It is assumed that external forcing synchronizes processes initiated by it. The concept of the synchronicity of processes is defined, based on their essential signs. The processes under study are decomposed on latent orthogonal components, which differ because of the coincidence and non-coincidence of the essential signs. The information from the original data is redistributed without distortion between these components. For computation of the components, algorithms were developed, using the Fourier transform on the basis of trigonometric functions. Theory and algorithms were applied to decompose on the components the Wolf numbers and the temperature series at 818 weather stations of the Northern Hemisphere from 1955 to 2010. By this approach, new properties of solar–terrestrial relations were revealed; the method characterizes the manifestation of the forcing and corresponds to the well-known notions of climatic processes. Therefore, the new method is informative, consistent; and it is suitable for the analysis of series under observation at this time.
non contemplantibus nobis quae videntur sed quae non videntur quae enim
videntur temporalia sunt quae autem non videntur aeterna sunt
(BSV. II Corinthios 4:18)
On the Earth, reasoning from experience, natural and climatic processes are significantly initiated and controlled by external forcing. It has a complex structure, but the Sun is a major contributor, acting directly and adjusting the other cosmic influences. Strengthening of the Sun's magnetic field is accompanied by an increase in the number of dark areas in the chromosphere, named sunspots. A linear combination of the sunspot numbers and sunspot groups has been chosen as a comprehensive indicator of solar activity. These combinations are called the Wolf numbers, and have no physical dimensions. Astronomers have been counting spots for about 400 years. As a result, a unique series of instrumental data about the cosmos has been obtained. See the classical treatises by Vitinsky (1973), Chizhevsky (1976), and Shurgin (1999).
In origin and due to inadequate measurements, natural and climatic processes consist of heterogeneous components, which are not always known. Besides, the basic values now studied were chosen long ago in past centuries to meet the challenges of technological development. For processes in nature, this traditional approach is not obvious. It should be expected that investigations will have a greater effect when they consider factors that are, in a sense, immanent to the system under study.
The development of options for component and factor analysis could help to make advances in this direction; see e.g. Lyubushin (2007). These approaches are universal but formal in character. Their efficiency is directly related to the anisotropy of the cloud of points representing the state of the system being studied in a multidimensional space. In addition, the results presented by the superposition of the original values are not always amenable to meaningful interpretation; this requires empirical regularities inherent in the system.
Distinctive features of the problem to be solved are taken into account more by different variants of informal classification, based on the optimization of objective functions of an empirical nature. In such approaches, the decomposition of the original set into subsets is performed. Each subset contains as strongly as possible related elements, and the relationship between elements of different subsets must be weaker. For example, Tartakovsky et al. (2015) performed this for climate classification by iteration of the phase of temperature series.
Based on observations, we can draw conclusions about the consistency of the Sun's forcing and certain climatic processes. It is well known that the cyclic motions in the solar system are manifested in the unceasing change of seasons, in daily warming and nocturnal cooling. These changes reflect a determinism that partially characterizes the climate for a certain time interval. There are other facts concerning possible determinism or the high correlation coefficients of the intensity of cosmic rays, the radio flux density, and series of Wolf numbers. We refer here to the publications by Alekseev et al. (2012), Budovyj et al. (2006), Buharov (1993), and Kuklin (1984).
Discussions are underway about how climate in general and the temperature particularly are sensitive to solar variability; see, e.g., Scafetta (2014). It should agree with this author in the sense that the progress can be expected in the way of sophisticated analysis of observational data, but in terms of some basic physical principles. The classic phenomenological approach involves the formalization of empirical data. Exaggerating the observed facts, we shall have to formulate the principle: “external forcing inherently initiates and synchronizes elementary processes in the geospheres”. Then we support this principle by a formal definition: “synchronicity of processes is manifested in the coincidence of their essential signs”, which it is necessary to define reasonably. Thus, the synchronism is selected as an essential factor in solar–terrestrial relations.
Below we present our theory and supporting algorithms to be applied for the synchronous analysis of series of average monthly Wolf numbers and the series of average monthly temperatures measured at 818 weather stations in the Northern Hemisphere of the Earth from 1955 to 2010.
The approach presented and the results of its application for climate problems and in the field of dendrochronology were partially reflected in the publications, e.g., of Tartakovsky (2015). It should be noted that the degree of synchronicity, as information about the transition of systems to a new state, was supported algorithmically and applied to solve geophysical problems as reported by Lyubushin (2007).
In our case, the novelty is that synchronicity was constructively defined, resulting in the decomposition of the original series. This computational procedure highlights the latent essence in the measured values and provides new information about the influence of the Sun on the temperature in the atmospheric surface layer, as is presented in this paper.
