JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C4, PAGES 8707-8725, APRIL 15, 2000
How can we improve a global oceantide model at a regional
scale? A test on the Yellow
Sea and the East China Sea
F. Lef'evre,C. Le Provost,andF.H. Lyard
Laboratoired'Etudesen G6ophysiqueet OcranographieSpatiales,Toulouse,France
Abstract. A globaloceanmodelhasbeendevelopedwithinthe contextof TOPEX/Poseidon
(T/P) onthebasisof a finiteelementhydrodynamic
modelingapproachcombinedwith data
assimilation
basedon the representer
method.The solutionproducedat globalscalerepresents
a spectacular
improvement
overwhatwasavailablebeforetheera of T/P. Typically,the mean
discrepancy
betweenthe maintidal components
anda hundredin situtide gaugesis 1.7 cm for
M2 and 1 cm or lessfor the othercomponents.
However,the accuraciesof this solutionandof
thatproduced
by differentauthorswithintheT/P tide workinggroupare all worsenear
coastlinesand over continentalshelves.This is the casefor our finite elementsolution(FES)
FES94.1 solutionover the Yellow Sea and the East China Sea (YS-ECS) where the
discrepancy
is 33 cm for theM2 tide and9 cm for the K• tide whencomparedto a setof 192
tide gaugesdistributedalongthecoastlines.
This is duelargelyto the complexgeometryof the
basinandthelimitedknowledgeof thebathymetry.
The aim of thispaperis to investigate
how
ourFES modelcanbe improvedwith a dedicatedapplicationfocusedon oneof the most
energeticcoastalbasins:the YS-ECS (- 180 Gigawattsfor M2, i.e. 8% of the globalenergy
dissipatedin the wholeocean).For thisapplication,a finite grid was implemented
with a
resolutiondownto 5 km alongthe coastsandoverthe continentalshelfbreak.Particular
attentionwaspaidto bathymetryby complementing
theETOP05 databasewith regionalmaps.
Sensitivityteststo thetuningof thebottomfrictionandthe nonlinearinteractions
betweenthe
diurnalandsemidiurnalcomponents
allow usto investigatethe impactof dissipation
parameterization
andto producean optimalsolution,withoutany dataassimilation,for nine
maincomponents
of the tidal spectrum(improvement
by a factorof 2). M2 distanceto the
observations
is now - 17.5 cm (globalvarianceof 92 cm), andK• is 4 cm (globalvarianceof
21 cm). As nonlinearcomponents
are significantoverthesebasins,the two mainquarterdiurnalcomponents
are calculated.The improvements
are explainedby five points:(1) the
choiceof a mixedfriction,(2) a refinementof themesh,(3) a choiceof reliableboundary
conditions,
(4)a refined
topography
and,(5)theuseofa specific
friction
coefficient
1.5xl0-3.
Surprisingly,
theenergybudgetfor theM2 component
leadsto a dissipationsimilarto the value
estimatedby the globalFES94.1. This is a majorresultof this study,whichleadsusto the
conclusion
that whenforcinga regionalmodelwith sealevel boundaryconditionsalongthe
openlimits,the simulatedvelocityfield adjuststo the tuningof frictioncoefficientsin orderto
dissipatethesameamountof energymadeavailableat its openboundaries,.
1. Introduction
Over most of the continental shelves and coastal seas, tides
are the main contributorto oceandynamics.This is a major
reasonwhy accuratepredictionsof sea level variationsand
currents are needed for commercial and engineering
applicationssuch as shipping and coastal management.
Specificmethodsare availableto performthese predictions
yearsin advancewith a high level of accuracyin the vicinity
of areas where observations
have been done before over a
conditions
suppliedeitherby in situtidegaugesdeployedover
a few months(but thesemeasurement
are costly),or by
referenceto basin-scale
or globalmodels.The standardglobal
modelhaslongbeenthe oneproducedby Schwiderski[1980].
The renewedinterestin globaltidesthat hasrecentlyemerged
with the adventof high-precision
satellitealtimetryLe Provost
et al. [1995]hasled to severalnew modelsfor predictingthe
tidal contributionto sea level variationsat the global ocean
scale. These models are able to give predictionswith an
accuracyof the orderof 3 cm or betterover the deep Shum
[1997]. However, their accuracyis stronglydegradedover
sufficientperiodof time. A largevarietyof numericalmodels
hasalsobeendevelopedto infer the characteristics
of the tides coastal and semi enclosed seas, as will be shown later in this
in areas of economicinterest in order to complementor
paper. Indeed, these altimetry-derivedmodels have been
minimize the number of in situ measurementcampaigns.
developedprimarily to eliminatethe tidal contributionto the
However, these regional and local models need boundary
sea surface topographyvariability for further use of the
altimetricdata for oceanopendynamicsinvestigations.
The
Copyright2000 by theAmericanGeophysical
Union.
methods used to extract tidal information
Paper number1999JC900281.
signalwere generallybasedon data binningover quite large
areas(typically3øx3ø) in orderto achievethe fight signal-to-
0148-0227/00/1999JC900281
$09.00
8707
from the altimetric
8708
LEF•VRE ET AL.:IMPROVINGREGIONALSCALEGLOBALOCEANTIDE MODELS
noise ratio and to overcomethe difficult aliasing problem
resultingfrom the longorbitperiodof altimetrysatellites(10
days for T/P and 35 days for ERS). Then it explainsthe
inadequacyof thesemethodsto get valuabletidal information
over shelves and coastal areas where tidal features are of
2030'
11rio
117030'
lPOø
122ø30'
12fiø
127ø30'
1,',i0ø
1320:
"
p•.'
'o'a•n
•L FJAPAN
SEA
smaller scales.In fact, many of thesenew altimetricmodels
used hydrodynamicsolutionsas an a priori, and thus have
nearly the accuracyof these hydrodynamicsolutionsover
shelves.For example,Andersen[1995], Eanesand Bettadpur
[ 1996], Le Provostet al. [ 1998] and Schrarnaand Ray [ 1994]
(in SR95.0) used the hydrodynamicfinite elementsolutions
(FES) FES94.1 of Le Provost et al. [1994] as a priori, and
•Bay
Sanchezand Pavlis [1995] usedthe Schwiderski'solutions.
New effortsare now underway to improvethe methodsfor
studyingtidesfrom satellitealtimetryovercoastalareas[Ray,
1998; Tierney, 1998]. The aim of this paperis to investigate
what can be expected from new dedicated efforts for
improvingthe globalhydrodynamic
modelsovercoastalseas.
We havechosento focusour studyon the Yellow Sea and the
East China Sea (YS-ECS). We had several reasons for
choosingthis area as a test. The main reasonis that the
FES94.1 solutionis indeedvery poor there,as will be shown
later,in contrastto othershallowseaslike the Europeanshelf,
MER
DIONAL
CHINA
;EA
where FES94.1 accuracywas correctAndersenet al. [1995].
Why is there sucha differencein quality betweenthesetwo
12•30'
115•
117•30'
120•
122•30'
125•
127•30'
130•
132•30,
areas within the same model? How can we improve the
solution in the YS-ECS? Tides in YS-ECS are large: Figure 1. Chart of the Yellow Sea and East China Sea with
Hydrographer of the Navy [1992] indicates that tidal thetide gaugesdatabank.
• •Inch•on•
Yangtze
Rive•
•• ghai
CHIN••STERN
CHINA
•EA•
elevations could reach ~ 5.6 m on the western coast of Taiwan
(near Chang Hua, 150kilometers southwestof Taipei),
3 metersin the HanchowBay (near Shanghai),3.7 m in the
PohaiBay (nearPekin),and 8.5 m in the IncheonBay (westof coasts
of Koreaandthewestward
coasts
of PohaiBay,thesea
Seoul, South Korea). Tidal currentscould reach very large bedtopography
is veryflat, coveredwith sanddunesand with
values:a Koreantidal power studyKorea Electric Company veryfew islands.Southof Shanghaiand alongthe Korean
[1978]
reports
thathighest
velocities
aremorethan3 ms'l at peninsula,
coastlines
are veryirregularwith numerous
bays
the mouth of Cheonsu Bay during ebb tides. These and inletsof varyingsizeswith coralreefsin shallowwater.
considerable elevations and velocities due to tides mean that
this area is one of the oceanicregionswheretidal dissipation
reachesveryhighvalues.Oneof thefirst estimations
of the M2
tidal dissipationin this areawas givenby Miller [1966], who
concludedfrom his study that 60 Gigawatts (GW) are
Aroundthis area,thereare severalpitsof morethan4000 m.
The extension
in latitudeexplainsthe observed
differences
in
temperature.
