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Research Article

Radiation Transfer Calculations and Assessment of

Global Warming by CO2

Hermann Harde

Experimental Physics and Materials Science, Helmut-Schmidt-University, Holstenhofweg 85, 22043 Hamburg, Germany

Correspondence should be addressed to Hermann Harde; harde@hsu-hh.de

Received 29 June 2016; Revised 3 October 2016; Accepted 1 November 2016; Published 20 March 2017

Academic Editor: Bin Yu

Copyright © Hermann Harde. is is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present detailed line-by-line radiation transfer calculations, which were performed under dierent atmospheric conditions

for the most important greenhouse gases water vapor, carbon dioxide, methane, and ozone. Particularly cloud eects, surface

temperature variations, and humidity changes as well as molecular lineshape eects are investigated to examine their specic

inuence on some basic climatologic parameters like the radiative forcing, the long wave absorptivity, and back-radiation as a

function of an increasing CO2concentration in the atmosphere. ese calculations are used to assess the CO2global warming

by means of an advanced two-layer climate model and to disclose some larger discrepancies in calculating the climate sensitivity.

Including solar and cloud eects as well as all relevant feedback processes our simulations give an equilibrium climate sensitivity of

𝑆= .∘C(temperatureincreaseatdoubledCO

2) and a solar sensitivity of 𝑆= .∘C (at .% increase of the total solar irradiance).

en CO2contributes % and the Sun % to global warming over the last century.

1. Introduction

e Fih Assessment Report (AR) [] of the Intergov-

ernmental Panel on Climate Change (IPCC), a list of all

abbreviations is found in the annex, announces new evidence

of an anthropogenic climate change based on many inde-

pendent scientic analyses from observations of the climate

system, paleoclimate archives, theoretical studies of climate

processes, and simulations using climate models. So, the

IPCC classies the human inuence as extremely likely to be

thedominantcauseoftheobservedwarmingsincethemid-

th century (AR-WG-SPM-D). Increasing emissions of

carbon dioxide (CO2) over the last century especially are

made responsible for this change, and the equilibrium cli-

mate sensitivity (ECS or 𝑆) as a measure for the Earth’s

temperature increase at doubled CO2concentration in the

atmosphere is specied to be likely in the range .∘Cto.

∘C

(high condence) (AR-WG-SPM, p).

Although in all these elds of climate sciences great

progress has been achieved over the last decades and our

knowledge about the Earth-atmosphere system (EASy) could

signicantly be improved, explanations of the observed

global warming over the last century in particular the anth-

ropogenic contributions to this warming are still quite con-

tradictorily discussed.

All the more it is surprising

(i) that many of the consulted analyses and also the AR

itself do not better and clearly distinguish between

an anthropogenic emission of CO2and a naturally

generated part, where the latter even contributes more

than % to the overall emission, and its generation

rate and the respective absorption rate sensitively

respondonglobaltemperaturevariations;

(ii) that the IPCC claims it would be extremely likely

that more than half of the observed increase in global

average surface temperature from to was

caused by the anthropogenic increase in greenhouse

gas concentrations and other anthropogenic forcings,

while contributions from natural forcing and an

internal variability both would only likely be in the

range of −.∘Cto.

∘C;

(iii) that the meanwhile well known delayed response of

CO2and methane (CH4)toseaandairtemperature

changes (see, e.g., Petit et al. []; Monnin et al. [];

Hindawi

International Journal of Atmospheric Sciences

Volume 2017, Article ID 9251034, 30 pages

https://doi.org/10.1155/2017/9251034

International Journal of Atmospheric Sciences

Caillon et al. []; Torn and Harte []; Humlum et

al. []; Salby []) are not considered in AR, and

respective consequences for an anthropogenic global

warming are not discussed;

(iv) that quite uncertain data about cloud feedbacks and

studies of the radiative forcing (RF) of greenhouse

(GH) gases are referred, which are mostly valid for

clear sky conditions, while the introduction of clouds

is usually omitted (AR-WG- Chap...);

(v) that the IPCC denies any noticeable solar inuence

on the actual climate, although strong evidence of an

increasing solar activity over the last century exists

(see, e.g., Hoyt & Schatten []; Willson & Mordvinov

[]; Shapiro et al. []; Ziskin & Shaviv []; Scafetta

& Willson []; Usoskin et al. []; Zhao & Feng [];

Soon et al. []);

(vi) that obviously important eects like convection and

evaporation feedback, which can contribute to sig-

nicant negative feedback (Harde- []), are not

considered in many analyses.

Nevertheless, despite these decits and simplications the

mean equilibrium climate sensitivity is specied with high

condence, and the GH gases are even assigned with very

high condence (%) to be responsible for the actual climate

changes.

