Linear or Non Linear Modeling for ENSO Dynamics?

30 Pages·2017·4.55 MB·English

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
Article
Linear or Non Linear Modeling for ENSO Dynamics?
MarcoBianucci1,AntoniettaCapotondi2,RiccardoMannella3andSilviaMerlino1
1 ISMAR-CNR,LaSpezia(IT);[email protected]
2 NOAA,Boulder,ColoradoXXX
3 PhysicsDepartment,PisaUniversity,Pisa(IT)2
* Correspondence:[email protected]
Abstract: 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
largelyusedLow-OrderModel(LOM)ofENSO,aswellasmethodologiesborrowedfromthefield
ofstatisticalmechanicstoidentifywhichaspectsofthesystemmaygiverisetononlinearitiesthat
areconsistentwiththeobservedENSOstatistics. Inparticular,weareinterestedinunderstanding
whetherthenonlinearitiesresideinthesystemdynamicsorinthefastatmosphericforcing. Our
results indicate that one important dynamical nonlinearity often introduced in the ROM cannot
justifyanon-gaussiansystembehavior,whilethenonlinearityintheatmosphericforcingcaninstead
produceastatisticssimilartotheobserved. Theimplicationsofthenon-GaussiancharacterofENSO
statisticsforthefrequencyofextremeElNinoeventsisthenexamined.
Keywords: Non Hamiltonian complex systems; Fokker Planck Equation; transport coefficients;
ElNiño-LaNiña;Lieevolution;projectionapproach
1. NonGaussian-NonLinearfeaturesoftheENSOstatistics
It is well established that some aspects of the El Niño Southern Oscillation (ENSO) have a
non-Gaussianstatistics,acharacteristicthatisindicativeofsomeunderlyingnonlinearprocess. The
histogramofthefrequencyoftheNiño3index(averagedseasurfacetemperature(SST)anomaliesin
theregion5◦S-5◦N,150◦W-90◦W)showsthatthelikelihoodoflargepositiveSSTanomaliesisgreater
than that of negative anomalies (Fig. 1). This non-Gaussian behavior is quantified by the positive
skewness: γ ∼0.7. Thehistogramalsoshowsapositivekurtosisγ ∼1.1thatclearlyindicatesthat
1 2
extremeeventsaremorelikelytooccurthanforanormaldistribution.
As previously mentioned, 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
whichnonlinearitiesplaythemostrelevantroleintheENSOevolution. WeknowthatENSOisthe
resultofcomplexlarge-scaleocean-atmosphereinteractionsthatalsoincludenon-linearprocesses,
someofwhicharisingfromtheocean-atmospherecouplingitself. However,thetimescaleseparation
betweentheENSOdynamicsandthetypicaldynamicsofatmosphericprocessesmakesitpossible
to reduce the ENSO system to a relatively simple Low Order Model (LOM), i.e. a set of Ordinary
Differential Equations (ODE). This LOM can be separated in a slow part, or “system of interest”,
associatedwiththeslowevolutionofoceanicvariableslikeSSTandthermoclinedepth,andafast“rest
ofthesystem”thatprimarilyincludesfastatmosphericprocesses,e.g. theMadden-JulianOscillation
(MJO), the Westerly Wind Bursts (WWBs), etc., which are considered important in forcing ENSO.
ENSOLOMsthathavebeenextensivelyconsideredintheliteratureincludetheRechargeOscillator
Model(ROM,[1–8]amongmanyothers)andtheDelayedOscillator(DO,e.g.,[9,10]). Inthesemodels
ENSOeventsaretriggeredbytheforcingprovidedbythefastatmosphericvariableswhichareusually
representedasastochasticperturbationofthesystemofinterest.
ThesimplifiedrepresentationofENSOintheLOMsprovidesasuitableframeworkforexamining
theimpactofthesystemnonlinearitiesontheENSOstatistics. Inparticular,animportantquestion
© 2018 by the author(s). Distributed under a Creative Commons CC BY license.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
2of30
iswhethertherelevantnonlinearitiesresideintheinteractionamongthevariablesofthesystemof
interest(dynamicalnonlinearities),orareassociatedwiththeinteractionofthesystemofinterestwith
theatmosphericfastforcing.
