Hierarchies of new invariants and conserved integrals in inviscid fluid flow
Bibliographic record
Abstract
A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flows in three dimensions. This approach describes invariants by means of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid velocity, which has the physical meaning of characterizing tensorial quantities that are frozen into the flow. Several new main results are obtained. First, simple algebraic and differential operators that can be applied recursively to derive a complete set of invariants starting from the basic known local and nonlocal invariants are constructed. Second, these operators are used to derive infinite hierarchies of local and nonlocal invariants for both adiabatic fluids and homentropic fluids that are either incompressible or compressible with barotropic and non-barotropic equations of state. Each hierarchy is complete in the sense that no further invariants can be generated from the basic local and nonlocal invariants. All of the resulting new invariants are generalizations of Ertel’s invariant, the Ertel–Rossby invariant, and Hollmann’s invariant. In particular, for an incompressible fluid flow in which the density is non-constant across different fluid streamlines, a new variant of Ertel’s invariant and several new variants of Hollmann’s invariant are derived, where the entropy gradient is replaced by the density gradient. Third, the physical meaning of these new invariants and the resulting conserved integrals is discussed, and their relationship to conserved helicities and cross-helicities is described.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".