Magnetar Giant Flares in Multipolar Magnetic Fields --- II. Flux Rope Eruptions With Current Sheets

We propose a physical mechanism to explain giant flares and radio afterglows in terms of a magnetospheric model containing both a helically twisted flux rope and a current sheet (CS). With the appearance of CS, we solve a mixed boundary value problem to get the magnetospheric field based on a domain decomposition method. We investigate properties of the equilibrium curve of the flux rope when the CS is present in background multipolar fields. In response to the variations at the magnetar surface, it quasi-statically evolves in stable equilibrium states. The loss of equilibrium occurs at a critical point and, beyond that point, it erupts catastrophically. New features show up when the CS is considered. Especially, we find two kinds of physical behaviors, i.e., catastrophic state transition and catastrophic escape. Magnetic energy would be released during state transitions. The released magnetic energy is sufficient to drive giant flares. The flux rope would go away from the magnetar quasi-statically, which is inconsistent with the radio afterglow. Fortunately, in the latter case, i.e., the catastrophic escape, the flux rope could escape the magnetar and go to infinity in a dynamical way. This is more consistent with radio afterglow observations of giant flares. We find that the minor radius of flux rope has important implications for its eruption. Flux ropes with larger minor radius are more prone to erupt. We stress that the CS provides an ideal place for magnetic reconnection, which would further enhance the energy release during eruptions.Comment: 31 pages, 7 figures, accepted by Ap

Magnetar Giant Flares in Multipolar Magnetic Fields --- I. Fully and Partially Open Eruptions of Flux Ropes

We propose a catastrophic eruption model for magnetar's enormous energy release during giant flares, in which a toroidal and helically twisted flux rope is embedded within a force-free magnetosphere. The flux rope stays in stable equilibrium states initially and evolves quasi-statically. Upon the loss of equilibrium point is reached, the flux rope cannot sustain the stable equilibrium states and erupts catastrophically. During the process, the magnetic energy stored in the magnetosphere is rapidly released as the result of destabilization of global magnetic topology. The magnetospheric energy that could be accumulated is of vital importance for the outbursts of magnetars. We carefully establish the fully open fields and partially open fields for various boundary conditions at the magnetar surface and study the relevant energy thresholds. By investigating the magnetic energy accumulated at the critical catastrophic point, we find that it is possible to drive fully open eruptions for dipole dominated background fields. Nevertheless, it is hard to generate fully open magnetic eruptions for multipolar background fields. Given the observational importance of the multipolar magnetic fields in the vicinity of the magnetar surface, it would be worthwhile to explore the possibility of the alternative eruption approach in multipolar background fields. Fortunately, we find that flux ropes may give rise to partially open eruptions in the multipolar fields, which involve only partial opening up of background fields. The energy release fractions are greater for cases with central-arcaded multipoles than those with central-caved multipoles emerged in background fields. Eruptions would fail only when the centrally-caved multipoles become extremely strong.Comment: 30 pages, 6 figures, accepted by Ap

Analytic properties of force-free jets in the Kerr spacetime -- III: uniform field solution

The structure of steady axisymmetric force-free magnetosphere of a Kerr black hole (BH) is governed by a second-order partial differential equation of $A_\phi$ depending on two "free" functions $\Omega(A_\phi)$ and $I(A_\phi)$, where $A_\phi$ is the $\phi$ component of the vector potential of the electromagnetic field, $\Omega$ is the angular velocity of the magnetic field lines and $I$ is the poloidal electric current. In this paper, we investigate the solution uniqueness. Taking asymptotically uniform field as an example, analytic studies imply that there are infinitely many solutions approaching uniform field at infinity, while only a unique one is found in general relativistic magnetohydrodynamic simulations. To settle down the disagreement, we reinvestigate the structure of the governing equation and numerically solve it with given constraint condition and boundary condition. We find that the constraint condition (field lines smoothly crossing the light surface (LS)) and boundary conditions at horizon and at infinity are connected via radiation conditions at horizon and at infinity, rather than being independent. With appropriate constraint condition and boundary condition, we numerically solve the governing equation and find a unique solution. Contrary to naive expectation, our numerical solution yields a discontinuity in the angular velocity of the field lines and a current sheet along the last field line crossing the event horizon. We also briefly discuss the applicability of the perturbation approach to solving the governing equation

Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible $H-$matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with nonstrictly diagonally dominant matrices and general $H-$matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general $H-$matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general $H-$matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper