Surface Ocean Enstrophy, Kinetic Energy Fluxes and Spectra from Satellite Altimetry

7.9 MB·

by  Hemant Khatri

SurfaceOceanEnstrophy,KineticEnergyFluxesandSpectrafromSatelliteAltimetry
Hemant Khatri1, Jai Sukhatme2,3, Abhishek Kumar4 and Mahendra K. Verma4
1. DepartmentofMathematics, ImperialCollegeLondon, LondonSW72AZ,UnitedKingdom.
2. CentreforAtmosphericandOceanicSciences, IndianInstituteofScience, Bangalore, India.
3. Divecha Centre for Climate Change, Indian Institute of Science, Bangalore, India.
4. Department of Physics, Indian Institute of Technology Kanpur, Kanpur, India.
Enstrophy,kineticenergy(KE)fluxesandspectraareestimatedindifferentpartsofthemid-latitudinaloceans
viaaltimetrydata.Tobeginwith,usinggeostrophiccurrentsderivedfromsea-surfaceheightanomalydatapro-
videdbyAVISO,weconfirmthepresenceofastronginversefluxofsurfaceKEatscaleslargerthanapproxi-
mately250km.Wethencomputeenstrophyfluxestohelpdevelopaclearerpictureoftheunderlyingdynamics
atsmallerscales,i.e.,250kmto100km. Here,weobservearobustenstrophycascadingregime,whereinthe
7 enstrophyshowsalargeforwardfluxandtheKEspectrafollowanapproximatek−3.5power-law.Giventhero-
1 tationalcharacteroftheflow,notonlyisthislargescaleinverseKEandsmallerscaleforwardenstrophytransfer
0 scenarioconsistentwithexpectationsfromidealizedstudiesofthree-dimensionalrapidly-rotatingandstrongly-
2 stratifiedturbulence, italsoagreeswithdetailedanalysesofspectraandfluxesintheupperlevelmidlatitude
troposphere.Decomposingthecurrentsintocomponentswithgreaterandlessthan100dayvariability(referred
n
toasseasonalandeddy,respectively),wefindthat,inadditiontotheeddy-eddycontribution,theseasonal-eddy
a
J and seasonal-seasonal fluxes play a significant role in the inverse (forward) flux of KE (enstrophy) at scales
larger(smaller)thanabout250km. Takentogether, wesuspect, itisquitepossiblethat, fromabout250km
7
to100km,thealtimeteriscapturingtherelativelysteepportionofasurfaceoceaniccounterpartoftheupper
2
troposphericNastrom-Gagespectrum.

]
h
p
- I. INTRODUCTION
o
a
. Forthepastdecade,satellitealtimetrydatahasbeenusedtoestimatetheinterscaletransferandspectraldistributionofsurface
s
c kineticenergy,henceforthabbreviatedasKE,intheoceans. Focusingonmesoscalesinmid-latitudinalregionswithhigheddy
i activity(forexample,neartheGulfStream,theKuroshioortheAgulhascurrents),thefluxofKEisseentobescaledependent.
s
y Inparticular,ithasbeennotedthatsurfaceKEtendstobetransferredupscaleforscaleslargerthanthelocaldeformationradius
h and downscale for smaller scales [1–4]. Mesoscale wavenumber spectra, on the other hand, are somewhat more diverse with
p spectral indices ranging from −5 to −3 depending on the region in consideration [5–7]. In fact, recent work using in-situ
[ 3
observationssuggeststhatthescalingchangeswithseason, andismodulatedbythestrengthofeddyactivity[8]. Interpreting

1 theseresultsintermsofthedynamicscapturedbythealtimeterandmorefundamentallythenatureoftheactualdynamicsofthe
v upperoceanhasbeenthesubjectofnumerousrecentinvestigations[seeforexamplethediscussionsin,9,10].
