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Statistical Study on the Nature of Solar Flux Emergence

We studied 101 flux emergence events ranging from small ephemeral regions to large emerging flux regions which were observed with Hinode Solar Optical Telescope filtergram. We investigated how the total magnetic flux of the emergence event controls the nature of emergence. To determine the modes of emergences, horizontal velocity fields of global motion of the magnetic patches in the flux emerging sites were measured by the local correlation tracking. Between two main polarities of the large emerging flux regions with more than around 2 \times 10^19 Mx, there were the converging flows of anti-polarity magnetic patches. On the other hand, small ephemeral regions showed no converging flow but simple diverging pattern. When we looked into the detailed features in the emerging sites, irrespective of the total flux and the spatial size, all the emergence events were observed to consist of single or multiple elementary emergence unit(s). The typical size of unitary emergence is 4 Mm and consistent with the simulation results. From the statistical study of the flux emergence events, the maximum spatial distance between two main polarities, the magnetic flux growth rate and the mean separation speed were found to follow the power-law functions of the total magnetic flux with the indices of 0.27, 0.57, and -0.16, respectively. From the discussion on the observed power-law relations, we got a physical view of solar flux emergence that emerging magnetic fields float and evolve balancing to the surrounding turbulent atmosphere. Key words: Sun: magnetic fields - Sun: emerging flux - Sun: photosphere - Sun: chromosphere


Introduction
The sites where the subsurface magnetic flux tubes emerge on solar surface are called emerging flux regions (Bruzek 1969;Zirin 1972). The typical simple emerging flux region has one main pair of the opposite magnetic concentrations at the both ends of the emerging site. The opposite magnetic concentrations of the main pair move away from each other with the speed of 1-2 km −1 (Zwaan et al. 1985;Brants 1985) in the developing phase of the region. Inside the developing emerging site there are many magnetic flux tube emerging on the photosphere. The newly emerged magnetic flux tubes at the emerging site appear as the dark granular lanes (Loughhead and Bray 1961;Brants and Steenbeek 1985) on the photosphere. At the both ends of the dark granular lane, there are the magnetic concentrations which are called the footpoints and observed as the bright points in G band image (Otsuji et al. 2007). With Hα the emerged flux tubes are observed as dark arch filaments (Bruzek 1967). The lifetime of an arch filament is 10-30 minutes (Bruzek 1967;Chou and Zirin 1988). The rise velocity of arch filaments is 10-15 km s −1 (Bruzek 1969;Chou and Zirin 1988). Otsuji et al. (2010) found the deceleration of the apex of the small-scale arch filaments in the chromosphere.
Emerging flux regions show a variety of size, lifetime, total magnetic flux and field strength. Especially the small emerging flux regions are called ephemeral active region (EAR; Harvey and Martin 1973). For convenience, in the following page we define larger (i.e. non-EAR) emerging flux regions as EFRs. EFRs are produced by fairly largescale flux emergence. They have a pair or more complex group of sunspots with definite penumbrae. The typical size of EFRs is more than 30 Mm (Bruzek 1967). EFRs show their emergence activities for several days and exist on the solar surface for a few month at the maximum. The total flux in an EFR increases with the rate of 10 20 Mx hr −1 and reaches to the order of 10 20 -10 22 Mx (Zwaan 1987). In the fairly developed EFR the field strength of main spot is around 3,000 Gauss (Brants and Zwaan 1982).
