Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology

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12.1 Cloud forcing 445

We now wish to interpret the net sensitivity coefﬁcient. Holding all other param-

eters constant, δ is the change in net energy absorbed by the climate system per

change in cloud cover fraction. From the above crude estimates of the sensitivity

coefﬁcients we conclude that of the two cloud effects, the effect on solar radia-

tion dominates the effect on long-wave radiation emitted to space when looking

on the global scale. Moreover, the sensitivity can be determined for any spatial

and temporal scale by using instantaneous satellite observations. However, it will

have signiﬁcance only on scales large enough to suppress the ‘noise’ of day-to-day

weather variations.

Inspection shows that (12.5) does not require any explicit cloud fraction infor-

mation. Since A

> A

, the sign of δ is determined by the ratio of the sensitivity

coefﬁcients appearing in the last factor. If this ratio is greater than 1, net cloud

warming is expected. If it is smaller than 1 net cloud cooling should result. Various

authors have investigated the effect of clouds on the climate system with the help

of satellite data. They concluded that clouds, on the whole, have a very substantial

cooling effect on the climate system. For further details we refer to Arking (1991)

where an extensive bibliography on the subject can be found.

Let us now consider the difference of ﬂux densities as stated by equations

f,l

= E

l,t

(C = 0) − E

l,t

(C), C

f,s

= E

s,t

(C = 0) (12.6)

The quantities E

l,t

(C = 0) and E

s,t

(C = 0) refer to cloudless conditions. These

differences, now commonly called cloud forcing, represent the effect of clouds on

the radiative ﬂux densities. The total cloud forcing effect C

is the sum of the cloud

forcing components for solar and long wave radiation, that is

= C

f,l

+ C

f,s

(12.7)

It is a simple exercise to show that cloud forcing and cloud sensitivity coefﬁcients

are related by

∂ E

l,t

∂C

=−

f,l

∂ E

s,t

∂C

f,s

,δ=

(12.8)

Various investigators have attempted to determine the cloud forcing C

. Ellis

(1978), for example, estimated that the mean annual, quasi-global (65

◦

S−65

◦

effect of clouds is a cooling of the climate system resulting in C

=−20 W m

−2

. The

long-wave effect is C

f,l

=22Wm

−2

and the short-wave C

f,s

=−42Wm

−2

. From

the ERBE satellite experiments Ramanathan et al. (1989) found C

=−17 W m

−2

which is not too different from the value given by Ellis. However, the individual parts

of cloud forcing differ substantially with C

f,l

=31 W m

−2

and C

f,s

=−48Wm

−2

From climatological data Ellis extracted and computed the individual sensitivity

factors and the ratio of sensitivity factors that are independent of cloud amount.

446 Inﬂuence of clouds on the climate of the Earth

The ratio of the sensitivity factors that can be obtained from the above empirical

formulas and the data provided by Ellis and Ramanathan are not too different.

However, the ratio proposed by Ohring et al. (1981) differs substantially from the

values presented by the other authors which may be partly due to the different

methods used. It is difﬁcult to decide whose value of the sensitivity coefﬁcient best

approximates reality.

12.2 Cloud feedback in climate models

As we know already, the global climate equilibrium is reached if E =0 whereby

E = E

l,t

−

(1 − A) (12.9)

If E < 0 then the system Earth–atmosphere will gain energy and heating will take

place. Whenever E differs from zero, primary forcing of the system occurs. This

may be due to changes of the solar constant, the global albedo, the upward long-

wave radiation or by a combination of these. According to the Milankovitch theory

a change of S

may have occurred over a period of 10

–10

years due to a change of

the Earth’s orbit. The variation of the solar constant may be responsible for changes

of the past climate. Here we will not dwell on this subject. Present measurements

show that the solar constant may vary by 0.1% due to changes between a minimum

and a maximum of the Sun’s activity. Variations of this order of magnitude have been

included in climate models to force climate changes. Global albedo changes may be

due to natural and anthropogenic causes such as the outburst of volcanoes, biomass

burning, agricultural and industrial activities. The observed increase of CO

and

other trace gases as well as changes in the aerosol content of the air deﬁnitely have

a non-negligible inﬂuence on the radiation budget of the atmosphere.

