On the Nature of Zonal Jet EOFs

16 Pages·2006·0.62 MB·English

15DECEMBER2006 MONAHAN AND FYFE 6409
On the Nature of Zonal Jet EOFs
ADAM H. MONAHAN
SchoolofEarthandOceanSciences,UniversityofVictoria,Victoria,BritishColumbia,andEarthSystemEvolutionProgram,
CanadianInstituteforAdvancedResearch,Toronto,Ontario,Canada
JOHN C. FYFE
CanadianCentreforClimateModellingandAnalysis,MeteorologicalServiceofCanada,UniversityofVictoria,Victoria,
BritishColumbia,Canada
(Manuscriptreceived28November2005,infinalform6April2006)
ABSTRACT
Analyticresultsareobtainedforthemeanandcovariancestructureofanidealizedzonaljetthatfluc-
tuatesinstrength,position,andwidth.Throughasystematicperturbationanalysis,theleadingempirical
orthogonalfunctions(EOFs)andprincipalcomponent(PC)timeseriesareobtained.TheseEOFsarebuilt
of linear combinations of basic patterns corresponding to monopole, dipole, and tripole structures. The
analyticresultsdemonstratethatingeneraltheindividualEOFmodescannotbeinterpretedintermsof
individual physical processes. In particular, while the dipole EOF (similar to the leading EOF of the
midlatitudezonalmeanzonalwind)describesfluctuationsinjetpositiontoleadingorder,itstimeseriesalso
containscontributionsfromfluctuationsinstrengthandwidth.NosimpleinterpretationsoftheotherEOFs
intermsofstrength,position,orwidthfluctuationsarepossible.Implicationsoftheseresultsfortheuseof
EOFanalysistodiagnosephysicalprocessesofvariabilityarediscussed.
1. Introduction respond to individual physical modes only in a very
limited class of physical systems (those governed by
Empirical orthogonal function (EOF) analysis, also
linear dynamics for which the linear operator com-
known as principal component analysis (PCA), is a
mutes with its adjoint). In general, observed geophysi-
standard technique for decomposing an observed geo-
cal flows do not belong to this class of systems (e.g.,
physical field into a set of orthogonal spatial patterns
FarrellandIoannou1996;Penland1996;Palmer1999).
with associated temporally uncorrelated time series.
In particular, if the underlying physical processes are
These spatial patterns (denoted the EOFs) are ob-
localized, nonstationary, not mutually orthogonal, or
tainedastheeigenvectorsofthecovariancematrix(or
nonlinearly coupled, they will generally be spread
sometimesthecorrelationmatrix)ofthefield,whilethe
across a number of EOF modes (e.g., Ambaum et al.
timeseries(denotedtheprincipalcomponents,orPCs)
2001;DommengetandLatif2002;Fyfe2003;Monahan
ariseastheprojectioncoefficientsofthecorresponding
et al. 2003; Fyfe and Lorenz 2005). Individual EOF
EOF pattern on the original field. It is common to in-
modes cannot in general be expected to correspond to
terpret individual EOF/PC pairs (together referred to
individual physical processes.
as a mode) as corresponding to distinct physical pro-
In particular, EOF analysis has been used to study
cesses, where the term physical process is used in this
the low-frequency (10–100 days) variability of the ex-
studytodenoteadegreeoffreedomofthesystemwith
tratropical atmosphere (e.g., Barnston and Livezey
a clear physical interpretation. It was emphasized by
1987; Thompson and Wallace 2000). In both hemi-
North(1984),however,thatindividualEOFmodescor-
spheres, throughout the troposphere, it is found that
the meridional spatial structure of the dominant EOF
modeofthezonalmeanzonalwindisadipolecentered
Corresponding author address: Adam H. Monahan, School of
at approximately the latitude of the core of the time-
EarthandOceanSciences,UniversityofVictoria,P.O.Box3055,
meanjet.Thisstructureisgenerallyinterpretedasrep-
STNCSC,VictoriaBCV8W3P6,Canada.
E-mail:[email protected] resenting meridional displacements of the eddy-driven
©2006AmericanMeteorologicalSociety
JCLI3960
6410 JOURNAL OF CLIMATE VOLUME19
jet (the so-called zonal index), while higher-order position, while the case of correlated fluctuations in
EOFs (when they are considered) are interpreted as strengthandwidthisconsideredinsection7.TheEOF
reflecting changes in jet strength or width (e.g., Feld- structure for simultaneous fluctuations in strength, po-
steinandLee1998;Feldstein2000;DeWeaverandNi- sition,andwidthisdiscussedinsection8.Adiscussion
gam2000;Codron2005;Vallisetal.2004).Wittmanet and conclusions are presented in section 9.
