Chuong V. Tran

Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence

We study two-dimensional (2D) turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form νμ(−Δ)μ. By “monoscale-like” we mean that the forcing is applied over a finite range of wavenumbers kmin≤k≤kmax, and that the ratio of enstrophy injection η≥0 to energy injection e≥0 is bounded by kmin2e≤η≤kmax2e. Such a forcing is frequently considered in theoretical and numerical studies of 2D turbulence. It is shown that for μ≥0 the asymptotic behaviour satisfies ∥u∥12≤kmax2∥u∥2, where ∥u∥2 and ∥u∥12 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for Navier–Stokes turbulence (μ=1), the time-mean enstrophy dissipation rate is bounded from above by 2ν1kmax2. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced 2D Navier–Stokes turbulence (μ=1) when it is forced in this manner. Inclusion of Ekman drag (μ=0) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified −3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity (μ<0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing.

On the dual cascade in two-dimensional turbulence

We study the dual cascade scenario for two-dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range [kmin, kmax] (with kmin > 0) such that the respective energy and enstrophy injection rates ǫ and η satisfy k 2 ǫ ≤ η ≤ k 2ǫ. The classical Kraichnan–Leith–Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scaleselectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For two-dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the acrosschannel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy β > 5 and α + β ≥ 8, where −α (−β) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analyzed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.

Revisiting Batchelor's theory of two-dimensional turbulence

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor ( Phys. Fluids , vol. 12, 1969, p. 233) is false. That theory, which predicts a χ 2/3 k −1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re , assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re . This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelors theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re , and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω 2 〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelors theory: just replace Batchelors enstrophy spectrum χ 2/3 k −1 with 〈ω 2 〉 k −1 (ln Re ) −1 ).

Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit

Batchelor ( Phys. Fluids , vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re . Here, by assuming power-law spectra, including the one predicted by Batchelors theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as

Late time evolution of unforced inviscid two-dimensional turbulence

\mbox{\it Re} \to \infty

Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows

, the dissipation approaches zero no slower than

Impeded inverse energy transfer in the Charney-Hasegawa-Mima model of quasi-geostrophic flows

(\ln\mbox{\it Re})^{-1/2}

Large-scale dynamics in two-dimensional Euler and surface quasigeostrophic flows

. The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation – but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

The number of degrees of freedom of three-dimensional Navier–Stokes turbulence

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 models 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 Batchelors (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 (~ t 1/6 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 k 1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ( k , t ) ∝ t 1/3 k 1 at large scales, together with ℰ( k , t ) ∝ t −2/3 k −5 over the vortex population, and finally ℰ( k , t ) ∝ t −1 k −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.

Enstrophy dissipation in freely evolving two-dimensional turbulence

Traveling-wave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number k as k−2. In the presence of viscosity ν>0, no shocks can develop, and smooth solutions remain so for all times t>0, eventually decaying to zero as t→∞. At peak energy dissipation, say t=t∗, the spectrum of such a smooth solution extends to a finite dissipation wave number kν and falls off more rapidly, presumably exponentially, for k>kν. The number N of Fourier modes within the so-called inertial range is proportional to kν. This represents the number of modes necessary to resolve the dissipation scale and can be thought of as the system’s number of degrees of freedom. The peak energy dissipation rate ϵ remains positive and becomes independent of ν in the inviscid limit. In this study, we carry out an analysis which verifies the dynamical...