An Analytical Model for CO 2 Forcing, Part II: State-Dependence and Spatial Variations

Clear-sky CO2 forcing is known to vary significantly over the globe, but the state dependence which controls this is not well understood. Here we extend the formalism of Seeley et al. (2020) to obtain a quantitatively accurate analytical model for spatially-varying CO2 forcing, which depends only on surface and stratospheric temperatures as well as column relative humidity. This model shows that CO2 forcing is primarily governed by surface-stratosphere temperature contrast, with the corollary that the meridional gradient in CO2 forcing is largely due to the meridional surface temperature gradient. The presence of H2O modulates this forcing gradient, however, by substantially reducing the forcing in the tropics, as well as introducing forcing variations due to spatially-varying column relative humidity. 11


Introduction
Changes in the Earth's CO 2 greenhouse effect (i.e. CO 2 radiative forcing) have been a primary 23 driver of past and present changes in Earth's climate, and are well simulated by state-of-the-art ra-  26 and has thus been a priority for radiation research, less emphasis has been placed on an intu-27 itive understanding of CO 2 forcing and its dependence on atmospheric state variables and hence 28 geography or climate. For instance, zonally averaged clear-sky CO 2 forcing exhibits a marked to depend somewhat on details of the fit (e.g. Seeley et al. 2020; Jeevanjee and Fueglistaler 2020b; Wilson and Gea-Banacloche 2012). Instead, we opt to determine these parameters via optimization 91 as described in Section 3. 92 We now write down the optical depth τ ν at a given wavenumber ν, with pressure broadening but (2) Here q is the CO 2 specific concentration and D = 1.5 is a 2-stream diffusion coefficient (e.g. The pressure p 0 (q) ≡ p em ( ν 0 , q) is an effective emission pressure at the center of the CO 2 band. 101 We show in Appendix B that a suitable CO 2 emission level for our purposes is τ CO 2 em = 0.5. With ν ± em (p, q) = ν 0 ± l ln Dqκ 0 p 2 2τ em gp ref .

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.000425 kg/kg, and for a final CO 2 concentration of q f = 4q i . Using a logarithmic axis for p em ( ν) 111 yields triangles in the ν − p plane, with the triangle in the q f case being taller and wider than that 112 from q i . Figure 1b also shows p em ( ν) but as calculated with a benchmark line-by-line code (see 113 calculation details in Section 3). To first order, the triangle picture is a reasonable approximation 114 to the benchmark result. 115 The simplicity of the idealized p em ( ν) triangles in Fig. 1a allows for a heuristic derivation of 116 the CO 2 forcing F (defined as minus the difference in outgoing longwave between the q f and q i then, their contributions to the forcing can be estimated once we know the spectral width ∆ ν over 130 which these contributions are made (Fig. 1a). Using (4), we find that this effective widening of 131 the CO 2 band from changing q i to q f is given by The logarithmic dependence of ∆ ν on q, which follows from (4), arises because τ ν ∼ qe −| ν− ν 0 |/l , 133 so for fixed p and τ ν = τ em , an arithmetic change in ν (which causes a uniform widening of the 134 CO 2 band) requires a geometric increase in q, because the ν-dependence of τ ν is exponential.

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Since the forcing is proportional to ∆ ν (Fig. 1a)  To set the parameters l and κ 0 and as a first, idealized test of (7), we calculate the instantaneous 182 TOA forcing F 4x from a quadrupling of CO 2 for our idealized single columns with variable sur-183 face temperature T s , isothermal stratosphere (Γ strat = 0), and for CO 2 as the only radiatively active 184 species (CO 2 -only). The results of this calculation, using both RFM as well as (7), are shown in Wilson and Gea-Banacloche (2012), and that all values in this range yield a reasonable fit.

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Next we optimize κ 0 . We do this by considering the same columns as in the previous paragraph 190 but with T s = 300 K and with variable Γ strat . These non-isothermal stratospheres allow us to 191 probe which κ 0 value yields the most appropriate emission pressure p 0 and hence T strat [cf. Eqns.

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(3),(5)]. A comparison of F 4x as computed by RFM and (7) for these columns and for various 193 values of κ 0 is shown in Fig. 2b. This panel shows that for typical values of −4 < Γ strat < 0 K/km, 194 the value κ 0 = 60 m 2 /kg provides an excellent fit. This is identical to the value obtained via a fit to  Table 1.  The simplicity of (7) allows us to identify the origin of these and other spatial variations in 226 F 4x . The only variables in (7) are T s and T strat , which are plotted in Fig. 3d To construct an analog to Fig. 1a, we first assume that the H 2 O emission on each side of the 256 CO 2 band has an (RH-dependent) emission temperature (continuing to make the emission level 257 approximation), and that under an increase in CO 2 it is this emission which will be swapped for 258 stratospheric emission. This idealization is depicted in Fig. 5a. We consider the 550 − 600 cm −1 259 spectral interval to be the the low wavenumber side of the CO 2 band, and quantities averaged over 260 or pertaining to this interval will be signified with a '-' ; similarly, we consider 750 − 800 cm −1 261 as the high wavenumber side, and quantities averaged over or pertaining to this interval will be 262 signified with a '+'.

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To turn the heuristic picture of Fig. 5a into a formula which generalizes (7), we will estimate 264 spectrally averaged optical depths τ ± est , which we can combine with an emission level τ H 2 O to find There is an implicit but strong assumption here that in spectrally averaging Eqn. ( The reference absorption coefficients κ ± ref are evaluated at distinct reference pressures and tem-  Table 1.

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The approximations (10) then allow for an analytical evaluation of τ ± est , as follows. We integrate 287 using temperature as our dummy integration variable, and set the lower bound of the integral to 288 the cold-point tropopause temperature T tp whose H 2 O concentrations are assumed negligible (here 289 and below we take the cold-point to demarcate the top of the troposphere). For τ − est , which we 290 model as being due to line absorption, such a calculation was already performed in Jeevanjee and 291 Fueglistaler (2020b), so we simply quote their Eqn. (12): depends on RH and has units of water vapor path, p ∞ v = 2.5 × 10 11 Pa 293 is a reference value for the saturation vapor pressure , and all 294 other symbols have their usual meaning.

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For τ + est , the self-broadening scaling (10b) makes for a different calculation. Denoting vapor 296 density by ρ v (kg/m 3 ) and noting that Note the dependence of T + em on RH 2 in (12b), characteristic of the continuum. Equations (12) are 300 the expressions we seek, and will be combined below with Eqns. (8) and (9)  We validate the expressions (12) for T ± em by comparing them to the spectral average of as calculated from RFM output for our single-columns with T s = 300 K, no CO 2 , and with varying 305 RH. The ground truth T em ( ν)d ν for T ± em is compared to our estimates (12) in Figure 6a,b, which 306 shows that Eqns. (12) do an excellent job of capturing the variation of T − em with RH, and do a good 307 job with T + em down to RH values of 0.025, at which a significant fraction of wavenumbers in the '+' spectral region become optically thin and thus have T em ( ν) = T s . In this case the min function 309 in (13) does not commute with the spectral averaging, violating the assumption behind (8). This 310 makes our expressions for T ± em an overestimate whenever the relevant spectral region contains a 311 mix of optically thick and thin wavelengths. This circumstance also occurs at the lower T s typical 312 of the extratropics, as we will see below.

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[As an aside, note that the T (τ ± est = τ em ) in Eqns. (12) are actually independent of T s , i.e. they With some confidence in our expressions (12) for T ± em , we now substitute T em from Eqn. (9) into 321 Eqn. (7) to obtain an expression for CO 2 forcing in the presence of H 2 O overlap: Note that as RH → 0, T em → T s so this equation indeed generalizes (7).

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As a preliminary test of (14) we take our single-column, T s = 300 K, variable RH calculations 324 (with q i = 280 ppmv) and compare F 4x as calculated from RFM with F 4x calculated from (14) 325 and (12). The result is shown in Fig. 6c and shows quite good agreement between the two, though 326 the errors in T + em at low RH discussed above do lead to small (∼ 0.5 W/m 2 ) errors in F 4x . We check this in Figure 8, which shows F 4x calculated from both RFM as well as Eqns. (7) and vapor paths for different wavenumbers, and thus in the spectral integral this transition is smooth and begins even at T s = 250 K (Fig. 8a). In our simple model, however, this transition can only

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It is important to note that this upper bound on CO 2 forcing does not imply an upper bound on 366 the total greenhouse effect from CO 2 , nor on any resultant warming. Increasing CO 2 will always 367 yield a positive forcing and hence some warming. We are making the much narrower statement 368 that at fixed CO 2 baseline concentration q i , the increase in total CO 2 greenhouse effect from 369 increasing q i to q f (i.e the forcing) has a T s -dependence which asymptotes to an upper bound. But to the meridional gradient in surface temperature (Fig. 3).

