1. Introduction

Changes to downwelling longwave radiation at Earth’s surface modify the surface energy balance, influencing the surface temperature, humidity, and hydrological cycle (Held and Soden 2006; Previdi 2010). Climate models predict large increases (∼20 W m⁻²) in downwelling longwave at the surface under even moderate future CO₂ emission scenarios (Roderick et al. 2014). However, the direct increase in downwelling longwave radiation due to increased CO₂ concentrations (termed the radiative forcing) is far smaller, in the range of 0–5 W m⁻² (Pendergrass et al. 2018; Colman 2015) for a doubling of CO₂ (depending on geographic location). Subsequent changes to the atmospheric state (primarily air temperature and moisture content), termed feedbacks, are responsible for the (much larger) remainder of the increase in downwelling longwave (Pendergrass et al. 2018). Disentangling the forcing and feedback contributions is vital in understanding to what extent surface processes can be considered to be forced by changes to downwelling longwave, as opposed to driving these changes (Vargas Zeppetello et al. 2019). This understanding is especially important in the development and interpretation of decoupled land surface, ice sheet, or ocean–sea ice models, which are typically forced by downwelling longwave fluxes extracted from a climate model or atmospheric reanalysis.

Heretofore, the disentanglement of longwave radiative forcing and feedbacks at the surface has been undertaken using a “radiative kernel” approach (Previdi 2010; Pendergrass et al. 2018; Vargas Zeppetello et al. 2019). The “kernel” expresses the change in the downwelling longwave radiation for a unit change in an atmospheric property (e.g., air temperature) at a given location and height. The radiative kernel methodology was initially developed to examine top-of-atmosphere forcing and feedbacks (Held and Soden 2000; Soden and Held 2006; Soden et al. 2008) before being adapted for the equivalent surface problem. The traditional kernel approach uses three-dimensional (3D), high-time-resolution climate model output, requiring significant computational resources to both calculate and apply the kernels (Pendergrass et al. 2018). This 3D approach is necessary for studying top-of-atmosphere sensitivity in which the vertical distribution of greenhouse gases is of the utmost importance (Hansen et al. 2005). However, for downwelling longwave at the surface, Vargas Zeppetello et al. (2019) demonstrate that feedbacks may be analyzed by vertically integrating the 3D kernels, and applying them to changes in near-surface properties only: in their analysis, this results in a decrease in the variance captured by the kernel method from 94% to 85%. In other words, it is surface air properties (specifically temperature and humidity) that predominantly control changes in downwelling longwave at the surface. The Vargas Zeppetello et al. (2019) approach removes the need for 3D climate model output to calculate feedbacks. However, 3D output is still required to calculate the kernels, and kernels do not exist for all important fields (e.g., clouds). Furthermore, previous studies such as Vargas Zeppetello et al. (2019) focused only on the feedbacks and did not consider the radiative forcing, implicitly assuming it to be small.

Here we develop an alternative methodology that provides first-ever explicit analytic expressions for all forcing and feedback terms relevant to the downwelling longwave at the surface. In this work, radiative forcing is taken to mean the instantaneous radiative forcing resulting directly from an increase in greenhouse gas concentration, and our feedbacks include all time scales (both rapid and slow) of responses to the change in forcing. This formulation differs from the more widely used effective radiative forcing (Myhre et al. 2013; Sherwood et al. 2015; Andrews et al. 2021), which quantifies the change in downwelling longwave after allowing for rapid adjustments of atmospheric temperatures, water vapor and clouds, but before longer time scale changes (i.e., slow feedbacks). Our methodology builds on the semi-analytic model for clear-sky downwelling longwave at the surface developed by Shakespeare and Roderick (2021, hereafter SR21). The SR21 model predicts the downwelling longwave at Earth’s surface as a function of near-surface air temperature, near-surface specific humidity, CO₂ concentration, surface pressure, and the e-folding height of water vapor (defined as the total column water vapor divided by the density of water vapor in the near-surface air). SR21 validated their model against independent subhourly surface observations, a reanalysis product, and climate model simulations. Here we extend the SR21 model to include clouds, and to predict changes in downwelling longwave radiation as a function of changes in each of the model parameters.

The paper is laid out as follows. In section 2a we provide an overview of the SR21 theory and derive explicit expressions for the longwave radiative forcing and feedbacks. We extend the results to include the effects of cloud in section 2b. In section 3 we apply the methodology to diagnose forcing and feedbacks in a set of five CMIP5 models. We conclude in section 4.

