DETECTION OF FLUX EMERGENCE, SPLITTING, MERGING, AND CANCELLATION OF NETWORK FIELD. I. SPLITTING AND MERGING

We use a clumping method for the detection of magnetic patches: each patch is picked up as a clump of marked pixels having magnetic strength beyond a given threshold (Parnell 2002). The adopted threshold of 1σ was obtained by fitting the histogram of the signed magnetic strength in each magnetogram, i.e., σ ∼ 5 G pixel ∼ 6.8 × 10¹⁴ Mx. This value is close to that of Parnell et al. (2009). A magnetic patch in a clumping method corresponds to a massif of magnetic patches in a downhill method and a curvature method that are used in some previous studies (Hagenaar et al. 1999; Welsch & Longcope 2003; DeForest et al. 2007). We pick up magnetic patches with sizes beyond 81 pixels for focusing our analysis on the network magnetic field. The validity of this choice is demonstrated in Figure 5, in which the network structure is seen when adopting the 81 pixel threshold.

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We track the motion of magnetic patches in consecutive images after their detection. Patches are marked as identical when they have a spatial overlapping in continuous images (Hagenaar 1999). It should be noted that the travel distance of magnetic patches in the data interval (∼1 minute) is up to nearly 1 pixel size, namely, $2 \ \rm {km}\ \rm {s}^{-1}\ \times$ 1 $\rm {min{\rm ute}} = 120\ \rm {km} \sim 1$ pixel. In a high-resolution magnetogram, more than one patch in a previous image often have spatial overlappings with one patch in a consecutive image and vice versa. To clear up this problem, we set two conditions when tracking patches. First, we investigate spatial overlappings of patches from those with larger flux contents. This is based on the concept that a smaller patch has a greater tendency to fall below the detection limit of the analysis by splitting and cancellation. Second, we select a patch with the most proximate flux content in case of overlappings of more than one patch. Tracking paths of detected patches become unique with these conditions.

Merging is a process where more than one patch of the same polarity converge and coalesce into one patch (Figure 3(a)). We define a merging as an event with two conditions. A merging event is defined as a feature that satisfies the following conditions: (1) that there are two or more parent patches in a previous magnetogram overlapping one daughter patch in the consecutive magnetogram, and (2) that more than one of the concerned patches in the previous magnetogram disappear in the time interval. Panels in the first row of Figure 6 show an example of detected mergings. Two patches converge in the first and second images. They coalesce into one patch in the third image.

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Splitting is a process where a single patch is divided into more than one patch (Figure 3(b)). A splitting event is defined as a feature that satisfies the following two conditions: (1) that there are one or more daughter patches in a magnetogram overlapping one parent patch in the previous magnetogram, and (2) that more than one of the concerned patches in the latter magnetogram appear in the time interval. Panels in the second row of Figure 6 show an example of the detected splittings. One patch stays in the first and second panel. It splits into two patches between the second and third panel.

The detailed explanation for the detection technique of emergence and cancellation will be given in the next paper in which many more events are detected by using another data set, making it possible for us to conduct a statistical study of them. Emergence and cancellation are defined as a pair of flux change events of each patch in opposite polarities within a certain distance. We pick up not only complete cancellations but also partial cancellations in this definition.

Panels in the third row of Figure 6 show an example of emergences. In the second panel, positive and negative patches appear but the positive patch is already coalesced with a pre-existing patch. These emerging patches continue to separate against each other and become distant in the last panel. Panels in the last row of Figure 6 show an example of partial cancellations. Opposite polarities are converging in the first and the second panel. They start canceling between the second and the third panel. The negative patch totally disappears in the last panel but the positive patch remains after the cancellation in this example.

Along with the total numbers of patches and their flux density averaged over the field of view and the observational period (in Mx cm⁻²), the frequencies of events in the magnetic flux density (in Mx cm⁻² s⁻¹) are summarized in Table 1. The total numbers of detected negative and positive patches are 1636 and 1637, respectively, enough for a statistical study. As for magnetic processes, merging and splitting are much more frequent than emergence and cancellation. The total numbers of merging for positive and negative patches are 493 and 482. The total numbers of splitting for positive and negative patches are 536 and 535. These are enough for a statistical study. The number of emergence is 3 and that of cancellation is 86, which are not enough for a statistical study.

