University of New Hampshire
University of New Hampshire Scholars' Repository
Earth Sciences Scholarship
Earth Sciences
3-1994
The lifetime of excess atmospheric carbon dioxide
B Moore
University of New Hampshire - Main Campus, [email protected]
Rob Braswell
University of New Hampshire - Main Campus, [email protected]
Follow this and additional works at: http://scholars.unh.edu/earthsci_facpub
Recommended Citation
Moore III, B., and B. H. Braswell (1994), The lifetime of excess atmospheric carbon dioxide, Global Biogeochem. Cycles, 8(1), 23–38,
doi:10.1029/93GB03392.
This Article is brought to you for free and open access by the Earth Sciences at University of New Hampshire Scholars' Repository. It has been accepted
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GLOBAL BIOGEOCHEMICAL
CYCLES, VOL. 8, NO. 1, PAGES 23-38, MARCH
1994
The lifetime of excessatmospheric carbon dioxide
Berrien Moore, III and B. H. Braswell
Institutefor the Studyof Earth,Oceans,andSpace,Universityof New Hampshire,Durham
Abstract. We explorethe effectsof a changingterrestrialbiosphereon the atmosphericresidencetime of CO2 usingthreesimpleoceancarboncyclemodelsand a modelof globalterrestrial carboncycling.We find differencesin modelbehaviorassociated
with theassumption
of an
activeterrestrialbiosphere(forestregrowth)and significantdifferencesif we assme a donordependentflux from theatmosphere
to the terrestrialcomponent(e.g.,a hypotheticalterrestrial
fertilizationflux). To avoid numericaldifficultiesassociatedwith treatingthe atmosphericCO2
decay(relaxation)curveasbeingwell approximated
by a weightedsumof exponentialfunctions,we definethe singlehalf-lifeasthe time it takesfor a modelatmosphere
to relaxfrom its
present-day
value half way to its equilibriumpCO2value.This scenario-based
approachalso
avoidsthe useof unit pulse(Dirac Delta) functionswhichcanprovetroublesomeor unrealistic
in thecontextof a terrestrialfertilizationassumption.
We alsodiscusssomeof the numerical
problemsassociated
with a conventional
lifetimecalculationwhichis basedon an exponential
model. We connectour analysisof the residencetime of CO2 andthe conceptof singlehalf-life
to the residencetime calculationswhichare basedon usingweightedsumsof exponentials.We
notethatthe singlehalf-life conceptfocusesupontheearlydeclineof CO2 undera cutoff/decay
scenario.If oneassumes
a terrestrialbiospherewith a fertilizationflux, thenour bestestimateis
thatthesinglehalf-life for excessCO2 lies within the rangeof 19 to 49 years,with a reasonable
averagebeing31 years.If we assumeonly regrowth,thenthe averagevalue for the singlehalflife for excessCO2 increasesto 72 years,andif we removethe terrestrialcomponentcompletely,thenit increases
furtherto 92 years.
1. Introduction
Since ancient times humanshave modified natural systems,
but only sincethe beginningof the industrialrevolutionhas human activity significantly altered biogeochemicalcycling at the
planetary scale. The magnitude of human disturbanceto the
biogeochemicalcyclesmay now be approachinga critical level;
the valuesof importantstatevariables,suchas the concentration
of atmosphericCO:, are movinginto a rangeunprecedented
during the past one million years.The pool of carbonin the atmosphere(in the form of CO:) increasedfrom about590 to almost
culture).For the period 1980-1989 an averageof 5.4 Pg C per
yearas CO: wasreleasedto the atmosphere
from the burningof
fossil fuels, and it is estimatedthat an averageof-0.6-2.6 Pg C
per year was emitted due to deforestationand land-usechange
during the same interval [e.g., Houghton and Skole, 1990;
Watsonet al., 1990; Bolin and Fung, 1992; Skole and Tucker,
1993, and Houghton,1993b). Figure 2 showsan estimateof these
two fluxes of anthropogenicCO 2from the mid-18th century to
the present.
The increasein the atmosphericCO2 concentrations(as well as
other radiatively active trace gases)due to human activity has
755PgC (1 PgC = lx10•5g C = 1 billionmetrictonsC) during producedseriousconcernregardingthe heatbalanceof the global
the 225 years between 1765 and 1991 as a result of fossil fuel
atmosphere.Specifically,the increasingconcentrations
of these
burningand forest clearing. The annualrate of increaseis curgaseswill lead to an intensificationof Earth's naturalgreenhouse
rently about2 ppm(V) per year (equivalentto roughly0.6% per effect [Shine et al. 1990; Watson etal., 1990; lsaksen et al.,
year). We have a direct record of this increase since 1958
1992]. Shifting this balancewill force the global climate system
[Keeling,1986] anda numberof indirectrecords(from ice cores) in ways which are not well understood,given the complex interof the increaseover the past two centuries,which show that the
actionsand feedbacksinvolved, but there is a generalconsensus
concentrationof CO9.has increasedby more than 25% (e.g., that globalpatternsof temperatureandprecipitationwill change,
Neftel etal., 1985; Raynaudand Barnola, I985; Friedli et al.,
though the magnitude,distribution and timing of these changes
1986; Siegenthalerand Oeschger,1987] since the mid-1700s are far from certain.The resultsof generalcirculationmodelsin(Figure 1). Moreover, from the ice core recordswe know that the
dicatethat globally averagedsurfacetemperatures
couldincrease
concentrationof carbondioxide was relatively constantfrom the
by asmuchas 1.5ø-4.5øC [e.g.,Mitchell eta/., 1990;Gateset al.,
beginningof the presentinterglacialperiod (-10,000 B.P.) to the •992] in a world with an atmosphericconcentration
of CO2 twice
onsetof increasesin the 18th century[Siegenthaler,1989].
thatof the preindustrialperiod (i.e., a world where the concentraThe primary human activitiescontributingto this changeare
tion wouldbe roughly550-580 ppm).
fossil fuel combustionand modificationsof global vegetation
The uncertaintyof futureclimatechangedoesnot rest solely
throughland use (e.g., biomassburning and conversionto agrion issues of physical-climate system dynamics and their
representationin general circulationmodels. Understanding'the
carboncycle (Figure 3) is a key to comprehendingthe changing
Copyright 1994 by the American GeophysicalUnion.
terrestrialbiosphere and to developing a reasonablerange of
future concentrationsof carbon dioxide and other greenhouse
Paper number 93GB03392.
0886-6236/94/93GB-03392510.00.
gases[e.g., Bacastow and Keeling, 1973; Bolin et al., 1979;
24
MOORE AND BRASW•F_LL: LIFETIME
OF EXCESS ATMOSPHERIC
CARBON DIOXIDE
ATMOSPHERIC CO2 DATA
FROM ICE CORES AND DIRECT MEASUREMENTS
360
340
A
320
3OO
280
260
1700
1750
1800
1850
1900
1950
2000
YEAR
Figure 1. Historicalatmospheric
CO2 concentrations
(pCO2).The squaresand trianglesrepresentdataderived
from ice core measurements
at Siple Station,Antarctica.The circlesare the annualaveragedatmosphericmeasuremenUfrom MaunaLoa Observatory(theKeelingrecord).
