Table Of ContentJournalofMarineSystems65(2007)376–399
www.elsevier.com/locate/jmarsys
An inverse model for calculation of global volume
transport from wind and hydrographic data
⁎
Peter C. Chu , Chenwu Fan
NavalOceanAnalysisandPredictionLaboratory,DepartmentofOceanography,NavalPostgraduateSchool,Monterey,CA93943,USA
Received24September2004;accepted9June2005
Availableonline28November2006
Abstract
TheP-vectorinversemethodhasbeensuccessfullyusedtoinverttheabsolutevelocityfromhydrographicdatafortheextra-
equatorial hemispheres, but not for the equatorial region since it is based on the geostrophic balance. A smooth interpolation
schemeacrosstheequatorisdevelopedinthisstudytobringtogetherthetwoalreadyknownsolutions(P-vectors)fortheextra-
equatorialhemispheres.Thismodelcontainsfourmajorcomponents:(a)theP-vectorinversemodeltoobtainthesolutionsforthe
two extra-equatorial hemispheres, (b) the objective method to determine the Ψ-values at individual islands, (c) the Poisson
equation-solvertoobtaintheΠ-valuesovertheequatorialregionfromthevolumetransportvorticityequation,and(d)thePoisson
equation-solver to obtain the Ψ and depth-integrated velocity field (U, V) over the globe from the Poisson Ψ-equation. The
Poissonequation-solverissimilartotheboxmodeldevelopedbyWunsch.Thus,thismethodcombinesthestrengthfrombothbox
andP-vector models.Thecalculated depth-integrated velocityand Ψ-field agreewell with earlier studies.
Published byElsevier B.V.
Keywords:Inversemodel;Volumetransportstreamfunction;Volumetransportvorticity;Wind-drivencirculation;Density-drivencirculation;Depth-
integratedvelocity;Poissonequation;P-vector
1. Introduction Sverdrup relation in Antarctic regions. Godfrey (1989)
used the Sverdrup model with climatological annual
Winds, density, and friction determine volume trans- winds(HellermanandRosenstein1983)tocalculatethe
port in oceans. The wind-driven volume transport has meandepth-integratedstreamfunctionfortheworldocean
beenestimatedusingtheSverdrup(1947)relationforthe under two assumptions: (1) the ocean is stagnant below
interiorocean,andusingtheStommel(1948)andMunk some depth, and (2) all major undersea topographic
(1950) linear frictional ocean model for the intensive featuressuchasmid-oceanridgesliebelowthatdepth.
westernboundarycurrents.LeetmaaandBunker(1978), The density-driven volume transport has been calcu-
Meyers (1980), andGodfrey and Golding (1981) latedbyseveralauthors.Thedensityfielddirectlydeter-
calculated similarly for the North Atlantic, tropical minesthegeostrophicvelocityrelativetothebottomflow.
Pacific, and Indian Oceans. Baker (1982) examined the Thebottomvelocity(u−H,v−H)isusuallycalculatedusing
the β-spiral (Stommeland Schott,1977),Box (Wunsch,
⁎ 1978),andP-vector(Chu,1995;Chuetal.,1998a,b;Chu,
Correspondingauthor.
E-mailaddress:[email protected](P.C.Chu). 2000;Chuetal.,2001a,b;Chu,2006)models.
0924-7963/$-seefrontmatter.PublishedbyElsevierB.V.
