1. Introduction

The equilibrium climate sensitivity (ECS) is defined as the equilibrium global mean surface warming in response to the doubling of CO₂ from the preindustrial level of 280 ppm. According to the Intergovernmental Panel on Climate Change (IPCC) Sixth Assessment Report (AR6), the likely range of ECS simulated by phase 6 of Coupled Model Intercomparison Project (CMIP6) climate models is between 2.5° and 4.0°C (Forster et al. 2021). The existence of such a large range of ECS, although narrower than the first assessment of ECS ranging from 1.5° to 4.5°C (Charney et al. 1979), causes uncertainties in global warming projections by climate models and limits our ability to foresee the severity of its impacts on nature and human civilization (Lehner et al. 2020; Lee et al. 2021).

A scientific question raised from the existence of uncertainties of ECS is why, under the same anthropogenic greenhouse gas increase scenario, different climate models would produce different amounts of global mean surface warming. Most studies focus on intermodel spreads in climate feedbacks, which control the difference in the climate sensitivity of different climate models and are regarded as the key to addressing the differences in the climate sensitivity in response to the same external forcing among climate models (e.g., Soden et al. 2008; Flato et al. 2013; Brient 2020; Sherwood et al. 2020; Zelinka et al. 2020; Forster et al. 2021). From the top of the atmosphere (TOA) perspective, the uncertainty in cloud feedback is regarded as the dominant source of the different climate sensitivities of climate models (Wetherald and Manabe 1988; Soden and Vecchi 2011; Andrews et al. 2012; Bony et al. 2015; Klein and Hall 2015; Zhao et al. 2016; Ceppi et al. 2017; Zelinka et al. 2017; Watanabe et al. 2018; Forster et al. 2021; Liang et al. 2022). The partial cancellation of the uncertainties in water vapor and lapse rate feedbacks makes their combined effect less important for the climate sensitivity differences among different climate models (Schneider et al. 1999; Held and Soden 2000; Vial et al. 2013; Po-Chedley et al. 2018; Colman and Soden 2021; Beer and Eisenman 2022; Feng et al. 2023). However, from the surface perspective, the uncertainties in ice–albedo and water vapor feedbacks, rather than cloud feedback, play more important roles in causing the uncertainties of global warming projections not only in terms of the global mean but also in terms of the spatial pattern (Hu et al. 2020; Sejas et al. 2021).

The global warming projection of each model is defined as the difference between the perturbation simulation forced by anthropogenic greenhouse gases and the control climate simulation of the same model in the absence of anthropogenic greenhouse gases. Despite the large intermodel spread of the control climate simulations, it is often implicitly assumed that uncertainties in global warming projections are exclusively due to differences in individual models’ climate responses to increases in anthropogenic greenhouse gases, with little concern for the differences in the control climate states. However, as reported in IPCC AR6 (Forster et al. 2021), climate sensitivity can be dependent on the climate state, especially climatological mean temperatures (Mauritsen et al. 2019 in climate models and Caballero and Huber 2013; Anagnostou et al. 2020 in paleoclimate proxy records). The state dependency is manifested as the nonlinear temperature response to anthropogenic greenhouse forcing (Bjordal et al. 2020; Rugenstein et al. 2020). In addition, the uncertainty in the climatological mean sea ice coverage is highly related to the uncertainty in ice–albedo feedback, as more extensive sea ice coverage contributes to stronger ice–albedo feedback due to an increased potential for the melting of sea ice (Screen and Simmonds 2010; Taylor et al. 2013; Caldeira and Cvijanovic 2014; Dommenget 2012, 2016; England et al. 2020). Another important aspect of state-dependent climate feedbacks is the masking effect of climate mean clouds for surface albedo changes’ contribution to planetary albedo (e.g., Soden et al. 2004; Zelinka et al. 2012; Tan et al. 2022). Hu et al. (2017) find that part of the uncertainties in climate feedbacks inherit the uncertainty from the models’ control climate state. Specifically, models that have a colder and drier climate state with greater ice coverage tend to have stronger ice–albedo feedback and experience greater warming and vice versa. The water vapor feedback also inherits diversity from the control climate, but in an opposite way, namely, models that have a warmer climate state generally exhibit a stronger water vapor feedback, resulting in stronger global warming (Yoshimori et al. 2011; Hu et al. 2020; Bastiaansen et al. 2021; Colman and Soden 2021). In addition, the recent study of He et al. (2023) provides evidence suggesting that the anthropogenic radiative forcing itself shows a large degree of climate mean state dependence.

