Forcing Dependence of Atmospheric Lapse Rate Changes Dominates Residual Polar Warming in Solar Radiation Management Climate Scenarios

Simulations of solar radiation management (SRM) geoengineering using comprehensive general circulation models show a residual surface warming at high latitudes. Previous work attributes this to the difference in forcing structure between the increase in greenhouse gases and decrease in insolation, but this neglects the role of the induced reduction in atmospheric energy transport. Here we show that the difference in vertical structure of temperature change between increasing CO2, decreasing insolation, and decreasing atmospheric energy transport is the dominant reason for the residual near‐surface warming at high latitudes. A single‐column model (SCM) is used to decompose the high‐latitude temperature change and shows the importance of the enhanced near‐surface warming from the CO2 increase in explaining the residual polar warming. This suite of models invites caution when attributing high‐latitude surface temperature changes to the lapse rate feedback, as various forcings and nonlocal processes affect the vertical structure of temperature change differently. Plain Language Summary Solar radiation management (SRM) geoengineering has been proposed as a way of counteracting the warming effects of increasing greenhouse gases by reflecting solar radiation. When the carbon dioxide concentration (CO2) is quadrupled and the solar constant is reduced in climate models to reach zero global mean surface temperature change, there is still residual warming in polar regions. Previous analyses suggested that it was caused by the latitudinal difference in forcing between the CO2 increase and insolation reduction. This work shows the importance of the differences in vertical structure of atmospheric temperature change between the CO2 increase and solar radiation reduction in explaining this residual polar warming. This underlines the importance of considering the vertical structure of temperature change caused by a given forcing when trying to understand what shapes the pattern of surface temperature change.


Introduction
Proposed solar radiation management (SRM) geoengineering schemes aim to counteract the radiative forcing and warming from anthropogenic emissions. One widely discussed method of SRM is to inject sulfate aerosols or their precursors in the stratosphere. Simulations of aerosol injection produce similar surface and tropospheric temperature changes to simulations that simply reduce the solar constant, although the stratosphere and the partitioning between direct and diffuse solar radiation reaching the surface differ between the two experiments (Kalidindi et al., 2015). The experiment G1 of the Geoengineering Model Intercomparison Project (GEOMIP) consists of reducing the solar constant to compensate for abruptly quadrupled CO 2 concentrations in fully coupled general circulation models (GCMs) (Kravitz et al., 2011). The temperature difference between the preindustrial and G1 experiments shows a residual polar warming: The surface air temperature change is positive near the poles and slightly negative in the tropics (Stocker et al., 2013). Figures 1a and 1c show the atmospheric and surface temperature change, respectively, between the geoengineered G1 experiment and preindustrial control climate from five comprehensive climate models (listed in the legend of Figure 1c). While there is a slight surface cooling in the tropics, the high latitudes of both hemispheres have from 0.5 K to 2 K of residual warming. This residual polar warming has important consequences for the shift in the intertropical convergence zone (ITCZ) and changes in atmospheric energy transport in solar geoengineered climates (Russotto & Ackerman, 2018). It is also relevant to our understanding of the polar amplification of surface temperature change and vertical structure of temperature change under increased CO 2 (Henry & Merlis, 2019;Manabe & Wetherald, 1975;Pithan & Mauritsen, 2014).
This residual polar warming is commonly explained by the difference in latitudinal forcing structure between the increase in greenhouse gases and the decrease in insolation, which leads to a positive top-ofatmosphere (TOA) forcing at the poles, a negative TOA forcing in the tropics, and a near-zero global-mean TOA forcing (Govindasamy & Caldeira, 2000). The tropically amplified CO 2 forcing results from the climatological atmospheric lapse rate being larger in the tropics than near the poles (Huang et al., 2016), although it is more latitudinally homogeneous than the forcing from reduced insolation (Hansen et al., 1997). This explanation, however, does not account for changes in atmospheric energy transport that result from latitudinally inhomogeneous TOA forcing. In Merlis and Henry (2018), we compute an analytic estimate of the effect of the different latitudinal structure of the solar and CO 2 forcings on the surface air temperature change in geoengineered climates that includes the effect of energy transport: We find that this effect accounts for approximately half of the total residual polar warming in the absence of regional climate feedback mechanisms.