Here we will focus on the assessment of one of the most

important quantities in climate sciences and its validation,

the ECS, which has to be scrutinized in more detail. Due to

its far reaching consequences for future climate predictions it

is particularly important to understand and to discover the

large discrepancies between dierent accounting methods

applied for this quantity. Also the weighting of some quite

dierent and even counteracting processes which control

ourclimate,butwhicharenotalwayswellunderstood,

has carefully to be investigated with its implications on the

climatesensitivity.Aquitecriticalreportofactuallypublished

ECS values and accounting methods expanded in AR has

been published by Lewis and Crok [].

In this contribution we will also retrace the main steps

of the IPCC’s preferred accounting system and compare this

with our own advanced two-layer climate model (LCM),

whichisespeciallyappropriatetocalculatetheinuenceof

increasing CO2concentrations on global warming as well

as the impact of solar variations on the climate (Harde-

[]). is model describes the atmosphere and the

ground as two layers acting simultaneously as absorbers and

Planck radiators, and it includes additional heat transfer

between these layers due to convection and evaporation. It

also considers short wave (sw) and long wave (lw) scattering

processes at the atmosphere and at clouds; it includes all

common feedback processes like water vapor, lapse rate,

and albedo feedback but additionally takes into account

temperature dependent sensible and latent heat uxes as

well as temperature induced and solar induced cloud cover

feedback.

e objective of our studies is not to present a new only

“true ECS” but to identify some of the dierent assumptions

and approximations with their far reaching consequences

in climate politics. It is without any doubt that the ECS

is the most important measure for the CO2inuence on

ourclimate,butitisalsoclearthatthisquantitydoesnot

distinguish between anthropogenic and natural CO2emis-

sions. erefore, as long as any natural variations in the CO2

concentrations are not accurately known, the ECS cannot be

used as a reliable indicator only for an anthropogenic global

warming. All this in mind the reader may have his own

reservations about the published data for this measure and

its signicance for a man-made climate change.

For the assessment of the ECS the IPCC favors the

concept of radiative forcing (RF), which is supposed to

be appropriate to describe the transition of the surface-

troposphere system from one equilibrium state to another

in response to an externally imposed perturbation. ere-

fore, in Section we briey outline some basic relations

characterizing this concept, before we present in Section

detailed line-by-line radiation transfer (LBL-RT) calculations

for the lw up- and downwelling uxes in the atmosphere (for

details see Harde [, , ]), this for clear sky, global mean

cloud cover, full cloudiness, dierent surface temperatures,

humidity, and even dierent lineshapes. ese calculations

were performed for the most important GH gases water vapor

(WV), carbon dioxide, methane, and ozone and are based

ontheHITRANdatabase[].Sincetheconcentration

of the GH gases and the atmospheric pressure are changing

with temperature and altitude, the atmosphere has to be

segmented into up to sublayers from ground to km

height and in some cases additionally into three climate

zones. When these computations are repeated for dierent

CO2concentrations at otherwise same conditions, from the

changing uxes on the one hand the CO2initiated RF as

the main parameter for the IPCC accounting scheme and on

the other hand the sw and lw absorptivities as well as the

back-radiated fraction of the atmospheric emission as the key

parameters in our LCM can be derived.

Section summarizes the main features of the LCM and

its calibration to satellite data, for the radiation and heat

uxes using the well known radiation and energy budget

scheme of Trenberth et al. [], for temperature changes due

to cloud cover variations applying the observations within the

International Satellite Cloud Climatology Project (ISCCP)

[]. With the respective key parameters of Section inte-

grated in the climate model we simulate the Earth’s surface

temperature and the lower tropospheric temperature as a

function of the CO2concentration. e temperature increase

at doubled CO2concentration then directly gives the CO2

climate sensitivity. Such simulations reproduce the direct or

basic ECS value (without feedback processes), as specied

by the IPCC, within better than %. Signicant dierences,

however, can be observed with the dierent feedback eects

included. Our investigations show the largest discrepancies

for the WV and cloud feedback, but they also disclose the

importance of one of the primary stabilizers of the whole

climate system, the evaporation feedback. erefore, these

processes and their contributions are extensively analyzed

under dierent humidity, surface temperature, lapse rate, and

cloud cover conditions.

International Journal of Atmospheric Sciences

In particular, these studies show that the observed cloud

changes within the ISCCP cannot exclusively be explained by

pure thermally induced cloud cover changes but obviously are

additionally controlled by a further cloud forcing mechanism.

Since there exists strong evidence that the solar activity

alsohasapowerfulinuenceonthecloudcover,itis

reasonable to postulate such a solar induced cloud feedback

(see, e.g., Svensmark []; Haigh []). is is investigated

in detail in Section , dierentiating between a pure thermal

impact of an increasing solar activity and a nonthermally

inducedsolarcloudfeedback.Animportantcriterionfor

a validation, which mechanism might control such cloud

changes, can be derived from model simulations, which

includethesolaranomalyoverthelastcenturyandcompare

this directly with the observed global warming over this

period. ese investigations indicate that due to the strong

cloud feedback the observed warming over the last century

can only satisfactorily be explained, attributing a signicant

fractiontotheincreasedsolaractivityoverthisperiod(see

also Ziskin & Shaviv []; Vahrenholt & L¨

uning []).

Our simulations predict a solar contribution of about %

and a CO2induced contribution of % to global warming

over the last century with an equilibrium climate sensitivity of

.∘C, which is almost a factor of ve smaller than published

in AR.

2. Radiative Forcing

e concept of RF is well established in climate sciences and

used to assess the global warming as a result of an external

perturbation of EASy (AR-WG-Chap.). is perturbation

can result from solar anomalies, from increased GH gas

concentrations or volcanic activities. In all cases a direct

proportionality of the Earth’s temperature increase 𝐸as

response to an external forcing isassumedintheform

𝐸=𝑆⋅, ()

with 𝑆as the climate sensitivity parameter, also known as

Planck sensitivity. e negative reciprocal of 𝑆is called the

Planck feedback (see AR-WG-Chap.), although this is not

afeedbackprocessinthecommonsenselikethewatervapor,

the albedo, or lapse rate feedback, which all describe an

internal amplication or attenuation of an external pertur-

bation. e Planck feedback rather represents the change of

the outgoing long wave radiation (OLR), or more precisely

the change of the total upwelling lw intensity up

total with the

ground temperature. Since an increasing amount of OLR with

the temperature causes a negative feedback for EASy, it can be

dened as the negative partial derivative as follows:

−OLR

𝐸=−up

total

𝐸=1

𝑆.(a)

Its value, deduced as an average from dierent climate

models, is given in AR-WG-Tab.. as −. W/m2/∘C, and,

thus, the Planck sensitivity becomes 𝑆=.W

−1 m2∘

C. A

somewhat simplied method to assess the Planck feedback

and sensitivity, which agrees within % with these values, is

to use the slope of the Stefan-Boltzmann law at an eective

Earth temperature e = K and emissivity e =1with

−up

total

𝐴

𝑇e =255 K=−4

e3

e =−4up

total

e ⇒ 1

𝑆(b)

and to assume the same temperature response for the surface

as for the atmosphere. is results in a Planck feedback

of −. W/m2/∘C and a climate sensitivity parameter 𝑆

=.W

−1 m2∘

C. is the Stefan-Boltzmann constant with

.⋅−8 W/m2/∘C4and up

total the OLR with . W/m2.

Eq. () is the basis for many climate models (see AR

[] and AR []) and even for more complex mod-

els like the Atmosphere-Ocean-General-Circulation Models

(AOGCMs) this fundamental relation is used to express

theinuenceofthegrowingGHgasconcentrationsonthe

surfacetemperature.So,forthesemodelsandalsoforthe

observationally based energy budget ECS estimates (see,

e.g., Forster & Gregory []; Lindzen & Choi [])

is a quite fundamental quantity, which cannot directly be

measured but has to be deduced from extensive spectroscopic

calculations or solar data, which on their part rely on many

individual observations and measurements. For quite actual

investigations of see also Feldman et al. [].

In this context it should be noticed that alternative

denitionsofRFhavebeendeveloped,eachwithitsown

advantages and limitations (see AR-WG-Chap.). Here

we only consider the instantaneous RF, which refers to an

instantaneous change in the net (down minus up) radiative

ux (sw plus lw) due to an imposed change. is forcing is

usually dened in terms of ux changes at the top of the

atmosphere (TOA) or at the tropopause.

In this contribution, particularly the inuence of CO2

on global warming is of interest. erefore, in the next

subsection we present some actual radiation transfer (RT)

calculations,fromwhichtheinstantaneousRFduetoincreas-

ing CO2in the atmosphere and also some related quantities,

which are of relevance for our two-layer climate model,

can be derived. Since for these model calculations it is not

sucient only to consider the net radiative uxes at the

tropopause, neglecting the downwelling absorption changes

in the troposphere and the upwelling absorption changes over

the stratosphere, we apply the RT concept from the surface to

TOA and vice versa. e sw absorption changes over the full

atmosphereandthisasafunctionoftheCO

2concentration

can be captured from our previous investigations (Harde-

[]).

As already outlined, an important reference for the

inuence of CO2is the temperature increase at doubled CO2

concentration under steady state conditions, which is known

astheequilibriumclimatesensitivityECSor𝑆,andthe

respective lw forcing may be designated as 2×CO2.Sinceat

TOA a lw downwelling ux is zero, the forcing at TOA can be

dened as dierence of the total outgoing intensities up

total at

single and doubled CO2concentration :

2×CO2=up

total ()−up

total (2),()

International Journal of Atmospheric Sciences

where here we primarily consider the lw radiative forcing.

up

total can be explained to consist of the nonabsorbed terrestrial

intensity (𝐸−

abs) plus the emitted intensity of the atmo-

sphere in upward direction up

𝐴; thus, it holds the following:

2×CO2=

𝐸−

abs ()+up

𝐴()−

𝐸+

abs (2)

−up

𝐴(2)

=up

𝐴−

abs

()

with

up

𝐴=up

𝐴()−up

𝐴(2),

abs =

abs ()−

abs (2).()

𝐸is the intensity of the incident terrestrial radiation and abs

theabsorbeduxintheatmosphere.

