Can Increased Atmospheric CO 2 Levels Trigger a Runaway Greenhouse?

Abstract

Recent one-dimensional (globally averaged) climate model calculations by Goldblatt et al. (2013) suggest that increased atmospheric CO₂ could conceivably trigger a runaway greenhouse on present Earth if CO₂ concentrations were approximately 100 times higher than they are today. The new prediction runs contrary to previous calculations by Kasting and Ackerman (1986), which indicated that CO₂ increases could not trigger a runaway, even at Venus-like CO₂ concentrations. Goldblatt et al. argued that this different behavior is a consequence of updated absorption coefficients for H₂O that make a runaway more likely. Here, we use a 1-D climate model with similar, up-to-date absorption coefficients, but employ a different methodology, to show that the older result is probably still valid, although our model nearly runs away at ∼12 preindustrial atmospheric levels of CO₂ when we use the most alarmist assumptions possible. However, we argue that Earth's real climate is probably stable given more realistic assumptions, although 3-D climate models will be required to verify this result. Potential CO₂ increases from fossil fuel burning are somewhat smaller than this, 10-fold or less, but such increases could still cause sufficient warming to make much of the planet uninhabitable by humans. Key Words: Planetary atmospheres—Habitability—Carbon dioxide—Environment. Astrobiology 14, 714–731.

1. Introduction

In his book Storms of My Grandchildren (Hansen, 2010), veteran climate scientist James Hansen speculated that humankind's continued burning of fossil fuels could conceivably trigger a runaway greenhouse, an atmospheric catastrophe in which surface temperatures become high enough to completely vaporize the oceans. On a water-rich planet like Earth, this phenomenon would occur if the surface temperature were to exceed the critical temperature for water (647 K for pure H₂O). Alternatively, some authors (Rennó, 1997; Goldblatt et al., 2013) prefer to define the runaway greenhouse as a situation in which the absorbed solar flux exceeds a limit on the outgoing IR flux. These two definitions are not equivalent, because an atmosphere that “runs away” may stop short of boiling off the oceans if a stable, high surface temperature solution exists below the critical temperature. That happens in some of the calculations described here. Runaway greenhouse atmospheres have finite lifetimes, because once the upper atmosphere becomes water-rich, the water can be lost in a few tens of millions of years by photodissociation, followed by escape of hydrogen to space (Ingersoll, 1969; Kasting, 1988). The water loss lifetime is longer than the lifetime of a fossil-fuel CO₂ pulse (Walker and Kasting, 1992), however, so water loss is not a major issue for the case of anthropogenic CO₂ increases.

Disturbingly, Hansen's conjecture has received support from recent climate calculations (Goldblatt et al., 2013), although the CO₂ concentration at which the runaway is predicted to occur, 30,000 ppmv, or ∼100 times the preindustrial atmospheric level (PAL), is a factor of 10 higher than that which might conceivably result from the burning of fossil fuels (not including the possible contribution from seafloor methane hydrates). This result contradicts previous predictions (Kasting and Ackerman, 1986, hereafter KA86) that surface liquid water would remain stable for CO₂ increases up to 100 bar. A second recent calculation of the effects of very high CO₂ levels on surface temperature also produced stable conditions for liquid water (Wordsworth and Pierrehumbert, 2013). However, this model used H₂O absorption coefficients from HITRAN, whereas both Goldblatt et al. and the present paper use coefficients from the more complete HITEMP database (see Section 2), so their calculation is not directly comparable to the other two. Wordsworth and Pierrehumbert also argued that high CO₂ concentrations should produce cold tropopause temperatures and thereby limit water loss from CO₂-rich atmospheres. But if the CO₂ additions are sufficient to produce near-runaway conditions, as in both the Goldblatt et al. model and the one outlined here, this point does not apply because both the lower atmosphere and the tropopause become water-dominated (Wordsworth and Pierrehumbert, 2013, Eq. 16), allowing water to be lost anyway. And, as pointed out above, it is surface temperature itself, not water loss, that is the chief cause for concern.

The old KA86 calculations actually deserve to be revisited for two reasons. First, Goldblatt et al. (2013), mentioned above, have shown that using the new HITEMP database for H₂O absorption (Rothman et al., 2010), as compared to the older HITRAN database (Rothman et al., 1998), results in much greater absorption of incoming solar radiation, thereby lowering the planet's albedo and destabilizing the climate. Second, the KA86 methodology assumed that the surface relative humidity (RH) remained constant at 0.8. As we demonstrate below, and as other authors have predicted (Ramanathan, 1981; Larson and Hartmann, 2003; Goldblatt et al., 2013), surface RH should increase as the surface temperature increases (as described in Section 2). This causes the water vapor feedback to be stronger in our current model than it was in KA86, which makes our model significantly more susceptible to the possibility of a runaway greenhouse. This feedback needs to be included to generate a worst-case scenario, as we are attempting to do here.

2. Methods

2.1. Radiative transfer model

The climate calculations described here were performed with a 1-D (horizontally averaged) global climate model first developed by Kasting et al. (1984) and described more fully by Pavlov et al. (2000) and Haqq-Misra et al. (2008). This model has recently been substantially updated (Kopparapu et al., 2013a; Ramirez et al., 2014) and has been updated yet again since these papers appeared (see below). The current version of the model divides the atmosphere into 100 unevenly spaced layers in log pressure extending from the ground to a specified low pressure at the top (10⁻⁴ bar for this paper). Radiative equilibrium is assumed for each layer in the stratosphere. At lower levels, if the radiative lapse rate within a layer exceeds the moist adiabatic lapse rate, then a convective adjustment is performed (Manabe and Wetherald, 1967). The model relaxes to a moist H₂O adiabat at higher temperatures or to a moist CO₂ adiabat when it is cold enough for CO₂ to condense (Kasting, 1991), defining a convective troposphere near the surface.

The planet is assumed to be flat, six different solar zenith angles were used, as in the work of Goldblatt et al. (2013), and the results were averaged by using Gaussian quadrature. A globally averaged incident solar flux value of 342 W/m² was assumed, which is consistent with a solar constant of 4 times that value, or 1368 W/m². In both the solar and thermal-IR parts of the radiation code, the diffuse flux was calculated by using a two-stream approximation (Toon et al., 1989). As in other recent papers from our group (Kopparapu et al., 2013a; Ramirez et al., 2014), we used KSPECTRUM (Wordsworth et al., 2010), a line-by-line radiative transfer program, to generate absorption spectra for CO₂ and H₂O. We then wrote another program that converts these spectra into correlated-k coefficients for use in our climate model (Kopparapu et al., 2013a; Ramirez et al., 2014).

Our correlated-k coefficients parameterize gaseous absorption across 38 solar spectral intervals ranging from 0.2 to 4.5 microns (∼2000–50,000 cm⁻¹) and 55 thermal-IR intervals extending from 0 to 15,000 cm⁻¹(>∼0.66 microns). Two different sets of CO₂ and H₂O coefficients are used in our calculations. At lower temperatures (≤310 K), foreign broadening is best represented by terrestrial air because both H₂O and CO₂ are minor constituents and the mole fractions of the remaining constituents (N₂, O₂, Ar, CH₄, N₂O, etc.) do not change appreciably. In this regime, we derived 16-term, mixed CO₂-H₂O coefficients using a grid of four temperatures (200, 250, 300, 350 K), six pressures (1×10⁻⁴ to 10 bar), four CO₂ mixing ratios (1×10⁻⁴ to 1×10⁻¹), and eight H₂O mixing ratios (1×10⁻⁸ to 1×10⁻¹). Mixed coefficients are much more time-consuming to calculate than are separate coefficients because they must be computed as a function of both CO₂ and H₂O mixing ratio, in addition to pressure and temperature. These mixed coefficients compute foreign broadening accurately because they are created with air as the background gas.

