Non‐Monotonic Feedback Dependence Under Abrupt CO2 Forcing Due To a North Atlantic Pattern Effect

Effective climate sensitivity (EffCS), commonly estimated from model simulations with abrupt 4×CO2 for 150 years, has been shown to depend on the CO2 forcing level. To understand this dependency systematically, we performed a series of simulations with a range of abrupt CO2 forcing in two climate models. Our results indicate that normalized EffCS values in these simulations are a non‐monotonic function of the CO2 forcing, decreasing between 3× and 4×CO2 in CESM1‐LE (2× and 3×CO2 in GISS‐E2.1‐G) and increasing at higher CO2 levels. The minimum EffCS value, caused by anomalously negative radiative feedbacks, arises mainly from sea‐surface temperature (SST) relative cooling in the tropical and subtropical North Atlantic. This cooling is associated with the formation of the North Atlantic Warming Hole and Atlantic Meridional Overturning Circulation collapse under CO2 forcing. Our findings imply that understanding changes in North Atlantic SST patterns is important for constraining near‐future and equilibrium global warming.

Moreover, previous studies on state dependence have been focused on how EffCS and feedbacks vary in response to changes in global-mean temperatures under different CO 2 forcing (Bloch-Johnson et al., 2021;Caballero & Huber, 2013;Meraner et al., 2013). Additionally, there has been enormous attention on the dependence of EffCS and feedbacks on the spatial patterns of surface warming (e.g., Andrews et al. (2015); Andrews et al. (2022); Zhou et al. (2016); Rugenstein et al. (2016Rugenstein et al. ( , 2020). Here, we systematically examine the dependence of EffCS on the level of abrupt CO 2 forcing, as well as its connection to the spatial patterns of surface warming and climate feedbacks. To accomplish this, we conduct and analyze GCM experiments with a range of abrupt CO 2 forcings including 2×, 3×, 4×, 5×, 6×, 7×, and 8×CO 2 relative to PI level (hereafter denoted as abrupt n×CO 2 experiments).

Models and Experiments
We use the original large ensemble version of the Community Earth System Model (CESM1-LE). CESM1-LE comprises the Community Atmosphere Model version 5 (CAM5, 30 vertical levels) and parallel ocean program version 2 (POP2, 60 vertical levels) with approximately 1° horizontal resolution in all model components (Kay et al., 2015). Some of the results are shown with the GISS-E2.1-G model (Kelley et al., 2020) in Supplementary Information S1. All experiments in this work are with abrupt CO 2 forcing.
We perform abrupt n×CO 2 experiments with the coupled version of the CESM1-LE and GISS-E2.1-G models (coupled runs) for 150 years with 2×, 3×, 4×, 5×, 6×, 7×, and 8×CO 2 forcing, with all other trace gases, aerosols, ozone concentrations, and solar forcing fixed at PI values. The response is defined as the difference between the n×CO 2 runs and the PI control run. The same experiments were analyzed in (Mitevski et al., 2021(Mitevski et al., , 2022. To estimate the effective radiative forcing (ERF) as per Forster et al. (2016), we perform prescribed pre-industrial SST and sea-ice runs for 30 years for each 2×, 3×, 4×, 5×, 6×, 7×, and 8×CO 2 . The ERF is then calculated as the global mean net top of the atmosphere (TOA) net radiation between PI and n×CO 2 , and it includes the stratospheric and tropospheric adjustments (Sherwood et al., 2015).
We also utilize atmosphere-only runs (AGCM) with prescribed monthly SST values taken from the 150-year abrupt n×CO 2 runs. The prescribed SST values are monthly data for 150 years. The CO 2 concentration, ozone concentrations, aerosols, solar forcing, and all other trace gases are fixed at pre-industrial values.
In addition to only prescribing SST values from the n×CO 2 runs, we also change the SST patterns. We use the pattern from 3×CO 2 in CESM1-LE and then scale the pattern by the global-mean warming amplitude from 4×CO 2 and 5×CO 2 . We do this by ΔSST( ) = SST3×CO 2 ( ) − SSTPI( ) t is monthly data from 150 years, x is longitude, and y is latitude. Next, we find the pattern S p as where ΔSST is the global mean monthly data for 150 years. Then we have and finally SST ×CO 2 ( ) = SST ( ) + ΔSST ′ ×CO 2 ( ).
