Climate sensitivity from radiative-convective equilibrium: A chalkboard approach a)

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 longterm 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). 3These 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. 5he 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. 8he 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 realworld 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. 11The 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. 12Simple 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 radiativeconvective 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 ¼ 1360 W=m 2 , and this flux is incident on an effective surface area pR 2 E (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 a, measured to be a % 0:30. 13ividing by the Earth's total surface area 4pR (1) The most basic physical constraint on climate is that of 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, for some "emission temperature" T em .Planetary energy balance then reads The observed value of S yields T em ¼ 255 K.This is much colder than the observed global average surface temperature T s ¼ 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
Perhaps the simplest model for estimating T s in the presence of GHGs is that of a single-layer atmosphere of unit emissivity, i.e., one that absorbs and emits as a perfect blackbody, pictured on the left hand side of Fig. 1 (ignoring convective processes for the moment).Such an atmosphere is characterized by an atmospheric temperature T a , blackbody emission rT a 4 from GHGs emanating both upwards to space and downwards to the surface, and transparency to sunlight. 14The 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 ¼ T em ¼ 255 K and T s ¼ 2 1=4 T em ¼ 303 K, overestimating T s by 15 K.
Despite this disagreement, this model is standard in introductory texts. 15Many 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 ¼ 288K, 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. 16o incorporate this into the simple one-layer model ( 4), we must add a convective heat flux F c to the surface energy balance (4b).An additional variable requires an additional equation, however.To obtain this, we invoke the fact that convective heat fluxes are extremely efficient at transferring heat to the atmosphere, with two simplifying consequences: The surface temperature becomes essentially continuous with the atmospheric temperature near the surface, and atmospheric temperature profiles 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,

À
dT dz where 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 C À dT dz % 6:5 K=km ðavg: lapse rateÞ: If we then assume a mean pressure p a ¼ 0:5 atm for our singlelayer atmosphere, corresponding to a height of z a % 5 km (Fig. 2), 17 then Eq. ( 6) yields the convective constraint, T s À T a ¼ Cz a ¼ 32:5 K: (7)   Our single-layer RCE model is then Solving these equations is again straightforward: Eq. (8a) again yields T a ¼ T em , so then Eq. (8c) yields This is remarkably close to the observed preindustrial value of 288 K. Furthermore, Eq. ( 9) tells us that this value stems from the radiative energy balance of the planet (as embodied in T em ), combined with the effects of convection on the temperature profile of the troposphere (as embodied in C).Thus, this single-layer RCE model appears to have significant pedagogical advantages over the single-layer radiative equilibrium model ( 4), but to our knowledge has not yet appeared in textbooks.It has appeared in the journal literature before, for instance, in the papers of Hansen 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.
While this simple one-layer RCE model yields a reasonable estimate of T s and also illustrates the essence of how convection couples the surface and atmospheric temperatures, we will also need a vertically resolved view of the atmosphere for what follows.Figure 2 shows the following temperature profile, obtained by integrating Eq. ( 6) and also converting to pressure coordinates Fig. 2. A simplified RCE temperature profile, given by Eq. ( 10) with T s ¼ 288 K, along with an isothermal stratosphere with temperature T strat ¼ 210 K.The RCE profile (10) reaches up to the tropopause at roughly 12 km or 200 hPa, where it intersects the isothermal stratospheric profile.Also shown is the representative height of the single-layer atmosphere p a ¼ 0:5 atm ¼ 500 hPa, as well as the characteristic cloud-top temperature where R d is the gas constant for dry air and p s is the surface pressure (see Appendix A for the definition of R d and the derivation of Eq. ( 10b)).Equation ( 10a) is the vertically resolved version of the convective constraint Eq. (8c).In general, however, this convective constraint is only valid below an altitude of 10-15 km; this region is known as the 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 ¼ 210 K, which is attached to the troposphere at the tropopause, where T a ¼ 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 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. 21This 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. 23In studying changes in surface temperature DT 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 ÀOLR as our variable for planetary thermal energy flux, since an increase in ÀOLR indicates increasing planetary thermal energy.) The first process is the decrease in OLR (increase in ÀOLR) due to an increase in q from an initial concentration q i to a final concentration q f , holding T s fixed; since the lapse rate C does not depend on q, this also fixes T a ðzÞ in Eq. (10a). 25This decrease in OLR due to the change in q is known as the CO 2 forcing, (F is the change in ÀOLR, so F > 0 for a CO 2 increase).
