ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-7-697-2016Why CO2 cools the middle atmosphere – a consolidating model perspectiveGoesslingHelge F.helge.goessling@awi.dehttps://orcid.org/0000-0001-9018-1383BathianySebastianhttps://orcid.org/0000-0001-9904-1619Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, GermanyWageningen University, Wageningen, NetherlandsHelge F. Goessling (helge.goessling@awi.de)29August20167369771511March201618March201627July20168August2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/articles/7/697/2016/esd-7-697-2016.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/7/697/2016/esd-7-697-2016.pdf
Complex models of the atmosphere show that increased carbon dioxide (CO2)
concentrations, while warming the surface and troposphere, lead to lower
temperatures in the stratosphere and mesosphere. This cooling, which is often
referred to as “stratospheric cooling”, is evident also in observations and
considered to be one of the fingerprints of anthropogenic global warming.
Although the responsible mechanisms have been identified, they have mostly
been discussed heuristically, incompletely, or in combination with other
effects such as ozone depletion, leaving the subject prone to misconceptions.
Here we use a one-dimensional window-grey radiation model of the atmosphere
to illustrate the physical essence of the mechanisms by which CO2 cools
the stratosphere and mesosphere: (i) the blocking effect, associated
with a cooling due to the fact that CO2 absorbs radiation 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 particularly
the case in the upper parts of the ozone layer). By contrast, in the grey
model without solar heating within the atmosphere, the cooling aloft is only
a transient blocking phenomenon that is completely compensated as the surface
attains its warmer equilibrium. Moreover, we quantify the relative
contribution of these effects by simulating the response to an abrupt
increase in CO2 (and chlorofluorocarbon) concentrations with an
atmospheric general circulation model. We find that the two permanent effects
contribute roughly equally to the CO2-induced cooling, with the indirect
solar effect dominating around the stratopause and the blocking effect
dominating otherwise.
Introduction
The laws of radiative transfer in the Earth's atmosphere are a key to
understanding our changing climate. With the absorption spectra of greenhouse
gases as one central starting point, climate models of increasing complexity
have been built during the last decades. These models show that increased
carbon dioxide (CO2) concentrations, while warming the surface and the
troposphere, lead to lower temperatures in the middle atmosphere (MA; the
stratosphere and the mesosphere) . Meanwhile, observations show a cooling trend in the
MA during the satellite era until the most recent years; the negative trend
is especially large in the upper stratosphere and in the mesosphere, although
uncertainties also increase with height .
Attribution studies have concluded that the depletion of stratospheric ozone
was probably the main driver of the cooling in the lower stratosphere
, especially in the Antarctic spring . In addition, the roles of volcanoes and atmospheric dynamics
, stratospheric water vapour , and climate variability
have been discussed. The increase of CO2 concentration
contributed to the decrease of lower stratospheric temperatures, but only to
a small extent . In the middle and upper stratosphere (and
beyond), the CO2 increase has probably been the most important reason for
the temperature decrease . As ozone concentrations are expected to recover in the future,
it seems likely that CO2-concentration trends will be of growing
importance also in the lower stratosphere .
The isolated effect of CO2 on temperatures in the MA is rarely explained.
Probably the most frequent argument found in textbooks
e.g., is related to the ozone layer where a
considerable part of the locally absorbed radiation is short-wave (solar)
radiation. Therefore, the temperature around the stratopause exceeds the
corresponding temperature in a hypothetical grey atmosphere by far. Because
the main absorption bands of CO2 are in the long-wave (LW) part and not in
the solar part of the spectrum, an increase in CO2 leads to increased
emission of LW radiation while the rate of solar heating remains unaltered.
The excess of emission compared to absorption leads to a cooling. It should
be stressed that the argument is related not to the depletion but to the mere
presence of ozone . Although this effect is a major
contributor to CO2-induced MA cooling (as we confirm below), this
explanation is less convincing in the middle and upper mesosphere where there
is unabated (observed and simulated) cooling despite the solar heating
becoming weaker with height. Neither can this effect explain an important
difference between CO2 and other long-lived greenhouse gases: while
methane and nitrous oxide have a much weaker effect on MA temperatures
compared to CO2, chlorofluorocarbons (CFCs) even tend to warm the lower
stratosphere (neglecting ozone depletion) .
More complete explanations discern not only between solar and LW radiation,
but treat the LW absorption spectra of greenhouse gases in more detail.
and point out that the balance of LW
emission and LW absorption must be considered: any greenhouse gas emits
simply according to its local temperature, but absorbs radiation emitted from
certain distances (represented by radiation mean free paths) depending on the
absorption spectrum of the gas and the atmospheric composition. At the
absorption bands of certain CFCs, the radiation mean free path of the
atmosphere is large because the bands are located in the spectral window
region. These CFCs thus absorb mainly radiation emitted from the warm surface
and lower troposphere, but emit with the low temperatures of the MA.
Consequently, increased CFC concentrations impose an LW warming tendency.
point out that, in contrast, the radiation mean free
path in the 15 µm band of CO2 is small, implying that the
radiation absorbed by CO2 in the MA mainly comes from the cold tropopause
region.
and address yet another aspect: the
MA responds much faster than the rather inert surface-troposphere system.
Hence, when greenhouse gases are added to the atmosphere, initially the MA
cools because the radiation mean free path of the atmosphere has been reduced
and the radiation arriving in the MA now stems from higher, colder levels of
the troposphere. After this first phase of MA temperature adjustment, the
surface and troposphere gradually warm, leading to increased upward LW
radiation at the tropopause which warms the lower stratosphere.
All these effects, though mentioned in the literature and included in complex
climate models, are rarely discussed together. Furthermore, they are usually
explained only heuristically and not formalized with a conceptual model. The
most popular educational model of the greenhouse effect, a global-mean grey
atmosphere model with one ground level and a one-layer atmosphere
e.g.,, can not explain CO2-induced MA cooling.
Although introduce an atmospheric window to their conceptual
model, they do not apply it to explain MA cooling. The same is true for more
complex conceptual models e.g.,, which are also not limited to the ingredients needed
to explain how CO2 cools the MA.
Our article aims to provide a consolidating model perspective on
CO2-induced MA cooling. The first part has an educational emphasis as we
demonstrate the physical essence of the mechanisms involved, using simple
variants of a vertically continuous global-mean radiation model: in
Sect. we derive the grey atmosphere model from the general
radiative transfer equation, in close analogy to such models in the
educational literature e.g.,. The grey atmosphere
model features no permanent MA cooling when applied in its pure form. In
Sect. we therefore extend the grey model to the simple
case of two LW bands of which one is fully transparent, i.e., the window-grey
case. The rigorous derivations in Sects. and
may be skipped by readers primarily interested in the
resulting explanations. In Sect. we explain
CO2-induced MA cooling using the window-grey radiation model. In
Sect. we provide a quantitative separation of the effects
based on simulations with the atmospheric general circulation model ECHAM6.
Based on these simulations we discuss what can, and what can not, be learnt
about the effect strengths based on the window-grey model. This is followed
by a summary and conclusions in Sect. . In addition to
our model derivations, Appendix A offers a simple analogy to understand the
blocking effect of CO2 without any equations. Moreover, we show the
relation of the vertically continuous model to discrete-layer models in
Appendix B and provide a formal response analysis in terms of partial
derivatives with respect to the parameters of the window-grey model in
Appendix C. Appendix D provides technical details on Fig. .
The grey-atmosphere model
(a) Normalized black body curves for 5800 K (the
approximate emission temperature of the Sun) and 288 K (the approximate
surface temperature of the Earth). (b) Representative absorption
spectrum of the Earth's atmosphere for a vertical column from the surface to
space, 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 Fig. 1.1 on p. 4.
