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 and 2 norms without any remapping steps. . . . . . . . . . . . . . . .3.9 Initial condition 3.52 and the numerical solution at t = T /2. . . . . . . . .3.10 Zoom in on the final frame up to a window size of 2 -13 . . . . . . . . . . .vii 3.11 Top: Advection of the Mandelbrot set on the sphere under the velocity field 3.50.Bottom: Zoom on window depicted at times t = 0 (top row) and t = 5 (bottom row) up to a width of 2 -12 . . . . . . . . . . . . . . . .4.1 Left: Error at the grid points for the solution 4.4.Right: Visualization of Cartesian gradient data obtained form using 4.3. . . . . . . . . . . . . .4.2 Left: Approximation of the L (S 2 ) error sampled on a grid of size 1000 2 .Error in the function value and the Cartesian gradient are given.Right: Visualization of Cartesian gradient data obtained form the Hermite interpolation . . . . . . . . . . . .
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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".