6
Asbaroclinicmodesareintensifiednearthesurface[11,12],ithasbeensuggestedthataltimetrydatamostlyrepresentsthe
6
9 firstbaroclinicmodeintheocean[5]. Indeed, energyisexpectedtoconcentrateinthefirstbaroclinicmodeduetoaninverse
7 transfer among the vertical modes [13]. Given this, at first sight, the observed inverse transfer of surface KE at large scales
0 was surprising, as classical quasigeostrophic (QG) baroclinic turbulence anticipates a forward cascade in the baroclinic mode
1. withenergyflowingtowardsthedeformationscale[14,15]. However,acarefulexaminationoftheenergybudgetinnumerical
0 simulationsrevealsthat,whileKEgoestolargerscales,thetotalenergyinthefirstbaroclinicmodedoesindeedflowdownscale
7 [16]; a feature that is comforting in the context of traditional theory. In fact, along with the two-layer QG study of Scott and
1 Arbic[16],inversetransferofKEhasalsobeendocumentedinmorecomprehensiveoceanmodels[3,4,17,18].
:
v Notingthesignificanceofsurfacebuoyancygradients(afactmissedintheaforementionedfirstbaroclinicmodeframework),
Xi it has been suggested that, surface QG (SQG) dynamics is a more appropriate framework for the oceans’ surface [19], and is
reflectedinthealtimetermeasurements[9]. Eventhoughthevarianceofbuoyancyistransferreddownscale[20,21],surfaceKE
r
a actually flows upscale in SQG dynamics [22, 23], consistent with the flux calculations using altimetry data. Of course, in the
QGlimit,acombinationofsurfaceandinteriormodesisonlynatural. Thereareongoingeffortstorepresentthevariabilityof
thesurfaceoceanandinterpretthealtimetrydataintheseterms[9,24,25].
Thus, muchoftheworkusingaltimeterdatahasfocussedontheinversetransferofKEatrelativelylargescales. Here, we
spend some time on the larger scales, but mainly concentrate on slightly smaller scales that are still properly resolved by the
data. Specifically,inadditiontotheKEflux,wealsocomputethespectralfluxofenstrophy. Infact,thisenstrophyfluxsheds
new light on the range of scales that span approximately 250 km to 100 km. We find that the enstrophy flux is strong and
directed to small scales over this range, and is accompanied by a KE spectrum that follows an approximate k−3.5 power-law.
This suggests that the rotational currents as derived from the altimeter are in an enstrophy cascading regime from about 250
km to 100 km, and in an inverse KE transfer regime for scales greater than about 250 km. In terms of an eddy and slowly
varyingorseasonaldecomposition(definedassmallerandlargerthan100daytimescalevariability, respectively), weobserve
2
FIG.1: Regions1-5chosenfortheanalysis. Thesearecenteredat−36◦,232◦(gridsize256×128);−38◦,78◦(gridsize256×64);33◦,
187◦(gridsize256×64);30◦,316◦(gridsize128×64)and−42◦,336◦(gridsize128×64),respectively.
thattheseasonal-seasonalandseasonal-eddyfluxesplayasignificantroleintheKE(enstrophy)fluxforscaleslarger(smaller)
thatabout250km. Finally,weinterpretthesefindingsinthecontextofidealizedstudiesofthree-dimensional,rapidlyrotating
andstratifiedturbulenceandalsocomparethemwithdetailedanalysesofmidlatitudeuppertroposphericKEspectraandfluxes.
II. DATAANALYSISANDMETHODOLOGY
A. DataDescription
Griddeddataofsea-surfaceheight(SSH)anomalies(MADTdelaytimegriddeddata)fromtheAVISOprojecthasbeenused
inouranalysis. Thedataspanning21years(1993-2013)isavailableataspatialresolutionof0.25◦×0.25◦,thusscalessmaller
thanapproximately50kmcannotberesolved. WecomputehorizontalcurrentsfromtheSSHanomalydatausinggeostrophic
balancerelations,andthelatitudinalvariationoftheCoriolisparameterhasbeenincludedinthecomputations.
B. GeographicalLocations
Thefivegeographicalregionschosenforanalysisarelocatedfarfromtheequatorsothatgeostrophicbalanceisexpectedto
bedominant. AsisseeninFigure1,theseregionsrepresentrelativelyuninterruptedstretchesintheNorthernandSouthernparts
ofthePacific,Atlantic,andtheSouthernIndianOcean. Inessence,weexpectthatthischoiceofdomainminimizesboundary
effects. Region1isthelargestwhichisabout4500kmlongand3500kmwide. Otherregionsarecomparativelysmaller.
