Another Look at Climate Sensitivity

14 Pages·2010·0.39 MB·English

ManuscriptpreparedforNonlin.ProcessesGeophys.
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Date: 1March2010
0
1 Another Look at Climate Sensitivity
0
2
r IlyaZaliapin1andMichaelGhil2
a
M 1DepartmentofMathematicsandStatistics,UniversityofNevada,Reno,USA.E-mail:[email protected].
2GeosciencesDepartmentandLaboratoiredeMe´te´orologieDynamique(CNRSandIPSL),EcoleNormaleSupe´rieure,Paris,

1 FRANCE,andDepartmentofAtmospheric&OceanicSciencesandInstituteofGeophysics&PlanetaryPhysics,University
ofCalifornia,LosAngeles,USA.E-mail:[email protected].

]
h
p
- Abstract. We revisitarecentclaimthattheEarth’sclimate feedback.
o
system is characterized by sensitive dependence to param-
a
Keywords:Climatesensitivity,energybalancemodels,
. eters; in particular, that the system exhibitsan asymmetric,
s
c globalwarming,stabilityanalysis,bifurcations
large-amplitude response to normally distributed feedback
i
s
forcing. Such a response would imply irreducible uncer-
y
h tainty in climate change predictions and thus have notable
p
implications for climate science and climate-related policy 1 Introductionandmotivation
[
making. We show that equilibrium climate sensitivity in

1
all generality does not support such an intrinsic indetermi- 1.1 Climatesensitivityanditsimplications
v
3 nacy; the latter appears only in essentially linear systems.
5
The main flaw in the analysis that led to this claim is in- Systems with feedbacks are an efficient mathematical tool
2
0 appropriatelinearizationofanintrinsicallynonlinearmodel; for modeling a wide range of natural phenomena; Earth’s
.
3 there is no room for physical interpretations or policy con- climate is one of the most prominent examples. Stability
0 clusionsbasedon thismathematicalerror. Sensitivedepen- and sensitivity of feedback models is, accordingly, a tradi-
0
1 dence nonetheless does exist in the climate system, as well tionaltopicoftheoreticalclimatestudies(Cess, 1976;Ghil,
: asinclimatemodels—albeitinaverydifferentsensefrom 1976; CrafoordandKa¨lle´n, 1978; Schlesinger, 1985, 1986;
v
i theoneclaimedinthelinearworkunderscrutiny—andwe Cessetal., 1989). RoeandBaker (2007) (RB07 hereafter)
X
illustrate it using a classical energy balance model (EBM) haverecentlyadvocatedexistenceofintrinsicallylargesen-
r
a withnonlinearfeedbacks.EBMsexhibittwosaddle-nodebi- sitivities in an equilibrium model with multiple feedbacks.
furcations,morerecentlycalled“tippingpoints,”whichgive Specifically, they argued that a small, normally distributed
rise to threedistinctsteady-stateclimates, two ofwhichare feedbackmay lead to large-magnitude,asymmetrically dis-
stable. Suchbistablebehavioris,furthermore,supportedby tributedvaluesofthesystem’sresponse.
results from more realistic, nonequilibriumclimate models. Suchaproperty,ifvalid,wouldhaveseriousimplications
Ina trulynonlinearsetting, indeterminacyinthe size ofthe forclimatedynamics(AllenandFrame,2007)andformod-
response is observed only in the vicinity of tipping points. elingofcomplexsystemsingeneral(WatkinsandFreeman,
We show, in fact, that small disturbances cannot result in 2008). In this paper, we revisit the dynamical behavior of
a large-amplitude response, unless the system is at or near ageneral,equilibriumclimatemodelwithgenuinelynonlin-
such a point. We discussbrieflyhowthe distanceto thebi- ear feedbacks, and focus subsequently on a simple energy-
furcationmayberelatedtothestrengthofEarth’sice-albedo balance model (EBM). We notice that the main, technical
partofRB07’sargumentiswell-knownintheclimatelitera-
Correspondenceto: IlyaZaliapin([email protected]) ture, cf. Schlesinger(1985, 1986), and thusitseems useful
2 I.ZaliapinandM.Ghil: ClimateSensitivity
toreviewtheassociatedassumptionsandpossibleinterpreta- ∆T toafixedforcing∆R. Thepurportedsensitivityisdue
tionsofthisresult. tothedivergenceoftheright-handside(rhs)ofEq.(4)[their
We rederive below in Section 1.2 the key equation of equation(S4)]asfapproachesunity.Figure1,whichisanal-
RB07 and commenton their purportedlynonlinearanalysis ogoustoFig. 1ofRB07,illustratesthiseffect.
