Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics
Bibliographic record
Abstract
The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon–Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density $h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$ , where $\boldsymbol{A}$ is the magnetic vector potential and $\boldsymbol{B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field $\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$ and the Godbillon–Vey helicity density $h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$ in general MHD flows in which either $h_{m}=0$ or $h_{m}\neq 0$ . A conservation law for $h_{\text{gv}}$ occurs in flows for which $h_{m}=0$ . For $h_{m}\neq 0$ the evolution equation for $h_{\text{gv}}$ contains a source term in which $h_{m}$ is coupled to $h_{\text{gv}}$ via the shear tensor of the background flow. The transport equation for $h_{\text{gv}}$ also depends on the electric field potential $\unicode[STIX]{x1D713}$ , which is related to the gauge for $\boldsymbol{A}$ , which takes its simplest form for the advected $\boldsymbol{A}$ gauge in which $\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$ where $\boldsymbol{u}$ is the fluid velocity. An application of the Godbillon–Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon–Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".