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A new consideration on the formation of sunspots

The formation of sunspots has been considered by many researchers in the past. After the discovery of magnetic fields in sunspots by George E. Hale, magnetic fields of sunspots have become one of the main issues of sunspots. Early studies of the formation of sunspots are summarized by Abetti,1 Cowling2, Kiepenheuer3; de Jager4; Menzel5, Bray and Loughhead6. A recent comprehensive review on sunspots was made by Solanki7.


Introduction
The formation of sunspots has been considered by many researchers in the past. After the discovery of magnetic fields in sunspots by George E. Hale, magnetic fields of sunspots have become one of the main issues of sunspots. Early studies of the formation of sunspots are summarized by Abetti, 1 Cowling 2 , Kiepenheuer 3 ; de Jager 4 ; Menzel 5 , Bray and Loughhead 6 . A recent comprehensive review on sunspots was made by Solanki 7 .
Among a few proposed theories, the rising magnetic flux tube model by Babcock 8 has long been accepted as a unique model in considering the formation of sunspots (hereafter, this model is referred as the RF model). In the model, sunspots appear as a pair of positive and negative spots as a magnetic flux tube breaks through the photospheric surface at two points. Thus, a pair of positive and negative forms the basic unit. In spite of a great observational and theoretical progress in studying sunspots during the last few decades, these studies have essentially based on the RF model.
Thus, a few long-standing problems, which are contradictory with the RF model, have remained unsolved until now. They are: i. The presence of magnetic monopole-like single spots.
ii. The delay of appearance of f-spot after p-spot.
iii. The unbalance of the magnetic flux between a p-spot and a fspot, In addition, there are two more recent observations, which are not easily be explained (McIntosh 9 ; see also Akasofu 10,11 ): They are: iv. Positive single spots tend to appear in a positive unipolar region (vice versa).
v. Pairs of spots tend to appear at or near the boundary of two unipolar magnetic regions, not in the middle of unipolar region.
Fortunately, the new efforts of attempting to understanding the observations (4,5) have provided important hints to study the above three problems (1, 2, 3).
The purpose of this paper is to suggest that the five problems (1-5) may be explainable all together by considering: (a) unipolar magnetic fields are one of the basic features in solar magnetism related to the solar internal dynamo, not just remnants of old active regions, (b) single spots are the basic unit of sunspots, not pairs of spots. (c) a spot induces a f-spot across the boundary of two unipolar magnetic fields (of the opposite polarity). This work is mostly based on the magnetic data based on the Kitt Peak Solar observatory.

Unipolar magnetic regions Morphology
Before examining the formation of sunspots, it is important to remind about the distribution of magnetic fields on the photospheric surface. Figure 1 shows an example. First of all, there are weak, but large-scale positive and negative fields. They have been called "large-scale fields", 9 and are unipolar magnetic regions. Secondly, there are much smaller, but concentrated fields. They are pores and small sunspots; note that positive ones are in positive unipolar regions (vice versa). Thirdly, some of these small positive and negative fields are concentrated together near the boundary of two unipolar regions (of the opposite polarity), again positive ones are located in a positive unipolar region (vice versa). Figure 2 shows a typical solar magnetic record in the latitudelongitude map. The region marked "1" are positive (red) and negative (blue) unipolar magnetic regions, respectively. The regions marked by "2" have pores, single spots and groups of single spots (positive and negative). A distinct group of pairs of positive and negative spots near the boundary of two unipolar magnetic regions are marked by "3". As can be seen in later examples, this general pattern is maintained throughout the sunspot cycle, although it becomes more complex when the sun becomes active.
In Figure 2, it may be recognized: (a) groups of positive pores, single spots and groups of spots tend to be present in a positive unipolar regions, and vice versa, (b) pairs of spots tend to be present at or near one of the boundary of two unipolar regions, not in the middle of unipolar regions, and further positive spots are in a Phys Astron Int J. 2018;2(5):408-418.
On the basis of Figure 1 & Figure 2, and a large number of others, including those which will be seen later. Figure 3A shows schematically the basic distribution of the unipolar regions. Positive and negative unipolar regions alternately distributed in longitude.

