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We propose a catastrophic eruption model for magnetar's enormous energyrelease during giant flares, in which a toroidal and helically twisted fluxrope is embedded within a force-free magnetosphere. The flux rope stays instable equilibrium states initially and evolves quasi-statically. Upon the lossof equilibrium point is reached, the flux rope cannot sustain the stableequilibrium states and erupts catastrophically. During the process, themagnetic energy stored in the magnetosphere is rapidly released as the resultof destabilization of global magnetic topology. The magnetospheric energy thatcould be accumulated is of vital importance for the outbursts of magnetars. Wecarefully establish the fully open fields and partially open fields for variousboundary conditions at the magnetar surface and study the relevant energythresholds. By investigating the magnetic energy accumulated at the criticalcatastrophic point, we find that it is possible to drive fully open eruptionsfor dipole dominated background fields. Nevertheless, it is hard to generatefully open magnetic eruptions for multipolar background fields. Given theobservational importance of the multipolar magnetic fields in the vicinity ofthe magnetar surface, it would be worthwhile to explore the possibility of thealternative eruption approach in multipolar background fields. Fortunately, wefind that flux ropes may give rise to partially open eruptions in themultipolar fields, which involve only partial opening up of background fields.The energy release fractions are greater for cases with central-arcadedmultipoles than those with central-caved multipoles emerged in backgroundfields. Eruptions would fail only when the centrally-caved multipoles becomeextremely strong.

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