3 Definition of the the tilt angle $\varphi$ and mean current helicity $<{\vec h}_{\vert\vert}>$

3.1 The tilt angle $\varphi$ of magnetic polarity axis

In calculation of the tilt angle, the magnetic flux-weighted center is determined as the position of each magnetic polarity. This position could be approximately indicated by the magnetic flux weighted center of each magnetic polarity in the line-of-sight as:

$$x_{\rm c}={\sum x(j,i)B_{\vert\vert}(j,i){\rm d}s\over \sum B_{\vert\vert}(j,i){\rm d}s},$$

$$y_{\rm c}={\sum y(j,i)B_{\vert\vert}(j,i){\rm d}s\over \sum B_{\vert\vert}(j,i){\rm d}s}$$

where, ${\rm d}s = {\rm d}x{\rm d}y$ is the area of each pixel. The tilt angle ($\varphi$) of an active region is defined as the angle between the line joining opposite polarities and the solar equator. The tangent of the tilt angle (tan $\varphi$) is given by $\delta y/\delta x$ (Gary & Hagyard 1990; Wang & Sheeley 1989; Howard 1989; Tian et al. 1999), where $\delta y$ and $\delta x$ are the distances between the leading polarity and the following polarity along the Y- and X-axis, respectively, in Cartesian coordinates in the heliographic plane.

We determinate the sign of the tilt angle of the magnetic polarity axis in an active region magnetic field such that: in the northern hemisphere, if the S polarity is closer to the solar equator, the tilt of the active region is positive ( $0<\varphi<180$ degree), otherwise the tilt is negative ( $0>\varphi>-180$ degree); in the southern hemisphere, the tilt is negative ( $0>\varphi>-180$ degree) when the N polarity is closer to the equator, and otherwise the tilt is positive ( $0<\varphi<180$ degree). Note that in the 22nd solar cycle, the leading spot should be S/N-polarity in the northern/southern hemisphere. Thus, active regions with a tilt angle $0<\varphi <90$ degree in the northern hemisphere, or $-90 <\varphi < 0$ degree in the southern hemisphere, all adhere to Joy's Law. Figure 1 shows the sign convention of the tilt angles ($\varphi$) in the 22nd solar cycle.

Figure 1: Convention for the sign of the tilt angle of the magnetic polarity axis. Active regions with a tilt angle $0<\varphi <90$ degree in the northern hemisphere, or $-90 <\varphi < 0$ degree in the northern hemisphere, follow Joy's law.

In order to obtain the flux-weighted center closer to the center of the sunspot and then the tilt of active regions, a magnetic field of 160 G was selected as a criterion of the minimum of magnetic strength, so that the effect of some other weak magnetic fields around main polarities would be minimized. It is useful for us to calculate the flux-weighted center and tilt angle of magnetic polarity axis by this method (Tian et al. 1999).

3.2 Mean current helicity $<\mathsfsl{h}_{\vert\vert}>$

Current helicity plays an important role in the study of twisted magnetic fields in the solar atmosphere. Its density can be split into two parts,

$$h_{\rm c} \vec{B}\cdot(\bigtriangledown \times \vec{B})$$ =

$$\vec{B}_{\vert\vert}\cdot(\bigtriangledown \times \vec{B})_{\vert\vert} +\vec{B}_{\bot}\cdot(\bigtriangledown \times \vec{B})_\bot$$ =

$$h_{\vert\vert}+h_{\bot},$$ = (3)

h_|| is the parallel component in the direction of the line of sight which can be inferred from photospheric vector magnetograms. $h_{\bot}$ is the transverse component which is hard to compute because we have magnetic measurement only at a single height in the solar atmosphere.

In the approximation of the force-free magnetic field: $\bigtriangledown \times \vec{B}=\alpha_{\rm f} \vec{B}$, there is,

$$\alpha_{\rm f}=\frac{(\bigtriangledown \times \vec{B})_\para... ...gledown \times \vec{B})_\parallel}{{\vec{B}_\parallel}^2}\cdot$$ (4)

Thus, the density of current helicity is:

$$h_{\rm c} \vec{B}\cdot(\bigtriangledown \times \vec{B})={\alpha_{\rm f}}{B^2}$$ =

$$\frac{\vec{B}^2}{\vec{B}^2_{\vert\vert}}(\vec{B}_\parallel \cdot(\bigtriangledown \times \vec{B})_\parallel).$$ = (5)

Here, we define

$$<h_{\vert\vert}>\,=\,<\vec{B}_\parallel \cdot(\bigtriangledow... ...uiv\ \mu_0 <{\vec B_{\vert\vert}}\cdot {\vec J_{\vert\vert}}>,$$ (6)

as mean of h_|| distribution over an active region, where $\mu_0=4\pi\times10^{-3}~{\rm GmA}^{-1}$, and the vertical current density J_|| can be expressed by

$$J_{\vert\vert}={1\over\mu_0}\left({\partial B_y\over \partial x}-{\partial B_x\over \partial y}\right)$$

(Wang et al. 1994).

In order to minimize the effect of observational error, the current helicity h_|| is calculated only in pixels as B_|| > 20 G, $B_\bot > 100$ G and current density J_||> 0.001 Am^-2, with the noise level of J_||(j,i). Eventually, Bao et al. (1998) determined an average of the current helicity for each active region. In this case, the uncertainty of magnetic fields affects <h_||> very little. We evaluated the error in 2$\sigma$ in Figs. 2 and 3. However, although an active region develops over days, the sign of < h_||>and the sign of the tilt angle of the active region do not change.