Linear or Nonlinear Modeling for ENSO Dynamics?

31 Pages·2017·4.77 MB·English

atmosphere
Article
Linear or Nonlinear Modeling for ENSO Dynamics?
MarcoBianucci1,* , AntoniettaCapotondi2,RiccardoMannella3andSilviaMerlino1
1 ISMAR-CNR,19032LaSpezia, Italy;[email protected]
2 NOAA-ESRL,PhysicalSciencesDivision,UniversityofColorado,CIRES,Boulder,CO80305,USA;
[email protected]
3 PhysicsDepartment,PisaUniversity,56127Pisa,Italy;[email protected]
* Correspondence:[email protected]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
Received:12September2018;Accepted:4November2018;Published:8November2018 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Abstract: The observed ENSO statistics exhibits a non-Gaussian behavior, which is indicative
of the presence of nonlinear processes. In this paper, we use the Recharge Oscillator Model
(ROM), a largely used Low-Order Model (LOM) of ENSO, as well as methodologies borrowed
from the field of statistical mechanics to identify which aspects of the system may give rise to
nonlinearitiesthatareconsistentwiththeobservedENSOstatistics. Inparticular,weareinterestedin
understandingwhetherthenonlinearitiesresideinthesystemdynamicsorinthefastatmospheric
forcing. OurresultsindicatethatoneimportantdynamicalnonlinearityoftenintroducedintheROM
cannotjustifyanon-Gaussiansystembehavior,whilethenonlinearityintheatmosphericforcingcan
insteadproduceastatisticssimilartotheobserved. Theimplicationsofthenon-Gaussiancharacter
ofENSOstatisticsforthefrequencyofextremeElNiñoeventsisthenexamined.
Keywords: non-Hamiltonian complex systems; Fokker–Planck equation; transport coefficients;
ElNiño-LaNiña;Lieevolution;projectionapproach
1. Non-Gaussian-NonlinearFeaturesoftheENSOStatistics
It is well established that some aspects of the El Niño Southern Oscillation (ENSO) have a
non-Gaussian statistics, a characteristic that is indicative of some underlying nonlinear process.
For example, the plot of the frequency of the Niño 3 index (averaged Sea Surface Temperature
(SST)anomaliesintheregion5◦ S–5◦ N,150◦ W–90◦ W)showsthatthelikelihoodoflargepositive
SSTanomaliesisgreaterthanthatofnegativeanomalies(Figure1),indicatingthattheSSTanomalies
achievedintheeasternequatorialPacificduringthematurephaseofanElNiñoarelargerthanthose
occurringduringlaNiñaevents. AnotherasymmetryisthedurationofthetransitionfromoneENSO
phasetotheother,witharapidtransitionfromElNiñotoLaNiña,butaslowertransitionfromLa
NiñatoElNiño[1]. ThedurationofElNiñoeventstendstobeshorterthanthatofLaNiñaones,
andtherangeofElNiñoflavorsislargerthanthatofLaNiña[2].
This non-Gaussian behavior is quantified by the positive skewness of the SST distribution:
γ ∼ 0.7. Apositivekurtosisγ ∼ 1.1alsoclearlyindicatesthatextremeeventsaremorelikelyto
1 2
occurthanforanormaldistribution.
The non-Gaussianity of the histogram arises from non-linear contributions in the equation
of motion governing the system, and it is important to understand which nonlinearities play the
mostrelevantroleintheENSOevolution. WeknowthatENSOistheresultofcomplexlarge-scale
ocean-atmosphereinteractionsthatalsoincludenon-linearprocesses. Forinstance, AnandJin[3]
proposed that the Nonlinear Dynamical Heating (NDH) in the ocean mixed layer is crucial for
generating intense El Niño events, mainly due to the vertical nonlinear advection. Asymmetries
inintensitycanalsoresultfromocean-atmospherecouplingprocesses(e.g.,[4])orfromthenonlinear
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Atmosphere2018,9,435 2of31
response of the zonal wind stress to ENSO SST anomalies [5,6]. In general, the complexity of the
ENSO,asformostofthelarge-scalephenomenaresultingfromtheocean-atmosphereinteraction,
makes it challenging to understand the general key features of the statistics and dynamics of the
events. Thus,tomimicthehighlycomplexnatureofENSOevents,asthediversityinspatialflavors
andthe“non-normal”growthoffluctuations[7],largemultivariatelinearmodelswithmultivariate
additivestochasticperturbationshavebeenexploited[8,9]. Here, wefocusinsteadonLowOrder
Models(LOMs),i.e.,thesetofOrdinaryDifferentialEquations(ODE)forENSO,perturbedbyafast
andchaoticforcing. Probably,amorerealisticmodelingofENSOshouldmergethesetwopictures
(multidimensionalityandnonlinearities). Adeeperdiscussionabouttheeffectiveandrespectiveroles
ofthedimensionandofthenonlinearitiesofanLOMtomodelthecomplexityofENSOgoesbeyond
thetaskofthepresentwork.
ThetropicalPacificcanbewelldescribedbyshallowwaterequationsdrivenatthesurfacebythe
easterlytradewinds,thusallowingforadrasticsimplificationofthesystemdynamics.
