## 1. Introduction

The change in atmospheric temperature due to radiative forcing is vertically and horizontally inhomogeneous. The zonal-mean warming is amplified near the surface at the poles and in the upper troposphere in the tropics. Polar amplification (PA) of surface air warming is a common feature of climate model projections (Manabe and Wetherald 1975; Pithan and Mauritsen 2014). Observations of the recent climate also show faster transient warming in the Arctic (IPCC 2013), while Antarctic amplification is delayed as a result of ocean heat uptake (Manabe et al. 1991). In the tropics, there is amplified warming in the upper troposphere relative to the surface in future climate projections that is comparable to that of a warmed moist adiabat (Santer et al. 2005).

There are several mechanisms that are thought to contribute to the pattern of temperature change, and the challenge is to understand the relative importance of each mechanism and the interactions between them. The main contributors to the PA of surface air warming are thought to be the surface albedo feedback (Winton 2006; Graversen et al. 2014), an increase in poleward energy transport (Alexeev and Jackson 2013; Lee 2014; Merlis and Henry 2018), and a destabilizing polar lapse rate feedback (Graversen et al. 2014; Pithan and Mauritsen 2014; Payne et al. 2015). The role of cloud and water vapor feedbacks on the PA of surface air warming is uncertain. According to Pithan and Mauritsen (2014), the ensemble-mean cloud change has a small positive impact on PA and water vapor counteracts PA via their radiative feedbacks in comprehensive climate models. Winton (2006) found that the total longwave feedback (which includes cloud, water vapor, and temperature) contributes to PA. Finally, the top-of-atmosphere (TOA) net radiation response to a vertically invariant temperature change, the Planck feedback, can have spatial structure that tends to amplify polar warming. The structure arises from the nonlinearity of the temperature dependence of the blackbody emission *E* of radiation given by the Stefan–Boltzmann law, ^{1} (Merlis 2014), and we show both in what follows.

Radiation also plays a key role in determining the vertical structure of temperature in the troposphere. In high latitudes, radiative cooling is thought to be critical in determining the mean stratification, particularly inversions (Curry 1983). In low latitudes, the troposphere’s stratification is close to moist adiabatic, but biases in the climatologies of comprehensive general circulation models (GCMs) and concomitant differences in projected changes may be related to discrepancies in the treatment of radiative processes between GCMs (Po-Chedley and Fu 2012; O’Gorman and Singh 2013). In addition, changes in cloud radiative effects leave a distinct fingerprint on tropical lapse rate changes in GCM simulations (Mauritsen et al. 2013). Hence radiation is key in determining the vertical structure of the troposphere at high latitudes and has a possible role in determining the tropical stratification. The nonlinearity of the Stefan–Boltzmann law may then impact the vertical structure of temperature change by increasing the temperature response of the cold upper layers of the troposphere relative to the warm lower layers.

Here, the role of the Stefan–Boltzmann law’s nonlinearity on the structure of radiatively forced temperature change is assessed. The Stefan–Boltzmann law has been examined as a stabilizing nonlinearity for the global climate sensitivity under large perturbations (Bloch-Johnson et al. 2015), while our focus is on the role of this nonlinearity on the pattern of global warming, rather than its amplitude. First, the spatial variations of the Planck feedback and CO_{2}’s radiative forcing for Earth’s atmosphere are compared using reanalysis data and a radiative kernel derived from a comprehensive GCM, as well as the Planck feedback and radiative forcing derived from a moist idealized GCM (section 2), which reveals a combined effect that would give rise to tropical amplification. We then compare the pattern of warming of an idealized aquaplanet atmospheric GCM (Frierson et al. 2006) with a gray radiation scheme to one that has a linearized version of the gray radiation scheme. It is linearized by replacing

## 2. Patterned warming from the Stefan–Boltzmann law

A consequence of the nonlinearity of

The simple argument for the role of the Planck feedback on polar amplification relies on the assumption that the radiative forcing is latitudinally uniform. However, it is important to note that the TOA longwave radiative forcing

