DTIC ADA480119: Japan Sea Thermohaline Structure and Circulation. Part 2: A Variational P-Vector Method

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2886 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME31
Japan Sea Thermohaline Structure and Circulation. Part II: A Variational
P-Vector Method
PETER C. CHU, JIAN LAN, AND CHENWU FAN
DepartmentofOceanography,NavalPostgraduateSchool,Monterey,California
(Manuscriptreceived11August2000,infinalform30January2001)
ABSTRACT
The second part of this work investigates the seasonal variabilities of the Japan/East Sea (JES) circulation
usingtheU.S.NavyGeneralizedDigitalEnvironmentalModel(GDEM)climatologicaltemperatureandsalinity
dataset(publicdomain)ona0.5(cid:56)(cid:51)0.5(cid:56)grid.AvariationalP-vectormethodwasdevelopedtoinvertthevelocity
field. The GDEM for the JES was built up on historical (1930–97) 136 509 temperature and 52 572 salinity
profiles. The climatologicalmean and seasonalvariability ofthe current systems are wellinverted, especially
theTsushimaWarmCurrentanditsbifurcation,theEastKoreanWarmCurrent(EKWC),theJapannearshore
branch, the confluence of the EKWC, and the North Korean Cold Current near the Korean coast and flows
northeastwardalongthesubpolarfront,andamesoscaleanticycloniceddyintheUlleng/TsushimaBasin.Fur-
thermore, this method has the capability to invert flow reasonably well across the shallow straits such as the
Tsushima/Korea,Tsugaru,andSoyaStraits.TheGDEMtemperatureandsalinityandtheinvertedvelocityfields
providebalancedinitialfieldsforJESnumericalmodelingandsimulation.
1. Introduction A typical summer monsoon pattern lasts nearly four
months (mid-May to mid-Sep).
The Japan Sea, known as the East Sea in Korea, has
TheJESgeneralcirculationhasbeeninvestigatedfor
asteepbottomtopography(Fig.1)thatmakesitaunique
several decades. The Tsushima Warm Current (TWC),
semi-enclosed ocean basin overlaid by a pronounced
dominatingthesurfacelayer,flowsinfromtheTsushima
monsoonsurfacewind.TheJapan/EastSea(JES)covers
Strait,andcarrieswarmwaterfromthesouthupto40(cid:56)N
an area of 106 km2, has a maximum depth in excessof
whereapolarfrontforms(SeungandYoon1995).Most
3700 m, and is isolated from open oceans except for
of the nearly homogeneous water in the deep part of
small (narrow and shallow) straits that connecttheJES
thebasiniscalledtheJapanSeaProperWater(Moriyasu
to the Pacific Ocean. The JES contains three majorba-
1972)andisoflowtemperatureandlowsalinity.Above
sinscalledtheJapanBasin(JB),Ulleng/TsushimaBasin
the proper water, warm and saline water that enters
(UTB),andYamatoBasin(YB),andahighcentralsea-
through the Tsushima Strait flows northeastward and
mountcalledtheYamatoRise(YR).TheJEShasagreat
flows out through the Tsugaru and Soya Straits.
scientific interest as a miniature prototype ocean. Its
basinwide circulation pattern, boundary currents, Sub- The TWC separates north of 35(cid:56)N into western and
polar Front (SPF), mesoscale eddy activities and deep eastern channels (Uda 1934; Kawabe 1982a,b; Chu et
water formation are similar to those in a large ocean. al.2001a;Chuetal.2001b).Theflowthroughthewest-
TheJESexperiencestwomonsoons,winterandsum- ernchannelcloselyfollowstheKoreancoast[calledthe
mer, every year. During the winter monsoon season, a EastKorean WarmCurrent(EKWC)]untilitbifurcates
cold northwest wind blows over the JES as a result of intotwobranchesnear37(cid:56)N.Theeasternbranchfollows
theSiberianHighPressureSystemlocatedovertheEast the SPF to the western coast of Hokkaido Island, and
Asian continent. Radiative cooling and persistent cold the western branch moves northward and forms a cy-
airadvectionmaintaincoldairovertheJES.Thenorth- cloniceddyattheEasternKoreanBay(EKB).Theflow
west–southeast oriented jet stream is positioned at the through the easternchannelfollowstheJapanesecoast,
JES. A typical winter monsoon pattern lasts nearly six called the ‘‘Nearshore Branch’’ by Yoon (1982). More
months (Nov to Apr). During the summer monsoon, a accurately, we may call it the Japan nearshore branch
warm and weaker southeast wind blows over the JES. (JNB).TheJNBisusuallyweakerthantheEKWC.The
TWC at both channels reduces with depth. The North
KoreaColdCurrent(NKCC)meetstheEKWCatabout
Correspondingauthoraddress:Dr.PeterC.Chu,Dept.ofOcean- 38(cid:56)N with some seasonal meridional migration. After
ography,NavalPostgraduateSchool,Monterey,CA93943.
