Bracco and McWilliams (J. Fluid Mech. 646, 2010) surveyed statistically stationary, randomly forced two-dimensional Navier-Stokes turbulence over Re≈1.5×103Re\approx 1.5\times10^{3} to 5.6×1065.6\times10^{6} and found that bulk and small-scale diagnostics (energy, enstrophy, palinstrophy, dissipation wavenumber, dissipation rates, vorticity structure functions) follow empirical power laws X∼ReλXX\sim Re^{\lambda_X} whose exponents are non-universal, differing with the forcing protocol, while normalized PDFs and intermittency measures stay Re-invariant. No theory predicts these exponents; extrapolation to the asymptotic Re→∞Re\to\infty regime of geophysical interest remains empirical, and existing rigorous dissipation bounds ignore coherent vortices. The open problem is to replace these empirical fits with a predictive, mechanism-based account: derive (or rigorously refute) the exponents and their forcing dependence, and establish whether the power laws and intermittency invariance persist as Re→∞Re\to\infty. The problem remains open.

Background

Two-dimensional (2D) turbulence, governed by the incompressible 2D Navier-Stokes equations for a streamfunction ψ\psi (velocity components u=−∂ψ/∂yu=-\partial\psi/\partial y, v=∂ψ/∂xv=\partial\psi/\partial x; vorticity ω=Δψ\omega=\Delta\psi), is the standard paradigm for 'thin' geophysical and astrophysical flows such as planetary atmospheres, ocean mesoscales, and soap films. When the flow is randomly forced at an intermediate wavenumber, energy cascades to larger scales (the inverse energy cascade) while enstrophy — one half of the spatially averaged square of vorticity — cascades to smaller scales (the forward enstrophy cascade). In the forward-enstrophy range, the classical Kraichnan-Leith-Batchelor (KLB) phenomenology predicts an energy spectrum E(k)∼k−3E(k)\sim k^{-3}, with a possible logarithmic correction, under assumptions of homogeneity, isotropy, and local cascade dynamics without persistent coherent structures.

In numerical practice the inverse cascade must be arrested by large-scale damping. Two standard mechanisms are hypoviscosity — a damping operator acting selectively on the largest scales (a negative-order Laplacian in spectral space) — and Ekman drag, a linear friction term −rω-r\omega modeling bottom (planetary boundary layer) friction. Small scales are damped by ordinary Newtonian viscosity ν\nu. The Reynolds number is defined as Re=U/(νkE)Re=U/(\nu k_E) with U=2EU=\sqrt{2E} the rms velocity, EE the spatially averaged kinetic energy, and kEk_E the energy-centroid wavenumber.

Bracco and McWilliams (2010) computed stationary solutions of forced 2D turbulence for two forcing/damping sets: stochastic forcing with amplitude a=0.14a=0.14 and decorrelation time tF=0.005t_F=0.005 with hypoviscous damping μ=1.5\mu=1.5 (covering Re≈1.5×103Re\approx 1.5\times10^{3} to 5.6×1065.6\times10^{6}), and fixed-phase, time-independent forcing with a=0.005a=0.005, tF=∞t_F=\infty, μ=1.65\mu=1.65 (covering Re≈1.7×103Re\approx 1.7\times10^{3} to 1.3×1051.3\times10^{5}); the fixed-phase set is a frozen random spatial field, subsumed by the source under its description of the survey as 'stationary solutions with large-scale, random forcing'. They found: (i) the energy spectrum approaches the KLB form k−3k^{-3} at large Re and the second-order velocity structure function is Re-independent; (ii) all other investigated diagnostics — energy EE, enstrophy ZZ, palinstrophy ⟨∣∇ω∣2⟩\langle|\nabla\omega|^{2}\rangle (mean squared vorticity gradient), the dissipation-related wavenumber kν=(χν/ν3)1/6k_\nu=(\chi_\nu/\nu^{3})^{1/6} with χν=ν⟨∣∇ω∣2⟩\chi_\nu=\nu\langle|\nabla\omega|^{2}\rangle the viscous enstrophy dissipation rate, the small-scale viscous energy dissipation rate ϵν=ν⟨ω2⟩\epsilon_\nu=\nu\langle\omega^{2}\rangle, the large-scale (hypoviscous) energy dissipation rate ϵμ\epsilon_\mu, the viscous and hypoviscous enstrophy dissipation rates χν\chi_\nu and χμ\chi_\mu (with ϵμ\epsilon_\mu and χμ\chi_\mu defined as in the cited source), and vorticity structure functions — follow empirical power laws X∼ReλXX\sim Re^{\lambda_X}; (iii) the exponents λX\lambda_X are non-universal: they differ between the two forcing sets (for example λE≈0.60±0.05\lambda_E\approx 0.60\pm0.05 for the finite-tFt_F set versus λE≈0.36±0.02\lambda_E\approx 0.36\pm0.02 for the fixed-phase set), which the authors attribute to differing populations of long-lived coherent vortices near the forcing scale; (iv) the normalized small-scale dissipation scales as ϵν/(kEU3)≈Re−0.9\epsilon_\nu/(k_EU^{3})\approx Re^{-0.9}, well below the rigorous upper bound ∼Re−1/2\sim Re^{-1/2} for steady 2D turbulence derived by Alexakis and Doering (2006) under homogeneity/isotropy-type assumptions that exclude coherent-vortex effects; and (v) normalized one-point probability density functions and kurtoses of ψ\psi, velocity, ω\omega, and ∇ω\nabla\omega are invariant with Re — no growing intermittency, in contrast with three-dimensional turbulence.

