Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades
Bibliographic record
Abstract
Abstract We study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids ( ${\it\alpha}$ -turbulence models) simulated at resolution $8192^{2}$ . We consider ${\it\alpha}=1$ (surface quasigeostrophic flow), ${\it\alpha}=2$ (2D Euler flow) and ${\it\alpha}=3$ . The forcing scale is well resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both ${\it\alpha}=1$ and ${\it\alpha}=2$ . The active scalar field for ${\it\alpha}=3$ contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction $-(7-{\it\alpha})/3$ in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One- and two-point p.d.f.s, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for ${\it\alpha}=1$ and ${\it\alpha}=2$ , while the ${\it\alpha}=3$ inverse cascade is much closer to Gaussian and non-intermittent. For ${\it\alpha}=3$ the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling $\mathscr{E}(k)\propto k^{-2}~({\it\alpha}=1)$ and $\mathscr{E}(k)\propto k^{-5/3}~({\it\alpha}=2)$ in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation ( ${\it\alpha}=1$ and ${\it\alpha}=2$ ) and non-realizability ( ${\it\alpha}=3$ ) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and non-intermittent statistics for ${\it\alpha}=1$ and ${\it\alpha}=2$ .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 teacher head, 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".