Earth’s OLR Is Approximately Linear

We first consider the empirical relation between OLR and surface temperature for present-day Earth. In doing so we focus on clear-sky regions and do not address the potential impact of clouds. Clouds reduce Earth’s OLR on average by about 30 W·m⁻², but their potential changes remain challenging to predict while their impact on Earth’s energy balance is additionally complicated by their countervailing reflection of solar radiation (17).

A histogram of the monthly mean OLR in cloud-free regions vs. near-surface temperature demonstrates that Earth’s thermal emission strongly deviates from the Stefan–Boltzmann law and instead is nearly linear (Fig. 1). A linear regression OLR=A+BTs, with A=−339.647 W·m⁻² and B=2.218 W·m−2·K−1 fitted to the data captures the vast majority of its variance (r2=97%). To first order the relation between OLR and surface temperature can therefore be approximated as linear between about 220 K and 300 K, with larger deviations from linearity above 300 K.

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Fig. 1.

Earth’s OLR strongly deviates from the thermal emission of a blackbody, σTs4. Instead, the dependence of OLR on surface temperature can be approximated as a linear function. Shown are monthly mean clear-sky OLR from a satellite data product and near-surface temperatures from reanalysis (Materials and Methods). Colors show the density of data points, and the blue curve is a simple linear regression (r2=0.97).

We can reproduce the main features of this relation by considering an idealized model of a single atmospheric column with 100% relative humidity in radiative–convective equilibrium, in which water vapor is the only greenhouse gas (Materials and Methods). We compute the column’s OLR with a line-by-line radiation code, using a modern spectroscopic database valid for cold climates as well as hot steam atmospheres in the runaway greenhouse limit (18).

Similar to the satellite data, we find that OLR is roughly linear over a wide range of temperatures (Fig. 2). To quantify this range we analyze the feedback in our calculations, by which we refer specifically to the net clear-sky longwave feedback, λ=dOLR/dTs. If the relation between OLR and surface temperature was perfectly linear, then λ would be constant. We find that λ stays within ±10% of 2.2 W·m−2·K−1 over a temperature range of 60 K, starting at 218 K and ending at 278 K. For comparison, the feedback of a blackbody with Earth’s emission temperature varies by ±10% over a temperature range of only 17 K (Fig. 2).

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Fig. 2.

OLR is an approximately linear function of surface temperature between 220 K and 280 K in an atmosphere with 100% relative humidity (blue). The linear range extends to even higher temperatures, 230–300 K, under more Earth-like conditions (gray). The thick lines are a linear fit (Left), which imply a constant feedback (Right) and show the range over which each feedback changes less than ±10%. In contrast, a blackbody would have a feedback that varies by ±10% over a significantly smaller range of temperatures (solid black line). Right Inset shows a feedback decomposition for the saturated H₂O atmosphere at 285 K: PLsurf is the surface Planck feedback, PLatm is the atmospheric Planck feedback, LR is the lapse rate feedback, and WV is the water vapor feedback (SI Appendix, section 4). The net feedback is dominated by the surface Planck feedback, while the other three feedbacks largely cancel.

To explain the remaining mismatch between our idealized model and the empirical results, Fig. 2 shows that λ remains nearly constant over an even wider range of temperatures, from 230 K up to 300 K, if we use a less idealized model with a bulk relative humidity of 50% and 400 ppm of CO2 (19). Given that the thermodynamics and radiative physics in our calculations are strongly nonlinear in temperature, the linearity of OLR is therefore an emergent property of Earth’s climate that is closely tied to the H₂O greenhouse effect. Other noncondensable greenhouse gases such as CO2 can modify this emergent property, but the effect of H₂O is distinct because a dry atmosphere with a CO2 greenhouse effect would exhibit a clearly nonlinear OLR (SI Appendix, Fig. S4). In the rest of this paper we therefore focus on our idealized model before considering how it is modified by CO2 and subsaturation.