Ice staysuntil March in Pohai Bay, while the
easternpart of the YS-ECS is warmerbecauseof the Kuroshio
currentwhosesurfacetemperature
is ~ 25.6C Su and Weng
dissipated
ontheYellowSeaalone,representing
up to 3.5% [1994]. These two seas are a severe test area for numerical
of the1.7Terawatts
(TW) heestimated
forglobaldissipationoceantide modeling.
bybottom
friction.Improved
knowledge
of oceantideshasled
In sectionI the modelusedis briefly described,and an
to newestimations
of thisdissipation:
Le ProvostandLyard analysis
of thequalityof tidegaugedatabase
is presented
for
[1998]estimated
thattheM2tidaldissipation
equals180GW further comparisonof tidal solutionswith sea truth. In
overthisarea,i.e., 11.1%of their2 TW globalestimatefor
bottom
frictiondissipation.
Kanthaet al. [1995]gotthesame
amountof dissipation.
Whichnumbers
arefight?Theyare
dependent
on thequalityof the tidal currentsolutions
andon
thechoiceof thefrictioncoefficient
usedto parameterize
this
section
2 we firstexaminecloselythephysicalandnumerical
parameters
affectingthe qualityof theapplication
of the finite
elementmodel:boundaryconditions,
modeof computation,
meshresolutionand tuningof the bottomfrictioncoefficient.
In section3, consideringthe best parametersfor the
dissipation.We detailbelowsomeof thesereasonsthat led us computations,
we presentand evaluatethe quality of the
elevenmajortidal solutions
thusproduced
(M2,S2,N2, K2 and
tochoose
theYS-ECSasa testareaforourstudy.
This area (see Figure1) is divided into four different 2N2forthesemidiurnal
components,
Ki, Ol, Pl andQ• for the
regions:
(1)at thelatitudeof Beijing,thePohaiBay(BoHai), diurnalcomponents
andM4 andMS4for the quarter-diurnal
Finally, in section4 the calculationof the tidal
(2)west of SouthKorea,the Yellow Sea (HuangHai), components).
(3) northof Taiwan,the EastChinaSea (DongHai), and velocitysolutions
allowsus to establish
an energybudgetas
(4) southof Taiwan,a partof themeridional
ChinaSea(Nan well as a sensitivitystudyof the velocityfields to bottom
Hai). This area is very complexbecause,of both the frictiontuning,validatedby (only) 12 mooredcurrentmeter
topography
and the localdynamicsof the tides.Indeed,it is measurements in order to better understand how to control
dottedwithsandbanks,
reefsandislands.
Moreover,
strong tidal bottom friction dissipationin our hydrodynamic
currents
arefoundnearthecoasts.
Northof Shanghai,
nearthe numerical model.
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
2. Brief Presentation
and in Situ Database
8709
month.We had to adjustthe tide gaugephasesof the Korean
databankto the Greenwichmeridian systembecausethey
of the Model
were referenced to 135øE.
2.1. HydrodynamicModel
In orderto comparethe calculatedsolutionwith the in situ
a root mean square(RMS) norm is used.
The basicequations
of the modelare the classicbarotropic data measurements
This
measures
the
difference A(3,q•) between the two
nonlinearshallowwaterequationsin a rotatingframe.Bottom
quantities
(where
N
is
the
numberof observations):
friction is parameterizedwith a quadraticChezy-like law
(coefficient
Cs).Tidal forcingis basedon the astronomical
developmentof the tidal potential and completedwith
corrections
takinginto accountthe effectof Earthtides,ocean
tide loadingand self-attraction
all summarizedin the global
tidal potentialYI (for a completeexpression
of this potential
refer, for instance,to Dronkers [1964], Hendershott [1972],
Francis and Mazzega [1990], Schureman[1958] and Le
Provost [ 1973].
The continuityequationis
V.½u)0.
where a• and G• are the amplitudeand the phaseof the
component
of indexi respectively.
In thefollowingall comparisons
are madeaccordingto this
RMS. As our computations
are purelyhydrodynamic
(no data
assimilation),the measurements
of the 192 tide gaugestations
are independentof our solutions,which allows coherent
comparisons.
3. Tests to Improve the Solutions
The momentumequationin its conservative
formis
-+ (V.u).u
+gV +m ^u: v(n- ga)-
eiG'
]Measurements--•i(
'•'(/9)eiG'
]Computatio
(3)
ullu,(2)
3.1. Strategy Adopted for the Study:
Choice of the Modeled
We wantto focuson thequestion'How canwe improvethe
globaloceantidal hydrodynamic
solutionFES94 overthe YS-
where
ECS
a
Area
the sea surfaceelevation;
b
theseabedtopography;
h
the instantaneous
depth(h = H+a);
u
thehorizontalbarotropicvelocity;
I-I
the globaltidal potential;
•
theEarthrotation(f=2f•sinq>is theCoriolis
parameter
with •p thelatitude);
g
thegravityconstant;
coastal
area?'
The
modeled
area extends
from
the
latitude20ø to 40øN and from the longitude113ø to 132øE
(see Figure1). The choiceof the boundarylimits is very
important,especiallyopen ocean lines along which good
boundaryconditionscan be a priori defined.The domain
follows the southwestcoastlinesof China, North Korea, South
Korea,andJapanas well as the islands.The openboundaries
are chosenaccordingto the bathymetry,the locationswherein
situ data are available,and open lines along which good
Cf theChezy-like
friction
coefficient;
elevationsolutionsare alreadyknown from previousstudies.
V
the nablavectoroperator.
These open boundariesare absolutelyfundamentalas they
mainly controlthe quality of the calculatedsolutions.In the
After a decomposition
in time in the spectraldomainthe area of the YS, the FES94 model is the best accurateglobal
hydrodynamic
equationslead to a set of threeequationsfor tidal model along coastlines.However, it is not the most
eachfrequency
with threecomplexunknowns:the seasurface accuratein deepocean.This is the reasonwe haveconsidered
elevationa, andthetwo components
of thehorizontalvelocity two globaltidal modelsto fix our openboundaryconditions:
(u and v). These equationsare solved with the Code the Center of Space Research global model (CSR3.0)
d'E16mentsFinis pour les Mar6es Oc6aniques(CEFMO) developedby Eanes and Bettadpur [1996] and the FES95
hydrodynamic
modelformulated
originallyby Le Provostand globalmodeldevelopedby Le Provostet al. [1998]. In order
Poncet [ 1978] and Le Provostet al. [ 1981].
to comparethe accuracyof these two models along the
The elliptic formulationof the problemallows a finite segments
of openboundaryconditionschosena priori (a first
elementresolutionupona variablegrid adaptedto solvethe simulation area that did not take into account the whole island
betterlocaldynamicsLe Provostand Vincent[1986].A more of Taiwanled to dramaticallypoorresults)we built a database
detailedexplanationaboutthe CEFMO codeis givenby Le of 14 tide gaugeslocatednear thesesegments.By usingthe
Provost et al [1994].
RMS alreadymentioned,
Table 1 givesthe comparisons
of the
two models with these in situ data.
2.2. In Situ data analysis
For the studiedarea we selected192 tide gauge stations
Table 1. RMS of CSR3.0 andFES95 Alongthe Segments
(locations
aregivenin Figure1).Thesestations
wereextracted
Conditions
from the InternationalHydrographicOffice databank oftheOpenBoundar•
Number of
RMS for
RMS for
International
Hydrographic
Office[1979]andfrom a Korean Wave
TideGau•ges CSR3.0,cm
FES95,cm
report(H.-J.Yoon,personal
communication,
1995).In order
to selectthesetidegauges
we examined
thedataoneby oneto
M2
16
12.06
10.28
determinetheir validity accordingto three criteria: (1) a
14
2.03
2.85
coherentlocation,(2) a spatialcoherence
with other tide K•
CSR is Centerof SpaceResearch;FES is finite elementsolutions.
gauges
in the area,and(3) a recordlengthlongerthan 1
8710
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
Table I showsthat FES95 is better for M2 (nearly 2 cm
betterthan CSR3.0) but worsefor K• (nearly0.8 cm worse
thanCSR3.0).One aim of our studyis ultimatelyto improve
ourFES95 modelalongthe coasts(as it alreadyworkswell in
deep ocean).We finally choseto build the open boundary
conditions
so as to keepthe continuityof the elevationalong
the open boundaryconditionsof both FES95 and the new
solutionto emergefrom this study.Indeed,the FES95 model
will be improvedby our localmodelby blendingthe new local
resultswith the globalsolutionFES95.
11__2ø30
' 115
ø 117•30' 120
ø 122ø30
' 125
ø 127ø30
' 130
ø 132ø30
'
b::::
'•:•5•½•F
• :½•:•:•:½;•;•g•;•
$• ::::::::::::::::::::::
3.2. Computation Mode Test
The original finite element model, as formulatedby Le
Provostand Poncet[ 1978], assumedthe existenceof a unique
dominant wave, in terms of velocities, inside the area.
;I
However, along the coastlinesof the Arctic Ocean, Lyard
[1997] showedthat it is importantto relax this assumption
over the region where diurnal velocitiescan reach higher
i. z '-...-.:
..............
