JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 103, NO. C5, PAGES 10,253-10,272, MAY 15, 1998
Semidiurnal tides observed in the western equatorial
Pacific during the Tropical Ocean-Global Atmosphere
Coupled Ocean-Atmosphere Response Experiment
Ming Feng,•'• Mark A. Merrifield,
• RobertPinkel,3 PeterHacker,
Albert J. Plueddemann,
4 Eric Firing,• RogerLukas,
•
and Charles Eriksen •
Abstract. The semidiurnal tide within a 100 km square region of the western
equatorial Pacific centeredat 1.8øS, 156.1øE is examinedusingshipboardsurvey
and mooring data collectedduring the Tropical Ocean Global Atmosphere Coupled Ocean-AtmosphereResponseExperiment (TOGA COARE). Baroclinicand
barotropic tidal amplitudesand phasesare estimatedfrom the surveyand mooring
observationsin the upper 300 m of the 1800 m deep ocean by specifyingtheir
horizontal and vertical structures. The barotropic tide is assumed to have zero
horizontal wavenumberover the domain, while a componentof the baroclinic tide
that is phase-lockedto the barotropic tide is determined by a searchingmethod
using plane wave fits to the data. The estimated barotropic tidal current is in good
agreementwith tide modelsderivedfrom TOPEX/POSEIDON observations.The
plane wave analysisindicatesa dominant mode one baroclinicwave propagating
toward
the northeast.
The
second vertical
mode can also be detected.
Given
the phase differencesbetween the M2 and S2 constituents in the barotropic and
baroclinic tides, the sourceof the baroclinic tidal signal is determined to be about
320 km southwestof the observingregion,at a seriesof islandsand shallowridges.
The cmnbinedestimatesof the barotropic and baroclinictides typically accountfor
only 40-60%of the observedsemidiurnalbandcurrentvariancein the mooringdata,
indicating the high degreeof temporal and spatial variability of the baroclinic tide
in this region. The resultsof this study suggest,however,that coherentbarotropic
and baroclinictidal signalscanbe successfully
distinguished
in the deepoceanusing
shipboardsurvey data, even when the data are limited to the upper 300 m.
1.
Introduction
Extensive
oceanic observations
were made in the west-
was to quantify air-seainteractionprocesses
[Webster
and Lukas, 1992];hencethe majority of the observa-
tions were made in the upper ocean. During TOGA
ern equatorial Pacific warm pool during the Tropical
COARE, an energetic semidiurnal tide was reported
Ocean Global AtmosphereCoupled Ocean-Atmosphere
[Picautet al., 1995; Pinkel et al., 1997]in the IntenResponseExperiment (TOGA COARE) IntensiveOb-
servingPeriod(lOP) fromNovember1992throughFebruary 1993. The primary purpose of this experiment
siveFlux Array (IFA), centeredat 1.75øS, 156øE (Figure 1). Tidal currentsof the orderof 20 cms-• in the
upper ocean with associatedvertical displacementsof
20 m in the thermoclineoccurduringsomespringtides.
The
energeticsof the tide are suchthat high-frequency
•School
of Ocean
andEarthScience
andTechnology,
University
of Hawaii, Honolulu.
undular bores or solitary wave packets were observed
2Now at Institute of Oceanography,ChineseAcademy of to be associated
with the baroclinictidal waves[Pinkel
Sciences,Qingdao, China.
aScrippsInstitution of Oceanography,University of California, San Diego.
4PhysicalOceanographyDepartment,WoodsHole OceanographicInstitution, WoodsHole, Massachusetts.
5Schoolof Oceanography,Universityof Washington,Seattle.
Copyright1998by the AmericanGeophysical
Union.
Papernumber98JC00264.
0148-0227/98/98JC-00264509.00
et al., 1997]. To be able to assess
the effectsof tidal
motions on various processesin the warm pool, we first
seek a basic description of the dominant spatial and
temporal patterns of the semidiurnal tide in the IFA.
A related goalof this study is to obtain a description
of the predictable component of the baroclinic tide in
order to removethis variability, as well as the barotropic
tide, from the shipboardobservations.During the IOP,
the R/VWecoma collectedrepeated acousticDoppler
current profiler (ADCP) and Seasoarobservationsof
10,253
10,254
FENG ET AL.: SEMIDIURNAL
o
TIDES IN THE WESTERN EQUATORIAL
PACIFIC
i
2øS
156øE
152øE
154øE
156øE
158øE
Figure 1. The topographyof the IntensiveFlux Array (IFA) region. The shadedareasare
islands. The depth contour interval is 1000 m. The heavy solid line in the upper right is the
standardbutterfly pattern of the R/V Wecoma.The positionsof the WoodsHole Oceanographic
Institution(WHOI), Universityof Washington(UW), and Tokaimooringsand the R/V Vickers
are alsoshown.The inseton the right showsthe cotidallinesfor the M2 (solid)and S2 (dashed)
barotropictide from the CRS sealevel data (with 1ø spacing).NI, NB, and SI representNew
Ireland, New Britain, and Solomon Islands, respectively.
the upper 300 m of the ocean along a butterfly pat-
note that
the mode
1 and 2 currents
are almost or-
tern in the centerof the IFA (Figure 1). The primary
objective of this survey was to measure the tempo-
thogonalin the upper300 m (FigureA1)). The model
results are comparedwith the TOPEX/POSEIDON
ral evolution of the zonal and meridional gradients in
(T/P) barotropictide predictions[Ray et al., 1994]
temperature,salinity, and velocity at the centerof the and with mooredmeasurements
within the IFA (FigIFA [Huyer et al., 1997]. Timescales
of interestrange ure 1). While T/P altimetrydata haveledto significant
from days to weeks,particularly associatedwith west- improvements
in barotropictide models[Shurnet al.,
erly wind bursts. Becauseeach circuit of the butterfly 1997],there are few deepoceanmeasurements
that can
took •1.5 days, the semidiurnaltide is aliasedin the be used to verify the modeledbarotropic tidal currents
mixed space-time survey data. In this paper, we at[Luytenand Storereel,1991].Thusthe resultsfromthis
tempt to constructa simpletidal modelin orderto de- paper give an independentverificationof the T/P tide
tide the shipboardobservationsso that the larger-scale, model in the observationregion.
In general,the baroclinictide is more difficult to charlower-frequencysignals can be analyzed with greater
acterizethan the barotropictide owingto short horizoncertainty.
At issueis whether a meaningfuldecompositioninto tal scales,the partition of energy into vertical modes,
barotropic and barocliniccomponentsis possibleusing and the strong variability of amplitude and phaseon
of only a few tidal periods[Wunsch,1975].
only upper ocean data (300 m in 1800 m total water timescales
depth). Here we show that repeated horizontal sam- Currents can cause spatial distortions and frequency
pling providesa meansto constructa model for the shifts of the baroclinictide. The strong background
barotropicand baroclinictides in the IFA. The model vertical shearsin the equatorial region have a signifiusesa wavenumbersearchingmethod and theoretical cant effecton baroclinicwavestructure[Boyd,1989].
There are severalexamplesof organizedbaroclinictidal
vertical modesfor determining the baroclinictide. The
spatial interpolation method of Candela et al. [1992] signalsin the open ocean [Hendry, 1977; Dushawet
is adapted.Mao [1997]extendedthis methodto deep- al., 1995]. For the COARE region,we seekto deterwater regimes.Our model useslinear variationsin time
and spaceto representthe mean field and plane waves
for the barotropic and baroclinic tides. The barotropic
and baroclinic tides are distinguishedby wavenumber:
the barotropic tide is assumedto have zero horizontal wavenumber, while the baroclinic tide has fixed
(nonzero)wavenumber.The separationbetweenbaroclinic modes is also based on their wavenumber
mine the main sourceregion of the baroclinictide, to
what degree the baroclinic tide is phase-lockedto the
barotropic tide, and how the vertical structure of the
baroclinictide changesin the presenceof the strong
horizontaland vertical shearsassociatedwith the equatorial current system.
In section2, we describebriefly the barotropictide
differdeterminedfrom the T/P tide model. In section3,
ence(TableA1), as well astheir verticalstructures(we we analyze the spatial and temporal variability of the
FENG ET AL.' SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
semidiurnalband from the mooring data in the IFA to
identify the coherentsignalsin the semidiurnaltide. In
section4, the baretropic and bareclinic tides are examined usingthe Wecomasurveydata. Plane wavemodels are presentedto describethe coherentsemidiurnal
tidal signals. Model resultsare discussedusingfits to
the data over 3 days (temporal resolution)and 15 days
and longer(tidal constituentresolution).The summary
and discussion follow in section 5.
1992
Wecoma
ADCP
WHOI
VMCM,ADCP
UW
PCM
Vickers
Doppler
Tokai
ADCP
:
10,255
1993
:
;
:
;
:
Figure 2. The instrument type and time coverageof
currentobservations
on the R/V Wecomaand the four
2. Baretropic
mooringsduring the IOP.
Tide in the IFA
Luther and Wunsch[1975]usedsealevelobservations
from island tide gaugesto derive maps for the M2, S2, mates in parenthesesfor the M2 tide are from an upN2, K1, and O1 tide constituentsin the central Pa- date of Schramaand Ray [1994]. The differencesbecific Ocean including the IFA region. Cartwright et tween the two estimates have not yet been resolved.
al. [1991]presentedtidal surfaceelevationsof all diurnal and semidiurnal
harmonic
constituents
over most
The baretropic tidal currents at the M2 and S2 frequenciesare dominated by the zonal component. The
dominant diurnal band constituents(O1 and K1) are
much
weakerthan the semidiurnaltide (Table 1).
altimetry. Cartwrightet al. [1992]presentedprograms
for convertingtidal elevationfields to fields of depthaveraged(baretropic)tidal currents,allowingpredic- 3. Semidiurnal Band Variability
of the world's oceans and seasfrom analysis of Geesat
tion of currentsfor arbitrary position and time. A sim-
From Mooring Data
ilar analysiswasmadeusingthe T/P satellitesealevel
The data used in this analysisinclude fixed position
data, whichcoverthe IOP period [Ray et al., 1994]
observations
in the upper 300 m of the water column
(hereinafterreferred to as the Cartwright, Ray, and
from
three
IFA
mooringsand the R/V Vickerswhich
Sanchez model or CRS model). The analysis of the
was
on
station
for
three legsof -•1 month duration durbaretropic tide in the present study makes use of the
ing
the
IOP
(Figure
1). The centralmooringof the IFA
CRS model results. Conversely,the IOP data provide
was developedby the WoodsHole OceanographicInstifurther verification of the CRS model results.
From the CRS tide model, the phasesof the M2 and tution (henceforththe WHOI mooring)and deployedat
S2 tides do not vary significantly over the survey do-
main (Figure 1). The M2 and S2 tides from the CRS
model have amplitudesof 27 cm and 22 cm and Greenwich epochs of 142ø and 155ø, respectively, at 2øS,
156øE. These amplitudes are nearly double those of
Luther and Wunsch[1975]. The phasesof the M2 and
S2 tides increase toward
the west-southwest
and the
1.75ø S, 156øE for the duration of the IOP. The meeting was outfitted with point sensorsfor velocity and
temperature in the upper 300 m of the water column
[Plueddemann
et al., 1993;WellerandAnderson,1996].
A profilingcurrent.meter (PCM) mooring[Eriksenet
al., 1982] was deployedby the Universityof Washington (henceforththe UW mooring)at the north end
south-southwest
(Figure 1), respectively,in agreement of the array, 156ø E, 1.26øS, measuring current above
195 m during the IOP. An uplooking ADCP mooring
with Luther and Wunsch[1975].
was
deployed at 156øE, 2øS by Japan's Tokai univerThe amplitude and phaseof the semidiurnalbaretropic tidal velocity are also computed from the CRS data sity (henceforththe Tokai mooring),which measured
at 2øS, 156øE (Table 1), which are representativeof velocity above 210 m during the IOP. A downlooking
the IFA regionin general[Ray et al., 1994]. The esti- Doppler sonar developedby the Ocean PhysicsGroup
of the ScrippsInstitution of Oceanography
wasinstalled
on the R/V Vickers.The Vickerswasallowedto drift
Table 1. Tidal amplitude and phase from the tidal off station by -•20 km beforerepositioning.For simplicity, we will refer to the Vickersdata as meeting data
modelof Ray et al. [1994]
in later analysis. All data are time-averaged(WHOI
u
v
and Tokai, I hour; Vickers,2 hours)exceptfor the UW
Amplitude,
Phase,
Amplitude,
Phase,
mms -•
G
mms -•
G
M2
35(29)
46(48)
5(6)
239(260)
S2
K1
O1
15
4
6
58
138
148
2
3
I
307
334
88
The values in the parentheses are from $chrama and
Ray [1994].
meeting where instantaneouscurrent profiles were collected at 3-hour intervals. The type of instrument and
the sampling time length for the fixed position data are
shown in Figure 2.
3.1.
Horizontal
Currents
A depth-averaged(from 5 to 260 m) powerspectrum
of velocityat the WHOI mooring(Figure 3) showsa
10,256
FENG ET AL.: SEMIDIURNAL
TIDES IN THE WESTERN EQUATORIAL
data. We focusour analysison the energeticsemidiurnal band, particularly at the dominant M2 and S2 tidal
constituents. The tidal energy in the S2 constituent
WHOI Power SpectrumDensity
104
PACIFIC
103
Semidiurnal
is about 40% of that in the M2 constituent
from the
spectral analysis. Given the often complicated current
102
structurein this region[Eldin et al., 1994],we do not
E90% attempt to accountfor Doppler shiftingeffectsand only
10•
'13
considerthe energy that falls in the nominal frequency
•'•10
o
bands.
A harmonic analysis is used to represent the semidiurnal band mooring currents as constituent ellipses
• 10-•
U
[Goldin,1972].The tidal ellipsesshowthat the M2 cur-
10-2
rents are always more energeticthan the S2 currents at
eachof the mooringsand at all depths(Plate 1). Typical current speedsare 4-6 cms-1 and 1-2 cms-1 for
lO-4
, , , , ,,i
,
.......
i
.......
10-•
10ø
10•
M2 and S2, respectively. The dominant diurnal band
cycle per day
constituents(not shown)are comparableor weakerthan
Figure 3. The depth-averagedpower spectrumof the the S2 amplitudes. For the M2 tide.,the direction of the
WHOI velocity. The 90% confidenceinterval is shown. ellipse semimajor axes changesnoticeably with depth.
At 50 and 100 m, the ellipsesare alignedin a northeastsouthwest orientation at the WHOI, Tokai, and Vickprominent peak in the semidiurnal band with weaker ers moorings,although at the UW mooring, the 100 m
peaks at the diurnal, 6-hour, and 2-day bands. Similar depth currents are oriented in a more east-west direcspectral characteristics are found for the current data tion. In the 150-250 m depth range, the orientation of
at the other mooringsand for the WHOI hydrographic the ellipses shifts to an east-west direction at each of
IIII
IIIII
III
iiiiii
I
82
I,
2 cm/s
Depths:
Depths:
50m
50m
100m
100m
1 50m
200m
1 50m
lO 30' S
200m
250m
,,,
1ø 30' S
250m
CRS Model
I
I
2 crn/s
CRS Model
,
...
WHOI
WHOI
I
Tokai
Vickers
2ø S
2øS
I
i
156 ø E
156 ø E
Plate 1. Tidal ellipsesfor the (left) M2 and (right) S2 constituentsat 50, 100, 150, and 200 m
depth at the four IFA moorings.The CRS barotropicmodel ellipseis shownin the inset
FENG ET AL.' SEMIDIURNAL
TIDES IN THE WESTERN EQUATORIAL
PACIFIC
10,257
UW
E,[100
Nov 92
Jan 93
Dec
Mar
Feb
WHOI
i
I
Jan 93
Feb
lOO
200
Nov 92
Dec
Mar
Tokai
lOO
200
Dec
Jan 93
Feb
Vickers
lOO
200
Dec
A
A
A
Jan 93
A
A
Feb
A
A
A
A
Peak SpringTides
o
lO
20
cm/s
Plate 2. The modulusof the semidiurnalband (1.8-2.1 cpd) tidal currents at the four IFA
moorings. The time of peak springtides is indicatedwith triangles. The red trianglesindicate
springtide when solitarywavepacketswereobservedin the IFA.
the moorings, in the same direction as the CRS estimate of the M2 barotropic tide. Ellipsesfor the S2 and
ulated both spatially and temporally as evidencedby an
estimateof semidiurnalband currentamplitudes(Plate
weaker tidal constituents
do not show as consistent a
2). Plate 2 showsthe modulusof band-pass-filtered
curpattern with depth. The variation in orientation of the rents (1.8-2.1 cpd), whichare further smoothedusinga
M2 semimajor axis with depth indicates an energetic low-passfilter (cutofffrequencyof 0.33 cpd) to emphabaroclinic tide relative to the barotropic component.
size the envelopeof the semidiurnalband currents. The
The ellipsesdepict the time-averagedtidal flow. In strongestcurrent events typically are found above 100
fact, the semidiurnalband current field is stronglymod- m depthwith peakamplitudes
exceeding
115cms-•. A
10,258
FENG ET AL.: SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
U
V
,
Temperature
.
50
100
.-• 15o
200
•*
250
I
..
300
o
I 0 0.20:4C20:6
0[8I 0 0.20.4C20.6
0.8I
Figure 4. Squaredcoherence
of the WHOI velocity(left, eastward;middle,northward)and
(right) temperature with the CRS barotropictide in the M2 and S2 bands. The solid lines
indicate the 95% significancelevel.
spring-neapcycleis apparent, although it is highly variable in amplitude and depth structure from one spring
tide to the next.
3.2.
Coherent
Vertical
Structure
The vertical structure of the semidiurnal tidal signal is further characterizedusingdata from the WHOI
mooring which had the most completecoverageof current and temperature of all the IFA moorings.We seek
to determine whether the tidal variability is coherent
or phase-lockedto the barotropic tide. We use 15-day
segmentsof WHOI data in calculating the coherence.
FollowingHendry [1977],the coherenceis definedas
The magnitudesof the coherentsignalin the u component are larger than the CRS results. Although coherent baroclinic tidal signalsare apparent in the current
data, it is not possibleto separatethe barotropic and
baroclinictides usingonly the shallowmooringdata.
The coherencesbetween temperature and the CRS
tide are significantbelow 120 m depth for M2 and be-
low 140 m for S2 (Figure4), consistentwith the first
verticaldisplacement
mode(AppendixA). The significant coherenceof the deeplayertemperaturedata indicates the phase-lockedcharacter of the baroclinic tide.
Below 150 m, the depth-averagedepochsare 341ø for
the M2 band and 391 ø for the S2 band with a standard
error of -•30 ø. The phasedifferencebetweenthe S2 and
ple, is the Fourier transform of temperature fluctuations M2 tides is then -•50 ø. The epoch differencein the S2
at a tidalfrequency,
• isthetransform
ofthebarotropicand M2 surfacetides is ((7s2- (7M2)0=13ø from the
tidal elevationfrom the CRS model, anglebracketsindi- CRS results.FollowingHendry[1977],(ks2- kM2)R=
cate an ensembleaverageand asteriskdenotesthe com- (Gs2-GM2)- (Gs2- GM2)0,whereR is the wavepropplex conjugation. Given the temporal intermittency of agation range. The observedphasedifferenceat depth
the semidiurnalband in this region (see Plate 2), we can be accountedfor by a mode 1 wave generated320
treat each ensembleas statistically independent for the km away when we take the wavenumber difference as
in Table A1. This encompasses
the islandregionto the
purposesof estimatingsignificancelevels.
For the u current component, the coherenceat the southwestof the IFA and northeastof New Ireland, the
M2 frequencyexceedsthe 95% significancelevel at all proposedgeneration site of the solitary wavesobserved
depthswith highervaluesat deeperdepths(Figure 4). duringsomespringtides [Pinkelet al., 1997].
At the S2 frequency,the coherences
alsoexceedthe sigWe next apply a modal decompositionto the WHOI
nificance level at depths between 60 m and 120 m and velocitydata in daily segments
[SiedlerandPaul, 1991];
below 160 m, although the valuesare generallylessthan that is, we do not resolve the M2 and S2 tides but confor the M2 tide. For the v component, significant co- sider a single semidiurnal band centeredat the M2 freherencesonly occur above 150 m for the M2 tide and quency.The first and second
verticalmodes(Appendix
between 100 and 220 m depth for the S2 tide. We note A) are least squaresfit to the harmonicamplitudesfor
that the predicted barotropic tidal current is weak in the each day. The modal amplitudesof the first and second
meridional direction as shownin Plate 1. The large co- baroclinicmodesare examinedonly for the v compoherencesbelow 180 m betweenthe u current component nent, for whichthe barotropiccurrentsare predictedto
and the CRS tide suggestthat at thesedepths,the tidal be weak (Plate 1). The amplitudeof the first modehas
current is dominated by the barotropic tide. The mean a clear fortnightly modulationthat corresponds
to the
epochsover this depth range of the u componentare barotropictide signalin the CRS sealevel (Figure5).
20ø and 78ø for the M2 andS2tides[Schureman,
1958], Short timescale variability is also evident in the mode
comparedto 46ø and 58ø for the CRS model(Table 1). 1 amplitudes, indicating variability that is not phase-
C2_< •p•, >2 / < •p•p,>< •, >, where
•P,forexam-
FENG ET AL.' SEMIDIURNAL
i
,
mode 0
I
,
TIDES IN THE WESTERN EQUATORIAL PACIFIC
,
10,259
locitiesof the two verucal modes(Table A1) indicate a
generationsite 200-300km awayfrom the WHOI mooring, consistentwith the valueestimatedfrom the WHOI
ß
2O
temperature data.
4. Survey Data Analysis
-2O
High spatial resolutiondata were collectedfrom the
R/V Wecoma
for threelegsduringthe IOP (Figure3):
November 13 to December2, 1992, December18, 1992
to January 9, 1993, and January 27 to February 15, 1993
•ode1 ......
[Huyeret al., 1997].The Wecomarepeateda butterfly
10
pattern every 1.5 days over a 130 km x 130 km region
with the crossoverpoint at 1.8ø S, 156.1øE. The shipboard measurementsinclude ADCP velocity collected
from -•18 to 300 m and averaged into 10 m depth bins
and Seasoartemperature and salinity profilesfrom the
E
-5
surface to 250 m with
a 2 m vertical
resolution.
All of
the Wecomadata are hourly averaged along the ship
track.
4.1. Barotropic/Baroclinic
10
-5
-10
10/26
11/10
11/25
12/10
1992
12/25
1/9
1/24
2/8
1/23
1993
Tidal Fits
Analysis of the IFA mooring data indicates that the
semidiurnal tide consistsof comparable baroclinic and
barotropic current amplitudes in the upper ocean and
that a significantportion of the baroclinic tide is coherent with the barotropic tide with a dominant northeastsouthwestcurrent orientation. To further quantify these
results, we examine the isopycnal and velocity survey
data from the Wecoma. Specifically,we seek a simple
description of the barotropic and baroclinic tides that
can explain a significantpercentageof the IFA semidiur-
Figure .5. Time seriesof sea surfaceelevationfrom
the CRS semidiurnalbarotropictide (mode0) and the naltidalcurrents
in theupperocean.Giventhenea}ly
modal variabilities
of modes 1 and 2 semidiurnal baro-
clinic tide in the v component at the WHOI mooring.
Note that minor semidiurnal constituents are also included in the sea level time series so that there are tem-
poral variations besidesthe spring-neapcycle.
locked to the surface tides. Although weak in general,
the mode 2 amplitudes are appreciableat times suchas
the peak after February 8, 1993. This mode2 peak lags
a mode 1 peak by 2-3 days. The computedphaseve-
constant phases of the M2 and S2 CRS tides over the
IFA (Figure 1), we assumea zero horizontalwavenumber for the barotropic tide. To capture the dominant
phase-lockedcomponent of the baroclinic tide, a series
of plane wave fits with variable wavenumberis applied
to the Wecomadata (Table 2). We note that the isopycnal data are dominated by the baroclinic tide, while
the current data contain both barotropic and baroclinic
signals. Therefore we begin by analyzing the isopycnal
data to determine the horizontal wavenumber vector,
Table 2. Summary of Tide Fits
Fit
Data
Barotropic
Baroclinic
Wavenumber
Mode
Period
1
2
3
4
5
isopycnal
isopycnal
current
current
current
yes
removed
removed
yes
yes
yes
yes
yes
search
fixed
fixed
fixed
fixed
1
0,1
1,2'
1,2'
15-day
15-day
15-day
15-day
3-day
Fits 1-4 use data in one leg (more than 15 days) in one realization,which are called 15-day analysesin the text. Fit
5 usesdata within two butterfly circuits(•3 days)in onerealization,whichis calleda 3-day analysis.
*Fits 4 and 5 use u and v data in all depth with the vertical modal shapesand polarization relation as constraints.
10,260
FENG ET AL.- SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
In the case of velocity, a barotropic tidal component
and therebythe directionof propagation,of the domiis now included in the model fit. The u and v current
nant baroclinicwave(fit I in Table 2).
in the samemanneras the isopycnal
The Wecomadensitydata are first convertedto ver- data are processed
tical displacements.To resolvethe M2 and S2 tides, data. The data fit is of the form
15-dayor longertime segments
in the threelegsare analyzed.Fromeachtime segment,the spatialmeanand
U(X,y, t) --- C1COS
•)M2cq- C2sin•)M2c
linear horizontalgradientson the timescaleof 3 days
q- C3COS
•)S2cq- C4sin q5S2c
or longerare removedusingplanefitting to reducethe
subtidalenergy.A planewavemodelof the M2 and S2
+ c• cos•bm2a+ c6sin •bm2a
baroclinic tides is then fit to the resultant data:
d(x,y,t)
--
½1COS
•M2c q- ½2sin •M2c
+
CaCOS
•bS•c+ c4sin •bS•c
whered is the isopycnaldepth, •M2c -- C0M2t-kxM2X-
kyM2Y,
•bs2c
- cos2t-kxs2X- kys2y,C0M2
andc0s2
are
the M2 and S2 angularfrequencies,
k•. and kv arezonal
and meridional wavenumbers,and cn, n=1,4, are constants to be obtained from the model-data fit. The sin-
gularvaluedecomposition
methodis usedto calculate
the constants
in eachdepthlayer[Candelaet al., 1992].
+
c7cos•bs2a
+ Cssin•bs2a
(2)
wherethe barotropictide (the terms with constantsof
cs- cs)is assumed
to havezerohorizontalwavenumber.
Similar to the isopycnalanalysis,the maximum baroclinic tidal amplitudesof both the u and v components
are alsonear the mode1 wavenumber(not shown).Figures8 and 9 showamplitude and phaseversusdepth for
leg 1 usinga fixed mode 1 wavenumberin the northeast
directionfor the baroclinictide at eachdepth (fit 3 in
Table 2). The calculationof standard errors in each
layermakesuseof the residualvariance[Uandelaet al.,
Depth layersare treated independentlysothat no vertical wavestructureis imposed.Resultsare obtainedas
wavenumbers
varyfrom-10to 10x10-2 radkm-• with
a resolutionof 0.2x 10-2 rad km-•.
For leg 1, maximum amplitudesof both the M2
and S2 tides are found near the wavenumberpair (kx,
kv)=(4,4)
x10-2 radkm
-• (Figure
6) indicating
propa-
70 rn
gationto the northeastwith a wavelength
of •110 km,
which is consistentwith a mode I wave (seeAppendix
-5
A). We notethat because
higher-mode
wavenumbers
are almostintegermultiplesof the modeI wavenumber,
somealiasingof high mode energymay occur. From
-10
-lO
-5
lO
1•
-lO
5
lO
lO
lO,
a bootstrapanalysis[Elton and Tibshirani,1986],the
90% confidence interval for the M2 tide maximum am-
5
plitudeat 210 m is between3.5 and 5.3 m. If we as-
o
sume that the area within the 3.5 m contour represents
the 90% confidenceinterval of the mode I wavenumber,
150 rn
' :::::::::.
-5-
then there is about 30ø uncertaintyin the wave direc-
tion (Figure6). We find similarresultsfor the other
two legs.Thesevariationsin the dominantwavedirection, alsodiscussed
by Mao [1997],may be attributed
-lO-lO
-10.
-10
lO
10
to a number of factors (i.e., multiple sourceregions,
wavesfrom distant sources,variable backgroundbuoy-
ancy,and currentconditions).In this study,we will
only considerthe phase-locked
signalin the dominant
,.:...-.:.:::::::.;:::.
,
5
•
o
o
-5
210 rn
-5
wave direction and treat the residual as inherently unpredictable.
-10
-10
Assuminga constantmodeI horizontalwavenumber
((4,4)x10
-2 radkm-•) with depth(fit 2 in Table2),
the waveamplitudesgenerallyincreasewith depthfrom
about 3 m at 70 m to more than 4 m at 210 m for the
M2 tide. The S2 tide has a similar depth structure but
smalleramplitudesthan M2 (Figure 7). Resultsfrom
the other two legs show similar amplitude and phase
characteristics.The phasesof the M2 and S2 tides are
verticallyconsistent
for the threelegswith meanvalues
of 176ø, 157ø, and 179ø for the M2 tide, 187ø, 181ø, and
203 ø for the S2 tide.
•.:•.2.:-:-':',x---.-..'..,'-.:-:
.....
:-•*.-....:*.:•1•::.:.,.
-.-:::•:::::,!:!:!:i::!
2
3
4
5
Figure 6. Amplitudesof the baroclinictide from the
isopycnaldata analysisin the wavenumberdomainfor
the (left) M2 and (right) S2 tides (fit 1). The mean
isopycnaldepthsare 70, 150, and 210 m. The unit of
the axesis 10-2 radkm-1, and the contourintervalis
0.5 m. The amplitudeslessthan 2 m are not plotted.
FENG ET AL.: SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
Phase
Amplitude
.
.,
10,261
.
,
xx• - M2
-- -- -- S2
-5O
I
!
-lOO
I
I
I
I
.c: -150
-200
,,
-250
-300
0
2
4
8 0
6
meter
90
'•
180
270
360
degree
Figure 7. Verticalmodalfit of the Wecoma
isopycnal
data for leg 1 (fit 2). The heavylinesare
amplitudesand phasesof the M2 and S2 tideswhenfixingthe wavenumberin the isopycnaldata
analysis,and the light linesare the mode I fits.
Amplitude
1992]. A weightedinversion,which usesthe standard
Phase
01
errors of each layer as priori noisecovariances,is applied
to fit the verticalmode[Dushawet al., 1995].
Wecoma
= - -CRS
The barotropic tide has almost constant amplitude
and phase with depth for the M2 and S2 tides in the u
component,in agreementwith the CRS estimate(Figure 8). Similar resultsare foundfor the other legs. The
amplitude of the barotropic tide in the v component
(not shownhere)is verysmallsothat the phasesare not
-200
stable. Becausethe barotropic and baroclinic tides are
not orthogonal over the sample space, energy leakage
between the barotropic and baroclinic tides is unavoidable, especially in the upper 100 m where the mode 1
velocity is largest. The resultsfor the barotropic tides
from the Wecomadata in Table 3 are the weightedaverage from 110,to 300 m. The phasedelay relative to the
-250
oI
,
:(•::-. '• '•<-'•:::•::i:::•i(•:i:::
-100
-.
•:•!i
'*i;
•.:.:.•:•:•:•:..-.•:"-::..•:•:•:•:•:•:•:•:r
================================
CRS resultsmay be causedby localfriction[Cartwright
½tal., 1992].
.
'
-
...
.•..
I'
• .j:•.•.-.'
-300
.............
..• 'r•..........
.i:::.-.•
.::::•i!ii::•?.•,
==========================
S2
.:i•:!:•?.'.'::'
In leg 1, the M2 baroclinic tidal amplitude in the
u componentis about 6 cms-1 near the surfaceand
decreases
sharplynear 100 m depth (Figure 9). A second smallerpeak (2 cms-i) occursin the 200-250m
depth range. For the v component, the M2 baroclinic
tidal amplitudeis also-•6 cms-1 nearthe surface,but
o
6
2
½m/$
-180
-90
0
degree
90
180
the amplitude decreasesmore linearly with depth than
for the u component although small scalefeatures are
present (Figure 9). The S2 baroclinictide has a verFigure 8. The barotropictidal amplitudeand phase tical structure similar to M2 but with smaller amplifor the (upper) M2 and (lower)S2 tidal constituents tudes. The theoretical and observed mode I vertical
calculatedfrom the Wecomadata in leg I when fit- structuresare similar, especiallywhen the background
ted usingzero wavenumber
barotropictidesand mode
in themodeequations
(FigI baroclinictides propagatingnortheast(fit 3). The currentshearis included
ure A1). It appearsthat the sharp vertical shearnear
shaded areas indicate the standard errors of the estimation in each layer. The dashedlines indicate the 100 m in the u component of the baroclinic tide results
estimationfrom the CRS tidal model(Table 1).
from the verticaladvectionof the meanflow (Figure9).
10,262
FENG ET AL.: SEMIDIURNAL
TIDES IN THE WESTERN EQUATORIAL
Phase
M2, Amplitude
PACIFIC
Phase
S2, Amplitude
,
,
,
,
u
-100
•,-lOO
..-15o
•-•..-150
I
-200
-200
!
-250
-250
-30o
-300
0
0
.
s
-100
-100
v
E
.c: -150
-1504
-200
-oo
].
300'
0
2
4
6
' ß
-180
cm/s
-3OO
-90
90
0
180
0
,
.
.
2
4
6
-180
-90
cm/s
degree
0 ß
9'0
180
degree
Figure 9. The baroclinictidal amplitudeand phaseof the M2 and the S2 tidal constituents
calculatedfrom the Wecomadata in leg I when fitted using zero wavenumberbarotropic tides
and mode i baroclinictides propagatingnortheast(fit 3). The shadedareasindicatethe standard
errors of the estimation in each layer. The dashedlines are the amplitudes and phaseswhen we
fit the baroclinic
tide with mode I structure.
The mode 1 phasesfor the M2 and S2 tides are calcu- consistentwith a northeastpropagatingwave[Levine
lated from the verticalmodal fit (Figure9). At depths and Richman,1989;Dushawet al., 1995]. The weaker
where the tidal amplitudes are small, the phasesfrom signalsin the S2 band are similar to the M2 results.
the data fit are not stable. The orientations of the semiThe temporal variability of the baroclinicenergyin
major axesof the tidal ellipsesfor the M2 and S2 first the first two modes is examined by fitting the tidal
vertical modes are largely in the northeast-southwest structuresto the data at all depthsat once(i.e., with
direction becauseof the similar magnitude and phase vertical modal shapeas constraints)insteadof analyzbetweenu and v (Figure 9). Note that we fit the u ing each depth bin separately(fit 4 in Table 2). The
and v componentsseparately in fit 3, that is, we do CRS barotropic tide in the M2 and S2 bands is renot imposethe polarization relation betweenu and v. moved from the data prior to the model fit. The first
The isopycnaldisplacements
of the Wecomaanalysisare two vertical modes are used to represent the baroclinic
nearly 180ø of phasewith both the velocityco•nponents tide. The modesare normalized by their surfacevalues
of the first vertical mode in the M2 band. This is also
and the polarization betweenthe u and v componentsis
Table 3. Amplitude and Phaseof the BarotropicTidal Velocity From the 15-day Analysisof the WecomaData
Leg I
Amplitude,
--1
mm
M2 u
v
S2 u
v
s
32
6
11
8
Leg 2
Leg 3
Mean
Phase,
Amplitude,
Phase,
Amplitude,
Phase,
Amplitude,
G
mms -1
G
mms -1
G
mms -1
Phase, .
G
45
103
76
255
35
4
18
7
48
169
73
272
34
6
19
6
61
266
74
327
34
5
16
7
51
179
79
285
FENG ET AL.' SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
10,263
current data from 20 to 300 m in each leg are used
in one fit so that the amplitudes and phases of modes
mode 0
I and 2 are solveddirectly from the fit (AppendixA).
Figure 10 showsthe temporal variability of the first two
modes of the baroclinic tide at the crossoverpoint from
the aboveanalysis(Table 4), togetherwith thoseof the
CRS barotropic tide in the u component. The phases
of both modesare consistentbetweenthe three legs. In
leg 3, the S2 amplitudeof mode I is too small (Table 4)
• 0
-5
i.
mode 1
to havean appreciablespring-neapbeating(Figure 10).
The mode 2 amplitude, however,is much strongerdur-
,
ing leg 3 than the other two legs. In addition, we also
include the semidiurnalband mooring data during the
ß
ß
ß
Wecomasurvey time as constraints on the Wecomadata
baroclinic tide fit. The CRS barotropic tide is first removed from the mooring data. The changesin amplitude and phase of the baroclinic tides due to the inclusionof the mooring data are minimal. For example,
-5
the amplitudes(phases)of the mode I M2 and S2 tides
ß
I
I
I
I
I
I
I
in leg I changeto 48 mms-1 (349øG)and 25 mms-•
(0øG) after inclusionof the mooringdata.
In fits 1-4• which use 15-day segmentsof data, the
spring-neapcycle is representedthrough the beating of
mode 2
ß
-5
.........
. ........
,
I
i
........
. ........
! ........
ß ........
•
........
I
i
I
i
i
1/9
1/24
2/8
the M2 and S2 constituents. To capture more shortterm temporal variability that is not representedonly
by the M2 and S2 frequencies•a 3-day fit is made to
the Wecomadata• which is similar to fit 4 except that
. .........
the S2 componentis not included(fit 5 in Table2). We
10/26
11/10
11/25
12/10
12/25
1992
focus on the modulation
1/23
Figure 10. Time seriesof the barotropic tide from the
CRS model (M2 and S2) and the modal variabilities
of the baroclinic tide at the crossoverpoint from the
Wecomadata analysisfor the +15-day time segments
(fit 4).
tide rather
than on changesfrom the dominant wave direction and
so fix the horizontal
1993
of the semidiurnal
wavenumber
to the northeast
di-
rection. The barotropic tide in the M2 and S2 bands
is removed from the data prior to the model fit as in
the fit 4. The central frequency of the semidiurnal tide
is assumed to have the M2 frequency. The model fit is
made over the entire depth range by assumingthat the
vertical structures are given by the first two baroclinic
modes(AppendixA). Thesebaroclinicmodestructures
usedso that we simultaneouslycalculatethe amplitudes
and phasesfor the u and v components,which are the
same for a northeast propagating wave. The equation
are calculatedin each 3-day time segment,i.e., changes
in mode structure due to temporal variations in the
background shear and stratification are taken into account. The phase speed of the baroclinic tide is now
a slowingvarying function of time. The variable mode
2 wavenumber.
Note that the mode 2 wavenumber is
structure yields a more distinct spring-neapcycle than
almostdoublethat of mode I (Table A1). All Wecoma if a constant mode structure is used. The timing of the
is the sameas equation(2), exceptthat we replacethe
barotropicwavenumber(whichis zero) with the mode
Table 4.
Wecoma
Amplitude and Phase of Mode 1 and 2 Baroclinic Tidal VelocitiesFrom the 15-Day Analysis of the
Data
Leg 1
Amplitude,
--1
mm
Mode
I M2
S2
Mode 2 M2
S2
s
54
29
5
3
Leg 2
Leg 3
Phase,
G
Amplitude,
mms -•
Phase,
(3
Am plit ude,
mms -1
Phase,
(3
353
10
210
175
26
16
7
5
357
35
119
170
45
7
17
11
358
5
128
186
10,264
FENG ET AL.' SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
mode 0
u2
x 100
where u is the semidiurnalband-passedmooring data
and • is the sum of barotropic tide and plane wave
• 0
baroclinic tidal model estimate. The barotropic tide is
specifiedby the CRS results. We considerfour separate
casesfor the baroclinic tide: model predictionsfor the
-5
15- and 3-day time segmentfits (fits 4 and 5) with and
without the mooring data. The Tokai data are excluded
from the mooring data used in the fit so as to provide
an independent test of the model. The Vickers data are
not used in the leg 3 calculationsbecauseof large and
variable ship drifts.
For each mooring, the model skill tends to be bet-
mode 1
ß
ß
ß
ter for the 3-day analysisthan the 15-dayanalysis(not
shown).In general,includingthe mooringdata in the fit
resultsin a small (5-10%) increasein skill for both the
3-day and 15-day analyses. Significantimprovements
(about 20% or more) occurabove100 m at the WHOI
mooring (Figure 13). The model skill for the WHOI
-5
mode 2
.
mooring is consistentwith the amount of variance coherent with the CRS surfacetide (Figure 4). This suggests that at the center of the IFA, the tide model is
accounting for much of the barotropic tide, as well as
the phase-lockedbaroclinic tidal energy in the upper
ß
• 0
ocean.
-5
10/26
11/10
11/25
12/10
1992
12/25
1/9
1/24
2/8
lOO
Figure 11. Same as Figure 10 except that the baroclinictide is for the 3-daytime segmentanalysis(fit 5).
The mooringdata have been includedto constrainthe
baroclinic
U variance
2/23
1993
9o
c
80
tide.
13. 70
springtide is the sameas in the 15-day analysis,and
the tidal phaseis in agreementwith the 15-dayanalysis,
especiallyduring the spring tide. The phaseestimates
in adjacent time segmentsin the 3-day analysisare in
good agreement.
The 3-day analysisis repeatedwith the mooringdata
6o
5o
100
suitsin slightchangesto the amplitudeand phase(Figure 11). The 3-day tide model (Figure 12) reducesthe
90
60
4.2.
50
With
Mooring
Data
The predictability of the tide in the IFA is quantified by skill of the barotropicplus plane-wavebaroclinic
model fits in describingthe tidal-band velocityobserved
at the IFA moorings.The skill, or measureof the percentageof varianceexplained,is specifiedby
i
,
,
:
:
:
:
11/25
12/10
12/25
1/9
1/24
•
,
'E 80
half that of the 15-day model (not shown). Including
the mooringdata doesnot significantlyaffectthe residual variance(Figure 12).
Comparison
i
v variance
included to determine whether this improves the datamodel comparisons. Including the mooring data re-
residual variance of the Wecomavelocity data to about
i
70
10/26
.......
11/10
2/8
1/23
Figure 12. Variance of the Wecomavelocity data reduced by the tidal model for the 3-day time segment
analysis(a) without and (b) with mooringconstraints.
The variancesare vertical mean weighted by the mode
I structure.
FENG ET AL.' SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
,
,
o
E
E•=
i
(uJ)qldoa
i
i
(m) t!•doo
i
10,265
10,266
FENG ET AL.- SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
The model skill is lower in general at the UW mooring than at the WHOI mooring, particularly in the u
componentwhere the skill is lower than the squaredcoherence between the observed UW current and the CRS
model (by 5-10%; comparingFigures13 and 14). Pre-
and TOPEX/POSEIDON data during COARE, submitted to Journal of Geophysical
Research,1997). The
skill for the Vickerscurrentsis good for legs I and 2.
A decreasein skill for leg 3 may be due to the larger
drift of the Vickersoff stationduringthis leg. The skill
for the Tokai currentsis near 60% in the surfacelayer
sumably, this is due in part to the tendencyfor the Wecoma data fit to represent the phase of the wavesin the v component while slightly lower in the u component.
central region rather than at the outskirts. In general, Note that we do not use the Tokai data in the model
the model skill seemsto be sensitiveto small changesin constraint.
the specifiedtidal phases. Also the water depth at the
Comparisonof the 3-day model and observedcurUW mooring is more than 2000 m, about 300 m deeper rent time seriesis made for the WHOI mooringat 52.5
than the WHOI site, so that there may be amplitude m depth (Figure 15), where the skills are lnoderate.
changesin the barotropic tides. A recent study based During the springtide centeredat day 331, the agreeon the T/P alongtrack measurements
suggests
that the ment in amplitude and phase is good for the u comnortheastward propagating baroclinic tidal signalsare ponent, whereas the model v current has about I hour
less coherentnorth of 1.5øS in this region (L. Gour- phase'shift. For the v component,the agreementbedeau, Internal tides observedat 2ø S-156øE by in situ tween model and observationis particularly poor near
(a)
WHOI
.-.100
I !
I
I
i
Feb
Mar
• 200
Nov 92
Dec
Jan 93
o
(b)
O:
lO
2o
cm/s (c)
u
v
I
i
!
•E100__
.=.
lOO--
200 - -
0
2
4
cm/s
6
0
2
4
cm/s
6
Plate 3. (a) The modulusof the semidiurnalband advectivecomponentof the tidal current
2
) estimatedfrom the WHOI mooring.(b) The standarddeviationof the observed
((• + •)1/•
semidiurnalbandu current(solid)and the advectivecomponent
ua. (c) SameasPlate 3b, except
for the v component.
FENG ET AL.' SEMIDIURNAL
U
TIDES IN THE WESTERN EQUATORIAL PACIFIC
V
10,2t57
uw eastcurrent
10
/ /:
('
;'
,'1 )
,
.......
-200' •
15
-
Noah
current
.......
10
- S2
--
o:. o
o:.
Figure 14. Squared coherencebetweenthe UW mooring current and the CRS barotropic tide.
the neap tide, days 320 and 336, when the observedcurrent is much more energeticthan predicted. From these
comparisons,it appears that the 3-day model still does
not capture all tide energy,someof which may relate to
variable propagation direction and higher modes. Similar mismatchesare found for the 15-day model.
A comparisonof the observedand 3-day model time
seriesat UW 60 m depth (Figure 16) showsthat, the
model greatly overpredictsthe u current during the
spring tide at day 328-330, thus resulting in low skill.
Moreover, the timing of the springneap is not captured
in the model. The reason for this anomalous spring
event at the UW mooringis not known. The model and
observed v currents are more similar, although some
small phase shifts do occur.
i
i
i
i
i
i
i
-1•20 322 324 326 328 330 332 334 336
Yearday1992
Figure 16. Comparisonof 3-day model (solid)and observed(dashed)semidiurnalcurrentat the UW mooring
60 m depth. The model is constrainedwith the mooring
data.
4.3.
Wave Field Energetics
On the basis of the model
results for the M2
con-
stituent, the potential and kinetic energiesin the resolved first baroclinic
mode can be estimated
and com-
pared to the barotropic tide. The potential energy of
the baroclinic tide is calculated by fitting the isopycnal
amplitudes(Figure 7) with the first vertical mode and
averagingover the three legs. The potential energy for
the barotropic tide is calculated usingthe CRS sealevel
data. The kinetic energiesfor both the barotropic and
baroclinic tides are calculatedusing the mean valuesin
Tables 3 and 4.
The mode 1 potential and kinetic energiesare 104 J
WHO! east current
15
m-2 and 116 J m-2, respectively.
At this low latitude,
.....
the two energy levels are predicted to be nearly equal
10
(i.e., (•:-f:)/(z•+
5
f:)•
1)[Wunsch,1975]. For
the barotropic tide, the potential and kinetic energies
are 189 J m-• and 572 J m-•, respectively.
The ratio
-10
-15
'
320
322
324
326
328
330
i
332
i
334
i
336
between the total energy in mode I and the barotropic
tide is 29%, consistentwith the estimation by Wunsch
[1975]for a locallygeneratedbaroclinictide.
The energy flux in the first baroclinic mode is 538
North current
W m-1, a relativelylargevaluecomparedto estimates
from east of Blake Escarpment in the western North
10
-,5
Atlantic Ocean[Hendry,1977]and north of the Hawaiian Ridge in the centralNorth PacificOcean[Dushaw
et al., 1995].
, ••
-10
-15
320
322
324
326
328
330
332
334
336
Year day 1992
Figure 15. Comparisonof 3-day model (solid) and
observed(dashed) semidiurnalcurrent at the WHOI
mooring 52.5 m depth. The model is constrainedwith
the mooring data.
5. Summary and Discussion
The barotropic and baroclinic components of the
semidiurnal
tide in the TOGA
COARE
IFA
domain
are analyzed in this research. The barotropic tide is
dominated by the zonal component of velocity at both
10,268
FENGET AL.' SEMIDIURNALTIDESIN THE WESTERNEQUATORIALPACIFIC
1996]. The currentin this regionwasstrongin the
upper300 rn with a multilayerstructure[Eldinet al.,
layerabove70m, thereisthewestthe upper300m, the baroclinictide appearsto include 1994].In the surface
contributions from the first two vertical modes. The
erly winddriveneastwardcurrentwith peakvelocityof
the M2 and S2 frequencies,
in agreementwith the CRS
tidal model. Although the observationsare limited to
first modeis moreenergeticthan the secondand propagatestowardthe northeast.On the basisof the relationshipsbetweenthe M2 and$2 tides,andbetweenthe
energyof the first and secondverticalmodes,a probable sourceof the baroclinictide in this regionis located
•0320km southwestof the observingregion,wherethere
is a seriesof islands and ridges.