The forcing F breaks the planetary energy balance, causing additional heat to accumulate in the system.Next, this heat accumulation causes a temperature increase DT s and, hence, a countervailing increase in OLR (decrease in ÀOLR), 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 ðdOLR=dT s ÞDT s , where the derivative is taken at fixed q ¼ q f .Consistent with the sign convention in Eq. ( 11), we then define the "feedback parameter" k as minus this derivative 26 Noting that ðdOLR=dT s ÞDT s ¼ F in equilibrium, we have DT s ¼ ÀF=k.When q f ¼ 2q i , we write the forcing as F 2Â , and the corresponding DT s is defined to be the equilibrium climate sensitivity (ECS), that is (The choice of a multiplicative rather than additive change in 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 k 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 dS=dT s term to the definition of k.We discuss these so-called "shortwave" feedbacks in Sec.VI.

B. Blackbody estimate of ECS
As an exercise, we will use Eq. ( 13) to estimate ECS using the blackbody approximation (2).The blackbody approximation allows us to calculate k, but we still need a value for the forcing F 2Â .For the moment, we obtain this by appealing to comprehensive radiative transfer calculations, which for decades 27 have found a fairly consistent value of We will provide a theoretical estimate of this number in Sec.IV.Turning now to the feedback parameter k, 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 ¼ 255 K, this yields According to Eq. ( 13), this yields ECS ¼ ð4 W=m 2 Þ= ð3:75 W=m 2 =KÞ % 1 K, three times smaller than the consensus value of 3 K.
To understand why F 2Â % 4 W=m 2 , as well as make a better estimate of k 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 ¼ 0:5 atm.This led to a 1-1 relationship between T em and T s , which allowed us to estimate k.
To refine this, we first define a spectrally resolved emission temperature T em ðÞ by the relation, where OLR is the spectrally resolved OLR in units of W=ðm 2 Á cm À1 Þ and satisfying OLR ¼ Ð OLR d, and Bð; TÞ is the Planck density in units of W=ðm 2 Á cm À1 Á srÞ.(The factor of p above accounts for integration over solid angle.) 28The spectral coordinate here is "wavenumber" , defined as inverse wavelength and, thus, proportional to frequency, with standard unit of cm À1 .The definition ( 16) of T em ðÞ, as the temperature whose Planck emission yields OLR , straightforwardly generalizes the blackbody definition (2).
The next task is to determine what level(s) in the atmosphere determine T em ðÞ, for a given .This task is aided by the following heuristic, illustrated in Fig. 3. Consider an atmospheric column with GHG molecules whose density q GHG [ðkg GHGÞ=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 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.
To formalize this, we need the notion of optical depth, defined as 30 s ðzÞ Effective area of absorbers Actual area of column : Here, j ref ðÞ are the mass absorption coefficients of our GHG, which give the effective cross-sectional area of our GHG molecules at wavenumber 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 q GHG dz 0 in the integrand in Eq. ( 17) gives the absorber mass per unit area (kg=m 2 ) in an atmospheric layer of differential depth dz 0 .Thus, the integral s 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 s < 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 s > 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 s % 1.
For simplicity, we further assume that all emission occurs at exactly s ¼ 1; we refer to this as the emission level approximation. 32With this in mind, we define our emission pressure p em ðÞ by the relation s ðp em ðÞÞ ¼ 1: (18) This equation will form the basis for reasoning about both CO 2 forcing as well as the H 2 O feedback, which are the subjects of Secs.IV and V.