.::,:....
•.....:*..•!..'i$•;•!i•.;.':3!
:•Sf.:...,.:'•
values than semidiurnal ones. So, in order to check the
validity of this assumptionin the area of the YS-ECS we
computedtheM2 solutionbothfor a casewhereM2 is assumed
'ß ,- ....
:.:.::::::;D%/•:•:;:i::::-':]:::"'!
:'"•.•
ß-'-:q'•.-:.::.
(•.:
A.::..-i
i-•-,*,•*..
•,;•...
•:•j..*...!x
..,.'.
:::
.... ...•/..
':::
dominant (dominant mode) and in a case where M2 is
dominantin a part of the areaand Kl is dominantin the other
part (compoundmode).The compoundmodeallowsus to take
care of the dominance
of either the diurnal or the semidiurnal
waves for the linearization
030'
115ø
117030'
,
II
...
120ø
122030'
125ø
127030'
130
132o30 '
...............................
• ..,.•
,,...•/.•,•••.•.•••••s•
of the friction coefficient. Table 2
0.0
10
20
50 100 200 5001000200030004000
gives the RMS calculated for the two different modes.
Surprisingly,the compoundmode is 4 times more accurate
Figure 2. Mesh of the global hydrodynamicsolutionsover
than
the YS-ECS.
the dominant
one.
The
formulation
of
the
friction
coefficientin the compoundmode is driven by both the M2
and the Kl velocitiesin eachpart of the modelizedarea. M2
and K• both play the role of dominant waves generating
turbulencein the oceanboundarylayer (referto Lyard [1999]
for further explanations). Thus, in the following, all
computations
arein compoundmode.
3.3.
Resolution
Test
For FES globaloceancomputations,
(1) and (2) are solved
over a triangulation grid that exceeds 300,000 nodes.
However, in the area of the YS-ECS
refinement tends to fit better the local Le Provost and Vincent
[1986],whichfixed the maximumdistanceAL for a triangle
side for
a correct
2It
the resolution of this
global meshis only 200 km in deep oceanand 10 km near
coasts (see Figure2). With respect to the finite element
approach,
we refinedthe meshin orderto improvethe global
hydrodynamictidal model performances
in this area. This
element
For instance,for the M2 tidal wave (selectedas a reference
for the mesh)with a depthof 1000 m, the maximumdistance
betweentwo nodes(P2-Lagrange)is - 150 km. However,this
criterioncouldnot be sufficientin areasof high topographic
gradient Luettichand Westerink[1995]. We also tried to fit
another criterion based on the horizontal gradient A/-/of
topographyat a pointof depthH. The maximumdistanceAL
for a triangleelementsidemustnotexceedthe value
simulation
of
a tidal
wave
propagationat a valuebelow:
AL=15w
2Ita/gH .
(4)
Table 2. RMS for theDominantMode andtheCompound
Mode
Mode
RMS, cm
Dominant mode
123.29
Compound
mode
30.24
aL = ••.
H
15 AH
(5)
Typically, for a depth of 1000m and a slope of 1% the
maximum distancebetween two nodes (P2-Lagrange)is
- 20km.
We selected the best o! the two above criteria to build our
mesh.As a result,as we can see on the mesh displayedin
Figure3 comparedwith the bathymetryin Figure4, the shelf
break of the boundarybetween the Pacific Ocean and the
Yellow Seais now muchbetterdefined.In practicaltermsthe
distancebetweentwo nodesis now -30 km in deep ocean
and 5 km along coastlines.So, in our new local model the
tidal solutionsare computedon a meshthat contains11,448
triangles. As the calculationsare performed in the P2
Lagrangeresolution,the numberof nodesis exactly23,880
(see Figure3). The tidal elevationsare calculatedon each
nodeof the meshgrid (i.e. the threeverticesof a triangleand
the three midpointsof each side). The tidal velocitiesare
computedon eachof the sevenGausspointsof onetriangular
element.
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
112ø:
.
11
•
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
8711
the bottomtopographyare essentialto a betterresolutionof
the tidal dissipation.The modelbathymetryusedin CEFMO
is extracted from the ETOP05 database. However, the first
I
simulationswere far from the in situ data. Smith [1993] has
showna real lack of accuracyof this database.One particular
problemis due to the fact that ETOP05 is basedupon the
analysis of shipboarddepth soundings,with sometimes
hundredsof kilometers between tracks in some areas, which
o
prevent many tectonic featuresfrom being seen with this
bathymetricchart. Besides, the spatial resolution is not
sufficient (the numerical resolution is 5'x5', but the actual
resolutionis commonlylessbecauseof the sparsedensityof
data), and is especiallyinsufficientnearthe coastsor in very
shallow water areas. A more accurate checking of the
topographywith marine maps of the French and English
Navies showed discrepancieshigher than 10 m along the
coasts.In accordancewith these maps, we correctedthe
ETOP05 bathymetryin order to suit better the depths of
shallowwater. Then, we computedthe M2 wave and the K]
wave with the ETOP05 bathymetry and the improved
bathymetryand comparedthe resultswith the tidal databank
(Table 3). The numericalresults are very clear. Indeed, the
improvementfor M2 is 36% and for K] is 30%, which
underscores
the real importanceof a goodbathymetry.
I
112ø30 '
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
Figure 3. Refined mesh.
Nevertheless, we were not able to quantify the
improvement
achievedafterrefiningthe meshon the marginal
seas of the YS-ECS
because of the intrinsic effects of the
refinement on the improvementof the bathymetry.The
National Geophysical Data Center' 5'x5' (-10km
of
resolutionat the equator)grid bathymetryNationalGeographic
Data Center[1987] is used for both the local and the global
models.In the finite element approach,the bathymetryis
interpolated
on eachGausspoint and then is more accurate
androugherwith therefinedmesh.As theglobalandthe local
solutions are not computed with the same bathymetry,
intercomparisonof the numerical result would not be
significant.
However,it is obviousthat the refiningof the meshalong
coastlinesmakes it possibleto take into accountvery small
structures
like baysor estuaries,whereastheglobalmeshdoes
not. Also, the local mesh,with its betterspatialresolution,in
particularalongcoastlines,betterfits the gradientsand the
irregularityof the topography.
We also take into account in our model the effect of Earth
tides, ocean tide loading and self-attraction.Hendershott
[1972] expressedthe loadingand self-attractionof the ocean
tides.Globalchartsof theseeffectswerecalculatedby Francis
and Mazzega [1990], on the basis of the tidal solutionsof
Schwiderski [1980] with a resolution of 1øxlø. We introduced
them as diagnosticforcingtermsin our finite elementmodel.
One possibleimprovementof our modelwould be to use our
112030 '
115 ø
117o30 '
120 ø
122o30 '
125 ø
115 ø
117o30 '
120 ø
122030 '
125 ø
127o30 '
130 ø
132o30 '
127o30 '
130 ø
132o30 '
i
3.4. Test on Bathymetry and Other Forcing Fields
Tidal solutions are very sensitive to the seafloor
topography.In particular, the local bathymetrydrives the
offshorepropagationof waves.Besides,the high gradientsof
the topographyare responsiblefor strongvariationsin the
velocityfield. In the YS, andespeciallyin the KoreanBay and
PohaiBay the localgeometryof the bathymetryis responsible
for the very high resonanceof the semidiurnaland diurnal
wavesnearcoastlines.This resultis supportedby Kang et al.
[1998] who analyzedthe high M2 amplitudein the Korean
Bay in termsof the distortionof the tidal wave againstthe
Koreancoasts.In addition,the precisionand the accuracyof
112o30 '
I
Figure 4. Bathymetryin meters.
8712
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
Table 3. RMS for M2 and K• with the ETOP05 and the
ImprovedBathymetries
Number
of Tide
Wave
Gauges
RMS for the RMS for the Percentageof
ETOP05
Improved
Improvement
Bathymetry, Bathymetry,
cm
cm
M2
192
44.82
28.58
36%
Ki
171
10.04
7.01
30%
o•
3
The frictioncoefficientis 3x10':•.
own a prioriFES solutionsto computeagainnew loadingand
self-attractionforcing terms. However, we have decidedto
wait until betterglobal solutions,improvedover continental
shelves,have become available to test this idea.
i
•01
i
0.0015
0.002
i
0.0025
0.003
Friction coefficient
Figure 6. RMS (in centimeters)of the calculatedsolutionas
a functionof the frictioncoefficientfor Ki.
3.5. Tuning of the Bottom Friction Coefficient
In the CEFMO
finite element model the bottom friction is
parameterized
with a quadraticbottomlaw. A Chezy-like 4. Results:Maps for 11 Components
coefficientC/is usedempiricallyto tune the friction.This of the Tidal Spectrum.
coefficient
is usuallytakenbetween
2.5x10
'3 and 3x10
-3 4.1. Harmonic Decompositionof the Tidal Spectrum
Dronkers [1964]. Following experimental studies on the
Europeancontinentalshelves,Pingtee and Griffiths [1987]
recommended
the
use of
a
friction
coefficient
between
2.30x10
'3 and2.60x10
-3.Fortheglobaltidalsimulation
of
FES94.1, Le Provost et al. [1994] took a friction coefficientof
Eleven componentsof the tidal spectrumwere simulated:
M2, S2, N2, K2, 2N2, K•, O•, P•, Q•, M4 and MS4. These
componentswere chosen by considering the harmonic
decompositionof the signal of the 192 tide gaugesdata
3x10-3 in the pelagicareasof the oceansurfaceand selected in 2.2. We calculated a mean of the tidal elevations
2.5x10
'3intheshallow
waterones.
Uptonowthiscoefficient,for each component (see histogram Figure7). Thus we
althoughdeterminedover a restrictedarea,has been applied determinedthe major tidal componentsin the diurnal and
semidiurnalspectrumoverthe YS-ECS.
globally.
As the studiedarea is mainly situatedin shallowwater, the
Testswereperformedoverthe YS-ECS with ninedifferent
friction coefficientsin order to find the best one compared effectsof the nonlinearityon tidal waves are reinforcedand
withthein situdatabank
(lx10-3,1.25x10
'3,1.5x10
-3,1.75x10' waves that are not presentin the deep ocean have to be
3, 2x10-3,2.25x10-3,
2.5x10-3,2.75x10-3
and 3x10-3).The consideredin shallowwater. Indeed,accordingto the results
tuningof the frictioncoefficientrevealsa thresholdeffectat of Parker [1991] and Le Provost [1991 ] the shallow water,
1.5x10
-3fortheM2waveandfortheK• wave(Figures
5 and constituentsof the tidal spectrum(for instance,M4 and MS4)
6). The improvement(in comparisonwith our tide gauge resultmainlyfromthe fact that whena waverunsinto shallow
databank) is -45.3% for M2 and -27.2% for K• when the water its troughis retardedmore than its crestand the wave
frictioncoefficient
decreases
from3x10'3to l.Sx10'3.Hereafter loses its simple harmonic form. The shallow water
the frictioncoefficientis takenat 1.5x10'3. This confirmsthat constituentsare classifiedas overtidesand compoundtides.
the optimalcoefficienton a regionalscalemay be different An overtidehas a frequencythat is an exactmultipleof the
elementaryconstituents.Compound tides have frequencies
from the onecommonlyusedfor the globalscale.
30
100
25
0
0.( )01
0.0015
0.002
0.0025
0.003
Waves
Friction coefficient
Figure 5. RMS (in centimeters)of the calculatedsolutionas
a function of the friction coefficientfor M2.
Figure 7. Harmonicdecomposition
of the tide gaugedata
(logarithmscale,heightin cm).
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
a •12ø3ø' 115ø 117ø3ø' 12øø 122ø3ø' 125ø 127ø3ø' 1•'øø 132ø:
8713
the principal lunar wave M2 (the data are extractedfrom the
192 tide gaugesselectedfor the study).
4.2.
Semidiurnal
Waves
Figures8 and 9 display the maps of the M2 and S2
components
in amplitudeand in phase(referredto 135øE).As
expected, our semidiurnal solutions display four
amphidromes.The tides are greatly amplified along the
•:'"'"'"'
':':(::'""':':'"'
:•5
"•'"'
12ø30
' 115
ø 117ø30
' 120
ø 122ø30
' 125
ø 127030
' 13•
ø 132ø:
, ..: .".:'",
•
•,
112030 '
115 ø
117ø30 '
30
112o30'
60
115ø
120 ø
122030 '
90
117o30'
120ø
-I
125 ø
127030 '
120
122030'
150
125ø
127030'
130 ø
132030 '
180
1,'10
ø
210
132ø
.. ::...:-::.:•.
•2o•0,
• •o
• •o•0,
10
20
•20 o
•22o•0,
•o
•2•0'
50
60
•0
ø
•ø•0'
I
0.0
b 112030' 11,
50
112ø30'
0.0
30
115ø
60
117ø30'
90
120
120ø
150
122ø30'
180
125ø
210
240
127ø30'
130ø
270
300
30
40
70
80
117
132ø30'
330
360
Figure8. Computed
M2 tidalchart:(a) amplitude
is in
centimeters,
(b) phases
arein degrees
(thefrictioncoefficient
is 1.5x10-3).
112ø30 '
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
thatequalthesumor difference
of thefrequencies
of two or
more elementarycomponents.For this reason we also 0.0 30 60 90 120 150 180 210 240 270 300 330 360
computed
thetwomajornonlinear
waves;thequarter-diurnal
components
M4andMS4.We summarize
thecharacteristics
of Figure 9. ComputedS2 tidal chart: (a) amplitude is in
the 11 tidal waveswe computedin Table4. The last column centimeters,(b) phasesare in degrees(the friction coefficient
indicatesthe ratioof theelevationof onewave comparedwith
is 1.5xl0-3).
8714
LEF•VREETAL.:IMPROVING
REGIONAL
SCALEGLOBAL
OCEANTIDEMODELS
Table 4. Major Tidal Components
of theYS-ECS
SymbolName
ofTidal
Component
Period,
solar
hours
Frequency,
desh-t
Coefficient
Ratio
(M:=100)
Me
Principallunar
12.42
28.984
100
S2
Principalsolar
12.00
30.0
34.8
Kt
Luni-solar diurnal
23.93
15.041
26.0
O•
Principallunar diurnal
25.82
13.943
19.7
N2
Largerlunar elliptic
12.66
28.439
16.1
K2
Luni-solar semidiurnal
11.97
30.082
10.0
Pt
Principalsolardiurnal
24.07
14.959
8.5
M4
Harmonic 2 of M2
6.21
57.968
5.0
MS4
Interaction between M2 and Kt
6.10
58.984
3.6
Qt
Largersolarelliptic
26.87
13.399
3.5
2N2
Second
orderlunarelliptic
12.91
27.895
3.2
YS-ECS is the Yellow Sea and East China Sea.
Koreancoastand along the part of the China coastthat is
solutions
of Kanget al. [1998].Thisfactis obviously
the
opposite
Taiwan.Most modelsdisplaytheseaspectsquite consequence
of thepreciseandaccurate
bathymetry
usedin
well. Indeed,the tidal modelsdeveloped
by Kang et al. [ the strait.However,
at the mouthof the YangsteRiverthe
1991], Choi [1984] and Kang et al. [1998] are similarto our tidalmechanism
is greatlyperturbed
by theinfluence
of the
local
solution
calculated
with
CEFMO.
The
four
fiverdischarge.
Indeed,
thefiver,witha waterdischarge
of
amphidromesare present in all M2 tidal solutions.In 928.2x109m3yr
-• (which
corresponds
to a yearly
averaged
particular,the highvaluesreachedby tidal elevations
in the flow rateof- 29,000m3s-•),
carriesa lot of material,
Korean Bay and along the coastlinesof China in front of --,500.108
t yr4 ofsediment
load,Zhang
etal. [1994],
which
Taiwan are virtually equivalent.Nevertheless,our model constantly
modifiesthetopography
of the straitandtherefore
showsan important
tidal elevation
in theBay of Shanghai thevaluesof tidalelevations.
Unfortunately,
we accessed
no
(more than 2 m), consistentwith in situ data, which is not valuable
tidegauges
in thisareatocompare
oursolutions
with
noticeable
in the otheravailablesolutions
exceptin the new in situ data.
115 ø
117 ø30'
1;'0 ø
122ø30'
125ø
127ø30'
130ø
132ø30'
.........
-':•....::.•
.•..**:....•,
ß::---:•i:'-"
:-"--'4-•'..•?•
;&'•.'.*':':::•:.':::i?:
.........
•::.'.:-.-.:-.-.
-:.:..%8-;::•::::•o
o
..... e,.•::•..-,z
-.•,,•-'-•.:-..-..-:5:.-'::---:
....:
:.•::-'.;...:,----:.-:..
-.•,-
•-.•'•'••
*•"
'"•'
'•''" *•-'?-•.-------!:'"
>.-;i
'.?:<?7
';.::.•'::.*.-'.:,'-'.-:•-"-:%;::-'-:.-"'-•.
':'•.•;t:/-:,-•,':"-'-:•:•-'-'?-::
:'-<'.:
:; '•
ß :.:5:•::.::,.---'
.-•.:.:::::•.:..-.:.--:--•½:•.:-•-:-.5:-;.?A::-::•.-.'::•5::s.:
112ø30
I
0.0
I
2.5
115 ø
117ø30'
120 ø
122ø30'
125 ø
127ø30'
130 ø
112030
'
132030'
•i•;•i•i::i::i$:.•iii•i•ii,"..:``..,.W•:::/•iii:;...:,...J•x.•;•;.:•.•...•*`•.
