An analytic theory for the degree of Arctic Amplification

Domain: earth_science
Published: 2024-06-13
Editorial outcome: minor_revision

Abstract
Arctic Amplification (AA), the amplified surface warming in the Arctic relative to the globe, is a salient feature of climate change. While the basic physical picture of AA has been depicted, how its degree is determined has not been clearly understood. Here, by deciphering atmospheric heat transport (AHT), we build a two-box energy-balance model of AA and derive that the degree of AA is a simple nonlinear function of the Arctic and global feedbacks, the meridional heterogeneity in radiative forcing, and the partial sensitivities of AHT to global mean and meridional gradient of warming. The formula captures the varying AA in climate models and attributes the spread to models’ feedback parameters and AHT physics. The formula clearly illustrates how essential physics mutually determine the degree of AA and limits its range within 1.5-3.5. Our results articulate AHT as both forcing and feedback to AA, highlight its fundamental role in forming a baseline AA that exist even with uniform feedbacks, and underscore its partial sensitivities instead of its total change as key parameters of AA. The lapse-rate feedback has been widely recognized as a major contributor to AA but its effect is fully offset by the water-vapor feedback.

Introduction
Robustly seen in paleoproxy records1,2, historical observations3–5, and model simulations6–9, the temperature response to climate change is amplified in the Arctic relative to the rest of Earth. This so-called Arctic Amplification (AA) not only affects the Arctic cryosphere and ecosystem but also influences global circulation and climate by modulating the meridional temperature gradient10–16. However, the degree of AA, commonly defined as the ratio in surface temperature changes between the Arctic and global mean, varies widely by a factor of two among climate models17. This large uncertainty is not unexpected, considering the complex processes and feedbacks contributing to AA17–19. Locally, surface warming in the Arctic is amplified by the positive albedo feedback due to sea ice loss20,21, the positive lapse-rate feedback due to surface-trapped warming18,22, and the less negative Planck feedback due to colder climate18. Remotely, the increase in meridional atmospheric heat transport (AHT), implied by enhanced meridional moisture gradient from global-scale warming, must be constrained by amplified Arctic warming so that it does not exceed the demand from the meridional gradient in radiative forcing23–28. Furthermore, the above effects are not independently additive but can interact with each other29–31.

The relation between AA and its contributing factors has been studied through diagnostic and modelling methods. For example, the contributions of individual factors to Arctic and global warming have been diagnosed based on energy budget equations32–36. The dependences of AA on regional feedbacks, ocean heat flux, and radiative forcing have been investigated using both the diffusive energy balance model (EBM)37–40 and comprehensive climate models41. For AA in a diffusive EBM with uniform feedbacks, analytic estimates have been proposed39. Despite these achievements, our understanding of how the degree of AA is determined remains limited as several outstanding questions remain: What are the essential physics that determine the degree of AA? What is the functional relationship between the degree of AA and the key physical parameters? How to understand the range of AA (i.e., why it is 1.5–3.5 instead of 10)? What causes the variation in the degree of AA among models?

Here, we answer the above questions by establishing an analytic theory for the degree of AA. Specifically, we build a two-box energy-balance model of AA and reveal that the degree of AA is a simple nonlinear function of five key physical parameters. The theoretical function conveys a concise picture of how the degree of AA emerges from essential physics and explains the varying degree of AA in individual climate models. Furthermore, it clearly interprets the intricate role of AHT in AA and reveals the close compensation between the effects of the lapse-rate and water-vapor feedbacks on AA.

Results
The degree of AA as a simple nonlinear function of key physical parameters

We start with the energy balance equations for the Arctic and global mean under climate change,

Arctic: FA + ΔOA + ΔAHT + Σi λAi ΔTA = 0, (1)

Global: FG + ΔOG + Σi λGi ΔTG = 0, (2)

where the superscripts A and G denote the Arctic and global mean respectively, F is the effective radiative forcing, Δ denotes the response to climate change, T is the surface temperature, λi represents individual feedback parameters (defined respectively with respect to the Arctic and global mean surface warming) from Planck, lapse rate, water vapor, albedo and clouds, ΔO refers to the change in oceanic heat flux, and ΔAHT represent the change in AHT into the Arctic.

The widely-used diagnostic method32–36 utilizes Eqs. (1, 2) to partition the total warming into contributions of individual factors as

ΔTA = FA / (-Σi λAi) + ΔOA / (-Σi λAi) + ΔAHT / (-Σi λAi) + (Σi λAi,alb + Σi λAi,LR + Σi λAi,WV + Σi λAi,cld + λ0P,A)ΔTA / (-Σi λAi), (3)

ΔTG = FG / (-Σi λGi) + ΔOG / (-Σi λGi) + (Σi λGi,alb + Σi λGi,LR + Σi λGi,WV + Σi λGi,cld)ΔTG / (-Σi λGi), (4)

where Σi λGi,P is the global mean Planck feedback and λ0P,A = Σi λAi,P - Σi λGi,P. The feedback λi is diagnosed as contributing to the total Arctic warming (ΔTA) by -λAi / Σi λAi ΔTA and to the total global warming (ΔTG) by -λGi / Σi λGi ΔTG. Such partition is helpful for understanding the relative importance of individual factors. However, with ΔTA and ΔTG used as input, it is silent on why the degree of AA varies among models, and more generally, how the degree of AA varies with these physical factors. Furthermore, the attributed warming may not reflect the real role of a specific factor. For example, if a feedback has the same strength between the Arctic and global mean (λAi = λGi), it should not physically contribute to AA but would be diagnosed as contributing more warming to the Arctic given ΔTA > ΔTG. The theory, by contrast, shows how the degree of AA is determined.

Here, we build a two-box energy-balance model of AA by deciphering the change in AHT and upon that derive an analytic formula between AA and its contributing factors (Fig. 1). The change in AHT, ΔAHT, is a unique factor involved in AA. It is neither a pure forcing (e.g., F) nor a pure feedback (e.g., λ) to AA. Instead, it consists of two parts—a forcing-like part as global-scale warming enhances meridional moisture gradient (dq*/dT is higher in warmer tropics) and increases AHT to amplify Arctic warming, and a negative feedback part as AA weakens meridional temperature gradient and reduces AHT. This motivates us to formulate ΔAHT as a function of global mean warming ΔTG and enhanced Arctic warming ΔTA - ΔTG,

ΔAHT ≈ aΔTG - b(ΔTA - ΔTG), (5)

and a reformulation of Eq. (5) gives

ΔAHT / ΔTG ≈ a - b(AA - 1), (6)

The above formula of ΔAHT is not only supported by the significant correlation between ΔAHT and AA-1 across models (r = -0.70; Fig. 2a) but also derivable from the theoretical diffusive formulation of AHT (Methods). The intermodel regression between ΔAHT / ΔTG and AA-1 estimates â ≅ 2.1 Wm-2K-1 and b^ ≅ 1.7 Wm-2K-1 (Fig. 2a). The diffusive theory derives a and b as functions of basic parameters of the climate systems and suggests consistent values (Methods). The hat symbol here denotes a general, constant estimate for the model ensemble instead of individual models. With constant estimates of a^ and b^, Eq. (5) reasonably captures ΔAHT projected by models (r = 0.71), explaining ~50% of the variance in ΔAHT (Fig. S1). The unexplained variance reflects the variations in the parameters a = ∂AHT/∂TG and b = -∂AHT/∂(TA - TG) among models, that is, the model-dependent sensitivities of AHT to global uniform warming and enhanced Arctic warming. We further estimate a and b for individual models according to the deviation of ΔAHT / ΔTG from the regression line, i.e., ΔAHT / ΔTG = a^ - b^(AA - 1), so that Eq. (5) reproduces ΔAHT in individual models (Methods).

To facilitate derivation, we have also written the change in ocean heat flux, ΔO, as a feedback to the temperature change, similar to ref. 42,

ΔO = λO ΔT, (7)

where λO is diagnosed as ΔO/ΔT for the globe and the Arctic. While this is mainly a mathematical treatment, it can be physically justified as ΔO is approximately proportional to ΔT over the transient response to climate change (Fig. S2).

Equations (1, 2) and (5–7) form a two-box energy-balance model of AA, in which the Arctic and the globe experience their respective forcing and feedbacks and interact through AHT (see Fig. 1 for a schematic). This two-box model yields a theoretical solution for the degree of AA as,

AA = ΔTA / ΔTG = 1 + a + λA - γλG / b - λA = IAA, (8)

where γ = FA / FG is the ratio in radiative forcing between the Arctic and the globe, λA = Σi λAi and λG = Σi λGi are the sums of feedbacks (Planck, albedo, lapse rate, water vapor, cloud, and ocean) for the Arctic and the globe respectively, a = ∂AHT/∂TG measures the increasing rate of AHT with global uniform warming and b = -∂AHT/∂(TA - TG) measures the decreasing rate of AHT with enhanced Arctic warming. Physically, the Arctic warms more than the globe because of the increasing AHT with global uniform warming and its less negative feedback, while the denominator b - λA measures the energy damping efficiency of the Arctic, reflected by the decreasing AHT with enhanced Arctic warming minus the Arctic feedback.

We proceed to validate the theory by applying it to capture the degree of AA in individual climate models (Methods). The responses to climate change (Δ) are computed from the differences between the 1980−1995 period in the historical simulations and the 2085−2100 period under the Shared Socioeconomic Pathways SSP2-4.5 in 32 models from the Coupled Model Intercomparison Project Phase 6 (CMIP6)43. The feedback parameters, λi, are diagnosed using the radiative-kernel method44,45 for both the globe and the Arctic. As shown in Fig. 2b, the Planck, albedo, lapse-rate feedbacks are less negative or more positive in the Arctic than in the globe (above the 1:1 line) and would contribute positively to AA; in contrast, the cloud and water-vapor feedbacks contribute negatively to AA. It is difficult to accurately estimate the parameter γ for individual models, so we adopt a constant value of γ^ = 0.6 based on the spatial pattern of the 2xCO2 radiative forcing46. The omission of a model-dependent γ may be justified as radiative forcing accounts for only a small portion of uncertainty in the absolute Arctic warming18,36. Using diagnosed model-dependent λA and λG but constant a^, b^ and γ^, the theoretical formula reasonably captures the degree of AA in individual models, with IAA = 1 + a^ + λA - γ^λG / b^ - λA correlated with the model-projected AA at r = 0.71 across models (Fig. 2c). The accuracy is further improved by using the model-dependent a and b. IAA = 1 + a + λA - γ^λG / b - λA is correlated with the model-projected AA at r = 0.92 across models (Fig. 2d).

A concise picture of how the degree of AA emerges from essential physics

The theoretical formula, IAA = 1 + a + λA - γλG / b - λA, conveys a concise picture of how the degree of AA is mutually determined by essential physics.

First, a baseline AA exists with even spatially uniform radiative forcing (γ = 1) and feedbacks (λA = λG). This baseline AA,

IAA,B = 1 + a / b - λG, (9)

reflects the fundamental role of AHT and is set by the ratio between the increasing rate of AHT with global uniform warming (a) and the decreasing rate of AHT with enhanced Arctic warming minus global climate feedback (b - λG). Its degree is estimated to be 1.67 ± 0.19 (± s.d.) if feedbacks are based on the value of b - λG (Fig. 2e). The Arctic influences AA by

FF = IAA - IAA,B = a + λA - γλG / b - λA - a / b - λG = (λA - λG) / b - λG - (λA - λG) / b - λA * (a + λA - γλG) / b - λG (10)

Specifically, the weaker radiative forcing in the Arctic relative to the globe reduces AA (γ = FA < 1 with λG < 0; cyan bar in Fig. 2e) but the less negative (or even positive) climate feedback in the Arctic amplifies AA (λA - λG > 0; golden bar in Fig. 2e). The effect of differential feedbacks dominates and the degree of AA is amplified from the baseline AA to 2.51 ± 0.49. We further quantify the contribution of individual feedback to the degree of AA by overriding its Arctic value with its global value in the formula (Methods). The Planck, albedo and ocean feedbacks increase the degree of AA respectively by 0.57 ± 0.03, 0.65 ± 0.28 and 0.35 ± 0.34, while the cloud feedback decreases AA by 0.49 ± 0.51.

The lapse-rate feedback, being negative in low latitudes with amplified upper-level warming but positive in the Arctic with surface-trapped warming, has been widely recognized as a major contributor to AA18. Furthermore, as lapse rate in the Arctic is affected by radiative forcing and AHT, the lapse-rate feedback presents dependency on other factors involved in AA29–31. Understanding the lapse-rate feedback and its intricacy seems to be a leading-order problem for understanding AA. On the other hand, it has been long recognized in the climate sensitivity research community that the lapse-rate feedback is compensated by the water-vapor feedback47. Here, we show that the contribution of the lapse-rate feedback to AA is fully, if not overly, compensated by that of the water-vapor feedback (Fig. 2e). As the surface-trapped warming simultaneously reduces the increase in the tropospheric water vapor (Fig. 3a, b), the more positive lapse-rate feedback in the Arctic is associated with a less positive water-vapor feedback, so the sum of these two feedbacks is nearly identical between the globe and the Arctic (Fig. 2b; Fig. 3c). Furthermore, models that simulate a larger tropospheric warming also present a larger tropospheric moistening (solid versus dashed line in Fig. 3a, b), so the lapse-rate and water-vapor feedbacks are negatively correlated across models for both the globe (r = -0.74) and the Arctic (r = -0.88), leading to small intermodel spread in their sums (Figs. 2b, 3c). Eventually, the sum of the lapse-rate and water-vapor feedbacks is 1.36 ± 0.11 Wm-2K-1 for the globe and 1.28 ± 0.07 Wm-2K-1 for the Arctic. Thus, the combined feedback contributes little to AA and accounts for little intermodel uncertainty in AA compared to other feedbacks (Fig. 2e).

The formula IAA = 1 + a + λA - γλG / b - λA articulates the intricate role of AHT in AA. In literature, the effect of AHT on AA is often diagnosed from ΔAHT (see Eq. (3)). ΔAHT, however, consists of a negative feedback part that depends on the degree of AA itself. The formula here highlights the partial sensitivities of AHT to global uniform warming (a = ∂AHT/∂TG) and meridional warming gradient (b = -∂AHT/∂(TA - TG)), instead of ΔAHT itself, as the fundamental parameters for understanding the role of AHT. Even without differential feedbacks, the physics of AHT leads to the baseline AA, IAA,B ≈ 1.67, which forms a major part of IAA. The skill of IAA in capturing AA is improved from r = 0.71 to r = 0.92 by using the model-dependent a and b, highlighting the important role of AHT in determining the degree of AA in individual models. The effect of AHT in individual models can be further understood from the following approximated formula (see derivation in Methods),

IAA = 1 + a + λA - γλG / b - λA ≈ 1 + a^ + d / b^ + λA - γλG / b - λA, (11)

where d = ΔAHT / ΔTG - ΔAHT^ / ΔTG is the difference between the model-simulated ΔAHT / ΔTG and the prior-estimated ΔAHT^ / ΔTG = a^ - b^(AA - 1) according to a^ and b^ (i.e., the regression line in Fig. 2a). The effect of AHT is represented by a^, b^, and d. A more positive d = ΔAHT / ΔTG - ΔAHT^ / ΔTG implies a higher a and/or a lower b that would favor a higher AA. This further explains why directly using ΔAHT to interpret the effect of AHT may be misleading. For example, as shown in Fig. 2a, M12 (CanESM5) projects a negative ΔAHT but has a positive d that would favor a larger AA.

The formula IAA = 1 + a + λA - γλG / b - λA provides a physical explanation for the likely range of AA, specifically, why AA sits roughly between 1.5 and 3.5 in climate models. The partial sensitivities of AHT to global uniform warming (a) and enhanced Arctic warming (b) are rooted in basic parameters of the climate system (Methods) and estimated to be a = 2.1 ± 0.3 Wm-2K-1 and b = 1.7 ± 0.3 Wm-2K-1 in models. The global and Arctic climate feedbacks, as diagnosed from the radiative-kernel method, are λG = -1.7 ± 0.6 Wm-2K-1 and λA = -0.4 ± 0.6 Wm-2K-1. Furthermore, the diffusive theory derives that a/b should be around one given the climatological distribution of specific humidity (Eq. (27); Methods), and the positive correlation between λA and λG (Fig. 2b) constrains λA - γλG to be 0.6 ± 0.5 Wm-2K-1. Applying these ranges in IAA suggests that it is very hard for the degree of AA to be below 1.5 or above 3.5.

Understand the variation and outliers of AA among climate models

As the theory well captures the degree of AA in individual models (Fig. 2d), we further apply it to understand the causes of the variation and outliers of AA. In Fig. 4a, the theory-predicted AA, IAA = 1 + a + λA - γ^λG / b - λA, is decomposed into the sum of the baseline AA, IAA,B Eq. (9) and the effect of differential forcing and feedback between Arctic and globe, FF Eq. (10). Models are ranked according to IAA and the magnitude of FF is indicated by the cool-to-warm color. The outlier models, three with the lowest IAA (M7, M8, M21) and three with the highest IAA (M12, M19, M32), are denoted. The variation in IAA are contributed by both IAA,B (1.3−2.1) and FF (0−1.8), but the effect of FF is more dominating. To understand why the degree of AA varies among models, we proceed to understand the variations in IAA,B and FF. According to Eq. (9), IAA,B is larger if a = ∂AHT/∂TG is larger, b = -∂AHT/∂(TA - TG) is smaller, and λG is weaker (less negative). A model that simulates a higher ΔAHT / ΔTG = a - b(AA - 1) than the prior-estimated ΔAHT^ / ΔTG = a^ - b^(AA - 1), should have a larger a and/or a smaller b. Thus, IAA,B would be larger if d = ΔAHT / ΔTG - ΔAHT^ / ΔTG is higher and λG is weaker. Indeed, as shown in Fig. 4b, the intermodel variation of IAA,B (indicated by cool-to-warm color of the diamond symbols) is consistently explained by its increases with d and λG. For the outlier models, M7 presents a small IAA,B due to its abnormally small d; M21 presents a small IAA,B due to its excessively strong λG (< -3 Wm-2); on the other hand, M12, 19, 32 present a large IAA,B due to their weak λG (> -1 Wm-2).

According to Eq. (10), FF would be larger when λG is weaker (less negative) but λA - λG is higher (more positive). We relate FF to λG and λA - λG instead to λA and λG, as there is a significant correlation between λA and λG (r = 0.6; black dots in Fig. 2b). As shown in Fig. 4c, the intermodel variation of FF (indicated by cool-to-warm color of the diamond symbols) is consistently explained by its increases with λG and λA - λG. For the outlier models, M7 and M8 present a very small FF as their λA - λG is abnormally low; M21 presents a very small FF despite normal λA - λG as its λG is excessively strong; on the other hand, M12, 19, 32 present a strong FF as their λG are abnormally weak.

The variations in λA - λG and λG are further attributed to the effects of individual feedbacks (λAi and λGi). Based on their baseline AA, FF, and d, λAi, λGi (Fig. 4d), we conclude the causes of the abnormal AA in the six outlier models as follows.

• M7 (CESM2): Small baseline AA due to abnormally low d; Small FF due to low λA - λG which is further attributed to the effects of λCld and λOcn.
• M8 (CIESM): Small FF due to abnormally low λA - λG, which is further attributed to the effects of λAlb and λOcn.
• M21 (INM-CM4-8): Small baseline AA and FF due to excessively strong λG, which is further attributed to the effect of λCld.
• M12 (CanESM5), M19 (HadGEM3-GC31-LL), M32 (UKESM1-0-LL): Large baseline AA and FF due to abnormally weak λG, which is further attributed to the effect of λCld.

Thus, the intermodel spread in the degree of AA can be understood from three key parameters, the dynamics of AHT (d), the differential feedbacks between Arctic and globe (λA - λG), and the global mean feedback (λG). Given the formula IAA = 1 + a + λA - γλG / b - λA, one might expect AA to increase with a stronger (more negative) λG. However, as λA and λG are correlated, a stronger λG implies a stronger λA and reduces IAA. By influencing λG, global mean cloud feedback strongly affects the degree of AA, as seen in the outlier models M21 and M12, 19 and 32.

