Wallace J.M., Hobbs P.V. Atmospheric Science. An Introductory Survey

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444 Climate Dynamics

This same framework will provide a basis for under-

standing the nature of “climate surprises”—instances

in which a modest (or gradual) climate forcing could,

at least in principle, give rise to a large and abrupt

climate change that would be irreversible.

In the interest of brevity we focus on variations

in only one climate variable, namely global-mean

surface air temperature T

, which we treat as being

governed by three factors: the incident solar radia-

tion, the planetary albedo, and the strength of

the greenhouse effect averaged over the Earth’s

surface. A number of auxiliary variables (e.g., the

concentrations of atmospheric water vapor and

ozone and the fractions of the surface area of

the Earth covered by clouds with tops in various

altitude ranges and by ice and snow) enter into the

determination of the albedo and the strength of

the greenhouse effect. We will treat the forcing and

the response as spatially uniform and consider only

globally averaged quantities.

The radiative forcing F is defined as the net

downward flux density or irradiance at the top of

the atmosphere

that would result if it were

applied instantaneously (i.e., without giving the

surface air temperature or the vertical tempera-

ture profile any time to adjust to it). For example, if

the flux density of solar radiation were to increase

by the amount dS, the net radiation at the top of

the atmosphere would initially be downward and

equal to dS. Surface air temperature T

would grad-

ually rise in response to this imbalance until the

outgoing flux of the atmosphere increased by the

amount dS, at which time the Earth system can be

said to have equilibrated with the forcing. When

the system reached its new equilibrium, T

would

have increased by the amount dT

in response to

the solar forcing dS.

In a similar manner, an increase in the greenhouse

effect can be expressed as a net downward irradiance

dG at the top of the atmosphere, equal to the initial

decrease in the upward irradiance of longwave radia-

tion due to the enhanced blanketing effect of the

atmosphere. As in the previous example, T

would

rise in response to the imbalance until the outgoing

irradiance at the top of the atmosphere became

equal to the incoming solar radiation. When the sys-

tem had fully adjusted to the change, T

would have

increased by the amount dT

in response to the

enhanced greenhouse forcing dG.

The sensitivity of T

to the radiative forcing F,

namely



 dT

dF, is called the climate sensitivity.

Consistent with its definition in terms of the total

derivative, the climate sensitivity takes into account

the changes in the auxiliary variables y

(the concen-

tration of atmospheric water vapor, the fraction of

the Earth’s surface covered by snow and ice, low

clouds, etc.) that influence T

.The various processes

that play a role in determining the climate sensitivity

can be evaluated by expanding the total derivative

using the chain rule, which yields

(10.5)

The partial derivative term







F is the climate sensi-

tivity



that would prevail in the absence of feedbacks

involving the auxiliary variables:

where T

is the equivalent blackbody temperature,

as estimated in Exercise 4.6.

Exercise 10.3 Estimate the sensitivity of the Earth’s

equivalent blackbody temperature to a change in the

solar radiation F

incident upon the the top of the

atmosphere.





T

F















The current operational definition of radiative forcing uses the tropopause rather than the top of the atmosphere as a reference

level with the proviso “after the stratosphere has come into a new thermal equilibrium state.” This definition is motivated by the following

considerations:

1. The tropospheric temperature profile, surface temperature, and the temperature of the ocean mixed layer are strongly coupled

because the tropical lapse rate is constrained to be close to moist adiabatic. In contrast, the stratosphere, with its stable stratification,

is decoupled from the underlying media. Hence, variations in surface temperature are related more strongly to the radiative forcing

at the tropopause than at the top of the atmosphere.

2. The stratosphere equilibrates with an abrupt change in radiative forcing within a matter of months, whereas the troposphere requires

decades to fully adjust because of the large thermal inertia of the oceans.

Once the stratosphere has come into thermal equilibrium with the new radiative forcing, the net irradiance through the tropopause and

the top of the atmosphere are the same.

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10.3 Climate Equilibria, Sensitivity, and Feedbacks 445

Solution: From the Stefan–Boltzmann law (Eq. 4.12)

or, taking the natural log of both sides,

Taking the differential yields

Hence,

Exercise 4.6 shows that for the Earth F

 239.4 W m

2

and T

 255 K. Hence,

Or, taking the reciprocal, it can be inferred that the

Earth’s equivalent black body temperature rises 1 K

for each 3.76 W m

2

of (downward) radiative forcing

at the top of the atmosphere. ■

The changes in the auxiliary variables in the last

term in (10.5) are a consequence of their tempera-

ture dependence. Hence,

Substituting into (10.5) we obtain

(10.6)

where

(10.7)

are dimensionless feedback factors associated with the

various feedback processes to be discussed in the fol-

lowing subsections. The feedback factor f

will be posi-

tive if the two derivatives of which it is comprised in

(10.7) are of like sign, and it will be negative if they are

















 0.266 K(W m

2

)

1



ln T



lnF













1/4

of opposing sign. [For example, if a rise in T

results in a

decrease in y

(as is the case for planetary albedo) and if

a decrease in y

favors a further rise in T

, then the feed-

back is positive.] The feedback factors are additive, i.e.,

(10.8)

where the algebraic sign of the feedback is taken into

account in the summation.

Solving (10.6) for the sensitivity of T

to the forcing

F, we obtain

(10.9)

Hence, the gain











due to the presence of climate

feedbacks is

(10.10)

provided that f  1.A value of f  1 corresponds to the

case of infinite sensitivity, in which even an infinitesimal

forcing may cause the climate system to diverge from

its present equilibrium state and seek a new one, as in

the mechanical analogs pictured in Fig. 3.17.

Exercise 10.4 Estimate the apparent climate sen-

sitivity







F based on estimates of the differences

in T

and F between current climate and the climate

at the time of the last glacial maximum (LGM)

around 20,000 years ago, using the relation

Solution: Global-mean surface air temperature is

estimated to have been 5 °C lower at the time of

the LGM than it is today. On the basis of ice core

data it is known that atmospheric CO

concentrations

were 180 ppmv, slightly less than half their current

values. On the basis of radiative transfer models it is

estimated that the climate forcing due to a doubling

of the CO

concentration is 3.7 W m

2

. On the basis

of what is known about the coverage of the continen-

tal ice sheets and the extent of sea ice at the time of

the LGM, it is estimated that the planetary albedo

was 0.01 higher at the time of the LGM than it is

today. Assuming that the flux density of solar radia-

tion at the time of the LGM is the same as the current

value (342 W m

2

), the corresponding increase in the





(current)  T

(LGM)

F (current)  F (LGM)



1  f









1  f

f 



P732951-Ch10.qxd 12/09/2005 09:12 PM Page 445

446 Climate Dynamics

flux of reflected solar radiation at the top of the

atmosphere is 342  0.01  3.4Wm

2

.Substituting

these values into the above expression yields

Comparing this result with the climate sensitivity in

the absence of feedbacks



 0.266 (W m

2

)

1

,as

computed in Exercise 10.3, and the apparent sensitiv-

ity of the climate system is enhanced by a factor of

0.70/0.266  2.7 due to the presence of feedbacks. ■

10.3.1 Transient versus Equilibrium Response

Because of the large heat capacity of the Earth sys-

tem (in particular, the oceans and the cryosphere),

global-mean surface air temperature exhibits a delayed

response to climate forcing. An equilibrium response to

an abrupt change in the forcing would be achieved only

after all components of the system have had adequate

time to adjust to the change. The adjustment time is

different for different components of the Earth system:

the atmosphere adjusts to changes in climate forcing

within a matter of a few months, the ocean mixed

layer requires a few years, the full depth of the ocean

requires centuries, and the continental ice sheets per-

haps even longer. The respective adjustment times

depend on both the heat capacity of the various compo-

nents of the Earth system and on the climate sensitivity.

We can gain some insight into the nature of this

adjustment process by considering the effect of

including a hypothetical ocean mixed layer whose

temperature adjusts instantaneously to changes in

surface air temperature. With this assumption we can

treat the ocean mixed layer as a slab with mean tem-

perature T  T

 T,where T

is the equilibrium

temperature before the forcing F is applied and T is

the time varying (i.e., transient) response to a change

Q in the climate forcing. Based on the energy

balance at the top of the atmosphere, we can write

(10.11)

where c is the heat capacity of the slab in units of

2

1

averaged over the surface area of the Earth

and



is the climate sensitivity dT

dF.The left-hand

side of (10.11) is the rate at which energy is stored in

dT



T



 Q





5 K

(3.7  3.4) W m

2

 0.70 K per W m

2

the slab, and the right hand side as the imbalance

between the forcing Q and the increase in outgoing

longwave radiation at the top of the atmosphere due to

the warming of the slab.

Now let us assume that the perturbation in the

forcing Q is “turned on” at time t  0 and main-

tained at a constant value after that time. In this case

(10.11) can be rewritten as

where c, and solved to obtain

(10.12)

Hence, after the forcing is turned on, T exponentially

approaches the equilibrium solution



Q, increasing

rapidly at first and then more gradually as the solu-

tion is approached. The e-folding timescale for the

approach to the equilibrium solution is proportional,

not only to the heat capacity of the mixed layer, but

also to the climate sensitivity. Hence the existence

of positive climate feedbacks lengthens the time it

takes the climate system to equilibrate with a change in

the forcing. It can be shown that for a linear rate of

increase in climate forcing, the response of T is also

linear and delayed by the same response time



 c



The time delay in the response to climate forcing

due to the heat capacity of the ocean mixed layer is on

the order of a decade or less (see Exercise 10.21). The

heat capacity of the oceans as a whole is about 50 times

larger than that of the ocean mixed layer; the continen-

tal ice sheets also possess a high effective heat capacity

because of the large magnitude of the latent heat of

fusion of water. If the atmosphere exchanged heat

freely with these large reservoirs, the response time, as

defined in the context of (10.11), would be on the

order of centuries or longer. Short-term perturbations

of the climate system such as volcanic eruptions would

be almost entirely damped out and the impacts of

longer term changes in forcing, such as those associ-

ated with greenhouse warming, would become appar-

ent only over the course of centuries.

The atmosphere does, in fact, exchange significant

quantities of heat with these large reservoirs, but the

rates of exchange are much slower than with the ocean

mixed layer. The thermohaline circulation ventilates

the deep layers of the ocean on timescales of centuries,

T



Q (1  e

t





)

dT



T





Q





See D. L. Hartmann, Global Physical Climatology, Academic Press, Section 12.6 (1994).

P732951-Ch10.qxd 12/09/2005 09:12 PM Page 446

10.3 Climate Equilibria, Sensitivity, and Feedbacks 447

and the mass of the continental ice sheets responds to

imbalances between accumulation and melting on a

similar timescale. Exchanges of energy with these

reservoirs are reflected in small, but measurable imbal-

ances in the net radiation at the top of the atmosphere.

Coincidentally, the warming of the oceans is reflected

in a rise of sea level due to the expansion of water as it

warms, and the shrinkage of the continental ice sheets

causes an additional sea level rise due to the increase

of the mass of the oceans. Hence, measurements of sea

level change can be used to constrain estimates of the

rate of energy exchange with these reservoirs.

Because the exchange of energy between the atmos-

phere and the larger reservoirs in the Earth system

takes place gradually, rather than instantaneously, the

response to climate forcing is felt almost immediately.

However, the full, equilibrium response to a prescribed

steady-state forcing is not realized until these large

reservoirs have had time to equilibrate with the forc-

ing. For example, if atmospheric CO

concentrations

were to continue to rise at their present rate until they

double (relative to preindustrial concentrations) some

time late in the 21st century and remain constant after

that, the transient response observed at the time of

the doubling would not be as large as the equilibrium

response observed several centuries later.

10.3.2 Climate Feedbacks

It is the sum of the feedback factors f

that determine

the climate sensitivity. Here we consider the contri-

butions of some of the more important individual

feedback factors to that sum, including ones that

might be negative.

a. Water vapor feedback

As a consequence of the Clausius–Clapeyron equa-

tion (see Section 3.7.3), the saturation vapor pressure

of water increases exponentially with temperature at a

rate of 7% K

1

in the temperature range of interest.

If the distribution of relative humidity were to remain

constant as the temperature rises, atmospheric water

vapor concentrations would increase with tempera-

ture at a roughly comparable rate. Higher concentra-

tions of water vapor, the atmosphere’s most important

greenhouse gas, favor higher surface air temperatures.

Based on relatively straightforward radiative transfer

calculations under the assumption of constant relative

humidity, the feedback factor for this process is esti-

mated to be 0.5, which, from (10.10), corresponds to

a gain of 2 (i.e., a doubling of the response of T

to a

prescribed forcing F) if no other feedback processes

were operative.

Due to the nonlinearity of the Clausius–Clapeyron

equation (3.92) the strength of the water vapor feed-

back increases with temperature. If the radiative

forcing ever became strong enough to raise the

tropical sea-surface temperature from its present

value of 28 °C to above 60 °C, the feedback factor

would approach unity, setting the stage for a runaway

greenhouse. Such a catastrophe, which may have

occurred on Venus, would ultimately lead to the

evaporation of the entire world’s oceans, creating a

massive atmosphere consisting mostly of steam, with

surface temperatures in excess of 1000 K!

The timescale of the atmospheric branch of the

hydrological cycle is so short that the water vapor

feedback can be considered to be virtually instan-

taneous. It sets in rapidly enough to come into play

even in the transient response to impulsive forcing

such as volcanic eruptions. A by-product of the

enhanced water vapor feedback is the intensification

of the hydrologic cycle. Climate models indicate that

in a warmer world, when it rains it rains harder, and

evaporation is also more rapid.

b. Cloud forcing and feedbacks

Clouds reflect a fraction of the solar radiation that

would otherwise be absorbed at the Earth’s surface

and they also contribute to the greenhouse effect. The

relative importance of these two competing effects is

largely determined by the height and by the optical

thickness of the clouds in the shortwave part of the

spectrum. For deep cloud layers, such as those typically

associated with tropical convection, the two effects

nearly cancel one another so that the net radiative

cloud forcing is small. In contrast, more reflective cloud

decks at the top of the planetary boundary layer,

whose tops are typically only 10 °C colder than the

underlying water or land surface, contribute much

more strongly to the albedo than to the greenhouse

effect, and thus produce a large negative net radiative

forcing (Fig. 10.37). If the areal coverage of these cloud

decks were to increase as the climate warmed, that

would constitute a negative feedback on global-mean

surface air temperature. However, of the coverage

were to decrease as the climate warms, that would

constitute a positive feedback.

