Comparison of Two Methods for Calculating Magnetic Helicity in the Solar Corona

Duo to the large magnetic Reynolds number, the magnetic helicity originating from the solar interior can be carried away through the photosphere into the corona. However, the relationship between the accumulated magnetic helicity flux through the photosphere and the magnetic helicity in the corona is still unclear. By selecting 36 newly emerging active regions in the 23rd solar cycle, we apply optical flow methods to derive the accumulated magnetic helicity through the photosphere ($H_m^p$) by using the sequential longitudinal magnetograms, use nonlinear force-free field extrapolation to obtain the 3D coronal magnetic field, and adopt finite volume methods to calculate the instantaneous relative magnetic helicity in the corona ($H_m^c$) by using vector magnetograms. It is found that the local correlation tracking (LCT)-based $H_m^p$ is larger than $H_m^c$ in $1"$, and that the Differential Affine Velocity Estimator-based $H_m^p$ is more consistent with $H_m^c$ than the LCT-based $H_m^p$. $H_m^p$ is more consistent with $H_m^c$ in evaluation from $2"$ than from $1"$. Moreover, $H_m^c - H_m^p$ systematically shows consistency with the Hemispheric Helicity Rule (over 55\%), no matter which resolution and method are used. These estimations suggest that the consistency of $H_m^c$ and $H_m^p$ is partly dependent on the resolution of the magnetograms and the calculation methods.