JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. C10, PAGES 18,361-18,379,OCTOBER 15, 1991
Variability of the Wind StressCurl Over the North Pacific'
Implicationsfor the OceanicResponse
ALAN D. CHAVE
1
AT&T Bell Laboratories,Murray Hill, New Jersey
DOUGLAS S. LUTHER AND JEAN H. FILLOUX
ScrippsInstitutionof Oceanography,
La Jolla, California
The subinertialfrequency-wavenumber
structureand spatialcoherenceof wind stresscurl, and their spatial
andtemporalvariability,are describedwith the emphasis
on characteristics
whichaffectthe ocean'sresponse
to
the wind stresscurl forcingfunction. Wind stresscurl over the North Pacificbetween25ø and 60øN was computedfor 3 years(1985-1988) from the Fleet NumericalOceanography
Centerwind product. The rangefrom
winterpeakto summertroughfor the meanandvarianceare typicallya factorof 3-4 and= 10 respectively,with
superimposed
interannualchangesof up to a factorof 2. Powerspectravary seasonally
andinterannually
in an
essentiallyfrequency-independent
mannerthatis consistent
with the curl variance.However,the spectrumover
the periodband5-100 daysoften fails a statisticaltest for whitenessat the 95% level. Zonal and meridional
wavenumberspectrawereestimatedfor 13 sitesdistributedaroundthe central-eastern
North Pacific usingthe
maximumlikelihoodmethod. Directionaltrendswith frequencyare comparableto thosefrom earlierstudiesif
3-year-longdatasegments
are analyzed,with approximatewavenumbersymmetryexceptfor the zonalterm at
mid-latitudesfor periodsshorterthan 10 days,whereeastwardpropagationis dominant. However,spectrafor
shorterdatasectionssometimesdisplayeastwardand westwardexcursionswhichare only weakly similarover
distancesof =1000 km and essentiallydissimilarover longerseparations.The spatialcorrelationstructureof
wind stresscurl is showntypicallyto have a main lobe of =1000 km and multipleintercorrelation
lobesseparatedby > 2000 km. The strengthandlocationof the intercorrelation
peaksvary slowlywith time. Theseresults
suggestthatcurl behavioris morecomplexthanwaspreviouslybelieved,thatthe useof long-termaveragesof
or simpleparameterizations
for the frequency-wavenumber
spectrumof wind stresscurl in modelstudiesmay be
unrealistic, and that more attention to actual curl characteristicsat the time oceanic measurementsare collected
will be requiredto reconcilemodelswith observations.
1. INTRODUCTION
simplycurl) at periodslongerthana day and for horizontalscales
largerthan a few hundredkilometers. In fact, the model studies
Much of the barotropicvariabilityof the deepoceanat periods cited above were all curl forced. The numerical result of Willeof daysto monthsis directly,atmospherically
forcedexceptnear
brand et al. [1980] employedgeostrophicwinds computedfrom
regionsdominatedby the instabilities
of strongmeancurrentsand
surfaceair pressuremapsas the input, while the remainingmodthe concomitant radiation of mesoscale eddies. This assertion is
els are basedon the stochasticapproximationwith the characterissupportedby several model studieswhich predict coherence
tics of curl specifiedby an analyticfrequency-wavenumber
specbetweenoceanicand atmosphericvariablesthat is either local
trum. In general,the resultsare sensitiveto both the frequency
[Willebrand et al., 1980; Muller and Frankignoul, 1981] or nonand the wavenumbercontentof curl. This is especiallytrue for
local [Brink, 1989; Satnelson,1989], dependingon topography,
the zonal wavenumberspectrumbecauseof the stronginfluence
the oceanicvariable under consideration,and the frequencyand
which the amountof power in westwardwavenumbershason the
vector wavenumberof the forcing which dictate whether an
excitationof free Rossbywaves. Thus, a gooddescriptionof the
evanescent
or free Rossbywave responseis possible.In addition,
spatial and temporalvariability of curl is essentialboth for the
observationalevidenceis accumulatingfor both local and nonlogenerationof realisticmodelsandfor the interpretation
of oceanic
cal coherenceof oceancurrentsand the wind forcing in areasof data.
weak eddy variability[Niiler and Koblinsky,1985;Koblinskyet
A numberof attemptshave been made to estimatethe spatial
al., 1989; Brink, 1989; Luther et al., 1990; Satnelson,1990].
andtemporalscalesof the windson a globalor hemisphericbasis
Philander [1978] and Frankignoul and Muller [1979] showed
outsidethe surfaceboundarylayer [e.g., Willson, 1975]. Together
that the dominantatmosphericforcingvariablefor extratropical
with some theoretical arguments,these results were used by
latitudesis the verticalcomponentof wind stresscurl (hereinafter
Frankignoul and Muller [1979] to constructsurfacefrequencywavenumberspectraof the atmospheric
variablesat mid-latitudes
for periodsof daysto months.However,suchestimateslack horiINowat Woods
HoleOceanographic
Institution,
Woods
Hole,Mas- zontal resolutionof small-scalewind features,which has imporsachusetts.
tant consequences
for computationof curl, and in any caseare not
basedon measurements
of the windsin the surfaceboundarylayer
which are mostrelevantto the oceanographer.Willebrand[ 1978]
Copyright1991 by theAmericanGeophysicalUnion.
inferred the frequency-wavenumber
characteristics
of the geostrophicwinds from synopticsurfacepressuredata collectedover
Papernumber91JC02152.
0148-0227/91/91JC-02152505.00
the oceansduring 1973-1976. His resultssuggesta symmetric
18,361
18,362
CHAVEetal.:WINDSTRESS
CURLVARIABILITY
zonal wavenumberspectrumfor curl at periodslonger than =10 present. When 6-month data sectionsare analyzed,eastwardor
days at mid-latitudeswith a corresponding
white frequencyspec- westwardpropagationof curl is noted at different times with no
trum. At shorterperiods,eastwardtravelingcyclonesand anticy- obviousseasonaldependence;
for longertime seriesthesemotions
clonesare dominant,yielding an increasinglyasymmetriczonal average away to produce wider apparent wavenumber peak
wavenumberspectrumfor curl as period decreasesand introduc- widthswith no preferredpropagationdirection. Section6 examing rednessinto the frequencyspectrum.Southof =35ø and north inesthe spatialcorrelationstructureof curl over the North Pacific,
of =50ø the shortperiod asymmetrydisappears.The meridional findingthatsmall-scale(= 1000 km) coherentpatchesseparated
by
wavenumberspectrumof curl is approximatelysymmetricwith a larger (_•2000km) distancesappearto be ubiquitous,so that simof curl's autocorrelation
are unreslight northwardbias at all periods. Finally, MacVeigh et al. ple unimodalparameterizations
[1987] computed curl from 4 years of European Center for alistic. The final sectionof the paperis a discussion
of the conseMedium Range Weather Forecast(ECMWF) productto examine quencesof these analysesfor modeling atmospherically-forced
both large-scalespatialvariability and grosswavenumberproper- oceanicmotionsat subinertialperiodsout to 100 days.
ties excluding directionality, showing some agreement with
Willebrand's
result.
2. THE WIND OBSERVATIONS
It appears that most of the available information on the
frequency-wavenumberstructureof curl at periods of days to
months is summarized by the work of Willebrand [1978] and
Frankignoul and Muller [1979]. However, a number of recent
investigationsof monthly mean oceanicwind data have appeared
in the literature. These studiesare typically concernedwith basin
scale spatial patternsand the annual or semiannualcomponents
based on empirical orthogonalfunction analysis covering the
North Atlantic [e.g., Ehret and O'Brien, 1989] and the North
Pacific [Rieneckerand Ehret, 1988]. Only Gallegos-Garciaet al.
[ 1981] estimatedcurl wavenumberpropertiesfrom monthlymean
data (with results that are somewhat discordant with those of
The raw datausedin this studyconsistof 3-yearsequences
of
the vector wind velocity at 405 locations distributedover the
North Pacificandobtainedfrom the Fleet NumericalOceanography Center. The FNOC windsare calculatediterativelyfrom surface atmosphericpressuredata blended with wind observations
usinga combinationof objectiveanalysisand prognosiswhich
includesageostrophiceffects and an adjustmentfor frictional
effectsin the surfaceboundarylayer. An early versionof the
FNOC algorithmis describedby Lewisand Grayson[1972]. The
FNOC productis reportedevery6 hourson a polarstereographic
grid at an altitudeof 19.5 m. The time serieswere nearlycomplete for the intervalJuly 1985 throughJune1988, with missing
dataconstituting
lessthan 1% of the total andcoveringonly brief
intervalswhichcouldeasilybe linearlyinterpolated.The sample
spacingis uniformon a stereographic
grid and henceunevenin
geographiccoordinates,but is typically 340 km nearthe centerof
the regionof interest(40øN, 162øW)with smaller(larger)values
MacVeigh et al. [1987]). However,thesestudiesshedno light on
the behaviorof curl at periodsshorterthantwo months.