At higher temperatures, though, foreign broadening is poorly represented by terrestrial air because the mole fractions of CO₂ and H₂O increase, while those of other species decrease. Thus, simply scaling the foreign broadening coefficients of terrestrial air to higher temperatures is inaccurate (Brown et al., 2007; Wordsworth and Pierrehumbert, 2013). In this high-temperature/high pCO₂ regime, we use separate 8-term coefficients for CO₂ and H₂O, which incorporate self-broadening in place of foreign broadening. These coefficients were derived in the same way as described by Kopparapu et al. (2013) and Ramirez et al. (2014). [They are not the same coefficients, though; rather, they are slightly stronger because we corrected an error in how the pressure broadening correction was performed. For hot, water-dominated atmospheres, this reduced the calculated asymptotic outgoing IR flux by about 4%, which revises the inner edge of the habitable zone in that paper from 0.99 to 1.01 AU. However, this result would imply that Earth is uninhabitable, which stems from the assumption of fully saturated atmospheres. A better estimate of 0.95 AU for the inner edge comes from the 3-D model of Leconte et al. (2013).] A similar self-broadening scheme was used by Wordsworth and Pierrehumbert (2013) to minimize errors in the high-CO₂/H₂O regime. We consider this approach to be justified because we are hoping to demonstrate that a runaway greenhouse is impossible to trigger on Earth even when using the most alarmist assumptions. At solar wavelengths, our H₂O coefficients are derived from the HITEMP 2010 database for pressures 0.1 bar and higher and from HITRAN 2008 at lower pressures. Our IR H₂O coefficients are derived by using HITRAN 2008 instead of HITEMP 2010 because using HITRAN is faster and because these water-rich atmospheres become opaque at temperatures well below those at which HITEMP starts to deviate appreciably from HITRAN (Goldblatt et al., 2013).

Depending on which set of CO₂-H₂O coefficients are used, we adjusted the surface albedo, A _S, in our (cloud-free) model so that it reproduced the observed surface temperature of modern Earth, T _S=288 K and modern atmospheric composition. As has been shown in previous climate calculations by our group (Kasting and Ackerman, 1986; Pavlov et al., 2000; Haqq-Misra et al., 2008), normalizing A _S in this manner simulates the net effect of cloud forcing without including clouds explicitly and ensures that the computed surface temperature does not drift toward warmer or cooler values. When using separate CO₂-H₂O coefficients, the required value of A _S is 0.315. These coefficients require a high surface albedo to offset the increased greenhouse effect caused by assuming self-broadening by CO₂ and H₂O in place of foreign broadening by air. For the mixed CO₂-H₂O coefficients, the required value of A _S decreases to 0.25, which is the same value obtained by Goldblatt et al. (2013) for modern Earth. This suggests that our computed greenhouse forcing and use of the mixed coefficients is similar to that found by their model.

Our calculations assume (a) that the stratosphere achieves radiative equilibrium—that is, the net emitted IR flux is equal to the net absorbed solar flux in each layer—and (b) that the surface temperature converges to a steady-state value. Rayleigh scattering is implemented as described by Kopparapu et al. (2013b). Rayleigh scattering is a key component in our model, especially at high CO₂ concentrations, because the scattering cross section for CO₂ is ∼2.5 times that of air (KA86).

2.2. Upper tropospheric relative humidity

A key issue that affects the response of Earth's climate to both CO₂ and solar luminosity increases is how RH should vary with surface temperature. The average RH near Earth's surface today is approximately 80% (Manabe and Wetherald, 1967). This value is determined by the surface heat budget, as described below. Above this height, RH is observed to decrease more or less linearly with pressure up to the upper troposphere (Manabe and Wetherald, 1967). The physics here is complex: Because the tropics cover a significant fraction of Earth's surface area, and because they receive an even larger fraction of Earth's solar insolation, they tend to dominate Earth's climate. Moist convection in the tropics produced deep cumulus convection cells, with rapidly rising air, over ∼10% of the surface. This air descends more slowly over the other 90% of the surface to complete the cycle. The air within the cumulus cells remains saturated as it rises, but it loses most of its moisture by cloud formation and precipitation. Once it spreads out at the tops of these cells and begins its descent, it warms adiabatically and becomes undersaturated. On top of this, the large-scale Hadley circulation produces downwelling air in the subtropics that is also strongly undersaturated. Pierrehumbert (1995) referred to these regions as “radiator fins” and argued that they are the main reason that a runaway greenhouse is avoided in the tropics today.

The physics of these processes is impossible to capture realistically in 1-D climate models; however, it can be simulated based on observations. In their classic paper, Manabe and Wetherald (1967) derived a fixed average tropospheric RH profile using empirical measurements taken for northern hemisphere summer and winter (Telegadas and London, 1954; Murgatroyd, 1960; Mastenbrook, 1963). We parameterized this tropospheric RH in the same way as in KA86: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}RH = R_ { { \rm surf } } \left[ \frac { P / P_ { \rm s } - 0.02 } { 0.98 } \right] ^ \Omega \tag { 1 } \end{align*} \end{document}

Here, R _surf (∼0.8) is the surface RH, RH is the RH at some height above the surface, P is the pressure at a given height, and P _s is the surface pressure. Ω=1 produces a standard Manabe-Wetherald RH distribution (Manabe and Wetherald, 1967). Other values for Ω are discussed below.

The RH values calculated with Eq. 1 with Ω=1 approach zero near the top of the troposphere and are thus not consistent with the higher values thought to be associated with much warmer atmospheres. This problem has been studied in the context of Venus' atmosphere, which is thought to have lost its water to space following photodissociation (Ingersoll, 1969; Donahue et al., 1982; Donahue and Hodges, 1992; Hamano et al., 2013). This implies that upper tropospheric RH must have been high for efficient photodissociation to have proceeded and for the diffusion limit on escape to have been overcome (Hunten, 1973; Walker, 1977). Thus, we search for a way to connect the undersaturated troposphere observed on modern Earth with the more fully saturated troposphere needed to explain water loss on Venus and very warm and wet atmospheres in general.

Recognizing this problem, KA86 modified the Manabe-Wetherald RH distribution by incorporating a suggestion by Cess (1976) and Coakley (1977). Cess had observed that northern hemisphere RH is slightly higher in the summer than in the winter. Coakley modified Eq. 1 with the following: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\Omega = 1 - 0.03 ( T_{\rm s} - 288 \ { \rm K} ) \tag{2}\end{align*} \end{document}

Here, T _s is the surface temperature. The latter expression yielded Ω=1.0 in the summer and Ω=1.4 in the winter. Because these relationships are both empirical, Eqs. 1 and 2 are at best only valid for conditions close to those on modern Earth. KA86 further modified this parameterization by replacing Eq. 2 with one based on the H₂O saturation mixing ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\Omega = 1 - \frac { q_ { \rm o } - q_0^ { 288 } } { 0.1 - q_0^ { 288 } } \tag { 3 } \end{align*} \end{document}

Here, q _o is the saturation H₂O mixing ratio at the surface, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q_0^{288} ( = 0.0166 )$$ \end{document} is the surface saturation mixing ratio at 288 K. When q ₀=0.0166, Ω=1, and Eq. 1 is the standard Manabe-Wetherald RH profile. At T _s=321 K, q ₀=0.1, so Ω=0, and R=R _surf throughout the troposphere. This, in turn, allows the stratosphere to become wet as the surface warms. If it did not, then condensation of H₂O would create atmospheric pressure imbalances that would likely be unphysical, as pointed out by Goldblatt et al. (2013).

2.3. Climate modeling procedures

In the present study, we performed two different types of climate model calculations, which we term “forward” and “inverse.” In forward calculations, we fix the solar flux and atmospheric CO₂ concentration and time-step the model until the surface temperature stabilizes at a fixed value. Convergence is checked by calculating the net radiative flux divergence in the stratosphere and making sure that it is below one part in 10³. For the inverse calculations, we initialize temperatures throughout the atmosphere at 200 K for most of our calculations, or to 215 K when comparing with the work of Goldblatt et al. (2013). However, the choice of stratospheric temperature had little effect on our results. Then, we specify a surface temperature and integrate a moist adiabat upward until it intersects that value. This produces a model with a moist adiabatic troposphere and an isothermal stratosphere, similar to the assumptions made by Goldblatt et al. (2013). We conducted forward calculations at lower surface temperatures but switched to inverse calculations once water became a major atmospheric constituent (above ∼340 K and 12 PAL CO₂).