The main objective of Part I of this three-part series of papers is to introduce a new kernel called the “energy gain kernel” (EGK). The EGK is formulated by considering the total energy perturbation (W m⁻²) in an atmosphere–surface column due to temperature warming in response to a unit forcing (1 W m⁻²) in a layer. The diagonal elements of EGK, whose values are all greater than the unit forcing, represent the local amplification of imposed energy perturbations in individual layers. The positive off-diagonal elements represent the energy perturbations gained in other layers. Both the local amplification and additional energy perturbations gained in other layers are due to temperature feedback generated through radiative thermal coupling across different layers in response to the imposed energy perturbations. One can easily apply EGK to external and internal energy perturbations associated with a nontemperature feedback agent, such as water vapor, clouds, and surface albedo, which will be generically referred to as “input energy perturbations.” The product of EGK and input energy perturbations corresponds to the amplified energy perturbations of the input energy perturbations by temperature feedback. Because the diagonal elements of the Planck feedback matrix correspond to the Stefan–Boltzmann feedback parameters of individual layers, the multiplication of the inverse of the diagonal matrix of the Planck feedback matrix with the product of the EGK and input energy perturbations is the radiative equilibrium temperature response to the input energy perturbation. Because both the EGK and Stefan–Boltzmann feedback parameters can be obtained directly from the climate mean state information, one can examine how the strength of energy gain through radiative thermal coupling in the vertical would vary spatially and across different climate models without relying on additional information obtained from observed trends or perturbed climate model simulations. Furthermore, the amplification of EGK to energy perturbations due to external forcing or induced by nontemperature feedback is independent of their strengths, polarity, and origins (e.g., changes in CO₂ concentration, or in water vapor or in atmospheric energy transport). Therefore, EGK facilitates a clear separation of the effects of climate mean state (represented by EGK) on ECS from those of nontemperature feedbacks, although EGK by itself does not offer insights into climate state dependence of individual nontemperature feedbacks. The information extracted from EGK as well as from Stefan–Boltzmann feedback parameters is expected to hold the key for the dependency of the climate sensitivity on the time mean climate state, namely, spatial patterns of climate mean temperature, water vapor, clouds, and surface pressure.

In Part II, our focus centers on examining the spatial pattern of the surface element of EGK and its relations with spatial variations in variables affecting the climate mean infrared opacity across atmospheric layers, namely, water vapor, clouds, and surface pressure. Given the wavelength dependence of peak emission radiance on temperature (as dictated by the Planck function), the alignment of the peak thermal emission radiance with the atmospheric absorption bands becomes temperature dependent. Consequently, the strength of EGK, in addition to the quantity of thermal infrared energy absorbers, is intrinsically linked with temperature, explaining why EGK can be stronger over midlatitude storm-track regions where surface temperature is colder than tropical rain belt under the same cloud coverage. In Part III, we will apply EGK to external energy perturbations and to energy perturbations due to individual nontemperature feedback variables derived from observed trend analysis. We will compare energy perturbations associated with individual processes at the surface and at the TOA before and after applying the EGK. The focus will be on the reconciliation for the dominance of the negative nature of the lapse rate feedback found in TOA-based feedback analysis methods and the positive nature of temperature feedback revealed by EGK. This would improve our understanding of the different roles of the radiative thermal coupling among different layers in maintaining energy balance at the surface and at the TOA. Following the same analysis presented in Part III, one can apply the EGK analysis to individual climate model simulations for isolating the contribution of the global warming projection uncertainty due to the differences in the model mean climate states from that due to the uncertainties in energy perturbations associated with individual climate feedbacks, including both their strength and spatial patterns.

The organization of Part I’s presentation is as follows. Presented in sections 2 and 3 are the rationale and the mathematical formulation of EGK, respectively. Section 4 presents examples of EGK to illustrate the characteristics of EGK and their variations with climate states. In section 5, we demonstrate how EGK serves as the core climate feedback process that acts to amplify external energy perturbations and internal energy perturbations induced by changes in nontemperature climate variables, such as water vapor, clouds, and surface albedo. The concluding remark is given in section 6.

2. Rationale for the energy gain kernel

The literature review in the introduction underscores the pressing need to separate the effects of a model’s control climate state on its climate sensitivity from the contribution of feedback processes. Among the existing feedback analysis methods, two methods show promising potential in effectively disentangling the control climate state information from the feedback information. The first one is the radiative kernel method (Soden et al. 2008). The radiative kernel method involves calculating a radiative kernel, which encapsulates the change in radiative flux at the TOA induced by a perturbation to an individual variable. The radiative kernel does not utilize simulations perturbed by an external forcing. Therefore, radiative kernels, in principle, are supposed only to contain control climate state information. However, except for the temperature kernel, the other kernels are obtained by introducing prescribed relationships linking the control climate state information to perturbations in variables under consideration. For example, the water vapor kernel is typically obtained by using perturbations in specific humidity that is equal to the increase in saturation specific humidity of 1 K warming from the control climate state with the same relative humidity as the control climate state (Soden et al. 2008). Obviously, different prescribed relationships linking the control climate state information to perturbations in specific humidity, such as assuming a 10% decrease in relative humidity instead of holding it constant, would lead to different water vapor kernels for the same climate model. The dependency of nontemperature radiative kernels for the same climate model on prescribed relationships suggests a certain level of arbitrariness in isolating the control climate state information from the feedback information, although different kernels would always, by design, result in the same partial radiative energy perturbations at the TOA as long as the same prescribed relationships are used when applying the kernels.