In the following we derive the vertical temperature profile of a grey
atmosphere, first with only the atmosphere in thermal equilibrium and then
assuming equilibrium also for the surface. We consider a vertically
continuous grey atmosphere with horizontally homogeneous (global-mean)
conditions. The grey atmosphere is transparent for solar radiation and
uniformly opaque for LW radiation. Splitting the electromagnetic spectrum
into a transparent solar band and an opaque LW band is a common approximation
that is naturally suggested by (i) the well separated emission spectra of the
Sun and the Earth (Fig. a) and (ii) the shape of the
absorption spectrum of the Earth's atmosphere (Fig. b). The
grey model accounts only for radiation while other processes of energy
transfer (most importantly convection) are neglected. Greenhouse gases are
assumed to be well-mixed and any effects from clouds or aerosols are
neglected. Assuming horizontally homogeneous conditions and the absence of
scattering we apply the two-stream approximation ,
meaning that we distinguish only upward and downward propagating radiation
(indicated by arrows in the subsequent equations). This leads to the
differential form of the radiative transfer equation e.g.,:
dL↑(z)dz=1μ[J-L↑(z)]ρ(z)k,
with radiance L, source term J, geometric height z, air density ρ,
mass absorption coefficient k, and μ=cos(θ) with the effective
angle of propagation θ. To remove any angular dependence from the
equations, we use the common assumption of an effective angle of propagation
of 60∘ relative to the vertical, i.e., μ=1/2seechap.
4.2. As we neglect scattering, J only consists of the
long-wave blackbody emission, which is isotropic. Integrating
Eq. () over the half sphere then yields
dF↑(z)dz=[σT(z)4-F↑(z)]ρ(z)k
with irradiance F (in W m-2) and the Stefan-Boltzmann constant
σ. Using the relative pressure deficit
h=1-p/psrf
as vertical coordinate, where p is pressure and psrf is surface
pressure, the radiative transfer equation reads
dF↑(h)dh=[σT(h)4-F↑(h)]α.
The absorption coefficient α is the only parameter of the grey model
and describes the atmospheric opacity in the LW band. Due to our definition
of the vertical coordinate h, α is independent of h (in fact, it
follows from hydrostatic balance that α=kpsrf/g).
Also, h is proportional to optical thickness: τ=αh .
Although we distinguish the parameter α from the vertical coordinate
h, our model is equivalent to similar approaches in popular textbooks of
radiative transfer which usually choose optical depth (τ, also called
optical thickness) as their vertical coordinate. For example, the optical
thickness between a height h and the top of the atmosphere, in our case
α(1-h), is identical to τ∞-τ in ,
to τ in , and to τ/μ‾ in .
In the latter two cases, the vertical axis points downwards, hence the
reversed sign. In , a parameter μ‾ still appears
in the equations as no assumption on the average direction of propagation is
made.
Equation () is the spectrally integrated grey-absorption-case of
Schwarzschild's equation and holds analogously for downwelling LW radiation
F↓. In radiative equilibrium F must be free of divergence
because other source or sink terms of heat are neglected. With
Ftoa↓=0, F↓ is hence determined by
F↑(h)-F↓(h)=Ftoa↑,
where the index toa stands for the top of the atmosphere (TOA). Thermal
equilibrium for a thin layer of air is given when
2ϵσT(h)4=ϵF↑(h)+F↓(h),
where ϵ=αdh is the emissivity, and hence also the
absorptivity, of the thin layer for LW radiation. Combining
Eqs. () and () yields
σT(h)4=F↑(h)-Ftoa↑2.
Substituting Eq. () into the radiative transfer equation
(Eq. ) gives
dF↑(h)dh=-α2Ftoa↑.
Because α is constant, Eq. () has the simple solution
F↑(h)=Ftoa↑[α2(1-h)+1].
With Eq. () it follows further from Eq. () that
F↓(h)=Ftoa↑α2(1-h).
Inserting Eqs. () and () into Eq. ()
leads to the vertical temperature profile of the equilibrated grey
atmosphere:
T(h)=Ftoa↑2σ(α(1-h)+1)4.
Evaluating Eq. () at the surface (h=0) gives
Ftoa↑=Fsrf↑α/2+1.
Assuming that the surface is a perfect black body for LW radiation, it is
Fsrf↑=σTsrf4.
With Eqs. () and (), the vertical temperature
profile described by Eq. () can be written as
T(h)=Tsrfα(1-h)+1α+24.
Equation () implies for the near-surface (h=0) air that
T(0)=Tsrfα+1α+24.
Hence, T(0)<Tsrf. The reason for this discontinuity at the
surface is that, in order to attain the same temperature as the surface, the
near-surface air would have to receive as much LW radiation from above as it
receives from the surface below, which is not the case (see
Eq. ). This is a result of the negligence of all mechanisms
of energy transfer other than radiation in the model; in reality, the
molecular and turbulent diffusion of heat removes the discontinuity, although
sharp temperature gradients right above the surface can still be observed
see for examplewhose Eq. 4.45 is identical to our
Eq. .
The vertical temperature profile can also be written by reference to the
effective radiative temperature of the planet, defined as
Teff=Ftoa↑σ4.
Inserting Teff into Eq. () gives
T(h)=Teffα2(1-h)+124.
Up to now we have not considered the surface energy balance but determined
the vertical temperature profile of an equilibrated grey atmosphere with
absorptivity α given an arbitrary surface temperature as lower
boundary condition. Equation () can thus be interpreted as
the quasi-instantaneous atmospheric temperature profile. In the
following, we consider the situation where not only the atmosphere but also
the surface is in thermal equilibrium. We refer to this situation as the
overall equilibrium and denote the corresponding variables with the index eq.
Assuming that no solar radiation is absorbed within the atmosphere, surface
equilibrium requires that
S=Fsrf,eq↑-Fsrf,eq↓,
where S is solar radiation absorbed at the surface. Inserting Eq. () into Eq. () gives
S=Ftoa,eq↑,
which is the overall equilibrium condition at the top of the atmosphere. It
follows with Eq. () that
Teff,eq=Sσ4.
In overall equilibrium, the vertical temperature profile
(Eq. ) hence becomes
Teq(h)=Teff,eqα(1-h)+124.
Apart from the different choices of the vertical coordinate, the temperature
profile given by Eq. () is identical to Eq. (12.21) in
, to Eq. (4.42) in , and to Eq. (3.47) in
.
Inserting h=1 into our Eq. () yields the temperature
at the TOA:
Ttoa,eq=Teff,eq124.
Finally, combining Eqs. () and ()
gives the corresponding equilibrium surface temperature:
Tsrf,eq=Teff,eqα+224.
Equations ()–() reveal that, in overall
thermal equilibrium, an increase in absorptivity of a grey atmosphere without
non-LW heat sources leads to a temperature increase everywhere except at the
TOA where the temperature is independent of α.
Vertical temperature profiles of a grey atmosphere in overall
equilibrium for different absorptivities α. The latter corresponds to
αo in the window-grey model with βw=0.
Teff,eq=255K. The circles at z=0 denote the
corresponding surface temperatures. Note that the vertical coordinate z is
only approximate height, calculated from h with a constant scale height H=8km such that h=1-e-z/H.
Figure shows solutions of Eq. () for
different values of α. In the limit of an almost completely
transparent atmosphere (α→0), the whole atmosphere attains
one single equilibrium temperature (Eq. ) and the
surface temperature attains the effective equilibrium radiative temperature
of the planet (Eq. ). Note that the vertically continuous model
derived here can be interpreted as a generalization of a discrete-layer model
(see Appendix B).
