1. Introduction

Since satellites have been available for weather and climate studies, outgoing longwave radiation (OLR) at the top of the atmosphere (TOA) has been easily observed and served as an important parameter for monitoring the state of Earth’s climate. One of the early discoveries from satellite observations is the quasi-linear relation of the latitudinal variation of OLR with the latitudinal variation of surface temperature (e.g., Budyko 1963). The empirical linear relation between OLR and surface temperature (denoted OLR–T_S hereafter) allows for the development of simple energy balance models without explicit consideration of thermal radiative energy transfer processes through the atmosphere. Such energy balance models have been widely used to understand the climate sensitivity in response to solar forcing, and to explain the climate oscillations between glaciation initiation and retreat and small ice-cap instability (e.g., Budyko 1969; Sellers 1969; Held and Suarez 1974; North 1975; Budyko 1977; North et al. 1981, 1983; North 1984).

Because of the absorption of the upward longwave (LW) radiation (i.e., infrared radiation) emitted by the surface by greenhouse gases such as CO₂ and atmospheric water vapor, a major portion of the OLR originates from an atmospheric layer at a considerable height above the surface, termed the radiating layer (Pierrehumbert 2011), rather than from the surface. The temperature of the radiating layer is called “effective radiation emission temperature” (Stull and Arhens 2000) or “equivalent blackbody temperature” (denoted T_EB hereafter, Wallace and Hobbs 2006) as the OLR would be proportional to the fourth power of T_EB per the Stefan–Boltzmann law. The height of the radiating layer is determined by the point where the atmosphere becomes optically transparent or where the mean optical path to space is unity (Goody and Yung 1989). Temperature generally decreases with height above the surface, implying that the LW emitted from the surface is greater than the OLR and thereby the OLR would increase with T_S at a much slower rate than the fourth power. In response to anthropogenic radiative forcing, the radiating layer moves farther up at a colder temperature as the troposphere moistens, which offsets part of the OLR increase due to the surface warming (Simpson 1927; Slingo and Webb 1997).

The concept of the effective radiation emission temperature or its difference from the much higher surface temperature is used for illustrating the greenhouse effect. However, it does not explain how slow the increasing rate of the OLR with T_S is, or why the latitudinal variation of the OLR loosely follows a linear relation with T_S, instead of a quadratic or cubic relation. Recent observational analysis (Koll and Cronin 2018, herein KC18) indicates that the linear slope of the clear-sky OLR with T_S is about 2.2 W m⁻² K⁻¹, slightly larger than the earlier estimate (2.08 W m⁻² K⁻¹) used in simple energy balance models (e.g., Budyko 1969; North 1975). KC18 obtained the OLR as a function of T_S from the output of a line-by-line radiative transfer model using prescribed vertical profiles of temperature and moisture that follow the moist adiabatic lapse rate with a uniform relative humidity in the troposphere. They found that the resultant OLR follows a quasi-linear relation with T_S and its slope is very close to the observed clear-sky OLR–T_S relation (2.2 ± 0.2 W m⁻² K⁻¹). They postulate that more abundance of water vapor at higher temperatures narrows the atmospheric window through which the surface thermal emission can escape to space. As a result, the contribution to the OLR from the surface emission gets smaller and smaller as T_S increases, implying the decoupling of the OLR from the surface’s emission at higher temperatures. Such decoupling leads to a quasi-linear OLR–T_S relation.

It is important to note that the slope of the observed quasi-linear OLR–T_S relation derived from the latitudinal variations of the OLR and surface temperature (e.g., Fig. 1 of KC18) is not the same as the climate sensitivity parameter (Bony et al. 2006), although they both have the same dimensionality. The OLR–T_S slope in the literature for climate sensitivity studies is derived from temporal variations or in response to external forcing in the global mean or regional (local) mean. Such OLR–T_S slope derived from temporal variations of the regional means varies spatially (e.g., Allan et al. 1999; Brown et al. 2016; Zhang et al. 2020; McKim et al. 2021). It is found that the spatial variability of the OLR–T_S slope derived from temporal variations of regional means can be largely explained by the spatial variability of the relative humidity in the troposphere above the boundary layer (Zhang et al. 2020). Therefore, the slope of the quasi-linear OLR–T_S relation derived from the latitudinal variations of the OLR and surface temperature would be equivalent to the OLR–T_S slope derived from temporal variations of the global mean or regional means under the same relative humidity in the free troposphere.

The scope of this study focuses on the explanation for the quasi linearity of the OLR–T_S and its slope derived from the latitudinal variations of the climatological and zonal mean OLR and surface temperature. The observed quasi-linear OLR–T_S relation exists for climatological mean vertical profiles of temperature and moistures (as well as clouds) that are determined by the energy balance among various radiative and nonradiative processes, rather than the prescribed vertical profiles considered in KC18. In other words, the observed OLR are fundamentally related to the incoming solar energy input, and so are climate variables that contribute to the OLR including surface temperatures, vertical profiles of temperatures, water vapor, and clouds. For a radiative–convective equilibrium (RCE) state, the spatial variation of the OLR has to be exactly equal to the net solar forcing. Therefore, a quasi-linear OLR–T_S relation in an RCE climate state is achieved by having T_S deviating from T_EB through radiative processes such as the greenhouse effect without changes in OLR. The observed quasi-linear OLR–T_S relation exists under the scenario that the OLR in low latitudes is less than the solar forcing but is greater than the solar forcing in high latitudes, i.e., energy surplus in tropics and deficit in polar regions at the TOA. Since the surplus in low latitudes and deficit in high latitudes of the net radiative energy fluxes at TOA are caused by the atmospheric/oceanic poleward energy transport, it is expected that nonradiative processes, such as poleward energy transport, also play important roles in the observed quasi-linear OLR–T_S relation. Cai (2006, hereafter C06) developed a four-box dry radiative–convective–transportive atmosphere–surface coupled model of a gray atmosphere for illustrating dynamic amplification of polar warming without water vapor and ice-albedo feedbacks. C06 provided evidence showing a quasi-linear OLR–T_S relation at both the low-latitude and high-latitude atmosphere–surface columns (blue curves in Fig. 5 of C06). Therefore, the atmospheric poleward energy transport alone could also lead to a quasi-linear OLR–T_S relation.