in Section 1.3. We then proceed in Section 2 with a more Roughly speaking, RB07 use the following argument: If
self-consistent version of sensitivity analysis for a nonlin- the derivativeof R(T) with respect to T is close to 0, then
ear model. This analysis is applied in Section 3 to a zero- thederivativeofT withrespecttoRisverylarge,andasmall
dimensionalEBM.ConcludingremarksfollowinSection4. changeintheradiationRcorrespondstoalargechangeinthe
temperatureT. Suchan argument,though,is onlyvalidfor
1.2 RoeandBaker’s(2007)linearanalysis
an essentially linear dependence R T. Our straightfor-
∝
ward analysis in Section 2 below shows that the sensitivity
We follow here RB07 and assume the following general
effect of Fig. 1 is absent in climate models in which gen-
setup. Let the net radiation R at the top of the atmosphere
uinelynonlinearfeedbacksR=R(T,α(T))arepresent.
berelatedtothecorrespondingaveragetemperatureT atthe
It is worth noticing that, since one seeks the temperature
Earth’s surface by R = R(T). Assume, furthermore, that
change ∆T that results from a change ∆R in the forc-
there existsa feedbackα = α(T), whichis affectedby the
ing, it might be preferable to consider the inverse function
temperaturechangeandwhichcan,inturn,affecttheradia-
T = T(R) or, more precisely, T = T(R,α(R)) and the
tivebalance.Hence,onecanwriteR=R(T,α(T)).
correspondingTaylorexpansion
Tostudyhowasmallchange∆Rintheradiationisrelated
∂T ∂T ∂α
to the correspondingtemperature change ∆T, one can use 2
∆T = ∆R+ ∆R+ (∆R) .
∂R ∂α∂R O
theTaylorexpansion(Arfken,1985)toobtain,as∆T tends
Themainconclusionsofouranalysiswillno(cid:0)tbeaffe(cid:1)ctedby
tozero,
theparticularchoiceofdirectorinverseexpansion,provided
∂R ∂R∂α
∆R= ∆T + ∆T + (∆T)2 . (1) thenonlinearitiesarecorrectlytakenintoaccount.
∂T ∂α∂T O
(cid:0) (cid:1)
Here, (x) is a functionsuch that (x) Cx as soon as 1.3 RoeandBaker’s“nonlinear”analysis
O O ≤
0<x<ǫforsomepositiveconstantsC andǫ.
Inthis section we addressthe analysis carriedoutby RB07
Introducingthenotations
inthesupplementalon-linematerials,pp.4-5,Section“Non-
1 ∂R ∂R∂α linearfeedbacks.” Themainconclusionofthatanalysiswas
= , andf = λ0 ,
λ0 ∂T − ∂α∂T that,givenrealisticparametervaluesfortheclimatesystem,
the effects of possible nonlinearities in the behavior of the
for the “reference sensitivity” λ0 and the “feedback factor”
f,weobtain functionR = R(T)arenegligibleanddonotaffectthesys-
tem’ssensitivity.Wepointoutheretwoseriousflawsintheir
1 f
∆R= − ∆T + (∆T)2 , (2) mathematicalreasoningthat,eachseparatelyandthetwoto-
λ0 O
gether,invalidatesuchaconclusion.
(cid:0) (cid:1)
whichreadilyleadsto First,andmostimportantly,despitetheirsection’stitle,the
analysiscarriedoutbyRB07isstilllinear. Indeed,theTay-
λ0 2
∆T = ∆R+ (∆T) , (3) lorexpansionintheirEq. (S7)isgivenby
1 f O
−
(cid:0) (cid:1) 1
′ ′′ 2
aslongasf =1. ∆R R∆T + R ∆T , (5)
6 ≈ 2
RB07dropthehigher-ordertermsin(3)toobtain
where ()′ stands for differentiationwith respect to T. But
·
λ0 RB07immediatelyinvertthisequationfor∆T subjecttothe
∆T = ∆R, (4)
1 f assumption
−
2
which is their equation (S4). This equation leads directly ∆T =∆T ∆T0,
totheirmainconclusion,namelythatanormallydistributed where ∆T0 is a constant. Hence, instead of solving the
feedbackfactorf resultsinanasymmetricsystem response quadraticEq.(5), RoeandBakersolvethefollowinglinear
I.ZaliapinandM.Ghil:ClimateSensitivity 3
approximation: higher-orderterms in the expansionof ∆T are vanishingly
small:
∆R=(R′+ 1R′′∆T0)∆T, (∆T)2 1−f∆T;
2 O ≪ λ0
(cid:0) (cid:1)
andthusobtainthekeyformula[theirEq. (S8)] and (b) the quantity in the rhs of this inequality is itself
nonzero.