So, due to () this forcing generally consists of two con-

tributions, the change in the upward atmospheric emission

and the change in the atmospheric absorption of terrestrial

radiation, both as response to a doubling of the CO2concen-

tration.

Expressing the downwelling atmospheric intensity down

𝐴

via the total emitted intensity total

𝐴and the fraction 𝐴,which

is directed downward (also called asymmetry factor of the

emitted atmospheric radiation) and which is generally found

from integrating the respective spectral intensities up,down

𝐴,̃

]

at TOA and the surface over the frequency or reciprocal

wavelength (wavenumber)

]with

𝐴=down

𝐴

total

𝐴=∫∞

0down

𝐴,̃

]

]

∫∞

0down

𝐴,̃

]

]+∫∞

0up

𝐴,̃

]

],()

the dierence down

𝐴between single and doubled concentra-

tion gives

down

𝐴=down

𝐴()−down

𝐴(2)

=total

𝐴()⋅

𝐴()−total

𝐴(2)⋅

𝐴(2)

=total

𝐴()⋅

𝐴()−total

𝐴()−total

𝐴

⋅

𝐴()−

𝐴

=total

𝐴()⋅

𝐴+total

𝐴⋅

𝐴()−total

𝐴

⋅

𝐴,

()

with total

𝐴and 𝐴as the respective dierences of total

𝐴and

𝐴at single and doubled CO2concentration.

en, with (), for the dierence of the upwelling atmo-

spheric intensities we can write

up

𝐴=total

𝐴()⋅1−

𝐴()−total

𝐴(2)

⋅1−

𝐴(2)

=total

𝐴()−total

𝐴(2)−total

𝐴()⋅

𝐴()

+total

𝐴(2)⋅

𝐴(2)

=total

𝐴−total

𝐴()⋅

𝐴−total

𝐴⋅

𝐴()

+total

𝐴⋅

𝐴

=total

𝐴⋅1−

𝐴()+

𝐴−total

𝐴()⋅

𝐴,

()

andtheradiativeforcingatdoubledCO

2concentration can

nally be expressed as follows:

2×CO2=total

𝐴⋅1−

𝐴()+

𝐴−total

𝐴()

⋅

𝐴−

abs.()

is CO2forcing can directly be derived from radiation

transfer calculations (Schwarzschild [, ]; Goody and

Yung []; Salby- []; Harde- []; Harde- []),

by which the up- and downwelling uxes as well as the

absorption and emission in the atmosphere are computed.

3. Radiation Transfer Calculations

Since it is obvious that the cloud cover has a strong inuence

ontheup-anddownwellinguxesintheatmosphereandalso

on the strength of the GH eect, we have performed line-by-

line radiation transfer (LBL-RT) calculations under dierent

cloudiness conditions, ground temperatures, and humidity to

evaluatetheinuenceofCO

2on global warming. We also

briey investigate the inuence of lineshape eects on RT

calculations and their consequences for the ECS.

Here we only present global RT calculation with averaged

values for the temperature, water vapor concentration, and

an average lapse rate, since separate computations for the

tropics mid- and high-latitudes with individual proles and

averaging over the climate zones with an area weighting factor

gave almost the same results (see also Harde []).

3.1. Clear Sky at Global Mean Temperature. Ta b l e s h o w s t h e

results of our LBL calculations under clear sky conditions

as a function of the CO2concentration, using a ground

temperature of 𝐸=.K=

∘Candaterrestrialintensity

of 𝐸=W/m

2.esevaluesareconsistentwiththoseused

by Trenberth et al. [].

Asanaverageoverthethreeclimatezonesthewatervapor

concentration at ground was assumed to be ,.ppm

and decreasing with altitude due to the Clausius-Clapeyron

equation (for details see Harde- []). ese data have

been derived from satellite measurements (Vey []) and

are almost a factor of two larger than those given by the

International Journal of Atmospheric Sciences

T : RT-calculations for dierent CO2concentrations at clear sky (𝐸=.W/m

2,𝐸= . K, water vapor at ground: .ppm,

CH4:.ppm,andO

3varying over the stratosphere with a maximum concentration of ppm).

CO2

ppm up

total

W/m2up

𝐴

W/m2down

𝐴

W/m2total

𝐴

W/m2abs

W/m2aLW

%𝐴

%

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

– . . −4.46 −4.41 −5.00 −1.27 −0.35

up

𝐴−

abs .

