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 $ CO 2$ 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.

## I. INTRODUCTION

Earth's equilibrium climate sensitivity (ECS), or the long-term global-mean surface temperature change due to a doubling of $ CO 2$, 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 Wetherald^{4} 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 ( $ H 2 O$) and carbon dioxide ( $ CO 2$). 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 $ CO 2$ radiative forcing^{6} and the $ H 2 O$ 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 Hartmann^{9} 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 $ CO 2$ and $ H 2 O$, which lead to analytical descriptions of both $ CO 2$ 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 $ T s$, 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 $ T s$ is the amount of sunlight Earth receives. The solar flux at Earth's orbit is $ S 0 = 1360 \u2009 W / m 2$, and this flux is incident on an effective surface area $ \pi R E 2$ (the projected area of the Earth onto a plane perpendicular to the Sun's rays, where *R _{E}* 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 $ \alpha \u2248 0.30$.

^{13}

*planetary energy balance*, which says that in steady-state

*S*must be balanced by outgoing thermal radiation, also known as the “outgoing longwave radiation,” or $ OLR$ (here “longwave” means thermal infrared). We estimate $ OLR$ as blackbody emission,

*S*yields $ T em = 255$ K. This is much colder than the observed global average surface temperature $ T s = 288$ K but is a reasonable estimate of an effective

*atmospheric*temperature, consistent with the fact that OLR largely emanates not from the surface, but from atmospheric greenhouse gases (GHGs, which are gases that absorb and emit thermal infrared radiation—most prominently water vapor and carbon dioxide). However, given the atmospheric $ T em$, how can we find $ T s$? How are surface and atmospheric temperatures related?

### B. Single-layer radiative equilibrium

^{14}The energy budgets at the top of the atmosphere (TOA) and surface then read (Fig. 1)

These equations can be immediately solved to yield $ T a = T em = 255$ K and $ T s = 2 1 / 4 T em = 303$ K, overestimating $ T s$ 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* $ T s = 288 K$, 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}

*T*(

*z*) lie more or less along a convective

*adiabat*. A convective adiabat is the temperature profile of a parcel lifted adiabatically from the surface; such a parcel will expand and cool as it rises along the

*z*-axis to lower pressures, much like the cool air escaping from the valve of a pressurized bicycle tire. For a dry air parcel, this profile is determined by the

*dry adiabatic lapse rate*,

*g*is the gravitational acceleration and $ C p$ is the specific heat of air (see Appendix A, which includes a derivation of Eq. (5) from undergraduate thermodynamics). Moisture, along with atmospheric dynamics besides convection, modifies this lapse rate somewhat and makes it variable over the globe, with a global average value of

^{17}then Eq. (6) yields the

*convective constraint*,

*et al.*

^{18}and Payne

*et al.*

^{19}

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 $ F c$ 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.

*troposphere*. Above the troposphere lie the stratosphere and other air masses, which are heated primarily by ultraviolet solar absorption rather than convection, and are closer to radiative equilibrium rather than RCE. For simplicity, the stratosphere is represented here by an isothermal layer with characteristic temperature $ T strat = 210$ K, which is attached to the troposphere at the

*tropopause*, where $ T a = T strat$ (Fig. 2). The physics governing the height and temperature of the tropopause and, hence, the characteristic temperature $ T strat$, is still a subject of active research.

^{20}

A profound implication of the convective constraint (10a) is that the tropospheric temperature profile $ T a$ is pegged to $ T s$, 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 $ CO 2$ specific concentration^{22} *q* (kg $ CO 2$/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 $ \Delta T s$, 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 $ T s$, and the other which occurs at fixed *q*.^{24}^{ } (Also, it will be natural sometimes to use $ \u2212 OLR$ as our variable for planetary thermal energy flux, since an increase in $ \u2212 OLR$ indicates increasing planetary thermal energy.)

*q*from an initial concentration $ q i$ to a final concentration $ q f$, holding $ T s$ fixed; since the lapse rate Γ does not depend on

*q*, this also fixes $ T a ( z )$ in Eq. (10a).

