“Simpson's Law” and the Spectral Cancellation of Climate Feedbacks

We spectrally resolve the conventional clear‐sky temperature and water vapor feedbacks in an idealized single‐column framework, and show that the well‐known partial compensation of these feedbacks is actually due to an almost perfect cancellation of the spectral feedbacks at wavenumbers where H2O is optically thick. This cancellation is a natural consequence of “Simpson's Law”, which says that H2O emission temperatures do not change with surface warming if relative humidity (RH) is fixed. We provide an explicit formulation and validation of Simpson's Law, and furthermore show that this spectral cancellation of feedbacks is naturally incorporated in the alternative RH‐based framework proposed by Held and Shell (2012, https://doi.org/10.1175/jcli-d-11-00721.1) and Ingram (2012, https://doi.org/10.1029/2011jd017221, 2013b, https://doi.org/10.1007/s00382-012-1294-3), thus bolstering the case for switching from conventional to RH‐based feedbacks. We also find a negligible RH‐based clear‐sky lapse rate feedback, suggesting that the impact of changing lapse rates depends crucially on whether relative or specific humidity is held fixed.


Introduction
The climate feedback parameter λ measures the response of net downward top-of-atmosphere radiation N to a change in surface temperature T s as   dN dT s W m K ( / / ) . 2 (1) Under radiative forcing ΔN = F, then, λ determines the climate response as ΔT s = −F/λ. As such, λ is a central quantity in climate science and has been intensely studied. Typically, λ is decomposed into different terms which aim to isolate the contributions from distinct physical processes. While particular definitions and methodologies have evolved over time, a "conventional" framework has emerged in which λ is decomposed as (e.g., Sherwood et al., 2020, and now writing λ as λ tot ): (RH) is a much better null hypothesis (e.g., Ferraro et al., 2015;Held & Soden, 2000;Romps, 2014;Sherwood et al., 2010;Soden et al., 2005;Zhang et al., 2020). This inappropriateness of the conventional Planck feedback can even lead to reference responses which are physically unrealizable (Held & Shell, 2012).
The conventional decomposition in Equation 2 also has practical drawbacks. Across models, λ wv and λ lapse exhibit significant spread but also a strong anti-correlation (Soden & Held, 2006;Soden et al., 2008). This means that the individual spread in λ wv and λ lapse largely cancels in the sum in Equation 2, and is thus not indicative of uncertainty in λ tot . Physically, this anti-correlation means that λ wv and λ lapse are not capturing independent physical processes, diminishing the utility of a decomposition such as Equation 2.
Such drawbacks led many studies to consider only the sum λ lapse + λ wv (e.g., Ingram, 2013a;Huybers, 2010;Sherwood et al., 2020;Soden & Held, 2006;Soden et al., 2008). This defines away the anti-correlation problem, and significantly reduces spread. But, there is a basic physical inconsistency in summing λ lapse and λ wv , in that λ wv is due to the entire specific humidity perturbation, whereas λ lapse is due only to temperature perturbations which are not vertically uniform. Furthermore, the anti-correlation between λ wv and λ lapse does not arise solely from colocated warming and moistening of the tropical upper-troposphere, as previously thought, but also from the nonlocal influence of tropical warming on the extratropical stratification (Po-Chedley et al., 2018). This further undermines the physical justification for summing λ lapse and λ wv .
This state of affairs led Held and Shell (2012) and Ingram (2012Ingram ( , 2013b to propose using RH as the moisture state variable for feedback analyses. This means that the Planck and LR feedbacks are to be computed while holding RH rather than specific humidity (q v ) fixed, and that the WV feedback is now only due to changes in RH rather than q v . These studies and others (e.g., Caldwell et al., 2016) showed that switching from conventional to RH-based feedbacks not only yields a more physical reference response (Planck feedback), but also greatly reduces the spread in and anti-correlation between the LR and WV feedbacks (spread in the Planck feedback is also reduced).
Given these advantages, some recent studies have adopted the RH-based formalism as their primary approach (e.g., Caldwell et al., 2016;Zelinka et al., 2020). Other influential studies have carried on with the conventional approach, however (e.g., Sherwood et al., 2020), leading to inconsistency in the literature. Furthermore, the underlying radiation physics of these two approaches remains underexplored. Ingram (2010) argued that λ wv must significantly offset λ planck and λ lapse due to what we will call "Simpson's Law": the fact that to first order, and under fixed RH, the outgoing longwave radiation (OLR) at H 2 O-dominated wavenumbers does not change with surface warming (first articulated by Simpson, 1928). If true, this implies that at such wavenumbers the total feedback should be roughly 0, and thus that (at such wavenumbers) the Planck, LR, and WV feedbacks should cancel almost exactly. These implications, however, have not been drawn out in detail or explicitly verified.
Accordingly, our goal in this paper is to highlight Simpson's Law and then demonstrate that spectrally resolved conventional feedbacks indeed largely cancel at optically thick, H 2 O-dominated wavenumbers. By contrast, we show that the RH-based formalism naturally incorporates this cancellation, providing a clearer view of the clear-sky feedbacks. We hope that this fundamental simplicity of the RH-based approach, and its consistency with the basic physics of Simpson's Law, will encourage more widespread use of RH-based feedbacks. Our spectrally resolved results also allow for more detailed interpretations of the various components of the RH-based feedbacks.
We begin in Section 2 by reviewing Simpson's Law and explicitly demonstrating it using line-by-line radiative transfer. After reviewing the definition of the feedbacks from Equation 2 in Section 3, we then apply Simpson's Law to understand the spectral cancellation of conventional feedbacks in Section 4. We conclude in Section 5.

