Energy and potential enstrophy flux constraints in quasi-geostrophic models

0.23 MB·English

by  Eleftherios Gkioulekas

Energy and potential enstrophy flux constraints in quasi-geostrophic models
EleftheriosGkioulekasa
aUniversityofTexas-PanAmerican,DepartmentofMathematics,1201WestUniversityDrive,Edinburg,TX78539-2999
Abstract
We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-
3
layerquasi-geostrophicmodels. Itsphysicalsignificanceisthatitcandiagnosewhetheranygivenmulti-layermodel
1
thatallowsco-existingdownscalecascadesofenergyandpotentialenstrophycanallowthedownscaleenergyfluxto
0
2 becomelargeenoughtoyieldamixedenergyspectrumwherethedominantk−3scalingisovertakenbyasubdominant
k−5/3contributionbeyondatransitionwavenumberk situatedintheinertialrange. Thevalidityofthefluxinequality
n t
implies that this scaling transition cannotoccurwithin the inertial range, whereas a violation of the flux inequality
a
J beyondsomewavenumberk impliestheexistenceofascalingtransitionnearthatwavenumber. Thisfluxinequality
t

1 holdsunconditionallyintwo-dimensionalNavier-Stokesturbulence,however,itisfarfromobviousthatitcontinuesto
2 holdinmulti-layerquasi-geostrophicmodels,becausethedissipationratespectraforenergyandpotentialenstrophy

nolongerrelateinatrivialway,asintwo-dimensionalNavier-Stokes. Wederivethegeneralformoftheenergyand
]
D potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this
result,weprovethatinasymmetricallycoupledmulti-layerquasi-geostrophicmodel,wherethedissipationtermsfor
C
eachlayerconsistofthesameFourier-diagonallinearoperatorappliedonthestreamfunctionfieldofonlythesame
.
n layer,thefluxinequalitycontinuestohold.Itfollowsthatanecessaryconditiontoviolatethefluxinequalityistheuse
li ofasymmetricdissipationwheredifferentoperatorsareusedondifferentlayers. We exploredissipationasymmetry
n
furtherin the context of a two-layer quasi-geostrophicmodel and derive upper bounds on the asymmetry that will
[
allowthefluxinequalitytocontinuetohold.AsymmetryisintroducedbothviaanextrapolatedEkmanterm,basedon

1 a1980modelbySalmon,andviadifferentialsmall-scaledissipation. Theresultsgivenaremathematicallyrigorous
v and require no phenomenologicalassumptions about the inertial range. Sufficient conditionsfor violating the flux
1
inequality,ontheotherhand,requirephenomenologicalhypotheses,andwillbeexploredinfuturework.
3
7
Keywords: two-dimensionalturbulence,quasi-geostrophicturbulence,two-layerquasi-geostrophicmodel,flux
4
inequality
.
1
0
3
1
: 1. Introduction
v
i
X Itisnowwell-knownthatintwo-dimensionalNavier-Stokesturbulence,mostoftheenergytendsto gotowards
r largesscalesandmostoftheenstrophytendstogotowardssmallscales,sometimesforminganupscaleinverseenergy
a cascade with energy spectrum scaling as k−5/3 and a downscale enstrophycascade with k−3 scaling [1–3], where k
isthe wavenumber. Kraichnan[1] argued,differentlyfrom Fjørtøft [4], thatthe directionof the two cascadescan
bejustifiedviaathermodynamicargumentinwhichweintroduce,withoutproof,theassumptionthattheenergyand
enstrophyfluxesshouldtendtoreverttheenergyspectrumfromacascadeconfigurationtotheabsoluteequilibrium
configuration. The existence of forcing and dissipation arrests this tendency, thus keeping the system locked in a
steady-stateforced-dissipativeconfigurationawayfromabsoluteequilibrium.
Less well-knownis the fact that there is a serious error with the original Fjørtøft argument: Fjørtøft claimed
that the twin detailed conservationlaws of energy and enstrophy alone imply that in every triad interaction group,
more energyis transferredupscale than downscale. However, a more rigorousanalysis shows that there exist triad
Emailaddress:[email protected](EleftheriosGkioulekas)
PreprintsubmittedtoPhysicaD January22,2013
interaction groups in which more energy is sent downscale than upscale, and it is not obvious, without additional
considerations,whichgroupisdominant[5,6]. Asidefromthismatter,thefundamentalproblemthatunderliesevery
other proof that utilizes only the twin conservation laws of enstrophyand energy, is that an additional assumption
needstobeintroducedto overcomethesymmetryoftheEulerequationsundertimereversal. Typicalassumptions,
suchasthetendencyofthe energyspectrumtoreverttoabsoluteequilibrium,orthetendencyofan energypeakto
spread,typifyadhocconstraintsimposedimplicitlyontheinitialconditionsthatareneededtobreakthetimereversal
symmetry[7]. InRef.[7]wecounterproposedaverysimpleandmathematicallyrigorousproofthatavoidstheneed
foranyadhocassumptionsbyconsideringthecombinedeffectoftheNavier-Stokesnonlinearityandthedissipation
terms. Theonlyassumptionusedbythisproofisthattheforcingspectrumisrestrictedtoafiniteinterval[k ,k ]of
1 2
wavenumbers,howevereventhatassumptioncanberelaxedtosomeextent,althoughnotentirelyeliminated[8,9].
The essence of the argument in Ref. [7] is to show that for every wavenumber k not in the forcing range, the
energyfluxΠ (k)andtheenstrophyfluxΠ (k)satisfytheinequalityk2Π (k)−Π (k)≤0.Here,Π (k)representsthe
E G E G E
amountofenergyperunitvolumetransferredfromthewavenumbersinthe(0,k)intervaltothewavenumbersinthe
(k,+∞)interval,andΠ (k)isdefinedsimilarlyfortheenstrophy. Fromthisinequalitywethenderivethefollowing
G
integralconstraintsforΠ (k)andΠ (k):
E G
k
qΠ (q)dq≤0, ∀k∈(k ,+∞), (1)
E 2
Z
0
+∞
q−3Π (q)dq≥0, ∀k∈(0,k ). (2)
G 1
Z
k
These constraints imply a predominantlyupscale transferof energyand a predominantlydownscale transferof en-
strophy. Theoriginalfluxinequalityk2Π (k)−Π (k) < 0itselfcanalsobedirectlyinterpretedasatightconstraint
E G
onthedownscaleenergyflux.
Thefluxinequalityisdirectlyrelevanttothecascadesuperpositionhypothesisthatwasinitiallyproposedinthe
contextoftwo-dimensionalNavier-Stokesturbulence[10, 11],accordingtowhich,forthecaseoffinitesmall-scale
dissipation viscosity, the downscale enstrophy cascade is accompanied with a hidden downscale energy cascade,
associatedwithanaccompanyingsmalldownscaleenergyflux. Westressthattheexistenceofthissmalldownscale
energyfluxisnotindoubt.Thehypothesisliesinthenotionthatitispartofasubdominantdownscaleenergycascade
withbothcascadescontributingak−3 andak−5/3 termtotheenergyspectrum E(k),thatarecombinedlinearly,with
similar linear combinationsof terms to the generalized structure functionsfor all orders. This linear superposition
principlecan be provedforthird-ordervelocity structure functions[12]. A generalconsequenceof this hypothesis
isthat, if the downscaleenergyfluxassociated withthe k−5/3 termisstrongenough,thena scalingtransitionin the
energyspectrumfromk−3 tok−5/3 shouldoccurnearatransitionwavenumberk ≈ η /ε ,withη thedownscale
t uv uv uv
enstrophyflux and εuv the downscale energyflux. The validity of the flux inequalpity for all wavenumbers k in the
downscaleinertialrangeoftwo-dimensionalNavier-Stokesturbulenceimpliesthatthedownscaleenergyflux ε is
uv
too weak to cause an observable scaling transition anywhere within the inertial range. On the other hand, it is far
fromobviousthatthefluxinequalitywillremainunconditionallyvalidin quasi-geostrophicmodels. A violationof
thefluxinequalitybeyondsomewavenumberk inquasi-geostrophicmodelswouldimplytheoccurrenceofascaling
t
transitionnearthatwavenumber.Arecentnumericalsimulationofatwo-layerquasi-geostrophicmodelhasindicated
thatascalingtransitionisindeedpossible[13].Coexistingdownscalecascadesofenergyandpotentialenstrophyhave
also beenobservedin morerealistic modelsof atmosphericturbulence,the mostrecentbeinga primitiveequations
numericalsimulation[14].
The goalof the presentpaperis to extendthe flux inequalityto quasi-geostrophicmodels. We will specifically
focusonverticaldiscretizationsofthequasi-geostrophicmodel,namelythen-layermodel,andthespecialcaseofthe
two-layermodel,withalllayershavingthesamethickness, intermsofpressurecoordinates,onbothmodels. From
aphysicalstandpoint,bothmodelssacrificethesurfacequasi-geostrophicdynamicsatthebottomboundary,butthey
are otherwise good models of atmospheric turbulence for scales down to an estimated length scale of 100km [15].
I should like to emphasize from the beginning that in spite of any mathematical or phenomenologicalsimilarities,
extending the flux inequality to quasi-geostrophic models is neither obvious nor straightforward. An overlooked
fundamentaldifferencebetweentwo-dimensionalNavier-Stokesturbulenceandquasi-geostrophicturbulenceisthat
thereare manymorepossibleconfigurationsforthe dissipationtermsinquasi-geostrophicmodelsthanthere arein
2
two-dimensionalNavier-Stokes. Dissipationtermsareusuallyignoredbecausephysicalintuitionalonemaysuggest
thattheyshouldnothavean effectonthe nonlineardynamicsininertialranges. Thisline ofreasoningignoresthat
theactualconfigurationofthedissipationtermscanstillhaveunexpectedeffectsonthemagnitudeoftheenergyand
potentialenstrophyfluxesintheinertialrange. Thesefluxeffectsaretheunderlyingmatterofinterestmotivatingthe
investigationinitiatedbythepresentpaper.
Theoriginalmotivationunderlyingtheaforementionednumericalinvestigation[13]ofthetwo-layerquasi-geostrophic
modelwastoshowthatitcanreproducetheNastrom-Gageenergyspectrumoftheatmosphere[16–19].However,the
Nastrom-Gagecontroversy,reviewedtosomeextentinpreviouspapers[6,20],isnotthemainconcernormotivation
of this paper. Our main interest in this problem stems from the following considerations: first, quasi-geostrophic
modelsaresimpleenoughthattheycouldbeaccessibletoinvestigationviatheoreticaltechniquesdevelopedfortwo-
dimensional turbulence [21–26]. Furthermore, the possibility of being able to study a downscale energy cascade
arising in the context of a two-dimensionalmodel is particularly exciting from the point of view of the turbulence
theorist,becauseittiesintotheopenquestionofwhythedownscaleenergycascadeofthree-dimensionalturbulence
has intermittencycorrectionsbut the inverse energycascade of two-dimensionalturbulence doesnot [27, 28]. Is it
antheeffectofdimensionnumberorcascadedirection? Inlightofsuchquestions,anobservabledownscaleenergy
cascadeinatwo-dimensionalsystemisinterestinginandofitself.
