Localness of the energy cascade for the sharp spectral filter: optimal rigorous decay rates, pointwise Hölder estimates, the Euclidean-ball cutoff, and the quantitative size of nonlocal triad contributions

Aluie and Eyink (Phys. Fluids 21, 115108, 2009) proved that for turbulent velocity fields with empirically observed Hölder/Besov scaling and a densely populated Kolmogorov spectrum, the Galilean-invariant subgrid-scale (SGS) energy flux of the sharp spectral (Fourier) filter is scale-local in both the ultraviolet and the infrared, and that wavenumber-band energy transfers are dominated by local triads when the bands have constant width on a logarithmic scale. Their rigorous bounds, however, have conservative exponents and slowly growing prefactors, and they left a precise set of problems open: proving sharp, possibly optimal decay rates for nonlocal contributions (matching the closure-predicted 4/3-type exponents); pointwise-in-space bounds in terms of the local Hölder exponent, available for graded filters but open for spectral projectors; extension of the rigorous arguments from polygonal wavenumber norms to the physical Euclidean-ball cutoff, obstructed by Fefferman's ball-multiplier theorem; and the quantitative empirical determination of nonlocal contributions in direct numerical simulation, including finite-Reynolds-number and bottleneck effects, intermittency corrections to the decay exponents, and the asymptotic dominance of local triads at large Reynolds number.

Background

Three-dimensional turbulent flows at high Reynolds number transfer kinetic energy from the large scales where it is injected to the small scales where it is dissipated — the Richardson–Kolmogorov energy cascade. In a Fourier representation on a periodic box, the nonlinear term couples wavenumbers satisfying the triad condition $\mathbf{k}+\mathbf{q}=\mathbf{p}$, so the cascade is carried by triadic interactions among wavenumber modes. The classical Kolmogorov (1941) picture assumes the cascade is scale-local: energy passes predominantly between wavenumbers of comparable magnitude. A sharp spectral filter at cutoff wavenumber $K$ keeps only Fourier modes with wavenumber below $K$; it is the scale decomposition implicitly used by most pseudospectral direct numerical simulations (DNS) and by spectral large-eddy simulation (LES). A graded filter instead uses a smooth transition in Fourier space, i.e. a rapidly decaying convolution kernel in physical space.

The natural pointwise measure of inter-scale energy transfer is the subgrid-scale (SGS) energy flux. For the sharp spectral filter, Aluie and Eyink write it as a trilinear form

$$\Pi_K(\mathbf{u},\mathbf{v},\mathbf{w}) \equiv -\partial_i u_j^{<K}\Big[(v_i w_j)^{<K}-v_i^{<K} w_j^{<K}\Big],$$

where the superscript $<K$ denotes projection onto wavenumbers below $K$; with $\mathbf{u}=\mathbf{v}=\mathbf{w}$ it is the transfer rate from $|\mathbf{k}|<K$ to $|\mathbf{k}|>K$ at each space point. Its domain average $\langle\Pi_K\rangle$ is the mean energy cascade rate $\varepsilon$, equal to the mean energy dissipation rate in a statistically stationary inertial range. This subtracted form is Galilean invariant: it removes the sweeping of small scales by the large-scale velocity, which otherwise contaminates the historically used unsubtracted flux and makes it nonlocal pointwise. The flux is called ultraviolet (UV) scale-local if replacing any argument by its band $\mathbf{u}^{[P]}$ above $K$ (with $P\gg K$) gives an asymptotically negligible contribution, and infrared (IR) scale-local if the same holds for bands $\mathbf{u}^{[Q]}$ with $Q\ll K$.

Whether these properties hold was contested for two decades. Early DNS shell-to-shell studies (Domaradzki and Rogallo 1990; Ohkitani and Kida 1992; Zhou 1993) and high-Reynolds-number simulations by Alexakis, Mininni and Pouquet (2005) and Mininni, Alexakis and Pouquet (2008) found the largest individual transfers in triads with one leg at the energy-containing scales, reporting a persistent 10–20% plateau in the nonlocal fraction of the flux for $K$ beyond the Taylor-scale wavenumber (the inverse of the Taylor microscale, the standard intermediate length scale of turbulence) — a picture of 'local transfer by nonlocal triads'. On the theory side, Kraichnan's Lagrangian-history closure theories — the abridged Lagrangian-history direct-interaction approximation (ALHDIA) and the test-field model (TFM), both from 1971 — predict that nonlocal contributions to the mean spectral flux decay as 4/3-type power laws in the scale separation, and L'vov and Falkovich (1992) derived the IR prediction $(Q/K)^{2-\zeta_2}$ from fusion-rule arguments (universal scaling relations among velocity correlation functions at widely separated scales), where $\zeta_p$ are the intermittency exponents of velocity structure functions ($\langle|\delta u(r)|^p\rangle\sim r^{\zeta_p}$; Kolmogorov values $\zeta_1=1/3$, $\zeta_2=2/3$; intermittency makes $\zeta_p$ strictly concave in $p$). Eyink (1994) moreover constructed a Fourier–Weierstrass velocity field with lacunary Fourier support (modes concentrated on a geometric sequence of wavenumbers rather than densely filling Fourier space) for which the sharp-filter spectral flux is dominated by nonlocal sweeping triads, showing that some scaling hypothesis on the velocity field is essential for any locality statement.