^{25}This decrease in $ OLR$ due to the change in

*q*is known as the $ CO 2$

*forcing*,

*increase*in $ OLR$ (decrease in $ \u2212 OLR$), which persists until the planetary energy balance (3a) is restored and a new equilibrium is established (how long this takes, and what happens along the way, is addressed in Sec. VII A). This increase in $ OLR$ due to increasing $ T s$ can be approximated as $ ( d OLR / d T s ) \Delta T s$, where the derivative is taken at fixed $ q = q f$. Consistent with the sign convention in Eq. (11), we then define the “feedback parameter”

*λ*as minus this derivative

^{26}

*equilibrium climate sensitivity*(ECS), that is

*q*for defining ECS will be discussed below.) Equation (13) is known as the

*forcing-feedback*decomposition of ECS. It allows us to study ECS by studying $F$ and

*λ*separately, which we will do in Secs. IV and V, respectively. Note that we have ignored the possibility that the absorbed solar radiation

*S*may also depend on $ T s$, which would contribute a $ d S / d T s$ term to the definition of

*λ*. We discuss these so-called “shortwave” feedbacks in Sec. VI.

### B. Blackbody estimate of ECS

*λ*, but we still need a value for the forcing $ F 2 \xd7$. For the moment, we obtain this by appealing to comprehensive radiative transfer calculations, which for decades

^{27}have found a fairly consistent value of

*λ*, a blackbody estimate for this can be obtained from Eq. (2), noting also that $ T em$ and $ T s$ vary in a 1–1 fashion according to Eq. (9). Recalling that $ T em = 255$ K, this yields

To understand why $ F 2 \xd7 \u2248 4 \u2009 W / m 2$, 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 $ T em$ of outgoing longwave radiation actually depends rather markedly on frequency, and that $ T em$ at a given frequency and $ T s$ 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

In Sec. II, we defined Earth's emission temperature $ T em$ in Eq. (2) and then assumed an average blackbody emission level $ p a = 0.5$ atm. This led to a 1–1 relationship between $ T em$ and $ T s$, which allowed us to estimate *λ*.

*π*above accounts for integration over solid angle.)

^{28}The spectral coordinate here is “wavenumber” $\nu $, defined as inverse wavelength and, thus, proportional to frequency, with standard unit of $ cm \u2212 1$. The definition (16) of $ T em ( \nu )$, as the temperature whose Planck emission yields $ OLR \nu $, straightforwardly generalizes the blackbody definition (2).

The next task is to determine what level(s) in the atmosphere determine $ T em ( \nu )$, for a given $\nu $. This task is aided by the following heuristic, illustrated in Fig. 3. Consider an atmospheric column with GHG molecules whose density $ \rho GHG$ [ $ ( kg \u2009 GHG ) / m 3$] decreases exponentially with height; this is true for both $ CO 2$ and $ H 2 O$.^{29} Now consider the emission to space (i.e., the contribution to the $ OLR \nu $ at a given $\nu $) 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.

*optical depth*, defined as

^{30}

*mass absorption coefficients*of our GHG, which give the effective cross-sectional area of our GHG molecules at wavenumber $\nu $ per unit mass (units $ m 2 / kg$), evaluated at a reference pressure $ p ref$. This effective cross-section depends strongly on frequency (e.g., Figs. 4(a) and 5(a) below) but also scales with pressure approximately as $ p / p ref$, hence the presence of this factor in Eq. (17).

^{31}The factor of $ \rho GHG d z \u2032$ in the integrand in Eq. (17) gives the absorber mass per unit area ( $ kg / m 2$) in an atmospheric layer of differential depth $ d z \u2032$. Thus, the integral $ \tau \nu $ in Eq. (17) can be interpreted as the ratio of the integrated effective area of absorbers above height

*z*to the actual area of the column, as noted in Eq. (17). Applying this notion to Fig. 3, we see that the top two layers with low emissions correspond to $ \tau \nu < 1$, where the total effective area of absorbers above is less than the actual area of the column (the “optically thin” regime). Similarly, the bottom two layers with low emissions correspond to $ \tau \nu > 1$, where the total effective area of absorbers above is greater than the actual area of the column (the “optically thick” regime). The sweet spot occurs around $ \tau \nu \u2248 1$.

