Variational principles for ideal MHD of steady incompressible flows via Lie-point symmetries with application to the magnetic structures of bipolar sunspots
Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
2 Egyptian Korean Faculty of Technological Industry and Energy, Beni-Suef Technological University, Beni-Suef, Egypt
Accepted: 7 July 2020
Published online: 20 July 2020
In this paper, Lie-point symmetries and conservation laws for the generalized Grad–Shafranov equation (GGSE) for symmetric magnetohydrodynamic (MHD) fluids with steady, inhomogeneous, incompressible flows are derived. The adjointness of that equation is discussed via a Lie operator. We noted that the GGSE can be self-adjoint, quasi-self-adjoint or nonlinear self-adjoint according to a physical magnetic flux quantity. With the aid of the derived symmetries, a general solution and construction for obtaining the solutions of the whole nonlinear MHD system of the considered flows are presented. Also, the linear stability of that general solution is discussed. To formulate variational principles for the problem, Lagrangians of first and second order are derived and used to obtain the conserved quantities for these flows. An application that may be of interest for magnetic structures of the bipolar sunspots groups is presented.
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