Long-term solar activity and terrestrial connections. Part I: theory

The research task described herein aims at the structuring of an analytical tool that traces the time course of geophysical phenomena, regional or global, and compares it to the course of long-term solar conditions, long-term meaning decades or a few centuries. The model is based on the premise that since in a last analysis the preponderance of atmospheric, hydrospheric, and, possibly, some aspects of geospheric phenomena are, or have been, powered by energy issuing from the sun – either now or in the past, the long-term behavior of such phenomena is ultimately “connected” to long-term changes occurring in the sun itself. Accordingly, the proposed research firstly derives and models a stable surrogate pattern for the long-term solar activity, secondly introduces a transfer-function algorithm for modeling the connection between the surrogate and terrestrial phenomena viewed as partners in the connection, and thirdly probes the connection outcome for episodic or unanticipated effects that may arise due to the fact that in the present context, the connection, should it exist, is very likely nonlinear. Part I of the study presents the theory of the concept, while Part II demonstrates the concept’s pertinence to a number of terrestrial phenomena.


Introduction
The particular aim of this essay is to examine the macrodynamic aspects of solar activity without invoking the particular agents that produce it (such as plasma physics, helioseismology, planet-induced tidal eects, etc), and to do this in view of the presently incomplete knowledge of the actual mechanisms involved. Consequently, the outcome of this research will be descriptive of behavior morphology, and not necessarily explanatory of causation as far as the phenomena in question are concerned. In this context, let it be remembered that Isaac Newton is often cited in support of the school which prefers to establish measurements rather than to ascertain causes,``hypotheses non ®ngo'', as he said.
However, he was far from suggesting this as a limitation of scienti®c research in general, as the following passage from his Opticks attests: To derive two or three Principles of Motion ...would be a very great step in Philosophy, though the Causes of those Principles were not yet discover'd: And therefore I scruple not to propose the Principles of Motion abovemention'd, they being of very general Extent, and leave their causes to be found out. (Samuel, 1952).
Our probing is based on the premise that since in a last analysis the preponderance of atmospheric, hydrospheric, and possibly some aspects of geospheric phenomena are or have been powered by energy issuing from the sun, their long-term behavior re¯ects long-term changes occurring in the sun itself. Hence, the proposed research 1. proposes and models a stable surrogate pattern for the long-term solar activity; 2. introduces a transfer-function algorithm for modeling the connection between this surrogate and terrestrial phenomena for which solar activity has relevance; 3. probes the connection outcome for episodic or unanticipated events, given that the connection, should it exist, is very likely nonlinear.