One caveat here is that we are only changing the SSTs, and holding sea-ice fixed at 3×CO 2 . Although sea-ice changes also cause albedo feedback changes, we find that imposing SSTs alone is sufficient to reproduce λ in the experiments (Haugstad et al., 2017).

Analysis
For each forcing experiment, we first estimate the long-term warming response from the Gregory method, namely, the x-intercept of regressing the change in net TOA radiation against surface air temperature over the 150 years 3 of 11 of the simulations (Gregory et al., 2004;Zelinka et al., 2020). We then obtain EffCS for each forcing scenario by normalizing the warming response by log 2 n for the n×CO 2 runs, assuming logarithmic CO 2 forcing, consistent with Bloch-Johnson et al. (2021), even though this is not a precise assumption (Byrne & Goldblatt, 2014;Etminan et al., 2016).
We calculate individual feedbacks with radiative kernels from Pendergrass et al. (2018). For each year, we multiply the spatially resolved kernels by the climate field anomalies of atmospheric temperature T, water vapor q, and surface albedo α. We regress these quantities on the surface temperature response, and the slope of this regression is the feedback. The cloud feedbacks are computed via the residual method (Soden & Held, 2006).

Non-Monotonic Effective Climate Sensitivity and Radiative Feedbacks
Results from CESM1-LE show that although the global-mean surface air temperature increases monotonically as CO 2 increases (Figure 1a), EffCS changes non-monotonically with CO 2 levels ( Figure 1b). That is, EffCS decreases between 3× and 4×CO 2 and then increases between 4× and 5×CO 2 , and at higher CO 2 forcing, with a minimum value at 4×CO 2 . We find the same non-monotonicity in the GISS-E2.1-G experiments except with a minimum EffCS at 3×CO 2 ( Figure S1 in Supplementary Information S1). In the rest of the paper, we will focus on the CESM1-LE simulations and note that the results hold for the GISS-E2.1-G simulations unless otherwise noted.
Changes in EffCS, in principle, are governed by changes in effective radiative forcing (ERF) and radiative feedbacks (λ). While the ERF, calculated from an additional 30-year fixed sea-surface temperature (SST) runs as per Forster et al. (2016), increases slightly more than the logarithm of the CO 2 concentration at higher CO 2 levels than 4×CO 2 (see Mitevski et al. (2022) for more detail), it is strongly monotonic with CO 2 and does not exhibit a minimum value (Figure 1c). On the other hand, the net radiative feedback parameter λ (Figure 1d), calculated from 150-year regressions of top-of-atmosphere (TOA) radiative response against surface air temperature change (Zelinka et al., 2020), exhibits a clearly non-monotonic behavior with respect to CO 2 levels, as for EffCS: λ becomes more negative (more stabilizing) between 3× and 4×CO 2 and less negative between 4× and 5×CO 2 , corresponding to the lowest EffCS at 4×CO 2 . Similar results are also found in the GISS-E2.1-G model experiments ( Figure S1 in Supplementary Information S1). These results suggest that EffCS depends not only nonlinearly on CO 2 , as found in previous studies (Bloch-Johnson et al., 2021;Caballero & Huber, 2013;Meraner et al., 2013), but also non-monotonically, and that the non-monotonicity is caused by the radiative feedbacks in our simulations. Hence the question is: what causes the non-monotonic changes in feedbacks?

Non-Monotonic λ Traced to Changes in Surface Warming Patterns
We hypothesize two reasons for the non-monotonic changes in λ with CO 2 : 1. The non-monotonic dependence in λ may arise from a nonlinear state-dependence of the feedbacks. As noted above, previous studies have found that radiative feedbacks change nonlinearly with global-mean surface temperature changes (i.e., feedback temperature dependence), mostly owing to the cloud and water vapor feedbacks (Bloch-Johnson et al., 2021;Caballero & Huber, 2013;Meraner et al., 2013;Seeley & Jeevanjee, 2021). Can the changes in global-mean surface temperature across the CO 2 levels in our simulations ( Figure 1a) explain the non-monotonic behavior of λ and, therefore, EffCS? 2. The non-monotonic dependence of λ may arise from a strong dependence of λ on the spatial pattern of SSTs.
