Freely decaying two-dimensional turbulence

Abstract

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 E_ω(k, t) ~ β^2/3k⁻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 Π_ω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Π_ω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Π_ω; in particular we propose E_ω(k, t) ~ Π_ω(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 Π_ω(k₁)^1/3. Thus E_ω(k) is a function of only k and Π_ω(k₁) in the inertial range, and dimensional analysis then yields E_ω ~ Π_ω(k₁)^2/3k⁻¹. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Π_ω varies as Π_ω(k)/Π_ω(k₁) = 1 − a⁻¹ ln(k/k₁), where a is a constant of order 1. This corresponds to ∂E_ω/∂t ~ k⁻¹. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Π_ω extend to wavenumbers much smaller than k₁, but this is masked in E_ω(k, t) by the presence of coherent vortices which also contribute to E_ω in this region. In fact, we find that kE_ω(k, t) ≈ H(k) + A(t), or ∂E_ω/∂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 ∂E_ω/∂t ~ k⁻¹ or Π_ω(k) ~ ln(k) is much broader than the observed E_ω ~ k⁻¹ region.