In some followup discussion with Barry Bickmore (by email) and Kevin C (at Skeptical Science) it became clear we were missing something in the analysis of Roy Spencer's climate model. Specifically, Bickmore's Figure 10:
differed significantly from Spencer's 20th century fit even though he was ostensibly using the same parameters. If you look at the year 1900 in particular, Bickmore's Figure 10 has a temperature anomaly slightly above zero, while Spencer's is pegged at -0.6 C. Bickmore did use a slightly different PDO index than Spencer for his 1000-year graph (figure 10) - but more importantly, he took Spencer's -0.6 C temperature as the starting point in the year 993 AD, rather than as a constraint on temperature in the year 1900, as it actually was in Spencer's analysis. It turns out that to actually match Spencer's 20th century temperature fit the starting temperature in 993 AD needs to be extraordinarily, far beyond impossibly, low. We'll get to the details shortly.
BYU geologist Barry Bickmore recently reviewed Roy Spencer's recent book, "The Great Global Warming Blunder", finding a number of true "blunders" by the author. In particular he found some very peculiar properties of the simplified physical model that Spencer made a central feature of the book, finding that Spencer's curve-fitting allow infinitely more solutions than the one Spencer somehow settled on, and a number of related issues.
I tangled with Spencer over an earlier model like this which he was promoting more than 3 years ago. What he didn't seem to realize about that first model was that it was essentially trivial, a linear two-box model with two time constants (a subject I explored in detail here a while back). I tried explaining this, but he seems not to have gotten my point that such a model inherently contains no interesting internal dynamics, just relaxation on some (in this case two) time scales. Which seems to go completely against the point I thought he was trying to make, that some sort of internal variability was responsible for decadal climate change.
So it was something of a surprise to me that Spencer based his "Great Blunder" book on an even more simplified version of this model, with just 1 effective time constant. He even tried to get a paper published using this essentially trivial model of Earth's climate. As Bickmore outlined in his part 1, the basic equation Spencer uses is:
(1) dT/dt = (Forcing(t) – Feedback(t))/Cp
where T is the temperature at a given time t, Forcing is a term representing the input of energy into the climate system (there is a standard definition for this in terms of radiation at the "top of the atmosphere") and Feedback is a term that itself depends on temperature as
(2) Feedback(t) = α (T(t) - Te)
with α a linear feedback parameter and Te an equilibrium temperature in the absence of forcing (Bickmore and apparently Spencer don't actually use absolute temperature T and equilibrium value Te, but rather write the equations in terms of the difference Δ T = T - Te, which amounts to the same thing, but obscures an important point we'll return to later).
The final term Cp is the total heat capacity involved. Each of forcing, feedback and heat capacity is potentially a global average, but would normally be expressed as a quantity per unit area, for example per square meter. Since the bulk of Earth's surface heat capacity that would respond to energy flux changes on a time-scale of a few years is embodied in the oceans (about 70% of the surface), Cp should be defined essentially as 0.7 times the heat capacity of water per cubic meter, multiplied by the relevant ocean depth in meters (h):
(3) Cp = 0.7*4.19*10^6 J/(m^3 K) * h = c * h
where c = 2.9 MJ/(m^3 K) (Spencer and Bickmore seem to have forgotten the factor of 0.7, so use a slightly larger value for c, which means their h values are probably smaller than the actual ocean depth such a heat capacity would be associated with).