On Some Solutions of Plasmastatic Equations
https://doi.org/10.1134/S2304487X20060103
Abstract
A system of MHD equilibrium equations is considered. Approaches used to solve this system lead to an overdetermined system of nonlinear partial differential equations. The overdetermined system is solved with analytical approaches based on the reduction of systems of partial differential equations to systems of ordinary differential equations with the subsequent solution of such systems. With such reductions, various quantities can be selected as an independent variable in the system of ordinary differential equations. An algorithm is described to reduce the overdetermined system of equations to systems of ordinary differential equations in which the independent variable ψ is such that ψ(x, y, z) = const is the level surface for solutions of the system of MHD equilibrium equations. It is shown that there is more than one way to choose such an independent variable. Exact solutions of the original system are obtained in the cases ψ = x/(1 – my – nz) (m = const, n = const) and ψ = x + my + nz + l (m = const, n = const, l = const). The exact solutions obtained depend on arbitrary constants. It is shown that the solution in the ψ = x + my + nz + l (m = const, n = const, l = const) can be use to construct a magnetic field of a given direction. One particular solution of the system of MHD equilibrium equations is presented for a vortex-free solenoidal magnetic field. It is noted that these exact solutions can be used as tests in numerical calculations.
About the Authors
O. N. Ul’yanov
Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences
Russian Federation
Russian Federation
620990
Yekaterinburg
L. I. Rubina
Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences
Russian Federation
Russian Federation
620990
Yekaterinburg
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Review
For citations:
Ul’yanov O.N., Rubina L.I. On Some Solutions of Plasmastatic Equations. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(1):12-18. (In Russ.) https://doi.org/10.1134/S2304487X20060103
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