Physics and Chemistry of the 31 (2006) 99–108 www.elsevier.com/locate/pce

Solar activity, Earth’s rotation rate and variations in the secular and semi-secular time scales

S. Duhau *

Facultad de Ciencias Exactas, Departamento de Fı´sica, Universidad de Buenos Aires, Argentina

Received 17 January 2005; accepted 18 March 2005 Available online 29 March 2006

Abstract

By applying the wavelet formalism to sudden storm commencements and aa geomagnetic indices and solar total irradiation, as a proxy data for solar sources of climate-forcing, we have searched the signatures of those variables on the Northern Hemisphere surface temperature. We have found that cyclical behaviour in surface temperature is not clearly related to none of these variables, so we have suggested that besides them surface temperature might be related to Earth’s rotation rate variations. Also it has been suggested that in the long-term Earth’s rotation rate variations might be excited by geomagnetic storm time variations which, in turn, depends on solar activity. The study of these phenomena and its relationships is addressed in the present paper. With this purpose we perform an analysis of the evolution during the last 350 years of the signals that conform the 11-year sunspot cycle maxima envelope for different time scales. The result is applied to analyze the relationship between long-term variations in sunspot number and excess of length of day; and to unearth the signature in surface temperature of this last from that of sudden storm commencement and aa geomagnetic indices and total solar irradiation. As solar dynamo experiments transient processes in all the time scales from seconds to centuries, Fourier analysis and its requirement for the process to be stationary is likely to produce spurious periodicities. We resort instead to wavelet formalism. By this representation we found that sunspot maxima envelope for the last 350 years may be described by means of the superposition of two cycles – a decadal and a semi-secular one – and a secular trend. And that the changing amplitude and phase of the cycles are well reconstructed using the superposition of two wavelets with nearby periods. The strong temporal changes in amplitude of the cycles facilitate to detect its phase in a given . It is found that a strong semi-secular cycle in the sunspot maxima envelope that started during the 1705 chaotic tran- sition having its maximum amplitude at the Dalton minimum, was mapped 94 years later in Earth’s rotation rate and simultaneously in surface temperature. Also, there was a decrease (increase) of 0.022 C for each millisecond of decrease (increase) in the Earth’s rotation period for the semi-secular cycle as well as the secular trend. 2006 Published by Elsevier Ltd.

Keywords: Earth rotational variations; Origin and modelling, of the magnetic field, ; Geomagnetic secular variation; Climatology

1. Introduction climate changes. They based this suggestion in the fact that trends towards colder with weakened general Besides anthropogenic and solar source of climate-forc- atmospheric circulation goes with slower Earth’s rotation ing, Lambeck and Cazenave (1976) proposed the angular rate. In agreement with this Courtillot et al. (1982) found momentum transferred to the atmosphere by the solid that variations in Earth rotation rate and global surface Earth as another possible source, in the long-term, of temperature time series from 1880 to 1975 are connected (cross-correlation coefficient = 0.85) with surface tempera- tures lagging Earth’s rotation rate by 5 year). Periods of * Fax: +54 51 011 4576 3357. increasing zonal troposphere circulation and increasing E-mail address: [email protected] surface temperatures are found that they correspond