Stratus and stratocumulus decks tend to occur in

regions of subsidence in which the sea surface and

P732951-Ch10.qxd 12/09/2005 09:12 PM Page 447

448 Climate Dynamics

the atmospheric boundary layer are cool relative to

the overlying free atmosphere (i.e., stably stratified).

It is not clear how much and in what sense the areal

coverage of these clouds would be likely to change in

response to global warming.

At the opposite end of the spectrum from boundary

layer stratus cloud decks, with respect to cloud forcing,

are high thin, subvisible cirrus cloud decks with very

cold tops, which have been detected in the upper

troposphere well above the tops of convective clouds

(Fig. 10.38). Subvisible cirrus warm the Earth’s surface

because they transmit downward solar radiation with-

out significant scattering or absorption, while blocking

a larger fraction of the outgoing longwave radiation

and reradiating it to space at very low temperatures.

The cloud forcing by subvisible cirrus is limited by their

small optical thickness which is generally less than 0.1.

Their areal coverage remains to be determined.

c. Ice-albedo feedback

Ice-albedo feedback has already been discussed in the

previous section in the context of coupled climate vari-

ability. Its importance depends on the fractional area of

the Earth covered by ice. At times like the present,

when perennial snow and ice are largely confined to

the polar regions, the decrease in fractional areal cov-

erage that would occur in response to an incremen-

tal change in global-mean surface temperature [the

counterpart of in (10.7)] is relatively modest.

If ice is assumed to cover the area poleward of

latitude



in which the Earth’s surface is colder than



some threshold temperature T* (presumably near the

freezing point), then

where m is the meridional temperature gradient in the

vicinity of the ice edge and the factor cos



reflects

the increasing length of latitude circles as the ice edge

moves equatorward. Provided that m does not vary

strongly with latitude, it follows that the incremental

increase in per unit change in T

decreases with

latitude, roughly in proportion to cos



.This simple

geometric relationship predicts that the strength of

the ice-albedo feedback should increase as the ice

edge advances toward lower latitudes and vice versa.

Simple energy balance model calculations indicate

that if the ice line were to advance far enough equa-

torward, the ice-albedo feedback would become so

strong that the denominator in (10.9) would vanish

and the climate sensitivity would become infinite. If

this situation were to be realized in nature, Earth

would abruptly become completely ice covered, as in

the Snowball Earth scenario described in Box 2.2.

While the sign of the ice-albedo feedback is certainly

positive, its magnitude is highly uncertain because it is

strongly seasonally dependent and it is coupled to the

behavior of clouds, land surface hydrology, and land

vegetation over high latitudes. Changes in albedo have

the largest impact on the surface energy balance during

 m cos



–140 –120 –100 –80 –60 –40 –20 0 20 40 60

W m

–2

Fig. 10.37 Annual mean net cloud radiative forcing, which is

estimated by comparing reflected solar radiation and outgoing

solar radiation, as in Fig. 4.35, but computing averages for all

pixels and cloud-free pixels separately at each grid point and sub-

tracting the latter from the former. For a more detailed explana-

tion see Exercise 10.22. [Based on data from the NASA Earth

Radiation Budget Experiment. Courtesy of Dennis L. Hartmann.]

5.0

4.0

3.2

2.2

1.00

54 55 56 57 58

Longitude (°E)

Altitude (km)

DLR

OLEX

Fig. 10.38 Thin layers of subvisible clouds in the tropical

upper troposphere as inferred from Lidar measurements from

an aircraft, at a wavelength of 1.064



m The colored shading

indicates the intensity of the light backscattered by cloud

droplets light blue shading indicates minimal scattering

and white indicates missing data. This observation was taken

over the Indian Ocean during a field experiment on board

aresearch aircraft. [From J. Geophys. Res., 107(D16),

Union. Reproduced/modified by permission of American

Geophysical Union. Courtesy of H. Flentje, Deutschen Zentrum

für Luftund Raumfahrt (DLR), Germany.]

P732951-Ch10.qxd 12/09/2005 09:12 PM Page 448

10.3 Climate Equilibria, Sensitivity, and Feedbacks 449

To illustrate the concepts of feedbacks, equilibria,

and abrupt climate change, let us consider the

climate of Daisyworld, an idealized spherical planet

with no atmosphere, warmed by a distant sun. The

surface temperature of Daisyworld is uniform and

in radiative equilibrium with the incoming solar

radiation. The surface of the planet is black, apart

from the part that is covered by perfectly white

daisies, which are distributed randomly over the

surface of the planet in small enough clumps that

the planet looks uniformly gray when viewed from

space. Hence, the albedo of Daisyworld is numeri-

cally equal to the areal coverage of daisies.

