Probing links between climate change and solar variability using wavelet analyses

Abstract

Possible links between the Earth’s surface temperature and solar variability are investigated using uni- and bivariate wavelet methods, in an attempt to attribute the fluctuations of the global mean surface temperature (GMST). Time-varying strengths of oscillations on different time scales in time series for several sets of GMST and solar activity indices are evaluated using continuous wavelet transform. Furthermore, possible relationships on different time scales between the GMST and solar activity time series are investigated using cross wavelet transform and wavelet coherence analyses. The results show that for the most prominent oscillations of solar activity on the ~11-year scale, no statistically significant response is found in any of the temperature time series. Little evidence is found supporting the postulated link on the ~60-year time scale between surface temperature and solar activity. The ~60-year oscillations in the climate system likely stem from internal variabilities of the coupled ocean-atmosphere system, especially the variabilities in the North Atlantic and the associated variabilities in the western tropical Pacific, rather than solar variability. The latitudinal differences in temperature response to solar activity depend on time scales. Our findings do not support the notion that solar variability has been playing a dominant role in recent climate change.

Graphical abstract

Keywords

climate change / global warming / solar variability / wavelet analysis

1 Introduction

There has been an increasing trend in the global mean surface temperature (GMST) since the mid-19th century (IPCC, 2014). The science community is nearly reaching a consensus that this warming, which is at a pace unprecedented in the past climate, is mainly attributable to the massive injection of anthropogenic greenhouse gases into the atmosphere. However, some different voices arise, arguing for a major role of solar variability in the warming of the Earth’s surface temperatures. In this study, the scientific viability of this claim is investigated.

Being the nearly single source of energy driving the Earth’s climate system, the Sun’s role in past and recent climate change is much debated. In a review by Eddy et al. (1982), it is concluded that the temperature change resulting from the variability of the solar irradiance, which is approximately 0.1%−0.2% of the total solar irradiance (TSI), is at most a few hundredths of 1 K, and is hardly appreciated compared with day-to-day weather changes. On the other hand, however, some evidence of the impact of the Sun’s 11-year cycles was put forward on the thermal and dynamical features of the atmosphere (Coughlin and Tung, 2004a, 2004b). Some mechanisms were proposed to explain why such a small magnitude of solar irradiance change can be amplified and manifest in some regions (Meehl et al., 2008, 2009; Weng, 2012a, 2021b). Nevertheless, when looking over a longer time, solar influence on climate is found very small (Schurer et al., 2014). A compelling rebuttal to the notion that it is the Sun that drives climate change is the finding that the recent climate trend is just opposite to the expected effect of solar forcing (Lockwood and Fröhlich, 2007).

The solar variability is known to be composed of several prominent cycles, and the cycles appear as peaks in power spectrum when time series of solar activity are subject to spectral analysis. In a recent study, Le Mouël et al. (2020) used the singular spectrum analysis (SSA) method to study spectral features of solar activity and climate time series. They argued, based on the apparent adjacency of spectral peaks in solar activity and climate time series, that the recent climate change is mostly attributable to solar variability. The argument incurred criticism by Cuypers et al. (2021), who identified several points of weakness in the Le Mouël et al. (2020) study.

In this study, we attempt to further investigate the issue of possible links between climate change and solar variability. We extend by going beyond the classical spectral method and employing the wavelet methods, which were developed in recent decades and have been proven useful in many disciplines. The wavelet approach has the advantageous capability of decomposing time series in time-frequency space. As such, intermittent and frequency-modulated oscillations can be disclosed. In addition, bivariate wavelet analysis methods are employed to probe links between time series. The robustness of the results obtained by the wavelet approach can be assessed by the confidence levels, which are given along with the analyses.

The rest of the paper is organized as follows. Section 2 describes the climate change and solar activity data sets selected for use in this study. Section 3 details the uni- and bivariate wavelet methods employed. The time varying feature of oscillations in climate and solar variability time series revealed by continuous wavelet transform are described in Sections 4 and 5, respectively. In Section 6 possible links between climate and solar activity are probed by using cross wavelet transform and wavelet coherence analyses. In Section 7 the globe is divided into zonal bands in order to study the latitudinal dependence of solar impact on climate. Finally, multiscale variabilities and connections with solar variability of the Atlantic Multidecadal Oscillation (AMO) is investigated in Section 8. Section 9 presents a showcase how the ~60-year oscillations in North Atlantic cast impacts on remote regions. Discussion and conclusions of our results are given in Section 10.