1474-7065/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.pce.2005.03.006 100 S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 to Earth’s mantle acceleration, while deceleration of this and Cazenave (1976) indicates that these changes in angu- last corresponds to decrease in zonal circulation and lar momentum of the crust are transferred in a few years to surface temperature. More recently, Mo¨rner (1999) sug- the atmosphere–hydrosphere system. Therefore, if long- gested that the angular momentum of the hydrosphere term variations in Earth’s rotation rate are the source of might play a role in climate warming (cooling) by the climate changes, long-term variations in sunspot maxima deceleration (acceleration) in Earth’s rotation and Duhau envelope will appear by about 94 years later in surface (2003c) found that a 70% of the long-term northern hemi- temperature. sphere surface temperature anomalies (NHT) increase Further evidences on the relationship between solar during the period 1868–1960 might be attributed to sudden activity, Earth’s rotation rate and surface temperature are commencement storm increases in intensity and frequency explored in the present work. Since other sources of cli- (as measured by SSC geomagnetic index (Duhau, 2003b)) mate-forcing have strong signatures in surface temperature and the remainder 30% to total solar irradiance (TSI), a careful study of the long-term evolution of the related but cyclical behaviour in NHT is not clearly related neither time series is necessary to unambiguously unearth these to SSC index nor to TSI, so she suggested that besides these long-term relationships between solar activity, Earth’s variables surface temperature is influenced by Earth’s rotation rate and surface temperature. rotation rate variations as suggested by Lambeck and There are evidences of the occurrence of transient phe- Cazenave. nomena in solar dynamo; even in time scales larger than It is well established that the origin of the short-period a decade (see e.g. Duhau and Chen, 2000a, 2002). Fourier variations (below the decadal scale) in the Earth’s rotation analysis assumes that the time series is the result of a rate is the transfer of angular momentum from the atmo- stationary process. Therefore, as there are transient phe- sphere to the solid Earth (see e.g. Eubanks, 1993; Dickey nomena in solar activity, Fourier analysis of solar and et al., 1994). For example, the 1982–1983 El Nin˜o/South- solar–terrestrial time series is likely to produce spurious ern Oscillation (ENSO) event was accompanied by the periodicities and inconsistency in the results. This problem largest interannual variation in the Earth’s rotation period has lead to a ‘‘cycle-mania’’ that obscures the searching for (Dickey et al., 1994; Chao, 1999), lagging Earth rotation solar–climate relationship (see e.g. Hoyt and Schatten, period to ENSO axial momentum by 2 months. To pre- 1997). A suitable tool to solve this problem, is the wavelet serve the total angular momentum of the atmosphere–solid transform formalism (see e.g. Rioul and Vetterli, 1991; Earth system during this event, the changes on atmosphere Torrence and Compo, 1998; Antoine, 1999). angular momentum were compensated by changes in Moreover, our main task is to accurately decompose the Earth’s angular momentum. However, a residual of 46 ± studied signals in the cycles that conform a quasi-periodic 6 ls (over a total of 0.5 ms) still remained. This residual state in the different time scales. The wavelet formalism is was suggested to be due to a contribution of oceanic specially suited for this task since a given cycle in the strong angular momentum. turbulence regimen on which solar dynamo system oper- For larger time scales Earth’s rotation rate variations ates, has strong variations in intensity that go with appre- are due to an interchange of momentum between the ciable changes in periodicity. Therefore as a cycle evolves in Earth’s core with the mantle (Vestine, 1953; Jackson time never repeat itself with the same shape. We may rep- et al., 1993). A mechanism by which geomagnetic storm resent this behaviour by the wavelet formalism by super- long-term variations might excite geomagnetic field and posing a few wavelet component with different Fourier motions in the liquid core and so Earth’s rotation rate vari- periods (Duhau, 2003a and references therein). We need ations by the topographic torque (Hide, 1969) excerpted by more than a Fourier period to be able to reproduce the the outer boundary of the liquid core on the mantle was change of its periodicity as the cycle evolves. So this for- proposed by Duhau and Martinez (1995). As geomagnetic malism provides a simple and precise tool to individualise storm intensity depends on solar activity level this mecha- the time scales on which operates a given cycle and com- nism implies that long-term variations in Earth’s rotation pare its signature in different time series. rate depends also on solar activity variations. In a strong turbulent regimen of a fluid the amplitude of Total angular momentum must be preserved in the cou- the cycles change so fast that their modulation is well rep- pled atmosphere–hydrosphere–mantle–core system. For resented by a resonant function, that just for simplicity we short period the atmosphere drives the Earth and for large have selected to be a gaussian (Sudan and Kiskenin, 1977; period the core–mantle system drives the atmosphere– Duhau and Hurtado de Mendoza, 1996). Therefore the hydrosphere system. The precise limit between the time Morlet wavelet functions that are gaussian wave packets scales that correspond to each case has not been deter- will be used in our analyses. mined yet. We start with the application of the wavelet formalism Duhau and Martinez (1995) found that the 5-year run- to a full description of the evolution during the last ning mean of Earth’s rotation rate followed variations in 350 years of the cycles that constitute the envelope of the the 11-year running mean Wolf sunspot number during 11-year sunspot cycle maxima. The wavelet formalism is the entire period – 250 years – along which the time series then applied to the excess of length of day (LOD) time ser- overlap, with a delay of 94 years. The result of Lambeck ies – defined as the difference of the Earth rotation period S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 101 with respect to its value at a given time in ms (see e.g. The Morlet wavelet function is