Simpson's Law
In this section we briefly review Simpson's Law, which dates back to Simpson (1928). Simpson's Law is the key ingredient in the "runaway greenhouse" effect (e.g., Goldblatt et al., 2013;Nakajima et al., 1992), and has also been used to explain the T s -dependence of OLR (Koll & Cronin, 2018), the rate of global mean precipitation change (Jeevanjee & Romps, 2018), and the strength of the water vapor feedback (Ingram, 2010).
A pedagogical treatment is given in Jeevanjee (2018). We emphasize at the outset that Simpson's "Law" does not hold exactly, but is rather a first-order approximation; we refer to it as a "Law" simply to emphasize the fundamental role it plays in the spectral structure of radiative feedbacks.
To arrive at Simpson's Law, we first note that if RH is uniform, then the vapor density ρ v (kg/m 3 ) is a function of temperature only, with no explicit pressure dependence: where e*(T) is saturation vapor pressure and all other symbols have their usual meaning. Viewing T as a vertical coordinate, then, implies that the profile ρ v (T) should be universal and independent of surface temperature, that is, "T s -invariant" (cf. Figure 1 of Jeevanjee & Romps, 2018).
This then implies that H 2 O optical depth at a given wavenumber should also be a T s -invariant function of T, at least to first order and under typical circumstance. To see this, we write H 2 O optical depth in temperature coordinates as where T tp is the tropopause temperature, κ is the mass absorption coefficient (m 2 /kg), and Γ the lapse rate.
(Such an expression neglects stratospheric water vapor and cannot be used when tropospheric T(z) is not single-valued, that is, when there is a temperature inversion. Future work could investigate the validity of Simpson's Law under such circumstances.) Though κ exhibits pressure and temperature dependencies due to collisional broadening and quantum effects (Pierrehumbert, 2010), and moist lapse rates Γ also vary in the vertical, these variations are expected to be weak compared to the strong exponential T-dependence of ρ v . Since ρ v is T s -invariant, we expect τ(T) to be so as well, at least to first order (cf. Figure S5 of Jeevanjee & Romps, 2018). Since cooling-to-space can be approximated as emanating from τ ≈ 1 for optically thick wavenumbers ν (e.g., Petty, 2006;Jeevanjee & Fueglistaler, 2020a), this suggests that the spectrally resolved OLR ν and corresponding emission temperature T em (ν), defined in terms of the Planck function B(ν, T) by should also be T s -invariant (so long as RH is fixed). This then yields Simpson's "Law": Simpson's "Law": At fixed RH, and for optically thick wavenumbers dominated by H 2 O absorption, emission temperatures and OLR are independent of surface temperature (to first order).
We explicitly verify Simpson's Law in Figure 1 by plotting T em (as diagnosed via Equation 5) as a function of wavenumber for a set of moist adiabatic columns at varying T s and with RH = 0.75 and no CO 2 , using the Reference Forward Model (RFM, details of these calculations are as given in Section 4). Atmospheric emission emanates from the optically thick sections of the H 2 O pure rotational band (0-800 cm −1 ) and vibration-rotational band (1,200-1,500 cm −1 ), while surface emission emanates through the optically thin water vapor "window" at 800-1,200 cm −1 (For further intuition for this structure, see Jeevanjee & Fueglistaler, 2020b). The optically thick wavenumbers show relatively little variation of T em with T s , validating Simpson's Law. Indeed, the average of dT em /dT s over 0 − 800 cm −1 at T s = 290 K is 0.2. Of course, the fact that dT em /dT s is not identically zero shows that Simpson's Law is only approximate, due to our neglect of pressure broadening and lapse-rate changes in deducing Simpson's Law above.
Simpson's Law is nonetheless a useful idealization, as it encapsulates the small changes in optically thick T em relative to the much larger changes in T em in the optically thin water vapor window (in the window, which remains optically thin for T s ≲ 290 K, we have T em ≈ T s and thus dT em /dT s ≈ 1). In particular, differentiating Equation 5  should be roughly zero at H 2 O-dominated wavenumbers. This then means that water vapor, Planck, and lapse rate feedbacks must cancel at those wavenumbers. A primary goal of this paper is to explicitly verify this. A further corollary is that the total feedback is nonzero primarily in the water vapor window, and thus that the window is the main channel through which OLR increase with T s (as emphasized by Koll & Cronin, 2018). We will sharpen and verify these claims in Section 4.

Feedback Formulation
With Simpson's Law in place we now turn to feedbacks. We begin by giving precise definitions of the Planck, LR, and WV feedbacks, in both the conventional and RH-based frameworks. Since the choice of moisture variable mostly impacts the clear-sky, longwave feedbacks, we consider these feedbacks only and do not consider λ cloud , λ albedo , or the shortwave component of λ wv in our analysis. See Held and Shell (2012), however, for a discussion of how the RH-based framework changes the relative importance of other feedbacks.
In a cloud-free atmosphere with H 2 O and CO 2 as the only greenhouse gases, the OLR is determined by their profiles, along with the surface temperature T s and atmospheric temperature profile T a (we suppress the vertical coordinate for clarity). A choice must be made, however, of which state variable to use for specifying H 2 O concentrations; we begin with the conventional choice of specific humidity q v , and later discuss the modification when using RH. We specify an atmosphere as an ordered triple (T s , T a , q v ), and the OLR is then a function of this ordered triple, that is, We suppress the dependence of OLR on CO 2 concentration since we consider feedbacks here, not forcings, and feedbacks are always computed with CO 2 concentrations held fixed. The relevant CO 2 concentrations will be specified in the next section.
Consider now an initial atmosphere s a v ( , , ) (This simple forward difference, or "one-sided" partial radiative perturbation (PRP), is not as accurate as the "two-sided" PRP methods or radiative kernel methods employed in comprehensive calculations (Colman & Mcavaney, 1997;Shell et al., 2008;Soden et al., 2008;Yoshimori et al., 2020), but suffices for our purposes here.) In the conventional (q v -based) framework, we then define the following individual feedbacks: The Planck feedback λ planck is minus the (ΔT s -normalized) OLR response to a uniform change in surface and atmospheric temperatures, with q v held fixed at the initial profile. The lapse-rate feedback λ lapse is minus the OLR response to the difference between the actual temperature response and the uniform Planck response, still holding q v fixed. The water vapor feedback λ wv is then minus the OLR response to the change in q v , holding temperatures fixed. Assuming linearity in the finite differences, we then have For RH-based feedbacks, we use Equation 8 but simply replace q v with RH, and denote the corresponding RH-based feedbacks with a tilde: .
Note that now the water vapor feedback   wv is due to RH changes, not q v changes. Our calculations below will be at fixed RH, so    wv 0 here by definition and we do not consider it further. This seems permissible because general circulation model studies find global-mean   wv to be small, namely   wv W m K   0 0 1 2 . / / (Held & Shell, 2012;Zelinka et al., 2020).
Since we expect increases in OLR to emanate from increased surface emission through the window (as suggested by Figure 1), we also introduce a "surface" feedback λ surf obtained by perturbing T s while holding the atmospheric temperature and water vapor profiles fixed: This feedback is identical in the q v -based and RH-based frameworks, and is equal to the "surface kernel" of radiative kernel analyses (cf. Figure 1 of Soden et al., 2008).
A key aspect of our analysis will be to consider spectrally resolved OLR and hence spectrally resolved versions of the feedbacks in Equations 7, 8, 10 and 11. These will be denoted with a subscript ν, with units W/m 2 /cm −1 /K.