The other method is the coupled atmosphere–surface climate feedback–response analysis method (CFRAM; Cai and Lu 2009; Lu and Cai 2009). Unlike the radiative kernel method, the formulation of the CFRAM analysis is based on energy balance in the atmosphere and at the surface (thereby at TOA as well) and it includes both radiative and nonradiative feedback processes. The CFRAM resolves the full temperature kernel across vertical layers (i.e., not just at the TOA), which is referred to as the Planck feedback matrix in Lu and Cai (2009). Similar to the TOA temperature kernel, the derivation of the full temperature kernel does not require a prescribed covariation relation of temperature with other climate variables. Given a scenario of CO₂ increasing, one can determine initial external energy perturbations using a radiative transfer model from the climate mean state. The next step of the CFRAM analysis is to diagnose (partial) energy perturbations due to individual nontemperature feedbacks inferred from the differences between perturbation and control climate simulations. In comparison with the kernel method, the full temperature kernel (i.e., the Planck feedback matrix) is not used to determine (partial) energy perturbations due to temperature changes in the CFRAM. Instead, one obtains individual partial temperature changes by multiplying the inverse of the Planck feedback matrix to vertical profiles of external energy perturbations and energy perturbations due to individual nontemperature feedbacks. Therefore, CFRAM does not explicitly require the information of temperature changes in climate feedback analysis as the Planck feedback matrix itself already implicitly encapsulates the information of temperature feedback. Because the Planck feedback matrix solely consists of the control climate state information and is unique for each climate model, partial temperature changes obtained from the CFRAM analysis can be regarded as the combined effect of the control climate state information and the nontemperature feedback information.

Sejas and Cai (2016) and Hu et al. (2018) formulated a surface version of the CFRAM, which is referred to as the surface feedback–response analysis method (SFRAM). In the SFRAM analysis, the air temperature feedback kernel, which is reduced from the Planck feedback matrix, measures the strength of the air temperature feedback onto the surface warming in response to energy perturbations at the surface and in the atmosphere due to the external forcing and nontemperature feedbacks (such as water vapor feedback and ice–albedo feedback). As the Planck feedback matrix, the air temperature feedback kernel also solely consists of the control climate state information and is unique for each climate model. By representing the continuous thermal radiative coupling between the atmosphere and surface, the air temperature feedback kernel reflects the air temperature feedback’s ability, through the radiative thermal coupling between the atmosphere and surface, to amplify the surface energy perturbation signal and to transfer energy perturbations in the atmosphere to the surface. The strength of the air temperature feedback kernel varies greatly with the climatological spatial distributions of air temperature and thermal radiation absorbers/emitters in the atmosphere, such as water vapor and cloud content. In the regions where the strength of the air temperature feedback kernel is stronger, the surface temperature change in response to the same energy perturbation becomes more pronounced. Conversely, in regions where the air temperature feedback kernel is weaker, the change in response to the same energy perturbation is less pronounced. It is found that the amplification by air temperature feedback kernel contributes approximately 76% of the total global mean surface warming, implying an amplification factor of roughly three times the surface warming (Sejas and Cai 2016; Hu et al. 2018). This finding highlights the significant role of the control climate state via influencing the strength of the air temperature feedback kernel in amplifying the overall surface temperature response to external forcing and individual nontemperature feedbacks.