The vertical structure of atmospheric temperature change is enhanced in the upper troposphere in the tropics and enhanced near the surface in high latitudes ( Figure 1a). We will use "top-heavy" and "bottom-heavy" to mean enhanced in the upper troposphere and near the surface, respectively. In the tropics, the atmosphere is close to radiative-convective equilibrium: Radiative cooling is balanced by convective heating. The vertical structure of temperature is approximately determined by the moist adiabat; hence, the lapse rate is uniquely a function of surface temperature and relative humidity. Therefore, the lapse rate change depends only on the surface temperature change, assuming no change in relative humidity (Xu & Emanuel, 1989). Near the poles, however, the atmosphere is close to "radiative-advective" equilibrium: Warming from atmospheric energy transport is balanced by radiative cooling. Cronin and Jansen (2016) use an analytic radiative-advective model of the high-latitude atmosphere to show that the lapse rate response differs depending on the nature of the forcing. In their model, a positive surface forcing (e.g., an increase in convergence of ocean heat transport or absorbed solar radiation at the surface) induces a destabilizing lapse rate change, a positive longwave radiative forcing induces a more destabilizing lapse rate change than the surface forcing, and an increase in atmospheric energy transport and/or solar atmospheric heating induces a weakly stabilizing or neutral lapse rate change. Moreover, they suggested that each additional feedback such as water vapor, clouds, or surface albedo would induce a different lapse rate response. This simple model led to the important insight that the high-latitude lapse rate change is forcing dependent. However, a change in atmospheric energy transport leads to a vertically uniform temperature change because of the simplified representation of the energy transport's vertical structure in the analytic radiative-advective model. Atmospheric energy transport convergence has been suggested to preferentially affect the middle troposphere in high latitudes in comprehensive climate models (Laliberté & Kushner, 2013). In this paper, we compare idealized and comprehensive GCM simulations of SRM. We quantify the contributions of forcings and feedbacks to the total high-latitude lapse rate change and concomitant surface temperature change using a single-column model (SCM). The central result is that the CO 2 forcing has a temperature response that is more surface-enhanced than the reductions in solar forcing and atmospheric energy transport, leading to a relatively large surface temperature response for the small total forcing.

Idealized GCM Experiment
We implement an SRM experiment using an idealized atmospheric GCM. A version of the Geophysical Fluid Dynamics Laboratory (GFDL) atmospheric GCM is used with no clouds, comprehensive clear-sky radiation, annual-mean insolation, and aquaplanet surface boundary conditions with no sea ice. This setup is similar to Merlis et al. (2013) and to the Model of an Idealized Moist Atmosphere (MiMA) (Jucker & Gerber, 2017). In order to compensate for the cooling radiative effect of clouds, the surface albedo is set to be an approximation of Earth's TOA albedo ( Figure S1 in the supporting information) instead of prescribing an idealized cloud distribution (Merlis et al., 2013) or uniformly increasing the surface albedo (Jucker & Gerber, 2017). The control simulation has a CO 2 concentration of 300 ppm, and the solar constant is 1,365 W m −2 . The solar constant in the SRM run is decreased to 1,317 W m −2 (a 3.5% reduction) in order to get zero-mean surface air temperature change when the CO 2 concentration is quadrupled. Figure S2a shows the atmospheric temperature for the control simulation, and it compares well with Earth's climate.
The surface boundary condition of the idealized atmospheric GCM 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 skin temperature is interactively computed using the surface radiative and turbulent fluxes, which are determined by bulk aerodynamic formulae. A k-profile scheme with a dynamically determined boundary layer height is used to parametrize 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. The model uses the comprehensive radiation scheme described in Anderson et al. (2004) with annual mean insolation and a solar constant equal to 1,365 W m −2 . The surface has no representation of sea ice other than the surface albedo distribution; hence, there is no surface albedo feedback. All simulations are run for 6,000 days with time averages over the last 3,000 days shown, when all climate states have reached a statistical steady state.
We perform four simulations. The control simulation has 300 ppm of CO 2 and a 1,365 W m −2 solar constant. The increased CO 2 simulation has 1,200 ppm of CO 2 . The reduced solar constant experiment has a 1,317 W m −2 solar constant. The SRM experiment has both increased CO 2 and a reduced solar constant. The value for the reduced solar constant was determined in order to get near-zero global surface air temperature change. Figure S2b shows the temperature difference between the control and increased CO 2 simulation, and Figure  S2c shows the temperature difference between the control and decreased solar constant simulation. Figure 1b shows the atmospheric temperature change between the control and SRM idealized GCM simulations, which has a similar structure to that of comprehensive GCMs (Figure 1a). The surface temperature change between the control and SRM run in the idealized GCM (black) is also reasonably close to the comprehensive GCMs (Figure 1c). In addition, the change in atmospheric energy transport is similar to that of comprehensive GCMs (a reduction of 0.1 PW in midlatitudes; Figures S3b and 1 of Russotto & Ackerman, 2018). This model's ability to reproduce the temperature and atmospheric energy transport changes from comprehensive GCM simulations suggest that processes present in this idealized model are sufficient to explain the ensemble-mean changes in SRM experiments. The idealized GCM underestimates the surface temperature change from comprehensive GCMs in the northern extratropics while slightly overestimating warming in the southern extratropics. This may be due to the absence of the surface albedo feedback (which enhances Northern Hemisphere warming) and ocean heat uptake (which moderates Southern Hemisphere warming) in the idealized GCM. We proceed to decompose the high-latitude temperature response in this GCM to identify the mechanism responsible for residual polar warming.