C. ComputationofSpectraandFluxes
Inthispaper, wecomputetheKEspectrumandfluxesofKEandenstrophy. Forthispurpose, thedataisrepresentedusing
Fouriermodesinbothspatialdirections,i.e.,
(cid:90)(cid:90)
U(k)= U(x,y)exp[−i(k x+k y)]dxdy, (1)
x y
where,k = (k ,k )andU = (u,v)containsthezonalandmeridionalcomponentsofthevelocity. Thetwo-dimensional(2D)
x y
Fouriertransformtechniquerequiresthedatatohaveuniformgridspacinginbothdirections,soalinearinterpolationschemeis
employedtogeneratethevelocitydataonarectangulargrid(theoriginaldataisonequidistantlatitudesandlongitudes).Inorder
tomakethevelocityfieldspatiallyperiodic,thedataismultipliedwitha2Dbumpfunction(exp[− 0.01 − 0.01 +0.02]where
1−x2 1−y2
(x,y)∈[−1,1])beforeperformingaFouriertransform,thisensuresthatthevelocitysmoothlygoestozeroattheboundaries.
3
InFourierspace,theKEequationisrepresentedas[1],
∂E(k)
=T(k)+F(k)−D(k), (2)
∂t
(cid:113)
whereE(k)istheKEofashellofwavenumberk = k2+k2,F(k)istheenergysupplyratetotheaboveshellbyforcing,
x y
andD(k)istheenergydissipationattheshell. T(k)istheenergysupplytothisshellvianonlineartransfer. Notethat,
1 (cid:88)
E(k)= |U(k(cid:48))|2. (3)
2
k−1<|k(cid:48)|≤k
In a statistically steady state, ∂E(k)/∂t = 0, and E(k) is approximately constant in time. The energy supply rate due to
non-linearityisbalancedbyF(k)−D(k). AusefulquantitycalledtheKEflux,Π(k),measurestheenergypassingthrougha
wavenumberofradiusk,anditisdefinedas,
(cid:90) k
Π(k)=− T(k(cid:48))dk(cid:48). (4)
0
In 2D flows, another quantity of interest is the enstrophy (ζ = (cid:82) ω2dr), where ω is the relative vorticity. The corresponding
2
enstrophyfluxisdenotedbyζ(k). WecomputetheKEandenstrophyfluxesusingtheformalismofDaretal.[26]andVerma
[27],andtherelevantformulaeread,
(cid:88) (cid:88)
Π(k)= δ Im([k(cid:48)·U(q)][U∗(k(cid:48))·U(p)]), (5)
k(cid:48),p+q
k(cid:48)>kp≤k
(cid:88) (cid:88)
ζ(k)= δ Im([k(cid:48)·U(q)][ω∗(k(cid:48))ω(p)]), (6)
k(cid:48),p+q
k(cid:48)>kp≤k
whereImstandsfortheimaginarypartoftheargumentandU∗isthecomplexconjugate.Theexpression(insidethesummation
operators)inequation5(6)representstheenergy(enstrophy)transferinatriad(k(cid:48) = p+q)wherek(cid:48) modereceivesenergy
(enstrophy)frommodespandq. Then,theexpressionisintegratedoverallsuchpossibletriadssatisfyingtheconditionk(cid:48) >k
andp≤k(notethat−(cid:82)kT(k(cid:48))dk(cid:48)=(cid:82)∞T(k(cid:48))dk(cid:48)).
0 k
III. RESULTS
Webeginbyconsideringthespectraandfluxesassociatedwithdailygeostrophiccurrents. Figure2showsKEspectraofthe
geostrophiccurrentsderivedfromSSHanomaliesinallfiveregions. Asseen,thesecurrentsshowanapproximatek−3.5scaling
(the best fits range from k−3.5 to k−3.6) over a range of 250 to 100 km in all regions except Region 1 (the Southern Pacific),
wheretheslopeissomewhatshallowerwithabestfitofk−2.9. Thus,thespectraweobtainforgeostrophiccurrentsaremorein
linewiththosereportedby[5],[28](globalextratropics),[3](Agulhasregion)and[29]&[30](neartheGulfStream),butdiffer
fromtheshallower−5 likescalingobservedby[6][seealso7].
3
The KE and enstrophy fluxes are shown in Figure 3. Note that we have computed the flux using daily data and the results
presentedareanaverageovertheentire21yearperiod. ThequalitativestructureoftheKEfluxconfirmsthefindingsof[1][see
also2,3,16]. Specifically,weobservearobustinversetransferofKEatlargescales(i.e.,greaterthanapproximately250km).