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 2Â .Fig. 3. Cartoon of emission to space from different atmospheric layers corresponding to different optical depths.Layer-wise emission to space maximizes at a sweet spot where the optical depth s % 1.
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. 37e begin with CO 2 reference mass absorption coefficients j CO 2 in the strongly absorbing and climatically relevant 2 bending mode centered on wavenumber 0 ¼ 667 cm À1 .Figure 4(a) displays this absorption spectrum. 38A key simplification we make is to ignore fine-scale spectral structure and note that on a broad scale, j CO 2 ðÞ may be parameterized as where j 0 represents a smoothed peak absorption coefficient and l is a "spectroscopic slope," which sets the rate at which absorption declines away from the peak.With Eq. ( 19) in hand, we next evaluate the optical depth integral (17) for CO 2 , for which q GHG ¼ qq where q is a constant CO 2 specific concentration and q is the density of air.Plugging this into Eq.( 17) and invoking hydrostatic balance dp=dz ¼ Àqg [see also Eq. ( A3)] then yields The pressure broadening factor in the integrand above, as well as the fact that the mass of CO 2 in a layer is given by qdp 0 =g, conspire to give s $ p 2 .
Invoking the emission level approximation (18), as well as the "exponential spectroscopy" (19), we can set s CO 2 ¼ 1 and invert Eq. ( 20) to find the emission levels p em ðÞ, These emissions levels are shown in Fig. 4(b) for ðq i ; q f Þ ¼ ð280; 1120Þ ppm.(A quadrupling of CO 2 is easier to visualize than a doubling.)The p em curves take the form of "emission triangles," which widen and deepen with increased CO 2 .The red and green highlighted regions show that an increase in 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 Note the logarithmic dependence on q, which we comment on further below.
To turn this insight into a formula for CO 2 forcing, Fig. 4(c) shows OLR ¼ pBð; T em Þ obtained by inserting the emission levels p em from Fig. 4(b) into the temperature profile Eq. (10b) and then applying the Planck function.The large values on the outside of the CO 2 band correspond to surface emission pBð; T s Þ, and the smaller values in the center correspond to stratospheric emission pBð; T strat Þ (i.e., where p em ðÞ Շ 200 hPa).Given the definition Eq. ( 11) of the forcing and the expression (22) for D, the forcing can then be read off from Fig. 4(c) as just the difference in the area under the solid and dashed curves, i.e., This formula can also be interpreted via Fig. 4(b) as a swap of stratospheric emission for surface emission over spectral regions of total width 2D.For an analytical derivation of Eq. ( 23), see Appendix B.
We are now in a position to make a back-of-the-envelope estimate of CO 2 forcing.Evaluating Eq. ( 23) for T s ¼ 288 K, T strat ¼ 210 K, and for q f ¼ 2q i yields the estimate F 2Â % 2l ln 2 pBð 0 ; 288 KÞ À pBð 0 ; 210 KÞ ½ % 4:15 W=m 2 ; (24) close to the standard value of 4 W=m 2 . 39The 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 ðq f =q i Þ factor in Eqs. ( 22) and ( 23).Unwinding the mathematics leading to Eq. ( 22), one finds that the ln ðq f =q i Þ factor stems from the exponential spectroscopy j CO 2 ðÞ in Eq. ( 19).The fact that j CO 2 ðÞ 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 (cf.Appendix B).Thus, multiplicative changes in q always change the width of the p em triangles in Fig. 4(b) by the same D, and the forcing is proportional to this width.

