The contribution of solar variability on recent global warming: testing four different proxies for solar activity

The contribution of solar variability on recent global warming:

testing four different proxies for solar activity

BSc Thesis

Helmer Gudde (6145353)

Examiner: J.H. van Boxel Co-assessor: E.E. van Loon 24-08-2018

Abstract

There are several drivers of the change in Earth’s temperature of the last 50 years, one of which is the sun. The sun’s contribution to this temperature change remains disputed, though. The aim of this research therefore is to determine the contribution of solar variability to contemporary global warming, testing different proxies for solar activity. The first proxy is total solar irradiance (TSI), the second is ultraviolet radiation (UVR), the third is galactic cosmic rays (GCR) and the fourth the sunspot number (SSN). These proxies were examined because of their relation to three sun-climate mechanisms.

To quantify the contributions of these proxies a correlation and multiple regression analysis is performed. Herewith different models, in which other potential drivers of recent global warming were added, were made.

This study found that solar variability, together with El Niño and aerosols, is the main contributor to earth’s temperature from 1957 until approximately 1964. Afterwards it steadily decreases but remains a contributing factor, its contribution fluctuating with an 11-year cycle. The greenhouse gases take over the main contribution since then and linearly increase their contribution up to 0.6 C in 2002. The different solar proxies used to determine the conclusions show marginal differences with respect to their contribution.

The contribution of solar variability to the temperature fluctuations is more substantial than its contribution to temperature itself, fluctuating again with an 11-year cycle. The greenhouse gases contribute negligibly little to temperature fluctuations. The differences between the contribution of the four solar proxies to temperature fluctuations are marginal.

Still, there are improvements to be made. The time interval in which solar variability is regarded a large contributor to global warming needs further study. Also, addition of albedo changing factors, a larger time frame, implementing of the leave one out method and a more advanced regression technique could improve understanding of this complex field of research.

Table of content

1. Introduction………...4

2. Methods and data………...5

2.1 Data………...6 2.2 Correlations……….11 2.3 Multiple Regression……….11 3. Results………...14 3.1 Correlations………...14 3.2 Multiple Regression………..15 4. Discussion………...21 5. Conclusion………...24 References………...25 Appendices………...28

1. Introduction

The global mean temperature rose approximately 1 °C since the beginning of the previous century and this rise has increased the last 3-4 decades. This increase in temperature is potentially severally damaging for life on earth and to prevent this from happening we have to answer the urgent question why this occurred. Many causes have been claimed to be the main drivers of this temperature rise: greenhouse gases, aerosols, land use and the sun being the most prominent ones (IPCC, 2013).

The sun has always been the main energy source for the earth’s atmosphere and the driver behind it’s dynamics. Early on scientists established a theoretical relationship between the sun and earth’s climate. Already in 1801 solar activity was coupled to the increase of wheat prices (Herschel, 1801). Much later Eddy (1976) showed that the temperature fluctuations of the last centuries strongly correlate with fluctuations in solar activity. Despite some inconsistencies (Pittock, 1979), the link for long term oscillations was convincing and subsequent studies, which were more refined, found the same parallels (e.g. Lassen & Friis-Christensen, 1995). Yet, the assumption that the sun has just as much influence on the temperature change of the 20th _{century as during the centuries before, diminished in power, because}

at the end of the 20th _{century the temperature kept rising, while the sun became ‘quiet’ (Lockwood & Fröhlich, 2007).}

Hence, the authoritative International Panel on Climate Change denounced in 2007 that the sun is climatologically less influential than other climate drivers (IPCC, 2007). However, the influence of the sun on the climate still remains the subject of debate (e.g. Van Geel & Ziegler, 2013; Stauning et al., 2014). To clarify this discussion, more insight is needed about the role of solar variability in the recent warming of the earth.

In this study an approach with four different proxies for solar activity was chosen to offer additional insight into this problem. The four proxies are the total solar irradiance (TSI), the sun’s ultraviolet radiation (UVR), the galactic cosmic rays (GCR) and the sunspot number (SSN). These proxies were chosen because they are related to certain sun-climate mechanisms. In this research we hope to improve the study of these mechanisms via quantification of these proxies. There are in general 3 mechanisms working between the sun and the earth, affecting the earth’s temperature (Solanki, Krivova & Haigh, 2013; Marsh & Svensmark, 2003). The first one is directly via irradiance, the second one is indirectly via ultraviolet-radiation and the third one is indirectly via galactic cosmic radiation (see fig. 1).