It stands to reason that changes induced by primary forcing modiﬁes the radiative

characteristics of the atmosphere and the Earth’s surface thus causing an additional

change of the net ﬂux density at the top of the atmosphere. This is known as the

feedback process. If the feedback increases the primary forcing of the net ﬂux

density at the top of the atmosphere, it is called a positive feedback. In contrast,

the feedback is called negative if it tends to reduce the primary net ﬂux density

change. A detailed study of the feedback process is very complicated partly due to

the fact that many open questions in connection with climate modeling still have

to be resolved.

However, it is possible to give a very simple linear analysis of the feedback

mechanisms which will help us to visualize the overall problem. In the subsequent

discussion we will follow the arguments given by Arking (1991) and Lenoble

(1993). They assume that without feedback the only response to a variation of the

12.2 Cloud feedback in climate models 447

net ﬂux density at the top of the atmosphere level would be a height-independent

temperature change which includes a change of the air temperature T

near the

surface of the Earth. The most simple formulation of this process is given by

T

=−k

E (12.10)

where k

is a positive climate sensitivity parameter. However, if feedback is assumed

to take place, this temperature change will be altered. Now we write

T



=−kE (12.11)

where k is another positive climate sensitivity parameter. By assuming that all acting

feedback processes are linear in T

and independent of each other, we may express

the modiﬁed net ﬂux density at the top of the atmosphere level after feedback has

occurred by



= E − T





(12.12)

The sensitivity coefﬁcients α

have the sign of the feedback. The response to the

perturbation E with feedback should be the same as the response to E



without

feedback. Thus from

T



=−kE =−k



(12.13)

we ﬁnd

k =

1 − k



1 −



with β

= k

(12.14)

For a particular atmospheric variable y

the sensitivity coefﬁcient α

may be

expressed by an equation of the form

=−

∂ E

∂y

(12.15)

It should be observed that only the determination of ∂ E/∂y

is a problem of radiative

transfer while the derivative dy

/dT

must be obtained with the help of a climate

model. If the variable y

refers to cloud cover fraction C we have the particular case

expressed by

=−

∂ E

∂C

(12.16)

12.2.1 Cloud feedback in response to doubling atmospheric CO

To give an impression of the complexity of the feedback problem we will brieﬂy

discuss the cloud feedback in response to doubling the atmospheric CO

and also list

several other mechanisms. There are a great number of general circulation models

448 Inﬂuence of clouds on the climate of the Earth

Climate system

Internal forcing

External forcing

Surface albedo

Water vapor

Cloud cover

...

∆T

= −kE

Fig. 12.1 Schematic diagram of the climate system depicting several feedback

processes.

(GCM) which can be used to investigate the effect of doubling the atmospheric CO

concentrations on the climate. Arking (1991) brieﬂy discusses the results provided

by the GISS (Hansen et al., 1984) and the GFDL (Wetherald and Manabe, 1988)

GCM models.

Both models have interacting clouds to the extent that the spatial distribution of

cloudiness (height and amount) may change as a reaction to climatic change. The

optical properties of the clouds remain ﬁxed. In principle, it is possible to include

changing optical cloud parameters in response to the climatic change as part of

the feedback process. The interpretation of results, however, becomes increasingly

difﬁcult due to phase changes and changing conﬁgurations of ice particles among

other things. Both models yield about the same change of the mean global surface

temperature of 4 K in response to doubling the amount of CO

. The overall con-

clusions derived from various other models are similar all showing an increase in

the surface temperature. However, due to different treatments of the feedbacks, the

models produce surface temperature changes ranging from about 1.5–5 K. Below

we will brieﬂy dwell on the uncertainties of the feedback effects.

Figure 12.1 shows schematically four internal feedback blocks and their

contribution to the climate sensitivity factor k. Let us also inspect Table 12.1 listing

the calculated feedback factors due to the modeling of several processes. While the

calculated individual feedback factors differ in both models, the sum of these is

nearly identical amounting to about 0.7. By substituting 0.7 for the sum over the β

12.2 Cloud feedback in climate models 449

Table 12.1 Feedback factors for several processes calculated with

the two GCM models GFDL and GISS

Feedback process GFDL GISS

Surface albedo 0.16 0.09

Water vapor 0.43 0.40

Cloud cover 0.11 0.22

Total feedback factor 0.70 0.71

into (12.14), we ﬁnd that the reponse is about 3.3 times what it would be without

feedback. For further details we refer to the original paper.

