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**Research article**
17 Apr 2018

**Research article** | 17 Apr 2018

The influence of internal variability on Earth's energy balance framework and implications for estimating climate sensitivity

^{1}Dept. of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA^{2}Max Planck Institute for Meteorology, Bundesstraße 53, 20146 Hamburg, Germany

^{1}Dept. of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA^{2}Max Planck Institute for Meteorology, Bundesstraße 53, 20146 Hamburg, Germany

**Correspondence**: Andrew E. Dessler (adessler@tamu.edu)

**Correspondence**: Andrew E. Dessler (adessler@tamu.edu)

Abstract

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Our climate is constrained by the balance between solar energy absorbed by the Earth and terrestrial energy radiated to space. This energy balance has been widely used to infer equilibrium climate sensitivity (ECS) from observations of 20th-century warming. Such estimates yield lower values than other methods, and these have been influential in pushing down the consensus ECS range in recent assessments. Here we test the method using a 100-member ensemble of the Max Planck Institute Earth System Model (MPI-ESM1.1) simulations of the period 1850–2005 with known forcing. We calculate ECS in each ensemble member using energy balance, yielding values ranging from 2.1 to 3.9 K. The spread in the ensemble is related to the central assumption in the energy budget framework: that global average surface temperature anomalies are indicative of anomalies in outgoing energy (either of terrestrial origin or reflected solar energy). We find that this assumption is not well supported over the historical temperature record in the model ensemble or more recent satellite observations. We find that framing energy balance in terms of 500 hPa tropical temperature better describes the planet's energy balance.

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Dessler, A. E., Mauritsen, T., and Stevens, B.: The influence of internal variability on Earth's energy balance framework and implications for estimating climate sensitivity, Atmos. Chem. Phys., 18, 5147–5155, https://doi.org/10.5194/acp-18-5147-2018, 2018.

1 The problem

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When an energy imbalance is imposed, such as by adding a greenhouse gas to the atmosphere, the climate will shift in such a way to eliminate the energy imbalance. This process is embodied in the traditional linearized energy balance equation:

$$\begin{array}{}\text{(1)}& {\displaystyle}{\displaystyle}R=F+\mathit{\lambda}{T}_{\mathrm{S}},\end{array}$$

where the forcing *F* is an imposed energy imbalance, *T*_{S} is the global average surface temperature, and
*λ* relates changes in *T*_{S} to a change in net top-of-atmosphere (TOA) flux (Gregory et al., 2002; Dessler
and Zelinka, 2014). *R* is the resulting TOA flux imbalance from the combined forcing and response. All quantities are
deviations from an equilibrium base state, usually the pre-industrial climate. Equilibrium climate sensitivity (hereafter
ECS, the equilibrium warming in response to a doubling of CO_{2}) is equal to $-{F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}/\mathit{\lambda}$,
where ${F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}$ is the forcing from doubled CO_{2}.

Many investigators (e.g., Gregory et al., 2002; Annan and Hargreaves, 2006; Otto et al., 2013; Lewis and Curry, 2015;
Aldrin et al., 2012; Skeie et al., 2014; Forster, 2016) have used Eq. (1) combined with estimates of *R*, *F*, and
*T*_{S} to estimate *λ*:

$$\begin{array}{}\text{(2)}& {\displaystyle}{\displaystyle}\mathit{\lambda}=\mathrm{\Delta}(R-F)/\mathrm{\Delta}{T}_{\mathrm{S}},\end{array}$$

where Δ indicates the change between the start of the historical period (usually the mid- to late
19th century) and a recent period. These calculations result in values of *λ* near −2 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ and
appear to rule out ECS larger than ∼ 4 K (Stevens et al., 2016). The substantial likelihood of an ECS below
2 K implied by these calculations led the IPCC Fifth Assessment Report to extend their lower bound on
*likely* values of ECS to 1.5 K (Collins et al., 2013).

We test this energy balance methodology through a perfect model experiment consisting of an analysis of a 100-member
ensemble of runs of the Max Planck Institute Earth System Model, MPI-ESM1.1. This is the latest coupled climate model from the Max Planck
Institute for Meteorology and consists of the ECHAM6.3 atmosphere and land model coupled to the MPI-OM ocean model. The
atmospheric resolution is T63 spectral truncation, corresponding to about 200 km, with 47 vertical levels, whereas
the ocean has a nominal resolution of about 1.5^{∘} and 40 vertical levels. MPI-ESM1.1 is a bug-fixed and improved
version of the MPI-ESM used during the fifth phase of the Coupled Model Intercomparison Project
(CMIP5; Giorgetta et al., 2013) and nearly identical to the MPI-ESM1.2 model being used to provide output to CMIP6, except that the historical forcings are from the MPI-ESM. Each
of the 100 members simulates the years 1850–2005 (Fig. 1) and uses the same evolution of historical natural and
anthropogenic forcings. The members differ only in their initial conditions – each starts from a different state sampled
from a 2000-year control simulation.

We calculate effective radiative forcing *F* for the ensemble by subtracting top-of-atmosphere flux *R* in a run with
climatological sea surface temperatures (SSTs) and a constant pre-industrial atmosphere from average *R* from an ensemble
of three runs using the same SSTs but the time-varying atmospheric composition used in the historical runs (Hansen et al.,
2005; Forster et al., 2016). The three-member ensemble begins with perturbed atmospheric states. We estimate
${F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}$ using the same approach in a set of fixed-SST runs in which CO_{2} increases at 1 % per
year, which yields a ${F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}$ value of 3.9 W m^{−2}.

We calculate *λ* using Eq. (2) for each ensemble member, producing values ranging from −1.88 to
−1.01 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (5–95 % range −1.63 to −1.17 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$), with an ensemble
median of −1.43 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (Fig. 2a). In this calculation, Δ(*R*−*F*) and Δ*T*_{S} are
the average difference between the first and last decade of each run. The spread in *λ* depends to some extent on
how the calculation is set up – if one used the difference between the averages of the first and last 20 years, for
example, the range in *λ* declines from 0.87 to 0.48 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$. Using longer averaging periods does
not further decrease the range.

We also calculate ECS = $-{F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}/\mathit{\lambda}$ for each ensemble member, producing values ranging from 2.08
to 3.87 K (5–95 % range 2.39 to 3.34 K) (Fig. 2b), with an ensemble median of 2.72 K. Thus,
our analysis shows that *λ* and ECS estimated from the historical record can vary widely simply due to internal
variability. Given that we have only a single realization of the 20th century, we should not consider estimates based on
the historical period to be precise – even with perfect observations. This supports previous work that also emphasized
the impact of internal variability on estimates of *λ* and ECS (Huber et al., 2014; Andrews et al., 2015; Zhou
et al., 2016; Gregory and Andrews, 2016).

Previous researchers have questioned whether the historical record provides an accurate measure of *λ* and ECS, and
we can check this by comparing the ensemble values to ECS estimates from a 2×CO_{2} run of the MPI-ESM1.2,
which is physically very close to MPI-ESM1.1. An abrupt 2×CO_{2} run yields an ECS of 2.93 K in
response to an abrupt doubling of CO_{2} (estimated by regressing years 100–1000 of a 1000-year run) – 8 %
larger than the ensemble median. This is in line with the 10 % difference in ECS estimated by Mauritsen and Pincus (2017) to arise from the average CMIP5 model
time-dependent feedback but smaller than suggested in other recent studies of ECS in transient climate runs (e.g.,
Armour, 2017; Proistosescu and Huybers, 2017).

Thus, there are a number of issues that need to be considered when interpreting estimates of *λ* and ECS derived
from the historical period. In addition to the precision and accuracy issues discussed above, it also includes the large
and evolving uncertainty in forcing over the 20th century (Forster, 2016), different forcing efficacies of greenhouse
gases and aerosols (Shindell, 2014; Kummer and Dessler, 2014), and geographically incomplete or inhomogeneous observations
(Richardson et al., 2016).