Mathematicalresults concerningthe flux inequalityin quasi-geostrophicmodelscan be organizedinto two cat-
egories: (a) sufficient conditions for the satisfaction of the flux inequality within the entire inertial range; and (b)
sufficientconditionsforviolatingthefluxinequalitybeyondsometransitionwavenumberk withintheinertialrange.
t
Resultsofthefirsttypecanbeprovedrigorouslywithoutadhocphenomenologicalassumptionsonthebehaviorof
theenergyandpotentialenstrophyspectra. Resultsofthesecondtyperequiretheintroductionofphenomenological
assumptionsaboutthedistributionofenergyandpotentialenstrophybetweenlayers. Consequently,thescopeofthis
paperhasbeenlimitedtowhatwecanproverigorously. Morepowerfulresultsthatcanbeobtainedbyintroducing
phenomenologicalhypotheseswillbeexploredinfuturepublications. Becausethedetailsofourargumentarevery
technical,wewillnowsummarizethemainargumentofthepaperasfollows.
Forthe generalizedcase ofan n-layermodel, we considerthe generalcase of a streamfunctiondissipationcon-
figuration,whereforeachlayerthedissipationtermsaregivenbyalineardifferentialoperatorappliedonthestream-
functionofthesamelayer,withoutentanglinganystreamfunctionsofanyotherlayers. Thedissipationratespectra
forbothenergyandpotentialenstrophyarederivedunderthisgeneralconfiguration. Then,wespecializetothecase
ofsymmetricstreamfunctiondissipation,whereweassumethatthecorrespondingdissipationoperatorsareidentical
layer-by-layer. We willshow thatundersymmetricstreamfunctiondissipation the fluxinequalityis satisfied forall
wavenumbersin the inertialanddissipation range. We notethatthis resultis non-trivialsince, beyondestablishing
cascadedirections, it also impliesboundson the subdominantdownscaleenergyflux, thataretightenoughto keep
theunderlyingdownscaleenergycascadehidden. Forthecaseofthetwo-layerquasi-geostrophicmodelweconsider
anasymmetricconfigurationofdissipationtermsandestablishresultsoftheformthatiftheasymmetryissufficiently
small, the fluxinequalitywillremainvalid. As waspreviouslyexplained,we limitourselvestoresultsof thisform
becausethisisasfarasonecangowithrigorousproofsfromfirstprinciples.
From a physical standpoint, asymmetry in the dissipation between the two layers usually originates from the
Ekman term, modeling the effect of friction with the surface boundary layer. However, for reasons that will be
discussedmoreextensivelyattheconclusionofthispaper, wewillintroduceanadditionalsourceofasymmetryvia
thesmall-scaledissipationtermsbyemployinganincreasedviscosityorhyperviscositycoefficientatthebottomlayer
relative to the coefficient at the top layer. We believe that this asymmetric small-scale dissipation can facilitate a
breakdownofthefluxinequality,therebyallowingthedownscaleenergyfluxratetobesufficientlystrongtoyieldthe
transitiontok−5/3scalingintheinertialrange.Wewillseethatasymmetricsmall-scaledissipationindeedtightensthe
boundsontheparameterspacewhereinthefluxinequalityissatisfied.
Another aspect of the dissipation term configuration, that will be shown to have significant impact on the flux
inequality, concerns the modeling of the Ekman term. In a typical formulation of the two-layer quasi-geostrophic
model,itisusuallyassumedthatEkmandissipationisdependentonlyonthestreamfunctionfieldofthebottompo-
tentialvorticitylayer. However,analternateformulationofthetwo-layerquasi-geostrophicmodelbySalmon[29],
requiresthattheEkmantermatthelowerlayerbedependentonthestreamfunctionfieldsofbothlayers. Toexplain
why,onemustrecallthatthetwo-layermodelisanextremeverticaldiscretizationofthefullquasi-geostrophicmodel,
whichconsists of a relative vorticityequation, a temperatureequation, and additionalconstrainingconditions. In a
3
generalmulti-layermodel,therelativevorticityequationsarediscretizedinhorizontallayersthatareinterlaced with
thediscretizationlayersofthetemperatureequations. Thus,forthecaseofthetwo-layermodelwehavealtogether
5 physically relevant layers: the surface boundary layer corresponding approximately to 1Atm, the lower relative
vorticitylayerat0.75Atm,thetemperaturemidlayerat0.5Atm,theupperrelativevorticitylayerat0.25Atm,andthe
top boundary layer at 0Atm. The potential vorticity equations are derived from the relative vorticity equations by
eliminatingthe temperaturefield fromthe system of equations, thereby placing the potentialvorticity field and the
correspondingstreamfunctionfield atthe 0.25Atmand0.75Atmlayers. AsnotedbyRef. [29], the Ekmandissipa-
tion term is dependenton the streamfunction field at the surface boundarylayer near 1Atm, which can be linearly
extrapolated from the streamfunction field at the lower and upper layer (0.75Atm and 0.25Atm correspondingly).
Consequently,eventhoughtheEkmantermisstillplacedonthelower-layer,owingtothelinearextrapolationofthe
surfacestreamfunctionfield,itisdependentonthestreamfunctionfieldofboththelowerandupperlayers.
It should be noted that for physical reasons, the potential vorticity layers need to remain fixed at 0.25Atm and
0.75Atm respectively. This correspondsto the physical assumption that the two fluid layers have equal thickness,
whichisanecessaryassumptionforatmosphericmodeling[30]. ThesurfacelayerdrivingEkmandissipation,onthe
otherhand, can be placed anywherebetween the surface layer at 1Atm and the lower streamfunctionfield layer at
0.75Atm. Whenthesurfacelayerandthelowerstreamfunctionlayercoincide,thiscorrespondstotheusualstandard
Ekmanterm. When the two layersdo not coincide, itcorrespondsto the moregeneralcase of extrapolatedEkman
dissipation.Forthepresentpaper,weretaingeneralitybyparameterizingtheplacementofthesurfaceboundarylayer
viaanadjustableparameterµ,andshowthatourmainpropositionsarevalidfortheentire rangeoftheparameterµ.
Wewillseethatanincreasingseparationbetweenthesurfacelayerandthebottompotentialvorticitylayertightensthe
boundsontheparameterspacewhereinthefluxinequalityissatisfied. Foroceanographicmodeling,aswellasforthe
purposeofsatisfyingbasicscientificcuriosity,itwouldbeinterestingtoconsidertwo-layerquasi-geostrophicmodels
withlayershavingunequalthickness. Duetomathematicalcomplications,wewillnotpursuethisgeneralizationin
the present paper. Nevertheless, the importance of symmetric vs. asymmetric Ekman dissipation in the context of
oceanographicmodellingisarelevantproblemthathasbeeninvestigatedbyapreviousstudy[31].
Admittedly, both Salmon’s idea of extrapolated Ekman dissipation and my idea of differential small-scale dis-
sipation can be considered controversial. On the other hand, in the context of investigating the flux inequality, it
is importantto be thoroughaboutconsideringeveryinterestingconfigurationof the dissipation terms, to determine
how much impact various choices of dissipation term configurations have on the robustness of the flux inequality.
Furthermore,aswillbecomeapparentfromtheresultsofthispaper,thedissipationconfigurationsexploredhereare
goodcandidatesforadissipationfilterthatcouldviolatethefluxinequalityandensureacontrolleddownscaleenergy
dissipationrateinnumericalsimulationsthatexceedstherestrictionsthataretypicalintwo-dimensionalturbulence.
The paperis organizedasfollows. Insection 2 we givethegoverningequationsforthe generalizedmulti-layer
model and discuss its conservation laws, the definition of the energy spectrum E(k), potential enstrophy spectrum
G(k), and theirrelationshipvia the streamfunctionspectrum C (k). In section 3, aftera brief recapitulationof the
αβ
flux inequality for the simple case of two-dimensional Navier-Stokes turbulence, we establish the flux inequality
forageneralizedmulti-layerquasi-geostrophicmodelundersymmetricstreamfunctiondissipation. Insection4,we
considerasymmetricdissipationconfigurationsforthespecialcaseofatwo-layerquasi-geostrophicmodel,wherewe
derivevarioussufficientconditionsforsatisfyingthefluxinequality. Conclusionsandabriefdiscussionaregivenin
section5.
2. Thegeneralizedmultilayermodelandconservationlaws
Following my previous paper [20], we write the governing equations for the generalized multi-layer model in
matrixform:
∂q
α +J(ψ ,q )=d + f , (3)
α α α α
∂t
d = D ψ . (4)
α αβ β
Xβ
Hereψ representsthestreamfunctionattheα-layer,q representsthepotentialvorticityattheα-layer,D isalinear
α α αβ
operatorencapsulatingthedissipationterms,and f istheforcingtermactingontheα-layer. Theindexαtakesthe
α
4
valuesα=1,2,...,nrepresentingthelayernumber,foramodelinvolvingnlayers. Sumsoverindices,suchasinthe
sumovertheindexβinthedissipationtermsabove,areassumedtorunoveralllayers1,2,...,n,unlessweindicate
otherwise.Itisalsoassumedthatthestreamfunctionψ andthepotentialvorticityq arerelatedviaalinearoperator
α α
L accordingto:
αβ
q (x,t)= L ψ (x,t). (5)
α αβ β
Xβ
The above equations encompass both the two-layer quasi-geostrophic model and the multilayer quasi-geostrophic
model,ontheassumptionthatweneglecttheβ-effect,arisingfromthelatitudinaldependenceoftheCoriolispseud-
oforce. This is a reasonableassumptionforEarth, especially if we restrictourinterestto a thin strip of the Earth’s
surface,orientedparalleltotheequator. Baroclinicityinstabilityisaccountedforbytheforcingterm f ,andimplicit
α
intheentireargumentistheassumptionthatitforcesthesystematlargescalesonly.Thisassumption,originallypro-
posedbySalmon[29,32],istheonlyphysicalassumptionimplicitinthetheoreticalframeworkofthefluxinequality,
andithasbeencorroboratednumerically[13,33].
