covariation of finite-amplitude wave activity and the zonal...
TRANSCRIPT
1
Covariationoffinite-amplitudewaveactivityand2
thezonalmeanflowinthemid-latitude3
troposphere.Part2:eddyforcingspectraandthe4
periodicbehaviorintheSouthernHemisphere5
summer6
7
8
9
10
LeiWangandNoboruNakamura*11
TheUniversityofChicago,Chicago,Illinois12
13
14
15
16
17
18
____________________19
Correspondingauthoraddress:NoboruNakamura,DepartmentoftheGeophysicalSciences,Universityof20Chicago,5734S.EllisAvenue,Chicago,IL60637.21
E-mail:[email protected] 22
2
Abstract23
PreviouslyinPart1ofthisstudytheauthorsusedlatitude-by-latitudebudgetsofthe24
verticallyintegratedfinite-amplitudewaveactivity(FAWA)todescribethecovariationof25
thezonal-meanstateandeddyamplitude.Intheaustralsummerwithin40- 55! S,FAWA26
exhibitsamarked20-30dayperiodicitydrivenmainlybythelow-levelmeridionaleddy27
heatflux,consistentwiththerecentlyidentifiedbaroclinicannularmode(BAM).28
Thepresentarticleexaminesthespectraofeddyheatfluxthatproducetheperiodic29
behaviorintheSouthernHemispherestormtrack.AnalysisoftheECMWFERA-Interim30
productrevealsthatthe20-30dayperiodicityinrawFAWAandeddyheatfluxis31
particularlyrobustduringthewarmseason.AdryGCMisshowntoreproducequalitatively32
BAM-likeeddyheatfluxspectraifthezonal-meanstateresemblesthatoftheaustral33
summerandifthesurfacethermaldampingissufficientlystrong.Theobservededdyheat34
fluxcospectrainsummercontainafewdominantfrequenciesforeachoftheenergy-35
containingzonalwavenumbers(4-6).ThecorrespondingFouriermodesareheat-36
transportingbutneutral,withslightlydifferentmeridionalstructures.Asthesemodes37
travelatdifferentphasespeedstheyinterferewitheachotherandproduceanamplitude38
modulationtotheeddyheatfluxwithaperiodicityconsistentwiththeBAM.The39
meridionallyconfinedbarocliniczoneinthemeanstateoftheaustralsummerprovidesa40
waveguidethatdirectsthemodepropagationandinterferencealongthelatitude41
circle.However,theprocessesthatgiverisetothequasi-discreteFouriermodesremainto42
beidentified. 43
3
1.Introduction44
Stronginteractionbetweenthezonalmeanflowandfinite-amplitudeeddiesfosters45
dynamicallyrichbehaviorsintheEarth’smidlatitudeatmosphere.TheSouthern46
Hemispheresummerexhibitsacoherentannularstructureinbothtime-meanflowand47
stormtrackdensity(HoskinsandHodges2005;Lee2014),thereforeitisanidealplaceto48
observesuchinteraction.49
InPart1(WangandNakamura2015;hereafterWN15),weformulatededdy-mean50
flowinteractionintermsoflatitude-by-latitude,verticallyintegratedbudgetsoffinite-51
amplitudewaveactivity(FAWA,NakamuraandZhu2010,hereafterNZ10)andthezonal-52
meanzonalwind.Further,weappliedittotheaustralsummerintheERA-Interim53
reanalysisproduct(Deeetal.2011).ApronouncedantiphasecovariationofFAWAandthe54
zonal-meanzonalwindwasfoundbetween40and 55! S.Whilesuchcovariationisexpected55
fromthenonaccelerationrelationunderadiabaticandfrictionlessdynamics(Charneyand56
Drazin1961anditsfinite-amplitudeextensioninNZ10),theaustralsummerstandsoutin57
thatthecovariationappearsquasi-periodicwithatimescalemuchlongerthanthetypical58
lifespanofsynopticeddies(seeFig.1ofWN15).WN15showthattheFAWApowerspectra59
exhibitadistinctivepeakaround25days(~0.04cpd)between40and 55! S[reproducedin60
Fig.1a;alltheobservationalanalysesinthisarticlearebasedontheERA-Interimreanalysis61
at 1.5!×1.5! horizontalresolution(Deeetal.2011)],althoughthecorrespondingspectraof62
thezonal-meanzonalwindareredanddominatedbyvariabilitiesattheflanksofthestorm63
tracklatitudes.64
4
Thepredominant20-30dayperiodicityinFAWAduringtheaustralsummeris65
consistentwiththerecentlyidentifiedbaroclinicannularmode(BAM)ineddykineticenergy66
(EKE)(ThompsonandWoodworth2014,hereafterTW14;ThompsonandBarnes2014,67
hereafterTB14).[SimilarperiodicitywasalsonotedearlierbyWebsterandKeller(1974,68
1975)throughlimitedballoondata,andbyChenetal.(1987)withtheFirstGARPGlobal69
Experiment(FGGE)data.]ThompsonandthecollaboratorsfindthattheperiodicityinEKE70
isassociatedwiththelow-levelmeridionaleddyheatflux,whichshowsawell-defined71
spectralpeakaround20-30daysduringtheaustralsummer.Asimilaranalysisforthe72
NorthernHemisphereshowsthatthespectralpeakinEKEandthelow-levelheatfluxis73
morerobustinsummer(ThompsonandLi2014,hereafterTL14).WhileTW14,TB14and74
TL14allusetheleadingempiricalorthogonalfunction(EOF)ofthezonallyaveragedEKEto75
defineBAM,wechoosetoworkwithFAWApartlybecauseitpossesses(unlikeEKE)a76
closedlatitude-by-latitudebudget,andpartlybecauseithasadirecttheoreticalconnection77
withthelow-levelmeridionaleddyheatflux(WN15,seealsosection2below).Despitethe78
differenceintheconservationproperties,theverticallyintegratedFAWAandthezonally79
averagedEKEat300hPabothdisplayaqualitativelysimilarspectrainthefrequency-80
latitudespacefortheaustralsummer(Fig.1aand1b):theybothexhibitadistinctive81
spectralpeakcenteredaround 46.5! Sand0.04cpd,whichcoincideswiththespectralpeak82
inthelow-levelmeridionaleddyheatflux(Fig.1c).83
Whentheanalysisisrepeatedfortheaustralwinter,thepicturechangessignificantly.84
Figures1d-1fshowthecorrespondingquantitiesforthemonthsofJune-September.FAWA85
nolongerpossessesacompactspectralpeakintheextratropicsandinsteadexhibits86
multiplepeaks,includingtwostrongonesathighlatitudes(~ 60! S)andasecondary87
5
maximumataslightlyhigherfrequency(~0.06cpd)atlowerlatitudes(~ 40! S)(Fig.1d).88
TheEKEspectraareappreciablydifferentfromFAWA:theyaresplitintomultiplepeaks89
overawidelatitudinalrangewiththestrongestpowerappearing 35!−45! Sandatlower90
frequencies(0.02-0.03cpd)(Fig.1e).Meanwhile,thespectraoflow-levelmeridionaleddy91
heatfluxaredominatedbylargevaluesinhighlatitudesathigherfrequencies(0.05-0.192
cpd)(Fig.1f).Therefore,unlikesummer,spectraofFAWA,EKEandeddyheatfluxappear93
quitedisparate.AsshowninAppendixA,theleadingEOFofEKEisqualitativelysimilar94
betweenthetwoseasons,butitexplainssubstantiallylessvarianceinwinterthanin95
summer(30versus45percent)anditspeakfrequencyislessseparatedfromtheseasonal96
timescale.Clearly,thestructureofeddyspectraandtheunderlyingdynamicsaremore97
complexinwinter,andtheBAM’speriodicityishardertodetectinrawdata.98
TofurtherelucidatethesimilarityanddifferencesbetweenFAWAandEKEasa99
diagnostic,weshowinFig.2thetime-latitudeplotsoftheanomaliesinverticallyintegrated100
FAWA(Fig.2a)andthezonallyaveragedEKEat300hPa(Fig.2b)fortheyear2012.Thetwo101
showsimilarpatternsbutthereareappreciabledifferencesduringwintermonths(days102
150-300):for example, some large wave activity events in high latitudes are missing in EKE. 103
Figure 2c shows the part of the EKE anomaly explained by the leading EOF (BAM), and its 104
amplitude (the BAM index, TW14, normalized to a unit variance) is shown as the blue curve in 105
Fig.2d. Also shown in Fig.2d is the volume integral of FAWA between 20! and 70! S[see106
equation(7)below](redcurve),againnormalizedtoaunitvariance.Thedomain-107
integratedFAWAcorrelateswiththeBAMindexremarkablywell(r=0.72).We thus argue 108
that, despite the difference in their latitudinal structures, both diagnostics capture the same low 109
6
frequency variability of the eddy amplitude when integrated over the extratropics1. 110
Howisitthatlarge-scaleeddyprefersaperiodicbehaviortoamorechaoticone,111
particularlyduringtheaustralsummer?TB14invokeanonlinearoscillatormodelofthe112
BAMbasedonthefeedbackbetweenthebaroclinicityoftheflowandmeridionaleddyheat113
flux.Theperiodicbehaviorofthismodelisreminiscentofthelimitcycleintheamplitudeof114
weaklynonlinearbaroclinicwaveswithweakdissipation(Pedlosky1970,1971).The115
timescaleofoscillationintheirmodeldependsonthe(empiricallydetermined)strengthof116
feedbackandtherelaxationtimescaleofbaroclinicity.AmbaumandNovak(2014)propose,117
inadifferentcontext,asimilarnonlinearoscillatormodelforstormtrackvariabilitythat118
incorporatesbothgrowthanddampingofeddiesaswellasrestorationofbaroclinicity119
(Eadygrowthrate).Intheirmodelthetimescaleofoscillationisdeterminedbythe120
specifiedrestorationrateforbaroclinicity.InthecontextoftheBAM,periodicityinthe121
nonlinearoscillatormodelsshouldbeinterpretedasagrowth-decaycycleofbaroclinic122
wavepacket,ratherthananindividualeddylifecyclethathasmuchshortertimescales(e.g.123
LeeandHeld1993).Themeridionaleddyheatfluxandgrowthratecharacterizethe124
averagepropertiesofeddieswithinthepacket,influencingeachotherthroughafeedback125
process.126
Thenonlinearoscillatormodelrepresentsazero-dimensionaleddyfluxclosurethat127
predictsamplitudevacillation,butobservationalevidencesuggeststhatitmightnotbean128
adequatemodelforBAM.Wechoosethedifferenceinthe850hPazonal-meanpotential129
1 Keep in mind that while the BAM index quantifies the leading EOF, FAWA here quantifies the entire eddy field. The difference should be small as long as the leading EOF dominates the eddy variance.
