G.H. Hale was the first to recognize magnetic fields of sunspots in 1908 [1], and he found many of the basic morphological features of sunspots we know today by 1938. He showed that sunspots consist of a pair of positive and negative spots and changes from NS pairs to SN pairs every 11 years. The knowledge of solar magnetic fields and sunspots was greatly advanced by [2] who succeeded in magnetically scanning the solar disk.

[3] summarized many of what were morphologically known about the solar magnetic fields and sunspots up to that time. Since then, there have been many extensive reviews in the past. One of extensive reviews of sunspots in recent years is given by [4]. Recently, most studies of sunspots are concentrated on individual aspects of sunspots in great details, rather than overall aspects. Thus, in this paper, we attempt to synthesize observed facts on the basis of the presence of single spots.

[5] proposed a theory of the formation of a pair of spots. He suggested that the nonuniform rotation of the sun tends to wind the solar dipole field lines around the sun, forming a thin tube of magnetic flux. When the magnetic tube is supposed to rise above the photospheric surface by magnetic buoyancy, its two cross-sections are identified as a pair of spots. His theory has been well accepted today either explicitly or inexplicitly; Figures 1a and b.

Figure 1: (a) A typical example of a pair of sunspots; ‘p’ indicates primary (appearing first) and ‘f’ (following, appearing later) [courtesy of the Kitt Peak Solar  Observatory]. (b) A schematic illustration of Babcock’s theory of the formation of a pair of spots.

On the other hand, there exist a number of spots, which have been called isolated, independent, solitary spots. [1] named them ‘unipolar spots’ (plates 99, 100 and 101 in this book), and Bray and Loughhead classified sunspots into two, ‘unipolar group’ and ‘bipolar group’. 

In this paper, the ‘unipolar group’ is called ‘single spots’ by contrasting them with ‘pairs of N/S spots’. An example of single spots is shown in Figure 2, in which there is no counterpart spot. Many standard textbooks on the sun and review papers on sunspots show single spots or one of a pair of spots [4]; a number of single spots can also be found in the NASA sunspot collection.

Figure 2 An example of single spots (NASA sunspot collection).

Thus, it is puzzling why single spots have hardly been discussed in the past, in spite of the fact that many standard textbooks and monographs present their images. This may be because it has been believed from the earliest days that spots are like a magnet (which has both the N and S poles together) and because Babcock’s theory can so intuitively be understood. Another reason may be that ‘magnetic monopoles’ are not supposed to exist, so that single spots are avoided to be considered. Thus, single sunspots have almost disregarded in the past as a “broken pipe” at best.

Nevertheless, single spots do exist and are contradictory to the widely accepted Babcock’s theory, because sunspots are considered to appear as a positive and negative pair. In this paper, our discussion is based on four stages of synthesis of observed facts.

In order to investigate single poles, it became necessary to reexamine magnetic fields on the solar disk as a first step. Figure 3 shows an example of the distribution of magnetic fields on the solar disk.

First of all, one can recognize weak positive and negative bands, aligned alternately in longitude (including the northern and southern Polar Unipolar Regions). They are unipolar magnetic regions. [6]considered that unipolar regions are old active regions, which are stretched out by the nonuniform rotation of the sun. This idea has been well accepted today.

Figure 3: The distribution of magnetic fields on the solar disk (Courtesy of the Kitt Peak Solar Observatory).

Secondly, Figure 3 shows also concentrated fields, positive ones, which are scattered in a positive unipolar region (vice versa); they include pores and single spots. There are also pairs of large clusters of single spots at positive/negative boundaries of neighboring unipolar regions, positive clusters in a positive unipolar region (vice versa), neither within unipolar regions nor anywhere else.

In examining in detail the relationship between unipolar regions and concentrated fields (pores and spots), the Kitt Peak maps of the photospheric magnetic fields (the latitude-longitude presentation, [Carrington map] in graded colors) are useful, because unipolar regions, pores and small spots are weak fields, and these color maps are more suitable to examine them than black/white ones). Each map is constructed during the period of one solar rotation (27 days), so that it takes one solar Carrington Rotation (CR) day to complete it.

Figure 4 Two examples of solar maps during active periods of the sun.The upper one shows the classification of the magnetic fields; “1” are unipolar regions, “2” are pores, single spots and “3” are clusters of single spots.

Figure 4 shows two examples of the solar map during active periods of the sun; they are chosen from the sunspot maximum periods, showing the complexity of magnetic fields on the solar disk. In this paper, such solar maps during three sunspot cycles (21, 22, and 23) are extensively examined. This subsection summarizes findings.

In Figure 4 (upper), “1” are unipolar regions, “2” pores, single spots and others, and “3” clusters of single spots. Several examples of the maps during different epochs of the solar cycle are shown in [7].

