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Calculation of solar energy, accumulated in the continental rocks

Pavel Kalenda1, Ivo Wandrol2, Karel Frydrýšek3, Vítězslav Kremlík4

1Pavel Kalenda, CoalExp, Pražmo 129, 73904 (Corresponding author – pkalenda@volny.cz)

2Ivo Wandrol, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava

3Karel Frydrýšek, VŠB-TU, Ostrava, 17 listopadu 15, 708 33 Ostrava-Poruba

4Vítězslav Kremlík, 547 01 Náchod, Czech Republic

Keywords: temperature; heat; accumulation; solar activity; climate changes

(Received 17 July. Accepted 18 August 2018 - NCGT, Vol. 6, No.3, 347-380. Erratum 30 June 2020 - NCGT

Journal Vol.8 No.2)

Two errors were found in this paper, published in the NCGT in 2018. Both eliminated each other and then the

data analysis and results of the paper remained unchanged. Changes are highlighted by yellow.

Abstract

We developed recursive procedure, which allows estimation of the part of solar energy accumulated in the

Earth´s crust and estimation of the half-time of the heat radiation/accumulation parameter. This kind of

parameter can show time during which one half of the accumulated energy is released back to space. The

theoretical relationships were verified by the long-term pedology measurements.

When we used the Wolf´s numbers as a proxy-solar irradiance parameter for the last 11000 years, we were

able to estimate the half-time of the heat parameter of the continental crust. The most probable value of this

parameter t1/2 is 270 years, which means that the amount of energy in the whole crust is now at its maximum,

because of the anomalously high solar activity starting after the Little Ice Age.

We estimated future accumulated solar energy in the crust based on three scenarios of solar activity. All of the

three results show a small increase in accumulated energy until 2060 and after that a smaller or higher drop in

accumulated energy, and therefore a decrease in the global surface temperature.

Key points of this manuscript:

1. The recursive procedure of temperature changes was developed.

2. The fluids were verified as the key factor for energy transportation.

3. The variations of accumulated solar energy in the crust correlate with climate.

Keywords: solar irradiation, heat accumulation, recursive calculation, OLR, climate, scenario

(Received 17 July. Accepted 17 August)

1. Introduction

The surface temperature of the Earth is determined by the solar irradiance, by the Earth´s albedo, by

the properties of the atmosphere, by the residual heat of the Earth emitted from the depths, but also by

the physical properties of the continental crust rocks, which can accumulate some part of solar energy

for some time and release it later to the space again. The climate models suppose that the solar

irradiance is almost constant (so called solar constant = 1 360,8 ± 0,5 W/m2 (Kopp and Lean, 2011)),

and it varies only during 11-year-long cycles of about 0,1% (Lee et al. 1995). Much more important

should be the annual variations of the solar irradiance due to the eccentricity of the Earth´s orbit. The

Milankowich theory (1930) of the alternations of hot climate periods and ice ages is based on these

changes of the Earth´s orbit and variations of the obliquity of the rotation axis. But the periods of such

changes are much longer than 11 years – tens of thousands of years. The Milankowich theory is now

well developed with the help of non-lineal models and we are able to estimate the reversal points of

climate changes (Paillard, 2013).

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Because the Sun is not a star with constant radiation and except of the 11-year-long solar cycle there

are much longer cycles, these variations in radiation should be detectable in the global surface

temperatures on the Earth. If the Earth were completely covered by the ocean, which has a big thermal

capacity and relatively short heat exchange periods (only several months), then the whole extra-energy

from the Sun would be released to the space in a short time and only short variations of global

temperature (and climate) would be observed. As the big part of the Earth´s surface is created by

continental rocks, part of incoming solar energy is accumulated in the crust and the back release only

depends on the physical properties of the rock of the continental crust.

The pedology uses the information of the accumulation properties of soil and near-surface sediments

for soil preparation and growing plants (Bedrna, 1989; Kutílek, 1990; Klabzuba, 2001; Hora, 2011).

Temperature measurements in the boreholes were used for paleoclimatological analyses (Čermák,

1971; Čermák et al., 2003; Majorowicz et al., 2006).

In this paper we will focus on derivation of recursive formulas which we will use to calculate

temperature at any depth depending on changes of the surface temperature (solar activity).