Recent studies have found a close coupling between SST patterns and radiative feedbacks in observations and model simulations, the so-called "pattern effect" (Dong et al., 2019;Sherwood et al., 2020;Zhou et al., 2016). If the SST pattern effect caused the non-monotonic response in λ, then what SST regions govern the global and local changes in our feedbacks?
To test the hypotheses, we run the atmospheric component of the coupled model CESM1-LE (CAM5) with specified SST boundary conditions, in order to examine the impacts of different surface warming on λ. First, we perform a set of 150-year long CAM5 simulations where we fix all radiative forcing agents at pre-industrial levels, and prescribe the time-varying SSTs produced by the corresponding coupled model n×CO 2 simulations. In these runs (denoted as "prescribed-SST"), TOA radiative fluxes and surface air temperature freely adjust to the underlying SSTs. Although not directly forced by CO 2 , we find that the prescribed-SST simulations accurately reproduce the values of λ from the corresponding coupled simulations (c.f. blue and black dots in Figure 2a, error bars shown in Figure S2 in Supplementary Information S1). This finding, consistent with other studies (Haugstad et al., 2017;Zhou et al., 2023), suggests that the dependence of λ on CO 2 forcing is primarily shaped by the SSTs induced by the CO 2 forcing and therefore confirms the validity of using prescribed-SST simulations to study radiative feedbacks to understand the coupled n×CO 2 results.
Next, we perform another set of prescribed-SST simulations with adjusted SST boundary conditions. To test hypothesis # 1, that is, whether λ responds non-monotonically to changes in global-mean surface temperatures, we conduct simulations where we scale the SST pattern from 3×CO 2 by the actual global-mean SST changes in coupled 4×CO 2 and 5×CO 2 , respectively. Such that these two runs have the same normalized global SST pattern (at every monthly time step) as the 3×CO 2 run but different global-mean SST values (denoted "prescribed-SST with 3×CO 2 pattern"). In these experiments, we find that the λ values do not reproduce those in the coupled and prescribed-SST simulations even though the same global-mean SST warming is prescribed (c.f. red and blue dots in Figure 2a), suggesting that the non-monotonic response in λ arises from changes in the spatial pattern of SSTs (hypothesis # 2) and not the changes in the global-mean values of SSTs (hypothesis # 1).
The above CAM5 prescribed-SST simulations highlight the role of SST patterns in driving the non-monotonic response in λ. To understand what regions contribute to this non-monotonicity, we show the spatial pattern of λ calculated as the local net TOA radiation regressed to global-mean surface air temperature response, shown in Figures 2b-2d. The spatial pattern of λ in the 4×CO 2 prescribed-SST run (4×CO 2 SST pattern), is shown in Figure 2b, corresponding to the globally averaged λ at 4×CO 2 shown by the blue dot in Figure 2a. The spatial pattern of λ in a 4×CO 2 run with 3×CO 2 SST pattern (red dot in Figure 2a) is shown in Figure 2c. Taking the difference between Figures 2b and 2c (panel d) shows substantially more negative feedback in the North Atlantic with the 4×CO 2 pattern, and not much change when we use the 3×CO 2 pattern, indicating that the anomalously low EffCS at 4×CO 2 in our coupled simulations is primarily associated with an anomalously negative λ in the North Atlantic. We show the same λ spatial patterns for 5×CO 2 runs in Figure S3 in Supplementary Information S1.

