Bounds for the number of degrees of freedom of incompressible magnetohydrodynamic turbulence in two and three dimensions
Bibliographic record
Abstract
We study incompressible magnetohydrodynamic turbulence in both two and three dimensions, with an emphasis on the number of degrees of freedom $N$. This number is estimated in terms of the magnetic Prandtl number $\mathrm{Pr}$, kinetic Reynolds number $\mathrm{Re}$, and magnetic Reynolds number $\mathrm{Rm}$. Here $\mathrm{Re}$ and $\mathrm{Rm}$ are dynamic in nature, defined in terms of the kinetic and magnetic energy dissipation rates (or averages of the velocity and magnetic field gradients), viscosity and magnetic diffusivity, and the system size. It is found that for the two-dimensional case, $N$ satisfies $N\ensuremath{\leqslant}\mathrm{Pr}\phantom{\rule{0.16em}{0ex}}{\mathrm{Re}}^{3/2}+{\mathrm{Rm}}^{3/2}$ for $\mathrm{Pr}>1$ and $N\ensuremath{\leqslant}{\mathrm{Re}}^{3/2}+{\mathrm{Pr}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{Rm}}^{3/2}$ for $\mathrm{Pr}\ensuremath{\leqslant}1$. In three dimensions, on the other hand, $N$ satisfies $N\ensuremath{\leqslant}{(\mathrm{Pr}\phantom{\rule{0.16em}{0ex}}{\mathrm{Re}}^{3/2}+{\mathrm{Rm}}^{3/2})}^{3/2}$ for $\mathrm{Pr}>1$ and $N\ensuremath{\leqslant}{({\mathrm{Re}}^{3/2}+{\mathrm{Pr}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{Rm}}^{3/2})}^{3/2}$ for $\mathrm{Pr}\ensuremath{\leqslant}1$. In the limit $\mathrm{Pr}\ensuremath{\rightarrow}0$, ${\mathrm{Re}}^{3/2}$ dominates ${\mathrm{Pr}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{Rm}}^{3/2}$, and the present estimate for $N$ appropriately reduces to ${\mathrm{Re}}^{9/4}$ as in the case of usual Navier-Stokes turbulence. For $\mathrm{Pr}\ensuremath{\approx}1$, our results imply the classical spectral scaling of the energy inertial range and dissipation wave number (in the form of upper bounds). These bounds are consistent with the existing predictions in the literature for turbulence with or without Alfv\'en wave effects. We discuss the possibility of solution regularity, with an emphasis on the two-dimensional case in the absence of either one or both of the dissipation terms.
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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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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".