Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries Article Swipe
Naoki Sato
,
Michio Yamada
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.10757
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.10757
This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface, and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.
Related Topics
Concepts
Curl (programming language)
Toroid
Vector field
Magnetohydrodynamics
Solenoidal vector field
Euler's formula
Mathematical analysis
Torus
Bounded function
Invariant (physics)
Mathematics
Vector potential
Physics
Classical mechanics
Magnetic field
Geometry
Mathematical physics
Computer science
Plasma
Quantum mechanics
Programming language
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2211.10757
- https://arxiv.org/pdf/2211.10757
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4309798097
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W4309798097Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2211.10757Digital Object Identifier
- Title
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Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometriesWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2022Year of publication
- Publication date
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2022-11-19Full publication date if available
- Authors
-
Naoki Sato, Michio YamadaList of authors in order
- Landing page
-
https://arxiv.org/abs/2211.10757Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2211.10757Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2211.10757Direct OA link when available
- Concepts
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Curl (programming language), Toroid, Vector field, Magnetohydrodynamics, Solenoidal vector field, Euler's formula, Mathematical analysis, Torus, Bounded function, Invariant (physics), Mathematics, Vector potential, Physics, Classical mechanics, Magnetic field, Geometry, Mathematical physics, Computer science, Plasma, Quantum mechanics, Programming languageTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.mechanics | 120 |
| abstract_inverted_index.solutions | 72 |
| abstract_inverted_index.surfaces. | 29 |
| abstract_inverted_index.concerning | 121 |
| abstract_inverted_index.continuous | 82, 137 |
| abstract_inverted_index.equilibria | 94 |
| abstract_inverted_index.identified | 92 |
| abstract_inverted_index.isometries | 84 |
| abstract_inverted_index.represents | 102, 145 |
| abstract_inverted_index.solenoidal | 9 |
| abstract_inverted_index.solutions. | 68 |
| abstract_inverted_index.tangential | 22 |
| abstract_inverted_index.translated | 37 |
| abstract_inverted_index.anisotropic | 96 |
| abstract_inverted_index.constructed | 87 |
| abstract_inverted_index.derivatives | 48 |
| abstract_inverted_index.determining | 42 |
| abstract_inverted_index.equilibrium | 126 |
| abstract_inverted_index.explicitly, | 88 |
| abstract_inverted_index.fundamental | 108 |
| abstract_inverted_index.isometries. | 139 |
| abstract_inverted_index.theoretical | 148 |
| abstract_inverted_index.differential | 56 |
| abstract_inverted_index.mathematical | 109 |
| abstract_inverted_index.second-order | 54 |
| abstract_inverted_index.stellarators. | 164 |
| abstract_inverted_index.configurations | 144 |
| abstract_inverted_index.two-dimensional | 51 |
| abstract_inverted_index.three-dimensional | 8 |
| abstract_inverted_index.magnetohydrodynamics | 117 |
| abstract_inverted_index.magnetohydrodynamics. | 97 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.17756026 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |