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An Analytical Solution for Investigating the Characteristics of Tidal Wave and Surge Propagation Associated with Non-Tropical and Tropical Cyclones in the Humen Estuary, Pearl River

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Zhuo Zhang
Zhuo Zhang
Zhuo Zhang
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Fei Guo
Fei Guo
Fei Guo
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Di Hu at Nanjing Normal University
Di Hu
Dong Zhang
Dong Zhang
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Abstract and Figures

The Humen Estuary, one of the largest outlets of the Pearl River, is a long and wide tidal channel with a considerable tidal flow every year. Storm surges, always superposing spring tide, travel from the estuary and endanger the safety of people living around the river. However, little research has quantified the relationship between the hydraulic characteristics and the geometry features in this estuary. In this regard, an analytical model, combined with a numerical model, is applied to investigate the characteristics of tidal waves and surge propagations in the estuary. Given the geometric, topographic, and tidal parameters at the mouth of the estuary, the tidal damping and wave celerity can be computed. The numerical results were used to calibrate and verify the analytical model. The results indicate that the analytical model can describe the astronomical tidal dynamics very well in correspondence with the numerical results. However, the analytical model cannot predict the tide well when a tropical cyclone-induced surge is superimposed on the astronomical tide. The reason is that this model does not take the wind stress and the pressure depression into account. After reducing Manning’s coefficient, we found that the analytical results could be close to the numerical results. Finally, we analyzed the characteristics of the tidal wave in the Humen Estuary using the analytical solution and its parameters.
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Water2021,13,2375.https://doi.org/10.3390/w13172375www.mdpi.com/journal/water
Article
AnAnalyticalSolutionforInvestigatingtheCharacteristicsof
TidalWaveandSurgePropagationAssociatedwith
Non‐TropicalandTropicalCyclonesintheHumenEstuary,
PearlRiver
ZhuoZhang
1,2
,FeiGuo
1,2,
*,DiHu
1,2
andDongZhang
1,2
1
KeyLaboratoryofVirtualGeographicEnvironment,NanjingNormalUniversity,MinistryofEducation,
Nanjing210023,China;mercury1214@126.com(Z.Z.);09374@njnu.edu.cn(D.H.);
zhangdong@njnu.edu.cn(D.Z.)
2
JiangsuCenterforCollaborativeInnovationinGeographicalInformationResourceDevelopmentandAp‐
plication,Nanjing210023,China
*Correspondence:guofei@njnu.edu.cn
Abstract:TheHumenEstuary,oneofthelargestoutletsofthePearlRiver,isalongandwidetidal
channelwithaconsiderabletidalfloweveryyear.Stormsurges,alwayssuperposingspringtide,
travelfromtheestuaryandendangerthesafetyofpeoplelivingaroundtheriver.
However,little
researchhasquantifiedtherelationshipbetweenthehydrauliccharacteristicsandthegeometryfea‐
turesinthisestuary.Inthisregard,ananalyticalmodel,combinedwithanumericalmodel,isap‐
pliedtoinvestigatethecharacteristicsoftidalwavesandsurgepropagationsintheestuary.Given
thegeometric,topographic,andtidalparametersatthemouthoftheestuary,thetidaldampingand
waveceleritycanbecomputed.Thenumericalresultswereusedtocalibrateandverifytheanalyt‐
icalmodel.Theresultsindicatethattheanalyticalmodelcandescribetheastronomicaltidaldy‐
namicsverywellincorrespondencewiththenumericalresults.However,theanalyticalmodelcan‐
notpredictthetidewellwhenatropicalcyclone‐inducedsurgeissuperimposedontheastronomi‐
caltide.Thereasonisthatthismodeldoesnottakethewindstressandthepressuredepressioninto
account.AfterreducingManning’scoefficient,wefoundthattheanalyticalresultscouldbecloseto
thenumericalresults.Finally,weanalyzedthecharacteristicsofthetidalwaveintheHumenEstu‐
aryusingtheanalyticalsolutionanditsparameters.