The primary objective of this study is to seek a general explanation for the quasi-linear OLR–T_S relation that remains valid regardless of the strength of the atmospheric window’s narrowing effect on planetary thermal emission at higher temperatures. Our goals are (i) to gain a better understanding of the reduction of the increasing rate of OLR with T_S in an RCE simulation by examining what causes the change in T_S, (ii) to explore the role of the poleward energy transport by atmospheric motions for a quasi-linear OLR–T_S relation, and (iii) to differentiate the contributions to the quasi-linear OLR–T_S relation from the greenhouse effect of water vapor (radiative processes), vertical convection (local nonradiative processes), and poleward energy transport (nonlocal and nonradiative processes).

The paper is organized as follows. In the next section, we introduce a new graphic analysis method based on the mathematical inequality equation for the reduction of the increasing rate of OLR with T_S. The new graphic analysis allows us to visualize the physical processes leading to a quasi-linear OLR–T_S relation. Section 3 describes the data, models, and methods for the study. Section 4 presents the results obtained from observations whereas section 5 contains the results from an RCE model and an idealized coupled GCM model, which both include water vapor radiative feedback and moist adiabatic convection adjustment. The RCE simulation is designed to duplicate the results of KC18 first in an RCE setting. The comparison of the observed quasi-linear OLR–T_S relation between the clear-sky OLR and all-sky OLR and the idealized GCM climate simulation is used to demonstrate the effects of poleward energy transport on the OLR–T_S relation. Discussion and conclusions are given in section 6. In the online supplementary information, we include the results of a series of online feedback suppression experiments, which help differentiate the contributions to the quasi-linear OLR–T_S relation from the greenhouse effect of water vapor radiative feedback, vertical convection, and poleward energy transport.

2. Graphic analysis for physical understanding of the reduction of the increasing rate of OLR with T_S

Before showing observational and modeling results, it is fruitful to examine the mathematical definition for the reduction of the increasing rate of OLR with T_S from the quartic relation to a quasi-linear one. Table 1 provides the definitions of the variables used in the graphic analysis.

Table 1.

A list of the variables used in the graphic analysis, where σ is the Stefan–Boltzmann constant and S is the net downward solar energy flux at the TOA.

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To apply (1) and (6) for gaining a physical understanding of the reduction of the increasing rate of OLR with T_S from the quartic relation to a quasi-linear one, we design a graphic (Fig. 1), named the EB–RCB–TS chart, to plot both OLR and the net downward solar energy flux at the TOA as a function of T_S, T_EB, and T_RCB. As illustrated in Fig. 1, both T_EB lines (dashed blue lines) and T_RCB lines (dashed magenta lines) tilt toward cold surface temperature (or negative slope) on an EB–RCB–TS chart. The tilting of T_EB lines and T_RCB lines reflects the strength of the greenhouse effect (i.e., the smallness of δT_EB > 0 and δT_RCB > 0). The stronger the greenhouse effect is, the warmer the surface temperature is and the stronger the meridional temperature gradient is, resulting in stronger tilting toward cold temperature or smaller positive values of δT_RCB and δT_EB. The difference in the tilting angles between T_RCB lines and T_EB lines is determined by the strength of poleward energy transport. The effect of the atmospheric poleward energy transport results in weaker titling of T_EB lines with respect to the vertical at cold temperatures than T_RCB lines, but stronger tilting of T_EB lines at warm temperatures than T_RCB lines, which is the graphic representation of δT_EB < δT_RCB or ${\tilde{\mathrm{\Delta }}}_{\text{EB}}>0$ as defined in (4). The stronger the poleward energy transport is, the greater the value of ${\tilde{\mathrm{\Delta }}}_{\text{EB}}>0$ is, and the larger the difference of the tilting angles between T_RCB lines and T_EB lines is, implying a smaller positive value of $\delta T_{\text{EB}}=\delta T_{\text{RCB}}-{\tilde{\mathrm{\Delta }}}_{\text{EB}}$.

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A schematic diagram illustrating the configuration of the EB–RCB–TS chart. The ordinate is for both OLR (dashed thick red line) and the net downward solar energy flux at the TOA (thick black line). The bottom abscissa is for T_S, and the top abscissa is for both T_EB and T_RCB. T_S lines (vertical dashed black lines) serve as tick marks of T_S under their OLR. T_EB lines (dashed blue) and T_RCB lines (dashed magenta) connect the intersection points of T_S lines with OLR and the net downward solar flux to the top abscissa and their intersection points with the top abscissa serve as tick marks of T_EB and T_RCB, respectively. Intervals of tick marks for T_EB lines and T_RCB lines decrease with temperature according to the Stefan–Boltzmann law. The solid black and blue segments on both the top and bottom abscissas serve as the reference length for the temperature difference of 10 K. Note that both dashed thick red lines and thick black lines are constructed following a linear relation with T_S for an illustrative purpose.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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Therefore, the EB–RCB–TS chart helps visualize the contributions to the quasi linearity of the OLR–T_S relation and its slope from both an increase in surface temperature and a decrease of (upper level) air temperature gradient. The quasi linearity of the OLR–T_S relation and its slope can be uniquely determined from the tilting of T_RCB lines and T_EB lines. Both stronger tilting of T_EB lines and T_RCB lines and the larger difference of the tilting angles of these two lines contribute to a gentler slope of OLR–T_S relation and a higher degree of linearity. In sections 4 and 5, we will use EB–RCB–TS charts to depict contributions to the reduction in the increasing rate of OLR with T_S or SLOPE from the greenhouse effect and atmospheric/oceanic poleward energy transport.