Most climate feedback studies in the literature are based on energy balance at the TOA, such as the partial radiative perturbation (PRP) method (Wetherald and Manabe 1988; Bony et al. 2015 and reference therein) and the radiative kernel method (Soden et al. 2008). In the TOA-based climate feedback framework, temperature feedback is consistently interpreted as negative, indicating that as temperatures rise, outgoing longwave radiation (OLR), or energy output from Earth’s climate system, increases. The studies of Sejas and Cai (2016) and Hu et al. (2018) reveal a dualistic nature of temperature feedback from the perspective of surface energy balance, contrasting with the consistent interpretation of temperature feedback as negative from the TOA perspective. One aspect of temperature feedback acts to amplify the surface component of imposed energy perturbations while also “spreading” the atmospheric components of imposed energy perturbations to the surface, both of which are achieved through changes in the downward longwave (LW) energy fluxes at the surface due to changes in air temperatures. As illustrated in Figs. 1c and 1d of Sejas and Cai (2016), air temperature feedback through thermal radiative coupling within an atmosphere–surface column always acts to amplify energy perturbations at the surface by inducing downward thermal radiative flux perturbations of the same sign. The response of air temperatures to energy perturbations in the atmosphere also results in downward thermal radiative flux perturbations, adding energy perturbations at the surface that have the same sign as energy perturbations in the atmosphere (Figs. 1a,b in Sejas and Cai 2016). When energy perturbations due to external forcing or due to individual nontemperature feedbacks have the same sign at the surface and in the atmosphere, or energy perturbations at the surface are greater than their counterparts in the atmosphere, which is often the case, this aspect is recognized as a positive feedback mechanism, as changes in downward radiation fluxes due to air temperature changes align with the sign of the original energy perturbation at the surface. An exception arises when the atmospheric components of the imposed energy perturbations not only have an opposite sign but also are considerably stronger than the surface, an occurrence which is rare but theoretically plausible. On the other hand, changes in surface temperature produce more output energy perturbations that always have the opposite sign as the imposed energy perturbations at the surface, a phenomenon referred to as the Planck feedback in the literature. This aspect of temperature feedback corresponds to a negative feedback mechanism, namely, increasing surface temperature leads to greater energy output from the surface through longwave thermal emission.

The SFRAM lacks the capability to explicitly discern the inherent dualistic nature of temperature feedback, although the results of SFRAM analysis reveal its existence. It is believed that disentangling this dualistic nature of temperature feedback would help gain a better understanding of the dependency of the climate sensitivity on the time mean climate state. This prompts us to embark on the endeavor of explicitly disentangling the dualistic nature by examining closely the Planck feedback matrix, which is the topic of the next section.

3. Mathematical formulation of the energy gain kernel

Following Lu and Cai (2009), the mathematical expression of the coupled atmosphere–surface CFRAM for a given horizontal location can be written as

$\left(\frac{\partial R_i}{\partial T_j}\right)(\mathrm{\Delta }{T}_j)=(\mathrm{\Delta }{F}_i).$(1)

In Eq. (1), $(\partial R_i/\partial T_j)$ is the Planck feedback matrix of (L + 1) × (L + 1), whose element at the ith row and jth column (W m⁻² K⁻¹), corresponds to the LW radiative flux divergence perturbation at the ith layer due to 1 K warming at the jth layer while holding other variables as their climate mean values (Fig. 1); (ΔF_i) is a column vector of length (L + 1) with i = 1 denoting the top layer of the atmosphere, L is the bottom atmospheric layer just above the surface, and (L + 1) is the surface layer, corresponding to the vertical profile of energy flux convergence perturbations (W m⁻²) associated with external forcing or individual nontemperature feedback variables such as the water vapor or ice–albedo feedback; and (ΔT_j) is a column vector of length (L + 1) with j = 1 denoting the top layer of the atmosphere, L is the bottom atmospheric layer just above the surface, and (L + 1) is the surface layer, corresponding to the vertical profile of temperature changes in response to (ΔF_i).

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The Planck feedback matrix at (a) (85.5°N, 60°E), (b) (24°N, 21°E), (c) (0°, 120°E), and (d) (85.5°S, 60°E). The elements (shadings; W m⁻² K⁻¹) of the Planck feedback matrix are the LW cooling rates at the level (hPa) indicated by the ordinate due to 1 K warming at the level (hPa) indicated by the abscissa.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0148.1

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To explicitly disentangle the climate mean state information and perturbed climate information for climate feedback analysis, we apply Eq. (2) to a generic energy flux convergence perturbation of 1 W m⁻² consecutively in a layer-by-layer fashion from the top layer of the atmosphere to the surface layer, which is referred to as the “unit forcing” hereafter. The resultant partial temperature changes would only contain the mean climate state information. Specifically, let us denote the vertical profile of the unit forcing as

$({\delta }_{i,j})=\left(\{\begin{array}{l}1\hspace{0.17em}\mathrm{W}{\mathrm{m}}^{-2}\text{for}i=j\hfill \\ 0\text{for}i\ne j\hspace{0.17em}\hfill \end{array}\right),$

which is the identity matrix of (L + 1) × (L + 1) (W m⁻²). Next, let us denote the corresponding (L + 1) vertical profiles of partial temperature changes as δT_i,j, representing the partial temperature changes in ith layer due to the unit forcing at jth layer. Applying Eq. (2) to (δ_i,j) yields

$(\delta T_{i,j})=(P_{i,j})({\delta }_{i,j}).$(3)

Therefore, the vertical profile of partial temperature changes in response to the unit forcing at the jth layer numerically is equal to the jth column of the inverse of the Planck feedback matrix (Fig. 2).