2.4. Surface relative humidity

Our surface RH parameterization is similar to that used by Kasting et al. (2013). It is based on the requirement that the net convective (latent+sensible) heat flux, F _net, must balance the net absorbed radiative (solar+thermal-IR) flux. This is not a new idea; rather, it is a simple physical principle that has been recognized by many other authors (Ramanathan, 1981; Boer, 1993; Trenberth, 1998; Allen and Ingram, 2002; Held and Soden, 2006; Soden and Held, 2006; Kasting et al., 2013). We start from the following commonly used expression (Garratt, 1992) for the latent heat flux, F _L, (in units of W/m²): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}F_L = LC_{\rm D} u \rho ( q_{\rm o} - q_1 ) \tag{4}\end{align*} \end{document}

Here, L is the latent heat of vaporization of water (2.5×10⁶ J/kg), C _D is the drag coefficient at layer 1 just above the surface, u is the mean horizontal wind speed near the surface in meters per second, ρ is the atmospheric sea level mass density in kilograms per cubic meter, q _o is the surface H₂O saturation mixing ratio, and q ₁ is the H₂O mixing ratio at layer 1. If layer 1 is close to the surface, then we can replace q ₁ with R _surf·q ₀. Defining the Bowen ratio, B, as the ratio of the sensible heat flux to the latent heat flux, we can then write \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}F_{{ \rm net}} = LC_{\rm D} u \rho q_{\rm o} ( 1 - R_{{ \rm surf}} ) ( 1 + B ) \tag{5}\end{align*} \end{document}

The drag coefficient (C _D) and mean surface wind speed (u) are both unknown and cannot be self-consistently calculated in our 1-D model. Thus, we make the simplifying assumption that the product C _D·u remains constant at all surface temperatures. This assumption is reasonable because C _D does not vary greatly over water surfaces (Arya, 2001) and because most of Earth's surface is ocean. The mean wind speed u may indeed vary with surface temperature, but not as fast as other terms in Eq. 5, as discussed further below. Evaluating Eq. 5 at the present mean surface temperature of 288 K allows us to solve for the product C _D·u: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}C_ { \rm D } \times u = \frac { F_ { { \rm net } } ^ { 288 } } { \rho_ { 288 } Lq_ { \rm o } ^ { 288 } \left( 1 - R_ { { \rm surf } } ^ { 288 } \right) \left( 1 + B_ { 288 } \right) } \tag { 6 } \end{align*} \end{document}

The relevant values for the quantities in Eq. 6 at 288 K are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$F_{{ \rm net}}^{288}$$ \end{document} =97 W/m², ρ ₂₈₈=1.225 kg/m³, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q_{\rm o}^{288}$$ \end{document} =0.0166, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$R_{{ \rm surf}}^{288}$$ \end{document} =0.8, and B ₂₈₈=0.59. Equation 6 can then be substituted into Eq. 5 to give \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}R_ { { \rm surf } } = 1 - \left( 1 - R_ { { \rm surf } } ^ { 288 } \right) \frac { F_ { { \rm net } } } { F_ { { \rm net } } ^ { 288 } } \times \frac { \rho_ { 288 } } { \rho } \times \frac { q_6^ { 288 } } { q_ { \rm o } } \times \frac { 1 + B_ { 288 } } { 1 + B } \tag { 7 } \end{align*} \end{document}

Here, F _net is the net radiative flux at the surface, ρ is the sea-level atmospheric mass density, and B is the Bowen ratio (the ratio of sensible to latent heat flux). Calculated values for the quantities in Eq. 7 at 288 K are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$F_{{ \rm net}}^{288} = 97 \ { \rm W} / { \rm m}^2$$ \end{document} , ρ ₂₈₈=1.225 kg/m³, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q_{\rm o}^{288} = 0.0166$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$R_{\rm surf}^{288} = 0.8$$ \end{document} , and B ₂₈₈=0.59.

A 3-D climate model would contain additional parameters (spatially variable wind speed and surface roughness) and thus might not produce the same result as Eq. 7. These complications are bypassed here by normalizing all parameters to present global average values. In reality, the mean surface wind speed, u, may decrease significantly at high CO₂ levels because the equator-to-pole temperature gradient is expected to decrease as the atmosphere becomes more radiatively opaque (e.g., Hay and Floegel, 2012). But neglecting this effect is acceptable for our purposes because a change of u in this sense would cause the surface RH to be lower than assumed here, thereby reducing the greenhouse effect and making the climate less likely to “run away.” Furthermore, the error is arguably small, as the second term in Eq. 7 is dominated by the q ₀ term in the denominator, which increases exponentially with temperature following the Clausius-Clapeyron equation. We also performed a sensitivity study by decreasing u to 1/10 of its value at 288 K and found this had a negligible effect on our results. Thus, we expect that 3-D climate models, when tested under high surface temperature conditions, should produce more or less the same result.

3. Results

3.1. Flux comparison with results of Goldblatt et al. for a pure-H₂O atmosphere

We began by performing inverse calculations to compare our model to that of Goldblatt et al. (2013) for the case of a pure H₂O atmosphere. The results are shown in Fig. 1. In Goldblatt et al., the reported net absorbed solar flux, F _S, for a pure H₂O atmosphere at surface temperatures >500 K was 294 W/m². The reported net outgoing IR flux, F _IR, for these warm atmospheres was 282 W/m², which is 12 W/m² lower than the absorbed flux. This value decreases to ∼280 W/m² as T _S rises to 1500 K. Because F _S>F _IR, a transient perturbation such as a CO₂ increase could, in principle, trigger a runaway greenhouse.

FIG. 1.

Top-of-atmosphere fluxes for a pure H₂O atmosphere with a surface albedo of 0.25, comparing our model (black) versus that of Goldblatt et al. (2013) (blue dashed). Steady-state climates are found when the net outgoing flux is zero; stable steady states exist when the slope of the net outgoing flux is positive. The bottom three panels show the differences between our calculated fluxes and those of Goldblatt et al. (Color graphics available online at www.liebertonline.com/ast)

A problem with this analysis is that the fluxes shown in Fig. 2 of Goldblatt et al. (2013) (repeated in our Fig. 1) do not match the values cited in their text; the asymptotic (high-temperature) value of F _S in their Fig. 2a appears to be ∼286 W/m², not 294 W/m², independent of the assumed surface albedo. This can be checked by recalculating F _S by using the data shown in the insert to their Fig. 2a. In a globally averaged climate model, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$F_ { \rm S } \equiv \frac { S } { 4 } ( 1 - A )$$ \end{document} The solar flux, S, assumed by Goldblatt et al. (2013) is 1368 W/m². At high surface temperatures, the planetary albedo, A, shown in the insert is ∼0.165; thus, the calculated value of F _S is 286 W/m². When this value is used in place of the quoted value of 294 W/m², F _S exceeds F _IR by only ∼5 W/m², so their model is very close to the borderline between a stable, subcritical solution and a runaway.

FIG. 2.

(A) Planetary albedo and (B) surface temperature versus CO₂ pressure for the simulations with our postulated RH feedback (solid), standard Manabe-Wetherald RH (dashed), and the sensitivity study (dots with dashes). The assumed surface albedo is 0.315. Separate (self-broadened) absorption coefficients for H₂O and CO₂ were used.

