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).
Rewriting equation 1 with these definitions we wind up with:
(4) dT/dt = (Forcing(t) - α (T(t) - Te))/(c h)
as the essential equation of Spencer's model. In the case where there is no forcing, Forcing(t) = 0, the model then reduces to:
(5) dT/dt = - (α / c h) (T(t) - Te)
Note that α has units of watts per square meter per degree, while c h has units of joules per square meter per degree. Since 1 W = 1 J/s, the ratio has units of inverse time (as it should to match the time derivative on the left-hand side). If we define a time constant
(6) τ = c h /α
Spencer's model equation (in the case of no forcing) becomes simply:
(7) dT/dt = -(T(t) - Te)/τ
This is one of the simplest possible first-order differential equations, and using the properties of derivatives of the exponential function it is easy to show the most general solution of this equation is:
(8) T(t) = Te + A e-t/τ
This means that, whatever the initial value of the temperature at time t = 0, say, this model forces the temperature to come exponentially close to the equilibrium temperature Te, with a characteristic time-constant τ. After one time period τ, the difference between T(t) and Te is a factor 1/e smaller than at first. After two time periods τ it's 1/e2 smaller, and so on.
In particular, Spencer's model has absolutely no internal dynamics other than a simple exponential decay to equilibrium.
So what is the relevant time-constant for Spencer's model? From eq. 6 and 3 we get:
(9) τ = 2.9 MJ/(m^3 K) * h/α * 1 year/31.6*10^6 s
= 0.092 h/α years
if h is expressed in meters and α in W/m^2K (note that the scale factor is about 0.13 for Bickmore and Spencer, without the 0.7 factor for ocean area). If ocean depth h = 700 m and α = 3.0 W/m^2K as Spencer apparently claimed to find from fitting this model to observed temperatures (with a certain forcing, which we'll get to) that gives a time constant τ = 21 years (or almost exactly 30 years without the ocean area factor).
You would get exactly the same time constant with h = 350 and α = 1.5, or h = 70 and α = 0.3. It is determined entirely by the ratio of ocean depth to the feedback parameter. In particular, a fit of temperatures using just this model (remember without any forcings) could not possibly determine both depth and feedback parameter, it could only determine the ratio of the two, because the response depends only on that ratio!
This single time-constant model (as it is so very simple) has in fact been used at least a few times before to try to model the observed temperature changes of the past century or so. Stephen Schwartz used just such a model in a 2007 paper, "Heat capacity, time constant, and sensitivity of Earth’s climate system", published in Journal of Geophysical Research volume 112, D24S05. That paper found a time constant of 5 years - but it was quickly criticized directly on the assumption that Earth's climate was governed by one single time constant. A later followup paper increased the likely time constant to 8.5 years. Lucia Liljegren has used essentially the same model, calling it "Lumpy", to fit historical temperatures with a single time-constant using the GISS Model E forcings, and found a time constant of 14.5 years. In that context Spencer's 30 year time constant seems a little long, but not out of the question - he was fitting a somewhat different "forcing" than Liljegren, for instance.
The mathematics for completely solving this simple single-time-constant model even with a forcing is hardly more complicated than the case with zero forcing. Let's abbreviate forcings to F(t):
(10) dT/dt = F(t)/c h - (T(t) - Te)/τ
and introduce a function A(t) so that:
(11) T(t) = Te + A(t) e-t/τ
Then taking a derivative gives:
(12) dT/dt = dA/dt e-t/τ - A(t) e-t/τ/τ
so (10) becomes:
(13) dA/dt = F(t) et/τ/c h
which is solved by a straightforward integral:
(14) A(t) = A0 + ∫-∞t F(s) es/τ ds / c h
So the full solution for temperature in Spencer's model can be expressed directly in this integral form (no "computer modeling" required at all):
(15) T(t) = Te + A0 e-t/τ + ∫-∞t F(s) e- (t - s)/τ ds / c h
The forcing F(t) appears here weighted exponentially in time, so that more recent (s close to t) values of forcing contribute more strongly, but there is some exponentially lower contribution with time-constant τ going back as long as you like. This expression is a convolution (essentially a smoothing) of the forcing with the exponential function, which we can express by a new function V(t):
(16) V(t) = ∫-∞t F(s) e- (t - s)/τ ds
so the temperature as a function of time in Spencer's model is in the end given simply by 3 terms:
(17) T(t) = Te + A0e-t/τ + V(t)/c h
i.e. temperature relative to equilibrium is an exponentially decaying transient term plus a convoluted form of the forcing function divided by heat capacity.
