Late time evolution of unforced inviscid two-dimensional turbulence

Abstract

We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.