research article a study on bottom friction coefficient in ...e bottom friction plays a signi cant...

Research ArticleA Study on Bottom Friction Coefficient in the BohaiYellow and East China Sea

Daosheng Wang1 Qiang Liu2 and Xianqing Lv1

1 Laboratory of Physical Oceanography Ocean University of China Qingdao 266100 China2 College of Engineering Ocean University of China Qingdao 266100 China

Correspondence should be addressed to Qiang Liu liuqiangouceducn

Received 9 April 2014 Revised 10 June 2014 Accepted 12 June 2014 Published 1 July 2014

Academic Editor Fatih Yaman

Copyright copy 2014 Daosheng Wang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The adjoint tidalmodel based on the theory of inverse problemhas been applied to investigate the effect of bottom friction coefficient(BFC) on the tidal simulation Using different schemes of BFC containing the constant different constant in different subdomaindepth-dependent form and spatial distribution obtained from data assimilation the M

2constituent in the Bohai Yellow and East

China Sea (BYECS) is simulated by assimilating TOPEXPoseidon altimeter data respectively The simulated result with spatiallyvarying BFC obtained from data assimilation is better than others Results and analysis of BFC in BYECS indicate that spatiallyvarying BFC obtained from data assimilation is the best fitted one meanwhile it could improve the accuracy in the simulation ofM2constituentThrough the analysis of the best fitted one new empirical formulas of BFC in BYECS are developed with which the

commendable simulated results of M2constituent in BYECS are obtained

1 Introduction

The bottom friction plays a significant role in the tidal phe-nomenon In numerical simulations of tide bottom frictionis generally parameterized by the bottom friction coefficient(BFC) In order to improve the simulation accuracy it isessential to determine the BFC correctly In previous studies[1ndash8] several methods were suggested to determine the BFCand some encouraging simulated results were achieved Leeand Jung [9] used a three-dimensional mode-splitting 120590-coordinate barotropic finite-difference model to examineM2tidal elevation and current in the Yellow Sea and East

China Sea and they treated the BFC as a constant in thewhole computing domain Zhao et al [10] simulated thesemidiurnal and diurnal tides and tidal currents in thewhole Eastern China Seas with different BFC in differentsubdomain Kang et al [11] carried out a fine grid tidalmodeling experiment to study the tidal phenomena in theYellow and East China Seas and they used the depth-dependent form of BFC He et al [12] set up a numericaladjoint model with TOPEXPoseidon (TP) altimeter data toinvestigate the shallow water tidal constituents in the Bohai

and Yellow Sea In their model the Bohai and Yellow Seawere divided into five sub-regions with different BFC Luand Zhang [13] used the adjoint method to assimilate TPaltimeter data into a 2-dimensional tidal model in the BohaiYellow and East China Sea (BYECS) and the spatially varyingBFC were estimated with the independent point strategy

Additionally open boundary conditions (OBCs) are cru-cial for the representation of tidal processes in the regionalocean model [14] Generally OBCs could be obtained fromthe larger scale model or by interpolating the existing obser-vation data near the locationHowever OBCs obtained by themethods mentioned above have to be adjusted by experienceto get ideal simulated results Based on the theory of inverseproblem the adjoint method is a powerful tool for parameterestimation [15] and thus OBCs could be optimized auto-matically Zhang and Lu [16] applied the four-dimensionalvariational data assimilation technology to simulate thethree-dimensional tidal currents in the marginal seas and theOBCs were optimized Guo et al [14] estimated the OBCsin Bohai Sea by an adjoint data assimilation approach withindependent point strategy and obtained good simulatedresult of M

2constituent Zhang and Wang [17] developed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 432529 7 pageshttpdxdoiorg1011552014432529

2 Mathematical Problems in Engineering

a new method based on the adjoint method to inverse theperiodic OBCs in two-dimensional tidal models and used itto simulate the M

2constituent in BYECS successfully

As mentioned above BFC is an important parameterfor tidal models and many schemes of BFC have been usedin previous study However so far there are few systematiccomparisons about the different schemes of BFC Becausedifferent numerical models and observations are used indifferent studies the simulated results in those papers inwhich the BFC are different could not be compared directlyIn this paper firstly the adjoint tidal model is employedto compare some different schemes of BFC At the sametime in order to reduce the influence of OBCs that are alsoimportant for tidal models we use the adjoint method tooptimize OBCs Based on the simulation of M

2constituent

in BYECS several different schemes of BFC including theconstant different constant in different subdomain depth-dependent form and spatial distribution obtained from dataassimilation are compared to find the best fitted oneThen wetry to analyze the best fitted one to set up new empirical for-mulas of BFC in BYECS with which the preferable simulatedresults could be obtained