Someregions(1and2)showaveryweakforwardtransferofKEatsmallscales, whileintheothers(Region3, 4and5), the
fluxcontinuestobenegative(thoughverysmallinmagnitude)evenatsmallscales. Infact, inalltheregionsconsidered, the
KEfluxcrosseszeroorbecomesverysmallbyabout200km. Notethat2πtimestheclimatologicalfirstbaroclinicdeformation
scaleinthesefiveregionsalsoliesbetween200and250km[31]. WhetherthisisindicativeofaKEinjectionscaleduetolinear
instabilityasputforthby[1], ormoreofacoincidenceisnotparticularlyclear. Indeed, amismatchbetweenthedeformation
andzero-crossingscalecanbeseenin[2]andhasalsobeenpointedoutby[17]inacomprehensiveoceanmodel.
Proceedingtotheenstrophy,alsoshowninFigure3,weseethatitischaracterizedbyalargeforwardfluxatscalessmaller
thanapproximately250km. Further, theenstrophyfluxdoesnotshowaninertialrange, ratheritincreaseswithprogressively
(cid:112)
smaller scales and peaks at approximately 150 km. Interestingly, we note that, in most of the regions, the scale (cid:104)E(cid:105)/(cid:104)ζ(cid:105)
(where(cid:104)·(cid:105)denotesadomainaverage)[32]—shownbythedashedverticallinesinFigure3—servesasareasonablemarker
fortheonsetoftheforwardenstrophyfluxregime.
4
FIG.2: SurfaceKEspectraforthegeostrophiccurrents(averagedover21yearsofdata). Theslopesobtainedbetweenwavenumbers0.004
and0.01(i.e.,250kmto100km)are:−2.85,−3.54,−3.48,−3.71and−3.61forregions1-5respectively.
A. Eddy(subseasonal)andslowlyvarying(seasonal)fluxes
An important difference between actual geophysical flows (the atmosphere and ocean) and idealized 3D rotating stratified
turbulence is the presence of nontrivial mean flows, and a hierarchy of prominent temporal scales. To get an idea of the KE
andenstrophyfluxcontributionsfromthefastandslowly-varyingcomponentsoftheflow,following[33],wefilterthederived
geostrophic currents. In particular, at every grid point, we consider the daily 21 year long time series and split this into two
parts: one that contains variability of less than 100 days (referred to as the eddy or transient component) and the other with
only larger than 100 day timescales (referred to as the slowly varying, or for brevity, as the seasonal component). To get a
feel for the physical character of the these decompositions, Figures 4 and 5 we show a snapshot of the zonal and meridional
velocities (during summer) for all of the five regions. Quite clearly, the slowly varying or seasonal u flow has a pronounced
zonalstructure,ascomparedtothemoreisotropiceddycomponent. Similarly,theseasonalv velocityhasalargerscaleandis
orientedinapreferentiallymeridionaldirectionascomparedtoitseddycomponent. Itisinterestingtonotethatthemeridional
(seasonalandeddy)velocityisalwayscomparableinstrengthtothezonalflow.
As with the original data (the total field), we compute the KE and enstrophy fluxes from the eddy field and the seasonal
component. Theseasonal-eddyfluxesarecomputedbysubtractingtheeddy-eddyandseasonal-seasonalcontributionsfromthe
totalflux.Figures6and7showthesefourterms(total:solidcurves,eddy-eddy:dashedcurves,seasonal-seasonal:dottedcurves
andseasonal-eddy: dash-dotcurves)forKEandenstrophyinthefiveregionsconsidered,respectively. FortheKE(Figure6),
weseethatthetotalfluxatlargescales(i.e.,greaterthan250km)hasastrongcontributionfromtheseasonal-eddyinteractions.
Infact,theeddy-eddytermisqualitativelyofthecorrectformbutquitesmallinmagnitude. Theseasonal-seasonalcontribution
is always upscale (except for a very small positive bump at small scales in Region 1), and thus, it too enhances the inverse
transferatlargescales. Fortheenstrophy(Figure7), weseethattheeddy-eddytermisreasonablystrong, andalongwiththe
seasonal-eddy flux (in Regions 3,4 and 5), or the seasonal-seasonal flux (in Region 1), or both (Region 2) leads to the strong
forwardenstrophycascadingregimeatsmallscales(i.e.,belowapproximately250km).
