Freely decaying two-dimensional turbulence

High-resolution direct numerical simulations are used to investigate freely decaying two-dimensional turbulence. We focus on the interplay between coherent vortices and vortex filaments, the second of which give rise to an inertial range. We find that Batchelor's prediction for the inertial-range enstrophy spectrum Eω(k, t) ~ β2/3k−1, where β is the enstrophy dissipation rate, is reasonably well satisfied once the turbulence is fully developed, but that the assumptions which underpin the usual interpretation of his theory are not valid. For example, the lack of a quasi-equilibrium cascade means the enstrophy flux Πω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Πω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Πω; in particular we propose Eω(k, t) ~ Πω(k1)2/3k−1, where k1 is the wavenumber marking the start of the observed k−1 region of the enstrophy spectrum. This provides a better collapse of the data and, unlike Batchelor's original theory, can be justified on theoretical grounds. The basis for our proposal is the observation that the straining of the vortex filaments, which fuels the enstrophy flux through the inertial range, comes almost exclusively from the strain field of the coherent vortices, and this can be characterized by Πω(k1)1/3. Thus Eω(k) is a function of only k and Πω(k1) in the inertial range, and dimensional analysis then yields Eω ~ Πω(k1)2/3k−1. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Πω varies as Πω(k)/Πω(k1) = 1 − a−1 ln(k/k1), where a is a constant of order 1. This corresponds to ∂Eω/∂t ~ k−1. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Πω extend to wavenumbers much smaller than k1, but this is masked in Eω(k, t) by the presence of coherent vortices which also contribute to Eω in this region. In fact, we find that kEω(k, t) ≈ H(k) + A(t), or ∂Eω/∂t ~ k−1 in this extended low-k region, where H(k) is almost independent of time and represents the signature of the coherent vortices. In short, the inertial range defined by ∂Eω/∂t ~ k−1 or Πω(k) ~ ln(k) is much broader than the observed Eω ~ k−1 region.