We discuss the importance of the Sun’s large-scale magnetic field to the Sun-Planetary environment. This paper narrows its focus down to the motion and evolution of the photospheric large-scale magnetic field which affects many environments throughout this region. For this purpose we utilize a newly developed Netlogo cellular automata model. The domain of this algorithmic model is the Sun’s photosphere. Within this computational space are placed two types of entities or agents; one may refer to them as bluebirds and cardinals; the former carries outward magnetic flux and the latter carries out inward magnetic flux. One may simply call them blue and red agents. The agents provide a granularity with discrete changes not present in smooth MHD models; they undergo three processes: birth, motion, and death within the photospheric domain. We discuss these processes, as well as how we are able to develop a model that restricts its domain to the photosphere and allows the deeper layers to be considered only through boundary conditions. We show the model’s ability to mimic a number of photospheric magnetic phenomena: the solar cycle (11-year) oscillations, the Waldmeier effect, unipolar magnetic regions (e.g. sectors and coronal holes), Maunder minima, and the march/rush to the poles involving the geometry of magnetic field reversals. We also discuss why the Sun sometimes appears as a magnetic monopole, which of course requires no alteration of Maxwell’s equations.
Magnetohydrodynamical (MHD) modeling within our Sun-Planet system is fraught with difficulties due to the diversity of environments within the vast dynamical systems influenced by the Sun’s radiative and particulate emissions. This may be well understood, when one considers the environmental changes that occur throughout the different gases and plasmas within our solar system. From the hot dense interior of our star to the low beta plasmas in the distant interplanetary regime, densities change from values in excess of
To gain an appreciation of the workings of this model, let us begin by considering the workings of simpler models of this type. Our particular model will be working with the calculation of magnetic field motions on the Sun’s surface, the photosphere, given a complete description of the existing magnetic fields at a certain time, and any changes associated with new field introduced into the photosphere. This is the equivalent of a Markov process. Namely, knowing the state of a system at time,
Interestingly, the current method, and the observed behaviors of solar magnetism are more weather-like in terms of exhibiting chaos in their behavior. That is, the subsequent behaviors have “nonlinear interactions.” For example, motions of field elements can result in their removal from the photosphere. This occurs in our model when two opposite field elements “collide.” It does not matter to the model, whether the collision results in the fields being pulled below the photosphere, or rather being ejected into the solar wind. They simply disappear from the domain of our model, the photosphere. We call this collision process between field entities, death, as a simple descriptor occurring when opposite sign field agents get too close to each other. A consequence of this nonlinear process is simple to understand for anyone who plays pool or billiards. A near miss of two balls results in quite a different behavior compared with a glancing strike. The model is ultra sensitive to small deviations of its initial conditions. This results in a “state trajectory” being highly sensitive to initial conditions, as in weather systems. We discuss a number of recurrent patterns of field structures in our model that appear similar to field structures observed by many solar physicists. The most famous of these patterns is the 11-year solar cycle. Other familiar patterns such as unipolar magnetic regions (UMRs, [
In addition to serving as the best observed source of solar magnetism, it is no coincidence that the Sun’s photosphere happens to radiate 99+% of the Sun’s energy into space. This makes the photospheric surface most significant to solar structures at a given time, but also where our knowledge of the solar structure is best “known,” or “tied down.” It is both a physical source of solar and interplanetary phenomena, plus a source of knowledge about happenings in the interplanetary environment. Hence it is no surprise that one may use the photospheric observed fields to serve as entities to fix our model’s boundary conditions on. One may simply call this “being pragmatic.” Hence, the model in this paper concentrates on understanding the motions of magnetic fields within this unique surface, the Sun’s photosphere.
The photosphere is quite different from our more familiar terrestrial surfaces, such as the oceanic surface. The photospheric boundary is less well-defined or sharp by having a significant scale height, within which its properties change. This makes it somewhat less than a true mathematical boundary, where one, for example, finds space cleanly divided in two by, for example, a plane located at
Another surprising feature of this remarkable outer boundary of the Sun, is that its pressure is markedly less than our more familiar 1 bar, or 1000 milli-bars of atmospheric pressure. The photosphere has a density similar to the Earth’s thin ionosphere. It is shocking that so thin a layer allows the photosphere to transform its particulate convective flux into radiative electromagnetic flux, such that virtually all of its energy flux radiates freely into space. This transformation is all the more remarkable, because of the wondrous energy flus involved: There are two main aspects which allows this transformation: the enormous turbulent velocities, several km/sec, near the sonic speed, and the enormous gravity of the photosphere, ~27x the Earth’s. These properties result in an enormous atomic collision frequency within the photosphere and yield the tiny scale-height of the photosphere yielding a sharp boundary. It is this sharp boundary that allows our model to work within the Sun’s relatively sharp 2D layer, as opposed to being dependant upon the 3D volume of the solar interior. Given these aspects, ascending in height from the photosphere, one finds a steep change into other layers: the chromosphere and the corona, from which only tiny amounts of energy escape, both outwardly into the solar wind and space, and others inwardly back into the Sun.
The photospheric magnetic field, in this author’s opinion, provides the Sun with the following: (1) a degree of granularity which affords the Sun with its ability to furnish changes on scales much vaster than the granulation and supergranulation patterns seen and (2) when the field is concentrated into small structures, such as spots, pores, or intergranular lanes, it opens “channels” to form which permit cold downdrafts to form and descend from the photosphere into the solar interior towards the base of the convection zone. These aspects provided the impetus to develop our 2D algorithmic approach via cellular automata as distinct entities that provide a granularity or coarseness to magnetic field motions within the photosphere. For the Sun, itself, rather than our model, it is the magnetic field that imbues the photosphere with its widely varying properties. As the great solar physicist, Robert Leighton, often said: “If the Sun did not have a magnetic field, it would be as uninteresting a star as most astronomers think it is.” In the absence of the Sun’s magnetic field, we observe virtually no semi-permanent large, spatially-varying structures that yield evident geometrical variations. We may clarify our use of granularity by pointing to some of its effects within the troposphere. If one lived within an atmosphere dominated by heat diffusion, one would find, for example, gradual temperature variations from hot to cold and vice-versa, rather than the more familiar weather patterns, where one finds a high degree of chaos (in the geophysical sense, namely both large-scale patterns and smaller scale stochastic variations), The larger-scale patterns have unique names, for example Hadley cell circulation, hurricanes/typhoons, pressure fronts, baroclinic instabilities, tornadoes, and so forth. Thus one finds some persistence in the atmospheric temperatures from day to day at any location, which we call coarseness. Namely, the patterns do not simply blend into molecular diffusion, but rather have motions and patterns on grander scales. Such behavior is typical of turbulence. Turbulence theory was originally developed for solar and stellar interior structures via mixing-length theory (MLT) and then evolved into general tropospheric usage by meteorologists. This theory has since been surpassed by more exact methods, which handle structures using computerized methodologies.
Returning to the Sun, in the absence of magnetic fields, upflows and downflows within the Sun’s convection zone do provide a tiny degree of granularity seen in its surface. These give the surface of the Sun a faint similarity to an orange peel, as seen in white light from the Earth. Yet, overall, the Sun’s surface, in the absence of magnetic fields, would resemble a totally desolate, unblemished desert that Leighton depicted rather than one with a more interesting terrain. The interesting terrain of the Sun on small scales, however, was captured by Robert Howard, when he described the Sun’s surface, seen at Mount Wilson, as “a can of worms.” This small-scale depiction accurately portrays the turbulent environment found within the Sun’s surface, owing to the roiling convection needed to resupply its surface with the energy it radiates freely into space. One might consider this tremendously wasteful except that we owe our existence to this grandiose generosity.
