NOTE-969: Eyink and Aluie — Localness of energy cascade I

Contribution

Two things. First, an exact decomposition of the kinetic energy (or any quadratic integral) into band-pass contributions at a geometric sequence of scales ℓ_n = ρ^(−n)L, using smooth filters. It generalizes the Germano identity, has no off-diagonal terms, and gives a band budget in which the fluxes telescope. Second, the observation that the fluxes in this budget are exactly the low-pass fluxes Π_ℓ that Eyink (2005) proved scale-local, so his bounds carry over to band-to-band transfer. The bounds themselves are restated here with heuristic sketches, not proved. A third part argues, with simulation support, why the rigorous exponent 2/3 is smaller than Kraichnan’s 4/3.

Key insight

The Galilean-invariant energy flux across a scale ℓ, Π_ℓ = −∇u_ℓ : τ_ℓ, depends on the velocity only through increments δu(r) over separations up to about ℓ. If increments are scale-local, as they are when they scale like r^σ with 0 < σ < 1, then so is the flux. The large scales enter only through their gradients, which are smaller than the gradients at ℓ by (ℓ/Δ)^(1−σ). The small scales enter only through their amplitudes, which are smaller by (δ/ℓ)^σ. Locality is a consequence of roughness of the right degree. A smooth field (σ ≥ 1) would lose infrared locality, and a field that does not decay with scale (σ ≤ 0) would lose ultraviolet locality.

Assumptions

• Filters. G is real, smooth, rapidly decaying, with ∫G = 1, ∫rG = 0 and ∫|r|²G = O(1). Its dilation is G_ℓ(r) = ℓ^(−d)G(r/ℓ). G ≥ 0 is needed to read u_ℓ as a local average and for the band energy densities to be non-negative pointwise. The simple form (11) of the band decomposition needs an S-type kernel, Ĝ(k) = 1 for |k| < 1 and 0 for |k| > ρ, which Appendix 1 shows can be C^∞ in k (so G decays faster than any power) but cannot be non-negative.
• Logarithmic bands. Scales ℓ_n = ρ^(−n)L with ρ > 1. The authors stress that the results are for graded filters on logarithmic bands; sharp spectral filters and linear bands are left to Part II.
• Incompressible Navier–Stokes, with ν → 0 and finite total energy, so viscous terms drop out of every band except the last.
• The scaling hypothesis (20): ‖δu(r)‖_p ∼ u_rms A_p (r/L)^(σ_p), σ_p = ζ_p/p, with 0 < σ_p < 1 and σ₃ = 1/3, “indicated” by experiments and simulations (Besov regularity). For the pointwise version: local Hölder continuity |δu(x; r)| = O(r^h) with 0 < h < 1.
• Non-zero mean flux: ⟨Π_ℓ⟩ of order ε across the range, needed to read the bounds as fractions. Called “completely realistic” in a constant-flux inertial range.
• For §III.B only: homogeneity or isotropy, so ⟨∇u⟩ = 0; ergodicity to trade space averages for time averages; a correlation coefficient between strain at Δ and stress at ℓ of order S(Δ)/S(ℓ); and a multiplicative-cascade or fusion-rule ansatz for multiscale structure functions.