Spectral Cancellation of Conventional Feedbacks
We now turn to spectrally resolved calculations of the various feedbacks defined above, for a variety of idealized atmospheric columns. We calculate OLR ν for these columns using the line-by-line Reference Forward Model (RFM, Dudhia, 2017), along with HiTRAN2016 spectroscopic data (Gordon et al., 2017) for H 2 O from 0 to 1,500 cm −1 and CO 2 from 500 to 850 cm −1 , using only the most common isotopologue for each gas.
We run RFM at a spectral resolution of 0.1 cm −1 and on 100 evenly spaced pressure levels between 1,000 and 10 hPa. Lineshapes follow the RFM default of a Voigt profile with 25 cm −1 cutoff, and H 2 O continuum effects are parameterized via RFM's implementation of the MT-CKD2.5 continuum (Mlawer et al., 2012).

All atmospheric columns have
,  s 289 f T K, RH = 0.75, and an isothermal stratosphere at T strat = 200K, with a uniform stratospheric q v set equal to its tropopause value. The lapse rates and radiatively active species vary between cases, as described below.

Constant Lapse Rate, H 2 O-Only Atmosphere
We begin by considering atmospheric columns with a constant lapse rate of 7 K/km and H 2 O as the only radiatively active species. This case avoids the complications due to the LR feedback and due to CO 2 , both of which we address below. We calculate the spectrally resolved conventional feedbacks   according to Equations 7 and 8; these are shown in Figure 2a.
The conventional Planck feedback   planck is strongly negative, as expected, but is not a good first approximation to the total feedback   tot ; there are large cancellations between   planck and the strongly positive water vapor feedback   wv . Indeed, as expected from Simpson's Law, at optically thick wavenumbers we have Planck and WV feedbacks cancel for optically thick ν (b) As in (a) but in the RH-based formalism. Now the Planck and total feedbacks are equal, and are well approximated by the surface feedback λ surf (c and d) As in (a and b), but for moist-adiabatic temperature profiles. We still find    tot 0 for optically thick ν, but now this implies a three-way cancellation of conventional feedbacks. The picture again simplifies for RH-based feedbacks, with a much smaller LR feedback in the RH-based formalism (e and f) As in (c and d), but now including 280 ppm of CO 2 . Now   tot exhibits CO 2 "radiator fins", that is, local extrema on either side of the CO 2 band. These extrema are overshadowed by other features in the conventional decomposition, but are highlighted in the RH-based decomposition.
Color-coded numbers give the spectral integrals λ of the corresponding spectrally resolved feedbacks   . Output is again smoothed over bins of width 10 cm −1 .
Thus, at most wavenumbers the conventional feedback decomposition splits the total feedback into equal and opposite terms, which are constrained to cancel by basic physics.
We now contrast this behavior with that of the RH-based formalism (Figure 2b). In this case the picture is markedly simpler: the Planck feedback takes place at constant RH and so Simpson's Law is manifest, yielding     planck 0 outside the window. In fact, since these idealized columns have no RH or lapse rate perturbations, we find that        planck tot identically (so only one of these curves is visible in Figure 2b). Thus, when RH is the moisture variable the reference response is a good null hypothesis (Roe, 2009); in fact, for this simple system, the reference response captures the total system response perfectly.
As mentioned earlier, the dominant contribution to λ tot seen in Figures 2a and 2b can be interpreted as an increase in surface cooling-to-space through the optically thin water vapor window. This can be made more precise by invoking the argument of Koll and Cronin (2018), who show that the effects of increasing atmospheric emissivity on surface emission and atmospheric emission cancel; the decreased emission-to-space from the surface is compensated for by increased emission from the near-surface atmosphere, which has the same temperature as the surface (Koll & Cronin, 2018, Equation 5). Hence, the effect of warming on OLR should be given by the Planck increase in surface emission, with atmospheric emissivity held fixed. This is just the λ surf term of Equation 11, so this yields the approximation The surface feedback   surf is shown in purple in Figure 2b, and we find that in this idealized case Equation 13 indeed holds, to an accuracy of about 10% in the spectral integral (errors in this approximation are due to deviations from Simpson 's Law). Equation 13 thus gives a straightforward way to interpret the dominant contribution to λ tot .

