By Andy May

The NCAR CERES EBAF satellite dataset has been adjusted to match upper ocean heat content changes. Thus, the EEI (Earth Energy Imbalance) from CERES EBAF (“Clouds and Earth’s Radiant Energy Systems, Energy Balanced and Filled”) is not estimated directly from satellite measurements. If upper ocean heat content were known accurately over a sufficiently long period, this would be fine, but it isn’t.

Even if CERES EBAF numbers are not accurate enough to derive EEI on their own, they are still useful. I’ve been going over the various CERES EBAF variables in some detail lately and noticed that one of the variables (“toa_net_all_mon”) provides the total net incoming radiation at the top of the atmosphere or TOA. All energy exchanged between space and the top of the atmosphere is via radiation so this variable can be used to estimate ECS (the Equilibrium Climate Sensitivity to a doubling of CO₂) if we assume that CO₂ and related anthropogenic greenhouse gases are the dominant factor in warming over the period studied.

Originally, ECS was defined as the ultimate warming due to an instantaneous doubling of atmospheric CO₂ (IPCC, 1992, p. 73). ECS has always been tied to climate models and was explicitly used to compare climate models to one another (IPCC, 1992, p. xxv) & (Gregory et al., 2004). An instantaneous doubling of CO₂ is almost impossible and, in any case, waiting for hundreds or thousands of years to determine the ultimate effect at a new equilibrium state is impractical, but it was a good single-number model metric, and an easy way to rank models from hot to cold.

However, as time went on, this convenient model metric began to be used as a predictive tool for the real world. People confused models with reality and claimed that the average model ECS told us how the world would warm. Attempts have been made to estimate ECS using real world measurements (see here and here), but due to the lack of adequate measurements over long enough time periods these estimates are considered inadequate. It is still difficult to estimate ECS using observations.

I used two approaches, when I applied AR6‑style feedback diagnostics to CERES EBAF observations, the resulting implied ECS values are physically implausible—5.7 to 7.1°C per doubling. Using a more conventional (AR5 and earlier) approach, I get more reasonable values between 1.8 and 2.4. This relatively simple exercise reveals a fundamental mismatch between the AR6 ECS framework and real‑world data.

Defining ECS

To understand what CERES can and cannot tell us, we need to revisit how ECS is currently defined and how the definition shifted after AR5 (IPCC, 2013). The definition of ECS used by the IPCC through AR5 probably originated with Gunnar Myhre (Myhre et al., 1998).

Eq. 1: ΔF_2xCO2 = α • ln(C / C₀)

Where ΔF_2xCO2 is the change in forcing due to a doubling of CO₂. Myhre et al. specifies α = 5.35 and C / C₀ = 2 (a doubling of the CO₂ concentration), so ΔF_2xCO2 is a fixed value of 3.71 W/m². All IPCC reports, including AR6, then define ECS as:

Eq. 2: ECS = -ΔF_2xCO2 /λ (IPCC, 2021, p. 993)

Myhre’s F_2xCO2 value of 3.71 W/m² is nearly identical to the value used in the IPCC AR5 report (IPCC, 2013, p. 818). The value of the net feedback parameter (λ, substituted for the AR6 α) can be determined as the slope of a fitted line through the change in net radiation at the top of the atmosphere (TOA) versus global mean surface temperature. Both F_2xCO2 and λ were fixed global averages before AR6 and ECS was computed from them using equation 2. The concept of ERF, a varying effective radiative forcing, was introduced in AR5, but not used, they still computed ECS with a fixed F, but in AR6 ERF was used and since it continuously varied, it could not be used to directly compute ECS without averaging it. In figure 1 I’ve plotted the CERES top of the atmosphere (TOA) net incoming radiation (both longwave and shortwave) on the y axis and the ERSST v5 SST anomaly on the x axis.

In the conventional scheme ΔF_2xCO2 is defined as a positive number in the sense that doubling CO₂ increases the energy retained in the climate system, thus it is warming. The TOA net radiation (“N” in figure 1) from CERES is also the downward net radiation. The global net feedback parameter (λ, units: W/m²/°C) is the negative of the slope of the best fit line through the CERES EBAF points plotted in figure 1 against the ERSST v5 SST anomaly on the x axis. The pink bands are the 95% confidence interval for the regression.