For the sake of simplifying our analysis, we assume that all fields are defined in an infinite two-dimensional
domain.ThenwecanwritetheFourierexpansionsforthestreamfunctionψandthepotentialvorticityqasfollows:
ψ (x,t)= ψˆ (k,t)exp(ik·x)dk, (6)
α α
ZR2
q (x,t)= qˆ (k,t)exp(ik·x)dk. (7)
α α
ZR2
WeassumethattheoperatorL isdiagonalinFourierspace.Thismeansthattherelationbetweenthestreamfunction
αβ
andthepotentialvorticity,inFourierspace,reads:
qˆ (k,t)= L (kkk)ψˆ (k,t). (8)
α αβ α
Xβ
Here kkk represents the 2-norm of the vector k. We also assume that L is symmetric with L = L . This
αβ αβ βα
impliesthatL (k) = L (k)forallwavenumbersk. Forquasi-geostrophicmodels,thematrix L (k)isnon-singular
αβ βα αβ
for all wavenumbers k > 0, due to being diagonally dominant, and we assume that to be the case in our abstract
formulationgivenabove.Consequently,thereisaninversematrixL−1(k)whichdefinestheinverseoperatorL−1. To
αβ αβ
accommodateapossiblesingularityatk=0weassumethatatwavenumberk =0,inFourierspace,thecorresponding
field component is 0 for all fields. This is equivalent to subtracting the mean field and considering only the field
fluctuationaroundthemean.
2.1. Conservationlaws
We will now show that the generalizedlayermodel, in the absenceof dissipation, conservesthe total energy E
andthetotalpotentialenstrophyG underverygeneralconditionsontheoperator L , Foranyarbitraryscalarfield
αβ
f(x,y)wewritethecorrespondingvolumeintegralusingthefollowingnotation:
hhfii= f(x,y)dxdy. (9)
ZZR2
We definethetotalenergy E overalllayers, andthelayer-by-layertotalpotentialenstrophyG forlayer α, as E =
α
− hhψ q ii andG = hhq2ii. The purposeof the minussign in ourdefinitionof the total energy E is to maintain
α α α α α
coPnsistencywiththenotationandsignconventionsusedbymypreviouspaper[20]. Specifically,wewillshowthat
thepotentialenstrophyisconservedonalayer-by-layerbasisunconditionallyregardlessofthedetailsoftheoperator
L .ConservationofthetotalenergyE,overalllayers,ontheotherhand,requiresthattheoperatorL besymmetric
αβ αβ
andself-adjoint. Bysymmetricwemeanthattheoperatorsatisfies L = L . To definethe self-adjointproperty,
αβ βα
consider two arbitrary two-dimensional scalar fields f(x,y) and g(x,y). We require that every component of the
operatorL mustsatisfyhhf(L g)ii = hh(L f)giiforanytwofields f(x,y)andg(x,y). Thisself-adjointproperty,
αβ αβ αβ
5
so defined, follows as an immediate consequence of our previous assumption that the operator L is diagonal in
αβ
Fourierspace.Intheproofgivenbelow,however,thereisnoneedtousethestrongerassumptionofdiagonality.
The proof is based on the following properties of the nonlinear Jacobian term. If a(x,y) and b(x,y) are two-
dimensionalscalarfieldsthatsatisfyahomogeneous(DirichletorNeumann)boundarycondition,thenwecanshow
thathhJ(a,b)ii=0,usingintegrationbyparts. Then,wenotethat,asanimmediateconsequenceoftheproductruleof
differentiation,giventhreetwo-dimensionalscalarfieldsa(x,y),b(x,y),andc(x,y)wehave
hhJ(ab,c)ii=hhaJ(b,c)ii+hhbJ(a,c)ii=0, (10)
fromwhichweobtaintheidentity
hhaJ(b,c)ii=hhbJ(c,a)ii=hhcJ(a,b)ii. (11)
Now,letusgoaheadanddropthedissipationandforcingtermsandwritethetime-derivativeofthepotentialvorticity
q asq˙ =−J(ψ ,q ). Then,thetimederivativeofthestreamfunctionψ reads:
α α α α α
ψ˙ = L−1q˙ =− L−1J(ψ ,q ). (12)
α αβ β αβ β β
Xβ Xβ
Differentiating the total potential enstrophy G for the α layer with respect to time and employing the identity
α
givenbyEq.(11)immediatelygives:
G˙ =2hhq q˙ ii=−2hhq J(ψ ,q )ii=−2hhψ J(q ,q )ii=0. (13)
α α α α α α α α α
Here,wenotethatfromthedefinitionoftheJacobian J(q ,q ) = 0. Thisestablishesthelayer-by-layerconservation
α α
law of potential enstrophy, unconditionally, as claimed. To show the energy conservationlaw, we differentiate the
totalenergyE withrespecttotimeandobtain:
E˙ =−(d/dt) hhψ q ii=− hhψ˙ q ii− hhψ q˙ ii (14)
α α α α α α
Xα Xα Xα
= hhq L−1J(ψ ,q )ii+ hhψ J(ψ ,q )ii (15)
α αβ β β α α α
Xαβ Xα
= hhJ(ψ ,q )L−1q ii+ hhq J(ψ ,ψ )ii (16)
β β αβ α α α α
Xαβ Xα
= hhJ(ψ ,q )L−1q ii= hhJ(ψ ,q )ψ ii (17)
β β βα α β β β
Xαβ Xβ
= hhJ(ψ ,ψ )q ii=0. (18)
β β β
Xβ
Note that the self-adjoint property is applied at Eq. (16), and the symmetric property is applied at Eq. (17). This
concludestheproof.