*all*emission occurs at exactly $ \tau \nu = 1$; we refer to this as the

*emission level approximation*.

^{32}With this in mind, we define our emission pressure $ p em ( \nu )$ by the relation

## IV. CO_{2} FORCING

This section constructs a simple analytic model for the $ CO 2$ forcing Eq. (11), with the aim of enabling a back-of-the-envelope estimate of the characteristic 4 $ W / m 2$ value for $ F 2 \xd7$. The approach here is to consider spectral variations in $ CO 2$ 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} Seeley^{34} 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 $ CO 2$ and $ H 2 O$ absorption.^{37}

^{38}A key simplification we make is to ignore fine-scale spectral structure and note that on a broad scale, $ \kappa CO 2 ( \nu )$ may be parameterized as

*l*is a “spectroscopic slope,” which sets the rate at which absorption declines away from the peak.

*q*is a constant $ CO 2$ specific concentration and

*ρ*is the density of air. Plugging this into Eq. (17) and invoking hydrostatic balance $ d p / d z = \u2212 \rho g$ [see also Eq. (A3)] then yields

*q*yields a decrease in surface emission, as well as “new” emission from portions of the stratosphere (around 20 hPa) that were not emitting significantly before. Also, from Eq. (21) one can deduce that the change in width of the emission triangles with

*q*is given by

*q*, which we comment on further below.

^{39}The magnitude of CO

_{2}forcing is thus set by the gross characteristics of CO

_{2}spectroscopy (as embodied in

*l*), as well as the difference in surface and stratospheric temperatures.

The formalism developed here also yields insight into the logarithmic scaling of $ CO 2$ forcing, evident in the $ ln \u2009 ( q f / q i )$ factor in Eqs. (22) and (23). Unwinding the mathematics leading to Eq. (22), one finds that the $ ln \u2009 ( q f / q i )$ factor stems from the exponential spectroscopy $ \kappa CO 2 ( \nu )$ in Eq. (19). The fact that $ \kappa CO 2 ( \nu )$ 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 $ p em$ triangles in Fig. 4(b) by the same $ \Delta \nu $, and the forcing is proportional to this width.

## V. THE WATER VAPOR FEEDBACK

*λ*. For

*λ*, $ H 2 O$ is the dominant GHG, as its absorption spectrum spans the entirety of the thermal infrared, as shown in Fig. 5(a) (compare the wavenumber range in Fig. 5 to that of Fig. 4; in analyzing

*λ*we neglect the effects of $ CO 2$ for simplicity). Similar to $ CO 2$, the $ H 2 O$ reference mass absorption coefficients $ \kappa H 2 O ( \nu )$ can be parameterized in terms of exponentials, this time with two exponentials peaked at $ \nu 1 = 150 \u2009 cm \u2212 1$ and $ \nu 2 = 1500 \u2009 cm \u2212 1$, as

^{40}

^{41}Here,

*p*is the partial pressure of water vapor, $ R v = 287 \u2009 J / ( kg \xb7 K )$ is the specific gas constant for water vapor, and

_{v}*L*the latent heat of vaporization (equal to $ 2.5 \xd7 10 6$ J/kg). The Clausius–Clapeyron expression is fundamental to our RCE picture of the atmosphere, as it determines how much the water vapor content of a parcel decreases (and, hence, how much condensation is produced) per degree of cooling.

^{42}Note that $ M v ref \u2009 exp \u2009 ( \u2212 L / R v T )$ is the column mass of water vapor above the isotherm with temperature

*T*, so Eq. (28) is just the pressure-weighted GHG mass above

*T*times the cross-sectional area per unit mass, consistent with the interpretation in Eq. (17). Water vapor emission temperatures $ T em$ can now be diagnosed by setting $ \tau H 2 O = 1$ and inverting Eq. (28) either numerically or analytically;

^{43}where $ \tau H 2 O ( \nu , T s ) < 1$, in the aforementioned optically thin region centered around 1000 $ cm \u2212 1$ (known as the infrared “window”), the emission level lies below the surface so we set $ T em = T s$. The results for $ T s = ( 280 , 290 , 300 )$ K are plotted in Fig. 5(b).

