Simple models for Earth's climate sensitivity (i.e. its temperature response to radiative forcing) are developed by combining the time-tested idealization of one-dimensional radiative-convective equilibrium (RCE) with simple yet quantitatively reasonable models for forcing and the water vapor feedback. Along the way, we introduce key paradigms including the emission level approximation, the forcing-feedback decomposition of climate sensitivity, and “Simpson's law” for water vapor thermal emission. We also discuss climate feedbacks unaccounted for in this RCE framework, as well as differing variants of climate sensitivity, all of which may be ripe for their own chalkboard treatments.
Earth's equilibrium climate sensitivity (ECS), or the long-term global-mean surface temperature change due to a doubling of , is arguably the most central quantity in climate science. First studied by Arrhenius in 1896,1 ECS sets the overall magnitude and, thus, the severity of global warming and remains a topic of intense interest to the present day.2
The consensus value of ECS has remained close to 3 K for decades, throughout many successive generations of model intercomparisons and literature reviews, most notably the assessment reports from the Intergovernmental Panel on Climate Change (IPCC).3 These assessments have been based largely on calculations with numerical models of increasing complexity, along with observations and paleoclimate reconstructions. However, the first credible estimate of ECS, given by Manabe and Wetherald4 in 1967 as 2–3 K, used a highly simplified one-dimensional representation of the climate system known as radiative-convective equilibrium (RCE; we refer to this paper hereafter as MW67). This model for Earth's climate makes some drastic simplifications, such as representing the atmosphere in terms of a single, global average column, but then treats other aspects of the climate system in detail, such as the frequency-dependent greenhouse gas radiative transfer of both water vapor ( ) and carbon dioxide ( ). These approximations, and insight they facilitated, have proved remarkably durable, and were recognized with Manabe's receipt of part of the 2021 Nobel Prize in Physics.5
The simplicity of Manabe's RCE approach and the robustness of the 3 K value for ECS suggest that much of the physics of ECS might actually be understood using a handful of basic physical principles, rather than lying hidden behind the intractable complexity of the climate system. Indeed, recent research has shown that essential topics, such as radiative forcing6 and the feedback,7 can be described to a reasonable approximation with simplified models amenable to analytic description. Taken together, such models allow for an analytic estimate of ECS, at least within the simplified context of RCE. A self-contained formulation of such an estimate is a primary goal of this paper.8
The utility of such an estimate and the motivations for documenting it here are many-fold. The most obvious motivation is for classroom teaching. Current texts in climate science at the advanced undergraduate or beginning graduate level, such as the excellent books by Hartmann9 and Pierrehumbert,10 treat the fundamentals extremely well and use them to build intuition but then tend to jump to empirical observations and numerical simulations to describe real-world phenomena. This is often the best that can be done, but reasonably quantitative chalkboard estimates (where possible) help fill this gap between fundamental theory and empirical observation. Such back-of-the-envelope reasoning is also helpful for everyday practitioners trying to understand and debug the behavior of complex climate models.11 The analytical approach taken here might also appeal more naturally to physicists, providing them a more suitable entry point for understanding or perhaps even contributing to climate science.
However, there are also more profound reasons for pursuing a simplified understanding of ECS and climate science more broadly.12 Simple models of the kind developed here embody our understanding of the subject at its most basic level. This philosophy is well known to physicists in the guise of Fermi problems and the Feynman lectures. Given the societal importance of climate change, as well as lingering skepticism about it in some quarters, a chalkboard explanation of ECS, even if only approximate, seems essential for demonstrating the depth of our understanding.
Given this motivation, we develop an estimate of ECS in the simplified framework of RCE pioneered by Manabe. We begin by establishing the paradigm of radiative-convective equilibrium as well as the forcing-feedback framework. We then turn to simple models of the greenhouse effect for both and , which lead to analytical descriptions of both forcing and the water vapor feedback. These ingredients are then combined, via the forcing-feedback framework, to yield an analytical estimate of ECS. We will find that this estimate is around 2 K, within the range found by MW67 but somewhat smaller than the consensus value of 3 K. This discrepancy is due to the approximations inherent in the RCE approach, which neglects clouds and changes in absorbed solar radiation. We then discuss these phenomena and the prospects for capturing their effects via principled estimates. We close by introducing other metrics of climate sensitivity that account for ocean heat uptake and the carbon cycle, and for which a chalkboard understanding might also be useful.