US Standard Atmosphere []. But they are in quite good

agreement with the Average Global Atmosphere (Ollila []),

which even species values % larger than ours. e CH4

concentration was set to . ppm and O3to vary over the

stratosphere with a maximum concentration of ppm at

kmaltitude.elapserateandpressurevariationwith

altitude agree with the US Standard Atmosphere.

Our calculations cover a spectral interval from to

cm−1, corresponding to .% of a Planck radiator at

𝐸, and they include more than , lines with spectral

intensities larger −24 cm−1/(molecules⋅cm−2). e spectral

resolution for calculations at CO2concentrations of

and ppm, which are used as reference data, was chosen

as . cm−1 (. GHz), and for the vertical direction

sublayers were distinguished, whereas the calculations at the

other concentrations were performed with half the spectral

and vertical resolution.

Comparison of the two last values in column of Table

shows that within rounding errors () is well conrmed.

Looking at the upwelling intensity up

𝐴oftheatmosphere(col-

umn ) we see that this is almost identical at single and dou-

bled CO2concentration, while the downwelling part down

𝐴

(column)andthetotalatmosphericradiationtotal

𝐴(column

) both are increasing by more than . W/m2.ismeans

that at clear sky conditions most of the additionally absorbed

terrestrial radiation at doubled CO2(see column ) is instan-

taneously directed back to the surface, and only a relatively

small contribution of . W/m2istrappedintheatmosphere.

Sinceinthiscasethersttwotermsin()almostexactly

compensate each other, the forcing under clear sky conditions

is in good approximation only determined by the absorption

dierence. us, from the simple radiative forcing model ()

it follows that also the ground temperature variations are

almost only determined by this additionally absorbed power.

In general, however, it should be clear that the surface

temperature also depends on the emission characteristic of

the atmosphere, which changes with the GH gas concentra-

f

Aclear sky

log t f

A

aLW clear sky

log t aLW

84

82

80

78

76

74

72

64

63

62

61

60

59

58

CO2-concentration (ppm)

f

A(%)

8007006005004003002001000

aLW (%)

F : Lw absorptivity aLW (black diamonds) and downwelling

fraction 𝐴of emitted atmospheric radiation (blue triangles) at clear

sky with respective logarithmic approximations.

tion as well as with the cloud cover and is determined by the

rst two terms in (). Only under clear sky they just feed most

of the additionally absorbed power back to the surface.

Inamoreadvancedmodel,asthiswillbediscussedlater

with a radiation and energy balance at TOA and at the surface,

including heat uxes of sensible and latent heat or from

neighbouringclimatezones,aswellasswandlwabsorption

losses, even at clear sky conditions the atmospheric emission

characteristic is of relevance, which can well be represented

by the fraction 𝐴ofthedownwardemittedradiation.

e relative absorption of terrestrial radiation by the GH

gases and its variation with increasing CO2concentration are

listed in column . It is normalized to the incident terrestrial

ux and in this way represents the lw absorptivity aLW as

dened in Harde- [] (4).isabsorptivityisrepre-

sented in Figure (black diamonds) together with the back-

radiated fraction 𝐴of the atmosphere (blue triangles). At

higher concentrations both quantities can well be represented

International Journal of Atmospheric Sciences

T : RT-calculations for CO2concentrations of and ppm at a surface temperature of .∘C (. K) and clear sky (𝐸=

W/m2, water vapor at ground: ,. ppm, CH4of . ppm and O3varying over the stratosphere with a maximum concentration of

ppm).

CO2

ppm up

total

W/m2up

𝐴

W/m2down

𝐴

W/m2total

𝐴

W/m2abs

W/m2aLW

%𝐴

%

. . . . . . .

. . . . . . .

. . −4.32 −3.64 −4.78 −1.14 −0.38

by logarithmic graphs (magenta line for aLW ,greenlinefor

𝐴), indicating that due to saturation eects then only the far

wings will further contribute to an increasing absorption or

emission.

3.2. RT Calculations at Clear Sky and Increased Surface Tem-

perature. Whereastheabovecalculationsreectasomewhat

articial situation, assuming clear sky and a global mean

temperature of ∘C, we know that with declining cloud cover

the average surface temperature is signicantly climbing up.

So, from observations within the International Satellite Cloud

Climatology Project (ISCCP) [] it is found that around

the global mean cloud cover of % a reducing cloudiness

of % causes a temperature increase of about .–.∘C, a

tendencywhichisalsoconrmedbydataofHartmann[].

erefore, to get a better understanding of how such

higher temperature modies the respective uxes in the

atmosphere and especially this aects the CO2radiative

forcing, we have performed additional calculations for a

surface temperature of .∘C. At this temperature and a

surface emissivity of % the terrestrial radiation climbs

up to W/m2. But with the higher temperature also the

humidity increases and therefore aects the absorptivity of

the GH gases. From the measured water vapor concentrations

as well as the temperatures at mid-latitudes and the tropics we

deduce at the higher temperature a growth of the vapor con-

centration of , ppm with a new total mean concentration

near the surface of , ppm.

