Why the Forcing from Carbon Dioxide Scales as the Logarithm of Its Concentration

1. Introduction

It is well known that the radiative forcing from carbon dioxide is approximately logarithmic in its concentration, producing about 4 W m⁻² of additional global-mean forcing for every doubling. There are, however, two different explanations in the literature for this logarithmic dependence. Given the dominant role that CO₂ plays in global warming, this mechanistic uncertainty merits resolution.

Perhaps the most widely accepted explanation is that the logarithmic behavior stems from the particular absorption spectrum of CO₂ (Pierrehumbert 2010, 2011; Wilson and Gea-Banacloche 2012; Jeevanjee et al. 2021). Many absorption bands of greenhouse gases can be approximated with an absorption coefficient κ (m² mol⁻¹) that decays exponentially from the band center as a function of frequency or wavenumber (Edwards and Menard 1964a,b) and the 15-μm band¹ of carbon dioxide is a particularly good example of this (Crisp et al. 1986). Heuristic arguments have been given as to why this would lead to a logarithmic forcing (Pierrehumbert 2010, 2011) and analytic calculations have even succeeded in reproducing the ≈4 W m⁻² of forcing from a doubling of CO₂ (Wilson and Gea-Banacloche 2012; Jeevanjee et al. 2021). Notably, this explanation does not depend on the value or even existence of a tropospheric lapse rate.

The competing explanation is what we might call the “lapse-rate theory,” which posits that the forcing stems from the troposphere’s lapse rate (Sloan and Wolfendale 2013; Huang and Bani Shahabadi 2014; North and Kim 2017). In brief, the lapse-rate theory states that the emission-to-space heights of individual wavenumbers move upward in the troposphere (to lower temperature) a distance that is proportional to the logarithm of the gas concentration, thereby generating a forcing that scales as the log of the concentration. Critically, this theory predicts that the forcing is proportional to the tropospheric lapse rate. Notably, this explanation does not require the absorption coefficients to be distributed logarithmically or in any other special way. We will see that the lapse-rate theory is neither necessary nor sufficient to explain the logarithmic forcing of carbon dioxide (see section 9) and that it generally cannot hold for a well-mixed greenhouse gas in Earth’s atmosphere (see section 10).

The aim of this paper is to set the explanation of Pierrehumbert (2010, see his Fig. 4.12) on a firm foundation by demonstrating the properties that lead to carbon dioxide’s logarithmic forcing, building a simplified analytic model to showcase the underlying processes, and showing that a line-by-line radiative transfer model gives a logarithmic forcing for the same reasons. This will occupy the bulk of paper, from sections 5 to 8. The reader who wants to quickly learn the basic mechanism may be satisfied with the overview in section 3 and could skip ahead to that section now.

2. Preliminaries

To reduce the problem to its essential elements, we will focus on the instantaneous top-of-atmosphere (TOA) forcing in dry atmospheres. These simplifications (TOA and dry) are acceptable because the logarithmic scaling of the CO₂ forcing does not depend on these choices. Figure 1 illustrates this point using radiative-transfer calculations in a three-dimensional snapshot from a cloud-resolving simulation of a tropical atmosphere.² Figure 1a shows, plotted as circles, the total, instantaneous, tropopause, longwave forcing from carbon dioxide (averaged over the domain of this snapshot) as a function of the carbon dioxide concentration; the best-fit line is overlaid to emphasize the linearity.

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Total instantaneous forcing F^tot calculated using RRTMG in a snapshot of a cloud-resolving RCE simulation plotted as circles for various CO₂ concentrations using (a) tropopause all-sky fluxes, (b) TOA all-sky fluxes, (c) TOA clear-sky fluxes (in which condensates have been zeroed), and (d) TOA dry-sky fluxes (in which condensates and water vapor have been zeroed). The lines are least squares fits. The r² values exceed 99% in all cases.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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Here and throughout, we will use the variable F to denote an instantaneous longwave forcing, which has dimensions of power per area and whose typically positive values indicate a net downwelling flux of radiative energy. With the exception of Fig. 1a, all forcings discussed in this paper will be calculated at the top of the atmosphere. We will dress F with various subscripts and superscripts to clarify what type of forcing is being used. For example, F^tot(q) will be the total forcing from the presence of a well-mixed greenhouse gas at a volume fraction of q. In particular, F^tot(q) is the TOA upwelling longwave flux with the greenhouse gas removed minus the TOA upwelling longwave flux with the greenhouse gas’s volume fraction set to q. In later sections, we will also discuss F^2×(q) ≡ F^tot(2q) − F^tot(q), which is the forcing from a doubling of the gas concentration.

From the linearity of the data on the logarithmic axis of Fig. 1a, we see that the all-sky forcing from CO₂ at the tropopause is logarithmic in its concentration to good approximation over a wide range of concentrations (i.e., the 10 doublings shown here). The instantaneous tropopause forcing is considered a better proxy for the stratosphere-adjusted forcing than the instantaneous top-of-atmosphere forcing (Hansen et al. 1997), but the logarithmic scaling is preserved in the TOA forcing as shown in Fig. 1b. Of the two forcings, the TOA forcing is easier to conceptualize because it only involves upwelling fluxes; therefore, the focus of this paper will be on the TOA forcing. In Fig. 1c, the clouds in the snapshot have been made transparent to infrared radiation, but this does not alter the logarithmic dependence. In Fig. 1d, the water vapor in the snapshot is also made transparent to radiation; again the logarithmic dependence remains. We see, therefore, that the logarithmic dependence of the CO₂ forcing is a phenomenon that does not depend on the presence of water vapor or clouds, or on whether the forcing is measured at the tropopause or TOA. Therefore, in the sections that follow, all forcings will be calculated at the top of the atmosphere and we will work exclusively with dry atmospheres. For an extension to moist atmospheres, see Jeevanjee et al. (2021).