Fig. 1. The different mechanisms affecting earth’s

Firstly, the direct effect of solar irradiance has to be taken into consideration. The sun emits electromagnetic radiation that reaches the top of Earth’s atmosphere. The total amount of this incident radiation over all wavelengths is the total solar irradiance (TSI). Variations in TSI can induce temperature shifts near the surface by changing Earth’s radiation budget (Solomon et al., 2007). Secondly, the indirect influence of UV-radiation (UVR) plays a role in heating up the earth. This mechanism relies on the fact that UVR produces ozone in the stratosphere which in turn affects the temperature of the stratosphere. Via dynamic coupling, the temperature of the troposphere also changes (North et al., 2012). Lastly, there is the influence of the sun on earth’s temperature via its effect on cloud formation. This mechanism starts off with the solar wind, blocking a part of the galactic cosmic radiation (GCR). Subsequently with less GCR, fewer low clouds are formed (Svensmark & Friis-Christensen, 1997). Thus, less solar radiation is reflected, causing a warming of the atmosphere. In short, when there is more solar wind the earth heats up (Marsh & Svensmark, 2003). The SSN is related to all three mechanisms. When more sunspots cover the sun, more energy is released and thus the TSI rises (Gil-Alana, 2014; Foukal et al., 2006). Furthermore, it is also coupled to UVR, because faculae surrounding the sun spots emit a lot of ultraviolet radiation (Lukianova & Mursula, 2010). Lastly, sunspots are expressions of the solar magnetic field and the solar magnetic field is highly related to the solar wind (Russel, 2001). Thus, the GCR-mechanism and the SSN are connected.

The principal aim of this study is to determine the contribution of solar variability to contemporary global warming, testing the four different proxies for solar activity. The secondary aim is to offer insight into the proportion in which these proxies contribute to recent global warming. Because the data of the GCR-proxy doesn’t extend further backwards than 1957, this was the starting date of this research. The main research question of this study therefore was: What is the contribution of solar variability to earth’s temperature since 1957, and in what proportion do the proxies representing this variability contribute to this?

Additionally, the fluctuations of temperature were considered, giving the following secondary research question: What is the contribution of solar variability to earth’s temperature fluctuations since 1957, and in what proportion do the proxies representing this variability contribute to this?

2. Methods and data

To answer this question, the four proxies were statistically tested and compared. The solar variables have been incorporated in a statistical model with other variables that represent drivers of global warming. This model was partially based on the model created by Besselink (2018). The other variables are greenhouse gases, aerosols and El Niño. These variables were chosen because they are important drivers of temperature change (IPCC, 2013; Besselink, 2018).Initially sea ice was also added to the model, but it turned out to be that its contribution was low and it was impossible to separate it from the radiative forcing of the greenhouse gases. The reason that the solar proxies were embedded in a bigger model was that an accurate estimation of their contribution is found in relationship to other climate drivers.

Multiple regression was used to build this model. Before running the multiple regression, some preparation work had to be done. This comprised a correlation of all the above-mentioned variables with each other and with Earth’s temperature over the time period 1957 until now. After that the multiple regression was executed. All this was done with Excel (version 2017). Lastly, after the multiple regression analysis some further quantifications were made. The contributions of the proxies relative to the other variables and the sample variance of the proxies were calculated. All these methods will also be implemented on the temperature fluctuations.

2.1 Data

Data were collected from various sources (see table 1) and shown graphically. The data were annual, monthly or daily and subsequently converted to annual numbers.

Data Source Units Type of data Period covered

Earth’s temperature NASA C Annual 1900-2015

Greenhouse gases CDIAC W m-2 _{Annual 1900-2015}

Aerosols CDIAC Dimensionless Monthly 1900-2014

El Niño NOAA C Annual 1900-2014

TSI Dasi-Espuig et al. W m-2 _{Daily 1700-2008}

UVR SOLAR2000 Solar Flux Units Daily 1947-2002

GCR BRI Dimensionless Hourly 1957-2017

SSN SILSO Dimensionless Annual 1700-2017

Earth’s temperature

The mean global temperature was extracted from NASA’s database (NASA, 2017).

Greenhouse gases

This data is a combination of the four greenhouse gases with the highest radiative forcing, namely CO2, CH4, N2O and

CFC’s (CDIAC, 2017a). Radiative forcing expresses the change in the energy balance at the edge of the atmosphere due to greenhouse gas emissions or another forcing agent.

Fig 2. Earth’s temperature

14.40 14.60 14.80 15.00 15.20 15.40 15.60 15.80 16.00 1900 1920 1940 1960 1980 2000 2020

Temperature

Fig 3. Greenhouse gases

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 1900 1920 1940 1960 1980 2000 2020

GHG

Aerosols

Radiation interacts with aerosols and this causes radiation extinction. A parameter in the equation describing the extinction is the optical depth. The optical depth at 550 nm was used as an indicator for aerosol concentration. The largest variations in aerosol concentrations originate from explosive volcanic eruptions in the tropics. Values of aerosol optical depth were retrieved from CDIAC (2017b).