Calculations for this higher ground temperature and

water vapor concentration are shown in Table .

Since for an evaluation of CO2with its inuence on the

radiation budget it is sucient to concentrate on the uxes at

single and doubled CO2concentration, here we only present

this reduced data set. Compared to the lower temperature

calculations the radiative forcing increases by . W/m2,

which is mainly due to changes in the upward atmospheric

emission with up

𝐴=.W/m

2, while the absorption changes

with abs =−. W/m2are even getting slightly smaller. Also

thedownwarddirecteduxchangewithdown

𝐴=−. W/m2

and the total atmospheric emission change with total

𝐴=

−. W/m2are reducing, although the absolute uxes and

the absorption are considerably larger, caused by the higher

temperature and water vapor.

e absorptivity as calculated under these modied

conditions is shown in Figure (black diamonds) together

with the back-radiated fraction 𝐴of the atmosphere (blue

triangles). At higher CO2concentrations both graphs can

f

Aclear sky

log t f

A

aLW clear sky

log t aLW

84

86

82

80

78

76

74

72

64

63

62

61

60

59

58

CO2-concentration (ppm)

f

A(%)

8007006005004003002001000

aLW (%)

F : Lw absorptivity aLW (black diamonds) and downwelling

fraction 𝐴of atmospheric radiation at 𝐸= .∘Candclearsky

with respective logarithmic approximations.

again well be approximated by logarithmic functions, similar

to Figure (magenta line for aLW , green line for 𝐴), only on

slightly shied scales.

3.3. LBL-RT Calculations at Global Mean Cloud Cover. Under

regular cloud cover signicant changes from clear sky condi-

tionshavetobeexpectedfortheup-anddownwellinguxes

due to the strong shielding eect of clouds not only for sw but

also for lw radiation. In our radiation transfer calculations

the clouds are considered as a single thinner layer, which

absorbs the incident lw radiation from up or down direction

with a cloud absorptivity aLC and which has a transmissivity

of 1−aLC, while lw scattering processes at the clouds are

here neglected. For the RT calculation only the resulting

attenuation of radiation, which passes the layer, is of rele-

vance, and for these calculations it is sucient to express the

total attenuation uniformly only by one parameter, the cloud

absorptivity. In addition to the transmitted radiation the

clouds are also emitting as a Planckian radiator with an emis-

sivity 𝜀LC and a temperature determined by the cloud altitude.

e Global Energy and Water Cycle Experiment

(GEWEX) Radiation Panel, which compares the available

global long-term cloud data products with the ISCCP

and consists of satellite measurement teams, species

the global total cloud amount between % and %

(Stubenrauch et al. []), where this relatively wide range

is mainly explained by dierent instrument sensitivity and

by retrieval methodology. For our further calculations we

assume a global mean cloud cover of 𝐶= % in agreement

International Journal of Atmospheric Sciences

T : RT-calculations for dierent CO2concentrations at mean cloud cover of %, a cloud altitude of . km, and cloud emissivity of %

(𝐸=W/m

2,𝐸= . K = .∘C, water vapor at ground of ,. ppm, CH4of . ppm, and O3varying over the stratosphere with a

maximum concentration of ppm).

CO2

ppm up

total

W/m2up

𝐴

W/m2down

𝐴

W/m2total

𝐴

W/m2abs

W/m2aLW

%𝐴

%

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . −3.45 −2.61 −2.93 −0.74 −0.35

up

𝐴−abs .

with the ISCCP []. e cloud altitude with 𝐶=.km

and the emissivity with 𝜀LC =%werechosentoalmost

exactly reproduce the widely accepted radiation and energy

budget scheme of Trenberth et al. [] (hereaer called

TFK-scheme) with a total upward ux of up

total = 239.2W/m2

and a nonabsorbed outgoing radiation from the surface with

up

total −up

𝐴=40W/m2.

Our RT calculations under otherwise identical conditions

as described in Section . and at a surface temperature 𝐸=

∘C are listed in Table . ey show signicant changes to

the radiation uxes under clear sky, for example, for down

𝐴,

more than W/m2or for total

𝐴even W/m2.Alsothe

total absorption, now additionally increased by the clouds, is

W/m2larger than at clear sky.

But respective dierences between single and doubled

CO2concentrations, which are here of main interest, in

general are getting smaller. So, the radiative forcing reduces

to 2×CO2= . W/m2andisalmostidenticalwiththe

gure used by the IPCC (Myhre et al. []), although here we

consider only the instantaneous lw RF at TOA. is forcing

istheresultoftheupwellingatmosphericcontributionof

up

𝐴= 0.84W/m2andontheotherhandareduced

absorption dierence of abs = −2.93W/m2compared to

−W/m

2at clear sky. Also changes in the back-radiation

diminish from −. W/m2to down

𝐴= −3.5W/m2and the

total emission even from −. W/m2to total

𝐴=−2.6W/m2.