Van Wijngaarden and Happer derive 3.0 W/m² for ΔF_2xCO2 in the midlatitudes at the TOA and pointed out that the forcing changes with altitude (Wijngaarden & Happer, 2020, Table 3). AR6 “assessed” a value of 3.93 (IPCC, 2021, p. 993) for the period from 1750-2019 and they call it “ERF” or effective radiative forcing. They explain how they get from the AR5 value of 3.71 to 3.93 in Table 7.4 (IPCC, 2021, p. 945). Basically, they believe that the forcing of CO₂ varies with temperature, stratospheric conditions, clouds etc. and try to compensate for these other factors. Using a CERES derived λ of -1.64 from figure 1 and these three values of ΔF_2xCO2, equation 2 gives the ECS values listed in figure 1. They are all between 1.8 and 2.4 and quite reasonable. AR6 “assesses” a λ of -1.16 (AR6, Table 7.10, p 978), which results in an ECS of 3.39 using equation 2.

Strictly speaking, if both F and λ vary independently, as claimed by AR6, it is very hard, if not impossible, to determine ECS from them. Equation 2 only works when F_2xCO2 is a fixed number, which is presumably why AR6 changed the ECS calculation and changed “F” to “ERF” or “effective radiative forcing” (IPCC, 2021, pp. 959, 1005). For a description of the new AR6 ECS calculation see (Sherwood et al., 2020) and for a critique of the method see (Lewis, 2023). They also introduce an “Effective Equilibrium climate sensitivity,” which is still is the surface temperature response to a doubling of CO₂, but can be different from ECS. “ECS” has become very bewildering.

The global mean ERSST v5 global sea surface temperature anomaly is related to ocean heat content or OHC, which is used to tune the CERES EBAF TOA net radiation measurement plotted on the y axis in figure 1. Thus, the two numbers plotted are not completely independent of one another. Both means are area-weighted by latitude and only values from populated ERSST v5 cells are used, thus land (~29% of the Earth) is ignored in this study. The CERES grid is a 1°x1° latitude/longitude grid, but it was aggregated to match the 2°x2° ERSST grid. A plot of the slopes, that is the change in downward total radiation flux from the TOA per the change in SST, for each 2°x2° grid cell, is shown in figure 2.

Since ΔF_2xCO2 in equation 2 is fixed, for any given value, ECS is a function of λ (the slope mapped in figure 2 is -λ). Figure 2 shows that both λ and ECS vary a lot with location, AR6 does a “spatial pattern effect” analysis to try and use these anomalies to determine both ERF (effective radiative forcing) and λ (IPCC, 2021, Section 7.3, page 941) & (Gregory et al., 2004). Normally, these studies involve comparing model results to observations spatially over the oceans because land observations are usually much more erratic (relative to model results) than ocean observations (IPCC, 2021, p. 942), although the eastern Pacific off South America is always a problem since it is cooling in recent years and the models predicted significant warming (IPCC, 2021, p. 990).

AR6 does consider observations. They study model-observation differences to evaluate model quality and to determine effective radiative forcing and λ. The large areal variability shown in figure 2 is problematic and may invalidate global λ estimates (including mine) over short periods of time. It suggests internal variability dominates the signal, at least over the period from 2001-2025, this is reinforced by known long-term ocean oscillations like the AMO. Such long-term internal variability, if not taken into account, may invalidate the AR6 pattern-effect methodology.

The Gregory Plot

Gregory et al. first described the new model-based “effective” forcing and climate sensitivity ideas used extensively in AR6 (Gregory et al., 2004). Gregory et al. devised a plot, since dubbed the “Gregory plot,” to evaluate model results. The plot is of modeled surface temperature versus the modeled change in downward flux where the intercept reflects ERF and the slope reflects the net feedback. I was curious what it would look like with real data, as opposed to model output. The plot is shown with CERES EBAF data and ERSST v5 SST data in figure 3.

Gregory et al. allows “ΔF_2xCO2” or “F” to vary, it is still positive downward and here taken as Forster’s “Best ERF” (Forster et al., 2023). The standard Gregory equation is:

Eq. 3: N = F – λΔT

Where:

• N is the net downward TOA flux (CERES, positive downward)
• F is the effective radiative forcing (Forster “Best ERF”, positive downward)
  

The version of the equation plotted is:

Eq. 4: N – F = -λΔT

In figure 3 the slope is λ and the sign does not have to be reversed. Other than this the only real difference between this plot and figure 1 is that F varies and is populated with the Forster et al. ERF values. Forster et al. constructed their ERF values following the AR6 “methods as closely as possible” (Forster et al., 2023).

