4 Relationship between the sign of the tilt angle and the sign of the mean current helicity $<{\vec h}_{\vert\vert}>$

We have calculated the tilt angle $\varphi$ in the paper by Tian et al. (1999) and current helicity < h_|| > in the paper by Bao & Zhang (1998) for the 286 active regions. Thus, we can investigate the relationship between the sign of the tilt angle of the magnetic polarity axis and the sign of the current helicity < h_|| >. Figure 2 illustrates this relationship for 203 bipolar active regions with relatively simple configurations of magnetic fields.

Figure 2: Relationship between the tilt angle $\varphi$ and the mean current helicity $<h_\parallel >$ = $<{\vec B}_\parallel\cdot(\bigtriangledown \times {\vec B})_\parallel>$ for 203 bipolar active regions. In these active regions, the magnetic configuration is simple. Each point denotes an active region. The length of error bars is determined 2$\sigma$.

Figure 3: The relationship between the tilt angles $\varphi$ and the mean current helicity $<(\bigtriangledown \times {\bf B})_\parallel \cdot {\vec B}_\parallel>$ for 83 bipolar active regions. In these active regions, magnetic configurations are more complicated. The sign convention of the tilt angle and the current helicity is same as in Fig. 2. The length of error bars is determined 2$\sigma$. Each point denotes an active region.

Figure 3 shows such relationship for the other 83 bipolar active regions with relatively more complicated magnetic fields than the 203 active regions. We find that in both kinds of bipolar active regions, about 60% of the regions have a positive/negative tilt angle and a negative/positive current helicity in the northern/southern hemisphere. In other words, there is a negative correlation between the sign of the tilt angle and the sign of the mean twist. These active regions adhere not only to Joy's Law (Hale et al. 1919), but also to the hemispheric helicity rule (Bao & Zhang 1998; Pevtsov et al. 1995). Thus, we call these active regions "normal active regions''.

If we consider an active region as an $\Omega$-flux tube rising through the convection zone (Babcock 1961) to the photosphere, the tube acquires writhe through the Coriolis force on internal flow (Fan et al. 1993). The tilt of active regions with respect to the equator is an observable manifestation of such writhe, at photospheric level. A positive/negative tilt angle corresponds to the right/left-handed writhe of the flux tube in the northern/southern hemisphere on the basis of our sign convention for the tilt angle in Fig. 1. On the whole, a negative/positive mean current helicity $<h_{\vert\vert}>\, =\, <(\bigtriangledown \times \vec{B})_\parallel\cdot \vec{B}_\parallel>$approximately reflects a left/right-handed mean twist of magnetic lines. Thus, these normal active regions are in a stable condition because of the minimization of total helicity.