∆T −λ0∆R ; (6) If one assumes, for instance, that (∆T)2
≈ 1−f − 21λ0R′′∆T0 C(∆T)2, where the precise meaning of gO(x) f(x)∼is
(cid:0) ∼ (cid:1)
the last step uses the, correct, fact that R′ = (1 f)/λ0. given by limx→0g(x)/f(x) = 1, then the assumptions
−
This equation artificially introduces a divergence point for (a,b) above hold for ∆T that satisfy both of the following
the temperature at f = 1 (λ0/2)R′′∆T0, which clearly conditions
−
cannot exist in a quadraticequation. Equation (6) is thus a
verycrudeapproximationthatsignificantlydeviatesfromthe 0<∆T ≪(1−f)/(λ0C) and 0<∆T <ǫ, (7)
true solutionto the fullquadraticequation(5) — whichwe
where C and ǫ are defined after Eq. (1). The first of these
discuss below in Section 2 — and thus cannot be used to
conditions implies that the range of temperatures within
justifygeneralstatementsaboutclimatemodels.
whichtheapproximation(4)worksvanishesasthefeedback
ThesecondflawintheRoeandBaker(2007)reasoningis
factor f approaches unity. Hence, all the results based on
that,usingthemodelandparametervaluestheysuggest,one
this approximation— including precisely the main conclu-
readilyfindsthat:
sionsofRB07—nolongerapplyoutsideavanishinglysmall
– R′ = 2,i.e.,globaltemperatureandradiationareneg- neighborhoodoff =1.0.
−
The asymptotic behavior we assumed above for
ativelycorrelated,whichishardlythecase forthecur-
(∆T)2 is not exotic. Consider for instance the
rent climate [e.g., HeldandSoden (2000)]. We notice
O
functionR = T2 intheneighborhoodofR = 0. ItsTaylor
thatthenegativesignofthecorrelationfollowsdirectly (cid:0) (cid:1)
expansion
from their Eq. (S10) and is not affected by particular
valuesofthemodelparameters.Furthermore,
2
∆R=2T∆T + (∆T)
O
– R′′ = 0.03,whichmeansthatthemodeltheyconsider (cid:0) (cid:1)
− canbeusedtoobtain,ignoringthesecond-orderterm,
is,indeed,essentiallylinear,andthusnotveryrealistic.
∆T ∆R/(2T). (8)
Although, in this part of their analysis, Roe and Baker as- ≈
sumedthatf =0.4,itiseasytocheckthat,forallf <0.95,
The last equation would seem to imply that the growth of
their model satisfies R′′/R′ < 0.1 and is therewith very
| | ∆T isinverselyproportionaltoT itself,sothechangeinT
closeto beinglinear. Toconclude,theeffectsonnonlinear-
shouldincreaseinfinitelyfastasT goesto0,aratherannoy-
itiesareindeednegligibleintheparticularmodelstudiedin
ingcontradiction.
thispartoftheRB07paper,sincethemodelisverycloseto
Thewayoutofthisconundrumistorealizethatthechange
beinglinear;onecannotextrapolate,therefore,theirconclu-
∆T givenbyEq.(8)isonlyvalidinasmallvicinityofT =0
sionstoclimatemodelswithsignificantnonlinearities.
andcannotbeextrapolatedtolargervalues.Ofcourse,weall
Wenextproceedwithamathematicallycorrectsensitivity knowthatthefunctionR=T2isnicelyboundedandsmooth
analysisof a generalclimate modelin the presenceoftruly
in the vicinity of 0, but it is essential to take into account
nonlinearfeedbacks.
the second term in its Taylor expansion in this vicinity to
obtaincorrectresults. We showinSection3belowthatthis
simpleexampledepictstheessentialdependenceoftheEarth
2 Aself-consistentsensitivityanalysis
surface temperature on the global solar radiative input, for
ItiseasilyseenfromthediscussioninSection1.2,especially conditionsclosetothoseofthecurrentEarthsystem.
fromEq.(2),thattherelationship(4)isacrudeapproxima- In summary, the linear approximation of the function
tion: it is valid only subject to the assumptions that (a) the R(T)derivedbyRB07fromitsTaylorexpansionisnotvalid
4 I.ZaliapinandM.Ghil: ClimateSensitivity
whenf approachesunity. Inthiscase—whichisprecisely sensitivityclaimedbyRB07doesnotexist;and(ii)sensitive
thesituationemphasizedbytheseauthors—thehigher-order dependencemay exist, in a very different sense, namely in
terms“hidden”inside (∆T)2 ,whichtheyneglected,are theneighborhoodofbifurcationpoints,asexplainedbelow.
O
indispensableforacorrect,self-consistentclimatesensitivity
(cid:0) (cid:1)
3.1 Modelformulation
analysis.