Importance of Spectral Window Regions

To understand how the near linearity of OLR arises, Fig. 3, Top shows the spectrally resolved top-of-atmosphere irradiances from our line-by-line calculations with 100% relative humidity. The OLR is equal to the spectral integral of irradiance, so Fig. 3 shows which wavenumbers contribute most to the increase of OLR with surface temperature.

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Fig. 3.

(Top) Thermal emission to space decouples from surface temperature in optically thick parts of the spectrum. At low temperatures this occurs in the H₂O rotation (<500 cm⁻¹) and vibration bands (∼1,600 cm⁻¹). Above 300 K, the window region becomes optically thick due to continuum absorption. Red curves show the surface’s blackbody emission at 240 K and 280 K. (Top Inset) The atmospheric water vapor path is an approximately single-valued function of temperature, which is why the atmosphere’s emission to space is approximately independent of surface temperature (main text). Gray dotted and dashed lines show analytical limits (SI Appendix, section 2). (Bottom) Spectrally resolved transmission between the surface and top of atmosphere, Tν, which shows the fraction of surface radiation that is directly emitted to space. The irradiance and transmission are smoothed using a median filter of width 10 cm⁻¹.

As surface temperature increases from 240 K to 320 K, the contribution from wavenumbers below 500 cm⁻¹ and above 1,500 cm⁻¹ to the OLR remains essentially constant. These parts of the spectrum correspond to the rotation and first roto-vibration bands of H₂O, which allow the H₂O molecule to absorb radiation very efficiently. Because the irradiance does not increase with temperature at these frequencies, the net increase in OLR with temperature is caused by the increased emission around 1,000 cm⁻¹. This part of the spectrum is the window region in which H₂O is only a weak absorber and transmission between surface and space is close to unity, at least until the window closes above 300 K (Fig. 3, Bottom).

The basic reason why optically thick parts of the spectrum stop contributing to the increase in OLR as Ts increases was described by Ingram (20); we summarize the argument here. At a given frequency ν the atmosphere’s optical thickness isτν=∫κvq*dp/g,[1]where κν is the absorption cross-section at that frequency and q* is the saturation-specific humidity. The specific humidity q* varies by many orders of magnitude between the surface and tropopause whereas κν varies far less (its moderate changes are largely due to pressure broadening), so one can approximately remove κν from the integral. This means τν≈κν×WVP, where WVP=∫q*dp/g is the water vapor path of the atmospheric column.

Next, Fig. 3, Top Inset shows that the water vapor path is an almost constant function of atmospheric temperature over a wide range of surface temperatures. This behavior is not just true for Earth, but also applies to atmospheres with other condensable gases (SI Appendix, section 2). It follows thatτν≈κν×WVP=κν×f(T),[2]so that, once the atmosphere is optically thick, the temperature of the emission level, where τν∼1, becomes independent of surface temperature. Fig. 3 confirms that, at optically thick frequencies, the emission to space varies little as the surface warms from 240 K to 320 K. Earth’s ability to shed more heat with warming therefore crucially depends on its spectral window regions.

A Simple Model of Longwave Feedback

The importance of window regions for Earth’s climate feedback allows us to formulate a simple model that explains why OLR is approximately linear with temperature. As long as the change in thermal emission with surface temperature outside window regions is small, we show that the feedback equals (SI Appendix, section 3)dOLRdTs=4σTs3×T¯,[3]which is simply the surface’s blackbody feedback, 4σTs3, times the average transmission between the surface and space, T¯. The average transmission is a spectral mean weighted by the derivative of the Planck function Bν,T¯=∫0∞TνdBνdT|Tsdν∫0∞dBνdT|Tsdν.[4]

Intuitively, T¯ measures how much of the increase in surface emission with warming escapes to space. If the entire spectrum is optically thin to thermal radiation, then T¯=1, whereas if the entire spectrum is opaque, then T¯=0.

