Controls of the transient climate response to emissions by physical feedbacks, heat uptake and carbon cycling

The increase in radiative forcing, ΔF(t), drives an increase in planetary heat uptake, N(t), plus a radiative response, ΔR(t), which is assumed to be equivalent to the product of the increase in global-mean surface air temperature, ΔT(t), and the climate feedback parameter, λ(t) (Gregory et al 2004, Knutti and Hegerl 2008, Andrews et al 2012, Forster et al 2013) by

$$\begin{equation} \Delta F(t) = \Delta R(t) +N(t) \equiv \lambda(t) \Delta T(t) +N(t) , \end{equation} \tag{ 5 }$$

where N(t) is the planetary heat flux into the climate system, which is dominated by the heat uptake by the ocean (Church et al 2011), and ΔF(t) is defined as positive into the ocean.

The dependence of surface warming on radiative forcing, ΔT(t)/ΔF(t), in (3) is then directly connected to the product of the inverse of the climate feedback, λ(t)⁻¹, and the planetary heat uptake divided by the radiative forcing, N(t)/ΔF(t),

$$\begin{equation} \frac{\Delta T(t)}{\Delta F(t)} = \frac{1}{\lambda(t)} \left( 1 - \frac{ N(t)}{\Delta F(t)} \right) , \end{equation} \tag{ 6 }$$

where (1 − N(t)/ΔF(t)) represents the fraction of the radiative forcing that warms the surface, rather than the ocean interior.

The dependence of radiative forcing on cumulative carbon emissions, ΔF(t)/I_em(t), in (3) may be expressed in terms of changes in the radiative forcing on changes in atmospheric CO₂, ΔF(t)/ΔI_atmos(t), and the airborne fraction, $\Delta I_{atmos}(t)/I_{em}(t)$ (Ehlert et al 2017),

$$\begin{equation} \frac{\Delta F(t)}{I_{em}(t)} = \left( \frac{\Delta F(t)}{\Delta I_{atmos}(t)} \right)\left( \frac{\Delta I_{atmos}(t)}{I_{em}(t)} \right) . \end{equation} \tag{ 7 }$$

The airborne fraction, $\Delta I_{atmos}(t)/I_{em}(t)$, is related to the changes in the oceanborne and landborne fractions (Jones et al 2013),

$$\begin{equation} \frac{\Delta I_{atmos}(t)}{I_{em}(t)} = 1 - \left( \frac{\Delta I_{ocean}(t)}{I_{em}(t)} + \frac{\Delta I_{ter}(t)}{I_{em}(t)} \right) , \end{equation} \tag{ 8 }$$

where the changes in the ocean and terrestrial inventories are denoted by ΔI_ocean(t) and ΔI_ter(t) respectively.

The sensitivity of the radiative forcing on atmospheric CO₂, ΔF(t)/ΔI_atmos(t), saturates with increasing atmospheric CO₂, with the radiative forcing represented by a logarithmic dependence,

$$\begin{equation} \Delta F(t) = a \ln( \mathrm{CO}_2(t)/\mathrm{CO}_2(t_0)), \end{equation} \tag{ 9 }$$

where a is a radiative forcing coefficient for CO₂ (Myhre et al 1998) and t₀ is the time of the pre-industrial. During emissions, the decrease in the sensitivity of the radiative forcing on atmospheric CO₂, ΔF(t)/ΔI_atmos(t), may equivalently be viewed in terms of the ocean acidifying with increasing atmospheric CO₂ and decreasing the amount of saturated carbon that the ocean can hold (Katavouta et al 2018).

Next we apply these theoretical relations (3) to (9) to understand how the TCRE is controlled.

Our analyses are applied to 9 CMIP6 and 7 CMIP5 models (table 1), which have been forced by an annual 1% rise in atmospheric CO₂ for 140 years starting from a pre-industrial control, following the 1pctCO2 experimental protocol (Eyring et al 2016). All the thermal and carbon diagnostics are performed on the forced model response with the unforced pre-industrial control (piControl) subtracted off to minimise model drift.