Let us assume that the series of experimental data
The series can be supplemented and extended by continuity to the whole real
axis periodically, using an even or odd manner. For such series there exist
both the discrete direct and inverse Fourier transform:
Possibly related to their common origin, the similarity of the series will
be significantly reduced if, for different numbers
For a pair of the series that characterizes the influence of the Sun and each
series associated with the climatic process, we introduce the “components
with coincident signs” (CS) and “components with non-coincident signs”
(NS). We refer to the procedure for pairwise decomposition of the series on
such latent components as “decomposition on the selected set”, hereinafter
“s-decomposition”. In this case, we select the series that describes the
solar activity for the entire planet. This Wolf number series
Let us obtain CS- and NS-components,
Note that s-decomposition leaves unchanged the original values of the Fourier
coefficients
We start from definitions (Eqs.
Thus, the CS- and NS-components of the original series are orthogonal,
resulting in the additive property of the scalar product of these components;
see expressions (
Let us make sure of the property of extreme correlation for the identical
components of the original series. We derive an equality relating the
correlation coefficients
In view of expressions (
From these expressions, it follows that s-decomposition extracts from a pair
of series
Considering that the average value of a function is proportional to the
value of its Fourier transform at zero frequency, we obtain the equalities
for the average values:
Bearing in mind the even continuation of the series and that the Wolf numbers
If the temperature scale has the zero value, the CS- and NS-components
(
Let us look at the normalized second initial moments for the original series and their CS- and NS-components. The initial moments do not include the centering operation, i.e., the average value is kept in series; it can have a physical sense as, e.g., a positive constant component of the solar activity. Naturally, the second initial moments are positive by definition.
Let the series
We used the series of the average monthly temperatures for the calculations. These were measured at 818 weather stations of the Northern Hemisphere from 1955 to 2010. This is the most complete archive of global data on surface temperature in the public domain. The archive is updated regularly in the Met Office Hadley Centre observations datasets. For the same time interval, the series of the monthly average Wolf numbers were taken on site at the Pulkovo Observatory.
In order to consolidate homogeneous information, the original data were subdivided into monthly groups. The terms of the series for a particular month in each successive year were selected and inserted in the group without changing the original order. In total there were 12 temperature series for each of the 818 weather stations; and 12 series of Wolf numbers, which are identical at all weather stations. Each group includes 56 terms.
The discrete Fourier transform is the base algorithm of the s-decomposition. Scafetta (2014) rightly wrote about the inadmissibility of the formal application of the discrete Fourier transform and the subsequent misinterpretation of the results. Doing discrete spectral analysis, make sure that the Fourier coefficients decrease rapidly within a finite interval of definition. This problem had solved, e.g., in Tartakovsky (1993) by means of the preliminary polynomial filtering and by optimal continuation of series beyond the domain of definition.
In this case, to calculate the Fourier coefficients, the series were
continued beyond the interval of definition, periodically in an even manner.
Moreover, to check and eliminate the influence of the series length on the
results, these series were interpolated from the original 56 up to 2
The designations remain the same for the newly formed series:
In this section, we analyze the foregoing reasoning and its consequences. First, we note that the properties mentioned in Sect. 3 were proved computationally with machine accuracy. This makes it possible to have confidence in the quality of the developed algorithms.
In the next step, the obtained algorithms were applied to the unique observational data. Initial moments, correlation coefficients, and mean values were calculated for these series. The computed dependencies were compared as far as possible with the climate geography and are discussed below.
The correlation coefficients of the temperature and Wolf number series and
their components (
Figure 1 shows that the fluctuations of
Coefficients
Positive correlations of CS-components and negative correlations of
NS-components characterize their opposite effect in the geosystem. The
consequence of this is a small correlation coefficient
Figure 2 shows the histograms of the original temperature series, their CS-
and NS-components for each month from the years studied. The temperature
partitioning interval was equal to 1
For each month, it was discovered that over a long and continuous range of
positive temperatures the histogram of the original temperature series
coincides with the histogram of CS-components. Moreover, in July and August
there is coincidence in the whole range of the temperature changes. For each
month there is also a continuous range of negative temperatures, where the
histogram of NS-components coincides with the histogram of the original
temperature series, excepting the range about
For all the months at zero temperature, there is a sharp peak of the
histogram of the NS-components. The Wolf numbers are always positive; there
are also dominant positive average values of the original temperature series.
In these conditions, according to the definition (Eqs.
In the histograms of the CS-components, the peak at zero is less noticeable, and is absent from June to September for the above reasons; i.e., a negative average temperature is less frequent than a positive one in the whole sample of 818 series.
Thus, the CS- and NS-components of the temperature series have a quite clear physical meaning – the distribution of the components on temperature intervals coincides with the same distribution of the original temperature within the above-mentioned ranges.