Plane wave modelsfor the barotropic and baroclinic
tides in the Wecomasurveydata are proposedin section
4. The modeled tidal phase, amplitude, and propaga-
50 cms-1. The eastwardEquatorialUndercurrent,with
zonalflowvaryingbetween
15 and45 cms-•, is found
below 120 m. Between these two currents is the rem-
nant of the SouthEquatorial Current that flowstoward
the west.Peakvelocityin thislayeris about30 cms-1.
To compute vertical mode structures, a mean den-
sity profileis formedfrom hydrographic
data collected
from several ships within the IFA. From 0 to 250 m,
the profile is the averageof the R/V Wecomasurvey
tion characteristics are consistent with the results from
data, whileat deeperdepthsan averageis madeof 130
deepCTD castsfromthe R/V Alisandthe R/V $hiyan
the CRS tidal modeland with the analysisof the WHOI
3 [TOGA COAREInternational
ProjectOffice,1993].
mooringdata. The modelsmatch•40-60% of the ob- The mean flow is taken from the horizontal mean of the
served semidiurnal current variance at the moorings
Wecomadata in the upper 300 m. Zero mean flow is
when the mooringdata are includedin the baroclinic
tide model fit. Neglectof a directionallyspreadbaroclinic wave spectrumand of higher baroclinicmodes
may contributeto lowskillvalues.We emphasize,
however, that the ability to distinguishcoherentbarotropic
and baroclinic tidal componentsin this region is due
assumed below 300 m.
Assumingthat the mean flow has larger horizontal
scalesthan the baroclinic tidal wavelengthof interest and a longertimescalethan the tidal waveperiods
[LeBlond
andMysak,1978;Boy&1989],theequations
for the near-equatorinternal wavesare (the Coriolis
primarilyto the surveydata. In particular,a similar terms are neglected)
descriptioncouldnot be obtainedfrom the available
Dou
1
mooringdata alone.
-- + wVz + - Px - O
(A1)
Dt
Po
In the westernequatorialPacific,the horizontalcurrent in the surfacelayer has complexstructure. There
Dov
1
-- + wVz+ - o
(A2)
are strongmeanshearsbetweenthe surfaceYoshidaJet,
Dt
Po
the South Equatorial Current, and the EquatorialUnDow
I
1
dercurrentduringthe IOP. The verticaladvectionof the
+- PzPg
(A3)
mean shearby the baroclinictide changesthe modal
Dt
Po
Po
structure(AppendixA). The comparisons
betweenthe
ux + Vy+ Wz- 0
(A4)
theoretical mode I structure and the velocity data fits
are significantlyimprovedwhen we includethe mean
flow shear in the modal calculation.
To our knowledge,this study is the first attempt to
capture the dominant features of the baroclinicand
barotropic tides in the deep ocean using only shallow shipboard survey measurements. A wavenumber
searchingmethodis usedto identifythe dominantpropagationdirectionof the mode I baroclinictide (fit 1).
The mode I wave found in this study has vertical and
horizontal structures that are consistent with theory
(fits 2 and 3). The plane wavefit is also consistent
with mooringobservations(fits 4 and 5). These resultssuggestthat a repeatedspatialsurveycan resolve
a narrow-band baroclinic signal even when the wave
length is closeto the surveyscale.
Appendix A' Vertical Mode Structure
and Tide Model Fitting Methods
Owing to the net surfaceheat and freshwater in-
Dop
--Dt
+ Wpoz- 0
(A5)
whereU(z) and V(z) are background
meanflowsin
the east and north directions, w is vertical velocity,
P is pressure,p is density, p0 is the vertical mean
density,g is the gravitationalacceleration,Do/Dr =
O/Or+ UO/Ox+ VO/Oy,poz= -N(z)2po/g,andN is
the buoyancyfrequency.After somemanipulation,we
have the equation for w,
Do
Dt2
(WVzz + WVz.)
Wyy + Wzz)+ N • (wxx+ Wyy)- 0
(A6)
Assuming
w hasthe waveform,w = W(z)exp[i(kx+
ly- wt)], the equationfor }V(z) becomes
Wzz+ a-2•W = 0
(A7)
puts[Webster
andLukas,1992],themixedlayerin the wherea = co- kU -1V and• = (N 2 - a2)(k2 +/2) +
conIFA was generallyshallowduring COARE. The pyc- a(kUzz+lVzz). Rigid-lidandflat bottomboundary
ditions
(W
=
0
at
z
=
0,-H)
and
a
shooting
method
noclinestarts at approximately70 m [Smythet al.,
FENG ET AL.: SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
10,269
are usedto solvefor W(z). Then the horizontalveloci-
Table
ties are
Modes Consideringthe Mean Flow Effect
U
z
i%
k2+12exp[i(kx
+ly- cot)]
v- k2i%
exp[i(kx
+ly- cot)]
Mode
(A8)
(A9)
where
% = kWz+ a-•(-12Uz + klVz)W
% = lWz+ a-•(-k•Vz + klUz)W
and the terms with WUz and WVz indicate the vertical
advectionsof the mean flow. In Appendix B, we will
give some detailed discussionabout the vertical advec-
A1.
Constituent
east. We note that the vertical
modes for the u and v
L,
k,
107.2
103.6
55.3
53.5
5.86
6.06
11.35
11.74
106.5
102.9
57.9
56.0
5.90
6.10
10.85
11.22
99.2
96.0
6.33
6.55
Leg 1
2.40
2
S2
M2
S2
1.24
-
Leg 2
I
M2
S2
M2
S2
2
2.38
1.30
-
Leg 3
I
M2
S2
M2
S2
2
2.22
1.26
-
56.5
11.12
54.6
11.50
c is the phasevelocity, L is the wavelength,and k is
the wavenumberin the three legsof the Wecomasurvey.
A.b. =dn
The first two modes for both u and v are
not orthgonal over the upper 300 m of the water column; however,the modes are sufficientlydifferent that
a model decompositionis pursued in section4. The inner product between modes I and 2 is -•0.27 for the u
c,
ms
-x km 10
-2tadkm
-x
M2
componentsdiffer becauseof the vertical shear. A notable changein the mode I u componentis a sharper
gradient near 100 m depth compared to the no mean
flow mode.
of the First Two Baroclinic
I
tion effects.
Figure A1 showsthe first 2 modesof the vertical displacement and horizontal velocity from the mean density and velocityprofilesof leg 1 of the Wecomasurvey.
The wavesare at M2 frequencyand toward the north-
Characteristics
(A10)
whereAn is an (Ln x M) modelmatrix, Ln is the number of observations
in layer n (n = 1,N), M is the
number of coefficients,bn is the vector of model coeffi-
cients,and dn is the vectorof observationdata at layer
modesand 0.19 for the v modes(hereeachmodeis nor- n. This equationcan be solvedby the singularvalue
malized so that the inner product of a mode with itself decompositionmethod. If F is the vertical mode strucis one). Table A1 givesthe phasespeed,wavelength, ture and b = {b• ... bs)', then solving
and wavenumber for the first two modes in the three
legs of the Wecomasurvey.
Ff = b
The plane wavefits of equations(1) and (2) in one
will give the modal constants,•. The modal fit can also
layer can be written as
Displacement
-50
./ .
(All)
Velocity
- l-l---
RAAGTRuu
T (Rug= GRAnG
• + R•, a priorimodel
covariance
matrix(Rnn)jj = ((Aj)•) anda prioridata
noisecovariance
matrix (R•)ii = ((ei)•), whichcomes
v mode I I
[[...... no-flow I[
•_...•'•.
•.-100
be appliedto the Fouriercoefficients
in the mooring
data analysis[Dushawet al., 1995]:• = Lb, whereL =
from the analysisin eachlayer).
When we use the vertical modal shape to constrain
the fit (fits 4 and 5), we solve
-150
Af = d.
-200
(A12)
A is the modelmatrix weightedby the modal shape.
For simplicity,whenthere is only oneverticalmode,
-250
A• Fx
-300
-200
0
200
-'
0
A-
'
dx
,
d-
'
.
Figure A1. The first two modesof vertical displaceAsFs
ds
mentandhorizontalvelocitycomputedfor legI in 0-300
m in the IFA domain. The insetin the left panelshows
the 2 modesoverthe full water depth (1744m). The Where Fn is the modal amplitude at depth n. The
of equations(All) and (A12) canbeproved
velocity modes are normalized with the values at the equivalency
surface.All modeshave been computedwith the mean by simplycomparingthe elementsin the two equations.
flow included except the "no-flow" modes.