V. THE WATER VAPOR FEEDBACK
Now that we have a handle on CO 2 forcing, we turn towards a better estimate of the feedback parameter k.For k, 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 k we neglect the effects of CO 2 for simplicity).Similar to CO 2 , the H 2 O reference mass absorption coefficients j H 2 O ðÞ can be parameterized in terms of exponentials, this time with two exponentials peaked at 1 ¼ 150 cm À1 and 2 ¼ 1500 cm À1 , as Note the absorption minimum at roughly 1000 cm À1 , which will play an important role in what follows. 40ext we evaluate the optical depth integral Eq. ( 17).Here, q GHG is now the water vapor density q v , which we write as where the first equality is the ideal gas law (A2) for water vapor and the second is just a definition of the relative humidity RH, which we take to have a fixed value of 0.75. 41Here, p v is the partial pressure of water vapor, R v ¼ 287 J=ðkg Á KÞ is the specific gas constant for water vapor, and is the Clausius-Clapeyron relation for the saturation partial pressure of water vapor, with reference pressure p ref v ¼ 2:5 Â10 11 Pa and L the latent heat of vaporization (equal to 2:5 Â 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.
Inserting these expressions into Eq.( 17) and changing the dummy integration variable from z 0 to T 0 (where the lower limit on T 0 is taken to be the tropopause and, thus, equals to where av =CL is a reference value for column water vapor mass (kg=m 2 ), and several approximations were made to make the integral tractable. 42Note that M ref v exp ðÀL=R v TÞ 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 s H 2 O ¼ 1 and inverting Eq. ( 28) either numerically or analytically; 43 where s H 2 O ð; T s Þ < 1, in the aforementioned optically thin region centered around 1000 cm À1 (known as the infrared "window"), the emission level lies below the surface so we set T em ¼ T s .The results for T s ¼ ð280; 290; 300Þ 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., dT em =dT s % 0. This can be deduced from Eq. ( 28), where vertical variations in s H 2 O 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, s H 2 O is a function of temperature alone, and s H 2 O ðÞ ¼ 1 will occur at approximately the same T em regardless of T s (assuming fixed RH).  25) with l 1 ¼ 56 cm À1 ; l 2 ¼ 40 cm À1 ; j 1 ¼ 130 m 2 =kg, and j 2 ¼ 8 m 2 =kg.These absorption coefficients have been simplified by neglecting continuum absorption.(b) Graph of emission temperatures T em diagnosed from Eq. (28) for T s ¼ ð280; 290; 300Þ K, which are insensitive to T s except in the optically thin infrared 'window' at 750 À 1200 cm À1 .
We formalize this fact, first formulated by Simpson 44  This result is of course only approximate, as the pressure broadening factor (which includes a mild T s -dependence via Eq.( 10b)) and other effects cause slight deviations, 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 ðÞ and, hence, OLR (by Eq. ( 16)) are independent of T s for optically thick , and if we consider the atmosphere perfectly transparent for the optically thin in the window region so that OLR is given by surface emission pBð; T s Þ for those wavenumbers, then the spectrally resolved feedback parameter k (satisfying In other words, the feedback is zero in the optically thick regions due to Simpson's law, and in the window region is just given by the surface blackbody response.Given this, it is now straightforward to spectrally integrate k and evaluate at T s ¼ 288 K to obtain k; all that is required is an estimate of the limits of the window region.This is typically 46 taken to be 800 < < 1200 cm À1 , although these limits are not precisely defined.For the sake of obtaining round numbers, we take the lower limit to be 750 cm À1 , which then yields our RCE estimate of k, 47 where the integral is computed numerically.This value of k, 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. 