Thus the photosphere predominantly has its structurally varying properties associated with its embedded magnetic field. The solar magnetic field carries on throughout the entire photosphere and does not simply disappear at the boundaries of sunspots and faculae. Magnetic fields are non-zero everywhere, except at infinitesimally small boundaries. Thus it is the large-scale non-zero structures that are the predominant subject of this paper. How are these magnetic fields transported from their origins in the photosphere, as fresh new features, often displayed dramatically as sunspots or faculae, then travel to their demise, as remnant structures, barely discernable as they melt into the background noise of the turbulent photosphere?
Within the convection zone, mixing is so efficient that without the channels provided by magnetism, no significant non-radial, spatially-varying properties would develop. Because the gases beneath the photosphere are effectively in local thermodynamic equilibrium (LTE), except for tiny turbulent differences which serve to transport energy, the SAHA equation for LTE is used, along with the chemical composition, so that one can uniquely determine the conversion of pressure, temperature, density, and ionization level to ascertain the various percentage gases and ionization state within and below the Sun’s surface. Hence, it is as Leighton described, that without a magnetic field, the Sun would be a lot duller star than it is. A solar weather forecast, in the absence of magnetic field, would always consist of the following:
In the absence of helioseismological observations, we predominantly would only see the Sun’s surface. So, it is there, and above the surface, in the chromosphere and corona, that interesting solar phenomena typically appear and are related to the Sun’s surface magnetic field. Despite MHD’s usefulness in the laboratory, where one does not need to consider energy transport on large scales, the meteorological equations do have applicability, as downflows undergo adiabatic heating, but owing to their superadiabatic surroundings, remain relative cool, and upflows transport great amounts of energy in the solar case; thus such effects are placed within stellar energy transport equations. Because of the interest and importance of the Sun’s magnetic field throughout the Sun-Planetary system, this paper will concentrate on understanding the Sun’s large-scale magnetic field.
Because of the thinning of gases above the photosphere, many rapid, yet vast, changes occur above the thin photospheric layer. Temporal and spatially variations of >100% are seen in flares and coronal mass ejections. In addition to affecting the majority of the plasma, they also issue forth particles on the tail end of the distribution function; low energy solar cosmic radiation sweeps through the solar system while the magnetic field carried by the plasma particles sweeps out the inner heliosphere from incoming galactic cosmic radiation. Hence various “space weather” effects are felt throughout many Sun-Planetary systems. Thus the rest of this paper focuses predominantly on what happens to the magnetic field in that surface layer of the Sun, the photosphere. Near the surface of our Sun, physical conditions vary greatly. Complex interactions of magnetic field, plasma, and radiation happen, giving rise to many spectacular and complex phenomena, such as erupting prominences, sunspots, and many activity-related effects from flares to the solar wind and energetic particle emissions, difficult to understand, yet a wonder to behold. Astronomically, the photosphere is that region of the Sun from which light emanates. This happens approximately near an optical depth of
Many properties of the Sun’s large-scale magnetic field may be found in the fairly comprehensive review paper [
Cellular automata models, however, have been able to make progress in some areas of nonlinear phenomena from forest fires to magnetic spin states and the list of successful CA modeling of physical phenomena, using Netlogo alone, appears to be impressive and growing [
Consideration about the correctness of an agent-based model is clearly dependent upon the correctness of the stated “set of rules,” given to the agents. Differential equations have been ubiquitous and the mainstay of the physics world from the 17th to the 20th century. The power of computer processing made any agent-based models too cumbersome to be useful until the present epoch. Further, the results do not result in any “beautiful equations,” as happens in analytic treatments; thus it is hard for scientists to appreciate the bunch of numbers that are the primary output of agent-based models. For these reasons many scientists are wary of such numerical models compared with analytical treatments. Nevertheless, to understand complex phenomena, it is readily apparent that analytical treatments may be totally inadequate in certain areas. Thus we hope to show that for large-scale solar magnetic fields, an agent-based model may be better able to mimic behaviors more effectively than any analytic method to date. It is with this in mind, that we discuss the present model’s behaviors.
In our magnetic field mapping model, we take the photosphere, to be the domain in which we model the Sun’s magnetic fields. Non-zero surface magnetic fields are carried, in our model, by cellular automata. In their absence, the magnetic field is assumed to be near zero. In addition we simplify the computational problems enormously by choosing a large “graininess” of the field entities; the field is 0 where there are no cellular automata, and where there is a cellular automaton present, the field is either out of the Sun (shown by a blue cellular automaton, CA) or into the Sun (shown by a red CA) with a quantized amount of magnetic flux.
Let us state briefly how our model works, and later discuss a number of solar phenomena that are illustrated by this model. We are guided in choosing the following rules based upon “locality,” namely that field entities’ behaviors (birth, death, and motions) are affected solely through the influences of nearest neighbors upon them. There are but a few effects that agents have on other agents: (1) while at the poles, field agents are connected via the Babcock-Leighton (B-L) subsurface field to lower latitudes which allows them to undertake two duties there: (A) the B-L subsurface field spawns new photospheric magnetic flux (the processes are outside the photospheric domain, and thus are not the subject of this paper) and (B) the B-L subsurface field attracts lower-latitude fields via the long-range magnetic tension; (2) nearby agents affect each other’s motion by short-range interactions we describe; (3) when opposite flux agents get too close, we invoke a “death process,” that is, the fields and agents disappear from the photosphere. One may consider this as magnetic loops being pulled into the Sun (or out of the Sun), and hence disappearing from the photosphere, or any number of scenarios; they are again outside the domain of this model.
Hence what goes on above or below the Sun’s surface is not generally considered except in a minimal fashion, when, for example, we discuss the subsurface “Babcock-Leighton” toroidal magnetic field. A recent version of the model may be found on the Netlogo Community website [
Field agents have a few properties; they possess a quantized amount of magnetic flux,
Let us now ask two questions and then discuss them. How can we model a highly restricted volume in space, namely the photosphere, and virtually ignore more than 99+% of the Sun, volumetrically, or by mass, outside our chosen photospheric domain? Although this appears puzzling, and counter-intuitive, the answer is actually relatively simple. The laws of physics are obeyed
In the current model we address the boundaries around the photosphere as follows: (1) we add active regions proportional to the Babcock-Leighton field, the North-South toroidal field they describe in their papers [
We now provide some sample runs with our model, thereby illustrating how our model mimics the following behaviors, as seen in the photosphere: Solar Cycle (~11-year) Oscillations, the Waldmeier effect (large cycles are shorter than small cycles), Unipolar Magnetic Regions/Sectors/Coronal Holes (large-scale field patterns), Maunder Minima (extended periods of time where low field levels exist in the photosphere), the March/Rush to the Poles (details of polar field reversal at high latitudes), and how the Sun sometimes appears as a magnetic monopole (wherein the Sun sometimes
To understand how the present model works, and what it does, we begin by inspecting Figure
Shown is the Field Mapping version 1p07 interface. This directs cellular automata to follow our algorithms written in the Netlogo 4.1.3 language. The blue slider bars on the left and below allow various parameters to be set, and the display in the center progresses through time, allowing large-scale solar fields to be “mapped.” The time history of the fields is displayed in the two plots to the right of the interface. The white monitors also provide instantaneous values of various parameters to be displayed. There are two graphs on the right; the lower one labeled Plot A, displays the polar field variations. In it are three colored curves: blue, red, and black. The blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole, namely, the mean field above 60 degrees latitude. The red curve, similarly counts the mean field in the Southern polar region. The black curve marks the Absolute Value Sum of the red and blue curves. So, this curve in black displays the amount of magnetic flux in the polar regions. It has been artificially placed 60 units downwards, for increased clarity, so 0 lies at the bottom of the graph. It essentially reaches a peak, during a solar cycle minimum. The very astute observer will observe a fourth black curve at the bottom of the graph. This last curve at the bottom of Plot A shows when the total polar field values reach a “peak.” This is taken as the start of a new cycle and is shown as a uniform tiny “blip” in this bottom curve. Thus one can count solar cycles with these blips. The second plot, towards the right of the display, is Plot B. It shows the mean latitude of all active regions born during a tick unit of time. In this manner one may observe the solar cycle “butterfly graph,” although only roughly, as the graph program does not average, and so forth.