Huang et al. (2016) showed radiative forcing estimates from homogeneous changes in greenhouse gas concentration using a contemporary reanalysis estimate of the atmospheric state and an accurate radiative transfer calculation. They subsequently related the TOA radiative forcing variations to climatological factors beyond the mean temperature and showed that the magnitude of the forcing is strongly dependent on surface and atmosphere variables, particularly the lapse rate. Figure 1a shows the fractional latitudinal variation of instantaneous radiative forcing from doubling CO_{2} (Huang et al. 2016, their Fig. 1c) and the fractional variation of the Planck feedback from the temperature radiative kernel derived from the Geophysical Fluid Dynamics Laboratory AM2 comprehensive atmospheric GCM (Soden et al. 2008, their Fig. 6) using the local definition of feedback. If the Planck feedback is the only active feedback and all energy transports are unchanged, the surface air temperature change can be calculated as _{2} would give a tropically amplified surface air warming that arises from the greater equator-to-pole variation of the forcing compared to the Planck feedback (Fig. 1a). The stronger equator-to-pole variation of the forcing arises primarily from the dependence of the forcing on the climatological lapse rate (Huang et al. 2016): the high latitudes have weak forcing because they are closer to the isothermal regime, where there is no greenhouse effect, than low latitudes. In summary, the climatological meridional temperature gradient gives rise to a spatially varying Planck feedback that would, in isolation, cause polar amplified warming. However, this is more than offset by the climatological temperature distribution’s effect on the radiative forcing, where both the mean temperature and lapse rate’s equator-to-pole decrease produce a tropically amplified forcing.

A similar argument to the one made for the role of the Planck feedback on polar amplification can be made for the vertical dependence of temperature change in response to radiative forcing. The effect of the nonlinearity of ^{2} This provides additional motivation for using a GCM to isolate the effect of the Stefan–Boltzmann nonlinearity. It is also worth mentioning that the perturbation longwave flux from increased longwave optical depth has vertical structure, which could be considered in a more detailed assessment than we perform here.

We aim to test the effect of the nonlinearity of

## 3. Idealized GCM

We use the moist idealized GCM described in Frierson et al. (2006) with the modifications and parameter values described in O’Gorman and Schneider (2008). The surface boundary condition is an aquaplanet with a slab mixed-layer ocean with the heat capacity of 1 m of water and no representation of ocean heat transport. The GCM’s spectral dynamical core has T42 spectral truncation for a nominal horizontal resolution of 2.8° × 2.8° and 30 vertical levels. The sea surface temperature (SST) is interactively computed using the surface radiative and turbulent fluxes, with the surface turbulent fluxes determined by bulk aerodynamic formulas. A *k*-profile scheme with a dynamically determined boundary layer height is used to parameterize the boundary layer turbulence. The GCM uses a simplified Betts–Miller convection scheme (Frierson 2007). The large-scale condensation is parameterized such that the relative humidity does not exceed one and the condensed water is assumed to immediately return to the surface. Incoming solar radiation is an idealized second Legendre polynomial function that is representative of Earth’s annual mean with no seasonal or diurnal cycle. For longwave radiation, the model has gray radiative transfer, with the longwave optical depth as in O’Gorman and Schneider (2008). The radiative fluxes are a function of temperature and pressure alone, so there are no water vapor or cloud feedbacks. The surface has no representation of sea ice and has a uniform surface albedo, so there is no surface albedo feedback. All simulations are 2000 days with time averages over the last 1000 days shown, when all climate states have reached a statistical steady state.

To calculate the value of the radiative forcing, simulations where the SST is prescribed to the time mean and zonal mean of the control simulation SST with both control and perturbed optical depth were performed. The perturbation in radiative flux at the top of the atmosphere after the stratospheric and tropospheric temperature has changed directly in response to the optical depth (as opposed to changes mediated by surface temperature variations) defines the troposphere-adjusted radiative forcing (Hansen et al. 2005), which is also known as the effective radiative forcing. This troposphere-adjusted forcing is a more accurate measure of the radiative forcing and alters our estimate of the lapse rate feedback compared to using instantaneous radiative forcing, although the difference in spatial structure between troposphere-adjusted and instantaneous radiative forcing remains small (not shown). This gray radiation model has less stratospheric cooling than comprehensive climate models. However, the focus of this study is the troposphere and any effect this different stratospheric cooling may have on the troposphere is taken into account by using this adjusted forcing. The latitudinal structure of the troposphere-adjusted radiative forcing of the idealized GCM closely resembles that of instantaneous CO_{2} radiative forcing from Huang et al. (2016) (Fig. 1, dashed lines).