12.2.2 Other trace gases

Besides CO

, the concentrations of several other trace gases are increasing. Because

of their strong infrared absorption bands, methane, nitrous oxide and several

chloroﬂuorocarbons are radiatively very important. Although ozone is increasing

in the troposphere it is decreasing in the stratosphere.

An increase of other gases such as CO, NO and NO

, although radiatively not

important, cannot be ignored since they may alter the atmospheric chemistry and

perturb the radiatively acting gases such as CH

and O

. There seems to be a general

agreement that the global effect of all trace gases is of the same order as the effect of

the CO

increase (Ramanathan et al. 1985; Ramanathan, 1987; Wang et al. 1986).

As the feedback models become more complete we should expect further changes

in the results. Deﬁciencies such as the omission of nonlinear effects and of the

complicated chemical processes will eventually be removed.

12.2.3 Liquid water and cloud microphysics feedback

We have previously stated that the optical parameters in the above GCM models

were ﬁxed. It has been argued quite early by Paltridge (1980) and later by others that

optical parameters are expected to change in response to CO

-induced atmospheric

warming. An increased temperature would increase the liquid water content of

clouds thus increasing the cloud optical thickness. Paltridge estimates that the

short-wave effect would dominate over the long-wave effect, resulting in a negative

feedback for this process alone and reducing the overall sensitivity of the climate

to CO

changes by about 40%. Subsequent investigations with one-dimensional

radiative–convective models also suggest that cloud liquid water would contribute

450 Inﬂuence of clouds on the climate of the Earth

negatively to cloud feedback. It stands to reason, if all else remains the same, the

optical thickness of the cloud would be proportional to the vertically integrated

liquid water. Additionally, we may argue with Bohren (1985) that changes in the

microphysical properties of clouds would be produced due to changes in atmo-

spheric temperature and humidity. If the size of the cloud particles increases along

with liquid water content, the magnitude of the negative feedback would decrease.

Detailed modeling is required to resolve the remaining questions.

12.2.4 Climatic impact due to aerosols

As is well-known, atmospheric aerosols have a strong impact on the greenhouse

effect of the Earth–atmosphere system due to aerosol extinction in the short-wave

and long-wave parts of the spectrum. As in case of clouds, the resulting effects

depend strongly on the particle concentration, on the optical properties of the

particles, and on the vertical as well as on the horizontal aerosol distributions.

In general, the effect of increasing concentrations of aerosol particles results in an

increase of the planetary albedo. The resulting cooling may counteract the green-

house warming of CO

and other trace gases as pointed out by Hansen and Lacis

(1990) and by others.

The single scattering albedo ω

of the aerosol particles plays an important role

in the radiation budget of the atmosphere since the aerosol effect can switch from

cooling to heating when ω

crosses a certain critical value. Due to the complex

composition of aerosol particles, the complex index of refraction is difﬁcult to

measure accurately which is a prerequisite for accurate Mie calculations of the

extinction parameters and the phase function.

The relatively small mass of stratospheric aerosols plays an important role in

the radiation budget because of their global distribution and their long lifetime.

Occasionally, volcanic eruptions increase the stratospheric aerosol concentration

over the whole globe for a period of years so that they are an important cause

for primary global climate forcing. The stratospheric aerosols, mostly sulfuric acid

particles, have a very small imaginary part of the complex index of refraction which

results in small values of the absorption coefﬁcient. For an average global albedo

of about 30%, they contribute to global cooling. In the stratosphere itself, however,

the main effect of the aerosols is an increase of the stratospheric temperature due

to the slight absorption of the particles themselves and due to the increase of the

photon pathlength in the absorbing gases. This warming was observed after the

El Chichon eruption as reported by Labitzke et al. (1983). Therefore, the cooling

effect of the stratospheric aerosols is limited to the troposphere and to the surface.

Tropospheric aerosols have a much larger optical depth than stratospheric

aerosols. However, tropospheric aerosols have a relatively short residence time

12.3 Problems 451

and vary strongly in type, spatially and temporally. This makes it very difﬁcult

to determine their mean radiative characteristics. Of course, they interact with the

solar and the infrared radiation ﬁeld. The inﬂuence of tropospheric aerosols on the

long-wave radiation budget is small because they are mostly located in the lower

tropospheric layers having a temperature not too far removed from the ground

temperature. Numerous papers on the radiative effects exist in the literature.