McCarthy and Babcock, 1986). A decrease in LOD means 2 W t p1=4eikgeg =2 1 an acceleration of the Earth’s rotation and vice versa. By 0ð Þ¼ ð Þ representing LOD time series by the same wavelet func- in the above equation k is a non-dimensional frequency. tions that we have used to represent Wolf sunspot number, Duhau and Chen (2000a, 2002) found that the accuracy R, a multi-resolution analysis of this last and LOD time of the representation by means of this formalism mostly de- series is performed. This procedure facilitates one of our pends on the value selected for k that, in our case, is 4. We main tasks that is finding the nature of the influence of have divided the time domain in octaves (Farge (1992)) (see solar activity in changes in Earth’s rotation rate. We have Table 1), based in the considerations given below. selected R to compare with LOD because it is the largest As R is the larger historical proxy data for solar activity proxy data for solar activity. we will determine the base functions to be applied to ana- Finally a simultaneous comparison between NHT with lyse the behaviour of solar activity related variables by LOD, STI and SSC and aa geomagnetic indices time series analysing this time series. The R time series spectrograms in the secular and the semi-secular time scales is performed for the last 352 year (heavy line) and for the last 157 year by which the influence of variations in Earth rotation per- (light line), respectively are shown in Fig. 1. Observing this iod on northern hemisphere surface temperature anomalies figure may identify five separated time scales. One of them is unearthed. corresponds to the Schwabe 11-year sunspot cycle (peaking at 10.7 year). The peaks at 121 year and at 252 year corre- 2. A wavelet representation of solar activity and related spond to the Gleissberg and the De Vries cycle respectively. variables We can distinguish two time scales between the Schwabe and the Gleissberg ones. One of these has a peak at 26 year From a visual inspection of R time series Feynman and in the 157 year time series and it does not appear in the Gabriel (1990) found that a phase catastrophe in the Gle- 352 year time series. And vice versa the 60 year peak that issberg cycle occurred in the solar dynamo system after is overlapped by the stronger Gleissberg spectrogram reso- the Maunder Minimum. Therefore they suggested that nant function in the 352 year time series does not appear in solar dynamo is chaotic and is operating in a region of dynamo variables close to the transition between period doubling and chaos. In fact, by means of the wavelet for- Table 1 malism, Duhau (2003a) found that a succession of quasi- Fourier periods of the Morlet wavelet functions from which the Wolf periodic behaviour and chaotic transitions might give the sunspot number yearly means (1650–2002) is rebuilt (see Fig. 2) evolution of solar dynamo system and that the average Scales Fourier period (year) level around which a given quasi-periodic cycle is under- Schwabe 2.7 3.8 5.3 7.6 10.7 15.1 gone changes drastically after each chaotic transition. Decadal 21.4 30.3 The solar dynamo system is operating in a strong turbu- Semi-secular 42.8 60.5 lent regime and them, as a wave grows takes enough energy Secular 85.5 123 171 242 342 to appreciably modify the background state and so, proper modes of oscillations strongly interacts between them. As a result a monochromatic wave, as Fourier formalism assumes, cannot be sustained by such a system. Instead, 1.5 the waves are well represented by a resonant functions cen- tred at the proper frequencies. For simplicity we may chose k=4 the resonant function to be a gaussian (Sudan and Kiske- nin, 1977) and so the Morlet wavelet transform that repre- 1.0 sents a monochromatic wave packet is well suited for our purposes. As the wave strongly interacts with its background and 0.5 also the ground-state changes strongly between chaotic Normalised Power transitions (from 35 to 90 and from 90 to 140 sport after the 1705 and 1923 episodes as shown in Fig. 2c) a proper mode changes appreciably in period for each quasi-periodic 0.0 states and also within them. For example the well-known 10 100 Gleissberg (1944) and the De Vries cycles, have periodici- Fourier Period (yr) ties (Ogurtsov et al., 2002) that vary with time in the 90– 140 and 170–260 year band, respectively. Therefore, we Fig. 1. Wavelet Spectrogram of sunspot number time series as a function of Fourier period for the last 157 year (light line) and for the last 352 year cannot represent a cycle by only one wavelet component (heavy line) normalised to be unity at the Gleissberg peak (Duhau and but by the superposition of a few components with neigh- Chen, 2000b). The in the heavy line are the power value at the bour periods (Duhau, 2003a and references therein). Fourier periods of Table 1, and k is defined as in Eq. (1). 102 S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 the shorter 157 year one. This illustrates that the intensity time series – (see light line in Fig. 1). Therefore, when try- of the corresponding cycles changes drastically as the ing to determine the non-linear relationship between R and dynamo system evolves. geomagnetic index aa, Duhau and Chen (2002) and Duhau From the above results, we have divided the time domain (2003a) have built the ‘decadal’ cycle by superposing the in the four time scales that are summarised in Table 1. The four wavelet functions that in Table 1 are split into two dif- shortest time scale is selected in order to have the 10.7 Fou- ferent time scales: the decadal and the semi-secular ones. rier period wavelet in the representation. Moreover we have The R decadal and semi-secular cycles and the secular selected seven octaves with six voices and eight octaves with trend are shown in Fig. 2a–c, respectively. The strong tem- seven voices to represent the 150 year and the 350 year poral changes in amplitude and period of the decadal time series, respectively. So, to rebuild them, the first 13 (Fig. 2a) and semi-secular (Fig. 2b) cycles allows us to and the complete set of 15 wavelet functions respectively unambiguously identify the phase of these cycles in each are superposed and added to the linear trend. In all the cases of the solar–terrestrial variables, on the contrary this is the largest point-to-point difference between the measured not possible for an stationary wave. The Gleissberg cycle and the rebuilt time series is less than 5% in a few points is clearly distinguishable in the secular trend (Fig. 2c) as but is quite smaller than this in the average. the quite regular oscillation around each constant level The semi-secular cycle is absent in the spectrogram for but the time series is too short to identify the De Vries cycle. the last 157 years – the time span of the aa geomagnetic SSC geomagnetic index time series is too short – 125 year

