Singularity formation of vortex sheets in 2D Euler equations using the characteristic mapping method
Bibliographic record
Abstract
The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers in 2D incompressible Euler flows are performed using the characteristic mapping method (CMM). This semi-Lagrangian method evolves the flow map using the gradient-augmented level set method (GALS). The semi-group structure of the flow map allows its decomposition into sub-maps (each over a finite time interval), and thus the precision can be controlled by choosing appropriate remapping times. Composing the flow map yields exponential resolution in linear time, a unique feature of CMM, and thus fine scale flow structures can be resolved in great detail. Here the roll-up process of vortex layers is studied varying the thickness of the layer showing its impact on the growth of palinstrophy and possible blow up of absolute vorticity. The curvature of the vortex sheet shows a singular-like behavior. The self-similar structure of the vortex core is investigated in the vanishing thickness limit. Conclusions on the non-uniqueness of weak solutions of 2D Euler for non-smooth initial data are drawn and the presence of flow singularities is revealed tracking them in the complex plane.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".