EARs have simple bipolar configuration. They have no penumbra in the sunspots. The typical size varies from 5 Mm to 30 Mm (Harvey and Martin 1973;Harvey et al. 1975;Hagennar 2001;Otsuji et al. 2007). EARs have short life time of hours or one day. The total flux in an EAR is up to 10 20 Mx with the increase rate of 10 19 Mx hr −1 . The magnetic field strength of main spot is from a few times 100 Gauss (Martin 1988) to 2,000 Gauss (Brants and Zwaan 1982).
Various simulation studies on flux emergence have been performed by many researchers (Shibata et al. 1989;Matsumoto and Shibata 1992;Fan 2001). They showed the simple bipolar emergence simulation which corresponds to the observation result. Matsumoto et al. (1993) and Magara and Longcope (2001) simulated the three-dimensional magnetohydrodynamics (MHD) of the emerging magnetic flux. Nozawa (2005) and Murray et al. (2006) also performed the MHD simulations of flux emergence with the sheared or twisted flux tube. They found that the flux tube with shear or twist emerges faster than that without any shear and twist. Nozawa et al. (1992) performed the MHD simulation of flux emergence in a sheet geometry. The initial stable flux sheet in the convective zone was perturbed with various wavelengths, which correspond to the convective motion. They found that irrespective of the wavelength of initial perturbation, a finite "most unstable wavelength" is excited. This wavelength (2-4 Mm) is inherent in the Parker instability (Parker 1966). As a result, the flux sheet is undulated and the apexes of the convex field line (Ω-loops) appear consecutively on the photosphere. Some of dipped field lines (U-loops) also emerge to form the regions called "bald patches (Titov et al. 1993)".
On the other hand, observational study on a large EFR and bald patches was performed by Pariat et al. (2004). They observed a fairly large (∼ 30 Mm) EFR with the magnetogram and found that the emerged field lines undulate vertically. They revealed that there are many bald patches between the main spots. These results confirmed that the emerging flux tube does not rise altogether at a time, but each Ω-loop component rises individually. They proposed this model as "resistive emergence model". The distance between two consecutive bald patches is in the range of 2-6 Mm, which is consistent with a theoretical argument on the flattening of emerging magnetic field just below the surface and its critical emergence length, first presented by Magara (2001).
Recently Isobe et al. (2007) further developed the simulation performed by Nozawa et al. (1992) and obtained the result in which the undulated field line caused reconnections with neighboring Ω-loops and created larger loops. Archontis and Hood (2009) performed the threedimensional MHD simulation of the emergence of undulating fieldlines. These reconnection events are interpreted as a sources of Ellerman bombs (Ellerman 1917;Kurokawa et al. 1982;Kitai 1983;Matsumoto et al. 2008a;Matsumoto et al. 2008b;Watanabe et al. 2008).
As stated above, the criteria of the bald patches formation are still uncertain. Furthermore, in the latest simulation, the footpoints of emerged flux loops showed converging motion toward the bald patches on the photosphere (Cheung et al. 2010). Although this converging motion was observed in preceding studies (Strous et al. 1996;Strous and Zwaan 1999;Bernasconi 2002;Cheung et al. 2008), statistical analysis on that phenomenon with respect to the size and magnetic characteristics of the flux emergences has not been done yet.
To clarify the criteria of forming bald patch and converging flow, we performed a statistical study about the nature of magnetic flux emergences using SOT. The flux emergence phenomena from small EARs to large EFRs observed by SOT were investigated on their morphological and magnetical characteristics. Furthermore, we derived the the relations between the total magnetic flux and the maximum spatial size, the flux growth rate and the mean separation speed of the emergence event to clarify how the total flux amount controls the entire evolution of the emergence.