2.2. Definitionofspectra
FollowingFrisch[34],wedefinespectrafortheenergyandpotentialenstrophyusingthebracketnotationintro-
duced in my previous paper [20]. Consider, in general, two arbitrary two-dimensional scalar fields a(x) and b(x).
Leta<k(x)andb<k(x)bethefieldsobtainedfroma(x)andb(x)bysettingtozero,inFourierspace, thecomponents
correspondingtowavenumberswhosenormisgreaterthan k. Formally,a<k(x)isdefinedas
H(k−kk k)
a<k(x)= dx dk 0 exp(ik ·(x−x ))a(x ), (19)
ZR2 0ZR2 0 4π2 0 0 0
withH(x)theHeavisidefunction,definedastheintegralofadeltafunction:
1, ifif x∈(0,+∞)
x
H(x)= δ(τ)dτ= 1/2, ifif x=0 . (20)
Z0  0, ifif x∈(−∞,0)
 6
Obviously,b<k(x)isdefinedsimilarly.Wenowusethetwofilteredfieldsa<k(x)andb<k(x)todefinethebracketha,bi
k
as:
d
ha,bi = dx a<k(x)b<k(x) (21)
k dk ZR2 D E
1
= dΩ(A) [aˆ∗(kAe)bˆ(kAe)+aˆ(kAe)bˆ∗(kAe)] . (22)
2ZA∈SO(2) D E
Here,aˆ(k)andbˆ(k)aretheFouriertransformsofa(x)andb(x),SO(2)isthesetofallnon-reflectingrotationmatrices
intwodimensions,dΩ(A)isthemeasureofasphericalintegral,eisatwo-dimensionalunitvector,andh·irepresents
takinganensembleaverage. Thestarsuperscriptrepresentstakingthecomplexconjugate. NotethatEq.(21)isthe
definitionof the bracket, and Eq. (22) followsfrom Eq. (21) as a consequence. The bracketsatisfies the following
properties:
ha,bi =hb,ai , (23)
k k
ha,b+ci =ha,bi +ha,ci , (24)
k k k
ha+b,ci =ha,ci +hb,ci . (25)
k k k
Moreover,every(αβ)-componentoftheoperatorL isself-adjointwithrespecttothebracket,whichgives
αβ
L a,b = a,L b = L (k)ha,bi , (26)
αβ αβ αβ k
k k
D E D E
andthesamepropertyisalsosatisfiedbyeverycomponentoftheinverseoperatorL−1:
αβ
L−1a,b = a,L−1b = L−1(k)ha,bi . (27)
αβ k αβ k αβ k
D E D E
Using the bracket, we define the energyspectrum E(k) = − hψ ,q i , and we also define the layer-by-layer
α α α k
potentialenstrophyspectrumGα(k) = hqα,qαik andthetotalpotePntialenstrophyspectrumG(k) = αGα(k). Unlike
the case of two-dimensional Navier-Stokes, where the enstrophy and energy spectra G(k) and E(Pk) are related via
a simple equation,G(k) = k2E(k), in the generalizedlayermodel, the potentialenstrophyspectrumand the energy
spectrumarerelatedindirectly,asshownbelow:
DefinethestreamfunctionspectrumC (k)= ψ ,ψ . Then,viathepropertiesofthebracketabove,theenergy
αβ α β
k
spectrumE(k)reads D E
E(k)=− hψ ,q i =− ψ , L ψ =− L (k) ψ ,ψ (28)
α α k * α αβ β+ αβ α β k
Xα Xα Xβ k Xαβ D E
=− L (k)C (k), (29)
αβ αβ
Xαβ
andthepotentialenstrophyspectrumG (k)reads
α
G(k)= hq ,q i = L ψ , L ψ (30)
α α k * αβ β αγ γ+
Xα Xα Xβ Xγ k
= L (k) ψ , L ψ = L (k)L (k) ψ ,ψ (31)
αβ β αγ γ αβ αγ β γ
* + k
Xαβ Xγ k Xαβγ D E
= L (k)L (k)C (k). (32)
αβ αγ βγ
Xαβγ
Thus,theyarerelatedonlyindirectlyviathestreamfunctionspectrumC (k).
αβ
We note that for α , β, C (k) may take positive or negative values. For the case α = β we define U (k) =
αβ α
hψ ,ψ i ,whichisalwayspositive(i.e.,U (k) ≥ 0). Thenwenotethat2|C (k)| ≤ U (k)+U (k). We canusethis
α α k α αβ α β
inequalitytoshowthatifthematrixL (k)satisfiesthediagonaldominancecondition
αβ
L (k)≥0, forα,β, (33)
αβ
7
L (k)≤0, (34)
αβ
Xβ
then the energy spectrum E(k) is always positive. We give the proof in Appendix A. Both the two-layer quasi-
geostrophic model and the multi-layer quasi-geostrophicmodel satisfy this diagonal dominance condition. As for
thelayer-by-layerpotentialenstrophyspectra G (k), it isimmediatelyobviousthattheyare unconditionallyalways
α
positive,regardlessoftheformofthematrix L (k),sincebydefinitionG (k)=hq ,q i .