IV. INTERPRETATIONANDCONCLUSION
By studying 21 years (1993-2013) of surface geostrophic currents derived from AVISO SSH anomalies in different midlat-
itudinal parts of the world’s oceans we find — in agreement with previous studies — that the spectral flux of rotational KE
5
FIG.3: KE(solid)andenstrophy(dash)fluxesforgeostrophiccurrents(averagedoverall21years)vswavenumber(k2=k2 +k2). Vertical
x y
(cid:112)
dashedlinesmarkthetransitionscale( (cid:104)E(cid:105)/(cid:104)ζ(cid:105))ineachregion.
exhibits an inverse transfer at scales larger than about 250 km. Further, at smaller scales, specifically, 250 km to 100 km, we
findastrongforwardfluxofenstrophyaccompaniedbyaKEspectrumthatapproximatelyfollowsak−3.5 power-law. TheKE
flux at these small scales is very weak, and in a few of the regions considered, it is in the forward direction. The transition
fromaninverseKEtoadominantforwardenstrophyfluxisroughlyinagreementwithasimpleprescriptionbasedonthetotal
enstrophy and KE in the domain. On splitting the original data into high (eddy) and low (seasonal) frequencies, we observed
thattheseasonal-seasonalandseasonal-eddyfluxesplayanimportantroleintheinverse(forward)transferofKE(enstrophy)
at scales greater (smaller) than 250 km. We now interpret these findings in the context of rapidly rotating, strongly stratified
three-dimensional(3D)turbulenceaswellasspectraandfluxanalysesfromthemidlatitudeuppertroposphere.
Specifically, idealized 3D rotating Boussinesq simulations suggest that rotational (or vortical) modes dominate the energy
budgetatlargescalesandexhibitarobustinversetransferofKEtolargerscales, andaforwardtransferofenstrophytosmall
scales[34–37]. Thesetransfers,akinto2DandQGturbulence,areaccompaniedbyKEspectrathatfollow−5 and−3power-
3
lawsintheupscaleKEanddownscaleenstrophyfluxdominatedregimes[38,39]. Giventherotationalnatureofthegeostrophic
currents, our observation of the upscale (downscale) KE (enstrophy) flux at scales greater (smaller) than 250 km is therefore
in accord with the aforementioned expectations. Indeed, the k−3.5 scaling is also close to the expected KE spectrum that
characterizestheenstrophyfluxdominatedregime[49].
With regard to the atmosphere, the forward enstrophy transfer regime of QG turbulence has been postulated to explain the
−3portionofmidlatitudeuppertroposphericKEspectrum[theso-calledNastrom-Gagespectrum,40]. Startingwith[41]and
[33], re-analysis products at progressively finer resolutions have been analyzed with a view towards seeing if the Nastrom-
Gage spectrum is captured by the respective models [see for example, 42–44], and if so, what are the associated energy and
enstrophyfluxesthatgoalongwithit[45,46]. Inall,thesestudiesdemonstratequiteclearlythatthe−3rangeoftheNastrom-
6
FIG.4:Snapshotofthezonalseasonalandeddyfieldsinthesummer(specificallyon16thJune,2005).R1-R5correspondtoRegion1-Region5
ofFigure1.
Gagespectrum(spanningapproximately3000-4000kmto500kmintheuppertroposphere)correspondstothedominanceof
rotational modes, and a forward enstrophy cascading regime. Further, at scales greater than the −3 range (i.e., greater than
approximately4000km),theuppertropospheresupportsaninverserotationalKEflux[45,46].
RegardingthesmallforwardfluxofrotationalKEatscalessmallerthanapproximately250kminafewregions,aspointed
out by [41] and [46], this is likely due to the limited resolution of the data. For example, a forward KE flux in the rotational
modeswasobservedinthecoarsedatausedby[41],butitvanishesinthemorerecentfinerscaleproductsanalyzedin[45]and
[46][see,47,forsimilarissuesinincompressible2Dturbulence]. Infact,muchliketheidealized3Drotatingstratifiedscenario
[34–36],theforwardtransferofKEatsmallscalesinatmosphericdata(i.e.,belowapproximately500kmandaccompaniedby
ashallowerKEspectrum)islikelyduetothedivergentcomponentoftheflow[45,46].