Keywords:tidaldamping;wavepropagation;analyticalmodel;surge;pearlriver
1.Introduction
Tidalwavesareamajorhydrodynamicfactorthatinfluencestheflooding,ecosys‐
tems,transport,andmorphologicalevolutioninestuaries.Incontrasttotidalwavesin
deepoceans,whicharedrivenbyastronomicalforceswithahighdegreeofpredictability,
wavespropagatingalongestuariestendtoadjusttheiramplitude,celerity,andshapedur‐
inginteractionswithtopography,friction,andothersecondaryfactorssuchaswindstress
andrunoff.Atidalwaveinashallowestuaryhasthecharacteristicsofamplificationand
damping,whichalternatealongtheestuary.Thetidalamplitudeisinfluencedbytwo
dominantprocesses:amplificationduetoconvergentcrosssectionswithadecreasein
depthinthelandwarddirectionanddampingduetobottomfriction.Iftheformerprocess
dominatesoverthelatter,thewaveisamplified;otherwise,thewaveisdamped.Thece‐
lerityofpropagationisalsoinfluencedbytheprocessesoftidaldampingandamplifica‐
tion.Ifthetidalwaveisdamped,thecelerityisdecreased.Conversely,ifthetidalwaveis
amplified,thecelerityisincreased[1].Moreover,thetidalwavecanbestronglydistorted
Citation:Zhang,Z.;Guo,F.;Hu,D.;
Zhang,D.AnAnalyticalSolutionfor
InvestigatingtheCharacteristicsof
TidalWaveandSurgePropagation
AssociatedwithNon‐Tropicaland
TropicalCyclonesintheHumen
Estuary,PearlRiver.Water2021,13,
2375.https://doi.org/10.3390/
w13172375
AcademicEditor:TonyWong
Received:1June2021
Accepted:24August2021
Published:29August2021
Publisher’sNote:MDPIstaysneu‐
tralwithregardtojurisdictional
claimsinpublishedmapsandinstitu‐
tionalaffiliations.
Copyright:©2021bytheauthors.Li‐
censeeMDPI,Basel,Switzerland.
Thisarticleisanopenaccessarticle
distributedunderthetermsandcon‐
ditionsoftheCreativeCommonsAt‐
tribution(CCBY)license(http://crea‐
tivecommons.org/licenses/by/4.0/).
Water2021,13,23752of14
asitpropagatesintotheestuarinesystem[2].Therefore,tidalwavepropagationinan
estuaryisaninterestingbutdifficultissuetosolvebecauseofthenonlinearinteraction
resultingfromthecomplexestuarineboundarylinesandvariablyshallowtopography.
Tounderstandthemechanismoftidalwavepropagationalonganestuary,vastef‐
fortshavebeenmadetoinduceandsolvetheestuaryhydrodynamicmodel.Generally,
theseworkscanbedividedintotwomethods:analyticalsolutionsandnumericalsimula‐
tions.Withadvancesincomputationaltechnology,tidalwavepropagationcanbeaccu‐
ratelysimulatedbynumericalmodels[2–5].Comparedwithanalyticalsolutions,numer‐
icalmodelsarebetteratprovidinghigh‐resolutiondataundervariouscontrollingcondi‐
tions,whichcanbecomparablewiththemeasurementsinarealestuary.However,the
numericalresultsshouldbefurtherinvestigatedandextractedthroughtheanalyticalso‐
lutiontoobtainadeeperunderstandingoftheeffectbymultiplecontrollingfactorswith
therespectoftidalwavepropagation.Specifically,tofigureouttherelationshipbetween
hydrodynamicsandthegeometryoftheestuary,weneedtheanalyticalsolutionnotonly
tofitthedatabutalsotoextractsomeimportantparameterssuchasvelocitynumber,
dampingnumber,celeritynumber,andtheirrelationshipswiththegeometryparameter
oftheestuary.Thus,ananalyticalsolutionisstillanirreplaceableinstrumentinanalyzing
thepropertiesoftidalwavespropagatinginanestuary.Thusfar,arangeofanalytical
solutionsbasedon1DSaint–VenantequationshavebeenderivedbyHunt[6](1964),Ippen
[7],andPrandle[8],whichareallforaprismaticestuaryandriver.Meanwhile,thesolu‐
tionsforamorewidelyconvergentestuaryhavebeenpresentedbyGodin[9],Jay[10],
LanzoniandSeminara[11],andVanRijn[12].Mostoftheaboveinvestigatorslinearized
theequationsbymeansoftheperturbationmethodinanEulerianframework.Incontrast,
Savenije[13]derivedarelativelysimplesolutioninaLagrangianframework.Furthermore,
Savenije[14]presentedafullyexplicitsolutionofthetidalequationsbysolvingthesetof
fourimplicitequationsderivedfromtheSaint–Venantequations.Withtheadvantageof
consideringthenonlineareffectofthebottomfriction,Savenije’sanalyticalsolutioniscon‐
stantlyimprovedandwidelyusedinanalyzingthewavepropagationontypicalalluvial
estuaries[15,16].
TheHumenEstuary,oneofthemajoraccessesforatidalwavefromtheSouthChina
SeaenteringthePearlRivernetwork,hasasignificanteffectonfloodingandinundation
inGuangzhoucity,whichisthecapitalandeconomiccenterofGuangdongProvince.