Methodology
It must be stated from the beginning that the basic nature of the analysis attempted here is purposely cross-disciplinary, partaking of both physical science and engineering. This stems from the conviction that theoretical tools which sent man to the moon and are now preparing equipment for the unmanned exploration of Mars, can also assist geoscience in deciphering messages from the sun, should they exist. However, regarding solarterrestrial connections, such geophysically basic occurrences as the El NinÄ o in the Paci®c, the temperature of either hemisphere, volcanos throughout the globe, and tree-rings in dierent locales, are occurrences which hardly interest most engineering analysts. By the same token, such potent tools of the latter as transfer-functions and system-response algorithms that deal with dynamic phenomena in general, do not people the tool kit of geophysisists at large. Against this background, our study takes the position that in order to proceed intelligently with regard to solar-terrestrial connections, the two types of skill need to complement each other on the task ± diculties notwithstanding (Key®tz, 1992(Key®tz, , 1994 ± and that the Kolmogorov algorithm is the medium that can bind the two together in the present context: its principal worth being that it implicitly captures the dynamics propelling a given phenomenon from the phenomenon's past course, even where no adequate theoretical knowledge of such dynamics presently exists. The algorithm's detailed derivation in Part I is for the bene®t of the geoscience element of the interdisciplinary team. The analysis begins with the observation that solar activity is not only in tune with the familiar 11-year cycle of the sunspot number, but, in addition, varies along a time-scale measured in centuries, as the Sporer and Maunder Minima demonstrate (Eddy, 1976;Hughes, 1977). One approach to examining the activity's longterm behavior will be to consider the sunspot number as its surrogate and, through it, structure a mathematical model of solar activity potentially useful to the study of whatever phenomena such activity may have a bearing upon.
The suggested method proceeds with the task of modeling the sunspot number in the long run based on the following tenet and its corollary: The Constrain Tenet. Any solar process involving the exchange or release of energy yt, has an upper bound , and a time rate of change dyadt that increases with both the current value of yt and the remaining potential À yt for further change. This renders both y and dyadt bounded. The modality is not new, having been introduced by P. F. Verhulst in 1844 in connection with population growth (Lotka, 1956).
The Corollary. Since the constrain tenet means that the process yt is nonlinear, the process cannot only give rise to a basic oscillation f 0 , but also cause: 1. the appearance of new incommensurable frequences f 1 Y f 2 Y F F F, 2. the entrainment of these frequencies so that (for example) f 1 , f 2 approach values f 0 , 3. the sudden appearance of aperiodic spectra, 4. the appearance of subharmonics f 1 a2, f 2 a4 successively (McDonald, 1976).
Put in terms of a dierential equation, the tenet requires that the process evolves as: where t is a factor, facilitative or inhibitory, established by the determinants of the process in question, which therefore may be called the mediating factor. The expression in Eq. (1) is a Bernoulli-type dierential equation when is time dependent, and becomes the well-known logistic law when is constant. The equation's general solution is: yt 1 1ay min À 1ay max expÀy max t dt 1ay max Y and t dt ln 1ay min À 1ay max 1ayt À 1ay max X 2 Depending on t, this basic expression can model a great variety of physical (as well as societal) phenomena and, being nonlinear, can lead to outcomes of unexpected complexity, including cases of the so-called deterministic chaos (May, 1976;Modis and Bebecker, 1992;Mullin and Price, 1989). The basic features of Eq.
(1) are that: 1X dyadt 0 when y 0 or y Y 2X dyadt max when 2y À 1a 2 dadt X 3 To compute the two extremes y min and y max for Eq. (2) from the data of a particular case, we may use the expedient of approximating either end y end by an exponential curve; it is easy to show that three equidistant points of such a curve will produce: y min or y max y end t 2 À y end t À y end t 2y end t À y end t À À y end t X 4 The two features in Eq.
(3) imply a cusp-like shape for dyadt, simple when t (constant), and more complicated when t varies in time. When, as in the present context, the mediating factor is a priori unknown, the morphology of dyadt may be approximated by the expression: as an intermediate step. This simple function is chosen because it allows for a single peak only and, therefore, for the capture of the very core of the growth process. The condition upon the parameter ensures that in the limit dyadt settles at a ®nite nonnegative level, as it should for the constrain tenet to be satis®ed. Given that the root of the derivative of dyadt in Eq. (5) de®nes the time t p of the dyadt peak as t p Àa2, it follows that when t p is known, Eq. (5) becomes: Since the parameters , k in Eq. (6) can be found from the data, yt and are computable using Eq.
(1) and, through them, the mediating factor t.