Figure 2a shows the time- and zonal-mean temperature field for the control simulation of the standard configuration of the idealized moist GCM. The temperature field is comparable to that of Earth’s annual mean, although the tropospheric lapse rate is everywhere positive, even in high latitudes, and the stratosphere is not realistically represented due to the use of gray radiation and the absence of ozone. We can, therefore, assess the role of the Stefan–Boltzmann nonlinearity on the pattern of radiatively forced temperature change for a regime similar to that of Earth’s climate.

The radiation source function, *A* and *B* are chosen to test the sensitivity of our result to these constants. The chosen values of *A* and *B* are displayed in the legend of Fig. 3. The values of *A* and *B* are chosen such that *A* with the same value of *B* with two more linearizations. With *A* for this value of *B*. In total, we present six linearizations. We note that none of the simulations reach a low enough temperature to have a negative value for longwave emission (*T*). While alternative linear radiation simulations with *B* is constant with latitude as the Planck feedback is then spatially uniform and thus has no effect on the horizontal structure of temperature change.

Figures 2a and 2b show the control temperature field for the nonlinear radiation simulation and a linear radiation simulation (*A* parameter (note the spread between the three simulations with *A* parameter sets the amount of temperature-independent emission of radiation, so a higher *A* means greater longwave emission and a colder control climate. The *B* parameter sets the amount of temperature change required for a certain change in emission and has a significant impact on the amount of global warming (Fig. 3b).

The radiative forcing due to an increase in greenhouse gas concentration is modeled by increasing the longwave optical depth. In this model, the optical depth at the surface varies in latitude as *ϕ* is the latitude. The control values for *p* is pressure and _{2} (see appendix). While the global-mean forcing is large in the perturbation simulations presented in this paper, the equator-to-pole variation in forcing is similar to estimates for Earth (Fig. 1a), although the forcing in the Southern Hemisphere is higher in the Earth estimates, which is partially due to the high altitude of the Antarctic continent and stratospheric temperature structure (Smith et al. 2018). It is also worth noting that we compare the forcing from a model with gray radiation, which has no cloud representation, to an all-sky Earth estimate of CO_{2} forcing.

Figures 2c and 2d show the temperature change field for the nonlinear radiation simulations and corresponding linear radiation simulations. Both temperature change patterns are broadly similar. Despite the idealized nature of the GCM, they are in good agreement with the ensemble-mean pattern of temperature change in the comprehensive GCMs shown in the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) (IPCC 2013, their Fig. 12.12); however, Figs. 2c and 2d represent temperature change between two equilibrium states while the IPCC figures represent the transient climate response, so the surface air temperature change in the Southern Hemisphere is not polar amplified in the IPCC figures. The idealized GCM has cooling in part of the stratosphere, although its structure and magnitude are unrealistic due to the simplicity of this model’s radiative transfer scheme.

Figure 3b shows the surface air temperature change between the control and increased longwave optical depth simulations for nonlinear radiation (black) and the different linearizations of *pattern* of warming. Figure 3c shows the surface air temperature change normalized by its global mean. For all linearizations, the simulated pattern of warming is quite similar to the nonlinear radiation simulation. These experiments then show that the linearization of the radiation scheme does not substantially affect the pattern of surface air warming. The amount of normalized surface temperature change at the poles varies by approximately 15% across the radiation parameters (0.3 K K^{−1} relative to a total normalized polar warming of 2 K K^{−1}). However, the range of linear radiation simulations includes both those with slightly more and those with less polar amplified warming. This result therefore contradicts the expectation that the nonlinearity of

## 4. Horizontal structure of warming

To compute the Planck feedback, a second temperature field is initialized at each time step in the GCM’s radiation scheme with a temperature value incremented by 1 K at the surface and for all of the GCM’s vertical levels. The Planck feedback is then computed as the time mean of the difference between the outgoing longwave radiation (OLR) corresponding to the GCM’s prognostic temperature field in the control simulation and the OLR corresponding to the temperature field incremented by one. For the linear radiation simulations, the Planck feedback is trivially equal to the *B* parameter of the