2 Why are estimates using the traditional energy balance approach imprecise?

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In this section, we explain the physical process by which internal variability leads to the large spread in *λ* and
ECS estimated from the ensemble. We begin by observing that Eqs. (1) and (2) parameterize *R*−*F* in terms of global average
surface temperature, *T*_{S}. In model runs with strong forcing driving large warming, such as abrupt
4×CO_{2} simulations, there is indeed a strong correlation between these variables (e.g., Gregory et al.,
2004). However, because *R*−*F* in such runs is dominated by a monotonic trend, correlations will exist with any geophysical
field that also exhibits a monotonic trend, regardless of whether there is a physical connection between the fields. Thus,
one should not take the correlation between *R*−*F* and *T*_{S} in these runs as proving causality.

If *T*_{S} is a good proxy for the response *R*−*F*, we would expect to also see a correlation in measurements
dominated by interannual variations. Observational data allow us to test this hypothesis. We use observations of *R* from
the Clouds and the Earth's Radiant Energy System (CERES) Energy Balanced and Filled product (ed. 4) (Loeb et al., 2018),
which cover the period March 2000 to July 2017. Our sign convention throughout the paper is that downward fluxes are
positive. Temperatures come from the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim)
(Dee et al., 2011). We assume forcing changes linearly over this time period and account for it by detrending Δ*R*
and Δ*T* anomaly time series using a linear least-squares fit to remove the long-term trend.

These data show that Δ*R* is poorly correlated with Δ*T*_{S} in response to interannual variability
(Fig. 3a), as has been noted many times in the literature; see, e.g., Sect. 5 of Forster (2016). In particular, the low
correlation coefficient tells us that Δ*T*_{S} explains little of the variance in Δ*R*. Using
explicit estimates of forcing or other temperature data sets (e.g., MERRA-2) yields the same result.

Global climate models that submitted output to CMIP5 (Taylor
et al., 2012) also show this poor correlation. To demonstrate this, we have calculated the correlation coefficient between
Δ*T*_{S} and Δ*R* in CMIP5 pre-industrial control runs (these are runs for which forcing *F*=0). To
facilitate comparison with the CERES data, as well as avoid any issues with long-term drift in the control runs, we break
each run into 17-year segments to match the length of the CERES data and calculate the correlation coefficient of
monthly anomalies of Δ*R* and Δ*T*_{S} for each segment. Figure 4 shows that the correlation between
Δ*R* and Δ*T*_{S} in the models is similar to that from the CERES analysis.

Recent work provides an explanation: the response of Δ(*R*−*F*) to a particular Δ*T*_{S} is determined
not only by the global average magnitude but also by the pattern of warming (Armour et al., 2013; Andrews et al., 2015;
Gregory and Andrews, 2016; Zhou et al., 2016, 2017; Andrews and Webb, 2018). During El Niño cycles that dominate the
observations in Fig. 3, the spatial pattern of warm and cool regions changes, leading to responses in Δ(*R*−*F*) that
do not scale cleanly with Δ*T*_{S} – something Stevens et al. (2016) refer to as “pattern effects”.

To demonstrate how this also generates the spread in *λ* in the model ensemble (Fig. 2a), we calculate the local
response *λ*_{r} in three equal-area regions (90–19.4^{∘} S,
19.4^{∘} S–19.4^{∘} N, 19.4–90^{∘} N). We define *λ*_{r} as the regional
analog to *λ* (Eq. 2):

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}{\mathit{\lambda}}_{\mathrm{r}}=\mathrm{\Delta}(R-F{)}_{\mathrm{r}}/\mathrm{\Delta}{T}_{\mathrm{S},\mathrm{r}},\end{array}$$

where the “r” subscript indicates a regional average value.

We find that *λ*_{r} varies between the regions (Fig. 5). This means that different ensemble members with
similar global average Δ*T*_{S} but different patterns of surface warming produce different values of
global average Δ(*R*−*F*), thereby leading to spread in the estimated *λ* among the ensemble members. We also
see strong variability in *λ*_{r} within each region, suggesting that how the warming is distributed within
the region also drives some of the spread in estimated *λ* in the ensemble.

This explanation is consistent with analyses showing that *λ* changes during transient runs as the pattern of
surface temperature evolves (Senior and Mitchell, 2000; Armour et al., 2013; Andrews et al., 2015; Gregory and Andrews,
2016; Stevens et al., 2016). In our model ensemble, however, the pattern changes are caused by internal variability rather
than differing regional heat capacities that cause some regions to warm more slowly than others during forced warming.