40 a Decadal 20

0

Cycle (spot) -20

-40

40 b semi-secular 20

0

Cycle (spot) -20

-40

200 c

150 1951 145

100 1923 1740 90

50 1705 Wolf Sunspot Number 35

0 1800 2000 Time (yr)

Fig. 2. The three components on which the envelope of 11-year sunspot cycle maxima may be decomposed is obtained from the Wolf sunspot number annuals means (the data are from Eddy (1976) for the interval 1650–1700 and from http//www.ngdc.gov for the interval 1700–2003). (a) The decadal and (b) the semi-secular cycles, respectively, and (c) the secular trend, this last is obtained by adding the secular wavelet components (see Table 1) to the linear trend. The dashed line in (c) represent the annual means of Wolf sunspot number, the horizontal lines are constant levels around which a quasi-periodic behaviour occurs. The numbers that are near to each level are its numerical value in spot. S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 103 long – to allow us to accurately separate the Gleissberg cycle 0.12 from it, this is the reason why in Table 1 we have included the Gleissberg and De Vries scales in only one scale, the sec- 1.0 ular one. 0.06 Slope (ms/yr)

3. A multi-resolution wavelet analysis of the relationship between excess of length of day and Wolf sunspot number 0.0 0.00 time series -0.06 The cycles and the steep changes associated to chaotic -1.0 transitions in solar activity might appear in all solar and Cross-correlation Function solar–terrestrial variables related to solar activity. Accord- -0.12 ing to the different nature of the involved systems for each 110100 one of the variables not all the solar signals will appear in Fourier Period (yr) the same way, if they appear at all. Therefore the relation- Fig. 4. The cross-correlation coefficient (heavy line), and the slope of the ship between solar activity related variables is likely to be regression line (slight line) between Wolf sunspot number and excess of non-linear (Duhau and Chen, 2002 and references therein). length of day wavelet components as a function of its Fourier period. The Hence to find out how R appear in LOD Duhau and Chen points in the light line indicate the Fourier periods of the 15 wavelets as (2000b) have performed a multi-resolution analysis of this given in Table 1 (from Duhau and Chen, 2000b). The is the value of the slope for the linear trend. The excess of length of day data is the same indices, a summary of their results that facilitates the com- than in Fig. 3. prehension of the relationship between LOD and climate indices related to solar activity is given below. To perform the multi-resolution analysis we have repre- • There is a very good correlation (cross-correlation coef- sented the two series by the Morlet wavelet transforms ficient – 0.9) at the 60.5 year. Fourier period, that has which Fourier periods are given in Table 1. As an example, the maximum slope modulus. This Fig. 3 shows the result for LOD. parameter decreases sharply afterward to reach a rela- We performed the cross-correlation between the 15 tive minimum at the 124 year. Fourier period and it wavelet functions which Fourier periods are given in Table increases monotonically when is reaching the 342 year 1 and that allow us to build the LOD and R time series. period, the largest one contained in the 350 year time The result is summarised in Fig. 4 where the cross-correla- series of LOD. tion coefficient (heavy line) and the slope of the related regression line (light line) as a function of the Fourier per- The fact that in the Schwabe time scale the signal is iod of each wavelet are plotted. almost absent in LOD would indicate that for scales below The most remarkable facts illustrated by Fig. 4 are the decadal these variations are not related to R variations. However, annuals means of these time series has been used • The Schwabe signal is almost absent in LOD. in the above analysis, therefore to reach to a definitive con- • A change of sign occurs when passing from the decadal, clusion about scales below the decadal we must analyse to the semi-secular time scales. data with a larger time resolution.

4 4

2 2

0 0

-2 -2

-4 -4 Excess of Length Day (ms) 1700 1800 1900 2000 Time (yr.)

