Interacting Vortices and Spin-Up in Two-Dimensional Turbulence

In simulations of decaying two-dimensional turbulence, Clercx and co-workers discovered the unexpected phenomenon of “spin-up.” (This is the spontaneous acquisition of angular momentum by a turbulent two-dimensional 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 a nonzero 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 Reynolds number.

Figure 1

Effect of aspect ratio.

Figure 2

Entropy vs energy. Square boundary.

Figure 3

Entropy vs energy. Rectangular boundary. Aspect $\mathrm{\text{ratio}}=1.20$.

Figure 4

Entropy vs energy. Rectangular boundary. Aspect $\mathrm{\text{ratio}}=1.25$.