7 Conclusions and discussions

We note the following results from this study:

(1) About 60% of studied bipolar active regions are normal active regions with "normal chirality''. It is inferred that the tilt angle of the magnetic polarity axis and the mean current helicity < h_|| > have opposite signs. In other words, there is a negative correlation between the sign of the tilt angle and the sign of the mean twist.

(2) About one-third of bipolar active regions are abnormal active regions with the "abnormal chirality''. In these active regions, the tilt angles and the current helicity have the same sign. These active regions tend to be located at particular longitudes in the heliosphere.

(3) Most of the 62 X-ray flares larger than M-class in the 22nd cycle also tend to occur in particular longitudinal bins, where active regions with "abnormal chirality'' appear frequently.

We know that total helicity H of a flux tube can be separated into the twist helicity $T_{\rm w}$ of magnetic lines in the tube and the writhe helicity $W_{\rm r}$ of the tube (Moffatt & Ricca 1992). Therefore, an active region with a "normal chirality'' (i.e., a left/right handed twist accompanied by a right/left handed writhe) is likely to be in a lower energy state due to a negative/positive twist helicity and a positive/negative writhe helicity. This is a relatively stable tube because the total helicity is minimized. An active region with an "abnormal chirality'' (i.e., a left or right handed twist and a same handed writhe) is likely to be in a higher energy state due to the same sign of twist and writhe helicity. There is much more chance of magnetic reconnection in this kind of active region. Therefore, we propose that the "abnormal chirality'' of active regions is a possible source of instability able to produce some eruptive phenomena as seen in the Fig. 6. Fisher et al. (2000) predict that the twist and writhe in $\delta-$spot active regions should have the same sign because of the kink hypothesis for $\delta-$spot active regions. The active region with an "abnormal chirality'' may represent these kinked regions. However, the relevance between the distribution of the flares and that of the active regions with "abnormal chirality'' is not high (shown in the Fig. 6). We believe that it is probably because the number of cases is relatively low for statistical significance.

From Figs. 2 and 3, we find that there is a roughly fixed percentage in each quarter. We believe this is not a coincidence and that there must be a physical reason. Canfield & Pevtsov (1998) use a different twist indicator, the force-free parameter $\alpha$, to investigate the relationship in sign between the tilt and the twist of magnetic fields. They obtain important and some what different results from ours. What is the reason? Is it because they use different twist parameters of magnetic fields? Is there another physical meaning? Figure 4 shows us that some longitudes keep the same or opposite sign between the tilt and twist for many rotations, that tells us that the twist of magnetic fields is produced at great depths, not on the surface. What is real origin of the twist of the magnetic fields?

This study could give us some information on the chirality of the magnetic fields in the sub-photospheric convection zone. We will further investigate the origin and development of the magnetic field twist and the tilt of active regions, which is an important respect of some dynamic models of sub-photospheric flux tubes and the dynamo theory as well. A following paper will discuss this question.

Acknowledgements

The authors are grateful to Drs. T. J. Wang, Y. H. Yan, and Y. Y. Deng for useful discussions, valuable comments and suggestions. They specially acknowlede Dr. Jiong Qiu who helped us to improve this paper. The authors are grateful to the referee for his/her helpful suggestions. This research is supported by NSFC Grant Nos. 19791090 and 10073013.