A correct analysis of the case when f approaches unity
Weconsiderhereahighlyidealizedtypeofmodelthatcon-
needstostartwithaTaylorexpansionthatkeepsthesecond-
nectstheEarth’stemperaturefieldtothesolarradiativeflux.
orderterm
The key idea on which these models are built is due to
1 f
2 3
∆R= − ∆T +a(∆T) + (∆T) , Budyko (1969) and Sellers (1969). They have been subse-
λ0 O
quently generalized and used for many studies of climate
where a = R′′/2. If (∆T)3 is much(cid:0)smaller(cid:1)than the
O stability and sensitivity; see HeldandSuarez (1974); North
othertwotermsontherhs,thenthetemperaturechangecan
(cid:0) (cid:1) (1975)andGhil(1976),amongothers.
beapproximatedbyasolutionofthequadraticequation
Theinterestandusefulnessofthese“toy”modelsresides
1 f
2 in two complementary features: (i) their simplicity, which
− ∆T +a(∆T) =∆R. (9)
λ0 allows a complete and thorough understanding of the key
Thereal-valuedsolutionstothelatterequation,iftheyexist, mechanismsinvolved;and(ii)thefactthattheirconclusions
aregivenby havebeenextensivelyconfirmedbystudiesusingmuchmore
2 detailedand presumablyrealistic models, includinggeneral
1 f 1 1 f 4∆R
∆T1,2 = − − + . circulationmodels(GCMs);see,forinstance,thereviewsof
2 aλ0 ±s aλ0 a 
(cid:18) (cid:19) Northetal.(1981)andGhilandChildress(1987).
 
Inparticular,whenf iscloseto1.0,then ThemainassumptionofEBMsisthattherateofchangeof
theglobalaveragetemperatureT is determinedonlybythe
∆R
∆T1,2 ≈± a . (10) net balance between the absorbed radiation Ri and emitted
r
One can see from Eq. (10) that the proximity of the feed- radiationRo:
back factor f to unity no longer plays an important role in dT
c =Ri(T) Ro(T). (11)
thequalitativebehavioroftheequilibriumtemperature.This dt −
pointisfurtherillustratedinFig.2thatshowstheclimatesys-
For simplicity, we follow here the zero-dimensional
tem’sresponse∆T asafunctionofthefeedbackfactorf for
(0-D) EBM version of CrafoordandKa¨lle´n (1978) and
different values of the nonlinearity parameter a. The most
GhilandChildress(1987),inwhichonlyglobal,coordinate-
important observation is that the climate response does not
independentquantitiesenter;thus
divergeatf = 1; moreover,theasymmetryoftheresponse
due to the changesin feedbackfactor f rapidlyvanishesas Ri =µQ0(1 α(T)), Ro =σg(T)T4. (12)
−
soonasthedependenceof∆Ron∆T becomesnonlinear.
Ingeneral,onecanconsideranarbitrarynumberofterms In the present formulation, the planetary ice-albedo feed-
intheTaylorexpansionofR(T). Theveryfactthatonere- backαdecreasesinanapproximatelylinearfashionwithT,
liesonthevalidityoftheTaylorexpansionimpliesthatR(T) within an intermediate range of temperatures, and is nearly
isboundedandsufficientlysmooth;inotherwords,adiver- constant for large and small T. Here Q0 is the reference
genceoftheequilibriumtemperatureduetoasmallchangein valueoftheglobalmeansolarradiativeinput,σistheStefan-
theforcingcontradictstheveryassumptionsonwhichRB07 Boltzmann constant, and g(T) is the graynessof the Earth,
basedtheirsensitivityanalysis. i.e. its deviation from black-body radiation σT4. The pa-
rameterµ 1.0multiplyingQ0 indicatesbyhowmuchthe
≈
globalinsolationdeviatesfromitsreferencevalue.
3 Sensitivityforenergybalancemodels(EBMs)
Wemodeltheice-albedofeedbackby
We consider here a classical climate model with nonlinear
1 tanh(κ(T Tc))
feedbacksto illustrate that, in such a model: (i) the type of α(T;κ)=c1+c2 − − . (13)
2
I.ZaliapinandM.Ghil:ClimateSensitivity 5
This parametrization represents a smooth interpolation be- theglobalmeansolarradiativeinputisQ0 = S/4 = 342.5,
tween the piecewise-linear formula of Sellers-type models, withthefactor1/4duetoEarth’ssphericity.
likethoseofGhil(1976)orCrafoordandKa¨lle´n(1978),and The parameterization of the ice-albedo feedback in
thepiecewise-constantformulaofBudyko-typemodels,like Eq. (13) assumes Tc = 273 K and c1 = 0.15, c2 = 0.7,
thoseofHeldandSuarez(1974)orNorth(1975). which ensures that α(T) is bounded between 0.15 and
Figure 3a shows four profiles of our ice-albedo feedback 0.85, as in Ghil (1976); see Fig. 3a. The greenhouseeffect
α(T) = α(T) as a function of T, depending on the value parametrization in Eq. (14) uses m = 0.4, which corre-
of thesteepnessparameterκ. Theprofileforκ 1 would sponds to 40% cloud cover, and T−6 = 1.9 10−15 K−6
≫ 0 ×
correspond roughly to a Budyko-type model, in which the (Sellers,1969;Ghil,1976). TheStefan-Boltzmannconstant
albedo α takes only two constant values, high and low, de- isσ 5.6697 10−8Wm−2K−4.