al. (2005) consider numerical simulations of the EOF
structure of an idealized midlatitude zonal jet [as in
2. The idealized Gaussian jet
Fyfe(2003)andFyfeandLorenz(2005)]characterized
by Gaussian fluctuations in strength, position, and Following Fyfe (2003), Fyfe and Lorenz (2005), and
width(denoted,respectively,aspulsing,wobbling,and Wittman et al. (2005), we consider our fundamental
bulging). It is shown that a meridional dipole arises as dynamical object to be a jet in the zonal mean zonal
the leading EOF of pure wobbling motion, and that wind with a simple Gaussian profile,
(cid:1) (cid:2)
neitherpulsingnorbulging(bothofwhicharesymmet-
(cid:5)(cid:1)(cid:4)(cid:2)(cid:1)t(cid:2)(cid:6)2
ric about the jet axis) produces dipole EOF patterns u(cid:1)(cid:1),t(cid:2)(cid:3)U(cid:1)t(cid:2)exp (cid:4) , (cid:1)1(cid:2)
(whichareasymmetricaboutthejetaxis).Ameridional 2(cid:3)2(cid:1)t(cid:2)
dipolewasalsofoundinthestudyofGerberandVallis where the U(t), (cid:7)(t), and (cid:8)(t) are the jet strength, po-
(2005) as the leading EOF of a one-dimensional spa-
sition, and width, respectively. We proceed to investi-
tially stochastic process that conserves momentum. gate the statistical structure of u((cid:9), t) by assuming
The present study takes as its starting point the ide-
U(cid:1)t(cid:2)(cid:3)U (cid:10)(cid:4)(cid:1)t(cid:2), (cid:1)2(cid:2)
alized midlatitude jet considered in Wittman et al. 0
(2005),andobtainsanalyticexpressionsforthe(covari- (cid:2)(cid:1)t(cid:2)(cid:3)(cid:1) (cid:10)(cid:5)(cid:1)t(cid:2), (cid:1)3(cid:2)
0
ancebased)EOFsandPCsintermsofthefluctuations
in jet strength, position, and width. These analytic re- (cid:3)(cid:4)1(cid:1)t(cid:2)(cid:3)(cid:3)0(cid:4)1(cid:5)1(cid:10)(cid:6)(cid:1)t(cid:2)(cid:6), (cid:1)4(cid:2)
sultsallowunambiguousdiagnosesoftherelationships where(cid:11)(t),(cid:12)(t),and(cid:13)(t)areindividuallyGaussiantime
between the EOF modes and the underlying physical series with mean zero, that is,
(cid:3) (cid:4)
processes.ItwillbeshownthatwhiletheleadingEOFs
aremadeupofasmallnumberofbasicspatialpatterns, 1 (cid:4)2
and are therefore simple in structure, the associated p(cid:1)(cid:4)(cid:2)(cid:3)(cid:14)2(cid:7)(cid:8)2exp (cid:4)2(cid:8)2 , (cid:1)5(cid:2)
timeseriesinextricablycoupletheunderlyingprocesses (cid:3) (cid:4)
ofjetvariability.Inparticular,adipoleisshowntoarise 1 (cid:5)2
as an EOF of the fluctuating jet under quite general p(cid:1)(cid:5)(cid:2)(cid:3)(cid:14)2(cid:7)w2exp (cid:4)2w2 , (cid:1)6(cid:2)
conditions,buttheassociatedPCtimeseriesmixesfluc- (cid:3) (cid:4)
tuations in jet strength, position, and width. Further- 1 (cid:6)2
more, because the EOFs associated with one physical p(cid:1)(cid:6)(cid:2)(cid:3)(cid:14)2(cid:7)(cid:9)2exp (cid:4)2(cid:9)2 . (cid:1)7(cid:2)
processarenotorthogonaltothoseassociatedwithan-
other,theseEOFswillbeseentobemixedwhenboth (Table 1 summarizes these and other symbols used in
processes are present simultaneously. It will be shown thisstudy.)Wedonotassumethatthejointdistribution
thatinthisidealized(butphysicallymotivated)system, p((cid:11), (cid:12), (cid:13)) can be factored as the product of the indi-
while some EOF modes may be associated with indi- vidual distributions p((cid:11))p((cid:12))p((cid:13)): in other words, we
vidualphysicalprocessestoaleading-orderapproxima- allow in general for dependence between these vari-
tion, this association cannot generally be made. In the ables.Forthepurposesofcalculatingmeansandcova-
physically motivated context of the fluctuating zonal riances of the zonal wind, while cross correlations be-
jet,thepresentstudyreinforcestheconclusionsofear- tween (cid:11)(t), (cid:12)(t), and (cid:13)(t) are important, the temporal
lierstudiesthatdemonstrateddifficultiesinassociating autocorrelationstructuresofthesetimeseriesareirrel-
individual physical processes with individual EOF evant. Note that the expression for the inverse width
modes (e.g., Ambaum et al. 2001; Dommenget and [Eq. (4)] allows (cid:8) to become negative, which is of
Latif 2002). courseunphysical;inpractice,thestandarddeviationof
Section 2 describes the idealized fluctuating midlati- (cid:13)is sufficiently small that the probability of negative
tudejetconsideredinthisstudy.TheEOFsofthejetin values of the jet width is negligible.
the case of pure fluctuations in strength, position, and Observational justification for these approximations
widthareconsideredrespectivelyinsections3,4,and5. is provided in Fig. 1, the top panel of which shows the
Section6describesthecovariancestructureinthepres- leading EOFs of daily National Centers for Environ-
ence of simultaneous fluctuations in both strength and mental Prediction–National Center for Atmospheric
15DECEMBER2006 MONAHAN AND FYFE 6411
TABLE1.Listofsymbolsusedinthisstudy.
Symbol Units Definition
t s Time
(cid:9) deg Latitude
u((cid:9),t) ms(cid:4)1 Jet
U(t) ms(cid:4)1 Jetpeak
U ms(cid:4)1 Time-meanjetpeak
0
(cid:11)(t) ms(cid:4)1 Fluctuationsinjetpeak:U(t)(cid:3)U (cid:10)(cid:11)(t)
0
(cid:18) ms(cid:4)1 Standarddeviationof(cid:11)
l — (cid:18)/U
0
(cid:8)(t) deg Jetwidth
(cid:8) deg Time-meanjetwidth
0
(cid:13)(t) — (Scaled)fluctuationsin(inverse)jetwidth:
(cid:8)(cid:4)1(t)(cid:3)(cid:8)(cid:4)1[1(cid:10)(cid:13)(t)]
0
(cid:19) — Standarddeviationof(cid:13)
(cid:7)(t) deg Jetcentrallatitude
(cid:9) deg Time-meanjetcentrallatitude
0
(cid:12)(t) deg Fluctuationsinjetcentrallatitude:
(cid:7)(t)(cid:3)(cid:9) (cid:10)(cid:12)(t)
0
w deg Standarddeviationof(cid:12)
h — w/(cid:8)
(cid:15) — Cor0relationof(cid:13)and(cid:11) FIG. 1. Leading EOFs of daily Southern Hemisphere winter
f((cid:9)) — ithbasisfunction[cf.Eqs.(11)–(13)] (May–September) 500-hPa zonal mean zonal wind (1958–2003).
Ci((cid:9),(cid:9)) (ms(cid:4)1)2 Jetcovariancefunction (top)FollowingthefittingprocedureinaccordwithEqs.(l)–(7).
E(i)(1(cid:9)) 2 — ithEOFspatialpattern (bottom)Notfollowingthefittingprocedure.Solidcurves:E(1).
(cid:22)(i)(t) ms(cid:4)1 ithPCtimeseries Dashedcurves:E(2).
(cid:23)(i) (ms(cid:4)1)2 ithPCtimeseriesvariance
sition.Ofcourse,angularmomentumconservationona
spherical domain implies that poleward (equatorward)
Research (NCEP–NCAR) reanalysis (Kalnay et al.
displacements of the jet should be associated with in-
1996) Southern Hemisphere winter (May–September)
creased(decreased)jetstrength;thelackofcorrelation
500-hPazonalmeanzonalwindafterfittingtothepro- betweenU(t)and(cid:7)(t)indicatesthatthisrelationshipis
file (1) (following the direct procedure in appendix B) weaker than that between U(t) and (cid:8)(t). Under these
andthenreconstructing(cid:12)(t)and(cid:13)(t)tobeindividually
assumptions the system is completely described in
Gaussian with observed variances. Additionally, in or- terms of the following best-fit parameters: U (cid:17) 23.3
0
der to preserve the observed correlation between ms(cid:4)1,(cid:18)(cid:17)2.7ms(cid:4)1,(cid:9) (cid:17)(cid:4)47.5degree(deg),w(cid:17)2.7
strength and inverse width, we set (cid:11)(t) (cid:3) (cid:15)(cid:13)(t) (cid:10) (cid:16)(t) deg,(cid:8)(cid:4)1(cid:17)0.095deg(cid:4)10,(cid:19)(cid:17)0.165,and(cid:15)(cid:17)13.1ms(cid:4)1.
