why co2 cools the middle atmosphere–a consolidating model … · 2020-06-08 · 698 h. f....

Earth Syst. Dynam., 7, 697–715, 2016www.earth-syst-dynam.net/7/697/2016/doi:10.5194/esd-7-697-2016© Author(s) 2016. CC Attribution 3.0 License.

Why CO2 cools the middle atmosphere – a consolidatingmodel perspective

Helge F. Goessling1 and Sebastian Bathiany2

1Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany2Wageningen University, Wageningen, Netherlands

Correspondence to: Helge F. Goessling ([email protected])

Received: 11 March 2016 – Published in Earth Syst. Dynam. Discuss.: 18 March 2016Revised: 27 July 2016 – Accepted: 8 August 2016 – Published: 29 August 2016

Abstract. Complex models of the atmosphere show that increased carbon dioxide (CO2) concentrations, whilewarming the surface and troposphere, lead to lower temperatures in the stratosphere and mesosphere. This cool-ing, which is often referred to as “stratospheric cooling”, is evident also in observations and considered to beone of the fingerprints of anthropogenic global warming. Although the responsible mechanisms have been iden-tified, they have mostly been discussed heuristically, incompletely, or in combination with other effects suchas ozone depletion, leaving the subject prone to misconceptions. Here we use a one-dimensional window-greyradiation model of the atmosphere to illustrate the physical essence of the mechanisms by which CO2 cools thestratosphere and mesosphere: (i) the blocking effect, associated with a cooling due to the fact that CO2 absorbsradiation at wavelengths where the atmosphere is already relatively opaque, and (ii) the indirect solar effect,associated with a cooling in places where an additional (solar) heating term is present (which on Earth is partic-ularly the case in the upper parts of the ozone layer). By contrast, in the grey model without solar heating withinthe atmosphere, the cooling aloft is only a transient blocking phenomenon that is completely compensated as thesurface attains its warmer equilibrium. Moreover, we quantify the relative contribution of these effects by sim-ulating the response to an abrupt increase in CO2 (and chlorofluorocarbon) concentrations with an atmosphericgeneral circulation model. We find that the two permanent effects contribute roughly equally to the CO2-inducedcooling, with the indirect solar effect dominating around the stratopause and the blocking effect dominatingotherwise.

1 Introduction

The laws of radiative transfer in the Earth’s atmosphere area key to understanding our changing climate. With the ab-sorption spectra of greenhouse gases as one central startingpoint, climate models of increasing complexity have beenbuilt during the last decades. These models show that in-creased carbon dioxide (CO2) concentrations, while warm-ing the surface and the troposphere, lead to lower temper-atures in the middle atmosphere (MA; the stratosphere andthe mesosphere) (Manabe and Strickler, 1964; Manabe andWetherald, 1967, 1975; Fels et al., 1980; Gillett et al., 2003).Meanwhile, observations show a cooling trend in the MAduring the satellite era until the most recent years; the neg-ative trend is especially large in the upper stratosphere and

in the mesosphere, although uncertainties also increase withheight (Beig et al., 2003; Randel et al., 2009; Liu and Weng,2009; Beig, 2011; Seidel et al., 2011; Thompson et al., 2012;Huang et al., 2014).

Attribution studies have concluded that the depletion ofstratospheric ozone was probably the main driver of the cool-ing in the lower stratosphere (Ramaswamy and Schwarzkopf,2002; Shine et al., 2003; Thompson and Solomon, 2005,2009; Forster et al., 2007a; Santer et al., 2012), especiallyin the Antarctic spring (Ramaswamy et al., 2001; Thomp-son and Solomon, 2009). In addition, the roles of volca-noes and atmospheric dynamics (Thompson and Solomon,2009), stratospheric water vapour (Ramaswamy et al., 2001;Shine et al., 2003; Maycock et al., 2011; Seidel et al., 2011),and climate variability (Seidel et al., 2011) have been dis-

Published by Copernicus Publications on behalf of the European Geosciences Union.

698 H. F. Goessling and S. Bathiany: Why CO2 cools the middle atmosphere

cussed. The increase of CO2 concentration contributed to thedecrease of lower stratospheric temperatures, but only to asmall extent (Shine et al., 2003). In the middle and upperstratosphere (and beyond), the CO2 increase has probablybeen the most important reason for the temperature decrease(Ramaswamy et al., 2001; Ramaswamy and Schwarzkopf,2002; Shine et al., 2003; Thompson and Solomon, 2005). Asozone concentrations are expected to recover in the future, itseems likely that CO2-concentration trends will be of grow-ing importance also in the lower stratosphere (Stolarski et al.,2010; Ferraro et al., 2015).

The isolated effect of CO2 on temperatures in the MA israrely explained. Probably the most frequent argument foundin textbooks (e.g., Pierrehumbert, 2010; Neelin, 2011) is re-lated to the ozone layer where a considerable part of thelocally absorbed radiation is short-wave (solar) radiation.Therefore, the temperature around the stratopause exceedsthe corresponding temperature in a hypothetical grey atmo-sphere by far. Because the main absorption bands of CO2 arein the long-wave (LW) part and not in the solar part of thespectrum, an increase in CO2 leads to increased emission ofLW radiation while the rate of solar heating remains unal-tered. The excess of emission compared to absorption leadsto a cooling. It should be stressed that the argument is relatednot to the depletion but to the mere presence of ozone (Ra-maswamy and Schwarzkopf, 2002). Although this effect is amajor contributor to CO2-induced MA cooling (as we con-firm below), this explanation is less convincing in the mid-dle and upper mesosphere where there is unabated (observedand simulated) cooling despite the solar heating becomingweaker with height. Neither can this effect explain an im-portant difference between CO2 and other long-lived green-house gases: while methane and nitrous oxide have a muchweaker effect on MA temperatures compared to CO2, chlo-rofluorocarbons (CFCs) even tend to warm the lower strato-sphere (neglecting ozone depletion) (Dickinson et al., 1978;Forster and Joshi, 2005).

More complete explanations discern not only between so-lar and LW radiation, but treat the LW absorption spectra ofgreenhouse gases in more detail. Ramaswamy et al. (2001)and Seidel et al. (2011) point out that the balance of LWemission and LW absorption must be considered: any green-house gas emits simply according to its local temperature,but absorbs radiation emitted from certain distances (repre-sented by radiation mean free paths) depending on the ab-sorption spectrum of the gas and the atmospheric composi-tion. At the absorption bands of certain CFCs, the radiationmean free path of the atmosphere is large because the bandsare located in the spectral window region. These CFCs thusabsorb mainly radiation emitted from the warm surface andlower troposphere, but emit with the low temperatures of theMA. Consequently, increased CFC concentrations impose anLW warming tendency. Ramaswamy et al. (2001) point outthat, in contrast, the radiation mean free path in the 15 µmband of CO2 is small, implying that the radiation absorbed

by CO2 in the MA mainly comes from the cold tropopauseregion.

Forster et al. (1997b) and Ramaswamy et al. (2001) ad-dress yet another aspect: the MA responds much faster thanthe rather inert surface-troposphere system. Hence, whengreenhouse gases are added to the atmosphere, initially theMA cools because the radiation mean free path of the atmo-sphere has been reduced and the radiation arriving in the MAnow stems from higher, colder levels of the troposphere. Af-ter this first phase of MA temperature adjustment, the surfaceand troposphere gradually warm, leading to increased up-ward LW radiation at the tropopause which warms the lowerstratosphere.

All these effects, though mentioned in the literature andincluded in complex climate models, are rarely discussed to-gether. Furthermore, they are usually explained only heuristi-cally and not formalized with a conceptual model. The mostpopular educational model of the greenhouse effect, a global-mean grey atmosphere model with one ground level and aone-layer atmosphere (e.g., Neelin, 2011; Liou, 2002), cannot explain CO2-induced MA cooling. Although Thomas andStamnes (1999) introduce an atmospheric window to theirconceptual model, they do not apply it to explain MA cool-ing. The same is true for more complex conceptual models(e.g., Pollack, 1969a, b; Sagan, 1969; Pujol and North, 2002),which are also not limited to the ingredients needed to ex-plain how CO2 cools the MA.

Our article aims to provide a consolidating model perspec-tive on CO2-induced MA cooling. The first part has an edu-cational emphasis as we demonstrate the physical essence ofthe mechanisms involved, using simple variants of a verti-cally continuous global-mean radiation model: in Sect. 2 wederive the grey atmosphere model from the general radiativetransfer equation, in close analogy to such models in the edu-cational literature (e.g., Goody and Yung, 1989; Thomas andStamnes, 1999). The grey atmosphere model features no per-manent MA cooling when applied in its pure form. In Sect. 3we therefore extend the grey model to the simple case of twoLW bands of which one is fully transparent, i.e., the window-grey case. The rigorous derivations in Sects. 2 and 3 may beskipped by readers primarily interested in the resulting expla-nations. In Sect. 4 we explain CO2-induced MA cooling us-ing the window-grey radiation model. In Sect. 5 we providea quantitative separation of the effects based on simulationswith the atmospheric general circulation model ECHAM6.Based on these simulations we discuss what can, and whatcan not, be learnt about the effect strengths based on thewindow-grey model. This is followed by a summary and con-clusions in Sect. 6. In addition to our model derivations, Ap-pendix A offers a simple analogy to understand the blockingeffect of CO2 without any equations. Moreover, we show therelation of the vertically continuous model to discrete-layermodels in Appendix B and provide a formal response analy-sis in terms of partial derivatives with respect to the parame-

Earth Syst. Dynam., 7, 697–715, 2016 www.earth-syst-dynam.net/7/697/2016/

H. F. Goessling and S. Bathiany: Why CO2 cools the middle atmosphere 699

ters of the window-grey model in Appendix C. Appendix Dprovides technical details on Fig. 1.

