Linear instability of a perturbed Lamb–Oseen vortex
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Bibliographic record
Abstract
Abstract This paper presents an investigation of the stability of a vortex with azimuthal velocity profile <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>V</mml:mi> <mml:mo>ˉ</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ε</mml:mi> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:math> . When ε = 0, the Lamb–Oseen vortex model is recovered. Although the Lamb–Oseen vortex supports propagating waves known as Kelvin waves, the flow is stable according to Rayleigh’s circulation criterion. In this paper, on the other hand, the modified vortex profile admits linearly unstable disturbances for ε > 0 and we investigate their characteristics. These may be either axisymmetric or non-axisymmetric, but we find that the axisymmetric perturbations have the largest growth rates. When their growth rates are small, it becomes very difficult to solve the linear equation governing the axisymmetric perturbations because the eigenfunctions have a rapid exponential growth as one moves outward radially from the vortex center. To deal with such cases, a modified Riccati transformation was employed and found to be effective in solving the associated eigenvalue problem.
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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 | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 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 it