2 Adjoint Tidal Model

21 Equations The governing equations are described underthe rectangular coordinate system Assuming that pressure ishydrostatic and density is constant the depth averaged two-dimensional tidal model is as follows

120597120577

120597119905

+

120597 [(ℎ + 120577) 119906]

120597119909

+

120597 [(ℎ + 120577) V]120597119910

= 0

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

minus 119891V +119896119906radic119906

2+ V2

ℎ + 120577

minus 119860(

1205972119906

1205971199092+

1205972119906

1205971199102) + 119892

120597120577

120597119909

= 0

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

+ 119891119906 +

119896Vradic1199062 + V2

ℎ + 120577

minus 119860(

1205972V1205971199092+

1205972V1205971199102) + 119892

120597120577

120597119910

= 0

(1)

where 119905 is time 119909 and 119910 are Cartesian coordinates ℎ isundisturbed water depth 120577 is sea surface elevation abovethe undisturbed sea level 119906 and V are velocity componentsin the east and north 119891 is the Coriolis parameter 119892 isthe acceleration due to gravity 119896 is the BFC and 119860 is thehorizontal eddy viscosity coefficient

With the adjoint method described in Lu and Zhang [13]the cost function is constructed as

119869 =

1

2

119870120577int

Σ

(120577 minus120577)

2

119889120590 (2)

where 119870120577is a constant and Σ is the set of the observation

locations

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 1 Bathymetry map of BYECS

And the adjoint model can be constructed as follows

120597120582

120597119905

+ 119906

120597120582

120597119909

+ V120597120582

120597119910

+

119896120583119906radic1199062+ V2

(ℎ + 120577)2

+

119896]Vradic1199062 + V2

(ℎ + 120577)2

+ 119892

120597120583

120597119909

+ 119892

120597]120597119910

= 119870120577(120577 minus

120577)

120597120583

120597119905

minus (119891 +

119896119906V(ℎ + 120577)radic119906

2+ V2

) ] minus 120583120597119906

120597119909

minus ]120597V120597119909

+

120597

120597119909

(120583119906) +

120597

120597119910

(120583V) + (ℎ + 120577)120597120582

120597119909

+ 119860(

1205972120583

1205971199092+

1205972120583

1205971199102) minus

119896 (21199062+ V2)

(ℎ + 120577)radic1199062+ V2

120583 = 0

120597]120597119905

+ (119891 minus

119896119906V(ℎ + 120577)radic119906

2+ V2

)120583 minus 120583

120597119906

120597119910

minus ]120597V120597119910

+

120597

120597119909

(]119906) +120597

120597119910

(]V) + (ℎ + 120577)120597120582

120597119910

+ 119860(

1205972]1205971199092+

1205972]1205971199102) minus

119896 (1199062+ 2V2)

(ℎ + 120577)radic1199062+ V2

] = 0

(3)

where 120577 is the simulated result 120577 is the observation and 120582 120583and ] denote the adjoint variables of 120577 119906 and V respectively

The finite difference schemes of (1) and (3) are similar tothose in Lu and Zhang [13]

22 Model Setting The computing area is BYECS(1175∘Endash131∘E 24∘Nndash41∘N) which is shown in Figure 1The horizontal resolution is 101015840times 101015840 The time step is 62103

Mathematical Problems in Engineering 3

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

Figure 2 Positions of TP altimeter tracks (ldquo∙rdquo) and tidal gauges(ldquoordquo) and open boundary (ldquo+rdquo)

seconds which is 1720 of the period of M2constituent The

eddy viscosity coefficient (119860) is 5000m2s The positions oftidal gauge stations the TP altimeter tracks and the openboundary are shown in Figure 2

3 Numerical Experiments and Result Analysis

31 Calculation Process of Numerical Experiments Initialconditions are that the sea surface elevation (120577) and thevelocities (119906 and V) are zero In addition the initial values ofOBCs are set to zero

The calculation process of the adjoint tidal model isdesigned as follows

(1) With the BFC given which is fixed in the wholecomputing process OBCs existed and other modelparameters run the forward tidal model

(2) The difference of water elevation between simulatedresults from step (1) and observations at the gridpoints on TP satellite tracks serves as the externalforce of the adjoint model Values of adjoint variablesare obtained through backward integration of theadjoint equations

(3) With the values of adjoint variables from the adjointmodel the OBCs could be adjusted by the methodmentioned in Cao et al [18]