7
temperaturebetween 55.5! Sand 40.5! Sasabulkmeasureofthelow-levelbaroclinicityof130
themeanstateinthemidlatitude.Figure3ashowsthespectraldensityofthisquantityfor131
summer(December-March).Thespectrumisredanditshowslittleevidenceofaspectral132
peakaroundtheBAMfrequencies(~0.04cpd).Wethenderivetheequationthatgoverns133
thisbaroclinicityparameterbyfirstformingthezonal-mean850hPapotentialtemperature134
equationat 55.5! Sand 40.5! Sandthentakingthedifferenceofthetwo.Wehavecomputed135
thetendencyandthemeridionaleddyfluxtermsofthisequationfromthe6hourlydataand136
savedthedatadaily.Figure3bshowsthespectrafortheseterms.Thetendency(black)and137
eddyforcing(red)havecomparablespectralintensity.Theirdifferenceismainlydueto138
turbulentheatfluxinthevertical,whichactsasadamperforthetemperaturevariability139
particularlyatlowfrequencies.Herebothspectrahaveapeakaround0.17–0.2cpd(5-7140
days),consistentwithsynopticweathersystemsbutnotwithBAM.Itisnotablethatdespite141
theclearspectralpeakintheeddyheatfluxaround~0.04cpd(Fig.1c),itsmeridional142
derivative(theredcurveinFig.3b)doesnotshowacomparablespectralpeak.This143
suggeststhatthedirectinfluenceoftheeddyheatfluxonbaroclinicityattheBAMfrequency144
islimitedatbest.Thecorrespondingspectraforwinter(June-September)areshownin145
Figs.3cand3d.Theoverallpowerofbaroclinicityisweakerinwinter(Figs.3aand3c).This146
isduepartlytotheweakermeridionaltemperaturegradient(seeFig.8abelow)andpartly147
toastrongerthermaldampingfrombelow(thedifferencebetweenthetwocurvesin148
Fig.3d).Againthebaroclinicityspectrumislargelyred(Fig.3c)andthetendencyandthe149
eddyforcingtermshavebroadspectrawithnodistinctivepeakattheBAMfrequency.The150
apparentlackofspectralcouplingbetweenbaroclinicityandeddyheatfluxcastsdoubton151
thevalidityofnonlinearoscillatormodel,whichassumesfeedbackbetweenthem.(Wehave152
8
alsousedthetroposphericmeanverticalshearofthezonal-meanzonalwindasasurrogate153
forbaroclinicity,andtheresultwasessentiallysimilar.)154
InthisarticleweproposeadifferentmechanisticinterpretationofBAM.Wefindthat155
onedistinctivepropertyoftheSHextratropicsisthattheeddyspectraaredominatedbya156
fewpeaksinthezonalwavenumber-frequencyspace.InFig.1, 46.5! Swasidentifiedasthe157
latitudeofmaximumeddyamplitudevariabilityduringsummer.Figure4showsthe158
spectraldensityofthe250hPageopotentialanomalyat 46.5! Sasfunctionsoffrequency159
andzonalwavenumber.[SeeSalby(1982)forarelatedresultbasedonFGGE.]Toaid160
visualizationthevaluesareinterpolatedfornon-integerzonalwavenumbers.Overall,the161
powerofgeopotentialanomalyextendsfromthelow-frequency/low-wavenumberquadrant162
tothehigh-frequency/high-wavenumberquadrant,suggestiveofeastward(downstream)163
grouppropagation(LeeandHeld1993;ChangandOrlanski1993).However,thissmooth164
structureofspectraispunctuatedbyoneormoredominantfrequencypeaksateachzonal165
wavenumber.Inwintertherearethreedominantpeaks:wavenumbers1and3at~0.02166
cpdandwavenumber4at~0.09cpd(Fig.2b).Insummer,however,therearemultiple167
peaksforeachindividualzonalwavenumberfrom4to6(Fig.2a).Thissuggeststhatthe168
geopotentialanomaliesat 46.5! SconsistofafewdominantFouriermodes(hereafterfor169
brevityas‘modes’).Aswewillseebelow,modeswiththesamezonalwavenumberbut170
distinctfrequenciesproduceamplitudemodulationofthezonal-meaneddyheatflux171
throughfrequencyinterference,withatimescaleconsistentwiththeBAM.Wewillalsouse172
aseriesofGCMsimulationstodemonstratethecriticalroleofthezonal-meanstateofthe173
australsummerandthesurfacethermaldampinginsettingupthemodalinterference.174
9
Thepaperisorganizedasfollows.Section2recapitulatesthebudgetsofvertically175
integratedFAWAasaframeworkforanalyzingthemidlatitudeatmosphericvariability,176
emphasizingtheimportanceofthelow-levelmeridionaleddyheatflux.Section3presents177
theobservededdyforcingspectraintheERA-Interimreanalysisandtheirseasonality.We178
willthenexploretheroleofthezonal-meanstateforshapingtheeddyforcingspectrawitha179
drydynamicalcoreofageneralcirculationmodel(GCM).WewillseethatthedryGCMis180
capableofproducingaBAM-likeeddyheatfluxspectraonlyifthezonalmeanstate181
resemblesthatoftheaustralsummerandifthesurfacethermaldampingrateischosen182
judiciously.Insection4weperformspectraldecompositionoftheeddyheatfluxusingthe183
ERA-Interimreanalysisanddemonstratethatitscospectracontaindistinctpeaksduringthe184
australsummer.Thepeaksdefineheat-transportingquasi-discretemodes,whose185
interferencemaybeinterpretedasamplitudemodulationofeddiesassociatedwiththeBAM.186
Section5summarizestheresults.187
2.BudgetofverticallyintegratedFAWA188
Theinteractionbetweenlarge-scaleeddiesandthezonal-meanzonalwindinthe189
Earth’smidlatitudeatmosphereisexpressedsuccinctlyinthetransformedEulerianmean190
formalismofAndrewsandMcIntyre(1976),whichhassubsequentlybeenextendedto191
finite-amplitudeRossbywavesandbalancededdies(NZ10).Inthequasigeostrophic192
frameworkthisiswrittenas193
∂u∂t
= f0 v− ∂∂ y′u ′v + !U = f0 v ∗+
1"ρ∇⋅F + !U , (1)194
10
∂A∗
∂t=−
1!ρ∇⋅F + "A . (2)195
Here u ( y,z,t) iszonal-meanzonalwind(overbarandprimedenotezonalmeanand196
deviationfromit,respectively)and A∗( y,z,t) isfinite-amplitudewaveactivity(FAWA)197
definedintermsofthemeridionaldisplacementofquasigeostrophicpotentialvorticity(PV)198
andquantifiesthepseudomomentumcarriedbyeddies[NZ10Eq.(11)]. !ρ= ρ0e−z/ H isthe199
densityoftheone-dimensionalbackgroundstatewithaconstantscaleheightH.The200
divergenceoftheEliassen-Palmfluxdensity2 F = − !ρ ′u ′v ,
!ρ f0
d !θ / dz′v ′θ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟appearsinboth201
equationsandactsastheagentofinteractionbetween u and A∗ .However,becauseofthe202
Coriolistorqueoftheresidualcirculation f0v∗ ,aswellasthezonal-meanfrictionaltorque203
!U andnonconservativesourcesandsinksofwaveactivity !A ,(1)and(2)stopshortofbeing204
aperfectaction-reactionrelation.205
WN15considerthedensity-weightedverticalaverage,denotedby206
⋅ =
0
∞
∫ e−z/ H ⋅( )dz0
∞
∫ e−z/ H dz ,of(1)and(2):207
∂∂t
u =−∂∂ y
′u ′v + !U , (3) 208
∂∂t
A∗ =∂∂ y
′u ′v +f0
H′v ′θ
d !θ / dz
⎛
⎝⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟z=0
+ "A . (4)209
In deriving (3), it is assumed v = 0 (no net mass flux across latitude). Because of the density 210
weighting, the vertical average primarily samples the troposphere (see a recent discussion in 211
2 Or equivalently the meridional flux of eddy PV through Taylor’s identity (Andrews et al. 1987);
θ is potential temperature and other notations are standard.
11
Huang and Nakamura 2016). By introducing an analogous surface FAWA, B∗( y,t) , on the lower 212
boundary based on the meridional displacement of potential temperature [NZ10 Eq. (41); WN15], 213
we have 214
∂∂t
B∗ =−f0
H′v ′θ
d !θ / dz
⎛
⎝⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟z=0
+ "B , (5) 215
where !B represents diabatic sources and sinks of surface wave activity B∗ , primarily through the 216
exchange of heat with the underlying land or sea surface. Note that A∗ ≥0 and B
∗ ≤0 due to the 217
opposite sign in the meridional gradients of the interior PV and of the surface potential 218
temperature (NZ10). In the absence of nonconservative processes, the sum of (3)-(5) yields 219
∂∂t
u =−∂∂t
A∗ + B∗( ), (6) 220
thus at each latitude, the barotropic component of the zonal-mean wind covaries with the sum of 221
the interior and surface wave activities. This is a vertically averaged version of the 222
nonacceleration theorem (Charney and Drazin 1961; NZ10; WN15). 223
Figure 5 summarizes the zonal momentum-wave activity cycle defined by (3)-(5) at each 224
latitude. The direction of the arrows in Fig.5 is typical of a baroclinically unstable, eddy-driven 225
jet. A poleward surface eddy heat flux transfers wave activity from B∗ to
A∗ , making the 226
former more negative and the latter more positive. This represents baroclinic instability and 227
vertical propagation of Rossby waves forced at the surface, among other things. The convergence 228
of eddy momentum flux drives u at the expense of
A∗ (i.e., barotropic decay of baroclinic 229
eddies; Randel and Stanford 1985). External sinks include surface drag on u and loss of
A∗ 230
through mixing (i.e. enstrophy dissipation) and radiative and Ekman damping (NZ10). On the 231
12
other hand, diabatic heating (latent heat of condensation) can be a source of
A∗ and offset some 232
of its dissipation. Surface thermal damping of B∗ also works as a source of wave activity 233
because B∗ is negative. In the quasigeostrophic framework form stress by topography also 234
constitutes the source of wave activity through the boundary potential temperature anomaly (and 235
hence B∗ ). 236
ComparedtotheglobalenergycycleofLorenz(1955),the zonal momentum-wave 237
activity cycle has two advantages for describing eddy-mean flow interaction: (1) unlike the 238
energy cycle its budget is closed at each latitude, allowing the meridional structure of the 239
interaction to be described; (2) the sole eddy forcing for the barotropic component of FAWA is 240
the convergence of the vertically averaged Eliassen-Palm flux, whereas each reservoir in the 241
energy cycle is driven by more than one flux whose direction depends on the choice of the 242
coordinate (see a detailed discussion in Plumb 1983).243
When averaged over a band of latitudes to even out the convergence and divergence of 244
vertically integrated eddy momentum flux, (4) and (5) become (the latitudinal average is denoted 245
by the square bracket): 246
∂∂t
A∗⎡⎣⎢
⎤⎦⎥=
f0
H′v ′θ
d !θ / dz
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ z=0
+ "A⎡⎣⎢⎤⎦⎥, (7) 247
∂∂t
B∗⎡⎣⎢⎤⎦⎥=−
f0
H′v ′θ
d !θ / dz
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ z=0
+ "B⎡⎣⎢⎤⎦⎥ . (8) 248
Intheconservativelimitwherethelasttermsin(7)and(8)arenegligible,theglobalbudget249
ofFAWAisgovernedsolelybythesurfaceeddyheatfluxandthedomainintegralofwave250
activity(sumoftheinteriorandsurfacecontributions)isconserved:251
13
∂∂t
A∗⎡⎣⎢
⎤⎦⎥+ B∗⎡⎣⎢
⎤⎦⎥( )= 0 . (9)252
ThislastconstraintmaybeusedtoextendtheCharney-Stern-Pedloskycriterionformodal253
baroclinicinstabilityforfiniteamplitude(CharneyandStern1962;Pedlosky1964;NZ10):254
forthedomainaveragedinteriorandsurfaceFAWAstogrowsimultaneouslywithout255
violating(9),theymusthaveoppositesign.ThisconditionismetfortheEarth’satmosphere256
since A∗ ≥0 and B
∗ ≤0 (NZ10).257
Inthenonconservativelimit,timeaveraging(denotedbycurlybracket)of(7)and258
(8)gives259
f0
H′v ′θ
d !θ / dz
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ z=0
⎧⎨⎪⎪⎪
⎩⎪⎪⎪
⎫⎬⎪⎪⎪
⎭⎪⎪⎪
= "B⎡⎣⎢⎤⎦⎥{ }=− "A⎡
⎣⎢⎤⎦⎥{ } . (10)260
Therefore,thetime-meanlatitudinal-meansurfaceeddyheatflux(polewardintheEarth’s261
atmosphere)isproportionaltothemeandampingrateofbothsurfaceandinteriorFAWAs.262
InthiscasetheinflowsandoutflowsbalanceforthetwobottomboxesinFig.5(eddy263
momentumfluxconvergenceandfrictionaldampingof u vanishuponlatitudinalaverage).264
3.Spectraofeddyfluxesandtheirseasonalvariation265
WN15comparethemagnitudeofthetermsin(3)-(5)fortheaustralsummerstorm266
track3.Theyfindthattheleft-handsideof(5)isnegligiblecomparedtotheright-handside267
termsandthat B∗ issmallcomparedto A∗ .Thenassumingthat !B representsthermal268
3 Following Nakamura and Solomon (2010), WN15 use the spherical coordinate to rewrite (3)-(5)
and apply the lower boundary condition halfway between the two lowest levels of analysis.
