math of climate change - 2007
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DanaMackenzie
2007 Publishedby
MathematicalSciencesResearchInstitute
Berkeley,CA
Anewdisciplineforanuncertaincentury
Climate
Change
Mathematicsof
Withgeneroussponsorshipfrom:
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In the spring o 007 MSRI organized a large public event, Climate Change: From
Global Models to Local Action, to examine some political and economic aspects
o climate: what we know, what we guess, how and how much our society can and
should respond to what we are learning. We chose the date to coincide with a visit byCongressman Jerry McNerney, whose background, as both a mathematician and an
advocate or alternative energy sources, made him a model participant or an event
that would combine the perspectives o several disciplines.
Te public panel discussion was ollowed on the next two days by a scientic
symposium, in which mathematicians rom many dierent elds mixed with
economists, climate modelers and others who have already been working on the
many questions involved. Tis booklet is a record o some o the discussions and
ideas in those meetings.
Te purpose o these events was to
connect the mathematical community
with the best current research andthinking about climate change, and
to point out the many dierent kinds
o mathematical challenges that are
presented by this issue. Society needs
to know more, and more accurately,
about what is happening with the earths
climate and to prepare or whatever
action is necessary and practical to
undertake. Mathematics and statistics
already play a central role in this as in
any sort o modeling eort. Likewise, computer science must have a say in the eort
to simulate Earths environment on the unprecedented scale o petabytes. With a
problem o this complexity, new mathematical tools will undoubtedly be needed
to organize and simpliy our thinking. Tus it seemed to us at MSRI important
to encourage direct discussions between those already in the eld and the many
mathematicians whose skills, and whose students skills, can bring new insights.
As Director o MSRI I organized the conerence, but as a non-expert I relied on
a number o others to make sure that the important scientic aspects were well-
covered, and to make sure that the conerence would represent the best current
science in the eld. I am particularly grateul to Inez Fung, Bill Collins and Chris
Jones or their scientic advice, and to Orville Schell or his advice and help in
arranging the public event. Nat Simons provided expert suggestions as well as
enthusiasm and great support throughout without him the event could neverhave happened.
David EisenbudDirector, Mathematical Sciences Research Institute, 1997-2007
forEword
Inez Fn let and David Eisenbd riht,
co-oranizers o the MSRI symposim on
climate chane.
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IntroductIon
When the history of climate change is written, the years 006 and 007 may be
seen as a turning pointa time when climate change ceased to be seen as a green issue and
became an everyone issue. In 006, Al Gores movie An Inconvenient ruth placed global
warming on Americas movie screens. In October 006, the British government released
the Stern Review, a rst attempt to quantiy the economic costs o climate change. Over a
period o our months, rom February to May 007, the Intergovernmental Panel on Climate
Change (IPCC) released its ourth report on climate change, which attracted much more
publicity than the previous three. In April 007, the United States Supreme Court ruled that
the Environmental Protection Agency has the authority to regulate carbon dioxide and other
greenhouse gases. In October 007, Gore and the IPCC shared the Nobel Peace Prize or
their eorts to build up and disseminate greater knowledge about man-made climate change,
and to lay the oundations or the measures that are needed to counteract such change.
Te increase in public discussion may reect an increasing comprehension that the scientic
debate over the reality o global warming has ended. (See sidebar, How Do We Know?) Te
IPCCs ourth assessment stated that warming o the climate is unequivocal and thatit was very likely (meaning more than 90 percent likely) that most o the warming is
anthropogenic. (See Figure 1.)
Tere are many uncertainties, however, in the specics o climate change and its impact.
Climate models tend to agree on the twenty-year projections, both in regard to their
sensitivity to variations in model physics as well as dierent
emissions scenarios1. Disagreement arises when projections are
carried out to the end o the century. For example, the equilibrium
response to a hypothetical scenario, involving an immediate
doubling o carbon dioxide, leads to varying predictions o warming
rom 1 degree Centigrade to a truly staggering 1 degrees. (Note
that these should not be interpreted as literal orecasts, because anovernight doubling is impossible.) Te dierence arises primarily
rom uncertainties in the climatic eedback processes represented in
the models, which tend to ampliy the direct eects by two or three
times.
Other aspects o climate change are even harder to predict accurately
than temperature. We can be certain that precipitation patterns
will change, and all the models indicate that some subtropical and
tropical regions will experience severe droughts.
But the models give contradictory predictions o where the droughts are likely to occur. As
another example, scientists reported in early 007 that glaciers in Greenland are melting
aster than any o the models in the IPCC report had predicted. Clearly, there are processesgoing on that we do not understand. Yet the extent o the polar ice caps is a critical variable
in climate models, because it triggers a eedback loop: the more the ice melts, the more
sunlight is absorbed by Earth (instead o being reected into space by the ice). Tis leads
to an increase in temperature, which in turn stimulates more melting o the ice cap. Tis
melting is o concern even to people who live ar away, because the melting o glaciers on
land is a major contributor to rising sea levels.
Dierent assumptions about emissions o greenhouse gases can be used as inputs into themodels.
FIguR E 1: Radiative Forcin Components
Anthropoenic hman-indced contribtions to lobal
climate chane are measred in watts per sqare meter
in other words, the increase in solar radiation that wold
prodce an eqivalent warmin eect. Some contribtionse.., the reenhose eect are positive, and others e..,
aerosols are neative. However, the net anthropoenic eect
since 1750, 1.6 watts per sqare meter, is nambiosly
positive, and also sinicantly reater than the amont o
warmin de to natral fctations in the sns brihtness
0.1 watts per sqare meter.
Image fromClimate Change 2007: The Physical Science Basis,Intergovernmental Panel on Climate Change.
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Te climate models used to make the IPCC projections are very complex systems o nonlinear
equations solved on large computers. Te IPCC relied on 4 dierent climate models in its
report. Many o them have been developed under the auspices o national meteorological
oces. Tough some models are superior to others, identiying them publicly is a ticklish
political issue. Because the models have dierent assumptions, we run the risk o comparing
apples to oranges. In many o its scenarios, the IPCC report simply averages all the models
together equally. It is not at all clear that this methodology is an optimal or even sound way to
integrate the data.
A serious limitation o the current models is their coarse scale. At present, even the highest-
resolution models chop up the world into pieces that are 10 to 50 kilometers wide.
Tis resolution is not ne enough to capture important details o topography,
such as mountain ranges, and it is also not ne enough to model individual
clouds, which play a complex and important role in the climate system.
(See Figure .) In practice, model parameters, especially those that representturbulent or ne-scale processes, are optimized or tuned in order to match
available observations. For example, the eect o clouds has to be added to
the model as an aggregate term, with all the uncertainties that implies. I the
climate models could be improved to 1-kilometer resolution, then clouds and
ner topography could be built into them; however, it has been estimated that
this would require a 10-petaop computer with 0 million core processors.
Tat kind o computing power is on its way, but it is not here yet. Even when
it arrives, its questionable whether climate modelers can take ull advantage o it. Many models
are legacy codes o a hal million lines or so that are not optimized or massively parallel
computation.
Finally, the mathematics o dynamical systems has taught us that uncertainty is an inevitablepart o predictions based on nonlinear physical models. Tis irreducible imprecision requires
us to use a variety o models, and run them with a diverse set o parameters, in order to capture
the real range o uncertainty in the climate system. It also means that climate modelers must
take care to communicate to policy makers that uncertainty is part o the story. As models
improve and more inormation becomes available, the model orecasts may change, and this
could lead to rustration among those needing to make decisions based on their predictions.
Tis rustration might be avoided i the original predictions are presented as a range o
possibilities rather than a single magic number.
Be that as it may, accurate and reliable prediction o global climate change is a key to policy
making. It is clear that policies should be based on predictions that are built on a sound
oundation. Mathematical scientists need to get involved, because the central questions acingthis research are mathematical in nature.
For example, clouds provide an important negative eedback mechanism that could reduce global warming. Asthe moisture in the atmosphere builds up due to warming, it could create more clouds, which would reect moresunlight back into space. However, this eect is by no means automatic; it depends on where the cloud is. High-altitude clouds radiate to space at a colder temperature and actually produce a net warming.
Figure 2: g
Climate models divide the worlds atmosphere and oceans
p into a very coarse rid. At present, even the best mode
do not have meshes ne enoh to simlate individal
tropical cyclones or the eect o montain ranes. Ftre
models may incorporate adaptive renements o the mes
size, as shown on the riht.
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From April 11 to April 13, 2007, the mAthemAticAl ScienceS reSeArch inStitute(mSri)convened a symposium, sponsored by the Sea Change Foundation, to assess how
mathematicians can address
the broader issues o climate
change and the narrower
issues o methodology lying
behind the climate models.