Returning to the interface seen in Figure
The second plot, above Plot A, is Plot B. Plot B shows the mean latitude of all active regions born during a specific period of time, called a “tick unit” in Netlogo language. In this manner one may observe the solar cycle “butterfly graph”. Unfortunately the Netlogo plotting software requires one value for each tick unit of time. Hence it does not allow multiple or 0 values for the ordinate. Hence one cannot graph butterfly diagrams in the conventional way (many or no values for a particular time). So the default value is 0, when no new active regions are born. Hence one observes “false marks” along the abscissa, which should be ignored. Thus the curve in Plot B is an imitation of the butterfly diagram, with the limited graphing method allowed in Netlogo.
Figure
Displayed are two graphs of polar field variations versus time for the first 8 solar cycles (a), and for 4000 tick time units (b). Each of the graphs is of the Plot A type using the Field Mapping model 1p07 run with the Netlogo 4.1.2 code version. Shown most prominently are three colored curves: blue, red, and black. The blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole, namely, the mean field above 60 degrees latitude. The red curve similarly counts the mean field in the Southern polar region. The two polar fields anti-correlate. The black curve marks the Absolute Value Sum of the red and blue curves, so it corresponds to the mean polar flux. It has been displaced downward, so the −60 value corresponds with 0 flux, for increased clarity. The black curve essentially reaches a peak, during a solar cycle minimum, when the polar fields reach their maximum values. Again, a fourth black curve is drawn, with little blips, and placed at the bottom of each plot showing when the actual peak in the polar field occurs. These provide a good indicator of the start of a solar cycle, so one can count solar cycles with these blips. The lower graph shows an extended period of low solar activity interspersed within typical periods of high activity. The extended periods of low activity (e.g. the one near 3000 time units) may be similar to Maunder Minima type phenomena. Examinations of the lengths between field minima, shown by the tick marks in the figure, readily display an inverse correlation with polar field magnitudes. Namely, when the fields are larger, the periods are shorter. This effect was discovered by Waldmeier.
From the quality of predictions based upon the polar field method [
We will see, from the field maps in this paper that the cyclical behavior of the solar activity cycle, as seen in the Sun’s polar field lines, may be regarded as the flotsam and jetsam of the solar cycle. These terms are nautical terms referring to the leftovers when a ship broke apart due to the ravages of the sea; the jetsam are those goods jettisoned overboard by the crew in their attempts to save them from a watery grave, whereas flotsam are those that floated away on their own, having been insecurely stowed. Their long-lost triplet, lagan, is the goods going to the bottom of the ocean. For the case of solar magnetic fields, we may regard the disappeared flux as lagan, and not distinguish between flotsam and jetsam, but simply apply these terms to fields that survive to collect in UMRs and the polar regions which then reform into the next cycle’s toroidal field and future solar activity. It does not matter not to the current model, since this is a photospheric model, whether this reformation happens by field being transported by shallow flows just below the photosphere, or at great depth, near the bottom of the Sun’s convection zone. Such events, outside the domain of this model are simply outside this model’s purview. As discussed earlier, this model tries to consider only those elements that happen in/or near the photosphere, and thus treats new flux that enters the photosphere as “updated boundary conditions.” We look forward to this model being updated to allow observed sunspot fields to serve as actual boundary conditions, as was done with Leighton’s model [
Another question which may arise is how are we able to model the photospheric field with a model which is incomplete, by not considering all the workings of the entire Sun? We do this as follows. The model is designed to sweep under the photospheric rug, metaphorically, the nature of how magnetic fields arrive in the photosphere. One may be surprised that one can construct a model this way. But yes, one can. One is always free to choose any volume of interest, apply the laws of physics to this volume, and consider the surface surrounding the volume as a boundary condition. This is how our choice of the photosphere as the domain of this model is designed; nothing more, nothing less. By restricting our domain, of course, we limit what we may learn or say about phenomena outside this volume. We are fundamentally trying to understand the photosphere, and what role the magnetic field and fluid motions therein play in affecting the large-scale photospheric magnetic field.
Let us continue our discussion of the magnetic flux calculations, as many have taken this as an important aspect of “flux transport models.” Our program may be regarded as a bit of a “toy model” compared with the actual Sun. Despite having the possibility of having greater computational ability, this program, as we run it, often uses a small finite number of agents, hence there is a need to increase the modeled Sun’s inefficiency to manufacture magnetic flux (one may change parameters, of course, to make the model more realistic to solar conditions, but we have not attempted to obtain a one-to-one relationship). The Sun creates a lot of magnetic flux and can well afford to waste 99.9% of its flux (as mentioned earlier, only 1 line per 1000 field lines arrives at the poles). We next examine and calculate the efficiency of polar field generation related to the wastage as follows.
We define an efficiency formula defined similar to Wang et al. [
Lowering the Sun’s ratio to our model’s, of 1 : 34, rather than 1 : 1,000, is a ratio of ~30. Our model’s greater efficiency has one primary advantage. It allows us feeble human beings to
Lest the reader be not as inclined by our reasoning, as we are; let us briefly add a bit more. One should not be at all troubled by our removing the wastage fields by considering the actual reality of “counting items,” as in the following. Imagine a straight field line just below the photosphere. Now we twist it, so one loop pops up above, so there are two new field sources (a+ and a−) in a small region. One can easily twist and pop two loops up. What happens to these little loops can be quite different. One little loop may simply be pulled back down; the other loop spread apart. There really is no gold standard in counting magnetic loops through a surface. The kinds of “counting processes” that we undertake of field lines through a mathematical surface (such as an idealized photosphere) simply provides us with a rough gauge of magnetic flux. This is simply what we do: we can count, but counting is nothing physical that the Sun can respond to. Counting is not a physical process. It is something that humans do, as we try to quantify physical attributes.
There are a number of parameters leading to interesting behavioral aspects of this model, which may be investigated. We shall be showing these in the next few sections. Let us first mention one of the ingredients in this model compared with earlier field models, like Leighton’s [
For any who are concerned about the ratio of our model’s reduced ratio of random fields to ordered fields compared with the Sun’s, let us mention one final analogy to understand this important point. Let us consider solar field lines to come in 100 flavors, say no. 1–no. 100, in each of two colors, red and blue. Only the red and blue lines labeled with a particular number, say no. 1, are counted; the rest are discarded. If we were to show the extra 99 field lines of higher flavors, that fail to go to the poles, it would be even more confusing to the patterns shown than it already is. This mathematical transformation would remove the numerical deficiency of the model we display, through a simple mathematical trick. This is why, in our mind, not fitting all the solar dynamo numbers (the Sun having a wastage of 99.9% and this model, 97%) is not of any significance. Our simplified “toy model” that we show here allows these field displays to be perceived by humans without the added complexity of having a significantly higher average field cancellation that the Sun provides.
Hence this simulation enhances the ratio of polar field lines to equatorial field, so that the most significant field lines may, hopefully, be adequately seen. Another way, of saying this is that the Sun’s active region fields are relatively strong compared with polar fields, and the inefficiency of the Sun’s dynamo is not adequately represented by the current field mapping model. If we were to make a direct one-to-one portrayal of field lines, we would need many more field lines in the active regions, and then have the model simulation spending a lot of time canceling many more, and running for longer periods to show the overall magnetic structure. We have compromised this aspect for an improved display.