Conventional and RH-Based Feedbacks in a Moist-Adiabatic, H 2 O-Only Atmosphere
Let us now incorporate the lapse-rate feedback, by replacing our constant lapse-rate temperature profiles with moist pseudo-adiabats based at  s 288 i T K and  s 289 f T K. We also introduce the conventional lapse rate feedback   lapse as defined in Equation 8b. This feedback, and the others calculated as before, are shown in Figure 2c.
Even though the lapse-rate feedback is present, Simpson's Law still operates: changes in moist adiabatic Γ(T) with T s are relatively small (Ingram, 2010), so by Equation 4 T em should still be insensitive to T s . Now, however, Simpson's Law implies that at optically thick ν, there should be a near-complete cancellation of three terms: This means that summing the LR and WV feedbacks only yields a partial cancellation even at optically thick wavenumbers, due to the aforementioned fact that λ wv is due to the entire q v perturbation but λ lapse is due only to part of the temperature perturbation (cf. Equations 8b and 8c). From a spectral point of view, then, little simplification arises from summing only the LR and WV feedbacks.
The RH-based feedbacks for these moist-adiabatic atmospheres are shown in Figure 2d. As before, the picture simplifies considerably: There is no RH-based water vapor feedback, and the RH-based Planck feedback is a good, if no longer perfect, approximation to the total feedback. A perhaps surprising result is that the RH-based lapse-rate feedback    lapse is small, even in a fully moist-adiabatic atmosphere. This is because in the RH-based framework, the lapse-rate temperature perturbation is made at constant RH, and thus Simpson's Law applies. This is consistent with the conclusions of Cess (1975), who finds that changes in lapse rate (at fixed RH) have little impact on global energy balance. Thus, the impact of changing lapse rates depends crucially on whether RH or q v is held fixed.
One consequence of    lapse being small is that the RH-based Planck feedback    planck is still a good null hypothesis for the OLR ν change. Another, related consequence is that the surface feedback approximation Equation 13 continues to hold (Figure 2d), again to about 10% in the spectral integral.