Figure 2
The distribution of the photospheric magnetic field map (longitudelatitude) during the solar rotation CR 2025. The regions marked by "1" indicate positive (red) and negative (blue) unipolar magnetic regions, respectively. The regions marked by "2" indicate pores, single spots, groups of single spots of positive and negative polarities. The region marked by "3" is the location of pairs of positive and negative pores and spots. There are also unipolar regions in the northern southern highest regions, the polar unipolar magnetic regions (The Kitt Peak Solar Observatory).
In this study, it is crucial to have a continuous color presentation of the magnetic fields, because all the above features have a wide and continuous range of the intensity (0-2000G) from very weak unipolar regions to most intense sunspots; The recent black-white presentation with a high resolution tends to misses the important features, which are essential in this paper; in particular, it is not easy to see the crucial relationship between the unipolar magnetic regions (which tend to have a low intensity) and sunspots; see Appendix.

Unipolar magnetic regions Morphology
It has long been considered that unipolar magnetic regions are remnants of old active regions ( Figure 3B) on the basis of a study of the sunspot cycle by Leighton 12 , in which active regions are elongated by the non-uniform rotation of the sun. In this section, it will be shown that unipolar regions are different from old active regions and thus they should be distinguished. Further, it is suggested that unipolar regions are one of the fundamental features of solar magnetism, which have the origin in the internal dynamo.

(a) Latitudinal extent
One of the important features which distinguish between unipolar regions and old active regions is their difference in the latitudinal extent. Unipolar regions extend from 10° to more than 60°, and often connect to the polar unipolar regions and also connect to unipolar regions of the same polarity across the equator, while the sunspot regions extend from 0° to at most 45° as the Butterfly diagram shows. Figure 4A shows an example of their connection to both the polar and equatorial regions. This polar connection may be related to the poleward migration of unipolar magnetic regions during a certain periods 13 and thus related to the reversal of the dipolar field of the sun and thus to the internal solar dynamo 14 ( Figure 4B).

(b) Development
However, an even more important difference between them is the fact that unipolar regions appear before sunspots are born in them at the early epoch of the sunspot cycle. It is unfortunate that such a fundamental importance of unipolar regions and their importance on the formation of sunspots have not been recognized in the past. Figure 5A represents the solar magnetic field during the earliest epoch and early epoch of a new sunspot cycle. It is particularly important to note that unipolar regions just begin to appear before any sunspots appear. This indicates clearly that unipolar magnetic regions are not remnants of old active regions. This fact becomes more clear in Figure 5B; it can be seen that a pair of unipolar magnetic regions appeared most clearly without sunspots, indicating the growth of unipolar regions without sunspots and thus before the birth of spots. Thus, they cannot be remnants of old active regions. Figure 5C shows two magnetic records of the rising and maximum epochs of the cycle. One can see that unipolar regions grow in intensity and latitudinal range (toward both higher and lower latitudes) as the cycle advances. Sunspot activities tend to distort the basic pattern of unipolar magnetic regions because of sunspot motions as the sunspot cycles advances. Figure 5D shows two records during the declining epoch of the cycle. As sunspot activities subside, the basic pattern begins to be restored. Based on a large number of records during more than two sunspot cycles, Figure 5F summarizes schematically the close relationship among single spots, pairs of spots and unipolar regions, including also the well-known other features of pairs of spots, such as the reversal of the polarity of spots in the northern and southern hemispheres and during the even and odd sunspot cycles, and also the p-spots are located a little equatorward of f-spots. 11 In the following, it will be shown that unipolar magnetic regions have a crucial importance in explaining the five problems (1-5).

Model
Based on the observed facts in this section, it can be said that the unipolar magnetic regions are important in reconsidering morphology of sunspots and that the unipolar regions seem to have the origin in the internal dynamo. Thus, it is worthwhile to refer here to the internal dynamo models of solar magnetism. Among many internal dynamo models, the theoretical model developed by Nakagawa 17 considers specifically the origin of positive and negative unipolar regions as shown in Figure 3C: he noted: "The unipolar regions in Figure 1 are identified with the variations of the magnetic field near the surface, so that the flux coil which appeared near the surface is identified as one polarity, while the submerged region is the opposite polarity."

