Nonlinear Evolution of Hydrodynamical Shear Flows in Two Dimensions
Bibliographic record
Abstract
We examine how perturbed shear flows evolve in two-dimensional, incompressible, inviscid hydrodynamical fluids, with the ultimate goal of understanding the dynamics of accretion disks. To linear order, vorticity waves are swung around by the background shear, and their velocities are amplified transiently before decaying. It has been speculated that sufficiently amplified modes might couple nonlinearly, leading to turbulence. Here we show how nonlinear coupling occurs in two dimensions. This coupling is remarkably simple because it only lasts for a short time interval, when one of the coupled modes is in mid-swing. We focus on the interaction between a swinging and an axisymmetric mode. There is instability provided that k_{y,swing}/k_{x,axi} < omega_{axi}/q, i.e., that the ratio of wavenumbers is less than the ratio of the axisymmetric mode's vorticity to the background vorticity. If this is the case, then when the swinging mode is in mid-swing it couples with the axisymmetric mode to produce a new leading swinging mode that has larger vorticity than itself; this new mode in turn produces an even larger leading mode, etc. Therefore all axisymmetric modes, regardless of how small in amplitude, are unstable to perturbations with sufficiently large azimuthal wavelength. We show that this shear instability occurs whenever the momentum transported by a perturbation has the sign required for it to diminish the background shear; only when this occurs can energy be extracted from the mean flow and hence added to the perturbation. For an accretion disk, this means that the instability transports angular momentum outwards while it operates.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".