2. Theory and methodology

In this section we develop a methodology for analyzing the sensitivity of the downwelling longwave radiation at the surface to a change in CO₂ concentration. Our objective is to be able to separate a modeled change in downwelling longwave into its forced component (i.e., the direct increase due to CO₂) and feedbacks from subsequent changes in atmospheric properties.

We first consider the sensitivity of the clear-sky longwave (section 2a) before extending the analysis to include clouds (section 2b).

3. Application to CMIP5 models

Here we apply the methodology developed in the previous section to diagnose the radiative forcing and feedbacks on the downwelling longwave in five randomly selected CMIP5 climate models. The five CMIP5 models used are the National Center for Atmospheric Research CCSM4 (Gent et al. 2011), NOAA Geophysical Fluid Dynamics Laboratory GFDL CM3 (Donner et al. 2011), Centre National de Recherches Météorologiques CNRM-CM5 (Voldoire et al. 2013), Japan Agency for Marine-Earth Science and Technology MIROC-ESM (Watanabe et al. 2011), and Beijing Climate Center Climate System Model BCC-CSM1-1 (Wu et al. 2014). In each model we examine the change in the time-mean downwelling longwave at the surface between an initial (2006–26) and final (2080–2100) period under the RCP8.5 scenario.

a. Model state

Figure 1 shows the mean state of each model in the initial period for the key parameters in our theory: temperature, specific humidity, water vapor e-folding height, and cloud fraction. The lower latitudes are warmer, wetter, and with water vapor more confined to the lower troposphere (smaller H). The five models are largely consistent in their realization of these three fields (the standard deviation is shown in the bottom row of Fig. 1). However, cloud fraction varies significantly between the models with CCSM4 having relatively less clouds (average of 0.48) and GFDL CM3 more (average of 0.76).

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Mean near-surface air temperature T, specific humidity q, water vapor height scale H, and cloud fraction C for the period 2006–26 from five CMIP5 models (see label on right) under the RCP8.5 scenario. The bottom row shows the standard deviation between the five models.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Figure 2 shows the change in the mean states between the initial and final periods for the same four parameters as in Fig. 1. Temperature changes are largest in the Arctic (the so-called Arctic amplification; e.g., Serreze and Francis 2006) and also vary significantly between models, especially in the Arctic and over the Southern Ocean.

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Changes in mean temperature T, specific humidity q, water vapor height scale H, and cloud fraction C for the period 2080–2100 compared with 2006–26 from five CMIP5 models (see label on right) under the RCP8.5 scenario. The changes in q, H, and C are expressed as fractional changes from the initial period (denoted “0”; 2006–26) per degree of temperature change. The bottom row shows the standard deviation between the five models.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Changes in specific humidity q in Fig. 2 (column 2) are shown as fractional changes per kelvin; that is, normalized by the initial value and the temperature change: $q_0^{-1}\mathrm{\Delta }q/\mathrm{\Delta }T$ where subscript 0 denotes the initial value. Since q = 0.622hP_s(T)/P for relative humidity h and saturation vapor pressure P_s(T), we may write (neglecting changes in P, as previously) that

$\frac{1}{q_0}\frac{\mathrm{\Delta }q}{\mathrm{\Delta }T}=c\left(T\right)+\frac{1}{h_0}\frac{\mathrm{\Delta }h}{\mathrm{\Delta }T},$(17)

where

$c\left(T\right)=\frac{1}{P_s\left(T\right)}\frac{\partial }{\partial T}{P}_s\left(T\right)$(18)

is the Clausius–Clapeyron scaling (K⁻¹). Thus, to the extent that relative humidity may be considered constant (i.e., Δh → 0), changes in specific humidity are directly related to the local change in temperature according to c(T). To confirm this relationship holds in the CMIP5 models, Fig. 3a displays a histogram of $q_0^{-1}\mathrm{\Delta }q/\mathrm{\Delta }T$ versus initial temperature T₀ in the models, overlaid by c(T) (black line).¹ The mean of the data at each temperature (red dashed line) indeed follows the Clausius–Clapeyron scaling. We observe that fractional change in q per unit temperature change is almost twice as large at very low temperature (0.12 at 220 K; i.e., Antarctica) compared to high temperature (0.06 at 300 K; equatorial latitudes). Given this scaling, we can approximate changes in specific humidity as Δq λ c(T)q₀ΔT. Figure 3d shows that this scaling based on ΔT can account for 88% (i.e., r² = 0.88) of the variance in Δq (implying that the remainder is due to relative humidity changes). Thus, it is reasonable to think of changes in specific humidity and water vapor feedback as directly linked to local changes in temperature [as we have implied in writing Eq. (8), and as has been argued by previous authors; e.g., Vargas Zeppetello et al. 2019].