The Earth system models couple the carbon cycle and climate together through sources and sinks of CO₂ being affected by atmospheric CO₂ and the change in climate (Ciais et al 2014). The sum of the changes in the atmospheric, ocean and terrestrial inventories of carbon balance the implied cumulative carbon emission, I_em(t) in PgC,

$$\begin{equation} I_{em}(t) = \Delta I_{atmos}(t) + \Delta I_{ocean}(t) + \Delta I_{ter}(t), \end{equation} \tag{ 10 }$$

where the inventory changes are evaluated from the time integral of the air-sea and air-land carbon fluxes. The model mean for the cumulative carbon emission, I_em(t), is 2790 and 2700 PgC for years 120 to 140 for the 9 CMIP6 and the 7 CMIP5 models respectively (table 2). There are differences in the cumulative carbon emission in each of the individual Earth system models from differences in their terrestrial and ocean cycling and uptake of carbon (Arora et al 2019) (figure 2(a)).

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Table 2. Model mean, intermodel standard deviation and coefficient of variation for the thermal and carbon variables for years 120 to 140 for the 9 CMIP6 and 7 CMIP5 Earth system models following a 1% annual increase in atmospheric CO₂. The coefficient of variation is defined by the intermodel standard deviation divided by the model mean, evaluated at the same time. The individual model responses are provided in table S1.

Variable:	ΔT	F	N	λ	N/F	Iem	$\Delta I_{atmos}/I_{em}$	$\Delta I_{ocean}/I_{em}$	$\Delta I_{land}/I_{em}$
Units:	K	W m−2	W m−2	(W m−2)K−1		PgC
CMIP6
mean, $\overline{x}$	4.55	7.27	2.42	1.16	0.33	2787	0.57	0.19	0.24
std, σx	1.04	0.45	0.35	0.45	0.05	213	0.04	0.02	0.05
$\sigma_x/ \overline{x}$	0.23	0.06	0.14	0.39	0.16	0.08	0.08	0.08	0.22\\ CMIP5
mean, $\overline{x}$	4.66	7.21	2.28	1.05	0.32	2702	0.59	0.20	0.21
std, σx	0.42	0.98	0.41	0.21	0.05	249	0.06	0.02	0.06
$\sigma_x/ \overline{x}$	0.09	0.14	0.18	0.20	0.16	0.09	0.10	0.09	0.31

The airborne fraction, $\Delta I_{atmos}(t)/I_{em}(t)$, initially falls to a minimum between years 50 and 80, and then increases in time reaching model means of 0.57 for the 9 CMIP6 models and 0.59 for the 7 CMIP5 models for years 120 to 140 (figure 2(b), table 2). This response is a consequence of the landborne fraction, $\Delta I_{ter}(t)/I_{em}(t)$, increasing in time to a maximum over a wide range from years 30 to 120, and then decreasing (figure 2(c)), as well as from the oceanborne fraction, $\Delta I_{ocean}(t)/I_{em}(t)$, increasing to a maximum around year 50 and then slightly decreasing (figure 2(d)). The response of the CMIP6 models is broadly similar to that of the CMIP5 models, although there is a greater range in the intermodel differences in the airborne, landborne and oceanborne fractions for CMIP5 (tables 2 and S1 (stacks.iop.org/ERL/15/0940c1/mmedia)).

For the thermal analyses, the planetary heat uptake, N(t), is provided from 1pctCO2 model output, but the radiative forcing, ΔF(t), radiative response, ΔR(t), and the climate feedback parameter, λ(t), need to be diagnosed. The effective radiative forcing, ΔF(t), is calculated using the logarithmic dependence in (9) with the radiative forcing coefficient $a = \Delta F_\mathrm{4xCO_2}/\ln 4$. The effective forcing due to a quadrupling of atmospheric CO₂, $\Delta F_\mathrm{4xCO_2}$, is diagnosed from the abrupt-4xCO₂ simulations using the y-intercept of a regression fit for N(t) versus ΔT(t) (Gregory et al 2004). To account for curvature in the N(t) versus ΔT(t) relationship (Andrews et al 2015), only the first twenty years of data are used to calculate the fits.

