**…with new results based on the latest data.**

In my readings, I came across an interesting article by Lin et al. (2010) . The model they develop is only a bit more complicated than the simple energy balance model we’ve previously discussed , as they use the current TOA radiative imbalance, along with the surface temperature change since pre-industrial times, in order to constrain the estimate of sensitivity. In their conclusions, they note:

Since for the modeled climate system (or for the climate variability on time scales about a century) the climate memory is generally within 1 to 10 years, the estimated total climate feedback coefficient f_tot would be in the range of -1.3 to -1.0 W/m^2/K for the estimated 0.85 W/m^2 TOA radiation imbalance. Thus, for the 2×CO2 climate (or 3.7 W/m^2 forcing), the estimated global warming would be in the range between 2.8K to 3.7 K. Since the best estimated memory length of the climate system is about 4 years owing to the time lag of the maximum autocorrelation beyond 0 lag of the GISS surface temperature data, the best estimates of f_m and f_tot would be 4.8 and -1.2 W/m^2/K, respectively, resulting in our estimated most likely warming of 3.1K if the radiation imbalance used is confirmed.

On the one hand, this would seem to be just another bit of evidence that sensitivity is around the 3K. On the other hand, Loeb et. al (2012) recently estimated the decadal TOA Imbalance to be 0.5 (+/- 0.43) W/m^2 (I should note Loeb is one of the Lin et. al co-authors), with that likely value substantially less than the 0.85 W/m^2 used in the 2010 paper. The Loeb et. al (2012) imbalance seems a bit high relative to an OHC-derived estimate, but I don’t have access to the paper to see the exact method. Regardless, I wanted to see what the new "best estimate" would be if I used their model, and simply replaced the 0.85 with 0.5 W/m^2.

### Summary of the model and method

Basically, the heart of Lin et al. is equation (5):

Many of these terms will look familiar relative to the energy balance models we’ve previously discussed. One interesting aspect here is that the single (**λ**) feedback term has now been broken into **f_s** (short-term feedback) and **f_m** (system memory feedback). **f_s **responds to the temperature perturbation from equilibrium (**T**) at the specific time, whereas **f_m** responds to the average temperature perturbation over the length of system memory **(t_m**) leading up to the current time. I want to note that I don’t believe the "short-term" feedback here is necessarily corresponding to the results of something like the Forster and Gregory (2006) analysis (which is also referred to as a "short-term" feedback), since the TOA flux observed will be a combination of the short-term feedback and the system memory feedback. However, Lin et. al (2010) references Spencer and Braswell (2009) as the source of **f_s** so I’m not exactly sure on this point. The other interesting aspect of this equation (actually in eq. 6) is the (1 – **μ**) term, which specifies the percent of radiation trapped in the system (thus assuming that the deep ocean heat uptake is proportional to net radiation at TOA).

The idea is that we can still easily see the ECS from this model. This is because once we stabilize F (at 2xCO2 for example, corresponding to 3.7 W/m^2/K), as **t->∞** the temperature will approach equilibrium to where the current **T** and the average system memory **T** will be equal, so that we can combine **f_s + f_m = f_tot** once again (**f_tot** corresponding to **λ **in the other equation).

Lin et al. use **f_s** = -6.0 W/m^2/K, and suggest 4 years is the best estimate for **t_m**, which I will also assume here. The forcing perturbation (**F**) they assume is linearly rising up to 1.8 w/m^2, and the constraints they place are 0.65 K for **T** after 120 years, and 0.85 (which I will later replace with .5) W/m/^2 as the current TOA imbalance. This means we are left with two unknowns: **f_m** (which is what we actually care about) and **μ**. If we actually cared about **μ**, we would require an accurate value for mixed layer depth (**D**), but since all we really care about is **f_m**, they point out that **μ **will simply compensate for **D** to give the best (**f_m**, **μ**) combination, where **f_m** will be the same regardless of the **D** chosen and only **μ **will change.

### New Results

As I lack some of the fancier math skills, I have used a simple "brute force"/computational method to derive the best estimate for **f_m** (and hence **f_tot**), iterating through logical values for **f_m** and **μ **to find out which pair best matches both our constraints (T_120 and TOA_120).