SCM Experiments
To decompose the high-latitude temperature change in the idealized GCM simulation, we use the SCM from the ClimLab python package for process-oriented climate modeling (Rose, 2018) to emulate the high-latitude troposphere of the idealized GCM. The temperature tendency budgets for atmospheric and surface temperature are given by the following equations: (1) ( 2) where t is time and p is pressure (with 40 pressure levels). The subscripts "rad", "conv", "adv", "cond", "SH", and "LH" refer to radiative, convective, advective, condensation, sensible heat flux, and latent heat flux temperature tendencies, respectively. The radiative, convective, sensible heat flux, and latent heat flux temperature tendencies are computed interactively. The RRTMG radiation scheme is used for the computation of shortwave and longwave radiative temperature tendencies. The surface albedo and control insolation are set such that the upwelling and downwelling TOA shortwave radiation match the idealized GCM simulation poleward of 80°. The horizontal atmospheric energy transport induces a temperature structure stable to convection, so including a convection parametrization has no effect. The surface sensible and latent heat fluxes are computed using bulk aerodynamic formulae with 5 × 10 −2 drag coefficient and 5 ms −1 near-surface wind speed (Rose, 2018). This SCM is similar to the high-latitude column model in Payne et al. (2015), but instead of parameterizing the atmospheric heat flux convergence and using a fixed distribution of specific humidity, we prescribe them from the idealized GCM. We are unaware of other analyses of SRM that use SCMs, a classic approach in climate science (Manabe & Wetherald, 1967).
Values from the idealized GCM experiments averaged poleward of 80°N are used to prescribe the specific humidity profile, which affects the radiation and surface latent heat flux. In addition, the time-mean advection and condensation temperature tendency profiles from the idealized GCM simulations are added as external temperature tendency terms to simulate the dry and moist components of atmospheric energy transport convergence, respectively. The advective temperature tendency term is calculated in the GCM as the difference in temperature tendency before and after running the dynamics module, and it, therefore, contains both the horizontal and vertical advection temperature tendencies. Because we prescribe atmospheric energy transport to the column model, we consider it to be a "forcing" in this context.
The climatological temperature profiles of the idealized GCM and SCM are similar ( Figure S4), though the SCM has an overly strong near-surface temperature inversion compared to the GCM. This may be due to the absence of boundary layer scheme in the SCM, which would smooth differences between the surface and lower atmospheric layers. Similarities between the temperature profiles simulated by the idealized GCM and by the SCM still hold when the latitudinal bounds of the high latitudes is set to 60°(see Figure S5).
We run four simulations: quadrupled CO 2 , reduced insolation, perturbed energy transport, and a simulation with all perturbations (4xCO 2 and reduced insolation, energy transport, and specific humidity). A summary of the specific parameter settings for each run are given in Table S1. The individual perturbations in the SCM add up to the full response to SRM ( Figure S8), which lets us assume that the full response to SRM is composed of a linear superposition of these individual changes. Figure 2a shows that the idealized GCM's vertical temperature change structure in high latitudes (black, "iGCM") is similar to that of five CMIP5 models in the Arctic (colors, listed in legend). Figure 2b shows the temperature change structure for the different SCM simulations, with points showing surface (or skin) temperature changes. The CO 2 -only simulation ("CO 2 ", red) has a bottom-heavy temperature change structure and a surface temperature increase of 3.1 K. The insolation reduction simulation ("Solar", yellow) has a more vertically uniform cooling structure and a surface temperature change of −1.5 K. The energy transport change ("ET", green) preferentially cools the lower atmosphere and leads to a −0.5 K surface temperature change. Finally, when all perturbations (CO 2 , insolation, water vapor, and energy transport) are included ("All", black), the surface temperature change is 1.1 K, as was simulated by the idealized GCM (1.1 K).