This non-universality continues a known pattern: the prefactor of the k−3k^{-3} spectrum in forced 2D turbulence is non-universal and depends on large-scale flow conditions (Kaneda and Ishihara 2001), stationary forced 2D turbulence statistics depend on the forcing and drag setup (Danilov and Gurarie 2001), and linear Ekman friction alters the intermittency of the direct cascade (Boffetta, Celani, Musacchio and Vergassola 2002); see Boffetta and Ecke (2012) for a review. What Bracco and McWilliams added is the quantitative, wide-range Re-scaling survey itself, and the explicit observation that its exponents are empirical inputs rather than predicted outputs of any theory.

Problem Statement

For statistically stationary, homogeneous, randomly forced two-dimensional Navier-Stokes turbulence with small-scale viscosity and hypoviscous large-scale damping — the configuration surveyed by Bracco and McWilliams (2010) — establish a predictive account of the Reynolds-number dependence of the flow statistics, replacing the empirical power-law fits of Bracco and McWilliams (2010). Concretely, with Re=U/(νkE)Re=U/(\nu k_E), U=2EU=\sqrt{2E}, and kEk_E the energy-centroid wavenumber:

  1. Derive a theory that predicts the observed exponents λX\lambda_X in X∼ReλXX\sim Re^{\lambda_X} for the bulk and small-scale diagnostics — energy EE, enstrophy ZZ, palinstrophy ⟨∣∇ω∣2⟩\langle|\nabla\omega|^{2}\rangle, dissipation-related wavenumber kνk_\nu, small-scale viscous and large-scale energy dissipation rates, enstrophy dissipation rates, and vorticity structure functions — together with their dependence on the forcing properties (amplitude, decorrelation time, spectral shape), including the extent to which the non-universal populations of large-scale coherent vortices mediate that dependence; or rigorously refute the power-law form itself.

  2. Determine, by systematic computation at Reynolds numbers beyond the published range and/or by asymptotic analysis, whether these power laws and the observed ReRe-invariance of the normalized one-point PDFs and intermittency measures persist in the limit Re→∞Re\to\infty and under qualitatively different forcing spectra.

An answer is complete when the exponents λX\lambda_X of the surveyed diagnostics, their dependence on the forcing properties, their asymptotic status as Re→∞Re\to\infty, and the persistence of the ReRe-invariance of the normalized one-point PDFs and intermittency measures in that limit and under qualitatively different forcing spectra are explained rather than merely tabulated — predicted by a mechanism-based argument rather than fitted to the data — or when the power-law form is rigorously refuted. Both branches apply to the enumerated diagnostics collectively: a derivation or refutation that covers only one or a few of them settles those diagnostics but does not by itself complete the task. Coherent-vortex-aware rigorous bounds on the dissipation rates (improving on the homogeneity/isotropy-based Alexakis-Doering-type bounds) and predictive closures for large-scale models of thin geophysical flows are adjacent directions motivated by the same physics, but they are not part of the completion criterion for this problem.

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