mathematics of global climate change

8/18/2019 Mathematics of Global Climate Change

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F.C. Ballesteros, Jr.,Ph.D.


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What is a climate model?Climate models use quantitative methods to

simulate the interactions ofthe atmosphere, oceans, land surface, and ice.

The are used for a variet of purposes fromstud of the dnamics of the climate sstem topro!ections of future climate.

"uch a model has to ans#er #hat happens totemperature, precipitation, humidit, #ind speedand direction, clouds, ice and other varia$les allaround the %lo$e over time


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Developin% the model&

"patial 'ridsDivide the earth(s

atmosphere into a )nitenum$er of $o*es +%ridcells.

-ssume that each varia$lehas the same valuethrou%hout the $o*.

Write a $ud%et for eacheach $o*, de)nin% thechan%es #ithin the $o*,and the o#s $et#een the$o*es.


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/*press chan%es in a %rid cell at a

%iven time step


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Continuit /quations

"ource& 0enderson1"ellers& - Climate 2odellin% Primer


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- samplin% of equationsFor ocean chemistr, air1sea C34 u* is&-ir1sea C34 e*chan%e u*5 ks(pCO2, ocean– pCO2,atmosphere)

For land $iolo% and chemistr, atmosphere1land C34 u* is&-tmosphere1land C34 e*chan%e u* 5

Photosnthesis 67espiration 6Decomposition5+'PP 67- 170

Time is also treated in discrete units.


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"olvin% the equationssolve for the values of the varia$les descri$ed $

these equations over time. This involvesinte%ratin% the set of di8erential equations

9 or more varia$les descri$ed $ the same num$er ofequations that descri$e chan%e #ith respect to time.+T,p, :, u, v, #, #ater, etc.. "o #e should $e a$le tosolve for the values of the varia$les throu%h time;

0o#ever, these equations can not $e solvedanalticall

need to use numerics& discreti


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"olvin% the equations


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=t is #orth notin% that...... For 'C2s, %rid cells are tpicall hundreds of miles across andoften there are thirt vertical laers for the atmosphere.

2an processes happen at smaller scales and must $e

appro*imatel included +a.>.a., parameteri


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"hortcomin%s of the'C2sBecause our understandin% of the various

natural processes included in the atmosphereis incomplete, current climate models +'C2&'lo$al Circulation 2odels do not accuratelrepresent the climate sstem.


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"hortcomin%s of the'C2s"ome climate models have $een ad!usted, or

cali$rated, to provide a reasona$le simulationof some aspects of recent climate.

cali$ratin% a model to ma>e its output loo>more li>e the real #orld does not provide a$asis for assumin% it #ill %enerate realisticpredictions of future climate

7ealistic predictions of future climate areassured onl if the climate model is validatedand run #ith an accurate set of inputs.


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"hortcomin%s of the'C2s- model is considered validated if it is

developed using one set of data and its output istested usin% another set of data.

if a climate model #as developed usin%o$servations from AA to A, it #ould $evalidated $ testin% its predictions a%ainsto$servations from AA to 4.

-t this time, no climate model has $eenvalidated +3(Eeefe Eueter, 4G, 2arshall=nstitute.


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"hortcomin%s of the'C2s The inputs required $ climate models include

$oth natural varia$les +e.%. Chan%es in solarradiation and human varia$les +e.%.%reenhouse %as and aerosol emission rates.

Hote& These inputs are not predicta$le over

the A1ear and lon%er periods of interest forclimate model studies.


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=f prediction is not possi$le,

#hat is the alternative?climate modellers fall $ac> to pro!ectin% +not predictin%

climate scenarios.

- climate scenario is the output of a climate model calculation

and, $ de)nition, is no $etter than the qualit of the inputdata and model. The =PCC addressed this uncertaint $usin% a ran%e of future emission rates and an arra of climatemodels.

... Consequentl .......

usin% a sin%le set of emission rates in the arra of modelsproduced as lar%e a ran%e in %lo$al avera%e temperature in4A as usin% the ran%e of emission rates in a sin%le model.