≈ ×
pendingonwhetherT < Tc orT > Tc. The otherprofiles
shown in the figure for smaller values of κ, correspond to 3.3 Sensitivityandbifurcationanalysis
Sellers-type models, in which there existsa transition ramp
between the high and low albedo values. Figure 3b shows 3.3.1 Twotypesofsensitivityanalysis
thecorrespondingshapesoftheradiativeinputRi =Ri(T).
The greenhouse effect is parametrized, as in We distinguishherebetweentwo typesof sensitivityanaly-
sis for the 0-DEBM (11). In the first type, we assume that
CrafoordandKa¨lle´n (1978) and GhilandChildress (1987),
byletting the system is driven out of an equilibrium state T = T0,
forwhich Ri(T0) Ro(T0) = 0, byan externalforce, and
6 −
g(T)=1 m tanh (T/T0) . (14) want to see whether and how it will return to a new equi-
−
libriumstate, whichmaybedifferentfromtheoriginalone.
(cid:0) (cid:1)
Substituting this greenhouse effect parametrization and the
Thisanalysisreferstothe“fast”dynamicsofthesystem,and
oneforthealbedointoEq.(11)leadstothefollowingEBM:
assumes that Ri(T) Ro(T) = 0 for T = T0; it is of-
− 6 6
cT˙ = µ Q0(1 α(T)) ten referred to as linear stability analysis, since it consid-
−
4 6 ers mainly small displacements from equilibrium at t = 0,
σT 1 m tanh (T/T0) , (15)
− − T(0)=T0+θ(0),whereθ(0)isoforderǫ,with0<ǫ 1,
whereT˙ = dT(cid:2)/dtdenotesth(cid:0)ederivati(cid:1)v(cid:3)eofglobaltempera- asdefinedinSection2. ≪
tureT withrespecttotimet.
Thesecondtypeofanalysisreferstothesystem’s“slow”
It is important to note that current concern, both
dynamics. We are interested in how the system evolves
scientific and public, is mostly with the greenhouse
alongabranchofequilibriumsolutionsastheexternalforce
effect, rather than with actual changes in insolation.
changessufficiently slowly for the system to track an equi-
But in a simple EBM model — whether globally aver-
libriumstate; hence,thissecondtypeofanalysisalwaysas-
aged, like in CrafoordandKa¨lle´n (1978) and here, or
sumes that the solution is in equilibrium with the forcing:
coordinate-dependent,as in Budyko (1969); Sellers (1969);
Ri(T) Ro(T)=0forallT ofinterest. Typically,wewant
HeldandSuarez (1974); North (1975) or Ghil (1976) — −
to know how sensitive model solutions are to such a slow
increasingµalwaysresultsinanetincreaseintheradiation
changein a given parameter, and so this type of analysis is
balance. It is thus convenient, and quite sufficient for the
called sensitivity analysis. In the problem at hand, we will
purpose at hand, to vary µ in the incoming radiation Ri, study — again following CrafoordandKa¨lle´n (1978) and
rather than some other parameter in the outgoing radiation
GhilandChildress (1987) — how changes in µ, and hence
Ro. WeshallreturntothispointinSection4. intheglobalinsolation,affectthemodel’sequilibria.
AremarkablepropertyoftheEBMgovernedbyEq.(11)
3.2 Modelparameters
is the existence of several stationary solutions that describe
The value S of the solar constant, which is the value of equilibriumclimatesoftheEarth(GhilandChildress,1987).
thesolarfluxnormallyincidentatthetopoftheatmosphere The existence and linear stability of these solutions result
alongastraightlineconnectingtheEarthandtheSun,isas- fromastraightforwardbifurcationanalysisofthe0-DEBM
sumedheretobeS = 1370Wm−1. Thereferencevalueof (11), as well as of its one-dimensional, latutude-dependent
6 I.ZaliapinandM.Ghil: ClimateSensitivity
counterparts(Ghil, 1976, 1994): there are two linearly sta- observed,sincetherearealwayssmall,randomperturbations
ble solutions — one that corresponds to the present cli- oftheclimatepresentinthesystem: justthinkofweatheras
mate and one thatcorrespondsto a muchcolder, “snowball representingsuchperturbations.