[where(cid:16)(t)isasmallresidualthatwillbeneglected].As 0
We now compare the leading EOFs obtained in this
discussedinFyfeandLorenz(2005),theobservedcor-
way with the leading EOFs of the unfit zonal mean
relation between strength and inverse width reflects
zonalwinds(Fig.1,bottom).Weseethatdespiteallthe
conservation of momentum:
ratherstringentapproximationsimposedby(1)–(7)the
(cid:5)
M(cid:1)t(cid:2)(cid:3) (cid:10) u(cid:1)(cid:1),t(cid:2)d(cid:1)(cid:3)(cid:14)2(cid:7)(cid:3)0U10(cid:10)(cid:10)(cid:6)(cid:4)(cid:1)(cid:1)tt(cid:2)(cid:2) (cid:1)8(cid:2) tizweodsmetosdoeflldeoaedsinaggEoOodFsjocbomofpcaarpetvuerirnygwtehlel:mtheeanideaanld-
(cid:4)(cid:10)
covariance structure of the original data.
(where the sphericity of the domain has been ne- Having specified the statistical structure of the fluc-
glected).Wenotethatthereisnomanifestdependence tuations in jet strength, position, and width, we can
between either strength and position or width and po- calculate the time mean of u((cid:9), t) as
(cid:5) (cid:6) (cid:7)
(cid:1)(cid:1)(cid:4)(cid:1) (cid:4)(cid:5)(cid:2)2(cid:1)1(cid:10)(cid:6)(cid:2)2
(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) p(cid:1)(cid:4),(cid:5),(cid:6)(cid:2)(cid:1)U (cid:10)(cid:4)(cid:2)exp (cid:4) 0 d(cid:4)d(cid:5)d(cid:6), (cid:1)9(cid:2)
0 2(cid:3)2
0
and the spatial covariance function of u((cid:9), t) as wheretheanglebrackets(cid:20)·(cid:21)denotetheexpectation(or
averaging)operator[andweusethesimplifiednotation
C(cid:1)(cid:1),(cid:1)(cid:2)(cid:3)(cid:20)u(cid:1)(cid:1)(cid:2)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:4)(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21), (cid:1)10(cid:2) (cid:20)u((cid:9))(cid:21) (cid:3) (cid:20)u((cid:9), t)(cid:21)]. Once the covariance function has
1 2 1 2 1 2
6412 JOURNAL OF CLIMATE VOLUME19
FIG.2.Plotsofthefunctionsf0((cid:9)),f1((cid:9)),andf2((cid:9))[Eqs.(11)–(13)]fromwhichtheleadingEOFsareconstructed,rescaledtobe
ofunitnorm.
been calculated, the eigenvalue problem for the EOFs theearthwillbeneglected:thejetwillbetakentoexist
can be posed as an integral equation as described in on an infinite domain.
appendixA.FortheanalysisofC((cid:9),(cid:9))thatfollowsit
1 2
is useful to define the functions 3. Fluctuations in jet strength alone
(cid:3) (cid:4) (cid:6) (cid:7)
1 1(cid:11)2 (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2 Consider first the case in which only fluctuations in
f0(cid:1)(cid:1)(cid:2)(cid:3) (cid:14)(cid:7)(cid:3) exp (cid:4) 2(cid:3)20 , (cid:1)11(cid:2) strength are nonzero (i.e., (cid:18)(cid:26) 0, w (cid:3) 0, (cid:19)(cid:3) 0). The
(cid:3) 0(cid:4) (cid:6) (cid:7) 0 (cid:6) (cid:7) time-mean je(cid:5)t is (cid:6) (cid:7)
1 1(cid:11)2 (cid:1)(cid:1)-(cid:1)(cid:2) (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2 (cid:10) (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
f1(cid:1)(cid:1)(cid:2)(cid:3) (cid:14)(cid:7)(cid:3)0 H1 (cid:14)2(cid:3)00 exp (cid:4) 2(cid:3)020 , (cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) (cid:4)(cid:10)p(cid:1)(cid:6)(cid:4)(cid:2)(cid:1)U0(cid:10)(cid:4)(cid:2)exp(cid:7) (cid:4) 2(cid:3)020 d(cid:4)
(cid:1)12(cid:2) (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
(cid:3) (cid:4) (cid:6) (cid:7) (cid:6) (cid:7) (cid:3)U exp (cid:4) 0 . (cid:1)14(cid:2)
f (cid:1)(cid:1)(cid:2)(cid:3) 1 1(cid:11)2H (cid:1)(cid:1)(cid:4)(cid:1)0(cid:2) exp (cid:4)(cid:1)(cid:1)(cid:4)(cid:1)0(cid:2)2 , 0 2(cid:3)02
2 3(cid:14)(cid:7)(cid:3) 2 (cid:14)2(cid:3) 2(cid:3)2
0 0 0
(cid:1)13(cid:2)
whereH (x)(cid:3)2xandH (x)(cid:3)4x2(cid:4)2aretheHermite
1 2
polynomialsoforder1and2(Arfken1985).Thefunc-
tions f((cid:9)) are normalized to have unit square norm:
i
(cid:24)(cid:25) f2((cid:9))d(cid:9)(cid:3) 1, i (cid:3) 0, 1, 2. Plots of these functions
–(cid:25) i
aregiveninFig.2.Itisworthnotingthatdespitetheir
resemblance to parabolic cylinder functions (e.g., Gill
1982),thesefunctionsarenotmutuallyorthogonal.By
symmetry,f ((cid:9))isorthogonaltobothf ((cid:9))andf ((cid:9)),
1 0 2
butf ((cid:9))andf ((cid:9))arenotmutuallyorthogonal,thatis,
(cid:24)(cid:25) f0((cid:9))f ((cid:9))d2(cid:9)(cid:3) (cid:4)1/(cid:14)3. Figure 3 provides a geo-
–(cid:25) 0 2
metric representation of the vector space spanned by
these functions.