Key results

• Subgrid budget (4), (5), (7). Filtering Navier–Stokes gives ∂_t u_ℓ + (u_ℓ·∇)u_ℓ = −∇p_ℓ − ∇·τ_ℓ + νΔu_ℓ with τ_ℓ = (uu)_ℓ − u_ℓu_ℓ. The flux Π_ℓ = −∇u_ℓ : τ_ℓ is a sink in the large-scale energy budget and a source in the small-scale one. All other non-viscous terms are divergences, i.e. transport in space.
• Band energy (8). Between scales ℓ < ℓ̃, k_[ℓ,ℓ̃] = ½τ̃(ū, ū), whose space integral is (2π)^(−d) ∫ d^dk ½|û(k)|² |Ĝ(kℓ)|² [1 − |Ĝ(kℓ̃)|²]. It is non-negative pointwise when G ≥ 0. Its budget (9) has a flux into scales below ℓ̃ and a flux out of scales below ℓ.
• Multiscale Germano identity (Appendix 2, (A-1)). For any f, g and N ≥ 0, with f^(N,n) = G_n ∗ G_(n+1) ∗ … ∗ G_N ∗ f, ∫fg = ∫f^(N,0)g^(N,0) + Σ_(n=1..N) ∫τ_(n−1)(f^(N,n), g^(N,n)) + ∫τ_N(f, g). Re-derived here: since f^(N,n−1) = G_(n−1) ∗ f^(N,n) and ∫ of a filtered quantity equals ∫ of the quantity, each band term is ∫f^(N,n)g^(N,n) − ∫f^(N,n−1)g^(N,n−1), and the sum telescopes. For an S-type kernel, f^(N,n) = f^n (the finer filters act as the identity on the coarser one’s support), which gives (11): ½∫|u|² = ½∫|u_0|² + Σ_(n=1..N) ½∫τ_(n−1)(u_n; u_n) + ½∫τ_N(u; u).
• Band budget. (d/dt)∫½|u_0|² = −∫Π_0; (d/dt)∫½τ_(n−1)(u_n; u_n) = ∫Π_(n−1) − ∫Π_n for n = 1, …, N; (d/dt)∫½τ_N(u; u) = ∫Π_N − ν∫τ_N(u_(i,j), u_(i,j)). The transfer into band n is T_n = Π_(n−1) − Π_n, so it is local if Π_(n−1) and Π_n are.
• Locality, defined (§III.A). Π_ℓ is infrared local if replacing u in any subset of its three arguments by u_Δ gives a result ≪ Π_ℓ when Δ ≫ ℓ, and ultraviolet local if replacing by u′_δ = u − u_δ gives a result ≪ Π_ℓ when δ ≪ ℓ. Scale-local means both.
• Increment estimates (14)–(22). ∇u_ℓ ∼ δu(ℓ)/ℓ, τ_ℓ ∼ δu(ℓ)², u′_ℓ ∼ δu(ℓ). Pointwise: δu_Δ(x; r) = O(rΔ^(h−1)), smaller than the full increment by (r/Δ)^(1−h); δu′_δ(x; r) = O(δ^h), smaller by (δ/r)^h. In L^p mean: ‖δu_Δ(r)‖_p = O(rΔ^(σ_p−1)) and ‖δu′_δ(r)‖_p = O(δ^(σ_p)), so scales above Δ contribute a fraction O((ℓ/Δ)^(1−σ_p)) and scales below δ a fraction O((δ/ℓ)^(σ_p)). Holds when: 0 < h < 1 (pointwise) or 0 < σ_p < 1 (mean).
• Flux bounds (23)–(24). ⟨Π_ℓ(u_Δ, u, u)⟩ = O(ε(ℓ/Δ)^(1−σ₃)) and ⟨Π_ℓ(u, u′_δ, u′_δ)⟩ = O(ε(δ/ℓ)^(2σ₃)), by Hölder’s inequality |⟨fgh⟩| ≤ ‖f‖₃‖g‖₃‖h‖₃. With σ₃ = 1/3 both exponents are exactly 2/3. Holds when: the scaling hypothesis holds as an upper bound; read as a fraction of the flux, when also ⟨Π_ℓ⟩ ≈ ε ≠ 0. Re-derived step by step in THEORY-754, where the prefactor for general σ₃ is (u_rms³/L)(ℓ/L)^(3σ₃−1), which is ε only at σ₃ = 1/3.
• Decorrelation estimates (§III.B, heuristic). With a correlation factor S(Δ)/S(ℓ) between large-scale strain and small-scale stress, ⟨Π_ℓ(u_Δ, u, u)⟩ ∼ (ℓ/Δ²)⟨δu²(Δ)δu(ℓ)⟩ ∼ O(ε(ℓ/Δ)^(1+ζ₁)) with ζ₃ = 1, and the same exponent for the ultraviolet term. Under K41, ζ₁ = 1/3, which is Kraichnan’s 4/3. Pointwise, isotropy of u′_δ inside the ball of radius ℓ would remove the leading part of the small-scale stress (the trace part does nothing against a traceless gradient), giving Π_ℓ(u, u′_δ, u′_δ) = O((δ/ℓ)
• Simulation (§III.C). 512³ pseudospectral DNS, Gaussian filters G_ℓ(r) = (π/ℓ²)^(3/2) exp(−π²r²/ℓ²), ℓ_K = 2π/K. Infrared, K = 100: fitted slopes 0.7 for the mean absolute flux and 1.1 for the mean signed flux, against 2/3 and 4/3. Ultraviolet, K = 4: 0.9 absolute and 1.9 signed. The gap between absolute and signed slopes is the cancellation the authors predict.
• Kernel construction (Appendix 1). A C^∞, compactly supported Ĝ with G and Ĝ both positive (square a compactly supported bump’s transform), and a C^∞ S-type Ĝ = 1 − (1/H̄)∫₀^k H, H a bump on (1, ρ). Every moment ∫|r|^(2n)G of an S-type kernel vanishes, so it takes negative values.