El Niño

For this study the Niño 3.4 index was used (NOAA, 2017). The Niño 3.4 index is defined as the temperature anomaly for the region 5N-5S, 120-170W (Trenberth, 1997).

Fig 4. Aerosols -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 1880 1900 1920 1940 1960 1980 2000 2020

Aerosols

El Niño

Total Solar Irradiance

This data was retrieved from Dasi-Espuig et al. (2016). It is created by the SATIRE (Spectral And Total Irradiance Reconstructions) empirical model, which is based upon the distribution and the brightness of magnetic structures on the solar disk.

Ultraviolet Radiation

UV-radiation data were extracted from SOLAR2000 (Floyd, Tobiska & Cebula, 2002). This is an empirical solar irradiance tool. F10.7 values were used as a proxy for UV-radiation. F10.7 is the solar radio flux at 10.7 cm and correlates well with ultraviolet radiation.

Fig 7. Ultraviolet radiation

0.0 50.0 100.0 150.0 200.0 250.0 300.0 1940 1950 1960 1970 1980 1990 2000 2010

UVR

Fig 6. Total Solar Irradiance

1360.4 1360.6 1360.8 1361.0 1361.2 1361.4 1361.6 1361.8 1362.0 1362.2 1362.4 1900 1920 1940 1960 1980 2000 2020

TSI

Galactic cosmic rays

Data was used from the Thule observatory in Greenland (BRI, 2017), where a neutron monitor is stationed. For the year 1977 though there was a data gap at Thule. A neutron monitor is a ground-based detector designed to measure GCR.

Sunspot number

The Sun Spot Number was obtained from the World Data Center SILSO at the Royal Observatory of Belgium in Brussels (SILSO, 2017).

Fig 8. Galactic Cosmic Rays

3500 3700 3900 4100 4300 4500 4700 4900 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

GCR

Fig 9. The sunspot number

0 500 1000 1500 2000 2500 1900 1920 1940 1960 1980 2000 2020

SSN

2.2 Correlations

Firstly, a cross correlation was performed. In other words, correlations of all the independent variables (IV) with each other. The independent variables that correlate with each other were not used together in the multiple regression. This was expected for the solar variables because they are all related to solar activity. Furthermore, a correlation of the independent variables with the dependent variable (DV) was also done before implementing the Multiple Regression. This gives us an early indication of the relationship of the independent variables with earth’s temperature.

2.3 Multiple Regression

Multiple regression examines the relationship between the independent variables and the temperature. Thereby, it creates a modelled temperature based upon the actual temperature and the independent variables’ values. It is founded on the following equation:

T = C0 + C1IV1 + C2IV2 + .... + Cn IVn (1)

Where T is Earth’s modelled temperature, IV the independent variable’s value, C the independent variable’s coefficient (and C0 the intercept) and n the number of independent variables.

Model comparison

The influence of the total of independent variables on the temperature was measured. This was done with several models. Each model contained all the climate drivers plus one solar variable. Also, a model was added with all the climate drivers, but without a solar variable. Thus, there were 5 models, called the A-models. The amount of influence can be derived from statistical parameters, such as the adjusted R2_{, the P-value and the SEE. The R}2_{is the proportion}

of the variance in the dependent variable (in this case temperature) that is accounted for by the independent variables. The adjusted R2 _{in particular was used because it is more appropriate when the number of independent variables vary}

(this is the case because the models without solar driver were added). The P-value expresses the significance and for each model tests the null hypothesis. The null hypothesis is the hypothesis in which all the coefficients of the independent variables are 0, hence there is no correlation between the independent variables and the dependent variable. A P-value less than 0.05 indicates that the null hypothesis can be rejected. A low P-value implies that the dependent variable is closely related to the total of independent variables. The SEE is the standard error of the estimate and is a measure of the accuracy of predictions made with the model. It is the square root of the mean of the squared errors. All these parameters together determine the fit of the model.

P-values independent variables

The P-values of the independent variables of each model will be considered. The P-value of the independent variables indicates that changes in the independent variable are associated with changes in the dependent variable. More specifically, the null hypothesis that the independent variables’ coefficient is zero can be rejected if the P-value is less than 0.05. If this is the case, the variable is statistically significant and probably a worthwhile addition to the model.