Application to explaining previous numerical results of AA

The formula, IAA = 1 + a + λA - γλG / b - λA, can analytically explain the degree of AA in previous numerical studies that manipulate feedbacks and radiative forcing. Reference 39 imposed uniform forcing and feedbacks in a diffusive EBM. The resultant AA is ~1.6, which is consistent with the baseline AA here. Reference 41 investigated AA under regionally confined radiative forcing in a comprehensive climate model. They found a strong AA with polar forcing (poleward of 60o) but a very weak AA with tropical or midlatitudinal forcing. These results can be analytically explained by our formula. Specifically, IAA ≈ a + b - γλG / b - λA predicts IAA ≈ 6 for polar forcing with γ ≈ 4 and IAA ≅ 1.7 for tropical or midlatitudinal forcing with γ = 0. Reference 40 conducted feedback-removing experiments in a diffusive EBM and showed that the presence of the water-vapor feedback warms the Arctic more than the globe although the water-vapor feedback is weaker in the Arctic. This puzzling result can be analytically explained by our formula. Specifically, given IAA = a + b - γλG / b - λA, the presence of the water-vapor feedback warms the global mean by ΔTG,wv = ΔTG - ΔTG,wv = -Fwv / λG ≈ 0.31FG and warms the Arctic by ΔTA,wv = ΔTA - ΔTA,wv = ΔTG IAA - ΔTG,wv IAA,wv ≈ 0.7FG (the subscript ,wv denotes the experiment with the water-vapor feedback set to zero). These successes highlight that, the degree of AA, emerging from controls of multiple factors, is well captured by our simple nonlinear formula.

Methods
Global climate models

Outputs of 32 global climate models in the Coupled Model Intercomparison Project Phase 6 (CMIP6) are used. Responses to climate change are computed from the differences in the climatology between the 1980−1995 period in the historical simulations and the 2085−2100 period in the Shared Socioeconomic Pathways (SSP) simulations. We have considered the low (SSP1−2.6), intermediate (SSP2-4.5) and high (SSP5-8.5) emission scenarios. To estimate the forced response to anthropogenic climate change, we take the mean of multiple realizations of each model, considering the effect of climate variability approximately canceled out among different realizations. The names of the 32 models and the numbers of realizations used are summarized in Table S1.

Feedback parameters quantified by the radiative-kernel method

The method of radiative kernel is applied to calculate the feedback parameters, λi, as,

λi = ΔRTOA,i / ΔT, (12)

where ΔRTOA,i is the annual-mean top-of-atmosphere (TOA) radiation anomaly induced by the change in the feedback variable and ΔT is the change in the annual-mean surface temperature. The radiative kernel contains the monthly and regionally dependent response of the TOA radiative fluxes to incremental changes in the feedback variables (e.g., albedo, air temperature, and specific humidity). ΔRTOA,i is estimated by multiplying the anomaly of a feedback variable with its corresponding radiative kernel and integrating the product vertically throughout the troposphere. Specifically, the Planck feedback is computed from the change in the TOA radiation due to surface temperature changes that propagate throughout the troposphere; the lapse-rate feedback is computed from the effect of temperature departures from the vertically uniform change; the water-vapor feedback is computed from the effect of changes in air specific humidity. The albedo feedback is computed from the effect of changes in albedo, which is diagnosed from the surface downward and upward shortwave radiation. The cloud feedback is computed from the change in cloud radiative effect (ΔCRE) minus the effect of non-cloud variables on ΔCRE (i.e. the difference in ΔRTOA,i using the all-sky and clear-sky kernels).

Change in atmospheric heat transport into the Arctic

The energy budget in the Arctic is influenced by the change in the atmospheric heat flux into the Arctic, termed as ΔAHT. Considering the short timescale of the atmospheric energy balance, ΔAHT can be computed from the area-weighted integration of the atmospheric energy convergence (i.e., the difference between the net surface and TOA energy fluxes) as,

ΔAHT = ∫∫Arctic (- ΔRTOA - RSFC + LH + SH) dA. (13)

ΔAHT is formulated as a function of the changes in global mean surface temperature ΔTG and meridional temperature gradient ΔTA - ΔTG as

ΔAHT ≈ aΔTG - b(ΔTA - ΔTG). (14)

A reformulation of Eq. (14) gives

ΔAHT / ΔTG ≈ a - b(AA - 1). (15)

This formula is supported by the significant correlation between ΔAHT / ΔTG and AA - 1 across models (Fig. 2a; r=-0.70). The intermodel regression estimate a^ ≈ 2.1 Wm-2K-1 and b^ ≈ 1.7 Wm-2K-1. The hat symbol denotes a constant estimate that works generally for the model ensemble instead of individual models. By using a^ and b^, Eq. (14) reasonably captures ΔAHT in individual models (r = 0.71), capturing ~50% of the variance in ΔAHT among models (Fig. S1). The deviation of ΔAHT from this estimate with constant a^ and b^ reflects the variations of a and b among models. It is however challenging to accurately estimate a and b for individual models. The spread of the climate responses among different SSPs (SSP1−2.6, SSP2-4.5, SSP5-8.5) do not always yield a reasonable regression line between ΔAHT / ΔTG and AA - 1 for estimating a and b (Fig. S4). This could be due to either the varied a and b among different SSPs or insufficient realizations to cleanly remove the contamination from internal variability.

We approximately estimate a and b in individual models as deviations from a^ and b^ so that Eq. (15) reproduces the model-projected ΔAHT. Specifically, the difference between the model-projected ΔAHT / ΔTG and the prior estimate ΔAHT^ / ΔTG = a^ - b^(AA - 1), that is, d = ΔAHT / ΔTG - ΔAHT^ / ΔTG is attributed to a and b as

a = a^ + (1 - δ)d, (16)

b = b^ - δd / (AA - 1). (17)

In the main text, we use δ = 0.5, that is, attributing d equally to a and b. With the model-dependent a and b, Eq. (14) fully reproduces ΔAHT (Fig. S1).

We show below that the value of IAA is insensitive to the exact attribution of d = ΔAHT / ΔTG - ΔAHT^ / ΔTG to a and b. If d is all attributed to a, a = a^ + d and b = b^, then we have,

IAA - 1 ≈ a^ + d / b^ - λA - λA - γλG / b^ - λA. (18)

If d is all attributed to b, a = a^ and b = b^ - d / (AA - 1), then we have,

IAA - 1 ≈ a^ + λA - γλG / b^ - d / (AA - 1) - λA ≈ a^ + λA - γλG / b^ - λA + d / b^ - λA (AA - 1) + d^2 / (b^ - λA)^2 (AA - 1)^2 + ... ≈ a^ + λA - γλG + d / b^ - λA (AA - 1) + d^2 / (b^ - λA)^2 (AA - 1)^2 + ... (19)

Substituting AA - 1 ≈ IAA - 1 = α^ + a0 / β^ + b0 = α^ / β^ (1 + a0 / α^ - b0 / β^) leads to

IAA - 1 ≈ α^ / β^ + d / β^ + b0 α^ / β^2 (1 + a0 / α^ - b0 / β^) - α^ / β^ b0 / β^ (AA - 1) ≈ α^ / β^ + d / β^ + b0 α^ / β^2 - α^ b0 / β^2 (AA - 1) = α^ / β^ + d / β^ - α^ b0 / β^2 (AA - 1) ≈ α^ / β^ + d / β^ = a^ + d + λA - γλG / b^ - λA. (20)

Theoretical derivation of the formula of ΔAHT Eq. (5)