2 Data

Historical GMST time series from several widely-used data sets of different sources and processing procedures are used in this study. The linear trends in the GMST time series are calculated using the least-squares method and subsequently removed, resulting in detrended time series (Fig.1) that are subject to further analysis. The Climatic Research Unit temperature version 5 data set (CRUTEM5; Fig.1(a); Osborn et al., 2021; data are available at the University of East Anglia website) is a gridded compilation of monthly near-surface air temperature anomalies over global land surfaces, serving as the land component of the Met Office Hadley Centre/Climatic Research Unit version 5 (HadCRUT5; Morice et al., 2021) global temperature data set, with updates and improvements made since its initial publication in the 1980s. Combining sea surface temperature (SST) measurements over the ocean with near-surface air temperature measurements from weather stations on land (CRUTEM5), the HadCRUT5 data set (Fig.1(b)) provides monthly average near-surface temperature anomalies from 1850 to 2018. The Berkeley Earth data set (BEST for short; Rohde and Hausfather, 2020; data available at berkeleyearth.org website) provides a comprehensive global land and ocean temperature record from 1850 to the present, integrating extensive land station data with spatially interpolated sea surface temperatures to offer a spatially complete and homogeneous temperature field. BEST has two versions differing in which temperature observations are used when the sea surface is covered by ice: one uses the air temperature above the ice (BESTair; Fig.1(c)), while the other uses the water temperature below the ice (BESToce; Fig.1(d)). There are appreciable temperature differences between the two versions, with the BESToce time series rising more steeply than that of the BESTair. The Goddard Institute for Space Studies Surface Temperature product data set (GISTEMP; Fig.1(e); Lenssen et al., 2019; data are available at NASA website) incorporates various uncertainty models for factors such as land station homogenization, ocean temperature products, spatial interpolation, coverage uncertainties, and parametric uncertainty within its methodology.

The GMST time series are compared with time series of solar activity indices, namely, the TSI, the sunspot numbers (SSN), and the polar faculae numbers (PF). The radiation energy received by the Earth depends critically on the TSI, which is observed to vary within a small range. In addition, SSN and PF numbers are also indicators of solar activity. Sunspots are linked to large-scale magnetic fields and active regions, whereas polar faculae are associated with small-scale magnetic fields near the solar poles. The TSI data are from Lean (2018), which presents state-of-the-art reconstructions of TSI variations since 850 CE, combining historical records and modern satellite observations to offer a continuous and accurate representation of solar energy input to Earth’s climate system (Fig.2(a)). The TSI remains nearly constant during the entire period of the Lean (2018) TSI data, as suggested by the rather small slope of the linear regression of the TSI time series. However, when the period after 1850 CE is subject to analysis, a linear trend of 3.5×10⁻³ W/m² per year is found (Fig.2(b)). To avoid possible influence of the linear trend, we have removed this TSI trend between 1850 and 2016 CE before further analyses. SSN number data are from World Data Center Sunspot Index and Long-term Solar Observations (SILSO), Royal Observatory of Belgium, Brussels (Fig.2(c); data available at the Solar Influences Data Analysis Center website). Polar faculae (of the Sun) numbers for 1837−1999 CE (Fig.2(d); Nagovitsyn et al., 2004) are downloaded from Gaoran.Ru website. The SSN and PF time series are also detrended.

3 Methods

In this study, continuous wavelet transform is applied on time series of temperature and solar activity to reveal their time-frequency evolution of variance. Cross wavelet transform and wavelet coherence, two bivariate extensions of single variable wavelet analysis methods, are employed to study the possible relationship between two time series on the time-frequency domain. The wavelet analysis methodology, similar to that used in Wang et al. (2022), is described as follows.