ESTIMATES OF CO2 FLUX
FROM FOSSIL FUELS AND DEFORESTATION
7i, , I , , i i I ,....i I,
I,
,
6-
FFE
mmmmmmmm
•m
B
5
I
m
mmmm
mmore
m•mmmm!
i1111II
ii II
me
mmmmmmmmm
mm
mmmmmmmmm
mmmm
o
1750
1800
185o
1900
1950
2000
YEAR
Figure2. The fossilfuelemissions
record(Fro)[RottyandMarland,1986;Marland,1989;Andreset al., 1993]
is accurate
to withinabout10%,buttheestimate
of bioticfluxdueto landusechange
(FB)[Houghton,
1993b]is
lesscertain,especially
for thelessrecentnumbers,
because
of a lackof a detailedhistoricalaccounting
of global
land usepatterns.It can be seenfrom this figure that until about1920, deforestation
contributedmore to the atmospheric
flux thanfossilfuel burning,whichdid not becomesignificantuntil the startof the currentindustrial
periodin the 1860s.
MOORE
AND BRASWELL:
LIFETIME
OF EXCESS ATMOSPHERIC
CARBON
DIOXIDE
25
Atmosphere 755
Deforestation
I-2
60
70
80
22
35
d Surface
;500
Biota 550
Warm
Fossil Fuel .:.-
.'
Soil
and
Detritus::
Rivers
0.5
1200
-. '
Biota
Recoverable
'
I0000
Surface
600
2
.2
Biota
;5.7;57
I
'' ......
".;':•
• esDissolved
Inorganic
Carbon
endI000 •.....
'."...."'""
'"'
Figure 3. •e globalc•bon cycle(adaptedfromMoore[1985]).•e valueswithincomp•en•
PgC, andthefluxes(a•ows) •e in unimof PgC/yr.
Bjb'rkstrom,!979; Bolin, 1981; Moore, 1985; Schlesinger,1991;
Bolin and Fung, 1992]. Conversely, our predictions about the
physical-climatesystemand climate changeare confoundedby
the fact that the carboncycle is still not adequatelyunderstoodor
quantified globally. We remain uncertainabout the role of the
oceansin carbondioxide exchange;they are withoutquestiona
sink for anthropogenic CO 2, but the strengthof this sink is
somewhatunclear [e.g., Keeling et al., 1989, Tans eta/., 1990,
Sarmiento,!991; Siegenthalerand Sarrniento,1993; Sarmiento,
•e in unitsof
valuable in the policy context; it is necessaryfor the calculation
of greenhousewarming potentials (GWP) [Lashof and Ahuja,
1990; Watson et al.,
Hasselmann, 1987).
1990.
See also Maier-Reimer
and
The GWPs provide an index of the relative climatic impacts
(costs)of the variousgreenhousegases.The index takesinto accountthe instantaneous
radiativeforcing of a singlemoleculeat
and the responsec(t) of the system(atmosphericconcentrationas
a functionof time) to an instantaneous
injectionof gas'
1993].
Uncertaintyalso centerson the role of terrestrialecosystems,
in which at least two factorsgovernthe level of carbonstorage.
First, and most obvious, is the anthropogenicalteration of the
Earth'ssurface,suchas throughthe conversionof forestto agriculture,which can result in a net releaseof CO2 to the atmosphere.Second,and more subtle,are the possiblechangesin net
ecosystemproduction(and hencecarbonstorage)resultingfrom
changesin atmospheric
CO2,otherglobalbiogeochemical
cycles,
and/orthe physical-climatesysterh.Ultimately, to addressadequatelysuchchangeswill requirea muchclearerunderstanding
of the nitrogenandphosphoruscycles,sincethey are the limiting
nutrientsin mostterrestrialecosystems,
butour knowledgeof the
way thesebiogeochemicalcyclesrelate to the carboncycle comparespoorly with our generalunderstanding
of the individualcycles themselves. In sum, these uncertainties are reflected in our
uncertaintyaboutthe atmosphericlifetime of CO2[e.g., Watson
et al,, 1990; Watson et al., !992].
The lifetimeof a traceatmospheric
constituent
may be thought
of asthe amountof time requiredfor somesignificantportionof
an excessquantityof the gasto be removedchemicallyor to be
redistributed
to anotherpartof the Earthsystem.The uncertainty
in the distribution of sourcesand sinks for carbon dioxide (and
for other greenhousegasessuchas methane)makesa determina-
tion of atmospheric lifetime difficult and ambiguous.
Unfortunately, the lifetime for CO2 (as well as for the other radiatively importantgases)is importantto know and particularly
_ l•ai'ci(t)dt
GWPi
I•arCr(t)dt
wherethe subscriptr refersto the referencemolecule.The curves
ci(t) are generally computedusing geochemicalbox-models.If
thereferencemoleculeis CO2, andthe radiativeforcingis taken
to be equal to one (i.e. all ais are relative to CO2), then the denominator becomes:
Tre
, =l•c(t)dt
(2)
In the simple or idealized context,where the concentrationcurve
decaysexponentially(i.e., c(t)=a-exp(-t/z)),then the integral (2)
is 'r, which is simplythe lengthof time requiredfor c(t) to decline
from any value alongits trajectoryto !/e of that value. Hencex is
oftencalledthe e-foldingtime. Generally,c(t) is not strictlyexponential,so the integralitself is calledthe residencetime (Tf,s).
On the basis of results of the convenient assumption of
c(t)=a'exp(-t/x),one often approximatesthe relaxation of the
concentrationby a weighted sum of exponentials(compare
Section 6).
We will showthatcalculatingintegral(2) presentstwo types
of difficulties. First, most model-derived estimates of the relax-
ation of the'concentrationof CO2 reveal a functionwhich is not
26
MOORE AND BRASWELL:
LIFETIME
OF EXCESS ATMOSPHERIC
always well approximated by weighted sums of exponentials
even thoughtheseare often usedto calculateTr½
s. Second,the
functionc(t) is quite sensitiveto assumptions
aboutthe terrestrial
biosphereand the relaxationexperiment.We turnour attentionto
the seconddifficulty first.
CARBON DIOXIDE
Box-diffusion Model with Polor Outcrops
(Siegentholer ond colleogues)
Atmosphere
2. Simple Global Carbon Cycle Model
From the most elementary viewpoint, and focusing solely
upon CO 2, the global carbon cycle can be treated as a one box
atmospherelinked to a submodelof terrestrialcarbondynamics
and to an oceancarbonsubmodel.This is preciselythe representation we employ in our experiments.
2.1. The Ocean
Carbon
.
t MixedLayer
........
r'
/ II1!
./','
Submodels
The net exchangeof carbonbetweenthe atmosphereand the
oceansis determinedto a greatextentby the carbonatechemistry
of the upper mixed layer, the convective transportof dissolved
Deep See
carbon in the water, the diffusion of carbon dioxide across the
Figure 5. Outcrop-diffusionmodel.Direct ventilationof the intermediateanddeepoceansat high latitudesis allowedby incorporatingoutcropsfor all sublayersinto the box-diffusionformulation [Siegenthaler,1983].
air-seaboundary,and the sinking of detrital carbonoriginating
from the biologicalproductionof marineorganisms[Bolin, 1981;
Broeckerand Peng, 1982; Moore, 1985; Sarmiento,1991). All of
theseprocessesgovernthe exchangeof carbondioxide between
the seasurfaceandthe atmosphere,and all havebeenrepresented
in modelsto varyingextents;however,all do not play an essential
or the same role in the perturbationproblem posed by the in-
which a constantcoefficientof diffusivity has been stimatedto
match an idealizedprofile of naturalcarbon-14or bomb carboncreasein carbon dioxide (Broecker, 1991; Sarmiento, 1991).