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Histograms of fractional changes in (a) specific humidity q, (b) height scale H, and (c) cloud fraction C per unit temperature change (see Fig. 2) against initial temperature T₀ (see Fig. 1) accumulated over the five CMIP5 models. Red dashed lines show the mean value at each temperature, with the shaded region representing one standard deviation either side. The black line in each panel indicates the Clausius–Clapeyron scaling c(T) [Eq. (18)] in (a), 0.07 − c(T) in (b), and −0.07 + c(T) in (c). Also shown are the histogram of CMIP5 model changes in (d) specific humidity q, (e) height scale H, and (f) cloud fraction C against predicted change using the Clausius–Clapeyron scaling as shown by the black line in (a)–(c). The red dashed line is the 1–1 line. The black dashed line is the line of best fit. The root-mean-square error (RMSE) and correlation coefficient (r) are shown in each plot.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Let us now consider the changes in height scale H and cloud fraction C. Given these parameters are both related to water vapor (and the condensation thereof) we might reasonably anticipate that their changes are also linked to some extent to the local thermodynamics of water. In other words, we expect that the height scale H depends on the saturation vapor pressure of water and other variables—for example, H = H[P_s(T), T, h, …]—implying that $H_0^{-1}\mathrm{\Delta }H/\mathrm{\Delta }T$ will involve at least one term that scales with c(T). Therefore, we repeat the same analysis used for specific humidity for both H and C.

The fractional changes of H and C per kelvin (Fig. 2; columns 3 and 4) show distinct latitudinal structure. Height scale H increases by ∼5% in the low-to-middle latitudes and decreases by a similar amount at polar latitudes. Changes in cloud cover are variable, but are generally negative (decrease) in low-to-middle latitudes and positive (increase) at high latitudes. As previously, we plot histograms of these changes against initial temperature (Figs. 3b,c).

Before proceeding, we note that we do not use the approximate scalings developed here (i.e., Fig. 3) for calculating q, H, and C feedbacks (λ) in the remainder of this section; rather, we directly use the ratio of the modeled changes Δq/ΔT, ΔH/ΔT, and ΔC/ΔT—the discussion below is included to aid the physical interpretation of these feedbacks.

Figure 3 shows that changes in height scale vary oppositely to specific humidity (they decrease with temperature) and indeed follow an offset Clausius–Clapeyron-like scaling: that is, ΔH ≃ [0.07 − c(T)]H₀ΔT (black line in Fig. 3b). This scaling accounts for 72% (i.e., r² = 0.72) of the variance in changes in height scale (Fig. 3e). Thus, as for specific humidity, changes in water vapor scale height are directly related to the local change in temperature (i.e., the scale height is largely a local feedback). Changes in cloud fraction (except at very low temperature) appear to follow the negative of the H scaling: ΔC ≃ −[0.07 − c(T)]C₀ΔT (black line in Fig. 3c). However, this scaling accounts for only 15% (i.e., r² = 0.15) of the variance in changes in cloud fraction (Fig. 3f). In other words, only a minority of the cloud cover changes can be directly linked to the local change in temperature. Therefore, the cloud feedback parameter [Eq. (16)] should not be literally interpreted as a direct local thermodynamic feedback to a local temperature change. However, local temperature changes could influence the cloud feedback in other ways—for example, through changes to the atmospheric stability.

