A study of singular modes associated with over-reflection and related phenomena
Bibliographic record
Abstract
Abstract This paper describes an investigation of the linear, diffusive critical layer for shear flows whose inviscid neutral modes have an algebraic branch point. As examples of flows exhibiting such singular behaviour, we treat both stratified shear flows and non-axisymmetric modes on vortices. For the stratified case, the coupled vorticity and energy equations are solved numerically. In this way, the density perturbation, which is unbounded in the absence of diffusion, is determined directly. As an example featuring a vortex, we consider helical modes on a modified Lamb–Oseen vortex whose velocity profile is perturbed in such a way that linear instability is possible. Both the axial and azimuthal velocity perturbations in the critical layer are determined. A characteristic shared by all the above problems is that they involve eigenfunctions that are oscillatory in some region. For forced waves in a stably stratified shear flow, we consider a larger range of parameters than previous investigators. We also examine some experiments and find that the Reynolds stress is sensitive to the actual density profile in the region between a wavy wall providing the forcing and the critical layer. For the stratified shear flows without forcing, much smaller Richardson numbers are involved and the modes considered are over-reflecting. For an unbounded $\tanh y$ mixing layer, we show that the Reynolds stress for a neutral mode must jump across the critical layer. With the presence of a horizontal boundary beneath the shear layer, on the other hand, over-reflection can occur, with the Reynolds stress vanishing on either side of the critical layer. In all cases, the variation of the Reynolds stress across the critical layer is determined.
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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 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 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".