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As in Fig. 1, but for the vertical profile (ordinate; hPa) of the warming (shadings; K) in response to the unit forcing (1 W m⁻²) at the level (hPa) indicated by the abscissa.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0148.1

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As indicated in Fig. 1, the diagonal elements of the Planck feedback matrix are all positive (i.e., $\partial R_i/\partial T_i>0$) and the off-diagonal elements are all negative (i.e., $\partial R_{j\ne i}/\partial T_i<0$). The former corresponds to the increase in the thermal energy emission (or radiative cooling) at the ith layer due to its 1 K warming, whereas the latter corresponds to the radiative heating to other layers from the increase in the thermal energy emission at the ith layer. The positiveness of $\partial R_i/\partial T_i$ is the condition that guarantees the existence of the inverse of the Planck feedback matrix (Fig. 2), which is numerically equal to the matrix (δT_i,j).

Physically speaking, in response to the unit forcing at the jth layer, temperatures of all layers in the atmosphere–surface column increase via the thermal radiative coupling imprinted by the Planck feedback matrix, and thus, all elements of the matrix (δT_i,j) are positive as indicated in Fig. 2. This together with the negative off-diagonal elements of the Planck feedback matrix implies $-{\displaystyle {\sum }_{k\ne i}(\partial R_i/\partial T_k)\delta T_{k,j}>0}$. Therefore, the term $-{\displaystyle {\sum }_{k\ne i}(\partial R_i/\partial T_k)\delta T_{k,j}}$ corresponds to the LW radiative heating perturbation (W m⁻²) at the ith layer induced by the temperature changes at all other layers but the ith layer in response to the unit of forcing at the jth layer. In other words, the convergence of the additional thermal energy emission from all other layers caused by δT_k≠i,j > 0 into the ith layer is equal to $-{\displaystyle {\sum }_{k\ne i}(\partial R_i/\partial T_k)\delta T_{k,j}}$.

In terms of the climate feedback concept, the left-hand side of Eq. (4) corresponds to the energy output of individual layers and thereby represents the negative nature of temperature feedback, which is referred to the Planck feedback in the literature, whose strength is inversely proportional to the cubic of climate mean temperatures of individual layers. The term $-{\displaystyle {\sum }_{k\ne i}(\partial R_i/\partial T_k)\delta T_{k,j}}$ can be regarded as the energy gain at the ith layer through temperature feedback as it is the LW energy flux convergence perturbation at the ith layer coming from the additional thermal emission by other layers due to their warming response to the same unit forcing. Therefore, the term $[{\delta }_{i,j}-{\displaystyle {\sum }_{k\ne i}(\partial R_i/\partial T_k)\delta T_{k,j}}]$ represents the total energy flux convergence perturbation at the ith layer, equaling the sum of the unit forcing and the energy gain.

The derivations above indicate that the original complete Planck feedback matrix encapsulates dualistic information: Its diagonal components represent thermal emission perturbations of individual layers (constituting the negative feedback aspect), while the off-diagonal elements signify information about energy gain (comprising the positive feedback aspect). By reformulating the radiative equilibrium equation as depicted in Eq. (4), we explicitly disentangle the negative feedback portion (on the left-hand side) from the positive feedback portion (embodied by the second term on the right-hand side). Without this separation, the left-hand side of the radiative equilibrium equation would be the net effect of the negative and positive temperature feedback parts, which by definition has to be negative, reflecting the increased net LW energy output due to the warming response to the unit forcing.

In light of the discussion aforementioned, we define the matrix on the right-hand side of Eq. (4) as the EGK, namely,

$\mathsf{G}=(G_{i,j})=\left({\delta }_{i,j}-{\displaystyle \sum _{k\ne i}\frac{\partial R_i}{\partial T_k}\delta T_{k,j}}\right).$(5)

Rewriting the radiative equilibrium balance condition Eq. (4) by utilizing the definition of the EGK given by Eq. (5) yields

$(G_{i,j})=\left(\frac{\partial R_i}{\partial T_i}\right){\left(\frac{\partial R_i}{\partial T_j}\right)}^{-1}.$(6)

Therefore, the EGK is equal to the product of the diagonal matrix of the Planck feedback matrix and the inverse of the full Planck feedback matrix. In other words, the EGK is the inverse of the Planck feedback matrix after factoring out the Planck feedback factors of individual layers, which measures the strength of the negative feedback aspect of temperature feedback. The positive nature of EGK is evidently illustrated in Fig. 3.