The most obvious behavior of our model is that it portrays the oscillatory 11-year solar cycle, (seen in Plot A). In addition to seeing oscillatory behavior, one sees the irregularity that has become legendary to those of us who marvel at the erratic behaviors that the Sun displays with little rhyme and reason. This has led many theories to fall into the waste-bin of theoretical ideas that many solar cycle predictors have had to trash, as they attempt to understand the Sun’s enigmatic behaviors, as they try to support the agencies whose enterprises are affected by solar activity. NASA, as well as other national and international agencies, has numerous technological achievements in space that may be affected by excessive solar activity. With increasing demands and dependency on these satellites, and so forth, knowledge of long-term solar behavior is becoming an increasingly important subject of practical as well as theoretical study. The long-term method by which such predictions are made has not yet reached a sufficiently agreed upon methodology [
Following our viewing of the most apparent solar cycle behavior, let us now consider a less well known effect. In addition to the variances in solar cycle amplitude we discussed, there is another effect, which Waldmeier [
For simplicity, consider just one type of species, although the results apply to the two equal species that we are dealing with. Consider also an approximately cyclical pattern, where we take the rate of polar field births,
We may consider the above aspects to gain some further understanding of the interplay between these the life and death of field entities. For simplicity (to allow understanding without the detailed tracking of both entities, since our model does that), we consider a 1D model, where we have polar and surface, red and blue agents; however since the number of each color is identical, we just consider the total number of red and blue surface and polar fields,
Now consider some of the elements of the Babcock-Leighton dynamo: the surface fields arise from a magnification process that amplifies the poloidal fields and converts them to toroidal field where they erupt into surface fields, hence we choose
If we now move on from these equations, we can consider the case for functional time dependence, rather than the uniformly oscillating situation we began with. Consider first a more general form of the above equations that allows the components to interact nonlinearly. One well-studied type of first order differential equations takes the Lotka-Volterra form:
Modifying our earlier equations to examine how the equations respond to temporal variations, we include a functional form,
For numerical simplicity, we choose
Shown are solutions to the differential equation (3.2.7-8). These solutions illustrate that as the amplitude decreases the period grows longer. This amplitude-time behavior for the solar cycle was noticed by Waldmeier [
In addition to the most evident features of the solar cycle on the face of the Sun, in white light, sunspots and faculae, on the global scale, difficult to observe features, however, are signs of the Sun’s polar fields. Observed by polar faculae [
We now often refer to large-scale unipolar regions on the Sun, whether at the poles or at lower latitudes universally, as “coronal holes”, owing to the manner in which unipolar field regions on the Sun allow the coronal plasma to vacate these field lines rather quickly. We now know that the patterns seldom, if ever, form mathematical idealized geometric structures. Nevertheless, early on, coronal holes had these earlier names. We now understand the phenomenon of the highest closed arches as forming when the Alfven velocity allows the magnetic field to travel downwards at the same speed that the coronal plasma is advected outwards. So a balance exists between the outward speed of the plasma and the inward tension pulling the field lines inward. It is an intricate race that allows the highest closed arches to lie about a solar radius above the Sun. In this manner, the “source surface” is that general height for these highest closed arches on average. Above these lines, the solar wind would extend field structures into interplanetary space, and inside of the height the field lines can reconnect to the Sun.
When running the current Solar Field Mapping program, the pattern of field on the solar display appears to show a variety of patterns and general motions of field quite similar to the backwards C-shaped UMRs and sectors discussed by the previous authors, when not too encumbered with new active regions. One often can see various numbers of the backwards C-shaped field structures emanating from active regions as the fields break up and drift towards opposite poles. One may consider that they are driven by the magnetic forces, and swept back by differential rotation. These may be some of the main driving influences, rather than diffusive properties. One cannot dismiss totally some of the other likely smaller influences, such as meridional flows and some amount of “diffusion” from turbulent mixing. To this author, it appears that magnetism from the Babcock-Leighton subsurface field provides a meridional (North-South) motion that may be regarded as the long-range magnetic force driving the 11-year oscillation, and the differential rotation draws the field out into a “streakline”, the technical fluid dynamical term that describes the patterns generated when a time-dependent wind blows over a “tracer” source leaving a “trail” of emissions, as in the case of a smokestack or in the case of the Sun, its magnetic fields, and fluid flows.
This viewpoint is similar to that previously espoused [
If the fields were symmetric (in their or our model), such fields might result in open flux, but not in the N-S plane, as they would not have a North-South dipole moment, and so, in the viewpoint of this paper, would not affect the N-S solar dipole, and hence would not tend to add to the Babcock-Leighton polar field. Of course, for the actual Sun, open flux can easily result in a large-scale magnetic torque on field lines from its interaction with the global field, and it is difficult to say what the effect would be of such a torque on the field regions below. Nevertheless, from the viewpoint in the present paper, what happens, in these perfectly symmetric cases, is that on the way to the opposite pole, the fields “interfere” with reverse sign fields, so cancellation would occur. If, of course, at a particular longitude, the North and South patterns interfere “constructively,” rather than canceling, the two patterns become symmetrical in the North-South direction, and a complete backwards-C results. Total constructive interference leads to backwards C-shaped structures similar to Babcock and Howard’s patterns. In addition to these patterns, Wang et al. at NRL have shown the patterns of field from magnetism diffusing and flowing from active regions in a variety of cases.
Let us now examine some of the random patterns that the present model generates. One can see a couple of patterns “sectors” in our Figure
This figure illustrates that large-scale fields are able to mimic the observed Unipolar Magnetic Regions (UMRs) discussed by Bumba and Howard. We see a number of the “backwards C-shaped” field structures emanating from active regions as the Bipolar Magnetic Region’s (BMRs) fields disperse, break up, and drift to one pole or the other, driven by magnetic forces which underlie the Sun’s surface. The fields are swept backwards by differential rotation as one may observe while watching the patterns evolve in time.
The “source surface” model [
In any case,
A deeper understanding of the manner in which active regions add field to the larger scale structures was greatly furthered by Wang et al. [
More recently a hybrid heliospheric modeling system was developed to forecast interplanetary properties directly from solar fields, in a multidecadal advance useful for solar-terrestrial research [
Figure
Top panel shows Plot A for 2000 ticks, and lower panels display maps during quiet intervals, as shown. During later phases (see Plot A at the top), long periods of low activity, activity are reminiscent of “Maunder Minimum” conditions occur. The Plot A type is again of the Sun’s polar fields as in the earlier graphs, so that the blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole, namely the mean field above 60 degrees latitude. The red curve, similarly counts the mean field in the Southern polar region. As with the Sun, the two polar fields anti-correlate with each other. The black curve again displays the amount of magnetic flux in the polar regions; it is thus a rough estimate of the amount of solar activity. Following such periods, the field recovers for every model run we have undertaken. One sees detailed circumpolar rivers of blue and red field line magnetic entities circumnavigating the solar poles during these intervals. The top curves in this figure show a plot similar to Plot A, with the three colored curves: blue, red, and black. The blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole, namely, the mean field above 60 degrees latitude. The red curve similarly counts the mean field in the Southern polar region. The black curve marks the Absolute Value Sum of the red and blue curves. So, this curve in black displays the amount of magnetic flux in the polar regions. It has been artificially placed 60 units downwards, for increased clarity. It essentially reaches a peak, during a solar cycle minimum.
Let us now examine the composite history of this model run for 2000 tick timesteps. First we note that as the activity lessens, the oscillation periods lengthen, as studied in the earlier section with regard to the Waldmeier effect. Figure
Let us now examine the composite history of this model run for 2000 tick timesteps. We focus first on the overall behavior and then some interesting aspects related to the quietest second period shown in Figure
The polar field transport rate per cycle is obtained using the peak-to-peak amplitude of about 46, (~±23), with ~23 field entities in a peak pole, requiring 46 to be transported there (23 to reverse the field, and 23 to reestablish the reverse polarity). The monitors in the upper right of the interface, display, respectively, the 24 solar cycles, the 72,880 total numbers of field entities (or 36,440 single species entities). The single species birth rate is thus 1,518 entities per cycle. The field transport efficiency then is the ratio of these numbers: FTE = 46/1518 = 3% = 1 : 33, virtually identical to our earlier estimate of 2.9% = 1 : 34, showing a remarkable uniformity at converting equatorial field to polar field transport. We are somewhat surprised at this uniformity, but that is likely due to our calculating long run averages, as opposed to cycle-to-cycle statistics. Again this number (3%) compares to the remarkable inefficiency of the Sun, for these calculations (1 : 1000, or 0.1%).
