Symmetric and Antisymmetric Components of Polar-Amplified Warming

1. Introduction

Climatological zonal-mean surface temperatures decrease from the equator toward both poles, a hemispherically symmetric signature much larger than the antisymmetric deviations therefrom. By symmetric or antisymmetric we refer to the average or difference, respectively, of each latitude with its mirror about the equator: for a given field χ, χ(φ) = χ_sym(φ) + χ_asym(φ), where φ is latitude, ${\chi }_{\text{sym}}\equiv (1/2)[\chi (\phi )+\chi (-\phi )]$ is the symmetric component, and ${\chi }_{\text{asym}}\equiv (1/2)[\chi (\phi )-\chi (-\phi )]$ is the antisymmetric component. Figure 1a illustrates this via a preindustrial control simulation in the Community Earth System Model version 1.0.4 (henceforth CESM1) general circulation model (GCM) whose formulation will be described below. Evidently, the symmetric annual-mean forcing of insolation and approximately symmetric forcing of CO₂ and other well-mixed greenhouse gases outweigh the antisymmetric components of Earth’s ocean basins, water vapor, orography, sea ice, clouds, and atmospheric and oceanic circulations.

(a) Climatological annual-mean, zonal-mean surface air temperature in a preindustrial control simulation, the hemispherically symmetric component thereof, the global mean thereof, and the global mean plus the hemispherically antisymmetric component, as indicated in the legend. The simulation was performed in a low-resolution configuration of CESM version 1.0.4, with results averaged over years 701–800. (b) Anomalous zonal-mean surface air temperature in an abrupt 4 × CO₂ simulation in the same model, averaged over four time periods as indicated in the legend. Solid curves are the full fields, and dotted curves are the corresponding symmetric component. Both panels have units in K.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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(a) Climatological annual-mean, zonal-mean surface air temperature in a preindustrial control simulation, the hemispherically symmetric component thereof, the global mean thereof, and the global mean plus the hemispherically antisymmetric component, as indicated in the legend. The simulation was performed in a low-resolution configuration of CESM version 1.0.4, with results averaged over years 701–800. (b) Anomalous zonal-mean surface air temperature in an abrupt 4 × CO₂ simulation in the same model, averaged over four time periods as indicated in the legend. Solid curves are the full fields, and dotted curves are the corresponding symmetric component. Both panels have units in K.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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(a) Climatological annual-mean, zonal-mean surface air temperature in a preindustrial control simulation, the hemispherically symmetric component thereof, the global mean thereof, and the global mean plus the hemispherically antisymmetric component, as indicated in the legend. The simulation was performed in a low-resolution configuration of CESM version 1.0.4, with results averaged over years 701–800. (b) Anomalous zonal-mean surface air temperature in an abrupt 4 × CO₂ simulation in the same model, averaged over four time periods as indicated in the legend. Solid curves are the full fields, and dotted curves are the corresponding symmetric component. Both panels have units in K.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Conversely, CO₂-forced zonal-mean surface warming—henceforth simply T—starts out appreciably antisymmetric: prevailing Southern Ocean upwelling (e.g., Armour et al. 2013; Marshall et al. 2015) impedes Antarctic warming for decades (likely reinforced by resulting changes in local lapse rates and clouds; Senior and Mitchell 2000; Rugenstein et al. 2020), while weakly negative to slightly positive radiative feedbacks in northern high latitudes (e.g., Stuecker et al. 2018) among other processes (Feldl et al. 2017; Russotto and Biasutti 2020; Henry et al. 2021) promote Arctic warming. This fast response typically gives way to a more symmetric warming pattern over subsequent centuries (Held et al. 2010), with Antarctic warming partially catching up to the Arctic in century-scale CMIP5 (Andrews et al. 2015) and CMIP6 (Dong et al. 2020) simulations. On longer time scales, polar amplification is comparable in the two hemispheres in multimillennial simulations in fully coupled GCMs (e.g., Danabasoglu and Gent 2009; Li et al. 2013; Rugenstein et al. 2019) and in runs to equilibrium in slab-ocean GCMs (e.g., Manabe et al. 1991; Armour et al. 2013) and diffusive moist energy balance models (MEBMs) (e.g., Merlis and Henry 2018; Armour et al. 2019). Figure 1b illustrates these behaviors via T and T_sym from an abrupt 4 × CO₂ simulation in CESM1 over each of four time periods (years 1–10, 21–100, 701–800, and 2901–3000): T and T_sym differ markedly in the first decade when T_asym is largest but gradually become more similar, with T ≈ T_sym and T_asym ≈ 0 to first approximation in the final period.

On millennial time scales changes in deep-ocean circulation become relevant (and can be non-monotonic, cf. Jansen et al. 2018), perturbing the prevailing antisymmetric transport of heat from the southern to the Northern Hemisphere by the Atlantic meridional overturning circulation. For example, in a 3000-yr GCM simulation with perturbed cloud albedos yielding a surface climate resembling the early Pliocene (∼4 Ma), a Pacific meridional overturning circulation emerges after ∼1500 years, increasing the heat convergence into the Northern Hemisphere (Burls et al. 2017).

Nevertheless, past studies indicate that T normalized by its global average—henceforth T^*—partially collapses toward a shared pattern at different time scales (cf. Fig. 4a of Armour et al. 2013). Such pattern scaling (e.g., Tebaldi and Arblaster 2014) also largely holds for zonally varying surface temperature responses across CO₂ values [e.g., Heede et al. (2020), though there is also considerable evidence for state-dependent climate sensitivity; e.g., Rohrschneider et al. (2019)]. Essentially, the present study combines pattern scaling and the symmetric/antisymmetric decomposition for T under 4 × CO₂, arguing that GCM-simulated $T_{\text{sym}}^{*}$ changes surprisingly little from the first decade to near-equilibrium while $T_{\text{asym}}^{*}$ weakens markedly. Taken at face value, this would imply that polar amplification (defined as the ratio of polar cap to globally averaged warming) at near-equilibrium in a given GCM can be meaningfully constrained from a single decade of forced change.