The window-grey atmosphere model
In reality, the atmosphere is not uniformly opaque for LW radiation, as
within the grey approximation, but interacts differently with LW radiation of
different wavelengths. To account for this in the simplest possible way, we
extend the grey model (Sect. ) by splitting the total LW
radiation F into two separate LW bands: an opaque band F1=O with opacity
αo>0, and a completely transparent (window) band F2=W
with opacity αw=0. With βw=1-βo
describing the fraction of LW radiation from the surface which is directly
emitted to space, the resulting window-grey model has only two parameters:
αo and βw. Thereby βw is
identical to the so-called transparency factor G in
, but independent of temperature in our case as we neglect
Wien's law. This approach represents the so-called window-grey or one-band
Oobleck case of a multiband model . In
contrast to the grey-atmosphere model, the window-grey model allows for the
existence of a spectral window, which can be interpreted as an idealization
of the region between 8 and 12 µm in the Earth's atmosphere
(Fig. b). The window-grey model is depicted in
Fig. .
Sketch of the window-grey atmosphere model. S: solar radiation
absorbed at the surface. O: radiation in the opaque LW band (↑:
upwelling, ↓: downwelling, srf: surface, toa: top of the
atmosphere). W: radiation emitted from the surface in the transparent LW
band (atmospheric window). p: pressure. Interpreting the number of arrows
in the opaque LW band as proportional to the radiative flux, the illustrated
case corresponds to an equilibrated atmosphere with αo=4.
The window-grey model remains analytically solvable because only radiation in
the opaque LW band needs to be considered within the atmosphere. The
resulting radiative transfer equation reads
dO↑(h)dh=[σT(h)4(1-βw)-O↑(h)]αo,
the energy balance equation reads
2ϵoσT(h)4(1-βw)=ϵo(O↑(h)+O↓(h)),
and the surface emission in the opaque band is
Osrf↑=(1-βw)σTsrf4.
Equations ()–() can be solved analogously to the
corresponding equations describing the grey case (Eqs. ,
, and ). This leads to the
quasi-instantaneous atmospheric temperature profile of the window-grey model.
It is
T(h)=Tsrfαo(1-h)+1αo+24.
Comparison with Eq. () reveals that, with the same surface
temperature prescribed as lower boundary condition, the vertical temperature
profiles in the grey and in the window-grey case are identical for
αo=α; the factor (1-βw) in
Eqs. ()–() has cancelled.
To determine the overall equilibrium state, the surface energy balance needs
to be incorporated. In overall equilibrium it is
S+Osrf,eq↓=Osrf,eq↑+Weq
where
Weq=βwσTsrf,eq4.
Here we omitted the indices denoting the orientation and
vertical position of W (as we already did for S) because the only
radiation in the window band is the one emitted upward from the surface, and
W remains unchanged throughout the atmosphere because αw=0.
With derivations analogous to the grey case (Sect. ), one
arrives at simple expressions for the overall equilibrium state. The surface
temperature for the window-grey model in overall equilibrium is
Tsrf,eq=Teff,eqαo+2αoβw+24.
The corresponding vertical temperature profile reads
Teq(h)=Teff,eqαo(1-h)+1αoβw+24,
which implies for the TOA temperature
Ttoa,eq=Teff,eq1αoβw+24.
The TOA temperature, sometimes called the skin temperature , can be thought of as the temperature an infinitely thin air
layer above the atmosphere would have in radiative equilibrium. Obviously,
with βw=0
Eqs. ()–() are reduced to the
grey case (compare Eqs. –).
The dependence of the overall equilibrium surface temperature on the
parameters αo and βw in the window-grey
model (Eq. ). βw=0 corresponds to the
grey case. Teff,eq=255K.
Equation () implies that an increased absorber amount
leads to an increased equilibrium surface temperature (Fig. ),
independent of whether the added molecules absorb in the already opaque part
of the LW spectrum (increasing αo) or in the window region
(decreasing βw, that is, “closing the atmospheric window”).
In the following section we discuss the sensitivity of atmospheric
temperatures to the model parameters.
The mechanisms of CO2-induced middle-atmosphere cooling
With the window-grey radiation model we are now equipped to investigate the
physical essence of CO2-induced MA cooling. In the window-grey model, the
response of temperature to changes in the parameters can be quantified with
partial derivatives. The different effects of CO2-induced MA cooling can
thereby be separated in a formal way. We present such an approach in
Appendix C, but constrain the discussion in the following main text largely
to the undifferentiated equations.
The blocking effect
The blocking effect is the result of a change in the long-wave radiative
balance when atmospheric CO2 is increased. Due to the optically thicker
atmosphere, less upwelling radiation in the non-window part of the spectrum
reaches high altitudes. The temperature at the TOA must thus be lower
because, as follows from Eq. (),
Ttoa=Otoa↑2σ(1-βw)4.
We first investigate the situation in which the assumption of thermal
equilibrium is kept for the atmosphere but dropped for the surface. This is a
reasonable assumption because 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 centennial timescales). In
reality, convection closely couples the surface with the troposphere, hence a
change in greenhouse gases first affects the middle atmosphere, after which
the slow surface-troposphere system adjusts. In the radiation model we
represent this timescale separation by letting the atmosphere respond while
keeping the surface temperature constant. The temperature profile for this
quasi-instantaneous response is given by Eq. () which is
valid even if the surface is not in thermal equilibrium.
Fig. shows the vertical temperature profile before
(blue curve) and after (orange curve) increasing α in the grey case
(for which α corresponds to αo with βw=0).
Inserting h=1 in Eq. () we arrive at the corresponding
quasi-instantaneous TOA temperature:
Ttoa=Tsrf1αo+24.
Equation () implies a cooling at the TOA for increased
αo. Furthermore, Eq. () implies that at
a certain height h^fast the sign of the fast temperature
response due to added greenhouse gases reverses. It is
h^fast=12.
This implies that at first the upper half of the atmosphere (with respect to
mass) is cooled while the lower half is warmed.
Both the upper-level cooling and the lower-level warming are due to enhanced
blocking, that is, a reduced mean free path of LW radiation in response to
increased absorptivity. In radiative equilibrium, the emission, determined by
the local temperature, and the absorption of radiation are locally balanced.
In the upper atmosphere, where downwelling radiation is subordinate, the
upwelling radiation received from below comes from higher (and thus colder)
levels when the absorptivity of the atmosphere is increased. Consequently,
the air cools until emission and absorption are in balance again. In
contrast, in lower levels near the ground, where most of the absorbed
upwelling radiation comes directly from the surface (with a fixed
temperature), the increased absorptivity mostly affects the downwelling
radiation which now comes from lower (and thus warmer) levels, resulting in
warming.
Vertical temperature profiles of a grey atmosphere for two
equilibrium states and one transient state. While the blue and red curves
show the same equilibria as the corresponding curves in
Fig. , the orange curve shows the transient state that occurs
after switching from α=2 to α=6, directly after
equilibration of the atmosphere but with Tsrf still unchanged.
Again α corresponds to αo in the window-grey model
with βw=0. Teff=255K. The circles at z=0 denote the corresponding surface temperatures. Note that the vertical
coordinate z is only approximate height, calculated from h with a
constant scale height H=8km such that h=1-e-z/H.
As long as the emission from the surface, determined by its temperature,
remains unchanged, the surface energy budget is imbalanced due to the
increased downwelling radiation. The surface will thus warm – which is the
common greenhouse effect – until a new overall equilibrium is attained.
During the gradual ascent of the surface temperature, accompanied by
increasing upwelling radiation, the whole atmosphere warms
(Eq. ), and the height at which the sign of the
temperature change reverses is shifted upwards. In the grey case, where
βw=0, this shift proceeds until the whole atmosphere except
the TOA is warmer than originally (red curve in Fig. ).
The upper-level cooling in response to increased absorptivity in a grey
atmosphere (and without absorption of solar radiation within the atmosphere)
is thus only a transient effect that vanishes when the new overall
equilibrium is reached. This is a consequence of the outgoing longwave
radiation (OLR) having to balance S, which is constant in the model. Even
in the very simple grey model a solar and a greenhouse forcing act
differently: while an increase in S would force the OLR to increase as
well, the OLR does not change when CO2 is increased, even though the
temperature is increased throughout the atmosphere.
The situation is different in the presence of an atmospheric window where a
part of the surface radiation is emitted directly to space, bypassing the
atmosphere. An atmospheric window implies a reduced sensitivity of the
surface temperature to the state of the atmosphere (its absorptivity in the
opaque band and the corresponding temperature profile, see
Eq. ) because the radiation in the opaque LW band becomes
less important in the surface energy budget (Eq. ) with
increasing window size. Consequently, in the presence of an atmospheric
window, a permanent cooling at the TOA remains after the surface has
equilibrated (Eq. ).
It becomes evident from Eq. () that, in contrast to
the surface, at the TOA the sign of the temperature response depends on the
spectral property of the added absorbers: if they absorb in the already
opaque part of the LW spectrum (increasing αo),
Ttoa,eq is decreased (MA cooling), but if they absorb in the
transparent part of the LW spectrum (decreasing βw, that is,
“closing the atmospheric window”), Ttoa,eq is increased (MA
warming) (Fig. ).
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. ). βw=0 corresponds to the grey case. Teff,eq=255K.
In fact, decreasing βw leads in overall equilibrium to a
temperature increase at every height in the atmosphere
(Fig. , top). In contrast, if molecules absorbing in the
opaque LW band are added, the sign of the equilibrium temperature response
reverses at a certain height h^eq, with cooling above and
warming below (see Eq. ; Fig. , bottom):
h^eq(βw)=1-βw2.
For βw=0, that is in the grey case, h^eq
becomes 1 (the corresponding geometric height z^eq becomes
∞), meaning that no cooling takes place.
Vertical temperature profiles in overall equilibrium for the
window-grey case with Teff,eq=255K for different
combinations of the parameters αo and βw
(Eq. ). The circles at z=0 denote the
corresponding surface temperatures. Note that the vertical coordinate z is
only approximate height, calculated from h with a constant scale height H=8km such that h=1-e-z/H.
We term the above described cooling in the upper parts of the atmosphere the
blocking effect of CO2-induced MA cooling. This presupposes that
the main consequence of adding CO2 to the atmosphere is, in terms of the
window-grey model, an increase of αo rather than a decrease
of βw. The permanent component of this effect, the
permanent blocking effect, is revealed by
Eqs. () and (). It has to be
distinguished from the instantaneous blocking effect, which consists of the
permanent blocking effect and a transient component. The instantaneous
blocking effect can be observed when atmospheric CO2 is altered but when
the surface temperature has not yet adjusted to the forcing (which is to some
extent also the case for present-day Earth). In the grey model the blocking
effect is only a transient phenomenon: the entire atmosphere has warmed
(except at the TOA) after the surface has equilibrated. In the window-grey
model the blocking effect has a permanent component that persists after the
surface has adjusted.
The blocking effect can be understood in terms of the interplay between the
sensitivity of the surface temperature to greenhouse-gases on the one hand
and the blocking of upwelling LW radiation by greenhouse gases on the other
hand: while an atmospheric window diminishes the sensitivity of the surface
temperature to αo (see Eq. ), the
blocking associated with αo is independent of the presence
or width of an atmospheric window (see Eq. ). Only in the
grey case, where the sensitivity of the surface temperature is at its maximum
(Eq. ), the surface temperature response is strong enough
to compensate for the blocking effect, resulting in an
αo-independent equilibrium TOA temperature (compare
Eq. ).
Another way of looking at the permanent blocking effect goes via the emission
spectrum of the planet viewed from space (i.e., the upwelling LW radiation at
the TOA). If the surface warms in response to an increased
αo, the radiation in the window region of the spectrum W
will be accordingly stronger, corresponding to a Planck curve at the
increased surface temperature. In overall equilibrium the radiation in the
opaque band Otoa↑ must be shifted to lower intensity to
compensate for W, given that the solar energy input is unchanged. It then
follows from Eq. () that the temperature at the TOA must decrease.
The same argument reveals why the TOA cooling due to enhanced blocking in a
grey atmosphere can only be a transient phenomenon: If no window exists,
Otoa↑ must attain its original intensity after
equilibration to balance the unchanged solar energy input.
The indirect solar effect
On Earth not all solar radiation transects the air unhindered, but some is
absorbed within the atmosphere and leads to increased temperatures,
particularly in the upper parts of the ozone layer. The solar heating can be
incorporated into Eq. () as an additional term S∗(h):
2ϵoσT∗(h)4(1-βw)=ϵoO↑(h)+O↓(h)+S∗(h).
Equation () is similar to Eq. (6.15) in ,
except that Neelin considers only the grey case (βw=0) and
neglects the downwelling LW radiation, constraining the validity of the
equation to the vicinity of the TOA.
Assuming that solar heating is confined to an infinitesimally thin layer at
h=h′, such that the equilibrium temperature everywhere else remains
unchanged and, thus, O↑(h′) and O↓(h′) are not
affected by the additional term, one arrives at
T∗(h′)4=T(h′)4+s∗(h′)αo(1-βw),
where T(h′) is the solution of Eq. () with S∗(h′)=0, that is, the window-grey solution of Eq. (), and
s∗(h′)=S∗(h′)/(2σdh) .
Equation () reveals the following: given that due to an
additional term in the local energy budget the atmospheric temperature at
some height is deflected from the window-grey solution, increasing the amount
of LW absorbers in the atmosphere results in a relaxation of the temperature
towards the window-grey solution. This holds both for increasing
αo and for decreasing βw. It must be kept in
mind though that the window-grey solution itself depends on
αo and βw (Eq. ),
making the relaxation towards the window-grey solution an additional effect.
Figure illustrates the indirect solar effect for the grey
case (i.e., for βw=0).
Vertical temperature profiles of a grey atmosphere that is
additionally locally heated (e.g., by absorption of solar radiation) at two
heights within the atmosphere. Apart from the heights at which the profiles
are locally deflected due to additional heating, the blue and red curves show
the same grey equilibria as the corresponding curves in
Fig. . Again α corresponds to αo in
the window-grey model with βw=0 . The heights at which
additional heating occurs (z1≈10km, z2≈50km) and the magnitude of the additional heating terms (specified
such that the temperature rise is 25K for α=2 at both
heights) are more or less arbitrarily chosen to demonstrate the effect.
Teff,eq=255K. The circles at z=0 denote the
corresponding surface temperatures. Note that the vertical coordinate z is
only approximate height, calculated from h with a constant scale height H=8km such that h=1-e-z/H.
If the additional term s∗ is positive, as it is the case for the
absorption of solar radiation by ozone, increasing the emissivity either by
increasing αo or by decreasing βw results in
local cooling. We call this effect the indirect solar effect of
CO2-induced MA cooling. The term “indirect” reminds us that this effect
is not due to any change in solar heating rates as might be caused by a
change in ozone concentrations. Instead, the mere presence of solar
absorption is a prerequisite for this effect. Like the permanent blocking
effect, the indirect solar effect is still at work when the system has
reached the new (more opaque) overall equilibrium. Note that the indirect
solar effect would also manifest if the opacity was changed only locally.