Simpleshallowwaterequationstogetherwiththeassumptionofalargetimescaleseparation
betweentheENSOdynamicsandthetypicaldynamicsofatmosphericprocessesmakeitpossibleto
reducethedescriptionofthedynamicsoftheENSO-relevantvariables,toarelativelysimpleLOM.
ThisLOMcanbeseparatedinaslowpart,or“systemofinterest”,associatedwiththeslowevolution
ofoceanicvariableslikeSSTandthermoclinedepth, andafast“restofthesystem”thatprimarily
includesfastatmosphericprocesses,e.g.,theMadden–JulianOscillation(MJO),theWesterlyWind
Bursts(WWBs),etc.,whichareconsideredimportantinforcingENSO.ENSOLOMsthathavebeen
extensivelyconsideredintheliteratureincludetheRechargeOscillatorModel(ROM,e.g.,[10–17]
amongmanyothers)andtheDelayedOscillator(DO,e.g.,[18,19]). Inthesemodels,ENSOeventsare
triggeredbytheforcingprovidedbythefastatmosphericvariables,whichareusuallyrepresentedasa
stochasticperturbationofthesystemofinterest.
ThesimplifiedrepresentationofENSOintheLOMsprovidesasuitableframeworkforexamining
theimpactofthesystemnonlinearitiesontheENSOstatistics. Inparticular,animportantquestion
is whether the relevant nonlinearities reside in the interaction among the variables of the system
ofinterest(dynamicalnonlinearities), assuggested, forexamplein[18], orareassociatedwiththe
interactionofthesystemofinterestwiththeatmosphericfastforcing, asindicated, amongothers,
by[6,20].
Thefastatmosphericforcingcanbeincludedintwoways: consideringthisforcingasstochastic
orassumingadeterministicmotionwithinitialconditionsdistributedaccordingtosomedistribution
(theGibbsideaofensemble). Theformerapproachisoftenusedintheoceanographicfield,wherethe
dynamicsisdividedintoaslowandaveryfastandchaoticcomponent,andthefastpartisconsidered
toberandom(usuallymodeledaswhitenoise)[21–25]. Thelatterapproachhasbeenlargelyused
in the context of the foundation of statistical mechanics and thermodynamics and, more recently,
alsointheoceanographicfieldbysomeoftheauthors[17,26,27]. Itaimsatstartingfrommorerealistic
deterministicsystems(insteadofthestochasticones),andbyusingprojection-perturbationprocedures,
itallowsmanagingalsothecasesinwhichthetimescaleseparationbetweenthedynamicsofthe
partofinterestandthatoftherestofthesystemisnotsolarge: thisisthereasonwhyinthiswork,
weshallfocusontheprojection-perturbationapproach,whichhasbeenappliedtothemulti-scale
ENSOsystem.
Bothapproacheshavetheirstrengthsandweaknesses,andalargeliteraturehasbeendevoted
tojustifybothfromaformalandtheoreticalpointofview(forareviewaboutthefoundationofthe
stochastic approach, see [28], and for the projection approach, see, e.g., [29–33]). It is far beyond
the scope of the present work to discuss the foundation of these approaches. However, both aim
atobtainingaPartialDifferentialEquation(PDE),ideallyaFokker–PlanckEquation(FPE),guiding
the dynamics of the Probability Density Function (PDF) for the variables of the part of interest of
thesystem.
Atmosphere2018,9,435 3of31
TheFPEhasbeenhistoricallyusedtodescribethePDFofBrownian-likeparticles[34,35].Actually,
theFPEhasbeenrecognizedasaveryusefultoolformodelingawidevarietyofstochasticphenomena
arising in physics, chemistry, biology, finance, traffic flow, etc. [36]. The importance of the FPE
cannot be overestimated, and in the classical books of stochastic methods, it is obtained from a
truncatingKramers–MoyalexpansionoftheMarkovianmasterequation[37–39]. FromanFPE,itis
possible to obtain statistical information about the quantity of interest. For example, the FPE is
the major approach for determining, both analytically and numerically, important quantities like
the First Passage Time (FPT; see for example [40]), in a variety of contexts ranging from civil and
mechanicalengineering,physicalchemistry[41–43],disorderedsystems[44,45],neuroscience[46,47],
biochemistry[48],biomedicine,mathematicalfinancing,imageprocessing,computerscience,etc.
Moregenerally,thepossibilityofusinganFPEtomodelthestatisticalbehavioroftherelevant
propertiesofthevariablesofinterestisoftencrucialbecausetheFPEistheonlynon-trivialpartial
differentialequationthatinmanycasescanbeapproachedforasolutionwithanalyticaltools,and
it looks like a Schrodinger equation [49]. In the present paper, we shall obtain a general FPE for
theperturbedlinear/non-linearROMrepresentingtheENSOdynamics,andwewillcomparesome
generalstatisticalfeaturesresultingfromtheFPE(suchasthestationaryPDF,therelaxationproperties
andtheaveragetimingoftheevents)withobservations,toinferthespecificnatureofthenonlinearities
oftheENSOmodeledbytheROM.