Conventional and RH-Based Feedbacks in a Moist-Adiabatic Atmosphere With H 2 O and CO 2
Next we consider the effects of CO 2 . We calculate feedbacks for the moist-adiabatic columns of the previous subsection, but now with 280 ppmv of radiatively active CO 2 .
The results are shown in Figures 2e and 2f. In the q v -based framework, there is still a marked cancellation between   planck ,   wv , and   lapse , but it now only occurs for ν which are outside the H 2 O window and outside the 575 − 775 cm −1 CO 2 band. Following Seeley and Jeevanjee (2020) we refer to the wings of the CO 2 band as "CO 2 radiator fins", as these wavenumbers radiate from the upper troposphere and are visible as local extrema in   tot at roughly 625 and 725 cm −1 . These wavenumbers radiate from fixed pressures (at fixed CO 2 ) rather than fixed temperatures (for CO 2 , τ ∼ p 2 ; Pierrehumbert, 2010), and thus do not obey Simpson's Law. They make non-negligible contributions to   tot , but are overshadowed by other features in   planck and   lapse , due to continued cancellation with   wv .
In the RH-based framework (Figure 2f), however, the picture is again much simpler. The non-Simpsonian CO 2 radiator fins remain, but are partially captured by the the RH-based Planck feedback    planck , which is thus still a reasonable first approximation to   tot (unlike the conventional   planck ). The rest of the CO 2 radiator fin contribution is due to enhanced upper-tropospheric warming from lapse-rate changes (Seeley & Jeevanjee, 2020), which is indeed the main feature in    lapse . The advantage of the RH-based formulation is that it highlights these features, rather than lumping them in with the larger   planck and   lapse which then cancel with λ ν wv .
Note that the negative contribution of the CO 2 radiator fins to spectrally resolved    lapse offsets the positive contribution from the window (which results from fixed-RH upper-tropospheric moistening helping to close the window), leading to an even smaller spectrally integrated   lapse . Thus, the conclusion from the previous H 2 O-only calculation-that the strength of the lapse rate feedback is highly dependent on the choice of moisture variable-is only reinforced by the addition of CO 2 . The presence of the CO 2 radiator fins also means that the surface approximation in Equation 13 breaks down in the CO 2 band. The surface feedback of Equation 11 still, however, gives a precise way of interpreting and accounting for the increase in surface emission through the window, which is a dominant contribution to λ tot in the present-day climate (Raghuraman et al., 2019;Seeley & Jeevanjee, 2020;Slingo & Webb, 1997).

Summary
This paper has shown that: 1. The well-known compensation of conventional, q v -based feedbacks is actually due to a near-perfect cancellation of these feedbacks at wavenumbers where H 2 O is optically thick, as dictated by Simpson's Law 2. This cancellation is incorporated more naturally in RH-based feedbacks, which more naturally manifest Simpson's Law Furthermore, because constant RH is our null hypothesis under surface warming, the RH-based Planck feedback   planck is a much better reference response (i.e., is closer to λ tot ) than the conventional Planck feedback. We also explicitly demonstrated that the increase in surface emission through the window is accurately captured by the surface feedback term of Equation 11, in line with the argument of Koll and Cronin (2018).
Our findings also add nuance to the interpretation of the lapse-rate feedback. The conventional view that λ lapse and λ wv should be summed is called in to question by the three-way cancellation of λ planck , λ lapse , and λ wv found here. Furthermore, we find (similar to Held & Shell, 2012;Zelinka et al., 2020) that   lapse can be an order of magnitude smaller than λ lapse , raising questions about the notion of a single, well-defined LR feedback.
One limitation of this study is its single-column framework with idealized temperature and moisture profiles. While these were chosen to represent global mean conditions, this framework assumed tight surface-troposphere coupling and thus cannot account for decoupled conditions with temperature inversions.
Indeed, the zonal mean analyses of Po-Chedley et al. (2018) found relatively large values of   lapse over the Southern Ocean, in contrast to our findings here. Future work could apply spectral feedback analyses to such decoupled atmospheres, to better understand these results. Future work could also consider how other greenhouse gases (e.g., ozone, methane) modulate the results shown here.
More broadly, however, our results suggest that RH-based feedbacks are not only more physical than conventional feedbacks from a thermodynamic point of view, as argued by Held and Shell (2012), but are also simpler from a radiative point of view. Notably, a similar tension between RH and q v -based points of view manifests in remote sensing applications, where it is long known that satellite-measured H 2 O brightness temperatures are more sensitive to RH than q v (Möller, 1961;Soden & Bretherton, 1993), yet more recent observational studies focus on q v (e.g., Dessler et al., 2008), perhaps because of its use in conventional feedback analyses. We hope that our explicit formulation and validation of Simpson's Law fosters a better appreciation of the emergent simplicity of H 2 O radiative transfer, and encourages the use of RH as moisture variable in such applications where this simplicity manifests, as is the case here.

Data Availability Statement
The simulation output used in this paper, as well as scripts for producing the figures, is available at http:// doi.org/10.5281/zenodo.4667563.