The purposeof this paper is to begin to fill this void in the literatureby examining the spatial and temporal characteristicsof
curl over the North Pacific during 1985-1988. The wind data are
obtainedfrom the Fleet Numerical OceanographyCenter (FNOC)
data product, which is more complete than the measurements
availablein earlier short-periodstudies. The primary motivation
for the work is a critical examinationof the variability on time
scalesshorterthan the seasonal,where the oceanicresponsemay
either be evanescentor resonantdependingon the wavenumber
characteristics
of the forcing. A secondarygoal is to serveas an
aid for the interpretationof an arrayof seafloorpressureandbarotropic currentdata collectedduring the BarotropicElectromagnetic and PressureExperiment(BEMPEX), which was designed
to study atmosphericforcing of the deep ocean [Luther et al.,
1987]. The BEMPEX data have already been found to be
strongly coherentwith local and/or nonlocal FNOC surface air
pressure,wind stress,and wind stresscurl [Luther et al., 1990;
A.D. Chave et al., The BarotropicElectromagneticand Pressure
Experiment,1, Barotropiccurrentresponseto atmosphericforcing, submittedto Journalof GeophysicalResearch,1991, here-
obtainthe FNOC product,it is difficult to quantifythe accuracy
or smoothingscalesof the resultingwind estimates.Ship observations of air pressure,wind velocity, and wind directionare a
major inputfor the FNOC product,andrecentstudiesindicatethat
significanterrorsoccur in such data with distressingfrequency
[e.g., Wilkersonand Earle, 1990]. However, the analysisprocedure is designedto correctfor bad data in a self-consistent
manner, and the relative successof this approachis attestedto by a
numberof intercomparisons
betweenthe FNOC windsand either
otherdataproductsor actualmeasurements.For example,Friehe
and Pazan [1978] found good agreementbetweenNOAA buoy
data and the FNOC product in the North Pacific near 35øN,
155øW. De Youngand Tang [1989] contrastedthe FNOC winds
with measurementsfrom an oil platform on the Grand Banks,
inafter referred to as Chave et al. ( 1991)].
In the next section, the FNOC data and estimates of curl
finding high correlations(=0.9) at periodslonger than 2 daysand
measurablediscrepanciesin directionat wind speedsunder 10 m
derived from them are outlined.
Section 3 constitutes an exami-
s-• andfor the summermonths.Brink[1989]compared
mea-
nation of the mean and variances of the curl field over the North
suredwindsobtainedwith mooredbuoysin the subtropicalNorth
Atlantic during the Frontal Air-Sea Interaction Experiment
(FASINEX) with three data products: the FNOC values, the
NOAA Analysisof the Tropical Ocean Lower Layer (ATOLL)
winds, and a simpler geostrophicestimate. Of the three, the
FNOC winds were the most highly correlatedwith the measured
Pacific for various averaging times with the purposeof both
emphasizingthe seasonalrange and relating the resultsto the
dominantatmosphericprocesseswhich controlNorth Pacific climate. Section4 discussesthe frequencyspectrumof curl in the
periodrangeof a few daysto a few months,showingthat whitenessat periodsgreaterthan 5-10 days is at best approximate.A
weak concentrationof variance near 30 days period observed
throughout the central-easternNorth Pacific is also described.
Section 5 presentsmaximum likelihood zonal and meridional
wavenumberspectrafor variousaveragingintervalswhich suggest that significanttemporaland spatialcurl inhomogeneityis
to the north (south).
Because of the complex data assimilationmethod used to
winds(192>
0.9) andappeared
to reproduce
themwithacceptable
accuracy.Halliwell and Cornilion [ 1990] performeda more thorough analysisof the in situ, FNOC, and ATOLL winds during
FASINEX and found that measuredaverageswere best reproducedby the FNOC values. However, Pazan et al. [1982] found
poor agreementof the FNOC productwith measuredwindsin the
CHAVEetal.:WIND STRESS
CURLVARIABILITY
equatorial Pacific where actual observationsare very limited.
Thus, it appearsthat the FNOC productyields usableestimatesof
surfacewinds over the ocean at periodslonger than a few days
and when wind velocities are moderate to large, especially in
regions where ship traffic is relatively dense, such as the midlatitude Noah
Pacific.
The 19.5 m FNOC winds were iteratively transformedto wind
stressat a heightof 10 m usingMonin-Obukhovsimilaritytheory
and a velocity-dependentneutral drag coefficient under the
assumptionof constantair density. No directionalcorrectionfor
boundarylayer rotationwas applied. The drag coefficientformulation is comparableto the standardLarge and Pond [ 1981] value
for windvelocities
over3 m s-I , buthasa constant
plusinverse
velocity relationshipin weak windsas discussed
by Trenberthet
18,363
3. MEAN AND VARIANCE OF CURL
Figure 1 showsthe curl meanfor the 3-year intervalJuly 1985
throughJune 1988. In the sequel,all contourmapswill be shown
on the original uniform FNOC polar stereographic
grid to avoid
interpolationerror, with selectedlatitudeand longitudelines and
outlinesof the continentsincludedas an aid to the eye. Specific
grid locationswill be referredto by a four digit numberwherethe
first (last) two digits correspondto the meridional(zonal) indices
on the axesof the figures. Both the spatialpatternsand the magnitudesof the curl meanin Figure 1 are similarto the recentcom-
pilationof Harrison[1989],whichis not surprising
giventhathe
usedessentially
the samedragcoefficientas in the presentstudy.
The major exceptionoccursnearthe west coastof North America
whereFigure 1 showsa southwarddeflectionof the zero contour;
al. [ 1989].
this is the result of a pervasivehigh-pressurecell over the southThe wind stresscurl was computedusing a two-dimensional
centeredfinite differenceoperatoron a sphere. Considera functionf(qb,X,),whereqbis longitudeand X,is latitude,and expandit
in a Taylorseriesfor smallincrements
œand15
f(qb+œ,•,+8) = f(qb,•,) + A'Vlf(qb,•,) +
«(A'Vl)2f(qb,•,) + ...
(•)
where
A=[aœcosX,,
aiS]
and the surfacegradientoperatoris givenby
a as the radius of the Earth.
To determine
presentdata. Slightlybetteragreementis seenbetweenFigure 1
and the mean computed from the ComprehensiveOceanAtmosphereData Set (COADS) by Rieneckerand Ehret [1988],
especiallynear North America. Figure 2 showsthe variancefor
the same3-yeartime interval. The varianceor totalpoweris largest immediatelysouthof the Aleutian Islands,reflectingthe influenceof the Aleutian low to the northand the subtropicalhigh to
the southeast
on averagestormtracks. Note that the variancecontoursshownin Figure 2 are at leastan order of magnitudelarger
than thosegiven by Rieneckerand Ehret; this is simply due to the
useof 6-hourlydata in this studyas opposedto monthlyaverages
for their result.
V• = acosX,
*' a
with
western U.S. land mass whose influence has been included in the
the first and
Figure 3 compares the 3-year averaged summer
(July-September),
fall
(October-December),
winter
(January-March),and spring(April-June) variancesfor curl. The
rangeexceedsa decadeat mid-latitudeswith the largestvaluesin
winter and the smallestonesin summer,but with the peak located
secondderivativesin (1), five equationsand hence six data are
needed,while the FNOC grid yieldsonly five data if a crosscentered on the point of interestis used. An additionaldatumcould somewherein the middle of the basinsouth of the Aleutian
be obtainedby samplingone of the four comersof the squarecirIslands. The variance reflects the climatology of the North
cumscribingthe cross,but there is no a priori reasonto choose
Pacific which is dominatedby the Aleutian low duringwinter and
one over the others. For this reason, the derivatives were esti-
matedby usingeachof the four comersin turn and averagingthe
resultsto minimize possiblebias errors. The standarderror com-
puted
from
the
four
derivative
estimates
was
typically
less
than
one
tenth
of
the
mean.
The
vertical
component
of
the
curl
of
the
.•
wind
stress
•Vx•'•
in
spherical
coordinates
from
•
= acosX,
1 I3,'rxfollows
- 3x(cosX,
•:,)1
(2) •
• •
20 "I•-
-
andyieldsestimatesat3251ocationsafterex
•_•l• / / "••/
• •///•0ø
deroftheoriginal
grid.This
approach
provides
explicit
control 1•
ofthe
derivative
error
(unlike
with
two
dimensional
polynomial
approximations),
does
not
require
afinal
rotation
togeographic
.• 16
coordinates,
and doesnot ignorethe mixed derivativein the sec-
ondorder
term
aswould
a more
conventional
finite
difference
• !4
approach.
It should
be notedthattheapplication
of spatial
derivative
operatorsto a smoothedatmosphericvariableresultsin truncation
of high wavenumbercomponentsand possiblereductionof the
wavenumber bandwidth [Willebrand, 1978]. This effect could be
40
B5
Zonal
30
25
Grid
Point
20
severefor estimatesof curl but is difficult to quantify on account
of the spatiallyvariableand data-dependent
smoothinginherentto Fig. 1. Contour map of time-averagedcurl for the interval July 1985
the FNOC product. Based on algorithmicdescriptionsprovided through June 1988 over the North Pacific Ocean shown on the FNOC
grid. Selectedlatitudeand longitudelinesand the outby Lewis and Grayson[1972], the smoothingscalevariesfrom polarstereographic
300 to 850 km. Consequently,inferencesaboutthe propertiesof
curl at wavelengths smaller than roughly 500 km must be
regardedasof questionable
reliability.
lines of North America, the Kamchatka Peninsula, Sakhalin Island, and
Japanare shownas aids to the eye. The axesare labeledwith the FNOC
meridionaland zonal grid points. The contourshave units of dynesper
cubic
centimeter
scaled
by109.
18,364
CHAVEetal.:WIND STRESS
CURLVARIABILITY
- '
• 24 -.
I • -
•
' c.•
the subtropicalhigh duringsummer[Teradaand Hanzawa, 1984].
>From late fall throughwinter, the low is centerednear 50øN,
175øE and extends acrossmost of the basin; it is particularly
intensebecauseof the frequentsoutheastward
passageof depressionsfrom the Bering Sea and Asia. At the sametime, the subtropicalhigh shrinksand is pushedwell to the southeast,
being
,._ /5 0ø
ß
500
centered near 30øN, 140øW. The effect of the low on storm
[
_
\
I
I
I
40
35
30
25
Zon•z•
Cr•;d
Point
0C
20
tracks and cyclogenesisis reflected in the fall and winter curl
variancemaps of Figure 3. By contrast,in the summerthe subtropicalhigh coversmostof the North Pacific andis centerednear
40øN, 150øW with the Aleutian low pushed well into polar
regions. This resultsin a substantialreductionin stormfrequency
and intensity,and a markeddecreasein the curl variance. The fall
and springvariancesare transitional,respectivelyreflectingthe
rapid growth of the low- and high- pressuresystems. Note the
differencesbetweenthe seasonalaveragesof Figure 3 and the 3year values in Figure 2. The latter most closely resemblethe
fall-winter values, which is not surprisinggiven the nonrobust
nature of the variance statistic.
Fig. 2. Contourmap of curl variancefor the interval July 1985 through
June 1988 over the North PacificOceanshownon the FNOC polar stereo-
Comparable seasonal changes in the curl mean are also
graphic
grid.Thecontours
haveunitsof dynes
2 cm-6 scaled
by 1018 observed.Duringthe summer,the zerocurl contourextendsfrom
the coast of Alaska at 55øN to 35øN in the west half of the basin.
Otherwiseplottedas in Figure 1.
fall
.
• _
• 24
•
.
7d •.
50 ø .
½ 24
•
/50ø,,.
22
• 20
ß•
20
18
••4
I
•
40
35
l?Qø ,..
I
30
Zonal
O•{d
1140 c
25
•12
170 ø ,,,...
40
20
35
30
•
25
\ 140 ½
20
Point
tv•nter
-
v
½ 24
'• 24
r• 22
r• 22
ß•
• 20
'to
20
18
_.
•16
.•
14
_....?
300
170 ø ....
40
35
Zo•a[
,
30
25
G?•ct
Po{•zt
\ 14oø•
20
ß
40
\140
170 ø
35
30
25
Zor•c•L
Gr•,ct
Po•;•zt
20
Fig. 3. Contourmapsof seasonalcurl variancefor the intervalJuly 1985 throughJune1988overthe North PacificOceanshown
on the FNOC polarstereographic
grid. Eachpanelgivesthevariancefor 3 monthsof the yearaveragedover3 years,with summer,
fall,
winter,
andspring
beginning
atthestartof July,October,
January,
andAprilrespectively.
Thecontours
haveunitsof dyn2
18
cm 6 scaled
by10 . Otherwise
plotted
asinFigure
1.
CHAVEetal.:WIND STRESS
CURLVARIABILITY
18,365
2328
•-
2027
f6
•10-
_
_
10-•?
-------I I I IllIll
10-•?
I I I 111111 I I I 111111 I I I II1111 I
10-3
10-2
10-½
-=i i ii11111 i i i iiiiii
I i i iiii1! I I I Iiiiii
10-3
10-•
10 o
Frequency (d-•)
10-,e
! i
10 0
Frequency (d- •)
1330
1830
10-•'3
10-•3
i
I
10
½ 10
_=_
==
95
95%
_
_
½
-•5
•
-½5
•'
•10
-16
•10
_
•0-•?
111111 i i i IIIIii
10-3
i i i IIII11
10-2
i i i 11!111 I i
10-½
10 o
Frequency (d- •)
10-•?
• I I I III111 I I I IIIIII
10 -3
I I I IIII1,1 I I IIIII11 .....
I I
10 -2
1O- ½
10 ø
Frequency (d- •)
Fig.4. Multiplewindowpowerspectra
of windstress
curlfor theintervalJuly1985through
June1988at theFNOCsitesas
labeled
withtheirmeridional
andzonalgridpointnumbers.
Theunitsoffrequency
arecpd,whilepower
isindyn2 cm-6 percpd.
Thetime-bandwidth
oftheestimate
is4,corresponding
toa spectral
bandwidth
of0.0072
cpd,andeachfrequency
has--14degrees
of freedom,
yielding
a double-sided
95%confidence
interval
of (0.54,2.48)
timestheestimated
powerforGaussian
data.The
bandwidth
oftheestimate
centered
ona 30dayperiod
isdelimited
bytheshort
horizontal
bar.Notetheapproximately
whitebackgroun
d forperiods
longer
thana fewdayswithweakpeakssuperimposed,
especially
at30days.
Positive curl is restricted to the environs of the Aleutian chain and
latitude North Pacific. Thus, year 2 standsout as being more
energeticthroughmost of the seasons.Interestingly,a weak E1
about 35øN in the center of the basin with northward curvature at
Nin0 event occurredin 1986-1987 [Kouskyand Leetmaa, 1989],
the eastand west ends,and the regionof positivecurl dominates which could'hayeaffectedthe climateover the mid-latitudeNorth
most of the mid-latitudeocean. The winter curl peak value is at Pacificin late1987to early1988duringyear3.
least4 timesthe summerone. The fall and springmonthsreflect
It shou.
ld be notedthat long-termaveragessuchas thoseof Figthetransition
between
these
extremes.
The3-yea.r
average
ofFig- ures 1 and 2 may be of limited value in inferringthe magnitudeof
ure 1 is intermediate between the seasonal extremes.
the ocean'stransientresponse
•toatmospheric
forcingin any given
The interval July •986 throughJune 1987 (hereinafteryear 2 season
or year. It is clearthatseasonal
variability
of curl is
butalsothatinterannual
changes
cannotbe ignored.
and nearlycoincidentwith the BEMPEX experiment)also pre- significant,
sentssomecontrastswith the two surroundingyears. During the
first 9 monthsthe mean is typically 50% largerand the varianceis
4. FREQUENCYSPECTRAOF CURL
doublethe 3-year values,especiallynear the middle of the Pacific
basin, while during the springof 1987 the differencesdecrease
Frequencyspectraof curl appearqualitativelyto be white for
somewhatexcept in the Gulf of Alaska. Year 1 (July 1985 periodslonger than a few days, with some additional,locationthroughJune 1986) is only slightlyweaker than year 2 in all sea- specific differencesbeing evident. The level of the spectrum
sons,while year 3 (July 1987 throughJune 1988) presentsuni- varies with both positionand seasonin an essentiallyfrequencyformly .weakercurl (by about a factor of 2) throughoutthe mid- independent
manner.For a givenlatitude,the poweris largest
north. By contrast,in the winter the zero curl contour lies at
18,366
CHAVEetal.:WIND STRESS
CURLVARIABILITY
nearthe middle of the basinand weakerby a factorof 2-4 at the a necessaryand sufficientcheckfor spectralwhiteness.The M
edges,whilefor a givenlongitude
thepowerdecreases
by overa
statistic is
decadefrom 55øN to 30øN. The seasonalrangeat a singleloca-
tion andfrequency
is typicallya factorof 2-4, with largerexcursions evident at more northern latitudes.
M = Nlog[•
i=1
viSi]+ •v/log[St]
i=1
(3)
of freedom
forthespectral
However,inspectionof high-resolution
spectrarevealssome whereN = • v i andv i is thedegrees
real featuressuperimposed
on the white backgroundwhich estimate
Si at the i-th frequency.For smallsamples,
Bartlett
deservecloserattention.Figure4 showspowerspectrafor four showed that M/C, where
locationsestimatedwith the multipleprolatewindowexpansion
of Thomson
[ 1982]using3 yearsof curldata. A multiplewindow
spectrum
is computed
by firstgenerating
a setof orthogonal
data
windowswhichareoptimalin a minimumspectralleakagesense
distributed
asZ2 fork-1 degrees
of freedom.
andwhichdependexplicitlyon thetime-bandwidth
product(reso- is approximately
1 [•1 1]
C= 1+ 3(k-l) i=•vi N
(4)
lution bandwidthtimes data serieslength). These windowsare Table 1 showsthe resultsof this test appliedto 13 spectracom-
eachappliedto thedataastaperspriorto Fouriertransformation,putedfrom3-yearcurltimeseries.A periodrangeof 5-100 days
and since the windows are orthogonal,the raw estimatesare wasusedto avoidanybiasfromtheweakannualandsemiannual
andfromthe short-period
roll-off. The resultis not
approximately
independent.
Theraw spectra
arecombined
using components
to thechoiceof short-period
cutoffoverthe5-10
a setof weigh,ts
chosen
to minimizebroadband
bias(spectral
leak- overlysensitive
the spectral
powerat
age).Theresulting
spectrum
is effectively
theconvolution
of the dayinterval.To ensuredataindependence,
frequencies
separated
by
more
than
the
resolution
bandwidth
was
unknowable
true spectrum
with a rectangular
frequencydomain
werepooled,and
windowof widthspecified
by thechosen
time-bandwidth
product. used. For eachsite,estimatesat 25 frequencies
Thisprocedure
results
inanincrease
instatistical
efficiency
com- since
Z224(0.95)=36.4,
it is clearthatseven
of theM values
pared
withconventional
band-averaging
withcomplete
control
of exceed
thethreshold,
andhence
thecorresponding
spectral
estispectral
leakage.
mates
arenotdrawn
froma homogeneous
variance
population
at
Rieneckerand Ehret [1988] fit deterministicannualand semi- the 95% confidence level.
At
the 90%
confidence level
(0.90)= 33.2),anadditional
twositesfailthetest.Thereis a
annualtermsto 20 yearsof curldata,andfoundthattheytypically (Z2•4
account for less than 10% of the variance in the central-eastern
definite
biastoward
theupper
endoftheZ2range
forallsaveone
North Pacific. Usingan F testfor deterministic
components
to a
of the sitesreportedin the table,andthe valuesof M/C do not
scatteraroundthe expectedmeanof 24. Note that if the spectra
werewhite,thenonly onesitemightbe expectedto fail the testat
the90% confidence
level. Of thefourspectrashownin Figure4,
onlysite1830passes
the whiteness
test,asappears
visuallyreasonable.For the remaininglocations,
theM testresultis consistentwith visualimpressions.
