Topological Invariants in Higher-Dimensional Magnetohydrodynamics
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| Format: | Preprint |
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2025
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| _version_ | 1866908664311316480 |
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| author | Sato, Naoki Abe, Ken Yamada, Michio |
| author_facet | Sato, Naoki Abe, Ken Yamada, Michio |
| contents | It is well known that the three-dimensional ideal magnetohydrodynamics (MHD) equations possess three magnetic invariants: (M) magnetic helicity, (C) cross helicity, and (P) the mean-square magnetic potential, in addition to the fundamental invariants of fluid motion. In this paper we construct higher-dimensional generalizations of these invariants for ideal MHD. Specifically, we identify generalized magnetic helicity and generalized cross helicity in all odd spatial dimensions $n=2m+1$, and families of invariants given by integrals of arbitrary functions of the scalar density $B^m/ν$ of the magnetic field $2$-form $B$, where $B^m$ denotes its $m$-fold wedge product and $ν$ the fluid-density top form, in all even spatial dimensions $n=2m$. We further establish the existence of invariants for symmetric solutions in arbitrary dimensions, generalizing the mean-square magnetic potential and showing that this invariant arises from symmetry rather than from even dimensionality, in contrast to the enstrophy invariant of the two-dimensional Euler equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological Invariants in Higher-Dimensional Magnetohydrodynamics Sato, Naoki Abe, Ken Yamada, Michio Mathematical Physics It is well known that the three-dimensional ideal magnetohydrodynamics (MHD) equations possess three magnetic invariants: (M) magnetic helicity, (C) cross helicity, and (P) the mean-square magnetic potential, in addition to the fundamental invariants of fluid motion. In this paper we construct higher-dimensional generalizations of these invariants for ideal MHD. Specifically, we identify generalized magnetic helicity and generalized cross helicity in all odd spatial dimensions $n=2m+1$, and families of invariants given by integrals of arbitrary functions of the scalar density $B^m/ν$ of the magnetic field $2$-form $B$, where $B^m$ denotes its $m$-fold wedge product and $ν$ the fluid-density top form, in all even spatial dimensions $n=2m$. We further establish the existence of invariants for symmetric solutions in arbitrary dimensions, generalizing the mean-square magnetic potential and showing that this invariant arises from symmetry rather than from even dimensionality, in contrast to the enstrophy invariant of the two-dimensional Euler equations. |
| title | Topological Invariants in Higher-Dimensional Magnetohydrodynamics |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2506.13251 |