doi:10.1016/j.jmarsys.2005.06.010
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1. REPORT DATE 3. DATES COVERED
SEP 2004 2. REPORT TYPE 00-00-2004 to 00-00-2004
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
An inverse model for calculation of global volume transport from wind
5b. GRANT NUMBER
and hydrographic data
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
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Naval Postgraduate School,Department of REPORT NUMBER
Oceanography,Monterey,CA,93943
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12. DISTRIBUTION/AVAILABILITY STATEMENT
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
The P-vector inverse method has been successfully used to invert the absolute velocity from hydrographic
data for the extraequatorial hemispheres, but not for the equatorial region since it is based on the
geostrophic balance. A smooth interpolation scheme across the equator is developed in this study to bring
together the two already known solutions (P-vectors) for the extraequatorial hemispheres. This model
contains four major components: (a) the P-vector inverse model to obtain the solutions for the two
extra-equatorial hemispheres, (b) the objective method to determine the Ψ-values at individual
islands, (c) the Poisson equation-solver to obtain the Π-values over the equatorial region from the
volume transport vorticity equation, and (d) the Poisson equation-solver to obtain the Ψ and
depth-integrated velocity field (U, V) over the globe from the Poisson Ψ-equation. The Poisson
equation-solver is similar to the box model developed by Wunsch. Thus, this method combines the strength
from both box and P-vector models. The calculated depth-integrated velocity and Ψ-field agree well
with earlier studies.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 24
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P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399 377
Since the geostrophic balance is used, these models The horizontal diffusivity A can be estimated by
h
provide the solutions for the extra-equatorial hemi- Smargrinsky parameterization,
spheres,butnotfortheequatorialregion.Animproved
D
inverse method is developed in this study to bring A ¼ DxDyjjVþðjVÞTj; ð6Þ
h
2
togetherthetwoknownsolutionsfortheextra-equatorial
hemispheresacrosstheequatorandtoestablishaglobal where
velocity dataset. The rest of the paper is outlined as " #
(cid:1) (cid:3) (cid:1) (cid:3) (cid:1) (cid:3) 1=2
follows. Section 2 describes the basic theory of the Au 2 1 Av Au 2 Av 2
model.Sections3–5describedepth-integratedvelocity, jjVþðjVÞTju Ax þ2 AxþAy þ Ay :
volume transport streamfunction, and volume transport
vorticity. Section 6 depicts theΨ-Poisson equation and
Here, the nondimensional parameter D varies from
itssolver.Section7depictsthemodelsensitivity.Section
0.1 to 0.2 (Mellor, 2003); we set D=0.15. The
8providestheglobalcirculationcharacteristics.Section horizontal grid in this study is 1°×1°, i.e., (Δx,Δy)
9presentstheconclusions. ∼100 km. Let the spatial variability of the velocity be
scaledby 0.1 ms−1, we have
2. Dynamics
0:1 m s−1
2.1. Basic equations jjVþðjVÞTjf2(cid:2) ¼2(cid:2)10−6s−1: ð7Þ
105m
Let (x, y, z) be the coordinates with x-axis in the
Substitutionof (7)into (6) leads to
zonal direction (eastward positive), y-axis in the
latitudinal direction (northward positive), and z-axis in
A ¼1:5(cid:2)103m2s−1: ð8Þ
thevertical(upwardpositive).Theunitvectorsalongthe h
three axes are represented by (i, j, k). For the extra-
equatorialregion,thelinearsteadystatesystemwiththe 2.2. Ekman number
Boussinesq approximationisgiven by
The Ekman number can identify the relative
A2u
−fðv−v Þ¼A þA j2u; ð1Þ importance of the horizontal gradient of the Reynolds
g zAz2 h stress versus the Coriolis force,
fðu−ugÞ¼AzAA2zv2þAhj2v; ð2Þ E ¼OðOjAðjhfjV2jVÞjÞ¼jfAjhL2; ð9Þ
Ap
¼−qg; ð3Þ where L is the characteristic horizontal length scale. In
Az
this study, the motion with L larger than 200 km is
considered. For extra-equatorial regions (north of 8°N
Au Av Aw and south of 8°S),
þ þ ¼0; ð4Þ
Ax Ay Az
jfjN0:2(cid:2)10−4s−1;
whereρisthein-situdensity;f=2Ωsinφ,istheCoriolis
parameter,ΩtheEarthrotationrate,andφthelatitude. and theEkman number isestimatedby
V=(u, v), is the horizontal velocity; w is the vertical
velocity; ∇=i∂/∂x+j∂/∂y, is the horizontal gradient Eb 1:5(cid:2)103m2s−1 ¼1:875(cid:2)10−3;
operator; V =(u , v ), is the geostrophic velocity ð0:2(cid:2)10−4s−1Þ(cid:2)ð2(cid:2)105mÞ2
g g g
representing thehorizontal pressure(p) gradients
whichshowsthatthehorizontalgradientoftheReynolds
1 Ap 1 Ap stresscanbeneglectedagainsttheCoriolisforce,i.e.,
u ¼− ; V ¼ ; ð5Þ
g f q0Ay g f q0Ax Ah ¼0; ð10aÞ
whereρ0isthecharacteristicvalue(1025kg/m3)ofthe forextra-equatorialregions.