First, I attempted to reproduce the Lin et. al results, and found that when assuming the 0.85 W/m^2, the "best" estimate for **f_m** was 4.7, yielding a **f_tot** of (-6.0 + 4.7) = -1.3 W/m^2/K, which is pretty close to the Lin et. al result.

When I updated the TOA imbalance constraint to use the new Loeb et. al (2012) value of 0.5 W/m^2, the best estimate for **f_m** becomes 4.1, yielding a **f_tot** of -1.9 W/m^2/K and an estimated sensitivity of (3.7 W/m^2 / 1.9 W/m^2/K) = *1.9K. *

For reference, if we assume an imbalance on the lower side (0.15 W/m^2), we get an **f_m** of 3.5 and a sensitivity of around 1.5K.

Anyhow, here we can see the evolution of temperatures and TOA imbalance in the two cases mentioned above (with the simple linear forcing increase):

Fig 1.

Fig 2.

The first graph should bring home the following point: even if the exact forcing history is known, the 120 years or so of temperature observations *cannot* differentiate between a low ECS (~1K) and a high ECS (~6K).

Paul_K noted this a while ago in an interesting series. The only reason we can attempt to constrain it here is because of the additional (albeit shaky) information about the TOA imbalance. Add in the fact that aerosol forcings are largely unknown (with each GCM uses a different history), and it should be clear that there is no such thing as an "observational" estimate based purely on recent temperature history. The best that can be done is to come up with a model (whether it be a GCM, or something more conceptual like this) that attempts to be physically realistic, and then see if it matches the temperature history without tinkering with the aerosol forcing.

### Limitations of this model

Other than the uncertainty surrounding the assumed parameters and constraints (f_s, TOA imbalance, forcing history, t_m, etc.), and the general simplification of the model, one concern is that inputting the forcing history from GISS does not seem to reproduce temperature record very well, particularly the quick response to volcanoes:

In this case the "system memory" may be too long, as we see the biggest difference is in the beginning of the record, where it takes temperatures in the model until about 1960 to recover from the volcanoes in the early part of the record, whereas the actual GISS temperatures have risen about 0.3 K by that point. Obviously, lowering the system memory feedback (and hence decreasing sensitivity) will mitigate this to some degree, but there’s also the fact that the model seems to assume the whole system is in equilibrium at T=0 (in 1880).

This assumption of equilibrium in 1880 is what I believe to be another limitation, since, based on actual temperature evolution, it seems unlikely that there was no TOA imbalance in 1880 (particularly since the early volcanoes do not create a drop in temperatures). If one were to assume that there was a minor TOA imbalance of ~ 0.25 W/m^2 in 1880, re-doing the same analysis yields an estimated sensitivity of *1.6K.*

### Conclusions

There are few truly “observationally-based” estimates of climate sensitivity (by which I mean those that do not use GCMs), and Foster and Gregory (06) was the only one I believe was present in AR4. FG06 has spawned Murphy et. al(2009) and carries with it plenty of criticism (see my “Radiation and Climate Sensitivity” link to the right, as well as this series by ScienceOfDoom).

Schwartz (2007) provides another such estimate based on the basic energy balance equation discussed, and generally shows a lower sensitivity, although this method has also encountered its share of criticism.

The Lin et. al (2010) is another attempt to diagnose sensitivity from a simple conceptual model + observations. I am curious to see if it is used in support of the likely 3 K estimate for climate sensitivity in IPCC AR5, perhaps in a chart similar to 9.20 in AR4. If so, it would be somewhat awkward – while technically L10 does come up with a likely sensitivity of 3.1K, this is strongly contingent upon the the energy imbalance of 0.85 W/m^2, with subsequent analysis and the future IPCC report (I’m assuming) agreeing that the TOA imbalance for the last decade was well below that. A simple update to this constraint and using the same method would place the sensitivity at < 2K, which is outside of the IPCC AR “likely” range. I may need to register as a reviewer of IPCC chapter 12 to see how this plays out!

Anyhow, my script for this post can be found here.

Troy: I seems to me that we should be able to learn everything we need to know about the depth of the mixed layer from seasonal changes in ocean temperature. With almost ten years of data from Argo, we should be able to follow the temperature change produced by seasonal forcing as it penetrates the upper layer of the ocean. There is probably a seasonal change even at the equator caused by our elliptical orbit.

Comment by Frank — March 4, 2012 @ 1:24 am