14
dampingof B∗ duetoexchangeofheatwiththeunderlyingsurface,onemayrewrite(5)as:269
f0
H′v ′θ
d !θ / dz
⎛
⎝⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟z=0
≈ "B≈− B∗
τH
> 0 . (11)270
Fromthetimeaverageof(11),WN15estimate τH ~0.9dayintheBAM-relatedlatitudes,271
closeto~1dayfoundbySwansonandPierrehumbert(1997)asthethermaldamping272
timescaleofsurfacetemperatureinthePacificstormtrackusingLagrangiantracer273
calculations.274
Sincethetendencyof B∗ isnegligible,covariationofthezonal-meanflowandFAWA275
isdictatedby(3)and(4).WN15conductaspectralanalysisof(3)and(4)andfindthatat276
highfrequencies(>0.1cpd) u and
A∗ arebothprimarilydrivenbytheeddymomentum277
fluxterm,whereastheeddyheatfluxforcingnearthesurfacehasapeakpoweraroundthe278
BAMfrequency(~0.04cpd)atwhichitdominatestheeddymomentumfluxforcingin(4)279
(Fig.5ofWN15).ThislastresultisreproducedinthebottompanelofFig.6a,whichshows280
thepowerspectraofthefirsttworight-handsidetermsof(4)at 46.5! Sforthewarmseason.281
The0.04cpdspectralpeakintheredcurvedrivestheBAMthroughthesecondtermonthe282
right-handsideof(4).ThebottompanelofFig.6bshowsthesamequantitiesforthecold283
season.Inwinterat 46.5! S,theeddyheatfluxspectrumismorered-noise-likeandlacksa284
clearpeak,andiscomparabletotheeddymomentumfluxforcingonlyinthelowfrequency285
limit.Wehaverepeatedtheanalysisatotherlatitudes–forexampleat 40! SatwhichFAWA286
maximizesinwinter–andfoundthattheresultsarequalitativelysimilar.Figure6also287
depictstheverticalstructureofthespectraldensitiesofthedensity-weightededdy288
momentumfluxconvergence(top)andthescalededdyheatflux(middle)atthesame289
15
latitude.Inbothseasonsthedensity-weightededdymomentumfluxconvergenceis290
concentratedintheuppertroposphere(seearecentdiscussioninAit-ChaalalandSchneider291
2014),althoughinwinterthemagnitudeissignificantlygreater.Theeddymomentumflux292
convergencespectraaremarkedbyafewdistinctpeaksinfrequencybutotherwisebroad293
(Fig.6,toppanels).Incontrast,thespectraofdensity-weightededdyheatfluxoccupy294
mostlythemiddle-tolowertroposphere(Fig.6,middlepanels).Eventhoughthespectral295
peakat~0.04cpdinthebottompanelofFig.6apertainstothenear-surfaceeddyheatflux296
(4),thisspectralpeakineddyheatfluxextendsverticallyupto~8km,suggestingthatthe297
20-30dayperiodicityintheeddyheatfluxispresentthroughtheentiretroposphere(Fig.6a298
middlepanel).Althoughinbothseasonstheeddyheatfluxanditscontributiontotheeddy299
PVfluxaremoreconfinedtolowfrequency(<0.25cpd)thantheeddymomentumflux300
contribution,the0.04cpdpeakintheeddyheatfluxclearlydistinguishesthetwospectra301
fromeachotherintheaustralsummer.302
The20-30daycycleinthelow-leveleddyheatfluxmeansthat,through(11), !B 303
shouldalsoexhibitsimilarperiodicity.ThisiscorroboratedbytherecentworkbyHerman304
(2015)whoshowsthatatmosphere-oceanturbulentheatflux(which !B represents)varies305
stronglyonthetimescaleoftheBAMintheaustralsummer.Howeverthisisnotanintrinsic306
timescaleofair-seaheatexchange,whichisdrivenbylarge-scaledynamicsofthe307
atmosphereandslavedtoitsmeridionaleddyheatflux,nottheotherwayround.Onthe308
otherhand,air-seaheatexchangedoesprovidearapidthermaldamping,whichkeepsthe309
surfacewaveactivityconsistentlyweak: 0<−B∗<< A∗ .Thismakesitdifficultto310
interprettheBAMvariationof
A∗ intermsofthetraditionalbarocliniclifecycles,inwhich311
16
B∗ wouldalsoundergoacomparablevariation(andsignchanges).312
TounderstandtheseasonalityoftheeddyfluxesinFig.6,particularlythelow313
frequencypeakintheeddyheatfluxthatonlyoccursinthewarmseason,weexplorethe314
roleofthezonal-meanstateinshapingtheeddyfluxspectra.Totheextentthatthelarge-315
scaleeddiesintheSouthernHemispherestormtrackobeythequasigeostrophicdynamics,316
thezonal-meanzonalwindsandthemeridionalgradientsofthezonal-meanPVinthe317
interiorandofthezonal-meanpotentialtemperaturenearthesurfaceareofparticular318
interest.319
Figure7showsthemeridionalcrosssectionsofthezonal-meanzonalwindand320
potentialtemperature(top),zonal-meanPVgradient(middle),andtheeddyheatflux321
(bottom)forthewarmandcoldseasonsoftheSH.Thezonal-meanzonalwindcapturesthe322
well-knownregimetransitionbetweentheextratropical(‘merged’)jetinsummer(topleft)323
tothesubtropicaljetinwinter(topright)(LeeandKim2003;LachmyandHarnik2014).324
Correspondingtothis,thePVgradientexhibitssignificantseasonalitynearthetropopause.325
Intheaustralsummer,thePVgradientisconcentratedinanarrowlatitudebandslightly326
polewardofthejetaxis(middleleft).Intheaustralwinter,thetropopause-levelPV327
gradientspreadsovertwomaxima:oneinthesubtropicsandtheotherinthemidlatitude,328
theformerbeingthestrongest.Anegativeflowcurvaturenearthejetaxisgenerally329
enhancesthePVgradientbutthevariationintheverticalshearisfoundtocontribute330
stronglytothePVgradientintheaustralsummer(notshown).Inaddition,strongPV331
gradientsappeararoundtheedgeoftheAntarcticaintheaustralwinter(middleright),a332
resultofthevastthermalcontrastbetweencontinentaliceandrelativelywarmocean333
17
surface.334
Theseasonalmarchofthemeridionalgradientsofthetropopause-levelPVandofthe335
low-level(850hPa)temperatureisshowninFig.8asafunctionoftimeandlatitude.Inthe336
warmseason(themergedjetregime)thePVgradientsmaximizearound 50−55! Swhereas337
inthecoldseason(thesubtropicaljetregime)abandofstrongPVgradientsalsoappears338
around 30! S(top).Thelow-leveltemperaturegradientsmaximizeat 45−50! Sinthewarm339
season,whereasinthecoldseasontheyexhibitverylargevaluesaroundtheedgeofthe340
Antarctica,surroundedbyrelativelyuniformvaluesintheextratropics(bottom).341
Theforegoinganalysisrevealsthatintheaustralsummer,boththelow-level342
baroclinicityandthetropopause-levelPVgradientsaremeridionallyconfinedtothe343
midlatitude,whereasintheaustralwinterthetwoquantitiesaremoremeridionallyspread344
andtheirmaximumvaluesdonotalignvertically.Thusduringtheaustralsummer,wecan345
expectbarocliniceddiestobegeneratedprimarilyinthemeridionallynarrowbaroclinic346
zoneinthemidlatitude.Furthermore,awell-definedjetinthezonal-meanwindinthe347
australsummerprovidescriticallinesfortheeddiesontheflanksandthushinderstheir348
radiationfromthebarocliniczone.Thusthemidlatitudeoftheaustralsummerbecomesa349
zonalwaveguideforthebarocliniceddies.Indeed,NakamuraandShimpo(2004)showthat350
thezonalcomponentoftheEliassen-Palmfluxisgreatlyenhancedduringtheaustral351
summer.Chang(2005)demonstratesthatintheaustralsummersynopticwavescantravel352
twoentireglobalzonalcircles.Consistentwiththispicture,thebottompanelsofFig.7353
showthatthepolewardeddyheatfluxintheaustralsummerissignificant(magnitude354
greaterthan10 mKs-1)onlywithinanarrowlatitudebandof 40−60! Sinthetroposphere.355
18
Whereasintheaustralwinterthesamerangeofeddyheatfluxisspreadfrom 30! S356
polewardinthelowertroposphere,withparticularlylargevaluesappearingneartheedge357
oftheAntarctica.Thestructureoftheeddyheatfluxisalsoconsistentwiththewave358
activityandheatfluxspectrainFig.1.Insummerbothspectraareconcentratedinthe359
midlatitude(Fig.1aand1c),whereasinwinterthegreatestwaveactivityisfoundat360
latitudesclosertotheAntarctica,althoughasecondaryFAWAmaximumappearsaround361
40! S(Fig.1dand1f).362
Figures7,8,and1illustratethattheseasonalchangesinthezonalmeanstateand363
thoseoftheeddystatisticsarerelated.However,causalityisnotimmediatelyevidentsince364
themeanflowandeddiesinfluenceeachother.Inthefollowing,weadoptadynamicalcore365
ofaGCMtoinvestigatetheroleofthemeanstateinshapingtheeddyforcingspectraforthe366
australsummer.367
a.DryHeld-SuarezGCMexperiment(HS94-SM)368
HereweusethedynamicalcoreoftheGeophysicalFluidDynamicsLaboratory369
atmosphericGCM.ThisisadryprimitiveequationmodelwithT85resolutionand20370
equallyspacedsigmalevels.ThesetupofthismodelresemblesthatdescribedintheHeld371
andSuarez(1994),withlinearfrictioninthebottomwithadampingrateof(1day)-1.In372
addition,surfacethermaldampingrateof(4day)-1isadopted.Tosimulateasummer373