NINO3 1950-2014 p (T)
eq
THREE-MONTH AVERAGE
0.5
0.4
0.3
0.2
0.1
-2.7 -1.8 -0.9 0 0.9 1.8 2.7
Figure1.Histogramofthefrequencyforthethree-monthaverageNiño3datafromNOAA[50].The
dashedlineisthebestfitofthestationaryPDFfoundin[17](stronglyskewedandwithaheavytail),
andsolidbluelineisanormaldistributionwiththesameaverageandvarianceoftheobservationdata.
Thecontentsareorganizedasfollows: Section2isdevotedtoreviewinganddiscussingtheway
theROMemergesasanLOMdescribingthedynamicsoftheanomaliesofthethermoclinedepthand
oftheSSToftheequatorialPacificOcean. Thissectionshowsfrombothaphysicalandobservational
pointofviewthattheROMcanprovideagoodrepresentationoftheENSOdynamics. InSection3,
westartdiscussingthenonlinearities,andwepresentafirstresultofthepresentpaper: comparing
observationswiththemodel,itishardtojustify/detectpossiblyinternaldynamicalnonlinearitiesfor
theROM(nonlinearcouplingbetweentheROMvariables). InSection4,weintroduceandjustifythe
nonlinearcouplingbetweentheunperturbedROMandthefast(butnotextremelyfast)forcing(mostly
Atmosphere2018,9,435 4of31
bytheatmosphere). InSection5,startingfromtheperturbedROM,weexploitrecentresults[17,26,27]
toobtaintheexplicitexpressionoftheFPEfortheROM.Thisismainlyareviewofpreviousresults,but
itisdevelopedratherextensivelyheretomakethepresentworkself-contained. InSection6,weuse
differentmethods(averagesoffirstmoments,covariancematrix,firstpassagetime,powerspectrum
andexpectationoftheprocesstimeincrements)toinfer,fromdata,thevaluesoftheFPEcoefficients,
comparingthemwiththetheoreticalones. Theaimistoassesswherethenonlinearitiesresponsiblefor
thenon-GaussianfeaturesoftheENSOstatisticsreside. Section7isdevotedtodiscussingtheresults
anddrawingconclusions.
2. TheDynamicsoftheComponentsoftheENSO:TheRechargeOscillatorModel
The El Niño Southern Oscillation, also known as ENSO, is a periodic fluctuation of the SST
(El Niño) and the sea level pressure (Southern Oscillation) across the equatorial Pacific Ocean.
ThefluctuationsariseoveranormalconditionofstrongSSTgradientbetweentheeast(coldtongue)
andthewest(warmpool)equatorialPacificOcean. ENSOisacomplicated,notyetfully-understood,
coupledocean-atmospherephenomenoninwhichbasin-widechangesinseasurfacetemperatures,
trade winds and atmospheric convection are intimately related. However, the particular ocean
and atmosphere conditions that generate, and in some way also define, the El Niño/La Niña
eventsallowtherepresentationofthekeyocean-atmosphericfeedbacksresponsibleforthiscomplex
phenomenoninasimplifiedfashion.
TheupperlayerofthetropicalPacificOcean,responsiblefortheenergyandmomentumexchange
withtheatmosphere,isaverylongstripofwaterwithameridionalwidthofafewhundredkilometers,
azonalextensionofabout10,000kmandanaveragedepthofabout120m,asdefinedbythedepth
ofthethermocline. Thewaterinthisstripflowsoverthedeeperandcolderocean,fromtheSouth
AmericancoaststothewarmAsiancoasts,mainlydrivenbytheeasterlytradewindsand“trapped”
intheconvergenceoceanzone,aroundtheequatoriallatitude,ultimatelyduetotheCorioliseffect.
ThismotionaccumulateswarmwaterinthewesternPacificOcean,andbecauseoftheclosed
easternboundary,itforcesthecoldanddeepwatertoriseupintheeasternPacifictocompensatethe
surfacewatermovingwestward. ThisprocessgeneratesthestrongSSTgradientbetweenthetwosides
ofthebasin,which,innormalconditions,isinequilibriumwiththeeasterlytradewinds.
DuringanElNiño(LaNiña)event,thisSSTgradientfromeasttowestisreduced(increased): we
considerthatanElNiño(LaNiña)eventisoccurringwhentheSSTanomalyontheeastequatorial
Pacificisgreater(lesser)than0.5(−0.5)degreesCelsiuswithrespecttotheaveragevalue. TheSST
anomaliesarerelatedtothethermoclinedepthanomalies. Thus,thekeyquantitiesthatcharacterize
ENSO in the equatorial Pacific Ocean are the SST anomaly (T ) in the eastern part of the domain,
E
usuallydescribedintermsoftheNiño3index(averageSSTinthearea5◦ S–5◦ N,150◦ W–90◦ W),
andthezonalaverageoftheanomalyofthethermoclinedepth(h).
ExploitingtheseverypeculiarphysicalconditionsoftheequatorialsurfacelayerofthePacific
Oceanandfromtwo-box(eastandwestside,respectively),two-strip(equatorialandoffequatorial,
respectively) approximations of the shallow water Navier–Stokes (NS) equations applied to the
tropicalPacificOcean,JinobtainedthefollowingverysimplesystemoflinearequationsforthePacific
equatorialthermoclinedepthanomaly[10,11]:
h˙ = −rh −ατ (1a)
W W s
h −h = τ (1b)
E W s
whereτ isthezonally-integratedwindstressanomaly,thesubscriptW (E)meansthatthevariables
s
refertothewest(east)sideofthebasin,thefrictiontermrcollectivelyrepresentsthedampingofthe
upperoceansystemthroughmixingandtheequatorialenergylosstotheboundarylayercurrentsat
theeastandwestsidesoftheoceanbasin,anditissetintherange6.3month(cid:46)r−1(cid:46)8.0months[10,11].