2 The grey-atmosphere model

In the following we derive the vertical temperature profile ofa grey atmosphere, first with only the atmosphere in thermalequilibrium and then assuming equilibrium also for the sur-face. We consider a vertically continuous grey atmospherewith horizontally homogeneous (global-mean) conditions.The grey atmosphere is transparent for solar radiation anduniformly opaque for LW radiation. Splitting the electromag-netic spectrum into a transparent solar band and an opaqueLW band is a common approximation that is naturally sug-gested by (i) the well separated emission spectra of the Sunand the Earth (Fig. 1a) and (ii) the shape of the absorp-tion spectrum of the Earth’s atmosphere (Fig. 1b). The greymodel accounts only for radiation while other processes ofenergy transfer (most importantly convection) are neglected.Greenhouse gases are assumed to be well-mixed and any ef-fects from clouds or aerosols are neglected. Assuming hor-izontally homogeneous conditions and the absence of scat-tering we apply the two-stream approximation (Liou, 2002;Pierrehumbert, 2010), meaning that we distinguish only up-ward and downward propagating radiation (indicated by ar-rows in the subsequent equations). This leads to the differen-tial form of the radiative transfer equation (e.g., Goody andYung, 1989; Pierrehumbert, 2010):

dL↑(z)dz

=1µ

[J −L↑(z)

]ρ(z)k, (1)

with radiance L, source term J , geometric height z, air den-sity ρ, mass absorption coefficient k, and µ= cos(θ ) withthe effective angle of propagation θ . To remove any angulardependence from the equations, we use the common assump-tion of an effective angle of propagation of 60◦ relative to thevertical, i.e., µ= 1/2 (see Pierrehumbert, 2010, chap. 4.2).As we neglect scattering, J only consists of the long-waveblackbody emission, which is isotropic. Integrating Eq. (1)over the half sphere then yields

dF↑(z)dz

=[σ T (z)4

−F↑(z)]ρ(z)k (2)

with irradiance F (in W m−2) and the Stefan-Boltzmann con-stant σ . Using the relative pressure deficit

h= 1−p/psrf (3)

as vertical coordinate, where p is pressure and psrf is surfacepressure, the radiative transfer equation reads

dF↑(h)dh

=[σ T (h)4

−F↑(h)]α . (4)

The absorption coefficient α is the only parameter of thegrey model and describes the atmospheric opacity in the LW

band. Due to our definition of the vertical coordinate h, α isindependent of h (in fact, it follows from hydrostatic bal-ance that α = k psrf/g). Also, h is proportional to opticalthickness: τ = αh . Although we distinguish the parameterα from the vertical coordinate h, our model is equivalent tosimilar approaches in popular textbooks of radiative trans-fer which usually choose optical depth (τ , also called opticalthickness) as their vertical coordinate. For example, the op-tical thickness between a height h and the top of the atmo-sphere, in our case α(1−h), is identical to τ∞− τ in Pierre-humbert (2010), to τ in Salby (1992), and to τ/µ in Thomasand Stamnes (1999). In the latter two cases, the vertical axispoints downwards, hence the reversed sign. In Thomas andStamnes (1999), a parameter µ still appears in the equationsas no assumption on the average direction of propagation ismade.

Equation (4) is the spectrally integrated grey-absorption-case of Schwarzschild’s equation and holds analogously fordownwelling LW radiation F↓. In radiative equilibrium F

must be free of divergence because other source or sink termsof heat are neglected. With F↓toa = 0, F↓ is hence determinedby

F↑(h)−F↓(h)= F↑toa , (5)

where the index toa stands for the top of the atmosphere(TOA). Thermal equilibrium for a thin layer of air is givenwhen

2ε σ T (h)4= ε

(F↑(h)+F↓(h)

), (6)

where ε = α dh is the emissivity, and hence also the absorp-tivity, of the thin layer for LW radiation. Combining Eqs. (5)and (6) yields

σ T (h)4= F↑(h)−

F↑

toa

2. (7)

Substituting Eq. (7) into the radiative transfer equation(Eq. 4) gives

dF↑(h)dh

=−α

2F↑

toa . (8)

Because α is constant, Eq. (8) has the simple solution

F↑(h)= F↑toa[α

2(1−h)+ 1

]. (9)

With Eq. (5) it follows further from Eq. (9) that

F↓(h)= F↑toaα

2(1−h) . (10)

Inserting Eqs. (9) and (10) into Eq. (6) leads to the verticaltemperature profile of the equilibrated grey atmosphere:

T (h)=4

√F↑

toa

2σ

(α (1−h)+ 1

). (11)

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A

bsor

ptio

n (%

)

A

bsor

ptio

n (%

)

λ

Bλ

(nor

mal

ized

)

0

0.2 0.3 0.5 0.7 1 1.5 2 3 5 7 10 15 20 30 50 70 100

Wavelength (μm)

288 K5800 K

(a) Black body curves

0

50

100(b) Surface−to−TOA

O2 O3O2

H2OH2O H2O H2O

H2OCO2

H2OCO2

CH4

CO2

N2OCO

O3

H2ON2OCH4

O3

CO2

O3

N2OH2O

0

50

100(c) Tropopause−to−TOA

Figure 1. (a) Normalized black body curves for 5800 K (the approximate emission temperature of the Sun) and 288 K (the approximatesurface temperature of the Earth). (b) Representative absorption spectrum of the Earth’s atmosphere for a vertical column from the surface tospace, assuming the atmosphere to be a homogeneous slab. (c) The same but for a vertical column from the tropopause (∼ 11 km) to space.Spectra based on HITRAN on the Web; see Appendix D for details. Figure after Goody and Yung (1989, Fig. 1.1 on p. 4).

Evaluating Eq. (9) at the surface (h= 0) gives

F↑

toa =F↑

srfα/2+ 1

. (12)

Assuming that the surface is a perfect black body for LWradiation, it is

F↑

srf = σ T4

srf . (13)

With Eqs. (12) and (), the vertical temperature profile de-scribed by Eq. (11) can be written as

T (h)= Tsrf4

√α (1−h)+ 1

α+ 2. (14)

Equation (14) implies for the near-surface (h= 0) air that

T (0)= Tsrf4

√α+ 1α+ 2

. (15)

Hence, T (0)< Tsrf. The reason for this discontinuity at thesurface is that, in order to attain the same temperature as thesurface, the near-surface air would have to receive as much

LW radiation from above as it receives from the surface be-low, which is not the case (see Eq. 5). This is a result of thenegligence of all mechanisms of energy transfer other thanradiation in the model; in reality, the molecular and turbulentdiffusion of heat removes the discontinuity, although sharptemperature gradients right above the surface can still be ob-served (see for example Pierrehumbert, 2010, whose Eq. 4.45is identical to our Eq. 15).

The vertical temperature profile can also be written by ref-erence to the effective radiative temperature of the planet,defined as

Teff =4

√F↑

toa

σ. (16)

Inserting Teff into Eq. (11) gives

T (h)= Teff4

√α

2(1−h)+

12. (17)

Up to now we have not considered the surface energybalance but determined the vertical temperature profile ofan equilibrated grey atmosphere with absorptivity α given



an arbitrary surface temperature as lower boundary condi-tion. Equation (14) can thus be interpreted as the quasi-instantaneous atmospheric temperature profile. In the fol-lowing, we consider the situation where not only the atmo-sphere but also the surface is in thermal equilibrium. We re-fer to this situation as the overall equilibrium and denote thecorresponding variables with the index eq.

Assuming that no solar radiation is absorbed within theatmosphere, surface equilibrium requires that

S = F↑

srf,eq−F↓

srf,eq , (18)

where S is solar radiation absorbed at the surface. InsertingEq. (5) into Eq. (18) gives

S = F↑

toa,eq , (19)

which is the overall equilibrium condition at the top of theatmosphere. It follows with Eq. (16) that

Teff,eq =4

√S

σ. (20)

In overall equilibrium, the vertical temperature profile(Eq. 17) hence becomes

Teq(h)= Teff,eq4

√α(1−h)+ 1

2. (21)

Apart from the different choices of the vertical coordi-nate, the temperature profile given by Eq. (21) is identicalto Eq. (12.21) in Thomas and Stamnes (1999), to Eq. (4.42)in Pierrehumbert (2010), and to Eq. (3.47) in Salby (1992).

Inserting h= 1 into our Eq. (21) yields the temperature atthe TOA:

Ttoa,eq = Teff,eq4

√12. (22)

Finally, combining Eqs. (21) and (15) gives the corre-sponding equilibrium surface temperature:

Tsrf,eq = Teff,eq4

√α+ 2

2. (23)

Equations (21)–(23) reveal that, in overall thermal equilib-rium, an increase in absorptivity of a grey atmosphere with-out non-LW heat sources leads to a temperature increase ev-erywhere except at the TOA where the temperature is inde-pendent of α.

Figure 2 shows solutions of Eq. (21) for different valuesof α. In the limit of an almost completely transparent at-mosphere (α→ 0), the whole atmosphere attains one singleequilibrium temperature (Eq. 22) and the surface temperatureattains the effective equilibrium radiative temperature of theplanet (Eq. 20). Note that the vertically continuous model de-rived here can be interpreted as a generalization of a discrete-layer model (see Appendix B).

200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

α = 0α = 2α = 6

T ef f ,eq

Figure 2. Vertical temperature profiles of a grey atmosphere inoverall equilibrium for different absorptivities α. The latter cor-responds to αo in the window-grey model with βw = 0. Teff,eq =255K. The circles at z= 0 denote the corresponding surface tem-peratures. Note that the vertical coordinate z is only approximateheight, calculated from h with a constant scale height H = 8kmsuch that h= 1− e−z/H .

3 The window-grey atmosphere model

In reality, the atmosphere is not uniformly opaque for LWradiation, as within the grey approximation, but interacts dif-ferently with LW radiation of different wavelengths. To ac-count for this in the simplest possible way, we extend thegrey model (Sect. 2) by splitting the total LW radiation Finto two separate LW bands: an opaque band F1 =O withopacity αo > 0, and a completely transparent (window) bandF2 =W with opacity αw = 0. With βw = 1−βo describingthe fraction of LW radiation from the surface which is di-rectly emitted to space, the resulting window-grey modelhas only two parameters: αo and βw. Thereby βw is iden-tical to the so-called transparency factor G in Thomas andStamnes (1999), but independent of temperature in our caseas we neglect Wien’s law. This approach represents the so-called window-grey or one-band Oobleck case of a multi-band model (Sagan, 1969; Pierrehumbert, 2010). In contrastto the grey-atmosphere model, the window-grey model al-lows for the existence of a spectral window, which can beinterpreted as an idealization of the region between 8 and12 µm in the Earth’s atmosphere (Fig. 1b). The window-greymodel is depicted in Fig. 3.

The window-grey model remains analytically solvable be-cause only radiation in the opaque LW band needs to be con-sidered within the atmosphere. The resulting radiative trans-fer equation reads



S W

O↑

toa

O↑

srf

Solarband

Opaquethermal band

Transparentthermal band

O↓

srf

h =

1 - p

/p

srf

0

0.5

1

z (k

m)

0

4

2

6810

20∞

Figure 3. Sketch of the window-grey atmosphere model. S: solarradiation absorbed at the surface. O: radiation in the opaque LWband (↑: upwelling, ↓: downwelling, srf: surface, toa: top of the at-mosphere).W : radiation emitted from the surface in the transparentLW band (atmospheric window). p: pressure. Interpreting the num-ber of arrows in the opaque LW band as proportional to the radiativeflux, the illustrated case corresponds to an equilibrated atmospherewith αo = 4.

dO↑(h)dh

=[σ T (h)4 (1−βw)−O↑(h)

]αo , (24)

the energy balance equation reads

2εo σ T (h)4 (1−βw)= εo (O↑(h)+O↓(h)) , (25)

and the surface emission in the opaque band is

O↑

srf = (1−βw)σ T 4srf . (26)

Equations (24)–(26) can be solved analogously to the cor-responding equations describing the grey case (Eqs. 4, 6, and). This leads to the quasi-instantaneous atmospheric temper-ature profile of the window-grey model. It is

T (h)= Tsrf4

√αo (1−h)+ 1

αo+ 2. (27)

Comparison with Eq. (14) reveals that, with the same sur-face temperature prescribed as lower boundary condition, thevertical temperature profiles in the grey and in the window-grey case are identical for αo = α; the factor (1−βw) inEqs. (24)–(26) has cancelled.

To determine the overall equilibrium state, the surface en-ergy balance needs to be incorporated. In overall equilibriumit is

S+O↓

srf,eq =O↑

srf,eq+Weq (28)

where

Weq = βw σ T4

srf,eq . (29)

Here we omitted the indices denoting the orientation andvertical position of W (as we already did for S) because theonly radiation in the window band is the one emitted upwardfrom the surface, and W remains unchanged throughout theatmosphere because αw = 0.

With derivations analogous to the grey case (Sect. 2), onearrives at simple expressions for the overall equilibrium state.The surface temperature for the window-grey model in over-all equilibrium is

Tsrf,eq = Teff,eq4

√αo+ 2αoβw+ 2

. (30)

The corresponding vertical temperature profile reads

Teq(h)= Teff,eq4

√αo (1−h)+ 1αoβw+ 2

, (31)

which implies for the TOA temperature

Ttoa,eq = Teff,eq4

√1

αoβw+ 2. (32)

The TOA temperature, sometimes called the skin temper-ature (Goody and Yung, 1989; Pierrehumbert, 2010), can bethought of as the temperature an infinitely thin air layer abovethe atmosphere would have in radiative equilibrium. Obvi-ously, with βw = 0 Eqs. (30)–(32) are reduced to the greycase (compare Eqs. 21–23).

Equation (30) implies that an increased absorber amountleads to an increased equilibrium surface temperature(Fig. 4), independent of whether the added molecules absorbin the already opaque part of the LW spectrum (increasingαo) or in the window region (decreasing βw, that is, “closingthe atmospheric window”). In the following section we dis-cuss the sensitivity of atmospheric temperatures to the modelparameters.

4 The mechanisms of CO2-inducedmiddle-atmosphere cooling

With the window-grey radiation model we are now equippedto investigate the physical essence of CO2-induced MA cool-ing. In the window-grey model, the response of temperatureto changes in the parameters can be quantified with partialderivatives. The different effects of CO2-induced MA cool-ing can thereby be separated in a formal way. We presentsuch an approach in Appendix C, but constrain the discus-sion in the following main text largely to the undifferentiatedequations.

4.1 The blocking effect

The blocking effect is the result of a change in the long-waveradiative balance when atmospheric CO2 is increased. Due to



250

300

350

400

450

αo

Tsrf (

K)

0.02 0.2 2 20 200

βw = 1

βw = 0.2

βw = 0

1.0 0.8 0.6 0.4 0.2 0.0

250

300

350

400

450

βw

Tsrf (

K)

αo = 0αo = 4

αo = 20

250

300

350

400

450

αo

Tsrf (

K)

0.02 0.2 2 20 200

βw = 1

βw = 0.2

βw = 0

1.0 0.8 0.6 0.4 0.2 0.0

250

300

350

400

450

βw

Tsrf (

K)

αo = 0αo = 4

αo = 20

Figure 4. The dependence of the overall equilibrium surface temperature on the parameters αo and βw in the window-grey model (Eq. 30).βw = 0 corresponds to the grey case. Teff,eq = 255K.

the optically thicker atmosphere, less upwelling radiation inthe non-window part of the spectrum reaches high altitudes.The temperature at the TOA must thus be lower because, asfollows from Eq. (25),

Ttoa =4

√O↑

toa

2σ (1−βw). (33)

We first investigate the situation in which the assump-tion of thermal equilibrium is kept for the atmosphere butdropped for the surface. This is a reasonable assumption be-cause the atmosphere adjusts quickly to energetic changes(on the order of months) while the response of the ocean-dominated surface is very slow (including decadal and cen-tennial timescales). In reality, convection closely couplesthe surface with the troposphere, hence a change in green-house gases first affects the middle atmosphere, after whichthe slow surface-troposphere system adjusts. In the radiationmodel we represent this timescale separation by letting theatmosphere respond while keeping the surface temperatureconstant. The temperature profile for this quasi-instantaneousresponse is given by Eq. (27) which is valid even if the sur-face is not in thermal equilibrium. Fig. 5 shows the verti-cal temperature profile before (blue curve) and after (orangecurve) increasing α in the grey case (for which α correspondsto αo with βw = 0).

Inserting h= 1 in Eq. (27) we arrive at the correspondingquasi-instantaneous TOA temperature:

Ttoa = Tsrf4

√1

αo+ 2. (34)

Equation (34) implies a cooling at the TOA for increasedαo. Furthermore, Eq. (27) implies that at a certain height hfastthe sign of the fast temperature response due to added green-house gases reverses. It is

hfast =12. (35)

This implies that at first the upper half of the atmosphere(with respect to mass) is cooled while the lower half iswarmed.

Both the upper-level cooling and the lower-level warmingare due to enhanced blocking, that is, a reduced mean freepath of LW radiation in response to increased absorptivity. Inradiative equilibrium, the emission, determined by the localtemperature, and the absorption of radiation are locally bal-anced. In the upper atmosphere, where downwelling radia-tion is subordinate, the upwelling radiation received from be-low comes from higher (and thus colder) levels when the ab-sorptivity of the atmosphere is increased. Consequently, theair cools until emission and absorption are in balance again.In contrast, in lower levels near the ground, where most ofthe absorbed upwelling radiation comes directly from thesurface (with a fixed temperature), the increased absorptivitymostly affects the downwelling radiation which now comesfrom lower (and thus warmer) levels, resulting in warming.