We repeated the pure H₂O atmosphere flux calculation in Fig. 2 of Goldblatt et al. (2013) using similar fixed vertical temperature profiles. The net absorbed solar flux in our model ranges from ∼283 W/m² at 500 K to 281 W/m² at ∼800 K and 286 W/m² at still higher temperatures (Fig. 1A). The highest of these values is the same as the asymptotic value of Goldblatt et al. (2013), as corrected by us above. Between 500 and 1000 K, our absorbed solar flux is about 2.5–5 W/m² smaller. We attribute this slight curvature above 500 K to numerical accuracy issues in interpolating between the HITEMP and HITRAN databases. Our outgoing IR flux at high surface temperatures is ∼280 W/m², which is close to the Goldblatt et al. value, although their peak outgoing longwave radiation at ∼300 K is 2 W/m² higher. [Our value for F _IR is 11 W/m² lower than the 291 W/m² reported by Kopparapu et al. (2013a) because we corrected an error in our H₂O, thermal-IR, k coefficients, as mentioned earlier.] Thus, like the Goldblatt et al. model, our model is unstable against runaway at surface temperatures below 500 K. However, unlike their model, our model is borderline stable between 500 and 1000 K. This stability will manifest itself in the high-CO₂ calculations described below because the highest temperature achieved in our simulations is ∼500 K. A plot of the differences between our calculated top-of-atmosphere fluxes and those of Goldblatt et al. is shown in Fig. 1. These differences are within±5 W/m² at surface temperatures between ∼300 and ∼1500 K. At surface temperatures above ∼1500 K, F _IR increases in both models because the dry adiabat achieves high-enough altitudes for significant near-IR and visible band emission (Goldblatt et al., 2013). This increase occurs at a somewhat higher temperature when coefficients from the HITEMP database are used, because of their more efficient absorption at these wavelengths. Although our climate model exhibits differences in stability at 500 K as compared to the Goldblatt et al. model, the difference at that temperature is only ∼2.5 W/m² (<1%), which will be outweighed in importance by other factors such as clouds and RH (as discussed below). Moreover, at temperatures lower than 500 K, our model is more unstable than that of Goldblatt et al., which is consistent with our goal of maximizing the runaway greenhouse potential.

Note that for pure H₂O atmospheres both our model (Fig. 1C) and that of Goldblatt et al. (2013) appear to be unstable against warming at surface temperatures comparable to that of modern Earth. Earth's climate is clearly not unstable, though; if it was, we would not be here. Clouds tend to cool Earth's climate; however, that effect is included implicitly in both these climate models through the adoption of a high surface albedo, so that is not the main difference between these models and Earth's real climate behavior. Stability is ensured by several factors, most importantly Rayleigh scattering by N₂ and O₂, which increases the planetary albedo, and by the “radiator fin” behavior of the descending branch of the Hadley cells (Pierrehumbert, 1995), described earlier. This shows that it can be misleading to use radiative flux calculations of pure H₂O atmospheres to evaluate climate stability against CO₂ increases. Stability must be tested by doing explicit high-CO₂ calculations, as described in the next section.

3.2. High-CO₂ calculations

We now turn to the question of whether increased CO₂ can trigger a runaway greenhouse on modern Earth. To answer this question, we performed forward calculations in which we increased the atmospheric CO₂ pressure incrementally from its present value, 5×10⁻⁴ bar in our units (see below), up to 100 bar, following a methodology similar to that of KA86. The results are shown in Figs. 2 –8. We performed two separate sets of calculations: one with the RH feedbacks from the previous section included and another with a standard Manabe-Wetherald RH profile (no change in surface RH and Ω=1 in Eq. 1). The third dashed-dot curve in Fig. 2 is the result of an RH sensitivity study elaborated in the Discussion. As in KA86, the quantity “CO₂ pressure” shown on the horizontal axis is the pressure that CO₂ would exert if no other gases were present (see Appendix B). The actual CO₂ partial pressure is lower than this because lighter gases, primarily N₂ and O₂, dilute the CO₂ by causing it to diffuse away from Earth's surface. Our choice of units ensures that CO₂ pressure varies linearly with added CO₂. We used the separate CO₂ and H₂O coefficients for this calculation because self-broadening should outstrip foreign broadening at higher temperatures and CO₂ concentrations and because this also ensures that the potential for triggering a runaway is maximized. As explained in Section 2, we assumed that surface RH increases with T _S in order to balance the surface energy budget (see Fig. 3).

FIG. 3.

Calculated surface RH versus surface temperature for a model calculation with the RH feedback of Eq. 7.

FIG. 4.

Temperature-altitude profiles for five representative atmospheres from the simulations shown in Fig. 2. (Color graphics available online at www.liebertonline.com/ast)

FIG. 5.

(A) Surface and (B) stratospheric saturation water vapor mixing ratio, fH₂O, versus CO₂ pressure for the simulations shown in Fig. 2. Above ∼305 K and 12 PAL CO₂ (6×10⁻³ bar), the atmosphere becomes water-dominated when the RH feedback is included (solid curves).

FIG. 6.

Top-of-atmosphere net absorbed solar flux, outgoing thermal-IR flux, and net outgoing flux as a function of surface temperature for the 0 PAL (black), 3 PAL (green), and 100 PAL (purple) CO₂ subsaturated cases of Goldblatt et al. (2013); 0.1 bar (blue) and 1 bar (red) CO₂ cases are also added. The assumed surface albedo is 0.315. The corresponding Goldblatt et al. (2013) curves and plots showing the differences between our calculations and theirs are shown in Fig. 7.

FIG. 7.

(Top panel from left to right) Top-of-atmosphere net absorbed solar, outgoing thermal, and net outgoing thermal fluxes as a function of surface temperature for the 0 PAL (black), 3 PAL (green), and 100 PAL (purple) CO₂ subsaturated cases of Goldblatt et al. (2013) of Fig 6. The assumed surface albedo is 0.315 for our model and 0.25 for that of Goldblatt et al. (2013). The curves in the top panel are those of Goldblatt et al. (2013). Dashed curves are the original Goldblatt et al. (2013) results, whereas the solid curves are the corrected solutions from their planetary albedo values. Flux differences (bottom panel) are our computed values minus those of Goldblatt et al. (2013). Positive values indicate that our model is warmer.

FIG. 8.

Planetary albedo as a function of surface temperature for the 0.25 surface albedo case in Fig. 4 of Goldblatt et al. (2013). Our planetary albedo values (solid curve) are comparable to those of Goldblatt et al. (2013) (dashed curve) throughout this temperature range.

The results for the RH feedback case in Fig. 2 are very different from those of KA86, although they start out the same at low surface temperatures. For the first CO₂ doubling, our model predicts 2.6 degrees of warming, near the middle of the range of values, 1.5–4.5 degrees, predicted by the Intergovernmental Panel on Climate Change for doubled CO₂. (Removing all RH feedbacks reduces this value slightly to 2.5 degrees.) But at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p { \rm CO}_2 \,\cong \,6 \times 10^{ - 3}$$ \end{document} bar of CO₂ (12 PAL) and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$T_S \, \cong \,305 \ K$$ \end{document} , our model makes an abrupt transition to a surface temperature>500 K. This transition marks the point at which the troposphere becomes highly saturated in our model. A sensitivity study with 1 solar zenith angle decreased the net absorbed solar flux by only∼4 W/m², but the resultant mean surface temperature at 12 PAL decreased to ∼470 K. This underscores the extreme sensitivity our results exhibit to small flux changes in a regime in which the net absorbed solar and outgoing fluxes are comparable to one another. An analogous, but much smaller, increase in T _S occurred at ∼60 PAL CO₂ in the model of KA86. Our 305 K transition is similar to that found by Rennó (1997), who used a 1-D model with an explicit convective parameterization scheme. Whether such an abrupt transition would actually occur is a question that should be studied with more realistic, 3-D climate models. Vertical profiles of temperature and H₂O mixing ratio for these calculations are shown in Figs. 4 and 5. Unlike the original KA86 calculations, which never exhibited water stratospheric volume mixing ratios higher than ∼0.1%, these new solutions are water-dominated (Fig. 5). Technically, the runaway greenhouse is not quite triggered because the critical temperature (647 K) is never exceeded, but this point may be moot, because at 500 K Earth would be completely sterilized and life would cease to exist.

To understand why our model stabilizes at 500 K for intermediate CO₂ concentrations instead of going runaway, it is useful to look at the corresponding inverse calculations (Fig. 6). These calculations are similar to those shown in Fig. 4 of Goldblatt et al. (2013), except that we extend our temperature range up to 540 K, whereas they stopped at 400 K. Figure 6 shows that our net absorbed solar flux decreases slightly as T _S is increased from 400 to 540 K. One reason is that the calculated planetary albedo decreases slightly over this range because of increased Rayleigh scattering from the thicker H₂O-dominated atmosphere. The other is because F _IR is still increasing to its asymptotic value of ∼280 W/m². These two effects cause our net outgoing flux to be close to zero at ∼500 K, thereby producing a neutrally stable solution.