Graphs models

Furthermore, the contributions (in C) of the independent variables to earth’s temperature change since 1957, the modelled temperature and the actual temperature were represented in graphs. The contributions’ calculation started off with equation (1). This equation gave the coefficients of the different independent variables. Afterwards the contributions were calculated with the following formula:

IVcontr = Ci • (IVi – IVref) (2)

where IVcontr is the contribution value of the independent variable, Ci the independent variables’ coefficient, IVi the

independent variables’ value and IVref the independent variables’ reference value. The reference value was determined

as the average of the first ten years of that particular independent variable. On the basis of the contribution values a graph was constructed. This was done for all the solar proxies.

The modelled temperature was calculated with the following formula:

Tmod = IVcontr,1 + IVcontr,2 + … + IVcontr,n + IVref (3)

On the basis of this equation a graph of the modelled temperature was made.

Relative contribution

In addition to the graphs we made further quantifications. The contributions of the proxies relative to the other drivers of the years in which the proxies showed maxima and minima were calculated. The years of maxima and minima were chosen because they indicate the amplitude of the proxies’ contribution, since they are wave-type functions. We called this the solar proxy’s relative contribution, SPrel. Mathematically it was constructed as follows, with SPcontr the solar

proxy’s contribution.

Sample variance

Another way to quantify the contribution of the proxies to earth’s temperature and temperature fluctuations is the sample variance. It expresses how varied the variables are. Mathematically, the sample variance (s2_{) is constructed as}

follows.

(5)

with X defined as a data point, X̅ the average of all data points and N the amount of data points.

Since the solar proxies are all sinusoidal around 0, the higher the variance, the higher the average amplitude, the higher it’s periodical contribution. Thus, it is a measure for strength of their contribution.

Temperature fluctuations

All above discussed methods were also executed over the fluctuations of temperature. The temperature fluctuations are the differences of the temperature relative to its polynomial trend line (appendix A). A third-degree polynomial was taken. This analysis was done to look at the relation between the solar proxies and earth’s temperature in a different way. Maybe the solar proxies contribute differently to the fluctuations of earth’s temperature than to temperature itself. Afterwards, the same procedure as above followed: a correlation was executed, the different models were compared, graphs were made, the relative contributions of the proxies were calculated, and the sample variance of the proxies determined. The models were called the B-models.

3. Results

3.1 Correlations

Cross correlation IV’s

The cross correlation unveiled the high correlation between the solar variables, as expected. Because of this finding the solar variables were not taken together into one multiple regression but taken separately into four different multiple regressions. The correlations between the other variables with each other or with the solar proxies is low.

Also noticeable is the anti-correlation of GCR with the other solar variables. This is because its function over time moves out of phase with the solar wind and therefore with solar activity (see fig. 8 and 9)

IV-DV correlation A-models

This correlation gave three important findings. First of all, as expected, the GHG’s are highly correlated with temperature. Secondly, none of the solar variables are strongly correlated with temperature, although UVR and GCR seem to show some correlation. Thirdly, GCR shows an anti-correlation with temperature. This is because it moves out of phase with solar activity, as previously mentioned.

temperature temperature 1.00 El Nino 0.15 aerosol -0.18 GHG 0.88 TSI 0.07 UVR 0.25 GCR -0.30 SSN 0.16

Table 3: a correlation of the independent

variables with the dependent variable of model A. The values are the correlation coefficients.

El Nino aerosol GHG TSI UVR GCR SSN

El Nino 1.00 aerosol 0.25 1.00 GHG 0.02 -0.01 1.00 TSI 0.19 -0.05 -0.16 1.00 UVR 0.11 -0.14 0.03 0.87 1.00 GCR -0.02 0.01 -0.12 -0.81 -0.88 1.00 SSN 0.07 -0.17 -0.06 0.89 0.97 -0.84 1.00

IV-DV correlation B-models

This correlation unveiled three main things. Firstly, the greenhouse gases are not correlated with the fluctuations of earth’s temperature. Secondly, all the solar proxies, El Niño and aerosols are weakly correlated with the temperature fluctuations. Thirdly, TSI, SSN and aerosols all correlate substantially better with temperature fluctuations than with temperature itself.

3.2 Multiple Regression

A-models

All A-models had a high adjusted R2_{. Thus, a large part of the variance of the temperature could be explained by the}

models’ independent variables. The SEE and P-values of all the models were very low, indicating that the models were highly significant. All in all, the models showed a good overall fit with the temperature data.

However, there were minor differences in overall fit between the models. A5 fitted well (adjusted R2 _{= 0.83, SEE =}

0.090 °C). However, it performed less (adjusted R2 _{and SEE smaller) than the models that do include a solar variable.}

A3 had the best fit. The adjusted R2 _{was the highest, the SEE the lowest and the P-value the lowest. A2 had the}

second-best fit, the A4 the third-second-best fit, A1 the fourth-second-best fit. The differences appear to be very small though.