To determine what magnetic processes dominate a flux balance in this region, we investigate total flux amounts of patches and those related to the magnetic processes per unit area per unit time. The results are shown in Table 1. The averaged flux densities are 2.53 Mx cm⁻² and 3.60 Mx cm⁻² for positive and negative patches, respectively. They correspond to the total flux amount of 1.63 × 10²⁰ Mx and 2.32 × 10²⁰ Mx for our field of view. The rate of averaged flux density involved in merging processes is 1.65 (3.53)  ×  10⁻³ Mx cm⁻² s⁻¹ for positive (negative) patches. We define the flux amount of merging process as that of parent patches. The flux replacement timescale by positive (negative) merging, τ_mrg = Φ_tot/(.∂Φ_tot/∂t|_mrg), is evaluated as 1.53 (1.02)  ×  10³ s from these values, which is much shorter than the estimated replacement timescale by cancellation and emergence reported in previous studies (Livi et al. 1985; Schrijver et al. 1998; Hagenaar 2001; Thornton & Parnell 2011). The rate of the averaged flux density involved in splitting processes is 1.48 (3.03) × 10⁻³ Mx cm⁻² s⁻¹ for positive (negative) patches. We define the flux amount of the splitting process as the sum of those of parent patches. In the same manner as merging, the flux replacement timescale by positive (negative) splitting is evaluated as 1.71 (1.19) × 10³ s.

Figure 7 shows a frequency distribution of the flux content. The red and blue dashed histograms denote those of positive and negative polarities, respectively. The solid histogram denotes the sum of them. The black dashed line indicates the fitting result of it. The distribution drops down below ϕ_th = 10^17.5 Mx, which is suggested as a detection limit in this study, below which only a limited number of the patches are detected. The dashed line is a fitted power-law result between 10^17.5 Mx and 10¹⁹ Mx. We make a least-squares fitting of the power-law function with a form, n(ϕ) = n₀(ϕ/ϕ₀)^−γ. We obtain n₀ = 2.42 × 10⁻³⁶ Mx⁻¹ cm⁻², ϕ₀ = 1.0 × 10¹⁸ Mx, and γ = 1.78 ± 0.05. The error means a 1σ error. The fitted power-law index is sensitive to the fitting range as reported by Parnell et al. (2009). The index varies from −1.78 to −1.91 with a minimum fitting range of 10^17.5–10¹⁸ Mx. The total error of our fitting of the power-law index becomes 0.18 when considering this error. This result is consistent with Parnell et al. (2009), which reports −1.85 ± 0.14 as a power-law index of a frequency distribution of the flux content.

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We investigate frequency distributions on flux content of merging and splitting for one patch, which are defined as frequency distributions of processes divided by a frequency distribution of flux content. The value of these distributions represents the probability of the processes. Further, the detection limit ϕ_th is taken into account in this study. We call them apparent probability distributions of processes.

Figure 8 shows the apparent probability distribution of merging. We make a least-squares fitting in a range of 10^17.5–10¹⁹ Mx, where the number of the merging event is enough for the fitting. The fitting form is

$$\begin{equation} \frac{\partial P_{\rm mrg}^{\rm APP}}{\partial t}=p_{0,{\rm mrg}} \left(\frac{\phi }{\phi _0} \right)^{\beta _{\rm mrg}}, \end{equation} \tag{ 1 }$$

where p_{0, mrg} is the reference probability, ϕ₀ is the reference flux content, and β_mrg is the power-law index of the probability distribution of merging. We obtained p_{0, mrg} = (2.52 ± 0.08) × 10⁻⁴ s⁻¹ and β_mrg = 0.28 ± 0.05 for the positive patches, p_{0, mrg} = (2.52. ± 0.08) × 10⁻⁴ s⁻¹ and β_mrg = 0.26 ± 0.05 for the negative patches, and p_{0, mrg} = (5.12 ± 0.11) × 10⁻⁴ s⁻¹ and β_mrg = 0.28 ± 0.04 for both patches with ϕ₀ = 10¹⁸ Mx. The errors mean a 1σ error of least-squares fitting.