The surface warming feedback factor (λ) is the slope from a regression of equation 4 in figure 3. The key problem is that if we assume an F of 3.71 (Myhre and AR5) and couple it with the Gregory AR6 compliant λ = -0.523, equation 2 gives us an ECS of 7.1 °C/2xCO₂, not very realistic. The Gregory method does not allow us to compute ECS directly because the ERF (effective forcing due to a doubling of CO₂) varies, but we can plug in a reasonable fixed F as a reality check, and when we do the resulting ECS is unreasonable and not what we observe in the real world.

Figure 3 overlays two Gregory plots. The red line is constructed so that the coefficient of F is forced to be one, which is the AR6 convention. In AR6 they assume that N = F + λT, in other words, that the net radiation at the top of the atmosphere is ERF + the feedback factor times the change in surface temperature. This is a physical law for blackbodies, but not necessarily for Earth over short (<100 years) periods of time where there are a lot of complicating factors involved, the most obvious being climate oscillations like ENSO and the AMO. The specific AR6-constrained regression (red line) is (N-F) on SST, which results in a λ of -0.523.

The blue line in figure 3 is a full unrestrained regression and tests the coefficient when both F and T are allowed to float, it results in a λ of -0.645. This full regression slope and the AR6-constrained slope are not significantly different statistically as shown by the confidence intervals in figure 3. Unfortunately, neither number passes a basic reality check. If we choose a Myhre/AR5 fixed F = 3.71 and couple it with the AR6 compliant λ value of -0.523 W/m²/°C we get an ECS of 7.1 °C/2xCO₂. Doing the same for the floating regression we get 5.75 °C/2xCO₂ using equation 2. Neither of these “reality-check ECS” values are reasonable.

AR6 did not do these calculations, instead they independently estimated the average 1750-2019 forcing as 3.93 W/m² and the feedback (our λ and their α) to be -1.16, which results in an ECS of 3.39 (IPCC, 2021, p. 993). The AR6 λ of -1.16 is almost twice the CERES Gregory derived λ of -0.523 and much less than the conventional CERES λ of -1.64 (figure 1), which is difficult to explain.

Discussion

The model results in Gregory et al. look nothing like figure 3. Neither do the “idealized 2xCO₂ response” plots in AR6 (IPCC, 2021, Ch. 7, Box 7.1, p 932). It could be that the time period used (2001-2025) is too short, but it is only five years short of the normal 30-year definition of a climate period. However, the very important AMO climate oscillation is 60-70 years long and we are currently at an AMO warming peak (May & Crok, 2024), so things may change radically over the next few decades.

CERES EBAF is not accurate enough on an absolute basis to measure the EEI but is probably directionally correct. Thus, the trend shown in figure 1 from 2001-2025 should be close to correct. The variable ERF used in figure 3 is problematic, it results in an unreasonable λ and may be conceptually flawed. The AR6 changes in the definition of ECS, λ, and ΔF_2xCO2 have muddied the water and, in my opinion, unnecessarily complicate their story. Further, by their own admission, the changes did not help, and the AR6 results have moved farther away from observations than in AR5 (IPCC, 2021, pp. 443-444) and here. It appears they are headed off in the wrong direction.

Works Cited

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May, A., & Crok, M. (2024, May 29). Carbon dioxide and a warming climate are not problems. American Journal of Economics and Sociology, 1-15. https://doi.org/10.1111/ajes.12579

Myhre, G., Highwood, E. J., Shine, K. P., & Stordal, F. (1998). New estimates of radiative forcing due to well mixed greenhouse gases. Geophysical Research Letters, 25(14), 2715-2718. https://doi.org/10.1029/98GL01908

Sherwood, S. C., Webb, M. J., Annan, J. D., Armour, K. C., J., P. M., Hargreaves, C., . . . Knutti, R. (2020, July 22). An Assessment of Earth’s Climate Sensitivity Using Multiple Lines of Evidence. Reviews of Geophysics, 58. https://doi.org/https://doi.org/10.1029/2019RG000678

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