Bibliographic record
Abstract
An analysis of the toroidal modes of a rotating fluid, by means of the differential equations of motion, is not readily tractable. A matrix representation of the equations on a suitable basis, however, simplifies the problem considerably and reveals many of its intricacies. Let Ω be the angular velocity of the star and () be the two integers that specify a spherical harmonic function. One readily finds the followings: 1) Because of the axial symmetry of equations of motion, all modes, including the toroidal ones, are designated by a definite azimuthal number m. 2) The analysis of equations of motion in the lowest order of Ω shows that Coriolis forces turn the neutral toroidal motions of () designation of the non-rotating fluid into a sequence of oscillatory modes with frequencies . This much is common knowledge. One can say more, however. a) Under the Coriolis forces, the eigendisplacement vectors remain purely toroidal and carry the identification (). They remain decoupled from other toroidal or poloidal motions belonging to different 's. b) The eigenfrequencies quoted above are still degenerate, as they carry no reference to a radial wave number. As a result the eigendisplacement vectors, as far as their radial dependencies go, remain indeterminate. 3) The analysis of the equation of motion in the next higher order of Ω reveals that the forces arising from asphericity of the fluid and the square of the Coriolis terms (in some sense) remove the radial degeneracy. The eigenfrequencies now carry three identifications (), say, of which s is a radial eigennumber. The eigendisplacement vectors become well determined. They still remain zero order and purely toroidal motions with a single () designation. 4) Two toroidal modes belonging to and get coupled only at the order. 5) A toroidal and a poloidal mode belonging to and , respectively, get coupled but again at the order. Mass and mass-current multipole moments of the modes that are responsible for the gravitational radiation, and bulk and shear viscosities that tend to damp the modes, are worked out in much detail.
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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 | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".