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Figure 9 shows the apparent probability distribution of splitting. The strong increase in the range larger than 10¹⁹ Mx is caused by the lack of a patch number in the analysis. On the other hand, there is a drop in the flux range near ϕ_th where the number of patches is enough for a statistical study. We interpret this dropping as an effect of splitting into the area below ϕ_th. This effect is evaluated in the discussion in Section 5.1. We see that the probability of splitting is almost constant as 1.0 × 10⁻³ s⁻¹, which means a timescale of 33 minutes, in the range enough above ϕ_th, 3.0 × 10¹⁸–1.0 × 10¹⁹ Mx. It means that the frequency of splitting is independent of the parents' flux content.

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We evaluate the probability density distributions of merging and splitting terms in a magneto-chemistry equation from the observational results. Because the probability distributions, which we obtained in this study, are obtained by integrating them in the flux content once, we have to put at least one assumption to evaluate them.

The merging function l(x, y) is given as follows. We obtain the form as a probability distribution from the definition of l(x, y) as

$$\begin{equation} \frac{\partial P_{\rm mrg}^{\rm APP}}{\partial t}=\int _{\phi _{\rm th}}^{\infty }n(x)l(\phi,x)\, d\phi. \end{equation} \tag{ 2 }$$

By comparing with the observational result, we obtain

$$\begin{equation} \displaystyle \int _{\phi _{\rm th}}^{\infty }n(x)l(\phi,x)dx=p_{0,{\rm mrg}}\left(\frac{\phi }{\phi _0} \right)^{\beta _{\rm mrg}} \ \ \ (\phi \ge \phi _{\rm th}). \end{equation} \tag{ 3 }$$

In the left-hand side of this equation, the variable ϕ appears only in the l(ϕ, x). We assume a simple form satisfying this relation, namely,

$$\begin{equation} l(x,y) \propto x^{\beta _{\rm mrg}}. \end{equation} \tag{ 4 }$$

From the symmetry of l(x, y) = l(y, x), this relation deduces

$$\begin{equation} l(x,y) = l_0 \left(\frac{x}{\phi _0} \right)^{\beta _{\rm mrg}} \left(\frac{y}{\phi _0} \right)^{\beta _{\rm mrg}}. \end{equation} \tag{ 5 }$$

Substituting it and the observational result of n(ϕ) into Equation (3), we obtain

$$\begin{equation} \displaystyle l_0=\frac{\displaystyle p_{0,{\rm mrg}}}{\displaystyle n_0 \phi _0^{\gamma -\beta _{\rm mrg}} \int _{\phi _{\rm th}}^{\infty }x^{-\gamma +\beta _{\rm mrg}}dx}. \end{equation} \tag{ 6 }$$

The upper value of the integration is limited on the actual Sun and we put it as ϕ_max. We substitute the value obtained in our study, namely, n₀ = 1.21 × 10⁻³⁶ Mx⁻¹ cm⁻², γ = 1.78, p_{0, mrg} = 2.56 × 10⁻⁴ s⁻¹, β_mrg = 0.28, ϕ_max = 10¹⁹ Mx, ϕ_th = 10^17.5 Mx, and ϕ₀ = 10¹⁸ Mx, and obtain l₀ = 7.24 × 10¹³ cm² s⁻¹.

The splitting function k(x, y) is given as follows. From our observations, the probability distribution of the splitting events ∂P^APP_splt(ϕ)/∂t is suggested to be independent of the parent patch flux:

$$\begin{equation} \frac{\partial P^{\rm APP}_{\rm splt}}{\partial t}(\phi)=k_0=\rm {constant} \ (\rm {for \ all} \ \it {\phi }). \end{equation} \tag{ 7 }$$

This claim is observationally supported at least in the range ϕ > ϕ_th (Figure 9). The drop off below ϕ_th is discussed immediately below. If the splitting ratio between the daughter patches is randomly determined, i.e.,

$$\begin{equation} \frac{\partial }{\partial x} [ k(x,\phi -x)] = 0 \ (\rm {for} \ 0 < \it {x} < \phi), \end{equation} \tag{ 8 }$$

then we obtain,

$$\begin{equation} k(x,y)=\frac{k_0}{x+y}. \end{equation} \tag{ 9 }$$

When the flux content of the daughter patch is below ϕ_th, such events are not recognized as a splitting event in our procedure. The probability distribution will be given as

$$\begin{equation} \frac{\partial P^{\rm APP}_{\rm splt}}{\partial t}(\phi) = \int _{\phi _{\rm {th}}}^{\phi -\phi _{\rm {th}}}k(x,\phi -x)\, dx = k_0 \left(1-\frac{2 \phi _{\rm {th}}}{\phi }\right). \end{equation} \tag{ 10 }$$