A conspicuous feature of Fig. 5(b) is that $ T em$ in the optically thick regions seems to be almost entirely insensitive to $ T s$, i.e., $ d T em / d T s \u2248 0$. This can be deduced from Eq. (28), where vertical variations in $ \tau H 2 O$ at a given $\nu $ are dominated by the temperature-dependent exponential, with the pressure-broadening factor playing only a secondary role. Thus, to a good approximation, $ \tau H 2 O$ is a function of temperature alone, and $ \tau H 2 O ( \nu ) = 1$ will occur at approximately the same $ T em$ *regardless of* $ T s$ (assuming fixed RH). We formalize this fact, first formulated by Simpson^{44} in 1928, as “Simpson's law”:

**Simpson's law**: At fixed RH, and for optically thick wavenumbers $\nu $ dominated by $ H 2 O$ absorption, emission temperatures are insensitive to surface temperature, i.e.,

^{45}but we, nonetheless, refer to it as a “law” as it plays a fundamental role in governing the strength of the water vapor feedback, as follows. If $ T em ( \nu )$ and, hence, $ OLR \nu $ (by Eq. (16)) are independent of $ T s$ for optically thick $\nu $, and if we consider the atmosphere perfectly transparent for the optically thin $\nu $ in the window region so that $ OLR \nu $ is given by surface emission $ \pi B ( \nu , T s )$ for those wavenumbers, then the spectrally resolved feedback parameter $ \lambda \nu $ (satisfying $ \lambda = \u222b \lambda \nu d \nu $) is given by

*λ*; all that is required is an estimate of the limits of the window region. This is typically

^{46}taken to be $ 800 < \nu < 1200 \u2009 cm \u2212 1$, although these limits are not precisely defined. For the sake of obtaining round numbers, we take the lower limit to be 750 $ cm \u2212 1$, which then yields our RCE estimate of

*λ*,

^{47}

*λ*, which was obtained here in an RCE context but in the literature is known as the “longwave clear-sky feedback” as it ignores cloud feedbacks and shortwave feedbacks, is rather universal and occurs ubiquitously throughout observational and modeling studies.

^{48}Furthermore, $ \lambda RCE$ embodies the water vapor feedback discovered by MW67; by holding RH rather than specific humidity fixed, Simpson's law becomes applicable and tells us that a

*significant portion of the longwave spectrum does not contribute to λ*because $ T em ( \nu $) is fixed (Fig. 5(b)). This significantly reduces

*λ*from the naive blackbody estimate (15) by a factor of about 2, consistent with MW67's early finding that the water vapor feedback doubles climate sensitivity.

^{49}

## 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. Clouds

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 clouds^{50}). In the longwave, certain aspects of the radiative effects of clouds can be described with a relatively simple formalism, as follows.^{51}

*f*of the Earth's surface ( $ f \u2248 0.18$ in the present-day global mean

^{52}), then one can write the “all-sky” (i.e., actual) OLR as

*f*of $ OLR clr$, replacing it with cloud-top emission $ \sigma T cld 4$.

*f*and $ T cld$ respond to warming. Fortunately, the latter question is answered quite simply by the so-called

*fixed anvil temperature*hypothesis,

^{53}which is related to Simpson's law and says that high clouds rise with global warming so as to keep $ T cld$ fixed, i.e.,

*f*is generally expected to decrease with warming (a feedback known as the “iris effect”),

^{54}but the magnitude of this decrease is uncertain and there is as yet insufficient theory to estimate it from first principles. The terms in Eq. (33) related to

*f*and $ d f / d T s$, thus, tend to compensate, and the value of $ \lambda all$ ends up not far from $ \lambda clr \u2248 \u2212 2 \u2009 W / m 2 / K$ but with larger error bars.