3$viousl, caution is required #hen dealin% #ith results in theli%ht of these uncertaintiesII


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'oin% $ac> to ho# models #or>.... The climate sstem consists of t#o inter1

connected su$1sstems& the atmosphere andthe oceans.

While the importance of the atmosphere in theclimate sstem is o$vious, it is the oceans thatcontain the over#helmin% share of the ener% in the

sstem.

Chan%e in the atmosphere can $e rapid, $ut chan%ein the oceans is slo#. -n calculation of futureclimate must ta>e this slo# chan%e in the oceansinto account.


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0o# models #or>....Heither the atmosphere nor the oceans are

homo%eneous. the most comple* climate models +e.%. H-"-(s 'oddard

=nstitute for "pace "tudies, or the E 2eteorolo%ical3Kce(s 0adle centre, divides the atmosphere into cellsthat are a$out 4 miles square and var in hei%ht froma fe# thousand feet close to the surface to several milesat the top of the troposphere.


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0o# models #or>....Conditions #ithin a sin%le cell are assumed uniform.

0o#ever, #eather and climate can $e ver di8erent over adistance of 4 miles, particularl in mountainous or coastal

re%ions.


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Conditions #ithin a sin%le cell are assumeduniform. 0o#ever, #eather and climate can$e ver di8erent over a distance of 4

miles, particularl in mountainous or coastalre%ions.

Lue


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0o# models #or>.....7unnin% a climate model also requires a set of

initial conditions, i.e., the #eather conditionsaround the %lo$e at a speci)c time.

climate is a chaotic sstem 1 small chan%es ininitial conditions can result in lar%e chan%es inoutput conditions.


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0o# models #or>.... The climate model is run, usin% standard numerical modellin%

techniques,

calculatin% the chan%es indicated $ the model(s equations over a shortincrement of timeM4 minutes in the most advanced 'C2sMfor one

cell, then usin% the output of that cell as inputs for its nei%h$ourin% cells.

The process is repeated until the chan%e in each cell around the %lo$ehas $een calculated. =n a perfect model, results for the initial cell at the end of the calculation #ould

$e the same as those determined at the start of the calculation. 0o#ever,climate models are far from perfect, requirin% the #hole process to $e repeatedand smoothed, a%ain usin% standard numerical calculation techniques until....

a consistent set of results is determined for the )rst time step.

The #hole process is repeated for the ne*t time step until the model isrun for the desired amount of time.


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"hortcomin%s of 'C2sHeed to distin%uish $et#een pro!ection@ and

prediction.@


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"hortcomin%s of 'C2sprediction de)ned as- pro$a$ilistic description or forecast of a future climate

outcome $ased on o$servations of past and currentclimate conditions and quantitative models of climateprocesses +Climate Chan%e "cience Pro%ram and"u$committee on 'lo$al Chan%e 7esearch +4N&"trate%ic Plan for the .". Climate Chan%e "cience

Pro%ram. P%. A9.


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"hortcomin%s of 'C2sWhile pro!ection is de)ned as

- description of the response of the climatesstem to an assumed level of future radiativeforcin%...; +Climate Chan%e "cience Pro%ramand "u$committee on 'lo$al Chan%e 7esearch+4N& "trate%ic Plan for the .". Climate

Chan%e "cience Pro%ram. P%. A9.


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"hortcomin%s of 'C2sClimate Opro!ections( are distin%uished from

climate Opredictions( in order to emphasi


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Pro!ection vs. Predictionclimate modelers are careful to ma>e the

distinction $et#een prediction and pro!ection,the media and political processes rarel do

the treat most model outputs as predictions.


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2edia 2isrepresentations of

Climate 2odel 7esults The %lo$all avera%ed surface temperature is projected to increase $ A.G to .QRC +4. toA.GRF over the period A to 4A. These

results are for the full ran%e of N "7/"scenarios, $ased on a num$er of climate models+=PCC, Nrd -ssessment 7eport.

+"7/" 6 "pecial 7eport on /mission "cenarios

The =PCC is un#illin% to assi%n pro$a$ilities to theli>elihood of occurrence of an of the "7/"scenarios, sain% onl that all are equall li>el orunli>el to occur.


