# Transient temperature change estimation under varying radiative forcing change scenarios, part two

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):

Fig. 1 Illustrating the method. a Global mean temperature evolution in a 4xCO2 step experiment (from the HadCM3 GCM; CMIP5 GCMs give qualitatively similar results). b Reconstruction method for years
1–5 of a 1pctCO2 experiment. Red, yellow, green, blue and purple curves temperature responses estimated for the forcing changes in years 1, 2, 3, 4 and 5 respectively. Each coloured curve is identical (for the case of the 1pctCO2 scenario) and is given by scaling the step experiment temperature response. Black curve reconstructed temperature response, given by the sum of the coloured curves (Eq. 1a).

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:

Fig. 2 Validation against 1 % experiment (all models). a,b Ensemble mean time-series (black GCM simulations, red simple model). c,d ensemble spread: mean over years 111–140 of the GCM simulations (y-axis) against simple model prediction (xaxis) (Each cross represents one GCM). a,c Temperature change/
K; b,d heat uptake/1022 J

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:

Figure 2. Temperature results for CMIP5 models that have performed the abrupt4xCO2 simulations (black dots). Also shown are fits to this data using the functions described in the text: θ1-exp, green; θ2-exp, blue; θ3-exp, brown; θ1D, red. The left vertical axis shows the fraction of equilibrium temperature change (i.e., ΔT/ΔT4×); the right vertical axis indicates the absolute change in global mean temperature. Fit parameters are listed in SOM tables S3–S5 (available at stacks.iop.org/ERL/8/034039/mmedia).

Figure 5. Results from CMIP5 models (black dots) running simulations of the 1pctCO2 protocol. Projections made by simulations based on curve fits to the abrupt4xCO2 simulations as described in the text: θ1-exp, green; θ2-exp, blue; θ3-exp, brown; θ1D, red. All but θ1-exp provide similar approximations to the temperature results for most of the fully coupled, three-dimensional climate model simulations. Note that the GFDL-ESM2G and GFDL-ESM2M models did not continue with increasing atmospheric CO2 content after reaching twice the preindustrial concentration.

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.