Equation(1b)describesthezonaltiltofthethermocline,whichisinbalancewiththeequatorialzonal
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windstressforcingoninterannualtimescales(e.g.,[10,51]). TheKelvinandRossbywaves,which
accomplishtheadjustmentprocesses,arefilteredout,inthispicture,becausethetimescaleofthemode
retained in thisapproximationis much longer than thebasin crossing time ofthese waves. Ifone
assumes that for a given steady wind stress forcing, the zonal mean thermocline anomaly of this
linearsystemisaboutzeroattheequilibriumstate,thatis,h +h =0(atequilibrium),thenfrom
E W
Equation(1),onefindsthatαshallbeabouthalfofr.
Theworkin[10,11]providedaphysicalbasisforthesimplelinearrelationshipsinEquation(1),
inthelongwave,two-stripapproximation.
Noticethatifthewindstressτ wereafastforcingcompletelyindependentofthedynamicsof
s
thethermoclinedepthanomaly,Equation(1)wouldgiverisetoaverysimplerelaxationprocesswith
relaxationtimegivenby1/r,which,ofcourse,isnotwhathappensinreality. Infact,asweshallsee
below, thewinddynamicshasaslowpartthatisstronglycorrelatedwiththedynamicsof h viaa
directlinearrelationshipwiththeotherENSOvariable: theSSTanomalyT .
E
AlinearODEforthethermodynamicsrelationshipbetweenthevariationoftheanomalousSST
andthethermoclinedepthwasproposedin[10,11,52,53]andrelatedsubsequentworks:
T˙ = −cT +γh +δ τ . (2)
E E E S s,E
where the first term on the right-hand side is a damping process with a collective damping rate c.
ThesecondtermdescribestheinfluenceonT ofthethermoclinedisplacement,andδ istheEkman
E S
pumpingcoefficientoftheverticaladvectivefeedbackprocessesduetothewindstressaveragedover
thedomain,whichactuallycanbeconsideredvanishing(see[10,52,53]fordetails): δ =0.
S
UsingEquation(1b),wecanrewriteEquation(2)as:
T˙ = −cT +γh +γτ. (3)
E E W s
The above equation relates the dynamics of T to that of h . However, as we have already
E W
observed,alsothedynamicsofh ,giveninEquation(1),dependsonthevaluesoftheT variable.
W E
This dependence is due to the fact that the wind stress τ is strongly influenced by the zonal SST
s
gradient. Ineffect,inmostoftheworksontheROM[52,53]andontheDO[18],itisexplicitlyassumed
thatthefollowinglinearrelationshipholdstrue:
τ = bT +g(t), (4)
s E
where g(t) is a fast “chaotic” fluctuating function of time. In this work, we define as “chaotic” a
unidimensionalfunctionoftime g(t) ifitsdynamicscanbeconsidered“almost”randomsoasto
allowustosubstituteitsafelywithagenuinestochasticprocessortouseasecond-orderperturbation
projectionapproach, ashereafterspecified. Inthefirstcase, weshouldexploitwell-knownworks
aboutlimittheorems(e.g.,[54])andintroduceamorespecificchaotichypothesis(e.g.,[55,56])that,
actually,hasbeenproventobesatisfiedonlybyhighlyidealizedmathematicalmodels. Inthesecond
case,therequestsarelessstringent,becauseitisenoughthatforanyn ∈N,thecorrelationfunction
(cid:104)g(t)g(t+t )...g(t+t )(cid:105)decays“quicklyenough”foreach1 ≤ k ≤ n,toavoiddivergencesinthe
1 n
neglectedtermsobtainedwiththeperturbationprocedure[29,57–59]. Theextensiveuseofthequotes
istostressthefactthat,asalreadystatedintheIntroduction,itisbeyondthefocusofthepresentwork
todiscussthedynamicalfoundationofstochasticprocesses/statisticalmechanics. Forfurtheruse,we
specifythatthesymbol(cid:104)...(cid:105)indicatesatimeaverage: (cid:104)a(cid:105) :=limT→∞(cid:82)0Ta(t)dt/T,butweshallexploit
theergodicassumptionandtheGibbsideatoturnthetimeaveragesintoensembleaverages.
Equation(4)impliesthattheslowcomponentofthewindstressandtheslowcomponentofT
E
areproportionaltoeachother. Figure1ofthepaper[60]supportsthisassumption,andwefurther
validate this hypothesis by analyzing the NOAA data [61], as detailed in Appendix A, where the
windstressanomalyτ isdividedintoslow(τ )andfast(τ )components. Theτ dataare
s s,slow s,fast s,slow
Atmosphere2018,9,435 6of31
obtainedfromtheone-yearaverageofτ data,whileτ ≡ τ −τ canbeidentifiedwithg(t)
s s,fast s s,slow
of(4).