For classroom instruction, this material might be considered a unit of perhaps eight lectures at the advanced undergraduate or beginning graduate level for either physics students or students of atmospheric and climate science. Suggestions for exercises are sprinkled throughout the main text and the endnotes.
II. RADIATIVE-CONVECTIVE EQUILIBRIUM (RCE)
We begin by building a very simple model for Earth's surface temperature , based on the idea of radiative-convective equilibrium (RCE), which we explain in detail below. Before turning to RCE, however, we must consider the even more basic notion of planetary energy balance.
A. Planetary energy balance
The zeroth order determinant of is the amount of sunlight Earth receives. The solar flux at Earth's orbit is , and this flux is incident on an effective surface area (the projected area of the Earth onto a plane perpendicular to the Sun's rays, where RE is Earth's radius). A significant fraction of this incident flux is reflected back to space, primarily via clouds and gaseous atmospheric (Rayleigh) scattering, as well as from bright surfaces such as deserts and ice caps. This fraction is known as Earth's albedo α, measured to be .13
B. Single-layer radiative equilibrium
These equations can be immediately solved to yield K and K, overestimating by 15 K.
Despite this disagreement, this model is standard in introductory texts.15 Many of these texts remedy this disagreement by introducing a non-unit emissivity for the atmosphere, but the value of this emissivity is typically set by requiring , thus eliminating the model's predictive power. Furthermore, this approach does not remedy the model's grossest approximation, which is not that of unit emissivity but rather of neglecting heat transfer by convection.
C. A radiative-convective equilibrium model
Convection brings water vapor and heat from the surface into the interior of the atmosphere, where the water vapor condenses to form cloud and rain drops. This condensation releases the latent heat of vaporization which was drawn from the surface when the molecules first evaporated, and this heat from condensation is then radiated out to space by atmospheric GHGs, closing the loop. These processes are depicted on the right-hand side of Fig. 1. Thus, planetary energy balance is not achieved through purely radiative means but is mediated by convection, with water vapor as the key middleman. The atmosphere is, thus, better thought of as in a state of radiative-convective equilibrium, or RCE for short.16
An important subtlety in the simple calculation above is that it was not necessary to solve the surface energy balance equation (8b); the convective flux there acts as a Lagrange multiplier, taking on whatever value is required to satisfy Eq. (8b) subject to the constraint (8c); the convective flux is, thus, analogous to the tension in a pendulum arm, which takes on whatever value is required to satisfy Newton's laws while keeping the pendulum bob at a fixed distance from its origin.
A profound implication of the convective constraint (10a) is that the tropospheric temperature profile is pegged to , and the two cannot be varied independently. In other words, the surface and atmosphere should be thought of as a unit, with a single degree of freedom between them.21 This tight surface-atmosphere coupling is a hallmark of RCE and has important consequences, as we will see below.
III. FORCING-FEEDBACK PRELIMINARIES
Now that we have a picture for Earth's energy flows and RCE, we can begin to think about climate sensitivity. An extremely useful paradigm for this is the forcing-feedback framework, which we describe next.
A. The forcing-feedback framework
In this paper, and in much (but certainly not all) climate modeling, the atmospheric specific concentration22 q (kg /kg air) is considered an external parameter, which is prescribed and does not respond to the internal dynamics of the system.23 In studying changes in surface temperature , which result from changes in q, it turns out to be extremely convenient to decompose the system's response into two distinct processes, one which occurs at fixed , and the other which occurs at fixed q.24 (Also, it will be natural sometimes to use as our variable for planetary thermal energy flux, since an increase in indicates increasing planetary thermal energy.)