'"-•"•,,•'•'
,•'•
'-•, •':•,T' •.:?:::•<
".•':
.....'.....:...•::....
ß
5
7.5
10
15
20
25
30
35
40
115ø
117030
'
120ø
122030
'
125ø
127030
'
130ø
132ø30
'
I
0.0
30
60
90
120 150 180 210 240 270 300 330 360
Figure
10.Computed
K•tidalchart:
(a)amplitude
isincentimeters,
(b)phases
areindegrees
(thefriction
coefficient
is
1.5x10-3).
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
o
o
o
o
o
8715
o
nearly40 cm). They penetratefartherinto the Yellow Sea, so
the different reflections of the wave occur at the back of Pohai
Bay. This is why the studiedarea highlightsthe dumping
mechanism
differences
semidiurnal
between
the
two
wave
families:
waves and diurnal waves. The size of the diurnal
wavesin the ECS and in, particular,in the YS impliesthat a
mixedtidal dissipationmechanismmustbe considered
there.
a 112øR0
' 115ø 117ø,3(3
' 12(3
ø 122øR(3
' 125ø 127ø,3(3
' 1RF)
ø 1R2ø•:10
'
o
o
-::.:•.•.';-.';:..
• :'::::•.:..•':•-•:•':•{•'•'
•'L
•::•
•
.........
-...>....-.-.....::-:-':
•:'•'••
'•i '•::'G'"'•:•:4•::1•i•'•
'•• "'"'--'••
'"'V""'
'iiii!!:? '•
o
112ø30
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
I
0.0
5
7.5
10
15
20
25
30
35
4O
o
o
,
112ø30 '
0.0
0.5
115 ø
1
117ø30 '
2
4
120 ø
6
122ø30 '
8
125 ø
10
127ø30 '
12
14
130 ø
16
132ø30 '
18
20
b 112ø'30'115ø 117030
' 123ø 122øR0
' 125ø 127ø`3(3
' 1`3•3
ø 1`32ø,30
'
o
o
.
112ø30 '
0.0
30
115 ø
60
117ø30 '
90
120
120 ø
150
122ø30 '
180
125 ø
210
127ø30 '
240
270
130 ø
300
:.,
o
132'30'
•'•''
'••';!!!
"•..'
....
•"
330
360
o
Figure 11. ComputedO1 tidal chart: (a) amplitudeis in
centimeters,(b) phasesare in degrees(the frictioncoefficient
•
'*•'••
• •-••..:••••::::::::•:•
•.•::::..
is 1.5x10-3).
o
4.3.
Diurnal
Waves
112ø30 '
115 ø 117ø30 '
120 ø 122o30 '
125 ø 127o30 '
130 ø 132o30 '
Figures10 and 11 show the maps of the K] and O]
componentsin amplitudeand in phase (referredto 135øE).
0.0
30
60
90
120
150
180
210
240
270
300
330
360
Our solutionsdisplaytwo amphidromes,which is consistent
with thefactthattheperiodof diurnaltidesis twicethe period Figure 12, ComputedM4 tidal chart: (a) amplitude is in
of semidiurnaltides. Contrary to semidiurnaltides, diurnal centimeters,(b) phasesarc in degrees(the friction coefficient
tidesare greatlyamplifiedin the LiatongBay (modelindicates is 1.5x10-3).
8716
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
interaction
between
thedominant
waveM: andtheprincipal
aI
secondary
wave S:. The computation
of thesetwo wavesis
morecomplex,
andnumerical
approximations
maydebase
the
resultsbecausethey are calculatedfrom the derivationof the
velocities
ofM: andS:.Moreover,
in published
archives,
only
coastaltide gaugespectranormallydisplayM4 and MS4,
whichprevents
dataobservations
of fixedopenboundary
conditions.
At leastthreenumerical
modelsinvestigated
M4
generation
onlyChoi[1990]andbothM4 andMS4generation
Kang et al. [1998] and Kantha et al. [1996] (who also
investigated
M6, 2MS6,2SM: and S4)in the areaof the YS-
ECSuntilnow.In ourmodel,openboundaries
weresetusing
datafromnearbytide gauges.Near coastlines
theextremities
of the boundaries
werefixedaccording
to in situdata,and,
laterlinearinterpolations
werecomputed
to setamplitude
and
phasealongthe boundaries.
In deepoceanthe boundaries
weresetto zero.This approximation
is consistent
with the fact
that in pelagicareas,M4 and MS4are virtuallyzero:the
bathymetry
is deepandthenonlineareffectonM: forM4 and
betweenM: and S: for MS4 cannotoccur.Near the coaststhis
112ø30'
I
I
0.0
0.5
115ø 117ø30'
120ø 122ø30'
125ø 127ø30'
....%:..:ii!:'=;i::i::i:i:.i•44i[:•!!•:•'•
'-•.•'ff•"•'•'
•••... •
••••Y••,-•.,,,•
I
2
4
6
8
10
•xe,,•
•.•,
12
14
130ø 132ø30'
16
18
20
b112iRO'
11
rio
117'
approximation
hasto be considered
morecarefully.
Indeed,
thesenonlinear
constituents
aresmallbutsignificant
in these
shallow water areas. They increaseas the tidal wave
propagates
through
PohaiBayandIncheonBayas shownon
thecharts.Fortunately,
theyaresmallat thecoastal
endof our
openboundaries.
This is why it is hopedthatthe solutions
givenin Figures
12 and 13 are not too dependent
on the
approximations
adopted
herefor fixingour openboundary
conditions.
The generalpatternsof our M4 and MS4 solutionsare in
goodagreement
with the existingsolutions
of Kanget al.
[1998]andKanthaet al. [1996]alongthecoasts.
Amplitude
amplifications
are observed
in the sameareas:alongthe
Koreancoasts(~20cm), in the SeohanBay (great
amplification
of 30 cm),in theYangtzeRivermouth,andin
theHanchow
Bay(> 20 cm).Theamphidromic
patterns
of the
_
M4 and MS4 solutionsare more difficult to comparewith
Kang's and Kantha's ones. However, the same number of
amphidromesis observedin the northem YS. The same
amphidrome
betweenTaiwanand Chinais present,as the 2
Table5. RMS forEachTidalComponent
Wave
112ø30'
I
0.0
I
30
1 5ø
I
60
117ø30'
120ø 122ø30'
125ø 127ø30'
' ...!-::•!•:•'::•-¾-•.'.%?-.k:•;.•:.•'•:.'""i
"•'• • - '
90
120 150 180 210 240
,•
270
130ø
132ø30'
300
330
Number
of Variance RMSon
Percentage
Tide
of
on Tide
Solution,
Gaul•es Gau•e,
cm cm
Accuracy
M2
189
87.06
15.82
96.70%
S2
189
30.39
13.15
81.26%
N:
20
14.44
5.86
83.53%
K:
33
8.75
2.04
94.59%
360
Figure 13. Computed
MS4tidal chart:(a) amplitude
is in
centimeters,
(b) phasesare in degrees(thefrictioncoefficient
is 1.5xl0'3).
4.4. Quarter-Diurnal Waves
Figures12 and 13 show the maps of the M4 and MS4
components
in amplitudeandphase(referredto 13•øE). These
solutionsare complexboth in amplitudeand phase.Indeed,
thesewavesare secondordernonlinearcomponents.
M4 is the
first harmonic of the dominant wave M•, and MS4 is the
2N:
9
3.16
1.77
68.74%
Kl
163
20.32
5.09
93.73%
O•
163
15.19
3.76
93.88%
P•
40
6.61
1.85
92.19%
Q•
163
2.67
0.70
93.22%
M4
15
6.01
5.74
62.15%
MS4
11
2.83
2.46
24.32%
LEF•VREETAL.:IMPROVINGREGIONALSCALEGLOBALOCEANTIDEMODELS
112ø:
I
117o30 '
122ø30 '
125 ø
127o30 '
130 ø
132o30 '
112o30'
115 ø
117o30'
120 ø
122o30'
125_
ø
8717
127o30'
130 ø
132o30 '
L
• ..................
;.-
100 c ws
112o30 '
115 ø
117o30 '
120 ø
122o30 '
125 ø
127o30 '
130 ø
132o30
112o30 '
115 ø
Figure 14. ComputedM2 tidal current(the frictioncoefficient
117o30 '
10
20
120 ø
30
40
122o30 '
50
60
125 ø
70
127o30 '
80
130 ø
132o30 '
90
is 1.5x10'3).
Figure16. Computed
M2maximum
velocities
(in cms-l, the
friction
coefficient
is 1.5x10-•).
amphidromes
in the easternpart of the middleof the Yellow
Sea.The majordifferencebetweenKang's solutions,Kantha's
solutionsand ours is the amphidromein the middle of the
Yellow Sea which existsin Kang' s and Kantha's solutions but not in ours(bothfor M4 andMS4).The complexity
of the
geometry,dueto manysmallislandsin the southof Koreaand
the approximation
of the bathymetryin these area, may
120
11
120 ø
122o30 '
125 ø
127o30 '
130 ø
132o30 '
explainthe differences.
