Covariation of finite-amplitude wave activity and the zonal …geosci.uchicago.edu/~nnn/Covariation-Part2.pdf81 latitude space for the austral summer (Fig.1a and 1b): they both exhibit

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

References716

Abernathey, R., and C. Wortham, 2015: Phase speed cross spectra of eddy heat fluxes in the 717

pacific. J. Phys. Oceanogr.,. 718

Ait-Chaalal, F., and T. Schneider, 2014: Why eddy momentum fluxes are concentrated in the 719

upper troposphere. J. Atmospheric Sci., 72, 1585–1604, doi:10.1175/JAS-D-14-0243.1. 720

Ambaum, M. H. P., and L. Novak, 2014: A nonlinear oscillator describing storm track variability. 721

Q. J. R. Meteorol. Soc., 140, 2680–2684, doi:10.1002/qj.2352. 722

Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: the 723

generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmospheric Sci., 724

33, 2031–2048, doi:10.1175/1520-0469(1976)033<2031:PWIHAV>2.0.CO;2. 725

Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle atmosphere dynamics. Academic 726

Press, Orlando, xi, 489 p. pp. 727

Chang, E. K. M., 2005: The role of wave packets in wave–mean flow interactions during 728

Southern Hemisphere summer. J. Atmospheric Sci., 62, 2467–2483, 729

doi:10.1175/JAS3491.1. 730

——, 2006: An idealized nonlinear model of the Northern Hemisphere winter storm tracks. J. 731

Atmospheric Sci., 63, 1818–1839, doi:10.1175/JAS3726.1. 732

——, and I. Orlanski, 1993: On the dynamics of a storm track. J. Atmospheric Sci., 50, 999–1015, 733

doi:10.1175/1520-0469(1993)050<0999:OTDOAS>2.0.CO;2. 734

36

Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteorol., 4, 735

136–162, doi:10.1175/1520-0469(1947)004<0136:TDOLWI>2.0.CO;2. 736

——, and P. G. Drazin, 1961: Propagation of planetary-scale disturbances from the lower into the 737

upper atmosphere. J. Geophys. Res., 66, 83–109, doi:10.1029/JZ066i001p00083. 738

——, and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. 739

Atmospheric Sci., 19, 159–172, doi:10.1175/1520-740

0469(1962)019<0159:OTSOIB>2.0.CO;2. 741

Chen, T.-C., M.-C. Yen, and D. P. Nune, 1987: Dynamic aspects of the Southern-Hemisphere 742

medium-scale waves during the southern summer season. J. Meteorol. Soc. Jpn. Ser II, 65, 743

401–421. 744

Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: configuration and performance of 745

the data assimilation system. Q. J. R. Meteorol. Soc., 137, 553–597, doi:10.1002/qj.828. 746

Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33–52, doi:10.1111/j.2153-747

3490.1949.tb01265.x. 748

Held, I. M., 1985: Pseudomomentum and the orthogonality of modes in shear flows. J. 749


0469(1985)042<2280:PATOOM>2.0.CO;2. 751

——, and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of 752

atmospheric general circulation models. Bull. Am. Meteorol. Soc., 75, 1825–1830, 753

doi:10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2. 754

37

——, and T. Schneider, 1999: The surface branch of the zonally averaged mass transport 755

circulation in the troposphere. J. Atmospheric Sci., 56, 1688–1697, doi:10.1175/1520-756

0469(1999)056<1688:TSBOTZ>2.0.CO;2. 757

Herman, A., 2015: Trends and variability of the atmosphere–ocean turbulent heat flux in the 758

extratropical Southern Hemisphere. Sci. Rep., 5, 14900, doi:10.1038/srep14900. 759

Hoskins, B. J., and K. I. Hodges, 2005: A new perspective on Southern Hemisphere storm tracks. 760

J. Clim., 18, 4108–4129, doi:10.1175/JCLI3570.1. 761

Huang, C. S. Y., and N. Nakamura, 2016: Local finite-amplitude wave activity as a diagnostic of 762

anomalous weather events. J. Atmospheric Sci., 73, 211–229, doi:10.1175/JAS-D-15-763

0194.1. 764

Lachmy, O., and N. Harnik, 2014: The transition to a subtropical jet regime and its maintenance. 765

J. Atmospheric Sci., 71, 1389–1409, doi:10.1175/JAS-D-13-0125.1. 766

Lee, R. W., 2014: Storm track biases and changes in a warming climate from an extratropical 767

cyclone perspective using CMIP5. University of Reading,. 768

Lee, S., and I. M. Held, 1993: Baroclinic wave packets in models and observations. J. 769


0469(1993)050<1413:BWPIMA>2.0.CO;2. 771

——, and H. Kim, 2003: The dynamical relationship between subtropical and eddy-driven jets. J. 772


0469(2003)060<1490:TDRBSA>2.0.CO;2. 774

38

Lindzen, R. S., B. Farrell, and D. Jacqmin, 1982: Vacillations due to wave interference: 775

applications to the atmosphere and to annulus experiments. J. Atmospheric Sci., 39, 14–23, 776

doi:10.1175/1520-0469(1982)039<0014:VDTWIA>2.0.CO;2. 777

Lorenz, D. J., 2014: Understanding midlatitude jet variability and change using rossby wave 778

chromatography: poleward-shifted jets in response to external forcing. J. Atmospheric Sci., 779