Te symposium consisted
o two parts. Several leading
politicians, business people
and academic experts
on energy and climate
convened or a panel
discussion (see Figure 3)
at San Franciscos Palace o
Fine Arts Teater on April
11, which drew a crowd o more than 300 people. On the ollowing two days, approximately
80 mathematicians and scientists attended a scientic symposium at the MSRI headquarters
in Berkeley. (See Appendix B.)
Inez Fung, the co-director o the Berkeley Institute or the Environment and one o the
authors o the IPCC report, started o the public event with a brie overview o the evidence
or global warming and the current state o knowledge about what will happen next. She
characterized the IPCC report, which acknowledges that climate change has been caused by
anthropogenic eects, as a bittersweet victory, because weve been saying the same thing or
0 years. She outlined the reasons why we know that the climate is warming (see Sidebar,How Do We Know?), and she discussed the main orecasts rom the IPCC report (see Sidebar,
A Bleak Future).
Aer Fungs introduction, MSRI director David Eisenbud introduced Congressman Jerry
McNerney (see Figure 4, page 7) and Caliornia Assembly Member Ira Ruskin (see Figure
5, page 7), who represents Silicon Valley. Aer brie remarks by McNerney and Ruskin,
Eisenbud summoned onto the stage a panel o
experts, which consisted o Daniel Kammen,
proessor o energy at the University o Caliornia
at Berkeley; Severin Borenstein, director o the
University o Caliornia Energy Institute; Nancy
McFadden, senior vice president o public aairs orPG&E Corporation; Doug Ogden, executive vice
president o the Energy Foundation in San Francisco;
Michael Peevey, president o the Caliornia Public
Utilities Commission; and Inez Fung, who had
already been introduced. Te legislators were given
an opportunity to pose questions to the experts, and
then the oor was opened to questions rom the
audience. Te ollowing section is based in large part
on the questions and answers that ensued.
How Do We Know?
The evidence For climaTe
changeClimate models and their projectionsor the utureespecially extendedout to 00are subject to avariety o uncertainties. Theseinclude imperections in the climatemodels, the limitations o ourcomputing power, and the inherentlyunpredictable nature o nonlinearequations. These uncertaintiesmust not be allowed to obscurethe central acts emphasized in thisyears IPCC report: Climate change
is happening, human activities areresponsible or most o the change,and the evidence indicates that it isaccelerating.
The basic acts that lead to thisconclusion are the ollowing:
. Carbon dioxide levels (and levelso other greenhouse gases, suchas methane) have been risingor at least hal a century. In act,they have risen by as much since960 as they did between the
last Ice Age and 960. (See Figure..) The current concentration ocarbon dioxide in the atmosphere,80 parts per million, is greaterthan it has been at any time inthe last 650,000 years, accordingto ice cores that contain trappedbubbles o earlier atmospheres.
. Carbon rom ossil uels is beingadded to the atmosphere. Weknow this because ossil uelscontain a lower ratio o theisotope carbon- to carbon-
than the atmosphere as a wholedoes, because they are derivedrom plant matter and plants havea preerence or the lighter isotopeo carbon. Tree-ring and ice-coredata show that the 1C:12C ratiobegan to decrease just at the sametime the overall levels o carbondioxide began to increase.
thE MSrI SyMpoSIuM on clIMatE changE
Fire 1.1 The concentration o carbon dioxide in the
atmosphere over the last 10,000 years main re and
over the last 50 years inset. Data rom the last 50 years
pink and red are based on direct measrement, and earlier
concentrations are inerred rom ice cores. At riht, the
concentrations are converted to an eqivalent increase in solar
radiation sin the year 1750 as a baseline.
Image fromClimate Change 2007: The Physical Science Basis, Intergovernmental Panel on Climate Change.
The panel at MSRIs pblic symposim on climate chane. Front row, let to riht: Nancy McFadden,
Do Oden, Michael Peevey, Inez Fn. Back row, let to riht: Daniel Kammen, Severin
Borenstein, Jerry McNerney, Ira Rskin, David Eisenbd.
FIguRE 3
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. Evidence rom ice cores shows avery strong correlation betweencarbon dioxide levels and globaltemperatures. (See Figure .)
When carbon dioxide levels go up,so does the temperature.
4. The physics behind thegreenhouse eect is not indispute. It has been knownor more than a century thatgases such as carbon dioxide,methane, and water vapor absorbinrared radiation coming romEarth (which would otherwiseescape to space) and re-radiatesome o its energy back towardEarth. Thereore an increase in
greenhouse gases must lead to anincrease in temperature, unlesssome other process comes alongto prevent it.
5. Finally, Earths surace temperaturehas increased sharply in recent
years, just as onewould expect.The observedwarming trendover the last00 years was0.74 degrees
per century, butover the last 50years the rateo increase hasnearly doubled,to . degrees percentury. The sixhottest years onrecord occurredin 998 (an ElNino year), 00,00, 004, 005,and 006. Thewarming eect
is now too largeto be explainedas a statisticalaberration. (SeeFigure ..)
What actions is Washington taking to reduce global warming?
At present, the U.S. government is trailing both public opinion in the U.S. and many other
world governments in addressing the climate-change problem. Nevertheless, there are
some grounds or optimism. At the United Nations Climate Change Conerence in Bali,in December 007, the U.S. agreed to a roadmap or uture negotiations that did not set
specic emissions targets. Also, in that same month Congress passed and President Bush
signed into law the Energy Independence and Security Act o 007, which will increase
automobile mileage standards to 35 miles per gallon by 00 (the rst change in the standards
in more than 30 years).
As o July 007, one hundred bills related to climate change
had been introduced in the current session o Congress. For
example, H.R. 809, the New Apollo Energy Act, would set a
target o decreasing greenhouse gas emissions to 80 percent
below 1990 levels by the year 050. It would institute a carbon
cap-and-trade program, commit $49 billion in Federal loan
guarantees or the development o clean energy technologies,
oer tax incentives or consumers to purchase plug-in hybrid
vehicles, increase unding or research and development o
clean energy technologies, and create a venture capital und to
help move new technologies to market.
It is also important or the U.S. government to make climate
change a part o its oreign policy, because climate change is
an problem o unprecedented international scope. H.R. 40,
the International Climate Cooperation Re-engagement Act,
would create an Oce on Global Climate Change within the
State Department and commit the U.S. to sending high-level
diplomats to uture international conerences on climate change.Tough proposals like H.R. 809 and H.R. 40 did not become
law this year, they represent an increased awareness o the
climate change issue on Capitol Hill.
How is Sacramento addressing global warming?
Te panelists emphasized that Caliornia can do little on its own to solve the
climate change problem, because it is global in scope. Nevertheless, in the absence
o concerted Federal action, Caliornia has played and can continue to play an
important role as a model or other states and even other countries.
clIMatE changE MItIgatIon
Fire 1.2 Over the last 400,000 years, lobal reenhose as
concentrations top and estimated temperatres bottom have been
extremely tihtly synchronized.
Fire 1.3 Direct measrements
o lobal mean temperatre leave little
dobt that a warmin trend exists, and is
acceleratin. The averae trends over the last
150 years red, the last 100 years ble, the
last 50 years orane and the last 5 years
yellow have otten proressively steeper.
FIguRE 4 (let). u.S. Conressman
Jerry McNerney at the MSRI
symposim.
FIguRE 5 riht. Caliornia
Assembly member Ira Rskin
speakin at the MSRI symposim.
Period Rate
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The iPcc rePorT: a Bleak
FuTure
In 007, the Intergovernmental Panel
on Climate Change (IPCC) releasedits Fourth Assessment Report on theworlds climate. The report has beenreleased in three sections, producedby three separate working groups. Ainal synthesis report was releasedin November.
Working Group I ocused on thephysical indicators o climate changeand projections o key climatevariables into the uture. This groupused the combined output o 4climate models to project surace
temperatures, precipitation, and sealevel changes out to the last decadeo this century, under six dierentemissions scenarios. The bestestimates o the temperature changerange rom .8 degrees Centigrade,in the most optimistic case (a worldwith high priority on sustainabledevelopment) to 4.0 degrees in themost pessimistic case (a business-as-usual world with intensive ossil-energy use). For comparison, thereport also includes one too good
to be true scenario, in which carbonemissions stay constant at 000levels. This scenario represents theminimum amount o climate changeto which we are already committed:about 0.6 degrees Centigrade.The IPCC report speciically avoidsany sort o doomsday scenarioinvolving a widespread breakdowno social institutions (though such ascenario might have made or juicier
headlines).