3. Data, models, and methods

b. Outputs of an idealized coupled GCM climate simulation

The outputs of an idealized GCM climate simulation reported by LC10 are used to demonstrate the important role of the poleward energy transport for the quasi-linear OLR–T_S relation. The key ingredients of the idealized coupled GCM include the NASA ARIES/GEOS dry dynamical core without topography, the Fu–Liou’s radiation transfer model (Fu and Liou 1992, 1993), a dry/moisture adiabatic convection adjustment parameterization, a simple surface energy balance model with a prescribed latitude dependent surface albedo and a simple boundary layer model that allows exchanges of momentum and sensible heat between atmosphere and surface. To include water vapor feedback in the idealized GCM that does not have a hydrological cycle, we parameterize water vapor as a function of temperature per the Clausius–Clapeyron relation by fixing the model’s relative humidity to a prescribed meridional and vertical profile. The horizontal resolution of the GCM is 4° (meridional) × 5° (zonal). There are a total of 41 sigma levels in the vertical with 19 layers above 100 hPa. In this study, we only use the outputs of the 30 000 days of the idealized GCM simulation made with the annual mean solar energy flux at the TOA and the standard CO₂ (330 ppm) concentration.

The outputs of the climatological mean (the 30 000-day mean) derived from the idealized GCM climate simulations used in this study include (i) latitudinal profiles of the net SW radiation fluxes at the TOA, the OLR, and surface pressure, (ii) latitude–vertical profiles of the (vertical) convergence of SW and LW radiation fluxes (or radiative heating/cooling rate in units of W m⁻²), the vertical convergence of vertical convective energy fluxes (or convective heating rate in units of W m⁻², Fig. 2a), and the horizontal convergence of poleward energy fluxes (or large-scale dynamic heating rate in units of W m⁻², Fig. 2b), and (iii) latitude–vertical profiles of temperature (K, contours of Fig. 2c) and atmospheric specific humidity (kg kg⁻¹, Fig. 2d). Note that the sum of Figs. 2a and 2b is exactly equal to the (vertical) divergence of the sum of SW and LW fluxes. We also note that the vertical summation of the convective heating rate from the surface to the top layer shown in Fig. 2a is exactly equal to zero at all latitudes and the vertical summation of the large-scale dynamic heating rate (black curve in Fig. 2b) is exactly equal to the difference between the OLR and the net solar energy fluxes at the TOA.

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Climatology of (a) the convergence of vertical convective energy fluxes (W m⁻²), (b) the convergence of poleward energy fluxes (W m⁻²), (c) temperature (K), and (d) atmospheric specific humidity (kg kg⁻¹) derived from the idealized GCM simulation. The ordinate is for sigma levels and the abscissa is for latitudes. The black curve in (b) corresponds to the vertically integrated convergence of poleward energy fluxes with the dashed line for its global mean value (ordinate on the right) whereas the white curve in (d) is the vertically integrated total water vapor in units of kg m⁻² (ordinate on the right). The contours in (c) correspond to the solution of the original GCM simulation whereas shadings are the equilibrium solution of the Control simulation using the equation [(S1)] reported in the supplementary information.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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c. Equilibrium climate simulations

To gain a quantitative understanding of the individual contributions to the quasi-linear OLR–T_S relation from the greenhouse effect of water vapor and poleward energy transport, we perform three equilibrium climate experiments listed in Table 1 using the model below:

$C_k{m}_{j,k}\frac{d{T}_{j,k}}{dt}=S_{j,k}(q_{j,k})-R_{j,k}(T_{j,k},q_{j,k})+\beta D_{j,k},$(7)

where

$\begin{array}{l}{C}_k=1004\mathrm{J}{\text{kg}}^{-1}{\mathrm{K}}^{-1}\text{,}\text{for}k=1,2,\dots ,40\text{and}{C}_{41}=4186\mathrm{J}{\text{kg}}^{-1}{\mathrm{K}}^{-1};\\ m_{j,k}=P_j({\sigma }_{k+1}-{\sigma }_k)/9.81\text{,}\text{for}k=1,2,\dots ,40\text{and}{m}_{j,41}=1000\text{kg}{\mathrm{m}}^{-2}\cdot \end{array}$(8)

In (7) and (8), j stands for latitude grids (j = 1, 2, …, 44 running from −86° to 86°), k in 1 ≤ k ≤ 40 for sigma levels (σ), and k = 41 for the surface layer with 0 < σ₁ < σ₂, …, < σ₄₁ = 1. The latitude and vertical grids are identical to those in the idealized GCM simulations. In (7), P_j is the surface pressure in units of Pa at the jth latitude derived from the climatological mean field of the idealized GCM simulations, and m_j,41 denotes the mass in the surface layer at each latitude grid, corresponding to 1-m-deep water column.

In (7), D_j,k is the vertical–latitudinal profile of the convergence of the poleward energy fluxes shown in Fig. 2b. The parameter β in front of D_j,k is an auxiliary parameter to “turn on” with β = 1 or “turn off” with β = 0 the atmospheric poleward energy transport. The terms S_j,k(q_j,k) and R_j,k(T_j,k, q_j,k) are, respectively, the vertical convergence of SW fluxes and divergence of LW fluxes obtained from the Fu–Liou’s radiation transfer model (Fu and Liou 1992, 1993) at latitude j using the same incoming solar energy flux, prescribed surface albedo, and vertical profile of ozone as LC10 but with different (either prescribed or parameterized as a function of T_j,k according to Table 2) vertical profiles of atmospheric water vapor equaling q_j,k. The other LW and SW radiation absorbers (e.g., CO₂, CH₄, and NO₂ as well as ozone) are set to be identical to those in LC10. In the supplementary information, we present the results of online feedback suppression experiments to explore the individual contributions to the quasi-linear OLR–T_S relation from the greenhouse effect of water vapor, vertical convection, and poleward energy transport by different combinations of using the outputs from the idealized GCM simulations shown in Fig. 2. The latitude–vertical profile of the equilibrium temperature, T_j,k, can be obtained by integrating (7) till the sum of the terms on the right-hand side is equal to zero. Obviously, the equilibrium temperature of (7) is independent of the parameters’ values on the left-hand side, which are given in (8).

Table 2.

A list of the three experiments using the model (7).