E-mail:[email protected] separation from the coast, the NKCC and the EKWC
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Japan Sea Thermohanline Structure and Circulation. Part II: A
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Variational P-Vector Method
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OCTOBER2001 CHU ET AL. 2887
FIG.1.GeographyandisobathsshowingthebottomtopographyoftheJapan/EastSea
(JES).
convergeandformastrongfront(i.e.,SPF)thatstretch- annual average of 1.3 Sv, a maximum of 2.2 Sv in
esinawest–eastdirectionacrossthebasin.TheNKCC October, and a minimum of 0.3 Sv in February (Sv (cid:91)
makes a cyclonic recirculation gyre in the north, but 106m3s(cid:50)1).Lateron,MittaandOgawa(1984)analyzed
most of the EKWC flows out through the outlets (Uda thehistoricalvelocitydataobtainedbythecurrentmea-
1934). The formation of NKCC and the separation of surements lasting one day in the summers of 1942 and
EKWCareduetoalocalforcingbywindandbuoyancy 1943 to investigate the current structure across the
flux (Seung 1992). Large meanders develop along the Tsushima/Korea Strait. The data show intense north-
front and are associated with warm and cold eddies. ward currents near the bottom in the Tsushima/Korea
TheseasonalvariabilityoftheTWCattheTsushima/ Strait. This leads to doubt about the validity of the dy-
Korea Strait largely impacts the JES physical condi- namical calculation with the level of no motion at the
tions. An accurate estimate of the volume transport at bottomofthestrait(MittaandOgawa1984).Toremove
this strait is important for the JES circulation and ther- the‘‘levelofnomotion’’assumption,weuseaninverse
mohaline structure. Two methods have been used to method to obtain absolute geostrophic velocity from
estimate the volume transport through the Tsushima/ hydrographic data.
Korea Strait: 1) dynamical calculation of the hydro- Usingthesealeveldifferenceacrossthestraitwithout
graphic data with alevelof ‘‘nomotion’’atthebottom considering the contribution from baroclinic motion,
and 2) calculation of the sea leveldifferenceacrossthe Kawabe (1982a) and Toba et al. (1982) found the vol-
strait. ume transport through the Tsushima/Korea Strait to be
Using the dynamical calculation approach, the vol- the same as the dynamical calculation (HidakaandSu-
ume transport through the Tsushima/Korea Strait is zuki1950;Yi1966).Consideringtheeffectofbaroclinic
characterized by the minimum in winter–spring, and motion and subtracting the sea level difference due to
maximum in summer–fall (e.g., Hidaka and Suzuki the baroclinic motion from the observed current data
1950; Yi 1966). For example, Yi (1966) estimated the (1988–90),Isobe(1994)foundthatthevolumetransport
2888 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME31
TABLE1.Glossaryofacronyms. 2. The Navy’s GDEM Dataset
Acronym Explanation Data for building the current version of GDEM cli-
matology for the JES were obtained from the MOODS
EKB EasternKoreanBay
EKWC EastKoreanWarmCurrent data. MOODS is a compilation of ocean dataobserved
GDEM U.S.NavyGeneralizedDigitalEnvironmentalModel worldwideconsistingof1)temperature-onlyprofiles,2)
JB JapanBasin both temperature and salinity profiles, 3) sound-speed
JES Japan/EastSea
profiles,and4)surfacetemperature(driftingbuoy).The
JNB Japannearshorebranch
main limitation of theMOODS datais itsirregulardis-
MOODS U.S. Navy Master Oceanographic Observational
DataSet tribution in time and space. Certain periods and areas
NKCC NorthKoreanColdCurrent areoversampled,whileotherslackenoughobservations
SPF SubpolarFront to gain any meaningful insights (Chu et al. 1997a,b).