The differences between the comprehensive GCMs (Figure 2a, colors), the idealized GCM (Figure 2a, black), and the SCM (Figure 2b, black) can be due to the different radiation schemes, to the time-averaging of boundary conditions in the SCM, and to the absence of climate components such as clouds, sea ice, and ocean circulation. Our CO 2 -only SCM simulation is different from the GCM simulation of Stuecker et al. (2018), where CO 2 forcing is only applied at high latitudes and all other components of the climate system (including energy transport) are free to change. Their diagnosis of the warming contribution from the radiative forcing (their Figure 4) is also different as it separately quantifies the contribution of the lapse rate, whereas the SCM accounts for the vertical structure of temperature change of each component Henry et al. (2020).
We calculate the forcing on the high-latitude atmospheric column for each simulation by calculating the change in outgoing longwave radiation (OLR) induced by the changes in surface and tropospheric temperature (here, the tropopause is set at 200 hPa). The temperature kernel of the column model is calculated by separately increasing the surface and each pressure level by 1 K and calculating the resulting OLR increase (see Figure S7 for kernel structure). The total feedback determines the surface temperature change per unit of forcing and is decomposed into the Planck, lapse rate, and water vapor feedbacks. The change in water vapor is small and induces a negligible change in surface temperature, so it is omitted here (see Figure S8). The Planck feedback is computed as the OLR change from a 1 K temperature increase at the surface and in the troposphere. Its value is −2.6 W m −2 K −1 , which is comparable to comprehensive GCM estimates in high latitudes (Feldl & Bordoni, 2016). The temperature feedback is computed as the OLR increase induced by the surface and tropospheric temperature change divided by the surface temperature change, and the lapse rate feedback as the temperature feedback minus the Planck feedback. The lapse rate feedback of the "All" experiment is 1.9 W m −2 K −1 , which is comparable to the high-latitude lapse rate feedback of the idealized GCM SRM experiment computed using aquaplanet temperature kernels (Feldl et al., 2017) (not shown).
The forcing and lapse rate feedback associated with each simulation are shown in Table 1. There is a 2.1 W m −2 positive TOA forcing from the difference between the CO 2 and solar forcings and a −1.4W m −2 reduction in atmospheric energy transport convergence (comparable to the change in high-latitude convergence of atmospheric energy transport in the idealized GCM). The relatively large surface temperature response in the "All" experiment (1.1 K) for a small forcing (0.63 W m −2 if the change in atmospheric energy transport is considered as a forcing on the high-latitude column) can be attributed to the very destabilizing lapse  rate feedback (1.9 W m −2 K −1 ). If we use the lapse rate feedback of the CO 2 -only simulation (0.79 W m −2 K −1 ), then the surface temperature change would be 0.4 K instead of 1.1 K. We are thus left with explaining this very destabilizing lapse rate feedback that provokes most of the residual polar warming in SRM simulations.
As shown in Figure 2b, the vertical structure of temperature change in the "All" experiment can be decomposed into the effect of individual forcings and feedbacks. The warming from the increase in CO 2 is very bottom-heavy (0.79 W m −2 K −1 lapse rate feedback), whereas the cooling from changes in insolation and energy transport are more vertically homogeneous. When this vertically homogeneous cooling is superimposed on the bottom-heavy warming from CO 2 , it decreases the OLR faster than it decreases the surface temperature. The vertical gradient in temperature is almost left unchanged (compare "CO 2 " and "All" in Figure 2b), the forcing on the atmospheric column is small, and the surface temperature change is 1.1 K. Given the importance of the lapse rate changes between forcing agents, we turn to a more idealized SCM to develop a theoretical understanding of forcing dependence of high-latitude lapse rate response.

Simplified Analytical Model
The analytical model of the high-latitude atmosphere in radiative-advective equilibrium (Cronin & Jansen, 2016) was used to show the forcing dependence of high-latitude lapse rate changes. In their model, an increase in greenhouse gases leads to a more bottom-heavy temperature change than an increase in atmospheric or surface forcing. Details including the climatological temperature and temperature changes of the analytical radiative-advective model are reproduced from Cronin and Jansen (2016) in Text S1.
The essence of the mechanism for the forcing dependence of the high-latitude lapse rate change is contained in the radiative equilibrium limit (no advection) of the Cronin and Jansen (2016) model, so we discuss this simpler case. We impose the convergence of atmospheric energy transport at the surface to keep the temperatures of this pure radiative equilibrium model similar to high latitudes. The parameters are F S = 120 W m −2 and τ 0 = 3, where F S is the surface forcing and τ 0 the total longwave optical depth. To further simplify the calculations, we do not include an atmospheric window and the optical depth decays as the square of the pressure normalized by the surface pressure.