Toaddressthisissueitisworthwhiletospecifybetterinwhichwaystatisticsenterinouranalysis
of a deterministic system, such as ENSO. In general statistics can enter the model in two ways:
consideringsomeforcingcomponentasstochastic(forexamplethefastpartofthewindstressthat
perturbstheoceandynamics),orassumingadeterministicmotionwithinitialconditionsdistributed
accordingtosomedistribution(theGibbsideaofensemble). Theformerapproachisoftenusedinthe
oceanographicfield,wherethedynamicsisdividedintoaslowandafastcomponent,andthefast
partisconsideredtoberandom(usuallymodeledaswhitenoise)[11–15]. Theintroductionofwhite
noisetomimicthedynamicsoffastchaoticperturbingforces,reducingthedeterministicdynamics
toaMarkovstochasticprocessis,ofcourse,verycommoninstatisticalmechanics. Thisprocedure
hasbeenformallyjustifiedbyoldwellknownworksaboutlimittheorems(e.g.[16])andalsobymore
recentones,introducingsomespecificchaotichypothesis(e.g.[17,18]). Actually,themathematical
theoremsprovedwereforveryspecialmathematicalmodelsystems,likeverychaoticsystemswhich,
fromaphysicalpointofview,lookedoftenhighlyidealizedtoberelevantforphysics. Inanycase,
onceassumedtoworkinastochasticframework,wellknownclassicaltheoremshelpustogeneralize
thisresult,showingthataverybroadclassofstochasticdynamicalsystemsconverge(inaweaksense)
todiffusionMarkovprocesses[19]. Morerecentworkshaveformallyshownthat,insomespecial
cases,alsoaprocesswithafinitecorrelationtimecanbereducedtoaMarkovstochasticprocessby
dividingthedynamicsintoMarkovpartitions(forareview,see,e.g.,thebookofDorfman[20]).
Inpracticalcases,however,itisnotusuallypossibletoprovethatthesystemweareinterestedin
satisfiesallthehypothesisrequestedbytheabovetheorems. Forexample,whenthefastperturbation
hasamultiplicativecharacterand/ortheperturbedsystemisnonlinear,itisnotformallyclearhowto
proceedtoobtainthisMarkovpartition. ItisobviousthatthisreductiontoasimpleMarkovprocessof
non-lineardynamicalsystemsperturbedbyafastchaoticperturbationthathasafinitetimescaleis
notingeneralpossible.
In oceanography, and specifically in ENSO research, the a priori assumption of infinite time
scaleseparationbetweenthedynamicsoftheoceanandtheatmospherevariablesmaynotalwaysbe
justified. Thus,inthisstudyweshallnotconsiderstochasticmodelsforENSObutjustdeterministic
onesandweintroduceastatisticaldescriptionofourdeterministicprocessbyusingtheGibbsidea
of ensemble. Therefore, we assume that the state of the system is defined, at any given time, by a
ProbabilityDensityFunction(PDF)onthewholephasespacespannedbythevariablesofthesystem.
2. ThedynamicsofthecomponentsoftheENSO
The El Niño Southern Oscillation, also known as ENSO, is a periodic fluctuation of the SST
(ElNiño)andtheseelevelpressure(SouthernOscillation)acrosstheequatorialPacificocean. The
fluctuationsariseoveranormalconditionofstrongSSTgradientbetweentheeast(coldtongue)and
the west (warm pool) equatorial Pacific ocean. ENSO is a complicated, not yet fully understood,
coupledocean-atmospherephenomenoninwhichbasin-widechangesinseasurfacetemperatures,
tradewinds,andatmosphericconvectionareintimatelyrelated. However,theparticularoceanand
atmosphereconditionsthatgenerate,andinsomewayalsodefine,theElNiño/LaNiñaevents,allow
therepresentationofthekeyocean-atmosphericfeedbacksresponsibleforthiscomplexphenomenon
inasimplifiedfashion.
TheupperlayerofthetropicalPacificocean,responsiblefortheenergyandmomentumexchange
withtheatmosphere,isaverylongstripofwaterwithameridionalwidthofafewhundredkilometers,
azonalextensionofabout10.000km,andanaveragedepthofabout120m,asdefinedbythedepth
ofthethermocline. Thewaterinthisstripflowsoverthedeeperandcolderocean,fromthesouth
AmericacoaststothewarmAsiancoasts,mainlydrivenbytheeasterlytrade-windsand"trapped"in
theconvergenceoceanzone,aroundtheequatoriallatitude,ultimatelyduetotheCorioliseffect.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
3of30
This motion accumulates warm water in the western Pacific ocean and because of the closed
easternboundaryitforcesthecoldanddeepwatertoriseupintheeasternPacifictocompensatethe
surfacewatermovingwestward. ThisprocessgeneratesthestrongSSTgradientbetweenthetwosides
ofthebasin,which,innormalconditions,isinequilibriumwiththeeasterlytradewinds.