3 A better way to describe energy balance

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Our analysis demonstrates limitations of the conventional energy balance framework (Eq. 1). It has been
previously noted that Δ*R* correlates better with tropospheric temperatures than Δ*T*_{S} (Murphy,
2010; Spencer and Braswell, 2010; Trenberth et al., 2015). Other analyses have also stressed the importance of
atmospheric temperatures – through its influence on lapse rate – as providing a fundamental control on the planet's
energy budget (Zhou et al., 2016; Ceppi and Gregory, 2017). Based on this, we test a new energy balance framework
constructed using the temperature of the tropical atmosphere:

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle}R-F=\mathrm{\Theta}{T}_{\mathrm{A}},\end{array}$$

where *T*_{A} is the tropical average (30^{∘} N–30^{∘} S) 500 hPa temperature and Θ
relates this quantity to *R*−*F*. *R* and *F* are the same global average quantities they were in Eq. (1). ECS can
be expressed in terms of Θ:

$$\begin{array}{}\text{(5)}& {\displaystyle}{\displaystyle}\text{ECS}\phantom{\rule{0.25em}{0ex}}=-{\displaystyle \frac{\mathrm{\Delta}{F}_{\mathrm{2}\times {\mathrm{CO}}_{\mathrm{2}}}}{\mathrm{\Theta}}}{\displaystyle \frac{\mathrm{\Delta}{T}_{\mathrm{S}}}{\mathrm{\Delta}{T}_{\mathrm{A}}}},\end{array}$$

where Δ*T*_{S} and Δ*T*_{A} are the equilibrium changes in these quantities in response to
doubled CO_{2}. The CMIP5 ensemble average ratio Δ*T*_{S}∕Δ*T*_{A} is 0.86 ± 0.10
(±1*σ*), where Δ represents the average difference between the first and last decades of the abrupt
4×CO_{2} runs.

Support for Eq. (4) can be found in the observations: Δ*R* shows a tighter correlation with Δ*T*_{A} than with Δ*T*_{S} in observations (Fig. 3a vs. Fig. 3b). CMIP5 models also show this
(Fig. 4). Given that the slope of these plots can be taken as estimates of Θ and *λ*, the tighter correlation
leads to more accurate estimates of Θ than *λ*, both in absolute and relative terms.

Turning to the model ensemble, we next demonstrate that Θ is a more precise metric than *λ*. We do this by
calculating Θ [$=\mathrm{\Delta}(R-F)/\mathrm{\Delta}{T}_{\mathrm{A}}$] in each ensemble member, yielding values ranging from −1.18
to −0.89 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (5–95 % range −1.16 to −0.92 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$), with an ensemble
median of −1.04 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (Fig. 2a). There is clearly less variability in Θ among the ensemble
members than for *λ*. This reflects less variability in the regional response Θ_{r} ($=\mathrm{\Delta}(R-F{)}_{\mathrm{r}}/\mathrm{\Delta}{T}_{\mathrm{A},\mathrm{r}}$) than in *λ*_{r} (Fig. 5), as well as less variability
within the regions. We therefore conclude that interannual variability has less of an impact on Θ than *λ*.

We can also reproduce this in a 2000-year control run (a run with fixed pre-industrial boundary conditions) of the
MPI-ESM1.1 model. Figure 6 shows *λ* calculated in a sliding 17-year window and confirms significant temporal
variability in *λ*. We can similarly calculate Θ, and the figure shows that temporal variability in Θ
is substantially smaller.

This result is also reproduced in the CMIP5 control models. Figure 7 plots the standard deviation (SD) of each CMIP5 model's set of
short-term *λ* divided by the SD of that model's set of short-term Θ (as described previously, we calculate
time series of short-term *λ* and Θ values for each model by regressing anomalies in a 17-year sliding
window of the control runs). All of the models fall above 1, demonstrating that there is less variability in the Θ
time series than in the *λ* time series in every climate model. This confirms that Θ is more robust with
respect to internal variability than *λ*. It also suggests that Θ estimated from the satellite data (Fig. 3)
should be considered a better estimate of the climate system's long-term value than *λ* estimated from the same data
set.