According to Fig. 4 of Goldblatt et al., their model is unstable against runaway at a CO₂ concentration of 100 PAL. The reasons for their result can be seen more clearly by recreating a portion of Fig. 4 of Goldblatt et al. (2013) and comparing our net absorbed and outgoing IR fluxes versus theirs for three different CO₂ concentrations: 0, 3, and 100 PAL (Fig. 7). At mean surface temperatures below∼315 K, the net absorbed solar fluxes are lower for our model than for that of Goldblatt et al. (2013) because our assumed surface albedo of 0.315 is significantly higher than theirs. In spite of this, the net absorbed solar fluxes in our model are always greater above ∼315 K because surface albedo effects become increasingly negligible as atmospheric opacity increases. Our stronger absorption at high surface temperatures is attributable to different surface RH parameterizations and to different assumptions about foreign broadening, which was just assumed by Goldblatt et al. (2013) to be terrestrial air for all atmospheres (Tyler Robinson, NASA Ames, private communication).

However, due to an error in the specification of solar zenith angles in the runs shown in Fig. 4 of Goldblatt et al. (2013), the net absorbed solar radiation is too high there by between 2.6 and 5.6 W/m² (C. Goldblatt, private communication). At 400 K, they obtain a net absorbed solar flux of 297 W/m² for a surface albedo of 0.25 (Fig. 7). However, for the corresponding planetary albedo of ∼0.15 (Fig. 8) and an incident solar flux of 1368 W/m², we calculate a net absorbed solar flux of ∼291 W/m² (Fig. 7), which should be sufficient to stabilize their atmosphere. Thus, instead of predicting a runaway greenhouse at 100 PAL CO₂, Goldblatt et al. (2013) should have found a stable solution with a surface temperature of ∼345 K. In Fig. 7, we have replotted their solar fluxes as recalculated from their published planetary albedos. Perhaps coincidentally, KA86 obtained a similar surface temperature at 100 PAL CO₂. To be sure, both our calculations and those of Goldblatt et al. (2013) are only borderline stable at high CO₂ levels and could be susceptible to runaway if solar absorption was just slightly stronger. Indeed, our calculation is more alarmist than theirs because the catastrophe occurs at a much lower CO₂ level: 12 PAL, as compared to 100 PAL in their model. However, we argue below that both calculations are overly pessimistic and that Earth's real climate is probably stable against a CO₂-induced runaway greenhouse.

To finish off this discussion, at CO₂ pressures above ∼0.1 bar, the continued addition of CO₂ actually causes surface temperatures to decrease (Figs. 2 and 6). This is because efficient Rayleigh scattering by CO₂ increases the planetary albedo, as pointed out previously by KA86. In KA86, the calculated surface temperature did not decrease as they do here, but that is because their surface temperatures were much lower at intermediate CO₂ concentrations. This effect was missed by Goldblatt et al. (2013) because they did not perform calculations for CO₂ concentrations>100 PAL (0.03 bar).

Another way of looking at these results is to consider the calculated mixing ratio of water vapor at the surface (Fig. 5A) and at the tropopause cold trap (Fig. 5B) in these calculations and compare with Fig. 2 of KA86. Our model atmosphere is more than 90% H₂O at CO₂ concentrations near 100 PAL, although that figure drops at still higher CO₂ levels. By contrast, the KA86 model predicted that tropopause H₂ concentrations never exceeded ∼10⁻³ by mass, or roughly twice that value by volume. The rate at which hydrogen escapes to space is directly proportional to the H₂O mixing ratio at the tropopause cold trap (Walker, 1977). So our current model is much more unstable against water loss than was that of KA86.

3.3. Mixed coefficient calculations

As noted previously, we are aware that our use of separate, self-broadened coefficients for CO₂ and H₂O should lead to an overestimate of absorption at thermal-IR wavelengths and thus an overestimate of the greenhouse effect. To estimate the magnitude of this error, and to better determine the response of Earth's climate to more modest CO₂ increases, we performed a sensitivity study using the mixed H₂O-CO₂ absorption coefficients. We again used six solar zenith angles to improve the accuracy of our solar calculation. The new model yields a slightly lower sensitivity to doubled CO₂, 2.3 K, and it warms somewhat more slowly at higher CO₂ concentrations. At a CO₂ pressure of 2×10⁻³ bar, surface temperatures with the new model were ∼1 K cooler than before. As CO₂ pressure increases above this value, the transition point to a moist atmosphere is delayed by 5 K (310 K at ∼9×10⁻³ bar or 18 PAL), which then occurs at the same temperature as that predicted by Goldblatt et al. (2013). These new estimates are also consistent with inferred maximum temperatures and pCO₂ levels during the mid-Cretaceous of ∼305 K and 14 PAL, respectively (Hay and Floegel, 2012). As mentioned in Section 2, the mixed coefficients were not used above this CO₂ level, as the assumption that foreign broadening can be treated as terrestrial air becomes increasingly invalid. However, this also suggests that the transition to a more water-rich atmosphere may not be as large or sudden as that predicted with the baseline model. For this reason alone, the real atmosphere is almost certainly less susceptible to runaway than our base calculation indicates.

3.4. Effect of clouds in high-CO₂ atmospheres

In reality, the climate at higher CO₂ levels would also be affected by cloud feedback. Clouds have a complex effect on climate; low water clouds tend to cool the surface, whereas high cirrus clouds warm it (Choi and Ho, 2006). Whether cloud feedback is positive or negative depends on which types of clouds change the most, and how. In doubled CO₂ experiments for modern Earth, many current 3-D climate models predict that cloud feedback is positive because cirrus clouds increase fastest as the climate warms slightly (Soden and Held, 2006). But researchers, using 1-D models (Kasting, 1988), have predicted that cloud feedback should be negative for much warmer atmospheres because clouds add little to their already-high IR opacity, whereas their effect on planetary albedo remains significant. Three-dimensional models yield contradictory results for atmospheres warmed by high levels of solar heating; Leconte et al. (2013) predicted positive cloud feedback, whereas Wolf and Toon (2014) predicted negative feedback.

We investigated cloud feedback using a methodology similar to that of Kasting (1988), except that we were careful to specify cirrus clouds at atmospheric temperatures below the freezing point and water clouds at temperatures above it; also, our cloud decks were thinner (∼1 km in depth) in order to maximize cirrus warming. This vertical resolution was achieved by increasing the number of layers in our model from 100 to 200. For water clouds, we assumed a log-normal particle size distribution and a mean radius of 15 microns, and published parameterizations were used to determine optical parameters (Slingo, 1989; Hu and Stamnes, 1993). For cirrus clouds, we used a gamma distribution of particle sizes (Hansen and Travis, 1974) with a mean radius of 30 microns. Optical properties were computed by using appropriate cirrus cloud parameterizations (Hong et al., 2009). The wavenumber-dependent optical depths (τ) were found from the following equation (e.g., Slingo, 1989; Platt, 1997): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\tau = \frac { 3Q_ { { \rm eff } } \cdot IWC \cdot \Delta z } { 4r \rho } \tag { 8 } \end{align*} \end{document}

Here, Q _eff is the wavenumber-dependent extinction efficiency, ρ is the mass density of ice, r is the particle radius, IWC is the ice water content (g/m³), and Δz is the vertical path length of the layer (m). Values for Q _eff and IWC were taken from the work of Yang et al. (1997) and Platt (1997), respectively. Q _eff was set equal to 2 across the thermal IR because extinction efficiencies for ice crystals>∼25 micron radius approach this value (Yang et al., 1997).

We compare our new results with those of Kasting (1988). The solid and dashed curves in Fig. 9 illustrate the effect of 100% and 50% cloud cover on both the net fluxes at the top of the atmosphere and planetary albedo for a 5 bar CO₂ atmosphere. Optical properties and flux data are listed in Tables 1 and 2, respectively. The first five rows are the cirrus cloud decks. As predicted by, for example, Kasting (1988), water clouds have little impact on the outgoing radiative flux at the top of the atmosphere, F _IR (Fig. 9A). However, high cirrus clouds exhibit more warming than originally shown by Kasting (1988), reducing F _IR to a minimum value of 182 W/m² at 8×10⁻⁴ bar and 100% cloud cover. In response, the net absorbed solar flux, F _S, decreases by a factor of ∼2.5, which corresponds to an increase in planetary albedo from 0.25 to ∼0.65 (Fig. 9B), exhibiting a similar trend to that of Kasting (1988). Our maximum planetary albedo value is lower than the corresponding maximum of 0.8 calculated by Kasting (1988) because that study used thicker clouds and smaller water droplets (5 microns), which are more strongly reflective.