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• The meridional forcing gradient is significantly modulated by the presence of H 2 O (Fig. 4), 381 where H 2 O replaces surface emission at the edges of the CO 2 band with colder atmospheric 382 emission (Fig. 5).

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• The T s -invariance of H 2 O emission temperatures T ± em implies an upper bound (with respect to 384 T s variations) on the CO 2 forcing per doubling (Fig. 8). This upper bound is likely realized 385 in the present-day tropics (Fig. 7), but is only relevant for q i close to preindustrial values, and 386 does not imply a saturation of the total CO 2 greenhouse effect.

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The present work could be extended in several ways. One important extension would be to-388 wards the calculation of the adjusted rather than instantaneous forcing, as the adjusted forcing 389 has long been recognized to be more directly related to surface warming (e.g. Hansen  should simply change the upwelling radiation which is blocked by additional CO 2 . This is already 407 well-known in the literature as the 'cloud-masking' of CO 2 forcing (e.g. H16), but might be 408 succinctly and quantitatively described by the substitution of cloud-top temperature for T s in (7).

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Although this work focuses on the spatial variations of CO 2 forcing, the physics of these varia- As

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Since the '+' wavenumber region is dominated by continuum absorption (as we will see), we 456 will adopt the above vapor pressure scaling for κ + est , as well as the explicit temperature scaling  It is interesting to note that the logarithmic slopes of κ lines and κ ctm are comparable for a given 480 wavenumber range, despite the naive expectation that κ lines scales with p (which varies by a fac-481 tor of 5 over the vertical range shown in Fig. A1) and κ ctm scales with p v (which varies by a 482 factor of 700). However, κ lines also exhibits a temperature scaling, which we ignore and which 483 accounts for much of the error in the slope of κ est in Fig. A1a. At the same time, κ ctm also ex-484 hibits a temperature scaling but with opposite sign which weakens its C-C scaling [Eqn. (A1)].

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These opposing temperature scalings for κ lines and κ ctm modify our naive expectations, and seem 486 to conspire to produce surprisingly similar overall logarithmic slopes. Whether or not this is a 487 coincidence, or is perhaps related to the hypothesis that continuum absorption is simply due to for some 'emission level optical depth' τ em which may depend on the parameters introduced above.

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This τ em may be thought of as characterizing the transition between surface and atmospheric emis-503 sion, or equivalently between 'optically thin' and 'optically thick' regimes. As such, we expect To determine τ em , we first analytically compute the OLR for our idealized gray gas, using Eqn.

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(B1) and assuming τ s 1 : where Γ(γ + 1) denotes Euler's Gamma function evaluated at γ + 1, and the tilde is introduced to 510 distinguish it from the atmospheric lapse rate. We may then combine Eqns. (B1), (B3), and (B5) 511 and solve for τ em , obtaining A plot of this curve is shown in Fig. 7. To determine τ em , then, we simply need appropriate 513 values for γ for CO 2 and H 2 O emission. For CO 2 , τ em only enters our theory quantitatively in It is interesting to note that the γ parameter of Eqn. (B2) was also found by Jeevanjee and 521 Fueglistaler (2020a) to determine the validity of the cooling-to-space approximation, which was 522 found to hold when γ 1. It is also interesting to note that in this limit, we may Taylor (Fig. 7).
Chung, E. S., and B. J. Soden, 2015b: An assessment of methods for computing radiative forcing     (7)  profiles of κ tot , κ lines , and κ ctm are calculated with RFM, whereas κ est is given by Eqns. (10)   6. (a,b) Validation of our simple expressions (12) for band-averaged H 2 O emission temperatures T ± em , as compared to band-averaged T (τ ν = τ em ) using τ ν as output by RFM. This comparison is made for idealized atmospheric columns with T s = 300, no CO 2 , and varying RH. (c) Validation of the simple model of Eqns. (12) and (7) for F 4x in the presence of H 2 O, as compared to F 4x calculated by RFM. This comparison is made for idealized atmospheric columns with T s = 300, q i = 280 ppmv, and varying RH. The simple expressions (12) predict T ± em very well except at low RH in the '+' region, leading to small (∼ 0.5 W/m 2 ) errors in F 4x at these RH values.    τ em = e −γ Euler ≈ 0.56, giving a canonical value for τ em which is indeed close to our value for τ CO 2 em and τ H 2 O em .