Model synthesis
The long-term behavior of solar activity can be brought into focus by ®ltering out the ubiquitous annual mean relative Wolf sunspot number and its 11-year periodicity.
The simplest means of eecting this is through the 11year moving average (Eddy, 1976), for which the symbol z 11 t will be used: the time history of which may be seen in Fig. 1.
The zt data on which the present analysis is based comes from McKinnon (1987) for the years 1700±1985, and from the monthly issues of Sky and telescope after 1985. It should be remarked that the reliability of the data deteriorates as one goes back in time (Eddy and Stuiver, 1977), but they are deemed accurate enough at least from 1800 to the present: the assessment is in agreement with the results of a study by Hoyt et al. (1994) who used 340 000 daily sunspot group counts from more than 350 observers active between 1610 and 1993.
Three prominent features characterize z 11 t over the course of the almost three hundred years for which data exist: 1. a continuously rising secular trend t, 2. long-term surging t, 3. a quasi-patterned transient rt.
Among them, t and t constitute the steady-state part of z 11 t. Of these two, the t part gives a strong indication of the`Gleissberg Cycle' (Eddy, 1976), represented by approximately two-and-a-half surges in the spans 1705±1811, 1812±1905, and 1906±?. Quite obviously, the course of each surge is of the type implied by Eq. (1). All three surges ride over what may be called a base-line which parallels the secular trend, and all three feature a de®nite bifurcation in the neighborhood of the surge peak. In addition, a higher-frequency transient rt is obvious in all three surge cases, causing a pair of subpeaks for each surge.

Secular trend t
While the upward trend of z 11 t over the last three centuries is quite obvious in Fig. 1, its increase with time cannot be open ended if the constrain tenet is to be satis®ed. However, since no peaking of t has been discerned over the length of the record, the analysis has to assume such a peaking at some future time t p and, based on it, determine the course of t using Eq. (6) in the form: The t p value can be found by applying linear regression iteratively to the 1700±1991 record and selecting the t p that minimizes the goodness-of-®t coecient 2 . The result, including year of the peak t p , peak value p , and duration t d , is given by: If, in violation of the constrain-tenet, the foregoing parameters were computed without imposing a peaking requirement, the result would be: This implies unbounded growth for t, due to the fact that the end of the data record used is the ascending part of the third surge presently at its peak. It is of interest, however, that the curve of Eq. (10) bottoms out in the year 2.4, or 1707 AD, which is the ending era of the Maunder Minimum.
For the base line ft, on the basis of which the surges are to be computed, the following simple de®nition was chosen: This is simply the bounded curve of Eq. (9) lowered by k units of z 11 t to between the 1815 and 1906 minima of z 11 t. The duration of t, and therefore ft, was established by ®rst ®nding the base line's value f1715 6X64 at the end of the Maunder Minimum, and computing the future year during which the descending branch of ft reaches the same level. To place the result in historical perspective, we observe that the contemporary trend t corresponds to the latest of Eddy's historical`maxima' of the sunspot activity's envelope using 14 g as a guide (Stuiver and Quay, 1980). Of these, the last six have occurred in historical times as seen in Table 1,``Columbian'' being our designation for the spurt of envelope activity that took place between the Sporer and Maunder Minima. Duration comparison between the two tabulations shows the ongoing Modern Maximum to be on par with the Medieval Maximum.

The surges t
The morphology of the surging part of z 11 t may be looked into either collectively or individually.
3.2.1 Collective surging. From a collective viewpoint, particularly dominant over the post-Maunder era appear to be two oscillations with periods of about 179 and 110 years (Williams, 1981). The ®rst oscillation in particular has attracted researchers' attention due to the fact that it approaches the behavior of orbital phenomena in the solar system: Jose (1965) has shown this periodicity to be prominent in the motion of the sun about the center of mass of the solar system, and Gribbin and Plagemann (1974) point out that 179 years is the period of total planetary alignment during which every planet is in conjunction with every other planet on the same side of the sun. Wood (1972), in analyzing the connection between sunspot number and the height of the planetary gravity tide on the sun's surface, observes that``the curve appears to repeat after about 170 or 180 years''. Cohen and Lintz (1974), on the other hand, suggest that``although the 179-year periodicity does exist in the sunspot cycle, it is not caused by a long-term planetary excitation but is, instead, a beat phenomenon''. By subjecting the 1844±1971 zt data to a maximum-entropy spectral analysis, they found three dominant frequencies in the spectrum: 110, 10.9, and 9.7 years. Of these the latter two``interfere to produce beats of roughly 187 years''. In our own work we found that the 178.7-and 110-year components do make for a good representation of the long-term surging in z 11 t. To arrive at this, the t were separated from z 11 t À ft and Eq. (8) was applied for each surge separately, leading to the cusps seen in Fig. 1, and the following parameters: In a follow-up step, the surge function j Y j IY IIY III, were expressed in terms of the aforesaid sinusoids: with the following results: Projecting into the future, this model suggests that the core (not counting the transient) of the next surge IV will peak in the year 2090 at the level of 43.9 units above the base line, as Fig. 2 portrays it. This renders the ongoing surge III the strongest in the span 1700±2100 of four centuries, with a core peak of 48.3 units above the base line. With the base line included, III and IV emerge about equal (79.8 units).

