A general Grad-Shafranov equation is obtained via differential forms, together with a scalar-field Lagrangian that yields the equation on-shell.
Grad-Shafranov equation in noncircular stationary axisymmetric spacetimes
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abstract
A formulation is developed for general relativistic ideal magnetohydrodynamics in stationary axisymmetric spacetimes. We reduce basic equations to a single second-order partial differential equation, the so-called Grad-Shafranov (GS) equation. Our formulation is most general in the sense that it is applicable even when a stationary axisymmetric spacetime is noncircular, that is, even when it is impossible to foliate a spacetime with two orthogonal families of two-surfaces. The GS equation for noncircular spacetimes is crucial for the study of relativistic stars with a toroidal magnetic field or meridional flow, such as magnetars, since the existence of a toroidal field or meridional flow violates the circularity of a spacetime. We also derive the wind equation in noncircular spacetimes, and discuss various limits of the GS equation.
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Direct comparison of Konno-99 perturbative and LORENE numerical methods for poloidal magnetized neutron stars shows perturbative validity for observed fields up to ~10^16 G and numerical resolution limits below ~10^10 G.
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General Grad-Shafranov Equation
A general Grad-Shafranov equation is obtained via differential forms, together with a scalar-field Lagrangian that yields the equation on-shell.
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Magnetized neutron stars: perturbative versus fully-numerical approaches
Direct comparison of Konno-99 perturbative and LORENE numerical methods for poloidal magnetized neutron stars shows perturbative validity for observed fields up to ~10^16 G and numerical resolution limits below ~10^10 G.