Fig. 3. The representation of excess of length of day annuals means time series obtained by adding to the linear trend the 15 Morlet wavelet components which Fourier periods are given in Table 1 (light line). The data (points) are from McCarthy and Babcock (1986) in the interval 1700–1976 and from ftp:// euler.jpl.nasa.gov/keof/combinations/2002/ (for details on this data see Gross, 2000) in the interval 1977–1997. 104 S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108

Regarding to the decadal scale Kirov et al. (2003) oscillation in LOD around year 1900. The strongest semi- found evidences that mediates the transfer secular cycle that occurred in R during the interval 1710– of angular momentum from the to the Earth and 1855 is mapped in LOD – changed of sign – 94 year later. its atmosphere. The simultaneity of the changes in solar The square-like behaviour of this strong oscillation in R as and terrestrial variables that they found proves that for compared with LOD is a direct consequence of the fact that decadal scale solar wind angular momentum is directly the decadal cycle is much stronger, as compared with the transferred to the atmosphere and them to the Earth. semi-secular variation in R that in LOD time series. Notice Indeed, as discussed by Duhau and Martinez (1995) a that we have resorted to the historical R time series because considerable time delay is needed for the solar wind to is the largest well-documented solar activity index. How- induce core motions and for the transference to these ever, in view of the mechanism proposed by Duhau and motions to the mantle–atmophere–hydrophere system. In Martinez (1995) by which solar magnetic field strength this framework, the change of sign in Fig. 4 might be variations may excite LOD variations, a geomagnetic index merely a consequence of the fact that the mechanisms such as Dst, it is likely to be related to LOD. In fact, time of excitation of earth rotation rate variations are different changes in this index are quite similar in shape to those in in the decadal than in larger scales. So this change of sign LOD (as may be seen in Fig. 2 from Duhau and Martinez, sharply defines the separation of time scales for which 1995). LOD is excited by atmosphere motions from that on The semi-secular cycle in R has its largest amplitude in which, by the contrary, LOD motions excite atmosphere the interval 1766–1826 and is suddenly interrupted at year motions. Therefore in the following we will restrict our- 1856 to restart with a weaker strength at year 1940 (see selves to the analysis of data for scales equal or larger Fig. 2b). The strong semi-secular cycle in R has occurred than the semi-secular. with opposite sign in LOD 94 years later as may be seen Geodynamo models predict, that there are two kinds of in Fig. 6. However its sudden interruption in R at 1940 is natural motions of the Earth’s core–mantle system, the tor- not followed by LOD (see Fig. 6). This behaviour reflects sional motions (with a period of 60 years) (Braginnsky, the fact that once a excites a similar cycle in 1964, 1984) and the magnetostrophic wave (with a period the Earth; its damping depends on a mechanism that is of 300 years) (Hide, 1969; Kuang and Bloxham, 1997). internal to the Earth itself. This illustrates the non-linear Therefore the two relative minima in Fig. 4 occur for values nature of the relationship between solar and solar–terres- very near to the periods of the natural motions of the core. trial variables that leads to inconsistency in the results This behaviour indicates that a cycle in the sun signal when the time series formalism applied to analysis them excites the corresponding signal on the Earth’s core–mantle assumes linearity. system only when the frequency of the solar signal reso- Notice that the slope of the regression curve for the De nates with that of a natural motion of the core. Vries scale is the same that the ratio (the star in Fig. 4) In Fig. 5 we compare the time series that results after fil- between the slope of the linear trend, this indicates that tering the Schwabe signal by eliminating the wavelet com- the secular trend in LOD is also closely related to that in ponent which Fourier period is in that time scales (as R, however within the secular scale, both, the phase shifts defined in Table 1) from LOD and R time series, respec- (see Duhau and Chen, 2000b) and the slope of the regres- tively. The semi-secular cycle predominates in LOD varia- sion line (Fig. 4) vary with the Fourier period of the wave- tions as may be seen by the bare eye in Fig. 5 in the strong let component, so an accurate reconstruction of the secular