TWC TsushimaWarmCurrent
Vertical resolution and data quality arealsohighlyvar-
UTB Ulleng/TsushimaBasin
YB YamatoBasin iable depending on instrument type and sampling ex-
YR YamatoRise pertise. The monthly distributions of the JES tempera-
ture (Fig. 2a) and salinity (Fig. 2b) stations show that
the number of temperature stations is 2–3 times more
than the number of salinity stations, and January has
was at a maximum in early winter and a minimum in
the least profiles and August the most. Yearly temper-
early spring with the annual range (maximum minus
ature (Fig. 3a) and salinity (Fig. 3b) profile numbers
minimum) of the seasonal variation as 0.7 Sv.
showtemporallyunevendistributionwithalmostnoob-
Most of these analyses of seasonal variability of the
servationsinthewholeJESincertainyears(e.g.,1944,
JES circulation and the volume transport at the Tsush-
1989 for temperature, and 1944, 1987–93 for salinity)
ima/KoreaStraitwerebasedontemporallyandspatially
andmanyobservationsinotheryears(e.g.,nearly6500
limited data. Using a more complete dataset, we may
temperature profiles in 1969, and 3700 salinityprofiles
improve these analyses and get statistically significant
in 1967). Spatial and temporal irregularitiesalongwith
results. The U.S. Navy Master Oceanographic Obser-
lackofdataincertainregionsmustbecarefullyweight-
vational Data Set (MOODS) contains 136 509 temper-
ed in order to avoid statistically induced variability.
atureand52572salinityprofilesduring1930–97.Based
Based on the MOODS data, the Navy’s global clima-
ontheMOODSdata,anunclassifiedGeneralizedDigital
tologicalmonthlymeantemperatureandsalinitydataset
Environmental Model (GDEM) was established with
was built (i.e., GDEM) with a four-dimensional (lati-
climatological annual and monthly mean temperature
tude, longitude, depth, and month) display.
and salinity fields on a 0.5(cid:56) (cid:51) 0.5(cid:56) grid.
The basic design concept of GDEM is the determi-
Chu et al. (2001a) used the P-vector inverse method
nation of a set of analytical curves that represent the
(Chu 1995, 2000; Chu et al. 1998a,b) to calculate the
mean vertical distributions of temperature and salinity
JESabsolutegeostrophicvelocityfromtheGDEMdata. forgridcells(0.5(cid:56)(cid:51)0.5(cid:56))throughtheaveragingofthe
The P-vector method contains two steps: 1) determi-
coefficients for the curves found for individualprofiles
nation of the velocity direction and2) determinationof
(Teagueetal.1990).Differentfamiliesofrepresentative
the velocity magnitude. Two necessary conditions for curves have been chosen for shallow, middepth, and
the inversion are easily implemented into this method: deep-depthranges,witheachchosensothatthenumber
1)theisopycnalsurfacedoesnotparalleltheisosurface of parameters required to yield a smooth, mean profile
of potential vorticity and 2) the velocity has a vertical overtherangewasminimized.AspointedoutbyTeague
spiral(Chuetal.1998a,b).TheP-vectormethodinverts etal.(1990),large-scaleoceanographicfeaturesaregen-
the circulation in the JES basin reasonably well (Chu erally found to be similarly representedinbothGDEM
et al. 2001a), but it fails to invert the velocity field at andtheNOAAClimatologicalAtlasoftheWorldOcean
shallow straits such as the Tsushima/Korea, Tsugaru, temperatureandsalinity.GDEMappearstorenderbetter
and Soya Straits. representations of seasonal variability and regions of
Inthisstudy,wedevelopavariationalP-vectormeth- high current shear because of a different smoothing
od to improve the inversion and to obtain realistic cir- method and a finer grid spacing. GDEM data contains
culationintheJESbasinaswellasintheshallowstraits. the monthly mean temperature and salinity (T, S) and
Theoutlineofthispaperisasfollows.Adescriptionof annual mean temperature and salinity (T, S) fields. In-
the GDEM data is given in section 2. The P-vector terested readers are referred to Teague et al. (1990) for
methodanditsdeficienciesarepresentedinsection3.The more information.