sea water density. The two coefficients (Az, Ah) are the Fortheequatorialregionsespeciallyneartheequator,
vertical and horizontal eddy diffusivities. |f| is very small. The Ekman number is not a small
378 P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399
Fig.1.Bathymetryofworldoceans.
parameter.Thehorizontalviscousforce,(A ∇2u,A ∇2v), where z=−H(x, y) represents the ocean bottom, and
h h
cannot be neglected against the Coriolis force in the z=0referstotheoceansurface.Depth-integrationof(1)
equatorialregion,thatis, and(2)fromtheoceanbottomtotheoceansurfaceleads
to
A p0; ð10bÞ
h Au Au
−fðV−VgÞ¼AzAzjz¼g−(cid:3)AzAzjz¼−H þAhj2U ð13Þ
fortheequatorialregions.
−2Ahju−H djH−Ahu−Hj2H;
3. Depth-integrated velocity
Av Av
Let (U, V) and (U , V ) be the depth-integrated fðU−UgÞ¼AzAzjz¼g−(cid:3)AzAzjz¼−H þAhj2V ð14Þ
horizontal velocity g g −2Ahjv−H djH−Ahv−Hj2H;
Z
0 where (u−H, v−H) are velocity components at the ocean
ðU;VÞ¼ ðu;vÞdz; ð11Þ bottom.
−H Theturbulentmomentumfluxattheoceansurfaceis
calculated by
and geostrophic velocity,
Z (cid:1) (cid:3)
0 Au Av ðs ;s Þ
ðUg;VgÞ¼ ðug;vgÞdz; ð12Þ Az Az;Az jz¼g ¼ xq y ; ð15Þ
−H 0
P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399 379
Fig.2.BoundaryconditionsofΨfortheglobalocean.
where (τ ,τ ) are the surface wind stress components. is the density-driven transport. Re-arranging (13) and
x y
The turbulent momentum flux at the ocean bottom is (14), we have
parameterized by
(cid:1) (cid:3) s
Az AAuz;qAAvzffiffiffiffiffijffizffi¼ffiffiffi−ffiffiHffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ahj2U þfV ¼fVdenþfVb−qx0þAhQ1; ð21Þ
¼CD u2−H þv2−Hðu−H;v−HÞ; ð16Þ −Ahj2V þfU ¼fUdenþfUbþqsy (cid:4)AhQ2; ð22Þ
0
where C =0.0025 isthedrag coefficient.
D
The thermal wind relation can be obtained from
where
vertical integration of the hydrostatic balance Eq. (3) (cid:3)
fromthebottom(−H)toanydepth(z)andthentheuse
of thegeostrophic Eq. (5) Q1uð2ju−H djH þu−Hj2HÞ;
Z (cid:3)
g z Aq
ug ¼u−H þfq0 −H AydzV; ð17Þ Q2uð2jv−H djH þv−Hj2HÞ;
Z
g z Aq and
vg ¼v−H−fq0 −H AxdzV: ð18Þ (cid:1) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:3)
C
Substitutionof(17)and(18)intothesecondequation Ub ¼ H− fD u2−H þv2−H u−H; ð23Þ
of (12) leads to
(cid:1) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:3)
ðUg;VgÞ¼ðUdenþHu−H;VdenþHv−HÞ; ð19Þ Vb ¼ H þCfD u2−H þv2−H v−H;
where
ðU ;V Þ are the transport due to the bottom currents, or simply
den de(cid:1)nZ Z Z Z (cid:3)
g 0 z Aq 0 z Aq called the bottom transport. With the known bottom
¼fq0 −H −H AydzVdz; − −H −H AxdzVdz ; (vUel,oVci)tycavnecbtoerd(uet−eHrm,vi−nHe)d,tfhroemdetphteh-winitnedg,radteednsviteyl,ocaintyd
ð20Þ topographic data.
380 P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399
Fig.3.ComputedΨ-valuesforeachcontinent/island:(a)annualmean,(b)January,and(c)July.
P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399 381
For the extra-equatorial regions, the horizontal dif- is the volume transport vorticity. Eq. (28) is called the
fusion can be neglected [see (8)]. Eqs. (22) and (21) PoissonΨ-equation.
become
5. Volume transport vorticity
U⁎ ¼U þU þ sy ; ð24Þ
den b fq
0 5.1. Volume transport vorticity equation
V⁎ ¼V þV − sx : ð25Þ
den b fq Summationofthedifferentiationof(21)withrespect
0
to y and the differentiation of (22) with respect to x
Withtheknown(u−H,v−H),thedepth-integratedflow gives the volume transport vorticityequation,
(U⁎,V⁎)maybedirectlycalculatedusing(24)and(25). (cid:1) (cid:3)
⁎ ⁎ b 1 As As
However, the computed (U , V ) field using (24) and j2C ¼ ðV−V −V Þ− y− x
(25) is quite noisy and cannot not be the final product. Ah(cid:1) den b(cid:3) Ahq0 Ax Ay
Thus, the subscript ‘⁎’ is used to represent the interim þ AQ2−AQ1 ; ð30Þ
depth-integratedvelocitycalculatedusing(24)and(25). Ax Ay
4. Volume transport streamfunction where β=df/dy, and (28) is used.
Integrationofthecontinuityequationwithrespectto
z from thebottomto thesurfaceyields, 5.2. Extra-equatorial region
AU AH AV AH
Ax þu−H Ax þ Ay þv−H Ay −w−H ¼0: ð26Þ For the extra-equatorial region, the horizontal
diffusion can be neglected [see (8)]. Substitution of A
h
With the assumption that the water flows following into 0 leads to
thebottom topography,
(cid:5) (cid:6)
AH AH Cu1 AðfVdenÞ−AðfUdenÞ
w−H ¼u−H Ax þv−H Ay ; f (cid:5) Ax Ay (cid:6) (cid:5) (cid:1) (cid:3) (cid:1) (cid:3)(cid:6)
1 AðfV Þ AðfU Þ 1 A s A s
Eq. (26) becomes þ b − b − x þ y :
f Ax Ay f Ax q Ax q
AU AV 0 0
þ ¼0; ð31Þ
Ax Ay
which leads to the definition of the volume transport Similarly, (30) becomes
streamfunctionΨ, (cid:1) (cid:3)
1 As As
U ¼−AAWy ; V ¼AAWx : ð27Þ bðV−Vden−VbÞ¼q0 Axy− Ayx ; ð32Þ
which isthe Sverdrup relation.
Subtractionofthedifferentiationof(22)withrespect
In (31), (U , V ) depend on ρ only; (τ ,τ ) are
to y from the differentiation of (21) with respect to x den den x y
windstresscomponents;and(U ,V )aredeterminedby
gives b b
the bottom current velocity (u−H, v−H). The P-vector
j2W¼P; ð28Þ inverse method (Chu, 1995; Chu et al., 1998a,b; Chu,
where 2000)isusedtodetermine(u−H,v−H)fromhydrographic
(cid:5) (cid:6) (cid:5) (cid:6) data(seeAppendixA).Inthisstudy,theclimatological
1 AðfV Þ AðfU Þ 1 AðfV Þ AðfU Þ
Cu den − den þ b − b : hydrographic data (Levitus et al., 1994) are used to
f Ax Ay f Ax Ay compute (U , V ) [see (20)]. The climatological
(cid:5) (cid:1) (cid:3) (cid:1) (cid:3)(cid:6) den den
1 A s A s surface wind stress (τ ,τ ) data are obtained from the
− x þ y x y
f Ax q Ax q ComprehensiveOcean-AtmosphereDataSet(COADS,
(cid:1) 0 (cid:3) 0
A AQ AQ da Silva et al., 1994). The bottom topography is
þ h 1þ 2 ; obtained from the Navy's Digital Bathymetry Data
f Ax Ay
Base 5-min (DBDB5) (Fig. 1). The volume transport
ð29Þ vorticityΠ isquite noisy.