This is not the case for the blocking effect, where integration over a finite
layer with perturbed opacity is needed.
Effect strengths
An essential question so far unanswered is how strong the above derived
effects are compared to each other. In this section we apply a complex
atmospheric model to give a quantitative answer to this question, and we
discuss the implications and limitations of the window-grey model in the
light of these results.
Simulations with a complex atmospheric model
To complement the findings obtained with the window-grey model, and to derive
meaningful estimates for the strength of the effects, we have conducted
simulations with the complex atmospheric general circulation model ECHAM6
. This model and its predecessors have been used for
comprehensive simulations, including future climate projections, in the
different phases of the Coupled Model Intercomparison Project (CMIP), which
are the backbone of the reports compiled by the Intergovernmental Panel on
Climate Chance (IPCC). These models, including ECHAM6, therefore have
sophisticated parameterizations for, e.g., radiation, convection, clouds,
boundary-layer turbulence, and gravity waves, and numerically solve the
governing equations of fluid dynamics on grids with steadily increasing
spatio-temporal resolution. The radiative transfer scheme used in ECHAM6,
which employs 16 LW bands, has been shown to give instantaneous clear-sky
responses to greenhouse-gas perturbations in close agreement with accurate
line-by-line calculations . To adequately resolve the
middle atmosphere, we have used the T63L95 configuration with relatively
coarse (∼ 2∘) horizontal but high (95 levels, top at 0.01 hPa)
vertical resolution. The distribution of ozone is prescribed by a
climatology.
The ocean and sea ice have been treated in a simple way similar to the
approach of . The ocean surface temperature and sea ice
concentration and thickness are prescribed with a realistic seasonal and
spatial pattern derived from observations. After every year the ocean surface
temperature pattern is updated uniformly according to the total energy
imbalance integrated over the global ocean surface (including sea ice) and
over the year, using a heat capacity that corresponds to a 50 m thick
mixed-layer ocean. Despite changing temperatures, the sea ice state pattern
is not updated, leading to discrepancies between the sea ice and ocean
states. This procedure also suppresses further changes to the surface
temperature pattern, such as polar warming amplification. However, this
allows for a rapid thermal equilibration of the surface with an exponential
timescale of ∼3 years, serving the purpose of this paper where the focus
is on the global-mean response.
ECHAM6 simulations.
Simulation IDSolar absorptionSST treatmentCO2aCFC-11/-12bGMSTcOLRdplanetary albedoin the atmosphere[K][W m-2][%]REFyesfree equilibration1×1×287.14241.4129.03CO2×2yesfree equilibration2×1×289.36241.5829.00CO2×2fixSSTyesprescribed from REF2×1×287.41238.1128.91CFC×15yesfree equilibration1×15×288.97241.9628.89CFC×15fixSSTyesprescribed from REF1×15×287.29238.8828.80REFnsnofree equilibration1×1×286.09227.4333.08CO2×2nsnofree equilibration2×1×288.31227.5233.03CFC×15nsnofree equilibration1×15×287.88227.5633.05
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-mean near-surface air
temperature, d outgoing longwave radiation at the top of the atmosphere.
We have conducted eight ECHAM6 simulations with differences in (i) the
treatment of solar radiation, (ii) the treatment of sea-surface temperatures
(SST), and (iii) the abundance of greenhouse gases
(Table ). In five simulations the solar radiation
follows the default behaviour, with some of the solar radiation absorbed by
gases within the atmosphere. This set includes one reference simulation where
pre-industrial greenhouse-gas concentrations are used and the SSTs are
allowed to run into equilibrium, and four sensitivity simulations. In two of
these the ocean is allowed to attain a new equilibrium, either with the
CO2 concentration doubled or with the Chlorofluorocarbon (CFC-11 and
CFC-12) concentrations increased by the factor 15, chosen such that the
surface warming is similar compared to the case of CO2 doubling. The other
two sensitivity simulations are identical with the previous two, except that
the SSTs are prescribed from the reference simulation.
In another set of three simulations the absorption of solar radiation by all
atmospheric gases (but not cloud droplets or ice) is turned off. This set
also includes a reference simulation and two sensitivity simulations with
increased CO2 and CFC concentrations. In these, the SSTs are again allowed
to run into equilibrium. All simulations are conducted over 22 years, but
only the last 10 years are used to compute averages for the analysis because
it takes a few years (in our setup) until an equilibrium is reached. This
experimental design allows us first to demonstrate the dependence of MA
temperature changes on the spectral properties of the added absorbers: CFCs
absorb mainly in the spectral window of the Earth's atmosphere, whereas
CO2 absorbs mainly at wavelengths where the atmosphere is already
relatively opaque. Second, we can quantify the effect strength for the two
permanent effects by which CO2 cools the MA, deduced above with the
window-grey model, and investigate how atmospheric temperatures respond to
the slow surface adjustment.
When the absorption of solar radiation by gases is switched off, the total
short-wave absorption in the atmosphere drops from 75 W m-2 in REF to
only 13 W m-2 in REFns, the residue being due to absorption
by tropospheric clouds. The lack of short-wave heating due to ozone leads to
a strong cooling of the MA. The local temperature maximum around the
stratopause completely disappears and temperatures drop to ∼ 160 K in
the upper stratosphere and in the mesosphere, in agreement with previous
studies . Tropospheric temperatures are only
slightly reduced by ∼ 1 K (Table and
Fig. left). This small temperature change is the result of a
compensation of different effects. After removing the absorption of solar
radiation, more short-wave radiation propagates downwards through the
atmosphere. A part of the previously absorbed radiation is then scattered and
the planetary albedo increases from 29 to 33 %. The rest is partly absorbed
in the troposphere by cloud droplets and ice crystals, and partly reaches the
Earth's surface where the downwelling solar radiation is increased by
42 W m-2. This instantaneous redistribution of short-wave fluxes tends
to warm the surface. However, the large cooling in the MA that follows also
leads to a decreased downwelling long-wave radiation at the surface which has
a cooling effect. To this extent, our result is in line with previous
simulations that quantified the effects of stratospheric ozone removal
. In contrast
to these studies, we still keep ozone as a greenhouse gas as only its
short-wave absorption is removed. However, this is not a very important
difference as the short-wave effect of ozone has been shown to be more
important than the long-wave effect . A similar cancellation of short- and long-wave effects on
the surface temperature seems to hold for other gases. The lack of absorption
by water vapour further shifts the heating from the lower troposphere to the
surface, without much impact on the surface temperature. An additional effect
that might be of relevance for the surface cooling in these simulations is
the reduction in specific humidity due to the atmospheric cooling.
Our simulations with perturbations in CO2 and CFCs are in the tradition of
several pioneering studies on the role of greenhouse gases , and confirm their results.
With standard treatment of solar radiation, CO2 doubling in ECHAM6 leads
to an increase in global annual-mean near-surface temperature by 2.2 K; this
is the climate sensitivity of our model setup. The tropospheric warming in
fact increases with height, reaching a maximum of 3.5 K in the upper
troposphere (Fig. middle, red solid curve). This pattern, which
is not captured by the window-grey model, results from the temperature
dependence of the moist-adiabatic lapse rate (the so-called lapse-rate
feedback) and is thus related to convective processes. Somewhat above the
tropopause the temperature response changes sign. In agreement with earlier
studies , the cooling then increases with height
and assumes a maximum cooling by 11.6 K around the stratopause region.
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) and without (dashed; REFns)
absorption of solar radiation in the atmosphere. Middle: temperature
difference to the corresponding reference runs in response to increased
CO2 or CFC concentrations (simulation IDs are explained in
Table ). Right: percentage of the permanent cooling
effect in response to CO2 doubling from the blocking and indirect solar
effects, estimated by dividing
CO2×2ns-REFns by
CO2×2-REF. Note that the vertical coordinate z is
only approximate height, calculated from h with a constant scale height H=8km such that h=1-e-z/H.