Claims

id	claim	strength	support
C1	Kinetic energy splits exactly into non-negative band contributions on a geometric sequence of scales, with no off-diagonal terms, and each band’s energy changes by Π_(n−1) − Π_n	strong for the identity; qualified for “non-negative”	(A-1) and (11), proved by telescoping; re-derived and checked numerically here. Pointwise non-negativity fails for the S-type kernels (11) needs (see Limitations)
C2	The mean energy flux across ℓ receives at most O((ℓ/Δ)^(1−σ₃)) from scales above Δ and O((δ/ℓ)^(2σ₃)) from scales below δ	strong as a conditional inequality	sketched in §III.A, proved in Eyink (2005); re-derived in THEORY-754 from the scaling hypothesis
C3	Turbulent energy transfer is therefore scale-local, “assuming only the scaling properties expected in a turbulent inertial-range”	moderate	C2 plus two hypotheses not proved for Navier–Stokes: 0 < σ_p < 1 and a non-zero mean flux
C4	Band-to-band transfers are local in both scale and space	moderate in scale (via C2); weak in space	space-locality rests on the pointwise Hölder sketch, which the authors say gives no fractional statement
C5	Kraichnan’s faster 4/3 decay is due to cancellation of signed non-local contributions under space averaging	moderate	heuristic decorrelation and fusion-rule argument (“plausible but non-rigorous”); one 512³ DNS whose signed slopes are 1.1 and 1.9
C6	Locality holds pointwise for individual realizations, and without it small-scale universality would have “scant basis”	weak as stated	the pointwise bounds are O(r^h) estimates at points of Hölder exponent h; the paper itself says they give no fractional statement
C7	The SGS flux defined with the subtracted stress is the unique pointwise Galilean-invariant definition	weak	asserted, not proved
C8	Because the decay is a power of the scale ratio, i.e. only exponential in band number, the cascade is “leaky” and very long inertial ranges may be needed for universality	moderate	follows from the size of the bounds; the inference to universality is the authors’

Method

1. Filter Navier–Stokes with a smooth kernel, and write the large- and small-scale energy budgets in Germano’s moment notation.
2. Band-pass by differences of filters, and prove the multiscale Germano identity by telescoping. With an S-type kernel this gives a diagonal band decomposition and a band budget with telescoping fluxes.
3. Express ∇u_ℓ, τ_ℓ and u′_ℓ through increments; bound large-scale increments by Taylor expansion of the smooth u_Δ, and small-scale ones by the size of u′_δ; combine by Hölder’s inequality (from Eyink 2005).
4. Estimate signed contributions with a decorrelation factor and a multiscale-structure-function ansatz; compare with absolute contributions in a DNS with Gaussian filters.