Observation and Data Selection
Hinode satellite has observed solar surface for over 4 years with SOT. SOT has Broadband Filter Imager (BFI) and Narrowband Filter Imager (NFI). Ca II H (3968.5Å) filtergrams were taken by BFI with the bandpass of 3Å. Fe I (6302Å) and Na I D (5896Å) polarimetric data in solar photosphere and chromosphere were observed with NFI. To search for the emerging flux phenomena, we used Hinode daily quicklook movies 1 . Among the possible candidates, we selected 101 emerging flux phenomena according to the criteria as follows: (1) conspicuous presence of separating bright points in Ca II H image and/or opposite polarities in Fe I or Na I D image, (2) location fairly inside the solar limb (α = arcsin(r/R) ≤ 70 • , where r is the distance from solar disk center to the location of the event and R is the solar radius) (3) observational time span longer than 1 hour, and (4) observational cadence higher than 10 minutes. The period in which we studied is between 2006 November 26 to 2010 August 23. The detailed observational information of the emerging flux is available in the associated electronic tables 2 .

Data Reduction
In this section we give the description of the data reduction with the data of EFR 20061201 (# 2 in the electronic table) as an example. First, dark-current subtraction and flat fielding were performed on the obtained SOT data in the standard manner. Then we processed every observed data as described below.
First, we used SOHO -Michelson Doppler Imager (MDI; Scherrer et al. 1995) magnetogram data to calibrate SOT polarimetric data. We compensated the differential rotation of two consecutive MDI data observed before and after the SOT observation and interpolated them by time to estimate the distribution of magnetic field at the time of SOT observation. Next we deteriorated SOT polarimetric data with the spatial resolution of MDI magnetogram (2 ′′ ). Using the deteriorated SOT polarimetric data and MDI magnetogram, we made a scatter plot of SOT polarimetric signal (Stokes V /I) to MDI field strength ( Figure  2). The correlation coefficient of SOT V /I to MDI field   strength was 0.95. We performed linear fitting on the scatter plot and obtained the conversion equation from SOT V /I to the photospheric field strength B p , expressed as In the case of Figure 2, the offset value B 0 and the scaling factor B 1 were 4.029 Gauss and 12528 Gauss, respectively. Note that the scatter of data points in the figure is mainly due to the Doppler effect arising from the satellite's orbital motion. With this method we converted the SOT polarimetric signals to the magnetic flux densities for all the samples.
In measuring the actual size of the solar features, we compensated the projection effect. Then we applied the subsonic filter of 3 minutes to the SOT Ca II H image sequence in order to suppress chromospheric oscillatory motions ( Figure 3b).

Analysis
As is stated in the introduction, mutually approaching anti-polarity patches were observed in undulating resistive emerging phenomena (Strous and Zwaan 1999). We studied the formation process of the converging motion between the opposite polarities in EFR and EAR by analysing the morphological, dynamical and magnetic evolutions of our SOT samples. In this section we introduce our analysis methods using the data of EFR 20061201.

Morphological Evolution
The morphological evolution of the magnetic flux emergence was analysed with two methods. One is the method of tracking magnetic elements using local correlation tracking (LCT), and the other using time-sliced diagram.

Local Correlation Tracking
The local correlation tracking (LCT) method is commonly used to derive the horizontal velocity field (November and Simon 1988;Berger et al. 1998;Matsumoto and Kitai 2010).
For LCT, we used flowmap.pro in SSW of IDL. Flowmap.pro calculates the two dimensional vector flowfield by following the subtiles in the time series of two dimensional images.
To examine the motion of the footpoints of flux tubes for all over the emerging site, we performed LCT on SOT magnetogram data and obtained velocity field of moving magnetic elements (Figure 3c). The size of tracking subtile for LCT was 0. ′′ 5. To reduce the velocity noise due to LCT error, the velocity fields were averaged both spatially and temporally over the zones of 1 ′′ × 1 ′′ and 10 minutes, respectively. The standard deviation of the velocity field inside the data cube of 1 ′′ × 1 ′′ × 10 minutes was ∼ 0.1 km s −1 .
Then we derived divergence of horizontal velocity field (Figure 3d). To emphasize global and sustained flows, the divergence maps were averaged both spatially and temporally. The spatial average was performed with the width of 10 ′′ for large EFR and 3 ′′ for small EAR. The threshold between large EFRs and small EARs was fixed to be 40 ′′ , which is the upper limit size of ephemeral active regions indicated by Harvey and Martin (1973). For EFR 20061201, the spatial averaging box size was 10 ′′ . The temporally averaging period was taken as 10 minute for all the events.

Time-sliced Diagram
To clarify the dynamics of footpoints more quantitatively, we made time-sliced diagram of SOT magnetograms ( Figure 3e). The spatial slit was located parallel to the axis of the EFR (shown in Figure 3a). From the time-sliced diagram, we derived the maximum distance d max between the main spots, the mean separating speed v of the main spots. Detailed footpoints motions, such as mutually approaching anti-polarities, were examined in this diagram.

Temporal Evolution of Magnetic Field
To investigate the temporal evolution of emerging magnetic flux, we measured the total flux within the emerging site. Positive and negative fluxes were summed up separately. The total flux of the emerging region Φ was derived by subtracting the fluxes at the initial time. We plotted the variation of positive and negative fluxes, respectively ( Figure 3f). From this plot, we derived the maximum amount of unsigned total magnetic flux Φ max and the unsigned flux growth rate dΦ/dt = Φ max /T . The growth rate is defined as the total magnetic flux Φ max divided by the continuously emerging period T . Note that the unbalance between positive and negative fluxes in the sample plot was due to the flow of the positive flux out of the field of view.

Results
First, we introduce the sample results for large EFR and small EAR at section 4.1 and 4.2, respectively. Then the statistical results are shown in section 4.3.