On decomposing the flow into eddy and seasonal components (less and greater than 100 day timescales, respectively), our
resultsaresomewhatanalogoustotheuppertroposphere. Specifically,intheatmosphere,thezonalmean-eddy(whichtranslates
toastationary-eddyorseasonal-eddydecomposition)fluxenhancedtheinverseKEtransfertolargescales[33,46]. Wefindthe
seasonal-eddytermtobeimportant,buttheseasonal-seasonalcontributiontoalsobesignificantintheinversetransfer. Infact,
inourdecomposition(basedonatimescaleof100days),thesetermsdominateovertheeddy-eddycontribution. Forenstrophy,
in the upper troposphere, [33] noted that the stationary-eddy fluxes are important (as we do here), but higher resolution data
employedin[46]suggeststhattheeddy-eddytermisthedominantcontributortotheforwardenstrophyflux.
Thus our findings using altimeter data, i.e., employing purely rotational geostrophic currents, are in fair accordance with
7
FIG.5: Snapshotofthemeridionalseasonalandeddyfieldsinthesummer(specificallyon16thJune,2005). R1-R5correspondtoRegion1-
Region5ofFigure1.
expectations from idealized simulations of rotating stratified flows as well as analyses of upper tropospheric re-analysis data.
ThequalitativesimilarityinrotationalKEfluxes,enstrophyfluxes,andKEspectrabetweenthesurfaceoceancurrentsandthe
neartropopauseatmosphericflowiscomfortingastheybothareexamplesofrapidly-rotatingandstrongly-stratifiedfluids. In
fact,inadditiontoaninverserotationalKEfluxatlargescales,webelieveitisquitepossiblethatthealtimeterdataisshowing
us an enstrophy cascading, and relatively steep spectral KE scaling range of a surface oceanic counterpart to the atmospheric
Nastrom-Gagespectrum. Quitenaturally,itwouldbeveryinterestingtoobtaindataatafinerscale,andseeiftheoceansurface
currents(rotationalanddivergenttogether)alsoexhibitatransitiontoshallowerspectra—liketheuppertroposphericNastrom-
Gagespectrum—withachangeinscalingatalengthscalesmallerthan100km.
Acknowledgments
We thank the AVISO project for making the SSH data freely available (http://www.aviso.altimetry.fr/en/
home.html).
[1] R.B.ScottandF.Wang,J.Phys.Oceanogr.35,1650(2005).
[2] R.Tulloch,J.Marshall,C.Hill,andK.S.Smith,J.Phys.Oceanogr.41,1057(2011).
8
FIG.6: Total(solid),eddy-eddy(dash),slowlyvarying-slowlyvarying(dot)andslowlyvarying-eddy(dash-dot)KEfluxesforgeostrophic
√
currents(averagedoverall21years)vswavenumber(k= k2+k2)inallthefiveregionsregions.
x
[3] B.K.Arbic,M.Mu¨ller,J.G.Richman,J.F.Shriver,A.J.Morten,R.B.Scott,G.Se´razin,andT.Penduff,J.Phys.Oceanogr.44,2050(2014).
[4] B.K.Arbic,R.B.Scott,G.R.Flierl,A.J.Morten,J.G.Richman,andJ.F.Shriver,J.Phys.Oceanogr.42,1577(2012).
[5] D.Stammer,J.Phys.Oceanogr.27,1743(1997).
[6] P.-Y.LeTraon,P.Klein,B.L.Hua,andG.Dibarboure,J.Phys.Oceanogr.38,1137(2008).
[7] Y.XuandL.-L.Fu,JournalofPhysicalOceanography42,2229(2012).
[8] J.Callies,R.Ferrari,J.M.Klymak,andJ.Gula,Nat.Commun.6(2015).
[9] G.Lapeyre,J.Phys.Oceanogr.39,2857(2009).
[10] R.FerrariandC.Wunsch,TellusA62,92(2010).
[11] C.Wunsch,J.Phys.Oceanogr.27,1770(1997).
[12] K.S.SmithandG.K.Vallis,J.Phys.Oceanogr.31,554(2001).
[13] L.-L.FuandG.R.Flierl,Dyn.Atmos.Oceans4,219(1980).