Individual surges.
While the use of the Eq. (6) expedient serves a purpose in that it gives an overview of the long-term activity, it is not conducive to probing the morphology of the surges' internal structure. For this, not only the core part t, but the entire surge t rt needs to be dealt with, i.e.: t rt z 11 t À ft X 15 After ®nding the total area e under the curve of the right-hand member, use of it and of st in Eq.
(1) produces the mediating factor t, that is the key structural element of the model, and the nature of which can be seen in Fig. 3. This is manifestly a trigger type of occurrence whose time behavior is made up of the combination of a growth term and a decay term operating concurrently, with the parameter values of each case seen in the following equation and the process pictured in Fig. 4.
Of interest is the fact that the observed dip at each t peak occurs near the crossing of the two 0 t branches in Eq. (16). The term 1 depicts a positive feedback process, which at some level triggers the negative feedback process represented by the negative feedback term 2 of the next surge's mediating factor. Interestingly enough, this modality replicates a relationship between solar activity and the earth's magnetic ®eld: the diurnal variation of the horizontal component of this ®eld on abnormally quiet days (AQD) at the time of sunspot minimum consistently anticipates the magnitude of the subsequent sunspot minimum (Brown, 1974). This pattern implies``that the size of solar activity maximum is determined at the very beginning of a cycle, or perhaps the very end of the preceding cycle from the`depth' of the solar minimum'' (Brown, 1974).   The restriction upon t places the projection at the year 1992 or later, while the constant k makes the average value of the dierence between columns A and B equal to zero. It is curve B that will be used in Part II of this study.
The following surge characteristics are presented at this point in order to eect comparisons: peak 1 trough peak 2 surge position position position duration I 0X301 0X528 0X783 1 106 years II 0X293 0X521 0X783 1 92 years III 0X400 0X540 0X637 1 125 yearsX 21

The transient rt
The relatively shorter period undulations rt exhibited by z 11 t, in addition to the long-span surges, are accounted for by the basic tenet's corollary and therefore have to be represented by a term r t in the mediating factor t. Accordingly, and for each surge separately, the residual: was regressed on the sum of the ®rst three subharmonics of the 11-year cycle and of the Gleissberg cycle (8th subharmonic) mentioned earlier: The reason for choosing this set of x -values is that the ®rst three subharmonics have been found, via Fourier analysis, in at least two widely separated terrestrial occurrences in which the solar factor is found implicated (Clark, 1988;Williams, et al. 1985). The eighth subharmonic was added heuristically, because its inclusion was found to triple the value of the regression's goodness-of-®t coecient 2 . This subharmonic is, of course, the Gleissberg cycle already mentioned. The x , x values found through regression are shown to be: 24 Figure 5 shows the complete model " z 11 t. It is worth noticing the similarity between " z 11 t and J.A. Eddy's annual mean sunspot numbers at maxima in the 11year cycle (Eddy, 1976, Fig. 8 Before leaving the topic of the long-term z 11 t surging of solar activity, we may point out the fact that the trigger-like nature of the``mediating factor'' is akin to the``eruption mechanism" that has been postulated for the 11-year sunspot cycle zt itself (Kiepenheuer, 1959). As Dicke puts it,``It has become clear by now that the rise and fall in the number of spots is due to a number of practically independent individual processes: thus the idea of a true periodic phenomenon was dropped in favor of the so-called``eruption hypothesis'' (Dicke, 1978).