We found these behaviors somewhat inadvertently in the aforementioned 4 × CO₂ simulation in CESM1 (which is described along with the other models and methodological choices in section 2), and this manuscript constitutes an attempt to better understand them. To assess their robustness, we analyze 11 additional GCMs from the LongRunMIP (Rugenstein et al. 2019) repository (section 3). To clarify their underlying physical mechanisms, we use an MEBM to first emulate the results from CESM1 and then identify the predominant factors determining $T_{\text{sym}}^{*}$ (section 4). To better understand their differences across these GCMs, we fit a three-box, two-time-scale model to two end-members of these 12, CESM1 and FAMOUS (section 5). And to assess their relevance across different forcings, we analyze 2, 8, and 16 × CO₂ simulations and the aforementioned Pliocene-like simulation in CESM1 (section 6). We conclude with summary and discussion (section 7), including comparison to the more traditional approach of studying polar amplification in either cap separately, potential means of further testing these behaviors, and implications for the real climate system. We view these results as suggestive, rather than definitive, and hope they motivate further studies of the hemispherically symmetric and antisymmetric components of surface warming and what controls them in models and the real world.

2. Methods

a. LongRunMIP and CESM1 4 × CO₂ simulations

LongRunMIP (Rugenstein et al. 2019) comprises increased-CO₂ simulations from CMIP5-class GCMs spanning from one thousand to several thousand years. We analyze 11 of the 12 available with ≥1000-yr integrations under 4 × CO₂, which are listed in Table 1. CESM104 from LongRunMIP was omitted because it is nearly the same as the above-noted CESM1 that we analyze separately. Nine of the LongRunMIP models ran under an abrupt 4 × CO₂ and two under a 1% increase per year to 4 × CO₂. The latter two (ECHAM5MPIOM and MIROC32) also ran shorter abrupt 4 × CO₂ simulations, and so for the first century when forcing and global-mean warming are modest under 1% yr⁻¹ we use the abrupt 4 × CO₂ simulation, switching to the 1% to 4 × CO₂ simulation for subsequent periods. Output was available regridded to a common 2.5° × 2.5° grid (cf. Table 2 of Rugenstein et al. 2019).

Table 1

Details of the LongRunMIP models and simulations used. (from left to right) Model name following Rugenstein et al. (2019) conventions, control simulation duration in years, simulation for which data over years 701–800 is taken, and simulation for which data over years 2901–3000 are taken (or “none” if not available for any simulation in that model).

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We include with the LongRunMIP models the 3000-yr 4 × CO₂ simulation in CESM1 referred to in the Introduction. This is version 1.0.4 of the model in its low-resolution configuration (Shields et al. 2012). It consists of the Community Atmosphere Model, version 4 with its spectral dynamical core truncated at T31 resolution (∼3.75° × 3.75°) and with 26 vertical levels coupled to the Parallel Ocean Program version 2 (POP2) with ∼3° horizontal resolution and 60 vertical levels.

We focus on temporal averages over four time periods (similar to those of Armour et al. 2013): years 1–10 and 21–100 (during which both the atmosphere and ocean are rapidly responding), 701–800 (during which the atmosphere is in a nearly statistically steady state but the ocean remains slowly varying), and 2901–3000 (at which time the deep ocean has nearly equilibrated). All 12 GCMs extend through year 800 and 5 through year 3000. We refer to the final period as near-equilibrium, recognizing that the climate response would likely meaningfully evolve beyond three millennia in most models given the deep ocean’s multimillennial, diffusive equilibration time scale (Jansen et al. 2018); the box model in section 5 highlights this.

We account for climate drift in each model’s preindustrial control simulation as follows. For the 11 LongRunMIP GCMs, for each time period we compute anomalies as the difference between the 4 × CO₂ simulation and the control at that time period if the control simulation extends that long. Otherwise, we subtract an average over the entire control simulation. For CESM1, drift in the control simulation is modest relative to the forced temperature responses, and so for convenience we report anomalies in all periods as differences with the control averaged over years 701–800. All major results presented are insensitive to reasonable methodological choices regarding control drift.

The five GCMs extending to year 3000 include the three farthest on the ends of the full 12-GCM distribution at years 701–800: FAMOUS on one end (highest global-mean warming, and second-weakest changes in both the symmetric and antisymmetric polar amplification indices defined below), versus GISSE2R and CESM1 (respectively, lowest and third lowest mean warming, largest and second-largest increase in symmetric amplification, and second-largest and largest decrease in antisymmetric amplification) on the other. Most likely then this subset usefully approximates the range generated by all 12 models had the others also run to year 3000.

3. 4 × CO₂ results in GCMs

Figure 2 shows the 4 × CO₂-forced T^*, $T_{\text{sym}}^{*}$, and $T_{\text{asym}}^{*}$ fields for each period specified above in 6 of the 12 GCMs; the remaining six are shown in Fig. 3. Printed in each T^* panel are that model’s global-mean warming (henceforth $\overline{T}$) for each time period, in each $T_{\text{sym}}^{*}$ panel that model’s PA_sym for each period and its percentage change from the first decade to the last available period, and in each $T_{\text{asym}}^{*}$ panel that model’s PA_asym for each period and its first-to-last percentage change. Across models, $\overline{T}$ spans 2.0–3.9 K in the first decade and increases monotonically afterward in all models, with values 3.0–9.0 K in years 21–100, 4.3–12.7 K in years 701–800, and 4.8–13.8 K in years 2901–3000. In the first decade only, in all 12 T^* minimizes over the Southern Ocean. For nearly all models and latitudes south of ∼40°S, T^* increases monotonically in time. The evolution of northern extratropical warming varies more across models, but in most T^* decreases with time over much of ∼40°–70°N.