Large EFR 20061201
In Figure 3a, there are two main spots aligned eastwest direction. The size of the main spots was 10 ′′ -20 ′′ . The field strength of the spots was ±1.8 × 10 3 Gauss at the maximum. Although the following spot is missing in the Ca II H image because of the field of view limitation (Figure 3b), there are sunspots locating at the same position with the preceding negative spots in the magnetogram. The velocity field shows prominent outward motions of the main spots ( Figure 3c). There are positive divergence areas inside the main spots (Figure 3d), which indicates the flux emergences. From the time-sliced diagram (Figure 3e), we can estimate the speed of main spots to be ∼ 0.3 km s −1 for each, thus the mean separating speed v is about 0.6 km s −1 . At the end of the observation period, the distance between two main spots increased to around 100 ′′ . We considered this value as the maximum distance d max . The emergence started at 16:00 UT on 1 December and lasted until the observation end at 24:00 UT on 2 December. Thus the active emergence period was taken as 32 hours. The maximum amount of total magnetic flux were 1.3 × 10 21 Mx for positive polarity and 3.1 × 10 21 Mx for negative polarity (Figure 3f). As the following positive spot flowed out of the field of view, measured positive flux was less than that of negative one. Thus we took the maximum negative flux as Φ max . Mean flux growth rate dΦ/dt for this event was 9.7 × 10 19 Mx hr −1 .
Let us look at the central part of the region where there are many small positive or negative magnetic patches (Figure 3a). These patches correspond to the Ca II H bright points (Figure 3b). Although the magnetic patches seem to move with apparently random velocities in Figure  3c, these patches are located in the converging region of divergence map (Figure 4d). Thus these magnetic patches accumulated and stagnated to the localized area. In fact, the time-sliced diagram shows that these magnetic patches actually approach to each other with speed of ∼ 1 km s −1 . So we identify these mutually approaching area as a U-loop formation in the EFR. In the following, we denote the area as the "stagnation zone (SZ)", where the anti-polarities mutually approach and accumulate.

Small EAR 20070213
We selected a small-scale magnetic emergence event which emerged on 13 February 2007 (# 9 in the electronic table) as the sample case of EAR. Figure 4a shows the magnetogram of the region. Two magnetic concentrations in the magnetogram correspond to the Ca II H bright points in Figure 4b. The field strength of the two magnetic concentrations was ±300 Gauss at the maximum. The velocity field of the small EAR shows separative and anti-clockwise rotational motion of two magnetic concentrations ( Figure 4c). Figure 4d is the divergence map derived from the velocity field and averaged spatially with 3 ′′ . The divergence map shows the positive area at the central region of the EAR, which indicates that there is no converging flow inside the emerging site. In the timesliced diagram we can see the simple separating motion of two main magnetic concentrations and no stagnation zone (Figure 4d). The maximum distance between the main concentration d max was 10 ′′ . The mean speed of separating motion for main concentrations v is about 1.5 km s −1 . The temporal evolution of total flux shows that the maximum amounts of total fluxes are 5. Mx hr −1 .

Statistical Result
Table 1 in Appendix 1 shows the measured quantities of all the samples. If there was no magnetic observation, Ca II H data were used to derive d max and v , and to judge the existence of SZs. There is no data of the total flux Φ max and the flux growth rate dΦ/dt for the observations without the magnetogram. Figure 5 shows the statistical characteristics of measured quantities.

Existence of SZ
In Figure 5a, the histogram of the maximum separation distance d max indicates that SZs are rarely found in small d max regions. The flux emergence phenomena with the separation size of more than 25 Mm always have SZs. Below the threshold of 25 Mm, there are both EFRs and EARs with/without SZs. To clarify the conditions to have SZs, we categorized all the events to three groups as follow. Group I: maximum separation distance d max ≥ 25 Mm and with SZs. Group II: maximum separation distance d max < 25 Mm and with SZs. Group III: maximum separation distance d max < 25 Mm and without SZ. Group I, II and III are indicated in the Figure 5 by white, gray and black bars/squares, respectively. Figure 5b shows the histogram of the maximum fluxes Φ max , where we present the maximum flux separately for these three groups. Group I has the maximum flux of Φ max ∼ 10 21 Mx, while group II and III have Φ max ∼ 10 20 Mx and ∼ 10 19 Mx, respectively. Figure 5c shows the relation between the maximum flux Φ max and the maximum separation distance d max . In the plot the three group I, II and III are clearly separated. The maximum separation distance d max depends on the maximum flux amount Φ max . The scatter plot implies a power-law relation of where d max is in Mm and Φ max is in Mx. We can see the trend in which the small EARs have low values of maximum flux while large EFRs have high values of maximum flux. The power-law relation is consistent with the result of Hagennar (2001), although the index of power was 0.18 instead of 0.27. From the result of categorization, we found that the SZ features are associated with the magnetic flux emergence of more than around 2 × 10 19 Mx.