b. Changes in downwelling longwave

Given the initial and difference fields for the five CMIP5 models (as shown in Figs. 1 and 2) we can compute the predicted change in downwelling longwave via our equations in section 2 [i.e., Eqs. (8) and (14)]. The optical depth τ and its derivatives in these equations can be calculated directly from the tabulated data for $\tau =\tau \left(q,\hspace{0.17em}\mathcal{H},\hspace{0.17em}{q}_C\right)$ given in SR21. The forcing and feedback terms for a given numerical model can then be directly calculated for the change between an initial ($T_0,\hspace{0.17em}{q}_0,\hspace{0.17em}{\mathcal{H}}_0,\hspace{0.17em}{q}_{C,0}$) and final state ($T_0+\mathrm{\Delta }T,\hspace{0.17em}{q}_0+\mathrm{\Delta }q,\hspace{0.17em}{\mathcal{H}}_0+\mathrm{\Delta }\mathcal{H},\hspace{0.17em}{q}_{C,0}+\mathrm{\Delta }{q}_C$), using (for greatest accuracy) the mean state ($T_0+\mathrm{\Delta }T/2,\hspace{0.17em}{q}_0+\mathrm{\Delta }q/2,\hspace{0.17em}{\mathcal{H}}_0+\mathrm{\Delta }\mathcal{H}/2,q_{C,0}+\hspace{0.17em}\mathrm{\Delta }{q}_C/2$) as the reference point for the calculation (i.e., T, q, τ, etc. in our equations are evaluated at these midpoint values). In our calculations, we use a CO₂ concentration of 400 ppm for the initial period and 800 ppm for the final period, which are the approximate mean values for the RCP8.5 scenario in CMIP5 models (Prather et al. 2013; Friedlingstein et al. 2006), although the actual concentration varies between models (and is not reported in the standard output).

We first consider the clear-sky longwave in section 1, followed by the all-sky longwave in section 2.

1) Clear sky

The first necessary step is to confirm that our methodology can reproduce the CMIP5 modeled changes in clear-sky downwelling longwave with sufficient accuracy. Figure 4 compares the model and calculated ΔL for each of the five CMIP5 models. The correlation between model and calculated values exceeds 0.96 for all models and the RMS error ranges from 1.6 to 2.9 W m⁻² (or about 10% of the mean change in longwave). We note that our correlations are higher than those previously reported using the 2D kernel method (0.92), and comparable to the 3D kernel method (0.97; Vargas Zeppetello et al. 2019). Our calculation tends to slightly overpredict the change in downwelling longwave in all models, especially at polar latitudes.

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Comparison of (column 1) CMIP5 model change and (column 2) calculated change in clear-sky downwelling longwave for five CMIP5 models (see label on right) under the RCP8.5 scenario, and (column 3) their difference (model minus calculated). (column 4) A histogram of model vs calculated values at each location. The 1–1 line (red dashed) and the line of best fit (black dashed) are also shown. The root-mean-square error (RMSE), global mean bias, and correlation coefficient are given on the plots.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Figure 5 shows the calculated radiative forcing (column 1) and total feedback (columns 2) for the clear-sky longwave in each CMIP5 model, and the standard deviation in these quantities between the different models (bottom row). The global mean values are summarized in Figs. 6a and 6b. The clear-sky radiative forcing averages 3.5 W m⁻². It is smallest near the equator (∼2 W m⁻²; humid, optically thick air) and larger near the poles (∼4 W m⁻²), and even larger in dry land areas (∼6 W m⁻²; e.g., Sahara; dry, optically thin air). This result reflects the fact that adding CO₂ has a larger effect on radiation when the air is optically thinner (farther from a blackbody), and no effect at all if the air is already behaving as a blackbody. The total downwelling longwave feedback (λ_LΔT) averages 23.4 W m⁻², almost an order of magnitude larger than the direct radiative forcing. It is highly variable between the different CMIP5 models; for example, the global mean value differs by 50% between BCC-CSM1-1 and MIROC-ESM (Fig. 6b). This variation in feedback λ_LΔT is largely due to the significant variation in temperature change (i.e., σ_ΔT) between the models (i.e., Fig. 2, column 1). Specifically, 79% of the variance in clear-sky downwelling longwave between different models (i.e., ${\sigma }_{\mathrm{\Delta }L}^2$) is explained by ${\overline{{\lambda }_L}}^2{\sigma }_{\mathrm{\Delta }T}^2$, where $\overline{{\lambda }_L}$ is the multimodel average feedback parameter—as may be determined by linearly regressing σ_ΔL against $\overline{{\lambda }_L}{\sigma }_{\mathrm{\Delta }T}$ (not shown).