The formula of ΔAHT ≈ aΔTG - b(ΔTA - ΔTG) can also be theoretically derived from the diffusive formulation of AHT. Specifically, according to the diffusive theory, AHT into the Arctic can be written as,

AHT = -βD T Cp ∂T / ∂θ |θo - βD q Lv ∂q / ∂θ |θo, (21)

where T and q are surface temperature and specific humidity respectively, D T and D q are the diffusivity for temperature and specific humidity, β is a constant, and ∂/∂θ |θo is the meridional gradient at the boundary θo. Under climate change, the change in AHT, ΔAHT, can be written as,

ΔAHT = -βD T Cp ∂ΔT / ∂θ |θo - βD q Lv ∂Δq / ∂θ |θo. (22)

ΔAHT = -βD T Cp ∂ΔT / ∂θ |θo - βD q Lv ∂Δq / ∂θ |θo = -βD T Cp ∂ΔT / ∂θ |θo - βD q Lv ∂Δq / ∂T ∂T / ∂θ |θo ΔTG = -βD T Cp ∂ΔT / ∂θ |θo - βD q Lv ∂q / ∂T ∂T / ∂θ |θo ΔTG, (23)

where the tilde symbol ̃ indicates the warming-weighted mean and ζ = ΔTA - ΔTG / ΔTG. In Eq. (23), we have used the relation ∂Δq / ∂T = R Lv / Tv^2, which is derived from the Clausius-Clapeyron relation and the assumption that relative humidity stays roughly constant under climate change.

Substituting Eqs. (22) and (23) into Eq. (21) lead to the same formula

ΔAHT = aΔTG - b(ΔTA - ΔTG), (24)

with

a = -βD q Lv R Lv / Tv^2 ∂q / ∂T |θo, (25)

b = βζ D q Lv R Lv / Tv^2 ∂q / ∂T |θo + D T Cp. (26)

The above derivation shows that the formula ΔAHT = aΔTG - b(ΔTA - ΔTG) can be theoretically derived from the diffusive formulation of AHT and the parameter a and b are functions of basic parameters of the climate system.

Approximate values of a and b can be estimated using Eqs. (25) and (26). According to the climatological distribution of surface (2m) specific humidity (Fig. S6a), ∂q/∂T |θo ≈ 7 g kg-1, qe |θo ≈ 2 g kg-1. According to the meridional structure of the warming pattern (Fig. S6b), ζ = ΔTA - ΔTG / ΔTG ≈ 1. For diffusivity, Ref. 50 estimated D T ≈ 1.8 km2 s-1 and D q ≈ 1.2 km2 s-1 based on an aquaplanet simulation, while Ref. 51 used a fixed diffusivity D T,q ≈ 1.06 km2 s-1 that best fit the results. The estimates from the intermodel regression, a^ ≈ 2.1 Wm-2K-1 and b^ ≈ 1.7 Wm-2K-1, are consistent with Eqs. (25) and (26) if D T ≈ 1.6 km2 s-1 and D q ≈ 1.0 km2 s-1. We note that the ratio between a and b is a function of the climatological distribution of specific humidity and D T = D q as

a / b = qe |θo + D T Cp / D q Lv R Lv / Tv^2 ∂q / ∂T |θo ≈ 2 + 5.7 D T / D q, (27)

whose value should be around (or more likely slightly above) 1.

Ocean feedback

The energy budgets in the Arctic and the globe are influenced by the changes in the ocean heat flux, termed as ΔO. Considering the short timescale of the ocean-mixed-layer energy balance, ΔO can be estimated from the change in the net surface fluxes as,

ΔO = Δ(SH + LH - RSFC), (28)

where SH and LH are the surface (upward) sensible and latent heat flux respectively and RSFC is the net downward radiation flux at the surface. To facilitate the derivation and simplify interpretation, we have written ΔO as a feedback,

ΔO = λO ΔT, (29)

where λO is diagnosed as ΔO/ΔT for the globe and the Arctic. While this is just a mathematical treatment, it can be physically justified as ΔO is approximately proportional to ΔT in individual models (Fig. S2). For the globe, λG,O = -0.41 ± 0.10 Wm-2K-1; For the Arctic, λA,O = -0.14 ± 0.37 Wm-2K-1, where ± indicates the s.d. among models. We note that λO is similar to the ocean heat exchange parameter κ in ref. 42 but subtly different from (and thus smaller in magnitude than) the parameter η in Ref. 52 which further considered a deep-ocean layer.

Decompose the degree of AA into contributions of individual physical factors

IAA is decomposed into a baseline AA and the net effect of differential radiative forcing and feedbacks between Arctic and globe, that is,

IAA = 1 + a + λA - γ^λG / b - λA = 1 + a / b - λG + (a + λA - γ^λG / b - λA - a / b - λG) (30)

The effects of differential forcing and feedbacks, FF, can be further separated. However, we should note that, due to the nonlinear format of FF, the separate effects of differential forcing and feedbacks depend on each other. For example, the effect of differential forcing may be estimated as (1 - γ^)λG / b - λG or (1 - γ^)λG / b - λA depending on whether the effect of differential feedbacks has already been included. Both estimates are negative but they differ by ~0.1. In Fig. 2e, the effect of differential forcing is computed with the effect of feedbacks already included as (1 - γ^)λG / b - λA.

The effect of differential feedbacks is estimated by replacing the total Arctic feedback with the total global feedback, that is,

a + λA - γ^λG / b - λA - a + λG - γ^λG / b - λG. (31)

The contribution of individual feedback λi is then estimated by replacing its Arctic value with its global value, that is,

a + λA - γ^λG / b - λA - a + λA,0 - γ^λG / b - λA,0, (32)

with λA,0 = λA - λAi + λGi.

Data availability

The CMIP6 outputs are available from the Earth System Grid Federation (ESGF) Portal at https://esgf-node.llnl.gov/search/cmip6/. The source data underlying the main figures is available at https://zenodo.org/records/10976607. Source data are provided with this paper.