The regression-based estimate of $\Delta F_\mathrm{4xCO_2}$ yields an incorrect value for the CNRM-ESM2.1 model, because in this model simulation for abrupt-4xCO₂ the quadrupling of CO₂ concentration takes about 15 years, rather than being instantaneous (see the discussion in Smith et al (2020)). Therefore for this model only, the forcing is estimated as the difference in N(t) between two fixed sea surface temperature simulations, piClim-4xCO2 and piClim-control (table 2 in Smith et al (2020)).

The radiative response, ΔR(t), is next diagnosed from ΔF(t)−N(t). The time-varying climate feedback parameter, λ(t), is then diagnosed from the least-squares regression slope of ΔR(t) against ΔT(t) (Gregory and Forster 2008), where the regression is calculated from the start of the time series to year t. For example, λ(t) for year 70 is based on the regression over the first 70 years, while λ(t) for year 140 is based on the regression over the entire 140 years.

The diagnostics for ΔR(t), ΔF(t), N(t) and ΔT(t) are smoothed to remove interannual variability using a moving average filter with a 10 year window.

The physical feedback parameter, λ(t), is further decomposed into contributions from the Planck feedback, the lapse rate, relative humidity, surface albedo and cloud feedbacks,

$$\begin{equation} \small\lambda(t) = \lambda_{Planck}(t)+ \lambda_{LR}(t) +\lambda_{RH}(t)+\lambda_{Alb}(t)+\lambda_{Cloud}(t) , \end{equation} \tag{ 11 }$$

where the subscripts identify each component. The radiative impact of water vapour is separated into two contributions: changes at constant relative humidity, counted as part of the temperature-driven response, and a residual contribution due to changes in relative humidity (Held and Shell 2012). The radiative decomposition is performed using CAM5 radiative kernels (Pendergrass et al 2017, Soden et al 2008).

In these Earth system models integrated under a 1% annual increase in atmospheric CO₂ scenario, the radiative forcing ΔF(t) is typically 7.3 W m⁻² for years 120 to 140 (figure 3(a)), which drives an increase in planetary heat uptake, N(t), of typically 2.4 W m⁻² and a radiative response, ΔR(t), of 4.9 W m⁻² (figure 3(b) and (c), table 2). The climate feedback parameter, λ(t), varies across the models from 0.7 to 2.5 W m⁻² K⁻¹ with a model mean of 1.2 W m⁻² K⁻¹ (figure 3(d)). There is a larger intermodel range in the radiative response and climate feedback parameter for the 9 CMIP6 models compared with for the 7 CMIP5 models (figure 3(c) and (d)), reaching twice the normalised spread based upon the ratio of the standard deviation over the model mean (table 2).

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The TCRE only slightly varies in time over the 140 years for most of the Earth system models (figure 4(a)), although sometimes there is a slight decrease in time. The evolution of the TCRE is usually viewed in terms of the product of the ratio of the surface warming and the change in atmospheric carbon, ΔT/ΔI_atmos, and the airborne fraction, $\Delta I_{atmos}/I_{em}$, in (2) (Matthews et al 2009), as illustrated in figures 4(b) and (c); this expression may be equivalently written in terms of the product of a climate sensitivity and carbon feedback parameters (Jones and Friedlingstein 2020).

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For the 9 CMIP6 models, the TCRE contribution from the ratio of the surface warming and the change in atmospheric carbon, ΔT/ΔI_atmos, typically involves a slight decline, while the airborne fraction, $\Delta I_{atmos}/I_{em}$, involves an initial decline and then a slight increase (figures 4(b) and (c), grey line). There is a broadly similar response for the 7 CMIP5 models, but slightly modified by ΔT/ΔI_atmos decreasing after typically year 40 and with a larger intermodel spread in the airborne fraction. While the near constancy of the TCRE is clear in these diagnostics, the control of the TCRE for different individual Earth system models is not particularly revealing using this identity (figure 4, table S1).

