dorsal/arxiv
View SchemaAn example of anti-dynamo conformal Arnold metric
| Authors | Garcia de Andrade |
|---|---|
| Categories | |
| ArXiv ID | physics/0703143 |
| URL | https://arxiv.org/abs/physics/0703143 |
Abstract
A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-{\lambda}z}dp^{2}+e^{{\lambda}z}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counterpart are computed. The curvature decay as z-coordinates increases without bounds. Some of the Riemann curvature components such as $R_{pzpz}$ also undergoes dissipation while component $R_{qzqz}$ increases without bounds. The remaining curvature component $R_{pqpq}$ is constant on the torus surface. The Riemann curvature invariant $K^{2}=R_{ijkl}R^{ijkl}$ is found to be 0.155 for the ${\lambda}=0.75$. A simple solution of Killing equations for Arnold metric yields a stretch Killing vector along one direction and compressed along other direction in order that the modulus of the Killing vector is not constant along the flow. The flow is shown to be untwisted. The stability of the two metrics are found by examining the sign of their curvature tensor components.
{
"annotation_id": "8b39b943-814e-4cbb-b40b-3a217ce19e7e",
"date_created": "2026-03-02T18:01:18.434000Z",
"date_modified": "2026-03-02T18:01:18.434000Z",
"file_hash": "683ac694c7428d7b600bc36a9903a854c7009b498a11a95f9575e73424f2b9a7",
"private": false,
"record": {
"abstract": "A 3D metric conformally related to Arnold cat fast dynamo metric:\n${ds_{A}}^{2}=e^{-{\\lambda}z}dp^{2}+e^{{\\lambda}z}dq^{2}+dz^{2}$ is shown to\npresent a behaviour of non-dynamos where the magnetic field exponentially decay\nin time. The Riemann-Christoffel connection and Riemann curvature tensor for\nthe Arnold and its conformal counterpart are computed. The curvature decay as\nz-coordinates increases without bounds. Some of the Riemann curvature\ncomponents such as $R_{pzpz}$ also undergoes dissipation while component\n$R_{qzqz}$ increases without bounds. The remaining curvature component\n$R_{pqpq}$ is constant on the torus surface. The Riemann curvature invariant\n$K^{2}=R_{ijkl}R^{ijkl}$ is found to be 0.155 for the ${\\lambda}=0.75$. A\nsimple solution of Killing equations for Arnold metric yields a stretch Killing\nvector along one direction and compressed along other direction in order that\nthe modulus of the Killing vector is not constant along the flow. The flow is\nshown to be untwisted. The stability of the two metrics are found by examining\nthe sign of their curvature tensor components.",
"arxiv_id": "physics/0703143",
"authors": [
"Garcia de Andrade"
],
"categories": [
"physics.flu-dyn",
"astro-ph",
"gr-qc",
"math.DG"
],
"title": "An example of anti-dynamo conformal Arnold metric",
"url": "https://arxiv.org/abs/physics/0703143"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "b6b42042-1f34-4507-bb99-2973f906ca45",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}