DuringanElNiño(LaNiña)eventthisSSTgradientfromeasttowestisreduced(increased):
weconsiderthataElNiño(LaNiña)eventisoccurringwhentheSSTanomalyontheeastequatorial
pacificisgreater(lesser)than0.5(-0.5)degreesCelsiuswithrespecttotheaveragevalue. TheSST
anomaliesarerelatedtothethermoclinedepthanomalies. Thusthekeyquantitiesthatcharacterize
ENSOintheequatorialPacificoceanaretheSSTanomaly(T)intheeasternpartofthedomain,usually
describedintermsoftheNiño3index(averageSSTinthearea5N-5S,150W-90W),andthezonal
averageoftheanomalyofthethermoclinedepth(h).
ExploitingtheseverypeculiarphysicalconditionsoftheequatorialsurfacelayerofthePacific
oceanandfromatwo-boxes(eastandwestside,respectively),twostrips(equatorialandoffequatorial,
respectively)approximationsoftheshallowwaterNavierStokes(NS)equationsappliedtothetropical
Pacificocean,JinobtainedthefollowingverysimplesystemoflinearequationsforthePacificEquatorial
thermoclinedepthanomaly[1,2]:
h˙ = −rh −ατ (1a)
W W s
h −h = τ (1b)
E W s
whereτ isthezonallyintegratedwindstressanomaly,thesubscriptW (E)meansthatthevariables
s
refertothewest(east)sideofthebasin,thefrictiontermrcollectivelyrepresentsthedampingofthe
upperoceansystemthroughmixingandtheequatorialenergylosstotheboundarylayercurrentsat
theeastandwestsidesoftheoceanbasinanditissetintherange6.3month(cid:46)r−1 (cid:46)8.0month[1,2].
Eq.(1b)describesthezonaltiltofthethermoclinewhichisinbalancewiththeequatorialzonalwind
stressforcingoninterannualtimescales(e.g.[1,21]). TheKelvinandRossbywaves,thataccomplish
theadjustmentprocesses,arefilteredout,inthispicture,becausethetimescaleofthemoderetainedin
thisapproximationismuchlongerthanthebasincrossingtimeoftheseswaves. Ifoneassumesthat
foragivensteadywindstressforcing,thezonalmeanthermoclineanomalyofthislinearsystemis
aboutzeroattheequilibriumstate,thatis,h +h =0(atequilibrium),thenfromEq.(1)onefinds
E W
thatαshallbeabouthalfofr.
Refs.[1,2]provideaphysicalbasisforthesimplelinearrelationshipsinEq.(1),inthelongwave,
twostripapproximation.
Noticethatifthewindstressτ wasafastforcingcompletelyindependentofthedynamicsofthe
s
thermoclinedepthanomaly,Eq.(1)wouldgiverisetoaverysimplerelaxationprocesswithrelaxation
timegivenby1/r,that,ofcourse,isnotwhathappensinreality. Infact,asweshallseebelow,the
winddynamicshasaslowpartthatisstronglycorrelatedwiththedynamicsofhviaadirectlinear
relationshipwiththeotherENSOvariable: theSSTanomalyT.
AlinearODEforthethermodynamicsrelationshipbetweenthevariationoftheanomalousSST
andthethermoclinedepthisproposedinRefs.[1,2,22,23]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 ofthethermoclinedisplacementandδ istheEkman
E S
pumpingcoefficientoftheverticaladvectivefeedbackprocessesduetothewindstressaveragedover
thedomain,thatactuallycanbeconsideredvanishing(see[1,22,23]fordetails): δ =0.
S
UsingEq.(1b)wecanrewriteEq.(2)as:
T˙ = −cT +γh +γτ. (3)
E E W s
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
4of30
TheaboveequationrelatesthedynamicsofT tothatofh . However,aswehavealreadyobserved,
E W
alsothedynamicsofh ,giveninEq.(1),dependsonthevaluesoftheT variable. Thisdependenceis
W E
duetothefactthatthewindstressτ isstronglyinfluencedbythezonalSSTgradient. Ineffect,in
s
mostoftheworksontheROM[22,23]andontheDO[9]itisexplicitlyassumedthatthefollowing
linearrelationshipholdstrue:
τ = bT +g(t), (4)
s E
where g(t) is a fast chaotic fluctuating function of time. The above equation means that the slow
componentofthewindstressandtheslowcomponentof T areproportionaltoeachother. Fig.1
E
ofthepaper[24]supportsthisassumption,andwefurthervalidatethishypothesisbyanalyzingthe
NOAAdata[25]asdetailedinAppendixA,wherethewindstressanomalyτ isdividedinaslow
s
(τ )andfast(τ )components. Theτ dataareobtainedfromtheoneyearaverageofτ data,
s,slow s,fast s,slow s
whileτ ≡ τ −τ canbeidentifiedwithg(t)of(4).
