Continuing from part one, this post looks at a specific method for estimating TCS (transient climate sensitivity), for any desired year and/or radiative forcing scenario, as predicted by any AOGCM climate model. And some associated topics.

The basic idea was devised by Good et. al (2010, 2011; links at end), and expanded upon by Caldeira and Myhrvold (2013), who fit various equations to the data. The basic idea is fairly simple, but clever, and integrates some nice mathematical solutions/approximations, including Gregory’s linear regression ECS estimation method. The basic idea is simply that if you have an idealized RF pulse or “step” increase (i.e. sudden, one-time increase, as with the instant 4X CO2 (= ~7.4 W/M^2) increase experiment in CMIP5), and run any given AOGCM for say 150-300 years from that point, you can record the temperature course resulting from the pulse, over that time (which will rise toward an asymptote determined by the climate sensitivity). That asymptote will be twice the ECS value (because the CO2 pulse is to 4X, not 2X, CO2). From these data one can fit various curves describing the T trend as a function of time. One then simply linearly scales that response curve to any more realistic RF increase of interest, corresponding to a 1.0% or 0.5% CO2 increase, or whatever. Lastly, if each year’s RF increase is considered as one small pulse, an overlay and summation of the temperature responses from all such, at each year, gives each year’s estimated temperature response, for however long the RF is increasing. The RF increase does not have to stop at any point, although it can. It can also increase or decrease at any rate over time.

The figure below from the paper, illustrate the method and the comparison (Fig. 1 of paper, original caption):

Good et al (2011), did this for nine AOGCMs, testing the method against the results of the CMIP5 1% per year CO2 increase experiment. This is interesting; they are testing whether the basic functional response to an instant, 400% CO2 increase, is similar to that from a 1% per year increase over 140 years. And lo and behold, the overall agreement was very high, both for the collection of models, and individually, for both surface T and heat content. Their Fig. 2 is shown below:

To me, this result is rather astounding, as it says that the time decay of the temperature response to a pulsed RF increase, is highly similar, no matter the magnitude of that increase. That is absolutely not a result I would have expected, given that the thermodynamic interaction between the ocean and the atmosphere is highly important and seemingly not likely to be in phase. Of course, this result does *not* prove this dynamic to be a reality–only that the AOGCM models tested consider, via their encoded physics, that the two responses to be highly similar in form, just differing in magnitude.

Caldeira and Myhrvold (2013) then extended this approach by fitting four different equation forms and evaluating best fits, via Akaike AIC and RMSE criteria. To do this they first used the Gregory ECS estimation method (ref at end) to define the temperature asymptote reached. They don’t give the details of their parameter estimation procedure, which must be some type of nonlinear optimization (and hence open to possible non-ML solutions), since the equation forms they tested were three (inverted) negative exponential forms and one other non-linear form (based on heat diffusion rates in the ocean). They also don’t provide any R^2 data indicating variance accounted for, but their figures (below) demonstrate that for all but one of their model forms (a one-parameter, inverted negative exponential) the fits are extremely good (and extremely similar) across most of the AOGCMs used in CMIP5:

So, both Good et al. (2011, 2012), and Caldeira et al. (2013) provide strong evidence that the physical processes involving surface temperature change, as encoded in AOGCMs, are likely very similar across extremely widely varying radiative forcing increases per unit time, from unrealistically huge, to (presumably) however small. Note that in both cases, a very large percentage (roughly, 40-60%) of the total temperature response (at equilibrium) occurs within the first decade (when normalized to the pulse magnitude). This seems to have implications for the importance of various feedbacks, an issue which is complicated by the fact that some of the models tested are Earth System Models, which include e.g. integrated carbon cycle feedbacks, while others do not. Certainly there will be major potential differences in carbon cycle feedbacks between an earth surface that has just increased 3 degrees C, instantly, versus one that has warmed only a tiny fraction of that amount.

TBC; the next post will demonstrate application to various delta RF scenarios.

**Refs:**

Caldeira and Myhrvold, (2013). Projections of the pace of warming following an abrupt increase in atmospheric carbon dioxide concentration. *Environ. Res. Lett.* 8: 034039, doi:10.1088/1748-9326/8/3/034039.

Good et al., (2011). A step‐response simple climate model to reconstruct and interpret AOGCM projections. *GRL,* 38, L01703, doi:10.1029/2010GL045208

Good et al., (2012). Abrupt CO2 experiments as tools for predicting and understanding CMIP5 representative concentration pathway projections. *Climate Dynamics* (2013) 40:1041–1053 DOI 10.1007/s00382-012-1410-4

Gregory et al., (2004) A new method for diagnosing radiative forcing and climate sensitivity. doi:10.1029/2003GL018747

See also: Hooss et al., (2001). A nonlinear impulse response model of the coupled carbon cycle-climate system (NICCS). *Climate Dynamics* 18.3-4: 189-202.

Bruce:

Bruce here. Wow. Does this mean the over six feet of snow received at Buffalo, NY will melt faster if an unaccounted for forcing constant is incompletely (or incorrectly) modeled?

You have been busy. You might be making all your fellow Bruce’s look bad. But that’s the price we lazy Bruce’s have to pay.

It was a big mistake when they decided to put W New York downwind of Lake Erie but left out the mountains.

Yeah, those dog gone mountains… what were they thinking?

I wonder how much ‘snow fence’ Buffalo would need to prevent this sort of phenomenon?