Instead we advocate that the TCRE may be interpreted in terms of a product of the thermal response, involving the surface warming dependence on radiative forcing, ΔT/ΔF, and a radiative forcing response, involving the dependence of radiative forcing on carbon emissions, ΔF/I_em in (3) (Williams et al 2016, Williams et al 2017), as illustrated in figure 5.

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Using this identity, the thermal response, ΔT/ΔF, is revealed to increase in time for all the models (figure 5(b)), while the radiative forcing response, ΔF/I_em, decreases in time (figure 5(c)). There is a broadly similar dependence in both the subsets of the CMIP6 and CMIP5 models. The intermodel spread is greater for the thermal response, ΔT/ΔF, and is instead smaller for the radiative forcing response, ΔF/I_em for the 9 CMIP6 models relative to the 7 CMIP5 models (figure 5(b) and (c); tables 3 and S2).

Table 3. Model mean, intermodel standard deviation and coefficient of variation for the TCRE and its components for years 120 to 140 for the 9 CMIP6 and 7 CMIP5 Earth system models following a 1% annual increase in atmospheric CO₂. The coefficient of variation is defined by the intermodel standard deviation divided by the model mean, evaluated at the same time. The individual model responses are provided in table S2.

Variable:	TCRE	ΔT/ΔIatmos	ΔT/ΔF	ΔF/Iem	λ−1	(1−N/	ΔF/ΔIatmos	$\Delta I_{atmos}/$
Units:	K EgC−1	K EgC−1	K (W m$^{-2})^{-1}$	(W m−2) (EgC)−1	K (W m$^{-2})^{-1}$	ΔF)	(W m−2) (EgC)−1	Iem
CMIP6
mean, $\overline{x}$	1.64	2.87	0.63	2.63	0.96	0.67	4.59	0.57
std, σx	0.41	0.65	0.17	0.27	0.31	0.05	0.30	0.04
$\sigma_x/ \overline{x}$	0.25	0.23	0.26	0.10	0.32	0.08	0.06	0.08
CMIP5
mean, $\overline{x}$	1.75	2.94	0.66	2.71	0.99	0.68	4.55	0.59
std, σx	0.28	0.27	0.10	0.56	0.21	0.05	0.62	0.06
$\sigma_x/ \overline{x}$	0.16	0.09	0.16	0.21	0.21	0.07	0.14	0.10

Hence, the near constancy of the TCRE is a consequence of nearly compensating contributions: a strengthening in the thermal response, ΔT/ΔF, offsetting a weakening in the radiative forcing response, ΔF/I_em.

The strengthening in the thermal response given by the surface warming dependence on radiative forcing, ΔT/ΔF (figure 6(a)), from (6) is due to two reinforcing contributions: (i) a weakening in the climate feedback parameter, λ, and (ii) an increase in the fraction of radiative forcing used to warm the surface, (1 − N(t)/ΔF(t)) (figure 6(b) and (c)). The decrease in λ over time is a well-documented feature of the response of coupled climate models to CO₂ forcing (Andrews et al 2015), resulting from sea surface warming patterns evolving in time (Armour et al 2013, Rugenstein et al 2016, Ceppi and Gregory 2017, Andrews and Webb 2018). The reducing fraction of radiative forcing taken up by the ocean interior is also well reported (Solomon et al 2009, Goodwin et al 2015, Williams et al 2016, Williams et al 2017). This thermal response is similar in pattern for both the CMIP6 and CMIP5 models. There is though a greater intermodel spread in these thermal responses and their contributions from λ⁻¹ for the 9 CMIP6 models relative to the 7 CMIP5 models (table 3).