Wenowproceedtodevelopanalyticexpressionsfor
thecovariancefunctionandEOFsofthefluctuatingjet
forprogressivelycomplexformsofvariability:first,in-
dividual fluctuations in strength, position, and width;
and second, simultaneous fluctuations in strength and
position, and in strength and width. While these indi- FIG.3.Geometricillustrationofthevectors(infunctionspace)
vidual examples do not describe the full covariance f0((cid:9)),f1((cid:9)),andf2((cid:9)).Becausef0((cid:9))andf2((cid:9))arenotorthogo-
nal, they cannot simultaneously be eigenvectors of a symmetric
structureofthefullyvariablejet,theyrepresentimpor-
functionsuchasthecovariance.Ifbothofthesevectorscontribute
tant limiting cases that can be used to understand the
totheleadingEOFs,theseEOFsmustbeorthogonallinearcom-
more complex case. In this analysis, the sphericity of binationsofthesevectors.
15DECEMBER2006 MONAHAN AND FYFE 6413
Inthiscasethetime-meanjetisidenticaltotheinstan- Notethatthisprovidesananalyticdemonstrationofthe
taneous jet with time-mean strength. The covariance result found numerically in Wittman et al. (2005).
function is
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:1)(cid:1) (cid:4)(cid:1)(cid:2)2 (cid:1)(cid:1) (cid:4)(cid:1)(cid:2)2 4. Fluctuations in jet position alone
C(cid:1)(cid:1),(cid:1)(cid:2)(cid:3)(cid:8)2exp (cid:4) 1 0 exp (cid:4) 2 0
1 2 2(cid:3)2 2(cid:3)2 Now consider the case in which only fluctuations in
0 0
jetpositionarenonzero(i.e.,(cid:18)(cid:3)0,w(cid:26)0,(cid:19)(cid:3)0).The
(cid:1)15(cid:2)
time-mean jet is
(cid:14) (cid:5) (cid:1) (cid:2)
(cid:3) (cid:7)(cid:3)0(cid:8)2f0(cid:1)(cid:1)1(cid:2)f0(cid:1)(cid:1)2(cid:2). (cid:1)16(cid:2) (cid:10) (cid:5)(cid:1)(cid:1)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:6)2
(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) p(cid:1)(cid:5)(cid:2)U exp (cid:4) 0 d(cid:5)
Thus, the integral equation defining the eigenvalue (cid:4)(cid:10) 0 (cid:6) 2(cid:3)02 (cid:7)
problem for the EOFs [Eq. (A1) in the appendix] is
cthhearfaucntecrtiiozendf0b(y(cid:9)a),kwehrnicehl tishattheisretfroivreialtlhyeseopnalyraebilgeenin- (cid:3)(cid:14)U(cid:3)020(cid:3)(cid:10)0 w2exp (cid:4)2(cid:1)(cid:1)(cid:1)(cid:3)02(cid:4)(cid:10)(cid:1)w0(cid:2)22(cid:2) . (cid:1)18(cid:2)
functionassociatedwithanonzeroeigenvalue(Arfken
In this case the time-mean jet differs from the instan-
1985). In the presence of Gaussian fluctuations in the
taneousjetatthetime-meanposition.Inparticular,the
jet strength alone, the leading (and only) EOF is the
jetwobblingarounditsmeanpositionproducesamean
monopole
jetthatisweakerandwiderthantheinstantaneousjet
E(cid:1)1(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)f (cid:1)(cid:1)(cid:2). (cid:1)17(cid:2) at the mean position. The covariance function is
0
(cid:6) (cid:7) (cid:3) (cid:4) (cid:6) (cid:7)
U2(cid:3) (cid:3)2(cid:10)w2 2w2 U2(cid:3)2 (cid:3)2
C(cid:1)x,y(cid:2)(cid:3)(cid:14)(cid:3)020(cid:10)02w2exp (cid:4)(cid:3)020(cid:10)2w2(cid:1)x2(cid:10)y2(cid:2) exp (cid:3)02(cid:10)2w2xy (cid:4)(cid:3)02(cid:10)0 w02exp (cid:4)(cid:3)02(cid:10)0w2(cid:1)x2(cid:10)y2(cid:2) ,
(cid:1)19(cid:2)
where we have defined the new coordinates (cid:1) (cid:4)(cid:1)
y(cid:3) 2 0. (cid:1)21(cid:2)
(cid:14)
2(cid:3)
0
(cid:1) (cid:4)(cid:1)
x(cid:3) (cid:14)1 0, (cid:1)20(cid:2) Defining the parameter h (cid:3) w/(cid:8)0 the covariance func-
2(cid:3) tion can be expressed
0
(cid:6) (cid:7)
U2 1(cid:10)h2
C(cid:1)x,y(cid:2)(cid:3)(cid:14)1(cid:10)0 2h2exp (cid:4)1(cid:10)2h2(cid:1)x2(cid:10)y2(cid:2)
(cid:1) (cid:3) (cid:4) (cid:6) (cid:7)(cid:2)
(cid:14)
2h2 1(cid:10)2h2 h4
(cid:27) exp xy (cid:4) exp (cid:1)x2(cid:10)y2(cid:2) . (cid:1)22(cid:2)
1(cid:10)2h2 1(cid:10)h2 (cid:1)1(cid:10)h2(cid:2)(cid:1)1(cid:10)2h2(cid:2)
Thiscovariancefunctionisnotobviouslyseparablefor give h2 (cid:3) 0.066), expanding the quantity in square
generalvaluesofh.However,inthelimitofsmallfluc- brackets in (22) in powers of h2 yields
tuationsinposition(h2K1;the500-hPaSHestimates
(cid:3) (cid:4) (cid:6) (cid:7)
(cid:14)
2h2 1(cid:10)2h2 h4 1 1
exp xy (cid:4) exp (cid:1)x2(cid:10)y2(cid:2) (cid:3) (cid:1)h2(cid:4)2h4(cid:2)H(cid:1)x(cid:2)H(cid:1)y(cid:2)(cid:10) h4H(cid:1)x(cid:2)H(cid:1)y(cid:2)(cid:10)O(cid:1)h6(cid:2),
1(cid:10)2h2 1(cid:10)h2 (cid:1)1(cid:10)h2(cid:2)(cid:1)1(cid:10)2h2(cid:2) 2 1 1 8 2 2
(cid:1)23(cid:2)
6414 JOURNAL OF CLIMATE VOLUME19
so
(cid:6) (cid:7) (cid:6) (cid:7)
U2 1(cid:10)h2 1 1
C(cid:1)x,y(cid:2)(cid:3)(cid:14)1(cid:10)0 2h2exp (cid:4)1(cid:10)2h2(cid:1)x2(cid:10)y2(cid:2) (cid:27) 2(cid:1)h2(cid:4)2h4(cid:2)H1(cid:1)x(cid:2)H1(cid:1)y(cid:2)(cid:10)8h4H2(cid:1)x(cid:2)H2(cid:1)y(cid:2)(cid:10)O(cid:1)h6(cid:2) . (cid:1)24(cid:2)
The covariance function is manifestly separable in EOF is the tripole f ((cid:9)), precisely in accordance with
2
terms of the functions the numerical results of Wittman et al. (2005).