Repeat steps (1)ndash(3) until the number of iteration stepsis exactly 100 For the setting of adjoint tidal model in thisstudy 100 iteration steps are sufficient because both thecost function and the difference between observations andsimulated results will decrease slowly after this step

32 Setting of Numerical Experiments In each numericalexperiment the BFC is fixed and the OBCs are optimized by

2

4

6

8

10

12

14

16

18

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

times10minus4

Figure 3 The BFC distribution in E4

assimilating TP altimeter data into the adjoint tidal modelso that we could compare the different schemes of BFCadequately without the possibility that the OBCs do notmatch the BFC Moreover the tide gauge data are used as anindependent check of the model fidelity

Refer to some schemes of BFC generally used in previousstudies and we design several numerical experiments tocompare them

E1 the BFC is treated as a constant (00015) in BYECS

E2 the BFC is depth-dependent form which is similarto that used by Kang et al [11] The BFC is definedby 119896 = 119892119862

2 where 119892 is gravity acceleration 119862 isChezy coefficient and the depth-dependent form ofthe Chezy coefficient are applied as 119862 = ℎ

16119899 with

119899 = 0023

E3 the scheme of BFC is the same as that employed inZhao et al [10] The BFC is taken to be 0001 at thewest of the line from (25∘151015840N 120∘451015840E) to (40∘001015840N124∘151015840E) 00035 in the Korean Strait and 00016 inother areas

E4 the space-varying BFC is obtained by assimilatingobservations using the adjoint method in Lu andZhang [13] The difference is that the initial conditionof BFC in this paper is 00015The spatial distributionof BFC is shown in Figure 3

33 Results of Numerical Experiments When the tide isstable the results of next period are used to do harmonicanalysis The mean absolute errors (MAEs) in amplitude andphase between simulation results and observations (TP dataand tidal gauge data) are shown in Table 1


Table 1 Differences between simulated results and observations(TP data and tidal gauge data)

EXPMAEs of TP data MAEs of tidal gauge data

Amplitude(cm) Phase lag (∘) Amplitude

(cm) Phase lag (∘)

E1 72 62 102 73E2 76 67 103 89E3 69 61 99 72E4 57 58 67 66

12

041

1

1

6060

0

270

300

240

0818

04

18

08

16 06

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

Figure 4 The cotidal chart obtained from E4 (the dashed linedenotes coamplitude line (m) and solid line denotes cophase line(degree))

From Table 1 one can find that E4 obtains the bestsimulated result From the MAEs in amplitude and phasebetween simulation and TP data it could be found thatE4 obtains the best assimilated results in the same steps ofassimilation And it is obvious thatMAEs between simulationand tidal gauge data are minimum We try to increasethe number of iteration steps in E1 E2 and E3 but noimprovements are achieved

The cotidal chart of M2constituent obtained in E4 is

shown in Figure 4 Compared with Lefevre et al [19] andFang et al [20] the cotidal chart seems to coincide with theobserved M

2constituent in BYECS fairly well It also proves

that E4 gets perfect simulated result As shown in Figure 4there are two amphidromic points in the Bohai Sea one ofwhich is near Qinhuangdao and the other is near the YellowRiver delta There are also two amphidromic points in theYellow Sea one of which is north of Chengshantou and theother is southeast of Qingdao

4 Discussion of BFC

41 Discussion from Numerical Results As shown by Table 1and Figure 4 it is obvious that E4with the space-varying BFCobtains the best simulated result

Mofjeld [21] used a turbulence closure model to inves-tigate the dependence on water depth of bottom stress andquadratic drag coefficient for a steady barotropic pressure-driven current in unstratified water when the current was theprimary source of turbulence He noted that the quadraticdrag coefficient was approximated reasonably well by aformula from nonrotating channel theory in which thecoefficient depended only on the ratio of the water depthto the bottom roughness Jenter and Madsen [22] studiedthe bottom stress in wind-stress depth-average coastal flowsand found that the drag tensor variation was a function ofwater depth wind stress and bottom roughness From theaforementioned studies it is seen that the BFC generallydepends on thewater depth andbottom roughness And thereis no doubt that the water depth and bottom roughness arediverse in different area and they vary spatially In additionKagan et al [23] studied the impact of the spatial variabilityin bottom roughness on tidal dynamics and energetics in theNorth European Basin and indicated that ignoring the spatialvariability in bottom roughness was only partially correctbecause it was liable to break down for the tidal energeticsTherefore the BFC should be spatially varying in fact It isnoticeable that the schemes of a constant BFC like in E1 is notreasonable enough The space-varying BFC obtained fromthe data assimilation seems to be more advisable in physics

In fact BFC in E2 is depth-dependent and thus it is alsospatially varying However the simulated results from E2 areworse than that from E1 and E3 and much worse than thatfrom E4 From Figure 3 the BFC in shallow water are largerthan those in deep water in the Bohai Sea and the Yellow SeaindividuallyMeanwhile the averagewater depth of the BohaiSea is 193m and the average BFC is 000082 while they were454m and 000081 for the Yellow Sea and 3347m and 00015for the East China Sea From the definition of BFC in E2 it isevident that the BFC and the depth are in inverse proportionin whole region In detail the average BFC of the Bohai Seais 00021 while it is 00017 for the Yellow Sea and 00011 forthe East China Sea We can find that the BFC in the BohaiSea and the Yellow Sea has the same changing trend withE4 but the value is larger Green and McCave [24] indicatedthat the form drag caused by the bottom topography wave-current interaction boundary-layer stratification and so onmay impact the BFC The water depth changes largely in theOkinawa trough so the form drag should be larger But inE2 the BFC in East China Sea is small and the East ChinaSea is the largest area in BYECS so the simulated result of E2is dissatisfactory We surmise that the scheme of BFC in E2may be reasonable in the shelf sea and not applicable in thearea of slope and troughThus it can be seen that the spatiallyvarying BFC from data assimilation is better than the depth-dependent form in BYECS especially in the East China Sea

In addition E3 obtains better result than E2 In E2 theBFC is 00019 at thewest of the line in E3 00012 in theKoreanStrait and 00014 in other areas meanwhile they are 00008


0 500 1000 1500 2000 2500Depth (m)

BFC

0 20 40 60 80 100 120 14005

1

15

2

times10minus3

times10minus3

06

08

12

14

16

18

2

1

Figure 5 BFC versus water depth

00014 and 00015 in E4 In the areas except the Korean Straitwhose area is small the BFC in E3 and E4 have the samechanging tendency and the average values are approximatelyequal However there is the opposite trend in E2 It seems toexplain that the BFC in E3 is better than that in E2 And itproves that the scheme of BFC in E4 is the best fitted one fromanother side

In conclusion the spatially varying BFC in E4 is the bestfitted BFC in BYECS

42 Further Exploration of BFC In this section the schemeofBFC in E4 is analyzed to investigate the relationship betweenBFC and water depth the change rate of seafloor topography(CRST) and bottom roughness

In this study CSRT is described as follows

119863ℎ =

10038161003816100381610038161003816ℎ119894119895minus ℎ119894+1119895

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894minus1119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895+1

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895minus1

10038161003816100381610038161003816

(4)

And bottom roughness is described as follows

119863119863ℎ = (ℎ119894119895minus ℎ119894+1119895

)

2

+ (ℎ119894119895minus ℎ119894minus1119895

)

2

+ (ℎ119894119895minus ℎ119894119895+1

)

2

+ (ℎ119894119895minus ℎ119894119895minus1

)

2

(5)

The correlation coefficient between BFC and water depthis 04540 while it is 03845 for CRST and 02520 for bottomroughness It is shown that water depth is the significantfactor that affects BFCWe demonstrate the BFC versus waterdepth in Figure 5 and could find that the BFC is a constantwhen water depth is larger than 100 meters However whenwater depth is less than 100meters the BFC varies complicat-edly So we focus on the study of quantitative relations of BFCwith water depth CRST and bottom roughness when waterdepth is less than 100 meters

Considering BFC is mainly affected by water depth infirst step we ignore the CSRT and bottom roughness to make

0 1 2 3 4 5 6 7 8

BFC

times10minus3

06

08

12

14

16

18

2

1

ln (Dh)

Figure 6 BFC versus ln(119863ℎ) when water depth is less than 100meters

the relationship simple and just investigate the quantitativerelationship of BFC and water depth From partial enlargeddrawing in Figure 5 it could be seen that there are twosections When water depth is less than 30 meters BFCdecreases with the water depth increasing while increasingfor larger than 30 meters The fitting function could beobtained as follows

119896 =

15363 times 10minus3 ℎ ge 100

(568850 + 09674ℎ) times 10minus5 30 le ℎ lt 100

(1000 minus 05413ℎ) times 10minus5 ℎ lt 30

(6)

From another perspective a linear function coulddescribe the relationship between BFC and water depthroughly and at the same time 119863ℎ and 119863119863ℎ are also consid-ered As seen in Figure 6 BFC increases linearly along withln(119863ℎ) by and large From Figure 7 it is shown that it isdifficult to use a formula to describe the relationship betweenBFC and ln(119863119863ℎ)Therefore considering the impact of ℎ119863ℎupon BFC we obtain the formula as follows