In the third chapter, we will calculate part of the solar energy accumulated in the continental crust

during the last 11000 years (Solanki et al., 2004), and we will evaluate the “half-life of the heat”

parameter for the accumulation/release of the energy in the continental crust as a whole. At the end,

we will estimate the global temperature development based on three scenarios of solar activity.

2. Recursive relations for temperature changes at the depth below the surface

Let us have a material cube of infinitesimal small dimensions at the depth h below the surface (half-

space), which is thermally bonded with the surrounding material (see Fig. 1).

Fig. 1. Scheme of the penetration of heat from the surface to the depth and its radiation from the cube in the

depth h.

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Changes in temperature inside the cube will depend firstly on the initial state of the temperature field

in the half-space, and on the temperature changes on the surface, which will be the same for the whole

surface of the half-space.

We adopt the following conventions for the next derivations:

. . . mean temperature at the depth

. . . temperature at the depth in time t

. . . time step = const

according to Majorowicz et al. (2006)

the formal beginning of the process

Then we can describe the changes in temperature inside the cube by the equation

where the function determines how the temperature in the monitored cube at the depth h is spread in

time to its surroundings, and the function determines the changes in temperature at the depth h

depending on the temperature on the surface in the time .

2.1. Equation for heat irradiation ( )

If the temperature at the depth in the material cube is different from the temperature in the

surrounding, then the temperature changes inside the cube are proportional to the

temperature difference between the cube and its surrounding, multiplied by the material constant .

The index h further marks the fact that the considered cube is at the depth h. This dependence is

known as Newton's law of cooling. It can be written as

where

material parameter - must be determined by measurement;

mean temperature in the cube – must be determined by long-term

measurements.

The solution of linear differential equations of the first order with constant coefficients /2/ using the

method of variation of constants is:

,

/1/

/2/

/3/

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where is the initial temperature at the depth h. We can suppose that for one step of iteration

, , and then we can write

where the function could be formally written as

After the substitution /5/ into /4/ we obtain

We can modify /6/ into

2.2. Relationships for the heat sources on the surface ( )

The Earth´s crust can be thought of as a half-space. The Fourier-Kirchhoff heat equation can be used

for the evaluation of the temperature distribution from the surface to the depths on condition that the

half-space is homogeneous and isotropic medium, which does not contain any heat source.

Berger´s model of thermoelasticity (Berger, 1975; Kalenda, et al., 2012; Frydrýšek et al., 2012)

considers elastic half-space with defined horizontal coordinates and , and with the vertical

coordinate (depth) . The model is designed as a 2-D model with the axes , , because all of the

variables are symmetric with respect to the horizontal axis . The additive surface temperature is

determined by the harmonic wave with the amplitude, the angular speedand

the wave number

We can solve the equation for determination of the thermal field at the depth h. The x-coordinate can

be neglected due to symmetry of the thermal field ( ) and for clarity, and the variable can be

changed to. Then the thermal field can be described by the equation (see also Mareš et al., 1990)

Carslaw and Jaeger (1959) derived in Chapter II (p. 50) a simplified solution of the equation for the

linear heat flow. This equation can be then simplified for conducting heat from the surface as

whereis the scalar function, which describes the temperature distribution between

borders of the stick or layer. The function can be written in the form

/4/

/5/

/6/

/7/

,

/8/

.

/9/

/10/

/11/

5

where are constants. After the substitution into /10/ we obtain

If is true according to /12/, then the function /11/ is the solution of /10/ and we can write

The function could be formally written as . If we sign

then /13/ could be written as

2.3. Total temperature changes at the depth h at the time t

The function at the depth according to the equation /7/ will have a form

The function will have a form according to the equation /14/. Then the relationship /1/ can be

rewritten for the time in the form

where and are material constants, which depend on the specific place and time

increment only. They should be evaluated by the measurement.

The relationships derived above show that when the material characteristics of the environment do not

change, or changes are very slow (from the calculation point of view), then the temperature changes at

the depth depend only on the temperature changes on the surface.

Based on Figure 1 and the equation /16/ it is evident that for the time, the temperature in the

monitored cube at the depth will change. The heat from the cube will transfer to the surroundings if

the temperature in the cube is higher than the average temperature in the surroundings, while part of

the heat difference from the surface will go to the depth. For the calculation, it is necessary to

determine the value of the coefficients and through the measurement of temperatures for a

reasonably long period in time steps.