3.1 Continuous wavelet transform

The continuous wavelet transform (CWT) is a widely used tool in identifying oscillatory features of time series in time-frequency space. The basic idea of CWT is to find sporadic periodicities by convolving the time series with a designed function, called wavelet, to suit specific applications. As in many studies in geophysics, we use the Morlet wavelet (Morlet et al., 1982a, 1982b) in the present study. The Morlet wavelet function {\psi }_0 (Torrence and Compo, 1998) is a function of dimensionless time \eta as expressed below:

(1){\psi }_0\left(\eta \right)= {\pi }^{-\frac{1}{4}}{e}^{i{\omega }_0\eta }{e}^{-\frac{1}{2}{\eta }^2}.

As a common practice in many climatic time series studies, the dimensionless frequency {\omega }_0 in Eq. (1) is set to 6 to strike a balance between time and frequency localization. For a time series X_n consisting of N observations taken at equal time interval \delta t, its CWT on the wavelet scale s can be expressed as

(2)W_n^X\left(s\right)= \sqrt{\frac{\delta t}{s}}{\sum }_{n^{\prime }=0}^{N-1}{x}_{n^{\prime }} {\psi }_0^{\ast }\left[\left(n^{\prime }-n\right)\frac{\delta t}{s}\right],

where the asterisk (*) denotes the complex conjugate. The wavelet power is defined as {\left|W_n^X\left(s\right)\right|}^2, and the complex argument of W_n^X\left(s\right) represents the local phase.

If the time series X_n bares a background power spectrum P_k^X, then the distribution of the wavelet power normalized by the standard deviation can be given by

(3)\frac{{\left|W_n^X\left(s\right)\right|}^2}{{\sigma }_X^2}\Rightarrow \frac{1}{2}{P}_k^X{\chi }_2^2.

3.2 Cross wavelet transform

The cross wavelet transform (XWT) can be used to identify regions of high common power and coherent phase lag in time-frequency space by analyzing the CWTs of two time series with speculated links (Grinsted et al., 2004).

The XWT of two time series is defined as the product of the CWT of the first time series and the complex conjugate of the CWT of the second:

(4)W_n^{XY}=W_n^X{W}_n^{Y\ast }.

The degree of correlation between the two time series is evaluated by the magnitude of the wavelet power W_n^{XY}. The complex angle of the XWT, \mathrm{a}\mathrm{r}\mathrm{g} \left(W_n^{XY}\right), represents the relative phase relationship of the two time series. If the two series are physically related, then constant or slowly varying phase lags between the two time series are expected.

The distribution of the cross wavelet power of two time series X_n and Y_n with power spectra P_k^X and P_k^Y is given by

(5)\left|W_n^X\left(s\right)W_n^{Y\ast }\left(s\right)\right| \Rightarrow \frac{Z_v\left(p\right)}{\nu }{\sigma }_X{\sigma }_Y\sqrt{P_k^X{P}_k^Y} .

In Eq. (5), Z_v\left(p\right) is the confidence level corresponding to the probability p, v the degree of freedom (set to 2 for the Morlet wavelet as its CWT is complex), and {\sigma }_X and {\sigma }_Y the standard deviations of X_n and Y_n, respectively. In this study we are interested in finding cross wavelet power significant on the 5% level, so we take Z₂(95%) = 3.999.

3.3 Wavelet transform coherence

The wavelet transform coherence (WTC) analysis was devised to assess the degree of coherence between two time series of speculated links in time-frequency space. As an extended functionality of XWT, WTC can identify regions of high coherence, even when the common power revealed by XWT analysis is low. WTC is measured by squared wavelet coherence, which is defined as (Torrence and Webster, 1999)

(6)R_n^2\left(s\right)= \frac{{\left|⟨s^{-1}{W}_n^{XY}\left(s\right)⟩\right|}^2}{⟨s^{-1}{\left|W_n^X\left(s\right)\right|}^2⟩\cdot ⟨s^{-1}{\left|W_n^Y\left(s\right)\right|}^2⟩} ,

where ⟨\cdot ⟩ is a smoothing operator operated in both time and scale. Without such smoothing the WTC would be identically 1 (Liu, 1994). Monte Carlo simulations are necessary to evaluate the statistical significance level of WTC against null hypotheses (Torrence and Webster, 1999; Grinsted et al., 2004).