14 [Oeschgereta/., 1975].
We investigatethe effect of using different (simple) represen2. The outcrop-diffusionmodel (OC, Figure 5) allows direct
tationsof ocean carbondynamicson the atmosphericconcentraventilationof the intermediateand deepoceansat high latitudes
tion of CO2 using three atmosphere-ocean
box models,either
by incorporatingoutcropsfor all sublayersinto the box-diffusion
standingaloneor in a global carbonmodel which includesa terformulation[Siegenthaler, 1983]. It is essentiallythe box-diffurestrial component. (For a discussionof more complexmodels sion model with the addition of direct connections between the
seeMaier-Reimer and Hasselmann[1987], Keeling et al. [1989],
atmosphereand the deeper ocean layers. In a similar manner a
and Maier-Reimer[1993].) The submodelsare as follows:
constantcoefficient of diffusivity is estimatedto match an ideal1. The box-diffusion model (BD, Figure 4) representsthe
ized profile of naturalcarbon-14or bomb carbon-14.
turnoverof carbonbelow 75 metersby a diffusion equationin
3. For the 12-box model (12B, Figure 6) the Atlantic and
Pacific-Indian Oceans are each divided into surface, intermediate,
Box-diffusion
Model
deep, and bottom water compartments.The Arctic and Antarctic
Oceansare divided into surface and deepwatercompartments.
The model is calibrated against multiple tracer distributions
[Bolin et. al., 1983].
(Oeschcjenond colleogues)
Atmosphere
ß
.
m
m
m
mm
mmmm
Mix,edayer
mm
mmm
mm
mm
m
m
m
m
m
m
m
m
m
m
mmm
mmmm
mt •m
•m
mm
mm
Deep Seo
The threemodelshave much in common:They are all diagnosticratherthanprognostic;each usescarbon-14in the parameterization process(in fact, carbon-14 is the basic clock for all of
the modelsand hencecontrolsmuch of their response);each includes ocean carbon chemistry (buffer or Revelle factor); and
they all includesomeform of oceanmixing. There are, however,
somemajor differences.Oceanbiologyis explicitly includedin
only one(12B); whereasit is simplyincorporated
into theparameterizationof the diffusive processin both the BD andOC models. As far as transportis concerned,in the box-diffusionand out-
crop-diffusionmodelsall of the physicsis capturedby a single
constanteddy diffusivity term; the Bolin model (12B) has both
advectionandeddydiffusivities.Deep waterformation(the sinking of cold, CO2-rich water at high latitudes)is not explicitly
considered in the box-diffusion
model. The treatment of ocean
chemistryalso varies in complexity: the 12B model includesa
detailedhandlingof the carbonate-borate
equilibriumsystem;the
Figure 4. Box-diffusion model. The turnoverof carbonbelow 75
m is representedby a diffusion equation.A constantcoefficient
of diffusivityis estimated
to matchan idealizedprofileof •4C
[Oeschgeret al., 1975].
BD model uses a constant buffer factor; and the OC model em-
ploysa quadraticthatis a fit to measureddata.Perhapsmostimportantly,the geometricalconfigurationsare quite different, as
can be seenin Figures4-6. In addition,thereare a hostof smaller
MOORE AND BRASWELL:
LIFETIME
OF EXCESS ATMOSPHERIC
CARBON DIOXIDE
27
12 Box Model of Bolin et al. (1983)
Atmosphere
,CA ,BA
,CAT,BAT
I
9AA,BAA
,Clp,BIp
:
---100-200m
i
Arctic
i
• •,
---
IOOOm
---
3500m
Antarctic
At I ant ic
Indian/Paci fic
Figure 6. Twelve-box model. The Atlantic and Pacific-Indianoceansare eachdivided into surface,intermediate,
deep, and bottom water components.The Arctic and Antarctic oceansare divided into surface and deep water
components.The soft tissueand carbonateformationin surfaceboxesis indicatedby B and C, with appropriate
subscriptsf9r region, and the fluxes to and decompositionin deeperlayers are indicatedby the vertical dashed
line and the branchingsolid arrow, respectively.The model is calibratedagainstmultiple tracer distributions
[Bolin et al., 1983].
(An interesting
discussion
of the•3Cconstraints
onoceanuptake
differences,includingthe specificationof oceanvolumeand surfacearea,parameterNation
procedures,
andcarbon-14profries.
As a result of thesedifferences,we find a range of responses
to the two simplestexperimentsthatcanbe performedwith these
threeatmosphere-ocean
submodels:
a calculationof theperturbation responseto a forcingby fossilfuel CO2 alone(Fj:•r,Figure7)
and to the forcing by CO2 from land use changeas well as the
fossilfuel-derivedflux (Fe+FrE, Figure 8). A principalreason
why all the modelsare so similar is the overarchingimportance
of ]4Cin settingthebasicrateswithin themodels[Moore, 1992].
of CO2 is provided by Broecker and Peng [1993].) The OC
model is the most efficient in taking up CO2, and this is the result
of the instantaneous
flux of CO2 to deeplayers.In general,however, all of the ocean-atmospheremodels undershootthe data
when forced by Frœ alone and overshootthe record when both
FrœandFe are used.Sincethereis evidence,basedon historicreconstructions
[Houghtoneta/., 1983; Houghtonand SkoIe,1990;
Houghtonet aI., 1991; Houghton, 1991; Houghton, 1993b; SkoIe
and Tucker, 1993] of an anthropogenicallyinducedflux from the
RESPONSE OF OCEAN MODELS TO
FFE FLUX ONLY
,
360
,
,
,
I
,
,
,
,
I
....
I
,
•
•
i
I
i
,
,
,
I
....
340
320
300
cDO
oo
o
280
260
....
1700
i
1750
....
•
1800
....
•
1850
....
'•
1900
....
•
1950
....
2000
Figure 7. The responses
of theocean-atmosphere
modelsto the historicfossilfuel forcing.All modelswere initializedat the 1744 ice corevaluefor pCO2. Circlesrepresenttheatmospheric
CO2record(compareFigure1).
MOORE AND BRASWELL: LIFETIME OF EXCESSATMOSPHERIC CARBON DIOXIDE
28
whichis generally
notbalanced
by increased
nibiosphere
totheatmosphere,
thediscrepancy
between
modeland drateformation,
uptake).
Consequently,
thereisthepossibility
thathigher
datain Figure8 hasgivenrisetothenotion
of a "missing
carbon trogen
mayleadtoanincrease
innetprimary
production
and
sink."Thisispartof themotivation
forconsidering
theterrestrialCO2levels
perhapsnet ecosystemproduction(carbonstorage).
component.