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As in Fig. 1, but for the EGK. The elements of EGK are dimensionless, but their numerical values (shadings) correspond to the total energy flux convergence perturbations (W m⁻²) at the level (hPa) indicated by the ordinate due to (1 W m⁻²) the coupled atmosphere–surface temperature response (or the temperature feedback) to the unit forcing at the level (hPa) indicated by the abscissa.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0148.1

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4. Illustration of the EGK

In this section, we use the examples of the EGK over four locations shown in Fig. 3 to elucidate how the climate state information contained in the Planck feedback matrix is transformed into the EGK by factoring out the Planck feedback factors of individual layers. These four locations are, respectively, at (85.5°N, 60°E), representative of the Arctic Ocean where the annual mean temperature is cold; (24°N, 21°E), representative of the Sahara Desert where the climate is hot and dry; (0°, 120°E), representative of the warm pool region of the western equatorial Pacific Ocean where surface temperature is warm with plenty of moisture and clouds; and (85.5°S, 60°E), representative of the Antarctic Plateau where the temperature is cold, moisture is scarce, and surface pressure is low (or elevation is high).

The data used to obtain these four examples of the EGK are the 20-yr (1980–99) mean of 3D temperature, moisture, cloud, and ozone fields as well as surface pressure fields derived from the ERA5 reanalysis (European Centre for Medium-Range Weather Forecasts; Hersbach et al. 2020), which is archived at the National Center for Atmospheric Research, Computational and Information Systems Laboratory, and can be downloaded at https://doi.org/10.5065/P8GT-0R61. The horizontal spatial resolution is 1.5° in longitude by 1.5° in latitude. There are a total of 37 pressure levels plus the surface layer with 14 levels above 200 hPa. The model used is Fu–Liou’s radiation transfer model (Fu and Liou 1992, 1993). The concentration levels of CO₂, CH₄, and N₂O are, respectively, 352.2, 1.7, and 0.3 ppm. The radiative transfer calculations at each horizontal grid point are carried out by using the pressure level data only at and above the surface layer.

In the literature, the phenomenon that the temperature change in response to the same strength of forcing is proportional to the inverse of the cubic of climate mean temperature is also referred to as the Stefan–Boltzmann feedback effect (Hartmann 1994) or the Planck feedback effect (Pithan and Mauritsen 2014). For this reason, we refer to the elements on the diagonal of the Planck feedback matrix as the Stefan–Boltzmann feedback parameters of individual layers. The Stefan–Boltzmann feedback parameters are defined as the LW emission perturbations (W m⁻² K⁻¹) by individual layers due to their 1 K warming while holding other variables, such as specific humidity and amount of cloud condensates, as their climate mean values. The Stefan–Boltzmann feedback parameters tend to decrease nonlinearly with the climate mean temperature (for a gray atmosphere, their decreasing rates with temperature are cubic). The Stefan–Boltzmann feedback parameters also tend to decrease with the amount of water vapor and cloud condensates, which collectively determine the infrared opacity of individual atmospheric layers. Consequently, the values of diagonal terms decrease with height and are greater over warmer (e.g., Fig. 1c versus Fig. 1a) and moister regions (Fig. 1c versus Fig. 1b). The off-diagonal elements are negative, representing the net LW heating at neighbor layers resulting from the 1 K warming at the diagonal layers. The strength and the vertical depth of LW heating rates in off-diagonal layers increase with climate mean moisture. The LW heating rates in the off-diagonal layers that are below the diagonal layer are generally greater than their counterparts above (Fig. 1) because water vapor tends to decrease with height. The strength and vertical depth of the LW heating rates are greater over warm (e.g., Fig. 1c versus Fig. 1a) and over moist regions (Fig. 1c versus Fig. 1b).

The inverse of the Planck feedback matrix (Fig. 2) represents the vertical profiles of the warming in response to the unit forcing in the diagonal layers. The vertical increase in warmings in the diagonal layers largely reflects the vertically weakening profile of the Stefan–Boltzmann feedback parameters of individual layers, which is proportional to the cubic of their climate mean temperatures. The Stefan–Boltzmann feedback effect explains that the warming in response to the same strength of forcing over cold layers is stronger than over warm layers. The warming signal in diagonal layers propagates into off-diagonal layers through the vertically thermal radiative coupling among different layers.

Due to the overwhelming influence of the Stefan–Boltzmann feedback effect, the representation of infrared opacity of individual layers in the inverse of the Stefan–Boltzmann feedback effect is not clearly conveyed. One way to reveal the effects of infrared opacity is to normalize the column vectors of the inverse of the Planck feedback matrix by their corresponding diagonal elements. It is seen from Fig. 4 that the vertical depth of larger values of normalized warming in response to the unit forcing in the diagonal layers is deeper in the regions where moisture is more abundant (the left panels versus the right panels). Because lower levels are more opaque for infrared energy, the vertical depth of larger values of normalized warming is deeper in the layers below the diagonal layers than above.