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The weakening in the radiative forcing response given by the radiative forcing dependence on cumulative carbon emissions, ΔF/I_em (figure 7(a)), from (7) is mainly due to a weakening in the radiative forcing dependence on atmospheric carbon, ΔF/ΔI_atmos, from a saturating effect (Myhre et al 1998) together with smaller contributions from the changes in the airborne fraction (Matthews et al 2009) (figure 7(b) and (c)). This response is also explained in terms of how far the ocean is from a carbon equilibrium with the atmosphere (Goodwin et al 2015, Williams et al 2017, Katavouta et al 2018). This radiative forcing response is similar in pattern for both the CMIP6 and CMIP5 models. There is though a smaller intermodel spread in both the radiative forcing dependence on atmospheric carbon and the changes in the airborne fraction responses for the 9 CMIP6 models relative to the 7 CMIP5 models (table 3).

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The normalised spread for the separate sets of 9 CMIP6 and 7 CMIP5 models are next assessed by diagnosing the coefficient of variation (also called the relative standard deviation) given by the intermodel standard deviation for each variable and dividing by the multi-model mean, all evaluated at the same time (Williams et al 2017, Katavouta et al 2019). The normalised spreads for the TCRE for years 120 to 140 are 0.25 and 0.16 for the subsets of 9 CMIP6 and 7 CMIP5 models respectively (figure 8(a), tables 3 and S2).

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For the CMIP6 subset of models, there is a larger normalised spread for ΔT/ΔF of 0.26 and a smaller normalised spread for ΔF/I_em of 0.10. Instead for the CMIP5 subset of models, there is the opposing response with a smaller normalised spread for ΔT/ΔF of 0.16 and larger normalised spread for ΔF/I_em of 0.21 (figure 8(a), table 3). Hence, for years 120 to 140, the intermodel spread for the TCRE is being controlled more strongly by the thermal response for the 9 CMIP6 models, but instead more strongly by the radiative and carbon responses for the 7 CMIP5 models.

If the normalised spread for each thermal and radiative forcing responses, ΔT/ΔF, and ΔF/I_em, were assumed to be independent of each other and combined in the same manner as random errors, then the normalised spread for the TCRE would be comparable to the square root of the sum of the squared contributions making up the TCRE in (3) (figure 8(a), grey line). This estimate of the normalised spread from the sum of these two contributions, ΔT/ΔF and ΔF/I_em, is always larger than the actual normalised spread for the TCRE. This inference of partial compensation is supported by the two contributions, ΔT/ΔF and ΔF/I_em, being negatively correlated to each other with a correlation coefficient value of typically -0.5 (figure 8(d), black line).

The normalised spread for the thermal response, ΔT/ΔF, ranges typically from 0.2 to 0.3 for both the CMIP6 and CMIP5 models (figure 8(b)), but the normalised spread becomes larger for the CMIP6 models than the CMIP5 models by years 120 to 140 (table 3). The normalised spread of ΔT/ΔF in both the CMIP6 and CMIP5 models is dominated by the contribution of the inverse of the climate feedback, λ⁻¹, rather than from the fraction of the radiative forcing used to warm the surface, 1 − N/ΔF, where N is effectively the ocean heat uptake, based upon (6). This response does differ though between these subsets of CMIP6 and CMIP5 models with a larger spread for the λ⁻¹ contribution for CMIP6 for years 120 to 140 (table 3).

While the intermodel spread in λ⁻¹ is more important than that in ocean heat uptake, N, there is a strong partial compensation in the changes in λ⁻¹, and the fraction of the radiative forcing used to warm the ocean interior, N/ΔF, with a correlation coefficient of typically -0.9 (figure 8(d), red line). Hence for a given radiative forcing, a larger physical climate feedback is associated with less ocean heat uptake, whereas a smaller physical climate feedback is associated with more ocean heat uptake.