Summary
In summary, the fact that the latitudinal extent and growth during the sunspot cycle, together with the fact that unipolar regions can be connected to the polar unipolar regions, as well as Nakawawa's internal dynamo model, suggest that unipolar magnetic regions are one of the basic features in solar magnetism related to the internal dynamo. Thus, unipolar magnetic regions are different from remnants of old active regions, 15,16 so that they should be distinguished from old remnants of active regions.

Single spots Pores and single spots
Although single magnetic monopole-like spots (or often called 'unipolar', 'isolated' or 'independent' spots) are illustrated in many books or review papers, 1,6,7,18 they have not been considered in recent years. Figure 6 shows two examples of a single spot. Since sunspots must appear as a pair of positive and negative spots in the RF model, the presence of single, isolated or independent spots has long been a puzzle, in addition to the fact that magnetic mono-poles cannot exist. Single spots we study here should not be confused with some of scattered single spots in old active regions during a late phase of active sunspot groups, the βγ classification in Bray & Loughhead 6 ; they will be referred later.

Figure 5F
Assembly of the available information regarding, single spots, sunspot pairs and their relationship among positive and negative unipolar regions, for both even and odd cycles. 11 Other known facts, such are the reversal of the polarity with respect the equator and sunspot cycle are also included.
Since magnetic monopoles cannot exist, we try to consider the formation of single pors and spots as a local phenomenon within unipolar magnetic regions, namely a positive single spot occurs in a positive unipolar region (vice versa). The clue for this consideration is the observation (4) that positive single spots appear in a positive unipolar region (vice versa). Figures 7A & 7B show two high resolution images of a single spot. They show that a single spot consists of several pores or small spots, 19 which indicate that a single spot is an assembly of several small spots, not a simple column of magnetic flux. There is a possibility that βγ spots may be disassembled small spots, rather that disintegration of a single column of the flux tube.
Before examining the formation of sunspots in unipolar magnetic regions, we examine unipolar magnetic regions with a high resolution image. Figure 8 shows an example, which may be compared with Figure 1. It can be seen that the high resolution image of unipolar magnetic regions shows a magnetic network structure. It has been considered that the network structure is formed by the convective flow in each cell, in which flows continuously sweep magnetic fields to its boundary and send them down. It has also been considered that some irregularities in the convection flow tend to accumulate the magnetic fields at a few locations more than other places along the boundary. This feature is schematically illustrated in Figure 9 by combining two convective cell structures (in a negative unipolar region) constructed by Clark and Johnson 20 .
In the left hand side of Figure 9, pores thus formed are assembled along the cell boundary to form small spots. Their assembly forms larger spots, as shown in the right hand side of Figure 9. This process is morphologically called 'coalescence'. 9 It is likely that the single spots in Figure 7A and 7B resulted from this process of coalescence. Therefore, the fact that positive monopole-like single spots are produced by the coalescence within a positive unipolar region (vice versa) can satisfy the condition of 0 div = B ,so long as unipolar magnetic regions are a product of the solar internal dynamo, as discussed in the above section.
In summary, the fact that a positive unipolar region is the birth place of a positive single spot (vice versa), not elsewhere, has advanced toward our understanding of this single spot formation. Therefore, the problem (1) and observation (4) may be explained in this way.  This point will become more important in considering a pair of spots. In the next section, it will be shown that the above-mentioned process occurs at the boundary of a positive and a negative unipolar magnetic region of the opposite polarity, forming a pairs of spots.

Pair of spots
Various ideas of the formation of pairs of spots before Babcock 8 was mentioned by Cowling 8 . After Babcock's paper, a large number of observations have been published by supporting the RF model. 22 Although Getling 23 showed recently that their observations do not support the RF model, we still face the problems (2, 3, 5) unsolved today. Now, the problem (1, 4) may now find its possible explanation in Section 3. In this section, it is suggested that this explanation of the problem (1) may be quite important in considering the problems (2 and 3). Actually, the problem 5 gives us some hint in considering the remaining problems.

