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To study the three-dimensional (3D) magnetic field topology and its long-termevolution associated with the X3.4 flare of 2006 December 13, we investigatethe coronal relative magnetic helicity in the flaring active region (AR) NOAA10930 during the time period of December 8-14. The coronal helicity iscalculated based on the 3D nonlinear force-free magnetic fields reconstructedby the weighted optimization method of Wiegelmann, and is compared with theamount of helicity injected through the photospheric surface of the AR. Thehelicity injection is determined from the magnetic helicity flux densityproposed by Pariat et al. using Solar and Heliospheric Observatory/MichelsonDoppler Imager magnetograms. The major findings of this study are thefollowing. (1) The time profile of the coronal helicity shows a goodcorrelation with that of the helicity accumulation by injection through thesurface. (2) The coronal helicity of the AR is estimated to be-4.3$\times$10$^{43}$ Mx$^{2}$ just before the X3.4 flare. (3) This flare ispreceded not only by a large increase of negative helicity,-3.2$\times$10$^{43}$ Mx$^{2}$, in the corona over ~1.5 days but also bynoticeable injections of positive helicity though the photospheric surfacearound the flaring magnetic polarity inversion line during the time period ofthe channel structure development. We conjecture that the occurrence of theX3.4 flare is involved with the positive helicity injection into an existingsystem of negative helicity.

Time Evolution of Coronal Magnetic Helicity in the Flaring Active Region NOAA 10930