For graded filters, Eyink (2005) and Eyink and Aluie (2009, Part I of this series) proved rigorous locality results, including bounds on the pointwise (in space) SGS flux in terms of the local Hölder exponent $h(\mathbf{x})$ of the velocity field. The sharp spectral filter has no decaying physical-space kernel, and the classical tool — boundedness of the Fourier multiplier for the Euclidean ball — fails by Fefferman's (1971) theorem that the ball multiplier is unbounded on $L^p$ for $p\neq 2$; this is why the Part II rigorous arguments are carried out with polygonal ($\ell_1$/$\ell_\infty$) wavenumber norms instead.

Aluie and Eyink (2009, Part II — the source of this problem) closed the main question for the sharp filter. Assuming only empirically observed properties of turbulent fields — Hölder/Besov regularity $u\in B^{\sigma_p}_{p,\infty}$ with $\sigma_p=\zeta_p/p$ (velocity increments scaling as $\langle|\delta u(r)|^p\rangle^{1/p}\sim r^{\sigma_p}$, where $B^{\sigma_p}_{p,\infty}$ is the Besov function-space scale for fields with such power-law increments) together with the strong spectral condition $E(k)\sim C\,u_{\rm rms}^2 L\,(kL)^{-(1+\zeta_2)}$ for unit-width wavenumber bins, a densely populated Kolmogorov spectrum that excludes lacunary counterexamples — they proved rigorous bounds establishing UV and IR scale-locality of the SGS spectral flux, showed that inter-band transfers are dominated by local triads when the bands have constant width on a logarithmic scale, and supported the results with $512^3$ DNS, including a hyperviscous run (dissipation acting at a higher power of the wavenumber than the physical Laplacian, which widens the effective inertial range) in which the nonlocal fraction decays like $K^{-1.32}$ (close to the closure-predicted $4/3$ rate) before asymptoting at 2–3%, an effect attributed to the spectral bottleneck (the pile-up of energy at the end of the inertial range). Their rigorous UV bound, however, decays only like $(K/P)^{2\sigma_3}$ (about $(K/P)^{2/3}$) and the IR bound like $(Q/K)^{1/2-\sigma_4}$ (about $(Q/K)^{1/6}$), both with slowly growing $K$-dependent prefactors — far from the expected 4/3-type rates — and they could not prove pointwise estimates in terms of $h(\mathbf{x})$.

Problem Statement

The common objective is to complete the quantitative theory of scale-locality for the sharp spectral filter: determine, rigorously and empirically, how the nonlocal triad contributions to the Galilean-invariant SGS spectral energy flux $\Pi_K$ decay with wavenumber separation, for incompressible turbulent velocity fields satisfying the empirical hypotheses of Aluie and Eyink (2009) (Hölder/Besov increment scaling $\langle|\delta u(r)|^p\rangle^{1/p}\sim r^{\sigma_p}$ with $\sigma_p=\zeta_p/p$, and the strong spectral condition $E(k)\sim C\,u_{\rm rms}^2 L\,(kL)^{-(1+\zeta_2)}$ for unit-width wavenumber bins). The open items left by that work are the following. These items stand independently: a thorough resolution of any single one of them is a complete answer to this task.

(1) Optimal rigorous decay rates. Prove sharper — ideally optimal — rigorous bounds on the UV contributions $\Pi_K(\mathbf{u},\mathbf{u},\mathbf{u}^{[P]})$ and $\Pi_K(\mathbf{u},\mathbf{u}^{[P]},\mathbf{u})$ (from $P\gg K$) and on the IR contributions $\Pi_K(\mathbf{u}^{[Q]},\mathbf{u},\mathbf{u})$, $\Pi_K(\mathbf{u},\mathbf{u}^{[Q]},\mathbf{u})$, $\Pi_K(\mathbf{u},\mathbf{u},\mathbf{u}^{[Q]})$ (from $Q\ll K$), removing or tightly controlling the slowly growing $K$-dependent prefactors. The targets are the physically expected rates: $(K/P)^{4/3}$-type decay for UV partial fluxes, $(Q/K)^{1+\zeta_1}$ (equal to $(Q/K)^{4/3}$ for Kolmogorov values) for the mean IR flux from decorrelation arguments, and $(Q/K)^{2/3}$ for the space-average of the absolute flux (the graded-filter rate) — or counterexamples, inside the class of fields satisfying the stated empirical hypotheses, showing which of these rates cannot be attained.

(2) Pointwise estimates. Establish (or disprove) pointwise-in-space bounds on the SGS spectral flux and its nonlocal parts in terms of the local Hölder exponent $h(\mathbf{x})$ of the velocity field, analogous to the bounds available for graded filters.

(3) Euclidean-ball cutoff. Extend the rigorous locality results from polygonal ($\ell_1$/$\ell_\infty$) wavenumber norms to the standard Euclidean-ball spectral cutoff $|\mathbf{k}|\le K$, overcoming or circumventing the Fefferman (1971) ball-multiplier obstruction, or prove that the extension fails in this generality.

(4) Quantitative empirical determination. Determine, by theory-guided high-Reynolds-number DNS (or experiment), the actual decay of nonlocal contributions to the spectral SGS energy flux with scale separation: resolve whether the residual nonlocal-fraction plateaus (10–20% beyond the Taylor-scale wavenumber) reported in earlier DNS are finite-Reynolds-number/bottleneck artifacts; discriminate among the nearly degenerate competing predictions $(Q/K)^{4/3}$, $(Q/K)^{1+\zeta_1}$ and $(Q/K)^{2-\zeta_2}$ for the mean flux and the $2/3$ rate for the space-averaged absolute flux, quantifying the intermittency corrections to the exponents and prefactors; and establish over what inertial-range lengths the asymptotic dominance of local-triad contributions becomes manifest. Item (4) is optional: no DNS or experimental data are provided with this task, and it is not required for a complete answer.

References

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