When you have known real forcings, for example the GISS Model E forcings used by Liljegren, for any given value of the time constant τ you can determine V(t), and then fitting the T(t) curve of equation 17 to observed temperatures constrains you to a real value for the effective ocean depth h. Varying both you can find a best fit for both parameters, h and τ.
This is where Spencer really ran into trouble, though, as can be seen from this figure in Bickmore's review:
The red line shows 24 "best-fit" curves for different values of ocean depth 'h' - and somehow they all lie on top of one another!
Spencer's problem here was that he introduced a third variable, β which he also tried to fit at the same time. Instead of using a known forcing F(t), he made an assumption that a measure of forcing was a certain ocean oscillation index, PDOI, and then allowed that to be scaled by an unknown factor β
The same equations as above apply with F(t) = β PDOI, and we can convolute to a V(t) = β Q(t) in the same way, for given time constant τ. That turns the solution equation for temperature (eq. 17) into:
(18) T(t) = Te + A0e-t/τ + β Q(t)/c h
The original three parameters α, β and h (Te and A0 are additional free variables we'll get to in a minute) appear in this equation for temperature only as ratios - τ = c h /α, and in the term β/c h multiplying Q(t).
That means, whatever is done regarding the other two free variables, using this model to fit temperatures cannot possibly constrain the absolute values of those original three. A fit constrains the ratio of α to h and β to h, but their actual scale is completely free. Spencer's claim that he ran these models with 100,000 different combinations should be highly embarrassing to him - he could have run them infinitely many times and this model simply cannot constrain the magnitude of the feedback parameter to within an arbitrary scale factor (with the same scale factor multiplying ocean depth). That is exemplified in Bickmore's graph of best-fit α, β values vs. h:
Spencer's claim that his model shows climate sensitivity to be low is truly embarrassing given that absolutely any nonzero value for α would give exactly the same temperature curve.
That's not quite the end of it - Spencer's other claim was that this model showed that 20th century temperatures were well modeled by the PDOI. But that depended on having those additional parameters Te and A0 to play around with, as Bickmore showed. In particular, the effect of freedom in choice of A0 is clearly seen in Bickmore's figure 8:
where different initial starting choices for the temperature in the year 1900 show their exponentially decaying contribution to temperatures, with 30-year time constant. That one arbitrary transient does much of the work in fitting the early part of the 20th century temperature curve, but it has no relation to the ostensible forcing involved at all.
Finally, as Bickmore's Figure 9 shows:
the choice of equilibrium temperature Te also has a strong effect on the fit. Changing Te to correspond to temperatures of the last half of the 19th century rather than the most recent 1961-1990 WMO "normal" period makes Spencer's ostensible fit from PDOI (the red curve) look very poor indeed. Why does he need an equilibrium temperature for his system that is already about 0.5 degrees C warmer than pre-industrial temperatures? Choosing that equilibrium temperature means most of 20th century warming is already built into Spencer's model - so using it to claim PDOI explains recent warming is, once again, just embarrassing.
I can understand preparing a scientific paper with faults of this sort - it's certainly easy to fool ourselves when we think we're on to something. But publishing a book on the subject? Doesn't Spencer have some smart colleagues he can run this stuff by first? Of course it really, really doesn't help that the book is also full of attacks on other climate scientists. Bickmore's most damning quote from Spencer's book suggests a certain hubris:
I find it difficult to believe that I am the first researcher to figure out what I describe in this book. Either I am smarter than the rest of the world’s climate scientists–which seems unlikely–or there are other scientists who also have evidence that global warming could be mostly natural, but have been hiding it. That is a serious charge, I know, but it is a conclusion that is difficult for me to avoid. (p. xxvii)
The first thing a true scientist should think of in a situation like this doesn't seem to have even occurred to Spencer. "What if I'm wrong?" He was.