Small aerosols act as cloud condensation nuclei (CCN) to form cloud droplets.

An increase of the number density of aerosols increases the number of droplets,

and, for a given liquid water content, decreases the droplet size. This effect which

is also known as the Twomey effect (Twomey, 1977) tends to increase the cloud

optical pathlength and cloud reﬂection. On the other hand, if the CCN contain

absorbing constituents they tend to reduce the reﬂectance. Thus the role of small

aerosol particles is quite complex.

12.3 Problems

12.1: To familiarize yourself with the terminology of this chapter, verify equations

(12.8).

Answers to problems

Chapter 1

1.1:

(T )dλ =

2hc



exp



λkT



− 1



−1

dλ

(a) λ

max

T = const = 2898 µmK

(b) λ

max

(T = 6000) = 0.48 µm, λ

max

(T = 300) = 9.66 µm

1.2:

(a) B

dν =

2ν

dν

(b) B

dν =

2hν

exp



−

hν



dν

1.4:

Lambert’s law of photometry: dφ =

IdA

cos α

I : radiance from dA

dφ: radiant ﬂux received at dA

E =

σ T

1 + (z/R)

1.5:

(a) E = σ T

1 − cos

+ h

+ 2ah cos α

(b) E = σ T

1 − cos

+ h

− 2ah cos α

Radiometer

Valley

h is the vertical distance between the radiometer

and the center of curvature.

452

Answers to problems 453

1.6:

(a) E

πa



∞

dν

(b) φ = 4π



∞

dν

(c)

u(r) =

2π



1 −





∞

dν

1.7:

0,ν

, E

net,ν

= S

0,ν

(sin ϑ

cos ϕ

i + sin ϑ

sin ϕ

j + cos ϑ

k).

The index 0 refers to the solar radiation and (i, j, k) are the unit vectors in the

directions of the axes of the Cartesian (x, y, z)-coordinate system.

Chapter 2

2.1:

C =

2.2:

4π

(µ, µ

)

+ µ



exp



−



− exp





exp



−

µ + µ

µµ



−

4π

(µ, µ

)

− µ



exp



−



− exp



−



with

(µ, µ

) =

∞



l=m

(−1)

l+m

(µ)P

(µ

), F

(µ, µ

∞



l=m

(µ)P

(µ

)

0 ≤ µ ≤ 1

2.3:

(a) I

−

(τ,µ

) =

4π

(µ

,µ

)

exp



−



(b) I

(τ,0) =

4π

(0,µ

)exp



−



, I

−

(τ,0) =

4π

(0,µ

)exp



−



(0, 0) =

4π

(0,µ

), I

−

(0, 0) = 0

The functions F

1,2

are given in the answer to Problem 2.2.

2.4:

For each integral we ﬁnd A(τ ).

454 Answers to problems

2.5:

(a) I

(τ,µ) = B

(b) τ =∞

2.6:

(a) E

(0) =∞, E

(0) =

n − 1

, n = 2, 3,...

(b) dE

/dx =−E

n−1

(x)

2.8:

J (τ,µ) =

3(3 − µ

)



−1

I (τ,µ)dµ +

3(3µ

− 1)



−1

I (τ,µ)µ

dµ

2.9:

(τ,µ) =



∞

J (τ



)exp



−



− τ



dτ



(a) I

−

(τ,µ) =



J (τ



)exp



−

τ − τ



dτ



(b) I

(τ ) =



∞

J (τ



n+1

(τ



− τ )dτ



+ (−1)



J (τ



n+1

(τ − τ



)dτ



net

= 2π



∞

J (τ



(τ



− τ )dτ



− 2π



J (τ



(τ − τ



)dτ



Chapter 3

3.7:

(a) J (0) =

√

3/(4π)S

,  = 4/

√

Chapter 4

4.5:

I = 0.65860. The quasi-exact value is I = 0.65882.

4.7:



j=−s



1 +



−

P(µ

,µ

)



(1 − δ

0, j

)Z(µ

) =

4π

P(µ

, −µ