As long as the emission from the surface, determined by itstemperature, remains unchanged, the surface energy budgetis imbalanced due to the increased downwelling radiation.The surface will thus warm – which is the common green-house effect – until a new overall equilibrium is attained.During the gradual ascent of the surface temperature, accom-panied by increasing upwelling radiation, the whole atmo-sphere warms (Eq. 27), and the height at which the sign of thetemperature change reverses is shifted upwards. In the greycase, where βw = 0, this shift proceeds until the whole atmo-sphere except the TOA is warmer than originally (red curvein Fig. 5). The upper-level cooling in response to increasedabsorptivity in a grey atmosphere (and without absorption ofsolar radiation within the atmosphere) is thus only a tran-sient effect that vanishes when the new overall equilibriumis reached. This is a consequence of the outgoing longwaveradiation (OLR) having to balance S, which is constant inthe model. Even in the very simple grey model a solar anda greenhouse forcing act differently: while an increase in S



200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

α = 2, T srf = T srf ,eq (α=2)α = 6, T srf = T srf ,eq (α=2)α = 6, T srf = T srf ,eq (α=6)

T ef f ,eq

Figure 5. Vertical temperature profiles of a grey atmosphere fortwo equilibrium states and one transient state. While the blue andred curves show the same equilibria as the corresponding curves inFig. 2, the orange curve shows the transient state that occurs afterswitching from α = 2 to α = 6, directly after equilibration of theatmosphere but with Tsrf still unchanged. Again α corresponds to αoin the window-grey model with βw = 0. Teff = 255K. The circles atz= 0 denote the corresponding surface temperatures. Note that thevertical coordinate z is only approximate height, calculated from h

with a constant scale height H = 8km such that h= 1− e−z/H .

would force the OLR to increase as well, the OLR does notchange when CO2 is increased, even though the temperatureis increased throughout the atmosphere.

The situation is different in the presence of an atmosphericwindow where a part of the surface radiation is emitted di-rectly to space, bypassing the atmosphere. An atmosphericwindow implies a reduced sensitivity of the surface temper-ature to the state of the atmosphere (its absorptivity in theopaque band and the corresponding temperature profile, seeEq. 30) because the radiation in the opaque LW band be-comes less important in the surface energy budget (Eq. 28)with increasing window size. Consequently, in the presenceof an atmospheric window, a permanent cooling at the TOAremains after the surface has equilibrated (Eq. 32).

It becomes evident from Eq. (32) that, in contrast to thesurface, at the TOA the sign of the temperature response de-pends on the spectral property of the added absorbers: if theyabsorb in the already opaque part of the LW spectrum (in-creasing αo), Ttoa,eq is decreased (MA cooling), but if theyabsorb in the transparent part of the LW spectrum (decreas-ing βw, that is, “closing the atmospheric window”), Ttoa,eq isincreased (MA warming) (Fig. 6).

In fact, decreasing βw leads in overall equilibrium toa temperature increase at every height in the atmosphere(Fig. 7, top). In contrast, if molecules absorbing in the opaque

LW band are added, the sign of the equilibrium temperatureresponse reverses at a certain height heq, with cooling aboveand warming below (see Eq. C6; Fig. 7, bottom):

heq(βw)= 1−βw

2. (36)

For βw = 0, that is in the grey case, heq becomes 1 (thecorresponding geometric height zeq becomes ∞), meaningthat no cooling takes place.

We term the above described cooling in the upper parts ofthe atmosphere the blocking effect of CO2-induced MA cool-ing. This presupposes that the main consequence of addingCO2 to the atmosphere is, in terms of the window-greymodel, an increase of αo rather than a decrease of βw. Thepermanent component of this effect, the permanent blockingeffect, is revealed by Eqs. (31) and (32). It has to be distin-guished from the instantaneous blocking effect, which con-sists of the permanent blocking effect and a transient com-ponent. The instantaneous blocking effect can be observedwhen atmospheric CO2 is altered but when the surface tem-perature has not yet adjusted to the forcing (which is to someextent also the case for present-day Earth). In the grey modelthe blocking effect is only a transient phenomenon: the entireatmosphere has warmed (except at the TOA) after the surfacehas equilibrated. In the window-grey model the blocking ef-fect has a permanent component that persists after the surfacehas adjusted.

The blocking effect can be understood in terms of the in-terplay between the sensitivity of the surface temperature togreenhouse-gases on the one hand and the blocking of up-welling LW radiation by greenhouse gases on the other hand:while an atmospheric window diminishes the sensitivity ofthe surface temperature to αo (see Eq. 30), the blocking as-sociated with αo is independent of the presence or width ofan atmospheric window (see Eq. 27). Only in the grey case,where the sensitivity of the surface temperature is at its max-imum (Eq. 30), the surface temperature response is strongenough to compensate for the blocking effect, resulting inan αo-independent equilibrium TOA temperature (compareEq. 32).

Another way of looking at the permanent blocking effectgoes via the emission spectrum of the planet viewed fromspace (i.e., the upwelling LW radiation at the TOA). If thesurface warms in response to an increased αo, the radiationin the window region of the spectrum W will be accordinglystronger, corresponding to a Planck curve at the increasedsurface temperature. In overall equilibrium the radiation inthe opaque band O↑toa must be shifted to lower intensity tocompensate for W , given that the solar energy input is un-changed. It then follows from Eq. (33) that the temperatureat the TOA must decrease. The same argument reveals whythe TOA cooling due to enhanced blocking in a grey atmo-sphere can only be a transient phenomenon: If no windowexists, O↑toa must attain its original intensity after equilibra-tion to balance the unchanged solar energy input.



050

100

150

200

αo

Ttoa (

K)

0.02 0.2 2 20 200

βw = 0

βw = 0.2

βw = 1

1.0 0.8 0.6 0.4 0.2 0.0

050

100

150

200

βw

Ttoa (

K)

αo = 20

αo = 4

αo = 0

050

100

150

200

αo

Ttoa [

K]

0.02 0.2 2 20 200

βw = 0

βw = 0.2

βw = 1

1.0 0.8 0.6 0.4 0.2 0.0

050

100

150

200

βw

Ttoa [

K]

αo = 20

αo = 4

αo = 0

Figure 6. The dependence of the overall equilibrium temperature at the top of the atmosphere on the parameters αo and βw in the window-grey model (Eq. 32). βw = 0 corresponds to the grey case. Teff,eq = 255K.

4.2 The indirect solar effect

On Earth not all solar radiation transects the air unhindered,but some is absorbed within the atmosphere and leads toincreased temperatures, particularly in the upper parts ofthe ozone layer. The solar heating can be incorporated intoEq. (25) as an additional term S∗(h):

2εo σ T∗(h)4(1−βw)= εo

(O↑(h)+O↓(h)

)+S∗(h) . (37)

Equation (37) is similar to Eq. (6.15) in Neelin (2011), ex-cept that Neelin considers only the grey case (βw = 0) andneglects the downwelling LW radiation, constraining the va-lidity of the equation to the vicinity of the TOA.

Assuming that solar heating is confined to an infinitesi-mally thin layer at h= h′, such that the equilibrium temper-ature everywhere else remains unchanged and, thus, O↑(h′)and O↓(h′) are not affected by the additional term, one ar-rives at

T ∗(h′)4= T (h′)4

+s∗(h′)

αo(1−βw), (38)

where T (h′) is the solution of Eq. (37) with S∗(h′)= 0,that is, the window-grey solution of Eq. (25), and s∗(h′)=S∗(h′)/(2σ dh) .

Equation (38) reveals the following: given that due to anadditional term in the local energy budget the atmospherictemperature at some height is deflected from the window-grey solution, increasing the amount of LW absorbers in theatmosphere results in a relaxation of the temperature towardsthe window-grey solution. This holds both for increasing αoand for decreasing βw. It must be kept in mind though that thewindow-grey solution itself depends on αo and βw (Eq. 31),making the relaxation towards the window-grey solution anadditional effect. Figure 8 illustrates the indirect solar effectfor the grey case (i.e., for βw = 0).

If the additional term s∗ is positive, as it is the case for theabsorption of solar radiation by ozone, increasing the emis-sivity either by increasing αo or by decreasing βw results in

local cooling. We call this effect the indirect solar effect ofCO2-induced MA cooling. The term “indirect” reminds usthat this effect is not due to any change in solar heating ratesas might be caused by a change in ozone concentrations. In-stead, the mere presence of solar absorption is a prerequisitefor this effect. Like the permanent blocking effect, the indi-rect solar effect is still at work when the system has reachedthe new (more opaque) overall equilibrium. Note that theindirect solar effect would also manifest if the opacity waschanged only locally. This is not the case for the blockingeffect, where integration over a finite layer with perturbedopacity is needed.