Formation of a pair of spots (a) Importance of the boundary of two unipolar regions
If magnetic buoyancy (or an upward flow) is the only reason for the rise of a magnetic flux tube in the RF model, a pair of positive and negative spots may appear together randomly at any location on the photosphere, even in the middle of unipolar regions. This is not the case. Thus, this is a serious problem for the RF model.
In considering the pair formation, we learned that a sunspot pair appears across the boundary of positive and negative unipolar regions, and further positive one in a positive unipolar region (and vice versa). [9][10][11] Thus, the boundary of two unipolar regions is a crucial fact in considering the pair formation; actually, this is collateral and consistent with the fact mentioned earlier, in that positive spots of the pair appear in a positive unipolar region (vice versa).
Thus, as we discuss later the delay of appearance of f-spots, there is a possibility that a positive p-spot in a positive unipolar magnetic region may induce a negative f-spot in a negative unipolar region across the boundary (vice versa), if a positive and a negative unipolar magnetic regions develop the connection by magnetic field lines. . Figure 10 is an example of the elliptical formation of sunspot pairs; note that as Figure 10 shows, larger spots consist of coalesced smaller spots. Many examples of the elliptical structure are shown in his paper, so that this elliptical formation is a typical way by which a spot pair develops. This is another fact which is not easy to explain by the RF model.  Figure 10 shows schematically a combination of positive and negative U-shaped structures at the boundary. It can be seen that the large negative spot may consists of several small spots. There is also a difference of the size between the two spots, which will be discussed later. For the blue and red U-shaped structures, see the text in the following.
If one of a positive U-shaped half-cell in the positive unipolar region at the boundary can induce the coalescence in a negative Ushaped structure, which is connected by magnetic field lines, two Ushaped structures of the opposite sign can form a pair of spots along the elliptical boundary. Figure 12 shows schematically how positive and negative U-shaped cells at the boundary form a pair of spots by combining the cell structure simulated by Clark & Johnson 20 . This is one way by which McIntosh's elliptical structure of a pair of spots can be explained. This inducing process will be discussed further in the following in connection with problems (2,3).. It may be the reason why a pair of spots tends to appear at the boundary of two unipolar magnetic region of the opposite polarity.   Figure 5F. Middle: The convective cells at the boundary. Positive and negative cells are combined at the boundary. The network structure at the boundary is thus different from that shown in Figure 9. Bottom: Two U-shaped structure shown in Figure 10.
Positive pores and small spots may coalesce along the positive Ushaped structure (which is a half of the boundary of a cell) to form a small positive single spots and further eventually a larger single spot (or spots) and vice versa. Eventually, spots in a positive U-shaped structure coalesce at its end and spots in a negative U-shaped structure coalesce at the end, forming a typical pair of spots, which consists simply of a pair of positive and negative spots. Figure 13 shows this transition and development. A number of such examples are shown by McIntosh 9 . Note that as mentioned earlier, both spots consist of many smaller spots.

Figure 13
Transition from an elliptical formation of small spots to a simplified typical pair of sunspots, which are actually consisted of several smaller spots. The elliptical formation is produced by combination of two U-shaped formation.

Delay of f-spots
According to the RF model, a p-spot and f-spot should appear simultaneously. However, it has long been known that a pair of spots does not appear at the same time; as so named, p (primary) spots appear earlier than f (following) spots. 3,4 There are only a few cases, in which f-spots appear first, so that such cases are rare. 9 Thus, this delay of f-spot is indeed an important fact in considering the pair formation. In Figure 10, p-spots (the right hand side) appeared first, and this pair developed in a period of less than 5 hours as a very faint elliptical form; because of its faintness, the elliptical structure (the cell boundary) may not be easily recognized, so that it may be seen only as two (p and f ) spots.
In the past, on the basis of the RF model, various attempts (such as the look-angle of the tube or branching of a tube below the photospheric surface) were made to explain the delay of the appearance of f-spots after p-spots, but those ideas cannot be confirmed by observations below the photospheric surface. The present way of explaining single spots and pairs of spots based on the observed features provides an opportunity to consider this problem which is contradictory to the RF model (thus, the problem cannot be explained without hypothesizing processes which cannot be proven at this time).
This inducing effect is possible, if some portions of positive and negative unipolar regions are connected by magnetic field lines. The formation of a p-side of the U-shaped half-cell could be communicated to f-side of U-shaped half-cell, because magnetic field lines are expected to be nearly equi-potential in the solar atmosphere, so that any electromagnetic changes associated with the coalescence (e.g. a converging flow, VxB ) in a p-side U will be communicated to a f-side U by Alfven waves, inducing a similar coalescence at the other end of the magnetic field lines, namely the formation of f-spots of the opposite polarity.
Thus, some delay might be expected as the communication time. This might be a possible explanation on the problem (2). In this respect, it may be noted that Kotov 24 showed that there is 10 12 A of electric currents around a single spot, since it can be caused by theVxB process (Figure 14). It is in this way that the problem (5) provides an opportunity to discuss the delay, because the RF model does not allow the delay.