48Furthermore, k 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 k because T em ð) is fixed (Fig. 5(b)).This significantly reduces k 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.Indeed, a feedback parameter of k RCE ¼ À2 W=m 2 =K, combined with F 2Â ¼ 4 W=m 2 , yields an estimated equilibrium climate sensitivity of This estimate is close to those obtained by MW67 (2.3-2.9K) though somewhat smaller than the consensus value of 3 K.Given the evidence that the neglected cloud and shortwave feedbacks are likely positive, however (see Sec. VI), it is unsurprising that ECS RCE is biased low, and it is perhaps best regarded as a lower bound on ECS.Indeed, 2 C is the lower bound of the very likely range of 2-5 C found in the recent sixth assessment report of the IPCC. 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 k 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. 51he longwave effects of clouds stem primarily from the high clouds arising from thunderstorms and mid-latitude winter storms; consistent with their height these clouds are very cold with a typical cloud-top temperature T cld ¼ 220 K (Fig. 2).Their Planck emission is, thus, much reduced relative to the clear-sky OLR one would see in their absence.If these high clouds cover a fraction f of the Earth's surface (f % 0:18 in the present-day global mean 52 ), then one can write the "all-sky" (i.e., actual) OLR as In other words, high clouds mask a fraction f of OLR clr , replacing it with cloud-top emission rT cld 4 .To differentiate Eq. ( 31) and obtain the all-sky longwave feedback k all , one needs to know how 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., dT cld dT s % 0: (32) Differentiating Eq. ( 31), thus, yields As for df =dT s , 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 df =dT s , thus, tend to compensate, and the value of k all ends up not far from k clr % À2 W=m 2 =K but with larger error bars. 55n 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 k SW cld $ 0:2 W=m 2 =K. 56This 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. 57Meanwhile, these clouds have also been described by simplified "mixed-layer models." 58However, 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 a [cf.Eq. ( 1)].This yields a positive feedback k albedo $ 0:3-0:4 W=m 2 =K. 59While 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 . 62Since 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 k SW wv $ 0:25 W=m 2 =K. 64A principled estimate of k SW wv 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
Putting the RCE (or longwave clear-sky) estimate ( 29) together with the cloud and shortwave feedbacks outlined above yields an estimate for the total feedback parameter This is close to the value of k ¼ À1:3 W=m 2 =K assessed by Sherwood et al. 66 from multiple lines of evidence, which indeed yields the consensus ECS value of ÀF 2Â =k ¼ ð4 W=m 2 Þ=ð1:3 W=m 2 =KÞ % 3 K.This more realistic value of k 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 k, 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. 67This model consists of a shallow mixed layer with depth h ml % 50 m and temperature anomaly T 0 ml equal to DT s , sitting atop a much larger deep ocean with global average depth h d % 2500 m and temperature anomaly T 0 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 jkjT 0 ml and also exports energy to the deep ocean, which we parameterize in linearized form as cðT 0 ml À T 0 d Þ (units of W=m 2 ).Here, c is the "deep ocean heat uptake efficiency," estimated from models at roughly 0.7 W=m 2 =K. 68Setting q w and C w as the densities and specific heat capacities of water, the corresponding equations are Because the deep ocean has a much larger depth and, hence, heat capacity than the mixed layer (h d ) h ml ), we expect T 0 ml to respond to F much faster than T 0 d does.Indeed, assuming T 0 d ¼ 0 in Eq. (35a) yields a linear, Fig. 6.Two-box model for the ocean.See the text for discussion.