The conclusion
thatcurl spectraare
oftennot formallywhitemustbe temperedwith cautionbecause
spectrum
devised
by Thomson
[1982]yieldsa significant
annual
term at only a singlesitein the Gulf of Alaska(2328 at 55øN,
146øW) and no significantsemiannual
terms. The remaining
sites,aswell asmanyotherswhichwereexamined,showonlythe
slightvarianceenhancement
at periodslongerthan--100 days
seenin Figure4, indicative
of theweakness
of thesemiannual
and
annualcomponents.
Weak spectralpeaksdo appearat shorter
periods
formanyeastern
NorthPacificsites,mostcommonly
near the Bartletttest is sensitiveto nonstationarity.However,it is not
immediatelytranslatesinto
30 daysas typifiedby sites2027 (45øN,147øW),1830 (41øN, clear that temporalnonstationarity
nonstationarity
of thespectrum,
although
it
162øW),and 1330(27øN,164øW)in Figure4. Thispeakis usu- frequency-dependent
ally significant
at aboutthe 80% level,andits shapereflectsthe will certainlyreducethe equivalentdegreesof freedom. The
bandwidthof the spectralestimateratherthanof the underlying Bartlett test is also conservative in the sense that the data are not
physicalprocess.The 30-daypeakis verycommonthroughoutranked;a trend(suchas a generallynonzerospectralslope)will
the area,andis notremovedwhenspectraarecomputedusingthe affectthe outcomeonly to the extentthat it altersthe variance
robustmethodsof Chaveet al. [ 1987], suggesting
thatit is not an rangeandnotby orderingthedata.
It is believedthat the flatnessof the low-frequencycurl specartifactof nonstationarity
in the curl time series.Note alsothat
trum
is due to dominanceby a whitenoiseextensionof a highthespectral
slope
forlocation
2328intheGulfofAlaska
is--f-
process
witha shortcorrelation
timescale,in thiscase
for periodsof 3-100 days,andthat a whitebackground
is not frequency
apparent
for location1330. The latteris common
for curlat lati- weatherfluctuations[Frankignouland Muller, 1979]. However,
tudesbelow--35øN,as is a weakerhigh-frequency
roll-off begin- real low-frequencyatmosphericfluctuationsmay also be
for the departures
from whiteness
seenin
ningat longerperiods
thanfor higherlatitudes.Thespectra
typi- significant,accounting
cally displayflatteningof the slopeabove1 cpd, suggesting
a Figures4 and5 anddetectedby the BartlettM test. In particular,
the weak 30 day peak probablyrepresents
a real low-frequency
noisefloorpossibly
dueto limitations
of theFNOCproduct.
atmospheric
Figure5 showssimilarmultiplewindowspectracoveringthe process.This may be importantfor understanding
samesitesas in Figure4 for only year 2. Sincethe time series
lengthis reducedby a factorof 3, the bandwidthof the estimate
risesproportionately,
andthebehaviorat periodslongerthan--90
days is obscuredby the weak annualand semiannualterms.
forcingof the ocean. For example,enhancedpowerin curl over
restrictedfrequencybandscouldresultin an increasedevanescent
or free wave response,dependingon both the frequencyand the
associated vector wavenumber characteristics.
Contourmapsof powerin a givenfrequencybandare qualitaHowever, increasedvarianceis evidentat 30 daysat all four sites,
and the varianceis enhancedat manyeasternNorth Pacificloca- tively like the variancemapsof Figures2 and3. Sincethe spectrum is almostwhite, suchmaps do not display any strongfretionscomparedwith the 3-yearaverage.
quencydependence.The seasonal
variationappearslike that in
Figure3 with somewhatlargerrangesin the eastthanin the west
arealsocomparable
to those
fact, Bartlett's M test [Bartlett, 1937] to check a populationof partsof the basin. Interyearchanges
independent
varianceestimates
for homogeneity
canbe appliedas in the simplevariance. Year 2 standsout in poweras compared
It appears
thattheusualassumption
thatthespectrum
of curlis
white at periodslongerthana few daysis only approximate.In
CHAVEetal.:WIND STRESS
CURLVARIABILITY
18,367
2328
2027
10-13
_
o-4
=-95%]
•"
95%
_
•"
-f6
•10
10-•?
•10
---I I I IIIIII
I I I IIIII1
10 -3
=-
I I I 111111I I IllJill
! 0 -2
! O-•
-f6
10-•?
I I
'• I I IIIIIII
I
10-3
!0ø
I
I IIIIII
I
I
10-2
I IllIll
I
I
I IIIIII
10-!
I
I
1o o
Frequency (el-j )
(d 1830
1330
10-13
10-13
10
-•4
o-4
==
95%
_
-•5
•..
•)10-
f6
•"
•10
-f6
_
_
_
_
_
10-•?
-• I I i iiiiii
I I i iiiiii
10 -a
i i i Illill
! 0 -2
I I I Illill
! O-•
10-•?
I I
-• I I I IIIIII
! 0ø
I I I IIIIII
10 -3
Frequency (d -
I I I IIIIII
! 0 -2
I I I IIIIII
! O-•
I I
!0ø
Frequency (el- •)
Fig. 5. Multiple windowpowerspectraof wind stresscurl for the singleyear intervalJuly 1986 throughJune 1987 at the FNOC
sitesaslabeled.The time-bandwidth
of theestimateis 4, corresponding
to a spectralbandwidthof 0.022 cpd,andeachfrequency
has--14degrees
of freedom,yieldinga double-sided
95% confidence
intervalof (0.54,2.48)timestheestimated
powerfor Gaussian
data. The bandwidthof the estimatecenteredon a 30 day periodis delimitedby the shorthorizontalbar. Note the strongerpresenceof the30 daypeak,especiallyat 1330,andtheexistence
of otherfine structure
onthewhitebackground
spectrum.
to the previousand the followingyears,especiallyin the central- cells in the form of a cross centered on the location of interest. In
eastern North Pacific.
eachcase,the frequencyspectrumwas first computedusingthe
multiple window technique,then the two dimensionalwavenum5. WAVENUMBER SPECTRA OF CURL
ber spectrumwascalculatedfor frequencies
spacedevenlyin the
The dominant atmosphericscalesat mid-latitudesare a few
thousandkilometersand hence only slightly smaller than the
Pacific basin. Because of this large scale, conventional
wavenumberestimatorslack adequateresolution to infer their
properties,a problemwhichis exacerbated
by spatialheterogeneity of power. For this reason,zonal and meridionalwavenumber
spectrafor curl were computedusing the maximum likelihood
method (MLM) of Capon [1969]. Unlike conventional
approaches,MLM employs a wavenumber spectral window
whoseshapeand sidelobestructureis data-dependent
and hence
changeswith the wavenumberin a well-defined,optimalmanner.
The resolutionof MLM dependsprimarilyon the incoherentnoise
presenton the sensorsand only weakly on the naturalbeampattern of the array, so it is typically much better than for conventional estimators. MLM spectrawere obtainedfor five-element
TABLE 1. M Testfor SpectralWhiteness
Site
M/C
90%
1934
35.7
x
1534
2332
25.5
25.1
1830
17.5
53.7
52.0
44.5
47.9
32.0
36.0
71.0
52.6
39.6
1330
2328
1628
2027
2324
1924
1524
1720
2019
x
x
x
x
x
x
x
x
95%
18,368
CHAVE
etal.:WINDSTRESS
CURLVARIABILITY
5-
s-0.4
3
6
.• -0.4
•
3
-0.6
5
5
'e
7 •,
'-- -0.8
.
10 •-
-1.,8
•
•
20
. •.•
•
•-1.4
30
•
• -1.6
5O
;•-1o
30
-1.6'
50
-1.8
•
ß
0
-1.2
•
• -1.8
100
-2.0
*-1
•
•
•
lOO
-2.0
-2.2
-2.2
-0.6
0.0
-0.6
0.6
0.0
0.6
E - F Wccu
ert,• wabe•
E-W Wavenumber (M'm.-•)
?/s ?- ?/s s
7/86-7/87
3
_3
.• -0.4
s
•-1o
•
?
-o.s
10•-
.
-1.2
20
-1.4
30
-1.6'
50
5
*-10
•
10•-.
ß
•
-1.2
t,,
•
-1.4
•
• -1.6
•
,20
•
30 •
50
• -1.8
-1.8
100
-2.0
•
•00
-2.0
-2.2
-2.2
-0.6
0.0
-0.6
0.6
E- W Wavenuwabew(M'm.-1)
0.0
0.6
E-W Wavenumber (Mwn.
-1)
Fig. 6. Zonal wavenumberspectrafor FNOC site 2328, locatedin the Gulf of Alaska, computedusingthe maximumlikelihood
method. The estimatesencompass
the time intervalJuly 1985 throughJune 1988 and the threeannualperiodscontainedtherein.
The frequencybandwidthis constantat 0.027 cpd for all of the spectra. The wavenumberhas units of cyclesper megameter.