For 4 × CO₂ simulations in 6 of the 12 GCMs analyzed: (left) full, (center) hemispherically symmetric component, and (right) antisymmetric component of zonal-mean surface air temperature change in years 1–10, 21–100, 701–800, and if available years 2901–3000, with each period as indicated in the legend in (e). Printed values at the bottom of each panel are (left) the mean warming during that period, (center) the symmetric polar amplification index for that period, and (right) the northern minus Southern Hemisphere polar amplification index for that period. Values at the top of each panel in the center column are the fractional change in the symmetric polar amplification index from the first to the last period, and at the top of each panel the right column are the same but for the northern–southern difference. (d)–(f) For CESM1, the asterisk signifies that these simulations are not from LongRunMIP (unlike the 11 other models). The remaining six models are shown in Fig. 3.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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For 4 × CO₂ simulations in 6 of the 12 GCMs analyzed: (left) full, (center) hemispherically symmetric component, and (right) antisymmetric component of zonal-mean surface air temperature change in years 1–10, 21–100, 701–800, and if available years 2901–3000, with each period as indicated in the legend in (e). Printed values at the bottom of each panel are (left) the mean warming during that period, (center) the symmetric polar amplification index for that period, and (right) the northern minus Southern Hemisphere polar amplification index for that period. Values at the top of each panel in the center column are the fractional change in the symmetric polar amplification index from the first to the last period, and at the top of each panel the right column are the same but for the northern–southern difference. (d)–(f) For CESM1, the asterisk signifies that these simulations are not from LongRunMIP (unlike the 11 other models). The remaining six models are shown in Fig. 3.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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For 4 × CO₂ simulations in 6 of the 12 GCMs analyzed: (left) full, (center) hemispherically symmetric component, and (right) antisymmetric component of zonal-mean surface air temperature change in years 1–10, 21–100, 701–800, and if available years 2901–3000, with each period as indicated in the legend in (e). Printed values at the bottom of each panel are (left) the mean warming during that period, (center) the symmetric polar amplification index for that period, and (right) the northern minus Southern Hemisphere polar amplification index for that period. Values at the top of each panel in the center column are the fractional change in the symmetric polar amplification index from the first to the last period, and at the top of each panel the right column are the same but for the northern–southern difference. (d)–(f) For CESM1, the asterisk signifies that these simulations are not from LongRunMIP (unlike the 11 other models). The remaining six models are shown in Fig. 3.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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As in Fig. 2, but for the remaining six GCMs.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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As in Fig. 2, but for the remaining six GCMs.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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As in Fig. 2, but for the remaining six GCMs.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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The $T_{\text{sym}}^{*}$ is polar-amplified in all models and time periods, and by eye it either changes modestly or becomes somewhat more polar amplified with time. Quantitatively, PA_sym spans 1.16–1.67 across models in the initial decade, 1.23–1.79 in years 21–100, 1.31–1.96 in years 701–800, and 1.49–2.05 in years 2901–3000. In one outlier, GISSE2R, it increases from the first decade to the last available period by 34%, and unique to this model the $T_{\text{sym}}^{*}$ field is similar for the first two periods but then seems to jump to one shared by the latter two periods. In the remaining 11 GCMs, PA_sym changes by ≤8% in 6 models (with negative change in FAMOUS, −5%, and HadCM3L, −6%), and increases by ≤22% in the remaining 5 models. As such, we consider a weak change to modest increase in PA_sym from decadal to millennial time scales under 4 × CO₂ to be an empirically robust response across these models.

The $T_{\text{asym}}^{*}$ field reflects greater Arctic than Antarctic warming initially but also a weakening in time of that difference in all models. Quantitatively, PA_asym is positive in the first decade in all models, spanning 0.89–1.65, and then decreases monotonically in 10 of 12 models, spanning 0.69–1.35 in years 21–100, −0.02 to +0.98 in years 701–800, and −0.15 to +0.73 in years 2901–3000. The two negative values, both in CESM1, indicate that Antarctic warming exceeds Arctic warming. The fractional change in PA_asym from the first decade to the last available period is negative in all models but varies considerably, −16% to −109%. We consider a modest to complete reduction in PA_asym to likewise be a robust response.¹

Given these two robust responses, empirically each model’s PA_sym in the initial decade provides a nontrivial albeit approximate lower bound on its near-equilibrium PA_sym value—and for models in which $T_{\text{asym}}^{*}$ weakens strongly leaving $T^{*}\approx T_{\text{sym}}^{*}$ at near-equilibrium, this extends to polar amplification in each hemispheric cap. By eye, indeed the full $T_{\text{sym}}^{*}$ field changes less with time in nearly all models than does T^* and less still than $T_{\text{asym}}^{*}$. We may quantify this using the ratio PA_asym/PA_sym, which ranges from −0.01 in CESM1 to +0.64 in IPSLCM5A for years 701–800 and −0.07 in CESM1 to 0.47 in FAMOUS for years 2901–3000. It becomes less positive between these periods in all five models run to years 2901–3000, at which point it is ≤10% in magnitude in CESM1, ECHAM5MPIOM, and GISSE2R. From Fig. 2 the correspondence between the initial $T_{\text{sym}}^{*}$ and final T^* fields is debatable for the two whose PA_asym/PA_sym ratios are not small (0.42 in MPIESM11 and 0.47 in FAMOUS), intermediate for GISSE2R (recall its aforementioned jump in $T_{\text{sym}}^{*}$ after the first century), but clear for CESM1 and ECHAM5MPIOM.

Having established these behaviors in full-physics GCMs, we turn to better understanding them via two simpler models: first with an MEBM to clarify the physical mechanisms underlying the robust $T_{\text{sym}}^{*}$ and $T_{\text{asym}}^{*}$ behaviors, and then with a box model to explore the GCM diversity in PA_sym and PA_asym evolutions.