5 Effect strengths

An essential question so far unanswered is how strong theabove derived effects are compared to each other. In this sec-tion we apply a complex atmospheric model to give a quan-titative answer to this question, and we discuss the implica-tions and limitations of the window-grey model in the lightof these results.

5.1 Simulations with a complex atmospheric model

To complement the findings obtained with the window-greymodel, and to derive meaningful estimates for the strength ofthe effects, we have conducted simulations with the complexatmospheric general circulation model ECHAM6 (Stevenset al., 2013). This model and its predecessors have been usedfor comprehensive simulations, including future climate pro-jections, in the different phases of the Coupled Model Inter-comparison Project (CMIP), which are the backbone of thereports compiled by the Intergovernmental Panel on ClimateChance (IPCC). These models, including ECHAM6, there-fore have sophisticated parameterizations for, e.g., radiation,convection, clouds, boundary-layer turbulence, and grav-ity waves, and numerically solve the governing equationsof fluid dynamics on grids with steadily increasing spatio-



200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

αo = 4βw = 0.3βw = 0.1

T ef f ,eq

200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

βw = 0.2αo = 2αo = 6

T ef f ,eq

200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

αo = 4βw = 0.3βw = 0.1

T ef f ,eq

200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

βw = 0.2αo = 2αo = 6

T ef f ,eq

Figure 7. Vertical temperature profiles in overall equilibrium forthe window-grey case with Teff,eq = 255K for different combina-tions of the parameters αo and βw (Eq. 31). The circles at z= 0denote the corresponding surface temperatures. Note that the verti-cal coordinate z is only approximate height, calculated from h witha constant scale height H = 8km such that h= 1− e−z/H .

temporal resolution. The radiative transfer scheme used inECHAM6, which employs 16 LW bands, has been shownto give instantaneous clear-sky responses to greenhouse-gasperturbations in close agreement with accurate line-by-linecalculations (Iacono et al., 2008). To adequately resolve themiddle atmosphere, we have used the T63L95 configurationwith relatively coarse (∼ 2◦) horizontal but high (95 levels,top at 0.01 hPa) vertical resolution. The distribution of ozoneis prescribed by a climatology.

The ocean and sea ice have been treated in a simple waysimilar to the approach of Dickinson et al. (1978). The oceansurface temperature and sea ice concentration and thicknessare prescribed with a realistic seasonal and spatial pattern de-

200 250 300 350

010

2030

4050

60

T (K)

z (k

m)

α = 2α = 6

T ef f ,eq

Figure 8. Vertical temperature profiles of a grey atmosphere that isadditionally locally heated (e.g., by absorption of solar radiation) attwo heights within the atmosphere. Apart from the heights at whichthe profiles are locally deflected due to additional heating, the blueand red curves show the same grey equilibria as the correspondingcurves in Fig. 2. Again α corresponds to αo in the window-greymodel with βw = 0 . The heights at which additional heating occurs(z1 ≈ 10km, z2 ≈ 50km) and the magnitude of the additional heat-ing terms (specified such that the temperature rise is 25K for α = 2at both heights) are more or less arbitrarily chosen to demonstratethe effect. Teff,eq = 255K. The circles at z= 0 denote the corre-sponding surface temperatures. Note that the vertical coordinate zis only approximate height, calculated from h with a constant scaleheight H = 8km such that h= 1− e−z/H .

rived from observations. After every year the ocean surfacetemperature pattern is updated uniformly according to the to-tal energy imbalance integrated over the global ocean surface(including sea ice) and over the year, using a heat capacitythat corresponds to a 50 m thick mixed-layer ocean. Despitechanging temperatures, the sea ice state pattern is not up-dated, leading to discrepancies between the sea ice and oceanstates. This procedure also suppresses further changes to thesurface temperature pattern, such as polar warming amplifi-cation. However, this allows for a rapid thermal equilibrationof the surface with an exponential timescale of ∼3 years,serving the purpose of this paper where the focus is on theglobal-mean response.

We have conducted eight ECHAM6 simulations with dif-ferences in (i) the treatment of solar radiation, (ii) the treat-ment of sea-surface temperatures (SST), and (iii) the abun-dance of greenhouse gases (Table 1). In five simulationsthe solar radiation follows the default behaviour, with someof the solar radiation absorbed by gases within the atmo-sphere. This set includes one reference simulation where pre-industrial greenhouse-gas concentrations are used and theSSTs are allowed to run into equilibrium, and four sensitiv-



Table 1. ECHAM6 simulations.

Simulation ID Solar absorption SST treatment CO2a CFC-11/-12b GMSTc OLRd planetary albedo

in the atmosphere [K] [W m−2] [%]

REF yes free equilibration 1× 1× 287.14 241.41 29.03CO2× 2 yes free equilibration 2× 1× 289.36 241.58 29.00CO2× 2fixSST yes prescribed from REF 2× 1× 287.41 238.11 28.91CFC× 15 yes free equilibration 1× 15× 288.97 241.96 28.89CFC× 15fixSST yes prescribed from REF 1× 15× 287.29 238.88 28.80REFns no free equilibration 1× 1× 286.09 227.43 33.08CO2× 2ns no free equilibration 2× 1× 288.31 227.52 33.03CFC× 15ns no free equilibration 1× 15× 287.88 227.56 33.05

a 1×: CO2 = 280 ppmv; 2×: CO2 = 560 ppmv, b 1×: CFC11= 0.2528 ppbv, CFC12= 0.4662 ppbv; 15×: CFC11= 3.792 ppbv, CFC12= 6.993 ppbv, c global annual-meannear-surface air temperature, d outgoing longwave radiation at the top of the atmosphere.

ity simulations. In two of these the ocean is allowed to attaina new equilibrium, either with the CO2 concentration dou-bled or with the Chlorofluorocarbon (CFC-11 and CFC-12)concentrations increased by the factor 15, chosen such thatthe surface warming is similar compared to the case of CO2doubling. The other two sensitivity simulations are identicalwith the previous two, except that the SSTs are prescribedfrom the reference simulation.

In another set of three simulations the absorption of so-lar radiation by all atmospheric gases (but not cloud dropletsor ice) is turned off. This set also includes a reference sim-ulation and two sensitivity simulations with increased CO2and CFC concentrations. In these, the SSTs are again al-lowed to run into equilibrium. All simulations are conductedover 22 years, but only the last 10 years are used to com-pute averages for the analysis because it takes a few years(in our setup) until an equilibrium is reached. This experi-mental design allows us first to demonstrate the dependenceof MA temperature changes on the spectral properties of theadded absorbers: CFCs absorb mainly in the spectral win-dow of the Earth’s atmosphere, whereas CO2 absorbs mainlyat wavelengths where the atmosphere is already relativelyopaque. Second, we can quantify the effect strength for thetwo permanent effects by which CO2 cools the MA, deducedabove with the window-grey model, and investigate how at-mospheric temperatures respond to the slow surface adjust-ment.

When the absorption of solar radiation by gases isswitched off, the total short-wave absorption in the atmo-sphere drops from 75 W m−2 in REF to only 13 W m−2 inREFns, the residue being due to absorption by troposphericclouds. The lack of short-wave heating due to ozone leadsto a strong cooling of the MA. The local temperature maxi-mum around the stratopause completely disappears and tem-peratures drop to ∼ 160 K in the upper stratosphere and inthe mesosphere, in agreement with previous studies (Man-abe and Strickler, 1964; Fels et al., 1980). Tropospheric tem-peratures are only slightly reduced by ∼ 1 K (Table 1 andFig. 9 left). This small temperature change is the result of

a compensation of different effects. After removing the ab-sorption of solar radiation, more short-wave radiation propa-gates downwards through the atmosphere. A part of the pre-viously absorbed radiation is then scattered and the plane-tary albedo increases from 29 to 33 %. The rest is partly ab-sorbed in the troposphere by cloud droplets and ice crystals,and partly reaches the Earth’s surface where the downwellingsolar radiation is increased by 42 W m−2. This instantaneousredistribution of short-wave fluxes tends to warm the surface.However, the large cooling in the MA that follows also leadsto a decreased downwelling long-wave radiation at the sur-face which has a cooling effect. To this extent, our result isin line with previous simulations that quantified the effectsof stratospheric ozone removal (Ramaswamy et al., 1992;Hansen et al., 1997; Forster et al., 1997a; Stuber et al., 2001).In contrast to these studies, we still keep ozone as a green-house gas as only its short-wave absorption is removed. How-ever, this is not a very important difference as the short-waveeffect of ozone has been shown to be more important than thelong-wave effect (Dickinson et al., 1978; Ramaswamy et al.,1992; Forster et al., 1997a). A similar cancellation of short-and long-wave effects on the surface temperature seems tohold for other gases. The lack of absorption by water vapourfurther shifts the heating from the lower troposphere to thesurface, without much impact on the surface temperature.An additional effect that might be of relevance for the sur-face cooling in these simulations is the reduction in specifichumidity due to the atmospheric cooling.