The possibleexistenceof a fertilizationeffect,the rapidresponseof the land biota-soilsystem,andthe fact that oneof the
majorfluxesto the atmosphere
(FB) is actuallydue to alteration
2.2. The Terrestrial Carbon Submodel
of the terrestrialbiospheremake it appropriate
to includeterresThe netexchange
of carbonbetweenterrestrial
vegetation
and trial dynamicsin our attemptto addressthe issueof the lifetime
CO2.Emanueleta/. [ 1984]hasdeveloped
theatmosphere
maybeconsidered
to bethesumof threefluxes: of excessatmospheric
grossphotosynthesis,
autotrophic
plant respiration,and het- a highly aggregatedmodel of globalcarboncyclingwhich is
in the
erotrophic
(soil microbial)respiration
[e.g.,Aber andMelillo, similarin scopeto the threesimpleoceanmodelspresented
1991]. (This latterrespirationcancoveralsothe anthropogenic previoussection.This model(Figure9) is composed
of a setof
perturbation
of biomassburningthoughthe humanrole is not eight coupleddifferentialequations(Table 1 containsthe terresusuallyconsidered
in thesebiologicalterms.)The grossfluxes trial components)which govern the flow of carbon between
betweenthe terrestrialsystemandthe atmosphere
are similarto reservoirsrepresenting
the atmosphere,
the surfaceocean,the
thosefor theoceans(100 Pg C per year,Figure3; seealsoBolin deep ocean,nonwoodyparts of trees,woody parts of trees,
et aI. [1979] andBolin [1981]),but theconsiderably
smallerter- groundvegetation,detritus/decomposers,
and soils.Its modular
restrialpoolsizeleadsto a muchfasterturnover
timethanfor the structuremeansthat one can replacerelativelyeasilyEmanuel's
two-boxocean(Figure9) with the othersimpleoceanmodelsthat
oceans[Bj6rkstrom,1979].
In the currentcontextEmanuel'smodelis of
Motivated,in part,by the issueof a missingcarbonsinkand we areconsidering.
partlyby physiology,
it hasbeensuggested
[e.g.,Bacastow
and interest,in part, becauseit is a convenientway to handleterresKeeling,1973] thatterrestrialvegetationmay be "fertilized"by trial carbon fluxes associatedwith forest clearing and reestabtheincreasing
concentration
of CO2. Thoughtheissueis contro- lishment.The model deals with theseactivitiesin the following
versial(Strainand Cure [1985],Baazaz[1990], Garbuttet al. way: (1) A specifiedmassof carbonis released(dueto land-use
[ 1990],BazzazandFajer [1992],Diaz et al. [1993], seealsoDai change)from 'woody'and 'nonwoody'partsof treesas a funcand Fung [ 1993]for a climate-based
hypothesis
andHoughton tion of time, in a ratio determinedby their relativesizes.(2) A
[1993a]for a landuse-based
hypothesis),
it is possiblethatterres- fraction of the carbonreleasedis transferredimmediately to the
anda fractionentersthedetritus/decomposers
pool.
trial ecosystemsare respondingto the rapid increasein atmo- atmosphere,
sphericCO2 concentration
by producing
morebiomassand/or The remainder is assumedto be in a very slow turnover pool
(3) All vegetation
poolsexperience
a lostoringmore soil carbon,therebybalancingthe globalcarbon (e.g.,timberproducts).
rateandwith a dynamibudget.For instance,theincreasing
concentration
of CO2 in the gisticgrowthrecoverywith a prescribed
(4) Uponclearing,the asymptotic
level
atmosphere
may raiseC/N ratiosby eithermakingplantsmore callyvaryingasymptote.
waterefficient(morecarbonfixedper H20 transpired)
or through for treesdecreases,and the asymptoticlevel for groundvegetaotherindirectbiochemicalmechanisms(e.g., enhancingcarbohy- tion increases.
RESPONSE OF OCEAN MODELS TO
FFE PLUS FB FLUX
.,
380-
,
•
•
I
!
,
,
,
I
,
,
•
,
I
•
t
,
,
I
,
,
,
•
I
,
,
,
,
.
360
~
340
320
300
.
o
280
260-,,,,
1700
•
....
1750
•''''
1800
,
....
1850
,
....
1900
'
....
1950
2000
YEAR
Figure
8.Theresponses
oftheocean-atmosphere
models
tothehistoric
fossil
fuelforcing
plusanestimate
of
biotic
(deforestation)
flux.Allmodels
wereinitialized
atthe1744icecorevalue
forpCO
2.Circles
represent
the
atmospheric
CO2record(compare
Figure1).
MOORE AND BRASWELL: LIFETIME OF EXCESS ATMOSPHERIC CARBON DIOXIDE
29
GLOBAL CARBON CYCLE MODEL OF EMANUEL, ET AL. (1984)
ATMOSPHERE
I/
MIXED LAYER
VEGETATION
r-•VEGETATION
DEEPOCEAN
Figure9. Globallyaveraged
carbon
cyclemodel[Emanuel
etal., 1984].Theatmosphere
exchanges
carbon
with
terrestrial
components,
andwiththeoceans.
Solidarrowsrepresent
equilibrium
(natural)carbonfluxes.Dashed
arrowsrepr6sent
fluxesassociated
withforestclearingactivities.
Returningto the globalcarboncycle,ourbasicapproach
is to
We remarkthat givena clearingandabandonment
time series,
theinterplayof thesedynamicsdetermines
the flux FB.We also modifyEmanuel'sglobalcarboncyclemodelin twoways.First,
note that the structureof the model allows an easy modification we replacehistwo-boxoceanmodelwith(in turn)thebox-diffumodel, and the 12-box ocean
to be made to include a fertilization effect, which we later discuss sionmodel, the outcrop-diffusion
model;and second,we developa simpletreatmentof the terresin detail.
Table1. Differential
Equations
Governing
theCarbon
Dynamics
in theEmanuel
etal. [1984]Model
Equation
Compartment
Atmosphere
Trees(NonwoodyFraction)
Trees(WoodyFraction)
GroundVegetation
Detritus/Decomposers
Soil Carbon
Assimilation Fluxes
Atmosphere
to Trees(NonwoodyFraction)
Atmosphere
to Trees(WoodyFraction)
Atmosphere
to GroundVegetation
•',: = VrG- prc•
F• = (F•O/F•O)F•:
El4 = VTC4 - pvC2
Definition
Variable
G
mass of carbon
flux from i toj
transfercoefficientfrom i toj
FFE
FB
FOA
fossilCOx flux
forestclearingC02 flux
net ocean- atmosphereflux
fractionof Fs divertedto atmosphere
F?
vr/Pr
vv/Pv
fractionof Fs divertedto detritus/ decomposers
soil to detritusflux associated
with Fs
initial steady-stateflux from i toj
post- disturbance
equilibriumfor C2
post- disturbance
equilibriumfor Cs
30
MOORE AND BRASWELL:
LWETIME
OF EXCESS ATMOSPHERIC
trial fertilization term. By comparingthe resultinghybrid models
(E-BD, E-OC, and E-12B; both with and without fertilization)
with resultsof the ocean-atmospheremodels (BD, OC, and 12B),
we effectively perform the experimentson the residencetime of
CO2 in the atmosphere
with the terrestrialbiosphereswitchedon
and off. We are thus able to investigatethe effectsof a fertilization assumptionplus forest regrowth or just forest regrowth on
the residencetime of CO2 in the atmosphere.The modeledforest
regrowth develops from a deforestationpattern for the period
1700 to the presentand is constrainedto producethe net flux in
Figure 2.