4. Moist energy balance model

Figure 4 shows T, T^*, $T_{\text{sym}}^{*}$, and $T_{\text{asym}}^{*}$ for each time period in the CESM1 4 × CO₂ simulation and the corresponding MEBM simulations; recall the MEBM simulations differ from one another only in the time period in CESM1 from which the radiative feedback parameter (λ) and ocean heat uptake (OHU; $\mathcal{O}$) were diagnosed as detailed in appendix. The MEBM captures the mean warming ($\overline{T}$ is within 0.3 K of CESM1 for all four periods) and raw warming patterns reasonably well and therefore T^*—in particular the gradual, modest increase in polar amplification in $T_{\text{sym}}^{*}$ and the steady but severe weakening of $T_{\text{asym}}^{*}$. High-latitude warming gradients are insufficiently sharp (a common feature of MEBMs with uniform diffusivity, e.g., Bonan et al. 2018), especially in the Arctic, yielding a moderate low bias in the Arctic amplification index by years 2901–3000 (1.62 in the MEBM versus 1.97 in CESM1) but less so for the Antarctic index due to compensating within-region T biases (2.16 in the MEBM versus 2.13 in CESM1 for years 2901–3000). More importantly, the MEBM suitably captures the fractional change in both PA_sym and PA_asym from CESM1: the MEBM PA_asym decreases by −132% (from 0.83 to −0.27, versus −109% in CESM1), and its PA_sym increases by 19% (from 1.59 to 1.89, versus 22% in CESM1). As such, we can use the MEBM to further probe the underlying physical mechanisms.

Surface air temperature response in the CESM1 4 × CO₂ simulation at the four selected time periods and in the moist energy balance model simulations meant to reproduce the CESM1 4 × CO₂ simulation at each of those time periods, as indicated by the text in (a). Panels show different temperatures: (a) raw (in K), (b) mean-normalized (unitless), (c) mean-normalized symmetric component (unitless), and (d) mean-normalized antisymmetric component (unitless). Note differing vertical axis spans in each panel.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Surface air temperature response in the CESM1 4 × CO₂ simulation at the four selected time periods and in the moist energy balance model simulations meant to reproduce the CESM1 4 × CO₂ simulation at each of those time periods, as indicated by the text in (a). Panels show different temperatures: (a) raw (in K), (b) mean-normalized (unitless), (c) mean-normalized symmetric component (unitless), and (d) mean-normalized antisymmetric component (unitless). Note differing vertical axis spans in each panel.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Surface air temperature response in the CESM1 4 × CO₂ simulation at the four selected time periods and in the moist energy balance model simulations meant to reproduce the CESM1 4 × CO₂ simulation at each of those time periods, as indicated by the text in (a). Panels show different temperatures: (a) raw (in K), (b) mean-normalized (unitless), (c) mean-normalized symmetric component (unitless), and (d) mean-normalized antisymmetric component (unitless). Note differing vertical axis spans in each panel.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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To test the role of anti-symmetries in λ and $\mathcal{O}$, Fig. 5 shows T^*, $T_{\text{sym}}^{*}$, and $T_{\text{asym}}^{*}$ from simulations with λ and $\mathcal{O}$ replaced by λ_sym and ${\mathcal{O}}_{\text{sym}}$. The resulting symmetric warming patterns (red curves) closely resemble the original ones (blue curves).² Quantitatively, $\overline{T}$ changes from the full MEBM simulation by ≤0.3 K and PA_sym by 0.04 or 3% in all periods. In a complementary simulation, the antisymmetric components are amplified rather than suppressed: we set ${\lambda }^{*}={\lambda }_{\text{sym}}+\alpha {\lambda }_{\text{asym}}$, where λ^* is the modified feedback parameter, $\mathcal{O}$ is modified likewise, and α = 3 (yellow curves in Fig. 5). Although this strongly modifies T and $T_{\text{asym}}^{*}$ in all periods, after the first decade $\overline{T}$ changes by at most 0.7 K, qualitatively the $T_{\text{sym}}^{*}$ pattern changes weakly, and quantitatively PA_sym changes by ≤0.03. These results suggest that $T_{\text{sym}}^{*}$ and PA_sym are largely insensitive to λ_sym and ${\mathcal{O}}_{\text{asym}}$, leading us to focus henceforth on λ_sym and ${\mathcal{O}}_{\text{sym}}$.

Mean-normalized (a) full, (b) symmetric, and (c) antisymmetric surface air temperature anomaly fields in MEBM simulations with the antisymmetric components of the radiative feedback parameter and ocean heat uptake fields multiplied by the factor α, with red curves for α = 0, blue for α = 1 (i.e., unchanged), and dark yellow for α = 3. Dotted, dash–dotted, dashed, and solid lines correspond to years 1–10, 21–100, 701–800, and 2901–3000, respectively, of the CESM1.0.4 abrupt 4 × CO₂ simulation. Note that the vertical axis range is identical in (a) and (b), but not in (c), while the vertical axis spacing is identical in all three panels.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Mean-normalized (a) full, (b) symmetric, and (c) antisymmetric surface air temperature anomaly fields in MEBM simulations with the antisymmetric components of the radiative feedback parameter and ocean heat uptake fields multiplied by the factor α, with red curves for α = 0, blue for α = 1 (i.e., unchanged), and dark yellow for α = 3. Dotted, dash–dotted, dashed, and solid lines correspond to years 1–10, 21–100, 701–800, and 2901–3000, respectively, of the CESM1.0.4 abrupt 4 × CO₂ simulation. Note that the vertical axis range is identical in (a) and (b), but not in (c), while the vertical axis spacing is identical in all three panels.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Mean-normalized (a) full, (b) symmetric, and (c) antisymmetric surface air temperature anomaly fields in MEBM simulations with the antisymmetric components of the radiative feedback parameter and ocean heat uptake fields multiplied by the factor α, with red curves for α = 0, blue for α = 1 (i.e., unchanged), and dark yellow for α = 3. Dotted, dash–dotted, dashed, and solid lines correspond to years 1–10, 21–100, 701–800, and 2901–3000, respectively, of the CESM1.0.4 abrupt 4 × CO₂ simulation. Note that the vertical axis range is identical in (a) and (b), but not in (c), while the vertical axis spacing is identical in all three panels.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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To test the role of the spatial patterns of λ_sym and ${\mathcal{O}}_{\text{sym}}$, Fig. 6 shows T_sym and $T_{\text{sym}}^{*}$ in simulations with λ and $\mathcal{O}$ replaced at all latitudes by their global averages, $\overline{\lambda }$ and $\overline{\mathcal{O}}$, respectively (purple curves).³ To focus on the biggest-picture behaviors, we restrict to simulations corresponding to the first decade (when $\overline{\lambda }=-2.0\text{W}{\text{m}}^{-2}{\text{K}}^{-1}$ and $\overline{\mathcal{O}}=3.9\text{W}{\text{m}}^{-2}$) and to years 2901–3000 (when $\overline{\lambda }=-1.3\text{W}{\text{m}}^{-2}{\text{K}}^{-1}$ and $\overline{\mathcal{O}}=-0.4\text{W}{\text{m}}^{-2}$ W m⁻²). With uniform $\mathcal{O}$ and λ, warming is reduced in the extratropics especially and globally averaged ($\overline{T}$ is 0.2 K cooler for the first decade and 1.7 K cooler for years 2901–3000), and T^* is less polar-amplified and changes less in time: PA_sym is reduced from 1.54 to 1.51 (a 2% decrease) in the first decade and from 1.89 to 1.60 (a 15% decrease) in years 2901–3000. In other words, the meridional patterns of ${\lambda }_{\text{sym}}$ and ${\mathcal{O}}_{\text{sym}}$ together make warming stronger in the mean and more polar-amplified compared to if they were uniform.