A Local Pattern Effect From the North Atlantic
While the stronger negative feedbacks appear to be located mainly in the North Atlantic, it is unclear whether they are driven by the local North Atlantic SST changes, or by remote SST impacts from other basins. In a prior study (Lin et al., 2019) the effect of North Atlantic SSTs on λ was connected to tropospheric stability response in abrupt 4×CO 2 runs across CMIP models. Our present focus is on the North Atlantic SSTs influence on λ across various CO 2 levels. In Figure 3, we show the normalized SST patterns from 3×, 4×, and 5×CO 2 simulations (panels a-c). We find that anomalous SST cooling primarily occurs in the North Atlantic: 4×CO 2 produces a strong cooling in the North Atlantic and less warming in the subtropical Atlantic (panel b), largely resembling the pattern of the North Atlantic Warming Hole (NAWH) (Chemke et al., 2020). However, this North Atlantic relative-cooling pattern to the global mean does not emerge at 3×CO 2 (panel a) and is much weaker at 5×CO 2 (panel c). Concurrently, we find that local feedbacks exhibit patterns that closely match the SST patterns (Figures 3f-3h). The majority of the negative feedback strengthening, resulting in a lower EffCS, is found at 4× relative to 3×CO 2 (Figure 3i) in the North Atlantic, which aligns with the local cooling pattern (Figure 3d). Conversely, most of the feedback weakening at 5× relative to 4×CO 2 (higher EffCS, Figure 3j) is also observed in the North Atlantic and corresponds with the local warming pattern (Figure 3e). These results suggest that the non-monotonic response of the feedbacks found in our simulations (Figure 2a) is predominately from feedback changes in the North Atlantic, associated with North Atlantic local SST changes. We note that significant feedback changes also occur in the tropical Pacific (Figures 3i and 3j), particularly the tropical Eastern Pacific, but these feedback changes are in the opposite sign to the global-mean feedback changes, and thus cannot account for the total feedback response we showed in Figure 2a. While some other regions may contribute to the negative feedback change (e.g., the tropical Western Pacific and the Southern Ocean), we find that the North Atlantic local λ (area between 0° and 60°N and 80°W to 10°E) explains up to two thirds of the total change in the global-mean λ ( Figure S4 in Supplementary Information S1). This suggests that most of the non-monotonicity at 4×CO 2 is due to the North Atlantic pattern effect.
To further understand the processes causing the λ non-monotonicity, we further decompose the net feedback parameter λ into the individual feedbacks using radiative kernels (Pendergrass et al., 2018) (Figure 4). In the North Atlantic at 4×CO 2 , the Planck feedback ( Figure 4e) is strongly positive as the local cooling reduces outgoing radiation, whereas the combined lapse rate and water vapor feedback (Figure 4h) and the cloud feedback ( Figure 4k) contribute negatively. The strong negative feedback at 4×CO 2 compared to 3×CO 2 in the subtropical North Atlantic is primarily due to the SW cloud feedback (Figures S4 and S5h in Supplementary Information S1); hence, it is one of the key contributors to the λ non-monotonicity at 4×CO 2 . Although our 4×CO 2 run shows little to no cooling in the tropical/subtropical North Atlantic (Figure 3b), this region warms less than the tropical average due to the formation of the North Atlantic warming hole, which increases local low-level cloud cover ( Figure S6a-S6c in Supplementary Information S1). This increase in low clouds coincides with an increase in the estimated inversion strength (EIS), which we find is primarily caused by local SST changes relative to the tropical average and not the upper tropospheric temperature changes ( Figure S6 in Supplementary Information S1). These results suggest that the anomalous subtropical SST changes between 3× and 4×CO 2 (Figure 3d), although weaker than the extratropical cooling, can efficiently change local cloud feedback and therefore global climate sensitivity. This EIS mechanism is consistent with the leading mechanism found in the tropical Pacific pattern effect (  with the North Atlantic SST changes (Lin et al., 2019), and causes the non-monotonic response in EffCS and λ across CO 2 levels in our experiments.
Additionally, it is important to note that two thirds of the difference in feedbacks between 4×CO 2 and 3×CO 2 comes from the North Atlantic and one third from the rest of the globe. At 4×CO 2 , there are strong responses in the individual feedbacks in the tropical Pacific (see Figure S5 in Supplementary Information S1 for albedo and longwave cloud feedbacks). However, the negative Planck feedback response in the tropical Pacific is compensated by the local positive feedback response from lapse rate, water vapor, and clouds (Figures 4b, 4e, 4h, and 4k), which makes the tropical Pacific less pronounced in the non-monotonic λ changes.