It is well understood that an atmosphere in pure radiative equilibrium is statically unstable; however, the argument for the perturbation temperature is fundamentally the same as for an atmosphere in radiative-advective equilibrium and easier to understand. In this model, increasing the total longwave optical depth is analogous to increasing atmospheric CO 2 , and decreasing the surface forcing is analogous to decreasing the TOA insolation (atmospheric absorption of solar radiation is ignored). 10.1029/2020GL087929 Figure 3a shows the temperature change from increasing the total longwave optical depth (red) and from decreasing the surface forcing (yellow). The total longwave optical depth is increased from 3 to 3.2 and induces an instantaneous reduction in OLR by 3.6 W m −2 , which we use for the magnitude of the reduction in surface forcing. The vertical structure of temperature change is more bottom-heavy for an increase in longwave optical depth ("CO 2 ", red) than for a decrease in surface forcing ("Solar", yellow). The forcing dependence of the lapse rate feedback thus does not depend on the presence of atmospheric energy transport convergence.
To explain this forcing dependence, we derive a simple expression for the temperature structure of the polar troposphere from the two-stream Schwartzchild equations for gray radiative transfer (Equations 1 and 2 in Text S1) with the simplifications described above: where F S is the surface forcing, τ 0 the total longwave optical depth, p the pressure, and p 0 the surface pressure. This equation shows that temperatures must change at all levels for a change in F S , but they do not change as p goes to zero for a change in τ 0 . Figure 3b shows the difference in 2σT 4 from changes in τ 0 and F S . When τ 0 is increased from 3 to 3.2, the change in 2σT 4 is zero at the TOA and δτ 0 F S at the surface. When F S is reduced from 120 W m −2 to 116.4 W m −2 , the change in 2σT 4 is δF S at the TOA and δF S (1 + τ 0 ) at the surface. Physically, an increase in greenhouse gases corresponds to a deepening of the atmosphere with respect to optical depth and the net upwards longwave radiative flux at the TOA between two equilibrium states is not affected, whereas a change in insolation affects the longwave radiative flux from the surface to the TOA. This reasoning applies for an atmosphere in pure radiative equilibrium, as well as an atmosphere in radiative-advective equilibrium (see Figure S9 for an analog of Figure 3 for an atmosphere in radiative-advective equilibrium).

Conclusion
In climate model simulations of SRM scenarios, where the solar constant is reduced to compensate for an increase in CO 2 , there is residual warming in polar regions ( Figure 1). We decompose the contributions of the CO 2 increase, insolation decrease, and energy transport change to the vertical structure of high-latitude temperature change in order to understand why this residual polar warming occurs. The high latitudes are close to radiative-advective equilibrium: The cooling from radiation is balanced by warming from atmospheric energy transport convergence. Where convection plays a dominant role, the tropospheric temperature change of the atmospheric column is close to independent of the forcing or feedback inducing the temperature change. Without convection, each forcing and feedback induces a different lapse rate response.
In the SRM experiment, the latitudinal structure of the forcing is such that the high latitudes have a positive TOA radiative forcing and a reduction in atmospheric energy transport convergence. If we consider the atmospheric energy transport convergence as a forcing on the high-latitude column, then the positive TOA forcing and reduction in atmospheric transport convergence add up to give a small forcing. However, the surface temperature change is relatively large, which is diagnosed to be a consequence of a destabilizing lapse rate feedback.
The vertical structure of the high-latitude temperature change of an idealized GCM is decomposed using an SCM (Figure 2b). It is shown that the warming from CO 2 alone is very bottom-heavy, whereas the cooling from a reduction in insolation and atmospheric energy transport are more vertically homogeneous. The combination of a bottom-heavy warming and a vertically homogeneous cooling gives a relatively large surface warming despite a small forcing. Using the no advection limit of an analytical model of the high-latitude atmosphere in radiative-advective equilibrium (Cronin & Jansen, 2016), we show that the difference in the vertical structure of temperature changes from increasing CO 2 and decreasing insolation result from different changes in the boundary conditions of the radiative flux (Figure 3). The increase in CO 2 deepens the atmosphere with respect to optical depth, whereas the change in insolation modifies the resulting longwave radiative fluxes through the whole atmosphere (surface to TOA).
The dominance of the forcing agent dependence of lapse rate changes in provoking residual polar warming in SRM simulations can be assessed by replacing the lapse rate feedback with that of CO 2 (Table 1) or considering models without this feedback (Merlis & Henry, 2018), both of which substantially underestimate the polar warming. An implication of the lapse rate dependence on forcing agent is that there will be residual polar surface warming even if the spatial distribution of scattering aerosols can be optimized to perfectly offset the local TOA greenhouse gas forcing, although not all SRM scenarios have that objective.

Data Availability Statement
Data sets and code for this research are available at https://zenodo.org/record/3711989 (Henry, 2020).