With the significantly higher efficiency of this model, compared with the Sun (3% versus 0.1%), we need to point out again, that the field lines calculated and displayed are not the tightly confined sunspot and active region fields, most often discussed by solar physicists interested in dynamic solar events such as flares. The field lines we study best in this model are not the most visible magnetic features, sunspots, but rather their weaker remnants, such as faculae and network fields, both rarely seen, but ever present. These may be of greater interest to some, since one finds they generally have a longer lasting global impact and help drive the Sun’s future activity behavior. Another interesting aspect that can be seen if one watches the model run long enough is that there is often a degree of asymmetry in the Sun’s polar field, usually some oppositely oriented field embedded within the polar field. This often disappears as the cycle progresses; however, we believe it to be a real effect, and that this asymmetry appears to us to be a tilting of the Sun’s dipole much like the Earth’s dipole is not centered on the Earth’s rotational axis, but currently some 11 degrees inclined to the axis; here we relate the tilt to Cowling’s [
With the Sun using only ~0.1% of its active region fields to regenerate its polar fields, this clearly makes the actual solar cycle a very inefficient process. Nevertheless, it is the photosphere’s job to transport the Sun’s energy outwards not to make magnetic fields. So, what may be described as “wasting 99.9% of its energy,” an inefficiency (to us, humans who are interested in the solar cycle), may be, to some higher vantage point, a marvelously efficient radiator. We really cannot judge such things; they just are what they are. In other words, we can simply agree that the Sun allows 99.9% of its active region magnetic flux to cancel before the remaining 0.1% makes its way to the opposite poles. The strength of active region fields compared with the background field may be seen in many field plots [
Another event corresponds with the following solar behavior. On the Sun, there are many temporal relationships between events. One that is often referred to is a “march” or “rush” to the poles of polar crown filaments. These motions often occur just before the time of polar field reversal during that phase of the solar cycle (solar maximum). These polar crown filaments, when seen above the Sun’s limbs, are referred to, most commonly, as prominences.
In times past, before the space age, they were most “prominent,” hence their name, at times of solar eclipses, when they displayed their rich, ruby-red light to terrestrial observers as they peaked above the moon’s limbs on the day of a solar eclipse. This color contrasted beautifully with the steely blue electron scattering of the photospheric light by the inner corona. The bluish-white corona displays the feeling of shivery-cold moonlight reflecting off icebergs above a freezing ocean, since eclipses often result in a temperature drop of many degrees. Such a feeling belies the corona’s heat of millions of degrees, and the prominences possess a ruby-red color, belying their cool, chromospheric temperatures, thousands of degrees lower than the dazzling white photosphere. Nature thus gives us fair warning not to judge it with our frail human senses. Instead, we must apply our equally frail mental facilities to interpret its mysteries. All we can do is pause to think and reflect how little we know.
Thus prominences appear in the ruby-red Hydrogen alpha line, at a wavelength of 656.281 nm. They are dark when contrasted against the solar disk (hence called filaments), because they absorb the light from the bright photospheric disk beyond. At times of eclipses, however, they are seen most prominently in ages past with their bright red appearance. Most commonly, although they are quiescent and scarcely, a tenth of solar radius above the photosphere, just lying peacefully like some kind of solar cow, nestled under the protective base of a helmet streamer, showing the inner corona’s base. These coronal streamers, of course, got their namesake, owing to the WW I shaped helmets warn by the Prussian soldiers to protect them from barrages; however, their shapely pointed tops probably made the soldiers more visible, rather than provided any real protection. Most commonly, in this modern era, however, the prominences only gain real fame, when they are ejected rapidly into the corona or solar wind, via coronal mass ejections (CMEs). These occur often with (or without) solar flares, marking their brightened appearance in the chromosphere, or infrequently with a white-light flare, in the photosphere.
The particular aspect of these features under discussion is their famous “march” or “rush” to the poles. Much as soldiers marched or rushed into battle, streamers all around the Sun gather en masse and march to the poles on queue, as if all called by a distant drummer on the Sun. We cannot hear the sound above the loud din emanating from the Sun’s surface, since the granules have material speeds approaching the sonic velocity. If we could hear sounds, it would be the loudest possible, essentially white-hot sound. We would go deaf instantly, but the march of all the magnetic features towards the Sun’s poles simultaneously would be a rather spectacular sight, and one that is not totally understood.
Gopalswamy et al. [
For the purposes of this paper, we see such fields parading to the poles at high latitudes, as rivers of field gradually moving poleward, just prior to field reversal. Figure
Again the top panel shows the polar field variations during the run for the first few “field reversals,” shown with timesteps at the bottom. Thus again at the top is a Plot A graph with the three colored curves: blue, red, and black. The blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole; the red curve similarly counts the mean field in the Southern polar region, and the black curve marks the Absolute Value Sum of the red and blue curves, placed 60 units downwards, for increased clarity. It essentially reaches a peak, during a solar cycle minimum. The lower panels are the field maps at these times, just prior to the complete reversal. One sees the new polar field rising from below, and the old polar field, on the verge of total annihilation. As mentioned in the text, the new fields are gradually marching to the poles. If one were to view the Sun’s limbs at these times, the border between old and new polar field would have a dark filament at high latitude, circling the poles, with the demarcation line being the old and new field boundaries.
At the pole, these field patterns just become increasingly stretched by differential rotation. Thus they may be a pattern drifting to the poles while at the same time rotating differentially, thus stretching increasingly in an East-West direction, and thus gradually decreasing the tilt angle of the magnetic striping of the field line. Thus it might resemble a thin sheet on the surface of the Sun, similar in geometry to the terrestrial aurora, which is circumpolar too, however circumpolar about the magnetic axis, not the axis of rotation. To us, the magnetic fields looked like rivers, with one polarity floating towards one pole, while the reverse polarity floated to the opposite pole. The pre-existing term “crown filament rush to the poles” seems adequate to describe this interesting behavior.
To obtain them from a model run, one must reset the value tick-end to either 0 to run continuously or to past the time one wants to examine for a longer run, for example, 300. In most examples the two polar fields do not reverse simultaneously, and there is even a slight difference at the poles, with magnetism at some longitudes disappearing after others. Such is the nature of the real solar cycle too. Such real phenomena are never as clear cut as our simple views suggest.
Perhaps because this subject is the most controversial, we are leaving it to the end. Wilcox [
We are raising this matter, even though we are not positively disposed to revamping basic laws of physics based upon solar observations. Yet, we do recognize that Helium was first discovered on the Sun, and that neutrino oscillations too revolved around the Sun. Lastly, of course, we take pride in Leighton’s wonderful comment that if the Sun had no magnetic field, it would be as boring a star as most astronomers think it is. Hence, we do not simply toss out puzzling observations of solar magnetic fields. Instead, we ask: What can we learn from them? Further, does the current model shed any light on the origin of these puzzling observations? Let us first review the findings.
Wilcox [
Before discussing the relevance of our model to these observations, let us first voice concerns we have about the solar observations that Wilcox reports on. We shall enumerate these reservations for clarity.
It appears that the detailed synoptic maps, made at Mt. Wilson at the time (from which Wilcox obtained his solar data), only show fields equatorward of 40 degrees latitude, owing to the difficulty in registering significant Zeeman splitting to obtain sufficient signal at that time. This allows the maps to have significant regions of the Sun from which positive or negative flux might emanate.