variationalPvectorispresentedinsection4.Theinverted
absolute geostrophic velocity field is discussedinsection
3. P-vector inverse method and its deficiency
5.ThevolumetransportsintheTsushima/Korea,Tsugaru,
a. Reduced physics
and Soya Straits are presented in section 6. In section 7
we present our conclusions. There are quite a few acro- As pointed out by Wunsch and Grant (1982), in de-
nyms used in this paper, listed in Table 1. termining large-scale circulation from hydrographic
OCTOBER2001 CHU ET AL. 2889
data,wecanbereasonablyconfidentoftheassumptions V (cid:53) r(x, y, z)P, (4)
of geostrophic balance, mass conservation, adiabatic,
where r is the proportionality. Applying the thermal
and no major cross-isopycnal mixing (except for water
wind relation at two different depths z and z , a setof
masses in contact with the atmosphere). Under these k m
algebraic equations for determining the parameter r is
conditions, the density of each fluid element would be
obtained:
conserved, which mathematically is given by
V·(cid:61)(cid:114)(cid:53) 0, (1) r(k)P(k) (cid:50) r(m)P(m) (cid:53) (cid:68)u
x x km
where (cid:114)is the potential density and V (cid:53) (u, (cid:121), w) is r(k)P(k) (cid:50) r(m)P(m) (cid:53) (cid:68)(cid:121) , (5)
the geostrophic velocity. The conservation of potential y y km
vorticityequationcanbeobtainedbydifferentiating(1) whicharetwolinearalgebraicequationsforr(k)andr(m)
withrespecttoz,usinggeostrophicandhydrostaticbal- [r(i) (cid:53) r(x, y, z)]. Here
i
ances,andincludingthelatitudinalvariationoftheCor- (cid:69)
iolis parameter to give g zk (cid:49)(cid:93)(cid:114)ˆ (cid:93)(cid:114)ˆ(cid:50)
((cid:68)u , (cid:68)(cid:121) ) (cid:53) , (cid:50) dz, (6)
V·(cid:61)q (cid:53) 0, (2) km km f(cid:114) (cid:93)y (cid:93)x
0 zm
where q (cid:53) f(cid:93)(cid:114)/(cid:93)z. Equations (1)and (2)implythatthe where (cid:114)ˆ is the in situ water density, and (cid:114) is the char-
velocity V is parallel to (cid:61)q (cid:51) (cid:61)(cid:114). acteristic value of the density. 0
The existence of a solution of (5) implies a nonzero
determinant of the coefficient matrix of (5), which is
b. Necessary conditions
necessary condition 2. This determinant is the sine of
StommelandSchott(1977)pointedoutthatthethree- the vertical turning angle between P(k) and P(m) (Chu
dimensional velocity field cannot be determined from h h
2000; Chu et al. 1998a,b; 2001a).
the density field alone when the q and (cid:114)surfaces co-
For water columns satisfying thetwonecessarycon-
incide. The first necessary condition for the validity of
ditions,wesolve(6)toobtainr(k)forthelevelz .There
this inverse method is as follows. are N (cid:50) 1 sets (m (cid:53) 1, 2, k (cid:50) 1, k (cid:49) 1, · · ·k, N) of
Condition 1: The (cid:114)surface is not parallel to the q sur- equations (5) for calculating r(k). Here N is the total
face, which mathematically requires number of vertical levels of the water column. The N
(cid:50) 1 sets of equations are compatibleunderthethermal
(cid:61)(cid:114)(cid:51) (cid:61)q (cid:177) 0.
wind constraint and should provide the same solution.
Stommel and Schott (1977) further pointed out that However, because of errors in measurements (instru-
the three-dimensional velocity field cannot be deter- mentation errors) and computations (truncationerrors),
mined from the (cid:114)field alone if the horizontal velocity the parameters r(k) may vary with m. A least squares
does not rotate with depth ((cid:98)spiral). The existence of error algorithm is used to minimize the error.
the (cid:98)spiral is the second necessary condition.