Adding CFCs instead of CO2 results (by design) in a similar tropospheric
response with a near-surface warming by 1.8 K, but temperatures in the MA
remain virtually unchanged (Fig. middle, black solid curve).
Again, this result agrees with previous studies . However, our simulations also show that the near-zero MA
temperature change is due to a cancellation of two effects: the indirect
solar effect is not wavelength dependent. More absorbers increase emission
more strongly than they increase absorption, thereby reducing the relative
importance of the solar heating term (Sect. ). This suggests
that another effect counteracts the cooling from the indirect solar effect.
The above considerations based on the window-grey model suggest that this
counteracting warming effect can be interpreted as an inverse blocking effect: instead of making the already opaque part of the spectrum
even more opaque, which mainly happens when CO2 is added, the increase of
CFC concentrations acts to narrow the atmospheric window, corresponding to a
decrease of βw in the window-grey model
(Figs. and top). In fact, the situation
corresponds not only to a decrease of βw, but also to a
simultaneous decrease of αo because the average opacity of
what should be translated into the single opaque band of the window-grey
model is decreased by the inclusion of the CFC-affected – still relatively
transparent – parts of the previous window band. Overall, the MA is more
strongly subjected to the radiation from the warm surface.
In the case without solar absorption by gases within the atmosphere, adding
CO2 or CFCs results in similar responses in the troposphere, but markedly
different responses in the MA (Fig. middle, dashed curves):
while the cooling in response to CO2 is roughly halved, the previously
neutral response to CFCs turns into a substantial warming by up to 3.5 K.
These results are consistent with the interpretation that the cooling due to
the indirect solar effect has been precluded, leaving only the response due
to the blocking effect (in the CO2 case) and the inverse blocking effect
(in the CFC case).
Under the assumption of linearity, this allows us to estimate the fractional
contributions of the two permanent effects to MA cooling (Fig.
right). According to our results the indirect solar effect contributes up to
∼ 70 % to the total permanent cooling around the stratopause where
solar heating is strongest. Outside this region the blocking effect gains
importance and begins to dominate the cooling in the middle stratosphere and
the middle mesosphere. The assumption of linearity is rather crude, so these
estimates should be taken with a grain of salt. In fact, it is probably not
possible to make a completely clean quantitative distinction, as the formal
analysis in Appendix suggests.
The window-grey model also suggests a transient MA cooling that adds to the
permanent cooling before the surface temperature has adjusted to the changed
radiative forcing. We can investigate this effect with the remaining two
simulations where the greenhouse gases are perturbed but SSTs are fixed to
the reference state (Table ). Interestingly, the initial
MA responses (Fig. middle, dotted curves) are nearly identical
with the corresponding equilibrium responses (solid curves) above
∼ 20 km. This means that, given a fixed atmospheric composition in
terms of well-mixed greenhouse gases, MA temperatures are almost independent
of the surface temperature.
In the window-grey model the increased surface temperature entails increased
upwelling LW radiation in both the window and the opaque band. In contrast,
the additional upwelling LW radiation of approx. 3.5 W m-2 (beyond the
tropopause) in CO2×2 compared to
CO2×22fixSST is constrained to transparent
parts of the spectrum and has thus no impact on MA temperatures. This result
is not specific to our simulations and in line with previous studies. In
particular, use a radiative-convective model and show
that the radiative forcing by CO2 depends on the definition of the
tropopause. While this affects the response of the surface-troposphere
system, the temperature profile above is not affected by the definition of
the tropopause (see their Fig. 8a). A similar argumentation applies to
changes in surface albedo: the latter would affect the surface temperature
directly and lead to an adjustment of the troposphere, but the effect decays
with height (, Fig. 19). Hence, the temperature in the MA
appears to be directly determined by the actual atmospheric composition and
not by the history of this composition (i.e., the concentration scenario) and
associated surface temperature changes.
A possible explanation for the discrepancy between complex models and the
window-grey model regarding the slow MA adjustment, as well as other
limitations of the window-grey model are discussed in the following section.
Limitations of the window-grey model
Given the simplicity of the window-grey model, quantitative statements are
difficult to make and the cases shown in Figs. ,
, , and are
quantitatively unrealistic. Here we nevertheless attempt to derive some crude
estimates based on the window-grey model, and discuss discrepancies to ECHAM6
in the light of obvious limitations of the window-grey model.
First we investigate the strength of the permanent blocking effect at the TOA
in relation to the surface response. It follows from
Eqs. () and ()
that
∂Ttoa,eq∂αo/∂Tsrf,eq∂αo=-12Tsrf,eqTtoa,eq3βw1-βw.
One can now insert typical temperatures prevailing at the Earth's surface
(∼ 290 K) and at the mesopause (∼ 180 K), and an estimate of
βw≈10–20 % (the actual values depend on the
optical thickness threshold used to derive βw from the
continuous absorption spectra, compare Fig. b–c). This simple
calculation yields response ratios of approximately only (-0.2)-(-0.5),
i.e., a larger temperature change at the surface than in the MA. This result
stands in sharp contrast to ECHAM6 where the surface warms much more than the
MA cools by the permanent blocking effect.
Probably the main reason for this discrepancy is that the effective width of
the atmospheric window is very different for the atmosphere as a whole and
for the atmosphere beyond the tropopause alone. The width of the atmospheric
window is however crucial for the atmospheric temperature profile and the
strength of the MA cooling effects both in absolute and relative terms.
Considering that βw≈90–95 % is more representative
for the largely water-free atmosphere beyond the tropopause (compare
Fig. ), Eq. () yields a response ratio of
(-20)-(-40), which is in much better agreement with the ECHAM6 results.
Regarding the transient component of the MA cooling, the window-grey model
predicts that the transient temperature adjustment decays with height (see
also Appendix C2, Eq. ), but it fails to explain the
virtual absence of a transient MA adjustment in ECHAM6. This might be linked
to another effect neglected in the window-grey model, namely the water vapour
feedback. Higher tropospheric temperatures imply higher water vapour
concentrations, leading to a pronounced temperature dependence of the
atmospheric opacity. The increased opacity as a result of tropospheric
warming entails that a secondary blocking effect may counteract the
slow reduction of the initial MA cooling associated with the transient
component of the blocking effect seen in the window-grey model. We therefore
speculate that the water vapour feedback might play a role to explain the
apparent insensitivity of MA temperatures to the surface temperature by
redirecting changes in upwelling LW radiation to parts of the spectrum that
are transparent in the MA.
Moreover, the height-dependent width of the atmospheric window 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 approximately constant lapse rate in the troposphere
e.g.,. The appearing radiative-convective equilibrium
in the troposphere is associated with an upward heat transport and increased
temperatures in the free troposphere and decreased temperatures at (and close
to) the surface. The redistribution of heat from the surface to the upper
troposphere by convection thus bypasses the lower levels where the
atmospheric window is small. Tropospheric convection is thus an efficient
process to attenuate the surface response to greenhouse forcing, but
convection is neglected in the window-grey model.
It is tempting to apply the window-grey model only to the MA, prescribing the
upward radiative flux in the opaque thermal band (O↑) at the
tropopause as a lower boundary condition. The omission of the troposphere
would have the advantages that convection does not play a significant role
anymore and that the complex influence of the unevenly distributed
atmospheric water (in all its aggregate phases) is strongly diminished.
A way to achieve this for the grey model is to apply Eq. () at the
tropopause (index tp) and to insert it into Eq. .