Surface air temperature response in MEBM simulations corresponding to years 1–10 (dotted curves) and years 2901–3000 (solid curves) of the CESM1 4 × CO₂ simulation, shown either (a) raw (in K) or (b) mean-normalized (unitless). Colors correspond to simulations with (red) both λ and $\mathcal{O}$ symmetrized but not uniform, (blue) λ set to its global-mean value at all latitudes, (yellow) $\mathcal{O}$ set to its global-mean value at all latitudes, or (purple) both $\mathcal{O}$ and λ set to their global-mean values at all latitudes.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Surface air temperature response in MEBM simulations corresponding to years 1–10 (dotted curves) and years 2901–3000 (solid curves) of the CESM1 4 × CO₂ simulation, shown either (a) raw (in K) or (b) mean-normalized (unitless). Colors correspond to simulations with (red) both λ and $\mathcal{O}$ symmetrized but not uniform, (blue) λ set to its global-mean value at all latitudes, (yellow) $\mathcal{O}$ set to its global-mean value at all latitudes, or (purple) both $\mathcal{O}$ and λ set to their global-mean values at all latitudes.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Surface air temperature response in MEBM simulations corresponding to years 1–10 (dotted curves) and years 2901–3000 (solid curves) of the CESM1 4 × CO₂ simulation, shown either (a) raw (in K) or (b) mean-normalized (unitless). Colors correspond to simulations with (red) both λ and $\mathcal{O}$ symmetrized but not uniform, (blue) λ set to its global-mean value at all latitudes, (yellow) $\mathcal{O}$ set to its global-mean value at all latitudes, or (purple) both $\mathcal{O}$ and λ set to their global-mean values at all latitudes.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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To distinguish the contributions of λ_sym versus ${\mathcal{O}}_{\text{sym}}$, Fig. 6 also includes simulations with only one of λ or $\mathcal{O}$ made uniform. For each time period, T_sym of the original simulation more closely resembles the uniform-$\mathcal{O}$ than uniform-λ simulation, which in turn more closely resembles the both-uniform simulation. Quantitatively, in years 2901–3000$\overline{T}$ is 0.2 K warmer than the λ = λ_sym, $\mathcal{O}={\mathcal{O}}_{\text{sym}}$ case when $\mathcal{O}$ is uniform versus 1.7 K cooler when λ is uniform; PA_sym is slightly increased with uniform $\mathcal{O}$, from 1.89 to 1.93 (2%) versus more strongly decreased with uniform λ, from 1.89 to 1.55 (−18%). In other words, the evolving spatial pattern of λ_sym (along with the mean of $\mathcal{O}$) acts to increase warming in the global mean and make it more polar-amplified, and these influences are stronger than those of the evolving spatial pattern of ${\mathcal{O}}_{\text{sym}}$ (along with the mean of λ) in determining T_sym and $T_{\text{sym}}^{*}$.

To interpret this strong influence of λ_sym, Fig. 7 shows λ, λ_sym, and λ_asym for each time period (as well as, for completeness in interpreting the various MEBM simulations, the corresponding $\mathcal{O}$ fields and those of the time-invariant radiative forcing). λ is negative at nearly all latitudes in all periods and is generally more negative in the tropics than high latitudes. In the first decade it has a pronounced global minimum of ∼−6 W m⁻² K⁻¹ near 50°S, just equatorward of the global maximum in $\mathcal{O}$ driven by Southern Ocean upwelling. After the first decade, it becomes less stabilizing in the global average (increasing climate sensitivity, cf. Armour et al. 2013) and at most latitudes south of ∼10°N at least somewhat. But the largest regional change is a vanishing of the sharp global minimum in λ by years 21–100. With much weaker changes in the Northern Hemisphere, these signals project onto λ_sym as well. λ_sym therefore becomes less stabilizing in the extratropics than tropics after the first decade, acting to increase polar amplification with time.⁴