2.4. Half-life of the heat radiation/accumulation parameter

We can define the half-life of the heat

, which is the equivalent of radioisotopes, as the

time during which one half of the heat from the cube at the depth will be radiated. From the equation

/15/ follows

. /17/

Using equation /16/ we can recalculate the surface temperature at any depth or conversely find

parameters and/or, which are consistent with the measured temperatures at these depths. We can

estimate from the parameters and the time

, which is necessary for the release of one half of

the accumulated energy back into the atmosphere.

2.5. Verification of theoretical equations by pedology measurements

/12/

/13/

/14/

/15/

/16/

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The temperature profile from the surface of the depth of soil has been monitored since 1961 at the

meteorological and climatological station Ljubljana-Bežigrad (46 ° 04'59 "N14 ° 31'59" E, 285 m

a.s.l) and there is an almost complete database of daily temperatures at the depths of 2 cm, 5 cm, 10

cm, 20 cm, 30 cm, 50 cm and 100 cm (Ljubljana – Bežigrad, 2013). A sample of these pedological

temperature measurements in years 2011 and 2012 is shown in Figure 2.

Fig. 2. The temperature records in the depths of 2 up to 100 cm in Ljubljana-Bežigrad.

As seen in Figure 2, the diurnal temperature curve smoothed with the depth and extremes showed the

phase shift (delay) with respect to extremes of temperatures on the surface. This is consistent with the

function of temperature changes in the vicinity of the surface for a homogeneous isotropic medium

(relation /9/ or Mareš a kol., 1990).

Fig. 3 - Comparison of the temperature development in the depth of 2 cm (turquoise) and 100 cm (red) and

recalculated temperatures from the depth of 2 cm to the depth of 100 cm (black). Residuals between measured

and recalculated temperatures at a depth of 100 cm are blue.

Figure 3 illustrates the recalculated temperature from the depth of 2 cm to the depth of 100 cm using a

recursive relation /16/ and the least squares method for the residues of two parameters and. It

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can be seen that although the temperature at the depth of 2 cm has a phase shift from the temperature

measured at the depth of 100 cm (correlation r = 0.88), after conversion it exhibits a high correlation (r

= 0.997) where both curves only shifted by the absolute value of 3.75 ° C, which is probably due

to a local geothermal gradient or a paleoclimate development. All recalculated temperatures

from the depth of 2 cm to the depths of 10 cm up to 100 cm show a high degree of correlation with the

correlation coefficient greater than 0.995 with the measured temperatures at these depths (see Table

1).

Table 1 – The results of the recalculation of the temperature curves from the depth of 2 cm into the greater

depths. Coefficients and according to the equation /16/ for transferring heat from a depth of 2 cm below

the surface to a depth,

– half-live time of heat at a given depth or heat transfer from the surface to this depth,

r with 2 cm – the correlation coefficient between temperature curve at a given depth and the temperature curve

in the depth of 2 cm (without recalculation),

r projection - the correlation coefficient between recalculated temperature from the depth of 2 cm to a given

depth and the measured temperature at the same depth,

shift (°C) - means temperature difference between the measured and calculated curve temperatures for the same

depth and the smallest residuals.

Table 1 shows that the heat penetrates the depth of 5 cm very rapidly, in a matter of hours, and half of

the heat is radiated (or arrives from the surface) in about 9 hours. The heat from the surface reaches to

the depth of 20 cm in 2 days and to the depth of 100 cm in almost 17 days. This is in conformity with

the equation /9/ and other measurements in boreholes (Mareš a kol., 1990; Čermák, 1971).

3. The accumulation of solar energy in the continental crust

3.1. Energy budget

According to the Energy budget made by NASA (Smil, 2000), 51% of solar irradiance is accumulated

in oceans and continents. The rest of energy is or reflected back to the space either is absorbed by

clouds or air. A similar model, which is extended by the greenhouse effect, can be found in the 5th

Assessment Report of IPCC (2013) or in the paper by Trenberth et al. (2009). The largest part of the

energy is accumulated in the oceans because of the large heat capacity of water compared to the rocks,

and due to the faster heat transport in the ocean, allowing the heat flow orders of magnitude larger than

in solids. That is why all the climate models foresee a large accumulation of heat in the oceans, which

can be accumulated in a few months, but, on the other hand, may be released during the same period.