B. Blackbody estimate of ECS
To understand why , as well as make a better estimate of λ and hence ECS, we need to move beyond the blackbody approximation and account for the spectral nature of Earth's greenhouse effect, i.e., that the emission temperature of outgoing longwave radiation actually depends rather markedly on frequency, and that at a given frequency and may not necessarily exhibit a 1–1 relationship as in Eq. (9). A key ingredient in understanding how these quantities do behave will be the emission level approximation, which we turn to next.
C. Emission level approximation
The next task is to determine what level(s) in the atmosphere determine , for a given . This task is aided by the following heuristic, illustrated in Fig. 3. Consider an atmospheric column with GHG molecules whose density [ ] decreases exponentially with height; this is true for both and .29 Now consider the emission to space (i.e., the contribution to the at a given ) from these molecules, as pictured in Fig. 3. The top two layers (Fig. 3(a)) have little difficulty emitting to space, because their view is unobstructed, but the density of emitters in these layers is relatively low, so the emission will also be low. In the third layer (Fig. 3(b)), the molecules' view of space is still unobstructed (just barely), and their density is higher, so their emission to space is higher. For layers four and five (Fig. 3(c)), there are plenty of emitters, but their view is almost totally obstructed, so their emission to space is again very low. Thus, emission to space is maximized around a “sweet spot” where the absorbers/emitters above have not yet totally obstructed the view of space, but the density is high enough for emission to be appreciable. This sweet spot will be our emission level.
IV. CO2 FORCING
This section constructs a simple analytic model for the forcing Eq. (11), with the aim of enabling a back-of-the-envelope estimate of the characteristic 4 value for . The approach here is to consider spectral variations in absorption but to do so in a simplified manner, so as to keep the model analytically tractable. The model we construct has precedent in relatively recent literature, e.g., the works of Wilson and Gea-Banacloche,33 Seeley34 Jeevanjee et al.,35 and Romps et al.,36 but has not yet appeared in textbooks. We present here a very simple version of the model, which can be generalized to include the effects of a non-isothermal stratosphere as well as spectral overlap between and absorption.37
The formalism developed here also yields insight into the logarithmic scaling of forcing, evident in the factor in Eqs. (22) and (23). Unwinding the mathematics leading to Eq. (22), one finds that the factor stems from the exponential spectroscopy in Eq. (19). The fact that appears multiplied by q in all the relevant physical quantities [e.g., Eqs. (20) and (21)] means that a multiplicative change in q is equivalent to an additive change in ν (cf. Appendix B). Thus, multiplicative changes in q always change the width of the triangles in Fig. 4(b) by the same , and the forcing is proportional to this width.
V. THE WATER VAPOR FEEDBACK
A conspicuous feature of Fig. 5(b) is that in the optically thick regions seems to be almost entirely insensitive to , i.e., . This can be deduced from Eq. (28), where vertical variations in at a given are dominated by the temperature-dependent exponential, with the pressure-broadening factor playing only a secondary role. Thus, to a good approximation, is a function of temperature alone, and will occur at approximately the same regardless of (assuming fixed RH). We formalize this fact, first formulated by Simpson44 in 1928, as “Simpson's law”:
VI. BEYOND RCE: ADDITIONAL FEEDBACKS
In this less detailed section, we sketch the phenomena unaccounted for in the RCE framework, discuss their impacts on λ and ECS as assessed with numerical simulations and observations, and discuss prospects for principled estimates similar in spirit to those presented above.
A major omission from the framework developed so far is clouds. Clouds exert enormous leverage over the climate system by absorbing and emitting longwave radiation essentially as blackbodies, and also by reflecting shortwave radiation (roughly half of Earth's albedo is due to clouds50). In the longwave, certain aspects of the radiative effects of clouds can be described with a relatively simple formalism, as follows.51
On the shortwave side, there are highly reflective subtropical marine low clouds whose areal coverage is thought to decrease with global warming, yielding an increase in absorbed sunlight with warming and, thus, a positive contribution to the total feedback parameter of .56 This decrease in coverage is often understood via changes in environmental variables known as “cloud-controlling factors” such as the local sea surface temperature and relative humidity; sophisticated analyses of these dependencies allow us to quantify the associated feedback.57 Meanwhile, these clouds have also been described by simplified “mixed-layer models.”58 However, these dots so far remain unconnected, and a first principles estimate of the tropical marine low cloud feedback also remains unformulated.