Daisies are able to grow only within some

finite range of surface temperatures T

 T  T

and their areal coverage is a maximum at some

intermediate temperature, as indicated by the red

curve in Figure 10.39. By mediating the planetary

albedo, the areal coverage of daisies, in turn,

affects the surface temperature of the planet.

From the Stefan–Boltzmann law (4.12), under

radiative equilibrium conditions

(10.13)

where



is the Stefan–Boltzmann constant, T is the

surface temperature of the planet, A is the fractional

areal coverage by daisies and F is the flux density of

solar radiation averaged over the surface the planet.

The effect of the areal coverage of daisies upon the

temperature of the planet is indicated by the blue

curves in Figure 10.39, each of which corresponds to

a different value of F.

Points P and P, where the blue and red curves

intersect, represent equilibrium states, for which the

fractional area coverage of daisies is consistent

with the temperature, and the temperature, in turn,

is consistent with the areal coverage of daisies.



 (1  A)F

10.2 Daisyworld

To make the plot as simple as possible, the blue curves are represented as straight lines. In accordance with the Stefan–Boltzmann law,

this plotting convention requires that temperature on the x axis be plotted on a non-linear scale with distance from the origin proportional

to T

and increments of distance



x proportional to T



T.For purposes of this qualitative discussion, the nonlinearity of the temperature

scale is unimportant.

Continued on next page

the summer season, when the insolation is strongest.

The presence of extensive stratus cloud decks over the

polar oceans during summer reduces the strength of

the ice-albedo feedback during this season and renders

it more difficult to estimate quantitatively.

The direct effects of changes in snow cover over

land are felt mainly in connection with the timing of

the spring thaw, but the indirect effects on surface

temperature via the ground hydrology linger into

summer. An earlier spring thaw increases the pos-

sibility that the soil will dry out during summer, allow-

ing for the possibility of higher daytime temperatures.

Higher summertime temperatures and longer growing

seasons enable shrubs and trees to encroach on areas

of tundra, increasing the surface roughness and reduc-

ing the albedo. Quantitatively modeling such a diverse

array of interactions is a formidable challenge.

d. Carbon dioxide feedback

In the context of greenhouse warming, CO

is the

dominant radiative forcing, but in the context of

glacial–interglacial cycles, changes in radiative

forcing due to variations in atmospheric CO

con-

centrations constitute a feedback. Just what caused

concentrations to increase and decrease nearly

in synchrony with temperature on a timescale of

thousands of years is still a matter of speculation.

Most of the proposed mechanisms involve long-

term changes in the oceanic thermohaline circula-

tion that could affect the rate at which the deeper

layers of the ocean are ventilated and organic car-

bon droppings from the “biological pump” dis-

cussed in Section 2.3 are recycled back to the

surface. If the ventilation rate slowed down signi-

ficantly, it is conceivable that enough organic car-

bon could be stored in the deep oceans to account

for the ~80 ppmv depression in atmospheric CO

concentrations during the ice ages relative to those

during interglacial epochs. Whatever their cause, it

is clear that the large positive feedback associated

with these fluctuations in atmospheric CO

concen-

trations significantly amplify the climate response

to orbital forcing.

P732951-Ch10.qxd 12/09/2005 09:12 PM Page 449

450 Climate Dynamics

P represents a stable equilibrium state in the sense

that a small perturbation in either temperature or

the areal coverage of daisies would induce a

response in the other variable that would tend to

reduce the amplitude of the perturbation. For exam-

ple, if T were raised by a small increment (without

changing F), as illustrated in the inset of Fig. 10.39,

the areal coverage of daisies and the planetary

albedo would increase, resulting in cooling. Cooling,

in turn, would reduce the positive amplitude of the

perturbation in the areal coverage of daisies, and the

state of the system would spiral inward toward P as

shown. Alternatively, if the areal coverage of daisies

were raised, the temperature of the planet would

drop in response to the increased planetary albedo,

and the cooling would decrease the amplitude of the

perturbation in the areal coverage of daisies. In both

cases, perturbations about the equilibrium state are

suppressed through the action of negative feedbacks.

By the same reasoning it follows that point P repre-

sents an unstable equilibrium state, about which small

perturbations in temperature or the areal coverage

of daisies would spontaneously amplify through the

action of positive feedbacks.

Now let us consider how the climate of

Daisyworld would respond to a slow increase in

the sun’s emission, such as is believed to have

occurred over the lifetime of the solar system. For

radiative equilibrium temperatures below T

there

would be no daisies and temperature would

increase with solar emission in accordance with

the Stefan–Boltzmann law. However, as soon as F

reaches , daisies begin to grow and the sensi-

tivity of Daisyworld’s climate system to further

increases in insolation is radically altered. At tem-

peratures just above T

the main response to

increasing insolation is an increase in the areal

coverage of daisies. The increase in planetary

albedo resulting from the proliferation of daisies

cancels most of the warming that would have

occurred in the absence of the daisies.