Our simulations with perturbations in CO2 and CFCs arein the tradition of several pioneering studies on the role ofgreenhouse gases (Manabe and Strickler, 1964; Manabe andWetherald, 1967, 1975; Dickinson et al., 1978; Fels et al.,1980), and confirm their results. With standard treatment ofsolar radiation, CO2 doubling in ECHAM6 leads to an in-crease in global annual-mean near-surface temperature by2.2 K; this is the climate sensitivity of our model setup. Thetropospheric warming in fact increases with height, reach-ing a maximum of 3.5 K in the upper troposphere (Fig. 9middle, red solid curve). This pattern, which is not captured



by the window-grey model, results from the temperaturedependence of the moist-adiabatic lapse rate (the so-calledlapse-rate feedback) and is thus related to convective pro-cesses. Somewhat above the tropopause the temperature re-sponse changes sign. In agreement with earlier studies (Man-abe and Wetherald, 1967; Fels et al., 1980), the cooling thenincreases with height and assumes a maximum cooling by11.6 K around the stratopause region.

Adding CFCs instead of CO2 results (by design) in asimilar tropospheric response with a near-surface warmingby 1.8 K, but temperatures in the MA remain virtually un-changed (Fig. 9 middle, black solid curve). Again, this resultagrees with previous studies (Dickinson et al., 1978; Forsterand Joshi, 2005). However, our simulations also show thatthe near-zero MA temperature change is due to a cancella-tion of two effects: the indirect solar effect is not wavelengthdependent. More absorbers increase emission more stronglythan they increase absorption, thereby reducing the relativeimportance of the solar heating term (Sect. 4.2). This sug-gests that another effect counteracts the cooling from the in-direct solar effect. The above considerations based on thewindow-grey model suggest that this counteracting warm-ing effect can be interpreted as an inverse blocking effect: in-stead of making the already opaque part of the spectrum evenmore opaque, which mainly happens when CO2 is added,the increase of CFC concentrations acts to narrow the atmo-spheric window, corresponding to a decrease of βw in thewindow-grey model (Figs. 7 and 6 top). In fact, the situa-tion corresponds not only to a decrease of βw, but also toa simultaneous decrease of αo because the average opacityof what should be translated into the single opaque band ofthe window-grey model is decreased by the inclusion of theCFC-affected – still relatively transparent – parts of the pre-vious window band. Overall, the MA is more strongly sub-jected to the radiation from the warm surface.

In the case without solar absorption by gases within the at-mosphere, adding CO2 or CFCs results in similar responsesin the troposphere, but markedly different responses in theMA (Fig. 9 middle, dashed curves): while the cooling inresponse to CO2 is roughly halved, the previously neutralresponse to CFCs turns into a substantial warming by upto 3.5 K. These results are consistent with the interpretationthat the cooling due to the indirect solar effect has been pre-cluded, leaving only the response due to the blocking effect(in the CO2 case) and the inverse blocking effect (in the CFCcase).

Under the assumption of linearity, this allows us to esti-mate the fractional contributions of the two permanent ef-fects to MA cooling (Fig. 9 right). According to our resultsthe indirect solar effect contributes up to ∼ 70 % to the totalpermanent cooling around the stratopause where solar heat-ing is strongest. Outside this region the blocking effect gainsimportance and begins to dominate the cooling in the middlestratosphere and the middle mesosphere. The assumption oflinearity is rather crude, so these estimates should be taken

with a grain of salt. In fact, it is probably not possible tomake a completely clean quantitative distinction, as the for-mal analysis in Appendix C3 suggests.

The window-grey model also suggests a transient MAcooling that adds to the permanent cooling before the surfacetemperature has adjusted to the changed radiative forcing.We can investigate this effect with the remaining two sim-ulations where the greenhouse gases are perturbed but SSTsare fixed to the reference state (Table 1). Interestingly, the ini-tial MA responses (Fig. 9 middle, dotted curves) are nearlyidentical with the corresponding equilibrium responses (solidcurves) above ∼ 20 km. This means that, given a fixed at-mospheric composition in terms of well-mixed greenhousegases, MA temperatures are almost independent of the sur-face temperature.

In the window-grey model the increased surface temper-ature entails increased upwelling LW radiation in both thewindow and the opaque band. In contrast, the additionalupwelling LW radiation of approx. 3.5 W m−2 (beyond thetropopause) in CO2× 2 compared to CO2× 22fixSST is con-strained to transparent parts of the spectrum and has thus noimpact on MA temperatures. This result is not specific to oursimulations and in line with previous studies. In particular,Forster et al. (1997b) use a radiative-convective model andshow that the radiative forcing by CO2 depends on the defini-tion of the tropopause. While this affects the response of thesurface-troposphere system, the temperature profile above isnot affected by the definition of the tropopause (see theirFig. 8a). A similar argumentation applies to changes in sur-face albedo: the latter would affect the surface temperaturedirectly and lead to an adjustment of the troposphere, butthe effect decays with height (Manabe and Wetherald, 1967,Fig. 19). Hence, the temperature in the MA appears to be di-rectly determined by the actual atmospheric composition andnot by the history of this composition (i.e., the concentrationscenario) and associated surface temperature changes.

A possible explanation for the discrepancy between com-plex models and the window-grey model regarding the slowMA adjustment, as well as other limitations of the window-grey model are discussed in the following section.

5.2 Limitations of the window-grey model

Given the simplicity of the window-grey model, quantita-tive statements are difficult to make and the cases shown inFigs. 2, 5, 7, and 8 are quantitatively unrealistic. Here wenevertheless attempt to derive some crude estimates basedon the window-grey model, and discuss discrepancies toECHAM6 in the light of obvious limitations of the window-grey model.

First we investigate the strength of the permanent block-ing effect at the TOA in relation to the surface response. Itfollows from Eqs. (30) and (32) that



150 200 250 300

020

4060

80

T (K)

z (k

m)

REFREFns

-10 -5 0 5

020

4060

80

ΔT (K)

z (k

m)

CO2x2 - REFCFCx15 - REFCO2x2ns - REFnsCFCx15ns - REFnsCO2x2fixSST - REFCFCx15fixSST - REF

0 20 40 60 80 100

020

4060

80

Percentage permanent cooling for CO2x2

z (k

m)

Permanent blockingeffect

Indirect solareffect

Figure 9. Results obtained with ECHAM6 coupled to a simplistic ocean model to allow for rapid thermal adjustment of the surface. Left:global annual-mean equilibrium temperature profiles for two reference runs under pre-industrial external forcing, with (solid; REF) andwithout (dashed; REFns) absorption of solar radiation in the atmosphere. Middle: temperature difference to the corresponding referenceruns in response to increased CO2 or CFC concentrations (simulation IDs are explained in Table 1). Right: percentage of the permanentcooling effect in response to CO2 doubling from the blocking and indirect solar effects, estimated by dividing CO2× 2ns−REFns byCO2×2−REF. Note that the vertical coordinate z is only approximate height, calculated from h with a constant scale heightH = 8km suchthat h= 1− e−z/H .

∂Ttoa,eq

∂αo/∂Tsrf,eq

∂αo=−

12

(Tsrf,eq

Ttoa,eq

)3βw

1−βw. (39)

One can now insert typical temperatures prevailing at theEarth’s surface (∼ 290 K) and at the mesopause (∼ 180 K),and an estimate of βw ≈ 10–20 % (the actual values dependon the optical thickness threshold used to derive βw from thecontinuous absorption spectra, compare Fig. 1b–c). This sim-ple calculation yields response ratios of approximately only(−0.2)− (−0.5), i.e., a larger temperature change at the sur-face than in the MA. This result stands in sharp contrast toECHAM6 where the surface warms much more than the MAcools by the permanent blocking effect.

Probably the main reason for this discrepancy is that theeffective width of the atmospheric window is very differentfor the atmosphere as a whole and for the atmosphere beyondthe tropopause alone. The width of the atmospheric windowis however crucial for the atmospheric temperature profileand the strength of the MA cooling effects both in absoluteand relative terms. Considering that βw ≈ 90–95 % is morerepresentative for the largely water-free atmosphere beyondthe tropopause (compare Fig. 1), Eq. (39) yields a responseratio of (−20)− (−40), which is in much better agreementwith the ECHAM6 results.

Regarding the transient component of the MA cooling,the window-grey model predicts that the transient tempera-ture adjustment decays with height (see also Appendix C2,Eq. C10), but it fails to explain the virtual absence of a tran-sient MA adjustment in ECHAM6. This might be linked toanother effect neglected in the window-grey model, namelythe water vapour feedback. Higher tropospheric temperaturesimply higher water vapour concentrations, leading to a pro-nounced temperature dependence of the atmospheric opac-

ity. The increased opacity as a result of tropospheric warm-ing entails that a secondary blocking effect may counteractthe slow reduction of the initial MA cooling associated withthe transient component of the blocking effect seen in thewindow-grey model. We therefore speculate that the watervapour feedback might play a role to explain the apparent in-sensitivity of MA temperatures to the surface temperature byredirecting changes in upwelling LW radiation to parts of thespectrum that are transparent in the MA.