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As in Fig. 1, but for the vertical profile (ordinate; hPa) of the normalized warming (shadings; %) in response to the unit forcing at the level (hPa) indicated by the abscissa. The normalized warming is defined as the ratio of the warming in off-diagonal layers to that in their corresponding diagonal layers.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0148.1

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The EGK is a physics-based way to normalize the inverse of the Planck feedback matrix by multiplying it with the diagonal matrix of the Planck feedback matrix. As discussed in section 3, the diagonal elements of the EGK (Fig. 3) are always greater than the imposed unit forcing (1 W m⁻²). The additional energy perturbations in diagonal layers correspond to the LW flux convergence perturbations induced by the coupled atmosphere–surface temperature response to the unit forcing. The off-diagonal elements of the EGK are all positive, representing the LW flux convergence perturbations in the off-diagonal layers induced by the coupled atmosphere–surface temperature response to the unit forcing in diagonal layers.

The LW flux convergence perturbations or energy gain, in both diagonal and off-diagonal layers induced by the coupled atmosphere–surface temperature response to the unit forcing, is proportional to the climate mean temperature and climate variables, such as water vapor, that determine the infrared opacity of individual atmospheric layers. This explains why the values of diagonal elements of the EGK, in general, tend to decrease with height and why both diagonal and off-diagonal elements are greater over warm places (e.g., Fig. 3b or Fig. 3c versus Fig. 3a or Fig. 3d) and over moist regions (Fig. 3c versus Fig. 3b). In particular, because of the warmness and abundance of water vapor over the warm pool region (Fig. 3c), the total energy perturbations in each layer below 800 hPa are all greater than 3 W m⁻², more than three times as large as the imposed forcing at these layers. Moreover, the energy perturbations at the surface layer (the bottom row in Fig. 3c whose mean surface pressure is 998 hPa) produced by the coupled atmosphere–surface temperature response to the unit forcing in these layers range from 1.2 W m⁻² (for the unit forcing at 800 hPa) to 2.5 W m⁻² (for the unit forcing at 975 hPa). In contrast, because of the scarcity of water vapor in the Sahara Desert, the total energy perturbation at the surface layer is only twice as large as the unit forcing despite its climatological mean surface temperature being as warm as that over the warm pool. It should be pointed out that besides the climatological mean temperature and water vapor, the climatological mean clouds and surface pressure also exert significant influence on the amplification of energy flux convergence perturbations by radiative thermal coupling among different layers, which will be one of the main topics in Part II of this three-part series of papers.

5. Physics-based feedback analysis framework centered at EGK

a. Formulation

Applying the EGK, Eq. (1) can be rewritten as

$\left(\frac{\partial R_i}{\partial T_i}\right)(\mathrm{\Delta }{T}_i)=(G_{i,j}){\displaystyle \sum _X\left[\mathrm{\Delta }{F}_j^{(X)}\right]},$(8)

where the superscript “X” denotes different origins of energy perturbations, which include external forcing and energy perturbations induced by changes in nontemperature variables such as water vapor and clouds or by changes in nonradiative processes such as surface sensible/latent heat fluxes, vertical convections, and poleward energy transport. The term inside the summation on the rhs of Eq. (8) represents amplified energy perturbations due to the process X by temperature feedback. The amplified energy perturbations at the ith layer originate from the two sources. The first one is $G_{i,i}\mathrm{\Delta }{F}_i^{(X)}$ that is equal to the product of the ith diagonal element of EGK (i.e., G_i,i) and the energy flux convergence perturbation at the ith layer directly caused by the process X [i.e., $\mathrm{\Delta }{F}_i^{(X)}$], representing the amplification of the original energy flux convergence perturbation at the ith layer through thermal radiative couplings among different layers. The second one is equal to ${\displaystyle {\sum }_{j\ne i}^{L+1}{G}_{i,j}\mathrm{\Delta }{F}_j^{(X)}}$, representing the energy gained at the ith layer from energy flux convergence perturbations of the same process X at other layers [i.e.,$\mathrm{\Delta }{F}_{j\ne i}^{(X)}$] through thermal radiative couplings among different layers.

Figure 5 summarizes the underlying processes associated with temperature feedback and outlines how to apply the EGK for obtaining the amplified energy perturbations associated with input energy perturbations and the temperature response to input energy perturbations. Here, input energy perturbations can be either external energy perturbations or internal energy perturbations due to individual nontemperature feedback processes’ response to external energy perturbations, namely, the terms $[\mathrm{\Delta }{F}_j^{(X)}]\hspace{0.17em}$ on the rhs of Eq. (8). As indicated by the solid garnet arrow in Fig. 5 and the rhs of Eq. (8), the product of the EGK and input energy perturbations yields the total input energy perturbations in individual layers initiated by external forcing and subsequent induced by nontemperature feedbacks which all get amplified by temperature feedback equally. The solid blue arrow in Fig. 5 is for the energy output perturbations, or the lhs of Eq. (8), corresponding to thermal energy emission perturbations due to the warming of individual layers in response to the total input energy perturbations. The thermal emission perturbations of individual layers are in radiative equilibrium balance with the amplified energy perturbations at individual layers.