Our model states that the feedback of a moist atmosphere is closely tied to the surface, which seems to contradict studies that attribute changes in OLR to changes in atmospheric lapse rate and water vapor as well as the Planck feedback (21, 22). Fig. 2, Right Inset shows why our model is valid despite such expectations: If we split the Planck feedback into its contributions from atmospheric and surface warming, we find that the atmospheric contribution largely cancels the lapse rate and water vapor feedbacks. This cancellation implies that the net feedback is dominated by the surface Planck feedback, which in turn is described by Eq. 3. To understand how Earth’s OLR changes with warming, it is therefore critical to understand how T¯ depends on temperature.

Fig. 4, Left shows the transmission T¯ in our calculations for Earth, as well as its equivalent for hypothetical colder planets in which CO2 and NH3 are condensable gases with unlimited surface reservoirs. The transmission for a water vapor greenhouse atmosphere is equal to unity below about 190 K. In this limit the atmosphere contains so little water vapor that the entire spectrum is transparent to thermal radiation, even inside the strong absorption bands of the H₂O molecule. Because the atmospheric water vapor content increases with warming, transmission decreases with temperature and it becomes negligible once the entire spectrum is opaque, above 320 K (Fig. 4).

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Fig. 4.

Our simple model reproduces Earth’s climate feedback, as well as the feedback in atmospheres dominated by other condensable gases. (Left) Transmission between surface and space, for three atmospheres dominated by different condensable species. (Right) Our model, 4σTs3×T¯ (gray lines), compared with 4σTs3 (black lines) and the feedback from our full radiative calculations (colored lines).

Fig. 4, Top Right compares our simple model, 4σTs3×T¯, to the actual feedback calculated from our line-by-line calculations. Our simple model reproduces both the shape and the magnitude of the feedback in a H₂O-dominated atmosphere. Fig. 4 therefore offers a simple explanation for why Earth’s OLR is approximately linear in surface temperature: The rapid increase of surface emission via 4σTs3 is strongly counteracted by the closing of spectral window regions due to the increase in atmospheric water vapor. Even though the cancellation is not exact, it always gives rise to a wide range of temperatures over which the feedback remains essentially constant (Fig. 4).

We gain additional insight by considering why T¯ changes with temperature. There are two mechanisms that dominate the greenhouse effect of water vapor. The first one is line absorption, which arises from the interaction of an isolated H₂O molecule with infrared radiation, and the second one is self-continuum absorption, which arises from collisions between H₂O molecules. Line absorption dominates at colder temperatures, whereas H₂O collisions increase with the square of water vapor concentration so that the continuum becomes significant at high temperatures.

Fig. 5, Left illustrates how these two absorption mechanisms combine to shape the transmission T¯. First, at cold temperatures the entire spectrum is optically thin and hence T¯ is equal to one. We denote the onset temperature at which line absorption starts to close off parts of the spectrum as T0, while T∞ denotes the runaway temperature at which line absorption closes off the entire spectrum. In between these two temperatures the transmission T¯ can be approximated as linear. The linearity of T¯ arises because molecules like H₂O and CO2 have absorption bands whose strength decreases exponentially away from band centers whereas the atmosphere’s water vapor path increases exponentially (SI Appendix, section 5). In contrast to line absorption, continuum absorption increases much more rapidly with temperature and thus acts to cut off T¯ at high temperatures (Fig. 5).

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Fig. 5.

Schematic for how the approximate linearity of OLR arises. The increasing surface Planck feedback (black, Right) is counteracted by the decreasing transmission due to the closing of spectral windows (blue, Left). The purple arrows (Right) indicate the range over which the feedback is approximately constant (within ±10%).

Fig. 5, Right illustrates how the product of 4σTs3 and T¯ gives rise to a broad range of temperatures over which the feedback can be approximated as constant. For example, if T¯ was exactly linear between T0 and T∞, the feedback would be constant to within ±10% over a temperature range of about 0.27×T∞ (SI Appendix, section 6). For a water vapor atmosphere like Earth’s T∞∼350 K (SI Appendix, Fig. S7) so the feedback would be constant over a range of ∼95 K. In reality this range is smaller because T¯ is not exactly linear and because continuum absorption rapidly closes off spectral windows at high temperatures (Fig. 3).