s,fast s s,slow
ExploitingEq.(4),fromEqs.(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[1,2],wecandisregardtheforcingterm−αg(t)inthe
firstequationofsystem(5)andfinally,weareledtothefollowingsimplifiedlinearROM:
h˙ = −rh −αbT
W W E
T˙ = γh −λT +g(t) (6)
E W E
wherewehaveincludedtheconstantγinthedefinitionofthefastpartg(t)ofthewindstressτ. The
s
resultofEq.(6)isreallyremarkablebecauseitisaverysimple,butstillinformative,descriptionofa
complexlargescaleocean/atmosphereprocess. WecanconsideritonthesamefootingastheOnsager
linear regression principle, chemical reaction rate models [26] and other linear results of standard
thermodynamicsandstatisticalmechanics,thatariseaslargetimeandspacescalephenomenafroma
underlyingcomplexfastandchaoticdynamics.
Noticethatthe“internal”dynamics(i.e.,forg(t) =0)oftheROMofEq.(6)islinear,andthisis
duetothelinearrelationshipbetweenh andT ofEq.(2).However,nonlinearitiesmaybepresentand
E E
theirinfluenceshouldbeconsidered. Infact,inRef.[9]itisarguedthatanonlinearcubicrelationship
betweentheanomaloussubsurfacetemperatureT andthethermoclinedepthanomalyshouldbe
sub
takenintoaccount:
T = ah −e∗h3 +O(h5), (7)
sub E E E
wherethevaluesforthecoefficients aandehavebeenroughlyestimatedin[9]as a ∼ 0.14◦Cm−1
and e∗ ∼ 3×10−5◦Cm−2. Fromthisnonlinearrelation,asmallcubictermin h wouldenteralso
E
inEq.(2)[9]. Giventhevaluesoftheaande∗ parameters,thecubictermshouldplaysomerolefor
h ≥ ±2σ (σ isthestandarddeviationofh thatisalittleover10m[27]),reachingthesamevalue
E hE hE √ E
ofthelineartermfor h = a/e∗ = 36m(about3σ ). Usingthesearguments, in[28]ithasbeen
E hE
alreadyshownthatanonlinearROM,linearlyinteractingwiththeatmosphere,hasanonnormal
stationaryPDF.However,aswecanseeinRef.[24],Figure5,bothobservationsandnumericalresults
showthattherelationshipbetween T and hismainlylinear,withasmalldispersionofthedata
sub
aroundthelinearfitandwithacoefficientthatisclosetoonewhenthequantitiesarenormalized
bythestandarddeviationofthequantitiesfromobservations. Inthisfigurethevaluesforh range
E
from±4σ ,thusitfollowsthatinEq.(7)thecoefficiente∗ ofthenonlineartermmustbereallysmall,
hE
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
5of30
smallerthanthevalueestimatedin[9]. Despitethat,fornowwetakeintoaccountthissmallnonlinear
term,thatonlycomesintoplayforverystrongENSO.
ThusweconsiderthefollowingnonlinearROM:
h˙ = −rh −αbT
W W E
T˙ = γU(cid:48)(h )−λT +g(t), (8)
E W E
inwhich
U(h ) = h2 /2+κh4 /4+O(h6 ) (9)
W W W W
whereκisasmallparameterorder10−4m−4andforagenericfunction f(x)wehavedefined f(cid:48)(x) =
df(x)/dx. Toverifyifthenonlinearityduetoanonvanishingvalueofκcanjustify,atleasttosome
extent, the non Gaussian features of the observed Niño3 index shown in Fig. 1, we try to find the
stationaryPDFthatcanbederivedfromEq.(8),consideringg(t)asanadditivefastfluctuatingforcing.