Earth” (Hoffmanetal., 1998) — separated by an unstable The fast decrease of the initial deviation θ(0) character-
one,whichliesabout10Kbelowthepresentclimate. izesstablesolutions;onlysuchequilibriacanbeobservedin
The existence of the three equilibria — two stable and practice.Thetwostablesolutionbranchesof(15)areshown
one unstable — has been confirmed by such results be- bysolidlinesinFig.5,whiletheunstablebranchisshownby
ing obtained by several distinct EBMs, of either Budyko- thedashedline. Thearrowsshowthedirectioninwhichthe
or Sellers-type (Northetal., 1981; Ghil, 1994). Nonlin- temperaturewillchangewhendrawnawayfromanequilib-
ear stability, to large perturbations in the initial state, has riumbyexternalforces. Thischange,whetherawayfromor
been investigated by introducing a variational principle for towardsthenearestequilibrium,isfastcomparedtotheone
the latitude-dependentEBMs ofSellers (Ghil, 1976) andof thatoccursalongeithersolutionbranch(Ghil,1976,1994).
Budyko (Northetal., 1981) type, and it confirmsthe linear
3.3.3 Bifurcationanalysis
stabilityresults.
Given the choice of model parameters, the present climate
3.3.2 Sensitivityanalysisfora0-DEBM
state corresponds to the upper stable solution of Eq. (15),
We analyze here the stability of the “slow,” quasi-adiabatic at µ = 1 (see Fig. 5). It lies quite close to the bifurcation
(in the statistical-physics sense) dynamics of model (15). point(µ,T) (0.9,280K), where the stable and unstable
≈
Theenergy-balanceconditionforsteady-statesolutionsRi = solutionsmerge.
Rotakestheform Theso-callednormalform ofthisbifurcationisgivenby
theequation
µQ0(1 α(T))=σT 4 1 m tanh (T/T0)6 . (16) X˙ =µ¯ X2, (17)
− − −
(cid:2) (cid:0) (cid:1)(cid:3) whereX isasuitablynormalizedformofT,andµ¯isanor-
Weassumehere,followingthepreviouslycitedEBMwork,
malizedformofµ. Equation(17)describesthedependence
that the main bifurcation parameter is µ; this happens to
between T and µ in a small neighborhood of the bifurca-
agree with the emphasis of RoeandBaker (2007) on cli-
tionpoint. Inparticular,thestableequilibriumbranchisde-
mate sensitivity as the dependence of mean temperature T
scribedby
onglobalsolarradiativeinput,denotedherebyQ=µQ0.
X =+√µ¯;
Figure4showstheabsorbedandemittedradiativefluxes,
Ri andRo,asfunctionsoftemperatureT forµ=0.5,1and thisresulthasexactlythesameformasthepositivesolution
2.0. One can see that Eq. (16) may have one or three solu- ofEq. (10), givenby ourself-consistentanalysisof climate
tionsdependingonthevalueofµ:onlythepresent,relatively sensitivityinthepresenceofgenuinenonlinearities,cf. Sec-
warmclimateforµ=2.0,onlythe“deep-freeze”climatefor tion2. Hence,thederivativedX/dµ¯,andthusdT/dµ,goes
µ = 0.5, andall three, includingthe intermediate, unstable toinfinityasthemodelapproachesthebifurcationpoint;this
oneforpresent-dayinsolationvalues,µ=1.0.Thesesteady- isexactlythesituationdiscussedearlierinSection2.
stateclimatevaluesareshownasafunctionoftheinsolation It is important to realize that the parabolic form of
parameterµinthebifurcationdiagramofFig.5. temperature dependence on insolation change is not an
The “fast” stability analysis (not presented here) shows accident due to the particularly simple form of EBMs.
that small deviations θ(0) from an equilibrium solution, WetheraldandManabe (1975) clearly showed, in a slightly
while all parameter values are kept fixed, may result in simplified GCM, that not only the mass-weighted temper-
twotypesofdynamics,dependingontheinitialequilibrium ature of their total atmosphere, but also the area-weighted
T0: fast increase or fast decrease of the initial deviation temperaturesofeachoftheirfivemodellevels,exhibitssuch
(GhilandChildress, 1987). The fast increase characterizes a parabolic dependence on fractional radiative input; see
unstableequilibria:asmalldeviationθ(0)fromsuchanequi- Fig.5intheirpaper.Moreover,theseauthorsemphasizethat
libriumT0forcesthesolutiontogofurtherandfurtheraway “AsstatedintheIntroduction,itisnot,however,reasonable
fromtheequilibrium. Inpractice,suchequilibriawillnotbe to conclude that the present results are more reliable than
I.ZaliapinandM.Ghil:ClimateSensitivity 7
the results from the one-dimensional studies mentioned lus, and is due to inappropriate,and unnecessary,lineariza-
abovesimplybecauseourmodeltreatstheeffectoftransport tionofanonlinearmodel.