To probe the mechanism of the logarithmic dependence, we will use a line-by-line radiative transfer model. Although the logarithmic dependence is robust across radiative transfer models, a line-by-line model provides spectral fluxes that are straightforward to interpret. The line-by-line model used here is the Reference Forward Model (RFM; Dudhia 2017), which we use with the HITRAN2016 spectroscopic database (Gordon et al. 2017) and flags MIX, CHI, and CTM to implement the line-mixing model of Strow et al. (1994), to apply the line-shape correction χ factor (Le Doucen et al. 1985; Cousin et al. 1985), and to include the CO₂ continuum. Here, and throughout the paper, air is defined to be 21% oxygen by volume with the remaining fraction apportioned among nitrogen and carbon dioxide. Nitrogen and oxygen are treated as transparent to infrared radiation. Since shortwave absorption by CO₂ alters its forcing by only about 4% (Myhre et al. 1998; Etminan et al. 2016), shortwave fluxes are ignored here for simplicity. Unless otherwise specified, the RFM is used with a vertical spacing of Δlog₁₀(p) = 0.05 and wavenumbers are sampled at a spacing of 0.01 cm⁻¹.

We focus here on the 15-μm band of CO₂ because it is the source of the vast majority of the forcing from CO₂ at modern atmospheric concentrations (Augustsson and Ramanathan 1977; Zhong and Haigh 2013; Mlynczak et al. 2016), as can be seen by comparing the top and middle panels of Fig. 2. One of the reasons the 15-μm band dominates the forcing is because the Planck distribution peaks near 15 μm at terrestrial temperatures. The bottom panel of Fig. 2 shows the Planck distribution for two different temperatures. The Planck distribution B (W m⁻² sr⁻¹ cm; emitted radiance per wavenumber interval) is defined as

$B\left(\tilde{\nu },T\right)=\frac{2h{\tilde{\nu }}^3{c}^2}{e^{hc\tilde{\nu }/k_{B}T}-1},$

where $\tilde{\nu }$ is the standard notation for wavenumber (the inverse of wavelength). With this definition, $\pi B\left(\tilde{\nu },T\right)d\tilde{\nu }$ is the power per area of photons emitted by a blackbody surface with wavenumbers between $\tilde{\nu }$ and $\tilde{\nu }+d\tilde{\nu }$.

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Functions of wavenumber $\tilde{\nu }$: (top) $F_{\tilde{\nu }}^{2\times }\left(\tilde{\nu },256\hspace{0.17em}\text{ppmv}\right)$, the spectral forcing, shown here smoothed by a 2 cm⁻¹ moving window, caused by a doubling of CO₂ from 256 to 512 ppmv in IsoStrat, a dry atmosphere with a troposphere sandwiched between an isothermal 205-K stratosphere (p < 10⁴ Pa) and a 289-K surface (p = 10⁵ Pa); (middle) carbon dioxide absorption coefficient κ calculated at a temperature of 289 K and a total air pressure of 1 bar given in units of m² per mole of carbon dioxide molecules; and (bottom) the solid-angle-integrated spectral blackbody Planck emission πB for two different temperatures (205 and 289 K).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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The middle panel of Fig. 2 shows the carbon dioxide absorption coefficient κ calculated at a temperature of 289 K and a total atmospheric pressure of 10⁵ Pa (1 bar). The 15-μm band, centered at 15 μm (667 cm⁻¹) and defined here to be the wavenumbers between 467 and 867 cm⁻¹, is a prominent feature in the absorption spectrum of carbon dioxide, which is another reason why this band dominates its forcing. At the very center of the band (around 667 cm⁻¹), a meter of dry surface air with today’s CO₂ concentration is virtually opaque; at the wings of the band, an entire dry atmospheric column is virtually transparent.

When talking about spectral forcings, we will put a subscript $\tilde{\nu }$ on the forcing variable (e.g., $F_{\tilde{\nu }}^{\text{tot}}$ or $F_{\tilde{\nu }}^{2\times }$) to denote that it is the forcing per wavenumber interval. In particular, $F_{\tilde{\nu }}^{2\times }\left(\tilde{\nu },q\right)d\tilde{\nu }$ is the contribution to F^2×(q) from wavenumbers in $\left[\tilde{\nu },\tilde{\nu }+d\tilde{\nu }\right]$. The top panel of Fig. 2 plots $F_{\tilde{\nu }}^{2\times }\left(\tilde{\nu },256\hspace{0.17em}\text{ppmv}\right)$ for carbon dioxide (calculated with the line-by-line model) in an idealized atmosphere that we will refer to as IsoStrat because the stratosphere is isothermal. All line-by-line calculations were performed using concentrations that are a power of 2 in parts per million by volume, so 256 ppmv is used here as the closest representative of a preindustrial value of ∼280 ppmv. The IsoStrat atmosphere has a 289-K blackbody surface overlain by a troposphere whose temperature drops from 289 K at the 10⁵-Pa surface to 205 K at the 10⁴-Pa tropopause, which is overlain by a 205-K isothermal stratosphere (see section 4 for a precise definition of IsoStrat and the other atmospheres used in this paper). Since this paper focuses on instantaneous forcings, the model atmospheres used here are static (i.e., they are independent of the greenhouse gas concentrations). As seen from Fig. 2, F^2× from CO₂ at modern concentrations is almost entirely due to changes in upwelling longwave fluxes in the 15-μm band, so that is where we will focus our attention in the sections that follow.