However,limitedmeasureddataintheHumenEstuaryrestricttheunderstandingoftidal
wavepropagationalongtheHumenEstuaryintheupstreamdirection.Althoughsome
previousnumericalsimulationshavebeenperformed[17,18],theirresultsarenotdirect
andclearenoughtoreflectthecharacteristicsofhowacertaincontrollingparameteraf‐
fectsotherparameters.AnotherquestioniswhetherSavenije’sanalyticalsolutioncanbe
appliedtothesituationundertheinteractionoftidesandsurgesduringtyphoonlandfall.
Ifnot,whatarethechangesinthehydrodynamicsystemandwhichparametersshould
beadjustedintheanalyticalsolution?
Withthequestionsmentionedabove,theobjectivesofthestudyareasfollows:1.to
composeandvalidateamethodofanalysisthatcombinesaclassicalanalyticalsolution
andthenumericalresultsandisabletocorrectlyreflectthecharacteristicsoftidepropa‐
gationintheHumenEstuary;2.tocheckandverifyawaytoadaptthemethodofanalysis
tothecontextoftidalwaveandsurgepropagationassociatedwithtropicalcyclones;and
3.toextractinsightintothehydrodynamicfeaturesandfactorsintheHumenEstuary.
Thus,weapplySavenije’sanalyticalsolutiontotheHumenEstuaryinconjunctionwith
numericalsimulationresultstodescribeandanalyzethecharacteristicsofthetidalwave
intheHumenEstuary,PearlRiver.Aseriesoftestsareconductedtoanswertheabove
questionsinthispaper.

Water2021,13,23753of14
Thesectionsareorganizedasfollows.First,somebackgroundinformationaboutthe
HumenEstuaryinthePearlRivernetworkisintroducedinSection2,followedbyabrief
introductionofSavenije’sanalyticalsolutioninSection3.InSection4,thenumerical
modelisverified.InSection5,calibrationandverificationforastronomictidesandstorm
surgesarecarriedout.InSection6,thediscussionfocusesonthecharacteristicsoftidal
wavesintheHumenEstuary.Finally,conclusionsaredrawninSection7.
2.StudyAreas
2.1.Overview
ThePearlRivernetwork,locatedinSouthChina,deliversalargeamountoffresh
water(rangingfrom20,000m
3
/sinthewetsummerto3600m
3
/sinthedrywinter)into
thenorthernSouthChinaSeathrougheightoutlets(Figure1).Fouroftheeightoutlets
(Humen,Jiaomen,Hongqimen,andHengmen)enterLingdingyangBay,whichhasa
trumpet‐likeshapewithawidthof5kmnearthenorthernendand35kmatthesouthern
end.Amongthefouroutlets,theHumenEstuaryisthelargestmouthwithalargeamount
oftidalinfluxandoutfluxbetweentheestuaryandoffshore.Itisobviousthatthetrumpet‐
likebaytendstoamplifythetideamplitudeinconjunctionwiththeconvergentestuary
whenthetidalwavepropagatesfromthedeepseaintotheHumenEstuaryintheland‐
warddirection.Thus,wefocusonthecharacteristicsofthetidalwavealongtheHumen
Estuary,specificallythesegmentfromthemouthupstreamtoapproximately1.5km
downstreamfromHuangpuBridge.
Figure1.ThelocationoftheoutletsofthePearlRivernetwork.
Water2021,13,23754of14
2.2.ShapeoftheHumenEstuary
Theshapesofmostalluvialestuariesaresimilarallovertheworld.Thewidthand
theareaofthecrosssectiondecreasesintheupstreamdirection,resultinginaconvergent
(funnel‐shaped)estuary.Themaingeometricparametersoftheconvergentestuarycanbe
describedbyexponentialfunctionsalongtheestuaryaxiswiththeoriginatthemouth:
𝐴
𝐴
𝑒𝑥𝑝 󰇛 𝑥
𝑎󰇜(1)
𝐵𝐵𝑒𝑥𝑝 󰇛 𝑥
𝑏󰇜(2)
ℎℎ𝑒𝑥𝑝 󰇛 𝑥
𝑑󰇜(3)
whereA,B,andharethetidalaveragecross‐sectionalarea,width,anddepth;A0,B0,and
h0arethesamevariablesattheestuarymouth;xisthedistancefromtheestuarymouth;
anda,b,anddareconvergentlengthsofthecross‐sectionalarea,width,anddepth.