As far as accuracy goes, we can compare Θ in the ensemble over the historical period to Θ in response to
much larger warming. The ensemble median of Θ from the historic period (Fig. 2),
−1.04 ± 0.01 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (5–95 % confidence interval), is close to the value obtained from an
analysis of the first 150 years of an abrupt 4×CO_{2} run of the same model,
Θ = −1.03 ± 0.04 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$, as well as Θ calculated from all 2600 years of
this run, Θ = −1.00 ± 0.01 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (values from the 4×CO_{2} runs are all
obtained using the Gregory method (Gregory et al., 2004) using annual average *R* and temperatures). On the other
hand, *λ* changes substantially in the 4×CO_{2} run as the climate warms:
*λ* = −1.36 ± 0.07 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ when calculated from the first 150 years,
but *λ* = −0.95 ± 0.01 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ from all 2600 years of that run.

We can verify this result in the CMIP5 abrupt 4×CO_{2} ensemble. It has been previously demonstrated that
plots of *R*−*F* vs. *T*_{S} do not trace straight lines as the climate warms (Andrews et al., 2015; Rugenstein
et al., 2016; Rose and Rayborn, 2016; Armour, 2017), so *λ* and ECS calculated in a single model run may depend on
the portion of the run selected. In the CMIP5 abrupt 4×CO_{2} ensemble, for example, average *λ*
calculated by regressing years 10–30 (*λ*_{10−30}) is more negative than *λ* calculated from years 30–150
(*λ*_{30−150}) by 0.49 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (Fig. 8).

Several explanations for this have been advanced, most prominently that *λ* is a function of the pattern of surface
warming (Senior and Mitchell, 2000; Armour et al., 2013; Andrews et al., 2015; Gregory and Andrews, 2016; Zhou et al.,
2016; Stevens et al., 2016). Using Θ largely eliminates this pattern effect: Θ_{10−30} and Θ_{30−150} have an average difference of 0.13 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ for the CMIP5 ensemble (Fig. 8). Thus, we find
additional evidence that Θ tends to be similar for different amounts and patterns of warming.

The lack of curvature in the Θ calculations means there is curvature in the relation between *T*_{A} and
*T*_{S} in the models. Thus, the pattern effect's impact on ECS calculations shifts from *λ* in the
traditional framework to the Δ*T*_{S}∕Δ*T*_{A} term in Eq. (4). This also emphasizes
the need to improve our understanding of the factors that control Δ*T*_{S}∕Δ*T*_{A}, as well
as how future patterns of surface warming will evolve.

There are several plausible reasons why *T*_{A} may control *R* better than *T*_{S}. It seems likely
that several of the feedbacks – e.g., lapse rate, water vapor, or longwave cloud – should be more strongly influenced by
atmospheric temperatures than *T*_{S}. More recently, it has been shown that atmospheric temperatures also play
a key role in regulating low clouds (Zhou et al., 2016, 2017), thereby influencing the shortwave cloud feedback. This is
also consistent with Ceppi and Gregory (2017), who identified a dependence of ECS on atmospheric stability in models. We
have not further investigated this – ultimately, our use of *T*_{A} in Eq. (4) is based on observations
(Murphy, 2010; Spencer and Braswell, 2010; Trenberth et al., 2015) that it correlates well with *R*. Other metrics, such
as global average atmospheric temperature, work almost as well. Clearly, further investigations on how to best describe the
Earth's energy balance are warranted.

Finally, one of our ultimate goals for this revised framework is to help produce better estimates of ECS. We are working
on a detailed analysis of ECS based on this framework, which is presently in review (Dessler and Forster, 2018), but we briefly show here
how the advantages of the revised energy balance framework may be leveraged to do this. Figure 9a shows Θ
calculated from control runs of 25 CMIP5 models. To calculate Θ in the control runs, we break each control run into
17-year segments and calculate monthly anomalies of Δ*R* and Δ*T*_{A} during each segment. Then, we
calculate Θ for each segment as the slope of the regression of Δ*R* vs. Δ*T*_{A} for that
segment. Thus, for each control run, we generate a large number of estimates of Θ. The value in Fig. 9a is the
average of these individual values.

