# Another study on solar influence

Another more serious weakness is a flawed approach to obtain their ‘climate sensitivity’, and especially so for ‘Zeq‘ in their Equation 4. They assume a linear relationship between the response and the forcing Zeq=288K/1365Wm-2. For one thing, the energy balance between radiative forcing and temperature response gives a non-linear relation between the forcing, F, and temperature to the fourth power, T4 (the Stefan-Boltzmann law). This is standard textbook climate physics as well as well-known physics. However, there is an additional shortcoming due to the fact that the equilibrium temperature is also affected by the ratio of the Earth’s geometrical cross-section to its surface area as well as how much is reflected, the planetary albedo (A). The textbook formulae for a simple radiative balance model is:

F (1-A)/4 = s T4, where ‘s’ here is the Boltzmann constant (~5.67 x 10-8 J/s m2K4).

(‘=’ moved after Scafetta pointed out this error. )

S&W’s sun-climate sensitivity (Zeq =0.21K/Wm-2), on which the given solar influence estimates predominantly depend, is thus based solely on a very crude calculation that contradicts the knowledge of climate physics. The “equilibrium” sensitivity of the global surface temperature to solar irradiance variations, which is calculated simply by dividing the absolute temperature on the earth’s surface (288K) by the solar constant (1365Wm-2), is based on the assumption that the climate response is linear in the whole temperature band starting at the zero point. This assumption is far from being true. S&W argue further that this sensitivity does not only represent the direct solar forcing, but includes all the feedback mechanisms. It is well known, that these feedbacks are highly non-linear. Let’s just mention the ice-albedo feedback, which is very different at (hypothetically) e.g. 100K surface temperature with probably ‘snowball earth’ and at 300K with no ice at all. In their formula for the calculation of the sun-related temperature change, the long-term changes are determined by Zeq, while their ‘climate transfer sensitivity to slow secular solar variations’ (ZS4) is only used to correct for a time-lag. The reason for this remains unclear.

In order to calculate the terrestrial response to more ephemeral solar variations, S&W introduce another type of ‘climate sensitivity’ which they calculate separately for each of two components representing frequency ranges 7.3-14.7 and 14.7-29.3 year ranges respectively. They take the ratios of the amplitude of band-passed filtered global temperatures to similarly band-passed filtered solar signal as the estimate for the ‘climate sensitivity’. This is a very unusual way of doing it, but S&W argue that similar approach has been used in another study. However, it’s not as simple as that calculating the climate senstivity (see here, here, here, and here). Hence, there are serious weaknesses regarding how the ‘climate sensitivities’ for the 11-year and the 22-year signals were estimated. For linear systems, different frequency bands may be associated with different forcings having different time scales, but chaotic systems and systems with convoluted response are usually characterised with broad power spectra. Furthermore, it’s easy to show that band-pass filtering of two unrelated series of random values can produce a range of different values for the ratio of their amplitudes just by chance (Fig. 2). As an aside, it is also easy to get an apparent coherence between two band-pass filtered stochastic series of finite extent which are unrelated by definition – a common weakness in many studies on solar-terrestrial climate connection. There is little doubt that the analysis involved noisy data.

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