ExploitingEquation(4),fromEquations(1b)and(3),weobtainthewell-knownlinearROM:
h˙ = −rh −αbT −αg(t)
W W E
T˙ = γh −λT +γg(t), (5)
E W E
withλ ≡ c−γb.
Consideringthatαisquitesmallerthanγ[10,11],wedisregardtheforcingterm−αg(t)inthe
firstequationofSystem(5),andfinally,weareledtothefollowingsimplifiedlinearROM:
h˙ = −rh −αbT
W W E
T˙ = γh −λT +g(t) (6)
E W E
where we have included the constant γ in the definition of the fast part g(t) of the wind stress τ.
s
The result of Equation (6) is really remarkable because it is a very simple, but still informative,
descriptionofacomplexlarge-scaleocean/atmosphereprocess. Wecanconsideritonthesamefooting
astheOnsagerlinearregressionprinciple,chemicalreactionratemodels[43]andotherlinearresults
of standard thermodynamics and statistical mechanics, which arise as large time and space scale
phenomenafromanunderlyingcomplexfastandchaoticdynamics.
3. IsaPossibleInternalNonlinearityRelevant?
Becauseofthelinearrelationshipbetweenh andT ofEquation(2),the“internal”dynamics(i.e.,
E E
forg(t) =0)oftheROMofEquation(6)is,byconstruction,linear. However,alinearrelationshipin
therealworldisalmostalwaysanapproximation;thusinthissection,weanalyzethepossibilitythata
nonlinearcorrectioniseffectivelypresentintheinteractionbetweenthethermoclinedepthanomaly
andthe SSTanomaly, and wetryto evaluateif thiscorrection cancontributeto thenon-Gaussian
featuresoftheENSO.Actually,in[18],itisarguedthatanonlinearcubicrelationshipbetweenthe
anomalous subsurface temperature T and the thermocline depth anomaly should be taken into
sub
account:
T = ah −e∗h3 +O(h5), (7)
sub E E E
wherethevaluesforthecoefficientsaande∗ havebeenroughlyestimatedin[18]asa ∼0.14◦Cm−1
ande∗ ∼3×10−5◦Cm−3. Fromthisnonlinearrelation,asmallcubicterminh wouldenteralsoin
E
Equation(2)[18]. Giventheverysmallvalueoftheratioe∗/a,thecubictermshouldplaysomerole
onlyforh ≥ ±2σ (σ isthestandarddeviationofh thatisalittleover10m[62]),reachingthesame
E hE hE √ E
valueofthelineartermforabouth ∼ a/e∗ = 36m(about3σ ). Usingthesearguments,in[63],
E hE
ithasbeenalreadyshownthatassuminganonlinearROMlinearlyinteractingwiththeatmosphere,
weobtainanon-normalstationaryPDF.However,aswecanseeinFigure.5of[60],bothobservations
and numerical results show that the relationship between T and h is far more linear than what
sub
wassuggestedin[18]. Infact,thisfigureshowsasmalldispersionofthedataaroundthelinearfit,
forarangeofvaluesofh thatspansbeyond±4σ . Thus,wehaveevidence,fromobservations,that
E hE
thenonlinearcorrectiontothelinearROMisquitesmallerthatthe(alreadysmall)valueestimated
in[18]. However,toevaluatetherelevanceofthissmallnonlineartermfortheENSOstatistics,wealso
havetoaccountforthebehaviorofthePDFforlargevaluesofh. ToestimatethePDFincludingthe
nonlinearcorrectionintheinteractionbetweenthethermoclinedepthanomalyandtheSSTanomaly,
weconsiderthefollowingnonlinearROM:
h˙ = −rh −αbT
W W E
T˙ = γU(cid:48)(h )−λT +g(t), (8)
E W E
Atmosphere2018,9,435 7of31
inwhich
U(h ) = h2 /2+κh4 /4+O(h6 ) (9)
W W W W
where for a generic function f(x), we have defined f(cid:48)(x) := df(x)/dx and κ is a parameter that,
accordingtotheaboveresults,ismuchsmallerthanthevalueoftheratioe∗/a,i.e.,κ <<10−4m−2.The
stationaryPDFthatcanbederivedfromEquation(8),consideringg(t)asanadditivefastfluctuating
forcing,isasfollows. Wechangevariables: v ≡ −rh −αbT ,x = h ,andEquation(8)becomes:
W E W
x˙ = v
v˙ = −αbγU(cid:48)(x)−λrx−(r+λ)v−αbg(t). (10)
Ifg(t)isassumedtobeawhitenoisewithacorrelationfunction(cid:104)g(0)g(t)(cid:105) =2dδ(t),Equation(10)
describesthedynamicsofaparticlewithcoordinatex,velocityv,inan“almost”harmonicpotential
givenby:
x2 x2 x4
V(x) = αbγU(x)+λr (cid:39) (λr+αbγ) +αbγκ (11)
2 2 4
andinteractingwithathermalbathwithfrictionr+λand“temperature”Θ = α2b2d/(r+λ). Itis
wellknownthatthestationaryPDFforsuchasystemisthe“canonical”one:
ρ (x,v) = Ze−H(Θx,v), (12)
s
where Z isthenormalizationconstantand H(x,v) ≡ V(x)+ v2 istheHamiltonian. Thus, Θisthe
2
varianceofthe“velocity”variablev: Θ = (cid:104)v2(cid:105) ≡ σ2. Asforthe x (= h )variable,wenoticethat,
s v W
giventhesmallvalueoftheκparameter,thenon-harmonictermαbγκx4/4inthepotentialV(x)of
Equation(11)givesanegligiblecontributiontothevariance;thus,wealsohaveΘ (cid:39) (λr+αbγ)(cid:104)x2(cid:105) :=
s
(λr+αbγ)σ2. Therefore,thestationaryPDFofEquation(12)isverywellapproximatedbyaGaussian
x
functionuntilthenonlineartermofthepotentialαbγκx4 entersintoplay,but,asalreadynoted,this
4
happens when x = h far exceeds at least two standard deviations, i.e., where the non-Gaussian
W
featurecouldnotreallyemergebecauseoftheextremelylowvalueofthePDFofEquation(12). Thus,
weareledtotheconclusionthatthepossiblynonlinearinteractionbetweentheinternalROMvariables
cannotinherentlybethecauseoftheobservednon-GaussianpropertiesoftheENSOstatistics.