0 LOD cross-correlation = 0.84

-2 0 -Rz (spot)

-4 -40

LOD (ms) Rz

-6 -80

1700 1800 1900 2000 2100 Time (yr.)

Fig. 5. Excess of length of day (LOD) variations (heavy line) after subtracting a linear trend of 0.0211 ms/year that is not of solar origin but is due to tidal friction – 0.0272 ± 0.056 ms/year, post-glacial rebound – 0.0061 ± 0.005 ms/year and atmospheric tides – 0.003 ms/year (Merrian, 1988), and Wolf Sunspot Number (R) time series shisfted 94 year ahead (light line). In both time series the Schwabe signal was filtered by substrantig the Schwabe wavelet functions (as defined in Table 1) from the respective time series. The data for R and LOD are the same as in Figs. 2 and 3 respectively. S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 105 Semi-secular cycle (spot) 2 20

1 10

0 0

-1 -10

Semi-secular cycle (ms) -2 -20 1700 1800 1900 2000 2100 Time (yr)

Fig. 6. The semi-secular cycle in excess length of day (LOD) (heavy line) and in solar sunspot number (R) shifted 94 year ahead (light line) (Duhau, 2003b). trend in LOD from that in R falls of the scope of the pres- ent paper. 40

4. A simultaneous wavelet analysis of northern hemispheric temperatures, solar total irradiance SSC geomagnetic index and excess of length of the day time series SSC

As summarised in Section 1, beside LOD we ought to consider TSI variations. Also the impact on the Earth’s 20 surface of –solar wind interaction related energy sources, and the modulation of cosmic energy by solar wind, must to be taken into account. Proxy data for these sources are the geomagnetic indices SSC and aa. 1900 1950 2000 For time scales larger than the semi-secular one aa is Time (yr) mostly indistinguishable from TSI (see Fig. 1 in Duhau, 2003c). To simplify the analysis we will consider that TSI Fig. 7. The long-term trend of the SSC geomagnetic index annual means roughly represent also aa. (light line) and after an iteration procedure (see text). The data to compute SSC was obtained from ISGI at http://www.cetp.ipsl.fr/~isgi/homepag1. The data from which the SSC geomagnetic index is com- htm. puted start at 1868 and end in 1993, i.e., we have data to compute this index for only the last 125 years, then if we wish to have a workable long-term trend (i.e., preserving series built from the wavelet representation and the one the details that makes the curve enough far from the linear obtained from the measured values at the ending times, trend) the semi-secular cycle must be added to the secular as may be seen in that figure, in the following we have elim- trend. Therefore, to go farther we have computed the inated from the built geomagnetic index SSC long-term ‘long-term trend’ by adding the semi-secular cycle to the trend the last 10 years. secular trend. We fitted a linear function on the three By the above iterative procedure we have found that the solar–terrestrial variables to the long-term trend in NHT best simultaneous linear fitting of SSC index, TSI and LOD and we forced the fitting to coincide with the value of to NHT time series is