Code availability

The code for the radiative-kernel method is available from Professor Yi Huang’s group page https://data.mendeley.com/datasets/3drx8fmmz9/1. The script for analyses and generating figures is available at https://zenodo.org/records/10976607.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors develop a theorefical framework to quanfitafively link the strength of Arcfic and non-Arcfic/global forcing and feedbacks, the response of atmospheric heat transport to these and the resulfing Arcfic amplificafion of climate change.\nThe idea is appealing and some of the results are promising, but I have reservafions on several of the lines of evidence suggested in the manuscript.\n1) Derivafion of ^a and ^b: The authors state “The theorefical diffusive formula of AHT esfimates the same values of ̂ and ̂ under a constant diffusivity of = 106 m2 s-1 (Methods)”\nThis sounds like an independent way of esfimafing the parameters on a theorefical basis, but reading the corresponding parts of the methods secfion, where “typical” values of large-scale climate state variables are used without further jusfificafion, suggests that these might just have been chosen to deliver the desired result. I suggest to underpin these calculafions by using climate model output to esfimate the values needed for the theorefical derivafion, or to omit this line of evidence.\n2) Derivafion of a and b (model-specific): These are not actually derived using the underlying regression for each parameter, but just fifted by assuming that each parameter accounts for half the deviafion from the inter-model mean parameters. The fact that this works equally well when aftribufing all the deviafion from the mean parameter to just one of either a or b might indicate arbitrary overfifting rather than robustness of the method.\nI do not understand why there are not enough climate states to derive these parameters individually – one could use PI-control, different 30-year intervals from historical, 1pct CO2, scenario or 4xCO2 runs as well. If that does not work out, maybe there is a problem with the method.\nI was further wondering if a should be temperature-dependent and could get a more theorefical foundafion based on the Clausius-Clapeyron relafion, and whether the ‘global’ parameter should instead be sub-Arcfic. Given the small surface area of the Arcfic, I do not expect the lafter to make a strong quanfitafive difference, but it would seem more in line with the conceptual sketch given in Fig1.", "reviewer_2": "This is a very interesfing exercise looking at what factors impact Arcfic Amplificafion (AA) in models. The arficle is worth publishing. Two major comments: this is in no way a 'predicfive' theory of AA. It only amounts to 'diagnosfics' of AA. If the authors want to keep 'predicfive' in the fitle and call it a predicfive theory in the arficle body, I will reject it. The conceptual model does not include any physical basis for how the increase in AHT happens in a warmer climate. My second comment is about the absence of fime in the analysis. It assumes everything is in equilibrium while the models are definitely not. But overall, the analysis is very interesfing and seems to be very solid.", "reviewer_3": "Arcfic amplificafion is an aspect of climate change that has widespread interest and has been the subject of many lines of inquiry concerning its mechanisms. This manuscript offers a new theorefical framework to the factors governing Arcfic amplificafion that is novel and grounded in a fundamental aspect of how the atmosphere transports heat to the pole. It's a very illuminafing new approach and I recommend publicafion upon a minor revision.\nThe manuscript deploys a well jusfified approach to atmospheric heat transport (AHT) to Arcfic Amplificafion (AA). AHT's changes with climate can be thought of as depending on two mean-state quanfifies: the global-mean surface temperature, which increases the heat transport, and the temperature contrast in lafitude, AA decreases this which is an offsefting factor. This is combined with imposed regional feedbacks and forcing to capture the behavior of comprehensive climate models.\nTwo overarching points that should be acknowledged in the manuscript:\nA. The lapse rate feedback is prescribed from the results of GCMs: the Arcfic lapse rate is affected by a combinafion of factors, including radiafive forcing and atmospheric heat transport. The theorefical understanding of this is clear about why different forcing and feedback have different lapse rate changes (e.g., Cronin and Jansen 2016), and there are feedback analysis approaches that are designed to separate this (e.g., Feldl et al 2020). So, in principle, the Arcfic box's feedback depends on how its solufion evolves. The empirical success of the authors' approach suggests this isn't a leading-order effect---good results despite the omission of this complicafing factor. But it's important to communicate this aspect of the new framework introduced.\nB. The need to impose feedbacks from climate model simulafion means this framework in pracfice is sfill heavily diagnosfic (vs. purely diagnosfic approaches described near L96). This means the new framework is not predicfive, despite the fitle and abstract's language.\nSome clarificafions:\n1. The theory's \"baseline AA\" (L178) has a similar dependence on the global feedback parameter as moist energy balance model theory (Merlis and Henry 2018)---it appears in the denominator. Are these the same? Is the value of 1.67 (L181) the same?\n2. Fig. 2e shows the baseline AA is the largest factor, larger than the feedback part (gray vs. yellow). Is it fair to conclude that the energy transport related part of AA is dominant? In other words, in the diagnosfic approaches that are crifiqued in the introducfion (Pithan & Mauritsen 2014), the AHT term is not dominant but one can't make firm conclusions based on that approach. Can one now make a firm conclusion on that basic quesfion?\nMinor presentafion revisions:\n- The diagram in Fig. 1 shows two regions: Arcfic and everywhere else, but the equafions suggest the feedback parameter for \"everywhere else\" is actually the global mean. Is there this double counfing where the diagnosed Arcfic feedback enters both lambda A and G?\n- L124 typo before which- paragraph ending L210: this sounds like a rehash of the central discussion of Held & Shell 2012 who suggest a different feedback decomposifion that assumes constant relafive humidity as the reference response, but applied to the zonal-mean. But others have presented zonal-mean feedbacks using that different decomposifion, so I don't think this is parficularly new, and maybe not worth emphasizing strongly parficularly given the ambiguity of what the Arcfic lapse rate is controlled by. For example, Hahn et al. 2021 did the diagnosfic energy budget approach with both feedback decomposifions cand found: \"As a result, the relafive contribufion of the lapse-rate feedback to Arcfic amplificafion is weakened in the fixed-RH framework, with stronger contribufions from the albedo feedback and poleward moisture transport.\" Using all of Fig. 3 on this point is overkill to me.\n- L272 typo with lambda superscript, repeated G\n- L293 highlighfing individual models by this coding is not the most effecfive way to communicate, perhaps point to the right side of Fig. 4 columns instead\n- L376: does this ocean heat uptake parameter agree with previous published results like Geoffroy et al. 2013 and work by J. Gregory? I had a ~50% more negafive number in mind\n- eqn. S9", "reviewer_10": "there's never a saturafion specific humidity defined or relafive humidity stated that it is assumed constant\n- Fig. 4 capfion uses FF, but that's not defined in the main text"}, "author_response": "Response to Reviewer #1:\nThe authors develop a theoretical framework to quantitatively link the strength of Arctic and non-Arctic/global forcing and feedbacks, the response of atmospheric heat transport to these and the resulting Arctic amplification of climate change. The idea is appealing and some of the results are promising, but I have reservations on several of the lines of evidence suggested in the manuscript.