Most models assume that the oceans do not serve as long-term heat storage, because a substantial

portion of the heat can be released back to the atmosphere after a few months, years, or a decade

(Yulaeva and Wallace, 1994, Trenberth et al., 2014; Kumar and Hoerling, 2003; McLean et al., 2009).

Therefore climate models consider that after the period of 30 years all of the energy exchange is stable

and "settled”.

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In the previous chapters, we analysed the transfer and accumulation of heat in the rock mass, and we

derived the recursive procedure, when we are able to reconstruct the temperature (heat) development

at a specific depth from the temperature variations on the surface, for common rocks of continental

crust. Just as we are able to estimate the "half-life of heat" parameter at a different place and at various

depths below the surface, we can estimate this parameter for the Earth's crust as a whole. This is based

on the consideration that the global temperatures at the surface reflect both short-term variations of

energy, coming from the sun, medium-term (several months) release of energy stored in the oceans,

and the long-term accumulation of energy throughout the Earth's crust. Therefore, we can try to

estimate the Outgoing Longwave Radiation (OLR) from the rocks of the continental crust, and the

most likely "half-life of heat" parameter of the continental crust as a whole.

3.2. Solar activity

To reconstruct the temperature curve at depth h, we need to know the long-time development of the

temperature at the surface, according to the relationship /1/ labelled as which is measured

with the step and the mean value of temperature at the depth h

. As direct temperature

measurements are known only at a few places on the Earth for about the last 500 years (Letfus, 1993),

and the Total Solar Irradiance (TSI) is only directly measured in the period of space flights (from

about 1980) (PMOD_WRC 2015; Fröhlich, 2006), we must find a proxy parameter that is proportional

to either global temperatures or the TSI, and it is known for a period of at least 10 times longer than

the supposed "half-life of heat” release parameter of the continental crust.

Therefore, we need the data series from at least several thousand years, and consistent over the whole

period. Such data series appears to be the proxy solar activity (Wolf numbers) that can be

reconstructed by several independent methods, and which is independent on the global temperature,

and even on other climate parameters, such as the concentration of GHG or flow in the oceans and

atmosphere. Wolf numbers can be reconstructed using at least two different methods (Usoskin, 2013):

Helio-modulation of cosmic rays, which modulates isotopes 1) 14C, and/or 2) 10Be. Usoskin et al.

(2002) or Solanki et al. (2004) reconstructed solar activity proxies (decade-averaged modulation

potential) based on the concentration 10Be, and comparison with the observed Wolf numbers in

historical times (Usoskin, 2013). For our purposes, we took a series of 11,000 Year Sunspot Number

Reconstruction by Solanki et al. (2004), which is available on the NOAA server.

3.3. Climate variations

The difference between the long-period component of OLR and the mean value of the surface

temperatures must be the same as the difference of the heat released at the depth h according to the

relation /7/. In other words, climatic variations should also reflect among others a long-term

accumulation of heat in the rocks. We can take, as a representative of the long-period component

OLR, directly the global temperature (or its differences from the mean temperature), or another

parameter, which contains global temperatures directly or indirectly, such as the water levels of the

world ocean or the glaciation area.

Comparing the two curves, i.e., which is represented in this case by the proxy of the solar activity,

converted to the depth h according to the relationship /14/, and the measured global temperatures or

the level of the world ocean, we should be able to estimate the parameter according to the

relationship /7/, for which we obtain the biggest correlation coefficient. Then we can calculate the

“half-life of the heat”

parameter according to the relationship /17/.

For our purposes, we chose as a climatic series the directly reconstructed global temperatures

“Glcru_eiv_composite” by Mann et al. (2008), which are available on the NOAA server. The

reconstructed temperatures with the time step of one year were recalculated into two series by

averaging - in a series of time increments of 10 years and the second series of time increments of 10

years, but with averaging window of 50 years. “Glcru_eiv_composite” series has been available since

500 AD, i.e. for the last 1500 years.

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3.4. Results of analyses

For the conversion of energy from the Earth's surface to the depth h, it is necessary to know two

parameters: and . For the analysis of the correlation between the radiated heat from the depth h

to the surface and surface temperatures, it is not necessary to know the parameter , since it only

moves the absolute level of temperature at the depth h compared to the surface of the constant value.