This negative feedback process is extraordi-

narily effective in stabilizing Daisyworld’s climate.

Consider the response to a small increase in F from

to , as indicated by the shaded wedge in

Fig. 10.39. In the absence of daisies the temperature

of the planet would rise from T

to T

. However,

when the daisy-albedo feedback is taken into

account, the temperature rises only to T

.It is-

evident from Fig. 10.39 and from Eq. (10.7) that the

remarkable stability of the climate is due to the

large value of dA



dT at temperatures just above T

As T approaches the optimal temperature for the

daisies, the strength of the feedback declines and

the climate sensitivity increases. When the insolation

reaches the value indicated by the sloping blue line

that is tangent to the red curve at Q,a climate catas-

trophe occurs. Beyond this point any further

increase in F, no matter how small, would result in a

complete collapse of the daisy population and an

abrupt increase in the temperature on the planet

from T

to T

. Hence, temperature T

with nearly

maximal daisy population and the much warmer

temperature T

with no daisies represent alternative

climate states on Daisyworld corresponding to the

same value of F =.Depending on the recent

history of the system, the climate could reside in

either state, and even an infinitesimally small per-

turbation in T or A would be sufficient to switch the

system from one state to the other. Such states are

referred to as multiple equilibria.



10.2 Continued

Stronger sun

P′

Daisy coverage (%)

100

Effect of temperature

on daisies

Effect of daisies

on temperature

Surface temperature

Fig. 10.39 Relationships between the areal coverage of

daisies (or, alternatively, the planetary albedo of Daisyworld)

and temperature, plotted on a non-linear scale. The red

curve indicates how the areal coverage of daisies depends

on the temperature, and the sloping blue lines indicate how

the temperature depends on the areal coverage of daisies.

Each blue line corresponds to a different value of the insola-

tion. Points P and P represent equilibrium states for the

same value of the insolation. P represents a stable equilib-

rium state and P an unstable equilibrium state. The inset in

the upper right corner is an enlargement of the region

around P showing qualitatively how the state of the system

would respond to a small positive perturbation in tempera-

ture, assuming that the insolation is held fixed.

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10.4 Greenhouse Warming 451

10.4 Greenhouse Warming

The notion that the burning of fossil fuels could result

in a buildup of carbon dioxide in the atmosphere,

leading to greenhouse warming, dates back over a

century.

23,24,25

Yet it was not until the late 1970s that

the potential seriousness of this issue began to become

widely recognized. In the ensuing years, greenhouse

warming has become a lightning rod for scientific and

political debate because important policy decisions

relating to fossil fuel emissions have to be made in the

face of scientific uncertainty. This section reviews what

is known, and what is not known, about greenhouse

warming, focusing on (1) the buildup of greenhouse

gases and the associated forcings, as defined in the pre-

vious section, (2) whether human-induced greenhouse

warming is already evident, and (3) the predicted

extent and impacts of human-induced greenhouse

warming.

10.4.1 The Buildup of Greenhouse Gases

Proof that greenhouse gases are actually building up

in the atmosphere was not forthcoming until the

1960s, when the CO

time series from the first moni-

toring stations (Fig. 1.3) became long enough to

reveal the presence of an upward trend. Averaged

over the record extending from 1958 up to the year

2000, the trend is roughly 1.3 ppmv per year; in

recent years it has been approaching 2 ppmv (0.5%

of the present concentration) per year. The trend is

believed to be attributable to fossil fuel burning for

the following reasons.

1. Analysis of gas bubbles trapped in ice cores

extracted from the Greenland and Antarctic ice

sheets indicates that atmospheric CO

started

to increase around the time of the industrial

revolution and it has roughly tracked the rate

of growth of fossil fuel consumption since that

time (Fig. 5.13).

2. Atmospheric CO

concentrations are higher by

several ppmv in the northern hemisphere, where

most of the strongest carbon sources are located.

3. Atmospheric oxygen is observed to be

decreasing at a rate of 3 ppmv per year,

consistent with the hypothesis that the CO

being added to the atmosphere is a product

of combustion.

4. The relative abundance of the radioactive

isotope

C and the stable isotope

C in

atmospheric CO

are declining.

C is virtually

absent in fossil fuels and

C is less abundant in

fossil fuels than in atmospheric CO

and in

carbon dissolved in the oceans.

Figure 10.40 compares the annual rate at which

carbon is being released by the consumption of fossil

fuels and the rate of increase of the mass of carbon in

the atmosphere. On average, the atmosphere is taking

up about half the carbon that is being released. Most

of the remainder is being taken up by the oceans.