NHT at year 1900. That time was selected to be far enough NHTLTðtÞ¼0:0157½SSCLTðtÞSSCLTðtoÞ from the beginning of the time series to allow as ensuring that the cone of influence does not affect its values (Tor- þ 0:103½STILTðtÞSTILTðtoÞ rence and Compo, 1998). And at the same time climate 0:022½LODLTðtoÞLODLTðtoÞ variables were still not affected by anthropogenic sources. þ NHT ðtoÞð2Þ All the data, but SSC geomagnetic index, start at/or LT prior to year 1650. Therefore, we have inverted the fitting where XLT is the long-term trend in the X variable and function (see Eq. (2) below) to built values of the SSC time to = 1900 year. series long-term trend prior to 1986 and iterates until to In Fig. 8 we have plotted the long-term trend in NHT find the set of three coefficients of the function that better anomalies (light line) and its fitting given from Eq. (2). fit the data values. In that way we have extended the reli- The contribution of each of the terms in Eq. (2) is plotted able SSC index long-term trend at the starting time of in Fig. 9. It may be seen that the relative magnitude of the historical NHT time series, 1986 year. The result is these contributions changes with time. At the starting time shown in Fig. 7. Due to the discrepancies between the time SSC and TSI are of the opposite sign and have almost the 106 S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108

0.2 C) °

0.0

-0.2

-0.4 Long-term Trend (

1600 1700 1800 1900 2000 Time (yr)

Fig. 8. The long-term trend in NH surface temperature anomalies since 1600–2000 (relative to the 1961–1990 mean) and the fitting function (Eq. (2)) (heavy line). The data used to compute the long-term trend are from Bradley and Jones (1993) reconstructed series record in the interval 1600–1857 and from P.D. Jones, T.J. Osborn, K.R. Briffa at the Climatic Research Unit Meteorological Office School of Environmental Sciences Bracknell, Berkshire University of East Anglia, United Kingdom and D.E. Parker at the Hadley Centre for Climate Prediction and Research, United Kingdom (July 2001) in the interval 1858–2000, respectively.

0.1 0.3 SSC C) ° C)

° 0.0 0.2 TSI -0.1 0.1

-0.2 0.0 Long-term Trend ( LOD Secular Trend (

-0.3 -0.1 1900 1950 2000 Time (yr) 1900 1950 Time (yr) Fig. 9. The contribution to the long-term trend in NH surface temper- ature anomalies of each of the terms in Eq. (2), namely SSC geomagnetic Fig. 10. The secular trend in NH surface temperature anomalies (heavy index (SSC), total solar irradiation (TSI) and excess of length of day line) and the linear fitting with the same coefficients than in Eq. (2) (dashed (LOD), respectively. line).