The normalised spread for the radiative forcing response, ΔF/I_em, is typically 0.1 for the 9 CMIP6 models and 0.2 for the 7 CMIP5 models (figure 8(c), table 3). This smaller normalised spread for CMIP6 is through smaller reinforcing contributions from the dependence of the radiative forcing on atmospheric carbon, ΔF/ΔI_atmos, and the airborne fraction, $\Delta I_{atmos}/I_{em}$ based upon (7); both terms have a weak positive correlation to each other of typically 0.5 or less (figure 8(d), blue line). The intermodel differences in the airborne fraction are themselves more dominated by the landborne fraction (figures 2(b)–(d)); the intermodel spread for the landborne fraction is a factor of 2 or 3 larger than that for the oceanborne faction for CMIP6 and CMIP5 respectively (tables 2 and S1).

A dominant cause of intermodel differences in the TCRE is from the thermal response, which is itself mainly controlled by intermodel differences in the physical climate feedback for both the 9 CMIP6 and 7 CMIP5 models. The overall radiative response and climate feedback parameter is positive, acting to cool the surface (figure 9(a)).

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The dominant contribution to the physical climate feedback, λ, in (11) is the Planck feedback (figure 9(a)) acting to cool the surface (with a positive λ in our sign convention), which is reinforced by smaller positive contributions from the lapse rate and relative humidity. There are negative contributions from the albedo and generally from clouds, both acting to increase surface warming. The intermodel spread in λ is dominated by intermodel differences in the shortwave and longwave effects of clouds with the net effect of clouds ranging from strongly negative (acting to enhance surface warming) to weakly positive (Ceppi et al 2017).

The analyses for the contribution to the feedback parameter, λ, between the sets of CMIP6 and CMIP5 models vary from being almost identical to very similar for the Planck feedback, the lapse rate and relative humidity contributions (figure 9(a)). There is a narrower spread for the albedo contribution for CMIP6 relative to CMIP5. However, there is a larger spread for the longwave and shortwave contributions from clouds for CMIP6 relative to CMIP5, which leads to the overall net climate feedback parameter having a larger positive range for CMIP6 (figure 9(a), tables 2 and S1).

The feedback parameter, λ, does evolve in time, consistent with figure 6(b), becoming smaller in magnitude, mainly due to the shortwave cloud contribution becoming more negative and acting to enhance surface warming (figure 9(b)). These analyses are in agreement with diagnostics of more extensive sets of CMIP5 and CMIP6 models (Andrews et al 2015, Ceppi and Gregory 2017, Ceppi et al 2017, Zelinka et al 2020).

The spread in the TCRE is now presented in a normalised fashion for each set of 9 CMIP6 and 7 CMIP5 models, where each model response is normalised by dividing by the model mean for years 120 to 140 (figure 10, left column). There is a slightly larger normalised spread for this subset of 9 CMIP6 compared with the subset of 7 CMIP5 models. This difference is probably a consequence of the choice and number of the models analysed, since other studies do not find a significant difference in the TCRE for CMIP5 and CMIP6 (Arora et al 2019, Jones and Friedlingstein 2020).

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There are robust differences in the controls of the intermodel differences in the TCRE as revealed by separating the TCRE into a thermal response involving the dependence of surface warming on radiative forcing, ΔT/ΔF, the dependence of the radiative forcing on atmospheric CO₂, ΔF/ΔI_atmos, and the airborne fraction, $\Delta I_{atmos}/I_{em}$ (figure 10, middle column). The normalised spread for the thermal response, ΔT/ΔF, is much greater for the 9 CMIP6 models than the 7 CMIP5 models. In contrast, the normalised spread for the dependence of the radiative forcing on atmospheric CO₂, ΔF/ΔI_atmos, and the airborne fraction, $\Delta I_{atmos}/I_{em}$ (figure 10, middle column) is smaller for the 9 CMIP6 models versus the 7 CMIP5 models.

The standard way of interpreting the TCRE in terms of the Transient Climate Response involving the dependence of surface warming on atmospheric carbon, ΔT/ΔI_atmos, and the airborne fraction, $\Delta I_{atmos}/I_{em}$ (figure 10, right column) shows a much larger spread for ΔT/ΔI_atmos for CMIP6 than CMIP5, and a comparable spread for the airborne fraction.