In accordance with Eq. (2.7) and (2.8), the increas-

ing storage of carbon in the oceans in the form of dis-

solved CO

in recent decades is increasing in the

concentration in H



ions, lowering the pH of sea

water. The increasing acidity, in turn, is driving the

The term greenhouse warming, as used in this section and in most of the climate literature, connotes human-induced warming due to

the buildup of CO

and other greenhouse gases in the atmosphere during the period since the industrial revolution.

In 1896 Svante Arrhenius published a seminal paper entitled On the Influence of Carbonic Acid in the Air upon the Temperature

of the Ground (Proc. Swedish Academy of Sciences, 22). This was the first attempt to quantify the affects of changes in atmospheric

concentrations upon the Earth’s surface temperature. Although Arrhenius’ calculations were based on scanty data, his conclusions

turned out to be surprisingly realistic.

Arrhenius was not the first to investigate the greenhouse effect. Fourier introduced the concept, comparing the atmosphere to a glass

bowl that is heated not so much by solar radiation as by longwave radiation emitted by the ground.

Svante August Arrhenius (1859–1927) Swedish chemist. Best known for his electrolytic theory of dissociation (i.e., that electrolytes,

when dissolved in water, become to varying degrees dissociated into electrically opposite positive and negative ions). He was an infant

prodigy, teaching himself to read at age three. In addition to his fundamental contributions to chemistry and his recognition of the “green-

house effect,” he suggested that life on Earth might have originated from spores transported from other planets. The story is told that

when Arrhenius took the podium to honor guests at the 1926 Nobel Awards dinner, he offered a toast to the guests from each country

honored at the ceremony. Because this was during the prohibition era, he toasted the Americans with water. By the end of the toasts,

Arrhenius was the only one still sober.

Jean Baptiste-Joseph, Baron, Fourier (1768-1830) French mathematician. Helped found mathematical physics and had a strong

influence on the theory of functions of a real variable. Accompanied Napoleon on his expedition to Egypt and was also known for his

research on antiquities.

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452 Climate Dynamics

equilibrium between and [Eq. (2.9)]

toward the left, thereby buffering some of the hydro-

gen ions.

If it were not replenished, the existing oceanic

reservior of ions would be sufficient to buffer

roughly three-quarters of the carbon that would be

released if the entire fossil fuel reservoir in Table 2.3

were consumed. If the acidity of the oceans increases

significantly, the oceanic reservoir of ions could

be replenished, at least to some degree, by the dissolu-

tion of carbonate sediments from the sea floor.

The

capacity and responsiveness of the oceans in taking up

and buffering CO

depend not only on the relevant

chemical reaction rates, but also on the rate of ventila-

tion of the deeper layers of the ocean by the thermo-

haline circulation: the more vigorous the ventilation,

the slower the buildup of carbon and acidity in the

surface waters and the slower the rate of increase of

atmospheric CO

The exchange of carbon between the atmosphere

and terrestrial biosphere also affects the rate of

buildup of atmospheric CO

.The prominent “spike”

in the uptake by the atmosphere in 1998 (Fig. 10.40)

is believed to be due to the CO

injected into the

atmosphere by the massive forest fire outbreaks that

occurred in Indonesia and Amazonia in association

with the 1997–1998 El Niño event. Clearing of the

2

HCO



tropical rain forests has released additional carbon in

recent decades, which has been compensated by an

expansion of forests in temperate latitudes.

Based on estimates of the annual rate of consump-

tion of fossil fuels and the rates of change of the

atmospheric and oceanic carbon inventories, it is pos-

sible to infer, as a residual, the net uptake or release of

carbon by the terrestrial biosphere. Current estimates

indicate that the terrestrial biosphere, as a whole, has

been taking up carbon and that the rate of uptake

increased substantially from the 1980s to the 1990s.

The identity and likely future behavior of this terres-

trial carbon sink are open questions at this point.

Projections of future atmospheric CO

concentra-

tions based on three different emissions scenarios

are shown in Fig. 10.41. Because it is assumed that

the partitioning of the carbon emissions between the

atmosphere and the other components of the Earth

1000

500

2000 2050 2100

emissions (GtC yr

–1

)CO

conc. (ppmv)

Year

Fig. 10.41 (Top) Projections of future CO

concentrations

based on the various emissions scenarios plotted in the bottom

panel. The red curves are based on the assumption of rapid eco-

nomic growth, world population peaking in the middle of the

21st century, and continued reliance on fossil fuels. The black

curves assume a more gradual increase in fossil fuel consump-

tion, sustained throughout the century. The green curves assume

a major shift in the world economy toward sustainability. It is

assumed that the oceans and the terrestrial biosphere will con-

tinue to take up roughly half the CO

injected into the atmo-

sphere by human activities. [Adapted from Intergovernmental

Panel on Climate Change, Climate Change 2001: The Scientific Basis,

Cambridge University Press, p. 11 (2001).]