In time-frequency representations of the XWT or WTC analyses, the phase differences between the two time series under study can be indicated by the pointing direction of the arrows. When the two time series are in phase the arrows point to the right, and when out of phase the arrows point to the left. When X_n leads Y_n by a quarter period the arrows point straight down, and when Y_n leads X_n by a quarter period the arrows point straight up. In a time-frequency analysis of the result obtained by the aforementioned wavelet analysis methods, certain regions are distorted by the zero-padding effects at both ends of the time series. These regions are called the cone of influence (COI), and in this study in all time-frequency plots we demarcate the boundary of the COI with a thick cone-shaped line and the region enclosed with a lighter shade. Caution is advised when interpreting the wavelet analysis results in the COI.

4 Multiscale variabilities in GMST time series

4.1 CRUTEM5

Time-frequency representation of the wavelet power of the CRUTEM5 GMST time series is given in Fig.3(a). There are some moderately active variabilities over the interannual range (2- to 9-year scale) throughout the whole time span between 1857 and 2022. Decadal oscillations (10- to 16-year scale) are seen in the time intervals 1857−1920 and 1960−2000. On the multidecadal scales, there is a lasting band of variability near the 20-year scale. Active oscillations are seen near the 60- and 80-year scales, which are just in-phase during the latter half of the 20th century.

4.2 HadCRUT5

The HadCRUT5 time series is a combination of air temperature observations over the ocean and the land component from the CRUTEM5. The CWT analysis of the HadCRUT5 time series (Fig.3(b)) resembles that of the CRUTEM5 on the decadal to multidecadal scales, with persisting variabilities on the decadal, bidecadal, 60-year and 80-year bands. However, a 40-year oscillation can be identified after 1950. In some early (1860−1885) and late (1990−2005) portions of the HadCRUT5 CWT plot the interannual variabilities are less significant than those in CRUTEM5, suggesting that the surface temperature over the ocean fluctuate less than those over land on the interannual time scale.

4.3 BEST

Although they differ in which temperature observations are used when the sea surface is covered by ice and have different paces of warming, the BESTair and BESToce GMST time series share many CWT features and are thus nearly indistinguishable (Fig.3(c)−Fig.3(d)). There is a band of high variability between the 20- to 30-year scale prior to 1940, and after 1940 it splits into two distinct bands, one near the 20-year scale and the other near the 40-year scale. The evolution of the 60- to 80-year variabilities in BESTair and BESToce highly resembles that in HadCRUT5 (Fig.3(b)).

4.4 GISTEMP

The variations of the GISTEMP GMST time series (Fig.3(e)) are active in the interannual and decadal bands. A band of oscillations near the 20-year scale persists through the entire time span. ~40-year oscillations are active after 1930, whereas ~60-year oscillations endure throughout the entire time series.

5 Multiscale variabilities in solar activity time series

5.1 Solar irradiance

The CWT of the Lean (2018) TSI time series is shown for the entire record length (850−2016 CE) in Fig.4(a), and for the period overlapping with the GMST time series (1850−2016 CE) in Fig.4(b). Strong signatures of the ~11-year solar cycle (the Schwabe cycle) can be seen only in parts of the time series (prior to 1000 and after 1900 CE). During the spells of solar activity minimums, e.g., the Sporer Minimum (1460−1550) and the Maunder Minimum (1645−1715), no elevated wavelet energy is seen around the ~11-year scale. Oscillations on time scales of ~120-year (the Gleissberg cycle; Peristykh and Damon, 2003) and ~200-year (the Suess or DeVries cycle, Wagner et al., 2001) are also evident in the CWT analysis of the Lean TSI, but are not significant on the 95% level. Some oscillatory activities are seen on multiples of the 11-year cycles, that is, near the 22-, 44-, and 88-year scales (Fig.4(b)). Interestingly, the 88-year cycles are larger in magnitude than the 22- and 44-year ones.

5.2 Sunspot number

The Schwabe cycles exhibit themselves as a continuous band of high wavelet energy in the CWT analysis of the SSN time series (Fig.4(c)). The frequency at which the Schwabe cycles are evident remains fairly steady near the 11-year band, except for the first couple of decades of the 19th century, when the frequency representation of the Schwabe cycles is somewhat distorted as a result of the anomalously low sunspot numbers during solar cycles 5 and 6 (1798−1823). The ~22-year cycles (the Hale cycles) are also discernible, but not as strong as the Schwabe cycles. ~40-year oscillations emerge in the 1900s. The Gleissberg cycles appear as a wide band of variability in the beginning of the time series, but the band of strong variability narrows to the longer period during its evolution, and the core Gleissberg variability lies near the 88-year scale.