condition,theradiativeequilibriumprofileTeq isdisplacedofftheequatorfollowingRing374
andPlumb(2007),alongwithamodificationofthestratospherictemperatureprofile375
followingtheappendixofMcGrawandBarnes(2016).WerefertothisexperimentasHS94-376
SM.377
19
Themodelisintegratedfor10000daysandthelast8000daysofdataareusedfor378
analysis.AsshowninFig.9a,HS94-SMproduceseddyheatfluxandmomentumflux379
convergencewhosespectraarebothbroadinfrequency.Unliketheaustralsummerinthe380
ERA(Fig.6,bottom),theeddyheatfluxcontributiontotheeddyPVfluxisconsistently381
smallerthantheeddymomentumfluxcontributionexceptatthelowestfrequencyatwhich382
theyarecomparable.Onemaybetemptedtoattributethisdiscrepancytothelackof383
hydrology.However,therearemultiplelevelsofconsequencestoremovingwaterfromthe384
GCM.Forexample,thedrymodelproducesasignificantlyweakerHadleycirculationthan385
themoistmodel(Schneideretal.2010)andasaresult,therealizedtime-meanzonal-mean386
statedeviatessignificantlyfromtheobservedmeanstate.Figure10depictsthemeanstate387
ofHS94-SM.Comparedtotheobservedaustralsummer(Fig.7left),thejetisabout25388
percentslowerandthetroposphericstaticstabilityissignificantlyweaker(top).Inthe389
meantime,HS94-SMalsoproducesanexcessivestaticstability(i.e.,temperatureinversion)390
nearthesurfaceassociatedwiththecoldadvectionbytheequatorwardboundaryflow391
(HeldandSchneider1999).Thestrongercontrastbetweenthetroposphericand392
stratosphericstaticstabilityleadstoanexaggeratedPVgradientintheuppertroposphere,393
whereastheoppositeverticalgradientinthelow-levelstaticstabilityproducesaspurious394
negativePVgradientinthemidlatitudelowertroposphere(middle).Further,thelow-level395
eddyheatfluxinHS94-SMisdisplacedequatorwardrelativetotheobservation(bottom).396
Giventhedeviationofthemeanstatefromobservation,onecannotstraightforwardly397
attributethediscrepancyofeddystatisticstothelackofhydrology.Toseparatetheeffects398
ofthemeanstateandofhydrologyontheeddyspectra,weneedtoconducttheexperiments399
differently.400
20
b.Towardtheaustralsummer(SHSM)401
Tosimulateaclimateusingthedrydynamicalcorewhilemaintainingtheobserved402
climatologicalmeanstate,weadoptthemethodofChang(2006).Givenatarget(observed)403
zonal-meantemperatureprofile,adiabaticheatingprofileissolvediterativelyuntilit404
balancesthetime-meaneddyheatflux.Theresultanttime-meanzonal-meantemperature405
profilecloselymatchesthetargetprofile.Byconstruction,theobtaineddiabaticheating406
profilebearsresemblancetotheobserveddiabaticheatingprofileintheaustralsummer.407
Weiteratetheheatingprofilefor40times,witheachsimulationrunfor600daysandthe408
last400daystakentocalculatetheremainingdifferencefromthetarget.Attheendofthis409
iteration,theglobalaveragezonal-meantemperaturebiasissufficientlysmall,i.e.below1K.410
Thustheeffectofmoistureonthemeanstateisincorporatedthroughheating,whereas411
eddiesaregovernedbydryprimitiveequationdynamics.Apartfromtheadjustmenttothe412
meanstate,themodelarchitectureandparametersettingareidenticaltoHS94-SM.Since413
themethodofChang(2006)onlyconstrainsthezonal-meantemperaturefield(andthe414
verticalshearinthezonal-meanzonalwindviathethermalwindbalance)butnotthe415
surfacewind,weiterateforthesurfacefrictioncoefficientinasimilarwayuntilthe416
obtainedtime-meanzonal-meansurfacezonalwindcloselymatchestheobservation.The417
resultingoptimaldampingratewasabout(1.4day)-1.WerefertothisexperimentasSHSM.418
Whenthemeanstateisadjustedtowardtheobservedaustralsummercondition,419
eddyheatfluxinthemidlatitudeexhibitsadistinctivelowfrequency(roughly20–30day)420
spectralpeak,whichwellsurpassesthepowerofeddymomentumfluxconvergence421
(Fig.9b).Althoughthespectralshapeoftheeddyheatfluxismoresimilartothatofthe422
21
observed(Fig.6abottom)thantheHM94-SM,theoverallpowerissignificantlysmaller:less423
than1/4oftheobservedvalue.Eddymomentumfluxconvergence,eventhoughits424
spectrumiscorrectlyseparatedfromtheeddyheatfluxspectrum,isdisproportionately425
smallathighfrequencies(itiscomparabletotheeddyheatflux).Thedeficienciesinthe426
eddyfluxmagnitudesintheGCMwithanadjustedmeanstatearealsonotedbyChang427
(2006),whoattributesthemtothelackofdiabaticeffectsoneddies.YamadaandPauluis428
(2015)alsodemonstratethattheEliassen-Palmfluxincreasessignificantlyinthemoist429
environment.Oneofthereviewerspointedoutthatitisalsoanaturalconsequenceofthe430
meanstatebeingnudgedtowardtheobservedprofilebyaforcing:eddiesdonotneedto431
workashardtomaintaintheobservedmeanstateandhencetheyremainatsmall432
amplitude.Inadditiontotheunderestimatedmagnitude,thespectralpeakoftheeddyheat433
fluxinFig.9bisnotsharpandthepeakfrequencyislowerthanthatofBAM(~0.02cpd).434
IntheHS94-SMandSHSMexperimentsabove,wehaveusedasurfacethermal435
dampingrateof(4day)-1,whichlinearlydecreasesto(40day)-1towardthefree436
troposphere.However,giventheanalysisin(11),astrongersurfacethermaldampingmay437
bemoreappropriateintheSouthernHemispherestormtrack.Strongsurfacethermal438
dampingwasalsocrucialfordecoupling(4)from(5).ThuswehaverepeatedtheSHSM439
experimentwithasurfacethermaldampingrateof(1day)-1.TheresultinFig.9cshowsthat440
thepeakoftheeddyheatfluxspectrumbecomessharperanditsfrequencyisclosertothat441
ofBAM(~0.03cpd).442
Figure11showsthespectraofverticallyintegratedFAWAforthetwoSHSM443
experimentsasfunctionsoffrequencyandlatitude.ThisissimilartoFig.1abutthelatitude444
22
increasesupwardhereduetotheexperimentalsymmetryabouttheequator.(Wedonot445
showtheFAWAspectraforHS94-SMbecausethespuriousnegativePVgradientsinthe446
lowertroposphereofthatmodelpreventFAWAfrombeingcalculatedaccurately.)Keepin447
mindthatbothrunshaveazonal-meanstatenearlyidenticaltotheobservedprofileforthe448
australsummer.Witha(4day)-1surfacethermaldamping,thespectralpeakofFAWA449
appearsat 58! and0.02cpdanditextendsbroadlyinthemeridionaldirectionandcovers450
theentireextratropics(Fig.11a).Itismeridionallyelongated,moresothantheobserved451
spectraofFAWA(Fig.1a).Howeverwitha(1day)-1damping,thespectralpeakbecomes452
muchmorecompact,locatedat 48! and0.03cpd(Fig.11b).Despiteanorderofmagnitude453
smallerpower,thespectraldistributioninFig.11bisqualitativelysimilartoFig.1a.454
Theaboveexperimentssuggestthattheprofileofthezonal-meanstateandthe455
surfacethermaldampingratedeterminetheshapeofthefrequencyspectraoftheeddyheat456
fluxintheaustralsummerstormtrack.Althoughthelackoflatentheatandtheforced457
adjustmenttowardtheobservedmeanprofileunderestimatethemagnitudeoftheeddy458
fluxes,thedrydynamicsqualitativelysupportsaBAM-likeoscillationintheeddyheatflux459
andFAWAaslongasthemeanstateresemblestheobservedaustralsummerconditionand460
thesurfacethermaldampingiscarefullychosen.461
4.Cospectraofeddyheatfluxandmodeinterference462
Toprovideananatomyoftheperiodicityinthemeridionaleddyheatflux,wewrite463
thedetrendedtimeseriesoftheeddymeridionalvelocity ′v andtemperature ′T atlatitude464
φ andaltitudezintermsoftheFouriermodesinlongitude λ andtimet:465
23
′v (λ,φ,z,t) = Re v̂kω (φ,z)
ω∑ ei(kλ−ωt )
k∑ ; ′T (λ,φ,z,t) = Re T̂kω (φ,z)
ω∑ ei(kλ−ωt )
k∑ , (12)466
wherethezonalwavenumberk>0andfrequency ω arediscretizedbytheperiodicityof467
longitudeandthelengthofdata;hereweassumethat ω isreal,thatis,modesareallneutral.468
v̂kω and T̂kω arethecomplexFouriercoefficients,andRedenotestherealpart.Itisreadily469
shownthat470
′v ′T (φ,z,t) =
12
Re v̂∗kωT̂k ′ω ei(ω− ′ω )t( )′ω∑
ω∑
k∑ , (13)471
wheretheasteriskdenotescomplexconjugate.Ifboth ′v and ′T involveonlyasingle472
frequency ω(k) foreachk,(13)becomes473
′v ′T =
12
Rek∑ v̂kω(k )
∗ T̂kω(k )( ) , (14)474
andhencetheeddyheatfluxdoesnotdependontime.Ifboth ′v and ′T havetwomodesfor475
eachkwithfrequencies ω1(k) and ω2 (k) ,476
′v ′T =12
Re v̂∗kω1(k )T̂kω1(k )( )k∑ +
12
Re v̂∗kω2 (k )T̂kω2 (k )( )k∑
+12
Re v̂∗kω1(k )T̂kω2 (k ) + v̂ kω2 (k )T̂kω1(k )∗( )ei(ω1(k )−ω2 (k ))t( )
k∑ .