Temperature fluctuations Temperature fluctuations 1 El Nino 0.23 aerosol -0.30 GHG -0.0035 TSI 0.31 UVR 0.34 GCR -0.29 SSN 0.36

Table 4: a correlation of the independent

variables with the dependent variable of model B. The values are the correlation coefficients.

Model Solar driver Drivers Observation points

Adjusted R2 _{SEE [°C] P-value}

A1 TSI El Niño, Aerosols, GHG 46 0,86 0,085 6,54E-16 A2 UVR El Niño, Aerosols, GHG 46 0,86 0,084 4,94E-16 A3 GCR El Niño, Aerosols, GHG 46 0,87 0,082 2,41E-16 A4 SSN El Niño, Aerosols, GHG 46 0,86 0,084 5,29E-16 A5 None El Niño, Aerosols, GHG 46 0,83 0,090 3,50E-16

B-models

All B-models had a low adjusted R2_{, implying that most of the variance of the temperature’s fluctuations could not be}

explained by the models’ independent variables. Furthermore, the SEE of all the models were low. Surprisingly though, they are about the same as those of the A-models. Also, the P-values were lower than 0.05. Therefore, this study determined that all the B-models were significant.

However, there were minor differences in overall fit between the models. B3 had the best fit. The adjusted R2 _{was the}

highest, the SEE the lowest and the P-value the lowest. The B4 had the second-best fit, B2 the third-best fit, B1 the fourth-best fit, and B5 the least-best fit.

Concluding, based on the three parameters the B-models are less trustworthy than the A-models, though statistically significant. The low adjusted R2 _{of the B-models though, could possibly be enhanced by increasing the degree of the}

polynomial (appendix A), thus improving the accuracy of the trendline. Model Solar driver Drivers Observation

points

Adjusted R2 _{SEE [°C] P-value}

B1 TSI El Niño, Aerosols, GHG 46 0.25 0.083 0.016 B2 UVR El Niño, Aerosols, GHG 46 0.26 0.083 0.013 B3 GCR El Niño, Aerosols, GHG 46 0.28 0.082 0.0083 B4 SSN El Niño, Aerosols, GHG 46 0.27 0.082 0.011 B5 None El Niño, Aerosols, GHG 46 0.20 0.085 0.026

P-values independent variables

All the P-values of the A-models were smaller than 0.05, thus the null hypothesis for every variable can be rejected and the changes in the independent variable are associated with changes in the temperature. Every variable is statistically significant and probably a worthwhile addition to the model.

When looking at the P-values of the B-models the question rises if it is statistically responsible to add the GHG’s to the B-models. Also, a scatterplot reveals the non-correlation between the GHG’s and temperature fluctuations (see appendix B). Furthermore, adding El Niño and the TSI variable to B1 and adding the UVR variable to B2 is questionable on the basis of an alpha of 0.05.

Independent variable B1 B2 B3 B4 El Niño 0.058 0.046 0.022 0.039 Aerosol 0.014 0.021 0.0077 0.024 GHG 0.85 0.88 0.71 0.97 Solar variable 0.085 0.063 0.035 0.048

Table 11. The P-values of the independent variables of each B-model. Independent

variable

A1 A2 A3 A4

El Niño 0.022 0.012 0.0038 0.010

Aerosol 0.0028 0.0052 0.00092 0.0064

GHG 2.55E-18 4.50E-18 6.81E-18 2.73E-18

Solar variable 0.0065 0.0048 0.0022 0.0053

Graphs Models

The graph below is the graph of A1 found after executing equation (2) and (3). It shows the temperature, the modelled temperature and the contributions of the different drivers to earth’s temperature change since 1957.

Fig. 10 shows that the contribution of the TSI proxy is 11-year cyclical and fluctuates around 0. El Niño’s contribution also fluctuates around 0 by approximately the same margins. The aerosol’s contribution tends to be slightly above zero with some negative peaks going below zero. Noteworthy is the coincidence of the negative peaks of the aerosol’s contribution and the negative peaks of earth’s temperature. The GHG’s contribution almost linearly increases 0.6 C from 1957 until 2002, unequivocally showing its rising dominance over the other driver’s contributions. The modelled temperature fairly closely follows the actual temperature data.

The temperature, the modelled temperature, the contribution to earth’s temperature change since 1957 of El Niño, aerosols and GHG’s were almost perfectly similar in A2, A3 and A4. That is why graphs of these variables were not presented in this work. However, there were slight differences in the solar proxies’ contributions belonging to these different models. They are presented below.