5.3 Solar polar faculae

The Schwabe cycles are the most pronounced variabilities persisting throughout the solar PF time series (Fig.4(d)), and the frequency is quite steady at 11-year. The Hale cycles are discernible only between 1900 and 1970. 40-year cycles are identifiable starting from the 1950s. Gleissberg cycles are evident near the 80-year scale throughout the PF time series.

6 Probing links between GMST and solar activity time series

6.1 GMST and total solar irradiance

Fig.5(a) presents the XWT analysis between the detrended CRUTEM5 GMST and the detrended TSI time series. There exists high energy near the ~11-year band (the Schwabe cycle) common in both time series, but such common high energy band is punctuated between 1910 and 1950. By inspecting the directions of the arrows in the time-frequency representation of common energy shared by the two time series, we can infer that the phase difference between the two time series changes significantly with time. The changing phase difference between the two time series suggests that the impact of the 11-year solar variability, if it does exist, is not stationary. The non-stationarity of such impacts might stem from the changing length of the Schwabe cycle (Friis-Christensen and Lassen, 1991).

The WTC analysis between these two time series is shown in Fig.5(b). There exists high level of coherence near the ~11-year band, but the coherence is limited to the period from 1880 to 1915. Note, however, that in this period the arrows approximately point to the left, which indicates that high TSI is associated with low temperature, and vice versa. This is contradictory to the expected warming (cooling) effect of positive (negative) TSI anomalies. Unless a physical mechanism that can explain the half-period delay in the radiative effect of TSI anomaly is established, caveats must be exercised when interpreting the high common wavelet energy and coherence between TSI and GMST time series. There appears a long period of wavelet coherence near the Hale cycle band (~22-year) between 1920 and 2000, but the cross wavelet power is not high according to XWT analysis (Fig.5(a)). A role of the Hale cycles on the ~22-year temperature variability can be thus excluded.

6.2 GMST and sunspot numbers

The XWT and WTC analyses between the CRUTEM5 GMST and SSN time series are shown in Fig.6(a) and Fig.6(b), respectively. High levels of common energy are observed near the Schwabe cycle scale during two intervals: 1870−1895 and 1950−1995 (Fig.6(a)); however, the phase relationships exhibit nearly opposite behavior between these intervals. Conversely, the WTC analysis does not detect a corresponding high level of wavelet coherence (Fig.6(b)). Significantly high level of wavelet coherence is identified near the Hale cycle scale between 1945 and 1970 (Fig.6(b)), yet this lacks a parallel elevation of common wavelet energy (Fig.6(a)). Consistent with the bivariate wavelet analyses of GMST and TSI, it would be premature to establish any robust physical link between the GMST and SSN time series based on these findings.

6.3 GMST and solar polar faculae

Fig.7(a) and Fig.7(b) show, respectively, the XWT and WTC analyses between the CRUTEM5 GMST and the solar PF number time series. Significantly high common energy is observed near the ~11-year band between 1975 and 1995, coinciding with significantly high wavelet coherence. An interval of significantly high common energy on the same scale is present between 1855 and 1895, but lacks corresponding high wavelet coherence. Significantly high wavelet coherence is evident between 1930 and 1965 near the Hale cycle timescale, despite a lack of significantly high common energy. Importantly, the phase relationships between the GMST and PF time series exhibit opposite behavior across these intervals. This contrasting phase relationship presents a challenge to the proposition of a direct physical mechanism linking solar PF and GMST variability.

7 Multiscale variabilities of zonal mean temperature and connection with solar variability

To further analyze the relationship between solar activity and temperature change around the world, we split the globe into 8 zonal bands (64°N−90°N, 44°N−64°N, 24°N−44°N, 0°−24°N, 0°−24°S, 24°S−44°S, 24°N−44°N, and 64°S−90°S) to study sensitivity of surface temperature to solar activity. Time-frequency features of temperature time series for each of the zonal bands are plotted in Fig.8.