Because of this smaller alteration in the atmospheric

emission the st term in () with total

𝐴⋅(1−

𝐴()+𝐴)

now reduces to −. W/m2,whereasthendtermwith

−total

𝐴()⋅𝐴=.W/m

2dominates. From the total forcing

of . W/m2in this case . W/m2goes back to the surface,

while . W/m2remains in the atmosphere.

To deduce the lw absorptivity at cloudiness from the radi-

ation transfer calculations, the absorbed intensity listed in

f

A66% clouds

log t f

A

aLW 66% clouds

log t aLW

83

81

79

77

75

63

62

61

60

59

CO2-concentration (ppm)

f

A(%)

8007006005004003002001000

aLW (%)

F : Lw absorptivity aLW (black diamonds) and downwelling

fraction 𝐴of emitted atmospheric radiation at % cloud cover

with respective logarithmic approximations.

column of Table , which now reects the total absorption

caused by the GH gases and the clouds, has to be reduced

by the cloud fraction. Such correction seems appropriate for

a direct comparison of the absorptivity changes caused by

the GH gases at clear sky and clouds, and it is necessary for

our two-layer climate model, where the cloud absorption is

already integrated in the model and represented by respective

parameters. Such a correction is done by comparing the

absorption, for example, at ppm with and without clouds

at the same ground temperature, and reducing column by

this dierence. Normalizing the corrected absorption to the

incident radiation then gives in good approximation the lw

absorptivity aLW oftheGHgasesinthepresenceofclouds.

FigureshowstherespectiveabsorptivityaLW (black

diamonds) and the back-radiated fraction 𝐴(blue triangles)

at a cloud cover of %. Again both quantities can well be

International Journal of Atmospheric Sciences

T : RT-calculations with MRT lineshape for CO2concentrations of and ppm at mean cloud cover of %, a cloud altitude of

. km, and cloud emissivity of .% (𝐸=W/m

2,𝐸= . K = .∘C, water vapor at ground of ,. ppm, CH4of . ppm, and O3

varying over the stratosphere with a maximum concentration of ppm).

CO2

ppm up

total

W/m2up

𝐴

W/m2down

𝐴

W/m2total

𝐴

W/m2abs

W/m2aLW

%𝐴

%

. . . . . . .

. . . . . . .

. . −3.18 −2.20 −2.82 −0.71 −0.34

represented by logarithmic graphs (magenta line for aLW ,

greenlinefor𝐴). While 𝐴slightly increases by .%,

𝐴remains constant. However, the absorptivity changes

almost decline by a factor of two to −.% at doubled CO2

concentration.

Simulations at higher cloud altitudes under otherwise

same constrains give a reduced upwelling ux up

total and a

further diminishing radiative forcing 2×CO2.

3.4. RT Calculations with MRT Lineshape. While the pre-

cedingcalculationsarebasedonthestandardmolecular

collision theory, considering collisional broadening of spec-

tral transitions, which are characterized by a Lorentzian

lineshapeorathigheraltitudesalsobyaVoigtprole,we

have also performed extensive calculations using a more

sophisticatedlineshapeasgivenbythemolecularresponse

theory (MRT) (Harde et al. [–]). is theory represents

a generalization and unication of the classical collision

theories of Lorentz and on the other hand of van Vleck and

Weisskopf, considering thermalization of molecules during a

collision, which in its limit is determined by the reciprocal of

the molecular transition frequency and controlled by Heisen-

berg’s uncertainty principle (Harde & Grischkowsky []).

e specialty applying MRT is that it not only describes the

near resonance absorption and emission behavior as already

adequately characterized by a Lorentzian, but also reects the

far wing response, which is mainly caused by the ultrafast

time response of molecules to an external electric eld during

collisions. So, the MRT lineshapes already represent very

well the continuum water vapor absorption extending up

to cm−1 and caused by the low frequency water vapor

absorption band around cm−1 (see Burch & Gryvnak

[]; Roberts et al. []).

is continuum water vapor background together with

the far wing contributions of the other molecules signicantly

modies the absolute up- and downwelling uxes. erefore,

tofurthersatisfytheradiationandenergybalanceoftheTFK-

scheme, in the radiation transfer calculations the cloud alti-

tude has to be increased to km and the cloud absorptivity =

emissivity reduced to .%.

e results of such a calculation at a mean cloud cover

of % and at otherwise same conditions as described in

Section . is shown in Table . To compare the alterations

caused by doubling the CO2concentration it is again su-

cient only to display the calculations for and ppm

CO2. It is interesting to see that the more sophisticated

lineshape and the continuum background absorption almost

have no inuence on 2×CO2, the RF at doubled CO2,which

agrees within better than % with the simpler calculations

shown in Table , neglecting the continuum absorption and

using a Lorentzian lineshape. Also changes in the other

uxes at doubled CO2concentration are consistent within

. W/m2; only the back-radiation changes to down

𝐴=

−3.18W/m2and is about . W/m2smaller; thus, the power

le in the atmosphere increases by this amount.