CARBON DIOXIDE
In orderto computethe missingflux Fv we specifythat the
rate of changeof the atmosphericpCO2 that is producedby the
modelsis equivalentto the measuredrate of change(i.e., R=0)
and is given by
dC
....
dt
dC•
dt
• F,
(10)
where dC•/dt is the net flux to the atmospherecalculatedby the
models,anddC/dt may be obtainedfrom the pCO2 record.Thus
the residualflux, for each time step,may be calculatedby a deconvolutionprocedure
3. Characterization Of The Biotic Response
F,(t) =
As discussedin section2.1, if the ocean-atmosphere
models
are forcedwith FrœandF s as input and the resultscomparedto
the recordof atmosphericpCO2, there is a discrepancy(i.e., an
overshoot;seeFigure 8). This residualR (the differencebetween
dC
dC•
dC
--dt
F•.œ+ •paFs+ Foa+ a5•Cs+ a'6•C6- (F•2 + F•3+ F•4) (11)
themodeloutput,
pCO2
M,andthepCO2data)canbedecomposed
where we have substitutedthe termsfrom Table 1 on the righthandside.The conceptis simple:we addtheflux Fr to therighthandsideof the derivativein (10) asthe modelproceeds.This is
R = (pCO2- pC02•a)= Rs + Rm + Rm+ Rœ•
(3) similar to the way in which deconvolutionhas been used to determine the biotic flux FB [e.g., Siegenthalerand Oeschger,
whereRs is causedby someunspecified,hypotheticalbiological
1987], giventhe fossilflux Fm. Figure 10 showsthe deconvolved
mechanism(e.g., the 'fertilization factor'), Rm resultsfrom inacmissingflux Fr for the ocean-atmosphere
modelsand Figure 11
curaciesin the models,Rm resultsfrom uncertaintyin the pCO2
shows
the
graph
of
F
r/Fnp
p
versus
C/C
i
with
thefittedcurvefor
data,andResis dueto uncertaintyin the Frtr andFs data.We will
the logisticparameterizationof Pt using the E-12B model. We
notattemptto characterize
thelatterthree,solet Re = Rm + Rex+
now havethe followingtoolsneededfor our investigation:
three
Res.The expressionthen becomes:
ocean-atmosphere
models, the correspondingocean-terrestrialas:
R = R s + Re
(4)
Thus the residual has been reduced to two terms: error associated
atmospheremodels, an estimateof the evolution of the terrestrial
componentas a resultof land use,and a setof parameterizations
for dealing with a hypotheticalbiotic responseto increasing
pCO2.
with someignoredphysiologicalor ecologicalprocessanderror
associatedwith uncertainties in information (data or model).
Further,we will make the following two assumptions:
Re<<Rs,
4. Single Half-Life
andtheresidualRsmay be associated
with an incrementFrin the
net flux of carbonfrom the atmosphereto the terrestrialbiota:
Beforeinvestigating
theresponse
of thesesimplecoupled
car-
boncyclemodels,we clarifyourconcerns
aboutcalculating
at-
Fr = pr'Fnpp
(5) toospheric
residence
timeof CO2andsuggest
a simple
interim
indexfor modelcomparisons.
The e-foldingtimeis reallyjust a
mathematical
way of sayingwhena decayingexponential
curve
will fall to 1/e of its originalvalue.The conventionaldetermina-
where (compareTable 1)
Fnp
p= Fi2+ F•3+ Fi4
(6) tionof lifetime(Tres;
compare
sections
1 and6) is conceptually
not muchmore substantive,
and its calculationmay presenta
To parameterizethe fertilizationfactor Pt, there are several numberof difficultieswhichreduceits utility.Thuswe introduce
functionalforms suggestedin the literature. We have selected a simpleindicatorof atmospheric
residence
time,thesinglehalfthreedifferentempiricalrelationships(logarithmic,logistic,and life (Tu2),whichis thetimerequired
for themodelatmosphere
to
linear) to describe the fertilization enhancement as a function of
elevatedCO 2
relaxfrom its presentvalue(or somefuturevalue,thoughthis
wouldalterT•/2) to half way to its new equilibriumvalue.This,
of course,
impliesthatwe will notdealwiththepulseresponse
of
Pr = a.log(C/CO
(7) themodels,
butwitha morescenario-based
approach
whichal-
p,= a.tanh(b(C/Ci+c))
(8) Fn;seeFigure
2) followed
bycutting
theforcing
tozero.From
lowsthemodelsto respond
to thehistorical
inputs(e.g.,Fm and
any cutoff time (in the past,present,or future) onward,one sirea-(C/CO + b
(9) ply records
thedecayof theconcentration
c(t)of atmospheric
CO2towardsits newequilibrium.
We acknowledge
thatthisconwhereC/Ci is the ratioof presentto initial pCO2, anda, b, andc ceptor indexfocuses
attention
upontheinitialresponse
andigare free parameters.Note that this treatmentwill not only vary noresthemid-termto verylongtermaspects
of c(t).
the functionaldependence,
but alsothe numberof free parameThereareonlytwo assumptions
involvedin thesecomputaters.To obtainthevaluesfor thoseparameters,
we mustcompute tionswhicharenecessary
to compareresultsacrossmodels:first,
Fr andFnppfrom
thecarbonmodels(E-12B,E-OC,andE-BD) themodels
areallrequired
toreproduce
thehistoric
pCO2record,
andperform
a least-squares
fitofp,toF,/F•.vp
(see(5)).
andsecond,'
though
notreally
essential,
theequilibrium
value
is
MOORE
AND
BRASWELL:
LIFETIME
OF EXCESS
ATMOSPHERIC
CARBON
DIOXIDE
31
DECONVOLVEDTERRESTRIALFLUX(Fr)
FOR OCF_AN-TERRESTRIAL
0.5
....
I ....
i
MODELS
• , , , I , , , , I ....
I ....
.
o
E-OC (bomb)
•
•
ø0ø5
-
x
-J
LI.
ø1
-
~
z_
•
E-OC(pre)
-1.5
-
E-BD (bomb)
-2
E-BD (pre)
E-12B
-2.5
....
1750
• ....
1800
• ....
• ....
1850
I ....
1900
• ....
1950
2000
2050
YEAR
Figure 10. Deconvolved
flux (Fr) determined
by theocean-terrestrial-atmosphere
models.This flux (generallya
sinkfor lateryears)represents
the additionalinputrequkedfor the modelsto reproducethepCO2record;it may
be associated
with the hypothetical
response
of the terrestrialbiosphereto increasingambientCO2concentrations.
zxc./c.= •0.(ZXCo/Co).
(•2)
approximatelythe samefor all models.This assumedequilibrium
pCO2 is a theoreticalvalue basedon the partitioningbetweenatThis factor of ten is called the Revelle factor. If we note that the
mosphereand ocean.Becauseof the carbonatebuffer systemthe
relationshipbetweena percentageincreasein atmosphericcarbon cumulativereleaseof CO2 from Fm and F• combinedis equal to
dioxide(ACa/Ca)and a percentageincreasein oceanicdissolved 377 Pg, and assumethat the initial values for the total oceanic
dissolvedinorganiccarbonand atmosphericC are 38200 Pg and
inorganiccarbon(ACo/Co)may be approximatedby
DECONVOLVED
E12B MODEL-
TERRESTRIAL
FLUX PER NPP
FIT TO LOGISTIC-
R=0.985
0.01
-O.Ol
-0.02
ß
ß
-0.03
ß
ß
ß
ß
-0.04
....
