A characteristic mapping method for the incompressible Euler equations on a sphere
Bibliographic record
Abstract
In this thesis, analytical and numerical aspects of the solution to the incompressible Euler equations on a two-dimensional sphere using the Characteristic Mapping (CM) method are presented.These equations dictate the time evolution of an incompressible, inviscid fluid from a prescribed initial condition.Their non-linear nature produces increasingly fine scales over time; posing a challenge for existing numerical methods.This problem is broached using the CM method, which considers the numerical quantity of interest to be the flow map generated by the fluid motion.The semigroup property of the flow map, facilitating its own evolution by means of composition, is leveraged to capture the fine scales manifest in the dynamics of the fluid.We begin with a presentation of the vorticity-stream formulation of the incompressible Euler equations on a sphere, from which the solution strategy is built.An implementation for solving the spherical Poisson equation using the double Fourier sphere method is presented.Thereafter, the CM method for linear transport on the 2-sphere is presented.The thesis is concluded with a discussion of combining these two numerical methods for the solution of the incompressible Euler equations.i Abrégé Dans cette thèse, les aspects analytiques et numériques du solution des équations d'Euler incompressibles sur une sphère deux dimensionelles en utilisant la méthode d'application des caractéristiques (AC) sont présentées.Ces équations dictent l'évolution d'une fluide incompressible et non-visqueux d'une état initiale donné.La nature nonlinéaire de l'écoulement se produit des échelles fines dans la fluid au fil du temps; posant des difficultés pour les méthodes numériques existantes.Cette problème est addressé en utilisant la méthode AC, qui considère la quantité d'intéresse d'être l'application du flot généré par la vitesse du fluide.La propriété semi-groupe de l'application du flot, facilitant sa propre évolution par composition, est utilisée pour résoudre des échelles fines inhérent aux dynamiques du fluide.On commence avec un présentation du vorticitécourant formulation des équations d'Euler incompressibles sur une sphère, à laquelle notre stratégie de solution est construite.Une implémentation de la double Fourier sphère pour la solution de l'équation Poisson sphérique est présentée.Par la suite, la méthode AC pour l'équation du transport linéaire sur la 2-sphère est présentée.La thèse se conclut avec une discussion des aspects de combiner ces deux méthodes pour la solution des équations d'Euler incompressibles.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".