same amplitude so nullifying between them, therefore LOD CO2 emission might be mostly of solar origin until 1970, contributes to the NHT anomalies long-term trend almost after that the fitting curve have remained below the NHT in a 100% for the first six decades. This explain the large secular trend. However we cannot arrive to a definitive 0.85 – cross-correlation coefficient found by Lambeck conclusion after climate modelling of the relevant solar- and Cazenave (1976) between variations in Earth’s rotation forcing variables would allow us to estimate the real mag- rate and global surface temperature time series from 1880 nitude of its effects on surface temperature. to 1975 in spite of the fact that the other relevant variables On the other hand, there is a general consensus that cli- were not taken into account. mate changes of anthropogenic origin have being increas- ing since the beginning of the 20th century following the 4.1. The secular trend exponential increases of CO2 with the fast industrialisation process (see e.g. Lean and Rind, 1999; see also Gonzalez- By inserting in Eq. (2) the secular trend of each variable Rouco et al., 2003). XST we have computed the linear fitting for that trend in However, other scenarios of regional nature have been NHT, the result of Fig. 10. The agreement between the sec- proposed for anthropogenic impact on climate changes, ular trend in NHT and the fitting function with the same namely: urbanisation and lad uses (Kalnay and Cai coefficients that allow a good fitting for the long-term trend (2003) and surface temperature changes related to CO2 (Eq. (2)) is surprisingly good and implies that the same direct impact on surface temperature over industrialised coefficients fits well also the semi-secular cycle. It may be areas (De Laat and Maurellkis, 2004)). seen that the steady increases of the secular trend since Duhau (2003a) found evidence that a descending chaotic 1900 that has been attributed to a sudden increases on transition has begun at year 1993, but, we still do not know S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108 107 neither the time nor the final average level that solar activ- NHTSSðtÞ¼0:0157SSCSSðtÞþ0:103STISSðtÞ ity will reach at the ending of that transition. The Sun is 0:022LOD ðtÞð3Þ releasing magneto-hydrodynamic energy to come back to SS a stable level of solar activity. If the strength of this source where XSS is the semi-secular cycle in time series X. of energy is truly related to sudden storm commencements, Fig. 11a and b shows the result of the fitting and the we might infer that the SSC geomagnetic index will contin- contribution of each of the terms on Eq. (3). ues increasing for the forthcoming decade since this stable level – of 90 spot – would be reached, as estimates by Duhau (2003a) only near year 2018. 5. Summary and conclusions In view of the results presented by Kalnay and Cai (2003) and De Laat and Maurellkis (2004), to perform a By a the Morlet wavelet formalism of the envelope of systematic analysis as that presented here on a regional the 350-year long R sunspot maxima time series we have bases might also help to found empirical evidences of the determined the signals that compose the quasi-periodic relative contribution of anthropogenic and solar activity states in solar dynamo system – this was done to unearth- sources to climate changes. ing the presence of these signals in related time series. We found that the historical time series of the involved solar– 4.2. The semi-secular cycle in Northern hemisphere terrestrial time series are given by the superposition of temperature two cycles, a decadal and a semi-secular ones, and a secular trend. As found above, to get the fitting function in this time From that representation we have compared LOD andR scale we may apply the same coefficients as given by time series between them and also we have proposed func- Eq. (1), as follows: tions that consist in a lineal superposition of the contribu- tion of SSC index, LOD and TSI – this last also as a proxy data for aa time series for time scales larger that the deca- dal – to fit the northern hemisphere surface temperature annual means. We have found that

C) a ˚ 0.1 • The same linear coefficients fit as well the semi-secular cycle as the secular trend in NHT. • As temporal changes in SSC index, TSI and LOD time 0.0 series are quite different the importance of the contribu- tion of each of them to NHT variations has strong tem- poral changes too. The long-term trend in LOD (defined as the superposition of the semi-secular cycle and the

Semi-secular Cycle ( -0.1 secular trend) has contributed to NHT anomalies almost in a 100% for the second half of the 19th century, this 1900 1950 2000 explain why Lambeck and Cazenave (1976) has find a Time (yr) close relationship between LOD and NHT long-term variations (cross-correlation 0.85) in spite of the fact 0.10 that they have not considered the other relevant vari- SSC ables. For the second half of the 20th century the contri- C) b ° bution of LOD almost nullifies and that from SSC index 0.05 predominates. • The semi-secular cycle, as compared with other solar sig- nals, is strongly amplified in LOD and appears 94 year 0.00 later than in R in agreement with previous evidence pre- TSI sented by Duhau and Martinez (1995). LOD -0.05 • Both the secular trend and the semi-secular cycle in LOD is clearly apparent in surface temperature changed Semi-secular Cycle ( in sign, which means that colder climates goes with -0.10 slower Earth’s rotation in agreement with Lambeck 1900 1950 2000 Time (yr) and Cazenave (1976).

Fig. 11. The semi-secular cycle in (a) NH surface temperature anomalies Summing up, we have presented evidence that solar (heavy line) and the curve that result from Eq. (3) (light line), and (b) the activity variation excites a semi-secular cycle in Earth rota- contributions from each of the terms in this equation, namely SSC geomagnetic index (SSC), total solar irradiation (TSI) and excess of length tion rate with a 94 years delay and that this cycle in Earth’s of day (LOD), respectively. rotation rate in turn forces surface temperature variations. 108 S. Duhau / Physics and Chemistry of the Earth 31 (2006) 99–108

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