constant coefficient ordinary differential equation with characteristic timescale, If we now consider timescales longer than s ml , we may set dT 0 ml =dt ¼ 0 in Eq. (35a), solve for T 0 ml , plug the result into Eq.(35b), and solve for the characteristic deep ocean timescale s d .This makes a nice exercise with result Thus, the vast difference in total heat capacity between the mixed layer and deep ocean indeed leads to two timescales for global warming: A "fast" timescale of about s ml % 3 years during which the mixed layer equilibrates (really a "quasi-equilibrium," since this equilibrium state will change as the deep ocean slowly responds), and a "slow" timescale of s d % 700 years during which the deep ocean equilibrates (a true equilibrium).On intermediate timescales in between s ml and s d in which the mixed layer is in quasi-equilibrium, we can assume both dT 0 ml =dt ¼ 0 and T 0 d % 0, which from Eq. (35a) yields This gives the transient warming which occurs before the deep ocean has responded.Indeed, if we specialize to the case of a CO 2 doubling, the temperature in Eq. ( 38) becomes a standard sensitivity metric known as the "transient climate response," or TCR, 69 This estimate is quite close to the value of 1.8 K found in the recent sixth assessment report of the IPCC. 70Comparing Eq. (39) with Eq. ( 13) shows that ECS > TCR: on the intermediate timescales during which T 0 ml ¼ 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 0 d % 0 on intermediate timescales.This approximation turns out to be a reasonable description of the present day, and the quasiequilibrium formula (38) (which is just a scaling of TCR) can be used to credibly model historical as well as near-term global warming. 71In 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. 72ote that the two-box model, while popular, is by no means canonical.Some recent work instead employs threebox models, and the older literature often employed diffusive models. 73Two-box, three-box, and diffusive models were all employed by the IPCC as emulators of more comprehensive models. 74All these approaches, however, require empirically determined parameters for heat transfer coefficients and diffusivities, analogous to our heat uptake efficiency c, 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 $2 K=1000 GtC, which in simulations turns out to be fairly invariant over time as well as insensitive to emissions scenario. 76The robustness of TCRE tells us that any identified temperature target (e.g., DT s ¼ 1:5 or 2 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. 77t is possible to use the TCR of Sec.VII A, as well as the logarithmic scaling of CO 2 forcing, to make a back-of-theenvelope estimate of TCRE as follows.Let C emit be our cumulative carbon emissions since the preindustrial era (GtC), C pre the preindustrial mass of CO 2 in the atmosphere (GtC), and a be the fraction of C emit still residing in the atmosphere (the "airborne fraction"), so that the total mass of CO 2 in the atmosphere is C pre þ aC emit .Recalling that in the quasi-equilibrium approximation, surface warming DT s can be obtained by scaling the TCR by F =F 2Â [cf.Eqs. ( 38) and ( 39)], we then have In the last equality, C emit drops out of this expression, giving some insight into why simulations find TCRE to be roughly invariant over time. 78Furthermore, evaluating Eq. ( 40) for C pre ¼ 590 GtC and the present day airborne fraction of a ¼ 0:4 indeed yields roughly 2 K/GtC, consistent with simulations. 79However, here we are simply using the observed value of a; we do not yet have ways to make principled estimates of a from basic carbon cycle dynamics, and thus to understand how and why a does or does not vary across time and emissions scenarios.The zero emissions commitment (ZEC), on the other hand, is complementary to TCRE and is defined to be the amount of residual warming occurring after emissions cease.Simulations 80 tend to show that ZEC is small relative to DT s , which is a requirement for net zero emissions goals to meaningfully limit DT s .Similar to the estimate for TCRE, we may scale TCR and also ECS to estimate ZEC, following an argument due to Tarshish. 81Let T ze and a ze be the temperature and airborne fraction of cumulative emissions when emissions cease, and T f and a f be the temperature and airborne fraction at final equilibrium, respectively.Normalizing ZEC T f À T ze by T ze , we have This tells us that the sign and magnitude of ZEC is determined by a competition between the chemical disequilibrium a f =a ze and thermal disequilibrium TCR=ECS when emissions cease.If the chemical disequilibrium is more pronounced than the thermal disequilibrium, i.e., if a f =a ze < TCR=ECS, then the decline in CO 2 forcing due to equilibrium carbon uptake will dominate over the reduction in the deep ocean heat sink, yielding ZEC < 0, and vice-versa if a f =a ze < TCR=ECS.Noting that a f % 0:2 is a characteristic value, which may be obtained from principles of carbonate chemistry, 82 the previously cited values a ze ¼ 0:4; TCR ¼ 1:8 K, and ECS ¼ 3 K then give ZEC=T ze % À0:17, 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 c, which entered into our estimate Eq. (39) of TCR, as well as the present day airborne fraction a % 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.