Figure 9b shows the ECS of these models, calculated from the first 150 years of the abrupt 4×CO_{2} runs
using the Gregory method. If we assume that models with more accurate simulation of short-term Θ produce more
accurate estimates of ECS (Brown and Caldeira, 2017; Wu and North, 2002), then we can use Fig. 9a and b to constrain
ECS. We find that the 15 models whose average short-term Θ falls within the uncertainty of Θ estimated from
CERES observations have ECS values ranging from 2.0 to 3.9 K, with an average of 2.9 K. This excludes many
of the highest-ECS models, a result consistent with other analyses (Cox et al., 2018; Lewis and Curry, 2015).

It would not have been possible to draw this conclusion with the conventional energy balance framework. Figure 9c shows
the comparison between *λ* from the control runs (calculated the same way Θ was calculated) and CERES
observations. Because of the much larger uncertainty in the observational estimate of short-term *λ*, almost all
models fall within the observational range, thereby prohibiting any constraint on the ECS range.

It may also be possible to use the relation between short-term and long-term Θ as an emergent constraint to convert short-term observations to the long-term response. There is some scatter in the relation in the CMIP5 ensemble, however, so more analysis of how these are related is likely required before ECS can be constrained in this way.

4 Conclusions

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We have estimated ECS in each of a 100-member climate model ensemble using the same energy balance constraint used by many investigators to estimate ECS from 20th-century historical observations. We find that the method is imprecise – the estimates of ECS range from 2.1 to 3.9 K (Fig. 2), with some ensemble members far from the model's true value of 2.9 K. Given that we only have a single ensemble of reality, one should recognize that estimates of ECS derived from the historical record may not be a good estimate of our climate system's true value.

The source of the imprecision relates to the construction of the traditional energy balance equation (Eq. 1). In
it, the response of TOA net flux (*R*−*F*) is parameterized in terms of global average surface temperature
(*T*_{S}). Recent research has suggested that the response is not just determined by the magnitude of
*T*_{S} but also includes other factors, such as the pattern of *T*_{S} (e.g., Armour et al., 2013; Andrews
et al., 2015; Gregory and Andrews, 2016; Zhou et al., 2017) or the lapse rate (e.g., Zhou et al., 2017; Ceppi and Gregory,
2017; Andrews and Webb, 2018). As a result, two ensemble members with the same Δ*T*_{S} can have different
climate responses, Δ(*R*−*F*), leading to spread in the inferred *λ*.

The lack of a direct relationship between *T*_{S} and radiation balance suggests that it may be profitable to
investigate alternative formulations. We test parameterizing the response in terms of 500 hPa tropical
temperature (Eq. 4) and find that it is superior in many ways. Ultimately, how investigators describe the energy
balance of the planet will depend on the problem and the available data. The surface temperature is indeed special, so the
traditional framework may be preferred for some problems. But investigators may find that the alternatives are superior
for certain problems, for instance constraining Earth's climate sensitivity.

Code and data availability

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Code and data availability.

The data set for the code is availeble at http://dx.doi.org/10.5281/zenodo.1206285, and the data set for the data is available at http://dx.doi.org/10.5281/zenodo.1206267.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was supported by NSF grant AGS-1661861 to Texas A&M University. This work was initiated while
Andrew E. Dessler was on faculty development leave from Texas A&M during the fall of 2016; he thanks Texas A&M and the
Max-Planck-Institut für Meteorologie for supporting this research. Computational resources were made available by
Deutsches Klimarechenzentrum (DKRZ) through support from German Federal Ministry of Education and Research (BMBF) and by
the Swiss National Supercomputing Centre (CSCS).

The article processing charges for this open-access

publication were covered by the Max Planck Society.

Edited by: Amanda Maycock

Reviewed by: two anonymous referees

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Short summary

One of the most important parameters in climate science is the equilibrium climate sensitivity (ECS). Estimates of this quantity based on 20th-century observations suggest low values of ECS (below 2 °C). We show that these calculations may be significantly in error. Together with other recent work on this problem, it seems probable that the ECS is larger than suggested by the 20th-century observations.

One of the most important parameters in climate science is the equilibrium climate sensitivity...

Atmospheric Chemistry and Physics

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