Concepts

• Smooth coarse-graining: convolution with a smooth, rapidly decaying, normalized kernel, which resolves the field in space and scale at once.
• S-type kernel: Ĝ(k) = 1 for |k| < 1 and 0 for |k| > ρ, smooth in between. Successive dilations by ρ then nest exactly. Never non-negative in physical space.
• Energy flux Π_ℓ: −∇u_ℓ : τ_ℓ(u, u), the subgrid-scale energy flux, a triple product Π_ℓ(u, v, w) = −∇u_ℓ : τ_ℓ(v, w) in which each argument can be filtered separately.
• Infrared / ultraviolet locality: the flux is insensitive to replacing any of its arguments by the part above Δ ≫ ℓ, or by the part below δ ≪ ℓ, respectively.
• Scale-local: both. A statement about which scales set the transfer across ℓ, not about where in space it happens.
• Ultralocal transfer: the authors’ name for “local transfer by non-local triads”, transfer between wavenumbers differing by a fixed k₀ ≈ 1/L, which they distinguish from locality at fixed ratio.
• Leakiness / diffuseness: the slow, power-law decay of non-local contributions with scale ratio.

Connections

It extends Eyink (2005), LIT-1252, from low-pass to band-pass decompositions, and generalizes Germano (1992)‘s identity. It answers Cheskidov, Constantin, Friedlander and Shvydkoy (2008), whose band decomposition keeps off-diagonal terms, and the DNS studies of Alexakis, Mininni and Pouquet that report non-local transfer, against Domaradzki and Carati’s, which report locality. The decorrelation argument follows Kraichnan (1966, 1971) and Tennekes and Lumley (1972). Part II (Aluie and Eyink, sharp spectral filters and linear bands) is not filed.