Figure 14
The electric current distribution around a single spot. 24

Unbalance of the magnetic flux
It is well known that there is often a significant unbalance of the magnetic flux between two spots of a pair. 4,6 Figure 10 is a good such example. This unbalance is not easy to explain in terms of a simple RF model, although in the past some efforts were made by considering a complicated branching of the tube, which cannot be observed deep in the photosphere. However, so long as we consider a single magnetic flux tube (the RF model), the unbalance is a contradictory problem for the RF model, namely the problem (3).
When a p-spot is forming in a positive unipolar region, it does not have to induce an identical f-spot just across its neighboring unipolar region, because the field distribution in the two unipolar magnetic regions across the boundary or in the surrounding regions is not a mirror image of each other and thus different ( Figure 11).
In this respect, as shown in Figure 15, X-ray observations of the corona show that some of the magnetic field lines in a pair of spots are connected to at least another pair of spots, even across the magnetic equator, 25 so that it is not necessary to consider the induction within just one pair of spots; another good example is seen in his Figure 5 of Aschwanden 26 . The unbalance of the magnetic flux is contradictory to the RF model, so that it is not possible to explain the problem 3, but the present idea of the formation of a pair of spots provides an opportunity to consider the problem 3 on the basis of the problem 5.

The first appearance of p-spot
Actually, instead of considering the delay of f-spot, it is more appropriate to consider why a p-spot, appears first before a f-spot. At this point, we cannot find definitive morphological observations in explaining why a p-spot appears first at the specific boundary of two unipolar regions, not the other side, except that the specific side of the unipolar region tends to become more active first. In this respect, it may be noted that spots are surrounded by faculae, indicating specific activity around single sunspots. 1 Svalgaard and Wilcox 27 noted this particular boundary and called it the "Hale boundary". This may be related to the structure of unipolar regions ( Figure 3C).

The sunspot cycle
Any idea of spot formation must explain the equatorward shift during the sunspot cycle. There are a number of dynamo theories on the Butterfly diagram, explaining the equatorward shift of the formation of sunspot pairs during the sunspot cycle. Since we are considering only morphologically the formation of sunspots in this paper, it may be noted that there is an interesting observation which shows that an east-west belt of 'torsional oscillation' (rotation or antiparallel flow) on the photosphere shifts from poleward to equatorward during each sunspot cycle; 28 Figure 16. McIntosh 9 showed that there is a counter flow along such a belt and that solar activities are high certain locations along the belt. How such a dynamical belt can stimulate the formation or coalescence of spots at the specific boundary of two unipolar regions is beyond the scope of this paper.
In this respect, McIntosh 9 noted that sunspots, which tend to show the Coriolis effect, exhibit drastic spot coalescence. Since the rotation speed of the sun at the equator is about 2 km/s and plasma velocity around sunspots is only about several km/s, one does not expect much the Coriolis effect on the photosphere. However, a rare example of vertical sunspot group was reported by Akasofu 29 ; Figure  17 shows both the vortical sunspot group and a nearby single spot. However, dynamical processes and their relation to the formation of sunspots are beyond the scope of this paper, except to mention that Parker 21 considered vortex motions of small spots in cexplaining the coalescence.

Conclusion
Although the RF model has long been well accepted, and there are a large number of supporting observations, 22 the three longstanding problems (1-3) and two new observations (4) and (5) seem to be difficult to explain, so long as we assume a rising magnetic flux tube. The present study began as an attempt to study the three (1-3) problems, but it is found to be very difficult to proceed on the basis of a simple RF model, since they are obviously contradictory to the RF model. Fortunately, the newer problems (4, 5) provided a hint in considering the problem (1, 2, 3) and further all of them together. This paper is the full paper of a paper published earlier as a short report. 30 It is hoped that this paper will initiate even more new ideas in considering the formation of sunspots in the future. It is suggested that the following points should further confirmed morphologically and theoretically, regardless of any theory.