Forthat,wechangevariables: v ≡ −rh −αbT ,x = h . Eq.(8)becomes:
W E W
x˙ = v
v˙ = −αbγU(cid:48)(x)−(r+λ)v−λrx−αbg(t). (10)
Ifg(t)isassumedtobeawhitenoisewithacorrelationfunction(cid:104)g(0)g(t)(cid:105) =2dδ(t),Eq.(10)describes
thedynamicsofaparticlewithcoordinatex,velocityv,ina"almost"harmonicpotentialgivenby
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.Asforthex(= h )variable,wenoticethat,given
s v W
thesmallvalueoftheκparameter,thenonharmonictermαbγκx4/4inthepotentialV(x)ofEq.(11)
givesanegligiblecontributiontothevariance,thuswealsohaveΘ (cid:39) (λr+αbγ)(cid:104)x2(cid:105) ≡ (λr+αbγ)σ2.
s x
ThereforethestationaryPDFofEq(12)isverywellapproximatedbyaGaussianfunctiontillx = h
W
farexceeds2standarddeviations,when,inanycaseanynonGaussianfeaturecouldnotreallyemerge
becauseofthelowprobability. ThusweareledtotheconclusionthattheobservednonGaussian
propertiesoftheENSOstatisticsarenotduetothepossiblynonlinearinteractionbetweentheinternal
ROMvariables.
Noticethatifweconsidertheexternalforcingg(t)asacorrelatednoise(i.e.,notwhite)orasa
weakdeterministicchaoticperturbation,usingaZwanzig’-likeprojectionapproachintheperturbation
version of [29–31] we obtain in any case a stationary PDF for the ROM variables that is similar to
theoneinEq.(12),apartfromsomesmallchangesintheparameterΘandinthecoefficientsofthe
potentialV(x). ThusalsointhesenonMarkoviancasesthesameargumentofthepreviouswhitenoise
caseapplies: thenonGaussianfeaturesofthePDFwouldbenegligibleandnotaspronuncedasseen
inobservations.
So far we have shown that analyzing the relations between the dynamics of h and T , it is
W E
hardtofindnonlinearitiesstrongenoughtojustifythenonGaussianbehavioroftheENSOstatistics.
Thus,fromnowonweshallassumethatthenonharmonicterminthepotentialU(h )ofEq.(9)can
W
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
6of30
be neglected. Namely, we return to the linear ROM of Eq. (6) and attempt to justify the observed
nonlinearitiesfocusingontheinteractionoftheslowcomponentsoftheROMofEq.(6)withg(t).
Itisworthwhileremarkingthatarecentstudy[32]hashighlightedtheimportanceofthenonlinear
advectivetermsintheupper-oceantemperatureequationsasasourceofENSOnonlinearitiesand
the resulting non-Gaussianity of eastern Pacific SST anomalies. These nonlinear advective terms,
whichhavebeencollectivelytermedNonlinearDynamicalHeating(NDH)areprimarilycontrolled
bytheanomaliesofthezonaladvectionterm(u(cid:48)∂ T(cid:48),see[32]forthenomenclature). However,the
x
asymmetricinfluenceofthenonlinearzonaladvectiontermuponwarmandcoldENSOeventsmay
belinkedtothedistinctivemultiplicativecharacteroftheanomalouswindforcingofENSO,whichis
strongerforpositivethannegativeanomalies. Nonlinearitiesinthesurfaceforcingandtheirinfluence
ontheSSTstatisticsaredescribedinthenextsection.
Beforedoingthat,however,wesimplifyalittleEq.(6)inthefollowingway:
h˙ = −ωT
E
T˙ = ωh−λT +g(t), (13)
E E
whereh ≡ (h +h )/2isthezonallyaveragedthermoclinedepthanomaly(directlyrelatedtothe
W E
anomalousheataccumulatedintheEquatorialPacificocean),achoicejustifiedbytheargumentsand
resultsof[3],shortlysummarizedhereafter. First,Eq.(1b)impliesthatthetiltofthethermoclinereacts
almostinstantaneouslytowindstress, butitismorerealistictoassumethattheadjustmentofthe
thermoclinedepthtakesafinitetime,approximatelythetimeittakesforaKelvinwavetopropagate
fromthewesterncentralPacifictotheeast. ThusEq.(1b)shouldbereplacedby
h˙ = −γ (h −h −τ ).
E h E W s
Second,repeatingthesamestepsthatleadusfromEq.(1b)toEq.(6),wegetanunperturbedROM
givenbythreedifferentialequationsthatbetterrepresentstheENSOdynamics,leadingtoabetter
agreementamongtheparameterofthemodelandobservations. Finally,studyingindetailthefeature
ofthislinear3-degreesoffreedomsystem,weseethatoneeigenstatehasafastdecaytime(shorterthan
onemonth),thus,thedynamicsisattractedtowarda2-dimensionalslowmanifold,wellrepresented
bythereducedROMofEq.(13).