αβ α α α k
3. Fluxinequalityforthen-layermodel
We now turn to the main issue of identifying sufficient conditions for satisfying the flux inequality k2Π (k)−
E
Π (k) ≤ 0forquasi-geostrophicmodels. LetusrecallthattheenergyfluxspectrumΠ (k)isdefinedastheamount
G E
ofenergytransferredfromthe(0,k)intervaltothe(k,+∞)intervalperunittimeandperunitvolume. Likewise,the
potentialenstrophyfluxspectrumΠ (k)istheamountofpotentialenstrophytransferredfromthe(0,k)intervaltothe
G
(k,+∞)interval,againperunittimeandvolume.Assumingaforced-dissipativeconfigurationatsteadystateandthat
thewavenumberkisnotintheforcingrange,theenergyandpotentialenstrophytransferredintothe(k,+∞)interval
eventuallyaredissipatedsomewhereinthatinterval.ItfollowsthatwemaywritethefluxspectraΠ (k)andΠ (k)as
E G
integralsoftheenergyandpotentialenstrophydissipationratespectraD (k)andD (k):
E G
+∞
Π (k)= D (q)dq, (35)
E E
Z
k
+∞
Π (k)= D (q)dq, (36)
G G
Z
k
whichimpliesthat
+∞ +∞
k2Π (k)−Π (k)= [k2D (q)−D (q)]dq= ∆(k,q)dq. (37)
E G E G
Z Z
k k
Weseethatasufficientconditionforestablishingthefluxinequalityistoshowthat∆(k,q) ≤ 0forallwavenumbers
k < q. Itisalsoeasytoseethat∆(k,q)> 0forallwavenumbersk < k < qissufficientforestablishingtheviolation
t
ofthefluxinequalityforallwavenumbersk>k.
t
Forthecaseoftwo-dimensionalNavier-Stokesturbulence,thedissipationratespectraD (k)andD (k)arerelated
E G
viaD (k)= k2D (k). Thisimmediatelygives∆(k,q)= k2D (q)−D (q)= (k2−q2)D (q)≤0forallwavenumbers
G E E G E
k < q(sinceD (k) ≥ 0),whichinturngivesthefluxinequalityk2Π (k)−Π (k) ≤ 0. Thephysicalinterpretationof
E E G
thisinequalityisthatwhenwestretchtheseparationofscalesinthedownscalerange,theenergydissipationrateat
small-scalesvanishesrapidly. Asaresult,mostoftheinjectedenergycannotcascadedownscalealthough,asnoted
previously[10,11],asmallamountofenergyisabletodoso. Aswehaveseenintheprevioussection,forthecase
ofquasi-geostrophicmodels,theenergyandpotentialenstrophydissipationratespectranolongerhaveadirectand
simplerelationwitheachother,sothevalidityofthefluxinequalityneedstobecarefullyre-examined.
Forthegeneralmulti-layerquasi-geostrophicmodel,therelationshipbetweenthepotentialvorticitiesq andthe
α
streamfunctionsψ isgivenby
α
q =∇2ψ +µ k2(ψ −ψ ),
1 1 1 R 2 1
qα =∇2ψα−λαkR2(ψα−ψα−1)+µαkR2(ψα+1−ψα), for1<α<n,
q =∇2ψ −λ k2(ψ −ψ ).
n n n R n n−1
Here,k istheRossbywavenumberandλ andµ arethenon-dimensionalFroudenumbers,givenby
R α α
1h ρ −ρ
λ = 1 2 1 , for1<α≤n,
α 2h ρ −ρ
α α α−1
1h ρ −ρ
µ = 1 2 1 , for1≤α<n,
α 2hαρα+1−ρα
8
withρ theaveragedensityoflayerα,andh theaverageheightoflayerα(inpressurecoordinates).Thedefinitionof
α α
thenon-dimensionalFroudenumberswasadjustedwitha1/2numericalfactor,fromtheonegivenbyEvensen[35],
toensureagreementwiththeformulationofthetwo-layerquasi-geostrophicmodelgivenbySalmon[29]forthecase
n=2. ThecomponentsofthecorrespondingmatrixL (k)aregivenby
αβ
−k2−µ k2, ifα=1
1 R
Lαα(k)= −k2−(λα+µα)kR2, if1<α<n
 −k2−λnkR2, ifα=n,
Lα,α+1(k)=µαkR2, for1≤α<n,
L (k)=λ k2, for1<α≤n.
α,α−1 α R
Inthepresentpaperwelimitourselvestothespecialcaseof asymmetricallycoupledmulti-layerquasi-geostrophic
model,wherewe assume thatthe layerthickness h isthe same foralllayers, therebyyieldinga symmetricmatrix
α
Lαβ(k)suchthatLα,α+1(k)= Lα+1,α(k)forall1≤α<n.
Toconsiderthefluxinequalityforthisgeneral n-layermodel,webeginwith writingthedissipationrates D (k)
E
andD (k)fortheenergyandpotentialenstrophyintermsofthestreamfunctionspectrumC (k). Weassumethatthe
G αβ
dissipationoperationD isdiagonalinFourierspaceandthatthe Fourierexpansionofthedissipationterm D ψ
αβ αβ β
reads:
(D ψ )(x,t)= D (kkk)ψˆ (k,t)exp(ik·x)dk. (38)
αβ β αβ β
ZR2
Then, in Appendix B we show thatthe energydissipationrate spectrum D (k) and the layer-by-layerpotentialen-
E
strophydissipationratespectraD (k)aregivenby
Gα
D (k)=2 D (k)C (k), (39)
E αβ αβ
Xαβ
D (k)=−2 L (k)D (k)C (k). (40)
Gα αβ αγ βγ
Xβγ
Notethatinorderforthedissipationtermstobetrulydissipative,thedissipationspectraD (k)andD (k)needtobe
E G
bothalwayspositiveforallwavenumbersk. Fromthegeneralformoftheaboveequationsthisisnotreadilyobvious.
