Parker theorem

Parker theorem, or the fundamental magnetostatic theorem, was formulated by physicist Eugene Parker in 1972.^[1] Parker's theorem describes how magnetic fields behave in perfectly conducting fluids, particularly in space plasmas. The theorem states that three-dimensional magnetic fields naturally form infinitesimally thin current sheets – regions where the magnetic field direction changes abruptly. These sheets arise from the fundamental interaction between magnetic fields that are "frozen" into the conducting fluid.^[2]

When different magnetic field regions come into contact, they cannot smoothly merge due to the perfect conductivity of the fluid. Instead, they form sharp boundaries where electric currents flow. This process is analogous to how non-mixing fluids like oil and water form distinct boundaries rather than mixing. The theorem's central claim is that such discontinuities are not exceptional but are the standard feature of magnetic field equilibria in perfectly conducting fluids.^[2]

^ Parker, E. N. (1 June 1972). "Topological Dissipation and the Small-Scale Fields in Turbulent Gases". The Astrophysical Journal. 174: 499. Bibcode:1972ApJ...174..499P. doi:10.1086/151512. ISSN 0004-637X. Retrieved 24 October 2024.
^ ^a ^b Low, B. C. (1 January 2023). "Topological nature of the Parker magnetostatic theorem". Physics of Plasmas. 30 (1). doi:10.1063/5.0124164. Retrieved 24 October 2024. state that a 3D magnetic field of complex topology, frozen into an electrically perfect fluid conductor, generally attains force equilibrium by spontaneously embedding infinitesimally thin current sheets (CSs) across which the field is tangentially discontinuous. In the absence of resistivity, the discrete electric current flowing in a magnetic tangential discontinuity (TD), given by Ampère's law, arises naturally. Magnetic flux surfaces move as fluid surfaces. In a 3D evolution to equilibrium, two flux systems separate in space can readily push an intervening third flux system completely out of their way to make direct contact at a TD. The crux of the theorem is that most equilibrium fields must embed TDs by the nonlinear action of the anisotropic Lorentz force under the frozen-in condition. That is, equilibria embedding TDs are the rule rather than the exception. This article incorporates text from this source, which is available under the CC BY 4.0 license.

[1] Parker, E. N. (1 June 1972). "Topological Dissipation and the Small-Scale Fields in Turbulent Gases". The Astrophysical Journal. 174: 499. Bibcode:1972ApJ...174..499P. doi:10.1086/151512. ISSN 0004-637X. Retrieved 24 October 2024.

[Low-2] Low, B. C. (1 January 2023). "Topological nature of the Parker magnetostatic theorem". Physics of Plasmas. 30 (1). doi:10.1063/5.0124164. Retrieved 24 October 2024. state that a 3D magnetic field of complex topology, frozen into an electrically perfect fluid conductor, generally attains force equilibrium by spontaneously embedding infinitesimally thin current sheets (CSs) across which the field is tangentially discontinuous. In the absence of resistivity, the discrete electric current flowing in a magnetic tangential discontinuity (TD), given by Ampère's law, arises naturally. Magnetic flux surfaces move as fluid surfaces. In a 3D evolution to equilibrium, two flux systems separate in space can readily push an intervening third flux system completely out of their way to make direct contact at a TD. The crux of the theorem is that most equilibrium fields must embed TDs by the nonlinear action of the anisotropic Lorentz force under the frozen-in condition. That is, equilibria embedding TDs are the rule rather than the exception. This article incorporates text from this source, which is available under the CC BY 4.0 license.

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