The derivative hz 11 t
As in most dynamic processes, the time-derivative hz 11 t of z 11 t can be instrumental in cases where z 11 t is a casual factor for the process at hand. It can be computed either from Eq. (1) or from the ®rst dierence of z 11 t as a surrogate which, according to Eq. (7), results in: zt 11hz 11 t zt À 11 Y 26 and oers the means of projecting the sunspot number 11 years ahead. Since Eq. (18) makes possible the extension of the model z 11 t beyond the present, a projection is possible with an accuracy commensurate with the ®delity of the model z 11 t; Fig. 6 shows the latter, revealing a regularity, which we discuss further later in this study.

Model implementation
The analytical means through which the present model is to be applied in seeking possible links between longterm solar activity on the one hand, and terrestrial phenomena on the other, range from the simple to the fairly sophisticated. In the sense that this means`c onnecting'' one function of time f t with another gt, the simplest relationship will be the direct one: or, when the data ranges warrant it, where " g and " f are the averages, r g and r f the standard deviations of the two functions, and g x tY f x t the normalized gt and f t. This simple transformation brings the two data ranges within common bounds, and when necessary, can be rendered more eective if before normalization, gt were``stretched'' to: Normalization allows for a meaningful quantitative comparison of the contribution of multiple inputs f 1 t F F F f n t aecting a common output gt. Normalization, therefore, will be applied routinely in the entries of Part II of the study.
Where a more complex dynamic input-output connection is involved or suspected than that of Eq. (27), the powerful Kolmogorov method of signal-processing and system-response may be applied to advantage (Kolmogorov, 1941). Signal-processing pertains to either the interpolation or extrapolation of f t, while system-response addresses underlying causality or coevolution; generally, both cases have to cope with contamination from extraneous noise. The method has proved most eective in signal detection and estimation, as well as in control system analysis and design. A simpli®ed derivation of the method by this author is given in the following.
Kolmogorov's formulation of the signal extrapolation case assumes that a function f t of t, where both t and f t are real variables, is given in terms of equally spaced samples f 0 being the current value, and that its value f k, k units of time into the future, is to be predicted. It is assumed that f t is weakly stationary and, therefore,