The black and blue dashed curves in Figure 9 indicate analytical curves with k₀ = 1.0 × 10⁻³ s⁻¹ and k₀ = 5.0 × 10⁻⁴ s⁻¹, respectively. This curve fits the drop of the observational line well, which supports the above assumptions. We obtain k(x, y) = k₀/(x + y) as a splitting function, where k₀ = 5.0 × 10⁻⁴ s⁻¹.

Since our observations show that merging and splitting are much more frequent than emergence and cancellation, it suggests that the former two have much influence on the maintenance of the power-law distribution. We show the time-independent solution by splitting, although we have not found the time-independent solution by merging and splitting. The magneto-chemistry equation is consistant with the source (emergence) term, merging terms, splitting terms, and cancellation terms (see the right-hand side of Equation (3) in Schrijver et al. 1997). The frequency of emergence, merging, splitting, and cancellation are represented by S(ϕ), l(x, y), k(x, y), and m(x, y), respectively. We use the magneto-chemistry equation only, including the splitting terms, by setting S(ϕ) = 0, l(x, y) = 0, m(x, y) = 0, namely,

$$\begin{equation} \frac{\partial n(\phi)}{\partial t} = 2\int _{\phi }^{\infty }n(x)k(\phi,x-\phi)\, dx - \int _{0}^{\phi }n(\phi)k(x,\phi -x)\, dx, \end{equation} \tag{ 11 }$$

where n(ϕ) is the frequency distribution of the flux content.³

After substituting k(x, y) = k₀/(x + y) into Equation (11) and differentiating with ϕ, we obtain

$$\begin{equation} \frac{\partial ^2n(\phi)}{\partial \phi \partial t} = - \frac{k_0}{\phi ^2}\frac{\partial }{\partial \phi }[\phi ^2n(\phi)]. \end{equation} \tag{ 12 }$$

This equation has a time-independent solution n(ϕ)∝ϕ⁻². This power-law index of the flux content is in good agreement with the observational result. This scale-free distribution comes from constancies of splitting, namely, that splitting has a constant timescale independent of the flux content and a constant probability of splitting the flux content. These constancies may come from convection dominating patch stability or flux tube instability with a constant timescale. We need further theoretical and observational studies to determine which of the hypotheses is the actual scenario on the solar surface.

We suggest a relationship model among the frequency distribution of the flux content, cancellation, and emergence. The important hypothesis in this model is that the frequency distribution of the flux content is rapidly maintained regardless of cancellation and emergence, which is supported by this study. Figure 10 shows a schematic view of this model. We put a power-law distribution of the flux content in units of patches Mx⁻¹ cm⁻² as

$$\begin{equation} n(\phi)=n_0 \left(\frac{\phi }{\phi _0} \right)^{-\gamma }, \end{equation} \tag{ 13 }$$

where ϕ₀ is a reference value of the flux content and n₀ is a reference frequency of the flux content. The power-law index, γ, is derived as 1.5 < γ < 2 by our observation and previous studies (Parnell et al. 2009; Zhang et al. 2010). The maximum flux content in the system (ϕ_max) is assumed to be much larger than the minimum (ϕ_min) in the following discussion. We calculate N(ϕ), a total patch number with a flux content larger than ϕ, by integrating a flux distribution from ϕ to ϕ_max as

$$\begin{equation} N(\phi)=\int _{\phi }^{\phi _{\rm max}}n(\phi ^{\prime }) \, d\phi ^{\prime } \approx \frac{n_0 \phi _0}{\gamma -1} \left(\frac{\phi }{\phi _0} \right)^{-\gamma +1}. \end{equation} \tag{ 14 }$$