TheparametersofEquations(1)–(3)canbeobtainedbyregressiononthetopographic
data.Fromthedigitalelevationmodel(DEM)ofthePearlRiverDelta,54crosssections
wereextractedtoobtainthecross‐sectionalarea,width,anddepth(Figure2).Then,the
datawereusedtoderivetheconvergentlengthsa,b,anddlistedinTable1.Figure3shows
theregressionlinesplottedonsemilogarithmiccoordinates.ItshowsthattheHumenEs‐
tuarycanbedividedintothreesections:thedownstreammouthsection(x=0–8.7km),
themiddlesection(x=8.7–30.4km),andtheupstreamsection(x=30.4–36km).Themouth
sectionisnearaprismaticestuary,andthemiddleandupstreamsectionsarethetypical
convergentestuary.
Table1.ShapecharacteristicsoftheHumenEstuary.
SubsectionsRange(km)A(km)B(km)D(km)
Mouth0–8.7166.7333.3333.3
Middle8.7–30.42571.440.0
Upstream30.4–36208.3208.3
Figure2.ThedeploymentofthecrosssectionsalongtheHumenEstuary.
Water2021,13,23755of14
Figure3.Theregressionlinesforthecross‐sectionalareaA(m
2
),widthB(m),anddepthh(m)along
theestuary.
3.AnalyticalSolution
ThetidaldynamicsinanestuarycanbedescribedbythefollowingSaint–Venant
equations:
𝜕𝑈
𝜕𝑡𝑈𝜕𝑈
𝜕𝑥𝑔𝜕𝑑
𝜕𝑥𝑔𝐼
𝑔𝑛
𝑈|𝑈|
𝑑
/
0
(4)
𝑟
𝜕𝐴
𝜕𝑡𝜕𝑄
𝜕𝑥0
(5)
whereUisthetidalflowcross‐sectionalvelocity,gistheaccelerationduetogravity,dis
theflowdepth,Iisthebottomslope,nistheManningcoefficient,andQisthetidalflow
discharge.𝑟 isdefinedastheratiobetweenthestoragewidthandthestreamwidth,
whichisoftenmorethanunityintheriverwithshallowplains[14](Savenije,2008).
AfterthescalingofEquations(4)and(5)andundertheassumptionofaharmonic
wavetravelingfromtheestuarymouthtotheupstreamreach,thefollowingdimension‐
lessparametercanbeobtained:
𝛾𝑐
𝜔𝑎
(6)
𝜒
𝑟
𝑓
𝑐
𝜔ℎ𝜁
(7)
𝜇1
𝑟
𝑣ℎ
𝜂𝑐
(8)
𝜆𝑐
𝑐
(9)
𝛿1
𝜂𝑑𝜂
𝑑𝑥𝑐
𝜔
(10)
Water2021,13,23756of14
wherec0istheclassicalwavecelerityofafrictionlessprogressivewave,ωisthefre‐
quencyofthewave,𝜁
isthedimensionlesstidalamplitude,and𝑓isthedimension‐
lessfrictionfactordefinedas
𝑓
𝑔𝑛
ℎ/󰇩14
3𝜁󰇪(11)
UsingtheabovedimensionlessparametersinEquations(6)–(11),thescalingformof
Equations(4)and(5)canbesolvedonthebasisoftheequationsforthephaselag:
𝑡𝑎𝑛󰇛𝜖󰇜𝜆
𝛾𝛿(12)
𝜇𝑠𝑖𝑛 󰇛𝜖󰇜
𝜆𝑐𝑜𝑠 󰇛𝜖󰇜
𝛾𝛿(13)
𝛿 𝜇
𝜇1󰇛𝛾
𝜒
𝜇𝜆󰇜(14)
𝜆1𝛿󰇛𝛾𝛿󰇜(15)
where𝜖isthephasedifferencebetweenhighwaterandhighwaterslack.Itisclearthat,
foraprogressivewave,𝜖0.5πandthat,forastandingwave,𝜖0.Theaboveequa‐
tionsshouldbeappliedtothefollowingconditions:(1)theratiobetweenthetidalampli‐
tudeandthewaterdepthislessthanunity,(2)therunoffismuchlessthanthetidalinflux
andoutflux,and(3)theincidentwavecanbesimplifiedasaharmonicwave.Equations
(12)–(15)canbesolvedbyaniterationprocesswhen γand χhavebeendetermined.