As explained elsewhere in this
report, individual numbers donot adequately summarize thecomplexity o climate models. Forinstance, the likely range or thebusiness-as-usual scenario is rom.4 to 6.4 degrees Centigrade. Thistranslates to a/3 probability that theactual temperature increase wouldlie within the stated range, and a 1/probability that it would be greateror less. The temperature increase is
In particular, the Caliornia
assembly last year passed
Assembly Bill 3, the Global
Warming Solutions Act o 006,
which committed Caliornia
to reducing its greenhouse gas
emissions in 00 to 1990 levels.
Te governor has proposed an
allocation o $36 million to create
the new positions required to
implement the actor example,
to determine what exactly is
meant by 1990 levels. Panelist
Borenstein commented that
we should ocus on ways o
meeting this target that are not
idiosyncratic to Caliornia, butcan be exported to the rest o the world.
On this years docket, the Caliornia legislature is considering bills to create green building
standards or the state government (A.B. 35); to provide unding or alternative uel research
(A.B. 118); and to create rebates on the cleanest new cars and surcharges on the dirtiest ones
(A.B. 1493). Te latter bill was voted down between the time o the symposium and the
writing o this document.
What are the most promising technologies for mitigation of climate change?
Panelist Kammen commented that we should not look or a single magic bullet, but should
look to a variety o technologies. At present, Germany, Spain, and Denmark are leading
the way in wind energy, but the U.S. has a large untapped potential (see Figure 6) and
placed more new wind generation capacity in
service than any other nation in 006. Te
photovoltaic industry is still trying to reduce
costs, but on a hot summer day solar power
can be generated more cheaply than the spot
market price o energy. Recently, scientists
invented spray-on solar panels, a material
like spray paint that can generate electricity
rom inrared light. Biouels continue to
be a major area o research, as scientists
try to nd crops that can be harvested to
produce uel more eciently than corn.Energy eciency is also an important eld
o research, with hybrid vehicles, plug-in
hybrids, and all-electric vehicles leading the
way. (See Figure 7.)
A serious issue or these new technologies, Kammen said, is how to move past the so-called
valley o death o tiny market share and high cost. Policy-makers need to set aside money
or these emerging technologies not only in the research stage, but also in the stage o
moving them to market. Te New Apollo Energy Act would be a step in that direction.
clIMatE changE MItIgatIon
FIguRE 6
The Tesla Roadster has been cited oten as a model or hih-perormance
all-electric vehicles. As o 007, no Roadsters are yet available or
prchase, bt reservations are bein taken.
FIguRE 7
Map o installed wind-enery capacity in the united States in 007, in meawatts. Wind
enery potential is reatest in the great Plains and Rocky Montain states, bt so ar the
exploitation o this resorce has been very neven.
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Will our efforts to reduce greenhouse gas emissions be overwhelmed by
the increasing emissions from China?
Yes, but that doesnt give America an excuse or inaction.
China is very dependent on coal energy, and built 9,000 megawatts o new coal-red plants
in 006enough in one year to cancel the entire greenhouse gas reductions pledged byEuropean nations under the Kyoto protocol. As long as the U.S. does not observe the Kyoto
treaty, China will be able to hide behind America.
Even so, China has made major commitments to improve its
energy eciency. Te eleventh Five-Year Plan calls or a 0
percent increase in energy eciency by 010. All o Chinas
leading 1000 enterprises, which together account or one-third o
the countrys energy usage, have made a commitment to increase
their energy eciency by 0 percent. China has also pledged to
derive 15 percent o its energy rom alternative uel by 00.
Although China has recently passed America as the worlds
largest emitter o greenhouse gases, panelist Ogden said that it
is important to realize that it also has a much larger population
base. Chinasper capita energy use is still only an eighth o ours.
It is dicult to tell the Chinese that they must cut back when they
have not yet reached the standard o living and energy use that
Americans enjoy.
Can renewable energy sources be integrated with the
rest of the power grid even though several of them are
intermittent, i.e., not always available?
Wind and solar energy, o course, are not under our control. Te wind doesnt blow when we
tell it to, and the sun shines only during the daytime (and even then it may be obscured byclouds). Fortuitously, the time o peak availability o solar energy coincides with the time o
peak demand. Wind energy can be load-shaped, by using natural gas, or example, to ll in
gaps in availability. Also, pricing schemes called demand response programs can help shi
the demand rom peak hours to other times o day. Battery storage may make it possible to
distribute energy availability more evenly. Finally, some alternative energy sources, such as
geothermal and biomass, donot have any intermittency problems.Panelist Peevey noted that the Caliornia Public Utilities Commission is committed by
statute to obtain 0 percent o our energy rom renewable sources by 010, and committed
by policy to obtain 33 percent rom renewables by 00. He expects that the state will in
act meet or come very close to the ormer target. Recent news reports, however, indicate
that Caliornia is still well below the target, with 1 percent o its energy coming rom
renewables.
What kinds of governmental regulation would PG&E like to see?
Panelist McFadden said there was no doubt that we need caps on carbon production.
However, she elt that it would be a heavy li to get such caps passed at a national level in
the short term. As an intermediate target, she suggested that the rest o the country should
improve its energy eciency as much as Caliornia has. Te per capita usage o energy in
Caliornia has remained constant in recent years, while increasing 50 percent in the United
States as a whole. Because Caliornia is a bellwether or the nation, Caliornia should
continue doing more to improve its energy eciency.
not uniormly distributed (see Figure.) but is greater over land andmuch greater in the Arctic. Increasesin precipitation are very likely inpolar regions and droughts are likelyin subtropical regions.
The consequences othese climate changeswere explored in theWorking Group IIreport. Many biologicaleects are alreadyapparent, such asearlier spring bloomingand shits in the rangeo species. Under eventhe optimistic scenario,
the report states thatabout 0 to 0 percento plant and animalspecies are likelyto be at increasedrisk o extinction. Amodest amount oglobal warminglessthan degreesCentigradewouldbe avorable or global
ood production. However, greatertemperature increases would havea negative eect, and in arid and
tropical regions, even a small risein temperature is expected todecrease crop productivity. Extremeweather events, such as loodsand hurricanes, will become morecommon.
Finally, Working Group III reportedon the potential o mitigationeorts to reduce greenhouse gasemissions. It concluded that optionswith net negative coststhosethat save more money over the longrun than they costcan already
reduce emissions by 7 to 0 percent,compared to the business as usualscenario. Further reductions, up to46 percent, can be achieved withstrong enough incentives, in theorm o carbon trading and carbontaxes or ees.
clIMatE changE MItIgatIon
Fire 2.1 The IPCCs climate chane simlations or the
decade 00-09 let and 090-99 riht, nder three
dierent scenarios. Note the neven distribtion o temperatre
chane, with especially dramatic increases in polar reions.
Image fromClimate Change 2007: The Physical Science Basis, Intergovernmental Panel on Climate Change.
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Is ethanol from corn a boondoggle?
Tis question elicited some disagreement. Panelist Kammen said that his studies show that
ethanol derived rom corn is marginally more ecient than gasoline. Kammen went on to
say, though, that it would be surprising i corn, which has been developed or generations
as a ood crop, happened to also be optimal or use as a uel. (See Figure 8.) Other options,including switchgrass or landll waste, will probably turn out to be better. Also, the net
emissions eect o any biouel will improve dramatically i the distillery runs on a cleaner
energy source. Tere would be no point in building an ethanol distillery and powering it
with a dirty coal-red generator.
Panelist Borenstein remained skeptical. Better ways to produce ethanol also cost a lot,
he argued. He elt that the excitement over ethanol is motivated primarily by the economic
sel-interest o the Midwestern states.
Besides carbon dioxide, methane has also been implicated as a greenhouse
gas. How serious a problem is it?
Molecule or molecule, the greenhouse eect o methane is 0 times stronger than that
o carbon dioxide, but it is not as big a problem or several reasons. Te absolute levels o
methane in the atmosphere are much lower than those o carbon dioxide (though they, too,
are rising ast). Second, methane remains active as a greenhouse gas or only about ten years
beore chemical reactions in the atmosphere break it down. Carbon dioxide, on the other
hand is eectively immortal.
Finally, methane is harder to regulate than carbon dioxide, because much o it comes rom
agricultural sources, such as cattle and rice paddies. However, in this country, the main
sources o methane are landlls and leakage rom coal mines and gas pipelines. Tereore,
an opportunity exists to control it, simply by reducing the amount o leakage and the
amount o waste we put in landlls.
What are the prospects for nuclear power?
Surely one o the most controversial outcomes o climate change has been the rehabilitationo nuclear power. In the audience or this symposium, opinions were deeply divided,
reecting the ambivalence o society as a whole toward nuclear power. (See Figure 9.)