119896 =


(05255 + 00068ℎ + 00731 ln (119863ℎ)) times 10minus3 ℎ lt100

(7)

Using formulas (6) and (7) two new schemes of BFC inBYECS are obtained and they are recorded as E5 and E6The differences between simulated results and observationsare shown in Table 2

From Tables 1 and 2 it could be found that the simulatedresults of E5 and E6 are better than those of others exceptE4 It indicates that the schemes of BFC obtained fromthe statistical relation could describe the BFC in BYECSpreferably and improve the result of numerical simulation

Through the analysis of the scheme of BFC in E4 we setup new empirical formulas of BFC in BYECS with whichthe commendable simulated results are obtained It should


0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D

Figure 7 BFC versus ln(119863119863ℎ)when water depth is less than 100meters





E5 65 59 85 65E6 67 60 84 61

be noted that the calculation of BFC in BYECS by the newempirical formulas just needs the bathymetric data So itcan be considered to be referenced in the simulation of M

2

constituent in BYECS

5 Conclusions

The adjoint tidal model based on the theory of inverseproblem has been applied to investigate the effect of BFCon the tidal simulation The M

2constituent in BYECS is

simulated by assimilating TP altimeter data with severaldifferent schemes of BFC the constant different constantin different subdomain depth-dependent form and spatialdistribution obtained from data assimilation Comparingwith the observations at tidal gauges it is found that thesimulated result with the spatially varying BFC is the bestand the MAEs in amplitude and phase are 67 cm and 66∘respectively while the least values in other experiments are99 cm and 72∘ Comparing with the observations at TPstations we found that the simulated result with spatiallyvarying BFC has advantages over others and the MAEsin amplitude and phase are 57 cm and 58∘ respectivelywhile in other experiments they are at least 69 cm and 61∘The simulated results and the analysis of BFC in BYECSsimultaneously indicate that spatially varying BFC obtainedfrom data assimilation is the best fitted one and it couldimprove the accuracy in the simulation of M

2constituent

Finally through the statistical analysis of the spatially varying

BFCobtained fromdata assimilation new empirical formulasof BFC in BYECS are obtained We found that the simulatedresults with new empirical formulas are better than tradi-tional schemes such as the constant different constant indifferent subdomain and depth-dependent form We believethat the new empirical formulas could be referenced in thesimulation of M

2constituent in BYECS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors deeply thank the reviewers and editor for theirconstructive criticism of an early version of the paper Partialsupport for this research was provided by the NationalNatural Science Foundation of China through Grants nos41072176 and 41371496 the National Science and TechnologySupport Program through Grant no 2013BAK05B04 theState Ministry of Science and Technology of China throughGrant no 2013AA122803 and the Fundamental ResearchFunds for the Central Universities 201362033 and 201262007

References

[1] C Chen H Huang R C Beardsley H Liu Q Xu and GCowles ldquoA finite volume numerical approach for coastal oceancirculation studies comparisons with finite difference modelsrdquoJournal of Geophysical Research C Oceans vol 112 no 3 ArticleID C03018 2007

[2] L S Quaresma and A Pichon ldquoModelling the barotropic tidealong the West-Iberian marginrdquo Journal of Marine Systems vol109-110 pp S3ndashS25 2013

[3] J Zhang and X Lu ldquoParameter estimation for a three-dimensional numerical barotropic tidal model with adjointmethodrdquo International Journal for Numerical Methods in Fluidsvol 57 no 1 pp 47ndash92 2008

[4] G D Egbert R D Ray and B G Bills ldquoNumerical modelingof the global semidiurnal tide in the present day and in the lastglacial maximumrdquo Journal of Geophysical Research C Oceansvol 109 no 3 Article ID C03003 2004

[5] H J Lee K T Jung J K So and J Y Chung ldquoA three-dimensional mixed finite-difference Galerkin function modelfor the oceanic circulation in the Yellow Sea and the East ChinaSea in the presence of M

2tiderdquo Continental Shelf Research vol

22 no 1 pp 67ndash91 2002[6] G Sannino A Bargagli and V Artale ldquoNumerical modeling of

the semidiurnal tidal exchange through the strait of GibraltarrdquoJournal of Geophysical Research C Oceans vol 109 no 5 2004

[7] A W Heemink E E A Mouthaan M R T Roest E AH Vollebregt K B Robaczewska and M Verlaan ldquoInverse3D shallow water flow modelling of the continental shelfrdquoContinental Shelf Research vol 22 no 3 pp 465ndash484 2002