that it has a ®nite second moment
if t 2 `I Y 31 if tf t s ff s Y 32 3. that it has a continuous covariance function ff s.
Assuming that all information available on f t is contained in the already-known samples means that the simplest way of ®nding its value f k k time-units into the future will be to express the latter as a linear combination of the former: Given that, by de®nition, the variance r 2 of the error between the true value of f k and its serial representation is: minimization of r 2 requires that its ®rst n 1 derivatives with respect to i be equated to zero. This results in n 1 equations in the unknowns i , However, because of the weak stationarity condition, Eq. (34), if kf k À s if tf t À s ff Às X 36 Furthermore, since f t is a real quantity, its autocovariance function ff has to be an even function: ff Às ff s X 37 Thus, the system of Eqs. (35) becomes: from which the unknowns i can be computed. While the analysis that led to Eq. (38) is based on the unit-time step, the unit's length has not been speci®ed: it can be one year, one decade, or any other convenient magnitude. The potential of this tool in projecting hz 11 t into the future is obvious.
Besides the case of signal extrapolation just seen, the Kolmogorov method can be used in the problem of system-response as well. In this case an output or response function is supposed to be the result of an input or forcing function being modi®ed by a linear operator. In the present context, gt, for instance, may be the result of the atmosphere's or hydrosphere's intervening between energy inputs from the sun and an ensuing climatic or other eect. Here again the representation: is used. Comparison with Eq. (33) shows that the solution given by Eq. (30) will be the result, except for the fact that the output vector in the right-hand side of Eq. (38) is now replaced by the vector: fg F being the cross-correlation function between f t and gt.
In the case of continuous time-functions the response relationship between output gt and input f t is given by the expression: where wt is the transfer function of the process (James et al., 1947). For the case of a discrete functional representation where both f t and gt are given in terms of equally spaced samples, Eq. (41) has the counterpart: Comparison of Eq. (41) with Eq. (34) shows that: meaning that the vector 0 Y À1 Y F F F Àn T is the system response (or transfer) function, between f t and gt in the serial case. In the present context, of course, f t z 11 t. Table 3 lists the values of ff F for zt, z 11 t, and hz 11 t, and Fig. 7 portrays the outcome. In the ®rst curve the 11-year cycle of the sunspot number dominates; in the second curve one half of the Gleissberg cycle is obvious, and in the third a new and unexpected oscillation whose period is t 8X25 years 44 manifests itself. This is a vivid example of nonlinear systems'``propensity to lapse into oscillations of new and incommensurable frequencies'' (McDonald, 1980), and has been previously detected in sunspot periodograms based on maximum entropy spectral analysis (Cohen et al., 1974).
One feature of the Kolmogorov algorithm as presented here is that its de®nition in Eqs. (33) and (39) does not provide for a constant term, having been developed originally in a communication-signal context. Care, therefore, needs to be exercized when applied to signals from other realms.
Equally challenging in the context of the present theme is the pertinence of the cross-correlation function fgF : it has to do with the case where the cause side of the presumed cause/eect relationship gt function of f t per diagram A: A is actually more complex as in diagram B, but the factor f c t has been omitted for lack of knowledge regarding the innards of the phenomenon at hand. In this circumstance application of the Kolmogorov algorithm on the system A instead of B will leave a residual in which the presence of f c t remains to be detected and extracted. This of course can be a dicult task, especially when incommensurable frequencies enter the picture. A case in point is terrestrial climate where not only the energy inputed by the sun is a factor, but contaminants, either manmade (Plantico et al., 1990) or subterranean (Schneider and Mass, 1975;Lamb, 1970), which are released into the atmosphere play a signi®cant role. Only in the case of known factors f t, f c t acting additively can the algorithm in Eq. (45) be changed explicitly to represent the case. For the record, Table 4 gives the actual and model values of the long-term solar activity's surrogate z 11 t.