^{55}

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 $ \lambda cld SW \u223c 0.2 \u2009 W / m 2 / K$.^{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 $ \lambda albedo \u223c 0.3 \u2013 0.4 \u2009 W / m 2 / K$.^{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 $ W / m 2$.^{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 $ \lambda wv SW \u223c 0.25 \u2009 W / m 2 / K$.^{64} A principled estimate of $ \lambda wv SW$ 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

*et al.*

^{66}from multiple lines of evidence, which indeed yields the consensus ECS value of $ \u2212 F 2 \xd7 / \lambda = ( 4 \u2009 W / m 2 ) / ( 1.3 \u2009 W / m 2 / K ) \u2248 3 \u2009 K$. This more realistic value of

*λ*will be a key ingredient in estimating other measures of climate sensitivity, which we take up in Sec. VII.

## 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 $F$ 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 $ CO 2$ concentration, and what does this evolution look like? To address this, we need a time-dependent model of the Earth's surface temperature $ T s$. 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 $ h ml \u2248 50 \u2009 m$ and temperature anomaly $ T \u2032 ml$ equal to $ \Delta T s$, sitting atop a much larger deep ocean with global average depth $ h d \u2248 2500 \u2009 m$ and temperature anomaly $ T \u2032 d$. The model is pictured in Fig. 6.

An important characteristic of this model is that if the mixed layer is warmed by a forcing $F$, it both radiates extra energy to space at a rate of $ | \lambda | T \u2032 ml$ and also *exports energy to the deep ocean*, which we parameterize in linearized form as $ \gamma ( T \u2032 ml \u2212 T \u2032 d )$ (units of $ W / m 2$). Here, *γ* is the “deep ocean heat uptake efficiency,” estimated from models at roughly 0.7 $ W / m 2 / K$.^{68} Setting $ \rho w$ and $ C w$ as the densities and specific heat capacities of water, the corresponding equations are

^{69}

^{70}Comparing Eq. (39) with Eq. (13) shows that $ ECS > TCR$: on the intermediate timescales during which $ T \u2032 ml = TCR$, the mixed layer is both radiating heat to space

*and*exporting heat to the deep ocean (Fig. 6) and can, thus, come to (quasi-)equilibrium at a lower temperature. This is of course not a true equilibrium state, and one can interpret the ratio $ TCR / ECS$ as a measure of the ocean's thermal disequilibrium; this measure will prove useful in Sec. VII B.

A key assumption in Eqs. (38) and (39) is that $ T \u2032 d \u2248 0$ 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 $ CO 2$ 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 $ CO 2$, 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 $ \u223c 2 \u2009 K / 1000 \u2009 GtC$, 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., $ \Delta T s = 1.5 \u2009 o r \u2009 2 \u2009 \xb0 C$) 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}

^{78}Furthermore, evaluating Eq. (40) for $ C pre = 590 \u2009 GtC$ and the present day airborne fraction of $ \alpha = 0.4$ indeed yields roughly 2 K/GtC, consistent with simulations.

^{79}However, here we are simply using the observed value of $\alpha $; we do not yet have ways to make principled estimates of $\alpha $ from basic carbon cycle dynamics, and thus to understand how and why $\alpha $ does or does not vary across time and emissions scenarios.

*after emissions cease*. Simulations

^{80}tend to show that ZEC is small relative to $ \Delta T s$, which is a requirement for net zero emissions goals to meaningfully limit $ \Delta T s$. Similar to the estimate for TCRE, we may scale TCR and also ECS to estimate ZEC, following an argument due to Tarshish.

^{81}Let $ T ze$ and $ \alpha ze$ be the temperature and airborne fraction of cumulative emissions when emissions cease, and $ T f$ and $ \alpha f$ be the temperature and airborne fraction at final equilibrium, respectively. Normalizing $ ZEC \u2261 T f \u2212 T ze$ by $ T ze$, we have

^{82}the previously cited values $ \alpha ze = 0.4 , \u2009 TCR = 1.8$ K, and $ ECS = 3$ K then give $ ZEC / T ze \u2248 \u2212 0.17$, which is indeed small (and, perhaps surprisingly, negative).