(cid:3) (cid:4)
Having obtained analytic expressions for the EOFs,
1(cid:10)h2
Q1(cid:1)x(cid:2)(cid:3)exp (cid:4)1(cid:10)2h2x2 H1(cid:1)x(cid:2) it is possible to(cid:5)calculate the PC time series as
(cid:10)
(cid:12)(cid:1)i(cid:2)(cid:1)t(cid:2)(cid:3) E(cid:1)i(cid:2)(cid:1)(cid:1)(cid:2)(cid:5)u(cid:1)(cid:1),t(cid:2)(cid:4)(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:6)d(cid:1). (cid:1)30(cid:2)
(cid:3)e(cid:4)x2H (cid:1)x(cid:2)(cid:5)1(cid:10)O(cid:1)h2(cid:2)(cid:6), (cid:1)25(cid:2) (cid:4)(cid:10)
(cid:3) 1 (cid:4)
1(cid:10)h2 To leading o(cid:3)rder, t(cid:4)hese are respectively
Q2(cid:1)x(cid:2)(cid:3)exp (cid:4)1(cid:10)2h2x2 H2(cid:1)x(cid:2) (cid:14)(cid:7) 1(cid:11)2
(cid:12)(cid:1)1(cid:2)(cid:1)t(cid:2)(cid:3) U (cid:5)(cid:1)t(cid:2)(cid:10)O(cid:1)h2(cid:2), (cid:1)31(cid:2)
2(cid:3) 0
(cid:3)e(cid:4)x2H2(cid:1)x(cid:2)(cid:5)1(cid:10)O(cid:1)h2(cid:2)(cid:6), (cid:1)26(cid:2) (cid:3)3(cid:14)0(cid:7)(cid:4)1(cid:11)2 U
where the O(h2) terms are corrections to the width of (cid:12)(cid:1)2(cid:2)(cid:1)t(cid:2)(cid:3) (cid:3) 4(cid:3)0 (cid:5)(cid:5)(cid:1)t(cid:2)2(cid:4)w2(cid:6)(cid:10)O(cid:1)h4(cid:2).
theGaussianenvelope,whichforh2K1alterthewidth 0 0
but not the shape of the functions Q (x) and Q (x). (cid:1)32(cid:2)
1 2
Changing coordinates back to (cid:9), (cid:9), the covariance WhilethePCtimeseriesareuncorrelatedbyconstruc-
1 2
function can be written tion,(cid:22)(1)(t)and(cid:22)(2)(t)areclearlynotindependent:their
(cid:14) (cid:6) joint distribution is parabolic. Statistical independence
(cid:7)(cid:3)U2
C(cid:1)(cid:1),(cid:1)(cid:2)(cid:3) 0 0 (cid:1)h2(cid:4)2h4(cid:2)f (cid:1)(cid:1)(cid:2)f (cid:1)(cid:1)(cid:2) oftwovariablesX1andX2requiresthattheconditional
1 2 2 1 1 1 2 distribution p(X |X ) is equal to the marginal distribu-
(cid:7) 2 1
3 tion p(X2) alone; that is, that knowledge of X1 has no
(cid:10) h4f (cid:1)(cid:1)(cid:2)f (cid:1)(cid:1)(cid:2)(cid:10)O(cid:1)h6(cid:2) (cid:5)1(cid:10)O(cid:1)h2(cid:2)(cid:6). effectonthedistributionofX (andviceversa).Thisis
4 2 1 2 2 2
clearlynotthecasefor(cid:22)(1)(t)and(cid:22)(2)(t).Fortheprob-
(cid:1)27(cid:2) lemunderconsideration,ateverytimet,thewindfield
is specified by the single number (cid:12)(t). Because varia-
As discussed in appendix A, because the functions
f ((cid:9))andf ((cid:9))aremutuallyorthogonal,itfollowsthat tionsinthejetprojectonaspectrumofEOFs,thePC
1 2 of all of these EOFs must be determined uniquely by
thesefunctionsarealsoEOFs.Furthermore,theorder-
the scalar time series (cid:12)(t), and cannot be mutually in-
ing of the EOFs is clear from the expansion in powers
dependent. This result was first obtained by Fyfe and
of h:
Lorenz (2005) using a Taylor series expansion of the
E(cid:1)1(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)f (cid:1)(cid:1)(cid:2), (cid:1)28(cid:2) wobbling jet; the above analysis formalizes the argu-
1
ment.
E(cid:1)2(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)f (cid:1)(cid:1)(cid:2) (cid:1)29(cid:2)
2
[where the O(h2) correction terms arising from the 5. Fluctuations in jet width alone
width of the Gaussian envelope have been neglected]. Consider the case in which only fluctuations in jet
For small fluctuations in jet position (relative to jet width are nonzero (i.e., (cid:18) (cid:3) 0, w (cid:3) 0, (cid:19) (cid:26) 0). The
width),thefirstEOFisthedipolef ((cid:9))andthesecond time-mean jet is
1
(cid:5) (cid:6) (cid:7) (cid:1) (cid:2)
(cid:10) (cid:1)1(cid:10)(cid:6)(cid:2)2 U (cid:3) (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) (cid:4)(cid:10)p(cid:1)(cid:6)(cid:2)U0exp (cid:4) 2(cid:3)02 (cid:1)(cid:1)(cid:4)(cid:1)0(cid:2)2 d(cid:6)(cid:3)(cid:14)(cid:3)02(cid:10)(cid:9)02(cid:1)(cid:1)0 (cid:4)(cid:1)0(cid:2)2exp (cid:4)2(cid:5)(cid:3)02(cid:10)(cid:9)2(cid:1)(cid:1)0(cid:4)(cid:1)0(cid:2)2(cid:6) .
(cid:1)33(cid:2)
As was the case with fluctuations in position, the mean jet is not equal to the instantaneous jet at mean width.