[14] R.Salmon,Geophys.Astrophys.FluidDyn.15,167(1980).
[15] J.-M.HoyerandR.Sadourny,J.Atmos.Sci.39,707(1982).
[16] R.B.ScottandB.K.Arbic,J.Phys.Oceanogr.37,673(2007).
[17] F.Schlo¨sserandC.Eden,Geophys.Res.Lett.34(2007),l02604.
[18] A.Venaille,G.K.Vallis,andK.S.Smith,J.Phys.Oceanogr.41,1605(2011).
[19] G.LapeyreandP.Klein,J.Phys.Oceanogr.36,165(2006).
[20] R.Pierrehumbert,I.Held,andK.Swanson,Chaos,Solitons&Fractals4,1111(1994).
[21] J.SukhatmeandR.Pierrehumbert,Chaos12,439(2002).
[22] K.Smith,G.Boccaletti,C.Henning,I.Marinov,C.Tam,I.Held,andG.Vallis,J.FluidMech.469,13(2002).
[23] X.Capet,P.Klein,B.L.Hua,G.Lapeyre,andJ.C.Mcwilliams,J.FluidMech.604,165(2008).
[24] R.B.ScottandD.G.Furnival,J.Phys.Oceanogr.42,165(2012).
[25] K.S.SmithandJ.Vanneste,J.Phys.Oceanogr.43,548(2013).
9
FIG. 7: Total (solid), eddy-eddy (dash), slowly varying-slowly varying (dot) and slowly varying-eddy (dash-dot) enstrophy fluxes for
√
geostrophiccurrents(averagedoverall21years)vswavenumber(k= k2+k2)inallthefiveregionsregions.
x
[26] G.Dar,M.K.Verma,andV.Eswaran,Phys.D:NonlinearPhenomena157,207(2001).
[27] M.K.Verma,Phys.Rep.401,229(2004).
[28] M.G.ScharffenbergandD.Stammer,J.Geophys.Res.:Oceans116(2011).
[29] D.-P.Wang,C.N.Flagg,K.Donohue,andH.T.Rossby,J.Phys.Oceanogr.40,840(2010).
[30] J.CalliesandR.Ferrari,J.Phys.Oceanogr.43,2456(2013).
[31] D.B.Chelton,R.A.Deszoeke,M.G.Schlax,K.ElNaggar,andN.Siwertz,J.Phys.Oceanogr.28,433(1998).
[32] S.D.DanilovandD.Gurarie,Phys.-Usp.43,863(2000).
[33] T.G.Shepherd,J.Atmos.Sci.44,1166(1987).
[34] P.Bartello,J.Atmos.Sci.52,4410(1995).
[35] Y.KitamuraandY.Matsuda,Geophys.Res.Lett.33(2006),l05809.
[36] J.SukhatmeandL.M.Smith,Geophys.Astrophys.FluidDyn.102,437(2008).
[37] A.Vallgren,E.Duesebio,andE.Lindborg,Phys.Rev.Lett.107,268501(2011).
[38] R.H.Kraichnan,Phys.Fluids10,1417(1967).
[39] J.Charney,J.Atmos.Sci.28,1087(1971).
[40] G.NastromandK.Gage,J.Atmos.Sci.42,950(1985).
[41] G.J.BoerandT.G.Shepherd,J.Atmos.Sci.40,164(1983).
[42] D.StraussandP.Ditlevsen,Tellus51A,749(1999).
[43] J.KoshykandK.Hamilton,J.Atmos.Sci.58,329(2001).
[44] K.Hamilton,Y.Takahashi,andW.Ohfuchi,J.Geophys.Res.113,D18110(2008).
[45] P.AugierandE.Lindborg,J.Atmos.Sci.70,2293(2013).
[46] B.Burgess,A.Erler,andT.G.Shepherd,J.Atmos.Sci.70,669(2013).
[47] G.Boffetta,J.FluidMech.589,253(2007).
[48] G.BoffettaandR.E.Ecke,Annu.Rev.FluidMech.44,427(2012).
10
[49] Itshouldbenotedthatthe−3exponent(eveninincompressible2Dturbulence)isfairlydelicate.Asdiscussedinthereviewby[48],itisnotuncommon
toobservepower-lawsfortheKEspectrumthatrangefrom−3to−3.5intheenstrophycascadingregime.