\nThank you for your insigh@ul review. Following your suggesBons, we have used climate model outputs to esBmate the values of 𝑎 and 𝑏 in the theoreBcal derivaBon and explained why our formula works well independent of the exact aDribuBon of 𝑑 = ∆\"#$ ∆'\"#$ to 𝑎 and 𝑏. We have provided point-by-point responses to your comments below.\n1) Derivation of ^a and ^b: The authors state “The theoretical diffusive formula of AHT estimates the same values of 𝑎̂ and 𝑏 ̂ under a constant diffusivity of 𝐷= 106 m2 s-1 (Methods)”\nThis sounds like an independent way of estimating the parameters on a theoretical basis, but reading the corresponding parts of the methods section, where “typical” values of large-scale climate state variables are used without further justification, suggests that these might just have been chosen to deliver the desired result. I suggest to underpin these calculations by using climate model output to estimate the values needed for the theoretical derivation, or to omit this line of evidence.\nFollowing your suggestion, we have used the climate model output to estimate the values needed for the theoretical estimate of 𝑎 and 𝑏. We have rewritten the derivation so it is more rigorous (see the Method section). Specifically, we have used the partial derivative to latitude radian instead of 𝛿, and considered the different diffusivities for temperature and moisture.\nWe would like to note that the purpose of this theoretical derivation is not to exactly reproduce the values of 𝑎+ and 𝑏, estimated from the intermodel regression, but to show that ∆𝐴𝐻𝑇 = 𝑎∆𝑇' −𝑏(∆𝑇"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have fundamentally addressed my comments from the first review, but I have a few remaining points that should be addressed before publicafion:\n1) I am unconvinced of the descripfion of AHT as somehow between forcing and feedback. While it is not a classic radiafive feedback, it clearly is a response of the climate system to the external forcing mediated by and scaling with surface temperature change. It should thus be understood as a feedback, not as part of the forcing.\n2) The derivafion of a and b from model output lacks detail and thus clarity. What T and q are used? I would guess tas and huss, i.e. 2m near-surface values, but this is not spelled out. If my guess is correct, are results similar when using 850 hPa values instead?\nLines 33-35: As there is some ambiguity in aftribufing the spread of AA to the different factors for individual models, this claim seems a bit overstated.\nLine 50: given the references to Paeleoclimate, climate change does not have to be warming\nl. 106 ff: The argument makes physical sense, but feedback analysis somefimes use global-mean temperature change even for regional feedback factors, in which case the argument would not hold. This should be discussed to avoid confusion.", "reviewer_2": "The arficle is worth publishing, it is an interesfing study offering (yet another) tool for diagnosing Arcfic Amplificafion (AA) in models. My main problem with the arficle is that the overall novelty of the arficle needs to be explained befter. I am happy to see that they removed the word \"predicfive\" from the arficle, but sfill, this is just a diagnosfic framework for AA, and it is not even close a \"theory\". I would strongly suggest not using big words for describing this analysis. I am glad the authors included the sentence explaining why AHT will increase in the warmer climate (higher sensifivity of dq*/dT in the tropics). This very important point was completely missing (I believe) in the first version. This has been employed before for explaining AA (e.g. Langen Alexeev, Climate Dynamics 2007). Having said that, sfill, there is no formula that will explicitly include any physics (e.g. Clausius-Clapeyron equafion) to quanfify AA dependence on sensifivity of AHT to changes in the temperature. I nofice other reviewers point that some other parts of the paper just reformulate previous analyses, which is fine, but then what is new and excifing here? But again, the arficle represents a nice and interesfing diagnosfic study of AA in models, which is why I think it is worth publishing. I'd give it a solid \"B\".", "reviewer_3": "Overall, the revised manuscript addresses the concerns I had with the presentafion (e.g., is it a predicfive or diagnosfic framework?) and cleaned up the typos I found. I recommend accept with an opfional minor revision.\nThe one outstanding point of clarificafion concerns the mofivafion for and discussion of Fig. 4b: the verfical axis is the difference between the atmospheric heat transport parameters 'a' and 'b'. The discussion of the theorefical formula is clear that Arcfic Amplificafion increases with increasing 'a' and decreases with decreasing 'b', but is there a proper mathemafical mofivafion for the difference 'a-b' as being the unique quanfity to characterize intermodel spread? This is worthy of a clear discussion because 'a' and 'b' are approximately the same magnitude (the authors added more on theorefical derivafions in the revised methods). Put another way, I don't see why the boundaries between the shaded colors in Fig. 4b are straight lines given the equafion has a/(b-lambda_G). Shouldn't there be some curvature from the inverse, 'b' appearing in the denominator?"}, "author_response": "Response to Reviewer #1:\nThe authors have fundamentally addressed my comments from the first review, but I have a few remaining points that should be addressed before publication:\nThank you for your insightful comments. We have provided point-by-point response below and revised the manuscript accordingly.\n1) I am unconvinced of the description of AHT as somehow between forcing and feedback. While it is not a classic radiative feedback, it clearly is a response of the climate system to the external forcing mediated by and scaling with surface temperature change. It should thus be understood as a feedback, not as part of the forcing.\nAccording to the formula ∆𝑨𝑯𝑻 ≅ 𝒂∆𝑻𝑮 −𝒃(∆𝑻𝑨 −∆𝑻𝑮) , the term 𝒃(∆𝑻𝑨 −∆𝑻𝑮) represents the reduced AHT due to AA and is a negative feedback to AA. On the other hand, the term 𝒂∆𝑻𝑮 represents the increase in AHT due to enhanced meridional moisture gradient from global uniform warming as 𝑑𝑞∗⁄𝑑𝑇 is higher in warmer tropics. 𝒂∆𝑻𝑮 is independent of AA and works to warm the Arctic. So, we think 𝒂∆𝑻𝑮 can be interpreted as a forcing to AA.\nAs you pointed out, this forcing part of ∆AHT is not a direct radiative forcing F. We have edited the related sentences (L117-121) as follows to avoid potential confusion.\nThe change in AHT, ∆𝐴𝐻𝑇, is a unique factor involved in AA. It is neither a pure forcing (e.g., 𝐹) nor a pure feedback (e.g., 𝜆 ) to AA. Instead, it consists of two parts -- a forcing-like part as global-scale warming enhances meridional moisture gradient (𝑑𝑞∗⁄𝑑𝑇 is higher in warmer tropics) and increases AHT to amplify Arctic warming, and a negative feedback part as AA weakens meridional temperature gradient and reduces AHT.\n2) The derivation of a and b from model output lacks detail and thus clarity. What T and q are used? I would guess tas and huss, i.e. 2m near-surface values, but this is not spelled out. If my guess is correct, are results similar when using 850 hPa values instead?\nYes, T and q here use 2m near-surface value. We have clarified this point in the text and in the figure caption. If one uses 850hPa T and q to derive 𝑎 and 𝑏, one should also use 850hPa T for the AHT formula and the energy-balance equations (Eqs. 1,2,5), and define AA and diagnose feedback parameters according to the 850hPa T. While this is doable and one can develop an analytic theory of AA with respect to the 850hPa temperature, it may not be preferred compared to that of surface temperature.