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4. The observed OLR–T_S relation

Although the observed linear OLR–T_S relation reported in the literature is mainly derived from the clear-sky OLR, we here in Fig. 3 show the OLR–T_S relation for both the clear-sky and total-sky OLR derived from the ERA5. It is seen that the points falling into the high kernel density zone appear to align up with the linear regression line for the OLR–T_S relation, especially for the clear-sky OLR. The linear slope for the clear-sky OLR–T_S relation is 2.07 W m⁻² K⁻¹, very close to the earlier estimate (2.08 W m⁻² K⁻¹) used in simple energy balance models (e.g., Budyko 1969; North 1975) and slightly smaller than the recent estimate of KC18 (2.2 W m⁻² K⁻¹). The linear slope for the total-sky OLR–T_S relation is 1.78 W m⁻² K⁻¹ with a smaller R-square value (0.81) than that for the clear-sky OLR–T_S relation (0.97). It should be pointed out that it is very evident that the total-sky OLR becomes nearly independent from T_S at high Ts (i.e., the tropics). It is found that the lack of dependence of the total-sky OLR with Ts at high Ts is apparent after applying either a zonal or a climatological mean (not shown here). However, the application of a combined zonal and climatological mean leads to the quasi-linear relationship of total-sky OLR and Ts of the present analysis.

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The Gaussian kernel density function (shadings; in logarithmic scale) for the observed OLR–T_S relation. (a) Clear-sky OLR (ordinate) vs T_S (abscissa) and (b) total-sky OLR (ordinate) vs T_S (abscissa). The total number of points used for estimating the Gaussian kernel density function is 33 343 920, corresponding to the 1° × 1° spatial resolution covering the entire globe for the 516 months in 1979–2021. The black solid lines are linear regression lines whose mathematical expressions and R-square values are given in the bottom-right corner of each panel. The dashed black lines correspond to the blackbody thermal emission at T_S calculated from the Stefan–Boltzmann law.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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As indicated in the EB–RCB–TS chart (Fig. 4), the OLR–T_S relation derived from the climatological and zonal mean of the OLR and T_S exists under the scenario that the OLR above warmer surface temperature (i.e., in low latitudes) is less than the solar forcing, or a radiative energy flux surplus at low latitudes’ TOA, while the OLR above the cold surface (i.e., in high latitudes) is greater the solar forcing, or a radiative energy flux deficit at high latitudes’ TOA. The monotonic increase of the difference between the net solar energy flux and OLR with temperature (Figs. 4b,d) confirms that the horizontal convergence of poleward energy transport increases from negative values over the warm place to positive values over the cold place in both Northern (NH) and Southern (SH) Hemispheres. The shorter interval of T_EB lines (dashed blue lines) than T_S lines (dashed black lines) illustrates δT_EB/δT_S < 1. The shorter interval between two adjacent T_EB lines in comparison with T_RCB lines (dashed magenta lines) illustrates δT_EB < δT_RCB or ${\tilde{\mathrm{\Delta }}}_{\text{EB}}>0$, which graphically illustrates the positive contribution of the poleward energy transport to the reduction of the increasing rate of OLR with T_S.

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The EB–RCB–TS chart of the OLR–T_S relation derived from the ERA5 climatological mean and zonal mean OLR–T_S for (a),(c) the clear-sky OLR and (b),(d) the total-sky OLR with the ordinate for radiation energy fluxes (W m⁻²) and the abscissa for temperature (K) in the(top) Northern Hemisphere (bottom) Southern Hemisphere. The solid black curves are for the net downward radiation flux at the TOA and dotted red curves are for the OLR. T_S lines (vertical dashed black lines) serve as tick marks of T_S under their OLR. T_EB lines (dashed blue) and T_RCB lines (dashed magenta) connect intersection points of T_S lines with OLR and the net downward solar flux to the top abscissa and their intersection points with the top abscissa serve as tick marks of T_EB and T_RCB, respectively. The dotted brown horizontal lines mark the value of the global mean net downward solar energy flux, which is (nearly) equal to the global mean OLR (with a small difference less than 0.5 W m⁻²). The point where the black, dashed red, and dotted brown lines intersect in (c) and (d) correspond approximately to the latitudes where the OLR, the net downward solar energy flux, and surface temperature are all close to their hemispheric (area-weighted) mean values.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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The comparison between the left and right columns of Fig. 4 indicates that the difference between the interval for T_RCB lines and that for T_EB lines, or ${\tilde{\mathrm{\Delta }}}_{\text{EB}}$ is greater for the total-sky OLR than for the clear-sky OLR. This can be explained by the fact that the clear-sky OLR does not fully include the information of the poleward energy transport, resulting in a more sloped meridional profile of the clear-sky OLR than the total-sky OLR. Specifically, the energy balance equation [(2)] can be rewritten as

${\text{OLR}}_{\text{clear-sky}}-S_{\text{net}}^{\text{TOA}}+({\text{OLR}}_{\text{total-sky}}-{\text{OLR}}_{\text{clear-sky}})=D.$(11)

As indicated by Fig. 5a, (OLR_total-sky − OLR_clear-sky) < 0, implying that the clear-sky OLR overestimates the convergence of the poleward energy transport in high latitudes but underestimates the divergence of the poleward energy transport in low latitudes in both hemispheres. It is seen from Fig. 5b, the negative slope of the convergence of the poleward energy transport inferred from the clear-sky OLR (blue lines) is weaker than that of the total-sky OLR (red lines) at cold temperatures in both hemispheres, resulting in ${(\delta T_{\text{EB}}/\delta T_S)}_{\text{clear-sky}}>{(\delta T_{\text{EB}}/\delta T_S)}_{\text{total-sky}}$ according to (3). This explains a weaker reduction of SLOPE for the clear-sky OLR than for the total-sky OLR in cold surface temperatures (orange curves, Fig. 5c versus Fig. 5d). In the tropical latitudes where T_S > 290 K, the negative slope of the convergence of the poleward energy transport inferred from the clear-sky OLR becomes slightly steeper than that of the total-sky OLR. According to (3), this results in a smaller value ${(\delta T_{\text{EB}}/\delta T_S)}_{\text{clear-sky}}$ than ${(\delta T_{\text{EB}}/\delta T_S)}_{\text{total-sky}}$ in the tropical latitudes, explaining a stronger reduction of SLOPE for the clear-sky OLR than for the total-sky OLR in the tropical latitudes. Therefore, the gentler slope of OLR–T_S relation for total-sky OLR than clear-sky OLR (blue curves in Figs. 5c,d) should not be interpreted as the result of the exclusion of the cloud radiative effects in clear-sky OLR. Instead, the gentler slope of OLR–T_S relation for total-sky OLR manifests an overall stronger reduction of SLOPE at all latitude bands when the effect of the poleward energy transport is taken into the consideration fully.