Fig. 10. A graph of A1 with all variables. The left y-axis is for the solid lines and the right axis is for the dashed lines.

-0.2 0 0.2 0.4 0.6 0.8 1 14.80 14.90 15.00 15.10 15.20 15.30 15.40 15.50 15.60 1950 1960 1970 1980 1990 2000 2010 Year (AD) T modelled T contr. TSI contr. el nino contr. aer contr. GHG ℃

_A1

The solar proxies’ contributions are 11-year cyclical and varying between -0.06 and 0.08 C. They show a low peak in 1969 and the amplitude of wave-type functions appear to increase until 2000. After 2000 the correlations seem to stop. There are little differences between them. Firstly, the TSI proxy’s contribution is more negative than the others. Secondly, the GCR proxy’s contribution shows a substantially higher peak in 1990 than the others.

The graph below was the graph of the B1 found after executing equation (2) and (3). It shows the temperature fluctuations and the contributions of the different drivers to the fluctuations of earth’s temperature since 1957. The modelled fluctuations of temperature were left out in this graph for the sake of clarity (see appendix C)

Fig 11. A graph of all the solar proxy contributions to earth’ temperature change since 1957 belonging to the four different A-models.

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 1950 1960 1970 1980 1990 2000 2010

Solar proxies' contributions (A-models)

contr. TSI (A1) contr. UVR (A2) contr. GCR (A3) contr. SSN (A4)

Year (AD)

℃

Fig 12. A graph of B1 with all the variables’ contributions.

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 1950 1960 1970 1980 1990 2000 2010

B1

The temperature fluctuations vary between -0.20 C and 0.20 C. GHG’s. GHG’s contribute nothing to the temperature fluctuations, although it increases marginally. The aerosols show a clear sign of great contribution in 1964 and 1992, in which the negative peaks of the aerosols coincide with the temperature fluctuations’ negative peaks. El Niño and the TSI proxy seem to follow the temperature fluctuations somewhat, but not clearly.

As was the case with the A-models, the temperature, the modelled temperature, the contribution to earth’s temperature fluctuations since 1957 of El Niño, aerosols and GHG’s were almost perfectly similar in B2, B3 and B4. That is why graphs of these variables were not presented in this work. The contributions of the solar proxies to the fluctuations of temperature were exactly the same as figure 11 except for their amplitude. Their amplitude ranged between -0.04 C and 0.05 C.

Relative contribution

The relative contribution of the solar proxies is substantial from 1957 until approximately 1964. After 1964 the solar proxies relative to the other drivers contribute significantly less. Still though they remain around 12%, slowly decreasing. Furthermore, we have to bear in mind that we are only considering the maxima and minima and that the relative contributions fluctuate between these maximum values. Moreover, some percentages diverge a bit. All the relative contributions are unexpectedly low in 1969. TSI’s relative contribution is unusually low in 1981. GCR’s relative contribution is remarkably high in 1990.

Year Relative contribution TSI proxy (A1) Relative contribution UVR proxy (A2) Relative contribution GCR proxy (A3) Relative contribution SSN proxy (A4) 1957 40.3% 48.8% 34.9% 47.4% 1964 32.1% 26.4% 25.7% 28.6% 1969 10.1% 5.1% 15.9% 11.8% 1975 12.5% 16.1% 13.9% 17.0% 1981 5.5% 14.5% 13.3% 12.3% 1986 11.2% 11.5% 9.7% 12.9% 1990 10.0% 9.3% 17.8% 9.1% 1996 10.5% 8.2% 6.3% 9.0%

Sample variance

The variance of the contributions for the A-models was the greatest for the GCR proxy, followed respectively the UVR proxy, the SSN proxy and TSI proxy. The variance of the contributions of the B-models was the greatest for the GCR proxy, followed respectively by the SSN variable, the UVR proxy and the TSI proxy.

4. Discussion

This study found that all the A-models were highly significant. They showed a good overall fit with the temperature data. This means that the factors aerosols, El Niño, greenhouse gases and a solar proxy statistically explain the variance of earth’s temperature since 1957. More importantly, this implies that embedding of the solar proxies in the A-models is statistically valid.

After calculation of the contribution of the different solar proxies to the change in earth’s temperature since 1957 graphs showed that the contributions were 11-year cyclical and varying between -0.06 and 0.08 degrees Celsius. Furthermore, they show a low peak in 1969. This corresponds to the small amplitude of the solar cycle at that time (see fig. 6 to 9). Moreover, the amplitude of the wave-type functions appears to increase a little until 2000. This coincides with the slow increase of TSI, sunspots and UVR and the slow decrease in GCR (see fig 6 to 9).