Having shown that feedback changes primarily come from the North Atlantic associated with local SST cooling, we finally return to the key pattern of North Atlantic SST cooling found in our simulations, the North Atlantic warming hole (NAWH). In the literature, the appearance of the NAWH has been attributed to the slowdown in the AMOC and linked to an atmospheric response (Caesar et al., 2018;Latif et al., 2022;Rahmstorf et al., 2015;Sévellec et al., 2017). Our previous work ( Figure S3 in Mitevski et al. (2021)) found the North Atlantic cooling (NAWH) in our experiments is primarily due to AMOC collapse. The AMOC collapses at 4×CO 2 in our GCM, and at all other higher CO 2 forcings. At higher CO 2 forcings (5×CO 2 and above), the AMOC collapse no longer produces anomalous North Atlantic cooling compared to the previous level of CO 2 forcing (e.g., 4×CO 2 ) because the AMOC collapse-induced SST cooling is further overwhelmed by the surrounding warming. Hence, the cooling over the NAWH is less pronounced at higher CO 2 forcings (Figure 3c) and has a smaller impact on the feedbacks (Figure 3h). The collapse of the AMOC under CO 2 forcing has been widely reported in climate models, including the GISS-E2.1-G model in this study (occurring at 3×CO 2 and higher) and many other CMIP5 and CMIP6 models ( Figure S3 in Mitevski et al. (2021)).

Discussion and Conclusion
In a series of n×CO 2 (n = 2, 3, 4, 5, 6, 7, 8) experiments, we find a non-monotonic response in the effective climate sensitivity (EffCS) to CO 2 forcing using two state-of-the-art, coupled climate models. EffCS becomes anomalously low at an intermediate level of CO 2 (4×CO 2 in CESM1-LE and 3×CO 2 in GISS-E2.1-G) but increases at higher CO 2 levels. This EffCS non-monotonicity is primarily linked to changes in radiative feedback λ due to tropical and subtropical North Atlantic cooling relative to the tropical mean; λ becomes anomalously negative when cooling emerges in the North Atlantic and forms a North Atlantic Warming Hole (NAWH).
The dependence of λ on sea-surface temperature (SST) patterns has been widely studied, with a focus on the time-evolution of those patterns (Andrews et al., 2015(Andrews et al., , 2022Dong et al., 2019;Sherwood et al., 2020;Zhou et al., 2016). For example, estimates of EffCS from the observed historical energy budget constraints are lower than those from long-term warming under CO 2 quadrupling, primarily owing to changes in the tropical Pacific SST patterns (Andrews et al., , 2022Dong et al., 2019;Gregory et al., 2020;Zhou et al., 2016). This "pattern effect" has been studied with a Green's function approach (Dong et al., 2019;Zhang et al., 2023;Zhou et al., 2017), which shows that the global feedback has a predominant dependence on tropical convective regions (Williams et al., 2023), and is less sensitive to the North Atlantic SSTs. This tropical Pacific SST pattern effect has been found to be a leading mechanism for the time evolution of EffCS estimates. However, our study proposes a North Atlantic pattern effect that accounts for changes in EffCS and feedbacks across different CO 2 forcing levels. This North Atlantic pattern effect shows that SST cooling (relative to the tropical mean) in the North Atlantic due to the formation of NAWH causes λ to become more negative and, therefore, lower EffCS. We note that the North Atlantic pattern effect proposed here operates on the dimension of increasing CO 2 forcing, instead of on the dimension of time evolution addressed in previous studies (Andrews et al., 2022;Andrews & Webb, 2018;Dong et al., 2019;Lin et al., 2019;Zhou et al., 2016). In particular, Lin et al. (2019) showed that the North Atlantic cooling pattern affects the evolution of λ in 150-year abrupt 4×CO 2 runs, whereas we have here considered how North Atlantic cooling impacts λ at different CO 2 forcing. Both findings highlight that the NAWH can influence λ depending on the timing of AMOC decline and feedback variations at other locations, and hence the NAWH is an important player in quantifying global warming.