[8] M U Altaf M Verlaan and A W Heemink ldquoEfficientidentification of uncertain parameters in a large-scale tidalmodel of the European continental shelf by proper orthogonaldecompositionrdquo International Journal for Numerical Methods inFluids vol 68 no 4 pp 422ndash450 2012


[9] J C Lee and K T Jung ldquoApplication of eddy viscosity closuremodels for the M

2tide and tidal currents in the Yellow Sea and

the East China SeardquoContinental Shelf Research vol 19 no 4 pp445ndash475 1999

[10] B Zhao G Fang and D Cao ldquoNumerical modeling on thetides and tidal currents in the eastern China Seasrdquo Yellow SeaResearch vol 5 pp 41ndash61 1993

[11] S K Kang S Lee and H Lie ldquoFine grid tidal modeling of theYellow and East China Seasrdquo Continental Shelf Research vol 18no 7 pp 739ndash772 1998

[12] Y He X Lu Z Qiu and J Zhao ldquoShallow water tidalconstituents in the Bohai Sea and the Yellow Sea from anumerical adjoint model with TOPEXPOSEIDON altimeterdatardquoContinental Shelf Research vol 24 no 13-14 pp 1521ndash15292004

[13] X Lu and J Zhang ldquoNumerical study on spatially varyingbottom friction coefficient of a 2D tidal model with adjointmethodrdquo Continental Shelf Research vol 26 no 16 pp 1905ndash1923 2006

[14] Z Guo A Cao and X Lv ldquoInverse estimation of openboundary conditions in the Bohai SeardquoMathematical Problemsin Engineering vol 2012 Article ID 628061 9 pages 2012

[15] J Zhang and H Chen ldquoSemi-idealized study on estimation ofpartly and fully space varying open boundary conditions fortidal modelsrdquo Abstract and Applied Analysis vol 2013 ArticleID 282593 14 pages 2013

[16] J Zhang and X Lu ldquoInversion of three-dimensional tidalcurrents in marginal seas by assimilating satellite altimetryrdquoComputer Methods in Applied Mechanics and Engineering vol199 no 49ndash52 pp 3125ndash3136 2010

[17] J Zhang and Y Wang ldquoA method for inversion of periodicopen boundary conditions in two-dimensional tidal modelsrdquoComputer Methods in Applied Mechanics and Engineering vol275 pp 20ndash38 2014

[18] A Cao Z Guo and X Lu ldquoInversion of two-dimensional tidalopen boundary conditions of M

2constituent in the Bohai and

Yellow Seasrdquo Chinese Journal of Oceanology and Limnology vol30 no 5 pp 868ndash875 2012

[19] F Lefevre C le Provost and F H Lyard ldquoHow can we improvea global ocean tidemodel at a regional scale a test on the YellowSea and the East China Seardquo Journal of Geophysical Research COceans vol 105 no 4 pp 8707ndash8725 2000

[20] G Fang Y Wang Z Wei B H Choi X Wang and J WangldquoEmpirical cotidal charts of the Bohai Yellow and East ChinaSeas from 10 years of TOPEXPoseidon altimetryrdquo Journal ofGeophysical Research C Oceans vol 109 no 11 Article IDC11006 2004

[21] H O Mofjeld ldquoDepth dependence of bottom stress andquadratic drag coefficient for barotropic pressure-driven cur-rentsrdquo Journal of Physical Oceanography vol 18 pp 1658ndash16691988

[22] H L Jenter and O S Madsen ldquoBottom stress in wind-drivendepth-averaged coastal flowsrdquo Journal of Physical Oceanogra-phy vol 19 pp 962ndash974 1989

[23] B A Kagan E V Sofina and E Rashidi ldquoInversion of two-dimensional tidal open boundary conditions of119872

2constituent

in the Bohai and Yellow Seasrdquo Ocean Dynamics vol 62 no 10ndash12 pp 1425ndash1442 2012

[24] M O Green and I N McCave ldquoSeabed drag coefficient undertidal currents in the eastern Irish Seardquo Journal of GeophysicalResearch vol 100 no 8 pp 16057ndash16069 1995

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a new method based on the adjoint method to inverse theperiodic OBCs in two-dimensional tidal models and used itto simulate the M

2constituent in BYECS successfully

As mentioned above BFC is an important parameterfor tidal models and many schemes of BFC have been usedin previous study However so far there are few systematiccomparisons about the different schemes of BFC Becausedifferent numerical models and observations are used indifferent studies the simulated results in those papers inwhich the BFC are different could not be compared directlyIn this paper firstly the adjoint tidal model is employedto compare some different schemes of BFC At the sametime in order to reduce the influence of OBCs that are alsoimportant for tidal models we use the adjoint method tooptimize OBCs Based on the simulation of M