Moreover, solar variability, together with El Niño and aerosols, appears to be the main contributor to earth’s temperature from 1957 until approximately 1964. Afterwards it steadily drops but remains an 11-year periodically contributing factor. In terms of relative contribution, it fluctuates between 0 and circa 12% since 1964. Since then the primary drivers seem to be the greenhouse gases as was expected based on correlation with temperature. Their contribution almost linearly inclines up to 0.6 C in 2002. Aerosols contribute to negative temperature peaks in 1992 and 1964.

The differences between the solar proxies’ contributions were marginal. Correlations between the proxies indicated the same. However, the solar proxies diverge after 2000. This could be due to a cooling down of the sun (Tapping et al. 2007) or a decrease in faculae because of a weaker magnetic field of the sunspots (Lukianova & Mursula, 2010). Both affect TSI and UVR more than they affect SSN.

Contribution Model A Sample Variance TSI proxy (A1) 0.00137

UVR proxy (A2) 0.00142 GCR proxy (A3) 0.00160 SSN proxy (A4) 0.00140

Table 8. The sample variance of the contribution of the solar

proxies of model A

Contribution Model B Sample Variance TSI proxy (B1) 0.000515 UVR proxy (B2) 0.000579 GCR proxy (B3) 0.000715 SSN proxy (B4) 0.000656

Table 9. The sample variance of the contribution of the solar

The variance of the contributions was the greatest for the GCR proxy, followed respectively the UVR proxy, the SSN variable and TSI proxy. This could entail that the GCR proxy, via the GCR-mechanism, has the greatest influence on earth’s temperature, followed by afore mentioned. However, it could be argued that the differences are trivially small. The B-models were also statistically significant, albeit much less so than the A-models. Nevertheless, also the temperature fluctuations since 1957 can be explained by the same explaining factors as the A-models. Embedding of the solar proxies in the B-models thus is statistically valid.

Calculation of the contribution of the different solar proxies to the fluctuations of earth’s temperature since 1957 graphs showed that the contributions were again 11-year cyclical and varying between -0.04 and 0.5 degrees Celsius. Moreover, the contributions to temperature’s fluctuations seem to be substantial, even more so than to temperature by itself, and constant in their fluctuation. This increase in contribution relative to temperature was already indicated by correlation. The contribution of the greenhouse gases seems to be none, as expected based on correlations. However, based on the P-value it is questionable if GHG’s in general should have been added to the B-models. The contributions of El Niño and aerosols are significant, with aerosols again clearly contributing to the negative peaks in 1992 and 1964.

The differences between the solar proxies’ contributions were again minimal.

The variance of the contributions was the greatest for the GCR proxy, followed respectively by the SSN variable, the UVR proxy and the TSI proxy. This could imply that the GCR proxy has the greatest contribution on the fluctuation of temperature, followed by afore mentioned. However, it could be argued that the discrepancies are trivially small.

Solar variability

The finding of this study that solar variability contributes lightly to earth’s temperature since approximately 1964 is supported by numerous studies (e.g. IPCC, 2013; Schurer, Tett & Hegerl, 2014; Solanki, Krivova & Haigh, 2013). Schurer, Tett and Hegerl (2014) even found that solar variability is a small contributor to climate the entire last millennium. This study partly contradicts this by finding because between 1957 and circa 1964 the influence of solar variability on temperature seems considerable (see fig. 10). Worth mentioning though, Shurer, Tett and Hegerl (2014) also used a simulation with high solar forcing (SOLAR SHAPIRO) which indicated a substantially higher solar contribution to temperature during this time interval.

Another study by Solanki, Krivova and Haigh (2013) found, also using multiple regression analysis and containing roughly the same drivers, a similar contribution of solar variability to earth’s temperature. However, they only used the TSI proxy to measure solar variability. Furthermore, the TSI proxy contributed in an 11-year cycle between -0.04 and 0.08 degrees Celsius, whereas our TSI proxy contributed in an 11-year cycle between -0.06 and 0.04 degrees Celsius. Furthermore, they found a slight positive trend in the TSI proxy. This was due to a longer temporal scale, namely from 1900 until 2007. From 1900 until approximately 1960 a positive trend can be recognized. Moreover, Solanki, Krivova and Haigh (2013) found that since 1960 the solar variability does not contribute to earth’s

temperature and GHG’s take over the contribution. In our study we determined that this is the case around 1964. Another little difference was that Solanki, Krivova and Haigh (2013) used volcanic aerosols instead off all aerosols. A study by Stauning (2014) though, found that solar variability plays a greater role in recent global warming. He based that on the flattening out of the temperature trend line since 2001, which coincides with a very low sun spot number (lowest in 100 years). However, the temperature measurements used in this study show no leveling-off of the trend line since 2001 (see fig. 2). The cause of this ambiguity concerning the trendline may well be the extended time interval. This study’s temperature data runs up to 2015, that of Stauning to 2013 and in the last 2 years the temperature has risen sufficiently. Furthermore, if such leveling-off would be the case, the question remains why the temperature rose between 1960 and 2001, while the sun became less active. However, if suddenly the sun contributes more to temperature, then it is unexpected that with the coming solar minimum the temperature is rising so quickly.

Nevertheless, as this and most other studies rule out the possibility of strong solar forcing on climate change, it still possibly affects regional and seasonal variability (Gray et al., 2010). More research in this field is needed.

The leave one out method

The leave-one-out method, as in Besselink (2018), could have been implemented in this study, thus showing more information about the contribution of the several proxies to temperature. It would have been interesting to see what overall fit extra models in this method would have. On the basis of these degrees of overall fit one could possibly see more clearly the effect of the solar proxies on temperature. For instance, what would the proxies contribute if the greatest contributor, the greenhouse gases, were omitted in the multiple regression?

Multiple regression

Multiple regression might not be a complete method and lacking some sufficient complexity. It shows statistical validation of the relation between the independent and dependent variables, but this relationship is not necessarily causal. Furthermore, it only covers linearity. Possible non-linear relationships are not included in this method, which is a limitation of multiple regression. However, it is questionable if non-linearities between the drivers do exist. The HadCM3 model with which Schurer, Tett and Hegerl (2014) simulated temperature also did not deal with non-linearities between the different forcing’s.

Time frame

This study’s time frame ended with the year 2002, while the sun’s influence on earth’s temperature might have changed considerably the last 16 years. Actually, the sun became substantially less active the last decade, while earth’s temperature is still increasing. Therefore, this last decade is of great interest for further study. Also, the study is limited by its starting date. Before 1957 the greenhouse gases were less important of a factor (see fig. 3) and the temperature

was increasing less (see fig. 2). It would be interesting to see what the relative influence of the sun would be since the beginning of the last century. On the other hand, temperature measurements were less accurate before 1950.

Albedo

Also, the models used in this study didn’t involve albedo changing factors. The latest IPCC report (IPCC, 2013) states that these together are one of the drivers of climate change, albeit less so than the greenhouse gases. Therefore, addition of these factors, if done carefully, could increase the validity of this research.

5. Conclusion

Solar variability is, together with El Niño and the aerosols, the main contributor to earth’s temperature from 1957 until approximately 1964. Afterwards it steadily decreases but remains a contributing factor, its contribution fluctuating with an 11-year cycle. In terms of relative contribution, it fluctuates between 0 and circa 12% since 1964. The greenhouse gases take over the main contribution since then and linearly increase their contribution up to 0.6 C in 2002. The different solar proxies used to determine the conclusions show marginal differences with respect to their contribution.

The contribution of solar variability to the temperature fluctuations is more substantial than its contribution to temperature itself, fluctuating again with an 11-year cycle. The greenhouse gases contribute negligibly little to temperature fluctuations. The differences between the contribution of the four solar proxies to temperature fluctuations are marginal.

Still, there are improvements to be made. The time interval in which solar variability significantly contributes to recent global warming is still under debate. More research is needed about this. Also, addition of albedo changing factors in a similar multiple regression analysis could improve the discussion. Furthermore, the time frame could have been extended. Moreover, the leave one out method could also be implemented, advancing further debate. Lastly, multiple regression has its shortcomings and more complex models should be applied to this field of research.

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Appendix A

Appendix B

Fig 13. The temperature, its trendline and an equation of the trendline

y = -1.5398605827E-05x3_{+ 9.1758571206E-02x}2_{- 1.8223912351E+02x +}

1.2064827367E+05 14.70 14.80 14.90 15.00 15.10 15.20 15.30 15.40 15.50 15.60 15.70 1950 1960 1970 1980 1990 2000 2010

Temperature and its trend line

temperature Poly. (temperature)

℃

Year (AD)

Fig 14. A scatterplot of the radiative forcing of the greenhouse gases versus the temperature fluctuations.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20

Scatterplot GHG vs. fluctuations temperature

Appendix C

Modelled fluctuations temperature (B1)

fluctuations T modelled fluctuations T

℃

Year (AD) Fig 15. The temperature fluctuations and the modelled temperature fluctuations of B1