While Eq. (41) appears to explain why $ ZEC$ 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 $ \alpha \u2248 0.4$, 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, $ CO 2$ 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### 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 Hobbs^{84} or the excellent lecture notes of Romps.^{85}

*dQ*is the heat gained by the system,

*U*is its internal energy,

*p*is its pressure, and

*V*is its volume. For an ideal gas, we have $ U = \rho V C v T$ as well as the ideal gas law, written in the form typical of atmospheric science,

*ρ*is the parcel density in $ kg / m 3 , \u2009 R d = 287 \u2009 J / kg / K$ is the specific gas constant for dry air (obtained by dividing the universal gas constant by the molar mass of air), and $ C v = ( 5 / 2 ) R d$ is the specific heat capacity at constant volume. Assuming an adiabatic process (

*dQ*= 0), we rearrange Eq. (A1) into

^{86}which says that the weight of a layer of air is balanced by the vertical pressure gradient across it,

*T*and

*p*can be integrated from an arbitrary pressure

*p*to surface pressure $ p s$ to obtain Eq. (10b).

### APPENDIX B: ANALYTIC DERIVATION OF FORMULA FOR $ CO 2$ FORCING

^{87}We will take advantage of the symmetry evident in Fig. 4 and simply double the integral for $ \nu > \nu 0$. We will also take advantage of the key fact that

*q*and $ \kappa CO 2 ( \nu )$ appear in Eq. (20) as a product; this, along with the exponential spectroscopy Eq. (19) and the fact that $ T em$ and $ p em$ are related by Eq. (10b), means that

*q*is equivalent to that of a uniform translation in $\nu $. Putting these ingredients together, the forcing Eq. (11) can now be evaluated as an integral over $\nu $ from band center $ \nu 0$ to the upper limit $ \nu + \u2248 750 \u2009 cm \u2212 1$, where we ignore spectral variations in the Planck density. This yields

^{1}

^{2}

^{3}

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).

^{5}

^{8}

For a similar but not explicitly pedagogical approach, see also Stevens and Kluft (2023).

^{9}

^{10}

^{11}

He (2022) is an example of this, in which the formalism for $ CO 2$ forcing developed here is applied to understand the spread in that quantity amongst comprehensive climate models.

^{12}

For a related point of view, see Emanuel (2020).

^{14}

See Sec. VI B and Note 65 for more on this approximation.

^{15}

For example, Petty (2006); Randall (2012); Vallis (2012); Coakley Jr., and Yang (2014); Hartmann (2015); Tziperman (2022).

^{16}

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.

^{17}

^{18}

^{19}

^{20}

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 $ T em / 2 4 = 214$ 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 $ T strat$ is governed by the physics of $ H 2 O$ radiative transfer (Seeley , 2019; Jeevanjee and Fueglistaler, 2020b), but more work on this topic is needed.

^{21}

At least to first approximation. See discussion in Jeevanjee (2022).

^{22}

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.

^{23}

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.

^{24}

There are, however, processes known as atmospheric *adjustments* which blur this distinction; see Sherwood (2015).

^{25}

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 $ T s$, precisely because the stratosphere is not coupled to the surface in the way the troposphere is. For further discussion, see Hansen (1997); Houghton (1994).

^{26}

Another useful consequence of this definition is that “positive,” amplifying feedbacks make positive contributions to *λ.*

^{27}

^{28}

*h*is Planck's constant,

*k*is Boltzmann's constant, and

_{b}*c*is the speed of light.

^{29}

Since $ CO 2$ is well-mixed, its density is proportional to the air density *ρ*, and *ρ* decreases exponentially with height (cf. Note 17). For $ H 2 O$, its density is dominated by its Clausius–Clapeyron exponential dependence on temperature (Eqs. (26) and (27)), and temperature is linear in height, so $ H 2 O$ density is exponential in height as well.

^{30}

We here assume a two-stream approximation with unit diffusion coefficient for simplicity (Pierrehumbert, 2010).

^{31}

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).

^{32}

For further analysis of the emission level approximation, with analytic calculations suitable for the classroom, see Appendix B of Jeevanjee (2021b) and Appendix B of Jeevanjee and Fueglistaler (2020a).

^{34}

^{35}

^{36}

^{37}

^{38}

For a pedagogical discussion of the structure of this spectrum and its computation, see Wilson and Gea-Banacloche (2012).