Power is expressedin decibelsdownfrom the peakat eachfrequencyat the peakmeridionalwavenumber.This meansthat larger
numberscorrespond
to weakerpower. Note the wider apparentpeakwidth for the 3-year estimatewhen comparedto the year 1
and 2 results.
logarithmicdomain. Zonal (meridional)sectionsas a functionof
frequencyare thenpresentedat the meridional(zonal) wavenumber corresponding
to the peakin power. By usingdifferentlength
datasections,informationon the varyingpropagationcharacteristics of curl over time can be inferred. Davis and Regier [1977]
have shown that MLM is an accuratemethod to locate the peak
wavenumberbut may not be the best methodto estimateother
propertiesof continuous
spectrasuchashigh-frequency
slope. As
a consequence,
emphasiswill be placed on grosswavenumber
propertiesratherthanon spectraldetails.
Caponand Goodman[1970] haveshownthatMLM spectraare
priate to determinewhen the peak power is different from the
power at zero wavenumberat a specifiedconfidencelevel. Using
the definitionof a confidenceinterval gives
2
•p - •o = 10log
x,•,;
_
I
or/2
(5)
where
•t, istheestimated
peak
power
and•o istheestimated
power at zero wavenumber,both expressedin decibels(dB). A
similar test can be applied to changesin relative power between
different
time sections at a selected wavenumber
if it is assumed
Z2 distributed
withn =2(m-k + 1) degrees
of freedom
assuming that the peak power to which the spectrahave been normalized
that the data are approximatelyGaussian,wherem is the number
of independent
frequencyestimatesand k is the numberof sensors. The wavenumberspectrumis ordinarilynormalizedby the
peak value at each frequencyand plotted in the logarithmic
domain;for testingthe significanceof the peak,it is oftenappro-
does not change. When consideringsignificancetests in the
sequel,it shouldbe rememberedthat the degreesof freedomestimate is basedon the assumptionthat the k sensorsare independent. Due to the smoothedform of the FNOC product,this is
only approximatelycorrect,and hence the estimateddegreesof
CHAVEetal.:WIND STRESS
CURLVARIABILITY
freedomwith k = 5 will be a lower limit. At the sametime, curl
18,369
At the northernmost
site of Figure6 (2328) thereis a weak
time seriesare clearlynonstationary,
andhencefrequency
esti- suggestion
of a widerpeakwidthin the3-yearspectrum
thanfor
matescomputedfrom themwill havefewerdegreesof freedom individual
yearspectra.Comparison
of the yearlyspectra
indithanconventional
wisdomwouldsuggest.Thismeansthatsignif- cates that this is due to substantial interannual variation in behavicancetestsshouldbe treatedasapproximations
thatareusefulfor ior;the3-yearestimate
reflectsthelong-term
variability.The 3qualitativecomparison
of spectrafrequency-by-frequency,
and year spectrumappearsto be nearlysymmetricat all frequencies,
will notbe followedrigorouslyin whatfollows.
with onlyweaktendencies
for eastward
propagation
at periods
Zonalandmeridionalspectrawerecomputed
for 13 locations shorterthan 10 daysand westward
propagation
at 10-50 days
on the FNOC grid distributedbetween25øN and 55øN and eastof whichareobscured
by thewidthof thepeak. However,theindithe180ømeridian.Figures
6-9 showzonalwavenumber
spectra vidualspectrasuggest
eastward
propagation
at periodsof 6-12
at 16cations
2328,2027, 1830,and1330(thesamesitesfor which daysin year1 andwestward
propagation
atperiods
of 13-50days
powerspectra,are
displayed
in Figures
4-5) for theentire3-year in year2 and15-22daysin year3. Other,weakersuggestions
of
and individual1-yearintervalsusingidenticalresolutionband- eastwardandwestwardpropagation
are alsoevident.
widths.The 3-yearspectra
have32 nominaldegrees
of freedom
The two mid-latitude
sites2027 and 1830(Figures7 and 8)
whiletheannualoneshave10nominaldegrees
of freedom;
using displaya markedly
stronger
tendency
for eastward
propagation
at
(5) requires=2.3 and=4.2 dB difference
of thepeakfromthezero periodsshorter
than10-15 days.Thisholdsbothfor the3-year
wavenbmber value at 1 standard deviation for an individual esti-
andindividual
1-year
estimates.
Interannual
differences
doexist;
mate. Comparable
valuesmaybeusedto assess
changes
between notethatmorewestward
propagation
is evidentat periodslonger
time sections.
than10daysduringyear2 ascompared
withyears1 and3 in Fig-
?/a 5 - ?/a e
3
3
5
co -1.0
-1.0
c• -1.2
c. -1.4
•
-1.6
20
o
-1.2
30
•
-1.4
5O
-i.,½
100
.q -2.0
o
30 •'
-1.6
5O
•o -1.8
20
-2.0
-2.2
-2.2
-0.6'
0.0
0.6'
-0.6
E-W Wavertttrrtber (Mrrt-•)
0.0
0.6'
E- W Wavert•zrrtber(Mrrt- •)
?/a ? - ?/a a
3
•o
-1.2
-i.6
-2.2
3
-0.6
,•
7
•
i0•,
•• -0.8
20 o
•
30 •'
•
50
• - 1.2
20
o
•
30
•
100
1
-0.6
•
5
-1.8
-2.0
,• -0.4
5
co -1.0
-i.4
• -i.•
•
5O
• -i.8
o
•
1oo
-2. Q
-2.2
0.0
0.6
E-W Wavenumber (M•r•-•)
-0.6
0.0
0.6
E-W Waver•uraber (Mrn•-•)
Fig.7,•Zonalwavenumber
spectra
forFNOCsite2027,located
inthemid-latitude
NorthPacific,
computed
using
themaximum
likelihood
method.
Theestimates
encompass
thetim
einterval
July
1985
through
June
1988
and
thethree
annual
periods
contained
therein.
The
frequency
bandwidth
isconstant
at0.027
cpd
forallofthespectra.
Thewavenumber
has
units
ofcycles
permegameter. Powerisexpressed
indecibels
downfromthepeakateachfrequency
atthepeakmeridional
wavenumber.
Notetheincreased
tendency
foreastward
propagation
atperiods
shorter
than10days
ascompared
withFigure
6,andtheinteryear
variability.
18,370
CHAVEetal.:WINDSTRESS
CURLVARIABILITY
?/a 5 - ?/a 6
?/85-?/88
-0.4
-0.8
• ½•,
-1.0 '•
10•..
-1.,8 •
30
-1.•
3
5
•-1o
.
-1.2
20
-1.4
30
-1.6'
50
50
-1.8
-1.8
-2.0
100
•
, •,•
100
-2.0
-2.2
-2.2
-0.6
-0.6
0.0
0.0
0.6
E- I• •avenumber
E- W Wavenumber (Mm- •)
(Mra- •)
7/86-7/87
-0.4
3
3
5
•-1o
•-1o
.
-1.2
20
-1.4
30
-1.6
•
20
•
•
-1.4
30
•
-1.6
5O
50
-1.8
-1.,9
100
-2.0
lOO
-2.0
-2.2
-2.2
-0.6
E-W
.
-1.2
0.0
0.6
Wavenumber (Mrn.-•)
-0.6
0.0
0.6
E-W Wavenumber (Mrn.-•)
Fig.8. Zonalwavenumber
spectra
forFNOCsite1830,located
in themid-latitude
NorthPacific,computed
usingthemaximum
likelihoodmethod.Theestimates
encompass
thetimeintervalJuly1985through
June1988andthethreeannualperiods
contained
therein.Thefrequency
bandwidth
isconstant
at0.027cpdforall of thespectra.Thewavenumber
hasunitsof cyclespermegameter. Poweris expressed
in decibels
downfromthepeakat eachfrequency
at thepeakmeridional
wavenumber.
Notethereduced
interyeardifferencesascomparedto Figure7.
ure 8. SomesimilaritybetweenFigures7 and 8 is evident;note roughly symmetricspectrumat longerperiods. At sitessouthof
especially
theshape
of thepeakdelineated
bythe3-dBcontour
in
year 2 between5 and 20 days. However,despitenearly equivalent latitudesand only a --1200-km east-westseparation,the differencesare generallylarge,especiallyduringyear 1.
The southernmostsite, 1330 (Figure 9), displays zonal
wavenumberspectrawhich are nearly symmetricat all frequencies and for all three years. The shortperiod (<10 days) differences between Figure 9 and Figures 7-8 reflect the reduced
incidenceof eastwardpropagatingstorm systemsat lower latitudes,but lessvariabilityis alsoseenat longperiods.
The latitudinaltrendsdisplayedin Figures6-9 are repeatedat
othersitesthroughout
the central-eastern
NorthPacificOcean. At
--30ø-35øN, the zonal wavenumberspectraare more symmetricat
all periods and possessa narrower peak width. Most of these
trendswere noticedpreviouslyby Willebrand [1978]. However,
considerablevariability over time is superimposed
on theseweak
averages,and at some sites it is difficult to discernthe pattern.
The interannualvariability of the zonal wavenumberspectrais
quite large. In addition,both Figures6-9 and resultsfrom other
sites suggesta spectrumwhich flattens for zonal wavenumber
magnitudes
lessthan0.2-0.6cyclespermegameter
(Mm-1),correspondingto wavelengthsgreater than 1700-5000 km, with a
steeperfalloff at largerwavenumbermagnitudes.
Meridionalwavenumber
spectracomputed
in an identicalman-
ner are shownfor all 3 years and individual yearsat grid points
ity anda correspondingly
wide peakwidthwhichobscures
propa- 2027 and 1330 in Figures 10-11. The variability with wavenumgationtrends. Between=30ø-35øN and --50øN there is a ten- ber is comparableto or larger than that for the zonal component,
det•cy for eastwardpropagationat periodsshorterthan 10-15 while the apparentpeak width is also similar. The three year
days with increasinglyshort scalesas period decreasesand a mean spectraat both sitesare nearly symmetricor weakly southlocations north of --50 ø, there is considerabledirectional variabil-
CHAVE
etal.:WINDSTRESS
CURL
VARIABILITY
3
,
•-lO
-12
.•, -0.4
3
5
,•
5
7
•,
10•-.