Size of Elementary Flux Emergence
We also measured the typical size of elementary structures of emergence d unit defined as the distance between two footpoints of individual Ω-loops at their emergence period. Some small footpoints of emerged loops might be finally transported to the border of supergranules by the local convection. Others cause cancellation between the opposite polarities and disappear, which enlarges the distance between two footpoints of individual Ω-loops, (i.e. d unit ). Thus the d unit varies with time. For the accurate descriptions, we adopted the d unit at the epoch when the Ω-loops were observed as Ca II H filaments (∼10 minutes after the start of the emergence; Otsuji et al. 2007). Figure 5d shows the scatter plot of d max and d unit , which suggest that d unit takes the values in the range of 2-6 Mm irrespective of d max . Thus elementary and unitary Ω-loops in any emerging flux region has a common size of around 4 Mm, which is consistent with the most unstable wavelength (2-4 Mm) of Parker instability and preceding observation/simulation studies (Magara 2001;Pariat et al. 2004;Isobe et al. 2007).

Magnetic Flux Evolution
The relation between the maximum flux Φ max and the flux growth rate dΦ/dt is shown in Figure 5e. In the scatter plot, the data points distribute along the relation of dΦ dt = Φ max T = 9.6 × 10 7 Φ 0.57 max , where T is the emergence duration, dΦ/dt is in Mx hr −1 and Φ max in Mx. From equation (3), we can derive the emergence duration T in the unit of hour as a function of Φ max , which is Equation (4) indicates that an emergence event with large maximum flux shows relatively rapid magnetic flux growth. According to equation (4), when a flux tube of Φ 0 emerges with T 0 , the flux tube with 2Φ 0 emerges with T 1 = 2 0.43 T 0 ≈ 1.4T 0 . T does not depend linearly but nonlinearly on Φ max . A tube with more magnetic flux emerges with less time than in the case of linear dependency. While the equation (4) is consistent with previous observations such as Zwaan (1987) and Hagennar (2001), the empirical relation (4) was first derived with wide range of magnetic parameters by Hinode high-resolution samples. Figure 5f presents the relation between Φ max and v , which indicates that the larger size EFRs show the sep-arating speed less than 1 km s −1 , while the small scale EARs footpoints separate with various speed up to 4 km s −1 . The mean separating speed v can be written as

Relation between Footpoints Separating Speed and Maximum Flux
From equations (2) and (4), equation (5) where v is in km s −1 and Φ max in Mx. Equation (6) indicates that the footpoints of emerged flux tube with less magnetic flux separate each other with larger speed. This relation is plotted with solid line in Figure 5e, which is consistent with the observed values.

Size and Flux Dependence of SZ Formation
The SZ features are associated with the magnetic flux emergence of more than around 2 × 10 19 Mx. Magara and Longcope (2003) performed three-dimensional MHD simulation of emerging magnetic flux and suggested that emerging field lines take the evolutionary path of a simple expansion if they emerge with a large aspect ratio (the ratio of their height to their footpoint distance); otherwise, field lines are inhibited from expanding and they show an undulating behavior (i.e. SZ formation). Our results provide a new criterion of the total magnetic flux regarding the formation of SZs.