0.95
I ....
1
I ....
1.05
I ....
1.1
I ....
1.15
I ....
1.2
• ....
1.25
1.3
c/ci
Figure 11. A fit of the funcQonp, to thedeconvolved
flux dividedby the netprimaryproductionflux as a function of relativeatmospheric
CO2 concentraQon.
This graphshowsthelogisticparameterization
(8).
32
MOORE AND BRASWELL: LIFETIME
OF EXCESS ATMOSPHERIC
Tables 2 and 3 showthe T•/2 valuescalculatedusingthe atmosphere-oceanmodels (12B, B D, and OC) standingalone (Table
2), using the coupledterrestrialmodels (E-12B, E-BD, and EOC; the latter two, again, with two parameterizationsfor the
oceansubmodels)with regrowthaloneassumed(Table 3, row 1)
and assumingbothregrowthand fertilization, which is described
with the three different schemesdiscussedin section 3 (Table 3,
Table 2. SingleHalf-Life Resultsfor theOceanAtmosphereModels
Model
Ti/2, yr
12B
BD1
OC1
BD2
OC2
81
116
42
79
37
CARBON DIOXIDE
rows 2-4). We can see that without the fertilization flux these
models yield single half-lives from as high as 116 years (BD,
preindustrial
•4Ccalibration)
toaslowas37 years(OC,bomb•4C
BD1,box-diffusion
(preindustrial
14Ccalibration);
OC1,outcropdiffusion(preindustrial);
BD2, boxdiffusion(bomb
calibration);OC2, outcropdiffusion(bomb);12B, twelve-box
model.
597 Pg respectively,simplealgebrayields an equilibriumpartial
pressureat 326 ppm. This number,which we usesimplyas a referencepoint,is the theoreticalasymptotefor the atmospheric
CO2
decaycurvesci(t). If we terminatethe forcingat a presentvalue
(e.g.,to= 354 ppm),Tu2 is definedto be thefirst yearaftercutoff
at whichc(t) < 326 ppm.We notethat smalldifferencescouldbe
obtainedby varying the Revelle value, by usingthe more complete mathematicaldescriptionof the carbonate-boratechemical
system[e.g., Bolin et al., 1979;Bolin, 1981), or by assumingdifferentconcentrations
of dissolvedinorganiccarbon.
calibration). The ranges would narrow considerably if the
Outcrop-Diffusion(OC) model were dropped.It also narrows
when the biosphereis includedwith regrowth alone assumed.
This simply addsa uniform sink to each system,which aids the
ocean-atmosphere
systemsthat have an ocean surfacebottleneck
problemin the removalof carbon(i.e. the OC and 12B systems).
The singlehalf-lives are obviouslyreduced;they are even more
significantlyreducedif one includesa fertilization sink (Table 3,
rows 2-4). In part, this is simplythe additionof a sink, aswas the
case in including forest regrowth; however, there is a further
compensating
factorthat the sink strengthis not Uniform.
Namely, the modeloceanswhichare moreefficientat takingup
excessCO2 requirea smallerF• flux to matchthe pCO2 record,
and vice versa.This is alsoreflectedin the regressioncoefficients
for the OC models(seeTable 4); Fr is closeenoughto zero that
random fluctuations in the residual nearly overwhelm the assumedlogarithmic trend.
Plots of the relaxation
5. ResultsUsing Single Half-Life
In this sectionwe explore the responseof the global carbon
cyclemodels(E-BD, E-OC, andE-12B) whentheinputs(FFeand
FB) arereducedinstantaneously
to zero undertwo separateconditions:when we allow Fr to operatefor all t and when thereis no
Fr. We comparetheseresultsto a similar calculationof single
half-lives involving only the three ocean-atmospheresystems
(BD, OC, 12B). Finally, to connectto the resultsreported by
IntergovernmentalPanel on Climate Change [Watson et al.,
1990], we computein a subsequent
sectionthe classicallifetimes
of CO2usingthe threeocean-atmosphere
modelsaswell as when
the terrestrial
curves associated with the ocean-atmo-
sphere results (Table 2), for all ocean models, are shown as
Figure 12. This shouldbe comparedto the somewhatsteeperinitial dropwhenthe terrestrialbiosphereis includedwith regrowth
alone and the even steeperslopeswhen fertilization is assumed
(Figure 13); in orderto compareFigures12 and !3 we includein
Figure !3 the relaxation dynamicsof the box-diffusion model
with the •4Cbombdatausedfor parameterization.
Note thatthe
functionaldifferencesfor different schemesto incorporatefertilization are unimportant;this is partly a result of the deconvolu-
tionmethodology
whichforcesthe dynamicsto fit the samepattern, namely the ice core record, which reducesthe effect of differences in the functional
sink is included.
forms.
Table 3. SingleHalf-Life Resultsfor theOcean-Atmosphere-Terrestrial
ModelsUnderVariousCO2Fertilization
Assumptions
Par ameterization
None (regrowthonly)a
Logarithmic
E- 12B
E-B D 1
E-OC 1
E-B D2
68
22
86
49
37
28
63
40
E-OC2
33
25
Linear
19
30
20
26
17
Logistic
20
38
23
33
20
Measurements
are in years.Notationfor theoceanmodelsis the sameasin Table 2.
a No fertilizationflux in this experiment.
Table 4. Coefficientof CorrelationR for the Fit of Fr to theCalculatedResidualCarbonFlux
Par ameterization
E- 12B
E-B D 1
E-OC 1
E-B D2
E-OC2
Logarithmic
.896
.914
.804
.869
Linear
.953
.956
.895
.931
.717
Logistic
.986
.986
.958
.975
.400
Notation
for the ocean models is the same as in Table 2.
.746
MOORE AND BRASWELL:
LIFE2IME
OF EXCESS ATMOSPHERIC
CARBON DIOXIDE
33
RELAXATION RESPONSE OF OCEAN MODELS
AFTER FB AND FFE FORCING ARE CUT
36O
350
340
330
(pC02)1
/2
320
oc (bomb)
310
OC (pre)
BD (pre)
BD (bomb)
300
12B
290
280
0
100
200
300
400
500
600
TIME SINCE CUTOFF (yr)
Figure 12. Response
of the ocean-atmosphere
modelsto a cessation
of all emissionsat t0=1990.The horizontal
arrowrepresents
thepCO2 level halfwaybetweenthe 1990valueandthetheoreticalequilibriumvalue.
sink, beyond forest regrowth. Thus the resultsmay be grouped
into the following three categories(with associatedaverageval-
In summary,there are small differencesin singlehalf-life resuits associated with the model ocean used and with the nature of
theparameterization
of themissingflux Fr, but theprincipalvari- ues)' (1)regrowth and fertilization(Tv2 = 27 years), (2) reationis predominantly
dueto the inclusionof a terrestrialcarbon growthand no fertilization(Tv2 = 57 years), and (3) no bio.
E-12B RELAXATION
SENSITIVITY
360
....
• ....
• ....
'
RESPONSE
TO FERTILIZATION
• ....
• ....
13.
O
• ....
LOGISTIC
[ ',,•,
.... LINEAR -
",
•3
• ....