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Line charts for (top) OLR and the differences between OLR and the net downward solar energy flux at the TOA (W m⁻² K⁻¹) and (bottom) SLOPE (W m⁻² K⁻¹; ordinate on the left for the blue curve) and ΔSLOPE (W m⁻² K⁻¹; ordinate on the right for the orange curve) as a function of the surface temperature (K; abscissa) derived from the ERA5 climatological mean and zonal mean OLR and T_S. (a) The OLR, (b) the difference between OLR and the net downward solar radiation at the TOA, (c) SLOPE and ΔSLOPE of the clear-sky OLR, and (d) SLOPE and ΔSLOPE of the total-sky OLR. The dotted blue curves in (c) and (d) are the cubic slope derived from the Stefan–Boltzmann relation, which serves as a reference for ΔSLOPE, SLOPE, $\overline{\text{SLOPE}}$, and NRMS_SLOPE where ΔSLOPE and SLOPE are defined in Table 1 and $\overline{\text{SLOPE}}$ and NRMS_SLOPE are defined in (9) and (10). The hemispheric (area-weighted) mean of each of the two red lines in (b) is very close to zero, implying the energy balance at the TOA of the ERA5. The definitions of SLOPE and ΔSLOPE are given in Table 1.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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It is seen from Table 3 that the analytically derived averaging slope of the OLR–T_s relation using the climatological mean and zonal mean data based on the definition (9) is very similar to that from the linear regression analysis of monthly data at all grids shown in Fig. 3; namely, clear-sky SLOPE is slightly above 2 W m⁻² K⁻¹ and total-sky SLOPE is about 1.7 W m⁻² K⁻¹. The values of the root-mean-square of the departure of the local slope from $\overline{\text{SLOPE}}$, i.e., NRMS_SLOPE defined in (10), is about 4.8% (10.7%) for the clear-sky (total-sky) OLR–T_S relation in the Northern Hemisphere. The values of NRMS_SLOPE for the OLR–T_S relation in the Southern Hemisphere are larger (about 8% for clear-sky and 18% for total-sky OLR). The difference in NRMS_SLOPE between the two hemispheres is consistent with their R-square difference. Because the reduction of SLOPE for the clear-sky OLR (orange curves in Fig. 5c) is nearly parallel to the slope of the Stefan–Boltzmann relation (dotted blue curve), the clear-sky OLR–T_S relation has a high degree of the quasi linearity (NRMS_SLOPE < 10%). The reduction of SLOPE for the total-sky OLR (orange curves in Fig. 5d) varies with surface temperature weakly (in reference to the slope of the Stefan–Boltzmann relation), implying a large value of NRMS_SLOPE for the quasi linearity of the total-sky OLR–T_S relation.

Table 3.

The slope of OLR–T_S relation in ERA5 observations and climate simulations.

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In comparison with the Northern Hemisphere, the larger value of NRMS_SLOPE for the total-sky OLR in the Southern Hemisphere mainly comes from high latitudes (south of 60°S) where the climatological and zonal mean surface temperature is below the freezing point. It is seen from Fig. 5 that the reduction of SLOPE for the total-sky OLR–T_S relation in both hemispheres is relatively uniform for the entire temperature range, about 2 W m⁻² K⁻¹. Because the slope of a quartic relation decreases rapidly as the temperature decreases, a similar reduction of SLOPE implies a much gentler slope of the OLR–T_S relation at colder temperatures. Because of the higher elevation, the surface temperature over the Antarctic is much colder than the Arctic. As a result, the slope of the total-sky OLR–T_S relation over the Antarctic is gentler ($\overline{\text{SLOPE}}\sim 1.2\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$) than the slope of the total-sky OLR–T_S relation over the warmer place (∼2 W m⁻² K⁻¹). Now let us turn our attention to the comparison of the clear-sky OLR–T_S relation between the two hemispheres. According to Fig. 5b, the negative slope of the convergence of the poleward energy transport inferred from the clear-sky OLR is weaker over the Antarctic regions than over the Arctic regions, leading to a larger (δT_EB)_clear-sky over the Antarctic. This helps neutralize the impact of (δT_S)_Antarctic > (δT_S)_Arctic on ${(\delta T_{\text{EB}}/\delta T_S)}_{\text{clear-sky}}$. This results in a similar pattern of the SLOPE reduction for clear-sky OLR between the two hemispheres (orange curves of Fig. 5c).

5. Results of equilibrium climate simulations

a. Equilibrium climate simulation of the REWV experiment

As indicated in Table 2, the REWV simulation is an RCE simulation made under active water vapor feedback by which the tropospheric water vapor follows the saturated specific humidity with a global uniform relative humidity of 50%. The idealized GCM simulation shown in Fig. 2 serves as the control run for the REWV simulation. Because of the absence of the poleward energy transport, the water vapor feedback in the REWV simulation is much stronger than the GCM simulation as evident from a much stronger meridional gradient of tropospheric water vapor in Fig. 6a in comparison with Fig. 2d. The stronger water vapor feedback in the tropics leads to a much warmer surface temperature, directly contributing to a much stronger meridional gradient of the temperature at the surface as well as in the troposphere (Fig. 6b versus Fig. 2c or Fig. 6c).

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Climatology of (a) atmospheric specific humidity (kg kg⁻¹), (b) temperature (K) derived from the REWV simulation, and (c) the difference between (b) and Fig. 2c. The ordinate is for sigma levels and the abscissa is for latitudes. The white curve in (a) is the vertically integrated total water vapor (kg m⁻²; ordinate on the right).