\nLines 33-35: As there is some ambiguity in attributing the spread of AA to the different factors for individual models, this claim seems a bit overstated.\nIn Fig. 4, we have attributed the intermodel variation of AA to the intermodel variation of the physical factors. First, we have decomposed the degree of AA into the baseline AA (𝐼&&) and the effect of differential forcing and feedback between the Arctic and the globe (𝐹𝐹) (Fig. 4a). Then, we show that the variation of 𝐼&& is well explained by the variation in 𝜆' and 𝑑 (Fig. 4b) and the variation in 𝐹𝐹 can be well explained by 𝜆& −𝜆' and 𝜆' (Fig. 4c). Finally, the intermodel variation in 𝜆 is further attributed to contributions of individual physical feedbacks (Fig. 4d).\nWe have replotted Fig. 4b using 𝑑 instead of 𝑎−𝑏. Unlike 𝑎 and 𝑏, 𝑑 can be accurately diagnosed for each model without ambiguity and a higher 𝑑 implies a larger 𝑎 and/or a smaller 𝑏. We have rewritten the text related to Fig. 2b to make this attribution clearer.\nConsidering we have assumed a constant parameter 𝛾 ≡ 𝜆, we have edited the sentence to “The formula captures the varying AA in climate models and attributes the spread to models’ feedback parameters and AHT physics”.\nLine 50: given the references to Paleoclimate, climate change does not have to be warming\nFollowing your comment, we have edited the sentence as “ Robustly seen in paleo proxy records1,2, historical observations3–5, and model simulations6–9, the temperature response to climate change is amplified in the Arctic relative to the rest of Earth.”\nl. 106 ff: The argument makes physical sense, but feedback analysis sometimes use global-mean temperature change even for regional feedback factors, in which case the argument would not hold. This should be discussed to avoid confusion.\nFollowing your comment, we have mentioned in both the text (L94) and figure legend (Fig. 2b) that the feedback parameters are defined with respect to the Arctic and global mean warming, respectively.\nThe conventional diagnostic framework partitions the absolute warming (e.g., ∆𝑇&) to contributions of individual factors according to the following energetic equations,\n𝐹& +∆𝑂& +∆𝐴𝐻𝑇+=𝜆&∆𝑇& = 0. (𝑅1)\nBy writing 𝜆& = 𝜆' +𝜆* , we have\n𝐹& +∆𝑂& +∆𝐴𝐻𝑇+(𝜆& +..+𝜆* )∆𝑇 = −𝜆'∆𝑇 (𝑅2)\nThen, by dividing −𝜆' on both sides, we have\n∆𝑇& = 𝐹& /−𝜆' + ∆𝑂& /−𝜆' + ∆𝐴𝐻𝑇+/−𝜆' + ∑ 𝝀𝑨∆𝑻𝑨 /−𝝀𝑮 (𝑅3)\nIn above derivation, 𝜆& must be defined with respect to the Arctic warming (multiplied by ∆𝑇& in the last term of RHS), otherwise one cannot move ∆𝑇& to the LHS and derive Eq. R3 which partitions the total warming ∆𝑇& to individual factors. Even if one ignores the derivation process and define 𝝀𝑨 with respect to global mean warming, the diagnosed contributions of a feedback (i.e., 𝝀∆𝑻 / 𝝀𝑮) should not be changed, as 𝜆∆𝑇 is the radiative anomaly associated with that feedback and it does not depend on whether the feedback parameter 𝜆 is define with respect to regional or global mean warming. We have included Eq. R3 in the main text.\nResponse to Reviewer #2:\nThe article is worth publishing, it is an interesting study offering (yet another) tool for diagnosing Arctic Amplification (AA) in models. My main problem with the article is that the overall novelty of the article needs to be explained better. I am happy to see that they removed the word \"predictive\" from the article, but still, this is just a diagnostic framework for AA, and it is not even close a \"theory\". I would strongly suggest not using big words for describing this analysis. I am glad the authors included the sentence explaining why AHT will increase in the warmer climate (higher sensitivity of dq*/dT in the tropics). This very important point was completely missing (I believe) in the first version. This has been employed before for explaining AA (e.g. Langen Alexeev, Climate Dynamics 2007). Having said that, still, there is no formula that will explicitly include any physics (e.g. Clausius-Clapeyron equation) to quantify AA dependence on sensitivity of AHT to changes in the temperature. I notice other reviewers point that some other parts of the paper just reformulate previous analyses, which is fine, but then what is new and exciting here? But again, the article represents a nice and interesting diagnostic study of AA in models, which is why I think it is worth publishing. I'd give it a solid \"B\".\nThank you for your insightful comments. The major contribution of this work is to establish an analytic formula for the degree of AA, which we believe works beyond a diagnostic framework. The formula allows us to predictively understand how AA will change if certain physical factors change, while a diagnostic framework cannot do that.\nFurthermore, the formula conveys a clear picture of how the degree of AA is mutually determined by multiple factors and articulates the intricate role of AHT. Such understanding cannot be gained from the existing diagnostic framework (see our discussion in L99-113). Besides that, the formula attributes the intermodel spread of AA to physical factors, which can be further investigated by modelling centers. Also, we have now added a paragraph to show that the formula can analytically explain the results of previous numerical studies which artificially manipulate the feedbacks and forcing of AA (L310-327).\nWe have explained these contributions throughout the paper and highlighted what the new insights are in the Summary and Discussion sections. We have added the reference of Langen Alexeev (2007) for the point that global-scale warming would increase AHT. But please note that AHT is a combination of forcing and feedback to AA and it can either increase or decrease in a warmer climate depending on the model.\nResponse to Reviewer #3:\nOverall, the revised manuscript addresses the concerns I had with the presentation (e.g., is it a predictive or diagnostic framework?) and cleaned up the typos I found. I recommend accept with an optional minor revision.\nThank you for your insightful comments. We have provided point-by-point response below and revised the manuscript accordingly.\nThe one outstanding point of clarification concerns the motivation for and discussion of Fig. 4b: the vertical axis is the difference between the atmospheric heat transport parameters 'a' and 'b'. The discussion of the theoretical formula is clear that Arctic Amplification increases with increasing 'a' and decreases with decreasing 'b', but is there a proper mathematical motivation for the difference 'a-b' as being the unique quantity to characterize intermodel spread? This is worthy of a clear discussion because 'a' and 'b' are approximately the same magnitude (the authors added more on theoretical derivations in the revised methods).\nWhile a/b should not be too far from 1, a-b still varies considerably among models, and its value can be either negative or positive. 𝑎−𝑏 is linked to 𝑑 = ∆𝐴𝐻𝑇 − ∆'𝐴𝐻𝑇 (i.e., the distance to the regression line in Fig. 2a) but may not be further simplified. A higher 𝑑 implies a larger 𝑎 and/or a smaller 𝑏, that is, a higher 𝑎−𝑏.\nPut another way, I don't see why the boundaries between the shaded colors in Fig. 4b are straight lines given the equation has a/(b-lambda_G). Shouldn't there be some curvature from the inverse, 'b' appearing in the denominator?\nIndeed, the exact relation between and 𝑎−𝑏 can be rather complicated with 𝑏 in the denominator. In the previous version, we have used the following approximate formula to plot the shading in Fig. 4b,\n𝑎 / (𝑏−𝜆') ≅ (𝑎X +𝑥/2) / (𝑏[ −𝑥/2−𝜆')\nwhere 𝑥 = (𝑎−𝑏)−(𝑎X −𝑏[).\nNow, inspired by your comment, we have related the baseline AA 1+ (𝑎/(-+*!)) to 𝑑 = ∆𝐴𝐻𝑇 − ∆'𝐴𝐻𝑇 (instead of 𝑎−𝑏) and 𝜆'. The benefit is that, unlike 𝑎−𝑏, 𝑑 can be accurately diagnosed for individual models. Please see Fig. 4b and the related paragraph (L268-276) for more details. The shading shows and helps visualize the dependency of the baseline AA on 𝑑 and 𝜆'. We have explicitly stated how the shading is calculated in the figure caption."}]

Decision letter