Sunspot number projection
We conclude Part I of this essay by demonstrating the model's ability to eect an 11-year projection of the annual sunspot number itself over the past 150 years; this via Eq. (33). A reliable prediction of the 11-year cycle``could be of great importance to many ®elds of human activity; telecommunications, space exploration, and climate forecasting being among them'' (Fyodorof et al., 1996). Over the years the list of attempts to produce a solution has been long: see for instance, Wilson (1985), McNish and Lincoln (1987), Withbroe (1989), and Friis-Christensen and Lassen (1991). Although the descriptive schemes have been many, and the numerical models range from simple to fairly elaborate, none has achieved the goal, including those based on a Fourier transform. The disappointment is due, of course, to the limited knowledge of the dynamics of the astrophysical process taking place. It is the view of this study that the Kolmogorov algorithm can help with the quest, and produce a better descriptor. For this, the suggested modus operandi proceeds in three successive steps: 1. Generate an 11-year Kolmogorov projection z t of zt based on three past equidistant states of it: zt À 33, zt À 22, and zt À 11.
2. Search the residual rt zt À z t for the presence of both the basic 11-year solar cycle and the longterm z 11 t or its derivative . 3. Finally, apply regression analysis to the residual rt as a function of the terms found in Step 2.
The ®rst step requires solving the system: Y 53 from which, based on Table 3 for the values of F of the autocorrelation of zt, the following transfer function is computed: i i À33 À0X00179 À22 0X25351 À11 0X97182Y 54 eecting the projection: Since z t and zt have a concurrent part, the former can be improved by making it: z t mzt n 56 and ®nding the parameters , through linear regression: This step serves to compensate for the absence of a constant term in the de®nition of the Kolmogorov algorithm mentioned earlier.
In the residual rt of the operation described by Eq. (56), the presence of hz 11 is discerned, and in the same residual's autocorrelation function (Fig. 8) the presence of the sunspot number's basic 11-year cycle is obvious. Hence, rt is set as: from which a new regression produces: Thus, the complete 11-year projection expression becomes: zt 11 z t rt 11 X 60 In this expression the hz 11 t 11 term present in rt 11 comes from the actual z 11 t up to the year 1995 and the model z 11 t beyond that.
The ®nal outcome may be seen in Fig. 9, while Table 5 lists the 1990± 2005 projections. The strength of the approach can be assessed by the fact that in the 1835±1995 record of ®fteen zt maxima and fourteen minima, the model missed only the 1968 maximum by 1 year and the 1913 minimum by 2 years, which is smaller than the 2.3 years suggested by Fyodorov et al. (1996) as the limit for useful prediction.
Equally important is the fact that over the same timespan the Kolmogorov term of the 11-year prediction captured 867 of the zt signal; this is based on the ratio of the standard deviation (37.6) of zt-predicted versus (41.5) of zt-actual. For the complete prediction algorithm the ratio is 0.94. As far as the sunspot number's peak values are concerned, the projection error is given in Table 6. The approach may be compared to the Fig. 8. Autocovariance of long-term derivative  Jo Ann Joselyn of the National Oceanic and Atmospheric Administration for the purpose of predicting the maximum of the next sunspot cycle: the panel considered 28 prediction methods of all kinds ± including numerology-based ones ± and settled on a maximum for the upcoming sunspot cycle no. 23 equal to that of the ongoing 22nd one. Considering that sunspot peaks are accompanied by the emission of highly energetic particles and magnetic storms that can blast the delicate electronics of military, scienti®c, and commercial satellites, and that the ultraviolet radiation from an active sun heats up Earth's upper atmosphere causing it to expand and exert extra drag on satellites, a real barn-burner of a maximum could create the need of new shuttle missions to reboost the orbiting Hubble Space Telescope and the planned space station'' (Glanz, 1997).

Conclusions
The analysis detailed in the foregoing sections brought out and modeled three main features of long-term solar activity based on the sunspot number as the activity's surrogate: a rising trend, long-term surging, and a patterned transient. Although the idea of using the sunspot number to gauge solar activity is not new (Brown, 1974), the casting of this activity in the role of a long-term dynamic surging agent is. Noteworthy is the fact that the model points to the present decade being the time of peaking of the latest surge III of long-term solar activity to date, after which a span of roughly ®ve decades of gradual subsiding is to follow. It is also signi®cant that the macrodynamics of a phenomenon as complex as the energy exchanges taking place in the sun's interior can be portrayed in terms of a representation as simple and compact as the mediating factor having the characteristics of a trigger mechanism. This concept, coupled with the system-response tool of the Kolmogorov formulation, make for an eective analytical tool for the study of at least the phenomenology of sun-related long-term terrestrial phenomena, as Part II of this study will demonstrate. It should be pointed out that the model's synthesis made use of long-term periodicities present in particular geophysical topics where solar energy certainly has a role. Beyond this, the mediating factor may, as an operational concept, have a bearing upon the search for the modus operandi of the complex dynamics of the interior of the sun itself (Cox et al., 1993). Finally, particular attention of the Kolmogorov algorithm by the applied scientist in any ®eld is warranted; this because of its premise that the dynamics of the studied phenomenon is contained in the historical data itself, even if no theory exists about such dynamics at the present time.