'•'
-0.8
•
•o -1.0
œ0 o
30 •
•-14
50
•
• -1.•
20
o
•
30
•
-1.4
• -1.•
•
•-18
50
• -1.8
100
-2.0
18,371
o
•
-2.2
100
-2.0
-2.2
-0.6
0.0
0.6
-0.6'
E-W Wctve•tzrn,
ber (Mvvz
-I)
0.0
0.6
E-W Wctvemzvvzbev
('M•-I,)
- 7/8 ?
3
.•
•
;•-1o
7
•,
10•.
.
-0.4
3
-0.6
5
'•'
-0.8
•
•o -1.0
-1.2
20 o
• -1.2
20
o
-1.4
30
•
•-1.4
30
•
•
• -1.•
•
-1.6
50
-1.8
50
• -1.8
100
-2.0
-2.2
•o -2.0
100
-2.2
-0.6
0.0
0.6
E-W Wave•tt•n.ber (Mvtz-•)
-0.6'
0.0
0.6
E- W Wctve•tzvvzber
(Mvvz-!)
Fig.9.Zonal
wavenumber
spectra
forFNOC
site1330,
located
inthetrade
wind
beltnorth
ofHawaii,
computed
using
themaximum
likelihood
method.
Theestimates
encompass
thetimeinterval
July1985
through
June
1988
andthethree
annual
periods
contained
therein.
Thefrequency
bandwidth
isconstant
at0.027
cpdforallofthespectra.
Thewavenumber
hasunits
ofcycles
per
megameter.
Power
isexpressed
in decibels
downfromthepeakateachfrequency
atthepeakmeridional
wavenumber.
Notethe
greatersymmetrycomparedwith Figures6-8.
ward,but individualyearresultsdisplaysomestrongasymme- hencehavedoublethe resolution
bandwidth
in the frequency
tries. For example,in Figure 10 the meridionalwavenumber domain. Figure 12 showstime sectionsof the zonal wavenumber
spectrumis fairly symmetricduringyear 1, but northwardtenden- spectrum
at fourperiods
forlocation
2027. Thevariability
is typ-
ciesoccurat all periods
duringyear2, andthewavenumber
peak icallyvery weakat the two longestperiodsshown,but becomes
widthis muchlargerin year 3 compared
with the previous2 moreimportant
at periods
shorter
than10 days.Noteespecially
years.Similarly,in Figure11, southward
departures
fromsym- the strongeastwardeventsin the summer-fallof 1985 (0.0-0.5
metryare observed
duringyear2 in particular.Comparable
variabilityin themeridional
wavenumber
spectrum
is observed
at the
remainingcentral-easternNorth Pacific siteswith no clear trends
yearsin the figure) and the winterof 1986-1987 (around1.25
yearsin thefigure)at 8 daysandduringyear3 (2.0-3.0 yearsin
thefigure)at 4 days.Somewhat
greater
variabilityat all periods
in propagation
direction
withlatitude.As for thezonalcompo- is seenat site 1830 (Figure13), located--1200km to the westof
nent,the meridionalwavenumber
spectrumappearsto be flat- 2027. In particular,
thereis morewestwardpowerat periods
tenedfor wavenumber
magnitudes
lessthan0.2-0.6 Mm-1 with a longerthan10 daysduringyear2 (1.0-2.0yearsin the figure).
morerapidfalloff for largermagnitudes.
Notealsothestrong
eastward
surgeat 32 daysduringthespring
To furtherillustratethe temporallyfluctuating
natureof curl of 1986(around1.0year). Thevariability
is largerin Figure13
wavenumber
spectra,estimates
wereobtainedfor 6-monthseg- evenat shortperiods;substantial
westwardtendencies
at 4 days
mentsof dataoverlapped
by 50% andsuccessively
covering
the are seenduringthe summer-fallof 1986 (around1.0 yearin the
entire3 year intervalof interest.Theseestimates
possess
the figure) and especiallyin the fall-winter of 1987 (around2.25
samenumberof degreesof freedomas the 1-yearspectraand yearsin the figure). Only limitedvisualsimilarityis apparent
18,372
CHAVE
etal.:WINDSTRESS
CURLVARIABILITY
7/a s - 7/a a
3
.• -0.4
•
5
•
?
3
-(9.6
5
-o.a
10 •.
•
-1.2
20 c•
• - 1.2
20
' e•
-1.4
30 •
•-1.4
30
•
;•-1o
.
-1.6
50
;•-1
•
0
• -1.•
•
c•
5O
• -1.8
-1.8
100
-2.0
••
100
-2.0
-2.2
-2.2
-0.6
N-S
ß
0.0
0.6
-0.6
Wavenumber (Mra- •)
0.0
0.6
N- $ Wavenumber (Mra- 1)
?/a ?- ?/s s
-0.4
-0.6
s
3
3
5
-0.8
••
5
7'•
-
20
50
-2.0
100
;•-1o
r•
•
20
•
e. -1.4
30
•
•'
50
-1.6
-1.8
lOO
-4 -2.0
-2.2
-2.2
-0.6
N-$
ß
-1.2
0.0
-0.6
0.6
N-S
Wavenumber (Mra- 1)
0.0
0.6
Wavenumbe• (Mm- lj
Fig. 10. Meridionalwavenumber
spectrafor FNOC site2027, locatedin themid-latitudeNorthPacific,computedusingthe maximumlikelihoodmethod.The estimates
encompass
thetime intervalJuly 1985throughJune1988andthethreeannualperiodscontainedtherein.The frequencybandwidthis constantat 0.027 cpdfor all of the spectra.The wavenumber
hasunitsof cyclesper
megameter.Poweris expressed
in decibelsdownfrom thepeakat eachfrequencyat thepeakzonalwavenumber.Note thegreater
symmetryascomparedto the zonalspectrumin Figure7.
betweenFigures 12 and 13 despitethe comparativelyshortdistanceseparating
the sites. No correlationof propagation
direction
with seasonis evidentin eitherFigure 12 or 13.
For pointsnorthof 50øN,moredirectionalvariabilitywith time
is observedat all periodsas comparedwith mid-latitudesites. At
pointssouthof 35øN, zonal spectraare observedwith variability
comparableto that in Figures12 and 13. Intervalsof strongwestward propagationat periodsshorterthan 10 daysare occasionally
the remaininggrid pointswere constructedfor selectedfrequencies. Only thosevalueswhich exceedthe zero significancelevel
at 95% confidenceare plotted, typically resultingin a comparatively sparsecontourmap. Thesecoherencemaps are equivalent
to the squareof the spatial autocorrelationfunction at each frequencywhen all of the coherencevaluesare plotted,and hence
they can be comparedwith the analyticformsusedin modelstud-
seen, in contrast to more northern locales. Evidence for seasonal
Figure 14 showsthe curl coherencemap for location2027 at a
period of 25 days for the entire 1985-1988 interval as well as
individual year results. The resolutionbandwidth is constant,
covering19-36 daysperiod,for all of the time sections,The largest peak is centeredon grid point 2027 where the coherenceis
unity, but additional,significant,nonlocalcoherencepeaksalso
appearsomedistanceaway with their locationsvarying weakly
with time. For example,the 3-yearaveragemap showssecondary
peaks centerednear grid points 1428, 2220, and 2230, all with
peak squaredcoherencevaluesof 0.3-0.4. The individualyear
estimatesdiffer significantly,with no substantialsecondarypeaks
dependence
is tenuousat best. Comparablevariabilityis seenfor
the meridionalcomponent.Willebrand [1978] inferreda largerwavenumberbandwidthfor the meridional wavenumbercomponent of curl from two point coherences;
this assertioncannotbe
supportedby the presentstudy.
6. SPATIAL CORRELATION OF CURL
As a final investigationof the propertiesof curl, contourmaps
of the squaredcoherenceof curl at a given locationwith curl at
ies.
CHAVEetal.:WIND STRESS
CURLVARIABILITY
-0.4
3
.• -0.4
5
7
10'•-
.
-1.2
20
-1.4
30
-1.6
50
•
-0.6
•
-1.0
3
5
-o.8
"•
•
•-1.4
•
• -1.•
-1.8
20
•
30
•
I
5O
• -1.8
100
-2.0
•
lOO
-2.0
i
-2.2
-2.2
-0.6
3
•
5
.•
7 •,
'• -0.8
10•.
.
0.0
0.6'
N- $ •av enumber (Mra- •)
N- S •av enumber (Mra- •)
•-1o
18,373
-0.4
3
5
•-10
•
ß
-1.2
20 o
• -1.,8
20
o
-1.4
30
•-1.4
30
•
-1.6'
•
5O
-1.8
100
-2.0
-2.2
•'
r•
-1.•
5O
• -1.8
o
.q -2.0
lOO
-2.2
-0.6
0.0
0.6
N- $ Wavenumber (Mra- •)
N-$
Wavenumber (Mra- •)
Fig. 11. Meridionalwavenumberspectrafor FNOC site 1330, locatedin the tradewind belt northof Hawaii, computedusingthe
maximumlikelihoodmethod. The estimatesencompass
the time intervalJuly 1985throughJune1988 andthe threeannualperiods
contained
therein. The frequencybandwidthis constantat 0.027 cpdfor all of the spectra.The wavenumber
hasunitsof cyclesper
megameter.Poweris expressed
in decibelsdownfromthepeakat eachfrequencyat thepeakzonalwavenumber.
in year 1 but strongcoherenceat 1428 and elsewherein years2
and 3. Figure 15 showssimilarcoherencemapsfor the samesite
with an identicalbandwidthcomputedfrom 6-monthoverlapped
datasectionsbeginningin July 1986. Note the variabilityof the
strengthof the main lobe centeredon site 2027, and especiallyits
extreme size in spring-summer1987. The secondarypeaks at
1428 and 2220 are most prominentduring the fall and winter
whencurl is at its strongest.The appearance
of more than a single coherencepeak appearsto be ubiquitous,and the examples
shown in Figures 14-15 are typical rather than extreme. The
locationsof the secondarypeaks,in distanceand azimuthfrom
theprincipalor primarypeaks,are a functionof position. Consistent changesof intercorrelationpatternwith latitudeor location
are not observed. However, there is an obvioussubstitutionprinciple;if the intercorrelation
patternat 1428 were computedfor all
3 years,it would have a peak at 2027 as is observedin reversefor
Figure 14.