Based on a close examination of unipolar regions during three sunspot cycles, it is found that unipolar regions grow and decay with the sunspot cycle. Further, it is found that unipolar regions appear before sunspots during the earliest epoch of the sunspot cycle. Further, the high latitude tip of unipolar regions extends to about 60°in latitude; most sunspots appear below 30°, as the Butterfly diagram indicates. Thus, unipolar magnetic regions are not decaying old active regions. A typical example of the solar magnetic maps during an earliest, maximum and late epochs of solar magnetic maps are shown in Figure 5.

Figure 5: The distribution of magnetic fields on the photosphere, from the top, an early, middle and end epochs of the sunspot cycle, showing that unipolar regions grow and decay with the sunspot cycle.

Furthermore, [8] found that unipolar regions often extend poleward and connect to the Polar Unipolar Regions and appear to be related to the reversal of the polarity of the Polar Unipolar Region. Figure 6b shows his results. Figure 6a shows an example of solar maps during the same period in Figure 6b, in which the northern unipolar regions extend poleward and are connected to the northern Polar Unipolar Region (vice versa). Although this phenomenon needs to study further, it is clear that unipolar regions are closely related to the 11-year cycle of the sun and thus to the basic dynamo process associate with the solar dipole field.

Figure 6: (a) An example of unipolar regions in the norther and southern hemispheres, which extends poleward and connect to the Polar Unipolar Regions, respectively. (b) Hathaway’s results (2010).

For these reasons, it is concluded that unipolar regions are one of the basic features associated with the basic solar magnetism (the internal dynamo process), not decaying old active regions; actually, single spots are born there as we see the following sections.

Figure 7 shows schematically the basic features of the solar magnetic field on the photosphere during a relatively quiet periods of the sun; this basic pattern is often considerably distorted during active periods, but Figure 7 is a good guide in studying complicated Figure 7 The magnetic fields on the solar disk and the basic pattern of unipolar magnetic fields on the solar disk maps (see Figure 6); [9] The concentrated fields will be discussed in the following sections.

Figure 7: A schematic presentation of the distribution of various magnetic fields on the solar disk.

Since Leighton’s work is generally believed, it is difficult to find a theory on the formation of unipolar regions. Figure 8 shows graphically the theory of unipolar regions by [10]. In it, unipolar regions are represented by coiled magnetic field lines, resulting from an azimuthal ring of electric currents. Positive and negative unipolar regions are represented by the degree of winding and also the outside and inside of the coil fields with respect to the photospheric surface.

Figure 8: A theoretical model of unipolar regions and sunspots by[10].

In summary, the above study of unipolar regions indicates that unipolar regions are not decaying old active regions as suggested by [6]. They grow and decay during the solar cycle are closely related to the Polar Unipolar Regions, far extending to the latitude of sunspot of about 30° or below. Thus, they should be considered in terms of the dynamo theory of solar magnetism.

The photospheric convection cells are known to accumulate some magnetic fields along their boundaries (namely, local concentrations along the boundaries). At the upper right conner of Figure 9, one of the convection cells considered by [11] is shown; it is old work (compared with recent extensive simulations of the convection), but shows the concept of phtospheric convection in the simplest way. The plasma flows around single spots are discussed in Synthesis 4.

It is likely that such irregularities of the convective motions leave some magnetic fields at the boundary of the convection cells. Figure 9 shows also schematically those magnetic fields at the focal points of the conversing convection in an assembly of convection cells (blue dots), forming the magnetic network, which appears as a uniform field (negative unipolar region in this case) in a low resolution images (Figure 3). Both pores and single spots are also schematically shown as dots and clusters of dots.

Figure 9: Schematic view of single spots in a negative unipolar region, which is formed within the network of pores in a negative unipolar region in this case. The single cell considered by Clark and Johnson (1967)  is also shown. The network is composed of their convection cells. Both pores and single spots are also shown.

Details of pores were given by [3], in which they noted that pores are simply small sunspots with no penumbra structure and that vast majority of them do not develop beyond this stage. They noted also that pores are related to individual granules and deep seated in the photosphere; see also[12] and[13]. Recently,[14] examined in fine details the relationship between granules and pores, umbral and penumbral structures, including flows toward pores. [15] made a detailed study of the relationship between the convection and magnetic fields; their pattern (their figure 44) may be compared with Figure 9. Their figure 60 suggests that pores have a short vertical structure; see also [16]. These observations suggest that pores are a thin and short column of magnetic field. Recently, [17] examined the rise of buoyant magnetic structure. The evolution of pores and plasma flows around them were studied by [18].

It is known that a single spot consists of a number of pores; Figure 10 shows two examples of single spots; [4] (section 5, figure 5.1) showed one example, which shows a fine structure of a single spot with pores, but without discussion of single spots. Thus, it is likely that a single spot consists of a cluster of several thin column of magnetic flux. (Although we do not discuss the decay period of sunspot in this paper, scattered spots during the decay period is not disintegration of a single column of magnetic flux, but may be scattering of pores).