4.NumericalModelandVerification
TheFiniteVolumeCommunityOceanModel(FVCOM),anunstructured‐gridfinite‐
volume,three‐dimensionalprimitiveequationcoastaloceanmodeldevelopedoriginally
byChenetal.[19]andupgradedbytheUMASS‐D/WHOImodeldevelopmentteam
[20,21],isappliedtosimulatethetideandsurgepropagationprocessinthePearlRiver
network.AsshowninFigure4,themodeldomaincoverstheentireregionofthePearl
Rivernetwork,eightestuaries,andthenearshore.Themodelcontains58,564nodesand
91,572elements.Horizontally,thespatialresolutionvariesfrom40to50mwithinthe
rivernetworkandfrom600to1000mneartheestuarinemouthandbayto5kmatthe
offshoreopenboundary.Tomodelthesurgeundervariousapproachingcyclonesinsum‐
merdays,thePearlRivernetworkmodelisnestedattheopenboundarywiththeSouth
ChinaSeamodel,whichisalsobasedonthesphericalcoordinateversionoftheFVCOM.
Vertically,themodelcontainsfiveuniformsigmalayerstocapturetheprocessofturbu‐
lencemomentumtransferringupwardsfromthebottom.
TheforcesdrivingthemodelincludetidalwavesfromtheSouthChinaSeaand
surgescausedbymeteorologicalforcessuchasstorms,cyclones,andrainfalls.Inthispa‐
per,weconsideronlytropicalcyclone.Thesemi‐empiricalparametriccyclonemodelis
integratedintotheoceanmodeltodrivethesurge.Theadvantageoftheparametriccy‐
clonemodelisthatitdoesnotrequireasmuchmeteorologicaldataasthedynamicmodel
todrivethemodel.Theonlydataneededcontainthetrackofthecyclone,thecenterpres‐
suredrop,theforwardspeed,andthemaximumwindradius,mostofwhichareavailable
onweatherforecastwebsitesexceptforthemaximumwindradius,whichshouldbeob‐
tainedusingastatisticalformulaandthenadjustedaccordingtothesurgeresults.Here,
thecycloneNo.1822(Table2),wellknownastheSuperTyphoonMangosteen(2018),is
usedasanexampletosimulatethestormsurgeinconjunctionwiththeastronomicaltide.
Thenumericalresultsarecomparedwiththemeasurementsanddemonstrategoodagree‐
mentinthehighsurgelevelbetweentheminFigure5.
Water2021,13,23757of14
Figure4.ThedomainandgridsofthePearlRivernetworkmodel.
(a)NanshaStation(b)WanqingshaStation
(c)HengmenStation(d)NeigangStation
Figure5.Verificationofthenumericalmodelinfourstations:Nansha,Wanqinsha,Henmeng,and
Neigang.
Water2021,13,23758of14
Table2.TheparametersoftheTyphoonMangosteen(twodaysbeforelandfall).
TimeLatitude(N) Longitude(E)Pressure(hPa)MaximumWindSpeed(m/s)
00:0014/0916126.990556.6
06:0014/0916.7125.790556.6
12:0014/0917.4124.190556.6
18:0014/0918122.390556.6
00:0015/0918120.594046.3
06:0015/0918.5119.794046.3
12:0015/0919.2118.395043.7
18:0015/0919.811795541.2
00:0016/0920.6115.396038.6
06:0016/0921.7113.596038.6
12:0016/0922.2111.697033.4
18:0016/0922.7109.798028.3
5.Results
WeusedtheHumenEstuarygeometrypresentedinTable1andthenumericalresults
on8–9September2018(aboutoneweekbeforethetyphoonlandfall)tocalibratetheana‐
lyticalmodel.Thecalibrationprocessareasfollows:wefirstdividedtheparameterrange
intosmallsubareaswithanincrementof0.1for𝑟and0.001fornwiththeirrangeslisted
inTable3.Therangeof𝑟andnwerereferredtoSavenijeetal.(2005;2008)[1,14].Then,
weusedtheseparameterstoobtainthetidalresults.Lastly,wecomparetheanalytical
solutionswiththenumericalresultsandcomputetheRMSE(rootmeansquareerror)for
everyanalyticalsolutioncorrespondingtotheparameters.TheonewiththeleastRMSE
isconsideredthebestfitsolution,anditsparametersaretheoptimalones.Thecalibration
parametersarepresentedinTable3.Ingeneral,theanalyticalsolutionfitsthenumerical
resultverywell(Figure6),andthecalibratedparametersarewithinthephysicallyproper
range.Incontrast,intheupstreamsection,thereappearstobemoredeviationthaninthe
othertwosections.Thereasonisprobablythatthelengthofthissectionistooshortto
matchamorereasonableconvergentlength.
Themodelwasfurtherverifiedwiththenumericalresultson16–17September2018.
ThereasonforchoosingthisperiodisthatSuperTyphoonMangosteenapproachedand
madelandfallonthecoastofGuangdongProvinceduringthistime.Thus,atfirst,weonly
simulatedtheastronomicaltidewithoutexertingawindforceinthenumericalmodel.