The covariance function is
15DECEMBER2006 MONAHAN AND FYFE 6415
(cid:1) (cid:2)
U2 x2(cid:10)y2
C(cid:1)x,y(cid:2)(cid:3)(cid:14)(cid:1)1(cid:10)(cid:9)2x20(cid:2)(cid:1)1(cid:10)(cid:9)2y2(cid:2)exp (cid:4)2(cid:5)1(cid:10)(cid:9)2(cid:1)x2(cid:10)y2(cid:2)(cid:6)
(cid:3)(cid:8) (cid:1) (cid:6) (cid:7)(cid:2)(cid:4)
(cid:9)4x2y2 1 (cid:9)2x2y2 1 1
(cid:27) 1(cid:10) (cid:4)exp (cid:4) (cid:10) , (cid:1)34(cid:2)
1(cid:10)(cid:9)2(cid:1)x2(cid:10)y2(cid:2) 21(cid:10)(cid:9)2(cid:1)x2(cid:10)y2(cid:2) 1(cid:10)(cid:9)2x2 1(cid:10)(cid:9)2y2
where Aswasthecasewithfluctuationsinpositionalone,this
covariancefunctionisnotobviouslyseparableforgen-
x(cid:3)(cid:1)(cid:1) (cid:4)(cid:1)(cid:2)(cid:11)(cid:3), (cid:1)35(cid:2)
1 0 0 eralvaluesof(cid:19).However,for(cid:19)2K1(the500-hPaSH
y(cid:3)(cid:1)(cid:1) (cid:4)(cid:1)(cid:2)(cid:11)(cid:3). (cid:1)36(cid:2) estimates give (cid:19)2 (cid:3) 0.027),
2 0 0
(cid:1) (cid:2)
U2 x2(cid:10)y2
C(cid:1)x,y(cid:2)(cid:3)(cid:14)(cid:1)1(cid:10)(cid:9)2x20(cid:2)(cid:1)1(cid:10)(cid:9)2y2(cid:2)exp (cid:4)2(cid:5)1(cid:10)(cid:9)2(cid:1)x2(cid:10)y2(cid:2)(cid:6) (cid:5)(cid:9)2x2y2(cid:10)O(cid:1)(cid:9)4(cid:2)(cid:6), (cid:1)37(cid:2)
so to O((cid:19)4),
(cid:6) (cid:7)(cid:6) (cid:7)
(cid:14)
3 (cid:7)(cid:3)U2 2 2
C(cid:1)(cid:1),(cid:1)(cid:2)(cid:3) 0 0(cid:9)2 f (cid:1)(cid:1)(cid:2)(cid:10) f (cid:1)(cid:1)(cid:2) f (cid:1)(cid:1)(cid:2)(cid:10) f (cid:1)(cid:1)(cid:2) (cid:10)O(cid:1)(cid:9)4(cid:2). (cid:1)38(cid:2)
1 2 4 2 1 (cid:14)3 0 1 2 2 (cid:14)3 0 2
The leading EOF for pure fluctuations in jet width is 6. Independent fluctuations in strength and
therefore position
2
E(cid:1)1(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)f (cid:1)(cid:1)(cid:2)(cid:10) f (cid:1)(cid:1)(cid:2) (cid:1)39(cid:2) Wenowturntothecaseinwhichonlyfluctuationsin
2 (cid:14) 0
3 widtharezero(i.e.,(cid:18)(cid:26)0,w(cid:26)0,(cid:19)(cid:3)0),andwhere,as
[againuptoO((cid:19)2)correctiontermsassociatedwiththe suggested by the observations, the fluctuations in
widthoftheGaussianenvelope].ThisEOF,illustrated strength and position are independent, that is,
in Fig. 4, is in excellent agreement with the leading (cid:3) (cid:4) (cid:3) (cid:4)
EOFofpurefluctuationsinjetwidthobtainednumeri- 1 (cid:4)2 (cid:5)2
p(cid:1)(cid:4),(cid:5)(cid:2)(cid:3)p(cid:1)(cid:4)(cid:2)p(cid:1)(cid:5)(cid:2)(cid:3) exp (cid:4) exp (cid:4) .
cally in Wittman et al. (2005). 2(cid:7)(cid:8)w 2(cid:8)2 2w2
Theorientationsinthespacespannedbyf ((cid:9)),f ((cid:9)),
0 1
and f ((cid:9)) of the leading EOFs for the cases of pure (cid:1)40(cid:2)
2
fluctuationsinjetstrength,position,andwidtharepre-
sented in Fig. 5. Then the time-mean jet is
(cid:5) (cid:5) (cid:1) (cid:2) (cid:6) (cid:7)
(cid:10) (cid:10) (cid:5)(cid:1)(cid:1)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:6)2 U (cid:3) (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) (cid:4)(cid:10) (cid:4)(cid:10)(cid:1)U0(cid:10)(cid:4)(cid:2)exp (cid:4) 2(cid:3)002 p(cid:1)(cid:4),(cid:5)(cid:2)d(cid:4)d(cid:5)(cid:3)(cid:14)(cid:3)020(cid:10)0w2exp (cid:4)2(cid:1)(cid:3)02(cid:10)w02(cid:2) . (cid:1)41(cid:2)
Not surprisingly, the mean jet depends on fluctuations ordinates x and y as in Eqs. (20) and (21), the covari-
in jet position but not in jet strength. Defining the co- ance function is
(cid:6) (cid:7)
U2 1(cid:10)h2
C(cid:1)x,y(cid:2)(cid:3)(cid:14)1(cid:10)0 2h2exp (cid:4)1(cid:10)2h2(cid:1)x2(cid:10)y2(cid:2)
(cid:1) (cid:3) (cid:4) (cid:6) (cid:7)(cid:2)
(cid:14)
2h2 1(cid:10)2h2 h4
(cid:27) (cid:1)1(cid:10)l2(cid:2)exp xy (cid:4) exp (cid:1)x2(cid:10)y2(cid:2) , (cid:1)42(cid:2)
1(cid:10)2h2 1(cid:10)h2 (cid:1)1(cid:10)h2(cid:2)(cid:1)1(cid:10)2h2(cid:2)
6416 JOURNAL OF CLIMATE VOLUME19
FIG.4.LeadingEOFofpureGaussianfluctuationsinjetwidth
[Eq.(39)].NormalizationasinFig.2.
where l (cid:3) (cid:18)/U . As was the case of fluctuations in jet
0
positionalone,thiscovariancefunctionbecomesmani-
festly separable when we assume that h is small and
expand in powers of h:
C(cid:1)x,y(cid:2)(cid:3)U02(cid:6)exp(cid:5)(cid:4)(cid:1)x2(cid:10)y2(cid:2)(cid:6) anFdIfG.((cid:9)5.)TohfethoerlieeandtaintigoEnsOiFnstfhoersthpeacceassepsaonfnpedurbeyfluf0c(t(cid:9)ua),tifo1n((cid:9)si)n,
2
1 jetstrength,position,andwidth.