Condition 2: The velocity (u, (cid:121)) should execute a (cid:98) d. Deficiency of the current P-vector method
spiral, which mathematically requires that, for at least
twodepths,z(cid:53)z andz(cid:53)z ,withhorizontalvelocities Let(u(P),(cid:121)(P))betheabsolutevelocitydeterminedby
k m
[u(k), (cid:121)(k)] and [u(m), (cid:121)(m)], the P-vector method, and (U(P), V(P)) be their vertical
(cid:41)u(k) (cid:121)(k)(cid:41) integrations,
(cid:177) 0. (cid:69)
u(m) (cid:121)(m)
0
(U(P), V(P)) (cid:53) (u(P), (cid:121)(P)) dz. (7)
Ifwecannotfindlevelszkandzmsuchthatthenecessary (cid:50)H
condition 2 is satisfied, the inverse method will fail to
Duetothelocaldeterminationoftheabsolutegeostrophic
get velocity in that water column.
velocity, the present P-vector method does not always
guarantee mass conservation over a domain (cid:115), that is,
c. P-vector method (cid:69)(cid:69) [ ]
(cid:93)U(P) (cid:93)V(P)
Consider the unit vector P (Chu 1995), defined by (cid:49) dx dy (cid:177) 0 (8)
(cid:93)x (cid:93)y
(cid:61)(cid:114)(cid:51) (cid:61)q (cid:115)
P (cid:53) . (3) ispossible.Suchadeficiencylargelyaffectsthequality
|(cid:61)(cid:114)(cid:51) (cid:61)q|
of the inversion. For example, the TWC in the shallow
The existence of this unit vector implies nonzero de- Tsushima/KoreaStraitwasnotwellinverted(Chuetal.
nominator of (3), which is the necessary condition 1. 2001a). Therefore, we develop a variational algorithm
Thevelocity,V(cid:53)(u,(cid:121),w),parallelstheunitvectorP, taking into account the mass conservation.
2890 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME31
FIG.2.SpatialdistributionsofMOODSstationsduring1930–97:(a)temperatureand(b)salinity.
OCTOBER2001 CHU ET AL. 2891
FIG.2.(Continued)
2892 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME31
FIG.3.TemporaldistributionsofMOODSstationsduring1930–97:(a)temperatureand
(b)salinity.
4. A variational algorithm b. Horizontal discretization
a. Formulation Let the dependent variable f(x, y) be defined on the
interval 0 (cid:35) x (cid:35) L , 0 (cid:35) y (cid:35) L . Use a uniform grid,
Let (U, V) be the vertically integrated velocity op- 0 (cid:53) x (cid:44) x (cid:44) · ·x· (cid:44) x (cid:53) Ly, and 0 (cid:53) y (cid:44) y (cid:44)
timally determined by minimizing the following func- · · · 1(cid:44) y 2 (cid:53) L with gNrixd spaxcing of ((cid:68)x1, (cid:68)y),2 as
tional (called the cost function), Ny y
(cid:69)(cid:69) shown in Fig. 4a. Here,
1 (cid:68)x(cid:53)x (cid:50)x (cid:53)L /N , (cid:68)y(cid:53)y (cid:50)y (cid:53)L /N .
J(U,V)(cid:53) [(U(cid:50)U(P))2(cid:49)(V(cid:50)V(P))2]dxdy (9) i(cid:49)1 i x x j(cid:49)1 j y y
2 (cid:115) Let (cid:114), (cid:108)be evaluated at the grid point (i, j), and the
with the constraint (mass conservation) integratedvelocitycomponentsU,V,U(P),V(P)beeval-
uatedatthestaggeredpoints,respectively(Fig.4b).The
(cid:93)U(P) (cid:93)V(P)
(cid:49) (cid:53) 0. (10) functional (11) is discretized as
(cid:93)x (cid:93)y 1 N(cid:79)x(cid:50)1 N(cid:79)y(cid:50)1
Lˆ (cid:53) [(U (cid:50) U(P))2 (cid:49) (V (cid:50) V(P))2](cid:68)x(cid:68)y
This problem becomes an unconstrained optimization 2 ij ij ij ij
i(cid:53)1 j(cid:53)1
using (cid:79) (cid:79)
1Nx(cid:50)1Ny(cid:50)1
L(U, V, (cid:108)) (cid:53) J(U(cid:69),(cid:69)V) [ ] (cid:49)2 (cid:79)i(cid:53)1 (cid:79)j(cid:53)1 (cid:108)ij(Uij(cid:49)Ui,j(cid:50)1(cid:50)Ui(cid:50)1,j(cid:50)Ui(cid:50)1,j(cid:50)1)(cid:68)y
(cid:49) (cid:108)(cid:93)U(P) (cid:49) (cid:93)V(P) dx dy, (11) (cid:49)1Nx(cid:50)1Ny(cid:50)1(cid:108)(V (cid:49)V (cid:50)V (cid:50)V )(cid:68)x.