Throughthissimulation,theanalyticalresultwascomparedwiththenumericalresult
assumingnocyclonelandfall.FromFigure7,itcanbeseenthatthecorrespondencewith
thenumericalresultisgoodforthetidalamplitude.However,thedeviationforthetravel
timeshowsthattheanalyticalmodeltendstooverestimatetheceleritycomparedwiththe
numericalmodel.Asimilarproblemconcerningthedistinctaccuraciesbetweenthetidal
amplitudeandthetraveltimewasalsoencounteredbyothersusingSavenije’sanalytical
model,whoattributedthisdeviationtotheeffectofthestoragewidthratio𝑟atdifferent
tidalamplitudes(Caietal.,2012).
Then,wesimulatedthetyphoon‐surgescenariobysettingthecyclonemodelandby
exertingthewindandpressureforcesonthesurfaceofthetidalflow.Theanalyticalresult
wascomparedwiththenumericalresult.ItisshowninFigure8thatthedifferencebetween
themcanreachamaximumofasmuchas0.6m.Thisresultisunsurprisingbecausethe
originalscopeoftheanalyticalmodeldoesnotincludestormsurges.Withrespecttothe
equation,awindstresstermshouldbesupplementedinthemomentumequationasfol‐
lows: 𝜕𝑈
𝜕𝑡𝑈𝜕𝑈
𝜕𝑥𝑔𝜕𝑑
𝜕𝑥𝑔𝐼𝑔𝑛𝑈|𝑈|
𝑑
τ0(16)
Water2021,13,23759of14
whereτisthewindstressalongthex‐coordinatedirectionwithapositivedirection
fromthemouthtotheupstream.
FromEquation(16),wecanfindthattheeffectofthepositivewindstressterm,which
causedamaximumsurge,isequivalenttoareductionintheManningcoefficient:
𝑔𝑛
𝑈|𝑈|
𝑑
τ
𝑔𝑛
𝑈|𝑈|
𝑑
(17)
𝑛
𝑛
τ
𝑑
𝑔𝑈|𝑈|
(18)
where𝑛isconsideredthereducedversionoftheoriginalManningcoefficientnunder
thecyclonesurgescenarios.
Therefore,wecanmaketheanalyticalsolutionclosetothenumericalresultbytuning
Manning’scoefficienttobeless.Theanalyticallycalculatedtidalamplitudesbeforeand
aftertuningthecoefficientareshowninFigure8,andalloftheparametersarelistedin
Table3.TheManningcoefficientconsideringthecyclonesurgeissignificantlylessthan
thatonlyconsideringtheastronomicaltide.Someareevenbelowthelowerlimitofthe
physicallyreasonablerange.Thismeansthattheeffectofthestormsurgeonthetidal
dampingissimilartothereductioninthebottomfriction.Inotherwords,thewindthat
blewintheupstreamdirectionbeforelandfallofthecyclonecounteractedpartofthefric‐
tionwhenthetidalwavetraveledupstreamalongtheriver.Foranapplicationoftheso‐
lutioninasurgescenarioandaphysicallyreasonableestimateoftheManningcoefficient
inthemeantime,anewsolutionbasedonSavenije’sanalyticalmodel,whichcontainsthe
windstressterm,shouldbederivedinthefuture.
(a)(b)
Figure6.Comparisonoftheanalyticalresultsfor(a)tidalamplitudeand(b)traveltime,andthenumericalsimulation
aftercalibrationduring8–9September2018.Thediamondsdenotesthenumericalsimulationresultsandthesameforthe
followingfigures.
Water2021,13,237510of14
(a)(b)
Figure7.Comparisonoftheanalyticalresultsfor(a)tidalamplitudeand(b)traveltime,andthenumericalsimulationfor
verificationunderastronomicaltideduring16–17September2018.
Figure8.Comparisonoftheanalyticaltidalamplitudebefore(solidline)andafter(dashline)tuning
Manning’scoefficient,andnumericalsimulationforverificationundertideandsurgeduring16–17
September2018.
Table3.Parametersusedforanalyticalmodel.