382 P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399
Fig.4.Globalvolumetransportstreamfunction(Ψ)computedfromtheinversemodel:(a)annualmean,(b)January,and(c)July.
P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399 383
5.3. Equatorial region (between 8°S and 8°N) boundary values of the vorticity Eq. (30). Here, the
forcing terms [righthand-side of (30)] are calculated
Let the volume transport vorticity Π calculated with the assumptions that (1) f=f(8°N) north of
using(31)at8°Nand8°Sasthenorthernandsouthern the equator, and f=f(8°S) south of the equator, and
Fig.5.Globaldepth-integratedvelocity(U,V)vectorscomputedfromtheinversemodel:(a)annualmean,(b)January,and(c)July.
384 P.C.Chu,C.Fan/JournalofMarineSystems65(2007)376–399
(2) (U, V) are calculated by (24) and (25). With the subjectivelysetupinsomeearlierstudies.Forexample,
given forcing terms and the northern and southern in calculating the geostrophic transport in the Pacific
boundary conditions and the cyclic eastern and Ocean,Reid(1997)setupΨ-valuetobe0forAntarctic,
western boundary conditions, the volume transport 135Sv(1Sv=106m3s−1)forAustralia,and130Svfor
vorticity Eq. (30) can be solved in the equatorial America.Incalculatingthegeostrophictransportinthe
regionbetween8°Nand8°S.ThecomputedΠ-fieldis SouthAtlanticOcean,Reid(1989)setupΨ-valuetobe
quite smooth. 0 for Antarctic, 132 Sv for Africa, and 130 Sv for
America. Such a treatment subjectively prescribes
6. Poisson Ψ-equation 130Sv throughtheDrakePassage and 132Svthrough
area between Africa and Antarctica.
6.1. Boundary conditions AnobjectivemethoddepictedinAppendix-Bisused
to determine Ψ-values at islands. Fig. 3 shows the
Withthecomputedglobalvolumetransportvorticity distribution of Ψ-value for each continent/island
(Π), the Poisson Ψ-equation (28) can be solved when computed from the annual, January, and July mean
the boundary conditions are given. No flow over the hydrographicandwinddata.Takingtheannualmeanas
Antarctic Continent leads to the southern boundary an example, we have: 0 for the American Continent,
condition 157.30 Sv for Antarctica, −21.74 Sv for Australia,
−27.17 Sv for Madagascar, and −21.74 Sv for New
W¼C ; at the southern boundary y¼y : ð33Þ Guinea.
1 s
7. Model sensitivity
Nohorizontalconvergenceofthe2-dimensionalflow
(U, V) at the North Pole (treated as an island) leads to
Withthegiven valuesattheboundariesandislands,
thenorthern boundarycondition
we solve the Poisson Ψ-equation (28) with climatolog-
icalannualandmonthlyΠ-fieldsandobtainannualand
W¼C ; at the northern boundary y¼y ; ð34Þ
2 n
where C and C are the constants to be determined.
1 2
Thecyclicboundaryconditionisappliedtothewestern
and the eastern boundaries (Fig. 2). We integrate ∂Ψ/
∂y=−U⁎withrespecttoyalongthewestern(oreastern)
boundary from the southern boundary (Ψ=0) to the
northernboundary toobtain
Z
y
⁎
Wj ðyÞ¼− U ðx ;yVÞdyV: ð35Þ
west west
ys
The cyclic boundary condition leads to
Wj ðyÞ¼Wj ðyÞ: ð36Þ
east west
Thus, thenorthern boundaryconditionis given by
Z
yn ⁎
W¼− U ðx ;yÞdy¼C : ð37Þ
west 2
ys
6.2. Ψ-Values at islands
Before solving the Poisson Ψ-equation (28) with
theboundaryconditions(33)–(35)and(37),weneedto
Fig.6.MonthlyvolumetransportthroughtheDrakePassagewitha
know the Ψ-values at all islands. These values were smallseasonalvariation.