Suitablevaluesforωandλ(the“friction”coefficient),canbeobtainedbyusingphenomenological
considerations when deriving the ROM from building block equations, and/or directly from a fit
to observations (see [3]). In literature, we find different values that range from 2π/48month−1 to
2π/36month−1 for ω andfrom1/12month−1 to1/6month−1 for λ. Weshallseeinthefollowing
thatwecanshrinktheseranges.
3. Themultiplicativenatureoftheforcing
Insomerecentworksithasbeenshownthattoaccountfortheinstabilityofthedynamics,the
skewnessandthetailoftheobservedstationaryPDFoftheENSO,theperturbationforcingg(t)ofthe
ROMcannotbesimplyadditive,butamultiplicativeterm,directlycorrelatedwiththeadditiveone,
shouldalsobeconsidered[5,6,8],leadingtoastochasticperturbationoftheform:
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. ManyfactssupportthehypothesisgivenbyEq.(14):
• themultiplicativenatureoftheENSOforcingprovidedbytheMadden-Julianoscillation(MJO)
anditshigherfrequencytail[33,34,andreferencestherein];
• multiplicativefastforcingstypicallyemergesfromthegeneralperturbationapproachestolarge
scaleoceandynamics[35,36];
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
7of30
Figure1. HistogramofthefrequencyforthethreemonthsaverageNINO3datafromNOAA[39].
DashedlineisthebestfitofthestationaryPDFfoundin[8](stronglyskewedandwithaheavytail),
solidbluelineisanormaldistributionwithsameaverageandvarianceoftheobservationdata.
• consideringthenonlinearnatureoftheheatbudgetequationforthesurfacemixedlayer[37]we
getmultiplicativefastfluctuationsinboththenetsurface/subsurfaceheatfluxandtheadvective
contributions[7,32,35,36].
Concerningthelastpoint,weagainnoticethattheequatorialPacificzonalsubsurfaceNDHu(cid:48)∂ T(cid:48),that
x
playsaroleintheasymmetryoftheElNiño-LaNiñaevents[32],maybelinkedtothemultiplicative
term(cid:101)βT ξ(t)ofEq.(14),assumingthatξ(t)andT aredirectlyrelatedtou(cid:48) and∂ T(cid:48),respectively.
E E x
Thus,insertingEq.(14)intheROMofEq.(13),weget:
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 [11–13], but we shall extend this notation
alsotodeterministic(i.e.,notstochastic)processes,sayingthat(1+βT )ξ(t)isaCAMforcing. In
E
Section5,startingfromobservations,weshalluseastatisticalinferencemethodfordiffusionprocesses
withnonlineardrifttoestimatetheeffectivecontributionoftheCAMforcingwithrespecttoamore
simpleadditivefastforcing.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
8of30
Ingeneral,itisreasonablethatatleastapartoftheadditiveforcingshouldnotbecorrelatedwith
themultiplicativeone,i.e.,thattherearedifferentindependentfastchaoticforcingthatcanbecollected
asanindependentadditivecontribution,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.
Wewanttostressagainthatwedonotmaketheassumptionthattheforcingtermξ(t)ofEq.(15)
hasastochasticnature. ξ(t)isadeterministic“chaotic”forcing,andstatisticsisintroducedbyusing
theGibbsconceptofensemble(ourPDF).
4. TheFokkerPlanckEquationguidingthestatisticsoftheROM
The ROM of Eq. (15) must be validated by observations of the ENSO, namely we must use
some data analysis approach to infer the values of the coefficients of the ROM of Eq. (15). Due to
themultiplicativecharacteroftheperturbation,thesystemisnotlinearthus,strictlyspeaking,well
consolidatedapproachesbasedontheGaussianpropertiesofthedatadispersion,suchastheLIM,
cannotbeusedinthiscase. However,asitwillbeshowninthenextSection,ifweobserveonly,orif
weareinterestedonlyinthefirst(themean)andthesecond(thevariance)moments,thenthesystem
ofEq.(15)lookslikelinear,witheigenvaluesthatareweaklyrenormalizedbytheβparameteranda
constantforcingthatdependslinearlyonβ. Ofcourse,replacingthenonlinearROMofEq.(15)witha
linearone,alltheinformationsconcerninglargedeviationsfromtheaverageswillbelost.