(a)–(c) Radiative forcing ($\mathcal{F}$; units: W m⁻²), (d)–(f) radiative feedback parameter (λ; units: W m⁻² K⁻¹), and (g)–(i) ocean heat uptake ($\mathcal{O}$, signed positive downward into the ocean; units: W m⁻²) fields used for the MEBM simulations for each time period, with radiative forcing constant across time periods. Columns show (left) The full fields, (center) the hemispherically symmetric component, and (right) the hemispherically antisymmetric component. For each row, the left and center panels have the same vertical axis ranges, and the right column has the same vertical axis spacing as the other columns (except for the bottom row), but not the same range.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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(a)–(c) Radiative forcing ($\mathcal{F}$; units: W m⁻²), (d)–(f) radiative feedback parameter (λ; units: W m⁻² K⁻¹), and (g)–(i) ocean heat uptake ($\mathcal{O}$, signed positive downward into the ocean; units: W m⁻²) fields used for the MEBM simulations for each time period, with radiative forcing constant across time periods. Columns show (left) The full fields, (center) the hemispherically symmetric component, and (right) the hemispherically antisymmetric component. For each row, the left and center panels have the same vertical axis ranges, and the right column has the same vertical axis spacing as the other columns (except for the bottom row), but not the same range.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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(a)–(c) Radiative forcing ($\mathcal{F}$; units: W m⁻²), (d)–(f) radiative feedback parameter (λ; units: W m⁻² K⁻¹), and (g)–(i) ocean heat uptake ($\mathcal{O}$, signed positive downward into the ocean; units: W m⁻²) fields used for the MEBM simulations for each time period, with radiative forcing constant across time periods. Columns show (left) The full fields, (center) the hemispherically symmetric component, and (right) the hemispherically antisymmetric component. For each row, the left and center panels have the same vertical axis ranges, and the right column has the same vertical axis spacing as the other columns (except for the bottom row), but not the same range.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Summarizing: the MEBM captures the CESM1 warming patterns reasonably well when forced with the latter’s λ and $\mathcal{O}$ fields; for T_sym and PA_sym to good approximation only the symmetric components of λ and $\mathcal{O}$ matter; of these the predominant influence on the time evolution of T_sym and PA_sym is that of λ_sym; and physically this stems from an initial strongly stabilizing radiative feedback over the Southern Ocean that vanishes after the first decade, making λ_sym become preferentially less stabilizing at high compared to low latitudes, increasing polar amplification. We infer that changes in $T_{\text{sym}}^{*}$ are modest to the extent that changes in the spatial pattern of λ_sym are themselves modest.

5. Box model of amplification indices

Having explored the mechanisms underlying the robust responses across the GCMs, we now investigate the cross-GCM discrepancies via a three-box, two-time-scale model applied to two of the end-member GCMs noted above, CESM1 and FAMOUS⁵ (Smith et al. 2008). Our three-box model is an extension of the well-known two-time-scale box model for global-mean warming (Held et al. 2010; Geoffroy et al. 2013; Rohrschneider et al. 2019) to region-mean warming (see also Geoffroy and Saint-Martin 2014) in the Arctic (60°–90°N), Antarctic (60°–90°S), and lower latitudes (60°S–60°N). Recalling that CESM1 has the second-least mean warming, second-most positive change in PA_sym (+22%), and most negative change in PA_asym (−109%) whereas FAMOUS has the most mean warming, second-least positive (−5%) change in PA_sym, and second-least negative (−27%) change in PA_asym, this diagnosis of the regional warming time scales points toward potential causes of the spread in $T_{\text{sym}}^{*}$ and $T_{\text{asym}}^{*}$ across the GCMs.

The physical basis for the two-time-scale model is the presence of a shallow, relatively rapidly evolving ocean layer and a deeper, more slowly evolving deep ocean (e.g., Held et al. 2010). The two-time-scale solution for a given region is given by

$T(t)=T_{\text{eq}}[a_{\text{f}}(1-e^{-t/{\tau }_{\text{f}}})+a_{\text{s}}(1-e^{-t/{\tau }_{\text{s}}})],$(2)

where T_eq is the equilibrium temperature change, τ_f and τ_s are the fast and slow warming time scales, respectively, and a_f and a_s are the fractional contributions of the fast and slow responses, respectively, to the equilibrium warming, with $a_{\text{f}}+a_{\text{s}}=1$. For each model and region, we fit T_eq, a_f, a_s, τ_f, and τ_s of (2) via nonlinear least squares (using the “curve_fit” function of the scipy Python package; Virtanen et al. 2020) applied to annual-mean time series of the region-mean surface temperature anomaly. Although they do not appear explicitly, note that the ocean heat uptake and anomalous atmospheric energy flux divergence fields implicitly influence the values of all five parameters. The resulting best-fit parameter values are listed in Table 2, and the GCM regional-mean, annual-mean time series (smoothed via a 10-year running mean) and corresponding box-model solutions are shown both raw and mean-normalized in Fig. 8.

Time series of 10-yr running mean of Arctic (red), Antarctic (yellow), and low-latitude (blue) box-average temperatures in the abrupt 4 × CO₂ simulation in (a),(c) CESM1 and (b),(d) FAMOUS. Overlain gray curves are the fits from the simple two-layer box model for each region. Rows show (top) raw fields and (bottom) the same time series but each normalized by the global-mean warming. The plot in (d) also includes as thin horizontal lines the predictions from the box model under the approximation of horizontally uniform fast and slow warming time scales, as described in the text.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Time series of 10-yr running mean of Arctic (red), Antarctic (yellow), and low-latitude (blue) box-average temperatures in the abrupt 4 × CO₂ simulation in (a),(c) CESM1 and (b),(d) FAMOUS. Overlain gray curves are the fits from the simple two-layer box model for each region. Rows show (top) raw fields and (bottom) the same time series but each normalized by the global-mean warming. The plot in (d) also includes as thin horizontal lines the predictions from the box model under the approximation of horizontally uniform fast and slow warming time scales, as described in the text.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Time series of 10-yr running mean of Arctic (red), Antarctic (yellow), and low-latitude (blue) box-average temperatures in the abrupt 4 × CO₂ simulation in (a),(c) CESM1 and (b),(d) FAMOUS. Overlain gray curves are the fits from the simple two-layer box model for each region. Rows show (top) raw fields and (bottom) the same time series but each normalized by the global-mean warming. The plot in (d) also includes as thin horizontal lines the predictions from the box model under the approximation of horizontally uniform fast and slow warming time scales, as described in the text.

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Table 2

Best-fit values of the five parameters in the two-time-scale model for the abrupt 4 × CO₂ simulations in CESM1 and in FAMOUS for each of the three regions of our box model and for the global mean. Units are in K for T_eq and years for τ_f and τ_s; a_f and a_s are dimensionless.