Moreover, the height-dependent width of the atmosphericwindow acts in concert with the effect of convection.Convection acts to reduce the lapse rate considerably, to∼ 6.5 K km−1 in the current climate, leading to an approxi-mately constant lapse rate in the troposphere (e.g., Manabeand Wetherald, 1967). The appearing radiative-convectiveequilibrium in the troposphere is associated with an upwardheat transport and increased temperatures in the free tropo-sphere and decreased temperatures at (and close to) the sur-face. The redistribution of heat from the surface to the up-per troposphere by convection thus bypasses the lower levelswhere the atmospheric window is small. Tropospheric con-vection is thus an efficient process to attenuate the surfaceresponse to greenhouse forcing, but convection is neglectedin the window-grey model.

It is tempting to apply the window-grey model only to theMA, prescribing the upward radiative flux in the opaque ther-mal band (O↑) at the tropopause as a lower boundary condi-tion. The omission of the troposphere would have the advan-tages that convection does not play a significant role anymoreand that the complex influence of the unevenly distributed at-mospheric water (in all its aggregate phases) is strongly di-minished.

A way to achieve this for the grey model is to apply Eq. (9)at the tropopause (index tp) and to insert it into Eq. 11. Trans-



ferred to the window-grey model, this yields

T (h)= 4

√√√√ O↑

tp

σ (1−βw)

(αo (1−h)+ 1

)(αo (1−htp)+ 2

) . (40)

One could now investigate how changes in O↑tp or htp af-fect the MA, but this would not be very conclusive becauseO↑

tp and htp respond in a complex manner to changes ingreenhouse-gas forcing. One could also follow a hybrid ap-proach by using values derived from a complex model likeECHAM6 for O↑tp and htp. However, in particular the deriva-

tion of O↑tp from a multi-band LW scheme would not bestraightforward. Moreover, the ECHAM6 results show thatabove a certain height the temperature profile will not re-spond to changes in the troposphere. In other words, O↑tp andhtp appear to change in such a way that the temperature pro-file above remains the same. Overall, applying the window-grey model only to the MA appears not to add to our expla-nation of why CO2 cools the MA.

6 Summary and conclusions

In this article we explain a well-known phenomenon thatis central to our general understanding of climate change –cooling of the middle atmosphere (MA) by CO2 – in a simplebut physically consistent way. We do so by applying a verti-cally continuous window-grey radiation model to the phe-nomenon. This way it is possible to distinguish two main ef-fects by which CO2 cools the MA.

First, enhanced blocking of upwelling LW radiation oper-ates towards lower MA temperatures. In principle, this block-ing effect has a transient component due to the slow warmingof the surface. This adjustment leads to intensified upwellingLW radiation and tends to reduce the initial MA cooling inthe window-grey model. While these effects exactly compen-sate each other in a grey atmosphere, leading to an equilib-rium TOA temperature that is independent of the atmosphericopacity, the blocking of upwelling LW radiation outweighs inthe presence of a spectral window because of the reduced sur-face temperature sensitivity, leaving lower equilibrium tem-peratures above a critical height after the adjustment. Hence,the blocking effect is permanent because the Earth’s atmo-sphere is not grey, i.e., uniformly opaque for LW radiationat any wavelength, but absorbs and emits LW radiation withvarying intensity depending on wavelength. The introductionof a spectral window into an otherwise uniformly opaque at-mosphere is the simplest possible means to capture the effectin a physical model.

The second permanent effect of CO2-induced MA coolingis the indirect solar effect. It owes its existence to the factthat there are heat sources within the atmosphere in addi-tion to LW radiation, most importantly solar radiation thatis absorbed in particular in the vicinity of the stratopause

by ozone. The additional heating term causes a deviation ofthe temperature profile from the window-grey solution. Thestrength of this deviation depends on the abundance of LWabsorbers because the relative importance of the constant ad-ditional heating term in the local energy budget decreaseswith increasing LW absorber abundance.

While the window-grey model allows for a fully analyt-ical treatment of CO2-induced MA cooling, it is not wellsuited to constrain the relative effect strengths. Uncertaintiesare large because the window-grey model entails a numberof gross simplifications, including in particular: the assump-tion of vertically well-mixed greenhouse gases (violated inparticular by water vapour); the simplistic LW band struc-ture; and the neglect of vertical heat transport by convection(and conduction at the surface). Additional simplificationsare the following: the neglect of Wien’s law; the two-streamapproximation; the neglect of the horizontal dimensions andthe associated differential heating and atmospheric dynam-ics (including gravity waves); the neglect of chemical pro-cesses; the implicit treatment of solar radiation; the neglect ofclouds, aerosols, and scattering in general; and the assump-tion of local thermodynamic equilibrium that does not holdin the upper mesosphere and beyond. Most of these factorsare discussed for example in Pierrehumbert (2010), and thosespecific to the mesosphere are reviewed in Mlynczak (2000).

Therefore, to quantify the effect strengths and to comple-ment the insights gained from the window-grey model, wehave conducted simulations with a much more complex at-mospheric model. The results indicate that the two perma-nent effects are similarly important, with the indirect solareffect dominating around the stratopause and the blockingeffect dominating away from the stratopause. The window-grey model also predicts a slow (re-)warming throughout theatmosphere in response to the slow surface warming. How-ever, this transient effect is negligible in the MA accordingto the simulations with the complex model, pointing to thelimitations of the window-grey model.

This article is meant to consolidate our understandingof why CO2 cools the middle atmosphere by filling a gapbetween reality and complex atmospheric models on theone side and somewhat scattered heuristic arguments on theother. The reconsideration of CO2-induced MA cooling asput forward here has a distinct educational element, withthe potential to convey the physical essence of the involvedmechanisms to a broader audience.

7 Data availability

The spectroscopic data underlying Fig. 1 are accessible viaHITRAN on the Web (http://hitran.iao.ru; see also AppendixD). The ECHAM6 simulation data are stored at the GermanClimate Computing Center (DKRZ) and can be obtained viathe authors.


http://hitran.iao.ru


Appendix A: An analogy for the blocking effect

While the explanations based on the window-grey model in-volve mathematical formalism, the following analogy mayfacilitate an intuitive understanding for the permanent andtransient components of the blocking effect.

Consider a building that is heated at a constant rate frominside. In steady state there is a higher temperature insidethe building compared to the fixed exterior temperature. Thewalls of the building represent an analogy to the Earth’s at-mosphere, with the outer surface as the top of the atmosphereand the inner surface as the atmosphere close to the Earth’ssurface. The temperature at the outer wall surface is higherthan the exterior temperature and the temperature at the innerwall surface is somewhat lower than the interior (room) tem-perature. These temperature differences maintain an exportof heat at the same rate at which the interior is heated. In thefollowing we assume that the walls have negligible heat ca-pacity whereas the interior reacts more inertly to disturbancesdue to a non-zero heat capacity.

We first assume that the building is insulated equally welleverywhere (corresponding to the grey case), resulting in auniform temperature of the outer surface. If now the heatresistance of the walls is instantaneously increased, at firstthe outer surface temperature drops and the inner surfacetemperature rises, while the interior temperature is still un-changed. In this situation less heat escapes from the buildingthan is released by the heating system. The imbalance leadsto a slow ascent of the interior temperature that continues un-til the outer surface temperature returns to its original value.The initial cooling of the outer surface temperature is anal-ogous to the quasi-instantaneous cooling that occurs in theupper half of the atmosphere in the grey model; the coolingis only transient and has no permanent component.

Assuming instead that there are parts of the building enve-lope that are more weakly insulated than the remainder, as istypically the case with windows, the outer surface tempera-ture in equilibrium is higher at the windows than it is at thewalls, and a larger fraction of the total energy escapes via thewindows compared to how much they contribute to the to-tal area of the building envelope. If now the heat resistanceof the walls is increased, the outer surface temperature ofthe wall is diminished not only temporarily, but some cool-ing remains also after the interior temperature has increasedto its new equilibrium value. In the new equilibrium, evenmore energy escapes through the windows and less throughthe walls. The permanent cooling of the outer surface tem-perature of the walls is analogous to the cooling in the higheratmosphere associated with the permanent blocking effect ofCO2-induced MA cooling.

The main difference between the building analogy and thewindow-grey radiation model is that the separation betweenwalls and windows in the former case is in geometrical space,whereas the separation into an opaque and a transparent ra-diation band in the latter case is in spectral space. Another

obvious difference is that the mechanism of energy transferis heat conduction in the walls of a building as opposed to ra-diation in the atmosphere. Nevertheless, we reckon the anal-ogy of an insulated building as a valid means to illustrate theblocking effect of CO2-induced MA cooling.

Appendix B: Relation to discrete-layer models

Without showing derivations we point out that the verticallycontinuous model(s) presented in the main text can be in-terpreted as a generalization of discrete-layer models. Thesimplest type of the latter, a model with only one grey atmo-spheric layer, is widely used to explain the greenhouse ef-fect in a conceptual way (e.g., Pierrehumbert, 2010; Neelin,2011). In the following we discuss only the grey case, but thewindow-grey case can be treated analogously.