The regions of the Sun’s polar field were not examined in Wilcox’s study, although he quotes support from Severny; however, there is no “hard data” in Wilcox’s report on this. Going back to Severny and colleague’s reports [
Concerning the fields in the Sun’s polar regions, even though they did not always appear on synoptic maps, we have been fortunate to have Robert Howard, an experienced and dedicated observer serving many years at Mt. Wilson, to describe the Sun’s polar fields quite adequately from 1960–1971, as well as during other years. So, let us mention at least what happened during that interesting time, 1965, and not focus on the remainder of the period. The interested reader will want to follow the entire history of the Sun’s polar fields at Mt. Wilson, and Howard [
Let us now briefly discuss the Sun’s polar fields. Interestingly, Severny reviewed Babcock’s work and stated that H. D. Babcock used the Mt. Wilson magnetograph at seven points in the Sun’s polar regions, and found that in 1957-8, the North polar field of the Sun reversed so suddenly, that both poles of the Sun had the same polarity for almost a full year. Severny also states that in 1959 the solar polar field was parallel to the Earth’s. Thus Severny was well aware of the sign convention used by solar physicists to measure solar magnetism. As mentioned earlier, the sign convention is not just an American solar magnetism standard; it is the universal standard for magnetic field signs in physics [
The Sun’s polar fields occupy those regions of the Sun poleward of
We may even call the polar regions of the Sun, the “ignorolatitudes” (modifying the term “ignorosphere,” used by atmospheric and space physicists to describe the mesosphere—the middle atmosphere at 50–85 km above the earth, which is not accessible by aircraft, too high, and not accessible by spacecraft, too low). The polar latitudes are the locations on the solar surface least studied by solar physicists, because there does not seem to be a lot going on there: no flares, no new activity centers, and so forth. Nevertheless, these regions may, in some ways be equally important to the balance of solar magnetism, as the active latitudes. The active latitudes are a great source of solar magnetism; however, the polar latitudes may be the graveyard of the many solar fields. Thus they deserve, perhaps, an equal amount of attention. We do not know what really happens there, except these regions are the source of the steadiest high speed solar winds, a key to understanding the solar dynamo, and useful as a predictor of future sunspot activity. As a consequence of this last point, they beg examination. How is it that these fields are a key component to understanding future solar activity, and yet more than 99% of the solar physicists do not study them?
Beyond the high latitude gaps in data, Wilcox shows three solar rotations of detailed Mt. Wilson data, and there is a period of about 10 days in Carrington rotation 1802, where the entire period seems to be devoid of data, thus missing about 1/3 of the data during this interval. Let us add to this the fact that the magnetograph can see at best just the visible hemisphere of the Sun (and most often only a fraction of that for greatest clarity), so the synoptic maps seen of the Sun’s field show an accumulated data display made from many, many different times (roughly 10 or more), not a snapshot over the Sun’s surface made at one time, from satellites peering at the Sun from all angles. Knowing the workings of the observatory, and that one was restricted to make observations as different longitudes of the Sun faced the Earth, the geometry is clarified.
Additionally, and this is typical of the magnetograph contours back in the 1960s, even when data exists, the majority of the solar disk, even when well observed, still has insufficient signal to allow non-zero contour lines to show over a broad portion of the Sun’s surface. In other words, at low latitudes where this data situated, most areas show zero data—not really because there is exactly a zero field present (although the Sun’s field is thought to exist in patches of high field and larger areas of weak, or near zero field), but all we can
“Observing window effect.” Lastly, we should be remiss if we did not mention the following aspect, which we call the “observing window effect.” Solar observations only see light that is emitted from the Sun’s photosphere, typically, and goes into the telescope observer window where it is processed. Hence, if regions on the Sun’s surface possess a magnetic field pointing into the Sun (at a particular time) but these regions happen to lie on rather darkened lanes (and there are lanes on the Sun that are relatively dark; there is a mottling of the Sun’s surface due to lowered temperatures (e.g., intergranular lanes, owing to the convective patterns), then it is likely that this field sense (the toward the Sun field) would be undercounted, since the magnetograph integrates the line splitting (into the observing window). Hence any solar observation is subject to “observer window” effects and irregularities in the Sun’s surface brightness, and other irregularities in its surface from an ideal sphere.
In the next few paragraphs, let us first put to rest the problems associated with the interplanetary fields from near Earth (not observations of the Sun’s surface). According to Wilcox, the Sun appeared virtually unipolar of one sign for an entire or even more than one solar rotation as observed from Earth using the Sun’s extended interplanetary field (IMF) polarity interpreted from geomagnetic observations. These are based on a highly accurate method, spearheaded single-handedly by Svalgaard [
Wilcox [
Year | month | day | Bartels no. | IMF sign: away +, towards X, mixed polarity * |
---|---|---|---|---|
1954 | 06 | 13 | 1656 | X+++++++XXX+++++++++X++++++ |
1954 | 07 | 10 | 1657 | +X++++XXX+++++++++++++++X++ |
1954 | 08 | 06 | 1658 | +++++++++X+++++++++++++++++ |
1954 | 09 | 02 | 1659 | +*+++++++++++++++++++++++++ |
1954 | 09 | 29 | 1660 | +++++++++++++XXX+X++++++*++ |
| ||||
1965 | 05 | 22 | 1804 | X*XXX+++++XXXXXXXXX++++XXX+ |
1965 | 06 | 18 | 1805 | +XX+XXX+++XXXXXXXX++XXXXXXX |
1965 | 07 | 15 | 1806 | XXX+XXXX++++XXXXXXXXXXXXXX+ |
1965 | 08 | 11 | 1807 | XXXXXXXX++++*XXXXXXXXXXXXXX |
| ||||
1967 | 05 | 21 | 1831 | X+X++XXX+**XXXX+XXXXXXXXXXX |
1967 | 06 | 17 | 1832 | XXX+XXX++XXXXX++XXXX++++XXX |
1967 | 07 | 14 | 1833 | XXXXXXXXXXXXXXXXXX+*X+++++X |
1967 | 08 | 10 | 1834 | XXXXXXXXXXXXXXXXXXXX*++++++ |
1967 | 09 | 06 | 1835 | +XXXXXXXXXXXXXXXX+*XX++++++ |
1967 | 10 | 03 | 1836 | XXXXXX+*XXXXXXXXXXXXX*+++XX |
1967 | 10 | 30 | 1837 | +XXXX+++XXXXX*XXXXXXXX++*++ |
| ||||
1987 | 01 | 18 | 2097 | ++++++++++++++++++++++++*++ |
1987 | 02 | 14 | 2098 | +++++++++++++++++++++++++++ |
1987 | 03 | 13 | 2099 | +++++++++++X+++++++X+*+++++ |
1987 | 04 | 09 | 2100 | ++++++++++++++++++*++*+++++ |
| ||||
1995 | 12 | 29 | 2218 | ++*XXXX*+*XXXXXXXXXXX*+X+++ |
1996 | 01 | 25 | 2219 | XXXXXXXXX*XX*X*XXXXXXX+*+XX |
1996 | 02 | 21 | 2220 | *XXXX*X*XX*XXXX++XXXXXXX+X* |
1996 | 03 | 19 | 2221 | *XXXXXXX*XX**XX+X*+++XXXXXX |
1996 | 04 | 15 | 2222 | XXXXXXXXXXX+++++X+X+XXXX+++ |
1996 | 05 | 12 | 2223 | +XXXXX+XXXX++XX*+*XXXXX*+++ |
| ||||
1997 | 02 | 06 | 2233 | **++X++XXXXXX**XXXXX+XXXXXX |
1997 | 03 | 05 | 2234 | +++++++XXXXXXX+++XXX*XXXXXX |
1997 | 04 | 01 | 2235 | +XXXXXXXXXX++++XXXXXXXX*X*X |
1997 | 04 | 28 | 2236 | ++XXXXXXXXXXX+++X+XXX*+++XX |
| ||||
2010 | 06 | 25 | 2414 | *XXXXXXXXXXXX*+++XX*++++XX+ |
2010 | 07 | 22 | 2415 | +XXXXXXXXXXXXXXX*X++++XX+++ |
2010 | 08 | 18 | 2416 | ++*XXXXXXXXXXXXXXXX+++*XXX* |
2010 | 09 | 14 | 2417 | +*XXX*X*XXXXXXXXXXXXX+++XX* |
Let us add some cold water to dampen our enthusiasm for this discordance with Maxwell’s Divergence
Putting aside our observational shortcomings, let us now consider some of the model’s behaviors that may shed light on the Sun’s strange field wanderings. Has the Sun gone on a bender, or off the deep end of a large swimming pool? Let us consider this seeming violation of one of Maxwell’s equations. Well, one might argue that Maxwell had 4 famous equations, and 3 out of 4 are a superb batting average. Of course, as physicists we cannot think that way. Instead, we are trained to acknowledge the many successes of these 4 equations, and that they help form the basis of special relativity, and quantum electrodynamics, the best tested set of equations in Physics. So, we must acknowledge this, and that something must be going on with the Sun that is indeed puzzling, but our classical bedrocks remain the inviolability of each and every Maxwell equation, particularly
So, we consider (
Rather than talk about the field of the Sun, in terms of the field on its surface at each latitude, and so forth, it is simplest to describe its spherical harmonic expansion, in terms of Legendre polynomials and their associated spherical harmonics,
The simplest way for the Sun to give this appearance is that a large solar quadrupole moment exists (a strong