(15)477
Inthiscasetheeddyheatfluxconsistsofsteadycomponentsassociatedwithindividual478
modesandanoscillatorycomponentarisingfromtheirinterference,withafrequency479
ω1(k)−ω2 (k) foreachk,representedbythelasttermin(15).Asmoremodesareincluded,480
thespectrumofeddyheatfluxbecomesbroader,sincetherangeof ω− ′ω in(13)increases.481
Notethatmultiplefrequenciesarerequiredforthesamewavenumbertoallowamplitude482
modulationoftheeddyheatflux.483
24
Figure12showsthreeidealizededdyspectra(forboth v̂kω and T̂kω )forafixedk484
(top)andthecorrespondingeddyheatfluxpowerspectra(bottom).Theeddyspectrahave485
twopeakseachfollowingstandardGaussiandistributionbutwithvaryingwidthsamongthe486
threecurves.Aslongasthepeaksaredistinct,thecorrespondingpeakinthepower487
spectrumofheatfluxisisolated.AstheGaussianwidthsincreaseandthetwospectral488
peaksoverlap,ared-noise-likelow-frequencyvariabilitygrowsintheheatfluxspectrum489
andeventuallymergeswiththepeak.Onemightsuspectthatthehighlypeakedspectrumin490
thelow-levelmeridionaleddyheatfluxassociatedwiththeBAMinFig.6a(bottom)may491
arisefromafrequencyinterferenceofasmallnumberofmodeswithdistinctfrequencies.492
Figure13showsthedensityofeddyheatfluxcospectraduringtheaustralsummer493
(December-March)fork=4,5,and6,analyzedfromtheECMWFERA-Interimreanalysis494
data.Theeddyheatfluxcospectrais Re(v̂kω∗ T̂kω ) ,whichquantifiesthetime-independent495
eddyheatfluxcontributedfromeachFouriermode.Thetoprowshowsthedensityof496
cospectraat850hPaasfunctionsoffrequencyandlatitude,andthebottomrowshowsthe497
samequantityat 46.5! Sasfunctionsoffrequencyandaltitude(pressure).Thetoppanels498
followthemannersofpreviousspectralanalyses[seeforexample,RandelandHeld(1991),499
Lorenz(2014),AbernatheyandWortham(2015)].Forareference,thetime-mean,zonal-500
meanzonalwindisindicatedbyabluecurveineachpanel.Inallpanels,theeddyheatflux501
cospectraaremarkedlybanded:foreachwavenumberthereareafewdistinctfrequencies502
atwhichthepoleward(negative)heatfluxmaximizes,andthesemaximaspan15-25503
degreesinlatitudeandtheentirecolumnofthetroposphere.Thispromptsustoreturnto504
thenotionofquasi-discreteFouriermodeshintedinFig.4.Figure13demonstratesthat505
25
thesemodestransportheatandindeedmakeupalargefractionofthetime-independent506
partofthemeridionaleddyheatflux.Furthermore,sincethereismorethanonedominant507
modeperzonalwavenumber,amplitudeoscillationmayariseineddyheatfluxasaresultof508
theirinterference.Itisworthpointingoutthatthefrequenciesofthespectralpeaksofthe509
heatfluxcospectrumintheobservation(topmiddlepanelofFig.13)aretheonesusedinthe510
idealizedspectrainFigure12.Noticethat,apartfromthemodesfork=6atlowlevels,the511
extentofeddyheatfluxcospectraisboundedbythefrequency(angularphasespeed)ofthe512
zonal-meanzonalwindindicatedbythebluecurves.Thusthemodesaretrappedinsidethe513
criticallines,corroboratingthewaveguidepictureintroducedearlier.Wehavealso514
subdividedtheentiredatalengthintoseveralsegmentsandrepeatedtheanalysisand515
confirmedthattheresultsarereproducibleforeachsegment,suggestingthatthesemodes516
arealwayspresentwithineachseason.517
UsingthesameformatasthebottomofFig.13,weplotinFig.14theeddyheatflux518
cospectraoftheGCMsimulationsperformedinsection3.The top row shows the eddy heat 519
flux cospectra at 46.9S for HS94-SM and the bottom row shows the same quantity at the same 520
latitude for SHSM with (1 day)-1 surface thermal damping. Curiously, the HS94-SM simulation, 521
which does not have a clear single spectral peak (BAM), displays far more bands in cospectra. 522
However, its bands are rather disorganized and the resultant interference does not produce a 523
single spectral peak. In fact, there are three comparable peaks in the low-frequency range of eddy 524
heat flux spectrum (Fig.9a). On the other hand, SHSM-1day, which shows a BAM-like spectral 525
peak in eddy heat flux (Fig.9c) and FAWA (Fig.10b), the number of bands are significantly 526
reduced. For k = 6 and 7 the cospectra are still banded but they are much more compact than the 527
corresponding cospectra of HS94-SM. For each wavenumber there is a highly concentrated main 528
26
band that is flanked by a few sidebands, except for k = 5 in which sidebands are weak and the 529
spectrum is dominated by a single fat band. For each zonal wavenumber the precise distribution 530
of cospectra is quite different from observation (Fig.13), but a main band flanked by a few 531
sidebands appears to be a common pattern, which is presumably efficient at producing a single 532
spectral peak in the eddy heat flux. 533
Figure15depictsthereconstructedspatialpatternofgeopotentialanomalyfork=5at534
twofrequencies(0.11and0.15cpd)thatcorrespondtotheobservedamplitudemaximain535
Fig.13.Theycorrespondto~10and~14ms-1intermsoftheequatorialzonalphasespeed536
(seetheappendixBforthedetailsofthismodeextraction).Figure15ashowsthevertical537
structureat 46.5! S.Thetwomodeshavesimilarverticalstructuresandamplitudes.They538
showwestwardtiltswithincreasingaltitude,similartoabaroclinicwavethatsustains539
polewardheatflux.Figure15bshowsthecorrespondinghorizontalstructuresofthemodes540
at250hPa.Bothmodesexhibitaweakdownsheartilt,suggestiveofaneddymomentum541
fluxintothejet,butthefastertravellingmodehasitspeakamplitudeslightlydisplaced542
polewardrelativetotheslowerone.Thisisconsistentwiththefactthatthecriticalline543
(indicatedbythewhiteline)ofthefastermodeisdisplacedpoleward(closertothejetaxis)544
relativetotheslowermode.Thepolewardshiftofthemodestructurewithincreasing545
frequencyisalsovisibleinthetoppanelsofFig.13.Asthesemodestravelatdifferentphase546
speeds,theirrepeatedconstructiveanddestructiveinterferencescreateamplitude547
vacillation.Thiscausesbotheddyheatfluxandeddymomentumfluxtovary,buttheeffect548
ofthelatteronthetotaleddymomentumfluxissmall.The eddy momentum flux arises 549
largely from the meridional tilting of phase lines of eddies, often during the decaying stage of 550
baroclinic lifecycles in which eddies are sheared out by the background shear. This means that it 551
27
takes superposition of many modes to create eddy momentum flux, since continuous shearing 552
cannot be described by a handful of discrete modes. Thus the eddy momentum flux tends to 553
spread over a broad frequency range, as demonstrated in Fig.6 as well as many previous studies 554
(Lorenz and Hartmann 2001, Thompson and Woodworth 2014). Theinterferencefrequencyis555
thedifferenceinthemodefrequencies(~0.04cpd),comparabletothefrequencyoftheBAM.556
IndeedthefrequencyseparationoftheneighboringmaximainFig.13isquiteevenand557
comparabletotheBAMfrequencyforallzonalwavenumbers.Thisreinforcesthestrong558
modulationoftheeddyheatfluxaroundthesinglespectralpeak.559
Figure16issimilartoFig.13butfortheaustralwinter(June-September).Here560
wavenumbers3-5areshown,astheeddyspectrashifttolongerwavesduringwinter(see561
Fig.4).Thelargestcontributiontotheeddyheatfluxisfromwavenumber3athighlatitudes562
andlowfrequencies.Thereisabandedstructure,butcomparedtothesummer,peaksvary563
significantlyinbothmagnitudeandtheirspacing.Thisisconsistentwiththelackof564
organizedpeakinthespectrumofeddyheatfluxintheextratropics(Fig.6b,bottom). 565
AlthoughtheFouriermodesareapurelymathematicalconstruct,theirresemblanceto566
unstablebaroclinicwaves(Charney1947;Eady1949)inFig.15promptsustoconsider567
waveactivitybudgetofeachFouriermode.Sinceeachmodeisneutral,gainandlossof568
waveactivitymustbeexactlyandcontinuouslybalanced.Forsmall-amplitudeeddieswave569
activitymaybepartitionedintocontributionsfromindividualmodes,sincelinearmodesare570
mutuallyorthogonalinthesenseofwaveactivity(Held1985).Inthatcase,(10)appliesto571
eachindividualmode(withouttimeaveraging,sinceallmodesareneutral)providedthat572
dampingisalsoalinearfunctionofwaveactivity.FormodesinFig.15,baroclinic573
28
conversionofwaveactivity(eddyheatflux)isexactlycompensatedbydamping.[Atlarge574
amplitudetheexchangeofFAWAbetweenmodesbecomessignificantduetoeddy-eddy575
interactions,so(10)appliesonlytothetotalfield.]Criticallinesdonotposemathematical576
singularitytothesemodesbecauseofdamping:themeridionaladvectionofeddyPV577
associatedwiththemodeisbalancedbydampingatthecriticalline.578
Giventhatbarocliniceddiesgrowanddecayconstantlyinthestormtrackregion(Lee579
andHeld1993,ChangandOrlanski1993),itmightappearcounterintuitivethatsuchan580
unsteadyflowmaybeexpressedassuperpositionofneutralmodes.Thereaderisreminded581
thattheFouriermodesarejustawayofdecomposingfinite-lengthdataatagivenlatitude582
andheightaccordingtothezonalwavenumberand(real)frequency.Theyarebydefinition583
neutralanddonotcreatetransientbehaviorineddyamplitudeormodifythemeanflowby584
themselves.Onlythesuperposition(interference)ofmultiplemodesgivesrisetothe585
spatiotemporalvariationineddyamplitudeanditsinteractionwiththemeanflow.The586
Fouriermodesmayormaynotbeassociatedwiththeeigenmodesofthelinearized587
dynamics.WesimplypointoutthatdecomposingdataintermsofdiscreteFouriermodes588
providesaconvenientinterpretationoftheBAMthroughinterferenceiftheeddyconsistsof589
asmallnumberofdiscretemodes.Todrivetheabovepointhome,supposealeading590
eigenmodewithzonalwavenumber k andfrequency ω isundergoing(nonlinear)591
amplitudemodulationatagivenlatitudeandheight:592
(1+ε cosω0t)cos(kλ −ωt), ω0 << ω , 0 < ε <1. (16)593
Since(16)canbealsowrittenas594
29
cos(kλ−ωt)+0.5ε cos(kλ−(ω−ω0 )t)+cos(kλ−(ω+ω0 )t)⎡⎣
⎤⎦ , (17)595
theFouriertransformof(16)willinevitablypickupthreediscretefrequencies ω and596
ω±ω0 ,whetherornot cos(kλ−(ω±ω0 )t) areeigenmodes.Tobesure,interferenceof597