Tobereallyinapositiontodistinguish,inthemodelofEq.(15),betweenthedynamicsofthe
ENSOvariablesofinterest(h,T ),andthatofthe“restofthesystem”,mimickedbythecollective
E
variableξ(t),itisimplicitlyassumedthatthereisatimescaleseparationand/oraweakinteraction
betweenthesetwosubsystems. Itisafactthatthedynamicsoftheatmosphereforcingtriggering
the ENSO, that is represented by the term proportional to the (cid:101) parameter in Eq. (15), has a time
scalethatisshorterthanthetimescaleoftheaveragedynamicsoftheENSO.Thus,wecansaythat
theautocorrelationtimeτofξ(t)issmallerthantheperiodandrelaxationtimeoftheunperturbed
√
ROM,givenby1/ ω2−λ2/4and2/λ,respectively. Noticethat“shorter"doesnotnecessarilymean
“extremelyshorter”,but,ifthisisthecase,wecanapproximateξ(t)byawhitenoise[17–20],with
(cid:104)ξ(0)ξ(t)(cid:105) = 2(cid:104)ξ2(cid:105)τδ(t). Itisawellknownresultthatifξ(t)isreplacedwithawhitenoise,Eq.(15)
becomesequivalenttothefollowingFokkerPlanckEquation(FPE)fortheProbabilityDensityFunction
(PDF)σ(h,T;t)oftheROMvariables(weshallusetheStratonovichintegrationruleforwhitenoise
dynamics[40],hereafterforthesakeofsimplicityweshalluseTforT and∂ ≡ ∂/∂t,∂ ≡ ∂/∂hand
E t h
∂ ≡ ∂/∂T):
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.
InthecasewherethetimescaleseparationbetweentheunperturbedROMandthefastforcingis
notsoextreme,aswehavealreadystressedwecannotapproximatethedeterministicforcingξ(t)with
awhitenoise,andwehavetoworkwithensembles(i.e.,PDF)ofwhichthetimeevolutionisguidedby
theLiouvilliancorrespondingtotheequationofmotionoftheperturbedROMofEq.(15). Starting
fromtheLiouvilleequation,assumingweakperturbationoftheROM(small(cid:101)values),andusinga
projectionperturbationapproach,itisthuspossibletoderiveaneffectiveFPEthatwellapproximates
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
9of30
thedynamicsofthePDF[8](seealsoAppendixBforashortsummaryofthisprojectionprocedure,
adaptedtothepresentcaseofEq.(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[8],
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
AsitisshowninAppendixB,the A andB coefficientsareproportionalto(cid:101)2,donotdependontheβ
i i
parameterandarelinearcombinationsoftheFouriertransformofthefunction ϕ(t),evaluatedatthe
√
frequencies2ΩandΩ−iλ/2,whereΩ ≡ ω2−λ2/4istheeffectivefrequencyoftheunperturbed
ROM.
FromtheFPEitispossibletogetalltherelevantstatisticalinformationabouttheENSOvariables,
including the T moments and the stationary PDF. From a formal point of view, it is worthwhile
mentioningthefactthattheFPEinEq.(18)isasecondorderPartialDifferentialEquation(PDE)with
discriminant [41] equal to B(h,T)2/2 ≥ 0, thus, for non vanishing B(h,T), this PDE is hyperbolic,
andnotparabolicastheFPEofEq.(17)stemmingfrom“true”stochasticMarkovianprocesses. The
diffusioncoefficientB(h,T)isasignatureofthefinite(i.e. “noninfinite")timescaleseparationbetween
thedynamicsofthesystemofinterestandthatofthebooster.
5. InferenceofthestatisticalfeaturesoftheROMfromobservations
5.1. LinearequationofmotionforthefirsttwomomentsoftheROMandthewhitenoiseapproximation
UsingtheFPEofEqs.(18)-(19),itiseasytoobtainasetofclosedequationsofmotionforthefirst
n-thmomentsoftheROM.Forthemomentsuptothesecondonesweget:
(cid:104)h˙(cid:105) = −ω(cid:104)T(cid:105)
(cid:104)T˙(cid:105) = (ω+β2A )(cid:104)h(cid:105)−(λ−β2A )(cid:104)T(cid:105)+βA (20a)
3 4 0
(cid:104)h˙2(cid:105) = −2ω(cid:104)hT(cid:105)
(cid:104)h˙T(cid:105) = (ω+β2A )(cid:104)h2(cid:105)
3
−(λ−β2D)(cid:104)hT(cid:105)−(ω−β2B )(cid:104)T2(cid:105)
4
+β(D+A −A )(cid:104)h(cid:105)−βB (cid:104)T(cid:105)+B
0 4 2 0
(cid:16) (cid:17)
(cid:104)T˙2(cid:105) =2(ω+2β2A )(cid:104)hT(cid:105)−2 λ−β22A (cid:104)T2(cid:105)
3 4
+2β(2A +A )(cid:104)T(cid:105)+2βA (cid:104)h(cid:105)+2A , (20b)
0 4 3 0
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0239.v1
10of30
fromwhich(hereafterthesubscript“s”standsfor“stationary”)
A (cid:18) β2A (cid:19)
(cid:104)T2(cid:105) = 0 1− 3
s λ−2β2A β2A +ω
4 3
(cid:20) (cid:18) (cid:19)(cid:21)
A A A A
≈ 0 1+β2 2 4 − 3 +O((cid:101)6) ≈ 0 +O((cid:101)4) (21)
λ λ ω λ
Thus,thewidthofthestationaryPDFcanbeconsideredindependentofβ.
Eq. (20a) is equivalent to the equation of motion of a linear dissipative oscillator, with bare
(cid:112)
frequency ω2+β2A andfrictioncoefficient(λ−β2A ),perturbedbytheconstantforceβA . The
3 4 0
weak perturbation assumption (small (cid:101)) and the fact that β ∼ 0.2 imply that ω+β2A ∼ ω and
3
(λ−β2A ) ∼ λ. Thus,theequationofmotionofthefirstmoment(Eq.(20a))isverysimilartothat
4
oftheunperturbed((cid:101) =0)ROM(apartfromtheconstantforcingβA thatcanbedisregarded,aswe
0
willshowshortly). Thisfactisreallyimportantbecausefromaphysicalpointofviewitgivesasound
meaningtothedefinitionofunperturbedROMofEq.(13): itisthedynamicalsystemthatcorresponds
tothe“slow”partoftheENSOdynamics.
Concerningtheconstantforce βA appearinginEq.(20a),itisnotreallymeasurable,infactit
0
is an “artifact” of the final approximation that, from the ROM of Eq. (6), leads to Eq. (15). This is
clearfromthefactthatthisconstantforcingyieldsanonvanishingaveragestationaryvalueforthe
thermoclinedepthanomaly: (cid:104)h(cid:105) ∼ −βA /ω. Tocurethisflawweshouldreplacehwithh−βA /ω
s 0 0
inEq.(15). However,becauseweshallfocusourattentiononT,wewillleavetheROMunchanged.
Concerningtheequationsofmotionofthesecondmoments(Eq.(20b)),weseethattheyweakly
dependontheBdiffusioncoefficient. Moreover,usingtherelationshipsofEqs.(B9),(B13)and(B14),
wehavethattheymainlydependon βand A ,thus,thedynamicsofthesecondmomentsshould
0
not depend too much on the time scale separation between the average ROM dynamics and that
ofthefastforcingξ. Estimatesofthetimescale1/λareintherangeof6-12months. Assumingan
exponentialdecayoftheautocorrelationfunctionofξ,i.e. ϕ(t) =exp(−t/τ),wecanverifythatfor
τupto∼ 3monththetransportcoefficient Ahasthesamestructureithasinthelimitofverylarge
timescaleseparation(τ → 0),namely A(T) ∼ A (1+βT)2 (seeFig.2)and,asitisshowninFig.3,
0
thedynamicsofthesecondmomentsisalmostindependentofτ. Theresultsofthissectionallowus
togettwomainimportantconclusions:
1. analyzing only the first and second moments/cumulants/cross-correlation functions of the
observationsdatawecannotidentify/detectthe(possible)nonlinearityduetotheinteraction
withtheatmosphere,intheENSOsystem;
2. comparingthefirstandsecondmoments/cumulants/cross-correlationfunctionsweobtainfrom
theFPEofEq.(18)withobservations,itwouldbereallyhardtodeterminehowsmallisthetime
scaleτofdecayingofthecorrelationfunctionoftheeffectivenoiseperturbingtheROM.Onthe
otherhand,asithasbeenshownin[8],alsoforτ →0theFPEofEq.(18)wellaccountsforthe
nonGaussianityoftheENSOstatistics. Thus,forthesakeofsimplicity,fromnowon,weshall
usetheFPEofEq.(17)thatisthewhitenoiselimitofEq.(18),evenifweareawareofthefact
thatthetimescaleseparationbetweenthedynamicsoftheaveragedROMandthatofthefast
atmosphereisnotso“extreme”.
Inthenextsub-sectionweshallanalyzethepoint1indetail.
5.2. ThecovariancematrixoftheROMandthecomparisonwiththeENSOdata
Given the results of the previous sub-section, we assume that to describe the statistics of the
ENSOwecanusethesimplifiedFPEofEq.(17),insteadofthemoregeneralEq.(18). ThisFPEmustbe
validatedwithdatafromobservations.