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2edia 2isrepresentations of

Climate 2odel 7esults2edia reports tpicall focused on the upper end

of this ran%e, did not e*plain that it #as $ased onclimate model pro!ections, or e*plain the

di8erence $et#een a pro!ection and a prediction.B 4A, %lo$al #armin% could raise the

avera%e temperature of the /arth as much asA de%rees SFahrenheit more than the avera%e

temperature in A, accordin% to a .H.1sponsored panel of hundreds of scientists +USAToda (Octo!er 2", 2###).


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=PCC Third -ssessment

7eport admits to thefollo#in%.... Discrepancies $et#een the vertical pro)le of temperature chan%e in the

troposphere seen in o$servations and those predicted models.

Uar%e uncertainties in estimates of internal climate varia$ilit +also referred to

as natural climate varia$ilit from models and o$servations.

Considera$le uncertaint in the reconstructions of solar and volcanic forcin%#hich are $ased on limited o$servational data for all $ut the last t#odecades.

Uar%e uncertainties in anthropo%enic forcin%s associated #ith the e8ects of

aerosols.

Uar%e di8erences in the response of di8erent models to the same forcin%.

3thers tpicall add uncertainties a$out the roles of clouds and oceancurrents in the climate sstem, and the sensitivit of the climate sstem tochan%es in %reenhouse %as concentrations, to the =PCC(s list.


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=PCC Third -ssessment

7eport admits to thefollo#in%.... Discrepancies $et#een the vertical pro)le of temperature chan%e in the

troposphere seen in o$servations and those predicted models.

Uar%e uncertainties in estimates of internal climate varia$ilit +also referred to

as natural climate varia$ilit from models and o$servations.

Considera$le uncertaint in the reconstructions of solar and volcanic forcin%#hich are $ased on limited o$servational data for all $ut the last t#odecades.

Uar%e uncertainties in anthropo%enic forcin%s associated #ith the e8ects of

aerosols.

Uar%e di8erences in the response of di8erent models to the same forcin%.

3thers tpicall add uncertainties a$out the roles of clouds and oceancurrents in the climate sstem, and the sensitivit of the climate sstem tochan%es in %reenhouse %as concentrations, to the =PCC(s list.


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- note on =PCC(s "7/".Pro$lem on the treatment of economic %ro#th in the

"7/" as pointed out $ =an Castles, former head ofthe -ustralian Bureau of "tatistics, and David

0enderson, Westminster Business "chool, Uondon ina series of letters to Dr. 7. Pachauri, Chair of the=PCC, and in presentations at =PCC e*pert meetin%s.se of mar>et e*chan%e rates rather than purchasin% po#er

parit in evaluatin% national incomes7esults that indicate that avera%e incomes in -sia could

%ro# as much as AG times durin% the 4Ast centur, rates of%ro#th that are far %reater than ever e*perienced, even $the fastest %ro#in% economies and

7ates of %ro#th in developin% #orld emissions andVor incomefrom A to 4 #hich #ere much lar%er than actualle*perienced.


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- note on =PCC(s "7/"."adl....

....the =PCC #ent on to use the "7/" scenarios inits Fourth -ssessment 7eport .


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-s a )nal note....We should distin%uish $et#een climate and #eather.$eather is #hat #e e*perience on a da1to1da or

seasonal $asis% Toda&s high temperature or theamount of rainfall in June are e*amples of #eather.

#eather is chan%ea$le and unpredictable more thana fe# das in advance

Climate is the lon%1term, tpicall thirt1ear, avera%eof #eather. The avera%e hi%h temperature for this dateor the avera%e rainfall for a #et season is an e*ampleof climate.


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'lo$al Warmin% PotentialConcept #as developed to compare the a$ilitof each '0' to trap heat relative to another%as

The 'WP of a '0' is the ratio of %lo$al#armin% or radiative forcin% from one unitmass of a '0' to that of one unit of C34

C0G traps over 4A times more heat permolecule than C34

H43 a$sor$s 49 times more heat per

molecule than C34


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On a global

basis, biogenicCH

4 is a much

larger source

than abiogenicCH

4 !


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That’s all folks…

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mathematics of global climate change

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