(cid:27) l2(cid:10) (cid:1)h2(cid:10)l2h2(cid:4)2h4(cid:2)H (cid:1)x(cid:2)H (cid:1)y(cid:2)
2 1 1
(cid:7)
1
(cid:10) h4H (cid:1)x(cid:2)H (cid:1)y(cid:2)(cid:10)h.o.t. (cid:5)1(cid:10)O(cid:1)h2(cid:2)(cid:6),
8 2 2
where we have also assumed that l is “small.” Trans-
(cid:1)43(cid:2) forming back to the original coordinates we obtain
(cid:6) (cid:7)
(cid:14) 1 3
C(cid:1)(cid:1),(cid:1)(cid:2)(cid:3) (cid:7)(cid:3)U2 l2f (cid:1)(cid:1)(cid:2)f (cid:1)(cid:1)(cid:2)(cid:10) (cid:1)h2(cid:10)l2h2(cid:2)f (cid:1)(cid:1)(cid:2)f (cid:1)(cid:1)(cid:2)(cid:10) h4f (cid:1)(cid:1)(cid:2)f (cid:1)(cid:1)(cid:2) (cid:10)h.o.t. (cid:1)44(cid:2)
1 2 0 0 0 1 0 2 2 1 1 1 2 8 2 1 2 2
FIG.6.ComponentsoftheEOFsspannedbyf0((cid:9))andf2((cid:9))forfluctuationsinbothjetstrengthandposition,
for(a)theEOFwiththelargervariance(cid:23)((cid:10))and(b)theEOFwiththesmallervariance(cid:23)((cid:4)).
15DECEMBER2006 MONAHAN AND FYFE 6417
(wherethehigher-ordertermsincludesmallcorrections
resulting from the width of the Gaussian envelope,
which will be neglected). The covariance function is
clearly separable in f((cid:9)), j (cid:3) 0, 1, 2. As noted earlier,
j
whilef ((cid:9))isorthogonaltobothf ((cid:9))andf ((cid:9)),f ((cid:9))
1 0 2 0
and f ((cid:9)) are not mutually orthogonal. Thus, while
2
f ((cid:9))isoneofthethreeleadingEOFsofthecovariance
1
function (44), the other two will be orthogonal linear
combinations of f ((cid:9)) and f ((cid:9)),
0 2
E(cid:1)(cid:28)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:13)(cid:1)(cid:28)(cid:2)f (cid:1)(cid:1)(cid:2)(cid:10)(cid:13)(cid:1)(cid:28)(cid:2)f (cid:1)(cid:1)(cid:2), (cid:1)45(cid:2)
0 0 2 2
where the plus (minus) superscript labels the EOF as-
sociated with the larger (smaller) variance. The EOF
problemcanthenberecastasatwo-dimensionaleigen-
value problem, details of which are given in appendix
A.Figure6displaysplotsof(cid:29)(0(cid:28))and(cid:29)(2(cid:28))asfunctions valFuIeGs.o7f.tHhyebrraitdioE3OhF4/s8lE2(e(cid:10)q)uoaflttohe0.c3o(vtahriinanscoelidfucnucrtvioen),(14(4t)hifcokr
oftheratio3h4/8l2.Forsmallvaluesoftheratio,thatis, curve),and3(dashedcurve).
forh4Kl2,(cid:29)((cid:10))(cid:9)1and(cid:29)((cid:10))(cid:9)0.TheEOFE((cid:10))((cid:9))is
0 2
therefore approximately equal to f ((cid:9)), the leading
0 tify the leading EOFs, but their ordering as well. It is
EOFforpurefluctuationsinjetstrength(uptoasign,
straightforward to show [appendix A, Eq. (A7)] that
always arbitrary for EOFs). For large values of the ra-
the ratio of the variance of the PC time series associ-
tio, that is, for h4 k l2, (cid:29)(2(cid:10)) (cid:9) 1 and (cid:29)(0(cid:10)) (cid:9) 0. In this atedwiththeeigenvectorE((cid:4))((cid:9))tothatoftheeigen-
limit, E((cid:10))((cid:9)) (cid:9) f ((cid:9)), the second EOF for pure fluc-
2 vectorf ((cid:9))is(atmost)oforderh2,sof ((cid:9))isalways
tuationsinjetposition.However,forintermediateval- a highe1r-order EOF than E((cid:4))((cid:9)). The1 ordering of
ues of the ratio, E((cid:10))((cid:9)) is necessarily a mixture of E((cid:10))((cid:9))relativetof ((cid:9))dependsontherelativemag-
f ((cid:9))andf ((cid:9)),thatis,ahybridstructurethatdoesnot 1
0 2 nitudes of the variance of the PC corresponding to
correspond to an EOF found in either of the cases of E((cid:10))((cid:9)):
purejetpositionorstrengthfluctuations(Fig.7).Forall (cid:6) (cid:8)(cid:3) (cid:4) (cid:7)
values of the ratio 3h4/8l2, E((cid:4))((cid:9)) is also a hybrid of (cid:14)(cid:7)(cid:3)U2 3h4 3h4 2
the vectors f ((cid:9)) and f ((cid:9)). Fluctuations in both (cid:14)(cid:1)(cid:10)(cid:2)(cid:3) 0 0 l2(cid:10) (cid:10) l2(cid:10) (cid:4)l2h4
0 2 2 8 8
strength and position are combined inextricably in the
structureoftheEOFs.ThestructureofthedipoleEOF (cid:1)46(cid:2)
of jet position variations, on the other hand, is unaf-
fected by fluctuations in the jet strength. (appendix A), and that of the PC corresponding to
Notethatthesecalculationsallowusnotonlytoiden- f ((cid:9)), (cid:14)(cid:30)(cid:8)U2h2/2:
1 0 0
(cid:10) 2(cid:14)(cid:1)(cid:10)(cid:2)
(cid:5)f1(cid:1)(cid:1)(cid:2),E(cid:1)(cid:10)(cid:2)(cid:1)(cid:1)(cid:2),E(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:2)(cid:6) if (cid:14)(cid:7)(cid:3)U2(cid:15)h2
(cid:5)E(cid:1)1(cid:2)(cid:1)(cid:1)(cid:2),E(cid:1)2(cid:2)(cid:1)(cid:1)(cid:2),E(cid:1)3(cid:2)(cid:1)(cid:1)(cid:2)(cid:6)(cid:3) 0 0 (cid:1)47(cid:2)
2(cid:14)(cid:1)(cid:10)(cid:2)
(cid:5)E(cid:1)(cid:10)(cid:2)(cid:1)(cid:1)(cid:2),f1(cid:1)(cid:1)(cid:2),E(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:2)(cid:6) if (cid:14)(cid:7)(cid:3)U2(cid:16)h2.