FIG. 9.

Effect of a single cloud layer on the net (A) absorbed solar (F _S) and outgoing (F _IR) fluxes at the top of the atmosphere, (B) planetary albedo, and (C) effective solar flux S _eff for a 5 bar CO₂, 1 bar N₂ atmosphere with surface temperature of 405 K and a surface albedo of 0.125. The horizontal scale shows the pressure at the center of the assumed, 1 km thick cloud deck. Solid and dashed curves are for 100% and 50% cloud cover, respectively. The vertical dashed lines mark the pressures associated with the location of the tropopause (left) and the freezing point of water (right). All pressures are calculated at the midpoints of the cloud deck. For cloud-free conditions, S _eff=1. Values of S _eff>1 mean that the cloud cools the surface; S _eff<1 means that the cloud warms the surface. (Color graphics available online at www.liebertonline.com/ast)

Table 1.

Cloud Decks and Associated Properties for 15 μm Water Droplets and 30 μm Cirrus Particles and for the 5 Bar CO₂ Atmosphere of Figure 3

Pressure (bar) ^a	Altitude (km) ^a	Temperature (K) ^a	Ice/Liquid water content (g/m³) ^a	Optical depth at 0.55 microns
2.57(−4)	68.4	202	1.0(−5)	6.4(−4)
7.84(−4)	63.2	220	1.36(−2)	0.9
3.6(−3)	55.3	240	1.66(−2)	1.13
1.68(−2)	46.7	260	1.59(−1)	10.6
3.78(−2)	41.8	272	4.63(−1)	30.6
2.07(−1)	30.6	303	9.87(−2)	16.1
8.25(−1)	20.45	334	3.46(−1)	52.8
2.75	10.62	366	1.019	179.8
4.86	5.62	384	1.696	266.1
8.162	0.78	402	2.698	439.7

Values are listed at the midpoint of the cloud deck. Read 1.0(5) as 1.0×10⁵.

Table 2.

Computed Fluxes for the Different Cloud Decks for the Same Atmosphere as Table 1

Pressure (bar) ^a	Altitude (km) ^a	F _IR	F _S	S _eff
2.57(−4)	68.4	257	257	1
7.84(−4)	63.2	182.1	226.1	0.805
3.6(−3)	55.3	200.6	220.6	0.909
1.68(−2)	46.7	215	137.7	1.56
3.78(−2)	41.8	234.4	114.9	2.04
2.07(−1)	30.6	256.1	170.2	1.50
8.25(−1)	20.45	256.3	185.9	1.38
2.75	10.62	256.3	210.6	1.22
4.86	5.62	256.3	223.4	1.15
8.162	0.78	256.3	232	1.10

Values are listed at the midpoint of the cloud deck.

The results of our cloud study are summarized in Fig. 9C, which shows the effective solar flux, S _eff=F _IR/F _S, needed to maintain the atmosphere in steady state. S _eff>1 indicates that the albedo effect of clouds exceeds their greenhouse effect, so that clouds cool the climate; S _eff<1 means that the cloud greenhouse effect dominates, so that clouds warm the climate. The two midlevel cloud decks exhibit net warming, with S _eff values below 1 and optical depths in the single digits (Tables 1–2). This result agrees with satellite observations suggesting that cirrus clouds warm the surface for visible/near-IR optical depths less than 10 (Choi and Ho, 2006). However, cirrus clouds at pressures>0.004 bar cool the surface, according to our calculations. Thus, the net effect of cirrus clouds in our model is to cool the surface, as the average S _eff value for these five cloud decks is nearly 30% above the clear sky value (Table 2). Therefore, the predicted increase in high clouds and the disappearance of low clouds at higher surface temperatures, as suggested by recent studies (Soden and Held, 2006; Leconte et al., 2013), does not necessarily yield positive cloud radiative forcing.

The above result should be compared with the fixed anvil temperature hypothesis (Kuang and Hartmann, 2007), which suggests that at higher temperatures the peak of radiative cooling will shift to higher altitudes as the IR opacity at lower levels of the atmosphere increases. This would imply that the peak of cloud formation should shift upward to less opaque altitudes. While such a change is physically reasonable, this increase in high clouds does not necessarily imply positive cloud feedback at such temperatures. If cirrus clouds keep rising as temperatures increase, they should eventually be located at very high altitudes where they would be physically and optically thin, which would greatly diminish their effect on the planetary radiation budget. Our cloud sensitivity test results are consistent with this hypothesis. Furthermore, if the cirrus clouds themselves thicken as the atmospheric opacity increases, or are low and warm enough (see Fig. 9), they will likely become more reflective than they are in the current climate. This suggests that overall cloud forcing may still be negative in moist and optically thick atmospheres, even if low liquid water cloud coverage decreases, as suggested in some studies (e.g., Brient and Bony, 2013; Leconte et al., 2013). Thus, we predict that in future 3-D studies of very high CO₂ levels, clouds will provide a stabilizing influence on climate.

4. Discussion

4.1. Sensitivity to relative humidity

Figure 2B shows that the response of our climate model to higher CO₂ levels is dictated largely by our assumptions about RH. When RH is allowed to increase, the climate is highly unstable. When RH is constrained at its present average value, the rise in surface temperature is much more restrained. Although our surface RH parameterization (Eq. 7) has a sound physical basis, our parameterization of upper tropospheric RH is more speculative. Equations 1 and 3 suggest that the atmosphere is nearly fully saturated by 321 K, as both atmospheric and surface RH exceed 92% at that temperature (Fig. 4). At 321 K, the mixing ratio of water vapor is only ∼10% at the surface and ∼4.5% in the troposphere (by volume) even though our parameterization predicts a sudden transition to a water-dominated atmosphere at an even lower, drier temperature than this (305 K, solid curve in Fig. 2). However, the upper atmosphere should only become water-dominated after the surface does (Eq. 16 in Wordsworth and Pierrehumbert, 2013), not at these relatively low temperatures with surfaces still dominated by dry air. This is because the rise in tropospheric RH should be slowed by atmospheric subsidence, which is not explicitly accounted for in this parameterization (discussed in Section 2). Thus, it is likely that the combination of Eqs. 1 and 3 overestimates the increase of tropospheric RH with surface temperature. We performed a sensitivity study to assess the effect of a slightly slower increase in RH (dot-dashed curve in Fig. 2B). For surface temperatures up to 320 K, maximum stratospheric RH was set to 50%, which is consistent with a recent general circulation model (GCM) study that analyzed a similar temperature range (Wolf and Toon, 2014). For even higher surface temperatures, Ω was set to an arbitrary value slightly larger than zero (1×10⁻²). At 321 K, this corresponded to a stratospheric RH of 80% (instead of 92%), increasing at still higher temperatures. Our results show that these relatively modest changes have an enormous impact, greatly decreasing both the amplitude of the temperature rise as well as the maximum temperatures achieved (∼400 K at 20 PAL CO₂) (Fig. 2). Thus, this sensitivity study suggests that a more realistic treatment of RH should find that Earth is stable against the runaway. Further improvements to this calculation will require a 3-D GCM that can self-consistently calculate RH in warm, moist atmospheres.