1960

1965

1970

1975

1980

1985

1990

1995

2000

Year

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Ocean and LandAtmosphere

GtC yr

–1

Fig. 10.40 Time series of the annual rate of release of car-

bon by the consumption of fossil fuels (in PgC or GtC per

year) based on industry records (red “staircase”), and the

annual rate of increase of the atmospheric carbon inventory

(irregular blue line), as inferred from the time derivative of

the time series of CO

concentration at Mauna Loa (Fig. 1.3).

[Reprinted with permission from Physics Today, 55(8), 30–36

Courtesy of Nicolas Gruber.]

Concern has been expressed that coral reefs may be vulnerable to an increase in the acidity of the oceans.

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10.4 Greenhouse Warming 453

system will not change with time, the concentration

curves in the top panel are proportional to the time

integral of the respective emission curves in the

lower panel: hence the monotonic increases in

projected CO

concentrations, even in the case of

the “green” scenario. In all three scenarios a value of

560 ppmv, equivalent to a doubling of preindustrial

concentrations, will eventually be attained. Yet it is

clear from these projections that current and future

decisions on energy policy will play an important role

in determining how long it will be before a doubling

is reached and how far beyond it CO

concentrations

will rise before they level off. In the more pessimistic

scenarios, concentrations reach 1000 ppmv by the

year 2200. Geological evidence indicates that con-

centrations that high have not occurred since the

Eocene epoch 50 million years ago, when tropical

animals and plants ranged as far northward as the

subarctic.

emissions will eventually decline, if for no

other reason than fossil fuel reservoirs will become

depleted. One might naively predict that the time

required for atmospheric concentrations to drop

back to preindustrial levels would be roughly compa-

rable to the residence time of CO

molecules in

the atmosphere, but this clearly is not the case. The

10-year residence time of atmospheric CO

reflects

cycling of carbon back and forth between the atmos-

phere and the marine biosphere and the leafy (as

opposed to woody) part of the terrestrial biosphere.

These biospheric reservoirs are very small in com-

parison to the atmospheric reservoir and they are not

very expandable. The relaxation of atmospheric CO

concentrations back toward preindustrial levels

depends on the uptake of carbon by the larger reser-

voirs in the Earth system, which takes place on a

timescale much longer than that of the human-

induced buildup of CO

is not the only greenhouse gas whose atmos-

pheric concentrations have been increasing during

the industrial era. Methane concentrations have

exhibited an even larger fractional increase since the

industrial revolution and nitrous oxide concentrations

have risen as well (Fig. 5.13). CFC concentrations

have risen dramatically since the mid-20th century,

but are leveling off and are expected to gradually

decrease over the next century in response to the

Montreal Protocol. HFCs, the replacements for CFCs,

are on the rise, along with a number of other long-

lived industrial gases. Stratospheric ozone concentra-

tions have declined since the mid-20th century in

response to the buildup of CFCs and are just begin-

ning to show signs of recovery. Meanwhile, tropos-

pheric ozone has been increasing in response to

worsening urban air pollution.

Even though their concentrations are much lower

than that of CO

, these trace gases contribute sub-

stantially to the greenhouse effect because some of

their absorption bands occupy regions of the spec-

trum that would otherwise be windows through which

terrestrial radiation could pass with relatively little

atmospheric absorption. On a per molecule basis, an

incremental increase in the concentrations of one of

these gases would thus have a much stronger impact

on the greenhouse effect than equivalent incremental

increase in CO

, which is so abundant that the atmo-

sphere is relatively opaque in the regions of the spec-

trum that correspond to its absorption lines.

To compare the relative contributions of various

trace gases to the greenhouse effect it is convenient

to define a greenhouse warming potential (GWP).

The GWP is defined as the mass of CO

that would

need to be instantaneously injected into the atmo-

sphere to produce an incremental increase in the

greenhouse effect equivalent to that caused by the

injection of 1 kg of a specified gas, integrated over a

specified time interval, taking into account the time-

dependent decay of the gas in question, as well as

that of CO

. In symbolic form

(10.14)

where T is the specified time interval, a

is the

radiative efficiency of gas x in units of W m

2

1

(i.e., the incremental change in radiative forcing

that would result from a small change in the con-

centration of the gas in question, taking into

account its present concentration and the concen-

trations of other greenhouse gases), c

is the fraction

of the mass of gas x injected into the atmosphere at

time t  0 that remains in the atmosphere at time t,

and and are the corresponding quantities

for CO

.The numerator of this expression is called

the absolute global warming potential (AGWP).

Greenhouse warming potentials for a few of the

important atmospheric greenhouse gases are

listed in Table 10.1. CFCs are especially potent in

terms of GWP because of their high radiative effi-

ciencies and long lifetimes. As the specified time

GWP 



(t)dt



(t)dt

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