discreteneutralmodeshasbeeninvokedpreviouslyasatheoryforstructuralvacillationof598
large-scaleeddies(Lindzenetal.1982;RotunnoandFantini1989).Howeverinthese599
studiesthediscretenessarisesnaturallyfromthefinitesizeofthedomainandisassociated600
withdifferentstructuresofeigenmodes[e.g.,symmetricandantisymmetricHoughmodes601
(Lindzenetal.1982)andEadyedgewavesimpingingonthetwohorizontalboundaries602
(RotunnoandFantini1989)].Inthepresentstudyitisnotentirelyclearhowdiscrete603
modesintheaustralsummerarise:thefinitewidthofthebarocliniczonemaysupporta604
normalmode-likebehaviorineddies(particularlythe850hPaeddyheatfluxinthetop605
panelsofFig.13iswellcontainedbetweenthecriticallines),butthemodesinFig.15have606
verysimilarstructures.Itisencouragingthatthequasi-discreteFouriermodescanbe607
reproducedatleastqualitativelyusingadrydynamicalcoreofGCM.Ifwecanexplainthe608
emergenceofafewquasi-discreteFouriermodesperzonalwavenumberintheaustral609
summer,thentheinterferenceideamayprovideasuccinctunderstandingoftheBAM.610
Otherwise,adirectattemptatunderstandingthefrequencyandamplitudeofmodulation611
ω0 , ε withoutregardtothemodespectramightprovemorefruitful.Whetherthenonlinear612
oscillatorideas(TB14,AmbaumandNovak2014)eventuallysucceedatthisseemstohinge613
onwhetherthedynamicalfeedbackbetweenthemeridionaleddyheatfluxandlow-level614
baroclinicitymaybefirmlyestablished.615
616
30
5.Summary617
BAMdefinedastheleadingEOFofEKEanomalyconstitutesasignificantcomponent618
ofthelow-frequencyvariabilityofeddyamplitudeintheextratropicaltroposphere(TW14,619
TB14,TL14).Its20-30dayperiodicityineddyamplitudeandtheassociatedmeridional620
eddyheatfluxisparticularlyrobustandvisibleeveninrawdataduringtheaustralsummer621
midlatitude.Thismaybeexploitedtoimprovethepredictabilityofweatherbeyondthe622
typical2-weeklimit(TB14).Wehaveexaminedthedynamicsofthisintra-seasonal623
variabilitybyapplyingthespectralanalysistotheverticallyaveragedzonalmomentum-624
FAWAcycle(WN15,NZ10,summarizedinFig.5)intheSouthernHemisphere.625
TherobustBAMsignalintheaustralsummer(45percentofeddyvariance626
explained)isaccompaniedbyconcentratedspectralpeaksintheverticallyaveragedFAWA627
andthelow-levelmeridionaleddyheatfluxinthefrequency-latitudeplane(Figs.1aand1c),628
wherethelatterbeingtheprimarydriveroftheformer[Eqn.(4)andFig.6a,bottom].In629
winterwhenBAMaccountsforonly30percentofvariance,thespectraofrawFAWAand630
eddyheatfluxarenotwellcorrelated(Figs.1dand1f).Stillthroughoutayearwefound631
highcorrelation(r=.72)betweenthevolumeintegralofFAWAandtheBAMindex(Fig.2).632
Ontheotherhand,littlesignatureofBAMisfoundinthespectraoftherawlow-level633
zonal-meanbaroclinicity,itstendencyandeddyforcing(Fig.3).Thiscastsareasonable634
doubtontherelevanceofthenonlinearoscillatormodel(TB14,AmbaumandNovak2014)635
inwhichthefeedbackbetweentheeddyheatfluxandbaroclinicityplaysakeyrole.We636
proposedanalternativeinterpretationofBAMbasedontheinterferenceofneutralFourier637
modeswiththesamezonalwavenumberbutwithdifferentfrequencies,basedonthe638
31
spectralanalysisofthelow-leveleddyheatflux(Figs.4,13).639
Wethenhypothesizedthatthezonal-meanstatehasastrongcontrolonthespectral640
propertiesofeddiestoinfluencethenatureofBAM:intheaustralsummerthephase641
propagationofthemodesispredominantlyalongthezonalwaveguide,setupbythenarrow642
uppertroposphericPVgradient,jetstream,andthelow-levelbaroclinicityallconcentrated643
inanarrowlatitudeband(Fig.7;Lee2014;NakamuraandShimpo2004).Thefocusingof644
wavesinthedirectionoflongitudewouldenhancetheinterferenceofmodesalongthe645
latitudecircle,makingBAMmorevisible.646
Totestthisidea,adrydynamicalcoreofGCMwasused.Wehavenudgedthezonal-647
meanstateoftheatmospheretowardtheobservedprofilewhileallowingeddytoevolve648
quasi-adiabatically.Keepingthemeanstatetotheobservedprofilewhilemaintaininga649
strongsurfacethermaldampinghelpedtoimprovetheshapeoftheeddyfluxspectra,650
althoughtheoverallpowerwastooweak.Thisisdueinparttothelackofdiabaticforcing651
oneddyandinparttothefactthattheforcingonthemeanstatereducestheneedfor652
additionalforcingfromeddytomaintaintheobservedstate.Furthermore,despitethe653
reasonablespectralshapeoftheeddyheatflux,intheforcedexperimentthepreciseform654
cospectradifferssignificantlyfromobservation.Stillthebandedstructureinthecospectra655
appearstoberesponsiblefortheBAM-likelowfrequencyperiodicity.656
Asitstands,althoughtheinterferenceoftheFouriermodesisrecognizedin657
observeddataanditprovidesanalternatedescriptionofthe20-30dayperiodicityinthe658
eddyamplitudeduringtheaustralsummer,theoriginoftheobserveddiscreteFourier659
modesremainstobeidentified.Thiswillbeasubjectoffuturestudies.660
32
Acknowledgments.661
WethankDaveThompsonforsuggestingtheBAMcalculationsfordifferentseasonsandfor662
carefullyreviewingthemanuscript,ananonymousreviewerforavaluablecritique,which663
greatlyimprovedthepresentationofthematerials,andMalteJansenforhelpfuldiscussions.664
ThisresearchhasbeensupportedbyNSFgrantAGS-1151790.TheERA-Interimdataset665
usedinthisstudywasobtainedfromtheECMWFdataserverat666
http://apps.ecmwf.int/datasets/data/interim-full-daily/withthehorizontalresolutionof667
1.5! .668
669
APPENDIXA670
BAMastheleadingEOFofEKEanomaly671
AlthoughtheemphasisofthepresentstudyistheapplicationoftheFAWAbudgettoBAMin672
theaustralsummer,asastartingpointwereproducethepreviousworkbycalculatingBAM673
withthemethodoriginallyintroducedbyThompsonandthecollaborators(TB14,TW14,674
TL14).Herewedefine‘summermonths’tobeDecember-Marchand‘wintermonths’tobe675
June-September.Bymakingthedata4monthslong,weincreasespectralresolutioninthe676
lowfrequencyrangewhereBAMappears.Thechoiceofthemonthsisalsoguidedbythe677
observationthatthereisawell-known,seasonaljumpinthelocationofthejetsinthe678
SouthernHemispherearoundApril-MayandSeptember-October.Thesummermonthsare679
33
withintheregimeofa‘mergedjet’inthemidlatitude,whereasthewintermonthsarewithin680
theregimeof‘thesubtropicaljet.’681
ForeachseasonwecomputedthefirstEOFofthezonal-meanEKEanomalyforthe682
SouthernHemispherefromtheERA-Interimreanalysis,followingthemethodofTB14.This683
isshowninthetoprowofFig.17(arbitraryunit).InbothseasonstheEOFhasitsamplitude684
concentratedinthemidlatitudeoftheuppertroposphere,butinwinteritsmeridional685
extentexpandssignificantlyintothesubtropics.Itturnsoutthatinsummer,thefirstEOF686
explainsnearly45percentofthetotalvariance,whereasinwinteritexplainsonly30687
percentofthevariance.688
ThebottomofFig.17showsthespectraofexpansioncoefficientoftheEOF(the‘BAM689
index’)forthecorrespondingseasons.Insummer,thespectrumhasadistinctivepeak690
around~0.04cpd,clearlydefiningtheBAMfrequency.Inwinterthespectralpeakshiftsto691
lowerfrequencyanditspeakwidens(0.02-0.03cpd).Infact,aroundtheleftshoulderofthe692
peakthespectralresolutionbeginstodeteriorate.Therefore,theBAMtimescaleislesswell693
separatedfromtheseasonaltimescaleinwinter,andthespectrumisclosertobecoming694
red.695
696
APPENDIXB697
AcompositeapproachtoextractFouriermodes698
34
TheFouriermodestructureisextractedastheverticalcrosssectionofgeopotential699
anomalyat46.5Sandasthehorizontalcrosssectionofgeopotentialanomalyat250hPa700
usingtheERA-Interimreanalysisproduct(1979-2014).Foreachyear,afterremovingthe701
zonalmeanandseasonalmean,weFouriertransformthegeopotentialanomalyfieldateach702
latitudeandheightinto (k,ω) -spectralspace.Subsequently,weretaintheFouriermodesfor703
(k,ω) = (5, 0.1111cpd) and (5, 0.1556cpd) .AsshowninFig.10,thesetwofrequenciesdefine704
thedominantpeaksinthezonalwavenumber5eddyfrequencyheatfluxcospectrainthe705
SouthernHemispheresummerat46.5S.Then,forthetwofrequenciesandforeachyear,706
wereconstructtheentireverticalandhorizontalmodestructuresinphysicalspace.707
Sincethemodestructuresdonotshareidenticalphaseinformationfromyeartoyear,708
theycannotbeaveragedamongmultipleyearsdirectly.Forthehorizontalmodestructure,709
wefurtherconductacompositeanalysisusing46.5Sasareferencelatitude.Thezonal710
phaseofthemodeforeachyearisshiftedsuchthatthelongitudesofthepeakamplitude711
alignforallyearsatthereferencelatitude.Finallywecomputethemultiyearcompositeby712
averaging.Similarly,weuse250hPaasareferencepressureleveltocompositethemulti-713
yearverticalmodestructure.714
715
35
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836
42
ListofFigures837
FIG.1.Spectraldensitiesofeddypropertiesasfunctionsoffrequencyandlatitude.Toprow838
[(a),(d)]:totalwaveactivity
A∗ + B∗ .SeeWN15andsection2forthedefinitionofwave839
activity.Contourintervalis4m2.Secondrow[(b),(e)]:eddykineticenergy ( ′u2 + ′v 2 ) / 2 at840
300hPa.Hereuandvarethezonalandmeridionalcomponentsofthehorizontalwind;841
overbarandprimedenotethezonalmeanandthedeparturefromit,respectively.Contour842
intervalis500m4s-2.Bottomrow[(c),(f)]:meridionaleddyheatflux ′v ′θ at850hPa.Here843
θ ispotentialtemperature.Contourintervalis4K2m2.Leftcolumn[(a)-(c)]:December-844
March.Rightcolumn[(d)-(f)]:June-September.BasedontheERA-Interimreanalysis845
1.5!×1.5! griddeddata(Deeetal.2011)for1979-2014.846
847
FIG.2.(a)Anomaly(departurefromtheannualmean)oftheverticallyaveragedwave848
activity
A∗ + B∗ asafunctionoftimeandlatitudefortheyear2012.(b)Sameas(a)but849
forthe300hPazonal-meanEKE.(c)Sameas(b)butforthepartoftheEKEanomaly850
‘explained’bythefirstEOF.(d)TheBAMindex(regressiononthefirstEOFofthezonal-851
mean300hPaEKEanomaly,blue),andthevolumeintegralofwaveactivitybetween 20!S 852
and 70!S (red).Bothcurvesarenormalizedtotheunitvariance.BasedontheERA-853
Interimreanalysis(1979-2014).854
855
43
FIG.3.Spectraldensitiesofbaroclinicitypropertiesat850hPafortheSouthernHemisphere856
summer(December-March).(a) θ (55.5!S)−θ (40.5!S) .(b)Sameas(a)butforthetendency857