4.2. Human health issues at modest CO₂ increases

Just because Earth's climate is probably stable does not imply that future CO₂ increases are without peril. In addition to long-term threats such as sea level rise, heat stress on humans is a significant issue, especially in the tropics (Sherwood and Huber, 2010). Sherwood and Huber (2010) argued that hyperthermia will be induced within hours when the local wet bulb temperature exceeds ∼35°C, two degrees below the human body temperature. According to their 3-D climate calculations, this wet bulb temperature would be reached seasonally and diurnally over some parts of the globe once the mean surface temperature exceeds 295 K. For T _S=300 K, hyperthermia would spread even farther and render most regions of the planet uninhabitable for humans. Our base model predicts that these thresholds would be reached at CO₂ levels of ∼1300 ppmv (4 PAL) and∼2600 ppmv (8 PAL), respectively (Fig. 10). Incidentally, this rise of temperature with pCO₂ agrees well with that of one recent 3-D study (Meraner et al., 2013). For the alternative (low-surface-albedo) model, the corresponding CO₂ levels are 2000 ppmv (6 PAL) and 3550 ppmv (10.8 PAL). Also shown in Fig. 10 are results from another recent 3-D climate model study by Caballero and Huber (2013). They computed significantly higher heat stress thresholds of∼2160 and ∼5000 ppmv CO₂, respectively, for modern surface boundary conditions. When they use Paleogene boundary conditions, though, with no ice and vegetated deserts, their predicted surface temperatures are intermediate between our base and alternative models. The Paleogene simulation may be a good analogue for what might happen as atmospheric CO₂ increases in the future. However, even the Paleogene 3-D calculation might underpredict surface temperatures at very high CO₂ concentrations because the climate model used, CCSM3, does not treat water vapor as a major constituent. Recall that our model assumes increased tropospheric RH at high surface temperatures, based on the idea that condensation of a major constituent should lead to pressure changes that, in turn, may make it more difficult for that constituent to condense.

FIG. 10.

Surface temperature as a function of CO₂ concentration (ppmv) for our model for both the separate coefficients (base model) and mixed coefficients (alternative model) simulations discussed in the text (black curves) and for the modern (blue dashed curve with squares) and Paleogene (green dashed curve) models of Caballero and Huber (2013). The dashed horizontal lines at 295 and 300 K represent published heat stress thresholds of Sherwood and Huber (2010). Localized thermal stress on humans begins to become significant at a global mean temperature of 295 K and becomes widespread above 300 K. (Color graphics available online at www.libertonline.com/ast)

Which, if either, of these two heat stress thresholds might be reached from burning fossil fuels? Estimates of economically recoverable fossil fuel reserves are in the range of 2000–5000 Gt C (Rogner, 1997; Mohr, 2010; Patzek and Croft, 2010; Rutledge, 2011). Values near the lower end of this range assume that recoverable coal is less abundant than previously believed (Rutledge, 2011). One preindustrial atmospheric level of CO₂ (330 ppmv) corresponds to ∼700 Gt C; thus, if one were to burn all 5000 Gt of carbon instantaneously, atmospheric CO₂ would increase by a factor of ∼8, to over 2600 ppmv, reaching at least the lower of the two postulated heat stress thresholds and making parts of the world uninhabitable by humans. The higher threshold could be reached if our 1-D climate calculations are correct. If the recoverable fossil fuel reserves are closer to 2000 Gt C, then CO₂ could still increase to 4 PAL in our simulation, reaching the lower heat stress threshold in our base model. Neither of these CO₂ estimates includes the vast quantities of methane gas hydrates thought to reside on the seafloor (Kvenvolden, 1988) and thus may be regarded as conservative.

An additional health threat is triggered in our model if atmospheric CO₂ exceeds ∼3600 ppmv (12 PAL) in our base model (or 18 PAL in our alternative model). Calculations (not shown) in which our 1-D photochemical model is used (Segura et al., 2003) suggest that the ozone layer would essentially disappear above this CO₂ level because the stratosphere would become wet and the ozone layer would be destroyed by catalysis caused by the by-products of H₂O photolysis. High stratospheric H₂O would also result in increased rates of water loss by way of photodissociation followed by hydrogen escape—a phenomenon sometimes termed a moist greenhouse (Kasting, 1988). However, atmospheric CO₂ concentrations would presumably be restored to more normal values by silicate weathering within a few million years (Walker et al., 1981), before substantial water loss could occur.

5. Conclusion

In summary, when using the most alarmist assumptions, Earth's climate does appear to be susceptible to a CO₂-induced runaway greenhouse, as predicted by Hansen (2010) and Goldblatt et al. (2013). Technically, our model stabilizes at surface temperatures below the critical temperature of water and hence does not represent a true runaway. However, this apparent stability would disappear if the calculated fluxes were only 1–2 W/m² higher, within the error of our radiative transfer method. Furthermore, the predicted surface temperature at moderately high CO₂ levels is so high, ∼500 K, that Earth would be sterilized anyway. Indeed, our own calculations suggest that such a phenomenon could be triggered at CO₂ concentrations as low as 12 PAL, as compared to the 100 PAL calculated by Goldblatt et al. But these calculations are performed with a choice of absorption coefficients and assumptions about RH that deliberately maximize the greenhouse effect. That said, several lines of evidence suggest that Earth is probably stable against the runaway. First, using “mixed” coefficients that simulate absorption more accurately for modern Earth conditions delays any climate catastrophe to at least 18 PAL CO₂, implying also that absorption is overestimated in the baseline calculations, including in the temperature rise to a water-rich atmosphere. Second, as originally found in KA86, cloud feedback in optically thick atmospheres is likely negative. And third, if atmospheric RH rises even slightly more slowly than predicted in our model, as seems likely, computed mean surface temperatures at the transition point can be ∼100 K lower than in our baseline calculations. These inferences should be validated by properly equipped 3-D GCMs that can self-consistently calculate RH in the high temperature regime.

Even though we argue that a runaway greenhouse probably would not be triggered by increased CO₂, life on Earth could be in great peril even for relatively modest CO₂ increases of 6–11 PAL. The threat to human health from unrestrained fossil fuel burning remains acute, with direct heat stress to humans posing the most immediate danger. If our calculations and those of other climate modelers (Sherwood and Huber, 2010; Caballero and Huber, 2013) are even approximately correct, holding atmospheric CO₂ increases to less than a factor of 6 above preindustrial values is essential to human health and survival.

Appendix A. Spectral Comparisons Versus Goldblatt et al. for Pure H₂O and Modern Earth Atmospheres

First, we compare our broadband surface downwelling and net outgoing thermal flux spectra against the line-by-line spectra of Goldblatt et al. (2013) for the 400 K pure water atmosphere shown in their Fig. 1 (Fig. A1). Internal comparisons of these spectra have revealed that the van Vleck–Weisskopf and Rautian profiles between ∼5 and 15 microns used by Goldblatt et al. (2013) produce more far-wing absorption than do our Voigt line profiles. This may explain our higher outgoing thermal flux between 10 and 15 microns when we compare our model against SMART (Meadows and Crisp, 1996), the radiative transfer model used by Goldblatt et al. (2013) (Fig. A1). However, this difference is offset by the smaller amount of absorption by the Goldblatt et al. (2013) model at shorter wavelengths (Fig. A1).

FIG. A1.

Net outgoing thermal flux spectrum in a pure water atmosphere for a 400 K surface temperature, comparing SMART (Meadows and Crisp, 1996) (blue) versus our model (green). The black curve is the blackbody emission flux at the specified surface temperature. The integrated area under both curves (including emission beyond the x axis limits shown) is ∼281 W/m². (Color graphics available online at www.liebertonline.com/ast)

Our models also differ in how the H₂O continuum is formulated. Whereas we use the BPS continuum (Paynter and Ramaswamy, 2011), Goldblatt et al. (2013) employed CKD continuum sub-Lorentzian line formulations (Clough et al., 1989) together with MT_CKD 2.4 continuum data (Mlawer et al., 2012). It is not clear which formulation is better, as empirical measurements for the continuum do not exist below ∼0.55 microns (D. Paynter, Princeton University, private communication). Although the continuum should extend to even shorter wavelengths, we have not attempted to extrapolate it here, which may cause us to underestimate absorption in the blue part of the visible spectrum. However, the continuum is thought to be proportional to the strength of the water vapor bands, which are weak at those shorter wavelengths (see Goldblatt et al., 2013, Supplementary Info, Fig. 3); thus, its contribution to the overall solar absorption should be rather small (D. Paynter, private communication). This is confirmed in the next section when we compare our solar fluxes for a pure H₂O atmosphere versus Goldblatt et al. (2013).