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For CESM1, T_eq is slightly higher for the Antarctic (17.8 K) than Arctic (15.1 K), both of which are ∼3 times higher than for lower latitudes (5.8 K). The fast response time scales for the Arctic (9.4 years) and lower latitudes (12.7 years) are comparable and an order of magnitude less than the Antarctic time scale (85.3 years). Equilibrium warming is weighted fairly evenly between the fast and slow responses (a_f = 0.59, 0.48, and 0.60 for the Arctic, Antarctic, and lower latitudes, respectively). The slow response time scales are all millennial—2223, 2564, and 1065 years for the Arctic, Antarctic, and lower latitudes, respectively. The two-time-scale fit captures the overall evolution for each region fairly well, though with too sharp a shoulder after the initial decades for the Arctic and lower latitudes (Fig. 8a). CESM1 also exhibits considerable centennial-time-scale variability particularly after ∼1800 years (roughly coinciding with the emergence of the Pacific meridional overturning circulation in the Pliocene-like simulation; Burls et al. 2017). For the fast response, the separation of the Antarctic time scale from the Arctic and lower latitudes is evident. For the slow response, it is evident that both caps would continue warming nontrivially beyond year 3000, which after all is only ∼1.2–1.3 times their slow response time scales. The global-mean-normalized time series (Fig. 8c) show the initial strong Arctic amplification and the Antarctic subsequently catching up by around ∼500–600 years.

The two-time-scale fit is even better for FAMOUS than for CESM1 (Fig. 8b) and highlights the striking result that the Antarctic fast response time scale is slightly shorter than the Arctic’s—τ_f = 14.1, 15.9, and 15.3 years for lower latitudes, Arctic, and Antarctic, respectively—unlike CESM1 and counter to physical intuition given the retarding influence of Southern Ocean upwelling. The predicted equilibrium warming is over 10 K higher in the Arctic (26.4 K) than Antarctic (16.1 K), which in turn is less than 4 K warmer than the lower latitudes (12.3 K). It is also weighted more toward the fast than slow response for all three regions, with a_f = 0.78, 0.71, and 0.66 for the Arctic, Antarctic, and lower latitudes, respectively. With comparable time scales and weightings for the fast response but much larger equilibrium warming in the Arctic, initial decades feature much greater Arctic than Antarctic warming. The slow response time scale is similar for lower latitudes and Arctic (433 and 471 years, respectively), and moderately longer for the Antarctic (588 years). As such, the Antarctic continues warming somewhat longer than the rest of the globe, which moderately weakens the antisymmetric amplification. Still, the Antarctic slow time scale is within ∼25%–35% of the others.

This is independent of time. Therefore so too are PA_sym and PA_asym—imperfect for the −27% decrease in PA_asym but capturing the modest −6% change in PA_sym well. Figure 8c shows the global-mean-normalized warming for each region for FAMOUS along with their predicted values from (4). The simple approximation (4) is biased low for each region, but in reasonable agreement with (4) the FAMOUS time series vary modestly in time, at most for the Arctic by ∼10% over the 3000 years.

Summarizing, for CESM1 there are three relevant time scales. In the initial decades, the Arctic warms rapidly but not the Antarctic, yielding large values of both PA_sym and PA_asym. The fast Antarctic warming transpires over subsequent decades to centuries, increasing PA_sym but weakening PA_asym. Over subsequent millennia, the slow responses emerge continuing to warm both polar caps, comparably to one another but more than lower latitudes, further increasing PA_sym while decreasing PA_asym. For FAMOUS, a surprisingly short time scale of Antarctic warming combined with much greater Arctic than Antarctic (or low-latitude) equilibrium warming combine to keep changes in both PA_sym and PA_asym modest from decadal to millennial time scales. These results show that the preferential initial Arctic versus Antarctic amplification, though robust, can arise via rather different processes in different GCMs.

6. Results across CO₂ levels and a Pliocene-like simulation in CESM1

Though bounding a GCM’s near-equilibrium PA_sym from a short integration would be useful, our ultimate concern is what can be inferred for the real climate system, for which an instantaneous quadrupling of CO₂ is not directly relevant to anthropogenic warming—in which the CO₂ increase is gradual and (one dearly hopes) remains well below a quadrupling—nor those paleoclimate states for which non-CO₂ forcings are of first-order importance. We therefore now present CESM1 simulations at 2–16 × CO₂ and the Pliocene-like simulation; these address the sensitivity of the results to CO₂ amount and to a strongly meridionally patterned, non-CO₂ forcing but do not directly address the issue of gradual rather than abrupt forcings, which we return to in the concluding discussion section below.

The left column of Fig. 9 shows T, T^*, $T_{\text{sym}}^{*}$, and $T_{\text{asym}}^{*}$ for each perturbed CO₂ simulation and time period. For T, warming occurs at all latitudes, is weakest and relatively flat at low latitudes, and increases nearly monotonically moving from low to high latitudes (peaking in the Southern Hemisphere from ∼65°S for 2 × CO₂ to ∼80°S for 16 × CO₂). Across CO₂ levels and time periods, low-latitude warming ranges from ∼2 to ∼11 K, peak SH high-latitude warming from ∼6 to ∼25 K, and peak NH warming at the North Pole from ∼7 to ∼34 K. $\overline{T}$ spans across periods 0.8–3.6, 2.0–6.9, 3.5–9.8, and 5.2–13.6 K for 2, 4, 8, and 16 × CO₂, respectively. For T^*, the patterns are most similar across CO₂ amounts and time scales in the tropics, moderately so in the northern extratropics, and least of all in the southern extratropics. The Arctic amplification index decreases in time, and the Antarctic index increases, in all cases, both with the largest changes under 2 × CO₂ (from 2.50 in the first decade to 1.97 for years 2901–3000 for the Arctic and from 0.75 to 2.57 for the Antarctic).