3. The basic mechanism

As a visual guide, we will use an analogy to a freight train to describe Earth’s radiation to space from within the 15-μm band. In this analogy, the train track runs flat along Earth’s surface until it slopes upward through the troposphere and into the stratosphere. The train straddles the troposphere with its head in or near the stratosphere and its tail on the surface. The cargo carried by the train are all the wavenumbers of the 15-μm band, with each train car carrying a same-sized set of those wavenumbers. Furthermore, each car has a spotlight pointing upward, sending infrared radiation directly to space with an intensity that depends on temperature: as a train car climbs upward to colder parts of the atmosphere, its light dims. All of the spotlights dim with ascent in the same way, so moving the train forward by one car has a simple effect on the emission to space: it effectively moves one train car from the warm surface to the cold stratosphere. Finally, if the train moves forward by, say, one train car every time the CO₂ concentration is doubled, then every doubling of CO₂ effectively moves one train car from the surface to the stratosphere, dimming its spotlight accordingly; indeed, we will see that the train moves in this way. What makes this analogy to a freight train particularly apropos is that, like a real freight train, each train car is the same length (there are no short cars or long cars), each car holds the same amount of stuff (in this case, equal-sized sets of wavenumbers), and all the cars move with the same speed.

For CO₂ in an otherwise transparent Earth-like atmosphere, there are five properties that combine to make this train analogy work (i.e., that make the forcing from CO₂ approximately logarithmic in its concentration q for q in the range of 4 to 4096 ppmv). These properties are as follows:

1. For carbon dioxide concentrations of 4 to 4096 ppmv—a range that encompasses all values experienced on Earth within the past 300 million years (Berner 2006) as well as any plausible values for the remainder of the Anthropocene—the forcing from carbon dioxide is dominated by the 15-μm band; this tells us that focusing on our 15-μm “freight train” is warranted.

2. For a given pressure and temperature, the wavenumbers in the 15-μm band are distributed approximately uniformly within an interval of log(κ); this tells us that the train cars hold same-sized sets of wavenumbers.

3. Even when accounting for the pressure and temperature dependencies of κ, log(p_em) ∼ −log(qκ) is valid to good approximation [i.e., Eq. (1) still holds]; this tells us that the train cars move together (as they should for a train!) and by how much they move for a given change in the carbon dioxide concentration.

4. For carbon dioxide concentrations ranging from 4 to 4096 ppmv, wavenumbers with the lowest κ in the 15-μm band have an optical depth much less than one at the surface while the wavenumbers with the highest κ have an optical depth of one near the tropopause or in the stratosphere; this tells us that rear of the train is on the surface and the head of the train is in or near the stratosphere.

5. For tropospheric temperatures, 15 μm is where the temperature dependence of the Planck distribution is nearly independent of wavenumber; this tells us that the spotlights all dim in the same way as they ascend.

Figure 3 illustrates how these facts combine to generate the logarithmic forcing of carbon dioxide. By virtue of property 1, we can focus on the 15-μm band, whose wavenumbers have been grouped into seven sets (or “train cars”), schematically represented by the black circles. The wavenumbers could be grouped into many more sets—even an uncountably infinite number—but seven sets are depicted here for simplicity of illustration.⁴ On the left side of Fig. 3, the wavenumbers are shown as uniformly distributed over an interval of log(κ) in accordance with property 2. By property 3, log(p_em) is linear in log(κ), so the wavenumbers are also distributed uniformly over an interval of log(p_em). (For wavenumbers that have an optical depth less than one at the surface, we can still calculate p_em by imagining that we replace the surface with a hydrostatic, semi-infinite atmospheric layer of the same temperature. This has no effect on the radiative fluxes from those wavenumbers above the surface, but it has the benefit of allowing us to visualize how the emission pressures of those wavenumbers approach the surface.) Consistent with property 4, the high values of log(p_em) are below the surface and the low values of log(p_em) are in the stratosphere. By property 3, log(p_em) is also linear in log(q), so each doubling of the CO₂ concentration q moves the train of wavenumbers to lower log(p_em) by the same amount. This is depicted in Fig. 3 with one doubling moving the train upward by one car length and two doublings moving the train upward by two car lengths.

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Explanation of why the forcing from carbon dioxide is logarithmic in its concentration. Each black circle denotes an equal-sized set of wavenumbers. (first axis) By property 1, we can focus on the 15-μm band, which, by property 2, has its wavenumbers uniformly distributed within an interval of log(κ). (second axis) By property 3, this implies that the wavenumbers are uniformly distributed within an interval of log(p_em). Consistent with property 4, the head of the “train” of wavenumbers is in the stratosphere and the rear of the train is in the surface. (third axis) By property 3, log(p_em) is linear in log(q), so multiplying q by successive multiplicative factors (e.g., doublings) moves the train to lower log(p_em) by the same amount. Here, a doubling is illustrated as moving the train up by one car. By property 5, the Planck distribution is approximately a function of temperature only, so a set of wavenumbers emitting to space from the warm surface temperature (red colors) has been effectively replaced by an equal-sized set of wavenumbers emitting to space from the cold stratosphere (blue colors). (fourth axis) Another doubling moves another equal-sized set of wavenumbers from the surface to the stratosphere. The size of the set of wavenumbers moved from the surface to the stratosphere is proportional to log(q), so the forcing is proportional to log(q). As a visual aid, the light-gray circles mark where the train was for q = q₀.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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As we have seen, the derivation of logarithmic forcing hinges on the validity of properties 1 through 5. Section 5 will explore these properties in more detail and show that they hold for carbon dioxide in an Earth-like atmosphere.