Figure 4a shows the area-weighted polar mean *B* values. We note that if the forcing is increased in a model that has an amplitude-independent ratio of polar to tropical warming, the points corresponding to the simulation with increased forcing will be further from the 1:1 line even though the pattern of warming is the same. Likewise, two models with identical warming patterns, but different global-mean climate sensitivity would appear to have different degrees of polar amplification on this diagram. Therefore, we normalize all points by the global-mean surface air temperature change in Fig. 4b and discuss this in what follows:

- All linear radiation simulations have a zero variable Planck feedback contribution by definition
. The nonlinear radiation simulations’ variable Planck contribution is above the 1:1 line and contributes to polar amplification according to this energy budget analysis (black x). We note that the variable Planck feedback term depends on the latitudinal profile of the Planck feedback and temperature change [Eq. (2)]. - The lapse rate–related temperature change is more positive in linear radiation simulations than in nonlinear radiation simulations over all latitudes (colored triangles are farther up and to the right compared to the black triangle, except for the extreme
case) due to the colder upper tropospheric layers having a larger magnitude temperature increase in the nonlinear radiation case (section 5). - Once normalized, the change in moist static energy transport is similar between linear radiation and nonlinear radiation simulations (except for the extreme
case). This is consistent with diffusive closures—the change in moist static energy transport is proportional to the pattern of temperature change (Fig. 3), all else being equal. - The forcing changes slightly between the different simulations—the effect of increasing the optical depth varies depending on the radiation scheme as a result of slight differences in the control simulation temperature field. However, the variation in the forcing term in Fig. 4 arises primarily from the generally higher value of
for the linearizations presented here, as evidenced by the fact that the spread of the linear radiation simulations’ terms form a line that intersects the origin. Therefore, the forcing term is generally smaller (closer to the origin) in the linear radiation simulations, as can be seen in Figs. 4a and 4b. The spread of the linear radiation simulations’ transport term also forms a line that intersects the origin in Fig. 4a, but these collapse together when normalized by the global-mean surface air temperature change (Fig. 4b), which suggests that the variation of does not affect the normalized transport term. The lapse rate feedback terms form a line that does not intersect with the origin, which suggests this spread does not primarily arise from the variation in .

In summary, the Planck feedback functions as expected: it promotes polar amplification in the nonlinear radiation simulations and has no effect on polar amplification in the linear radiation simulations. According to the budget analysis, this difference could be offset by the change in MSE flux convergence, radiative forcing and/or lapse rate feedback to arrive at a comparable warming pattern (Fig. 3c). Figure 4b shows there are only modest differences in the MSE flux convergence change between the linear and nonlinear radiation simulations. The radiative forcing in the linear radiation simulations contributes to slightly less tropical warming than in the nonlinear radiation simulation, but not enough to compensate for the Planck feedback. The only other degree of freedom in this GCM is the lapse rate feedback. The difference in the variable Planck feedback contribution between the nonlinear and linear radiation simulations is offset by changes in the lapse rate feedback: the tendency of the lapse rate feedback in the linear radiation simulations is generally toward more polar amplification than in the nonlinear radiation simulation. The structure of the lapse rate feedback is discussed in more detail in the next section.

## 5. Global-mean feedbacks and vertical structure of warming

The perturbed simulations have about 6 K of global warming that results from a combination of a large global-mean radiative forcing

Figures 6a and 6b show the area-weighted polar mean

The interpretation that the difference in lapse rate arises from the cold-altitudes-warm-more mechanism (described in section 2) is supported by two lines of evidence. First, the changes in longwave radiative flux at different levels are approximately the same for the linear and nonlinear radiation simulations, as shown in Fig. 7. Figure 7a shows the change in net longwave radiative flux between the control and increased optical depth experiments for the nonlinear radiation configuration. Figure 7b shows the mean of the differences between the change in longwave radiative flux in the nonlinear radiation simulations and the linear radiation simulations. This mean is insensitive to the details of which linear radiation simulations are used to form it, as long as the chosen simulations span the range of *B* values. In the tropics, the difference in the change in longwave radiative flux decreases with altitude, which induces a more bottom-heavy warming in the nonlinear radiation simulation and is opposite to how the