For this insensitivity to a dierent collision and line-

shapetheoryweseethreemainreasons:rst,becauseof

the larger absolute ux variations, caused by the dierent

wing absorption and WV continuum, the outgoing uxes

were recalibrated to the TFK-scheme via cloud altitude

andabsorptivitytoensurecomparableabsoluteuxesin

agreement with the observations. Second, we only consider

ux dierences at single and doubled CO2concentration for

the same shape, so that discrepancies to another shape are

not directly observed and smaller absolute variations with the

CO2concentration are to some extent cancel out. ird, the

interference of the CO2spectrum with strong water vapor

lines and the underlying WV continuum further attenuates

any CO2lineshape eects.

So, calculations based on the classical collision theory

obviously reproduce quite reliable data of any radiation

changes in the atmosphere. Since the far wings of the CO2

lines are found to decay more rapidly than a Lorentzian and

thus should contribute less to the total absorption (see, e.g.,

Edwards and Strow []), such calculations even simulate

slightly worse conditions and result in a more conservative

assessment of global warming by CO2.ForactualCO

2line-

shape studies see also Happer [], and for RT calculations

including CO2line mixing see Mlynczak et al. [], and Ozak

et al. [].

3.5. RT Calculations for Total Cloud Cover. With a further

increasing cloud cover and thus a stronger shielding of the

sw radiation also the surface temperature further drops, and

owing to the ISCCP observations at % cloudiness then

a reduced temperature of approximately .∘Ccomparedto

mean cloudiness is expected.

Respective radiation transfer calculations at % cloud

cover, a surface temperature of .∘Cwithaterrestrial

intensity of W/m2, and a further reduced water vapor

concentration at ground of , ppm, otherwise using the

same conditions as in Section ., are shown in Table .

e total outgoing radiation up

total drops by . W/m2and

the total atmospheric emission total

𝐴by . W/m2compared

International Journal of Atmospheric Sciences

T : RT-calculations for dierent CO2concentrations at % cloudiness, a cloud altitude of .km, and cloud emissivity of %

(𝐸=W/m

2,𝐸= . K = .∘C, water vapor: ,. ppm, CH4:.ppm,andO

3varying over the stratosphere with a maximum

concentration of ppm).

CO2

ppm up

total

W/m2up

𝐴

W/m2down

𝐴

W/m2total

𝐴

W/m2abs

W/m2aLW

%𝐴

%

. . . . . . .

. . . . . . .

. . −2.93 −1.90 −1.93 −0.50 −0.34

f

A100% clouds

log t f

A

aLW 100% clouds

log t aLW

82

81

80

79

77

78

63

64

62

61

60

59

CO2-concentration (ppm)

f

A(%)

8007006005004003002001000

aLW (%)

F : Lw absorptivity aLW (black diamonds) and downwelling

fraction 𝐴(blue triangles) of emitted atmospheric radiation at

% cloudiness and a ground temperature of .∘Cwithrespective

logarithmic approximations.

to a cloud cover of %. Nevertheless, the forcing due to

doubled CO2reduces to 2×CO2= . W/m2,consistingof

a slightly increasing contribution up

𝐴= 1.0W/m2and a

decreasing absorption abs = −1.9 W/m2.Alsotheabsolute

downwelling radiation down

𝐴further decreases by . W/m2,

and its alteration at doubled CO2concentration reduces to

−. W/m2, so almost nothing remains in the atmosphere.

e absorptivity aLW as listed in column is again

found by comparing the absorption at ppm with clouds

(. W/m2) and without clouds (. W/m2), reducing

column by this dierence and then normalizing the cor-

rected absorption to the incident radiation. aLW is shown

together with the parameter 𝐴in Figure . Both graphs can

againwellberepresentedbylogarithmicplots.

e changes in absorptivity from single to doubled

CO2with aLW =−.% further decline compared to %

cloudiness or clear sky, and also 𝐴is slightly reducing to

−.%.

So,fromthesecalculationsitisclearthattheup-and

downwelling uxes and also the absorption in the atmosphere

are signicantly varying with the cloud clover and, therefore,

they will also signicantly inuence the global mean surface

temperature. But it can also well be recognized, that changes

with the CO2concentration by trend get smaller with increas-

ing cloudiness.

4. Assessment of CO2Global Warming

Dierent to general circulation models, which try to predict

local climate variations over some time period and, there-

fore, have to solve complex coupled nonlinear dierential

equations with a large number of parameters, making these

calculations extremely time