In Caliornia, at least, the legal status o nuclear power is clear. Under state law, no new
nuclear power plants can be built in Caliornia unless and until the state government
certies that there is a sae way to dispose o the waste. With the ate o the Yucca Mountain
nuclear repository still in limbo, it is clear that there will be no new investment in nuclear
power in Caliornia or the oreseeable uture.
Economically, nuclear power is less well understood than any other energy source. Te
costs o waste management and protection against terrorism need to be actored into the
clIMatE changE MItIgatIon
energy economics
Electric uel costs or dierentenergy sources are diicult tocompare. For natural gas, thegreatest expense is the uel itsel.For nuclear power, the cost o uelis relatively small but the cost obuilding the plant, running it saely,and decommissioning it is muchhigher. In addition, the costs maydepend on location; natural gas,or instance, is cheaper in Texas,and geothermal energy is not evenavailable in New England. Thecapital costs or nuclear power are
particularly uncertain because nonew plants have been ordered since977.
Nevertheless, it can be useul tocompare the levelized cost oelectricity, which amortizes thecost o an electric plant over itsentire (estimated) lietime. TheDepartment o Energy estimates theollowing costs or new plants thatwould come online in 05 and in00 (costs are given in cents perkilowatt-hour):
Year 2015 2030
Coal 5.6 5.4
Natural gas 5.5 5.7
Wind 6.9 6. (*)
Nuclear 6. 5.9
Biomass 6.4 (*)
Solar Thermal . (*)
Geothermal 6. (*)
(*) These igures are speciically or aplant located in the Northwest.
Source: Annual Energy Outlook 2007,Figures 56 and 6.
As these igures show, the costs oseveral renewable energy sourcesare expected to come down to the
10
E85 as pmps sell a blend o 85 percent
ethanol and 15 percent asoline. Ethanol has
been hihly toted in some places as a el
that can redce reenhose as prodction.However, the technoloy is problematic at
present, becase the distillation o ethanol
itsel reqires enery that may come rom a
reenhose as-prodcin power plant.
FIguRE 8
Photo:
JohnCross,Minneso
taStateUniversity.
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point where they are nearly, but notquite, competitive with conventionalsources. However, solar energyremains prohibitively expensive.
In the reerence scenario o theAnnual Energy Outlook report,renewable energy sources will notgain any ground as a percentage othe market between now and 00.(See Figure ..) They provided 9percent o the U.S. overall output oenergy in 005, and they are orecastto provide 9 percent in 00 as well.Coal power is projected to increaserom 50 to 57 percent o the market.Meanwhile, the total amount oenergy sold will increase rom 660
billion kWh to 568 billion. Thus thetotal output o energy rom coalwill increase by 60 percentanespecially worrisome outcome orthe climate, because coal plantsproduce the most greenhouse gases.
As noted in the report, changesin uel prices or in environmentalpolicies could aect all o theseprojections.
cost o nuclear power, but no one has any
idea how large these costs will be. Also, the
nuclear industry gets a subsidy rom the
government, in the orm o protection rom
insurance claims resulting rom a catastrophic accident. Depending on your point o view,
the value o this subsidy may be anywhere rom zero to innity.
Even putting aside these unknowns, nuclear energy has an uncommonly large range o costs.
(See Sidebar, Energy Economics.) Te cost o nuclear power presently ranges rom 3 cents to
1 cents per kilowatt-hour. I America is going to embark on an ambitious new program o
nuclear construction, we need to understand the reasons or this broad range o economies,
standardize the designs, and choose designs that are cheaper and saer.
All in all, nuclear power is back on the table. But it seems unlikely that America, aer
shunning it or more than 0 years, is ready or the kind o huge ramp-up that would be
required to have a signicant impact on greenhouse gas emissions. Te problems o saety
and waste disposal are not mere public relations.
What is the status of carbon sequestration?
Sequestration reers to the process o burying carbon in the ground, the ocean, or in
vegetation and soils. One method o sequestration involves injecting pressurized carbon
dioxide into an oil eld. Tis procedure can help companies extractmore oil rom it, so the process is sometimes called enhanced oil
recovery. Once injected, the carbon dioxide willhopeully
remain isolated indenitely rom the atmosphere. Whether this is
true in act remains an open scientic question.
Carbon sequestration is attractive to large oil companies because it
requires a minimal change rom business as usual (and, in act,
can be seen as improving business). Several sequestration projects
are already in place, in exas, in the North Sea, and in Norway.
BP has recently announced plans or a new clean energy plant in
Caliornia, which would separate petroleum cokea very dirty
uel that is currently shipped to China or burninginto hydrogencompounds and carbon dioxide. Te hydrogen compounds would
be burned cleanly, while the carbon dioxide would be sequestered.
What are the prospects for carbon cap-and-trade agreements and
carbon taxes?
Although carbon cap-and-trade agreements may be a useul and even essential mechanism,
panelist Borenstein said that he does not consider them a solution by themselves to the
problem o greenhouse gases. Somebody, somewhere, has to cut back on the production o
carbon. Another unresolved question is how to enorce the agreements so that no one can
cheat on them.
clIMatE changE MItIgatIon
Fire 3.1 Sorces o united States
electric power, historically and projected
throh 030. Note that enery sed or
transportation or or heatin does not
appear in this re.
FIguRE 9
The Three-Mile Island nclear power plant in Pennsylvania is a symbol
o nclear enerys trobled past. For many years, no new nclear
plants have been bilt in the u.S. becase o concerns abot saety and
storae o spent el. With climate chane now loomin as a reater
threat, even some ormer opponents o nclear power are beinnin toreconsider this carbon-netral enery option.
11
Electricity Generation by Fuel, 1980-2030
billion kilowatthours3500
3000
2500
2000
1500
1000
500
0
1980 1990 2000 2005 2010 2020 2030
2,094
5,478
Electricity Demand
ProjectionsHistory
Coal
Natural Gas
Nuclear
Renewables
Petroleum
20301980
Annual Energy Outlook 2007
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maThemaTics and renewaBle energy
How can mathematics contribute to the development orenewable or alternative energy sources?
Tis question was not discussed specically at the symposium.
However, some areas where mathematicians are currently
making contributions include:
Fuel cells. Te membranes in a uel cell are made o a porous
eon-like material, which allows ions to pass through. Te
process o pore ormation involves dierential equations that
have not been solved beore. A good mathematical model o
the pores might bring down the cost o uel cells, by reducing
the amount o platinum required as a catalyst.1
Wind energy. Te mathematical problems in designing a
wind turbine are similar to those in designing airplane wings.However, to maximize energy eciency, these wings have
to push the limits o size and weight. Large, lightweight wings
tend to utter, so engineers need methods to predict and
automatically compensate or this behavior.
Carbon sequestration. Mathematical models o porous media
are used to predict how long carbon dioxide will remain
underground. One recent study showed that abandoned
oil wells may compromise the ability o an oil eld to store
carbon dioxide.3
Nuclear energy. Mathematicians are helping to design the
next generation o reactors. For example, researchers use
computational uid dynamics to model the ow o coolant
past a uel pin. Tey have showed that wrapping wire around
the pins, like a stripe on a barber pole, can improve the mixing
o coolant and bring down the temperature o the pins.4
Wave energy. Harnessing energy rom ocean waves is
still a technology in its inancy. Engineers used nonlinear
optimization, a mathematical technique, to design a generator
that produces energy rom the relative oscillation between two
oats. Te product is expected to go on the market in 010.5
1 Keith Promislow, NSF Award Abstract # 0708804.2 A. Balakrishnan, NSF Award Abstract # 0400730. Barry Cipra, Geosciences Conerence Tackles Global Issues, SIAM News,
June 007. P. Fischer et. al., Large Eddy Simulation o Wire-Wrapped Fuel Pins I: Hydro-
dynamics in a Periodic Array. Joint American Topical Meeting onMathematics and Computation and Supercomputing in NuclearApplications, 007.
Scott Beatty, Capturing wave energy o the coast o BC a profle o anintern, MITACS Connections, May 007. The product is the SyncWave PowerResonator.
Nevertheless, carbon cap-and-trade agreements are popular
in Washington because they use market orces. In the present
political environment, Congressman McNerney said, it is simply
impossible to talk about carbon taxes, even i they are calledees. Te minute he arrived in Washington, his opponents
began painting him as an advocate o carbon taxes, even though
McNerney had never advocated them. Assembly Member Ruskin
strongly echoed this last point. He recalled a conversation with
an environmentalist in Europe who said that his country had
wasted ten years debating a carbon tax. We need to debate
things that are possible, Ruskin concluded.
How will climate change affect developing countries?
It seems certain that some o the eects o climate change will
hit developing countries hardest. For example, subtropical and
tropical regions are more likely to be subjected to drought. Low-
lying island nations will be threatened by rising sea levels. Most
importantly, poorer countries will not have the resources to
adapt to climate change, while wealthier countries will. For all o
these reasons, plus simple cost-eectiveness, investing in energy
eciency is the airest and most universal approach to mitigating
climate change.
What can individuals do about climate change?
For individuals as or countries, the most cost-eective solutionis to reduce consumption through energy eciencyor
example, changing rom incandescent to compact uorescent
lightbulbs. Several Caliornia communities provide good
examples o action a local level. For example, Palm Desert has
reduced its energy usage by 30 percent.
While individual and local conservation eorts are important,
panelist Fung noted that there is one other remedy that citizens
should be ready to use: the vote. Te problem is so large that
we need state and government-level action. Tat means voting,
Fung said. Congressman McNerney noted that the League o
Conservation Voters Dirty Dozen list had proven very eective
in the 006 election. Assembly Member Ruskin added that
McNerney was being too modest, because he had personally
deeated one o the Dirty Dozen incumbents in 006.
clIMatE changE MItIgatIon
12
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theScientificWorkShopportionofthe mSri SympoSiumon
Climate Change convened in Berkeley on April 1 and
13. (See gure 10.) In this session, the ocus shied
rom local action to global models, and rom energy
policy to climate issues.Te symposium was organized into six groups o
lectures (see Appendix A), which provide the source
material or this section. In addition, discussion groups
were ormed to identiy research problems in climate
models that would be amenable to mathematical
research. Te ollowing two sections, Research
opics in Climate Change and Opportunities and
Challenges or Mathematical Sciences, are based in
part on the reports o the discussion groups.
Te questions below give a representative, though not exhaustive, sample o the issues
discussed in the lectures.
What goes into a climate model?
Te main components o a climate model are the atmosphere, the ocean, land, and ice. As shown
in Figure 11, the atmosphere model incorporates our main dierential equations, which relate
the motion o air to the physical inputs. First, the momentum equation relates the acceleration o
any parcel o air to the orces on it: the pressure gradient, gravity, and riction. Tis equation also
includes the Coriolis orce, rom Earths rotation, and a nonlinear inertial term. Te conservation
o mass equation says that matter is neither created nor destroyed. Te energy equation says
that the energy o a unit o atmosphere can change in two waysby changing the temperature
or by advection (conveying the warm or cold air somewhere else). Te net o these two eects is
governed by our energy inputs: short-wave radiation rom the Sun, long-wave radiation rom
Earth, sensible heat, and latent heat (the heat stored or released in water when it changes phase).Finally, a separate water vapor equation says that the amount o water in the atmosphere changes
by advection as well as by evaporation or condensation. Tis equation determines the water
vapor content o the atmosphere, which in turn aects its density and pressure, and in this way
eeds back into the momentum and mass equations.
Uncertainties in these equations enter on the physics side. How much energy is coming
in rom the sun? How much is reected into space by clouds or aerosols? What is involved in
the turbulent mixing in the atmosphere, which governs the ormation o clouds? Te eect
o convective mixing is added in as an extra term in the momentum, energy, and resh water
vapor equations. Every climate model does this dierently.
Te ocean models likewise contain equations or momentum, mass, and energy, plus a ourth
equation describing the salinity. Te ocean exchanges momentum, energy, and water with theatmosphere, so these equations are linked to the previous our. Salinity aects the ocean in much the same way that water content
aects the atmosphere: it changes the waters density, which in turn changes the pressure gradient. wo very important parts o the
ocean model are the wind-driven ocean currents at the surace, and the thermohaline circulation, which takes place deep in the
ocean. (See Figure 1.) Te thermo part o this word reects the act that cool water tends to sink, and warm water tends to rise.
Te haline part reers to salinity, and the act that saltier, denser water tends to sink, while less dense resh water tends to rise. Te
interplay o these two actors creates a worldwide conveyor belt o water that redistributes heat rom the equator to the poles, and is
believed to have a strong moderating eect on our climate.
Like the atmosphere models, the ocean models are complicated by convective mixing. Tey also have to deal with the complicated
geometry o coastlines and the ocean oor. Te rearrangement o continents has had a huge eect on ancient climates. However, on
a time scale o hundreds or thousands o years, the arrangement o land masses can be assumed constant.
clIMatE changE ModElIng
13
Eqations o a typical climate model. The rst dierential
eqation refects conservation o momentm, and the
second expresses the conservation o mass. This eqation
is copled to the rst by the ideal as law line 3. The thir
dierential eqation line 4 models the enery fx, with
short-wave radiation SW and lon-wave radiation LW.
The latter term incldes the eects o carbon dioxide CO2
and other reenhose ases gHg. The nal dierential
eqation tracks the motion o water in the atmosphere,
and mst be copled to an ocean model. Several o
these eqations also inclde terms red boxes that
model convection in the atmosphere, which is still poorly
nderstood becase it occrs at sch a small scale the
scale o individal clods.
FIguRE 11: ATMOSPHERE
FIguRE 10
MSRIs scientic symposim on climate chane broht abot 80
mathematicians and climate researchers to Chern Hall. Christopher
Jones ar let, acin camera was instrmental in oranizin the
two-day symposim.
u+ u u + 2 u = 1p +gk + F + (u)
+ (u) = 0
p =RT;= (T,q)
T+ u T = SW
+LW
+SH + LH + (T)
SW =f (clouds, aerosols . . .)
LW = f (T,q,CO2, GHG . . . )
q + u q = Evap Condensation + (q)
convective mixing
t
^
t
t
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Figure 13 represents in schematic orm the various processes that enter into a climate
model. As described above, o the processes illustrated here, the most challenging or
modelers to get right are the clouds (which are too small-scale or the current generatio
o models to describe accurately) and the ocean currents, including the vertical motion
Te modeling o the solid phase o water presents its own peculiar problems. (See
Sidebar, Te Sea Ice Conundrum.)
Finally, there are signicant unanswered questions about the amount o incoming solar
radiationthe solar constant, which is currently estimated at 136 watts per square
meterand how constant it really is. Te total amount o anthropogenic orcing o the
climate since 1800 is estimated at 1.6 wat
per square meter. Tus, even a tenth o o
percent variation in the solar constant
would equal the entire human impact on
the world climate. At present, there is no
evidence that the solar constant has varie
that much in the last 00 years. However
it may have varied by that much in thepast. What will happen to it in the uture
beyond the expertise o climate modeler
who have to ask solar physicists or the
answer.
What is left out of climate models?
Figure 13 omits one important ingredien
in the climate: the entire carbon cycle.
In act, this gure represents the status
o climate models about 5 years ago,
when the work behind the ourth IPCC
Assessment was being done. At that timethe concentrations o greenhouse gases
like carbon dioxide and methane had
to be added in as an exogenous orcing
term. Newer models are beginning to
incorporate the carbon cycle: the eects
o plants and animals, the eects o ossil
uel burning, and the dozens o chemical
reactions that convert one orm o carbon
to another in the ocean.
Another omission will be even more
challenging to repair: None o the modelcontain any humans. Again, in the
IPCC simulations the results o human
activities (primarily the production o
greenhouse gases) are simply added in
by at. However, such an approach is
not completely satisactory. Even in the
absence o deliberate governmental policies, the change in climate will produce change
in human behavior. Dierent crops will be planted, dierent regions o the world will
become suitable or unsuitable or agriculture, and so on. A truly integrated model shou
include these eects (see Sidebar, Climate and the Indian Rice Crop, page 16).
clIMatE changE ModElIng
1
The sea ice conundrum
One o the most dramatic yet leastunderstood eects o global warming istaking place in the Arctic Ocean, where
both observational data and climate modelspoint to a rapid melting o the polar icecap. The extent o the ice cap at the peako its summer melting has been decreasingby 8 percent per year since 979. Thearea covered by sea ice in the winter hasnot decreased as rapidly, because the icepack tends to recover during that season.However, the thickness o the ice cap inwinter is decreasing. As the amount orecovery during the winter decreases, theextent o the ice pack in summer will alsotend to decrease.
It is well known that melting sea ice causesan ampliying eedback loop, called the ice-albedo eedback, which tends to exacerbateglobal warming. Melting ice leaves moreopen water exposed, which in turn absorbsmore solar energy rather than reectingit into space. All o the climate modelsincorporate this eedback loop, and as aresult they predict much steeper temperatureincreases in the Arctic than worldwide (SeeFigure 4.).
Unortunately, sea
ice is also one o theleast well-understoodingredients in theclimate change puzzle.Not only is the amounto warming expected inthe Arctic greater thanthe rest o the world,but the uncertaintyin this orecast is alsogreater. In the IPCCclimate models, whilethe equatorial regionsace a to 4-degree increase by the end
o the century, the North Pole region ispredicted to warm up by 4 to degrees.And the dierent models or the extent osea ice vary extravagantly (see Figure 4.).Some o them show the summer ice packvirtually disappearing in the Arctic Oceanby mid-century, while others predict onlya moderate decrease. This igure is more aconession o our ignorance than a reliableprediction. (Actual observations in thisigure are shown by the heavy red line.)
Fire 4.1 Predicted winter temperatre increase by mid-
centry 040-59 aainst 1980-99. Winter warmin in the Arctic is
at least doble the lobal mean and peaks at more than 16 derees
Centirade in some places.
Fire 4.2 Projected smmer ice extent in the Arctic, in the
bsiness-as-sal scenario. The heavy red line represents observed data;
the remainin lines represent 18 dierent climate models sed by theIPCC. Dierent models disaree widely, de to dierent assmptions
abot sea ice physics. However, the rate o retreat o ice in the 1st
centry is sinicantly correlated with the mean ice extent in the late
0th centry.
Image fromClimate Change 2007: The Physical Science Basis,
Intergovernmental Panel on Climate Change.
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Furthermore, climate modelers realize that in the new environment, decision
makers will be consulting their models more and more requently, and they will ask
dierent sorts o questions. Instead o How much will the temperature rise? they
will ask How much will it cost? or What are the impacts? In other words, climate
variables will eventually have to be restated in economic or social-justice terms. Some
preliminary eorts to do this have been made. For example, the Stern Review in the
U.K. was an attempt to delineate the economic impacts o climate change. Another
example, presented at this meeting, was Max
Aufammers study o the eects o climate
change on agriculture. Nevertheless, a true
integration o climate and economic models
remains in the uture.
How might economics enter into
climate change models and strategies?
In order to ormulate the results o climate
change in economic terms, modelers will haveto learn rom the great advances economists
have made in quantiying risk. However,
some conerence attendees expressed concern
at a too narrow, market-centric approach to
dening risk. First, such an approach might
not give adequate weight to the interests o people who do not participate in nancial
markets, such as native peoples, developing countries, or unborn generations.3 Also, a
purely economic approach might downplay the importance o outcomes such as species
extinctions.
Possibly a separate issue, but nevertheless important, is the question o how we can
economically reach a desired emissions target. Can we get there using market orces and
cap-and-trade agreements? Do we need a carbon tax? Some attendees suggested using
game-theory approaches to design agreements that would be sel-enorcingin other
words, to give both parties an economic incentive to abide by the agreement.
How does a climate
model differ from a
weather model?
Te physical equations in a
climate model are similar
to those in a weather
model, and some speakers
argued that there is noreal dierence between
them. However, the time
scales involved are vastly
dierent, and the nature o
the questions asked o them
is dierent as well. Weather
models track the evolution
o weather systems, and lose
3 The Stern Review took a very hard-line position on this issue, arguing that all generations should be treatedequally, which implies a discount rate o 0 percent. Other economists have questioned this assumptionand argued that it leads to an unrealistically high estimate o the current cost o climate change.
clIMatE changE ModElIng
1
The models perorm so erratically orseveral reasons. First, they vary up to 50percent in their estimates o cloudiness.That translates to a variation o 40 wattsper square meter in the amount o
solar energy reaching the suraceanuncertainty that swamps the greenhousegas eect. (Remember that theanthropogenic change in carbon dioxideaccounts or .6 watts per square meter.)In addition, none o the models treat seaice in a physically realistic way. They havejust begun to incorporate the thickness oice as well as its extent, and they do notyet include an estimate o the loe sizedistribution. As loes get smaller, there ismore contact between ice and water andhence more rapid melting in summer (andreezing in winter). In general, climate
models treat sea ice as a homogeneousand continuous medium, but bothassumptions are wrong. Sea ice varies inthickness and composition, and it is highlyractured.
Why is it important to model sea icecorrectly? First, the regional impacts arehuge. The opening o the long-soughtNorthwest Passage in the Arctic Oceancould be an economic boon to Canada, oran ecological nightmare. The liestyles onative populations would be threatenedby the retreat o the ice. Species like the
polar bear, which depends on the ice,would suer even more.
In addition, the extent o sea ice hasglobal ramiications. No, the melting osea ice does not raise the sea level (apopular misconception), because theice and water are already in hydrostaticbalance. But the melting o land ice wouldcause sea levels to rise. The increase intemperatures caused by the ice-albedoeedback aects glaciers on land, tooand indeed, observations show that theice on Greenland is melting even aster
than predicted. Finally, melting o seaice also reduces the salinity o the ocean,an important ingredient in all climatemodels. An extreme possibility would bea shutdown o the oceans temperatureand salinity-driven circulation. This is thecirculation that brings warmth rom thetropics to the mid-latitudes, by poweringocean currents such as the Gul Stream.Such a shutdown would produce anegative eedback, and would alter theclimate dramatically.
FIguRE 1
The thermohaline circlation o sea water, driven by
dierences in temperatre and salinity, has a major impact
on world climate. The shtdown o this circlation is oten
cited as a tippin point that cold lead to dramatic lobal
coolin.
FIguRE 13
This diaram illstrates the variety and complexity o interactions that enter into the crrent
eneration o lobal climate models. Note that the crrent models omit one key inredient: the
eedback between the climate and hman activities.
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accuracy aer our or ve days, and there is no real point in running them beyond a ew month
On the other hand, or climate predictions we are interested in running the models decades, even
100 years into the uture.
Given the dierence in time scales, an uncharitable observer might wonder whether a climate
model is anything more than an expensive random number generator. Te answer is yes and nbecause the type o question one asks o a climate model is dierent.
Climate is, by denition, the statistics o the weather o an area over a long period o time,
including the long-term mean, the variability, and the extremes. Te mean climate is what
remains o weather aer you average out all the uctuations. Numerical weather orecasting
is, by contrast, all about the uctuationshow the weather yesterday is going to change today,
tomorrow, and so on. Te goal o a mathematical model is to capture the average state, and eve
a probability distribution o deviations rom the average, and is not to predict, say, the wind in
Edinburgh on December 13, 080. I one imagines boiling water in a pot, weather prediction
is analogous to describing the location o the bubbles, while climate describes the temperature
in the pot. From another perspective, weather prediction is an initial-value problem whereas
initialization is less important in the climate. Predicting tomorrows weather is based on
inormation about todays. But in climate change the predictions are about seeing what happen
under dierent orcing scenarios. For instance, how will the climate system respond to the
doubling o CO2, or a change in the amount o energy rom the Sun?
However, there remain some serious issues with climate models that make them a good deal
less predictable than the temperature o the heated water (in the analogy above). First, climate
models cannot be completely tested and validated, while weather models are validated every
day. Te climate orcing o a century ago is poorly known, and observations o what actually
happened are sparse. Climate models are assessed by plugging past orcing data (e.g. aerosols
rom volcanic eruptions) into them and comparing the predicted climate states with available
observations.
Unortunately, though, some models are already using the observations to estimate modelparameters. Tat makes it impossible to validate the model independently with past data. Finally
even i a model tested out satisactorily against the more static, pre-1950 climate, it would not
necessarily give correct answers or the changing climate o today because the processes that
are important or the uture climate, such as sea ice and glacial dynamics, may not be operating
in the same way in the static early 0th century. Another dierence between weather and
climate models is that weather models are constantly assimilating new data to update the initial
conditions or the next prediction. omorrows orecast will be based on a combination o today
orecast and the new observations accumulated over the next 4 hours by weather instruments
and satellites. Tis allows the weather models to get back on track quickly aer an unsuccessul
prediction. On the other hand, i a climate model gets o track by 030, its predictions or 100
may be completely invalid.
Why do different climate models disagree?
First, it is worth pointing out that there are signicant areas o agreement. All the climate model
agree that global warming is a reality, and their predictions or 030 are also in rough agreement
Teir predictions or 100, however, span a wide range.
One reason or the wide range is that the models prioritize dierently the processes on the
physical side o the equationsparticularly the processes that are not well understood, such
as convective mixing in the atmosphere and ocean, and the ormation o clouds, and hence
represent them dierently. o some extent, this divergence among models is a good thing. Most
clIMatE changE ModElIng
climaTe change and The
rice harvesT in india
A recent paper published in theProceedings o the National Academy
o Sciences1 exemplies the insightsthat can be obtained rom anintegrated model that combines
climate and economy. Global climatemodels show that the efect ogreenhouse gases is reduced, tosome extent, by industrial haze inthe atmosphere. Aerosols absorbsolar radiation and release it back tospace, thus reducing the energy thatreaches Earths surace rom the sun.
The PNAS study highlights aneconomic system where greenhousegases and aerosols have a comple-mentary, not ofsetting, impact: the
Indian rice market (see Figure 5.).Rice grows better when nighttimetemperatures are cool, whichsuggests that greenhouse gaseswould reduce rice output, while theIndo-Asian haze would increaseit. On the other hand, rice requiresplenty o rain during the monsoonseason. But the Indo-Asian hazetends to reduce rainall, by reducingthe temperature gradient betweenthe southern and northern IndianOcean. Thus a purely climatic
viewpoint leads to ambiguousconclusions or the efect o aerosols.
M. Auhammer, V. Ramanathan, andJ.Vincent. Integrated model showsthat atmospheric brown clouds andgreenhouse gases have reduced riceharvests in India, Proc. Natl. Acad. Sci. 03
(006), no. 5, 9668-967.
Fire 5.1 Rice harvest, Kashmir, Pahalam, India.
1
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climate modelers agree that there is no such thing as a best model, and it is useul to have a
variety o models to sample the space o dierent possibilities. In act, when weather models
are run on the time scale o months, to make seasonal predictions, an ensemble o several
models will usually perorm better than any individual one.
In addition, a certain amount o tuning o the models is standard practice. Ideally, this is
justied on the grounds that climate scientists can use the observations o dierent climate
variables (e.g. cloud top height and sea surace temperature) to deduce the best parametric
relationships linking them. But in practice, tuning is, as one participant said, a subterranean
process where all thats reported is the outcome. Some models are tuned to the point where
they actually violate well-known laws o physics. It might be desirable to discard or discount
models that are known to be less trustworthy, but politically this is hardly easible.
Finally, another reason that models dier is that the climate system itsel is inherently
unpredictable. Precipitation is especially dicult to orecast accurately (see Sidebar, Rainfall:
Beyond Its Warmer, So Its Moister). Even a mathematically exact model started rom two
slightly dierent initial conditions may not be able to issue similar precipitation orecast a
season ahead, because precipitation processes, such as evaporation and condensation, are
inherently non-linear. At best, it would oer a range o possibilities and a most likely caseand indeed, this is the way that the IPCC presents its model results. Te chaotic dynamics
within the climate system make it impossible to do better.
Tis inherent uncertainty may explain why a suite o models will outperorm a single model.
A well-designed ensemble might be able to sample dierent parts o parameter space or
model space and in this way more clearly outline the uncertainties in the climate orecast.
Tere was a very strong consensus at the symposium that communicating the uncertainty
in the model predictions was a dicult and important challenge or climate modelers. One
speaker worried that as the models improve, they will inevitably give slightly dierent answers
rom the old ones, and it will look to the public as i the climate modelers are changing their
mindswhen in act the new predictions may lie within the error bars o the old predictions.
Tis is not merely an academic concern, as proved by some o the press coverage o the
ourth IPCC report. Te media made a uss over the act that the predicted rise in sea levels
was not as great as in the third IPCC assessment. Did this mean that global warming was not
going to be as bad as predicted? Not at all. It meant that the uncertainty had been improved,
and in act the modelers had been more honestthey had no longer attempted to quantiy
the uncertainty in sea ice melting, because the process is not well enough understood. An
improved product turned into a black eye or the modelers, as they were orced to explain
that they werent backing down on the dire consequences o global warming.
How can we combine the results of different models?
In general, the IPCC averages the outcomes o the dierent models and reports an ensemble
mean, along with error bars representing a 66 percent condence interval. Such an approachwould be statistically valid i the models represented independent random samples rom a
clIMatE changE ModElIng
However, a combined climatic-economic analysis tells a diferentstory. When early-season rainalls allshort, armers respond by shitingthe acreage planted in rice to othercrops. In this way, economic actors
enhanced the impact o the aerosols.The article concluded that, over theperiod rom 985 to 998, aerosols ledto a 0.6 percent reduction in the riceharvest, compared to the harvest ina simulated climate without aerosols.The combination o aerosols andgreenhouse gases reduced the riceharvest by 4.4 percent over the sameperiod o time. These results coincidedwith a period when Indias riceproduction, which had grown rapidlyin the 970s and early 980s, beganto grow more slowly and eventuallyleveled of. The study suggests thatthe increasing levels o aerosols andgreenhouse gases in the atmospherewere responsible.
The interaction between climate andhuman behavior, driven by economicactors, was crucial or understandingthe efects o the aerosols. Most o theefect isnt on the plants themselves,but on the armers shiting to othercrops, Aufhammer said. In spite othe titles description o an integratedmodel, the interaction betweenclimate and economy in his paper wasairly simplistic. Aufhammer simplytook the outputs rom a climate modeland plugged them into a regressionequation to predict the armersresponse. In the uture, he says, climatescientists and economists should worktogether on the same model. Insteado merely downloading data, we needa spirit o true collaboration acrossdisciplines, he said.
1
Fire 6.1 Precipitation changes or the decade 090-099,relative to 980- 999. Business-as-usual scenario, December-February (let) and June-August (right). White regions indicate
where ewer than two-thirds o the climate models used orthe IPCC report agreed on the direction o change; shading
indicates where more than 90 percent o them agreed.
Image fromClimate Change 2007: The Physical Science Basis, IntergovernmentalPanel on Climate Change.
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single model space. However, the existing 4 models used in the IPCC report are in no
way a rationally designed, systematic exploration o model space. Tey are a sample o
convenience. Moreover, as some participants pointed out, there is a danger o throwing
out the physical baby with the statistical bathwater. Dierences between models may resu
rom physical phenomena that are represented correctly in one model and incorrectly in
another. Obviously, it will be a challenge to modelers to try to distinguish chance eectsrom dierences with real, physical causes.
One speaker illustrated the problem with averaging by a colorul parable. Tree
statisticians are asked whether it is sae to cross a river. Tey construct separate models
o the river, and each one nds that the river is deeper than 6 eet in some place. But they
disagree on where. So they average their models, and nd that in the ensemble mean,
the river never gets deeper than 3 eet. As one might guess, the parable ends with the
statisticians drowning. (See Figure 14.)
All in all, there must be a better way than taking a mean. A weighted average, which takes
into account each models strengths and weaknesses, might be an improvement. Even
better would be a Bayesian (machine-learning) approach, described by one speaker. In thi
approach, one model is omitted rom the ensemble, and then treated as a new model tha
changes the a posteriori probability o various climate outcomes. Ten a dierent model
is le out o the ensemble, and the process is repeated. Aer this process is repeated many
times, one can bootstrap up to a reasonable weighting o the dierent models.
How can we downscale global models in order to obtain local predictions?
How can we upscale local effects to incorporate them in global models?Several modelers elt that the issue o unresolved processes or sub-grid processes
was crucial. Tey are besieged with questions like, What will happen to this species?
or How will this aect the water supply in that state? For elected ocials, what really
matters is what will happen in their community or their constituency. I the climate
modelers shrug their shoulders and say they dont know, they will lose credibility (even
i thats the honest answer).
Te one obvious solution is more computing power, in order to resolve the models down
to smaller and smaller grid sizes. As computers have steadily increased in power, the
resolution o climate models has improved as well. For example, as seen in Figure 15, the
grid sizes in the IPCCs our assessment reports, over a period o less than two decades,
have shrunk rom 500 kilometers to 110 kilometers. Even so, the grids o all global models
are too coarse to resolve individual clouds, or even to represent a hurricane realistically.
Besides increasing computer power, there are several other options or modeling subgrid
processes. One is adaptive
mesh renement, in
which the size o the grid
is reduced in regions that
require more detailsay, a
storm system or a mountain
clIMatE changE ModElIng
rainFall: Beyond iTs
warmer, so iT s moisTer
The publics attention in discussions oclimate change has always tended toocus on the increase in temperature.Indeed, the most popular term ormany years was not climate changebut global warming. However, someo the most disruptive eects oclimate change are likely to involveprecipitation: severe storms, loods, ordroughts.
It makes sense that an increase in globaltemperatures should lead to an increasein global precipitation. Warmer air canhold more water vapor, and with morewater vapor in the atmosphere thereshould be more clouds and eventuallymore rainall. However, common sensecan be misleading. Where water vapor isconcerned, its not necessarily true thatwhat goes up must come down. Thewarmer air could simply hold onto theextra water. For this reason, the IGCCreport predicts only a to percentincrease in global precipitation perdegree o global warming. However,satellite observations disagree: Over thelast 0 years, the precipitation increasehas been closer to 7 percent per degree
o warming.
Precipitation also has a much morecomplex pattern o local and regionaleects than temperature. Indeed, it ishard to ind any place in the world thatwill have a decrease in temperaturebetween now and 00. Butprecipitation will decrease dramaticallyin some places, while increasing inothers (see Figure 6., page 7). Evenunder the conservative assumptionso the climate models, many areas arepredicted to have precipitation changes
well over 0 percent.