0 0
(cid:3) (cid:4)
Asitisthefirstofthesetwocasesthatisrelevanttothe (cid:14)(cid:7) 1(cid:11)2
midlatitude tropospheric jets, we will assume this or- (cid:12)(cid:1)1(cid:2)(cid:1)t(cid:2)(cid:3) 2(cid:3) (cid:5)U0(cid:10)(cid:4)(cid:1)t(cid:2)(cid:6)(cid:5)(cid:1)t(cid:2)(cid:10)h.o.t. (cid:1)48(cid:2)
0
dering for the remainder of this section.
WhilethespatialpatternofthedipoleEOFarisingin
the case of pure jet position fluctuations is unaffected Variationsof(cid:11)(t)changetheoverallamplitudeofu((cid:9),t),
by the addition of jet strength fluctuations, the associ- so they must project on the dipole EOF. Clearly, the
ated principal component time series couples both time series of the dipole EOF cannot be simply inter-
forms of variability: pretedasreflectingvariabilityinjetpositionalone(al-
6418 JOURNAL OF CLIMATE VOLUME19
FIG.8.Scatterplotsofnumericallycalculated(cid:22)(1)(t)/U0vs(cid:22)(2)(t)/U0forh(cid:3)0.26and3h4/8l2(cid:3)0.1,0.25,1,2.5,and10.Darkdots
denotethosepointsforwhich(cid:11)(cid:31)0;lightdotsdenotethosepointsforwhich(cid:11) 0.
though for relatively weak fluctuations in jet strength nent time series, (cid:22)(2)(t), is even more complicated be-
the differences will be small). cause E(2)((cid:9)) is a hybrid of f ((cid:9)) and f ((cid:9)):
0 2
The interpretation of the second principal compo-
(cid:1) (cid:2) (cid:3) (cid:4)
(cid:12)(cid:1)2(cid:2)(cid:1)t(cid:2)(cid:3)(cid:13)(cid:1)(cid:10)(cid:2)(cid:1)(cid:14)(cid:7)(cid:3)(cid:2)1(cid:11)2 (cid:4)(cid:1)t(cid:2)(cid:4)U0 (cid:5)(cid:5)2(cid:1)t(cid:2)(cid:4)w2(cid:6)(cid:4)(cid:5)2(cid:1)t(cid:2)(cid:4)(cid:1)t(cid:2) (cid:10)(cid:13)(cid:1)(cid:10)(cid:2) 3(cid:14)(cid:7) 1(cid:11)2(cid:5)U0(cid:10)(cid:4)(cid:1)t(cid:2)(cid:6)(cid:5)(cid:5)(cid:1)t(cid:2)2(cid:4)w2(cid:6)(cid:10)h.o.t.
0 0 4(cid:3)2 4(cid:3)2 2 (cid:3) 4(cid:3)
0 0 0 0
(cid:1)49(cid:2)
RegardlessofthedegreeofalignmentofE(2)((cid:9))along E(1)((cid:9)) and E(2)((cid:9)). It is also clear that both fluctua-
either f ((cid:9)) or f ((cid:9)), the PC time series (cid:22)(2) is an in- tions in strength and position generally project along
0 2
extricablemixtureofvariabilityinbothjetstrengthand E(2)((cid:9)),precludingitsinterpretationintermsofeither
position.Onlyinthelimitingcasesoflkh2andlKh2 individually.
canthistimeseriesreasonablybeinterpretedasreflect-
ing (to leading order) fluctuations in strength or posi-
7. Dependent fluctuations in strength and width
tion, respectively.
To illustrate this coupling of strength and position Consider now the case in which only fluctuations in
fluctuationsinthePCtimeseries,104realizationsofthe positionarezero(i.e.,(cid:18)(cid:26)0,w(cid:26)0,(cid:19)(cid:26)0),andwhere,
field(1)withh(cid:3)w/(cid:8) (cid:3)0.26weremadeforarangeof as suggested by the observations, the fluctuations in
0
valuesofl(cid:3)(cid:18)/U [selectedsuchthatf ((cid:9))remainsthe strength and inverse width are correlated, that is,
0 1
leading EOF], and the time series (cid:22)(1) (t) and (cid:22)(2) (t) (cid:11)(t) (cid:3) (cid:15)(cid:13)(t). Then,
(cid:1) (cid:2)
were calculated. Scatterplots of these time series are
(cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
plotted in Fig. 8, conditioned on the sign of (cid:11)(t) (dark u(cid:1)(cid:1),t(cid:2)(cid:3)(cid:5)U (cid:10)(cid:17)(cid:6)(cid:1)t(cid:2)(cid:6)exp (cid:4) 0 (cid:5)1(cid:10)(cid:6)(cid:1)t(cid:2)(cid:6)2 .
for(cid:11)(cid:31)0,lightfor(cid:11) 0).ItisevidentinFig.8that,for 0 2(cid:3)2
0
sufficientlyweakfluctuationsinjetstrength,thedistri-
(cid:1)50(cid:2)
bution clusters around the parabolic curve associated
with the projection of position fluctuations on both The time-mean jet is
(cid:5) (cid:3) (cid:4)
(cid:10) 1 (cid:6)2
(cid:20)u(cid:1)(cid:1)(cid:2)(cid:21)(cid:3) (cid:4)(cid:10)(cid:14)2(cid:7)(cid:9)2exp (cid:4)2(cid:9)2 u(cid:1)(cid:1),t(cid:2)
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:3) (cid:17)(cid:9)2(cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2 1 (cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2
(cid:3)(cid:14)(cid:3)2(cid:10)(cid:9)20(cid:1)(cid:1)(cid:4)(cid:1)(cid:2)2 U0(cid:4)(cid:3)2(cid:10)(cid:9)2(cid:1)(cid:1)(cid:4)0(cid:1)(cid:2)2 exp (cid:4)2(cid:3)2(cid:10)(cid:9)2(cid:1)(cid:1)(cid:4)0 (cid:1)(cid:2)2 . (cid:1)51(cid:2)
0 0 0 0 0 0
As before, the mean jet is distinct from the instantaneous jet with mean width. Calculating the covariance
function, we find that