Transferred to the window-grey model, this yields
T(h)=Otp↑σ(1-βw)(αo(1-h)+1)(αo(1-htp)+2)4.
One could now investigate how changes in Otp↑ or
htp affect the MA, but this would not be very conclusive because
Otp↑ and htp respond in a complex manner to
changes in greenhouse-gas forcing. One could also follow a hybrid approach by
using values derived from a complex model like ECHAM6 for
Otp↑ and htp. However, in particular the
derivation of Otp↑ from a multi-band LW scheme would not
be straightforward. Moreover, the ECHAM6 results show that above a certain
height the temperature profile will not respond to changes in the
troposphere. In other words, Otp↑ and htp
appear to change in such a way that the temperature profile above remains the
same. Overall, applying the window-grey model only to the MA appears not to
add to our explanation of why CO2 cools the MA.
Summary and conclusions
In this article we explain a well-known phenomenon that is central to our
general understanding of climate change – cooling of the middle atmosphere
(MA) by CO2 – in a simple but physically consistent way. We do so by
applying a vertically continuous window-grey radiation model to the
phenomenon. This way it is possible to distinguish two main effects by which
CO2 cools the MA.
First, enhanced blocking of upwelling LW radiation operates towards lower MA
temperatures. In principle, this blocking effect has a transient
component due to the slow warming of the surface. This adjustment leads to
intensified upwelling LW radiation and tends to reduce the initial MA cooling
in the window-grey model. While these effects exactly compensate each other
in a grey atmosphere, leading to an equilibrium TOA temperature that is
independent of the atmospheric opacity, the blocking of upwelling LW
radiation outweighs in the presence of a spectral window because of the
reduced surface temperature sensitivity, leaving lower equilibrium
temperatures above a critical height after the adjustment. Hence, the
blocking effect is permanent because the Earth's atmosphere is not grey,
i.e., uniformly opaque for LW radiation at any wavelength, but absorbs and
emits LW radiation with varying intensity depending on wavelength. The
introduction of a spectral window into an otherwise uniformly opaque
atmosphere is the simplest possible means to capture the effect in a physical
model.
The second permanent effect of CO2-induced MA cooling is the
indirect solar effect. It owes its existence to the fact that there
are heat sources within the atmosphere in addition to LW radiation, most
importantly solar radiation that is absorbed in particular in the vicinity of
the stratopause by ozone. The additional heating term causes a deviation of
the temperature profile from the window-grey solution. The strength of this
deviation depends on the abundance of LW absorbers because the relative
importance of the constant additional heating term in the local energy budget
decreases with increasing LW absorber abundance.
While the window-grey model allows for a fully analytical treatment of
CO2-induced MA cooling, it is not well suited to constrain the relative
effect strengths. Uncertainties are large because the window-grey model
entails a number of gross simplifications, including in particular: the
assumption of vertically well-mixed greenhouse gases (violated in particular
by water vapour); the simplistic LW band structure; and the neglect of
vertical heat transport by convection (and conduction at the surface).
Additional simplifications are the following: the neglect of Wien's law; the two-stream
approximation; the neglect of the horizontal dimensions and the associated
differential heating and atmospheric dynamics (including gravity waves); the
neglect of chemical processes; the implicit treatment of solar radiation; the
neglect of clouds, aerosols, and scattering in general; and the assumption of
local thermodynamic equilibrium that does not hold in the upper mesosphere
and beyond. Most of these factors are discussed for example in
, and those specific to the mesosphere are reviewed in
.
Therefore, to quantify the effect strengths and to complement the insights
gained from the window-grey model, we have conducted simulations with a much
more complex atmospheric model. The results indicate that the two permanent
effects are similarly important, with the indirect solar effect dominating
around the stratopause and the blocking effect dominating away from the
stratopause. The window-grey model also predicts a slow (re-)warming
throughout the atmosphere in response to the slow surface warming. However,
this transient effect is negligible in the MA according to the simulations
with the complex model, pointing to the limitations of the window-grey model.
This article is meant to consolidate our understanding of why CO2 cools
the middle atmosphere by filling a gap between reality and complex
atmospheric models on the one side and somewhat scattered heuristic arguments
on the other. The reconsideration of CO2-induced MA cooling as put forward
here has a distinct educational element, with the potential to convey the
physical essence of the involved mechanisms to a broader audience.
Data availability
The spectroscopic data underlying Fig. 1 are accessible via HITRAN on the Web (http://hitran.iao.ru; see also Appendix D). The ECHAM6
simulation data are stored at the German Climate Computing Center (DKRZ) and
can be obtained via the authors.
An analogy for the blocking effect
While the explanations based on the window-grey model involve mathematical
formalism, the following analogy may facilitate an intuitive understanding
for the permanent and transient components of the blocking effect.
Consider a building that is heated at a constant rate from inside. In steady
state there is a higher temperature inside the building compared to the fixed
exterior temperature. The walls of the building represent an analogy to the
Earth's atmosphere, with the outer surface as the top of the atmosphere and
the inner surface as the atmosphere close to the Earth's surface. The
temperature at the outer wall surface is higher than the exterior temperature
and the temperature at the inner wall surface is somewhat lower than the
interior (room) temperature. These temperature differences maintain an export
of heat at the same rate at which the interior is heated. In the following we
assume that the walls have negligible heat capacity whereas the interior
reacts more inertly to disturbances due to a non-zero heat capacity.
We first assume that the building is insulated equally well everywhere
(corresponding to the grey case), resulting in a uniform temperature of the
outer surface. If now the heat resistance of the walls is instantaneously
increased, at first the outer surface temperature drops and the inner surface
temperature rises, while the interior temperature is still unchanged. In this
situation less heat escapes from the building than is released by the heating
system. The imbalance leads to a slow ascent of the interior temperature that
continues until the outer surface temperature returns to its original value.
The initial cooling of the outer surface temperature is analogous to the
quasi-instantaneous cooling that occurs in the upper half of the atmosphere
in the grey model; the cooling is only transient and has no permanent
component.
Assuming instead that there are parts of the building envelope that are more
weakly insulated than the remainder, as is typically the case with windows,
the outer surface temperature in equilibrium is higher at the windows than it
is at the walls, and a larger fraction of the total energy escapes via the
windows compared to how much they contribute to the total area of the
building envelope. If now the heat resistance of the walls is increased, the
outer surface temperature of the wall is diminished not only temporarily, but
some cooling remains also after the interior temperature has increased to its
new equilibrium value. In the new equilibrium, even more energy escapes
through the windows and less through the walls. The permanent cooling of the
outer surface temperature of the walls is analogous to the cooling in the
higher atmosphere associated with the permanent blocking effect of
CO2-induced MA cooling.
The main difference between the building analogy and the window-grey
radiation model is that the separation between walls and windows in the
former case is in geometrical space, whereas the separation into an opaque
and a transparent radiation band in the latter case is in spectral space.
Another obvious difference is that the mechanism of energy transfer is heat
conduction in the walls of a building as opposed to radiation in the
atmosphere. Nevertheless, we reckon the analogy of an insulated building as a
valid means to illustrate the blocking effect of CO2-induced MA cooling.
Relation to discrete-layer models
Without showing derivations we point out that the vertically continuous
model(s) presented in the main text can be interpreted as a generalization of
discrete-layer models. The simplest type of the latter, a model with only one
grey atmospheric layer, is widely used to explain the greenhouse effect in a
conceptual way e.g.,. In the following we
discuss only the grey case, but the window-grey case can be treated
analogously.