2constituent

in BYECS several different schemes of BFC including theconstant different constant in different subdomain depth-dependent form and spatial distribution obtained from dataassimilation are compared to find the best fitted oneThen wetry to analyze the best fitted one to set up new empirical for-mulas of BFC in BYECS with which the preferable simulatedresults could be obtained

2 Adjoint Tidal Model

21 Equations The governing equations are described underthe rectangular coordinate system Assuming that pressure ishydrostatic and density is constant the depth averaged two-dimensional tidal model is as follows

120597120577

120597119905

+

120597 [(ℎ + 120577) 119906]

120597119909

+

120597 [(ℎ + 120577) V]120597119910

= 0

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

minus 119891V +119896119906radic119906

2+ V2

ℎ + 120577

minus 119860(

1205972119906

1205971199092+

1205972119906

1205971199102) + 119892

120597120577

120597119909

= 0

120597V120597119905

+ 119906

120597V120597119909

+ V120597V120597119910

+ 119891119906 +

119896Vradic1199062 + V2

ℎ + 120577

minus 119860(

1205972V1205971199092+

1205972V1205971199102) + 119892

120597120577

120597119910

= 0

(1)

where 119905 is time 119909 and 119910 are Cartesian coordinates ℎ isundisturbed water depth 120577 is sea surface elevation abovethe undisturbed sea level 119906 and V are velocity componentsin the east and north 119891 is the Coriolis parameter 119892 isthe acceleration due to gravity 119896 is the BFC and 119860 is thehorizontal eddy viscosity coefficient

With the adjoint method described in Lu and Zhang [13]the cost function is constructed as

119869 =

1

2

119870120577int

Σ

(120577 minus120577)

2

119889120590 (2)

where 119870120577is a constant and Σ is the set of the observation

locations

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 1 Bathymetry map of BYECS

And the adjoint model can be constructed as follows

120597120582

120597119905

+ 119906

120597120582

120597119909

+ V120597120582

120597119910

+

119896120583119906radic1199062+ V2

(ℎ + 120577)2

+

119896]Vradic1199062 + V2

(ℎ + 120577)2

+ 119892

120597120583

120597119909

+ 119892

120597]120597119910

= 119870120577(120577 minus

120577)

120597120583

120597119905

minus (119891 +

119896119906V(ℎ + 120577)radic119906

2+ V2

) ] minus 120583120597119906

120597119909

minus ]120597V120597119909

+

120597

120597119909

(120583119906) +

120597

120597119910

(120583V) + (ℎ + 120577)120597120582

120597119909

+ 119860(

1205972120583

1205971199092+

1205972120583

1205971199102) minus

119896 (21199062+ V2)

(ℎ + 120577)radic1199062+ V2

120583 = 0

120597]120597119905

+ (119891 minus

119896119906V(ℎ + 120577)radic119906

2+ V2

)120583 minus 120583

120597119906

120597119910

minus ]120597V120597119910

+

120597

120597119909

(]119906) +120597

120597119910

(]V) + (ℎ + 120577)120597120582

120597119910

+ 119860(

1205972]1205971199092+

1205972]1205971199102) minus

119896 (1199062+ 2V2)

(ℎ + 120577)radic1199062+ V2

] = 0

(3)

where 120577 is the simulated result 120577 is the observation and 120582 120583and ] denote the adjoint variables of 120577 119906 and V respectively

The finite difference schemes of (1) and (3) are similar tothose in Lu and Zhang [13]

22 Model Setting The computing area is BYECS(1175∘Endash131∘E 24∘Nndash41∘N) which is shown in Figure 1The horizontal resolution is 101015840times 101015840 The time step is 62103


Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40












2

4

6

8

10

12

14

16

18

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

times10minus4







16119899 with

119899 = 0023









E1 72 62 102 73E2 76 67 103 89E3 69 61 99 72E4 57 58 67 66

12

041

1

1

6060

0

270

300

240

0818

04

18

08

16 06

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40







4 Discussion of BFC






0 500 1000 1500 2000 2500Depth (m)

BFC

0 20 40 60 80 100 120 14005

1

15

2

times10minus3

times10minus3

06

08

12

14

16

18

2

1






119863ℎ =

10038161003816100381610038161003816ℎ119894119895minus ℎ119894+1119895

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894minus1119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895+1

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895minus1

10038161003816100381610038161003816

(4)