SubsectionsStorageWidthRatio
r
s
ValueRangeManning’sCoefficientn(m
−1/3
s)
AstronomicalSpringTideConsideringCycloneSurgeValueRange
Mouth11–20.0050.0050.017–0.06
Middle1.81–20.0310.0180.017–0.06
Upstream 1.51–20.0350.0150.017–0.06
6.Discussion
6.1.TheCharacteristicsoftheTidalWavePropagationintheHumenEstuary
Savenije’ssolutiondescribesthevariationofthefourdimensionlessparametersϵ,
μ, δ,andλasafunctionoftheshapenumberγandthefrictionnumberχ,whichisde‐
pictedinFigure9.Asshowninthesefigures,thesolutionscanbegenerallydividedinto
twofamilies.Fortheareaswithasmallshapenumberγ(γ<2),alloftheparameters
dependonfrictionnumberχ.Thephaselagϵdecreaseswithanincreaseinχ,andthe
tidalwaveisamixtureoftheprogressivewaveandstandingwave,indicatingariverine
estuary.Incontrast,fortheareaswithalargeshapenumberγ(γ>3.5),thefrictionnum‐
berhaslittleinfluenceonthefourdimensionlessparameters.Thephaselagϵincreases
withanincreaseofχ,andthetidalwaveismuchclosertothestandingwave,indicating
anoceanicestuary.Intheintermediateareawithin2<γ<3.5,thecriticalγfortrans‐
formingfromtheprogressivewavetothestandingwaveincreaseswiththeincreasein χ.
Thehydrodynamicanalysis,aswiththegeometricanalysis,indicatesthatthereare
threedistinctsegmentsforthemainstreamoftheHumenestuary.Figure9showsthe
Water2021,13,237511of14
parametersforthethreesegmentsoftheestuary.Inthemouthsection(0–8.7km),asa
resultofthenearlyprismaticestuary,thissectionisfeaturedbytheriverineestuarydue
tothesmallγ.Moreover,thefrictioninthissectionisalsoverysmall.Boththesmallγ
and χmakethetidalwaveinthissectiontravelasaseriesofprogressivewaves,with
smallamplificationofthewaveamplitudeintheupstreamdirection.Inthemiddlesection
(8.7–30.4km),γiswithintheintermediateareaandbecomessmallupstreamwhile χbe‐
comeslargeinthesamedirection.Thesectionalcurveiscrossedanddividedintotwo
partsbytheidealrivercurve.Abovetheidealrivercurve,theeffectofcross‐sectional
convergencedominatesoverthefriction;thus,thetidalenergyisaccumulatedandthe
tideisamplifiedinthedownstreampartofthemiddlesection.Belowtheidealrivercurve,
thefrictioneffectdominatesoverthecross‐sectionalconvergence;thus,thetidalenergyis
dissipatedandthetideisdamped.Theanalysiscorrespondswiththeresultinthelast
section.Intheupstreamsection(30.4–36km),thelargeγmakesthissectionmoresimilar
toanoceanicestuary.Theconvergenceeffectissuperiortofrictionaccordingtothevalue
of
𝛿
;thus,thetideisamplified.Thewaveinthissectionistypicalofastandingwave,
whichisnearlyindependenttothefriction.Intheprocessofcalibration,thissectionis
insensitivetoManning’scoefficient.
Toinvestigatethedifferencemadebythestormsurgecomparedwiththenormal
tidalwave,wedepictedthecalibratedparametercurvesinthemiddlesectionunderthe
stormsurge.Asexpected,thecurvesunderthesurgemoveslightlytowardthezonewith
lessfrictionfromtheoriginalcurvesundertheastronomicaltide.Thisisinagreementwith
theanalysisoftheresultsintheformersection.Previousstudieshaveindicatedthatrunoff
canincreasethefrictionforwavepropagationinanestuary;however,thisstudyshowsthat
stormsurgecanreducethefrictionforthewavepropagationwhileexaggeratingthewave
amplitude.
(a)(b)

(c)(d)
Water2021,13,237512of14
Figure9.Therelationbetweenfourparameters:(a)velocitynumber,(b)dampingnumber,(c)celeritynumber,and(d)
phaselag.Theestuaryshapenumber γfordifferentfrictionnumbers χisindicatedbydifferentlinetypes.Theboldlines
indicatetheparametersfortheHumenEstuaryunderastronomicaltide(red)andtideplussurge(blue),withthenumbers
indicatingthedistancefromtheestuarymouthinkilometers.Thedrawnlinewiththecrossesrepresentstheidealestuary.