Visual representations of the first few spherical harmonics. Red portions represent regions where the function is positive, and green, negative, respectively. The right hand side of the figure shows
It is clear that the
The Sun may fool us as follows. When we view the Sun from Earth, we are at low heliographic latitudes. There are two ways we underestimate the Sun’s polar fields: (1) the high latitudes show only a much smaller projected angle (the cosine angle of normal of the surface to the observer), and (2) the Zeeman splitting is roughly proportional to the projected field along the line of sight, providing another cosine factor. Since the Sun is tilted by 7.25 degrees from the ecliptic, this angle gets at
More explicitly, what happens is that the quadrupole term (2,0) (see Figure
Nevertheless, although it stifled monopole moments, perhaps it did this more than the Sun does, and thus perhaps we were “throwing the baby out with the bath water.” Namely, perhaps a more realistic solar model would be obtained by including some of these field drifts, which induce a quadrupole moment. We now consider this by running our model with a significantly reduced quadrupole moment. Figure
The Sun appearing as a monopole. This model is obtained by using a 0 value for the “quad-blaster” setting. The lower graph shows the polar fields, again a Plot A type graph of the polar field variation seen in many of the preceding figures. The Plot A graph has the three colored curves: blue, red, and black. The blue curve displays the amount and sign of the magnetic flux in the northern magnetic pole; the red curve similarly counts the mean field in the Southern polar region, and the black curve marks the Absolute Value Sum of the red and blue curves, placed 60 units downwards, for increased clarity. The polar fields overall, are observed drifting well into the positive arena (thus with both polar regions positive, with the assumption of Maxwell’s divergence B condition, one may assume that the equatorial field of the Sun balances the magnetic flux), indicative of a quadrupole moment, and these three field maps are shown by the three graphs above indicated by the times. These are all forays of one sign, but either sign is equally likely. This upward drift indicates increasingly positive or blue polar fields (at the poles) and negative or red equatorial fields. The solar fields calculated by the model are
Thus, to reinforce this aspect, we need to know what the polar field of the Sun actually does, before deciding what the best factor is for this parameter. Nevertheless, this lack of definition just serves to illustrate the point we tried to make earlier, that the polar fields are amongst the least well observed magnetic fields in the Sun’s photosphere.
In running our model, we see, in addition to the solar cycle related
These events support the model’s behavior of occasional “apparent monopole-like behavior,” as observed from the ecliptic viewpoint of Earth, which is preferentially situated near the Sun’s equator. We have incorporated a term in the model, called quad-blaster, which is just a term tending to reduce the quadrupole moment of the Sun, but the parameter only works moderately well, and the quadrupole moment still wanders around owing to the random walk nature of the surface fields in our model. If quad-blaster is greater than 0, and the nominal value is 100, then the term tends to reduce the quadrupole moment and tends to remove any false monopole term. If quad-blaster is less than 0, then the term can magnify an apparent monopole term, and if quad-blaster equals 0, then there is no effect on the monopole term. We should like to mention one more recent bit of news on this subject supporting the quadrupole interpretation. Yang et al. [
It is this author’s viewpoint, that the best eigenmodes are the UMRs of Bumba and Howard [
The boundary conditions for the Sun are more problematic, since the global aspects of the Sun are not so well understood. These are governed by global studies: the inflow (turbulent torques and pressures into the photosphere; however stellar models can provide these), and outflow equations: effluent angular momentum (governed by the outflow of solar wind versus latitude on open field lines, etc.). The global Sun needs to be examined, not just the microscopic aspects. The turbulence in a river is controlled by how much energy is available in the stretch of a river governed by the amount of water moving towards lowered gravitational potential, so that the recent rainfall affects the turbulent state, via the Reynolds number. In a similar manner, the turbulence in the solar atmosphere may be sensitive to the local and global state of its magnetism. Field inhibition [
While discussing unusual solar behaviors, another aspect of this model that we think might occur or have occurred on the Sun is that sometimes the polar (dipole) fields do not reverse. They can head towards zero, seemingly about to reverse, but then goes right back to their former sign or they may have just a short, small reversal before heading back. Such behavior does not seem like our solar dynamo, but we only have about a century of observations of the Sun’s fields, begun by George E. Hale. It is hoped that some of these odd behaviors will be better understood, and if unrealistic, the model will be improved.
This section considers two broad aspects: how the solar field mapping model relates to (1) broader aspects of geophysics, namely, those areas affected by solar activity: the solar wind, through the magnetosphere, ionosphere, and the various layers in the Earth’s atmosphere, and (2) physical models and improvements.
The reasoning behind using the model for “broader aspects of geophysics” is that knowing the relationships between the Sun as a source and as a driver of various solar-terrestrial indices may be useful in the subdisciplines mentioned. The relationships between indices are at the heart of this, and using this model to improve that connection can only strengthen our understanding and usage of these interactions. In this section, we shall mention aspects of this model that one might study, through which the model can be further investigated, and also many more areas where it could well stand improvement in. The author shall be delighted to comment on, and help where possible, others undertaking such endeavors, not just with this model we refer to as Solar Field Mapping Model 1.07a, but other advances. Working with this model is as much fun as it is work, since Netlogo is such an easy language to program in.
Let us first discuss the use of indices as a future direction, global and daily indices. We may take the daily indices to indicate the daily count of a global index; however, there is not generally any easy transformation from a “global index” to a daily one, because each index has its own unique transformation qualities, rather than the same exact mathematical structure. To be more explicit, daily indices often require some complex weighting function of, or over, physical solar parameters and this process represents a some type of conformal mapping, for example, a radiative transfer function (or another type of) integration over solar atmospheric parameters. Such parametric functions may be photospheric, chromospheric, or coronal, but each index refers to a unique and unknown “mapping.” Often such an “inversion” represents trying to find a solution to an “inverse” problem in the sciences. Usually is not a unique answer. Consider a simple example: find the density within the Earth, knowing
In the same vein, the solar field mapping model does not calculate sunspot number, really just field entities, and then performs simple statistics on these, to enable observables, like the polar field (abs-count = n-s-pole to be calculated which might relate to polar faculae), or the “Active Region Count”, (AR-count) in the upper right monitors. There are a plethora of solar-terrestrial/space weather parameters/indices that solar- and geophysicists have tracked in their attempts to monitor the Sun. The oldest is, of course, sunspot number, plus the plage and facular indices, the Calcium K index, and the facular count. After this, there are various geomagnetic indices, such as Kp, and its linear complements: Ap and aa indices. Additionally, one might be interested in modern observables, more directly related to the Sun’s output, many commonly employed by space researchers, with direct connections to the Sun at short wavelengths (e.g., solar EUV, UV, etc.) and various chromospheric parameters, Mg II, Calcium K2, or F10.7, used often as a proxy for solar EUV, and useful for space orbital drag calculations, for example, or total solar irradiance (TSI). This list is not complete. Modern updates to these involving specific spectral observations (for the Sun), or ionospheric and atmospheric layers (for the Earth), and magnetospheric and interplanetary areas (for space) may be possible broad ways to consider the ranges of this research.