Noticethatifweconsidertheexternalforcingg(t)asacorrelatednoise(i.e.,notwhite)orasa
weakdeterministicchaoticperturbationwithafinitetimescale(namely,withcorrelationfunction
thatdecaysinafinitetime),usingaZwanzig-likeprojectionapproachintheperturbationversion
of[64–66],weobtainastationaryPDFfortheROMvariablesthatissimilartotheoneinEquation(12),
apartfromsomesmallchangesintheparameterΘandinthecoefficientsofthepotentialV(x). Thus,
alsointhesenon-Markoviancases,thesameargumentofthepreviouswhitenoisecaseapplies: the
non-GaussianfeaturesofthePDFwouldbenegligibleandnotaspronouncedasobserved.
Sofar,wehaveshownthatanalyzingtherelationsbetweenthedynamicsofh andT ,itishard
W E
tofindnonlinearitiesstrongenoughtojustifythenon-GaussianbehavioroftheENSOstatistics. Thus,
fromnowon, weshallassumethatthenon-harmonicterminthepotentialU(h ) ofEquation(9)
W
can be neglected. Namely, we return to the linear ROM of Equation (6) and attempt to justify the
observednonlinearitiesfocusingontheinteractionoftheslowcomponentsoftheROMofEquation(6)
withg(t).
Itisworthwhiletoremarkthatarecentstudy[67]hashighlightedtheimportanceofthenonlinear
advectivetermsintheupper-oceantemperatureequationsasasourceofENSOnonlinearitiesand
the resulting non-Gaussianity of eastern Pacific SST anomalies. These nonlinear advective terms,
whichhavebeencollectivelytermedNonlinearDynamicalHeating(NDH),areprimarilycontrolled
by the anomalies of the zonal advection term (u(cid:48)∂ T(cid:48); see [67] for the nomenclature). However,
x
theasymmetricinfluenceofthenonlinearzonaladvectiontermuponwarmandcoldENSOevents
Atmosphere2018,9,435 8of31
may be linked to the distinctive multiplicative character of the anomalous wind forcing of ENSO,
whichisstrongerforpositivethannegativeanomalies. Nonlinearitiesinthesurfaceforcingandtheir
influenceontheSSTstatisticsaredescribedinthenextsection.
Beforedoingthat,however,wesimplifyEquation(6)alittleinthefollowingway:
h˙ = −ωT
E
T˙ = ωh−λT +g(t), (13)
E E
whereh ≡ (h +h )/2isthezonally-averagedthermoclinedepthanomaly(directlyrelatedtothe
W E
anomalousheataccumulatedintheEquatorialPacificOcean),achoicejustifiedbytheargumentsand
resultsof[12],shortlysummarizedhereafter. First,Equation(1b)impliesthatthetiltofthethermocline
reactsalmostinstantaneouslytowindstress, butitismorerealistictoassumethattheadjustment
of the thermocline depth takes a finite time, approximately the time it takes for a Kelvin wave to
propagatefromthewesterncentralPacifictotheeast. Thus,Equation(1b)shouldbereplacedby:
h˙ = −γ (h −h −τ ).
E h E W s
Second, repeating the same steps that led us from Equation (1b) to Equation (6), we get an
unperturbedROMgivenbythreedifferentialequationsthatbetterrepresenttheENSOdynamics,
leadingtoabetteragreementamongtheparameterofthemodelandobservations. Finally,studying
indetailthefeatureofthislinearthree-degreesoffreedomsystem,weseethatoneeigenstatehasafast
decaytime(shorterthanonemonth);thus,thedynamicsisattractedtowardatwo-dimensionalslow
manifold,wellrepresentedbythereducedROMofEquation(13).
Suitablevaluesforωandλ(the“friction”coefficient)canbeobtainedbyusingphenomenological
considerationswhenderivingtheROMfrombuildingblockequationsand/ordirectlyfromafitto
observations(see[12]). Intheliterature,wefinddifferentvaluesthatrangefrom2π/48month−1–2π
/36month−1forωandfrom1/12month−1–1/6month−1forλ. Weshallseeinthefollowingthatwe
canshrinktheseranges.