In the northern polar region (64°N−90°N; Fig.8(a)) there are sporadic significant activities in the 2- to 8-year band. The Schwabe cycles are evident only after 1970. A 12- to 16-year oscillation is present prior to the 1960s. In addition, a 30- to 40-year oscillation persists the entire time series. The Gleissberg cycles are manifested as high wavelet energy in a wide band between 60- and 100-year scale throughout the time series.

Interannual variabilities (2- to 8-year scale) are strong in the northern mid-latitude regions (44°N−64°N; Fig.8(b)). The Schwabe cycles are evident only prior to the 1950s. High variabilities near the ~20-year band are also found throughout the time series. The ~60-year cycles are persistent in the entire time series, and in the second half of the time series they merge with the cycles of even longer periods.

In the northern subtropical region (24°N−44°N; Fig.8(c)) interannual variabilities are not as strong as in the polar and high-latitude regions. The Schwabe cycles are evident only after the 1950s. Throughout the time series, notable variations are present in the range of 20 to 30-year cycles. There are also persisting cycles near the ~60-year range.

In the tropical regions (0°−24°N and 0°−24°S; Fig.8(d) and Fig.8(e)) interannual variabilities are strong, reflecting the dominance of the El Niño-Southern Oscillation (ENSO; McPhaden et al., 2006) dynamics in the tropics. The Schwabe cycles are blurred by the strong ENSO variabilities and some evolving multi-decadal oscillations at 20- to 30-year scales. The Gleissberg cycles are split into the ~60-year and centurial bands.

In the southern subtropical regions (24°S−44°S; Fig.8(f)) and the southern mid-latitude regions (44°S−64°S; Fig.8(g)) interannual variabilities are suppressed. Active oscillations on the 20- to 40-year band are conspicuous. The ~60-year cycles are weak, but the ~88-yr cycles emerge in the southern mid-latitudes.

In the southern polar region (64°S−90°S; Fig.8(h)) there are sporadic significant activities in the interannual range. The Schwabe cycles are evident in the first half of the 20th century. The Hale cycles are present throughout the entire time series. The ~60-year cycles are absent.

The relationship between zonal mean surface temperature and solar activities is investigated by subjecting each zonal mean temperature time series and the Lean TSI time series to XWT and WTC analyses, and the results are given in Fig.9 and Fig.10, respectively. Common wavelet energy between each zonal mean temperature and the TSI time series is found to be high near the Schwabe cycle in many time intervals and in the low frequency (period greater than 60-year) range in all 8 zonal regions (Fig.9). However, statistically significant high wavelet coherence at the Schwabe cycle scale is only found by WTC analysis between 1975 and 2003 in the northern subtropical region (Fig.10(c)), between 1995 and 2010 in the southern subtropical region (Fig.10(f)), and between 1940 and 1955 in the southern polar region (Fig.10(h)). Combining the XWT and WTC analysis results, little evidence is found of a link between TSI and zonal mean temperature on the ~11-year scale. On the ~60-year and lower frequency scales, there exists high common wavelet energy between the TSI and zonal mean temperature time series in all zonal bands (Fig.9). Nevertheless, the wavelet coherence between the time series is very low (Fig.10). It is therefore hard to link the low frequency climate oscillations in any zonal band to solar variabilities.

8 Multiscale variability of Atlantic multidecadal oscillation and connection with solar variability

Although the origin of the oscillations of frequency lower than 60-year of the global surface temperature (Fig.3) is still a topic of debate, it is well accepted that such oscillation is most evident in the North Atlantic region (Schlesinger and Ramankutty, 1994). To verify the possible links between the solar activity and secular temperature time series, as postulated by, for example, Le Mouël et al. (2020), we further apply the bivariate wavelet methods to time series of the North Atlantic temperature time series to compare against solar variabilities.

Fig.11(a) shows the time series of temperature averaged in a region (0°−65°N, 80°W−0°) of the North Atlantic Ocean from the GISTEMP data set, which is considered the center of action of the AMO, and its CWT result is given in Fig.11(b). Compared with the CWT of the global GISTEMP time series (Fig.3(e)), the Schwabe cycles are more evident and the Gleissberg cycles are stronger and more regular in the North Atlantic temperature time series (Fig.11(b)). Our CWT result well captures the Atlantic Multidecadal Oscillation identified by Schlesinger and Ramankutty (1994).

On the XWT plot (Fig.12(a)) there is long-enduring high common wavelet energy near the 11-year scale between the Lean TSI and the GISTEMP North Atlantic temperature time series, except for two short spells (1905−1910 and 1920−1935). However, coherence between the two time series appears low around the 11-year scale, and the changing arrow directions implicate that the phase difference between the two time series changes significantly along with time. Nevertheless, very strong coherence appears near the 22-year band between 1940 and 2005 (Fig.12(b)). This strong coherence region in the time-frequency domain is accompanied by high common energy, albeit the common energy is not high enough to be significant on the 95% level, and nearly constant phase difference. Some influence of the 22-year Hale cycles on the North Atlantic temperature is hence indicated and is worth further investigation. Although the common wavelet energy is moderately high (Fig.12(a)) on the low frequency range (periods longer than 60-year), little wavelet coherence is found in the corresponding range by WTC analysis (Fig.12(b)). A link between solar activity and surface temperature on the ~60-year scale seems far-fetched.

9 Influence of the North Atlantic temperature on the Western Tropical Pacific

Recent studies have revealed more geographically wide-spread impact of the North Atlantic Ocean temperature variabilities. For example, the Western Tropical Pacific Ocean (WTP) has been identified as a key region influenced by the North Atlantic (Sun et al., 2017). Fig.13(a) shows the time series of temperature averaged in the WTP region (0°−25°N, 130°E−170°E) from the GISTEMP data set, and its CWT result is given in Fig.13(b). As for the AMO, vigorous oscillations are seen near the 60-year band in CWT of the WTP regional averaged surface temperature time series. Further bivariate wavelet analyses expose high common wavelet power (Fig.14(a)) and wavelet coherence (Fig.14(b)), both being significant on the 95% confidence level throughout the entire time span, on the 60-year scale in time-frequency domain between the AMO and WTP surface temperature time series. The phase difference between the two time series on the 60-year scale remains fairly constant (close to zero), indicating that the WTP temperature fluctuates in synchrony with that of the North Atlantic.

10 Discussion and conclusions

To our best knowledge, this paper details the first attempt to look for the links between climate change and solar variability by studying time-varying relationships between climate and solar variabilities on multiple time scales.

Continuous wavelet analyses of the GMST time series show, regardless of which data set is analyzed, that intermittent bands of high wavelet energy are identifiable on the interannual, decadal, and multidecadal ranges, though none of them are significant on the 95% level. The bands undergo frequency modulations along with time, likely reflecting the changing modes of the climate system. The only band all GMST data sets share lies between the 60- and 120-year range, the range of the Gleissberg cycles, with its core oscillations near the 80-year scale. Oscillations of higher frequencies differ among the different GMST data sets.

Continuous wavelet analyses of the solar activity time series find that the Schwabe cycles in both SSN and PF numbers have been active ever since the observational records began. However, wavelet analyses of Lean (2018) TSI data set show that the 11-year cycles of the TSI are comparatively weak during solar activity minimums and have been active since the beginning of the 20th century. This active period in recent times coincides with the large anthropogenic emission of greenhouse gases into the atmosphere, and thus complicates the attribution of climate change to solar and anthropogenic origins.

Significantly high wavelet energy on the 11-year scale shared by the GMST time series and the TSI and SSN time series has been found for some portion of the studied period. This is the basis of the conclusion that the GMST is influenced by the Sun’s 11-year cycles by Le Mouël et al. (2020) rests on. Nevertheless, it is at the risk of being naïve to draw a link just because there exist coinciding spectral peaks of energy. A closer look at the phase difference between the temperature and solar activity time series (Fig.5−Fig.7) would identify drastic changes in phase differences between the time series, which contradicts the notion that two physically linked time series should have constant or slowly varying phase differences (Grinsted et al., 2004). Unless the changing phase difference can be readily reconciled, a physical mechanism linking the temperature change and the Schwabe cycles of solar activity seems difficult to establish. The 11-year signatures are more likely internal variability of the climate system than being forced by solar variability (Moore et al., 2006).

Another point made by Le Mouël et al. (2020) is that the ~60-year oscillations of the climate system are caused by solar activity on the same scale. Consistent with Cuypers et al. (2021), we find little evidence supporting this speculation based on our bivariate wavelet analyses of climate and solar variabilities. To date, the mechanism underlying the 60-year internal variability in the climate system is not yet fully understood. Nonetheless, it is widely accepted that such fluctuations are linked to inherent oscillations within the climate system (Knudsen et al., 2011), particularly those associated with oceanic circulation patterns. These natural oscillations can induce significant temperature anomalies over multidecadal to centennial timescales.

Extensive research has highlighted significant multidecadal to centennial variability in the Atlantic basin, intimately connected to the Atlantic Meridional Overturning Circulation (AMOC) (Li et al., 2013; Sun et al., 2015; Zhang et al., 2019). Prior investigations have established a robust dynamical link between multidecadal changes in AMOC strength and decadal North Atlantic Oscillation (NAO; Hurrell et al., 2001) variability. Positive NAO phases, characterized by intensified Icelandic Low and Azores High pressure systems, enhance atmospheric forcing on the ocean. The cumulative impact of this forcing triggers multidecadal variations in AMOC strength and SST anomalies, with the AMOC lagging the NAO by a couple of decades (Li et al., 2013). The substantial inertia of a strengthened AMOC amplifies ocean heat transport to the subpolar/polar regions, inducing warming and weakening westerly winds. This delayed negative feedback mechanism from the AMOC to the NAO sustains the oscillatory cycle, albeit in a reversed sense. This conceptual framework effectively elucidates the robust dynamical coupling between NAO and AMOC, as well as the quasi-periodic multidecadal oscillation. To further quantify these processes, Sun et al. (2015) proposed a delayed oscillator paradigm that captures the air-sea interactions underlying the 60-year quasi-periodic variability in the Atlantic.

Recent research indicates that the 60-year oscillation in sea surface temperature associated with the AMO, extends beyond the North Atlantic to remote regions like the Western Tropical Pacific (Sun et al., 2017). A teleconnecting mechanism is established to explain the link between the two distant regions. Specifically, an atmospheric teleconnection is generated as a result of the AMO warm SST anomalies, weakening the Aleutian low and subtropical North Pacific westerlies. The consequent wind changes induce warming of the subtropical North Pacific (SNP) SST through the wind-evaporation-SST effect. In response to the SNP warming, the surface winds converge toward this region from the tropics, leading to an anomalous cyclonic circulation and low pressure over the WTP. This WTP low pressure anomaly further evolves into an SST warming pattern via a positive feedback loop involving changes in sea level pressure, cloud cover, and longwave radiation.

Our bivariate wavelet analyses of the surface temperature time series of the North Atlantic and Western Tropical Pacific confirm a close link between the two regions (Fig.14). These findings suggest that the AMO can cast impact far beyond the North Atlantic, potentially influencing global climate variability. Integrating the findings of the present study, which examines sun-climate relationships and the AMO in the North Atlantic, with the work of Sun et al. (2017) on the AMO’s Pacific teleconnections, could offer a more comprehensive perspective on the AMO’s impact on global climate. Furthermore, additional studies have highlighted the effects of AMOC variations on global temperature, using various AMOC indicators and observational data (Sun et al., 2021). These previous studies enhance our understanding of the remote impacts of AMOC fluctuations and establish a foundation for comprehending the cycles in the Global Mean Surface Temperature (GMST).

By construction, the wavelet methods can decompose a time series in time-frequency space so that the time-varying features of oscillations on different time scales within the time series can be revealed. This is a substantial advantage over the spectral methods that can only obtain one estimate of the power spectrum density averaged over the entire time series. Furthermore, the bivariate wavelet methods used here enable us to reveal hidden, scale-dependent, time-varying relationships, including their phase differences, between two time series. Another advantage of the wavelet methods used here is that statistical confidence levels can always be given, so the robustness of the results can be readily estimated. The wavelet methods are thus especially suitable for the task of the present study, that is, to probe possible links between climate change and solar variabilities.

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