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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Unlike the observations, the OLR of the climate simulation without considering poleward energy transport overlaps with the net solar energy fluxes at the TOA exactly, implying that no change in T_EB as evident by the overlapping of T_EB lines (dashed blue) and T_REB lines (dashed magenta) on the EB–REB–TS chart for the REWV simulation (Fig. 7a). As a result, the slope reduction of the OLR–T_S relation from the OLR–T_EB relation in the REWV simulation is achieved only through changes in T_S without changes in T_EB. The enhanced meridional gradient of surface temperature by the stronger greenhouse effect of the water vapor feedback in the tropics is also evidenced from the smallness of the ratio of δT_EB to δT_S at higher temperatures, implying a strong reduction of SLOPE (orange curves in Fig. 8a). The sudden rise of the reduction of SLOPE at warmer surface temperatures above 290 K is caused by the nonlinear dependence of saturation water vapor pressure with T_S per the Clausius–Clapeyron relation. More water vapor implies a much stronger greenhouse effect of water vapor at higher temperatures, causing a stronger increase of δT_S, leading to a much smaller ratio of δT_EB to δT_S, as evident by the association of warmer surface temperatures with highly packed T_EB lines (Fig. 7a, meaning δT_EB/δT_S ≪ 1). As a result, the slope reduction of the OLR–T_S relation is much stronger at warmer surface temperatures above 290 K (orange curve in Fig. 8a). As summarized in Table 3, the OLR–T_S relation in the REWV simulation is close to a quasi-linear one with $\overline{\text{SLOPE}}\hspace{0.17em}=2.11\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and NRMS_SLOPE = 11. 5%.

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As in Fig. 4, but for the results obtained from (a) the REWV simulation, (b) the RE0 simulations, (c) the DRE simulation, and (d) the idealized GCM simulation. The points where the black, dashed red, and dotted brown lines intersect in (c) and (d) correspond approximately to the latitudes where the OLR, the net downward solar energy flux, and surface temperature are all close to their global (area-weighted) mean values. Note that the vertical range of the black lines in all panels is identical and they are very close to their counterparts in Fig. 4 in terms of their latitudinal profiles.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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Line charts for SLOPE (W m⁻² K⁻¹; ordinate on the left for the blue curve) and ΔSLOPE (W m⁻² K⁻¹; ordinate on the right for the orange curve) as a function of the surface temperature (K; abscissa) derived from the results of (a) the REWV simulation, (b) the RE0 simulations, (c) the DRE simulation, and (d) the idealized GCM simulation. The dotted blue curve is the cubic slope derived from the Stefan–Boltzmann relation, which serves as a reference for ΔSLOPE, SLOPE, $\overline{\text{SLOPE}}$, and NRMS_SLOPE where ΔSLOPE and SLOPE are defined in Table 1 and $\overline{\text{SLOPE}}$ and NRMS_SLOPE are defined in Eqs. (9) and (10).

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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b. Equilibrium climate simulations of the RE0, DRE, and GCM experiments

The RE0 climate simulation is designed to examine the impact of the lack of stronger greenhouse effect of water vapor in the tropics (or at higher temperatures) by using a global uniform vertical profile of water vapor under the same RCE setting as its control experiment, the REWV simulation. The difference between Figs. 9a and 6b or Fig. 9c indicates that the meridional gradient of T_S is substantially weaker under the uniform greenhouse effect of water vapor, explaining a larger ratio of δT_EB to δT_S on the EB–REB–TS chart for the RE0 simulation (Fig. 7b), a much weaker reduction of SLOPE (orange curve in Fig. 8b), a more sloped OLR–T_S relation (blue curve in Fig. 8b). As summarized in Table 3, the OLR–T_S relation in RE0 simulation is less linear with $\overline{\text{SLOPE}}\hspace{0.17em}=3.24\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and NRMS_SLOPE = 17.75%. The larger value of NRMS_SLOPE and steeper slope of the OLR–T_S relation derived from the RE0 simulation than their counterparts of the REWV simulation are evidence for the important role of the equatorward increasing latitudinal profile of water vapor for both stronger SLOPE reduction and quasi linearity of OLR–T_S relation. Nevertheless, the uniform greenhouse effect of water vapor alone can also lead to a quasi-linear OLR–T_S relation whose mean slope is much steeper with a large local deviation from the mean slope.

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Climatology of temperature (K) derived from (a) the RE0 simulation and (b) the DRE simulation, (c) the difference between (a) and Fig. 6b, and (d) the difference between (b) and (a). The ordinate is for sigma levels and the abscissa is for latitudes.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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The DRE experiment is designed to examine if the presence of the atmospheric poleward energy transport alone (i.e., in the absence of a stronger greenhouse effect of water vapor in the tropics) would yield a quasi-linear OLR–T_S relation. For this purpose, the RE0 simulation serves as the control experiment for the DRE simulation. The difference between Figs. 9b and 9a results from the inclusion of the poleward energy transport in the DRE simulation, which leads to a weakening of the meridional gradient of temperatures at the surface and in the troposphere (Fig. 9d). The reduction in the meridional gradient of temperature is stronger in the troposphere than the surface, confirming the inequality (6), or ${\tilde{\mathrm{\Delta }}}_{\text{EB}}/{\tilde{\mathrm{\Delta }}}_S>1$ and explaining a smaller ratio of δT_EB to δT_S than the ratio δT_RCB to δT_S (Fig. 7c). The comparison of Fig. 8c with Fig. 8a indicates the reduction of SLOPE by the atmospheric poleward energy transport alone is quite comparable to that by greenhouse effects of the water vapor feedback alone. The slope reduction of the OLR–T_S relation in the DRE simulation is through the effective rerouting of the OLR to be emitted from the warm place to the cold place, which is vividly illustrated by the increase (decrease) of about 60 W m⁻² (50 W m⁻²) of the OLR at the coldest (warmest) place (Fig. 7c) in comparison with its counterpart in Fig. 7a or in Fig. 7b. As summarized in Table 3, the OLR–T_S relation in the DRE simulation is close to a quasi-linear one with $\overline{\text{SLOPE}}\hspace{0.17em}=2.38\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and NRMS_SLOPE = 21.2%. Therefore, unlike the greenhouse effect of a global uniform water vapor profile (i.e., the RE0 simulation), the effect of poleward energy transport alone can lead to a quasi-linear OLR–T_S relation that has a similar mean gentle slope as that due to the greenhouse effect of the water vapor feedback (i.e., the REWV simulation), although its degree of quasi linearity is not as good as that derived from the REWV simulation.

