Scenarios for magnetic X-point collapse in 2D incompressible dissipationless extended magnetohydrodynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908501510455296 |
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| author | Brizard, Alain J. |
| author_facet | Brizard, Alain J. |
| contents | The equations of 2D incompressible dissipationless extended magnetohydrodynamics (XMHD) extend the equations of incompressible Hall MHD (HMHD) by retaining finite-electron inertia. These XMHD equations couple the fluid velocity ${\bf V} = \wh{\sf z}\btimes\nablaϕ+ V_{z}\,\wh{\sf z}$ with the magnetic field ${\bf B} = \nablaψ\btimes\wh{\sf z} + B_{z}\,\wh{\sf z}$ in a process that is known to support dissipationless solutions that exhibit finite-time singularities associated with magnetic X-point collapse in the magnetic plane $(B_{x} = \partialψ/\partial y, B_{y} = -\,\partialψ/\partial x)$. Here, by adopting a 2D self-similar model for the four XMHD fields $(ϕ,ψ,V_{z},B_{z})$, we obtain five coupled ordinary differential equations that are solved in terms of the Jacobi elliptic functions based on an orbital classification associated with particle motion in a quartic potential. Excellent agreement is found when these analytical solutions are compared with numerical solutions, including the precise time of a magnetic X-point collapse. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07311 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scenarios for magnetic X-point collapse in 2D incompressible dissipationless extended magnetohydrodynamics Brizard, Alain J. Plasma Physics Mathematical Physics The equations of 2D incompressible dissipationless extended magnetohydrodynamics (XMHD) extend the equations of incompressible Hall MHD (HMHD) by retaining finite-electron inertia. These XMHD equations couple the fluid velocity ${\bf V} = \wh{\sf z}\btimes\nablaϕ+ V_{z}\,\wh{\sf z}$ with the magnetic field ${\bf B} = \nablaψ\btimes\wh{\sf z} + B_{z}\,\wh{\sf z}$ in a process that is known to support dissipationless solutions that exhibit finite-time singularities associated with magnetic X-point collapse in the magnetic plane $(B_{x} = \partialψ/\partial y, B_{y} = -\,\partialψ/\partial x)$. Here, by adopting a 2D self-similar model for the four XMHD fields $(ϕ,ψ,V_{z},B_{z})$, we obtain five coupled ordinary differential equations that are solved in terms of the Jacobi elliptic functions based on an orbital classification associated with particle motion in a quartic potential. Excellent agreement is found when these analytical solutions are compared with numerical solutions, including the precise time of a magnetic X-point collapse. |
| title | Scenarios for magnetic X-point collapse in 2D incompressible dissipationless extended magnetohydrodynamics |
| topic | Plasma Physics Mathematical Physics |
| url | https://arxiv.org/abs/2504.07311 |