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A schematic illustrating the underlying processes associated with temperature feedback. The unfilled red arrow represents input energy perturbations initiated by external forcing and induced by subsequent nontemperature feedbacks. The unfilled blue arrow represents the part of thermal energy emission (output energy) perturbations that pass through the temperature feedback loop via thermal radiative coupling within different layers. The resulting amplified energy perturbations by temperature feedback correspond to the total input energy perturbations to the climate system, represented by a solid garnet arrow. The solid blue arrow is the total thermal emission perturbations of individual layers that are in balance with total input energy perturbations.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0148.1

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The equilibrium temperature changes in response to external forcing, which is often referred to as the ECS, can then be determined by multiplying the total input energy perturbations with the inverse of the diagonal matrix of the Planck feedback matrix, namely,

$(\mathrm{\Delta }{T}_i)={\left(\frac{\partial R_i}{\partial T_i}\right)}^{-1}(G_{i,j}){\displaystyle \sum _X\left[\mathrm{\Delta }{F}_j^{(X)}\right]}.$(9)

6. Concluding remarks

In this paper, we present the mathematical formulation of a kernel, referred to as the energy gain kernel (EGK) for climate feedback studies. The formulation of the EGK is based on the coupled atmosphere–surface climate feedback–response analysis method (CFRAM; Cai and Lu 2009; Lu and Cai 2009). The CFRAM allows for calculating (partial) coupled atmosphere–surface temperature response to each external and nontemperature feedback energy perturbation individually. Their sum corresponds to the total temperature change in response to external energy perturbations. The kernel used in the CFRAM is the Planck feedback matrix. The EGK is the product of the diagonal matrix of the Planck feedback matrix and the inverse of the full Planck feedback matrix.

We demonstrate that the EGK, through a sequence of otherwise uncomplicated mathematical manipulations and the accompanying graphical representations, provides climate state information regarding the strength of the coupled atmosphere–surface temperature response to the unit forcing (1 W m⁻²). The EGK represents the energy gain effect of radiative thermal coupling among different layers resulting from the coupled atmosphere–surface temperature response. Specifically, the elements of the EGK correspond to the total energy flux convergence perturbations via the coupled atmosphere–surface temperature changes in response to the unit forcing in individual layers. The diagonal elements of the EGK are always greater than the imposed unit energy perturbation in diagonal layers (i.e., layers represented by the diagonal index). The additional energy perturbations (with respect to 1 W m⁻²) of diagonal elements correspond to the LW flux convergence perturbation in diagonal layers induced by the coupled atmosphere–surface temperature response. The off-diagonal elements of the EGK are all positive definite, representing the LW flux convergence perturbations across other layers induced by the coupled atmosphere–surface temperature response to the same unit forcing in diagonal layers. The LW flux convergence in both diagonal and off-diagonal layers induced from the coupled atmosphere–surface temperature response to unit forcing is determined collectively by the climate mean temperature and variables that affect the infrared opacity of individual atmospheric layers at the mean climate state, such as water vapor, clouds, and surface pressure.

According to Eq. (8) or Eq. (9), the total energy perturbation associated with each individual nontemperature feedback is equal to the product of EGK and the original energy perturbation solely due to changes in the nontemperature feedback variable under consideration. Because EGK depends solely on climate mean state, its amplification to energy perturbations due to external forcing or induced by nontemperature feedbacks is independent of their strength and origins. Therefore, the amplification by temperature feedback makes the effects of each of external forcing and nontemperature feedbacks climate mean state dependent. Mathematically, one can express the intermodel spread of the total energy perturbation associated with each individual nontemperature feedback as a linear combination of the intermodel spread of EGK and the intermodel spread of the original energy perturbation solely due to changes in the nontemperature feedback variable under consideration, plus their products. Such decomposition of the intermodel spread of the total energy perturbation associated with each individual nontemperature feedback would help gain insights into climate state dependence of external forcing and individual nontemperature feedbacks. The ability to diagnose the strength of the energy gain factor at the surface using the EGK solely based on climate mean state information offers a solution to the quandary of effectively and objectively separating control climate state information from climate perturbation in climate feedback studies. Because the EGK contains critical climate mean state information stemming collectively from mean temperature, water vapor, clouds, and surface pressure, we envision that the intermodel variability of the EGK would hold the solution to the inquiry of why, under the same anthropogenic greenhouse gas increase scenario, distinct climate models yield varying degrees of global mean surface warming.

Data availability statement.

The ERA5 reanalysis is downloaded at https://doi.org/10.5065/P8GT-0R61. Upon request, all model codes and data analysis codes will be made available to researchers interested in reproducing the results. The authors declare no competing interests.