We can now understand why less than 100% relative humidity and the addition of CO2 extend the linear range of OLR to even higher temperatures (Fig. 2). Low relative humidity reduces the atmosphere’s water vapor path and delays the closing of spectral windows, so T¯ effectively shifts toward higher temperatures. The addition of CO2 requires two modifications to our simple model: CO2 closes off parts of the spectral window, so T¯ decreases and the net feedback becomes smaller at low temperatures (Fig. 2). At the same time, inside the CO2 absorption bands the atmosphere’s emission temperature near τν∼1 is not constant with surface temperature anymore. To see why, we can approximate CO2 as a uniformly mixed greenhouse gas with constant absorption cross-section,τν≈∫κvqCO2 dp/g≈κvqCO2p/g=κv×f(p).[5]

Thermal emission from the CO2 bands therefore originates from a roughly constant pressure level. Surface warming raises the atmosphere’s temperature at a fixed pressure so, unlike for a pure-H2O atmosphere, CO2’s absorption bands allow the atmosphere to emit more with surface warming. This effect can be expressed as an addition to the net feedback dOLR/dTs=4σTs3T¯+(CO2 term). Because the CO2 term is always positive, it helps extends the linear range of OLR toward somewhat higher temperatures before eventually vanishing at around 350 K (Fig. 2), at which point H2O becomes opaque even inside the CO2 bands. However, except for temperatures much higher than 300 K for which the linearity of OLR already breaks down, the correction from the CO2 term is modest and Earth’s feedback is dominated by the influence of water vapor rather than that of CO2.

Application to Earth and Other Planets

Our results lend increased confidence to the robustness of clear-sky feedbacks in global climate models (GCMs). It is well known that clear-sky feedbacks roughly double Earth’s climate sensitivity (23) and that the magnitude of these feedbacks is highly consistent across models (21). This agreement is not obvious, however, given that GCMs exhibit various temperature and relative humidity biases and differ with respect to satellite data as well as other GCMs (24). Because a linear OLR entails a constant feedback, our results imply that the magnitude of the net clear-sky longwave feedback in GCMs is insensitive to moderate biases (SI Appendix, Fig. S5). Our results thus underline that even GCMs with biased mean states can adequately capture the clear-sky feedback of present-day Earth. This logic, however, no longer holds under hot conditions. Above surface temperatures of ∼300 K the longwave feedback rapidly diminishes and linearity breaks down (Fig. 2). Such conditions would have been widespread during past warm climates such as the Eocene hothouse and could occur regionally under strong global warming. Model biases under such conditions will amplify, making it difficult to accurately simulate past climates or to constrain the worst-case outcomes of future warming.

Similarly, a number of recent studies have pointed out the potential importance of nonlinearities in Earth’s radiative balance as well as the importance of regionally varying climate feedbacks for global warming (25⇓⇓⇓–29). For Earth’s present-day climate, our results underline that cold and warm regions contribute roughly equally to changes in clear-sky OLR (Figs. 1 and 2). Strong nonlinearities and regional differences therefore arise from processes not included in our simple model, such as clouds, changes in surface albedo, or changes in relative humidity.

The physics in our model are general and capture the feedback in atmospheres dominated by other condensable gases, such as hypothetical cold atmospheres in which CO2 or NH3 can condense (Fig. 4). The same processes that render Earth’s OLR essentially linear over a wide range of temperatures thus also shape the climates of these worlds. We note that present-day Titan, where CH4 is a condensing gas, is too cold for its greenhouse effect to be dominated by CH4. Instead its greenhouse effect is largely due to collision-induced absorption between N2, H2, and CH4 plus absorption by photochemical hazes, with only a minor contribution from the CH4−CH4 continuum (30). Nevertheless, our results suggest that extrasolar planets with exotic condensable greenhouse gases, such as hot rocky planets covered with lava oceans and with atmospheres made of outgassed silicate–vapor species (31, 32), would have radiative balances surprisingly similar to Earth’s. Future space telescopes could thus study these worlds as hot analogs of Earth’s H2O-dominated climate.