Unlike the DRE equilibrium simulation, the GCM simulation is made under active water vapor feedback, resulting in more abundance of water vapor in the tropics than in high latitudes (Fig. 2d). In addition, the GCM simulation is made under active vertical convective feedback whose heating rate is shown in Fig. 2a. The inclusion of vertical convections in the GCM simulation helps warm the very cold tropopause above the tropical latitudes in the DRE simulation (Fig. 2c versus Fig. 9b). Therefore, the slope reduction of the OLR–T_S relation in the GCM simulation is achieved through the combined effect of the greenhouse effect of water vapor feedback, vertical and horizontal energy redistribution by vertical convection and large-scale circulations.

The EB–RCB–TS chart for the OLR–T_S relation obtained from the GCM simulation (Fig. 7d) shows a similar pattern as the observational counterpart (Figs. 4b,d). As in observations, the intervals of T_EB lines (dashed blue) are shorter than T_RCB lines (dashed magenta) for the same interval of δT_S, meaning that the relatively longer intervals of T_EB lines in Fig. 7d than their counterparts in Figs. 4b and 4d are the sign indicating a smaller value of ${\tilde{\mathrm{\Delta }}}_{\text{EB}}>0$ or weaker poleward energy transport in the GCM climate simulation. Again, the slope reduction of the OLR–T_S relation in the GCM simulation is through the effective rerouting of the OLR to be emitted from the warm place to the cold place (Fig. 7d versus Figs. 7a,b). The absence of land–ocean contrast, topography, and a hydrological cycle (or poleward moist energy transport) in the idealized GCM is responsible for about 33% reduction in the poleward energy transport (3.55 PW in the GCM simulation) in comparison with the observations (about 5.3 PW). This explains a noticeably weaker slope reduction of the OLR–T_S relation in comparison with its observational counterpart (the orange curve in Fig. 8d versus the orange curves in Figs. 5b,d). As a result, the slope of the OLR–T_S relation in the idealized GCM simulation is greater than the observation (the blue curve in Fig. 8d versus the blue curves in Fig. 5d). As indicated in Table 3, the $\overline{\text{SLOPE}}$ of the OLR–T_S relation in the GCM is 2.05 W m⁻² K⁻¹, which is gentler than the slope of the OLR–T_S relation in the simulations that exclude the poleward energy transport (REWV) or the greenhouse effect of water vapor feedback (DRE). The degree of the quasi linearity of the OLR–T_S relation (NRMS_SLOPE = 12%) is within the variation of NRMS_SLOPE for the observed quasi-linear OLR–T_S relation between the two hemispheres.

6. Conclusions and discussion

The slope of the quasi-linear relation between planetary outgoing longwave radiation (OLR) and surface temperature (T_S), which is referred to as the climate sensitivity parameter in the literature (e.g., Bony et al. 2006), is an important parameter measuring the sensitivity of the Earth climate system. In this study, we aim to gain a physical understanding of the quasi-linear OLR–T_S relation and its slope from observation analysis, climate simulations using RCE and GCM models, and a series of online feedback suppression experiments. The primary objective of this study is to seek a general explanation for the quasi-linear OLR–T_S relation that remains valid regardless of the strength of the atmospheric window’s narrowing effect on planetary thermal emission at higher temperatures.

The novelty of this study is a new graphic analysis that reflects the mathematical analysis of the term causing the reduction of the increasing rate of OLR with T_S from the Stefan–Boltzmann law. The new graphic analysis introduces two auxiliary lines and adds the net downward solar energy flux at the TOA to the conventional OLR–T_S chart, named as EB–RCB–TS chart. The two auxiliary lines are called T_EB lines and T_RCB lines. The former connects the OLR to its blackbody emission temperature (T_EB) and the latter connects numerical values of the net downward solar energy flux at the TOA to the blackbody temperature of the OLR in a radiative–convective equilibrium state (T_RCB). The newly introduced EB–RCB–TS chart helps visualize the contributions to the quasi linearity of the OLR–T_S relation and its slope from both an increase of surface temperature and a decrease of (upper level) air temperature gradient. Specifically, both T_EB lines and T_RCB lines tilt toward cold surface temperature (or negative slope) on an EB–RCB–TS chart. The tilting of T_RCB lines reflects the strength of the greenhouse effect whereas the difference in the tilting angles between T_EB lines and T_RCB lines is determined by the strength of poleward energy transport. The stronger the greenhouse effect is, the warmer the surface temperature is, the stronger the meridional temperature gradient is, and the stronger their tilting is. The stronger the poleward energy transport is, the weaker the meridional gradient of T_EB, and the larger the difference of the tilting angles between T_RCB lines and T_EB lines is. The quasi linearity of the OLR–T_S relation and its slope can be uniquely determined from the tilting of T_RCB lines and T_EB lines. Both stronger tilting of T_EB lines and T_RCB lines from the vertical and the larger difference of the tilting angles of these two lines contribute to a gentler slope of OLR–T_S relation and a higher degree of linearity.

The observed quasi-linear OLR–T_S relation manifests a climate footprint of radiative (such as the greenhouse effect) and nonradiative processes (poleward energy transport). The combined effect of the water vapor feedback and poleward energy transport on the reduction of the increasing rate of OLR with T_S can be succinctly summarized in Fig. 10. The mathematical implication of a stronger reduction is a higher degree of the quasi linearity of the OLR–T_S relation. The greenhouse effect of the water vapor feedback acts to increase the meridional gradient of surface temperature, which is represented by the pattern of (δT_S)₁ < (δT_S)₂ < (δT_S)₃ along the increasing direction of the greenhouse effect in Fig. 10. The poleward energy transport acts to decrease the meridional gradient of (upper level) atmospheric temperatures, causing the flattening of the meridional profile of OLR as represented by the pattern of (δOLR)₃ > (δOLR)₂ > (δOLR)₁ along the increasing direction of poleward energy transport in Fig. 10. Radiative processes alone can lead to a quasi-linear OLR–T_S relation that is more steeply sloped. Nonradiative processes make the quasi-linear OLR–T_S relation less sloped and further increase the degree of linearity. The by-product of the nonradiative flattening of the meridional profile of atmospheric temperature, through downward thermal radiation, is a reduction of the meridional surface temperature gradient (represented by the positively tilted contour lines of the surface temperature in Fig. 10), albeit weaker than the reduction in the atmosphere. It follows that nonradiative processes act to increase the climate sensitivity determined by radiative processes alone.