The NAWH has been proposed to arise from the reduction of surface meridional ocean heat transport (Chemke et al., 2020) or AMOC slowdown that reduces transient warming due to increased ocean heat uptake (Caesar et al., 2020;Palter, 2015;Rugenstein et al., 2013;Trossman et al., 2016;Winton et al., 2013). In our study, we find that the NAWH can further reduce EffCS and transient warming by causing more negative feedback (more efficient radiative damping at the top of the atmosphere). The fact that the NAWH has been observed in the historical period and is projected to persist in future scenarios with increasing GHG (Chemke et al., 2020;Gervais et al., 2018;Keil et al., 2020;Liu et al., 2020;Menary & Wood, 2018;Ren & Liu, 2021) suggests a considerable damping effect on global warming from the North Atlantic.
We further analyzed two subsets of CMIP6 models with and without NAWH in the abrupt-4×CO 2 runs ( Figure  S7 in Supplementary Information S1). Models with a NAWH in the abrupt-4×CO 2 scenario also show more surface cooling in the North Atlantic in transient 21st-century simulations (under both SSP5-8.5 and SSP2-4.5 scenarios) than models without NAWH. This suggests that uncertainty in the projected long-term North Atlantic SST patterns in response to abrupt CO 2 forcing also persists in transient projections. Thus, understanding North Atlantic SST changes is crucial for constraining global climate change at both transient and equilibrium timescales.
One caveat to our findings is that the AMOC collapse in our models occurs at 3× and 4×CO 2 , which are relatively low CO 2 values, where the collapse can induce a substantial cooling in the North Atlantic. When the AMOC collapses at a low CO 2 value, the North Atlantic cooling is strong, leading to a considerable non-monotonicity in EffCS. However, if the AMOC collapses at a higher CO 2 value, such as 5×CO 2 , then the overwhelming CO 2 warming from the surrounding areas results in a weaker North Atlantic SST cooling or delayed warming pattern. In this case, the EffCS non-monotonicity would be smaller than the one reported in this study. Hence our results suggest that future changes in AMOC and NAWH may add additional uncertainty to EffCS and transient 21st century warming projections.
We also acknowledge that our results are based on only two GCMs, and that 150-year model runs are not fully equilibrated. The North Atlantic SST cooling and λ response in 150 years could be transient, as previous studies have shown that AMOC can resuscitate when models are run longer (e.g., Rind et al. (2018); Bonan et al. (2022)). However, understanding AMOC changes under CO 2 forcing on centennial to millennial timescales is beyond the scope of this study. It is also important to note that the feedback dependence on CO 2 forcing we have reported here differs from the temperature dependence proposed in other studies (e.g., Bloch-Johnson et al. (2021)) because we have here shown that λ depends on the SST pattern (4×CO 2 blue vs. red dots in Figure 2a) rather than the global mean surface temperature.
The fact that EffCS is nonlinear and even non-monotonic with respect to CO 2 levels complicates equilibrium climate sensitivity constraints using models, observations, the paleoclimate record, and process-based understanding. While the non-constant λ across different CO 2 levels has been mainly attributed to feedback temperature dependence within models (Bloch-Johnson et al., 2021;Mauritsen et al., 2019;Meraner et al., 2013;Sherwood et al., 2020;Zhu & Poulsen, 2020) and paleoclimate records (Anagnostou et al., 2016(Anagnostou et al., , 2020Farnsworth et al., 2019;Friedrich et al., 2016;Shaffer et al., 2016;Zhu et al., 2019), we here have shown that the SST pattern also plays a role. Our study adds additional evidence of EffCS state dependence and pattern effects, and adds to the growing body of evidence pointing to the North Atlantic as an important region for understanding climate sensitivity and feedbacks. This work was supported by NASA FINESST Grant 80NSSC20K1657. YD was supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR's Cooperative Programs for the Advancement of Earth System Science (CPAESS) under award NA210AR4310383. MR was supported by the National Science Foundation under Grant 2202916. The work of LMP is supported, in part, by a grant from the US National Science Foundation to Columbia University. We thank the high-performance computing resources provided by NASA's Advanced Supercomputing (NAS) Division and the NASA Center for Climate Simulation (NCCS).