However,exceptfor this point the
solutions
are in agreement
andshowthe greatimportance
of
the diurnal waves in the area of YS-ECS.
4.5. Numerical
Results
In orderto quantifythe qualityof our localtidal solutions,
we computedthe RMS for eachtidal wave (Table5). The first
column of Table 5 gives the name of the tidal wave. The
secondcolumngives the numberof tide gaugesused to
calculatethe RMS. The third column gives the RMS
calculatedon the tide gaugein centimeters.
The fourthcolumn
gives the RMS of the calculatedsolutionalso in centimeters.
Thefifth columngivesa percentage
of accuracy
(POA) of the
tidalelevationaccording
to thetidegaugemeasurements.
The
POA is a goodindicationof the accuracyof the solutionfor
::::::::::::::::::::::
everycalculatedcomponent:
,•/•
•
..........
.....
.. ,;,y
.......
•..
): x]00.
POA
--(gMS
measurements
)2
_
(gMSmode
I
(gMSmeasurements)
2
....
(6)
Table 5 shows that the major componentsof the
semidiurnal
group
(M:)anddiurnal
group
(K•)arewell
explainedwhencomparedwith the in situdata.S: andN: are
1•CI •/S
12030'
115 ø
117030 '
120 ø
122030 '
125 ø
127030 '
130 ø
132030 '
lessaccurate
thantheotherwaves:thisresultscertainlyfrom
thefactthattheboundary
conditions
usedforall thecomputed
Figure 15. ComputedK• tidal current(the frictioncoefficient componentsare extractedfrom the global solutionFES95.
is 1.5x10'3).
They are not accuratefor S: and N2 in particularat the
8718
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
112o30'
115ø
117o30'
120ø
122o30'
125ø
127o30'
130ø
132o30'
112o30,
112o30 '
115 ø
117o30 '
120 ø
122o30 '
125 ø
127o30 '
130 ø
132o30 '
112o30 '
2
6
4
8
10
12
14
16
18
0.6
115 ø
0.7
0.8
117o30'
120ø
122o30'
125ø
127o30'
130ø
132o30'
117o30 '
120 ø
122o30 '
125 ø
127o30 '
130 ø
132o30 '
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Figure17. Computed
K] maximum
velocities
(incms-],the Figure 18. Classificationof tidal currentvelocities:ratio of
friction
coefficient
is 1.5xl0-3).
maximum velocitiesKi/M:.
boundary
betweentheYS andtheSeaof Japan,thusadversely the tidal spectrum(significanterror in phase).It must be
affectingthe solutionsin the whole area. However,as said pointedoutalsothatforthesecomponents
thewavelengths
are
before,in orderto keepthe continuitybetweenthe deepocean shorter and would need much more data for validation.
(solutionFES95) and the local area(our localmodel)we have
kept the FES95 boundaryconditionsin this study.Moreover,
it mustbe noticedthatexceptfor the fourmajorwavesM2, S2, 5. Tidal Energy
Ki and O], the comparison(between in situ data and
5.1. Barotropic Velocities
computedsolutions)of the otherwavesis basedon few tide
In orderto calculatethe tidal energybudgetwe had to
gauges.This alters the analysisand the resultsof these
comparisons,
in particular,for the nonlinearwavesM4 and computethe velocitiesof the tidal spectrumcomponents.
MS4whichare,in general,
apparently
notwell extracted
from Figures14 and 15 displaythe calculated
M: and K] tidal
Table 6. EnergyBudgetsforM2As a Functionof theFrictionCoefficient
FrictionCoefficient Astronomic
Forcing LoadingandSelfRateof Work, GW
AttractionForcing
M2Dissipation
VersusM2, GW
M2Dissipation
VersusK], GW
GlobalDissipation,
GW
-126.894
-44.511
-183.474
Rate of Work, GW
lx10"•
-7.499
-4.57
1.25x10'3
-6.988
-5.058
- 126.439
-45.245
- 183.729
1.5x10"•
-6.586
-5.399
-125.272
-45.720
- 182.978
1.75x10"•
-6.260
-5.646
-123.776
-46.039
-181.720
2x10"•
-5.989
-5.830
- 122.136
-46.255
- 180.209
2.25x10"•
-5.759
-5.969
-120.451
-46.402
-178.581
2.5x10"•
-5.561
-6.075
-118.773
-46.502
-176.776
2.75x10"•
-5.388
-6.157
-117.130
-46.569
-175.243
3x10"•
GW is Gigawatt.
-5.237
-6.220
-115.538
-46.610
-173.605
'
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
8719
Table 7. EnergyBudgetsfor K• As a Functionof theFrictionCoefficient
Friction
Coefficient Astronomic
Forcing Loading
andSelf- K•Dissipation
RateofWork,GW Attraction
Forcing Versus
K•,GW
K•Dissipation
Versus
M2,GW
GlobalDissipation,
GW
Rate of Work, GW
lx10'3
-0.970
-0.956
-0.911
-8.110
-10.946
1.25x10'3
-0.969
-0.957
-0.930
-8.252
- 11.108
1.5xl0'3
-0.968
-0.958
-0.951
-8.373
-11.249
1.75x10'3
-0.968
-0.958
-0.973
-8.483
- 11.381
2xl0'3
-0.968
-0.958
-0.997
-8.587
-11.510
2.25x10'3
-0.967
-0.958
-1.023
-8.688
-11.636
2.5xl0'3
-0.967
-0.959
-1.049
-8.786
-11.761
2.75x10'3
-0.967
-0.959
-1.076
-8.882
-11.884
3x10-3
-0.967
-0.960
-1.103
-8.977
- 12.006
ellipseson a regular grid extractedfrom FES. The ellipse
charts give the magnitudeand the direction of the tidal
currents.The major axes (and minor axes, respectively)
represent the maximum (and minimum) velocities. The
velocitiesof semidiurnalcurrentsare high in the strait of
Taiwan,aroundthe entranceto HanchowBay, in IncheonBay
virtually zero, which is verifiedby Fang [1994]. _Theregions
of strongdiurnal tidal currentsare confinedto the boundary
between
theYS andECS, wherethecurrentis ---20cms'•
(see Figure 17). Moreover, strongdiurnal currentsoccur in
LiatongBay and Pohai Bay, where shallow watersare found
(- 20cms-l),aswellasat thecrossing
of thestraitof Luzon
andin Seohan
Bay.Theyareof theorderof 100cms-• (see to thesouthof Taiwan(morethan20 crr#s).Fang [1994]
Figure16), which is indicativeof a strongdissipation.These
resultsare in goodagreementwith Fang [1994]. However,in
particularalongthe Ryuku islands,whichdelimit the Y S and
the Pacific Ocean, the velocities of the M2 tidal wave are
112ø30'
115ø
2080'
• • 5ø
117ø30'
120ø
122ø30'
125ø
127ø30'
130ø
132ø30'
already showed these results. In the Korea Strait, diurnal
velocities
are---10- 20cms'• although
waters
arenotshallow.
However, both the Korea Strait model by Fang and Yang
[1988a] and the ECS model by Fang and Yang [1988b]
112ø30'
115ø
117ø30'
120ø
122ø30'
125ø
127ø30'
130ø
132ø30'
• • 7ø30'
• 20ø
• 22ø•0'
• 25ø
• 27ø30'
• 30ø
• 82080'
:'
.......................
•...................
..........
.................
::::::::.
....................
....,....•....•..•.•::••
.:
.........................
•::,:-:::•.:•
.....
• • 7ø30'
• 20ø
• 22080'
-.5-.45-.4-.35-.3-.25-.2-.15-.1
• 25ø
127ø• '
• 80ø
-.05 0.0
• 32ø30'
• • 2ø30'
• • 5ø
-.5-.45-.4-.35-.3-.25-.2-.15-.1
-.05 0.0
Figure 19. Tidaldissipation
of theM2tidalwaveversus:(a) thefrictionof M2and(b) thefrictionof K•, (c) is theglobal
tidaldissipation
oftheM2tidalwave
(energies
areinW m'2andthefriction
coefficient
is 1.5xl0-3).