Therefore the correlation coefficient is not dependent on it. We used therefore coefficient = 1. On

the other hand, the coefficient has a real physical meaning and shows how quickly the heat

accumulates in the Earth's crust, and/or how quickly it is radiated back to the surface. We calculated

for different coefficients the correlation coefficient between the two time series: 1) Of the

reconstructed surface temperature by Mann et al. (2008) and 2) volume of the heat release from the

depth h to the surface (OLR) according to /1/ (see Fig. 4).

Fig. 4 Correlation coefficient between reconstructed global temperature CRU_composite (Mann et al., 2008) and

released heat from the crust for various half-life of heat as a parameter.

It is evident that the correlation coefficient between the smoothed temperatures within the window of

50 years is higher than for the temperature smoothed within a 10-year window, and reaches the

maximum r = 0.86, indicating a statistical dependence between the two rows at a significance level of

15%. The higher correlation coefficient for the longer window shows that the values of the proxy-

Wolf numbers in increments of 10 years are physically smoothed, and that the samples did not allow

obtaining a higher accuracy either in amplitude or time. Therefore, the smoother curve of the

reconstructed temperatures in the 50-year window corresponds better with the primarily smoothed

curve of the proxy-Wolf numbers. The resulting curve of heat stored in the continental crust (in

relative units) for the maximum correlation coefficient (i.e. for

years) together with the

reconstructed temperature curve, is shown in Figure 5.

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Fig. 5 Comparison between accumulated heat from solar activity and reconstructed global temperature in 50-year

time window (according to Mann et al., 2008).

We can also compare the same curve of the accumulated heat in the continental crust, with the global

sea level; reconstructed by Jevrejeva et al. (2009) (see Fig. 6).

Fig. 6 Comparison between accumulated heat from solar activity and reconstructed global sea level (Jevrejeva et.

al 2009 – Fig. 3b).

Correspondence between the reconstructed global ocean level and the accumulated heat in the Earth's

crust appears to be better than for the reconstructed temperatures by Mann et al. (2008), because the

first derivative of the two curves matches more. Visible differences are only in time of the ocean level

drops after the biggest explosions of volcanoes, especially Samalas, Kuwae and Tambora, which is, of

course, a matter of the model used for the reconstruction of the global ocean level.

3.5. Discussion

The temperature variations on the surface are transmitted into the depths very slowly due to the low

heat conductivity of rocks (Čermák, 1971; Čermák et al., 2003). Heat is transferred faster by help of

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fluids. Thus, we can estimate that the whole continental crust plays crucial role during the

accumulation of heat and its radiation back into space. Fluids play a significant role in this process.

The heat transfer from surface to depths can be made not only by sharing or by convection of fluids,

but also by other mechanisms, such as the thermoelastic wave with ratcheting (Kalenda et al., 2012).

The interaction between planets, the Sun, atmosphere, hydrosphere, and geosphere is well known

(Pulinets and Ouzounov, 2011; Mörner ed., 2015). For example, the variations of seismic activity have

the same periodicity as the Sun or the whole Solar System. There are periods of the Schwabe cycle

(≈11 years), the Hale cycle (≈22 years), and the Van (Jose) cycle (≈179 years) (Jose, 1965; Kalenda et

al., 2012). The solar activity reflects relative positions of the planets, and thus the movement of the

Sun around the Solar System barycenter (Charvátová, 1990). There is a close link between the

orogeny and cool climatic periods (Kalenda et al., 2012). It is therefore not surprising that many

authors have found similar periods of planets in the Solar System and climate changes. These links

were described in the entire issue of the PRP journal (Mörner 2015). For one of shortest periods we

can point to a period of 60 years, which shows the binding activity of the sun and the giant planets of

the Solar System (Scafetta, 2010, 2011 and 2012). This period is observable both at the global sea

level (Jevrejeva et al., 2009), and from the PDO and AMO climatic parameters (Klyashtorin, 2001;

Akasofu, 2009). Most importantly, the 60-year period is also marked at the aurora borealis, which

obviously does not have any connection with the climate on the Earth, but it reflects the solar wind

variations, and thus the variations of solar activity (Křivský and Pejml, 1988). Therefore, we modeled

60-year oscillations on the curve of the accumulated heat at the depths, although this period cannot be

seen on proxy data – the Wolf numbers.