of θ (55.5!S)−θ (40.5!S) .(c)Sameas(b)butforthedifferencein (acosφ)−1∂(cosφ ′v ′θ ) ∂φ 858
between 55.5! Sand 40.5! S,whereaistheEarthradiusand φ islatitude.Basedonthe859
1979-2014ERA-Interimreanalysis.860
861
FIG.4.Spectraldensitiesofthe250hPageopotentialat 46.5! Sasfunctionsoffrequencyand862
zonalwavenumber.Apositive(negative)frequencymeansthattheeddiesarepropagating863
eastward(westward).(a)December-February.(b)June-August. ±0.011cycleperday864
(CPD)isthelowestfrequencyresolvedbythedata.BasedontheERA-Interimreanalysis865
1979-2014.Valuesareinterpolatedfornon-integerzonalwavenumberstoaidvisualization.866
Contourintervalis 1.5×103m6s−2 .867
868
FIG.5Zonalmomentum-waveactivitycycleasexpressedinEqns.(3)-(5).Thefatgray869
arrowsindicateeddyfluxeswhereasthethinblackarrowsindicatesourcesandsinks.The870
directionsofthearrowsarerepresentativeofthelatitudesforabaroclinicallyunstable,871
eddy-drivenjet.Seetextfordetails.872
873
FIG.6Spectraldensitiesoftheeddyfluxesat 46.5! S.Top:convergenceofeddymomentum874
fluxdensity (!Hacos2φ)−1∂(e−z/ H ′u ′v cos2φ) / ∂φ ,where a istheradiusoftheEarth, φ is875
44
latitude, H = 7km and !H ≡ e−nΔz/ H
n=1
n=49
∑ Δz = 6.5km (nindicatestheverticallevelsofanalysis876
and Δz =1km .)Contourinterval: 6.0×10−19s−2 .Middle:scalededdyheatfluxdensity877
( f !H−1)(e−z/ H ′v ′θ )(d !θ / dz)−1 ,wheretheCoriolisparameterisevaluatedat 46.5! S.Contour878
interval: 2.0×10−11m2s−2 .Bottom:Redcurve:theeddyheatfluxcontributiontothe879
barotropiccomponentoftheeddyPVflux.Bluecurve:theeddyvorticityfluxcontribution880
tothebarotropiccomponentoftheeddyPVflux.Theverticalintegralofthetoppanels881
correspondstothebluecurvesinthebottompanels,whereasthevaluesofthemiddle882
panelsatz=0.5kmcorrespondtotheredcurves.Left:December-March.Right:June-883
September.Therangeoffrequencyshownis0.0167-0.5cpd.Seetextfordetails.884
885
FIG.7.Zonal-meanclimatologyfortheDecember-March(left)andJune-September(right).886
Top: u{ } (blackcontours,contourinterval:5ms-1)and θ{ } (shading,contourinterval:10887
K).Middle:meridionalgradientsofquasigeostrophicPV(contourinterval:2.0×10-11m-1s-1).888
Bottom:eddyheatflux ′v ′T{ } (contourinterval:2.5mKs-1).BasedontheERA-Interim889
reanalysis1979-2014.890
891
FIG.8.Top:Seasonalcycleofthemeridionalgradientsofthezonal-meanquasigeostrophic892
PVatthe10kmpressurepseudoheight(contourinterval:4.0×10-11m-1s-1).Bottom:893
Seasonalcycleofthemeridionalgradientsofthezonal-meanpotentialtemperaturegradient894
at850hPa(contourinterval:5×10-7Km-1).Averageof1979-2014basedontheERA-895
45
Interimreanalysis,withaweaktimefilteringtosuppressexcessivenoise.Thewhitelines896
indicate 46.5! S.897
898
FIG.9.Spectraofeddyfluxesat 46.5! S.Red:scalededdyheatflux.Blue:eddymomentum899
fluxconvergence.Magenta:totaleddyforcing(verticallyintegratededdyPVflux).900
ConventionsarethesameasthebottomofFig.4.(a):HS94-SM.(b):SHSMwith4day901
thermaldamping.(c):SHSMwith1daythermaldamping.Seetextfordetails.902
903
FIG.10.Zonal-meanclimatologyforHS94-SM.TheconventionisidenticaltoFig.7.The904
regionbelowz=1kmismaskedduetotheinterpolationfromsigmacoordinatetopseudo-905
heightcoordinate.906
907
FIG.11.Spectraldensityoftheverticallyintegratedwaveactivity
A∗ + B∗ asfunctionsof908
frequencyandlatitude.(Note:latitudeincreasesupward.)(a)SHSM-4day.(b)SHSM-1day.909
Contourintervalis1m2.Seetextfordetails.910
911
FIG.12.Idealizededdyheatfluxcospectra(top)andtheeddyheatfluxpowerspectra912
(bottom).Seetextfordetails.913
914
46
FIG.13.Top:eddyheatfluxcospectra Re(v̂∗kωT̂kω ) at850hPaforDecember-Marchasa915
functionoffrequencyandlatitude.Left:zonalwavenumber4.Center:zonalwavenumber5.916
Right:zonalwavenumber6.Bottom:thecorrespondingeddyheatfluxcospectraat 46.5! S917
asafunctionoffrequencyandpressure.Thislatitudeisindicatedbytheblacklineinthe918
toppanels.Inallpanels,thebluecurveindicatesthetime-meanzonal-meanzonalwindin919
termsofangularfrequency.Contourinterval: 0.008mKs−1 .BasedonERA-Interim1979-920
2014.921
922
FIG.14.Top:eddyheatfluxcospectrafordifferentzonalwavenumbersat 46.9! SforHS94-923
SM,asafunctionoffrequencyandpressure.Bottom:SameastopexceptforSHSM-1day.924
Panelsfromlefttoright:zonalwavenumber5,6,and7.Conventionsarethesameasthe925
bottomofFig.13.926
927
FIG.15.StructureoftwoleadingFouriermodes(zonalwavenumber:5)duringtheaustral928
summer.Top:geopotentialanomalyat 46.5! Sasafunctionoflongitudeandpressure.929
Bottom:geopotentialanomalyat250hPaasafunctionoflongitudeandlatitude.Left930
column:zonalphasespeed=10ms-1.Rightcolumn:zonalphasespeed=14ms-1.Contour931
intervals: 20m2s−2 .Whitelinesinthebottompanelsindicatecriticallines.Basedonthe932
ERA-Interimreanalysis(1979-2014).SeeappendixBforthecalculationmethodforthe933
modestructure.934
935
47
FIG.16.SameasFig.13butforJune-Septemberandzonalwavenumbers3,4,and5.936
937
FIG.17.Toprow:LeadingEOFofzonallyaveragedEKEanomaly.Bottomrow:Spectral938
densityoftheBAMindex(expansioncoefficientoftheleadingEOF)intheunitofm4s-2.Left939
column:December-March.Rightcolumn:June-September.Basedonthemethodoutlined940
inTB14withthe1979-2014ERAInterimreanalysis.NotethatinDecember-March,the941
leadingEOFexplains~45percentofvariance,whereasinJune-Septemberitexplainsonly942
30percent. 943
48
944
FIG.1.Spectraldensitiesofeddypropertiesasfunctionsoffrequencyandlatitude.Toprow945[(a),(d)]:totalwaveactivity
A∗ + B∗ .SeeWN15andsection2forthedefinitionofwave946
activity.Contourintervalis4m2.Secondrow[(b),(e)]:eddykineticenergy ( ′u2 + ′v 2 ) / 2 at947
300hPa.Hereuandvarethezonalandmeridionalcomponentsofthehorizontalwind;948overbarandprimedenotethezonalmeanandthedeparturefromit,respectively.Contour949intervalis500m4s-2.Bottomrow[(c),(f)]:meridionaleddyheatflux ′v ′θ at850hPa.Here950 θ ispotentialtemperature.Contourintervalis4K2m2.Leftcolumn[(a)-(c)]:December-951March.Rightcolumn[(d)-(f)]:June-September.BasedontheERA-Interimreanalysis952
1.5!×1.5! griddeddata(Deeetal.2011)for1979-2014. 953
latit
ude
EKE (summer)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
5000
10000EKE (winter)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
5000
10000
frequency (CPD)
latit
ude
Heat Flux (summer)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
20
40
60
80
frequency (CPD)
Heat Flux (winter)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
20
40
60
80
latit
ude
FAWA (summer)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
20
40
60
80
FAWA (winter)
0.05 0.1 0.15 0.2−70
−60
−50
−40
−30
−20
0
20
40
60
80
(a) (d)
(b) (e)
(c) (f)
Sunday, August 14, 16
49
954
FIG.2.(a)Anomaly(departurefromtheannualmean)oftheverticallyaveragedwave955activity
A∗ + B∗ asafunctionoftimeandlatitudefortheyear2012.Unit:ms-1.(b)Same956
as(a)butforthe300hPazonal-meanEKE.Unit:m2s-2.(c)Sameas(b)butforthepartof957theEKEanomaly‘explained’bythefirstEOF.Unit:m2s-2.(d)TheBAMindex(regression958onthefirstEOFofthezonal-mean300hPaEKEanomaly,blue),andthevolumeintegralof959waveactivitybetween 20!S and 70!S (red).Bothcurvesarenormalizedtotheunit960variance.BasedontheERA-Interimreanalysis(1979-2014). 961
latit
ude
Vertically−averaged FAWA during year 2012
50 100 150 200 250 300 350−70
−60
−50
−40
−30
−20
−5
0
5
latit
ude
Zonal−mean EKE at 300 hPa during year 2012
50 100 150 200 250 300 350−70
−60
−50
−40
−30
−20
−200
−100
0
100
200
latit
ude
BAM explained zonal−mean EKE at 300 hPa during year 2012
50 100 150 200 250 300 350−70
−60
−50
−40
−30
−20
−200
−100
0
100
200
50 100 150 200 250 300 350
−2
0
2
Total FAWA Index and BAM Index (corr=0.72)
norm
alize
d am
plitu
de
time (day)
20
40
60
BAM−index (based on EOF)FAWA−index (based on eq.(7)(8))
(a)
(b)
(c)
(d)
Sunday, August 14, 16
50
962
FIG.3.Spectraldensitiesofbaroclinicitypropertiesat850hPafortheSouthernHemisphere.963Toprow[(a),(c)]: θ (55.5!S)−θ (40.5!S) .Bottomrow[(b),(d)]:thetendencyof964
θ (55.5!S)−θ (40.5!S) (black)andthedifferencein (acosφ)−1∂(cosφ ′v ′θ ) ∂φ between 55.5!965Sand 40.5! S(red),whereaistheEarth’sradiusand φ islatitude.Leftcolumn[(a),(b)]:966December-March.Rightcolumn:[(c),(d)].Basedonthe1979-2014ERA-Interimreanalysis.967
968
969
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3(a) summer
K2 S2
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2x 10−10 (c) summer
frequency (CPD)
K2
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3(b) winter
K2 S2
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2x 10−10 (d) winter
frequency (CPD)
K2
(a)
(b)
(c)
(d)
Monday, August 15, 16
51
(a) Summer (b) Winter 970
971
FIG.4.Spectraldensitiesofthe250hPageopotentialat 46.5! Sasfunctionsoffrequencyand972zonalwavenumber.Apositive(negative)frequencymeansthattheeddiesarepropagating973eastward(westward).(a)December-February.(b)June-August. ±0.011cycleperday974(CPD)isthelowestfrequencyresolvedbythedata.BasedontheERA-Interimreanalysis9751979-2014.Valuesareinterpolatedfornon-integerzonalwavenumberstoaidvisualization.976Contourintervalis 1.5×103m6s−2 . 977
frequency (CPD)
zona
l wav
enum
ber
250hPa Geopotential SH summer at 46.5 S
−0.1 0 0.1 0.2 0.30
1
2
3
4
5
6
7
0
0.5
1
1.5
2
2.5
3x 104
frequency (CPD)
zona
l wav
enum
ber
250hPa Geopotential SH winter at 46.5 S
−0.1 0 0.1 0.2 0.30
1
2
3
4
5
6
7
0
0.5
1
1.5
2
2.5
3x 104
52
978
FIG.5Zonalmomentum-waveactivitycycleasexpressedinEqns.(3)-(5).Thefatgray979arrowsindicateeddyfluxeswhereasthethinblackarrowsindicatesourcesandsinks.The980directionsofthearrowsarerepresentativeofthelatitudesforabaroclinicallyunstable,981eddy-drivenjet.Seetextfordetails. 982
u
A∗ ≥ 0 B∗ ≤ 0
−∂∂y
′u ′v
f0H
′v ′θd !θ / dz
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟z=0
surface frictiongravity wave drag, etc.