In an n layer grey-atmosphere model with uniform layer emissivity
ϵl, from the radiative balances at every atmospheric layer
it follows that, given an arbitrary surface temperature, the equilibrium
temperature at layer i is
Ti=Tsrfϵl(n-i)+1ϵl(n-1)+24,
where i=1 is the lowest and i=n the highest atmospheric layer. The
overall equilibrium situation is obtained when Tsrf in
Eq. () is replaced by the value it attains in overall
equilibrium, which is
Tsrf,eq=Teff,eq1+nϵl2-ϵl4.
For α/2∈N the vertically continuous grey model is
equivalent to a discrete grey model with n=α/2 atmospheric layers,
each with emissivity ϵl=1. The heights h that
correspond to the discrete levels i are then determined by
hi=i-12n.
Although providing a very suitable conceptual tool to understand the
greenhouse effect, the discrete-layer grey-atmosphere model (just like its
continuous analogue) obviously can not explain greenhouse-gas induced MA
cooling. For such an explanation it is again necessary either to introduce
non-uniform opacity for LW radiation (e.g., by introducing an atmospheric
window), or to introduce an additional (solar) heating term.
Formal response analysis
To supplement the discussion in the main text, in this appendix we quantify
the response of temperature to changes in the parameters of the vertically
continuous window-grey atmosphere 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 the indirect solar effect but
include it into the formalism later.
In the following a response is simply the partial derivative of
temperature with respect to either αo or βw.
Different responses are discerned based on the conditions introduced into the
derivatives. We distinguish between a fast (quasi-instantaneous) response
F where the surface temperature is kept fixed at its previous
equilibrium value, and a subsequent slow response S during which
also the surface attains its new equilibrium temperature. The overall
equilibrium response E can thus be written as
E=F+S.
During the slow transition from F to E, the current
response C(t) at time t deviates from E by the
transient response T(t):
C(t)=E+T(t),
with
T(t)=(f(t)-1)S,
where f(t)∈[0,1] is that fraction of the slow response that has already
taken effect at time t, with f(0)=0 and
f(t→∞)=1. The transient response is thus defined as
the part of the quasi-instantaneous response that is later compensated by the
adjustment to surface warming.
Surface response
Differentiating Eq. () with respect to
αo and βw gives the overall equilibrium
responses Eαo and
Eβw at the surface:
Eαo,srf≡∂Tsrf,eq∂αo=Teff4αo+2αoβw+241αo+2-βwαoβw+2
and
Eβw,srf≡∂Tsrf,eq∂βw=-Teffαo4αo+2(αoβw+2)54.
Excluding the trivial cases αo=0 and βw=1 , Eqs. () and () imply that
Eαo,srf>0 and
Eβw,srf<0. That is, the surface
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 and
Sβw,srf=(f(t)-1)Eβw,srf : the transient response fully
compensates for the equilibrium response initially (where f=0), but
vanishes for t→∞.
Atmospheric response
The temperature response of the continuous window-grey atmosphere in overall
equilibrium to αo and βw as a function of
height is obtained by differentiating Eq. () with
respect to the two model parameters, giving
Eαo(h)≡∂Teq(h)∂αo=Teff,eq4αo(1-h)+1αoβw+24⋅1-hαo(1-h)+1-βwαoβw+2
and
Eβw(h)≡∂Teq(h)∂βw=-Teff,eqαo4αo(1-h)+1(αoβw+2)54.
Differentiating the quasi-instantaneous temperature profile given by
Eq. () in overall equilibrium at h=1 (i.e., at the
TOA) with respect to αo leads to a form that supports the
interpretation of the permanent blocking effect as the interplay between the
sensitivity of the surface temperature to greenhouse-gases on the one hand
and the blocking of upwelling LW radiation by greenhouse gases on the other
hand:
Eαo,toa=1αo+24∂Tsrf,eq∂αo-Tsrf,eq4(αo+2).
Here the surface sensitivity is represented by the minuend in the brackets
whereas the blocking effect is represented by the subtrahend in the brackets.
Differentiating Eq. () with respect to
αo under the constraint Tsrf=const and
inserting Eq. () gives the fast temperature response as a
function of height:
Fαo(h)≡∂T(h)∂αoTsrf=Tsrf,eq=const=Teff,eq4αo(1-h)+1αoβw+241αo+(1-h)-1-1αo+2.
Note that changing the width of the atmospheric window entails no fast
response, i.e., Fβw(h)=0.
The transient part of the response follows from Eqs. () and
() with Eqs. ()–() as
Tαo(h,t)≡(1-f(t))(Fαo(h)-Eαo(h))=(1-f(t))Teff,eq4αo(1-h)+1αoβw+24⋅βwαoβw+2-1αo+2.
The transient part of the response therefore becomes smaller with height.
Comparison with Eq. () further shows that |Tαo(h,t)|<|Eαo,srf| , that is, the transient
cooling at any height in the atmosphere is always weaker than the equilibrium
warming of the surface. Note that Tβw(h,t)
follows directly from Eβw(h) because
Fβw(h)=0.
Inclusion of the indirect solar effect
Considering overall equilibrium, and simplifying the annotation by leaving
away h′, differentiation of Eq. () with respect to the
two model parameters gives
Eαo∗≡∂Teq∗∂αo=TeqTeq∗3Eαo-s∗4Teq∗3αo2(1-βw)
and
Eβw∗≡∂Teq∗∂βo=TeqTeq∗3Eβo+s∗4Teq∗3αo(1-βw)2,
where Eαo and Eβw
are the window-grey overall equilibrium responses given by
Eqs. () and ().
To include the indirect solar effect I into the formalism of
Eqs. ()–(), one can extend Eq. () using
Eqs. (), (), and ()
as follows:
C∗(t)=E+XEI+I︸E∗+T(t)+XTI(t)︸T∗(t)
with
Iαo=-s∗4Teq∗3αo2(1-βw),Iβw=s∗4Teq∗3αo(1-βw)2,XEI=TeqTeq∗3-1E,XTI(t)=TeqTeq∗3-1T(t),
where the terms in Eqs. ()–() follow naturally from
Eqs. () and (). Equation ()
results from Eq. () with the analogues of
Eqs. () and () for the 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∗).
The terms XEI and XTI are
interaction (or synergy) terms that result from the fact that E,
I, and T(t) are not linearly additive. Due to these
terms, the quantitative attribution of a total response to the different
mechanisms is not unambiguously possible.
Supporting information on Fig. 1
The absorption spectra have been computed with HITRAN on the Web
(http://hitran.iao.ru). The atmosphere from surface to space
(Fig. b) was approximated with an 8000 m thick homogeneous gas mixture at 260 K and
1013.25 hPa with the following composition (with respect to volume):
4000 ppm H2O, 300 ppm CO2, 0.4 ppm O3, 0.3 ppm N2O, 1.7 ppm
CH4, 209 000 ppm O2, and the remainder N2. The middle atmosphere
(Fig. c) was approximated with an 8000 m thick homogeneous
gas mixture at 220 K and 202.65 hPa with the same composition except for
H2O (4 ppm) and O3 (2 ppm) (and correspondingly N2). Some Gaussian
smoothing was applied to the spectra.
Acknowledgements
The foundations for this work have been laid during the authors' employment
at the Max Planck Institute for Meteorology, Hamburg, Germany. The ECHAM6
simulations have been conducted at the German Climate Computing
Center (DKRZ). We thank Stephan
Bakan and Thomas Rackow for helpful discussions, and Sebastian Rast for
technical support with ECHAM6. Thanks are also extended to Angus Ferraro, Max
Popp, an anonymous reviewer, and Michel Crucifix for very constructive
comments.The article processing charges for
this open-access publication were covered by a Research
Centre of the Helmholtz Association.
Edited by: M. Crucifix Reviewed
by: M. Popp and one anonymous referee
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