## Friday, December 28, 2007

### 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.

crandles said...

Off topic of that paper but perhaps just about on the topic of estimating climate sensitivity.

Wondered if you were aware of CPDN work in progress.

http://www.climateprediction.net/board/viewtopic.php?p=71930#71930

eg "Earlier this year ROSALIND WEST completed her fourth year MPhys project in which she used CPDN data to investigate using the seasonal cycle in temperature to constrain climate sensitivity. She found quite different results for the slab and the coupled experiments and is in the process of writing this up for publication in the peer-reviewed literature."

James Annan said...

Hi Chris,

Thanks for that - I don't seem to be on their Christmas Card list for some reason :-)

ICE said...

hi,

thanks a lot for this post.
One more question, then, if you have time, about climate sensitivity:

reading a book about climate change (here in france), i found this (by P.Morel and M.Chahine), which is probably well-known (?) but new to me:

they use differences in seasonal variations in the energy budget of each hemisphere to derive a rough "atmophere-only" Climate Sensitivity.

With data from ERBE, it yields (for nothern hemisph.):
DF (summer - winter) = 21 Wm-2
DT (summer - winter) = 11.7° K
CS=0.56 K/W.m-2

and for the southern hemisph.:
DF (summer - winter) = 9 Wm-2
DT (summer - winter) = 5.1 K
CS=0.57 K/W.m-2

they argue that this should hold, as long as GW remains weak compared to seasonal temperature variations; and that this atmo-only CS is compatible with model-derived (and more complete) CS, of more of less 1K/W.m-2.

Do you think this is a useful constraint on CS, specially regarding estimations like the one by Chylek et al.( CS=0.3-0.4) ?

James Annan said...

Well, one thing to note about that sort of calculation is that the seasonal climate does not reach equilibrium (note that July/Aug and Jan/Feb are peak and trough in temperature, even though the radiation forcing peaks are late Dec and June). Several people have looked at using the seasonal cycle to estimate sensitivity, and got rather weak results since the basic mechanisms are rather different - the season cycle is strongly influenced by circulation changes.

I also have another comment with ref to Chris' post: I think it is already known that flux corrections in 3D models alter the relationship between seasonal cycle and sensitivity (as they effectively allow the ocean heat uptake to be a new free parameter). I assume the coupled CPDN runs use flux corrections, so it will be interesting to see how Rosalind extends our knowledge about this.

crandles said...

AFAIK Slab model used flux corrections. Coupled models ran for long periods in spin up to get close to equilibrium then a control was used to remove remaining drift. I wonder if this means the coupled models don't have flux corrections?

http://www.climateprediction.net/board/viewtopic.php?t=7808

Any answers to any of the questions there would be gratefully received. (Well by me anyway, don't know about the CPDN team ;) )

James Annan said...

I would be surprised if the coupled models do not have flux corrections, as changing the atmosphere parameters will change the radiation balance so much that the equilibrium climate is awful (the Hadley Centre have certainly taken this approach in their coupled runs). OTOH what I was talking about was some vaguely-remembered stuff probably relating to coupled models both with and without flux corrections, so perhaps does not apply anyway in comparing slab to coupled. As you say, a slab model certainly has flux corrections.

crandles said...

Silly me, I should have remembered Nick's 'Bag of Tricks'

http://www.climateprediction.net/board/viewtopic.php?p=49085#49085

Or Nick's presentation video
http://www.climateprediction.net/science/pubs/OpenDay2006/nick_f.wmv

http://www.climateprediction.net/science/pubs/OpenDay2006/NF_OpenDay2006.pdf

So yes there are flux adjustments and these were derived from the slab runs.

viento said...

"The fundamental problem is that the expression dH/dt = k.T...."

A somewhat similar approximation was used by Rahmstorf (Science 315,368). Instead of dH/dt he uses dSLH/dt, where SLH is sea-level-height. SLH containes also the contribution of glacier melt, but in the 19th and 20th century dSLH/dt should be to a good approximation proportional to dH/dt at short timescales.

Micajah said...

What do you think of the equations recently published by Ferenc Miskolczi?

I confess to knowing so little that I won't even try to describe what his equations may mean, but it seems to have something to do with the effect on temperatures of increasing GHGs.

From what little I've read so far, it seems that a very basic part of the equations for measuring sensitivity to GHG may be wrong. Could that be?

James Annan said...

Hi Micajah,

I've seen that paper, and although it's not my area I get a strong sense that it's nonsense. Unknown authors don't overturn decades of science overnight, and the suggestion that a semi-infinite atmosphere is somehow radically different from one with a definite boundary (which seems to be the main claim), and moreover that no-one has been aware of this up to now, seems implausible to say the least.

Hank Roberts said...

Amusing:

Chylek's 2008 paper has been cited exactly one time to date