Unortunately, the dierent climatemodels used or the IPCC reportdisagree strongly on the regionaldetails (see Figure 6.). Given the extento disagreement, can we say anythingsolid about rainall?
F.J. Wentz et.al., How Much More Rain WillGlobal Warming Bring?Science 37 (007),33-35.
1
Fire 6.2 Two o the models thatwere used in the IPCC orecast
disagree on the precise locationand magnitude o precipitation
increases or decreases. Never-theless, the overall message o the
models is airly consistent, with increased precipitation in the tropics and decreased precipitation in thesubtropics. (Units in the igure are 0. mm o rain per day, with increases in green and decreases in red.
Image fromClimate Change 2007: The Physical Science Basis, Intergovernmental Panel on Climate Change
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range. As one speaker pointed out, this
has to be done with caution because
it can lead to unrealistic artiacts
along grid boundaries. Methods do
exist or understanding what causes
these artiacts and controlling them.Another speaker discussed Levy noise,
which would allow or a more realistic
depiction o atmospheric turbulence.
Everybody is looking orward to
petaop computers, which might bring
cloud-scale processes into the picture
or the rst time. However, as explained
next, Moores Law has some caveats.
What kid rests ca weaticipate rm ext-eerati
cmpters?
In a word, the world is going parallel.4
Moores Law (which says that the number
o transistors on a chip doubles every
year and a hal) is still going strong, but
clock speeds are not keeping up, because
the heat density inside todays chips is
getting too high. Parallel processing
is a way to compensate or the lack o
improvement in speed. Te most powerulprocessors today are, ironically, made or
computer games, and they typically have eight
cores. Programming or these machines is not
easy, and it may not become easy until new
languages are invented.
It is not clear that climate modelers are ready or the new computing environment. Teir
programs typically have hal a million lines o code, and it will be a non-routine task to
convert them to work on parallel processors. Climate modelers will have to think about
what algorithms can work
eciently on parallel
processors. For example,
adaptive mesh renement,
though it is desirable or
other reasons, is very
tricky to implement on
a parallel machine. In all
likelihood, it is not the
climate modelers who
will have to solve these
problems but the postdocs and graduate students whom they hire. But this talent will not4 This section is based on a presentation by Kathy Yelick, Architectural Trends and Programming Model
Strategies or Large-Scale Machines.
clIMatE changE ModElIng
In act, according to David Neelin,the situation is not as bad as it looks.The predictions do ollow a patternthat makes physical sense, which hecalls the rich-get-richer model o
precipitation. The regions that willsee the greatest rainall increase areprecisely the ones that get the mostrainall now, the tropical latitudes.And the big decreases in rainall willoccur at the edge o those regions,where increased advection will bringdry weather in rom the subtropics.Thus the models agree on the physicalprocesses. They disagree on theprecise location o the wet and dryspots because o dierences in windcirculation rom model to model.
In a ew regions the models didproduce consistent predictions. Inthe Caribbean, nine or even all teno the ten models in Neelins surveyagreed that there will be a more than0 percent drop in precipitation. Andindeed, 50-year precipitation recordsin the Caribbean already show apronounced decrease (See Figure 6.).Neelin concluded that these regionsneed to take the climate orecasts veryseriously.
Climate modelers do need a betterunderstanding o the convectivethreshold, the point where a moistcolumn o air starts to precipitate. Theonset o convection is usually describedby quasi-equilibrium models, butaccording to Neelin, these make theprocess appear too smooth. The resultis too many gentle rain showers andnot enough extreme weather events.He presented an alternative model,developed in conjunction with OlePeters, which describes convection
in a similar way to other thresholdphenomena in statistical mechanics.A rainstorm is like an avalanche, witha slow buildup and a ast release, sothe statistical requency o mild andintense rainalls should resemble thato small and large avalanches. Thoughstill relatively untested, Neelin andPeters interdisciplinary approach mightind a place in uture climate models.
1
Fire 6.3 One region where the IPCC models
substantially agree is the Caribbean Sea, whereprecipitation records over the last 50 yearsalready show a signiicant decrease in rainall,
which is expected to continue. Red shadingindicates the observed amount o decrease
over the past 50 years, measured in unitso 0. mm per day.
Parable o the statisticians ater Lenny Smith. Three statisticians
independently orecast that the river is nsae to cross, bt averae o their
proles o the river bottom red dashes indicates that it is sae. Smith told
this story to illstrate the daners o relyin on ensemble means, instead
o critically examinin each model on its own merits.
FIguRE 14
HadCM3
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come cheap. Climate modelers will have
to compete with the big money being
oered to these programmers by game
companies.
o run a global circulation model with
a 1-kilometer grid size, which would
be detailed enough to allow or the
modeling o clouds, a back-o-the-
envelope calculation suggests that
climate scientists will need a 10-petaop
computer, with 100 terabytes o memory
and 0 million processors. Both IBM and
Japan have set targets o developing a 10-
petaop computer by 01.
What ideas ca mathematicias
ctribte that cimatemdeers
dt eve kw abt et?Te main purpose o this workshop
was to inorm mathematicians about
climate modelingnot vice versa. It is
hoped that the active participation o
mathematicians will lead to new insights
or new ways o doing business that the
climate scientists have not anticipated.
Te two ollowing sections contain a much
more thorough discussion o the possible
role o mathematicians. However, two
points may be worth mentioning herebecause they came up repeatedly at the
symposium:
Mathematicians like to work rom simpler models to more complex
ones. Many o the mathematicians in the audience expressed serious
reservations about being able to carry out serious mathematical
investigations on models o such complexity. Mathematicians should
not try to re-invent the wheel by designing their own models. However,
climatologists do use smaller models, so-called Earth models o
intermediate complexity, or intuition-building. Tey also use simpler
models or process studies aimed at revealing phenomena and
cause-eect relations in more localized settings. For mathematicians
interested in working on climate change, these models may make a
good entry point.
Inormal discussions seemed to show that climate modelers have a
ew misconceptions about dynamical systems. Even i a dynamical
system is inherently unpredictable because o chaos, some aspects o
its behaviorthe probability distribution o its statesmay in act be
tractable. Te technology transer o ideas rom stochastic dynamical
systems to climate models has not happened yet.
Cimate mdei
A large amount o eort continues to go into modeling o climate. Te
clIMatE changE ModElIng
Each eneration o climate models has sed
ner and ner rids. The labels FAR, SAR, TAR,
and 4AR reer to the rst, second, third, and
orth assessment reports o the IPCC. The
next eneration may nally be able to model
individal storms. However, to achieve this level
o renement, climate scientists will have to
adapt their prorams to rn in a new, massively
parallel comptin environment.
20
FIguRE 15
motion o uid is described by partial dierential
equations (PDEs), using the ramework o
continuum mechanics. Forcing terms, or external
inputs, or the uid equations come rom thephysics, chemistry, and biology o the atmosphere,
ocean, land, and cryosphere. Te main questions
are what eects to include in the model, how to
include them accurately and eciently (when their
eects range over several orders o magnitude),
and how much o an impact they will have on the
prediction. It is important to note that there is no
clear consensus o what needs to be included in the
model.
It is not out o the realm o possibility to remove
Newtonian physics rom the equations entirely.
Tis radical proposal has a precedent in molecular
biology, where the most successul models
o protein olding do not model the protein
molecules rom rst principles, but instead use an
empirical approach based on learning rom data.
A particular challenge lies in the act that the
continuum model response, and the orcing terms
going into the model, vary in time scales o hours
and days, while the predictions we need involve
the coarse behavior in time windows o decades.
Many established climate models have their origin
as numerical weather prediction models, which
do remarkably well at short-term prediction.
However, models or long-term prediction need
to include slow processes, or which there are ew
observations.
Analysis of data
An enormous amount o climate data continues
to be collected at a wide range o locations, rom
diverse platorms, and using dierent methods.
Tey need to be synthesized into coherent
rameworks and linked to standard climatevariables. For example, work needs to be done
to determine how satellite measurements, which
integrate over a column or noodle o atmosphere,
correspond to events at the surace. Te data
should guide modelers in deriving the proper
representation o climate processes, and the models
should indicate what other measurements should
be collected to gain urther insight into the system.
In this way, a mutually benecial eedback would
occur between models and theory. Another active
rESEarch topIcS In
A partial list of active research
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clIMatE changE
areas in climate change is given below.
research area is to use data, possibly in conjunction with
models, to ngerprint the dierent actors that contribute to
climate change.
Cmptatia methds ad patrms
Once a model is chosen, and initial and boundary conditions
are specied, numerical methods are used to simulate the
climate. Tere is a wide variety o computational approaches
to integrating time-depen
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