Chylek on sensitivity

Another one of this strange collection of papers has an odd attempt to estimate climate sensitivity by Chylek et al, which has recently been published.

They start from the familiar zero-dimensional energy balance

(1) dH/dt = F - l.T

where H is the total heat anomaly (dominated by ocean heat uptake), F is the forcing anomaly and T the surface temperature anomaly (note that my notation is slightly different from theirs). Here l = 1/S is the radiative feedback parameter, the inverse of climate sensitivity S. At equilibrium, the LHS of equation (1) is zero and so the formula T = F/l gives the equilibrium temperature change for a given forcing.

Chylek cites Raper et al (2002) to justify the substitution

(2) dH/dt = k.T

where k is a constant. This is slightly different from the "effective heat capacity" approach of Schwartz: in this expression, the rate of heat uptake is assumed proportional to the surface temperature anomaly, whereas for Schwartz, the heat content is assumed proportional to surface temp anomaly - with heat content being the integral of heat uptake over time. This expression that Chylek uses may be a reasonable approximation during an experiment in which the forcing is increasing steadily (as Raper et al state), but is clearly not valid in general, as we shall see shortly. (For reference, Raper et al apply their analysis to the standard test case in which the CO2 level increases at 1% per year and all other forcing are held fixed.)

Substituting (2) in (1), we get

(3) k.T = F - l.T

which can be differentiated (or finite-differenced) with respect to time to get

(4) k.DT = DF - l.DT

where the D indicates the decadal trend (it's a greek Delta in Chylek et al).

This can be rearranged to give

(5) S = 1/l = DT/(DF - k.DT)

which is the expression that Chylek use to diagnose S using data from the past decade.

But it is immediately obvious that this expression will have some problems when used to diagnose S over an arbitrary interval. For example, if we take a decade over which the net forcing does not change, then DF = 0 and from equation (5) we get

(6) S = DT/(0 - k.DT) = -1/k

unless DT is also zero (in which case the expression is undefined) which is implausible to say the least. So something odd is going on here. Let's look a little closer...

Stepping back to equation (3), we can rearrange it to give:

(7) T = F/(l+k)

which implies that (in order for the equation to be valid) the surface temperature must always be directly proportional to the forcing - with constant of proportionality 1/(l+k) - and thus is instantaneously in equilibrium with the forcing, with no lag.

However, we started off with the zero-dimensional energy balance (1) which as I already pointed out has the equilibrium solution

(8) T = F/l

These latter two expressions (7) and (8) are only compatible if k=0 (which it certainly isn't, and Chylek never claim it is).

So we have a contradiction here. The fundamental problem is that the expression dH/dt = k.T is only a plausible approximation under certain circumstances. Chylek's formula (5) cannot be used as a general expression to diagnose the sensitivity of the climate system, as it gives nonsensical answers when forcing does not follow a steadily-increasing profile such as that of the CMIP (1% pa) experiments. There is no justification in their paper as to why their expression should be accurate when applied over the last decade, and I'd bet dollars to doughnuts that it fails to usefully diagnose sensitivity when applied to GCM output such as that in the AR4 database (I've not bothered to do this).

So in the absence of any evidence that their method actually works, it's hard to take their results seriously. I haven't even started to talk about how sensitive their calculation is to natural variability noise, which they have not accounted for either.