Earth’s outgoing longwave radiation linear due to H2O greenhouse effect
See allHide authors and affiliations
Edited by Dennis L. Hartmann, University of Washington, Seattle, WA, and approved August 23, 2018 (received for review June 8, 2018)

Significance
Earth’s climate is set by a balance between incoming solar and outgoing infrared radiation. The physical processes that influence this balance are complex and nonlinear, yet models and satellite measurements counterintuitively show that Earth’s infrared radiation is simply a linear function of surface temperature. Here we explain why: Linearity is due to the cancellation of two nonlinear processes and always arises in an atmosphere dominated by a condensable greenhouse gas. Our work explains a fundamental property of Earth’s climate and has implications for climate change as well as the climates of extrasolar planets with exotic greenhouse gases.
Abstract
Satellite measurements and radiative calculations show that Earth’s outgoing longwave radiation (OLR) is an essentially linear function of surface temperature over a wide range of temperatures (≳60 K). Linearity implies that radiative forcing has the same impact in warmer as in colder climates and is thus of fundamental importance for understanding past and future climate change. Although the evidence for a nearly linear relation was first pointed out more than 50 y ago, it is still unclear why this relation is valid and when it breaks down. Here we present a simple semianalytical model that explains Earth’s linear OLR as an emergent property of an atmosphere whose greenhouse effect is dominated by a condensable gas. Linearity arises from a competition between the surface’s increasing thermal emission and the narrowing of spectral window regions with warming and breaks down at high temperatures once continuum absorption cuts off spectral windows. Our model provides a way of understanding the longwave contribution to Earth’s climate sensitivity and suggests that extrasolar planets with other condensable greenhouse gases could have climate dynamics similar to Earth’s.
Earth’s climate is set by a balance between incoming solar and outgoing longwave radiation (OLR). Changes in
The processes that determine Earth’s OLR are inherently nonlinear, so a linear approximation might seem valid only for small perturbations in temperature. Nevertheless, multiple lines of evidence going back to the 1950s indicate that a linear relation is justified over a surprisingly wide range of temperatures. Early ground-based and satellite measurements of radiative fluxes suggested that OLR is linear in temperature over a range of more than 50 K (5⇓–7). Similarly, pioneering radiative transfer calculations around the same time found that OLR is linear over a range of about 70 K (8).
Although these results date back more than half a century, it is unclear why linearity holds across such a wide range of temperatures. Early radiative calculations pointed out that Earth’s OLR has to increase less rapidly with temperature than suggested by the Stefan–Boltzmann law,
Compounding the puzzle further, any linear relation has to break down eventually. At high temperatures Earth’s OLR approaches the runaway greenhouse limit, in which OLR becomes independent of surface temperature (10, 12⇓⇓⇓–16). Similarly, at sufficiently cold temperatures the water vapor feedback has to become negligible and Earth’s OLR should approximately follow the Stefan–Boltzmann law.
The linear relation between OLR and surface temperature is thus a fundamental yet poorly understood feature of Earth’s climate, with a number of consequences: Linearity implies that Earth’s longwave climate feedback,
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
Earth’s OLR strongly deviates from the thermal emission of a blackbody,
We can reproduce the main features of this relation by considering an idealized model of a single atmospheric column with
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,
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
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
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.
(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 (
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
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 that
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)
Intuitively,
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
Fig. 4, Left shows the transmission
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,
Fig. 4, Top Right compares our simple model,
We gain additional insight by considering why
Fig. 5, Left illustrates how these two absorption mechanisms combine to shape the transmission
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
Fig. 5, Right illustrates how the product of
We can now understand why less than 100% relative humidity and the addition of
Thermal emission from the
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
Materials and Methods
Datasets.
We use monthly mean clear-sky OLR from the Clouds and Earth’s Radiant Energy Systems-Energy Balanced and Filled (CERES-EBAF, v. 4) satellite data product (33), and near-surface air temperatures from the National Centers for Environmental Prediction (NCEP) reanalysis (34). The height of the air temperatures corresponds to
Line-by-Line Code.
We use our own line-by-line radiation code, PyRads. PyRads is written almost entirely in Python and is freely available for research and teaching. The only exception is the continuum model, for which we use the Mlawer–Tobin–Clough–Kneizys–Davies (MTCKD) model (see below). We validate PyRads against the line-by-line calculations in ref. 16 (SI Appendix, section 1).