explicitlyratherthanbyparameterization.[...]Nevertheless, Our analysis complements, reinforces and goes beyond
itseemstobe significantthatboththeone-dimensionaland thatofHannartetal.(2009),whoalsoshowedthattheclaim
three-dimensional models yield qualitatively similar results of RB07 “results from a mathematical artifact.” We notice
inmanyrespects.” simplythatHannartandcolleaguesdidnotevenquestionthe
In fact, rigorous mathematical results demonstrate that linearapproximationframeworkofRB07andstillconcluded
the saddle-node bifurcation whose normal form is given thattheclaimsofirreducibilityofthespreadintheenvelope
by Eq. (17) occurs in several systems of nonlinear partial ofclimatesensitivityarenotsupportedbytheRB07analysis.
differential equations, such as the Navier-Stokes equations
Tosummarize,whilethegeneralhumanconcernaboutcli-
(Constantinetal.,1989;Temam,1997),andnotonlyinordi-
mate sensitivity expressed by RB07 should be reasonably
narydifferentialequations,likeEqs.(11)and(15)above.We
shared by many, their scientific conclusions do not follow
emphasize,though,thatthisdoesnotcause thetemperature
fromtheirmodelanditsresults,whencorrectlyanalyzed,as
to increase rapidly due to small changes in insolation: the
donehereinSection2. Noraretheseconclusionssupported
presenceofthebifurcationpointwillresultinsmall,positive
byothermodelsofgreaterdetailandrealism,whenproperly
changesof globaltemperatureforslow, positivechangesin
investigated. Accordingly,conclusionsaboutthe likelihood
µ,whileitmaythrowtheclimatesystemintothedeep-freeze
ofextremewarmingresultingfromsmallchangesinanthro-
stateforslow,negativechangesinµ.
pogenicforcingcanhardlybeusedtosupportpoliticalpro-
posals [e.g., AllenandFrame (2007)] that claim to provide
4 Discussion futuredirectionsfortheclimate-relatedsciences.
Still, thispaper’sanalysis doesnotprecludein any sense
4.1 Howsensitiveisclimate?
the Earth’s temperature from rising significantly in com-
ing years. The methods illustrated here can only be used
Making projections of climate change for the next decades
to study climate sensitivity in the vicinity of a given state;
andcenturies,evaluatingthehumaninfluenceonfutureEarth
theycannotbeappliedto investigateclimate evolutionover
temperatures, and making normative decisions about cur-
tens of years, for example in response to large increases
rent and future anthropogenicimpacts on climate are enor-
in greenhouse gases or to other major changes in the forc-
mous tasks that requiresolid scientific expertise, as well as
ing, whether natural or anthropogenic. This latter prob-
responsible moral reasoning. Well-founded approaches to
lem requires global interdisciplinary efforts and, in partic-
handle the moral aspects of the problem are still being de-
ular, the analysis of the entire hierarchy of climate models
bated [e.g., HillerbrandandGhil (2008)]. It is that much
(SchneiderandDickinson, 1974), from conceptual to inter-
more importantto master existing tools for acquiring accu-
mediatetofullycoupledGCMs(GhilandRobertson,2000).
rate and reliable scientific evidence from the available data
Italsorequiresamuchmorecarefulstudyofrandomeffects
and models. Severalof these tools come from the realm of
thanhasbeendoneheretofore(Ghiletal.,2008).
nonlinear and complex dynamical systems (Lorenz, 1963;
Smale, 1967; RuelleandTakens, 1971; GhilandChildress, ItseemstousthatRoeandBaker’stitlequestion”WhyIs
1987;Ghil,1994;Ghiletal.,2008). ClimateSensitivitySoUnpredictable?”stillremainsopen.
A straightforward analysis, carried out in Section 2 of
thispaper,showsthatapropertreatmentofthehigher-order 4.2 Wherearethe“tippingpoints”?
termsinaclimatemodelwithnonlinearfeedbacksdoesnot
revealtheexaggeratedsensitivitytoforcingthatwasusedin The S-shaped diagram of Fig. 5 — see also Fig. 10.6 in
RB07 to advocate intrinsic unpredictability of climate pro- GhilandChildress(1987)andFig. 4inGhil(1994)—was
jections. We emphasize that the error in Roe and Baker’s usedheretoshowthesmoothnessandboundednessoftem-
analysisisnotrelatedtotheirchoiceofmodelformulationor perature changes as a function of insolation changes, away
ofthemodelparametersnortotheirinterpretationofmodel fromasaddle-nodebifurcation,likethatofEq.(10)inSec-
results. Theproblemispurelyamatterofelementarycalcu- tion2orofEq.(17)inSection 3.3.3.