The $T_{\text{sym}}^{*}$ is quite similar across CO₂ values and time periods, though least for the 2 × CO₂ first and last periods. For all CO₂ values the pattern becomes slightly more polar amplified in time, with low-latitude values decreasing and high-latitude values increasing. Quantitatively, PA_sym spans 1.62–2.29 across all CO₂ amounts and time scales (respectively occurring in years 1–10 and 2901–3000 under 2 × CO₂). In other words, PA_sym starts smallest and ends up largest in 2 × CO₂, increasing less with time (from the first to last period, by +41%, 22%, 18%, and 8% for 2–16×, respectively), and to a smaller near-equilibrium value (2.29, 2.05, 1.96, and 1.83, respectively) as CO₂ increases. The value of $T_{\text{asym}}^{*}$ varies appreciably across the four time scales, reflecting the gradual catching-up of Antarctic and Southern Ocean warming with (and for 2 and 4 × CO₂, surpassing) the initially rapid Arctic warming. Similar to PA_sym but with signs reversed, PA_asym is most positive in the first decade under 2 × CO₂ but then becomes most negative at near-equilibrium (0.9 and −0.3 in the first decade and years 2901–3000, respectively), and it decreases less with time as CO₂ increases (from the first to last period, by −132%, −109%, −81%, and −72% for 2–16×, respectively). Particularly in the first century, $T_{\text{asym}}^{*}$ groups together more by time scale than by CO₂ value (cf. curves with the same line markings to those with the same color). This likely reflects the intrinsic time scales of the underlying physical processes—no matter how large a radiative forcing, prevailing Southern Ocean upwelling inhibits initial local surface warming, while the deep ocean equilibration that acts to homogenize subsurface warming between the hemispheres takes millennia.

The right column of Fig. 9 shows T, T^*, $T_{\text{sym}}^{*}$, and $T_{\text{asym}}^{*}$ in the Pliocene-like simulation, with the corresponding 2–16 × CO₂ values underlain for comparison. Recalling that cloud albedo is increased 15°S–15°N and decreased poleward thereof, the warming is unsurprisingly more polar-amplified in all time periods than under CO₂. Mean warming is 1.6, 3.2, 4.6, and 5.5 K, respectively, in the four periods. The first decade’s mean-normalized fields, with $T_{\text{sym}}^{*}$ particularly cool at low latitudes and $T_{\text{asym}}^{*}$ large near the poles, sit separate from the three subsequent periods across which they are very similar. As for CO₂, the Arctic amplification index is initially large (3.2) and the Antarctic index smaller (1.9), but by years 21–100 they are nearly the same: 2.4, 2.2, and 2.4 for the Arctic in the three time periods versus 2.3, 2.4, and 2.4, respectively. The $T_{\text{sym}}^{*}$ is more polar-amplified than the CO₂ cases, but like the CO₂ cases it changes modestly in time. Quantitatively, PA_sym = 2.6 in the first decade and 2.4 in the remaining three periods, and PA_asym = 0.6 in the first decade and essentially vanishes (within −0.1 to +0.1) thereafter.

Zonal-mean surface warming in (a),(c),(e),(g) 2, 4, 8, and 16 × CO₂ simulations and (b),(d),(f),(h) the Pliocene-like simulation in CESM1. For CO₂, colors are according to the legend in (a) and line styles according to the legend in (c). For the Pliocene-like simulation, the legend is in (b). Shown are (a),(b) raw warming (in K) and the mean-normalized (c),(d) full field, (e),(f) symmetric component, and (g),(h) antisymmetric component. Vertical axis range is identical in (c)–(f), which also have the same vertical axis spacing as (g) and (h).

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Zonal-mean surface warming in (a),(c),(e),(g) 2, 4, 8, and 16 × CO₂ simulations and (b),(d),(f),(h) the Pliocene-like simulation in CESM1. For CO₂, colors are according to the legend in (a) and line styles according to the legend in (c). For the Pliocene-like simulation, the legend is in (b). Shown are (a),(b) raw warming (in K) and the mean-normalized (c),(d) full field, (e),(f) symmetric component, and (g),(h) antisymmetric component. Vertical axis range is identical in (c)–(f), which also have the same vertical axis spacing as (g) and (h).

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Zonal-mean surface warming in (a),(c),(e),(g) 2, 4, 8, and 16 × CO₂ simulations and (b),(d),(f),(h) the Pliocene-like simulation in CESM1. For CO₂, colors are according to the legend in (a) and line styles according to the legend in (c). For the Pliocene-like simulation, the legend is in (b). Shown are (a),(b) raw warming (in K) and the mean-normalized (c),(d) full field, (e),(f) symmetric component, and (g),(h) antisymmetric component. Vertical axis range is identical in (c)–(f), which also have the same vertical axis spacing as (g) and (h).

Citation: Journal of Climate 35, 20; 10.1175/JCLI-D-20-0972.1

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Summarizing, across CO₂ values in CESM1 the $T_{\text{sym}}^{*}$ pattern is qualitatively consistent, more so than $T_{\text{asym}}^{*}$ due to relative Arctic warming increasing with CO₂. Quantitatively, PA_sym modestly increase with time for each CO₂ amount, but less so as CO₂ increases. Similarly, PA_asym decreases (i.e., becomes more negative) in time at all CO₂ amounts, but less so as CO₂ increases. Under the polar-amplified forcing of the Pliocene-like simulation, unsurprisingly surface warming is itself more polar-amplified than for the quasi-uniform CO₂ forcing, but it changes weakly after the first decade. These results help to contextualize the cross-model spread in PA_sym and PA_asym under 4 × CO₂. They suggest that the fractional changes in PA_sym and PA_asym depend to a nontrivial extent on the forcing magnitude itself—consider that CESM1’s 41% increase in PA_sym under 2 × CO₂ is appreciably larger than all 12 GCMs under 4 × CO₂ (at most +34%). Similarly, the 72%–132% range in the decrease of PA_asym for CESM1 across CO₂ amounts is not vastly smaller than the range across LongRunMIP models under 4 × CO₂ of 16%–109%. At the same time, consistency of PA_sym and PA_asym at least after the first decade is even stronger under more meridionally structured forcing.

7. Conclusions