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A schematic diagram for illustrating the combined effect of the water vapor feedback and poleward energy transport on the reduction of the increasing rate of OLR with surface temperature. Red lines represent contours of the equator-to-pole surface temperature contrast with (δT_S)₁ < (δT_S)₂ < (δT_S)₃ along the increasing direction of the water vapor feedback, whereas blue lines represent contours of the equator-to-pole OLR contrast with (δOLR)₃ > (δOLR)₂ > (δOLR)₁ along the increasing direction of the poleward energy transport. The positively tilted red lines represent the dynamical amplification of polar warming. The dashed magenta arrow points to the direction of the maximum reduction of the slope of the OLR–T_S relation from the slope of the Stefan–Boltzmann relation by the combined effect.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-22-0261.1

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Below are the specific main findings of this study:

• (i) The slope of the OLR–T_S relation derived from the climatological mean and zonal mean total-sky OLR (1.7 W m⁻² K⁻¹) is highly consistent with the regression analysis of 43 years of the ERA5’s monthly 2D total-sky OLR with their surface temperatures (about 1.78 W m⁻² K⁻¹). The relatively small value of the normalized root-mean-square value for measuring the quasi linearity (about 10% for the Northern Hemisphere and 18% for the Southern Hemisphere) confirms the quasi linearity of the total-sky OLR–T_S.

• (ii) The relatively steeper slope derived from the climatological mean and zonal mean clear-sky OLR (2.09 W m⁻² K⁻¹), which is also highly consistent with the regression analysis (2.09 W m⁻² K⁻¹), is mainly due to the fact that clear-sky OLR only includes part of information about the strengthen of atmospheric/oceanic poleward energy transport, rather than the exclusion of the cloud radiative effects in clear-sky OLR.

• (iii) The RCE simulations with active water vapor feedback and without active water vapor feedback (i.e., using a global uniform vertical profile of water vapor) all reveal a quasi-linear relation. The quasi-linear relation in the RCE climate state with the active water vapor feedback yields a nearly identical slope of the quasi-linear OLR–T_S relation (about 2.11 W m⁻² K⁻¹) with a similarly high degree of the quasi linearity (about 10%) as that reported in KC18.

• (iv) The slope of the quasi-linear OLR–T_S relation obtained using a global uniform vertical profile of water vapor is steeper with a moderate degree of the quasi linearity (about 20%). The results obtained using a global uniform vertical profile of water vapor are suggestive of the narrowing of the atmospheric window for the surface’s thermal emission at higher temperatures is not a critical condition for the existence of a quasi-linear OLR–T_S relation.

• (v) The inclusion of the poleward energy transport always leads to a quasi-linear OLR–T_S relation that has a gentler slope with a higher degree of quasi linearity for the same strength of the greenhouse effect and vertical convection. As a result of the poleward energy transport, a significant portion of the OLR (about 20%–25% in observations) to be emitted in tropical latitudes where surface temperature is the warmest is actually emitted in high latitudes where surface temperature is coldest, causing the further reduction of the increasing rate of the OLR with T_S. This further supports the general explanation that does not require the narrowing of the “atmospheric window” of planetary thermal radiation at higher temperatures for the quasi-linear OLR–T_S relation to exist.

The combination of the greenhouse effect and atmospheric circulation responsible for a quasi-linear OLR–T_S relation is also the mechanism that leads to the dynamic amplification of polar warming. As elicited in C06, Cai and Lu (2007), LC10, and Cai and Tung (2012), the anthropogenic radiative forcing or orbital solar forcing perturbations alone always tend to increase the meridional gradient of surface and air temperatures. Water vapor feedback that is much stronger in the tropics further enhances the increase in the meridional gradient of surface and air temperature by external forcing. The stronger air temperature gradient caused by radiative energy perturbations leads to a stronger atmospheric poleward energy transport, acting to cancel out part of the increase of air temperature gradient by radiative energy perturbations. Through the enhanced (reduced) downward thermal emission to the surface in high (low) latitudes, the increase of surface temperature gradient can be reversed, leading to dynamic amplification of polar warming (as indicated by the positively tilted red lines in Fig. 10).

The latitudinal variation of albedo is partially but indirectly included in our analysis as we use the net downward solar energy flux at the TOA to infer the strength of the poleward energy transport by atmospheric/oceanic circulations. Besides increasing the meridional gradient of the new downward solar energy flux at the TOA, the ice-albedo effect also acts to increase the meridional temperature gradient in high latitudes. This would favor a stronger slope reduction. This may be the factor for a relatively uniform slope reduction across the entire range of the surface temperature for the observed (total-sky) OLR–T_S relation. Because the slope of a quartic relation decreases rapidly as the temperature decreases, a similar reduction of SLOPE implies a much gentler slope of the OLR–T_S relation at colder temperatures. This explains why the slope of the observed (total-sky) OLR–T_S relation is gentler over the polar regions. In particular, because of the very cold surface temperature at higher elevations, the $\overline{\text{SLOPE}}$ of the total-sky OLR–T_S relation over the Antarctic is about 71% (∼1.2 W m⁻² K⁻¹) of the global mean value (∼1.68 W m⁻² K⁻¹).

It is important to point out that the application of either a climatological or zonal mean does not lead to quasi linearity of the OLR–Ts relation in the tropics, where a scatterplot of local values shows Ts to be nearly independent of total-sky OLR. Furthermore, the slope of the quasi-linear OLR–T_S relation derived from the climatological and zonal mean OLR and surface temperature may be different from the slope of the quasi-linear OLR–T_S relation derived from the temporal variations of regional mean and/or global mean OLR and T_S, such as the trends in response to external forcing. For example, based on the results of a simple 4-box climate model in C06, the slope of the quasi-linear OLR–T_S relation derived from the changes in response to external forcing is negative in the tropics and positive in high latitudes in the presence of the poleward energy transport.

Data availability statement.

The ERA5 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 to reproduce the results. The authors declare no competing interests.