Bearing on the record

• THEORY-754 is filed from this reading, Active: the conditional mean bounds C2, re-derived step by step. It holds them as an inequality from a hypothesis, not as a fact about turbulence.
• THEORY-661 and CLAIM-157: scale locality is a different organizing principle from near-decomposability. The paper does not mention near-decomposability; the comparison is the record’s. Near-decomposability is a block pattern with a gap: strong inside, weak (order ε) across. Scale locality is a banded pattern: coupling strongest between neighbouring scales and decaying as a power of the scale ratio, with no gap. In the band budget, the whole throughput ε crosses every inertial scale. So no contiguous group of bands is weakly coupled to the rest, and CLAIM-157’s test would find no block among scales. What the paper establishes is that the coupling decays with distance in scale; what it does not establish is any weak coupling. This supports the premise QUESTION-219 and CLAIM-157’s note of 2026-10-19 take from A3, in that form. Those entries say “unread” and can now cite this reading. They should also say that the proofs are Eyink (2005)‘s and that locality is conditional on the scaling hypothesis.
• QUESTION-219. The paper meets A5’s precondition for any flux test, “a balance law for a specified transferred quantity”: energy, with an exact band budget and Galilean-invariant fluxes. It does not touch the question itself. Π_ℓ is a field at each scale of one velocity field. It is not a unit, has no spatial boundary, and nothing in the paper concerns persistence under replacement of constituents, self-restoration or dependence on internal organization (A5’s tests 1–4). The bands are chosen by the analyst (G, ρ, L), not found. “Flux as a form of unity” is A3’s and A5’s reading, and nothing in this paper argues for it.
• A3’s description in the LIT (“a proof that coupling across a hierarchy of scales is local”) overstates this paper’s share. The proof is Eyink (2005)‘s and is conditional. What this paper adds is the band decomposition, the transfer to band-pass fluxes, and the decorrelation account. Eyink (2005) should be read before anything rests on the pointwise statements.
• CLAIM-331 (higher levels slower). Not addressed. The paper says nothing about time scales across bands beyond strain rates S(ℓ) ∼ δu(ℓ)/ℓ in the heuristic section.
• THEORY-257 and QUESTION-218. The bands are not dynamically closed: each band’s energy is driven by the flux from above. The paper offers no closed coarse variable; large-eddy closure is a separate literature.
• Information transfer across scales. Nothing here; A3’s caution against identifying energy and information transfer (LIT-1243) stands.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• The rigorous bounds are not proved here. §III.A gives order-of- magnitude sketches (“For rigorous details, see Eyink (2005)”). The abstract’s “we derive rigorous upper bounds” and the introduction’s “by deriving rigorous upper bounds” overstate what this paper does. Its contribution to the bounds is noticing that its band fluxes are Eyink’s Π_ℓ.
• Conditional, not established for Navier–Stokes. “Assuming only the scaling properties expected” is two assumptions: 0 < σ_p < 1, and a non-zero mean flux. Neither is proved for Navier–Stokes solutions. The bounds are rigorous inequalities from them.
• Positivity and S-type kernels. §II.B says the decomposition (11) “contains only non-negative, diagonal terms”. The integrated band terms are non-negative when |Ĝ| decreases with |k|, as the Appendix 1 construction does. But (11) needs an S-type kernel, which Appendix 1 shows cannot be non-negative, so the pointwise positivity stated for G ≥ 0 does not apply to it. Checked here on a one-dimensional random field: with the Appendix 1 S-type kernel (ρ = 2), the band energy density τ₀(u₁, u₁) was negative at 38% of points, while the integrated terms were positive and the identity held to 2 × 10⁻¹⁵. With a Gaussian kernel and the general form (A-1) the density stayed positive.
• Normalization by ε. (23) and (24) are written with ε. That is right only when σ₃ = 1/3 exactly; the paper says it assumed this. “Very close to 2/3” is exactly 2/3 under that assumption.
• Pointwise locality. The conclusions’ “Fortunately, this is not the case” (that non-local triads are not sizable pointwise) rests on Hölder estimates which, by the paper’s own §III.A, “cannot make precise statements about the fractional contributions”. The pointwise claim C6 is stronger than what is shown.
• Triadic decomposition. “It follows that transfer for non-local modal triads is negligibly small” is asserted without defining triadic contributions for smooth filters.
• The simulation supports less than “confirmed”. One 512³ run, one filter (Gaussian, not S-type), one K each way. The ultraviolet comparison measures ∇u_ℓ,ℓ : u′u′, the form derived for S-type kernels, with a Gaussian. The signed slopes 1.1 and 1.9 bracket 4/3. The absolute-flux slopes (0.7, 0.9) do respect the 2/3 bounds.
• Re-checks. On a synthetic one-dimensional Gaussian field with σ₂ = 1/3 (65 536 points), the L² share of u_Δ in an increment at separation r fell as (r/Δ)^0.62 for Δ/r from 2 to 16, and that of u′_δ as (δ/r)^0.36 for r/δ from 2 to 32. The predictions are 2/3 and 1/3. This checks the increment estimates for one field, not anything about Navier–Stokes.

Open questions

• Do Navier–Stokes solutions satisfy 0 < σ_p < 1 with a non-zero mean flux? That is the open regularity question behind Onsager’s 1/3, and only a proof about the equations would close it.
• Can the 4/3 exponent be made rigorous, by bounding the correlation of strain at Δ with stress at ℓ?
• Part II’s question: does locality survive sharp spectral filters and linear bands, where “local transfer by non-local triads” is reported?
• For the record: is there a coarse-graining of the scale chain itself, a closed description of a range of bands, and would it be a block? The paper gives the budget such a question needs and does not ask it.