstract

In simulations of decaying two-dimensional turbulence, Clercx et al [1,2] discovered the unexpected phenomenon of "spin-up". (This is the spontaneous acquisition of angular momentum by a turbulent 2-D fluid in a rigid container.) Here we show that this phenomenon can readily be interpreted in terms of statistical models of two-dimensional turbulence. When the net vorticity is zero in a bounded system there are two distinct types of statistical equilibrium. The first has the expected property that its angular momentum is zero. However, the second type has non-zero angular momentum even though its circulation vanishes. The relative probability of the two types of equilibrium depends strongly on the shape of the boundary and weakly on the energy. The angular momentum predicted for the second type of equilibrium is in good agreement with that found in simulations at high Reynold's number. Two-dimensional turbulence, (which may to some extent be realized in magnetized plasmas, in rotating and stratified fluids, and in thin films), has been extensively studied by direct numerical simulation. In high Reynolds number simulations of decaying 2-D turbulence, Clercx et al [1,2] discovered the remarkable phenomenon of "spin-up". This is the spontaneous acquisition of angular momentum by a turbulent 2-D fluid in a rigid container -a process that depends on the shape of the container.It has often been proposed [3,4,5,6], originally by Onsager [7], that 2-D turbulence might be interpreted in terms of the statistical equilibria of interacting point vorticesor equivalently of interacting charged rods in a parallel magnetic field [8,9,10]. Here we show that this, and other statistical models, readily account for spin-up in a bounded domain -both qualitatively in its dependence on boundary shape and quantitatively in the amount of angular momentum it generates.

Vorticity Generation by Rough Walls in 2D Decaying Turbulence

“…Particularly active areas of research are wall effects at high Reynolds numbers [15,20,22], or two-dimensional (2D) turbulence in bounded domains of various geometries [10][11][12]18,25,28,32,34]. As for B Imre M. Jánosi imre.janosi@ttk.elte.hu Gábor Tóth tooth@caesar.elte.hu the latter subject, Lattice Boltzmann (LB) method has proven to be specifically well suited to related numerical studies [1,3,8].…”

“…As for B Imre M. Jánosi imre.janosi@ttk.elte.hu Gábor Tóth tooth@caesar.elte.hu the latter subject, Lattice Boltzmann (LB) method has proven to be specifically well suited to related numerical studies [1,3,8]. A notable aspect of interaction between vortices and rigid walls is the so called "spin-up" phenomenon, where measures of total vorticity such as the enstrophy or angular momentum exhibit locally increasing trends during an overall spontaneous decay [10][11][12]28].…”

Universal statistics of point vortex turbulence

“…In other words, at low energies, distinct solutions of the sinhPoisson equation emerge from (e, β h ) = (0, β 1 ), (0, β 2 ), .... The mean-field theory can be used to show (Chavanis & Sommeria 1996;Taylor et al 2009) that, of these solutions, the first has maximum entropy and will therefore be selected by the dynamics. In the E → 0 limit the first solution has streamfunction Numerical solutions of (3.26-3.27) have been obtained, using the algorithm of McDonald (1974), in domains A and B for a range of energies starting close to E = 0.…”

Two-dimensional decaying turbulence in confined geometries

“…Taylor and his co-workers pointed out in their theoretical work how the the shape of the boundary can influence the interactions between vortices and the wall. 1 In the last two decades, numerical computational studies of decaying 2D turbulence in a container have revealed some interesting results. It was pointed out, for instance, that the decay process of 2D turbulence in a container differs significantly from the one which takes place in a double-periodic domain.…”