B. Shortwave feedbacks
In addition to changes in sunlight reflected by clouds, there are other significant shortwave feedbacks (i.e., changes in absorbed solar radiation S with warming) left unaccounted for in the RCE framework. Perhaps the largest of these is the surface-albedo feedback, due primarily to decreasing snow and ice cover with warming, which manifests as changes in albedo α [cf. Eq. (1)]. This yields a positive feedback .59 While highly idealized models of the ice-albedo feedback have existed for decades,60 and comprehensive modeling studies reveal a close connection between this feedback and the seasonal cycle,61 again this gap has not been bridged and a chalkboard estimate of the surface-albedo feedback has yet to be formulated.
Another shortwave feedback, which receives less attention but is not insignificant, is that due to shortwave absorption by water vapor. Though often neglected in introductory treatments such as that of Sec. II, it turns out that water vapor absorbs a rather significant amount of near-infrared sunlight—around 80 .62 Since the mass of water vapor in the atmosphere increases with warming [at a rate roughly dictated by the Clausius–Clapeyron relation (27)],63 water vapor shortwave absorption also increases, reducing the amount of (near-infrared) sunlight reflected out to space and, thus, increasing S. The end result is a positive shortwave water vapor feedback .64 A principled estimate of may be fairly easy to obtain, leveraging the fact that water vapor shortwave absorption should be a fixed function of temperature (i.e., it obeys its own version of Simpson's law), in analogy to water vapor longwave emission.65
C. The total feedback
VII. BEYOND ECS: OTHER MEASURES OF CLIMATE SENSITIVITY
In this final section, we look beyond ECS to other measures of climate sensitivity. We will find that ECS is a quite idealized notion, and that other measures of climate sensitivity are more relevant for present-day warming and for understanding and defining emissions targets. However, we will also see that ECS, as well as its key ingredients and λ, naturally appear in these other measures. Thus, the basic understanding of ECS developed here is necessary for understanding these other measures.
A. The deep ocean and timescales of global warming
The equilibrium climate sensitivity is exactly that: An equilibrium quantity. However, how long does it take the climate system to equilibrate with a given concentration, and what does this evolution look like? To address this, we need a time-dependent model of the Earth's surface temperature . We proceed by neglecting the dynamics of the land surface (since the Earth is roughly 2/3 ocean covered) and invoking the popular two-layer or two-box model for the ocean.67 This model consists of a shallow mixed layer with depth and temperature anomaly equal to , sitting atop a much larger deep ocean with global average depth and temperature anomaly . The model is pictured in Fig. 6.
An important characteristic of this model is that if the mixed layer is warmed by a forcing , it both radiates extra energy to space at a rate of and also exports energy to the deep ocean, which we parameterize in linearized form as (units of ). Here, γ is the “deep ocean heat uptake efficiency,” estimated from models at roughly 0.7 .68 Setting and as the densities and specific heat capacities of water, the corresponding equations are
A key assumption in Eqs. (38) and (39) is that on intermediate timescales. This approximation turns out to be a reasonable description of the present day, and the quasi-equilibrium formula (38) (which is just a scaling of TCR) can be used to credibly model historical as well as near-term global warming.71 In these ways, TCR is a more relevant metric for present day climate change than ECS, which instead assumes that both the mixed layer and deep ocean have reached a mutual equilibrium, which from Eq. (37) would take many hundreds of years.72
Note that the two-box model, while popular, is by no means canonical. Some recent work instead employs three-box models, and the older literature often employed diffusive models.73 Two-box, three-box, and diffusive models were all employed by the IPCC as emulators of more comprehensive models.74 All these approaches, however, require empirically determined parameters for heat transfer coefficients and diffusivities, analogous to our heat uptake efficiency γ, and principled estimates for these quantities are still lacking.75
B. The carbon cycle and measures of carbon-climate sensitivity
In addition to only describing very long-term warming, another limitation of ECS is that it assumes that the perturbed concentration q is constant while the Earth system equilibrates. If we stop burning fossil fuels, however, q will not remain constant; the real Earth has an active carbon cycle in the land and ocean, both of which absorb significant amounts of anthropogenic , which would cause q to decrease over time. Full consideration of these dynamics leads to two additional measures of climate sensitivity which are fundamental for both climate change science as well as policy: the transient climate response to cumulative emissions (TCRE) and the zero emissions commitment (ZEC).
TCRE is defined to be the warming at a given time divided by the cumulative emissions released prior to that time, in Kelvins per gigaton of carbon (K/GtC). TCRE is found to have a characteristic value of , which in simulations turns out to be fairly invariant over time as well as insensitive to emissions scenario.76 The robustness of TCRE tells us that any identified temperature target (e.g., ) automatically reduces to a cumulative emissions target (e.g., 750 or 1000 GtC), which can only be met if we cease emissions prior to reaching the target. This leads directly to the notion of net zero emissions.77
While Eq. (41) appears to explain why is small, it is only a proximal explanation. The heat uptake efficiency γ, which entered into our estimate Eq. (39) of TCR, as well as the present day airborne fraction , were evaluated via simulations or observations rather than theoretically, so we do not yet have fully principled estimates for TCRE or ZEC. Such estimates would necessarily draw upon physics from across the Earth system, including the dynamics of ocean heat uptake, ocean carbon uptake, fertilization of the biosphere, and more. A chalkboard explanation of these quantities, thus, poses a grand challenge to climate science; the simple models presented here are simply a first step towards that goal.
This article grew out of notes prepared for lectures hosted by the Princeton Geosciences department, Princeton Environmental Institute (PEI), and the Princeton Program in Atmosphere and Ocean Sciences in January 2018.83 The author would like to thank Robert Socolow and Mike Celia of PEI for the encouragement to develop and deliver these lectures; Isaac Held, Stephan Fueglistaler, and Yi Ming for continued support and constructive feedback; Nathaniel Tarshish, Brett McKim, and Sam Schulz for detailed comments; and Tapio Schneider and Brett McKim for encouragement to publish.
Conflict of Interest
No conflict of interest to disclose.
APPENDIX A: ADIABATIC TEMPERATURE PROFILES
We here derive expression (5) for the dry adiabatic lapse rate. For a textbook treatment of this topic, as well as extension to include the effects of moisture, see, e.g., the textbook by Wallace and Hobbs84 or the excellent lecture notes of Romps.85
APPENDIX B: ANALYTIC DERIVATION OF FORMULA FOR FORCING
The report of Charney (1979) was the first such model intercomparison report. Meehl (2020) reviewed ECS over the many generation of IPCC reports. The latest IPCC assessment of ECS is given in Chapter 7 of the sixth assessment report from working group I, Forster (2021), Sec. 7.5; see also the comprehensive ECS review in Sherwood (2020).
For a similar but not explicitly pedagogical approach, see also Stevens and Kluft (2023).
He (2022) is an example of this, in which the formalism for forcing developed here is applied to understand the spread in that quantity amongst comprehensive climate models.
For a related point of view, see Emanuel (2020).
See Sec. VI B and Note 65 for more on this approximation.
Thermodynamically speaking, this is a steady-state, not an equilibrium, as the net fluxes of e.g., solar radiation, thermal radiation, water vapor at the Earth's surface, etc., are nonzero. Accordingly, the Earth does not come into thermal equilibrium with either the Sun or outer space but tends towards a temperature in between them. Nonetheless, the use of the term “equilibrium” to describe a steady-state in climate is ubiquitous, so we adopt this terminology here.
Stratospheric temperatures are sometimes thought to be governed by the so-called “skin temperature,” which arises by considering an optically thin layer of atmosphere sitting atop the troposphere; a standard argument yields a skin temperature of K (Pierrehumbert, 2011; Hartmann, 2015). This argument ignores the strong dependence of tropopause and stratospheric temperatures on atmospheric composition, however (e.g., Manabe and Strickler, 1964). Recent work instead postulates that is governed by the physics of radiative transfer (Seeley , 2019; Jeevanjee and Fueglistaler, 2020b), but more work on this topic is needed.