where the specific heat at constant pressure C p ¼ C v þ R d % 1000 J=kg=K.Further assuming hydrostatic balance, 86 which says that the weight of a layer of air is balanced by the vertical pressure gradient across it, and rearranging then yields Eq. ( 5) Now consider an arbitrary but constant lapse rate C.Then, we have where Eq. (A3) was used in the second equality and Eq.(A2) in the third.This relation between 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
The forcing F is usually obtained by numerical integration in across the CO 2 band shown in Fig. 4. Here, we will instead pursue an analytical integration over , following the thesis of Seeley. 87 We will take advantage of the symmetry evident in Fig. 4 and simply double the integral for > 0 .We will also take advantage of the key fact that q and j CO 2 ðÞ 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 This equation encapsulates the fact, mentioned towards the end of Sec.IV, that a multiplicative change in q is equivalent to that of a uniform translation in .Putting these ingredients together, the forcing Eq. ( 11) can now be evaluated as an integral over from band center 0 to the upper limit þ % 750 cm À1 , where we ignore spectral variations in the Planck density.This yields dT em d ln q d by Eq: ð16Þ; where in the last line we invoked the fundamental theorem of calculus, noting also that 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. 17It is a nice exercise to derive this height analytically, as follows: Assume an isothermal atmosphere with T ¼ T em , which from Eqs. (A3) and (A2) yields the exponential dependence p ¼ ð1 atmÞ exp ðÀz=HÞ; )   where H R d T em =g % 7:5 km is the atmospheric "scale height."Setting p a ¼ 0:5 atm then yields z a ¼ H ln 2 % 5 km.Payne et al. (2015). 20Stratospheric 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 = ffiffi ffi 2 4 p ¼ 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 et al., 2019;  Jeevanjee and Fueglistaler, 2020b), but more work on this topic is needed. 21At least to first approximation.See discussion in Jeevanjee et al. (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 et al. (2015). 25An 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  et al. (1997); Houghton et al. (1994). 26Another useful consequence of this definition is that "positive," amplifying feedbacks make positive contributions to k. 27  Charney et al. (1979); Myhre et al. (1998); Ramaswamy et al. (2019). 28We will not need the specific form of Bð; TÞ, but for completeness, it is given by Bð; TÞ ¼ where h is Planck's constant, k b is Boltzmann's constant, and c is the speed of light. 29Since CO 2 is well-mixed, its density is proportional to the air density q, and q 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. 30We here assume a two-stream approximation with unit diffusion coefficient for simplicity (Pierrehumbert, 2010). 31This 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). 32For further analysis of the emission level approximation, with analytic calculations suitable for the classroom, see Appendix B of Jeevanjee et al.  (2021b) and Appendix B of Jeevanjee and Fueglistaler (2020a). 33Wilson and Gea-Banacloche (2012). 34Seeley (2018). 35  Koll et al. (2022). 41This fixed relative humidity assumption, popularized by Manabe and  Wetherald (1967), has been well justified by decades of subsequent simulations and observation (Jeevanjee et al., 2022; Colman and Soden, 2021). 42In particular, we multiply the integrand by T av =T 0 , 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 pðT 0 Þ % pðTÞ so it can be pulled outside the integral.This expression and these approximations are discussed in more detail in Jeevanjee and Fueglistaler (2020b).
(2021a) for further discussion. 45These deviations can be estimated by calculating dT em =dT s implicitly from Eq. ( 28) and invoking Eq. ( 10b).This makes a nice exercise, with result

Fig. 1 .
Fig. 1.A cartoon of radiative-convective equilibrium.See the text for explanation.
in 1928, as "Simpson's law":Simpson's law: At fixed RH, and for optically thick wavenumbers dominated by H 2 O absorption, emission temperatures are insensitive to surface temperature, i.e., 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.SeeJeevanjee et al. (2021b);Huang  et al. (2016).40TheH 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.SeePierrehumbert (2010);Shine et al. (2012); Jeevanjee et al. (2021b).36Rompsetal.(2022).37Jeevanjeeetal.(2021b).38Forapedagogicaldiscussion of the structure of this spectrum and its computation, seeWilson and Gea-Banacloche (2012).39