•,
340
[
FUNCTION
• ....
• -OCEAN
ONLY
'•'•
(pcø2)1
/2
•.•.
'..'•.,... 0 DATA
!
300
.- - - '
' -
/
-
/
/
/
ß
280
260
....
1700
•',,,
1800
,'•
1900
....
•,
2000
,' ,','
i"',",',
2100
,'•
2200
,',',",
•,
2300
,"',',
• ....
2400
'
2500
Figure 13. Responseof the E-12B (Emanuelterrestrialsubmodelplustwelve-boxocean)modelto the emissions
cutoff scenario.Curvesare shownwhich representthe inclusionof threepossibleparameterizations
of fertilization flux, as well as the curve for no fertilization.For comparisonwe also includethe curve for the ocean-only
(12B) case.
34
MOORE
AND BRASWELL:
LIFE•E
OF EXCESS ATMOSPHERIC
sphere( T1/2 = 71 years). If we drop the OC model, which (as
mentionedin section2.1) mixes atmospheric
CO2 instantaneously
to deepwatersandperhapsoverestimatesthe rate of oceanuptake,
and calculateaveragesacrossthe 12B model and two versionsof
the BD model, the single half-lives increasesomewhatas fol-
lows:(1) regrowthandfertilization,(Tu2 = 31 years), (2) regrowthand no fertilization,(Tu2 = 72 years),and (3) no biosphere, (Tu2 = 92 years). The questionof short-termatmosphericCO 2 retentionis effectivelybracketedby thesevalues.
CARBON
DIOXIDE
The mathematicalconstructrequiresthe following:the calculated concentrationc(t) of atmosphericCO2 following a pulseat
time zero in a given atmosphere-ocean
model; a temporalinterval
(the periodof consideration,[0, N•,]); an upperboundXm•
x on the
time constant;and the numberof terms,say N, to be usedin the
approximationprocess(the numberis not important,only the issue of goodnessof fit really matters). One then finds, by constrained least squares fit, non-negative numbers (weights)
a•..... aN, where a•+...+as=l, and numbers (e-folding times)
xl .....xN such that
6. Exponential Characterizations Of Lifetime
We now turn to a discussion
of the traditional
c(t)=Ziaiexp(-t/'q)' (O<t<N•
and0<xi••)
sphericlifetime and point out someof the difficulties that are inherentin this methodwhen applied to a responsewhich is not an
exponential decay. For this discussionwe use the definition of
Lashof and Ahuja [1990]. The underlying idea in the LashofAhuja definition of atmosphericlifetime is the classicalconcept
of exponentialdecay
c(t)= exp(-t/'c)
(13)
where 'c is the decay constant(e-folding) time for the material.
There is an immediateconsequenceof usingthis simpleexpression.Namely, the concentrationc(t) goesrapidly to zero as t goes
to infinity and the integralin (2) is x. To usethis basicconceptof
exponentialdecay,Lashof andAhuja (seealsoMaier-Reirnerand
Hasselmann [1987] must accommodatetwo facts about (most)
models of atmosphericCO2. Given a unit pulse of carbondioxide, the decayis neitherexponentialnor doesit go to zero. Their
accommodationis reasonable:fit the responseto a weightedsum
of exponentialdecay functionsand fix a finite time window for
the fit, which in effect ignoresthe atmosphericconcentrationafter
a given length of time.
By integrationof (14) the atmospheric
lifetimeTm is thensimply
theweightedaverageof the x's(an averagee-foldingtime), i.e.,
15)
Lashof and Ahuja used Maier-Reimer and Hasselman's [ 1987]
resultsandpresenteda CO2residencetime valueof 230 years.
Using thb three atmosphere-ocean
submodels,we inject an
amountof CO2 into the atmospherecomponentof the models
equalto 25% of the amountpresentlyin our atmosphere.
Each
systemthenrelaxesto different steadystatevalues(Figure 14):
approximately10-20%of thepulseremaining,depending
on the
specificationof equilibriumcarbonatechemistryin the model.
This relaxationcurveis approximated
by a weightedsumof exponentials(i.e., (14)) underassumptions
of N•,, N, andXm•. The
resultsare summarizedin Table 5. Some of the resultsare, at first
glance, counterintuitive;the most efficient oceanmodel (OC) can
yield a longer calculatedatmosphericlifetime than the lessefficient ocean models. This is an artifact of the relative quickness
that the OC model reachesan asymptoticvalue. Simply stated,
the greaterthe overlap of the fitting window (0 < t < N,•) with a
IMPULSE RESPONSE OF OCEAN MODELS
TO A PULSEOF 150 Pg C
i
i
i
I
•
•
•
I
•
•
•
I
,
•
•
I
BD (pre)
-- OC (pre)
BD (bomb)
....... OC (bomb)
0.8
.....
12B
0.6
0.4
02
0
'
0
'
I
200
(14))
notion of atmo-
'
'
'
I
'
'
'
400
I
600
'
'
'
I
'
800
'
1000
TIME (Yr)
Figure 14:Response
of theocean-atmosphere
modelsto aninstantaneous
pulseof CO2 equivalent
to 25% of the
presentvalue.The origin of the time axis is t=0 becauseno historicalemissionsare used.
MOORE AND BRASWFJ•L: LIFE•
OF EXCESS ATMOSPHERIC CARBON DIOXIDE
35
Table 5. Sensitivityof Tres
for the Ocean-Atmosphere
Models
Tresfor Each OceanModel, yr
Numerical Parameters,yr
N•,
BD
OC
12B
500
1000
5000
500
1000
5000
500
1000
5000
500
1000
5000
132.7
262.1
1133.0
170.9
253.8
667.5
204.9
287.4
250.6
230.1
320.7
490.5
114.3
202.9
930.1
123.0
205.4
871.0
160.3
271.2
567.2
1147.0
236.8
231.6
612.0
243.4
256.3
262.8
262.8
259.9
348.8
283.9
1ooo
5 oo
-
-
-
10o0
10oo
-
-
-
1000
5000
1015.0
-
963.7
50
50
50
100
100
100
200
200
200
500
500
500
X'ma
x
Missingvaluesindicate
poorfit (Xe > 0.1).
period where the change (or the derivative) in the simulated atmosphericconcentrationis near zero, the greater the necessity
that the fitting term must contain exponentialsthat are as flat as
possible(i.e., very large x's).
For the purposesof comparisonwe usedNw valuesof 50, 100,
200, 500, and 1000 years(Table 5). In general,increasingthe interval (for a given model, and leaving 'r.maxfixed) leads to an increasein the value of. Tresor a failure of the fitting routine since
the larger interval includesmore of the curve that is relatively
fiat. This sensitivityis why the choice of a reasonablevalue for
Nw is important. Unfortunately there is no clear guideline.
Moreover, this sensitivityto Nw is alsoconnectedwith the sensitivity of Tres to the assumedXma
x.
To the extent that c(t) approaches0 for t --> N•, then the least
squares
routinewill attemptto returnveryhighxi values(because
the functionis not decayingrapidly). Consequently,one needsto
place a size constrainton the x's (i.e., 'r.max).Unfortunately,for a
given window (N•) the calculatedresidencetime is stronglydependenton this constraint(Table 5). This is partly a numerical
artifact of the over-determinedinversionprocess,in which min-
imizingerrorcanleadto instabilities(for example,seethe X 2
valuesin Table 6). Figure 15 showsa plot of c(t) for two different
valuesof '•max'The fitted curvesare very different beyondthe
window N•.