diabatic heating
enstrophy dissipation (mixing)radiative and Ekman damping
surface thermal dampingform stress (mountain torque)
53
(a) Summer (b) Winter 983
984
FIG.6Spectraldensitiesoftheeddyfluxesat 46.5! S.Top:convergenceofeddymomentum985fluxdensity (
!Hacos2φ)−1∂(e−z/ H ′u ′v cos2φ) / ∂φ ,where a istheradiusoftheEarth, φ is986latitude, H = 7km and
!H ≡ e−nΔz/ H
n=1
n=49
∑ Δz = 6.5km (nindicatestheverticallevelsofanalysis987and Δz =1km .)Contourinterval: 6.0×10−19s−2 .Middle:scalededdyheatfluxdensity988
( f !H−1)(e−z/ H ′v ′θ )(d !θ / dz)−1 ,wheretheCoriolisparameterisevaluatedat 46.5! S.Contour989interval: 2.0×10−11m2s−2 .Bottom:Redcurve:theeddyheatfluxcontributiontothe990barotropiccomponentoftheeddyPVflux.Bluecurve:theeddyvorticityfluxcontribution991tothebarotropiccomponentoftheeddyPVflux.Theverticalintegralofthetoppanels992correspondstothebluecurvesinthebottompanels,whereasthevaluesofthemiddle993panelsatz=0.5kmcorrespondtotheredcurves.Left:December-March.Right:June-994September.Therangeoffrequencyshownis0.0167-0.5cpd.Seetextfordetails. 995
eddy momentum flux convergence
heig
ht (k
m)
0.1 0.2 0.3 0.4 0.50
5
10
0
0.5
1x 10−17
scaled eddy heat flux
heig
ht (k
m)
0.1 0.2 0.3 0.4 0.50
5
10
0
2
4x 10−10
0.1 0.2 0.3 0.4 0.50
2
4
6
8x 10−10 barotropic component
m2 /s
2
frequency (CPD)
eddy momentum flux convergence
heig
ht (k
m)
0.1 0.2 0.3 0.4 0.50
5
10
0
0.5
1x 10−17
scaled eddy heat flux
heig
ht (k
m)
0.1 0.2 0.3 0.4 0.50
5
10
0
2
4x 10−10
0.1 0.2 0.3 0.4 0.50
2
4
6
8x 10−10 barotropic component
m2 /s
2
frequency (CPD)
54
996
FIG.7.Zonal-meanclimatologyfortheDecember-March(left)andJune-September(right).997Top: {u} (blackcontours,contourinterval:5ms-1)and {θ} (shading,contourinterval:10K).998Middle:meridionalgradientsofquasigeostrophicPV(contourinterval:2.0×10-11m-1s-1).999Bottom:eddyheatflux { ′v ′T }(contourinterval:2.5mKs-1).BasedontheERA-Interim1000reanalysis1979-2014. 1001
55
1002
FIG.8.Top:Seasonalcycleofthemeridionalgradientsofthezonal-meanquasigeostrophic1003PVatthe10kmpressurepseudoheight(contourinterval:4.0×10-11m-1s-1).Bottom:1004Seasonalcycleofthemeridionalgradientsofthezonal-meanpotentialtemperaturegradient1005at850hPa(contourinterval:5×10-7Km-1).Averageof1979-2014basedontheERA-1006Interimreanalysis,withaweaktimefilteringtosuppressexcessivenoise.Thewhitelines1007indicate 46.5! S. 1008
56
(a) HS94-SM 1009
1010
(b) SHSM-4 day 1011
1012
(c) SHSM-1 day 1013
1014
FIG.9.Spectraofeddyfluxesat 46.5! S.Red:scalededdyheatflux.Blue:eddymomentum1015fluxconvergence.ConventionsarethesameasthebottomofFig.6.(a):HS94-SM.(b):1016SHSMwith(4day)-1surfacethermaldamping.(c):SHSMwith(1day)-1surfacethermal1017damping.Seetextfordetails.1018 1019
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
0.5
1
1.5
2
2.5
3
x 10−10
frequency (CPD)
m2 /s
2
eddy forcing from momentum flux eddy forcing from heat flux
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
1
2
3
4
5
6
7
8x 10−11
frequency (CPD)
m2 /s
2
eddy forcing from momentum flux eddy forcing from heat flux
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
1
2
3
4
5
6
7
8x 10−11
frequency (CPD)
m2 /s
2
eddy forcing from momentum flux eddy forcing from heat flux
57
1020
1021
FIG.10.Zonal-meanclimatologyforHS94-SM.TheconventionisidenticaltoFig.7.The1022regionbelowz=1kmismaskedduetotheinterpolationfromsigmacoordinatetopseudo-1023heightcoordinate.1024
1025
58
1026
FIG.11.Spectraldensityoftheverticallyintegratedwaveactivity
A∗ + B∗ asfunctionsof1027frequencyandlatitude.(Note:latitudeincreasesupward.)(a)SHSM-4day.(b)SHSM-1day.1028Contourintervalis1m2.Seetextfordetails. 1029
frequency (CPD)
latit
ude
U power spectrum:extSHSM2Dtdamp01ufric14
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)la
titud
e
<A>+B power spectrum:extSHSM2Dtdamp01ufric14
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)
latit
ude
<A>+B & U co−spectrum
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)
latit
ude
<A>+B & U coherence squared
0.05 0.1 0.15 0.2 0.25 0.320
30
40
50
60
70
0
0.2
0.4
0.6
0.8
1
(a) SHSM-4day (b) SHSM-1day
frequency (CPD)
latit
ude
U power spectrum:extSHSM
2D
tdamp04
ufric14
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)
latit
ude
<A>+B power spectrum:extSHSM
2D
tdamp04
ufric14
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)
latit
ude
<A>+B & U co−spectrum
0.05 0.1 0.1520
30
40
50
60
70
0
5
10
15
frequency (CPD)
latit
ude
<A>+B & U coherence squared
0.05 0.1 0.15 0.2 0.25 0.320
30
40
50
60
70
0
0.2
0.4
0.6
0.8
1
Tuesday, August 16, 16
59
1030
FIG.12.Idealizededdyheatfluxcospectra(top)andtheeddyheatfluxpowerspectra1031(bottom).Seetextfordetails.1032
1033
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
frequency (CPD)
spec
tral i
nten
sity
Idealized eddy frequency distribution
m = 0.003 CPD m = 0.006 CPD m = 0.009 CPD
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
frequency (CPD)
norm
aliz
ed s
pect
ral i
nten
sity
Corresponding power spectrum
60
1034
FIG.13.Top:eddyheatfluxcospectra Re(v̂∗kωT̂kω ) at850hPaforDecember-Marchasa1035functionoffrequencyandlatitude.Left:zonalwavenumber4.Center:zonalwavenumber5.1036Right:zonalwavenumber6.Bottom:thecorrespondingeddyheatfluxcospectraat 46.5! S1037asafunctionoffrequencyandpressure.Thislatitudeisindicatedbytheblacklineinthe1038toppanels.Inallpanels,thebluecurveindicatesthetime-meanzonal-meanzonalwindin1039termsofangularfrequency.Contourinterval: 0.008mKs−1 .BasedonERA-Interim1979-10402014.1041
1042
latit
ude
zonal wavenumber 4
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0zonal wavenumber 5
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0zonal wavenumber 6
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0
frequency (CPD)
Pres
sure
(hPa
)
zonal wavenumber 4
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
frequency (CPD)
zonal wavenumber 5
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
frequency (CPD)
zonal wavenumber 6
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
61
1043
FIG.14.Top:eddyheatfluxcospectraat 48! SfortheHS94-SMexperimentasafunctionof1044frequencyandpressure.Bottom:SameastopexceptfortheSHSM-1dayexperiment.Panels1045fromlefttoright:zonalwavenumber5,6,and7.Conventionsarethesameasthebottomof1046Fig.13. 1047
(a) HS94-SMpr
essu
re (h
Pa)
pres
sure
(hPa
)la
titud
e
at 860 hPa, zonal wavenumber 5
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
at 860 hPa, zonal wavenumber 6
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
at 860 hPa, zonal wavenumber 7
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
latit
ude
at 48 S, zonal wavenumber 5
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
at 48 S, zonal wavenumber 6
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
at 48 S, zonal wavenumber 7
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
latit
ude
at 860 hPa, zonal wavenumber 5
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
at 860 hPa, zonal wavenumber 6
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
at 860 hPa, zonal wavenumber 7
0 0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
latit
ude
at 48 S, zonal wavenumber 5
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
at 48 S, zonal wavenumber 6
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
frequency (CPD)
at 48 S, zonal wavenumber 7
0 0.1 0.2 0.3
200
400
600
800
1000−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
(b) SHSM 1 day
62
1048
FIG.15.StructureoftwoleadingFouriermodes(zonalwavenumber:5)duringtheaustral1049summer.Top:geopotentialanomalyat 46.5! Sasafunctionoflongitudeandpressure.1050Bottom:geopotentialanomalyat250hPaasafunctionoflongitudeandlatitude.Left1051column:zonalphasespeed=10ms-1.Rightcolumn:zonalphasespeed=14ms-1.Contour1052intervals: 20m2s−2 .Whitelinesinthebottompanelsindicatecriticallines.Basedonthe1053ERA-Interimreanalysis(1979-2014).SeeappendixBforthecalculationmethodforthe1054modestructure.1055
1056
1057
1058
longitude
pres
sure
mode with phase speed ~ 10m/s
0 20 40 60
100
200
300
400
500
600
700
800
900
1000 −250
−200
−150
−100
−50
0
50
100
150
200
250
longitude
pres
sure
phase speed around 14m/s
0 20 40 60
100
200
300
400
500
600
700
800
900
1000 −250
−200
−150
−100
−50
0
50
100
150
200
250
longitude
latit
ude
phase speed around 10m/s
0 20 40 60−80
−70
−60
−50
−40
−30
−20
−250
−200
−150
−100
−50
0
50
100
150
200
250
longitude
latit
ude
phase speed around 14m/s
0 20 40 60−80
−70
−60
−50
−40
−30
−20
−250
−200
−150
−100
−50
0
50
100
150
200
250
63
1059
FIG.16.SameasFig.13butforJune-Septemberandzonalwavenumbers3,4,and5.1060
1061
latit
ude
zonal wavenumber 3
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0zonal wavenumber 4
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0zonal wavenumber 5
0.1 0.2 0.3−65
−60
−55
−50
−45
−40
−35
−0.15
−0.1
−0.05
0
frequency (CPD)
Pres
sure
(hPa
)
zonal wavenumber 3
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
frequency (CPD)
zonal wavenumber 4
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
frequency (CPD)
zonal wavenumber 5
0.1 0.2 0.3
200
400
600
800
1000−0.15
−0.1
−0.05
0
64
1062
FIG.17.Toprow:LeadingEOFofzonallyaveragedEKEanomaly.Bottomrow:Spectral1063densityoftheBAMindex(expansioncoefficientoftheleadingEOF)intheunitofm4s-2.Left1064column:December-March.Rightcolumn:June-September.Basedonthemethodoutlined1065inTB14withthe1979-2014ERAInterimreanalysis.NotethatinDecember-March,the1066leadingEOFexplains~45percentofvariance,whereasinJune-Septemberitexplainsonly106730percentofthevariance.1068
1069
1070
1071
1072
Leading mode of EKE (Dec.−Mar.)
latitude
pres
sure
−60 −50 −40 −30
200
400
600
800
1000
Leading mode of EKE (June−Sept.)
latitude
pres
sure
−60 −50 −40 −30
200
400
600
800
1000
0.05 0.1 0.15 0.20
1
2
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6 x 10−5Power Spectrum of BAM index (Dec.−Mar.)
frequency (CPD)
Pow
er S
pect
rum
variance explained: 44.6%
0.05 0.1 0.15 0.20
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6 x 10−5Power Spectrum of BAM index (June−Sept.)
frequency (CPD)
Pow
er S
pect
rum
variance explained: 29.9%