71, 2370–2389, doi:10.1175/JAS-D-13-0200.1. 780

Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. 781

Tellus, 7, 157–167, doi:10.1111/j.2153-3490.1955.tb01148.x. 782

McGraw, M., and E. A. Barnes, 2016: Seasonal sensitivity of the eddy-driven jet to tropospheric 783

heating in an idealized AGCM. Submitted. J. Clim.,. 784

Nakamura, H., and A. Shimpo, 2004: Seasonal variations in the Southern Hemisphere storm 785

tracks and jet streams as revealed in a reanalysis dataset. J. Clim., 17, 1828–1844, 786

doi:10.1175/1520-0442(2004)017<1828:SVITSH>2.0.CO;2. 787

Nakamura, N., and D. Zhu, 2010: Finite-amplitude wave activity and diffusive flux of potential 788

vorticity in eddy-mean flow interaction. J. Atmospheric Sci., 67, 2701–2716, 789

doi:10.1175/2010JAS3432.1. 790

Pedlosky, J., 1964: The stability of currents in the atmosphere and the ocean: Part I. J. 791


0469(1964)021<0201:TSOCIT>2.0.CO;2. 793

——, 1970: Finite-amplitude baroclinic waves. J. Atmospheric Sci., 27, 15–30, doi:10.1175/1520-794

39

0469(1970)027<0015:FABW>2.0.CO;2. 795

——, 1971: Finite-amplitude baroclinic waves with small dissipation. J. Atmospheric Sci., 28, 796

587–597, doi:10.1175/1520-0469(1971)028<0587:FABWWS>2.0.CO;2. 797

Plumb, R. A., 1983: A new look at the energy cycle. J. Atmospheric Sci., 40, 1669–1688, 798

doi:10.1175/1520-0469(1983)040<1669:ANLATE>2.0.CO;2. 799

Randel, W. J., and J. L. Stanford, 1985: An observational study of medium-scale wave dynamics 800

in the Southern Hemisphere summer. Part I: wave structure and energetics. J. Atmospheric 801

Sci., 42, 1172–1188, doi:10.1175/1520-0469(1985)042<1172:AOSOMS>2.0.CO;2. 802

——, and I. M. Held, 1991: Phase speed spectra of transient eddy fluxes and critical layer 803

absorption. J. Atmospheric Sci., 48, 688–697, doi:10.1175/1520-804

0469(1991)048<0688:PSSOTE>2.0.CO;2. 805

Ring, M. J., and R. A. Plumb, 2007: Forced annular mode patterns in a simple atmospheric 806

general circulation model. J. Atmospheric Sci., 64, 3611–3626, doi:10.1175/JAS4031.1. 807

Rotunno, R., and M. Fantini, 1989: Petterssen’s “type b”cyclogenesis in terms of discrete, neutral 808

Eady modes. J. Atmospheric Sci., 46, 3599–3604, doi:10.1175/1520-809

0469(1989)046<3599:PBITOD>2.0.CO;2. 810

Salby, M. L., 1982: A ubiquitous wavenumber-5 anomaly in the Southern Hemisphere during 811

FGGE. Mon. Weather Rev., 110, 1712–1721, doi:10.1175/1520-812

0493(1982)110<1712:AUWAIT>2.0.CO;2. 813

40

Schneider, T., P. A. O’Gorman, and X. Levine, 2010: Water vapor and the dynamics of climate 814

changes. Rev. Geophys., 48, doi:10.1029/2009RG000302. http://arxiv.org/abs/0908.4410. 815

Swanson, K. L., and R. T. Pierrehumbert, 1997: Lower-tropospheric heat transport in the pacific 816

storm track. J. Atmospheric Sci., 54, 1533–1543, doi:10.1175/1520-817

0469(1997)054<1533:LTHTIT>2.0.CO;2. 818

Thompson, D. W. J., and J. D. Woodworth, 2013: Barotropic and baroclinic annular variability in 819

the Southern Hemisphere. J. Atmospheric Sci., 131108125536009, doi:10.1175/JAS-D-13-820

0185.1. 821

Thompson, D. W. J., and E. A. Barnes, 2014: Periodic variability in the large-scale Southern 822

Hemisphere atmospheric circulation. Science, 343, 641–645, doi:10.1126/science.1247660. 823

——, and Y. Li, 2014: Baroclinic and barotropic annular variability in the Northern Hemisphere. 824

J. Atmospheric Sci., 72, 1117–1136, doi:10.1175/JAS-D-14-0104.1. 825

Wang, L., and N. Nakamura, 2015: Covariation of finite-amplitude wave activity and the zonal 826

mean flow in the midlatitude troposphere. Part 1: Theory and application to the Southern 827

Hemisphere summer. Geophys. Res. Lett., 2015GL065830, doi:10.1002/2015GL065830. 828

Webster, P. J., and J. L. Keller, 1974: Strong long-period tropospheric and stratospheric rhythm in 829

the Southern Hemisphere. Nature, 248, 212–213, doi:10.1038/248212a0. 830

——, and ——, 1975: Atmospheric variations: vacillations and index cycles. J. Atmospheric Sci., 831

32, 1283–1301, doi:10.1175/1520-0469(1975)032<1283:AVVAIC>2.0.CO;2. 832

41

Yamada, R., and O. Pauluis, 2015: Momentum balance and Eliassen–Palm flux on moist 833

isentropic surfaces. J. Atmospheric Sci., 73, 1293–1314, doi:10.1175/JAS-D-15-0229.1. 834

835

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


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


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


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


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)


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)


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


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


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


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


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)


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)


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


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


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

3

4

5

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

1

2

3

4

5

6 x 10−5Power Spectrum of BAM index (June−Sept.)

frequency (CPD)

Pow

er S

pect

rum

variance explained: 29.9%

Documents

Covariation of finite-amplitude wave activity and the zonal …geosci.uchicago.edu/~nnn/Covariation-Part2.pdf81 latitude space for the austral summer (Fig.1a and 1b): they both exhibit