It appearsthat the numerical sensitivityof Tres may not have
been sufficiently well appreciated;however,Lashof and Ahuja
[1990] do note this sensitivity to one aspectof the calculation
when they discusschangingthe constraintto '•i --<1000. In their
studythis Xmax
sensitivitywas dealt with in a different way. They
did not fit a c(t) term but rather usedparametervalues (basedon
Green'sfunction) from a Maier-Reimer and Hasselmanstudy
(1000,362.9,73.6,17.3,1.9)
(0.131,0.201,0.321,0.249,0.098)
In other words,they took as given a weightedexponentialform
for c(t). It was notedthat the large%=1000 valueeffectivelydiscountslong-termCO2 retention.(This valuewas a fitting constraint used by Maier-Reimer and Hasselmann.)Specifically,
changing'r.
o from 1000 to 3000 andrefittingthe ai valuesled to a
morethandoublingof residencetime fi'om230 to 500 years.This
is consistent
with our results.
In an attemptto connecttheseresultswith the earlier single
half-life calculationswe perform the sameexperimentwith the
scenario-based relaxation of the BD and the E-BD models, with
terrestrialbiosphereandno fertilizationandwith a CO2-fertilized
terrestrialbiosphere(Table 7). We find the sameproblemsassociated with the calculationof Try. Often the combinationsthat
clearly reduceCO2 concentrationthe fastest,suchas including
fertilization,havelongerlifetimes.Therewas a similarcounterintuitive resultmentionedwhen discussingthe resultsof the atmosphere-ocean
systems(Table 5); the mostefficientoceanmodel
Table 6. Goodness of Fit for Selected Runs From Table 5
X 2 of Fit for Each Ocean Model
Numerical Parameters,yr
,
N•,
50
50
50
1000
1000
1000
Vmax
500
1000
5000
500
1000
5000
BD
OC
12B
1.9e-7
1.2e-5
1.1e-5
3.92
0.21
1.1e-3
1.2e-5
1.1e-5
9.7e-6
12.9
0.27
0.20
7.1e-8
3.8e- 8
3.8e-8
1.34
0.10
1.4e-2
36
MOORE
,AND BRASWELL:
LIFE•
OF EXCESS ATMOSPHERIC
CARBON
DIOXiDE
IMPULSERESPONSEC(t) AND FITTEDCURVES
FOR BD MODEL; Nw=50; TauMax=500,5000
,•,
• ,,, •
I ,,
•,
•
,•
I
•
•,
,
I
,•
, ,•
•
I
•
c(t)
.......
fit: TauMax=500
--fit:
TauMax=5000
-
,,,
0.6
.
0.4
0.2
ß
-.
-..
0
200
..
..
400
600
8O0
1000
TIME (Yr)
Figure 15. Responseof the box-diffusionmodel to an instantaneous
pulseand two curvesfitted by the least
squaresmethodto theresponsec(O. For thesecurves,Nw=50,N=4, andtherearetwo differentvaluesof %ax(500
and 5000).
(OC) often yieldeda longercalculatedatmospheric
lifetime than
7. Conclusions
the less efficient ocean models.
It is necessaryfor policy applicationsto have a consistent
There are a numberof difficultieswith the exponential-based method for estimatingthe relative contributionsof the anthromethod that should be addressed.On the basis of these results, pogenic, radiatively active trace gases to potential climate
we may concludethat mostof the T•, valuesare, in somesense, change.The GWP is one method,but we have shownin section6
determinedby the constraints
placedon theparameters
xi, andby that the estimationof the integralin the denominatorof the GWP
theassumption
aboutNw.In otherwords,Trescouldbe considered formula (1) is ambiguous.As an interim step,we have introduced
arbitrary unless there is some otherwise logical means for a simpleindicatorof the Earth system'sability to removeexcess
choosing(basedon physicalconsiderations)
reasonablefitting atmosphericCO2, the singlehalf-life (T•/2). This indicatoris free
constraints.
from the numerous
difficulties
ciated with a conventional
instabilities
calculation.
asso-
Thus one could
envisiona modified GWP, in which the singlehalf-livesof the
radiativelyimportantgasesreplacethe integralin (1).
Table 7. Sensitivityof Tres
for the E-BD OceanAtmosphere-Terrestrial
Model
We have found that the inclusion of an active (donor and recipient controlled) terrestrialbiosphereconsiderablyaltersmod-
,
Tres,yr
Numerical Parameters,yr
and numerical
lifetime
eled estimatesof the effective lifetime of atmosphericCO2.
Moreover, since this factor is the dominant sourceof variation,
N•
'•max
Biosphere
Included
50
Fertilized
Biosphere
500
351.8
258.2
50
1000
50
5000
561.6
1591.6
375.4
391.9
401.1
2128.7
273.7
512.6
2334.8
379.9
560.3
1354.1
528.2
1186.7
619.1
840.6
2135.3
100
500
100
1000
100
200
5000
500
200
1000
200
5000
500
500
500
500
1000
5000
Missingvaluesindicatepoorfit (X'2> 0.1).
465.5
-
we groupthe resultsinto threecategories:biospherewith fertil-
ization
flux(T•/2= 27years),
biosphere
without
fertilization
flux
( rl/2 = 57 years),andno biosphere
(purelyanoceanatmosphere
system;Table 2), then Tu2 = 71 years.If we droptheoutcropdiffusionmodel,whichis extremelyefficientin takingup CO2
from theatmosphere,
thentheseresultschangein detailbutnot in
pattern'biospherewith fertilizationflux (Tu2 = 31 years),
biospherewithoutfertilizationflux (T•/9. = 72 years),and no
,
1
,
biosphere(purely an ocean atmospheresystem;Table 2), then
Ti/2 = 92 years.
The inclusionof an active(donorandrecipientcontrolled)terrestrialbiosphereclearly and considerablyaltersour estimatesof
theeffectivelifetimeof atmospheric
CO2. Severalimportantand
difficult scientificquestionsremain. Does this terrestrialsink exist?Will it continueto operateor hasit begunto saturate?What
is the effect of changesin other biogeochemicalcycles?What if
MOORE AND BRASWELL:LnqsrLMEOF EXCESSATMOSPHERICCARBON DIOXIDE
37
andatmospheric
carbondioxide,
climatebeginsto shift;doesthat increaseor decreasethe terres- Houghton,R. A., Tropicaldeforestation
Cli• Change, 19, 99-118, 1991.
trial sink?How will the oceansrespond?Theseare alsoimportant
Houghton,R. A., Is carbonaccumulating
in theNortherntemperatezone?
policyquestions.
GlobalBiogeocherr•Cycles,
7, 611-617, 1993a.
Acknowledgments.
Thisworkwassupported,
in part,by grantsfromthe Houghton,R. A.,Emissionsof carbonfrom land-usechange,Proceedings
from The 1993 Global ChangeInstitute:The CarbonCycle,UCAR
NationalAeronautical and SpaceAdministration,the Environmental
Office for InterdisciplinaryEarth Studies,Boulder, Colo., in press,
Protection
Agency,andthe ScienceandEngineering
EducationDivision
1993b.
of theDepartmentof Energy.
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