Testing Hypotheses about Sun-Climate Complexity Linking

We reexamine observational evidence presented in support of the hypothesis of a sun-climate complexity linking by N. Scafetta and B. J. West, Phys. Rev. Lett. 90, 248701 (2003), which contended that the integrated solar flare index (SFI) and the global temperature anomaly (GTA) both follow Lévy walk statistics with the same waiting-time exponent $\mu \approx 2.1$. However, their analysis does not account for trends in the signal, cannot deal correctly with infinite variance processes (Lévy flights), and suffers from considering only the second moment. Our analysis shows that properly detrended, the integrated SFI is well described as a Lévy flight, and the integrated GTA as a persistent fractional Brownian motion. These very different stochastic properties of the solar and climate records do not support the hypothesis of a sun-climate complexity linking.

Figure 1

(a) Monthly values of global temperature anomalies from the year 1880 until today. The thick line is a fourth order polynomial trend. (b) Daily values of the solar flare index over a few solar cycles. (c) PDFs of the increments $\Delta \widehat{x}(\Delta t,t)\equiv \widehat{x}(t+\Delta t)-\widehat{x}(t)$ rescaled with respect to a self-similarity exponent $\delta =0.65$. The various symbols refer to different $\Delta t$ in the range from 1 to 64 months. Here the fourth order polynomial shown in (a) is subtracted before making the PDFs. (d) PDFs of wavelet coefficients $T_{\psi }[y](\Delta t,t)$ of the signal $y(t)$ [see Eq. (2)] rescaled with respect to a self-similarity exponent $\delta =1$. The symbols refer to different $\Delta t$ in the range from 4 to 128 d.

Figure 2

(a) GTA signal $T(t)$ together with a synthetic data set $x_s(t)={\int }^t[w_{\delta }(t^{\prime })+P(t^{\prime })]d{t}^{\prime }$ obtained superposing a fractional Gaussian noise $w_{\delta }(t)$ with $\delta =0.65$ onto the polynomial $P(t)$ shown in Fig. 1a. The synthetic signal is shifted up two degrees Celcius to avoid overlap between the graphs. Panels (b)–(c) show results of DEA ($\exp [S]$ against $t$ is shown as circles) and SDA ($D$ against $t$ is shown as triangles). (b) DEA and SDA applied to the detrended integrated GTA signal $\widehat{x}(t)={\int }^t[T(t^{\prime })-P(t^{\prime })]d{t}^{\prime }$. The dotted line has slope 0.70. (c) DEA and SDA applied directly (without detrending) to the integrated GTA signal $x(t)={\int }^{t}T(t^{\prime })d{t}^{\prime }$. The steepest dotted line has slope 0.90 and the other dotted line has slope 0.80. (d) DEA and SDA applied to a synthetic fBm ${\int }^t{w}_{\delta }(t^{\prime })d{t}^{\prime }$ with $\delta =0.65$. The dotted line has slope 0.65. (e) DEA and SDA applied to the synthetic GTA data ${\widehat{x}}_s={\int }^t[w_{\delta }(t^{\prime })+P(t^{\prime })]d{t}^{\prime }$ shown in (a). The steepest dotted line has slope 0.90 and the other dotted line has slope 0.77.

Figure 3

(a) Empirical moments for the temperature signal $x(t)$ where the fourth order polynomial shown in Fig. 1a is subtracted. (b) The scaling function $\zeta (q)$ calculated from the empirical moments in (a). (c) Wavelet-based empirical moments (WBEMs) for the solar flare signal $y(t)$, using time scales up to 500 d. Inset: The same for time scales up to ${10}^4\mathrm{d}$. (d) The scaling function calculated from the WBEMs in (c). Circles: Using linear fits to the WBEMs in the range 1–500 d. Triangles: Using linear fits in the range $1–{10}^4\mathrm{d}$. (e) WBEMs for a simulated Lévy walk with $\mu =2.5$ ($\delta =0.67$). (f) The scaling function calculated from the WBEMs in (c) (circles), and for $\mu =2.05$ (upper dashed line) and $\mu =2.95$ (lower dashed line). From this figure it can be observed that ${\widehat{H}}_D\equiv \zeta (2)/2$ takes values consistent with Eq. (1).