Math of Climate Change - 2007

8/14/2019 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.


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.


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


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.


11/28

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.


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


12/28

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.


12


13/28

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


14/28

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).


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.


15/28

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.


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.


16/28

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 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


17/28

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


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.


18/28

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


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

NCAR_PCM1


19/28

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.


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


20/28

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


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


21/28

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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Math of Climate Change - 2007

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