The temperatures from weather stations at different locations can fall into the same interval of the histogram by construction. From this, it follows that the observed properties of the histograms are not local and relate to the whole temperature field of the Northern Hemisphere that is formed under the influence of the Sun. How these properties are realized for each weather station separately, it is a subject for further research?
The monthly progress of the second initial moments
Let us consider the location of these 400 stations. Among them are 31
stations located in the zone of influence of the warm North Atlantic Flow
(Table A1). This moves in the northern part of the Atlantic Ocean and is an
extension of the Gulf Stream. The next 11 stations are located on the
North-West Coast of America (Table A2) under the influence of the North
Pacific warm flow. Off the coast of North-West Canada and South Alaska, this
flow turns into the Alaska Flow. The remaining 358 stations are located from
0.5 to 50
Throughout the year, a steady flow of solar energy can occur by direct
heating of the Earth's surface and atmosphere according to climate zone, as
well as by a steady heat transfer. Conceivably, in this case, the transfer
is related to the warm ocean flow from the South to the North.
Interestingly, the annual course of the second initial moments of
CS-components of Wolf numbers
There are two types of dependencies of moments
The same properties are apparent in both Figs. 3 and 4. In the January panel (Fig. 4, black bars) to the left of the jump, in the negative temperature zone, are located 418 stations. The remaining 400 stations are to the right of the jump, i.e., in the positive temperature zone, across the whole year (Fig. 4, orange bars). In Fig. 3, the second initial moments of these 400 stations are located in the upper range (orange colour) and throughout the year also. The distance between ranges varies with the size of the jump in the course of the year in Fig. 3, including the zero distance in July and August.
During the course of the year, the stations gradually move from the left to the right zone excepting July and August Within this period, the left zone is not revealed. Then the reverse process begins, ending in December This month, based on the number of stations in zones, is similar to February, and March is similar to November This similarity is reduced in an April–October pair, and then further between May and September.
The jumps of the average temperature from negative to positive occur due to
natural causes. On the contrary, solar activity
Thus, obtained as a result of s-decomposition, the second initial moments
This work is based, as a consequence of experience, on the hypothesis that synchronicity is an essential feature of solar–terrestrial relations. The Sun's external forcing initiates climatic processes on the Earth and should therefore be manifested in the similarity of their essential signs. Solar activity is characterized by the Wolf numbers and is considered as an integral indicator of forcing. Use is made of the orthogonal CS- and NS-components of Wolf numbers and the processes under study that differ in the coincidence and non-coincidence of their essential signs. These components are the latent essence of the phenomenon, and between them the information from the original data is redistributed without any distortion. To calculate these components, algorithms based on the Fourier transform were developed.
The theory is applied to decompose the Wolf numbers and temperature series from 818 weather stations of the Northern Hemisphere from 1955 to 2010. We obtained the following results.
The CS- or NS-components of the Wolf numbers and temperature series have significant correlation coefficients in the range from weak to strong values for small samples, typical for the minimum period of stability of the climate.
The histograms of the original temperature series coincide with the
histograms of their components over long continuous ranges of temperatures,
excluding the range of about
The second initial moments of the CS-components of the Wolf numbers display the climate geography, and fall into two ranges, the width and the distance between which is about 30 % of the possible changes.
The relationship between the CS-components of the Wolf numbers and temperature series can be interpreted as an inflow of energy from the Sun, and between the NS-component – as an energy outflow. The distribution of the inflow and outflow of solar energy over the weather stations undergoes jumps from about 27 % in January to 39 % in May, excluding July and August.
This new approach is informative; it describes the manifestation of the
forcing and corresponds to the known concepts of natural and climatic
processes. It deserves wide application and the search for other matches or
mismatches. The results are convincing that the things, “which are
seen”
This research has been funded by the Russian Academy of Sciences.
Monthly progress of the correlation coefficients of the
second initial moments
Weather stations from the zone of influence of the warm North Atlantic Flow.
Weather stations of the North-West Coast of America.
Correlation coefficients of Wolf numbers and temperature series,
of their CS- and NS-components. Red colour denotes
Histograms of the temperature series. Green circles and lining
denote histograms of the average temperatures; red lines are histograms of
CS-components; black lines denote the histograms of NS-components. The
normalized rms-difference of histograms does not exceed 3–4 % along range
matching. The relative frequencies on ordinate are in the power scale with
exponent equal to
Normalized second initial moments of CS-components of Wolf numbers from 1955
to 2010 at weather stations. Yellow and orange colours denote 358 stations
from 0.5 to 50
Normalized average monthly temperatures for 56 years at 818
weather stations in the Northern Hemisphere: black bars denote the polar and
temperate zone; orange bars denote tropical and subtropical zone; green
lines are moments