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The calculated change in clear-sky downwelling longwave decomposed into (column 1) radiative forcing R and (column 2) total feedback λ_LΔT, and the calculated change in all-sky downwelling longwave decomposed into (column 3) radiative forcing R′ and (column 4) total feedback ${\lambda }_L^{\prime }\mathrm{\Delta }T$. Rows 1–5 each correspond to one CMIP5 model (see label on right) under the RCP8.5 scenario. The bottom row shows the standard deviation between the five models.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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(a),(d) Calculated global mean radiative forcing, (b),(e) total feedback, and (c),(f) total feedback parameter for (top) clear sky and (bottom) all sky for five CMIP5 climate models. The multimodel average is also shown. The total feedback parameter is decomposed into the individual contributions (see legend).

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Indeed, the total feedback parameter for clear skies λ_L is similar for all models, as shown in Fig. 7 (column 1). It is largest near the equator (∼10 W m⁻² K⁻¹) and smallest near the poles (∼4 W m⁻² K⁻¹), with a global mean value of 6.5 W m⁻² K⁻¹. Columns 2–4 of Fig. 7 show the individual feedback contributions under clear skies from temperature (column 2), water vapor (column 3), and height scale (column 4)—which respectively contribute 4, 2, and 0.5 W m⁻² K⁻¹ to the global mean (Fig. 6c). Note that the color scales are different for each column of Fig. 7. The temperature feedback dominates at all latitudes, but with significant additional contributions from water vapor at low-to-middle latitudes, and height scale at high latitudes. The high-latitude height scale feedback is the only consistently negative feedback on the downwelling longwave (associated with a decrease in the water vapor height scale at these latitudes; see Fig. 2). The height scale feedback also has a strong positive component in the North Atlantic and Southern Ocean in some of the CMIP5 models (notably GFDL CM3 and CCSM4). Thus, despite being a relatively minor component of the total longwave feedback, these strong positive signals result in the height feedback being responsible for the majority of the variation in total longwave feedback parameter between models (see the standard deviation shown in the bottom row of Fig. 7). Comparison with Fig. 2 shows that these elevated positive feedbacks are associated with GFDL CM3 and CCSM4 exhibiting a large increase in water vapor height scale in the North Atlantic and Southern Ocean.

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Calculated clear-sky feedback parameters: (column 1) total feedback λ_L and each individual feedback: (column 2) temperature λ_T, (column 3) water vapor λ_q, and (column 4) height scale λ_H. Rows 1–5 each correspond to one CMIP5 model (see label on right). The bottom row shows the standard deviation between the five models.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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2) All sky

We now consider the all-sky downwelling longwave at the surface for the same set of models. As in the previous section, we first verify our theory can reproduce the CMIP5 modeled changes in all-sky downwelling longwave with sufficient accuracy. Figure 8 compares the model and calculated ΔL′ for each of the five CMIP5 models. The correlation between model and calculated values exceeds 0.95 for all models and the RMS error ranges from 2.3 to 3.5 W m⁻²—slightly larger, but comparable to the clear-sky results. However, contrary to the clear-sky results, our all-sky calculation tends to slightly underpredict the change in downwelling longwave in most models.

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Comparison of (column 1) CMIP5 model and (column 2) calculated change in all-sky downwelling longwave for five CMIP5 models (see label on right) under the RCP8.5 scenario, and (column 3) their difference (model minus calculated). (column 4) A histogram of model vs calculated values at each location. The 1–1 line (red dashed) and the line of best fit (black dashed) are also shown. The root-mean-square error (RMSE), global mean bias, and correlation coefficient are given on the plots.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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The calculated radiative forcing and total feedback for the all-sky longwave in each CMIP5 model are shown in the same figure as the clear-sky results (Fig. 5, columns 3 and 4). Similarly, the global-mean all-sky forcing, total feedback, and feedback parameters are summarized in Figs. 6d, 6e, and 6f. The spatial structure of the all-sky feedback parameters is shown in Fig. 9. The masking effect of clouds is immediately apparent in the significant reduction in radiative forcing (global average 1.5 W m⁻² for all sky vs 3.5 W m⁻² for clear sky), water vapor feedback parameter (1 vs 2 W m⁻² K⁻¹), and height scale feedback parameter (0.2 vs 0.5 W m⁻² K⁻¹). However, as anticipated, the additional blackbody radiation from the clouds increases the temperature feedback parameter (4.5 vs 4 W m⁻² K⁻¹), but this increase is insufficient to compensate for the decrease in water vapor and height feedbacks. Thus, the total downwelling longwave feedback decreases slightly in the all-sky case. The direct cloud feedback is close to zero in the global average (Fig. 6f). However, it is highly variable both spatially within models, and between models (Fig. 9, column 5), mirroring the differences in cloud fraction changes (see Fig. 2). The cloud feedback accounts for almost all the variation in the total all-sky longwave feedback parameter between models (see bottom row of Fig. 9). In some locations—notably over the low-to-middle latitude ocean—strong positive cloud feedback is a major contributor to the total feedback (comparable to the temperature feedback).