Finally, curl is clearly not independentof eitherthe wind stress
componentsor the atmosphericpressure,nor is the coherence
betweenthem necessarilysmall at low frequencies,althoughthe
largestvalues may be nonlocal. To illustratethe magnitudeof
curl coherencewith other atmosphericvariables,Figures16 and
17 show mapsof the squaredcoherencebetweencurl at FNOC
site 2027 and zonal wind stress,meridional wind stress,and sur-
face air pressure. The wind stress maps (Figure 16) show
double-lobed structureselongated zonally (meridionally) and
spacedmeridionally(zonally) for the curl coherencewith zonal
(meridional) wind stress. The peak squaredcoherenceexceeds
0.8 for the zonal componentand 0.7 for the meridionalone. The
phaseof the curl to zonal wind stresscoherence(not shown) is
--180 ø for the north lobe and =0 ø for the south lobe, while the
phaseof the curl to meridionalwind stresscoherence(not shown)
is =0 ø for the east lobe and = 180 ø for the central lobe with north-
southvariation;all of thesephasesare as expectedfrom (2). The
multiple correlationpeaksof curl with itself in Figures 14 and 15
are consistentwith the elongatedcoherencelobesand sharpspa-
18,374
CHAVEetal.:WIND STRESS
CURLVARIABILITY
32
"•
15d
d
3.0
\.,.
3.0
2.5
)•
2,.5
'•2
2.0
0.,5
0.0
-0.6
0.0
0
0.5
)•/•l
,
/
-0.6
0.6
0.0
E-14r Wctver•tzraber(Mrr•-1,)
E-It r Wct'u
e• • rn,be• ('Mrn,- l,)
8
4
d
3.0
3.0
2..5
2,.5
'•
2.0
1.5
1.5
0.5
0.0
.'•
-0.6
0.0
0.6
E- • Wa•e•mber
0.0
-0.6
0.0
0.6
E-I½ Wctver•tzraber(Mm -1)
Fig. 12.Time sections
of thezonalwavenumber
spectrum
for fourperiodsat FNOC site2027,locatedin themid-latitude
North
Pacific.Frequency-wavenumber
spectra
werecomputed
for 6-monthtimesections
beginning
in July 1985(corresponding
to time
0) with an offset of 3 monthsbetweenruns; the zonal wavenumberspectrumat the meridionalwavenumberpeak was then
extracted
to generate
theplots.Thecenterperiodsareindicated
aboveeachspectrum,
andthefrequency
bandwidth
is0.055cpdin
eachinstance.The wavenumber
hasunitsof cyclesper megameter.Poweris expressed
in decibelsdownfrom the maximumat
eachperiod.
studieshave shown. While the grossseasonalchangesof mean
and variancehave been noted in earlier work, interannualchanges
in total power have receivedlittle emphasis. Furthermore,spectral fine structurewhich resultsin enhancedpower in limited frequencybandsbut over substantialareashas not previouslybeen
reported. It shouldbe notedthat a strongbarotropicresponseis
seenat --30 daysin the BEMPEX data(Chaveet al., 1991) which
may be due in part to the observedincreasein curl power at that
period. Both the interannualchangesand spectralfine structure
value in excess of 0.7 centered over the eastern limb. These
are factor of 2 fluctuationsfrom the long-term averagepower,
results are typical of curl throughoutthe central-easternNorth which makes them only slightly weaker than normal seasonal
Pacific.
fluctuations. The existenceof seasonalvariability of the forcing
impliesa similarchangein the ocean'slinear response.In fact,
7. DISCUSSION
the expectedseasonalvariation of the atmospherewas used by
Theresultspresented
in thispaperindicatethatthefrequency- Dickson et al. [1982] to concludethat seasonalchangesin deep-
tial gradientsseenin Figure 16. This is becausestrongcurl-curl
coherencecoincideswith large spatial gradientsof the curl to
wind stresscoherence.Note that the local coherenceat grid point
2027 is insignificant.Weak localcoherenceof curl with the wind
stressat low frequencieswas predictedon the basisof approximate wavenumberspectralsymmetryby Willebrand [1978] and
Frankignoul and Muller [1979]. The surfaceair pressuremap
(Figure 17) showsa zonally extended,massivecoherentpatch
over mostof the North Pacific, but with a peak squaredcoherence
wavenumberbehaviorof curl is more complexthan previous oceankinetic energy in the easternAtlantic were the result of
CHAVEetal.:WINDSTRESS
CURLVARIABILITY
32
18,375
15d
d
3.0
3.0
2.5
'•,
2.0
1.5
•1.0
0.5
0.•
0.0
0.0
-0.6
0.0
-0.6
0.6
E-W Waver•u,e'r•bew
(Mra -•)
0.6'
E-W
4d
3.0
0.0
8d
xd
'
2.5
3.0
25
'•,2.0
'•,20 •o
E.o
O.5
O5
0.0
O0
-0.6
0.0
0.6'
-0.6
0.0
0.6'
Fig. 13. Timesections
of thezonalwavenumber
spectrum
at FNOCsite1830,located
in themid-latitude
NorthPacific.
Frequency-wavenumber
spectra
werecomputed
for 6 monthtimeseriesbeginning
in July1985(corresponding
to time0) withan
offset
of 3 months
between
runs;
ttiezonalwavenumber
spectrum
atthemeridional
wavenumber
peakwasthenextracted
to
generatetheplots. The inclusivetime periodsare indicatedaboveeachspectrum,andthefrequencybandwidthis 0.055 cpdin each
instance.The wavenumberhas unitsof cyclesper megameter.Power is expressedin decibelsdown from the maximumat each
period. Note the similaritiesanddifferenceswith Figure 12.
direct atmospheric
forcing, Deep-oceanseasonalkinetic energy shownin Eigures6-11 are not very differentfrom the frequency-
fluctuations
havealsobeennoted
byNiilerandKoblinsky
[1985] dependentmean wavenumbersinferredby Willebrand [1978]. It
for the NorthPacific,whileKoblinsky
et al. [1989]citeexaniples appearsthat the peak widths for the MLM spectraare larger,
of interannualchangesin oceanicenergylevels that are correlated althoughcomparisonsare difficult becausethe standarderror Of
withthestrength
of curl. Thus,it appears
thatchanges
in the the mean wavenumber computed by Willebrand is not i'eally
forcing by a factor of 2 will producesignificantchangesin the equivalent
to spectral
bandwidth.
However,
if theyearlyaverage
in Figures
6-11 arecompared
withWillebrand's
results,
ocean'sresponse.However,the BEMPEX datacontainlittle evi- spectra
dencefor seasonalchangesin barotropickinetic energydespite then differencesarise as asymmetrybecomesapparentfor some
Thisis evenmoreobviousin the6-monthspectral
their location (41øN, 163øW) near the peak in curl variance, datasections:
suggestingthat other factorsthan curl power, suchas fluctuating time sections of Figures 12-13. Note also that differences
propagationdirection,also play an importantrole in determining betweenzonalandmeridionalvariability,andespeciallythe bandwidth as denoted by the main lobe width, are not readily
the ocean'sresponse.
Perhapsthe mostimportantresultof this work is the realization detectablebecauseof wavenumbervariability. These observathat it is variabilitythat determinesmany of the
that wavenumbervariability for curl may be substantial.This has tions SUggest
of curl, that the lengthof the analysisinterval
implicationsfor comparisonsof models with oceanic data. For grosscharacteristics
example,the peak wavenumbersfor the 3-year averagedspectra will affect the estimation of these characteristics, and that care is
18,376
CHAVEetal.:WIND STRESS
CURLVARIABILITY
,
,. _
50ø
½ 24
'ca
•
22
..
_
20
.g
,
_
,
40
35
30
Zonal
Cr•d
25
20
I
40
Point
I
170ø1
,' ...
I
35
30
25
Zonal
Cr•d
Point
35
30
25
Grid
?oi•t
\ 140 ø
20
?/ss-?/s?
40
35
Zonal
30
25
Grid
Point
20
40
Zonal
20
Fig. 14.Mapsof thesquared
coherence
between
curlatFNOCgridpoint2027andcurlovertheNorthPacificOceanata periodof
25daysforthe3-yearinterval
1985-i988
andindividual
yearsasnoted
above
eachplot.Thecoherence
wascomputed
using
the
multiplewindowmethodandpossesses
a bandwidth
of 0.044cpdfor all fourestimates.
Because
thedegrees
of freedom
aredifferentforthe3- and1-yearspectra,
thezerocoherence
levelsat95%significance
are0.15and0.18,respectively.