PyRads computes opacities on a large grid in spectral and pressure/temperature space and then integrates the longwave radiative transfer equations over this grid. Many line-by-line codes use additional techniques to reduce the numerical cost of resolving each individual absorption line. However, modern computers have sufficiently large memory that our approach is feasible. For example, it takes about 1 min to compute the OLR for a single absorbing gas on a 2017 MacBook Pro. Opacities are calculated based on the PyTran script, which is developed by Raymond Pierrehumbert and available online at geosci.uchicago.edu/∼rtp1/PrinciplesPlanetaryClimate/Courseware/PlanetaryClimateCourseware/ChapterScripts/Chapter4Scripts/Chapter4Scripts.html.
Atmospheric Structure and Relative Humidity.
We use the formulation of the moist adiabat from ref. 35, which is valid in both the dilute (dry atmosphere) and the nondilute (steam atmosphere) limits (35). We cap the troposphere with an isothermal stratosphere, where the amount of water vapor in the stratosphere is equal to its value at the tropopause. The stratosphere is set to be colder than the coldest surface temperature we consider (for
Our vertical resolution is 60 grid points, evenly distributed in log space between
Spectral Database and Resolution.
We use the HITRAN 2016 database (18), with a Lorenz line profile assumed for all lines. Because we do not use a Voigt line shape, we do not resolve the cores of absorption lines. However, our validation shows that we reproduce OLR to within the same degree of accuracy as achieved by other line-by-line radiation codes (SI Appendix, section 1). To be consistent with the definition of the continuum in the MTCKD model (36), we truncate lines 25 cm−1 away from the line center. For
Continuum Absorption.
We compute
Data Availability.
The CERES and NCEP datasets are publicly available at ceres.larc.nasa.gov and esrl.noaa.gov/psd/data.
Acknowledgments
We thank Nick Lutsko for discussions and comments and two anonymous reviewers for insightful feedback. D.D.B.K. was supported by a James McDonnell Foundation Postdoctoral fellowship. T.W.C. was supported by NSF Grant AGS-1623218, “Using a hierarchy of models to constrain the temperature-dependence of climate sensitivity.”
Footnotes
- ↵1To whom correspondence should be addressed. Email: dkoll{at}mit.edu.
Author contributions: D.D.B.K. and T.W.C. designed research; D.D.B.K. performed research; D.D.B.K. and T.W.C. analyzed data; and D.D.B.K. and T.W.C. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Data deposition: The PyRads radiation code used in this study has been deposited on GitHub and is available at https://github.com/ddbkoll/PyRADS.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1809868115/-/DCSupplemental.
Published under the PNAS license.
References
- ↵
- ↵
- ↵
- ↵
- ↵
- Budyko MI
- ↵
- Held IM,
- Suarez MJ
- ↵
- Budyko MI
- ↵
- ↵
- ↵
- Pierrehumbert RT
- ↵
- Huang Yi,
- Shahabadi MB
- ↵
- Simpson GC
- ↵
- Komabayasi M
- ↵
- ↵
- ↵
- Goldblatt C,
- Robinson TD,
- Zahnle KJ,
- Crisp D
- ↵
- Kiehl JT,
- Trenberth KE
- ↵
- Gordon IE, et al.
- ↵
- Alexander R,
- Aumann HH,
- Manning EM
- ↵
- Ingram W
- ↵
- Soden BJ,
- Held IM
- ↵
- Soden BJ, et al.
- ↵
- ↵
- ↵
- Armour KC,
- Bitz CM,
- Roe GH
- ↵
- Feldl N,
- Roe GH
- ↵
- Rose BEJ,
- Armour KC,
- Battisti DS,
- Feldl N,
- Koll DDB
- ↵
- ↵Knutti R, Maria A. Rugenstein A (2015) Feedbacks, climate sensitivity and the limits of linear models. Philos Trans R Soc A 373:20150146.
- ↵
- McKay CP,
- Pollack JB,
- Courtin R
- ↵
- Schaefer L,
- Bruce F
- ↵
- Miguel Y,
- Kaltenegger L,
- Fegley B,
- Schaefer L
- ↵
- Wielicki BA, et al.
- ↵
- Kalnay E, et al.
- ↵
- ↵
Citation Manager Formats
Article Classifications
- Physical Sciences
- Earth, Atmospheric, and Planetary Sciences