The temperature increase of a body of water is:
T = Joules/(Specific Heat water x Mass)
The accumulation of 1 Watt per sq meter on a 100 metre column of water for one year gives an expected temperature increase of
T = 32 x 10^6/(4.2 x 10^8)
= 0.08 C
Given that about one third attenuation of radiation from top-of-atmosphere to the surface, and a duration of solar cycle of 11 years, the increase in temperature due to the solar cycle will be:
Ta = 0.08 x 11 x 0.3 = 0.26 C
The expectation of the temperature increase for the direct forcing (no accumulation) using the Plank relationship of 0.3C/W would be 0.09 C. So the gain is:
Gain = Accumulated/Direct = 0.26/(0.3×0.3) = 3
For a longer accumulation of solar anomaly, from a succession of strong solar cycles such as we saw late last century, the apparent amplification will be more. From the AR correlation of surface temperature you get an estimate of gain of 10. But this is only apparent amplification, as the system is accumulative, the calculated gain increases with the duration of the forcing. For long time scales, gain (and hence solar sensitivity) approaches infinity — a singularity — and ceases to be useful. Hence the term ‘supersensitivity’. For long periods the non-linearity of the Stephan-Boltzmann law will become dominant.
Sensitivity cannot be represented in Watts/K (or K/Watt). It will be in units of rate like K/Watt/Year.
Extend this calculation for 1000 years and a small solar forcing can cause a transition between ice ages with no other input. The role of GHGs, water vapor and albedo in this theory is to maintain the heat state of the system, e.g. solar forcing increases temperature increase which causes CO2 concentrations to change. But this does not mean an increase in CO2 ‘necessarily’ increases temperature, because the system is being heated by accumulation of solar anomaly. The reason that a forcing from CO2 has apparently very low sensitivity, but solar very high, would be due to other issues that I haven’t worked through fully yet (coming soon).