6.2.ComparisonwithOtherEstuaries
TheHumenEstuaryissocomplicatedthatitcanbedividedintothreesections.Any
sectiondemonstratesitsdistinctfeaturescomparedwiththeothers.Tobetterunderstand
theproperty,wecomparedthemwithothertypicalestuaries,whichhavebeenanalyzed
bySavenijeetal.[14](2008).Thefeatureofthemouthsection(0–8.7km)intheHumen
estuarycanbecomparablewiththeHauEstuariy,especiallytheupstreamsection(57–160
km).Theyareallriverineestuarieswithsmallshapenumbers.Thus,theyhaveariverine
characterwithalongconvergencelengthandahighphaselag.Thedifferenceisthatthe
mouthsectionoftheHumenEstuaryhaslessfrictionthantheHauEstuary.Therefore,the
tideisamplifiedfortheformeranddampedforthelatter,andthecelerityinthemouth
sectionoftheHumenEstuaryislargerthanthatoftheHauestuary.Themiddlesection(8.7–
30.4km)oftheHumenEstuaryiscomparablewiththeScheldeEstuary,withcloseshape
numberslocatedintheintermediatezone.Bothoftheircurvescrosstheidealestuarycurve,
whichindicatesatransitionpointbetweentidaldampingandamplification.Thedifference
isthatthedepthoftheHumenEstuaryisdecreasedintheupstreamdirectionandthatitis
theoppositefortheScheldeEstuary.Moreover,thelengthofthemiddleHumenestuaryis
muchlessthanthatoftheScheldeEstuary,butthecurvelengthissimilar.Thismeansthat
theparametervariationrateofthemiddleHumenEstuaryisgreaterthanthatoftheSchelde
Estuary.
(a)(b)
Figure10.PositioningoftheHumen(redline),Schelde(yellowsquares),andHau(pinktriangles)Estuariesinthedamp‐
ingnumber(a)andceleritynumber(b)diagrams,withthenumbersattheinflectionpointsindicatingthedistancefrom
theestuarymouth(inkilometers).TheotherlinesarethesameasthoseinFigure9.
7.Conclusions
Inthisstudy,Savenije’ssolutionwasappliedforthefirsttimetotheHumenEstuary.
Duetothescarcityandlowresolutionofthemeasureddata,weusedthenumericalresults
tocalibrateandverifytheanalyticalmodel.Themeritofusingananalyticalmodelisthat
itcanprovidedirectinsightintorelationshipsamongthetidalproperties,suchasvelocity
amplitude,thetidaldampingrate,andwavecelerity;thegeometryindicator;friction;and
thetidalforces.Thisanalyticalmodeldemonstratedgoodaccuracywhenitwasapplied
toanestuarywherethetidedominatesincomparisonwiththeriverdischarge.Theresults
indicatethattheanalyticalmodelcanpredicttheastronomicaltidewellintheHumen
Estuaryaftercalibration.However,itcannotpredictthetidewellwhenatropicalcyclone‐
inducedsurgeissuperimposedontotheastronomicaltide.Thereasonmaylieinthefact
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thatSavenije’ssolutiondoesnottakethewindforcesintoaccount.AfterreducingMan‐
ning’scoefficient,wefoundthattheanalyticalresultcouldbeclosetothenumericalre‐
sults.Thismeansthatthelossofthewindforcingcanpartlybecompensatedbyadjusting
thefriction.
ByanalyzingthetidalwavepropagationalongtheHumenEstuary,wefoundthat
thecharacteristicsofthethreesections—themouthsection(0–8.7km),themiddlesection
(8.7–30.4km),andtheupstreamsection(30.4–36km)—arenotalikeatall.Themouthsec‐
tionisatypicalriverineestuarywithanearlyconstantcrosssection.Thetidalwavetrav‐
elingthereisintheformofaprogressivewaveandwithanearlyclassicalcelerityina
frictionlessstate.Incontrast,theupstreamsectionisatypicaloceanicestuarywithashort
convergencelength.Thetidalwavethereisintheformofastandingwavewithanearly
infinitecelerity.Themiddlesectionisintheintermediatezoneofthetwosolutionfami‐
lies.Thetidalwaveisfirstamplifiedandthendamped,withatransitionpointthatisthe
locationofthemaximumtidalamplitude.Finally,acomparisonwasconductedbetween
theHumenEstuaryandotherestuaries.
AuthorContributions:Conceptualization,Z.Z.andF.G.;methodology,Z.Z.;validation,Z.Z.;for‐
malanalysis,D.Z.;investigation,Z.Z.;resources,D.H.;datacuration,D.H.;writing—originaldraft
preparation,Z.Z.;writing—reviewandediting,F.G.;visualization,D.Z.;supervision,D.Z.;project
administration,F.G.;fundingacquisition,F.G.Allauthorshavereadandagreedtothepublished
versionofthemanuscript.
Funding:ThispaperwassupportedbytheNationalKeyR&DProgramofChina(grantNo.
2018YFB0505500and2018YFB0505502)andbytheNationalNaturalScienceFoundationofChina
(grantNo.41771421,41771447,and41571386).
InstitutionalReviewBoardStatement:Notapplicable.
InformedConsentStatement:Notapplicable.
DataAvailabilityStatement:Noreportanydata.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
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