Second, we can eliminate potentially offsetting local vertical tendencies from advection and convection by using different model configurations. To better understand the impact of linearizing

In the Rad Eq configuration of the GCM, the convection and advection processes are deactivated. The vertical temperature profile is thus determined only by how the longwave radiation of each column balances the absorbed solar radiation. The vertical structure of temperature change in the radiative equilibrium configuration decreases by about a factor of 2 from the surface to the

The RCE configuration follows the method described in O’Gorman and Schneider (2008). Convection lessens the difference in lapse rate feedback between the linear and nonlinear radiation cases in the tropics to approximately

We note that the control temperature in our simulations does not have a high-latitude inversion, unlike the Earth’s climatology. Having a high-latitude inversion would make the climatological vertical structure of temperature more homogeneous and this would reduce the high-latitude lapse rate feedback’s sensitivity to linearizing radiation because the temperature at the emission level would be more similar to the surface.

Presenting the lapse rate feedbacks across this range of atmospheric model configurations shows that the initial difference in lapse rate feedback between the linear and nonlinear radiation simulations in the radiative equilibrium configuration, where cold altitudes warm more as a consequence of the

## 6. Conclusions

Isolating the factors governing the inhomogeneous pattern of atmospheric temperature change is a central problem in climate dynamics, with implications for the atmospheric general circulation and regional climate changes. Here, we assess the role of the nonlinearity of the Stefan–Boltzmann law

We further examine the role of the nonlinearity of

Finally, we show that the nonlinearity of the Stefan–Boltzmann law affects the vertical structure of atmospheric temperature change in a manner that tends to make the lapse rate feedback more stabilizing across all latitudes. This suggests that the Stefan–Boltzmann nonlinearity is responsible for increasing the temperature response of the colder upper troposphere and reducing the temperature response of the warmer lower troposphere. We confirm this by examining the lapse rate feedbacks of pure radiative and radiative–convective configurations of the GCM. The difference in the lapse rate feedbacks between the linear and nonlinear radiation simulations in the pure radiative configuration is well understood: the cold upper layers of the troposphere warm more in the nonlinear radiation simulations to reach a similar perturbation longwave flux. This difference then propagates to the full GCM configuration, although it is attenuated by advection and convection. The sensitivity of the magnitude of the tropical upper-tropospheric warming to the treatment of radiative transfer in these idealized GCM simulations is noteworthy and suggests further examination in radiation’s role on the tropical stratification.

This work was supported by Fonds de Recherche du Québec—Nature et Technologies (FRQNT) Nouveau Chercheur award and a Natural Sciences and Research Council (NSERC) Discovery grant, as well as a Compute Canada allocation. We thank Yi Huang for providing the CO_{2} radiative forcing, Brian Soden for providing the AM2 radiative kernel, Paul O’Gorman for providing the RCE code, and Kyle Armour and Tim Cronin for helpful discussions. We thank an anonymous reviewer for encouraging the use of adjusted radiative forcing. We also thank two anonymous reviewers and Karen Shell for constructive reviews.

# APPENDIX

## Dependence of Feedback Analysis on Forcing Amplitude

In Fig. A1, we show the lapse rate feedbacks from all simulations as in Fig. 6c (solid lines), along with lapse rate feedbacks computed with smaller forcing (dashed lines), where perturbed simulations have 1.1x control optical depth instead of 1.4x control optical depth. The simulations with smaller forcing have noisier lapse rate feedbacks, but they are not systematically different than simulations with larger forcing. The noisiness of the results with smaller forcing is the reason we chose to present the simulations with larger forcing.

Our feedback calculation technique assesses the

We have also computed the Planck feedback with

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^{1}

This can account for the differing conclusions between Langen et al. (2012) and Pithan and Mauritsen (2014) about the role of the water vapor feedback for polar amplification.

^{2}

We thank Karen Shell for pointing this out.