In an n layer grey-atmosphere model with uniform layeremissivity εl, from the radiative balances at every atmo-spheric layer it follows that, given an arbitrary surface tem-perature, the equilibrium temperature at layer i is

Ti = Tsrf4

√εl (n− i)+ 1εl (n− 1)+ 2

, (B1)

where i = 1 is the lowest and i = n the highest atmosphericlayer. The overall equilibrium situation is obtained when Tsrfin Eq. (B1) is replaced by the value it attains in overall equi-librium, which is

Tsrf,eq = Teff,eq4

√1+

nεl

2− εl. (B2)

For α/2 ∈N the vertically continuous grey model is equiv-alent to a discrete grey model with n= α/2 atmospheric lay-ers, each with emissivity εl = 1. The heights h that corre-spond to the discrete levels i are then determined by

hi =i− 1

2n

. (B3)

Although providing a very suitable conceptual tool tounderstand the greenhouse effect, the discrete-layer grey-atmosphere model (just like its continuous analogue) obvi-ously can not explain greenhouse-gas induced MA cooling.For such an explanation it is again necessary either to in-troduce non-uniform opacity for LW radiation (e.g., by in-troducing an atmospheric window), or to introduce an addi-tional (solar) heating term.

Appendix C: Formal response analysis

To supplement the discussion in the main text, in this ap-pendix we quantify the response of temperature to changesin the parameters of the vertically continuous window-greyatmosphere model in terms of partial derivatives. We thereby



also separate the simultaneously occurring effects of CO2-induced MA cooling in a formal way. We start without theindirect solar effect but include it into the formalism later.

In the following a response is simply the partial deriva-tive of temperature with respect to either αo or βw. Differentresponses are discerned based on the conditions introducedinto the derivatives. We distinguish between a fast (quasi-instantaneous) response F where the surface temperature iskept fixed at its previous equilibrium value, and a subsequentslow response S during which also the surface attains its newequilibrium temperature. The overall equilibrium response Ecan thus be written as

E = F +S . (C1)

During the slow transition from F to E , the current re-sponse C(t) at time t deviates from E by the transient re-sponse T (t):

C(t)= E + T (t) , (C2)

with

T (t)=(f (t)− 1

)S , (C3)

where f (t) ∈ [0,1] is that fraction of the slow responsethat has already taken effect at time t , with f (0)= 0 andf (t→∞)= 1. The transient response is thus defined as thepart of the quasi-instantaneous response that is later compen-sated by the adjustment to surface warming.

C1 Surface response

Differentiating Eq. (30) with respect to αo and βw gives theoverall equilibrium responses Eαo and Eβw at the surface:

Eαo,srf ≡∂Tsrf,eq

∂αo=

Teff

44

√αo+ 2αoβw+ 2

(1

αo+ 2−

βw

αoβw+ 2

)(C4)

and

Eβw,srf ≡∂Tsrf,eq

∂βw=−Teff αo

44

√αo+ 2

(αoβw+ 2)5 . (C5)

Excluding the trivial cases αo = 0 and βw = 1 , Eqs. (C4)and (C5) imply that Eαo,srf > 0 and Eβw,srf < 0. That is, thesurface warms when greenhouse gases are added.

As there is by definition no fast response at the surface,i.e., Fαo,srf,Fβw,srf = 0 , it is Sαo,srf = (f (t)− 1)Eαo,srf andSβw,srf = (f (t)− 1)Eβw,srf : the transient response fully com-pensates for the equilibrium response initially (where f = 0),but vanishes for t→∞.

C2 Atmospheric response

The temperature response of the continuous window-grey at-mosphere in overall equilibrium to αo and βw as a functionof height is obtained by differentiating Eq. (31) with respectto the two model parameters, giving

Eαo (h)≡∂Teq(h)∂αo

=Teff,eq

44

√αo(1−h)+ 1αoβw+ 2

·

(1−h

αo(1−h)+ 1−

βw

αoβw+ 2

)(C6)

and

Eβw (h)≡∂Teq(h)∂βw

=−Teff,eq αo

44

√αo(1−h)+ 1(αoβw+ 2)5 . (C7)

Differentiating the quasi-instantaneous temperature profilegiven by Eq. (27) in overall equilibrium at h= 1 (i.e., at theTOA) with respect to αo leads to a form that supports theinterpretation of the permanent blocking effect as the inter-play between the sensitivity of the surface temperature togreenhouse-gases on the one hand and the blocking of up-welling LW radiation by greenhouse gases on the other hand:

Eαo,toa =4

√1

αo+ 2

(∂Tsrf,eq

∂αo−

Tsrf,eq

4(αo+ 2)

). (C8)

Here the surface sensitivity is represented by the minuendin the brackets whereas the blocking effect is represented bythe subtrahend in the brackets.

Differentiating Eq. (27) with respect to αo under the con-straint Tsrf = const and inserting Eq. (30) gives the fast tem-perature response as a function of height:

Fαo (h)≡∂T (h)∂αo

∣∣∣∣Tsrf=Tsrf,eq=const

=Teff,eq

44

√αo(1−h)+ 1αoβw+ 2

(1

αo+ (1−h)−1 −1

αo+ 2

).

(C9)

Note that changing the width of the atmospheric windowentails no fast response, i.e., Fβw (h)= 0.

The transient part of the response follows from Eqs. (C6)and (C9) with Eqs. (C1)–(C3) as

Tαo (h, t)≡(1− f (t)

)(Fαo (h)− Eαo (h)

)=(1− f (t)

)Teff,eq

4

4

√αo(1−h)+ 1αoβw+ 2

·

(βw

αoβw+ 2−

1αo+ 2

). (C10)

The transient part of the response therefore becomessmaller with height. Comparison with Eq. (C4) further shows



that |Tαo (h, t)|< |Eαo,srf| , that is, the transient cooling at anyheight in the atmosphere is always weaker than the equilib-rium warming of the surface. Note that Tβw (h, t) follows di-rectly from Eβw (h) because Fβw (h)= 0.

C3 Inclusion of the indirect solar effect

Considering overall equilibrium, and simplifying the anno-tation by leaving away h′, differentiation of Eq. (38) withrespect to the two model parameters gives

E∗αo≡∂T ∗eq

∂αo=

(Teq

T ∗eq

)3

Eαo −s∗

4T ∗eq3α2

o(1−βw)(C11)

and

E∗βw≡∂T ∗eq

∂βo=

(Teq

T ∗eq

)3

Eβo +s∗

4T ∗eq3αo(1−βw)2

, (C12)

where Eαo and Eβw are the window-grey overall equilibriumresponses given by Eqs. (C6) and (C7).

To include the indirect solar effect I into the formalismof Eqs. (C1)–(C3), one can extend Eq. (C2) using Eqs. (38),(C11), and (C12) as follows:

C∗(t)= E +XEI + I︸︷︷︸E∗

+ T (t)+XT I (t)︸︷︷︸T ∗(t)

(C13)

with

Iαo =−s∗

4T ∗eq3α2

o(1−βw), (C14)

Iβw =s∗

4T ∗eq3αo(1−βw)2

, (C15)

XEI =

(Teq

T ∗eq

)3

− 1

E , (C16)

XT I (t)=

(Teq

T ∗eq

)3

− 1

T (t) , (C17)

where the terms in Eqs. (C14)–(C16) follow naturallyfrom Eqs. (C11) and (C12). Equation (C17) results fromEq. (C13) with the analogues of Eqs. (C11) and (C12) forthe fast response (i.e., with E∗ and E replaced by F∗ and F)and the definition of T ∗(t) :

T ∗(t)= (1− f (t))(F∗− E∗) . (C18)

The termsXEI andXT I are interaction (or synergy) termsthat result from the fact that E , I, and T (t) are not linearlyadditive. Due to these terms, the quantitative attribution of atotal response to the different mechanisms is not unambigu-ously possible.

Appendix D: Supporting information on Fig. 1

The absorption spectra have been computed with HITRAN onthe Web (http://hitran.iao.ru). The atmosphere from surfaceto space (Fig. 1b) was approximated with an 8000 m thickhomogeneous gas mixture at 260 K and 1013.25 hPa with thefollowing composition (with respect to volume): 4000 ppmH2O, 300 ppm CO2, 0.4 ppm O3, 0.3 ppm N2O, 1.7 ppmCH4, 209 000 ppm O2, and the remainder N2. The middle at-mosphere (Fig. 1c) was approximated with an 8000 m thickhomogeneous gas mixture at 220 K and 202.65 hPa with thesame composition except for H2O (4 ppm) and O3 (2 ppm)(and correspondingly N2). Some Gaussian smoothing wasapplied to the spectra.


http://hitran.iao.ru


Acknowledgements. The foundations for this work have beenlaid during the authors’ employment at the Max Planck Institutefor Meteorology, Hamburg, Germany. The ECHAM6 simulationshave been conducted at the German Climate Computing Center(DKRZ). We thank Stephan Bakan and Thomas Rackow forhelpful discussions, and Sebastian Rast for technical support withECHAM6. Thanks are also extended to Angus Ferraro, Max Popp,an anonymous reviewer, and Michel Crucifix for very constructivecomments.

The article processing charges for this open-accesspublication were covered by a ResearchCentre of the Helmholtz Association.

Edited by: M. CrucifixReviewed by: M. Popp and one anonymous referee

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