As we can estimate the future solar activity according to the gravitational influence of a planet

(Kalenda and Málek, 2008), we can suggest three possible scenarios for the development of solar

activity in the next 80 years – Table 2.

Table 2 – The estimations of average Wolf´s numbers in 10-year window

A

B

C

2025

20

16

40

2035

16

10

40

2045

16

24

40

2055

10

29

40

2065

26

29

40

2075

26

24

40

2085

53

24

40

2095

68

15

40

In scenario A, we assume that solar activity between the years 1622-1658 corresponds to solar activity

between the years 1980-2016. The average monthly number of sunspots will be at its maximum in the

following years: 2029 (20), 2041 (16), 2051 (10), 2062 (26), 2074 (26), 2085 (53) and 2096 (68).

In scenario B, we use the prediction of the future solar activity according to R.J. Salvador (2013) with

maxima: 2029 (27), 2038 (16), 2048 (40), 2058 (49), 2069 (49), 2079 (40) a 2093 (26), which we

smooth in a 10-year time window by the coefficient 1.7 to be comparable with the Wolf´s number

according to Solanki et al. (2004) (see Table 2).

In scenario C, we assume that the solar activity is the same as now, i.e. the Wolf´s numbers are 40 for

the whole period until 2100. The result for scenario C can be seen in Figs. 5 and 6 (the real data of

solar activity is used until 2015). We can see a so called “hiatus” between 2000 and 2030, which is

confirmed on real data now. The global temperature should rise between 2030 and 2060, but less than

between 1970 and 2000. Since 2060 we suppose the next “hiatus” of global temperatures. Moreover,

12

we suppose at least one volcano eruption with the VEI at least 5 (Kalenda and Neumann 2012), which

can drop the global temperature by about 0.5°C and more.

We make a projection to 2100 of the accumulated heat in the crust according to all three scenarios for

the development of solar activity, together with developments of global temperatures by Mann et al.

(2008) (see Fig. 7).

Fig. 7 The development of global temperatures according to scenarios of solar activity A, B and C until 2100

(relative units). Green – real global temperatures according to Mann et al. (2008).

The current observed "hiatus" in temperatures between 1998 and 2015 corresponds fully to all three

scenarios of developments in global temperatures, because solar activity has been decreasing generally

since 1995. This "hiatus" is only assumed by 1 of 102 scenarios of the IPCC (Christy, 2015). The

period 2020 - 2040 will prove or disprove our hypothesis, based on the accumulation of heat in the

rocks of the continental crust and its release back into space after some time.

4. Conclusion

If we admit that part of the incident energy from the Sun accumulates in the rocks of the continental

crust, we can estimate, based on the solar activity and a climatological parameter containing heat or

temperature, what the material parameters of these rocks are. The greatest correlation coefficient

between the number of the proxy-Wolf numbers (Solanki et al., 2004) and global temperature

anomalies in the window of 50 years in the increments of 10 years (Mann et al., 2008) has been

detected for the "half-life of heat" parameter

= 270 years. For this parameter, the correlation

coefficient between the proxy-Wolf numbers and reconstructed temperature reached r = 0.86, which is

a sign of the fact that there is a link between them.

An even closer link exists between the proxy solar activity and the reconstructed global sea level

(Jevrejeva et al. 2009). The first derivative of both curves also corresponds, which is a sign of one

integrated physical mechanism that combines both series.

On the basis of the three scenarios of future developments of solar activity over the next 80 years, we

have been able to predict the development of future global temperatures or global sea level. It has been

shown that the rate of global temperatures increase will slow down (conservative scenario C), or will

completely stop (more likely scenarios A and B). The increases in global temperatures until 2100 will

not in any case exceed 1.5 ° C, even if people emit CO2 into the atmosphere in the same way as

recently. The modern temperature maximum seems to be the result of accumulated energy in the

13

Earth´s crust after anomalously high solar activity, the biggest for at least for last 1000 years. The

accumulation started right after Little Ice Age, i.e. after Maunder minimum of solar activity, when

people were only on a way to the first industrial revolution (Jevrejeva et al., 2008).

The analysis of the development of the heat stored in the continental crust shows that the currently

existing climate changes are caused by nature origin, not mankind (Mörner, 2015).

When we sit by the tiled stove, we feel the heat for a long time after the fire has gone out...

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