Modal constantscan be deriveddirectly from equation
10,270
FENGET AL.: SEMIDIURNAL
TIDESIN THE WESTERNEQUATORIAL
PACIFIC
(A12). Thisis usedin the lasttwofitsin Table2. Note of individual isotherms becauseof the large upwelling
signalsthat occuron longertimescales
that we canimposethe polarizationrelationby combin- and downwelling
ing the equations
for u andv. Whenthe mooringdata
in the IFA which cause individual isotherms to advect
are used as constraints in the data-model comparison, out of the depth range of interest. In addition, isopyof the
the mooringdata are treated as the sameas the survey cnal trackingis not useddueto the sparseness
at the WHOI mooring.
data and addedto equation(A12). Includinga second salinitymeasurements
The standard deviation of Z is •4 m between 70 and
mode is simply addinga similar columnfor mode2 in
A.
200 m depthwith an increaseto 7 m near 230 m depth
(FigureB1). Comparedto the first bareclinicmode
(FigureA1), theamplitude
oftheestimated
profiledoes
Appendix B' Vertical Advection
of Background Shears
One concernin interpretingthe tidal variability from
Eulerian measurements is that vertical advection of the
not decreasewith amplitude above 100 m depth, and
the observedincreasein amplitude below 200 m depth
doesnot occur in mode 1. A complexdemodulationof
Z showsdisplacementsof 10-15 m during somespring
low-frequencycurrentfield by the bareclinictide may be
a significantcontributionto the tidal variance,particularly for the COARE regionwherebackgroundshears
are large. Isopycnaldisplacements
during springtide
events of order 20 m have been observedin the region
tides(e.g.,near day 410) (FigureB1). High-amplitude
eventsbelow 140 m (near day 360) without a correspondingresponse
at shallowerdepthsagainemphasize
that a singlebareclinicmodecanaccountfor onlya por-
[Pinkelet al., 1996;Picautet al., 1995].Thusattempts
sis of the horizontal current data. A similar mode fit to
tion of the variability, consistentwith the modal analy-
to determine modal structure based on the horizontal
the displacementestimatesis not made giventhe similarity of modesI and 2 for this depth range.
The amplitudeof the semidiurnalband currentsdue
nentto the tidal current(ua,va), fixeddepththermistor to the advection of lower frequency shears is taken
tidal current may be misleading.
To estimate the magnitude of this advectivecompo-
as u•(zo,t) = U(zo + Z,t)- U(zo,t), and v•(zo,t) =
meetingare usedto inferverticaldisplacements,
Z(t), V(zo + Z,t)- V(zo,t), whereU and V are the lowat the depthsof the currentmeasurements
(z0). Z is passedcurrents(cutoff of 0.33 cpd). Linear interpoobtainedby integrating
w = (OT/Ot)(OT/Oz)
-• Ver- lation is used to infer currents between measurement
tical displacementsin the semidiurnalband are then depths.A depth-timeplot of currentamplitude(Plate
computedby band-pass
filtering(1.8-2.1cpd). We ig- 3, similar to Plate 2) showsthat the advectivecomponore the horizontal advection of horizontal temperature
nentis weak(< 5 cms-•) for mostof the deployment.
gradientswhichwe assumeto be weakrelativeto the A spring-neapcycleis not apparent,althoughthe peadvectionof the verticalgradients.We favorthis fixed riod of strongestcurrentamplitudeoccurrednear 100m
depth approachrather than trackingthe displacement depth during the springtide just beforemid-February.
measurements between 52.5 and 260 m from the WHOI
-40
-60
,
(a)
ib) '
'
'
,
-80
,
,
-40
, -60
, -80
- 1O0
- 1O0
-140[
-160[
-140
-160
1
-200[
-24% •
,• ;
Meters
300 3•0 3,•0 3•0 3•0 4(•0 4•0
Year day 1992
Figure B1. (a) Standard
deviation
oftheestimated
isopycnal
displacement
in thesemidiurnal
band.(b) Timedependence
oftheisopycnal
displacement
in thesemidiurnal
bandversus
depth.
FENG ET AL.: SEMIDIURNAL TIDES IN THE WESTERN EQUATORIAL PACIFIC
10,271
Becausebackgroundshearsassociatedwith the u cur- Huyer, A., P.M. Kosro, R. Lukas, and P. Hacker, Upper oceanthermohalinefieldsnear 2ø S,156ø E during the
rent componentare strongerthan for the v component,
Tropical Ocean-Global Atmosphere-CoupledOceanUa is greaterthan Va (Plates3b and 3c). Again, comAtmosphere Response Experiment, November 1992 to
pared to the standard deviation of semidiurnalband
February 1993, J. Geophys. Res., 102, 12749-12784, 1997.
currents, Ua and Va are weak at all depths. The max- LeBlond, P.H., and L.A. Mysak, Waves in the Ocean, 602
imum measured standard deviation of Ua near 100 m
pp., Elsevier, New York, 1978.
Levine, M.D., and J.G. Richman, Extracting the internal
may accountfor a relative maximum in the standard
tide from data: Methods and observations from the Mixed
deviation of the horizontalu current (Plate 3b). In
Layer Dynamics Experiment, J. Geophys. Res., 9d, 8125general,however,it doesnot appearthat the advective
8134, 1989.
signalwill have a strongeffecton the analysisof the Luther, D.S., and C. Wunsch, Tidal charts of the central
horizontal
currents.
Acknowledgments.
The authors would like to thank
Richard Ray for providing the barotropic tide harmonic
constantsat 2øS, 156øE from his T/P tide model. Dou-
glasLuther, Feifei Jin, and Bo Qiu gavehelpfuladviceon
Pacific Ocean, J. Phys. Oceanogr.,15, 222-230, 1975.
Luyten, J.R., and H.M. Stommel, Comparison of M2 tidal
currents observed by some deep moored current meters
with those of the Schwiderskiand Laplace models, Deep
Sea Res., 38, S573-S589, 1991.
Mao, M., Analysisof 3-dimensionalcurrent structuresusing
ship-mountedADCP, M.S. thesis, 105 pp., Sch. of Ocean
and Earth Sci. and Technol., Univ. of Hawaii, Honolulu,
an early version of this manuscript. Dr. Robert Weller
kindly providedthe WHOI mooringVMCM current data.
1997.
David Stoneand Richard Eanesassistedwith the T/P alPicaut,
J., A. J. Busalachi, M. J. McPhaden, L. Gourdeau,
timetry data. Julie Ranada, Xiaomei Zhou, and Sharon
F. I. Gonzalez, and E. C. Hackert, Open-ocean validation
DeCarlo processedthe R/V Wecomadata. Ying Xia and
of TOPEX/POSEIDON sea level in the westernequatoXian Zhou providedhelp in the computingwork. We would
rial Pacific, J. Geophys. Res., 100, 25109-25127, 1995.
like to thank all the peoplewho helped collectthe Wecoma,
Vickers,and mooring data during the IOP. M. Feng would Pinkel, R., M. Merrifield, M. McPhaden, J. Picaut, S. Rutledge, D. Siegel,and L. Washburn, Solitary wavesin the
like to thank his colleagueswho participatedin the Univerwestern equatorial Pacific Ocean, Geophys. Res. Lett.,
sity of Hawaii ADCP groupweeklymeetingfor stimulating
helpfulideas. M. Merrifieldwassupportedby NSF (OCE9415979). This work was also supportedby the U.S. and
Chinese TOGA COARE programs.
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10,272
FENG ET AL.: SEMIDIURNAL
TIDES IN THE WESTERN EQUATORIAL
PACIFIC
A.J. Plueddemann,
PhysicalOceanography
Department,
WoodsHole
M. Feng,Instituteof Oceanography,
ChineseAcademyof Sciences,
OceanographicInstitution, Woods Hole, MA 02543. (e-mail:
Qingdao,PeoplesRepublicof China.
E. Firing, P. Hacker, R. Lukas, and M.A. Merrifield, Schoolof [email protected])
OceanandEarthScienceandTechnology,
Universityof Hawaii, 1000
Pope Road, Honolulu,HI 96822. (e-mail: [email protected];
[email protected];
[email protected]; (ReceivedDecember31, 1996; revisedNovember11, 1997;
acceptedDecember3, 1997.)
[email protected])