Figures 10: Two example of single spots, which are an assembly of pores (NASA Sunspot Collection).

[3]. [19]described this assembly process as “coalescence”.[20]considered the coalescence process as vortex attraction. This point will be discussed in the following.

If a pore is likely a short column of magnetic field [15], it must be surrounding electric currents. Indeed, [21] showed  a circular currents of 10¹² A around a single spot; Figure 11. [4] mentioned that the current intensity is 6-7 mA/m². Thus, there is a possibility that the currents among pores attract together. Sunspots are hardly discussed in terms of electric currents these days.

Thus, one possibility for the formation of a pore and single spot is that:

 (1) A small eddy with a conversing flow (V) at the boundary of convection cell generates a circular current (V x B) around it;

(2) Its (J x B) force increases the conversing;

(3) Circular currents attract further each other, together with vortex attraction [20].

Figure 11: The circular currents around a single spot [21]. The related vectors V, B and J are added.

Since this convergence/coalescence process occurs only in a small area within a large positive or negative unipolar region, it is likely that there is no significant magnetic flux re-arrangement in its upper part unipolar regions and in neighboring unipolar regions. This situation may be similar to the situation envisaged by [22], as shown in Figure 12, although he produced it in a different context. This consideration of the formation of single spots avoids the problem of a single spot as a ‘magnetic monopole’ and thus there is no need to look for its counterpart in a variety of forms.

Figure 12: The local convergence of photospheric magnetic flux, taken from Cranmer (2009, his figure 4), in which he used it for a different context.

From the above discussion, it is obvious that positive pores and positive small spots grow in positive unipolar regions (vice versa).

The transition from pores to single spots appears to be continuous, depending on the number of pores and their size/intensity. The most crucial fact on the formation of pores and single spots is attractive effects of vortexes and electric currents.

Here, we attempt to bring together a number of observations of flows around single spots.

(1) Outward from the top

The first one is the well-known Evershed flow; plasma flows out from the top of a spot; this flow was discovered by Evershed [3]. [23] showed that the velocity is a few hundred m/s at all highs in a spot.

(2) Downward flow from the top

The second is a downward flow from the top of spots ([4], figure 7.1); see also [15]. [24] made a detailed observation of flows around granular light bridge, upward flow in the bridge, a downward speed of 10 km/s just outside of umbra. On the other hand, [25] found supersonic downflow in a sunspot.

(3)  Converging flow near the bottom

There are many subsurface observations of flows under active regions (cf.[26]; [27]; [24]; [28]; Braun, 2016,[7]. Further, [26], figure 4) showed that there is a large-scale inward flow toward a spot in the upper level of the photosphere (depth of 5 Mm), but an outward flow from the spot in the deeper part. These observations based on a helioseismic observation, so that it shows the flow distribution as a function of depth. [26] showed that the flow from the surface down to 5Mm is converging and below it is outward.  For the most extensive review, see [29].

There is also a shear flow, which is discussed in Section 5.3.

By bringing together all these flow observations, the choice is made by considering a cyclonic flow pattern shown in Figure 13.

Figure 13: A possible cyclonic flow pattern of plasma around a single spot.

This choice is partly due to the fact that sunspots show a cyclonic feature [19]; this feature has not got enough attention, perhaps because of the fact that the solar rotational speed is just comparable with various photospheric motions. Thus, an extensive search is made to find the cyclonic feature from a large number of sunspot photographs taken at the Kitt Peak Solar Observatory.

Figure 14 shows a rather rare case [30], in which the cyclonic effects around sunspots in the norther hemisphere are so clearly exhibited; this example may be considered as a partial proof of the presence of Coriolis force to the proposed convergence flow discussed in the above. One can see that pores and single spots in a large ‘spot’ (a cluster of single spots, Synthesis 4) may correspond to cumulus clouds in a large hurricane and are rearranged to form a cyclonic structure by the Coriolis force.

Figure 14: Cyclonic structure of sunspots. There is a single spot at the upper left corner. [30]. (Courtesy of the Kitt Peak Solar Observatory).

Another  example of cyclonic flows around sunspots was studied by[31]; Figure15. Note an intense cyclonic flow around the negative spot.

Figure 15: An example of intense cyclonic flow[31].

Importance of the boundary of neighboring unipolar regions

The most important fact about a pair of spots is that a distinct pair of spots forms at the neighboring unipolar regions (positive and negative). Secondly, a ‘spot’ of the pair consists of is a cluster of single spots by coalescence [19]. Obviously from what we have learned, a positive cluster is formed in positive unipolar regions (vice versa).

Figure 16 shows schematically this situation at the boundary of the two unipolar regions in the simplest case, in which the two clusters are connected by magnetic field lines.