4. TheMultiplicativeNatureoftheForcing
In some recent works, it has been shown that to account for the instability of the dynamics,
theskewnessandthetailoftheobservedstationaryPDFoftheENSO,theperturbationforcingg(t)of
theROMcannotbesimplyadditive,butamultiplicativeterm,directlycorrelatedwiththeadditive
one,shouldalsobeconsidered[14,15,17],leadingtoaperturbationoftheform:
g(t) = (cid:101)(1+βT )ξ(t), (14)
E
where ξ(t) is a fast chaotic fluctuating term and (cid:101) is a parameter that controls the strength of the
interaction. ManyfactssupportthehypothesisgivenbyEquation(14):
• themultiplicativenatureoftheENSOforcingprovidedbytheMadden–JulianOscillation(MJO)
anditshigherfrequencytail([68,69]andthereferencestherein);
• multiplicative fast forcings typically emerge from the general perturbation approaches to
large-scaleoceandynamics[70,71];
• consideringthenonlinearnatureoftheheatbudgetequationforthesurfacemixedlayer[72],we
getmultiplicativefastfluctuationsinboththenetsurface/subsurfaceheatfluxandtheadvective
contributions[16,67,70,71].
Concerning the last point, we again notice that the equatorial Pacific zonal subsurface NDH
u(cid:48)∂ T(cid:48),whichplaysaroleintheasymmetryoftheElNiño-LaNiñaevents[67],maybelinkedtothe
x
multiplicativeterm(cid:101)βT ξ(t)ofEquation(14),assumingthatξ(t)andT aredirectlyrelatedtou(cid:48) and
E E
∂ T(cid:48),respectively.
x
Thus,insertingEquation(14)intotheROMofEquation(13),weget:
Atmosphere2018,9,435 9of31
h˙ = −ωT
E
T˙ = ωh−λT +(cid:101)(1+βT )ξ(t), (15)
E E E
thatisthefinalmodelweshalltakeintoaccounthereafter.
When ξ(t) is a white noise, the term (1 + βT )ξ(t) is often named
E
Correlated-Additive-Multiplicative (CAM) noise [21–23], but we shall extend this notation
alsotodeterministic(i.e.,notstochastic)processes,sayingthat (1+βT )ξ(t)isaCAMforcing. In
E
Section6,startingfromobservations,weshalluseastatisticalinferencemethodfordiffusionprocesses
withnonlineardrifttoestimatetheeffectivecontributionoftheCAMforcingwithrespecttoamore
simpleadditivefastforcing.
Ingeneral, itisreasonablethatatleastapartoftheadditiveforcingshouldnotbecorrelated
withthemultiplicativeone,i.e.,thattherearedifferentindependentfastchaoticforcingsthatcanbe
collectedasanindependentadditivecontribution,uncorrelatedtothepreviousone,consideredabove:
g(t) = f(t)+(1+βT )ξ(t), (16)
E
where f(t) is uncorrelated with ξ(t). However, for the sake of simplicity, we do not consider this
possibilityhere. ThismoregeneralROMwillbeexaminedinasubsequentpaper.
We want to stress again that we do not make the assumption that the forcing term ξ(t) of
Equation (15) has a stochastic nature. ξ(t) is a deterministic “chaotic” forcing, and statistics is
introducedbyusingtheGibbsconceptofensemble(ourPDF).
5. TheFokker–PlanckEquationGuidingtheStatisticsoftheROM
The ROM of Equation (15) must be validated by observations of the ENSO, namely we must
usesomedataanalysisapproachtoinferthevaluesofthecoefficientsoftheROMofEquation(15).
Duetothemultiplicativecharacteroftheperturbation,thesystemisnotlinear;thus,strictlyspeaking,
wellconsolidatedapproachesbasedontheGaussianpropertiesofthedatadispersion,suchasthe
LinearInversionModel(LIM),cannotbeusedinthiscase. However,aswillbeshowninthenext
section, if we observe only, or if we are interested only in the first (the mean) and the second (the
variance) moments, then the system of Equation (15) looks to be linear, with eigenvalues that are
weaklyrenormalizedbytheβparameter.Ofcourse,replacingthenonlinearROMofEquation(15)with
alinearone,alltheinformationconcerninglargedeviationsfromtheaverageswillbelost. Because
weareinterestedinevaluatingtheroleofthenonlinearitiesintheENSOdynamics,modeledbythe
perturbedROM,wewouldneedanFPEforthePDFoftheSSTanomaliesthatfullyretainsthenonlinear
natureoftheinteractionbetweentheENSOvariablesofinterest(h,T )andthecollectivevariable
E
ξ(t)ofEquation(15). Inthisway,infact,acomparisonbetweentheFPEpredictionsandobservations
shouldpossiblyvalidatethemodelandallowonetoestimatetherelativeweightofthemultiplicative
perturbation.Inthissection,webrieflyrecallandusesomeoftheresultsof[17,26,27,32,33]tointroduce
anexplicitFPEforthePDFoftheROMsystem.