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Calculated all-sky feedback parameters: (column 1) total feedback ${\lambda }_L^{\prime }$ and each individual feedback: (column 2) temperature ${\lambda }_T^{\prime }$, (column 3) water vapor ${\lambda }_q^{\prime }$, (column 4) height scale ${\lambda }_H^{\prime }$, and (column 5) cloud ${\lambda }_C^{\prime }$. Rows 1–5 each correspond to one CMIP5 model (see label on right). The bottom row shows the standard deviation between the five models.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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3) Summary

Figure 10 summarizes the diagnosed changes in clear-sky and all-sky downwelling longwave in the analyzed CMIP5 models for the global (Figs. 10a,d), low-latitude (Figs. 10b,e), and high-latitude (Figs. 10c,f) regions. The directly modeled changes are also included (solid gray bar) for comparison, showing that our method is able to replicate the modeled longwave with sufficient accuracy to distinguish between the different CMIP5 models.

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Calculated (color bar) and modeled (solid gray bar) changes in (top) clear-sky and (bottom) all-sky downwelling longwave for five CMIP5 climate models (a),(d) globally, (b),(e) at latitudes below 30°N/S, and (c),(f) at latitudes above 60°N/S. The multimodel average is also shown. The calculated change is decomposed into the individual contributions (color coded; see legend). The gray outline overlying the colored bar shows the total calculated change for comparison with the CMIP5 model value (as shown by the solid gray bar below the colored bar).

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Larger global mean downwelling longwave in some models (e.g., MIROC-ESM and GFDL CM3) is due to larger increases in temperature in these models; that is, the feedback parameters are comparable to the other models (e.g., Figs. 6c,f) but the total feedback is larger (Fig. 10). Thus, the enhanced downwelling longwave in these models is forced by enhanced temperature and not the other way around.

Figure 11 displays the zonal mean of the feedback parameters diagnosed from the five CMIP5 models. As noted previously, with the exception of the cloud feedback, the intermodel spread in feedback parameters is small (∼10%). For clear-sky conditions (Fig. 11a), the total feedback exceeds 9 W m⁻² K⁻¹ at the equator and is around 2–4 W m⁻² K⁻¹ at the poles, with the dominant contributor being the temperature feedback. It is notable that the temperature feedback diagnosed here is significantly larger than that diagnosed in previous work; for example, Vargas Zeppetello et al. (2019) report a zonal average clear-sky temperature feedback that never exceeds ∼3 W m⁻² K⁻¹ (see their Fig. 1b) whereas ours reaches 5.6 W m⁻² K⁻¹ at the equator. The reasons for this discrepancy are unclear. As a result, our total clear-sky feedback exceeds the Planck feedback (i.e., 4σT³; black dashed line in Fig. 11) everywhere equatorward of 70°N/S, implying that the net longwave feedback is positive (i.e., warming the surface). Poleward of 70°N/S, the reverse is true: the net (clear-sky) longwave feedback is negative, cooling the surface.

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Zonal mean feedback parameters for (a) clear sky and (b) all sky. The multimodel mean is shown by the solid line, with the maximum and minimum values from individual models shown by the shaded regions. For reference, the multimodel mean Planck feedback 4σT³ is shown as a dashed black line.

Citation: Journal of Climate 35, 12; 10.1175/JCLI-D-21-0865.1

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Turning to the all-sky case (Fig. 11b), we observe that the height scale and cloud feedbacks—accounting for intermodel spread—are roughly equal and opposite at most latitudes (consistent with the correlations shown in Fig. 3). As such, the total all-sky feedback parameter is mostly composed of the temperature feedback, with a smaller contribution from the water vapor feedback. The total all-sky feedback parameter equals or exceeds the Planck feedback at all latitudes, implying the net longwave feedback is positive (or zero) everywhere. Notably, the net longwave feedback is close to zero in the Antarctic, but significantly positive in the Arctic, consistent with the observed asymmetry of polar warming (i.e., Arctic amplification).

4. Discussion

We have presented a new semi-analytic methodology for decomposing modeled changes in downwelling longwave radiation at the surface into direct radiative forcing due to changes in CO₂ concentration, and “feedbacks” due to subsequent changes in near-surface temperature, specific humidity, the vertical extent of water vapor, and cloud cover. Our diagnosed zonal mean downwelling longwave feedback (Fig. 11b) equals or exceeds the Planck feedback at all latitudes, implying that the net longwave feedback acts to heat the surface.

Our novel methodology provides explicit analytic expressions for the forcing and feedbacks for both clear-sky conditions [i.e., Eq. (8)], based on the downwelling longwave model of Shakespeare and Roderick (2021), and all-sky conditions [i.e., Eq. (16)] based on our extension to the SR21 model to include clouds. Previous methods relied on the computationally expensive calculation of radiative kernels from high-time-resolution climate model output (Previdi 2010; Pendergrass et al. 2018; Vargas Zeppetello et al. 2019). By contrast, our method is model-independent, involves inexpensive calculations using single-level and vertically integrated time-mean model output, and the analytic expressions provide a more transparent view of the thermodynamics at play. Despite its relative simplicity, the magnitude of the error in our calculated change in downwelling longwave is comparable to that from previous kernel methods [e.g., Pendergrass et al. (2018) report a global mean absolute error of 1.4 W m⁻² in clear-sky longwave at the surface; here we report 1.2–2.6 W m⁻² depending on the CMIP5 model considered]. The correlation between the calculated and modeled changes using our method also exceeds 0.95 in all cases, comparable to the 3D kernel method, and better than the vertically integrated kernel method (Vargas Zeppetello et al. 2019). We note that the errors in our calculated change in longwave between two time periods are smaller than the errors in the calculated mean longwave itself for either of those time periods (which is typically ∼10 W m⁻²; e.g., see SR21). This result implies that there exists a systematic (time mean) error in the longwave calculations (as noted by SR21), which cancels when the difference is computed. However, there remain smaller but nonnegligible biases in the prediction of the change in longwave of ∼0.5–3 W m⁻² depending on model. Intriguingly, these biases tend to change sign from a slight overestimate of the change in longwave in the clear-sky case, and a similar magnitude underestimate in the all-sky case (with the exception of the CCSM4 model; e.g., Figs. 10a,d).

Compared to previous feedback analyses, we have identified a new vertical structure “height scale” feedback associated with changes to the vertical extent of water vapor (i.e., a change in specific humidity at height). Previous kernel methods (Previdi 2010; Pendergrass et al. 2018) would have implicitly included this contribution in the vertical structure of their three-dimensional water vapor and temperature kernels. The height scale contribution is small in the global average but locally significant, especially for the clear-sky longwave (e.g., Fig. 6). Omission of this effect by Vargas Zeppetello et al. (2019) in their vertically integrated kernel method may explain its reduced ability to capture the geographic variations in changes in downwelling longwave, compared to the 3D kernel method (e.g., their Fig. 2).

We emphasize that the radiative forcing we have defined here is the instantaneous radiative forcing, meaning we have calculated it allowing only CO₂ concentration to change and keeping atmospheric properties fixed, as per Eq. (4). As such, our radiative forcing differs from the more common “effective” radiative forcing (Myhre et al. 2013; Sherwood et al. 2015; Andrews et al. 2021) which permits rapid adjustments of atmospheric properties, prior to evaluating the change in downwelling longwave.

