Discussion on the spectral coherence between planetary, solar and climate oscillations: a reply to some critiques

Abstract

During the last few years a number of works have proposed that planetary harmonics regulate solar oscillations. Also the Earth’s climate seems to present a signature of multiple astronomical harmonics. Herein I address some critiques claiming that planetary harmonics would not appear in the data. I will show that careful and improved analysis of the available data do support the planetary theory of solar and climate variation also in the critiqued cases. In particular, I show that: (1) high-resolution cosmogenic ¹⁰Be and ¹⁴C solar activity proxy records both during the Holocene and during the Marine Interglacial Stage 9.3 (MIS 9.3), 325–336 kyear ago, present four common spectral peaks (confidence level ⪆95 %) at about 103, 115, 130 and 150 years (this is the frequency band that generates Maunder and Dalton like grand solar minima) that can be deduced from a simple solar model based on a generic non-linear coupling between planetary and solar harmonics; (2) time-frequency analysis and advanced minimum variance distortion-less response (MVDR) magnitude squared coherence analysis confirm the existence of persistent astronomical harmonics in the climate records at the decadal and multidecadal scales when used with an appropriate window lenght (L≈110 years) to guarantee a sufficient spectral resolution to solve at least the major astronomical harmonics. The optimum theoretical window length deducible from astronomical considerations alone is, however, L⪆178.4 years because the planetary frequencies are harmonics of such a period. However, this length is larger than the available 164-year temperature signal. Thus, the best coherence test can be currently made only using a single window as long as the temperature instrumental record and comparing directly the temperature and astronomical spectra as done in Scafetta (J. Atmos. Sol. Terr. Phys. 72(13):951–970, 2010) and reconfirmed here. The existence of a spectral coherence between planetary, solar and climatic oscillations is confirmed at the following periods: 5.2 year, 5.93 year, 6.62 year, 7.42 year, 9.1 year (main lunar tidal cycle), 10.4 year (related to the 9.93–10.87–11.86 year solar cycle harmonics), 13.8-15.0 year, ∼20 year, ∼30 year and ∼61 year, 103 year, 115 year, 130 year, 150 year and about 1000 year. This work responds to the critiques of Cauquoin et al. (Astron. Astrophys. 561:A132, 2014), who ignored alternative planetary theories of solar variations, and of Holm (J. Atmos. Sol. Terr. Phys. 110–111:23–27, 2014a), who used inadequate physical and time frequency analyses of the data.

Appendix

1.1 A.1 The three frequency solar model

Here I summarize the functions used for constructing the planetary/solar three frequency solar model discussed in Sect. 2 and shown in Fig. 2. This appendix reproduces the Appendix in Scafetta (2012c) for the reader convenience.

The three basic proposed harmonics are:

$$\begin{aligned} h_1(t) =&0.83 \cos \biggl(2\pi \frac{t-2000.475}{9.929656} \biggr), \end{aligned}$$

(11)

$$\begin{aligned} h_2(t) =&1.0 \cos \biggl(2\pi \frac{t-2002.364}{10.87} \biggr), \end{aligned}$$

(12)

$$\begin{aligned} h_3(t) =&0.55 \cos \biggl(2\pi \frac{t-1999.381}{11.862242} \biggr), \end{aligned}$$

(13)

where the relative amplitudes are weighted on the sunspot number record since 1749. Three frequencies derive from the spectrum of the sunspot record (see Fig. 2A) where the two side harmonics at 9.93 and 11.86 year period are theoretically deduced from the tidal oscillations generated by Jupiter and Saturn. The three phases are deduced: from the conjunction date of Jupiter and Saturn, t=2000.475; the perihelion date of Jupiter, t=1999.381; and by regression on the sunspot cycle, t=2002.634.

The basic harmonic model is

$$\begin{aligned} & h_{123}(t)=h_1(t)+h_2(t)+h_3(t), \end{aligned}$$

(14)

$$\begin{aligned} & f_{123}(t)= h_{123}(t) \quad \mathrm{if} \ h_{123}(t)\geq 0, \end{aligned}$$

(15)

$$\begin{aligned} & f_{123}(t)= 0 \quad \mathrm{if}\ h_{123}(t)<0, \end{aligned}$$

(16)

which is depicted in Fig. 2B.

The chosen beat function modulations in generic relative units and their sum are:

$$\begin{aligned} & b_{12}(t)=0.60 \cos \biggl(2\pi \frac{t-1980.528}{114.783} \biggr), \end{aligned}$$

(17)

$$\begin{aligned} & b_{13}(t)=0.40 \cos \biggl(2\pi \frac{t-2067.044}{60.9484} \biggr), \end{aligned}$$

(18)

$$\begin{aligned} & b_{23}(t)=0.45 \cos \biggl(2\pi \frac{t-2035.043}{129.951} \biggr), \end{aligned}$$

(19)

$$\begin{aligned} & b_{123}(t)=b_{12}(t)+b_{13}(t)+b_{23}(t)+1. \end{aligned}$$

(20)

The three relative amplitudes are roughly estimated against Eq. (15). The millennial modulating function is

$$ g_m(t)=A~\cos \biggl(2\pi~\frac{t-2059.686}{983.401} \biggr)+B. $$

(21)

The parameters A and B may be changed according to the application. The two proposed modulated solar/planetary functions are

$$\begin{aligned} & F_{123}(t)=g_m(t) f_{123}(t)\quad \mathrm{with}~A=0.2, B=0.8, \end{aligned}$$

(22)

$$\begin{aligned} & B_{123}(t)=g_m(t) b_{123}(t)\quad \mathrm{with}~A=0.3,B=0.7. \end{aligned}$$

(23)

See Scafetta (2012c) for more details and for a supplement file with the actual data.

1.2 A.2 Scafetta (2010) astronomical—temperature spectral coherence

For convenience of the reader Fig. 12 reproduces figures 6B and 9A of Scafetta (2010). Figure 12B shows MEM evaluations of numerous climatic records such as the global surface temperature (G), the northern and southern global surface temperatures (GN and GS), the global, northern and southern land surface temperatures (L, LN, LS) and the global, northern and southern ocean surface temperatures (O, ON, OS). The green bars are the main solar, astronomical and lunar expected harmonics (cf. Fig. 5). It is easy to notice a coherence between the astronomical harmonics and the MEM spectral peaks at multiple frequencies. Figure 12A directly compares the temperature average periods (red) against the astronomical average periods (blue). A χ² test output among the various frequencies is shown suggesting that the coherence confidence is at the 96 %. Additional calculations and evidences are provided in Scafetta (2010).

Fig. 12

Reproduction of figure 6B and 9A of Scafetta (2010) showing [A] the χ² spectral coherence test and [B] the direct comparison between the MEM curve of several climatic records and the astronomical, solar and lunar harmonics (green bars). See the original paper for details

1.3 A.3 Kepler’s diagram of Jupiter–Saturn conjunctions

Figure 13 shows the original diagram of Jupiter–Saturn conjunctions prepared by Kepler (1606). It highlights the date and the constellation position of the great conjunctions, that occur every 20 years, from 1583 to 1763. The 60-year trigon pattern, that involves three consecutive conjunctions, is clearly visible together with its slow millennial rotation. The 20, 60 and 800–1000 year oscillations associated to the movement of Jupiter and Saturn were well known since antiquity and used to construct some kind of astrological-based climate models (Kepler 1606; Iyengar 2009; Ma’sar 2000; Temple 1998). See Scafetta (2012a) for additional details.

Fig. 13

The original diagram of Jupiter–Saturn conjunctions prepared by Kepler (1606)