However,forsimplerconfigurationsofthedissipationoperators,theaboveexpressionsforD (k)andD (k)simplify
E G
considerably, thereby making it possible to establish that they are both always positive. These expressions also
underscorethemaindifferencebetweentwo-dimensionalNavier-Stokesturbulenceandquasi-geostrophicturbulence
and the reason why the flux inequality becomes a non-trivial problem in the later case. Unlike two-dimensional
turbulence,andinspite ofthetwin conservationlawsofenergyandpotentialenstrophy,thedissipationrates D (k)
E
andD (k)arenolongerrelatedbyanysimplerelationoftheform D (k)=k2D (k).
G G E
Werestrictourattentiontothecasewherethedissipationoperatorsateverylayerinvolveonlythestreamfunction
ofthe correspondinglayer, with noexplicitinterlayerterms. Thiscan be arrangedin termsofa linearoperator D
α
appliedtothestreamfunctionψ . IfD (k)isthespectrumofthepositive-definiteoperatorD ,thenforthecaseofa
α α α
dissipationtermd =D ψ ,wehaveD (k)=δ D (k),withδ givenby
α α α αβ αβ β αβ
1, ifα=β
δ = . (41)
αβ ( 0, ifα,β
Wedesignatethiscaseasstreamfunction-dissipation.TheD (k)andD (k)simplifyas:
E Gα
D (k)=2 D (k)C (k)=2 δ D (k)C (k)=2 D (k)C (k)=2 D (k)U (k), (42)
E αβ αβ αβ β αβ α αα α α
Xαβ Xαβ Xα Xα
D (k)=−2 L (k)D (k)C (k)=−2 L (k)δ D (k)C (k)=−2 L (k)D (k)C (k). (43)
Gα αβ αγ βγ αβ αγ γ βγ αβ β αβ
Xβγ Xβγ Xβ
9
Note that for D (k) ≥ 0, it follows that D (k) ≥ 0, but it is not obvious that the same result extends to D (k).
α E Gα
However,ifwefurtherassumethatthesameoperatorisusedforalllayers,i.e. D (k)= D(k),thenwehavethemore
α
specialized case of symmetric streamfunction-dissipation, and the dissipation rate spectra D (k) and D (k) can be
E G
simplifiedfurthertogive:
D (k)=2 D (k)U (k)=2D(k) U (k)=2D(k)U(k), (44)
E α α α
Xα Xα
D (k)= D (k)=−2 L (k)D (k)C (k)=2D(k) − L (k)C (k) =2D(k)E(k). (45)
G Gα αβ β αβ  αβ αβ 
Xα Xαβ  Xαβ 
Now,D(k)≥0impliesbothD (k)≥0andD (k)≥0.
E G
Itfollowsthat,undersymmetricstreamfunctiondissipation,∆(k,q)isgivenby
∆(k,q)=k2D (q)−D (q)=k2D(q)U(q)−D(q)E(q)= D(q)[k2U(q)−E(q)], (46)
E G
andsinceD(q)≥ 0,thevalidityofthefluxinequalityisdependentonthesignofthefactork2U(q)−E(q). Thatsign
isinturnintimatelyrelatedwiththeexpressionγ (k,q)definedas:
α
γ (k,q)=k2+ L (q). (47)
α αβ
Xβ
Notethatforthecaseoftwo-dimensionalNavier-Stokes,L(q)becomesa1×1matrixwithL (q)=q2,thusγ (k,q)=
11 α
k2−q2,whichisnegativewhenk<q. Formoregeneralizedn-layerquasi-geostrophicmodels,theexpressionγ (k,q)
α
continuestobegivenbyγ (k,q) = k2−q2 whichremainsnegativewhenk < qforalllayersα. We willnowshow
α
that:
Proposition1. Inageneralizedn-layermodel,undersymmetric streamfunctiondissipationd = +Dψ withspec-
α α
trumD(k)≥0,weassumethatL (q)≥0whenα,β,andL (q)= L (q),andγ (k,q)≤0whenk<qforallα. It
αβ αβ βα α
followsthat:
∆(k,q)≤ D(q) γ (k,q)U (q)≤0.
α α
Xα
Proof. WebeginbyrecallingfromAppendix A,thatE(q)canberewrittenas
1
E(q)=− L (q)U (q)− L (q)[2C (q)−U (q)−U (q)]. (48)
αβ α αβ αβ α β
2
Xαβ Xαβ
α,β
Itfollowsthatk2U(q)−E(q)satisfies:
1
k2U(q)−E(q)=k2 U (q)+ L (q)U (q)+ L (q)[2C (q)−U (q)−U (q)] (49)
α αβ α αβ αβ α β
2
Xα Xαβ Xαβ
α,β
≤k2 U (q)+ L (q)U (q)= k2+ L (q) U (q) (50)
α αβ α αβ α
Xα Xαβ Xα (cid:18) Xβ (cid:19)
= γ (k,q)U (q). (51)
α α
Xα
TheinequalityusestheassumptionL (q)≥0combinedwiththetriangleinequality2C (q)≤U (q)+U (q)ofthe
αβ αβ α β
streamfunctionspectra. Itfollowsthat
∆(k,q)= D(q)[k2U(q)−E(q)]≤ D(q) γ (k,q)U (q)≤0, (52)
α α
Xα
sinceD(q)≥0,U (q)≥0,andγ (k,q)≤0,therebyconcludingtheproof.
α α
10