At least to first approximation. See discussion in Jeevanjee (2022).
This is the ratio of the mass of a given substance in a parcel to the total mass of the parcel. For further discussion, see Pierrehumbert (2011), p. 87.
While such a perspective ignores the interactive carbon cycle dynamics of the atmosphere, land, and ocean, it is, nonetheless, a useful starting point for understanding the impacts of fossil-fuel combustion on Earth's climate. We discuss how carbon cycle dynamics modify the picture presented here in Sec. VI.
There are, however, processes known as atmospheric adjustments which blur this distinction; see Sherwood (2015).
An important exception to this reasoning is the stratospheric adjustment, which is the direct response of stratospheric temperatures to a change in q resulting from the increased emission to space depicted in Fig. 4(b) (green dashed lines). This change in stratospheric temperatures occurs independently of changes in , precisely because the stratosphere is not coupled to the surface in the way the troposphere is. For further discussion, see Hansen (1997); Houghton (1994).
Another useful consequence of this definition is that “positive,” amplifying feedbacks make positive contributions to λ.
Since is well-mixed, its density is proportional to the air density ρ, and ρ decreases exponentially with height (cf. Note 17). For , its density is dominated by its Clausius–Clapeyron exponential dependence on temperature (Eqs. (26) and (27)), and temperature is linear in height, so density is exponential in height as well.
We here assume a two-stream approximation with unit diffusion coefficient for simplicity (Pierrehumbert, 2010).
This pressure scaling is due to collisional pressure broadening away from spectral line centers and is also accompanied by additional, typically less pronounced temperature scalings. See Pierrehumbert (2010).
For a pedagogical discussion of the structure of this spectrum and its computation, see Wilson and Gea-Banacloche (2012).
In particular, we multiply the integrand by , which does not deviate too far from 1 but which allows the exponential to be integrated. Since the integrand is dominated by values of the exponential evaluated near T, we also approximate so it can be pulled outside the integral. This expression and these approximations are discussed in more detail in Jeevanjee and Fueglistaler (2020b).
Forster (2021), Table 7.13.
We obtain f = 0.18 as an average of the AIRS and ISCCP satellite data values for global mean high cloud fraction given in Fig. 4.5 of Siebesma (2020).
See Trenberth and Fasullo (2009). It is an interesting exercise with the RCE model Eq. (8) to show that subtracting this from the surface energy budget and adding it to the atmospheric energy budget does not affect , consistent with the interpretation that the atmosphere and surface function as a unit so repartitioning energy between them do not affect their temperatures.
Strictly speaking, TCR is defined via simulations in which is increased from its preindustrial value at 1% year until the concentration has doubled, and TCR is defined as the warming at the time of doubling (around year 70). However, it has been shown that the two-box expression (39) for TCR captures the actual TCR fairly well (Held et al., 2010; Geoffroy et al., 2013).
Forster (2021), Sec. 7.5.5.
Indeed, it is a simple exercise to show that imposing the full equilibrium conditions in Eqs. (35) for yields .
Though there has been some work in this direction; see e.g., Marshall and Zanna (2014).
Friedlingstein (2020) provide values of and . It is also a nice exercise to estimate by multiplying the mass of the atmosphere by the specific concentration corresponding to the number concentration of 280 ppm. Note also that for present-day of roughly 700 GtC, we have , so the approximation used in Eq. (40) is marginally acceptable, but will fail for larger .
The original notes contain additional material beyond that presented here; see Jeevanjee (2018).
There is some sleight-of-hand here, as hydrostatic balance typically applies to the pressure and density of the quiescent environment, rather than a rising parcel. For most applications (such as ours), this approximation is permissible, but if we take it to extremes (such as applying the dry adiabatic lapse rate Eq. (5) over large enough distances to generate negative temperatures) it can lead to nonsensical results; for further discussion see Romps (2020).
See Sec. 2.3 of Seeley (2018).
See Note 88.