4. Model atmospheres

We will describe here the five model atmospheres used throughout the paper. All of the atmospheres are dry, transparent to shortwave radiation, hydrostatically balanced with Earth’s gravitational constant, and have temperature profiles that are piecewise linear in the logarithm of pressure. The first three atmospheres are IsoAtmo (a cold isothermal atmosphere sitting on top of a warm surface), IsoStrat (a cold isothermal stratosphere that is separated from the warm surface by a troposphere with an Earth-like lapse rate), and StdAtmo (an atmosphere in which both the troposphere and stratosphere have Earth-like lapse rates).

Figure 4 plots the temperature profiles of those three atmospheres as black curves. The gray curve in each of the panels is the global area-weighted mean of atmospheric temperature during year 2020 in the ERA5 reanalysis (Hersbach et al. 2020), plotted from the surface (at 10⁵ Pa), through the cold-point tropopause (at 10⁴ Pa), and up to the top of the stratosphere (at 10² Pa). Noting that the ordinate is the logarithm of pressure, we see that the global-mean temperature can be approximated as two pieces linear in log(p) that connect a near-surface air temperature of 289 K, to a tropopause temperature of 205 K, and back up to a stratopause temperature of 261 K.

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The temperature profiles of three model atmospheres: (from left to right) IsoAtmo, IsoStrat, and StdAtmo. The area-weighted mean of atmospheric temperature during year 2020 in the ERA5 (gray).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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IsoAtmo, shown in the left panel, places a cold isothermal atmosphere directly on top of the warm surface. Here, as throughout the paper, the surface is represented by a semi-infinite atmospheric layer with the same gas composition as the other layers; this is done purely for the purposes of visualization and has no impact on the radiative fluxes at pressures below 10⁵ Pa. In reality, placing a cold isothermal layer directly on top of a warm isothermal layer would induce vigorous convection, but IsoAtmo is a well-defined atmosphere that will serve an important purpose: it will demonstrate that the logarithmic forcing persists even without a continuously varying temperature profile.

IsoStrat, shown in the middle panel, has a troposphere with a realistic lapse rate, but has a stratosphere that is isothermal. The advantage of this atmosphere is conceptual simplicity: the surface and stratosphere are isothermal, and so the movement of sets of wavenumbers from the surface to the stratosphere has a straightforward radiative effect. Fortunately, this simplicity is largely retained in the real atmosphere because the stratosphere has a mean lapse rate—defined here as dT/dlog(p)—whose magnitude is only one-third as large as the troposphere’s: dT/d log(p) is 84 K in the troposphere and −28 K in the stratosphere. Therefore, we will find that the logarithmic nature of the forcing is retained in StdAtmo, which is the atmosphere shown in the right panel of Fig. 4.

These three atmospheres, plus two others, are defined in Fig. 5. The HotStrat and DeepTrop atmospheres are modifications of IsoStrat in which the stratosphere is given the same temperature as the surface (HotStrat) or the troposphere is made much deeper (DeepTrop). HotStrat and DeepTrop will be used in sections 9 and 10.

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For x ≡ log₁₀(p/Pa), the temperature (in K) for each atmosphere (columns) within each range of x (rows).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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5. The five properties

a. Property 1: Dominance of the 15-μm band

We saw in the top panel of Fig. 2 that F^2×(256 ppmv) in IsoStrat is dominated by the 15-μm band. Figure 6 shows, using the line-by-line model, that the 15-μm band dominates over a wide range of CO₂ concentrations in StdAtmo (the most realistic of the model atmospheres). The solid curve shows the fraction of F^tot generated by wavenumbers in the 15-μm band (recall that F^tot is the TOA upwelling flux without the CO₂ minus the TOA upwelling flux with the CO₂). The dashed curve shows the fraction of F^2× (change in F^tot from a doubling of CO₂) that is caused by wavenumbers in the 15-μm band. Since F^2×(q) is defined as the change in forcing from q to 2q, the value at 2048 ppmv corresponds to doubling from 2048 to 4096 ppmv. We see that, for concentrations ranging from 4 to 4096 ppmv, the 15-μm band is responsible for the majority of the forcing, ranging from 89% to 97% for F^tot and from 61% to 98% for F^2×. Between 2048 and 4096 ppmv, the bands of CO₂ centered at 15, 10, 7.6, 5, and 4.3 μm contribute 61%, 27%, 6%, 5%, and 1% to F^2×, respectively, indicating that bands at shorter wavelengths start to make substantial contributions to the forcing at these large concentrations (Zhong and Haigh 2013). At preindustrial and current concentrations, the 15-μm band is responsible for 96% of F^tot and 82%–86% of F^2×.

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Fraction of F^tot in StdAtmo that is generated by the 15-μm band (solid). Fraction of F^2× in StdAtmo that is generated by the 15-μm band (dashed).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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b. Property2: Uniform distribution of log(κ)

The top-left panel of Fig. 7 shows, in the solid curve, the probability distribution function (PDF) of the log(κ) values within the 15-μm band calculated at the surface (1 bar and 289 K) using the line-by-line model. The dashed curve depicts a uniform distribution for illustration. We see that the line-by-line PDF approximates a uniform distribution over a remarkable six orders of magnitude. The bottom-left panel of Fig. 7 shows the corresponding cumulative distribution function (CDF) of the κ values (i.e., the integral of the PDF). The integral of a uniform distribution, shown by the dashed curve, is an upward-sloping line with values bounded below and above by zero and one, respectively. We see that the line-by-line CDF approximates this closely. The right column plots the PDF and CDF in the stratosphere (0.01 bar and 233 K). Note that the distribution remains approximately uniform, but has shifted to lower κ by a factor equal to the ratio of pressures; this behavior will be explained in section 5c.

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(top left) Probability distribution function (PDF) of the values of log₁₀[κ/(m² mol⁻¹)] within the 15-μm band calculated at 1 bar and 289 K using the line-by-line model. The dashed curve depicts a uniform distribution for illustration. (bottom left) The corresponding cumulative distribution functions (CDFs). (right) As in the left column, but for 0.01 bar and 233 K (the temperature there in StdAtmo).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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The uniform distribution is also apparent when looking at the spectrum as a function of wavenumber. The top-left part of Fig. 8 shows the absorption coefficient $\kappa \left(\tilde{\nu }\right)$ for the 15-μm band plotted on a log axis and calculated using the line-by-line model at 1 bar and 289 K. Property 2 is apparent from the approximately linear decay of log(κ) as a function of wavenumbers from the center of the band.

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(top left) Plots of the absorption coefficient $\kappa \left(\tilde{\nu }\right)$ of carbon dioxide at 1 bar and 289 K (solid) and the resorted ${\kappa }^{\prime }\left(\tilde{\nu }\right)$ defined in Eq. (2) (dashed). (bottom left) The Planck distributions at temperatures of 205 and 289 K. (top right) The simple model’s absorption coefficient $\kappa \left(\tilde{\nu }\right)$, which is exponential in $\tilde{\nu }$. (bottom right) The simple model’s Planck distributions, which have no explicit dependence on wavenumber.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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Let us now resort those wavenumbers in order of increasing κ, as is done in the k distribution method (e.g., Stephens 1984), which is a standard technique in radiation parameterization. Let the function CDF be the cumulative distribution of κ values within the 15-μm band at this pressure and temperature, which was shown in the bottom-left panel of Fig. 7. In other words, CDF(κ) is the fraction of the 400-cm⁻¹-wide interval of wavenumbers that have an absorption coefficient less than κ. The inverse of this, CDF⁻¹, maps the interval [0, 1] to absorption coefficients in a monotonically increasing way. We can then define ${\kappa }^{\prime }\left(\tilde{\nu }\right)$ as

$\kappa \prime \left(\tilde{\nu }\right)={\text{CDF}}^{-1}\left(\frac{\tilde{\nu }-{\tilde{\nu }}_1}{{\tilde{\nu }}_2-{\tilde{\nu }}_1}\right),$(2)

where ${\tilde{\nu }}_{\text{1}}=\text{467}\hspace{0.17em}{\text{cm}}^{-\text{1}}$ and ${\tilde{\nu }}_{\text{2}}=\text{867}\hspace{0.17em}{\text{cm}}^{-\text{1}}$ are the boundaries of the 15-μm band. By construction, this function is simply a resorting of the original absorption coefficients. It is plotted as the dashed curve in Fig. 8, and it is nothing more than the curve in the bottom panel of Fig. 7 with the axes swapped. Note that log(κ′) is nearly linear in wavenumber.

The lower-left part of Fig. 8 plots the Planck distribution within the 15-μm band at temperatures of 205 and 289 K. We see that the difference between the two Planck distributions is nearly independent of wavenumber, consistent with property 5. Section 5f will show that this is sufficient for us to ignore the wavenumber dependence of the Planck distribution for the purposes of understanding the forcing. Once this approximation is made, the only physical property that depends on wavenumber is κ. Therefore, we are free to resort the wavenumbers so long as we keep track of the corresponding κ values. In particular, we can replace κ with κ′ in calculations of the forcing without affecting the answer.

To go a step further, we can approximate the spectrum shown in the left panel of Fig. 8 with the κ shown in the right panel. This represents $\kappa \left(p_0,\tilde{\nu }\right)$ at p₀ = 10⁵ Pa and it is written mathematically as

$\kappa \left(p_0,\tilde{\nu }\right)=\{\begin{array}{ll}{\kappa }_0{e}^{b\tilde{\nu }}\hfill & {\tilde{\nu }}_1<\tilde{\nu }<{\tilde{\nu }}_2\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}$(3)

with ${\tilde{\nu }}_{\text{1}}=\text{467}\hspace{0.17em}{\text{cm}}^{-\text{1}}$ and ${\tilde{\nu }}_{\text{2}}=\text{867}\hspace{0.17em}{\text{cm}}^{-\text{1}}$. Fitting Eq. (3) to the κ′ in the left panel using least squares, we find best-fit coefficients of κ₀ = 8.4 × 10⁻¹⁵ m² mol⁻¹ and b = 0.04 cm. Equation (3) is nearly all of the spectroscopic information that is needed to construct a model of the forcing from carbon dioxide. The only remaining detail is to specify how κ varies with pressure.

c. Linearity of κ in pressure

In section 3, we ignored any pressure or temperature dependence of κ when we derived relation (1). We will now show that relation (1) continues to hold even with those dependencies.

The value of κ at any particular wavenumber has contributions from spectral lines representing many different vibrational and rotational transitions of CO₂, and each of those lines has a shape that varies with temperature and pressure. To get an indication of how this sum of contributions might change with pressure and temperature, we can use the Lorentz line shape. The Lorentz line shape can be derived from first principles with some approximations (e.g., Van Vleck and Weisskopf 1945) and is supported empirically within about 100 line widths of the line center (Pierrehumbert 2010). Although line-by-line models use a line shape that deviates from this, especially far from the line center, the Lorentz line shape is adequate for our discussion here.

With a Lorentz line shape, the absorption coefficient due to a single line centered at ${\tilde{\nu }}_c$ is

$\kappa =\frac{S\left(T\right)}{\pi \gamma }\hspace{0.17em}\frac{{\gamma }^2}{{\gamma }^2+{\left(\tilde{\nu }-{\tilde{\nu }}_c\right)}^2},$

where the line width γ can be represented by

$\gamma ={\gamma }_0\hspace{0.17em}\frac{p}{p_0}{\left(\frac{T_0}{T}\right)}^n$

with an n that varies from line to line but is typically around 0.5 (Pierrehumbert 2010). This line shape is plotted in the left panel of Fig. 9 using γ = 0.1 cm⁻¹.

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(left) Absorption coefficient κ for a single spectral line with a Lorentz line shape with width γ = 0.1 cm⁻¹. The absorption coefficient is plotted as a function of distance (in wavenumber) from the line center, and it is plotted as normalized by the value at line center. (right) The change in log(κ) for this spectral line per change in log(p), keeping temperature constant. Where the value is one, κ scales linearly with pressure.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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Although both the line intensity S and line width γ depend on temperature, it is the dependence of γ on pressure that tends to dominate the changes in line shapes as we move up and down in the atmosphere (Pierrehumbert 2010). Far from the line center ($\tilde{\nu }-{\tilde{\nu }}_c\gg \gamma$), when pressure is varied isothermally, κ varies linearly with pressure. The right panel of Fig. 9 plots ∂log(κ)/∂log(p) to illustrate this point: everywhere except for the vicinity of the line center, this derivative is unity, indicating a proportionality with respect to pressure.

To the extent that the prominent spectral lines of CO₂ are sufficiently well spaced, this result tells us that most of the 15-μm band should scale linearly with pressure. Thus, it might be sufficient to approximate κ as linear in pressure throughout the entire band. It is not obvious a priori that this should work since there are also temperature dependencies of the individual line shapes and line strengths, but it does work in many cases and it is a common approximation (Pierrehumbert 2010). To demonstrate the approximate linearity in pressure, Fig. 10 plots κ calculated from the line-by-line model as a function of pressure level for 4000 equally spaced wavenumbers in each of three different atmospheres (IsoAtmo, IsoStrat, and StdAtmo). To avoid the figures becoming saturated with color, the individual curves are plotted using a translucent color. We see that, regardless of the thermal structure of the atmosphere, the vast majority of the wavenumbers have κ values that increase quasi-linearly as we move to higher pressure in the atmosphere. The lower-right panel quantifies this by plotting three histograms (one for each atmosphere) of the slopes of the best-fit lines to each of the 4000 curves. Less than 2% (1%, 3%) of the slopes in IsoAtmo (IsoStrat, StdAtmo) case are negative. We see that the slopes are clumped around unity, demonstrating the quasi-linear dependence of κ on pressure.

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Translucent plots of $\kappa \left[\tilde{\nu },p,T\left(p\right)\right]$ as a function of p for 4000 equally spaced samples of $\tilde{\nu }$ from 467 to 867 cm⁻¹ for the (top left) IsoAtmo, (top right) IsoStrat, and (bottom left) StdAtmo atmosphere. The bar in $\overline{{\log }_{10}\left(\kappa \right)}$ represents an average over log₁₀(p/Pa) from 2 to 5. (bottom right) For each case, the histogram of the best-fit slopes for each of the 4000 wavenumbers.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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Since the κ values are approximately linear in p, we can generalize Eq. (3) to give a simplified model for κ at any pressure:

$\kappa \left(p,\tilde{\nu }\right)=\{\begin{array}{ll}\frac{p}{p_0}\hspace{0.17em}{\kappa }_0{e}^{b\tilde{\nu }}\hfill & {\tilde{\nu }}_1<\tilde{\nu }<{\tilde{\nu }}_2\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$(4)

This equation, together with the constants given at the end of section 5b, is all of the spectroscopic information we need about carbon dioxide to understand its forcing for concentrations in the range of 4 to 4096 ppmv.

e. Property 4: Head in the stratosphere and rear in the surface

With Eq. (7), we can evaluate whether property 4 is true, that is, if the head of the wavenumber train is in or near the stratosphere and if the rear of the wavenumber train is in the surface. Using the values for b and κ₀ obtained in section 5b, we can calculate the minimum and maximum emission pressures from Eq. (7) as $p_{\text{em}}\left({\tilde{\nu }}_2,q\right)$ and $p_{\text{em}}\left({\tilde{\nu }}_1,q\right)$, which correspond to the head and rear of the train, respectively. Figure 11 plots these as functions of q from 4 to 4096 ppmv. At all of these concentrations, the rear is in the surface. At modern concentrations, the head is in the stratosphere. At 4 ppmv, the head of the train is in the upper troposphere at 300 mbar, but we will see in section 7 that this is close enough to the tropopause to make F^2× nearly as large (75% as big) as it is when the head of the train is well ensconced in the stratosphere.

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The locations of the train’s head and rear, $p_{\text{em}}\left({\tilde{\nu }}_2,q\right)$ and $p_{\text{em}}\left({\tilde{\nu }}_1,q\right)$, as functions of the CO₂ concentration q.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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Only those wavenumbers emitting to space from above the surface contribute to F^tot. Those are the wavenumbers whose $\kappa \left(p_0,\tilde{\nu }\right)$ exceeds 2gm₀/fqp₀. For q = 4 (256, 4096) ppmv, these are the wavenumbers with $\kappa \left(p_0,\tilde{\nu }\right)$ exceeding 0.9 (0.01, 8 × 10⁻⁴) m² mol⁻¹. The middle panel of Fig. 2 can be used to identify the wavenumbers in the original (unsorted) spectral space that contribute to F^tot at each of these concentrations.

f. Property 5: Wavenumber independence of the Planck distribution

If we approximate all of a wavenumber’s radiation to space as coming from its emission pressure, then the spectral forcing from a doubling of concentration is

$F_{\tilde{\nu }}^{2\times }\left(\tilde{\nu },q\right)\approx \pi B\left\{\tilde{\nu },T\left[p_{\text{em}}\left(\tilde{\nu },q\right)\right]\right\}-\pi B\left\{\tilde{\nu },T\left[p_{\text{em}}\left(\tilde{\nu },2q\right)\right]\right\}.$(8)

We see that $F_{\tilde{\nu }}^{2\times }$ does not depend on the overall magnitude of B, but on how much it changes with respect to temperature. Likewise, F^2×, which is simply an integral of $F_{\tilde{\nu }}^{2\times }$, depends only on how B changes with temperature and not its overall magnitude. Therefore, if the change in B with temperature is independent of the wavenumber to good approximation, then we can approximate B as having no explicit dependence on wavenumber (as was depicted in the right column of Fig. 8).

In Earth’s troposphere, the globally averaged temperatures range from 205 to 289 K. We need to know, therefore, whether $B\left(\tilde{\nu },289\hspace{0.17em}\text{K}\right)-B\left(\tilde{\nu },205\hspace{0.17em}\text{K}\right)$, for the various $\tilde{\nu }$ within the band, can be approximated as $B\left({\tilde{\nu }}_0,289\hspace{0.17em}\text{K}\right)-B\left({\tilde{\nu }}_0,205\hspace{0.17em}\text{K}\right)$, where ${\tilde{\nu }}_0$ is the wavenumber in the middle of the band. For this to be true, we would need dB/dT at the midtropospheric temperature of 247 K (the average of 205 and 289 K) to be independent of $\tilde{\nu }$ in the vicinity of $\tilde{\nu }={\tilde{\nu }}_0$. In other words, we would need $d^{\text{2}}B/dT\hspace{0.17em}d\tilde{\nu }=0$ at T = 247 K and $\tilde{\nu }=1/\left(15\hspace{0.17em}\mu \text{m}\right)=667\hspace{0.17em}{\text{cm}}^{-\text{1}}$. By differentiation of the Planck distribution, we find that $d^{\text{2}}B/dT\hspace{0.17em}d\tilde{\nu }=0$ when $\tilde{\nu }\hspace{0.17em}=\mathit{xkT}/hc$, where x is the solution to x = 4(1 − e^−x)/(1 + e^−x), which evaluates to x ≈ 3.83. For the midtropospheric temperature of T = 247 K, this gives $\tilde{\nu }=658\hspace{0.17em}{\text{cm}}^{-1}=1/\left(15.2\hspace{0.17em}\mu \text{m}\right)$, confirming that the 15-μm band is ideally situated for property 5 to be valid.

We can also check property 5 numerically. For any particular wavenumber $\tilde{\nu }$ in the band, the relative error (RE) caused by ignoring the wavenumber dependence of B is

$\text{RE}=\frac{B\left({\tilde{\nu }}_0,289\hspace{0.17em}\text{K}\right)-B\left({\tilde{\nu }}_0,205\hspace{0.17em}\text{K}\right)}{B\left(\tilde{\nu },289\hspace{0.17em}\text{K}\right)-B\left(\tilde{\nu },205\hspace{0.17em}\text{K}\right)}-1.$

For a band that is 400 cm⁻¹ wide like the 15-μm band of CO₂, the root-mean-square (RMS) of this relative error (RMSRE) across all the wavenumbers in the band is

$\text{RMSRE}={\sqrt{\frac{1}{400{\text{ cm}}^{-1}}{\displaystyle {\int }_{{\tilde{\nu }}_0-200\hspace{0.17em}{\text{cm}}^{-1}}^{{\tilde{\nu }}_0+200\hspace{0.17em}{\text{cm}}^{-1}}d\tilde{\nu }}\left[\frac{B\left({\tilde{\nu }}_0,289\hspace{0.17em}\text{K}\right)-B\left({\tilde{\nu }}_0,205\hspace{0.17em}\text{K}\right)}{B\left(\tilde{\nu },289\hspace{0.17em}\text{K}\right)-B\left(\tilde{\nu },205\hspace{0.17em}\text{K}\right)}-1\right]}}^2.$(9)

This RMSRE is plotted in Fig. 12 as a function of ${\tilde{\nu }}_0$. We see that the center of the 15-μm CO₂ band is located almost exactly where the RMSRE is minimized. For the 15-μm band, replacing $\tilde{\nu }$ with ${\tilde{\nu }}_0=\text{667}\hspace{0.17em}{\text{cm}}^{-\text{1}}$ in the Planck distribution generates a root-mean-square relative error of only 7%. Since the purpose of this paper is understanding the origin of carbon dioxide’s logarithmic forcing, this error is perfectly acceptable. In recognition of property 5, we will henceforth write the Planck distribution as a function only of temperature [i.e., as B(T)] with the understanding that it is evaluated at a wavenumber of ${\tilde{\nu }}_0=\text{667}\hspace{0.17em}{\text{cm}}^{-\text{1}}$, that is,

$B=B\left(T\right)=\frac{2h{\tilde{\nu }}_0^3{c}^2}{e^{hc{\tilde{\nu }}_0/k_{B}T}-1}.$(10)

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Error [RMSRE from Eq. (9)] produced by ignoring the explicit wavenumber dependence of the Planck distribution, plotted as a function of the center of a hypothetical 400 cm⁻¹-wide band. The 15-μm band of carbon dioxide is located close to where this error is minimized.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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6. The log(p) axis

To visualize a greenhouse gas’s radiative forcing, we need to derive the weighting function for emission to space, but, to do that, we must first choose an appropriate vertical axis. We learned in section 3 that, by virtue of relation (1) and the particular spectrum of CO₂, the wavenumbers in the 15-μm band are uniformly distributed within an interval of log(p_em). This suggests that the most natural vertical coordinate for understanding the radiative forcing from carbon dioxide is not pressure p or height z, but the logarithm of pressure log(p). In fact, the case for using log(p) as the vertical coordinate goes even deeper: not only is it natural to use for CO₂, but the log(p) axis is also a natural vertical axis to use for any generic greenhouse gas with κ ∝ p.

To see why, note that the spectral flux to space of infrared radiation can be written as

$\text{Spectral}\hspace{0.17em}\text{flux}\hspace{0.17em}\text{to}\hspace{0.17em}\text{space}={\displaystyle {\int }_0^{\infty }\pi }B{e}^{-\tau }\hspace{0.17em}d\tau .$(11)

From the expression for τ in Eq. (5), we know that τ ∝ p² so long as κ ∝ p. This then implies that dτ = 2τd log(p). This relation allows us to transform the expression in Eq. (11) from an integral over τ to an integral over log(p):

$\text{Spectral}\hspace{0.17em}\text{flux}\hspace{0.17em}\text{to}\hspace{0.17em}\text{space}={\displaystyle {\int }_{-\infty }^{\infty }\pi }B\underset{\equiv {\varphi }_{\log \left(p\right)}}{\underbrace{2e^{-\tau }\tau }}\hspace{0.17em}d\hspace{0.17em}\log \left(p\right),$(12)

where we see that

${\varphi }_{\log \left(p\right)}\left(\tilde{\nu },p,q\right)\equiv 2e^{-\tau }\tau$(13)

is the spectral weighting function for emission to space on the axis of log(p). In other words, ${\varphi }_{\log (p)}\left(\tilde{\nu },p,q\right)d\hspace{0.17em}\log \left(p\right)$ is the fraction of emission to space at wavenumber $\tilde{\nu }$ that emanates from between log(p) and log(p) + d log(p). Integrating ϕ_log(p) over all log(p) gives a value of one: this can be checked by noting that ϕ_log(p)d log(p) = e^−τdτ. Figure 13 plots ϕ_log(p) for several different concentrations q that all differ by factors of 2. Note that Fig. 13 is not specific to carbon dioxide: it applies to any atmosphere with a single well-mixed greenhouse gas whose absorption coefficients are proportional to pressure.

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Weighting function ${\varphi }_{\log \left(p\right)}\left(\tilde{\nu },p,q\right)$ of emission to space for any atmosphere with a single well-mixed greenhouse gas whose absorption coefficients are proportional to pressure. The rightmost curve has q = q₀ and so peaks where $\log \left(p\right)=\log \left[p_{\text{em}}\left(\tilde{\nu },q_0\right)\right]$. Curves for eight other values of q are also plotted with the value of q labeled just under the peak of each curve.

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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7. Simple model of the forcing

To calculate the broadband weighting function ψ, we need to know the details of the function $p_{\text{em}}\left(\tilde{\nu },q\right)$. In our simple model of carbon dioxide, $p_{\text{em}}\left(\tilde{\nu },q\right)$ takes the form of Eq. (7), which encodes the fact that the wavenumbers of carbon dioxide’s 15-μm band have a uniform distribution on log(κ). Using Eqs. (7), (15), and (17) to calculate ψ (see the appendix for details), we get

$\psi \left(p,q\right)=\frac{2}{b}\left[e^{-p^2/p_{\text{em}}{\left({\tilde{\nu }}_1,q\right)}^2}-e^{-p^2/p_{\text{em}}{\left({\tilde{\nu }}_2,q\right)}^2}\right].$(18)

This is plotted in Fig. 14 for q = 256 ppmv as the curve labeled by n = 2. Note that the magnitude of ψ is 2/b = 50 cm⁻¹. Mathematically, Eq. (18) is closely related to a boxcar function: if the exponents of 2 are replaced with an integer n that is then taken to infinity, we obtain a boxcar function, that is,

$\underset{n\to \infty }{\lim }\left[e^{-p^n/p_{\text{em}}{\left({\tilde{\nu }}_1,q\right)}^n}-e^{-p^n/p_{\text{em}}{\left({\tilde{\nu }}_2,q\right)}^n}\right]=\{\begin{array}{ll}1\hfill & p_{\text{em}}\left({\tilde{\nu }}_2,q\right)<p<p_{\text{em}}\left({\tilde{\nu }}_1,q\right)\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$(19)

The same boxcar function for ψ would be obtained if the factors of 2 in Eq. (17) for ϕ were replaced with n and taken to infinity; in that limit, $\varphi \left(\tilde{\nu },p,q\right)=\delta \left\{\log \left(p\right)-\log \left[p_{\text{em}}\left(\tilde{\nu },q\right)\right]\right\}$ and wavenumbers emit to space exactly from their p_em. As is evident from Fig. 14, the n = ∞ boxcar limit is a decent approximation for the real n = 2 case, at least for the purposes of understanding the behavior of the forcing and its overall magnitude.

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Plot of ψ, the broadband weighting function for emission to space, from Eq. (18) (blue). Plots of Eq. (18) with the exponents of 2 replaced with other positive integers (gray). Plot of Eq. (18) with the exponents of 2 replaced with ∞ (black).

Citation: Journal of Climate 35, 13; 10.1175/JCLI-D-21-0275.1

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