Schrijver et al. (1997) evaluated a collision rate of opposite patches from a total patch number density with assumptions of a constant velocity and a randomness of patch motion along a network. They obtained the collision frequency, ν, as

$$\begin{equation} \nu = \frac{v_0}{4 \sqrt{\rho }}N_t^2, \end{equation} \tag{ 15 }$$

where v₀, ρ, and N_t mean a typical velocity of patches, a number density of network cells, and a number density of patches, respectively. We multiply 1/2 taking the double counting into account. We obtained that the frequencies of merging and splitting are larger than that of cancellation in this study. It can be deduced that the frequency distribution is maintained rapidly by merging and splitting compared to the timescale of cancellation. This enables us to treat that the number density of patches is time independent in the evaluation of cancellation and apply the same analogy to the number density expanded in the dimension of the flux content. The collision frequency,.∂N(ϕ)/∂t|_col, is evaluated as

$$\begin{equation} \left. \frac{\partial N(\phi)}{\partial t} \right|_{\rm col}= - \frac{v_0 n_0^2 \phi _0^2}{4(\gamma -1)^2 \sqrt{\rho }} \left(\frac{\phi }{\phi _0} \right)^{-2\gamma +2}. \end{equation} \tag{ 16 }$$

Note that this total collision number becomes time independent with an assumption of maintenance of a power-law flux distribution. We assume that the total number of cancellation, .∂N(ϕ)/∂t|_cnc, equals the total number of collision events of the opposite polarities, .∂N(ϕ)/∂t|_col, namely,

$$\begin{equation} \left. \frac{\partial N(\phi)}{\partial t} \right|_{\rm {cnc}} = \left. \frac{\partial N(\phi)}{\partial t} \right|_{\rm {col}}. \end{equation} \tag{ 17 }$$

The frequency distribution of cancellation, .∂n(ϕ)/∂t|_cnc is given by a differentiating Equation (16) with ϕ, namely,

$$\begin{equation} \left. \frac{\partial n(\phi)}{\partial t} \right|_{\rm {cnc}}= \left. \frac{\partial ^2 N (\phi)}{\partial \phi \partial t} \right|_{\rm {cnc}} = - \frac{v_0 n_0^2 \phi _0}{2(\gamma -1) \sqrt{\rho }} \left(\frac{\phi }{\phi _0} \right)^{-2\gamma +1}. \end{equation} \tag{ 18 }$$

This assumption means that there are no patches passing through the patches of opposite polarity once they collide, which is difficult to check and we justify this assumption from the comparison of the obtained frequency distribution of emergence in this model and that in the observation. We also evaluate a frequency distribution of emergence with assumptions of a small amount of flux supply from outside of the system and re-emergences of canceled fluxes. These assumptions lead to the relationship that a frequency distribution of emergence nearly equates that of cancellation,

$$\begin{equation} \left. \frac{\partial n(\phi)}{\partial t} \right|_{\rm {emrg}} \approx - \left. \frac{\partial n(\phi)}{\partial t} \right|_{\rm {cnc}} = \frac{v_0 n_0^2 \phi _0}{2(\gamma -1)\sqrt{\rho }} \left(\frac{\phi }{\phi _0} \right)^{-2\gamma +1}. \end{equation} \tag{ 19 }$$

We compare the power-law index and the absolute value of the frequency distribution of emergence in this model with the observational results. Based on the above discussion, our observation suggests a frequency distribution of emergence as .∂n/∂t|_emrg ∼ 4.3 × 10⁻³⁵ × (ϕ/1.0 × 10¹⁶ Mx)^−2.6 Mx⁻¹ cm⁻² s⁻¹, where we adopted ρ = 1.0 × 10⁻¹⁹ cm⁻² (Hagenaar et al. 1997) and v₀ = 2.0 km s⁻¹. Thornton & Parnell (2011) reports the power-law frequency of emergence as .∂n/∂t|_emrg ∼ 2.5 × 10⁻³⁵ × (ϕ/1.0 × 10¹⁶ Mx)^−2.7 Mx⁻¹ cm⁻² s⁻¹. The steepness of the power-law distribution and the absolute value in our model are in good agreement with the observational results.