^{39}

This estimate ignores several effects, each of which generate 20%–30% corrections but which offset each other. These effects include the overlapping of $ CO 2$ absorption by $ H 2 O$ absorption, the masking of $ CO 2$ forcing by clouds, and stratospheric adjustment. See Jeevanjee (2021b); Huang (2016).

^{40}

The $ H 2 O$ absorption coefficients shown here have been simplified by neglecting so-called “continuum” absorption. Continuum absorption significantly affects absorption in the infrared window at warmer surface temperatures of 300 K and above. See Pierrehumbert (2010); Shine (2012); Koll (2022).

^{41}

This fixed relative humidity assumption, popularized by Manabe and Wetherald (1967), has been well justified by decades of subsequent simulations and observation (Jeevanjee , 2022; Colman and Soden, 2021).

^{42}

In particular, we multiply the integrand by $ T av / T \u2032$, 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 $ p ( T \u2032 ) \u2248 p ( T )$ so it can be pulled outside the integral. This expression and these approximations are discussed in more detail in Jeevanjee and Fueglistaler (2020b).

^{43}

*τ*= 1 using the Lambert W function to obtain

^{44}

See, e.g., Simpson (1928). See also Ingram (2010); Jeevanjee (2021a) for further discussion.

^{46}

^{47}

This approach has precedent in the work of Koll and Cronin (2018); Ingram (2010).

^{49}

Forster (2021), Table 7.13.

^{50}

^{52}

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).

^{55}

For example, Andrews (2015); Flynn and Mauritsen (2020).

^{56}

^{57}

^{59}

^{60}

^{61}

^{62}

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 $ T s$, consistent with the interpretation that the atmosphere and surface function as a unit so repartitioning energy between them do not affect their temperatures.

^{63}

^{64}

^{65}

^{66}

^{68}

^{69}

Strictly speaking, TCR is defined via simulations in which $ CO 2$ 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).

^{70}

Forster (2021), Sec. 7.5.5.

^{71}

^{72}

Indeed, it is a simple exercise to show that imposing the full equilibrium conditions $ d T \u2032 ml / d t = d T \u2032 d / d t = 0$ in Eqs. (35) for $ F = F 2 \xd7$ yields $ T \u2032 ml = T \u2032 d = F 2 \xd7 / | \lambda | = ECS$.

^{73}

Three-box models were employed in, e.g., Proistosescu and Huybers (2017); Cummins (2020); Tsutsui (2020). Diffusive models were employed in, e.g., Hansen (1984); Siegenthaler and Oeschger (1984); Wigley and Schlesinger (1985).

^{74}

See Chapter 7 of the IPCC sixth assessment report (Forster , 2021), and in particular, cross-chapter box 7.1. Two popular emulators employed there are MAGICC (Meinshausen , 2011), which has a diffusive deep ocean, and FaIR (Leach , 2021), which has a three-box ocean.

^{75}

Though there has been some work in this direction; see e.g., Marshall and Zanna (2014).

^{76}

^{77}

^{78}

Additional insight can be found in the theoretical approaches of, e.g., Goodwin (2015); MacDougall and Friedlingstein (2015); Seshadri (2017).

^{79}

Friedlingstein (2020) provide values of $ C pre$ and $\alpha $. It is also a nice exercise to estimate $ C pre$ by multiplying the mass of the atmosphere $ 4 \pi R E 2 p s / g$ by the specific $ CO 2$ concentration corresponding to the number concentration of 280 ppm. Note also that for present-day $ C emit$ of roughly 700 GtC, we have $ \alpha C emit / C pre \u2248 0.45$, so the approximation $ ln \u2009 ( 1 + \alpha C emit / C pre ) \u2248 \alpha C emit / C pre$ used in Eq. (40) is marginally acceptable, but will fail for larger $ C emit$.

^{80}

^{81}

^{82}

^{83}

The original notes contain additional material beyond that presented here; see Jeevanjee (2018).

^{84}

^{85}

^{86}

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).

^{87}

See Sec. 2.3 of Seeley (2018).

^{88}

We used the reference forward model (Dudhia, 2017) along with spectroscopic data from HiTRAN2016 (Gordon , 2017).

^{89}

See Note ^{88}.

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