We also compared our model against Goldblatt et al. (2013) for a representative modern Earth atmosphere in order to illustrate the radiative differences between mixed and separate CO₂-H₂O coefficients in the low-pCO₂ regime (Figs. A2 –A5). Convolving separate absorption coefficients derived for pure H₂O and CO₂ makes the implicit assumption that foreign broadening by terrestrial air (N₂ and O₂) can be approximated by H₂O and CO₂ self-broadening. As explained in Section 2, this assumption may be reasonably accurate for CO₂- and H₂O-dominated atmospheres but results in an overestimate of thermal-IR absorption for atmospheres dominated by N₂, as self-broadening is generally stronger than foreign broadening. We compensate for our ∼10 W/m² lower top-of-atmosphere outgoing thermal radiation flux (Fig. A2) by assuming a high surface albedo (see Section 2), which results in ∼10 W/m² less solar absorption being absorbed (Fig. A3).

FIG. A2.

Upwelling and downwelling vertical IR flux comparisons between our model (solid red curves) and SMART (green dashed curves) for a modern Earth atmosphere. H₂O and CO₂ are self-broadened. Upwelling thermal fluxes agree to within ∼10 W/m² (∼4%), and downwelling thermal fluxes agree to within 2–3 W/m² (1%) at most heights. (Color graphics available online at www.liebertonline.com/ast)

FIG. A3.

Upwelling and downwelling vertical shortwave flux comparisons between our model (solid red curves) and SMART (green dashed curves) for the modern Earth atmosphere in Fig. A2. H₂O and CO₂ are self-broadened, as in Fig. A2. The assumed surface albedo is 0.274. (Color graphics available online at www.liebertonline.com/ast)

FIG. A4.

Upwelling and downwelling vertical IR flux comparisons between our model (solid blue curves) and SMART (red dashed curves) for a modern Earth atmosphere. Terrestrial air is assumed for foreign broadening. The calculated outgoing longwave radiation (OLR) in our model is 266 W/m², only 3 W/m² higher than that computed by SMART. (Color graphics available online at www.liebertonline.com/ast)

FIG. A5.

Upwelling and downwelling vertical shortwave flux comparisons between our model (solid blue curves) and SMART (red dashed curves) for the modern Earth atmosphere in Fig. A4. Terrestrial air is assumed for foreign broadening as in Fig. A4. The assumed surface albedo is 0.274. The top-of-atmosphere net absorbed flux is 260 W/m² for our model, whereas SMART obtains 258 W/m². (Color graphics available online at www.liebertonline.com/ast)

We then repeated the modern Earth comparison using our mixed CO₂-H₂O coefficients with more accurate foreign broadening in this low-pCO₂ regime (see Section 2). These coefficients produce a lower surface albedo (see Section 2), allowing our model to agree closely with the SMART radiative transfer model used by Goldblatt et al. (2013) for modern Earth (Figs. A4 and A5).

Appendix B. Relationship between CO₂ Pressure and CO₂ Volume Mixing Ratio

The CO₂ pressures referred to in the main text are not actual partial pressures but the surface pressures that would exist if CO₂ did not mix with the remaining atmospheric gases. Although Dalton's law states that the total atmospheric pressure is equal to the sum of the pressures of the individual gases, in an unbounded planetary atmosphere, vertical mixing reduces the mole fractions of heavier species near the surface.

Mathematically, the CO₂ pressure, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p {\rm CO}_2 {\prime}$$ \end{document} , is related to the column masses of CO₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$M_{{\rm CO}_2}$$ \end{document} , and (non-CO₂-containing) air, M _Air, by the following expressions: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}p{ \rm CO}_2 { \prime} = M_{{ \rm CO}_{2}} \cdot g = N_{{ \rm CO}_{2}} m_{{ \rm CO}_{2}} \cdot g \tag{{ \rm B1}}\end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}p{ \rm Air}{ \prime} = M_{{ \rm Air}} \cdot g = N_{{ \rm Air}} m_{{ \rm Air}} \cdot g \tag{{ \rm B}2}\end{align*} \end{document}

The prime on pCO₂ and pAir is to indicate that these are not partial pressures but their isolated surface pressures. Here, g is gravity, N_i is the column density of species i (where i is either CO₂ or air), and m_i is its molecular mass.

The total column number density is conserved when these gases are mixed, so that the total column number density, N _tot, is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}N_{{ \rm tot}} = N_{{ \rm CO}_{2}} + N_{{ \rm Air}} \tag{{ \rm B}3}\end{align*} \end{document}

The volume mixing ratio for CO₂, fCO₂, is equal to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$N_{{\rm CO}_2}$$ \end{document} /N _tot. Substituting in values from Eqs. B1 and B2 yields \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}f { \rm CO } _2 = \frac { p { \rm CO } _2 { \prime } } { \left( p { \rm CO } _2 { \prime } + \frac { 44 } { m_ { { \rm Air } } } p_ { { \rm Air } } { \prime } \,\,\, \right) } \tag { { \rm B } 4 } \end{align*} \end{document}

where we have used the fact that the molecular mass of CO₂ is 44 g/mol. We use a value of 29 g/mol for m _Air and 1 bar for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p_{{ \rm Air}} {\prime}\ \ $$ \end{document} , which represents the combined total pressure of N₂, O₂, and Ar. Then fCO₂ is used to compute a new mixing ratio fCO₂ ^new with respect to moist air. Mixing ratios with respect to moist air are recalculated for all noncondensible species.

Note added in proof:

A recently published 3-D climate model calculation (Russell et al., 2013) investigates CO₂ pressures as high as 256 PAL CO₂ (∼0.128 bar in our units). Their calculated mean surface temperatures closely resemble those of KA86. Both studies conclude that a runaway greenhouse is impossible to trigger for any plausible anthropogenic increase in CO₂. This result is not entirely surprising, as Russell et al. used an older water continuum and absorption coefficients, which would likely lead to lower surface temperatures than those found here. Curiously, Russell et al. predict a decrease in average surface relative humidity with increasing surface temperature—a result they attribute to drier continents. Surface relative humidity over the oceans remains essentially unchanged in their model. More 3-D climate modeling is needed to investigate the differences between their results and those shown here.

Footnotes

Acknowledgments

The authors thank David Pollard for help with the surface RH parameterization and David Archer for updates on current fossil fuel inventories. We are also grateful to Tyler D. Robinson for using SMART to perform flux comparisons against our model. We thank both Colin Goldblatt and Max Popp for very helpful and detailed reviews. We also thank Colin Goldblatt again for alerting us to the increased absorption afforded by HITEMP in very moist, runaway atmospheres. We are also grateful to Vincent Eymet for having written KSPECTRUM. Finally, we would like to acknowledge the Virtual Planetary Lab, especially Ray Pierrehumbert and Vikki Meadows for providing comments which improved the quality of this manuscript. Funding for this work was provided by the NASA Exobiology and Evolutionary Biology Program and by the NASA Astrobiology Institute.

Abbreviations

GCM, general circulation model; PAL, preindustrial atmospheric level; RH, relative humidity.

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Can Increased Atmospheric CO 2 Levels Trigger a Runaway Greenhouse?

Abstract

1. Introduction

2. Methods

2.1. Radiative transfer model

2.2. Upper tropospheric relative humidity

2.3. Climate modeling procedures

2.4. Surface relative humidity

3. Results

3.1. Flux comparison with results of Goldblatt et al. for a pure-H2O atmosphere

3.2. High-CO2 calculations

3.3. Mixed coefficient calculations

3.4. Effect of clouds in high-CO2 atmospheres

4. Discussion

4.1. Sensitivity to relative humidity

4.2. Human health issues at modest CO2 increases

5. Conclusion

Appendix A. Spectral Comparisons Versus Goldblatt et al. for Pure H2O and Modern Earth Atmospheres

Appendix B. Relationship between CO2 Pressure and CO2 Volume Mixing Ratio

Note added in proof:

Footnotes

Acknowledgments

Abbreviations

References

3.1. Flux comparison with results of Goldblatt et al. for a pure-H₂O atmosphere

3.2. High-CO₂ calculations

3.4. Effect of clouds in high-CO₂ atmospheres

4.2. Human health issues at modest CO₂ increases

Appendix A. Spectral Comparisons Versus Goldblatt et al. for Pure H₂O and Modern Earth Atmospheres

Appendix B. Relationship between CO₂ Pressure and CO₂ Volume Mixing Ratio