Tobereallyinapositiontodistinguish,inthemodelofEquation(15),betweenthedynamicsof
theENSOvariables(h,T )andthatoftheperturbationξ(t),itisimplicitlyassumedthatthereisatime
E
scaleseparationand/oraweakinteractionbetweenthesetwosubsystems.Itisafactthatthedynamics
of the atmosphere forcing triggering the ENSO, which is represented by the term proportional to
the(cid:101)parameterinEquation(15),hasatimescalethatisshorterthanthetimescaleoftheaverage
dynamicsoftheENSO.Thus,wecansaythattheautocorrelationtimeτ ofξ(t)issmallerthanthe
√
periodandrelaxationtimeoftheunperturbedROM,givenby1/ ω2−λ2/4and2/λ,respectively.
Noticethat“shorter”doesnotnecessarilymean“extremelyshorter”,butifthisisthecase,undersome
conditionsaboutthefeaturesofthedynamicsoftheξ variable,wecanapproximateξ(t)byawhite
noise[28,55,56,73],with(cid:104)ξ(0)ξ(t)(cid:105) =2(cid:104)ξ2(cid:105)τδ(t). Itisawell-knownresultthatifξ(t)isreplacedwith
awhitenoise,Equation(15)becomesequivalenttothefollowingFokker–PlanckEquation(FPE)for
theProbabilityDensityFunction(PDF)σ(h,T;t)oftheROMvariables(weshallusetheStratonovich
Atmosphere2018,9,435 10of31
integrationruleforwhitenoisedynamics[74];hereafter,forthesakeofsimplicity,weshalluseTfor
T and∂ ≡ ∂/∂t,∂ ≡ ∂/∂hand∂ ≡ ∂/∂T):
E t h T
(cid:8)
∂ σ(h,T;t) = ω∂ T−ω∂ h+λ ∂ T+Dβ∂ (1+βT)
t h T T T
+∂ D(1+βT)2∂ (cid:9)σ(h,T;t)
T T
(cid:8)
= ω∂ T−ω∂ h+λ ∂ T−Dβ∂ (1+βT)
h T T T
+∂2 D(1+βT)2(cid:9)σ(h,T;t) (17)
T
whereD ≡ (cid:101)2(cid:104)ξ2(cid:105)τisthe“standard”diffusioncoefficient.
InthecasewherethetimescaleseparationbetweentheunperturbedROMandthefastforcing
isnotsoextreme,aswehavealreadystressed,wecannotapproximatethedeterministicforcingξ(t)
with a white noise, and we have to work with ensembles (i.e., PDF) for which the time evolution
is guided by the Liouvillian corresponding to the equation of motion of the perturbed ROM of
Equation(15). StartingfromtheLiouvilleequation,assumingweakperturbationoftheROM(small
(cid:101)values)andusingaprojectionperturbationapproach,itisthuspossibletoderiveaneffectiveFPE
thatwellapproximatesthedynamicsofthePDF[17](seealsoAppendixBforashortsummaryofthe
projectionprocedure,adaptedtothepresentcaseofEquation(15)):
(cid:8)
∂ σ(h,T;t) = ω∂ T−ω∂ h
t h T
+(λ+Dβ2)∂ T+Dβ∂
T T
(cid:9)
+∂ A(h,T)∂ +∂ B(h,T)∂ σ(h,T;t), (18)
T T T h
where the decay time τ of the auto-correlation function of ξ is defined in the following way: if
(cid:82)∞
ϕ(t) ≡ (cid:104)ξ(t)ξ(0)(cid:105)/(cid:104)ξ2(cid:105)isthenormalizedautocorrelationfunctionofξ,thenτ ≡ ϕ(t)dt.Asin[17],
0
thediffusionfunctions A(h,T)andB(h,T)aresecondorderpolynomialsoftheROMvariables(h,T):
A(h,T) =A +βA h+βA T
0 1 2
+β2A hT+β2A T2
3 4
B(h,T) =B +βB h+βB T
0 1 2
+β2B hT+β2B T2. (19)
3 4
AsisshownintheAppendixB,theA andB coefficientsareproportionalto(cid:101)2,donotdependon
i i
theβparameterandarelinearcombinationsoftheFouriertransformofthefunction ϕ(t),evaluated
√
at the frequencies 2Ω and Ω−iλ/2, where Ω ≡ ω2−λ2/4 is the effective frequency of the
unperturbedROM.
FromtheFPE,itispossibletogetalltherelevantstatisticalinformationabouttheENSOvariables,
includingtheTmomentsandthestationaryPDF.Fromaformalpointofview,theFPEinEquation(18)
isasecondorderPartialDifferentialEquation(PDE)withthediscriminant[75]equaltoB(h,T)2/2≥0;
thus, for non-vanishing B(h,T), this PDE is hyperbolic, and not a parabolic one like the FPE of
Equation(17)derivedfroma“true”stochasticMarkovianprocess. ThediffusioncoefficientB(h,T)is
asignatureofthefinite(i.e.,“noninfinite”)timescaleseparationbetweenthedynamicsofthesystem
ofinterestandthatofthebooster.
6. InferenceoftheStatisticalFeaturesoftheROMfromObservations
6.1. LinearEquationofMotionfortheFirstTwoMomentsoftheROMandtheWhiteNoiseApproximation
UsingtheFPEofEquations(18)and(19),itiseasytoobtainasetofclosedequationsofmotion
forthefirstn-thmomentsoftheROM.Forthemomentsuptothesecondones,weget: