A new description of Earth's wobble modes using Clairaut coordinates: 1. Theory
Bibliographic record
Abstract
This paper presents a novel mathematical reformulation of the theory of the free wobble/nutation of an axisymmetric reference earth model in hydrostatic equilibrium, using the linear momentum description. The new features of this work consist in the use of (i) Clairaut coordinates (rather than spherical polars), (ii) standard spherical harmonics (rather than generalized spherical surface harmonics), (iii) linear operators (rather than J-square symbols) to represent the effects of rotational and ellipticity coupling between dependent variables of different harmonic degree and (iv) a set of dependent variables all of which are continuous across material boundaries. The resulting infinite system of coupled ordinary differential equations is given explicitly, for an elastic solid mantle and inner core, an inviscid outer core and no magnetic field. The formulation is done to second order in the Earth's ellipticity. To this order it is shown that for wobble modes (in which the lowest harmonic in the displacement field is degree 1 toroidal, with azimuthal order <it>m</it> = ±1), it is sufficient to truncate the chain of coupled displacement fields at the toroidal harmonic of degree 5 in the solid parts of the earth model. In the liquid core, however, the harmonic expansion of displacement can in principle continue to indefinitely high degree at this order of accuracy. The full equations are shown to yield correct results in three simple cases amenable to analytic solution: a general earth model in rigid rotation, the tiltover mode in a homogeneous solid earth model and the tiltover and Chandler periods for an incompressible homogeneous solid earth model. Numerical results, from programmes based on this formulation, are presented in part II of this paper.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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 teacher head, 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".