8 I.ZaliapinandM.Ghil: ClimateSensitivity
ThisS-shapedcurveneverthelessrevealstheexistenceof hypothetical tipping points on the “warm” side have been
sensitive dependence of Earth’s temperature on insolation identified by Lentonetal. (2008) and references therein,
changes,oronotherchangesinEarth’snetradiationbudget, amongmanyothers. But onlyfewof these havebeenstud-
such as may be caused by increasing levels of greenhouse ied with the same degree of mathematical and physicalde-
gases,ontheonehand,orofaerosols,ontheother.Thissen- tail as the ones of Fig. 5 here. One worthwhile example is
sitive dependenceis quite differentfrom the one advocated thatoftheoceans’buoyancy-driven,orthermohaline,circu-
by RB07. Namely, if the parameter µ were to slightly de- lation (Stommel, 1961; Bryan, 1986; QuonandGhil, 1992;
crease—ratherthanincrease,asitseemstohavedonesince ThualandMcWilliams,1992;DijkstraandGhil,2005).
the mid-1970s, in the sense described in the last paragraph
Accordingly, humankind must be careful — in pursu-
of Section 3.1 — then the climate system would be pushed
ing its recent interest in geoengineering (Crutzen, 2006;
pastthebifurcationpointatµ 0.9. Theonlywayforthe
MacCracken, 2006) — to stay a course that runs between
≈
global temperature to go would be down, all the way to a
tippingpointsonthewarm,aswellasonthe“cold”sideof
deep-freezeEarth,withmuchlowertemperaturesthanthose
ourcurrentclimate. Inanycase, theexistence,positionand
ofrecent,Quaternaryiceages.
properties of such tipping points need to be established by
It has become common in recent discourse about po-
physicallycarefulandmathematicallyrigorousstudies. The
tentially irreversible climate change to talk about “tipping
“margins of maneuver” seem reasonably wide, at least on
points”; e.g., Lentonetal. (2008). The term was originally
thetimescaleoftenstohundredsofyears,butthisdoesnot
introducedintothesocialsciencesbyGladwell(2000)tode-
eliminatethepossibilitytoeventuallyreachonesuchtipping
noteapointatwhichapreviouslyrarephenomenonbecomes
point,andthusweareleddirectlytothenextquestion.
dramaticallymorecommon. Inthephysicalsciences, ithas
beenidentifiedwithashiftfromonestableequilibriumtoan-
4.3 Howclosearewetoacoldtippingpoint?
otherone,i.e.,toasaddle-nodebifurcation,asseeninFig.5
hereandexplainedinSection3.3.3above.
Let us assume for the moment that the dangers of further
In the EBM context of Fig. 5, it would require an enor-
warming will lead humanity to actually stop, and possibly
mous, almosttwofoldincreasein theinsolationinorderfor
reverse, the currenttrend of an increasinglypositive netra-
adeep-freeze–typeequilibriumtoreachthebifurcationpoint
diationbalance. Given, on the other hand, the dangersof a
at µ 1.85 and jump from there to T 350 K, a tem-
≈ ≈ snowball Earth, one might want to estimate then the close-
peraturethatsoundsequallyunpleasant. Withinthebroader
nessoftheclimatesystemtothetop-leftbifurcationpointin
contextoftherecentdebatesonhowtoexitasnowball-Earth
Fig.5here.
state,verylarge,andpossiblyimplausibleincreasesinCO2
The GCM simulations of WetheraldandManabe (1975)
levelswouldberequired(Pierrehumbert,2004).
(see again their Fig. 5) suggest that this point might lie no
Indeed, the likelihood to actually reach the tipping point
fartherthan5%belowthecurrentvalueofthesolarconstant.
to the left ofthe currentclimate in Fig. 5 seemsto be quite
Atthesametime, theSunhasbeenmuchfainter4Gyrago
small. Mechanisms for entering a snowball-Earth climate
(by approximatyely25–30%) than today, without the Earth
have been recently studied with a number of fairly realistic
ending up in a deep freeze, exceptpossibly much later. So
climate models (Hydeetal., 2000; Donnadieuetal., 2004;
howclosearewetothistippingpoint?
PoulsenandJacob, 2004). Both modeling and independent
geologicalevidencesuggestthatEarth’sclimatecansustain Figure6hereshowsstableandunstableequilibriumsolu-
significantfluctuationsofthesolarradiativeinput,andhence tions for different profiles of the ice-albedo feedback, α =
of global temperature, withoutentering the snowball Earth, α(T;κ);thisprofileisdeterminedbythevalueofthesteep-
andevidenceforEartheverhavingbeeninsuchastateisstill nessparameterκ (cf. Fig. 3a). The figuresuggeststhatthe
controversial. steepertherampoftheice-albedofeedbackfunction,i.e. the
Nevertheless,theexistenceoftheupper-lefttippingpoint largerκ, the furtherawaythe bifurcationmightlie. Infact,
shown in Fig. 5 is confirmed by numerous model studies, foraverysmoothdependenceofthealbedoontemperature,
includingGCMs, and we have alreadycited some evidence i.e. for a very small κ, there is no bifurcation at all (not
also for the lower-right tipping point in the figure. Several shown):verysmallvaluesofκproduceonlyasingle-valued,
I.ZaliapinandM.Ghil:ClimateSensitivity 9
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