8720
112030 '
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
115 ø
117030 '
120 ø
122o30 '
125 ø
127030 '
130 ø
132030 '
c
-dissipate
at theentranceof the Yellow Sea,wherethe Pacific
Ocean narrows,and at the front end of Liatong Bay. As the
two typesof wave are complementary
in the studiedarea,
although we have 5 orders of magnitudebetween the
velocitiesof Kl andM2, the assumption
of mixeddissipation
is
confirmedagain.
o
o
5.2. Tidal Energy Budget
The global dissipationis quantifiedfor the astronomical
waves of the tidal spectrumaccordingto the tuning of the
friction coefficient.As we have alreadyconcludedabove, a
mixed frictioncoefficientis necessaryover the YS-ECS. So,
for one wave the global energybudgetis the sum of (1) the
astronomicforcingrate of work (contributionof the moonand
the sun), (2) the loading and self attractionrate of work
(perturbationof the astronomicpotential),(3) the dissipation
o
o
.
of the wave consideredversus the friction coefficient of the
112ø30 '
115 ø
117o30 '
120 ø
122o30 '
125 ø
-.5-.45-.4-.35-.3-.25-.2-.15-.1
127o30 '
-.05
130 ø
132ø30 '
0.0
dominantwave (M2 if the computedwave is semidiurnaland
Kl if it is diurnal)and (4) the dissipationof this wave versus
the friction coefficientof the secondarywave (K• if the
computedwave is semidiurnaland M2 if it is diurnal). These
budgets were evaluated with the tidal velocities and the
friction coefficientscalculatedon each Gausspoint of the
meshshownin Figure3. Table 6 givesthe dissipationfor the
M2 wave, andTable 7 givesthe dissipationfor the Kl wave as
a function of the nine studied values of the friction coefficient.
which is also confirmedby Fang [1994]. BetweenJapanand
the Ryuku islandsstrongdiurnal currentsare presentin our
solutions.Thesemay be an artefactof computationratherthan
a real tidal phenomenon.
Indeed,theyoccurat 30øN which is
Figures19 for M2 and Figures20 for Kl give the distribution
of the tidal dissipationfor the friction coefficientof the
dominantwaveand for the frictionof the secondary
waveas
well as theglobaltidal dissipationfor eachwave.The regions
of main tidal dissipationare correlatedwith mostregionsof
high values of tidal velocities. Table 8 gives the tidal
dissipation for the other computed waves for a friction
coefficientof 1.5x10-3.
For the semidiurnalwave M2 the globaldissipationin the
modeledareais - 182 GW as computedwith our local model
comparedwith Choi's previousestimateof 119 GW [Choi,
1980]. Choi's estimatemay be lower than ours estimations
the critical latitude of diurnal waves.
because he modeled a smaller area. Indeed, he limited it
Figure 19. (continued)
discoveredthe samevelocities.As proposedby Fang [1994],
a numericalmodel covetingthe JapanSea, the Korea Strait,
and the ECS can providebetterestimatesto checkthis point.
In the rest of the studied area the velocities of the diumal
component
K, areweaker
butnotnull(about
a fewcmS-l)
In the Y S, the semidiurnal and diurnal velocities are
complementary(see Figure18). As the velocities are
representative
of the tidal dissipation,thesecomputations
lead
us to conclude that when the semidiurnal tidal waves enter the
YS-ECS, they mainly dissipatein the shallow water of the
continentalshelf. On the contrary,the diurnal waves mainly
mainly alongthe shelf slopeof- 200 m depthline (between
Taiwan and Japan) and Taiwan. On the other hand, our
domaintakesinto accounta part of the westernPacificOcean
and the northernpart of the Meridional China Sea. Thus we
include the whole dynamic phenomenon,which occurs,in
particular,aroundthe islandof Taiwan.
Table8. Energy
Budgets
fortheOther
Tidal
Wave
AsaFunction
ofaFriction
Coefficient
of1.5x
I0-3
Wave
Astronomic
ForcingRateof
LoadingandSelfAttraction
Forcing
WaveDissipation
VersusM2,GW
WaveDissipation
VersusK•, GW
GlobalDissipation,
GW
Work, GW
Rate of Work, GW
S2
-1.264
-0.971
-50.425
-7.218
-59.875
N2
-0.122
-0.266
- 13.967
-2.012
- 16.367
K2
-0.088
-0.070
-2.735
-0.383
-3.277
2N2
0.000
-0.005
-0.219
-0.030
-0.253
O•
0.136
-0.560
- 1.358
-4.168
-5.950
P•
-0.089
-0.116
-0.187
-0.610
-1.002
QI
0.019
-0.022
-0.050
-0.148
-0.201
LEF-•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
8721
112ø30 '
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
112ø30 '
115 ø
117'•30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
112ø30 '
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
112ø30 '
115 ø
117ø30 '
120 ø
122ø30 '
125 ø
127ø30 '
130 ø
132ø30 '
a
.
o
o
o
o
-.05 -.045-.04-.035-.03-.025-.02-.01
5-.01 -.005
112ø30 '
115 ø
-.05-.045-.04-.035-.03-.025-.02-.01
0.0
117ø30 '
120 ø
122ø30 '
-.05-.045-.04-.035-.03-.025-.02-.01
125 ø
127ø30 '
130 ø
5-.01 -.0050.0
132ø30 '
5-.01 -.0050.0
Figure 20. Tidal dissipation
of theK• tidal waveversus(a) thefrictionof K] and(b) thefrictionof M2, (c) is the global
tidaldissipation
oftheK]tidalwave(energies
areinW m'2andthefriction
coefficient
is 1.5x10-3).
8722
12o30'
LEF•VRE ET AL.: IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
I rio
117ø30'
120 o
12'2øR0'
12fiø
127030'
1510ø
132 o'
Measurement
.001
.002
.003
1
-1
,.ol
112ø30'
115ø
117ø30'
120ø
122ø30'
125ø
127ø30'
130ø
132ø30'
Figure 21. Locationof thecurrentmeasurements
usedin the
Comparison
with velocityfields.
Surprisingly,the dissipationremainsconstant,both for the
M2 wave and the K• wave, for varyingvaluesof the friction
coefficient.Moreover,the dissipationcomputedwith our local
modelis exactlythe samethat was calculatedby Le Provost
and Lyard [1998] with the global hydrodynamicmodel,
althoughthe topography,the mesh,and the frictioncoefficient
were refined
and tuned.
The
reason for this is that the
computedvelocityfield adaptsto the variationof the friction
coefficientfor the model to dissipatethe amountof energy
supplied at the open boundaries,which is fixed by the
boundaryconditions.
5.3. Sensitivityof the VelocityFields
The constant dissipation, as the friction coefficient is
varied, led us to studythe sensitivityof the tidal elevations
andthetidal velocitiesto thetuningof thefriction.To quantify
both the elevations and the velocities, we introduced two
means(one for the elevationsand the otherfor the velocities).
As we computedthe elevationsand the velocitieson a finite
elementmesh, we gave a weight equal to the surfaceof the
triangle where the values are computedto take into account
the variablegeometryof our meshcoefficient(seeTable 9 for
M2 and Table 10 for K•). It appearsthat the velocitiesare
twice as sensitiveto the tuning of frictioncoefficientas the
elevation.For instance,althoughtheM2 elevations
decrease
by
16.5% when the friction coefficientincreasesfrom lx10 -3 to
3xl0-3,theM2minimum
andmaximum
velocities
decrease
by
-- 29%. We checkedthatthe tidal dissipationremainsconstant
(decreased
onlyby 5.4%) whenthecoefficient
is multipliedby
3. As the tidal dissipationis proportionalto the friction
1--1
0
1-1
0
1
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
8723
Table9. Elevation,
Velocities,
andDissipation
As a Function
oftheFriction
Coefficient
fortheM2TidalWave
FrictionCoefficient
Cf.U3
Mean Elevation, Mean Minimum
Mean Maximum
Global
cm
Velocity,
cms'l
Velocity,
cms'1
Dissipation,
GW (X104m3s-3)
I x10'3
78.7
10.4
34.2
-183.474
4.01
1.25x10'3
76.2
9.8
32.2
- 183.729
4.16
1.5x10'3
74.0
9.3
30.5
-182.978
4.24
1.75x10'3
72.2
8.9
29.1
- 181.720
4.30
2x10'3
70.6
8.5
27.9
- 180.209
4.32
2.25xl0'3
69.2
8.2
26.8
-178.581
4.34
2.5x10'3
67.9
7.9
25.9
- 176.776
4.34
2.75x10'3
66.8
7.6
25.1
-175.243
4.33
3x10'3
65.7
7.4
24.3
-173.605
4.32
Percentage
ofvariation
fromlx10"•to3x10'3:
mean
elevation,
16.5%;
meanminimum
velocity,
28.8%'meanmaximum
velocity,
28.9%;
global
dissipation,
5.4%;andCf.U
3( thatisthefriction
coefficient
multiplied
bythecubic
velocity),
7.2%.
coefficientmultipliedby the cube root of the tidal velocity finite element model computes barotropic velocities, we
[e.g.,seeLe ProvostandLyard,1998],we verifiedthatthis assumethat the in situ datausedcorrespondto the barotropic:
productremainsconstant
(maximumincrease:
7.2%). The we did not compare our results with the bottom velocity
same resultsare verified with the K• wave except for the
measurements,
which are generallyon the bottomboundary
product
termof thefrictioncoefficient
withthecubeof the layer.