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We calculate the flux replacement time by cancellation and emergence. The total flux amount in the system is calculated from Equation (13) as

$$\begin{equation} \Phi _{\rm tot} = \int _{\phi _{\rm min}}^{\phi _{\rm max}} \phi ^{\prime } \, n(\phi ^{\prime }) \, d\phi ^{\prime } \approx \frac{n_0 \phi _0^2}{2-\gamma } \left(\frac{\phi _{\rm max }}{\phi _0} \right)^{-\gamma +2}. \end{equation} \tag{ 20 }$$

Since γ < 2, this result shows that the total flux is dominated by patches with a larger flux content. On the other hand, we obtain a total flux loss amount by cancellation, .∂Φ_tot(ϕ)/∂t|_cnc, from Equation (18) as

$$\begin{eqnarray} \left. \frac{\partial \Phi _{\rm tot }}{\partial t} \right|_{\rm {cnc}} &=& \int _{\phi _{\rm {min}}}^{\phi _{\rm {max}}}\phi ^{\prime } \left. \frac{\partial n(\phi ^{\prime })}{\partial t} \right|_{\rm {cnc}} \, d\phi ^{\prime } \nonumber\\ & \approx & - \frac{v_0 n_0^2 \phi _0^3}{2(\gamma - 1)(2 \gamma - 3) \sqrt{\rho }} \left(\frac{\phi _{\rm {min}}}{\phi _0} \right)^{-2\gamma +3}. \nonumber\\ \end{eqnarray} \tag{ 21 }$$

The total flux supply by recycled emergence is also evaluated in the same manner from Equation (19) as

$$\begin{eqnarray} \left. \frac{\partial \Phi _{\rm tot }}{\partial t} \right|_{\rm {emrg}} &=& \int _{\phi _{\rm {min}}}^{\phi _{\rm {max}}}\phi ^{\prime } \left. \frac{\partial n(\phi ^{\prime })}{\partial t} \right|_{\rm {emrg}} \, d\phi ^{\prime } \nonumber\\ & \approx & \frac{v_0 n_0^2 \phi _0^3}{2 (\gamma - 1)(2 \gamma - 3) \sqrt{\rho }} \left(\frac{\phi _{\rm {min}}}{\phi _0} \right)^{-2\gamma +3}. \end{eqnarray} \tag{ 22 }$$

Then flux replacement time is evaluated as

$$\begin{eqnarray} \tau _{\rm {replace}} & = &\Phi _{\rm tot}/(\left. \partial \Phi _{\rm tot}/\partial t \right|_{\rm cnc}) =\Phi _{\rm tot}/(\left. \partial \Phi _{\rm tot}/\partial t \right|_{\rm emrg}) \nonumber \\ & = &\frac{2(\gamma - 1) (2 \gamma - 3)}{(2 - \gamma) v_0 n_0 \phi _0 \sqrt{\rho }} \left(\frac{\phi _{\rm {max}}}{\phi _0} \right)^{-\gamma +2} \left(\frac{\phi _{\rm {min}}}{\phi _0} \right)^{2 \gamma -3}. \nonumber\\ \end{eqnarray} \tag{ 23 }$$

This result is qualitatively consistent with the previous result that the flux replacement time becomes shorter with higher resolution (Martin et al. 1985; Schrijver et al. 1998; Hagenaar 2001).

We summarize our interpretation from the discussion. In our interpretation, a power-law frequency distribution of the flux content is rapidly maintained by merging and splitting, which is supported by comparing the change rate of the flux amount in each process. In addition, we have found that a power-law frequency distribution with an index of −2 is a time-independent solution of splitting. Cancellation is caused by convective dominant motion and frequency distribution of cancellation should naturally become a steep power-law distribution. Emergence is interpreted as re-emergence of submerged flux by cancellation, which is consistent with a steep frequency distribution of emergence (Thornton & Parnell 2011). This should be one of the possibilities but the important point of this model is that the apparent flux transport through the photospheric layer becomes much more drastic when investigating emergence and cancellation, although the injected flux amount from the deeper layer is small or even zero. A recent paper, Meyer et al. (2011), reports their numerical simulations of the magnetic carpet on the photosphere. They show that a power-law distribution of the flux content is maintained with an input of a steep power-law frequency of emergence. The induced frequency of cancellation becomes a steep power-law one, which is consistent with our model.

What should be investigated for a further understanding of flux transport is a statistical investigation of cancellation on the photosphere. Another important question is whether there is a stable solution of our model with four magnetic processes or not. We will investigate these questions in this series of papers.