119863119863ℎ = (ℎ119894119895minus ℎ119894+1119895

)

2

+ (ℎ119894119895minus ℎ119894minus1119895

)

2

+ (ℎ119894119895minus ℎ119894119895+1

)

2

+ (ℎ119894119895minus ℎ119894119895minus1

)

2

(5)



0 1 2 3 4 5 6 7 8

BFC

times10minus3

06

08

12

14

16

18

2

1

ln (Dh)



119896 =




(6)


119896 =



(7)





0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D






E5 65 59 85 65E6 67 60 84 61


2


5 Conclusions




2constituent






Acknowledgments


References































2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40












2

4

6

8

10

12

14

16

18

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40

times10minus4







16119899 with

119899 = 0023









E1 72 62 102 73E2 76 67 103 89E3 69 61 99 72E4 57 58 67 66

12

041

1

1

6060

0

270

300

240

0818

04

18

08

16 06

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40







4 Discussion of BFC






0 500 1000 1500 2000 2500Depth (m)

BFC

0 20 40 60 80 100 120 14005

1

15

2

times10minus3

times10minus3

06

08

12

14

16

18

2

1






119863ℎ =

10038161003816100381610038161003816ℎ119894119895minus ℎ119894+1119895

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894minus1119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895+1

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895minus1

10038161003816100381610038161003816

(4)


119863119863ℎ = (ℎ119894119895minus ℎ119894+1119895

)

2

+ (ℎ119894119895minus ℎ119894minus1119895

)

2

+ (ℎ119894119895minus ℎ119894119895+1

)

2

+ (ℎ119894119895minus ℎ119894119895minus1

)

2

(5)



0 1 2 3 4 5 6 7 8

BFC

times10minus3

06

08

12

14

16

18

2

1

ln (Dh)



119896 =




(6)


119896 =



(7)





0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D






E5 65 59 85 65E6 67 60 84 61


2


5 Conclusions




2constituent






Acknowledgments


References































2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra














E1 72 62 102 73E2 76 67 103 89E3 69 61 99 72E4 57 58 67 66

12

041

1

1

6060

0

270

300

240

0818

04

18

08

16 06

Longitude (E)

Latit

ude (

N)

118 120 122 124 126 128 13024

26

28

30

32

34

36

38

40







4 Discussion of BFC






0 500 1000 1500 2000 2500Depth (m)

BFC

0 20 40 60 80 100 120 14005

1

15

2

times10minus3

times10minus3

06

08

12

14

16

18

2

1






119863ℎ =

10038161003816100381610038161003816ℎ119894119895minus ℎ119894+1119895

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894minus1119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895+1

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895minus1

10038161003816100381610038161003816

(4)


119863119863ℎ = (ℎ119894119895minus ℎ119894+1119895

)

2

+ (ℎ119894119895minus ℎ119894minus1119895

)

2

+ (ℎ119894119895minus ℎ119894119895+1

)

2

+ (ℎ119894119895minus ℎ119894119895minus1

)

2

(5)



0 1 2 3 4 5 6 7 8

BFC

times10minus3

06

08

12

14

16

18

2

1

ln (Dh)



119896 =




(6)


119896 =



(7)





0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D






E5 65 59 85 65E6 67 60 84 61


2


5 Conclusions




2constituent






Acknowledgments


References































2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 2500Depth (m)

BFC

0 20 40 60 80 100 120 14005

1

15

2

times10minus3

times10minus3

06

08

12

14

16

18

2

1






119863ℎ =

10038161003816100381610038161003816ℎ119894119895minus ℎ119894+1119895

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894minus1119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895+1

10038161003816100381610038161003816+

10038161003816100381610038161003816ℎ119894119895minus ℎ119894119895minus1

10038161003816100381610038161003816

(4)


119863119863ℎ = (ℎ119894119895minus ℎ119894+1119895

)

2

+ (ℎ119894119895minus ℎ119894minus1119895

)

2

+ (ℎ119894119895minus ℎ119894119895+1

)

2

+ (ℎ119894119895minus ℎ119894119895minus1

)

2

(5)



0 1 2 3 4 5 6 7 8

BFC

times10minus3

06

08

12

14

16

18

2

1

ln (Dh)



119896 =




(6)


119896 =



(7)





0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D






E5 65 59 85 65E6 67 60 84 61


2


5 Conclusions




2constituent






Acknowledgments


References































2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15

BFC

times10minus3

02

04

06

08

12

14

1

16

18

2

Dh)ln (D






E5 65 59 85 65E6 67 60 84 61


2


5 Conclusions




2constituent






Acknowledgments


References































2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























2constituent










Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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