Despiteconsideringthat velocitymeasurements
correspond
to the barotropiccomponentof the flows, which is a huge
bottom friction than elevations both for the diurnal and assumption [see Wang and Huang, 1994], reasonable
semidiurnal
wavesof thetidal spectrum.
A fine tuningof this agreementwas found betweenthe observationsof the current
frictioncoefficient
mightthenbe controlled
by comparison
to measurementsand the computedvelocitiesfor the optimal
in situ data velocities rather than sea level variations (if tuning of the friction coefficient(seeTable 12) sincethe best
velocity.
It is clearthat velocitiesare more sensitiveto the tuningof
with 1.5x10
'3.Moreover,
thisstatement
is
available). To verify this statement,we comparedour RMS is calculated
verified qualitatively. Figure22 displays the tidal current
mainlyat the boundary
of the YS-ECS(seeFigure21) are ellipses for the 12 measurementsand for the associated
velocities
(witha friction
coefficient
of l xl 0-3,2xl0taken from Choi [1984], which extractedthem from Trump computed
velocities with 12 in situ data of current measurementslocated
The onlymeasurement
notcorrelated
with
and Butt [1981] and Larsen et al. [1985], and five 3, and3x10-3).
measurements
aretakenfromthepaperof Kanget al. [1998], computationis the positionM2. Indeed, this is a surface
which extractedthem from Harkema and Hsueh [1987]. The
measurement
neara coastin shallowwaterwherethe shearing
measurements
reported
in Table11 arereferenced
to 0øE.We stressescertainlyreach high values. On the other hand, the
keptthesamenamesforthecurrentmeterinstruments.
As our measurementson the eleven other positions show good
Table10. Elevation,
Velocities
andDissipation
As a Function
of theFrictionCoefficient
fortheK• TidalWave
Friction
Coefficient
Mean
Elevation,
Mean
Minimum Mean
Maximum Global
cm
Velocity,
cms-I
Velocity,
cms-•
Dissipation,
GW (x10
4m3s-3)
Cf.U
3
I x10'3
23.4
1.9
6.6
- 10.946
2.87
1.25x10-3
22.8
1.9
6.3
-11.108
3.09
1.5x10-3
22.4
1.8
6.8
-11.249
3.25
1.75x10
'3
22.0
1.7
5.8
-11.381
3.40
2x10'3
21.7
1.6
5.6
-11.510
3.53
2.25x10'3
21.4
1.6
5.5
-11.636
3.65
2.5x10-3
21.2
1.6
5.3
-11.761
3.77
2.75xl0'3
21.0
1.5
5.2
-11.884
3.87
3x10'3
20.8
1.5
5.1
-12.006
3.98
Percentage
ofvariation
fromlx10-3to3x10':•:
mean
elevation,
11.1%;
mean
minimum
velocity,
21.1%;
mean
maximum
velocity,
22.7%;
global
dissipation,
8.8%;
andCf.U
3,7.2%.
8724
Table
LEF•VRE ET AL.: IMPROVING REGIONAL SCALEGLOBALOCEAN TIDE MODELS
11. Measurements
of In Situ Current Meter Data
Name
Depth,
m
Eastern
velocity Eastern
Velocity Northern
Velocit• Northern
Velocity
Amplitude,
cms'• Phase,
de•rees Amplitude,
cms' Phase,
de•rees
M2
4
84
141
83
146
M4
2
47
153
48
281
M5
20
47
192
51
288
M7
5
40
184
43
295
MS
23
42
216
41
314
CM7
20
31
249
23
11
OR
23
45
184
57
282
I
48
13
128
44
241
F
70
5
233
40
221
D
41
15
229
26
177
B
38
19
236
33
94
SB
110
12
260
7
21
correlationwith the computations.Graphically,we can see the mesh(twice as accuratealongcoastlinesand much more
thatthe computedvelocitiesthatbestfit the measurements
are preciseon high gradientsof depths),(3) a choiceof reliable
boundedby the value calculatedwith the frictioncoefficient boundaryconditions(extractedfrom the FES95 global model
equalto lx10-3andthevaluecalculated
with2x10
'3.This in orderto ensurethe continuitywith this model),(4) a refined
confirms that the best RMS found was calculated with the
topography
(improvementof 36% for M2 and 30% for Kl) and
frictioncoefficient1.5xl0-3.
(5) the use of a specificfrictioncoefficient
1.5x10
-3
(improvementby 45.2% for M2 comparedwith the typical
value3x10'3).
It is clearthatsignificant
improvements
to a
6. Conclusions
global model like FES95 can be made over the different
A local tidal hydrodynamic
modelbasedon the shallow coastalareasby dedicatedstudiesalongthesefive points.The
waterequations
anda finiteelementapproach
was developed analysisof the harmonicdecomposition
of the tidal spectrum
to improve the global tidal model FES95. A tide gauge based on the 192 tide gaugesprovidesthe selectionof the
databank comprising 192 gauges was built in order to main tidal wave componentsneededfor a correctdescription
comparethe computedsolutionswith in situ data.The tidal of the tides over the YS-ECS. With the optimizedconditions
elevationsare improved by a factor of 2, both for the describedabove,besidesM2 and Kl, nine other components
semidiurnaland the diurnal waves. These improvementsare
were computed:seven astronomical(classical)constituents,
explainedby five points:(1) the choiceof a mixed friction S2,N2, K2.and 2N2 (for the semidiurnalcomponents),
Ol, P•,
coefficient(improvement
by a factorof 4), (2) a refinementof and Qi (for the diurnal components)and two nonlinear
constituents, M4 and MS4 (for the quarter-diumal
components).Indeed, thesetwo nonlinearwaves, which are
Table 12. ComparisonBetweenMeasurements
of Current
usuallynot presentin the tidal spectrumof the deep ocean,
Meter andComputedVelocitiesfor the M2 Wave As a
reachhigh elevationvaluesin the area of the YS-ECS. This
Function of the Friction Coefficient
demonstrates
the ability of our modelto simulatenot only the
different astronomical constituents but also the nonlinear ones.
Friction
RMS for the
Coefficient Eastern
Velocity,
cm
s
RMS for the
RMS for the
Northern
Velocity,
Global
Velocity,
-1
cm s
cm
s
As strongcurrentsoccurin this area,tidal velocitieswere also
computedas well as the tidal dissipationfor each of the 11
waves. For the M2 and Ki tidal waves, elevation, velocities,
and tidal dissipationwere calculatedfor nine differentvalues
I x 10-3
11.90
19.25
16.00
of thefrictioncoefficient
ranging
fromlx10-3to 3x10'3.On
1.25x10'3
10.8
19.5
15.79
1.5x10-3
10.11
19.90
15.78
averagefor M2 (or K•), the elevationsdecreaseby 16.5% (or
11.1%), the maximumvelocitiesby 28.9% (or 22.7%) and the
global dissipationby 5.4% (or 8.8%). The velocitiesare as
1.75x10-3
9.68
20.30
15.90
twice sensitive as the elevation when the friction coefficient is
2x10-3
9.42
20.75
16.12
2.25x10-3
9.26
21.22
16.37
varying. On the contrary, the tidal dissipation remains
virtuallyconstant.The velocityfield adaptsto the variationof
the frictioncoefficient.This sensitivityanalysisdemonstrates
the benefitsof tuninga tidal modelon the velocityfield rather
2.5x10-3
9.17
21.68
16.64
2.75x10-3
9.13
22.11
16.92
3x10-3
9.15
22.52
17.19
than the elevations; however, it is evident that few in situ
velocitydata are availableand that separating
the barotropic
flow fromits barocliniccomponents
remainsa majorproblem.
LEF•VRE ET AL.' IMPROVING REGIONAL SCALE GLOBAL OCEAN TIDE MODELS
This approach
appliedto the areaof the YS-ECS will be
extended to other areas of shallow waters where the FES95
8725
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Le Provost,C., G. Rougier,andA. Poncet,Numericalmodelingof
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constituents
of thetides,withapplication
to theEnglish
areaof theMalaysianSea.By developing
localfiniteelement theharmonic
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solutions
andblendingthemin the globalsolutionwe hopeto
Le Provost,C., M.L. Genco,F. Lyard,P. Vincent,andP. Canceil,
providea tidal solutionaccurateboth in deep oceanand Spectroscopyof the world ocean tides from a finite element
shallow waters.
hydrodynamic
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Le Provost,
C., A.F. Bennett,andD.E. Cartwright,Oceantidesfor
andfromTOPEX/Poseidon,
Science,267, 639-642, 1995.
Le Provost,C., F. Lyard, J.M. Molines, M.L. Genco, and F.
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(ReceivedFebruary3, 1999; revisedJuly23, 1999;
acceptedSeptember30, 1999.)