Large-scale multifractality and lack of self-similar decay for Burgers and three-dimensional Navier–Stokes turbulence
Bibliographic record
Abstract
We study decaying turbulence in the one-dimensional (1-D) Burgers equation (Burgulence) and 3-D Navier–Stokes (NS) turbulence. We first investigate the decay in time $t$ of the energy $E(t)$ in Burgulence, for a fractional Brownian initial potential, with Hurst exponent $H$ , and demonstrate rigorously a self-similar time decay of $E(t)$ , previously determined heuristically. This is a consequence of the non-trivial boundedness of the energy for any positive time. We define a spatially forgetful oblivious fractional Brownian motion (OFBM), with Hurst exponent $H$ , and prove that Burgulence, with an OFBM as initial potential $\varphi _0(x)$ , is not only intermittent, but it also displays a, hitherto unanticipated, large-scale bifractality or multifractality; the latter occurs if we combine OFBMs, with a distribution of $H\hbox{-}$ values. This is the first rigorous proof of genuine multifractality for turbulence in a nonlinear hydrodynamical partial differential equation. We then present direct numerical simulations (DNSs) of freely decaying turbulence, capturing some aspects of this multifractality. For Burgulence, we investigate such decay for two cases: (a) $\varphi _0(x)$ a multifractal random walk that crosses over to a fractional Brownian motion beyond a cross-over scale $\mathcal{L}$ , tuned to go from small- to large-scale multifractality; (b) initial energy spectra $E_0(k)$ , with wavenumber $k$ , having one or more power-law regions, which lead, respectively, to self-similar and non-self-similar energy decay. Our analogous DNSs of the 3-D NS equations also uncover self-similar and non-self-similar energy decay. Challenges confronting the detection of genuine large-scale multifractality, in numerical and experimental studies of NS and Magnetohydrodynamics turbulence, are highlighted.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".