Vargas Zeppetello et al. (2019) argue that at least 60% of the clear-sky downwelling longwave response is driven by changes in surface temperature. Based on our results, we can further extend this important conclusion. In our multimodel average, the increase in clear-sky (all-sky) downwelling longwave is 27.0 W m⁻² (23.5), of which 3.5 W m⁻² (1.4) is the direct radiative forcing due to CO₂, 15.7 W m⁻² (18.3) temperature feedback, 7.2 W m⁻² (2.9) water vapor feedback and 0.5 W m⁻² (0.3) height scale feedback (see Fig. 10). There is an additional 0.6 W m⁻² cloud feedback in the all-sky case. We have further shown (Fig. 3) that 88% of the spatial variance in water vapor changes and 72% of the spatial variance in height scale changes may be linked to local changes in temperature. Thus, in the global average, (15.7 + 0.88 × 7.2 + 0.72 × 0.5)/27 = 83% of the clear-sky downwelling longwave response can be attributed to near-surface temperature changes (the number increases to 90% for the all-sky longwave), with the remainder mostly due to direct radiative forcing. Thus, we re-emphasize the conclusions of Vargas Zeppetello et al. (2019) that changes in downwelling longwave should not be treated as an independent forcing of the surface. Instead, it is vital that the surface is coupled to the atmosphere above, such that downwelling longwave radiation at the surface responds to changes in surface temperature (i.e., as in a coupled climate model). However, many land surface (e.g., Rodell et al. 2004), ice sheet (e.g., van den Broeke et al. 2008), and ocean–sea ice models (e.g., Kiss et al. 2020) instead prescribe the downwelling longwave, while allowing the surface temperature to vary largely independently. Yang and Roderick (2019) analyze satellite data over the global oceans to show that treating the downwelling longwave radiation at the surface as independent of the surface temperature is a flawed approach (their Fig. 8), and reasonable estimates for evaporation can only be found when the downwelling longwave radiation is treated as being coupled to the surface temperature (their Fig. 11). Those results, along with the new results from this study, emphasize the importance of parameterizing the downwelling longwave radiation as being coupled to the surface temperature. Indeed, our methodology could be used to parameterize the longwave forcing in ocean and land surface models, since we have explicit expressions for the downwelling longwave in terms of near-surface air properties [e.g., Eqs. (1) and (3)], and robust relationships to describe how these properties vary with near-surface air temperature (e.g., Fig. 3).

Here we have also extended the SR21 model to include the impacts of clouds on the downwelling longwave radiation, and investigated the impact of clouds on longwave forcing and feedbacks. The crude model of clouds we have introduced assumes that they radiate as a blackbody at the near-surface air temperature and mask the clear-sky longwave for the fraction of sky that they cover. These assumptions are poor for anything other than very low clouds: in practice, clouds will radiate at colder temperatures the higher their altitude according to the local lapse rate. Therefore, unsurprisingly, our crude model overestimates the all-sky downwelling longwave in the CMIP5 models at a given time by 8–23 W m⁻² (depending on model), but with correlations above 0.91 for all five CMIP5 models (not shown). The strong correlation is because the overestimate takes the form of a nearly spatially uniform offset. Since this offset cancels when taking the difference in downwelling longwave between two time periods, it does not affect our forcing/feedback calculation, and hence the crude cloud model leads to satisfactory results.

Our analytic expressions for longwave forcing and feedbacks with [Eq. (16)] and without [Eq. (8)] clouds provide a transparent view of the effects of clouds on changes in downwelling longwave (also see Fig. 6). The biggest effect is the masking of the clear sky, which reduces both the radiative forcing and the clear-sky feedbacks according to the fraction of cloud cover (48%–76% in the global mean, depending on model). The second effect is an enhancement of the temperature feedback due to the clouds behaving as a blackbody at (approximately) the near-surface air temperature for the fraction of sky that they cover. The third effect—the longwave “cloud feedback”—involves changes in cloud cover modifying effects 1 and 2. As noted in previous studies, these changes in cloud cover are only weakly (if at all) linked to local changes in near-surface temperature (Previdi 2010; Vargas Zeppetello et al. 2019), although they are weakly correlated (Fig. 3f). The cloud feedback is relatively small in the global mean, accounting for only 2% of the total change in all-sky downwelling longwave (Fig. 10d). However, locally it can be much larger: averaging 11% of the change in longwave for latitudes above 60°, and accounting for ∼50% of the longwave feedback in some midlatitude ocean regions in certain models (Fig. 9).

In summary, we have developed a simple and transparent framework for diagnosing forcing and feedback effects in changes to downwelling longwave, which performs comparably to previous more complex methods. The optical depth data required to implement the method is provided via an online repository (see the data availability statement below).

Data availability statement.

The optical depth data required to implement our method is provided via an online repository at https://doi.org/10.5281/zenodo.5034082, with a detailed description available in Shakespeare and Roderick (2021). Access to CMIP5 data was obtained through the National Computational Infrastructure (NCI), Canberra, Australia, which is funded by the Australian government.