Valuessmallerthan
thiswerenotincluded
in thecontouring
process,
andtheremaining
contours
areatintervals
of 0.1 beginning
at0.2. Thecoherence
is 1 at 2027. Notethevariability
of boththemainlobeshape(centered
ongridpoint2027)andthestrength
of theintercorrelation
peaks.
requiredwhen inferring curl propertiesfor inclusionin models. trate this. it should be remembered that the summer to winter
In particular, for a given time interval, differencesbetween range in power is typically a factor of 2-4, so that curl power
wavenumberspectraare observedover distancesof --1200 km, as fluctuationsinferredfrom directionalchangesin theseexamples
seenin Figures12 and 13, suggesting
thataveragecurlmodelsfor are comparableto the seasonalrange. Neitherof thesepostulated
anoceanbasinarenotentirelyapplicable.
shorttermchangesin the ocean'sresponse
to atmospheric
forcing
As an ex.ample,
it is interesting
to contemplate
thepossibility would necessarilybe predictedif longer-termaveragewavenumthata winterof very strongcurl amplitudecouldresultin only a ber spectrawereusedto forcea model. This hasobviousimplicfi-
weakoceanic
response
if thezonalwavenumber
spectrum
were tionsfor specificdatato modelcomparisons.Theseresultssuginordinately
eastward,assuming
that a resonant
response
is typically strongerthan an evanescent
one. The fall-winter of
1987-1988 at site 1830 for 15 days periodin Figure 13 (see
2.2-2.7 yearsin the figure)is a goodcandidatetime periodfor
suchbehavior.Of course,the reverseis alsopossible,andwest-
gest that more attentionshouldbe paid to the characteristics
of
curlduringthetimeoceanic
measurements
arebeingtakenrather
than to long-term curl statistics. However, this might be mitigated somewhatby the weak tendencyfor long-termaverage
wavenumberspectrato display wider peak widths than shorter
ones. The widerpeakwidth will resultin an increasein westward
Rossbywave excitationwhenno corresponding
powerchange powerthatwill at leastpartiallycompensate
for thelackof specioccurs.The fall-winterof 1986-1987at 1830for a 15 or 32 day ficity.
A number of differences between the curl wavenumber obserperiodin Figure 13 (see 1.3-1.7 yearsin the figure) might illus-
ward curl fluctuationsmight result in substantialincreasesin
CHAVE
eta!.:
WINDSTRESS
CURLVARIABILITY
18,377
• 24
•
r• 22
22
' .•20
.,., 20
18
-
ao
½16
40
35
30
Zona. l Grid
25
20
40
Point
35
30
25
Zonal
Grid
Point
?-
20
?
/50 a
• 24
• 24
•
22
.• 20
!00 o
,'•12,
ß
! ?Oø I
40
35
Zonal
30
25
Grid
Point
40
20
35
Zonal
,' •..
I
30
25
Grid
Point
\ 140 •'
20
Fig.15.Mapsofthesquared
coherence
between
curlatFNOCgridpoint2027andcurlovertheNorthPacific
Ocean
ata period
of
25 daysforthefourseasons
beginning
in July1986,asnotedaboveeachplot.Thecoherence
wascomputed
usingthemultiple
window
method
andpossesses
a bandwidth
of0.044cpd,identical
tothatforFigure14. Thezerocoherence
levelat95%significance
is0.39;values
smaller
thanthiswerenotincluded
in thecon.touring
process.
Theremaining
contours
are•atintervals
of0,1
beginning
at0.5. Thecoherence
is 1 at2027.Notethevariability
ofboththemainlobeshape
(centered
ongridpoint2027)and
thestrengthof theintercorrelation
peaks.
Note also that neither these observations nor those of
vationsof thispaperandbothpriorobservations
andmodelshave
beennoted. First,the averagespectralshape,and especiallythe MacVeighet al. [1987] agreewith the curl wavenumberparameby
high-wavenumber
slope,are not comparableto thosecomputed terizationderived largely from troposphericmeasurements
by Gallegos-Garciaet al. [1981] usinga conventional(Fourier- Frankignouland Muller [1979] and adoptedfor the Muller and
forcing simulation.North of
based) wavenumber estimator. Their composite wavenumber Frankignoul[1981] atmospheric
is symmetric
andhasa spectral
slope
of k4 from
estimatefrom 11 yearsof monthlywind observations
was nearly 35ø,theirmodel
symmetric,did not displaywavenumberindependence
at large zero wavenumberto that characteristicof baroclinicinstabilityin
(equivalent
wavelength
of 5000km), fallsoff with
scales,and falls off approximately
like k-'/-'. By contra•t, theatmosphere
to a high-wavenumber
cutoffcorresponding
MacVeigh et al. [1987], also using a conventionalwavenumber inversewavenumber
estimatorappliedat unspecifiedfrequencies,
got a white resultfor to a scale smaller than the oceanic motions of interest, and zero
(<5000 km) powerthan
scales
largerthan--1200km (0.83Mm-1 in theunitsof Figures beyond.This containsmore short-scale
6-14) and•a =k-4 slope•atshorter
wavelengths.
No significant real
variationwith seasonwas reported. This is in qualitativeagree-
curl based on actual wavenumber estimates.
As a conse-
quence,moreshortRossbywaveswill be presentin theirmodel
mentwiththeresults
of thispaper.However,
theflatpartof the than in the real ocean,resultingin an increasedwavenumber
MacVeighet al. spectrumappearsto have a widerbut variable bandwidth and reduced coherence between oceanic and atmowidth, possiblybecausetheir conventionalwavenumberestimator spheric variables. However, assuming a high-frequency
could
notadequately
resolve
large-scale
features.
Noattempt
has wavenumbercutoffof 2•r/500km appropriatefor the FNOC prodbeen made to computean averagespectralslopein the present uct, the Frankignouland Muller spectralmodelyields a white
study on accountof marked directionalvariability. However, noiselevelof 1.04x10-14dyn2 cm-6 d whichis in goodagreethat while theremay be a
both the zonal and meridional wavenumberspectraclearly are ment with Figures4-5. This suggests
in thewavenumber
distribution
of power,its integral
quiteredfor wavenumbers
largerthan0.2-10.6
Mm-1 (wave- discrepancy
lengthsof 1700-5000 km).
is in agreement.
18,378
CHAVEetal.:WIND STRESS
CURLVARIABILITY
curl
? with
ps
24
22
30
ZonzL
C?t
25
20
Point
Fig. 17. Map of the squaredcoherence
betweencurl at FNOC grid point
2027 andthesurfaceair pressure
overtheNorthPacificOceanat a period
curl
of 25 daysfor July 1986 to June 1987 inclusive. The coherencewas com-
putedusingthe multiplewindowmethodwith a bandwidthof 0.022 cpd,
yielding=14 degreesof freedomat eachfrequencyfor a zerocoherence
level of 0.39 at 95% confidence.
Values smaller than this were not
includedin thecontouringprocess.
22
ß
scalein the zonal (=1000 km) than in the meridional(=300 km)
directions.Samelsonused a simplerdoubleexponentialwith
comparablee-foldingscaleswhich was basedon spatdallyaveragedcurl coherence
maps. While Samelson'smapsdo not show
any secondary
lobes,it is likely thatthe spatialaveragingwould
(
remove them. Note that the different zonal and meridional scales
.½.
are not supportedby this study. Both of thesefunctionshave a
singlepeakat the origin,andcannotreproduce
the secondary
lobesseenin Figures14-15. However,thepresence
of secondary
peakswill haveimportantramifications
for the oceanicresponse,
40
35
Zonal
30
Grid
25
20
Point
Fig. 16. Maps of the squaredcoherence
betweencurl at FNOC grid point
2027 and the zonal (top) and meridional(bottom)wind stressover the
NorthPacific
oceanat a period
of 25 daysforJuly1986to June1987
inclusive. The coherencewas computedusing the multiple window
method'with
a bandwidth
of0.022cpd,yielding
=14degrees
offreedom
at
eachfrequencyfor a zerocoherencelevel of 0.39 at 95% confidence.Valuessmallerthanthis werenot includedin the contouringprocess,andthe
remainingcontoursare at intervalsof 0.1 beginningat 0.5. The phaseof
thecoherence
is consistent
with therelationship
expectedfrom(2); seetext
for discussion.
especiallyif they are locatedto the east of the oceanicmeasure-
mentandhencecanresonantly
excitelongRossbywaves,assuming the frequency-wavenumber
characteristics
are appropriate.
This suggests
that the Brink and Samelsonmodelsprobably
underrepresent
theoceanicresponse
to atmospheric
forcing.Similarly, the Muller and Frankignoul[1981] modeldoesnot include
the effectof curl intercorrelation.
The complexspatialstructure
of curlandits temporalfluctuations
arejustadditional
manifestations of a spatdally,inhomogeneous
and variable wavenumber
spectrum,and can be accountedfor either in the spatial or
wavenumberdomains. However,simpleparameterizations
chosenfor analyticsimplicityareprobablyinadequate
to reproduce
thereal propertiesof curl.
Acknowledgments.The authorsare gratefulto David J. Thomsonof
Finally,the observation
thatthe curl intercorrelation
possesses Bell Laboratories
for suggesting
theBartlettM testandprovidinggeneral
advice. This work wassupported
at SIO by NationalScience
botha relativelytightmainlobeanda numberof nonlocalpeaks statistical
FoundationgrantsOCE-8420578, OCE-8800783,andOCE-8922948.
hasfurtherimplications
for the oceanicresponse.Sincethe spatialvariationof thesquared
coherence
of curlat a givenfrequency
REFERENCES
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CURLVARIABILITY
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J.H. Filloux andD.S. Luther,ScrippsInstitutionof Oceanography,
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Code 0230, La Jolla, CA 92093.
(ReceivedMay 13, 1991;
revisedAugust 15, 1991;
acceptedAugust 15, 1991.)