Flux Dependence of the Spatial Size of Flux Emergences
Another notable result is that the flux emergence phenomena with/without SZ follow the relation between the total flux and the maximum spatial size as described in equation (2). This relation is derived from wider range of magnetic parameters than the previous studies (Bruzek 1967;Zwaan 1987;Harvey and Martin 1973;Harvey et al. 1975). There authors gave the result for only narrow range of magnetic parameters. We present a comprehensive result on this relation compared to the previous works.
Let us try to derive the power-law relation between the maximum flux Φ max and the maximum separation distance d max , from the viewpoint of simple dimensional analysis. Figure  6 shows the schematic image of flux emergence from the convection layer. First, d max is estimated as follows. Initially horizontal flux tube in the convection layer will rise with the typical length λ = 10 ∼ 20H P where H P is the local pressure scale height. The maximum separation distance d max between two main spots depends on the most unstable wavelength of Parker instability λ at the initial depth of the flux tube, Thus d max is proportional to H P at the depth where initially the flux tube locates. Next the total flux Φ max is estimated as follows. From equipartition arguments, the magnetic and kinetic energy in the solar convection layer will balance to each other, where B, ρ and v conv are the field strength inside the flux tube, the mass density around the flux tube and the mean convection velocity. From mixing length theory (Stix 1989), v is given as From equations (9) and (10), Now we assume that the solar convection layer can be approximated by an adiabatically stratified atmosphere (Foukal 2004), T and γ are temperature and adiabatic index γ = c P /c V , where c P and c V are the specific heats at constant pressure and volume, respectively. The local scale height H P is proportional to the temperature T , thus From equation (11) and (13), Let us think about the flux tube width w. If the w is much larger or smaller than the local mixing length l ∝ H P , the flux tube will be disintegrated by the convection flows or accumulated at the convection boundary. Thus the flux tube width is expected to be comparable to the mixing length, Thus the total flux Φ max can be estimated as From equation (8) and (16), Thus the power-law index of the relation between the maximum flux Φ max and the maximum separation distance d max is derived to be α 1 = 2(γ−1) 5γ−4 . For example, α 1 is 0.30 with the adiabatic index γ = 5/3 (ideal gas case realised in deep convective layers). With γ ∼ 4/3 at near the solar surface where the ionization status is changing rapidly (Bhatnagar and Livingston 2005), α 1 is 0.25. These calculated values are comparable to the observed value α 1 = 0.27. From the argument above, we get the view of the emergence depicted as in Figure 6. Magnetic tubes of large flux are anchored in deep layers and appear with large separation between two main spots on the solar photosphere.

Flux Growth Rate
We derived the relation between the maximum flux Φ max and the flux growth rate dΦ/dt to be equation (3). By a simple model of an emerging tube with uniform magnetic flux density, let us try to derive the power-law relation between the maximum flux and the flux growth rate, Figure 7 shows the schematic image of the flux tubes just beneath the photosphere. Around the photosphere, the plasma β is almost 1. Thus the magnetic pressure B 2 /8π ∼ P = const. where P is the gas pressure at the photosphere. So we can think that flux densities are nearly constant irrespective of the spatial size or total flux of the magnetic tube, The rise velocity v of the flux tube is estimated as follows. When the apex of flux tube reaches to the underneath the solar surface, the rise motion is suppressed and the tube top becomes flattened (Magara 2001). Our observation (Section 4.3.2) showed that the emergence occurs in unitary form irrespective of total magnetic flux. In these situations the rise velocity v from the photosphere will not depend upon the total magnetic flux of the tube, v = const. (20) Assuming that the aspect ratio of flux tube width h/w is constant just beneath the photosphere, where w and h are horizontal and vertical width of the flux tube, respectively. The flux growth rate and the total flux are described as respectively. From equation (22) and (23), the relation between Φ max and dΦ/dt are written as Thus the power-law index is which is consistent with the observed value 0.57. In this discussion we have not considered the factors such as magnetic field stratification inside the flux tube, realistic aspect ratio of tube and so on. If we include these factors then we can get more realistic interpretation of equation (3).

Summary
We investigated the morphological, dynamical and magnetical characteristics of various flux emergence phenomena using high-resolution Hinode SOT data. To estimate the magnetic field density of SOT data we used SOHO MDI magnetogram data for the calibration. From 101 samples of flux emergence events, we derived the total flux, flux growth rate, maximum separation and mean separation speed. The SZ features are associated with the magnetic flux emergence of more than around 2×10 19 Mx. The magnetic flux growth rate, emergence duration and mean separation speed were found to follow the powerlaw functions of the total magnetic flux with the indices of 0.57, 0.43 and -0.16, respectively. The typical size of elementary emergence structures is around 4 Mm, which is consistent with the most unstable wavelength (2-4 Mm) of Parker instability. The mean separating speed v decreases with larger magnetic flux.
We got a physical view of solar flux emergence that emerging magnetic fields float and evolve balancing to the surrounding turbulent atmosphere from the discussion on the observed power-law relations. These observational results should be verified by future numerical studies. Possible influence of twisting or pre-existing magnetic field could be studied with the data of horizontal magnetic field in the emerging site and will be reported in the near future.