(cid:93)x (cid:93)y 2 ij ij i(cid:50)1,j i,j(cid:50)1 i(cid:50)1,j(cid:50)1
(cid:115) i(cid:53)1 j(cid:53)1
where (cid:108)is the Lagrangian parameter. (12)
OCTOBER2001 CHU ET AL. 2893
FIG.4.Staggeredgridusedforthecomputation:(a)griddistributionand(b)staggeredgridsfor(u,(cid:121))and
standardgridsfor((cid:108),(cid:114)).
c. Combined local–global determination 1
(U (cid:49) U (cid:50) U (cid:50) U )
(cid:68)x ij i,j(cid:50)1 i(cid:50)1,j i(cid:50)1,j(cid:50)1
Minimization of Lˆ becomes a combination of local 1
determinationatthestaggeredgrid(i,j)forvelocity((cid:98)- (cid:49) (cid:68)y(Vij (cid:49) Vi(cid:50)1,j (cid:50) Vi,j(cid:50)1 (cid:50) Vi(cid:50)1,j(cid:50)1) fi 0, (14)
spiral approach)
whichkeepsthemassconservationatanyboxcentered
U fi U(P), V fi V(P) (13) at the nonstaggered grid for (cid:114)and (cid:108)(Fig. 4b). Thus,
ij ij ij ij this variational P-vector method can be treated as a (cid:98)-
and global determination (box model) spiral box model.
2894 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME31
TABLE2.Invertedmonthlyvariationofvolumetransport(Sv)atthreemajorstraits.Thepositive/negativevaluesmeaninflow/outflow.
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Soya (cid:50)0.6 (cid:50)0.7 (cid:50)0.9 (cid:50)1.1 (cid:50)1.0 (cid:50)1.1 (cid:50)1.4 (cid:50)1.2 (cid:50)0.8 (cid:50)0.7 (cid:50)0.5 (cid:50)0.5
Tsugaru (cid:50)1.5 (cid:50)1.4 (cid:50)1.5 (cid:50)1.5 (cid:50)1.3 (cid:50)1.3 (cid:50)1.5 (cid:50)1.5 (cid:50)1.2 (cid:50)1.5 (cid:50)1.4 (cid:50)1.6
Tsushima 2.1 2.1 2.4 2.6 2.3 2.4 2.9 2.7 2.0 2.2 1.9 2.1
d. Optimal determination of vertically integrated the given value of (cid:108) into (16) and (17), we obtain the
ij
velocity optimal estimation of U and V .
ij ij
The variational problem is the determination of
(U , V , (cid:108) ) through minimizing thecostfunctionL, e. Determination of the bottom velocity
ij ij ij
(cid:93)Lˆ (cid:93)Lˆ (cid:93)Lˆ Let h (cid:53) h(x, y) be the bottom topography and (u ,
(cid:93)Uij (cid:53) 0, (cid:93)Vij (cid:53) 0, (cid:93)(cid:108)ij (cid:53) 0. (15) (cid:121)re(cid:50)lha)tiboent(h6e)btootttohme tvweololceivtye.lsAzppalnydin(cid:50)ghth,ewtehehramvaelwi(cid:50)nhd
Substitution of (12) into (15) leads to (cid:69)
g z (cid:49)(cid:93)(cid:114)ˆ (cid:93)(cid:114)ˆ(cid:50)
Uij (cid:53) Ui(jP)(cid:50)2(cid:68)1x((cid:108)ij(cid:49)(cid:108)i,j(cid:49)1(cid:50)(cid:108)i(cid:49)1,j(cid:50)(cid:108)i(cid:49)1,j(cid:49)1) (16) (u, (cid:121))z (cid:50) (u, (cid:121))(cid:50)h (cid:53) f(cid:114)0 (cid:50)h (cid:93)y, (cid:50)(cid:93)x dz(cid:57). (22)
Vertical integration of (22) from the bottom (z (cid:53) (cid:50)h)
1
V (cid:53) V(P)(cid:50) ((cid:108) (cid:49)(cid:108) (cid:50)(cid:108) (cid:50)(cid:108) ) (17) to the surface (z (cid:53) 0) leads to
ij ij 2(cid:68)y ij i(cid:49)1,j i,j(cid:49)1 i(cid:49)1,j(cid:49)1
1
1 (u, (cid:121)) (cid:53) (U, V)
(U (cid:49) U (cid:50) U (cid:50) U ) (cid:50)h h
(cid:68)x ij i,j(cid:50)1 i(cid:50)1,j i(cid:50)1,j(cid:50)1 (cid:69) (cid:69)
(cid:49) (cid:68)1y(Vij (cid:49) Vi(cid:50)1,j (cid:50) Vi,j(cid:50)1 (cid:50) Vi(cid:50)1,j(cid:50)1) (cid:53) 0. (18) (cid:50) fhg(cid:114)0 (cid:50)0h dz (cid:50)zh (cid:49)(cid:93)(cid:93)(cid:114)yˆ, (cid:50)(cid:93)(cid:93)(cid:114)xˆ(cid:50) dz(cid:57). (23)
Substitution of (16) and (17) into (18) leads to a linear With theestimatedbottomvelocity(u,(cid:121))(cid:50)h,weusethe
algebraic equation for the Lagrange parameter, thermal wind relation (22) to obtain the absolute ve-
locity from the density field.