Figure 16: A schematic illustration at the boundary of neighboring unipolar regions, showing two clusters in each side of positive and negative unipolar regions. They  are connected by a magnetic field line.

Figure 17 shows an example of the above situation by an observed image; it is interesting to note that large ‘cells’ show only their halves at the boundary. [19] noted that a pair of spots (a cluster of single spots) tends to occur along an elliptical shape, and both sides converge toward each end to form a pair of spots. Figure 16 (right) is an example of such cases; the final configuration of it is shown in Figure 1a. [19] suggested a large ‘cell’ for the elliptical formation.

Figure 17 Left: Magnetic field distribution at the boundary of neighboring unipolar regions. Right: An example of elliptical formation of a pair of spots; The final form of this spots is shown in Figure 1a. (Courtesy of the Kitt Peak Solar Observatory).

Figure 18 shows an example of what is generally called “a pair of spots”, which is actually a pair of clusters in a high resolution image. First of all, the left side consists of a large number of pores and single spots, while in the righthand side there is a large spot (coalesced single spots, at least three independent single spots). They seem to be almost individually connected at random on the counterparts on the other side.

Figure 18: An example of “a pair of sunspots”, which is actually a pair of clusters. (NASA sunspot collection).

Two clusters across the boundary of neighboring unipolar regions are often connected by magnetic field lines above the photospheric surface. It is likely that the magnetic connection between two active segments occurs in the way[32] presented; on the basis of X-ray images of magnetic field lines, he described: “---these field lines usually interact by changing their flux linkage, much as they do in a vacuum”. That is to say, for a given magnetic field distribution of two active segments at the boundary of neighboring unipolar regions, the magnetic connection may occur almost like in a potential field.

Figure 19 shows that what[32] described. A few sunspot pairs in both hemispheres are magnetically connected among themselves; thus, pairs are appeared to be connected even across the equator. Thus, the pattern is very different from what we expect from Figure 1b. In general, the clusters in both sides are very different in the distribution and size. Thus, it is difficult to consider that the clusters in both sides are inherently connected under the photospheric surface before emerging, another observed contradiction to the tube theory.

Figure 19: The image shows that sunspot pairs are interconnected [32].

Any theory of the formation of sunspots must explain the equatorward shift of spots as the solar cycle advances, namely the Butterfly diagram. Since we are dealing with a new morphological theory (not the assumed equatorward winding of the thin magnetic flux tube around the sun), it is necessary to find an observed fact to the equatorward shift. [33] found a large-scale latitudinal tortional oscillation (shear flows), which shifts equatorward during each solar cycle; Figure 20 (upper).[19] showed that there occur sunspots and flare activities along the oscillation belt; Figure 20 (lower).

Thus, there must be a close relationship between the boundaries of unipolar regions and the torsional oscillation belt, causing shear flow. It is expected that the cyclonic flow mentioned in subsection 3.3 may be related to this interaction; the shear flow and the converging flow may be closely related to form eddies and the cyclonic flow, as we discussed in Synthesis 3. More recently, torsional oscillation was studied by [34].

Figure 20 Upper: The equatorward shifting of the torsional oscillation [33]. Lower: The torsional oscillation belt (arrows) and solar activities (flares) indicated by dots. It shows that various solar activities, including the formation of spots and solar flares, occur along torsional oscillation [19].

In this paper, it is shown that a study of single spots in the four stages of synthesis has led us to three findings.

• Unipolar magnetic regions are related to the basic solar magnetism, which grow and decay with the sunspot cycle, not just decaying old active region.
• Positive single spots are formed in positive unipolar regions by coalescence of positive pores (vice versa).
• There is a cyclonic flow around a single spot.
• A pair of ‘spots’ is formed at the boundary of two (N/S) unipolar regions. The ‘spot’ consists of a dense cluster of single spots and large spots (coalesced single spots), and is not likely a cross section of a single tube; the pair of the two clusters is not symmetric at all.

These results all together suggest that a single spot is the basic element of spots, not a pair of spots.

In this synthesis, the following four issues need further studies as unsolved problems.

• Unipolar regions should be considered as a major parts of the dynamo theory associated with the solar cycle; sunspots are born there.
• Causes of coalescence of pores into a single spot and single spots to a large spot.
• Causes of the cyclonic flow (the converging flow and eddy) pattern of plasma in forming a pore and single spot.
• The relationship between the tortional oscillation and the boundary of unipolar regions.

The author would like to thank Dr. Kees De Jager for his comments on an early draft of this paper. I would like to thank for the Kitt Peak Solar Observatory for the solar data used in this paper; they were most helpful when I visited them. NSO/Kitt Peak data used here were produced by cooperatively by NSAF/NSO, NASA/GSFC, and NOAA/SEC. The author claims that there is no conflict of interest in this paper.

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