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Abstract: Abridged We develop a theoretical framework to construct axisymmetricmagnetic equilibria in stars, consisting of both poloidal and toroidal magneticfield components. In a stationary axisymmetric configuration, the poloidalcurrent is a function of the poloidal magnetic flux only, and thus shouldvanish on field lines extending outside of the star. Non-zero poloidal currentis limited to a set of toroid-shape flux surfaces fully enclosed inside thestar. If we demand that there are no current sheets, then on the separatrixdelineating the regions of zero and finite toroidal magnetic field both thepoloidal flux function and its derivative should match. Thus, for a givenmagnetic field in the bulk of the star, the elliptical Grad-Shafranov equationthat describes magnetic field structure inside the toroid is an ill-posedproblem, with both Dirichlet and Newman boundary conditions and {\it a priori}unknown distribution of toroidal and poloidal electric currents. We discuss aprocedure which allows to solve this ill-posed problem by adjusting the unknowncurrent functions. We find a poloidal current-carrying solution that leaves theshape of the flux function and, correspondingly, the toroidal component of theelectric current the same as in the case of no poloidal current. The equilibriadiscussed in this paper may have arbitrary large toroidal magnetic field, andmay include a set of stable equilibria. The method developed here can also beapplied to magnetic structure of differentially rotating stars, as well as tocalculate velocity field in incompressible isolated fluid vortex with a swirl.



Author: Maxim Lyutikov Purdue

Source: https://arxiv.org/



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