a (cid:108) (cid:49) a (cid:108) (cid:49) a (cid:108) (cid:49) a (cid:108) (cid:49) a (cid:108)
11 i(cid:50)1,j(cid:50)1 21 i,j(cid:50)1 31 i(cid:49)1,j(cid:50)1 12 i(cid:50)1,j 22 i,j
(cid:49) a (cid:108) (cid:49) a (cid:108) (cid:49) a (cid:108) (cid:49) a (cid:108) f. Volume transport streamfunction
32 i(cid:49)1,j 13 i(cid:50)1,j(cid:49)1 23 i,j(cid:49)1 33 i(cid:49)1,j(cid:49)1
(cid:53) S , (19) Due to the continuity (10), the volume transport
ij streamfunction ((cid:67)) can be defined by
where i (cid:53) 2, 3, · · ·, N (cid:50) 1; j (cid:53) 2, 3, · · ·, N (cid:50)
1; and x y (cid:93)(cid:67) (cid:93)(cid:67)
U (cid:53) (cid:50) , V (cid:53) (24)
(cid:93)y (cid:93)x
(cid:49) (cid:50)
1 1 1
a11 (cid:53) a13 (cid:53) a31 (cid:53) a33 (cid:53) (cid:50)4 (cid:68)x2 (cid:49) (cid:68)y2 , and satisfies the Poisson equation
(cid:49) (cid:50) (cid:49)(cid:93)2 (cid:93)2(cid:50) (cid:93)V (cid:93)U
a22 (cid:53) (cid:68)1x2 (cid:49) (cid:68)1y2 , (cid:93)x2 (cid:49) (cid:93)y2 (cid:67) (cid:53) (cid:93)x (cid:50) (cid:93)y. (25)
(cid:49) (cid:50)
1 1 1
a (cid:53) a (cid:53) (cid:50)a (cid:53) (cid:50)a (cid:53) (cid:50) (20) 5. Absolute geostrophic velocity
21 23 12 32 2 (cid:68)x2 (cid:68)y2
a. Annual mean
and
Figure 5 shows the inverted horizontal velocity vec-
1
S (cid:53) (U(P) (cid:49) U(P) (cid:50) U(P) (cid:50) U(P) ) tors at depths 0, 50, 100, 150, 200, and 300 m respec-
ij 2(cid:68)x ij i,j(cid:50)1 i(cid:50)1,j i(cid:50)1,j(cid:50)1 tively. The variational P-vector method inverts the ve-
1 locity well at the three major straits: Tsushima/Korea,
(cid:49) 2(cid:68)y(Vi(jP) (cid:49) Vi((cid:50)P)1,j (cid:50) Vi(,Pj(cid:50))1 (cid:50) Vi((cid:50)P)1,j(cid:50)1). (21) Tsugaru, and Soya Straits. However, the flow in Tatar
Strait is not well resolved due to the poor data quality.
Thealternativedirectionimplicitmethod(Pressetal. We take the velocity field at depths 0, 50, and 100
1986) is used to obtain the value of the Lagrange pa- m (150, 200, and 300 m) to represent the upper (inter-
rameter at the grid point, (cid:108), solving (19). Substituting mediate)-layer circulation features. The inverted flow
ij