Now I shall point out weaknesses in the current model, namely, ways that it might stand improvement, or at least further investigation. I shall enumerate these and give brief identifying names so as to improve clarity, in any later discussions.
Momentum conservation: the model has a number of simplifying assumptions; one is the innocent choice that field entities have constant speed. This appears to conflict with momentum conservation since if field agents turn relative to one another and still retain the same speed, there is no guarantee that they conserve momentum. This sounds troubling, but this problem is really minor when we consider that the lower ends of any magnetic feature in the photosphere interacts with much denser material below the photosphere. Thus we may invoke that the lower density material absorbs the magnetic field tensions so as to conserve momentum. Still this is an area that deserves further investigation.
Coordinate system: the coordinate system could well be transformed to an equal area synoptic chart configuration. This would be an improvement to this model; however, Netlogo does not at present employ this geometry.
Field agent parameters: how the model chooses the birth of field could be modified, and there are various ways they might be improved or modify this model’s choice. Some might require major renovation others minor adjustments, depending upon the type of change considered. Our choice affects how field agents spread away from the active regions in which they were spawned. Their motions are consequently dependent upon the random velocity (heading) of their initial motion. This clearly introduces elements of (apparent noise) into our deterministic system. This, of course, means that the noise is
Hale boundary active regions: one might consider rather than adding sunspot fields with random longitudes, adding them at preferential longitudes associated with the appropriate Hale boundary in the source surface or current sheet boundaries mapped down to the Sun. This stems from the work of Svalgaard and Wilcox [
Observational feedback to model: here we envision putting in active region information, and generating field entities that are put into the model. As these move about on the solar surface, one imagines some comparison of the agents changed positions to observational data. For example, this could be using the agents to obtain spherical harmonic coefficients,
Another aspect of this is to use actual new active region fields, as opposed to computer generated ones, and then ascertain how large-scale fields emanate from these regions. This surely will result in significant findings. We may also learn better how to predict changes in solar activity, better than we do at present.
Coronal field and solar wind implications: it would be of considerable interest to ascertain how these results would reflect in coronal field patterns and motions. One may see how closed and open fields change forms, and whether these variations are consistent with what is observed and lead towards improved modeling aspects, or are the results too disparate, and so point to some key failings in the model, from which it cannot recover, without being totally gutted.
We have developed an algorithm that allows a calculation of photospheric field motions, given the placement of field in the photosphere. The photospheric placement is done by the present program using a pseudo-random placement in accord with the various “rules” developed by Hale, and others related to the solar cycle, with parameters such as the polar field strength, solar cycle phase, and so forth. Despite uncertainties in our model’s algorithms, we have accomplished this by making certain assumptions that allow us to find specific solutions to the motions of the Sun’s surface fields. Clearly, this is a daring task, since the fields appear to move on the photospheric surface in rather chaotic ways. It remains to be seen then, to what extent the solutions bear any resemblance to observed solar field behaviors. When the strengths and weaknesses of this model are evaluated, then subsequently, one may advance the methodology towards improved understandings and/or solutions. It may, of course, be that the methodology is a dismal failure, but then perhaps it will shed light on dismissing the ideas contained herein that are intractable.
To summarize the model, a photospheric field mapping program 1.07 was developed using Netlogo algorithms involving cellular automata entities. These entities were of two types: bird species or breeds: either bluebirds or cardinals, for the two signed magnetic flux directions, inward or outward. The model tracks photospheric field motions, once given the initial pseudo-random placement of photospheric fields, based upon solar cycle parameters: polar field amplitude, phase of the solar cycle, and so forth. The model is online at the Netlogo community website and may be run either there and/or further developed by the interested user. The author shall try to be available to help the interested user.
The model is able to mimic the following: Solar Cycle Oscillations, the Waldmeier effect, Unipolar Magnetic Regions/Sectors/Coronal Holes, Maunder Minima, March/Rush to the Poles, or Rivers of Magnetism prior to Polar Field Reversal, and that the Sun sometimes appears as a monopole, but most likely has a magnetic quadrupole form giving rise to preferentially one sign at the solar equator. We then discuss various changes/improvements to the program that we might hope to see.
We are surprised that as simple a model as this is, that numerous photospheric phenomena are able to be displayed without specific knowledge of flows hidden deep within the solar interior; namely, the model does not have access to the dynamic structure of the Sun’s interior (state variables such as density, pressure, temperature, composition, ionization, magnetic field distributions, global flows, etc. versus height and their temporal variations). We do not know what this implies but are nonetheless happy for any serendipitous behaviors. There are at least two possibilities concerning the interior information. It may be that the interior information is making its presence known predominantly through the arrival of sunspots in the photosphere, which we place randomly, but the Sun does this through processes not fully understood, and that these sources of new magnetic flux are, in fact, the “solar interior information that allows the dynamo to happen.” It may be that the surface motions are able to “sweep aside” much of the interior behavior, except for the amount and location of new magnetic flux eruptions, as well as the Babcock-Leighton magnetic tension force. Thus our model sweeps other factors aside much as the Earth’s oceans reveal little of the undersea terrain and what lies beneath for at least a few oceanic wavelengths.
These are questions our model brushes aside. Since this is a discussion, however, we consider this second possibility further, not because it is more likely, but because it is less likely, and therefore less popular, that somehow the surface “sweeps aside” much of the interior information.
How can a physical entity sweep aside its roots? It somehow leaves the photosphere behaving as if the internal structure mattered little. Aside from the ocean analogy, there are more behaviors of this kind: black holes, it is thought, with the “no hair theorem” states, that above the event horizon of a black hole, only a few elementary properties are felt “outside” owing to what exists “inside,” namely, their gross mass, electric charge, and angular momentum. The internal aspects inside the black hole are brushed aside, and only the most elemental properties are allowed to “escape.” The sins of the black hole are buried inside the event horizon.
Our personal viewpoint of the successfulness of the current model is the following: that any model may consider any volume and fit reasonable boundary conditions to the surfaces surrounding that volume, and if one handles the boundary conditions correctly and the physics within that volume correctly, then one
To briefly summarize our current viewpoint, which appears similar to some of the earlier work [
A whole field of mathematical science has begun around phenomena which occur in new groupings, which otherwise are inexplicable. This is the field of “self-organized criticality” (SOC). One of the earliest papers in this field is that of Bak et al. [
On the other hand, it may be that the photosphere stands as a unique place in the solar atmosphere. It is a driver for much that occurs throughout the convection zone. We know that the interior of a star cannot be calculated without the “outer boundary condition.” Modeled stellar surface conditions provide key parameters for stellar models of their deep interiors. Without knowledge of the radiative aspects of the photosphere, the deep interior cannot be constructed; the upper boundary conditions are vital to the whole solution. Even for the Earth’s atmosphere, the outer boundary conditions are vital. This is important for stars as it is for the Earth’s atmosphere.
With regard to the Sun, Brandenburg et al. [
Alternatively, this model appears to shed light concerning large-scale photospheric field motions. We recognize that the surface of the Sun is more complicated than counting apples. So let us end by saying that we hope that the current simplified model may be improved, and that at present, we are encouraged with the model’s preliminary behaviors. We have
We see our Solar Field Mapping Model as an opportunity to allow solar knowledge to grow, and try to make a “testable” model, as opposed to one separated from reality. Our early work in Source Surface theory [
The author appreciates comments from Hans Mayr, Norman Ness, Eugene Parker, and Leif Svalgaard.