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On he assessmen o channel deepening impac s in mic o-meso idal
es ua ies: A sys ema ic analysis
Guille mo Ma ín-Llanes ∗, Alejand o López-Ruiz
Depa amen o de Ingenie ía Ae oespacial y Mecánica de Fluidos, Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092, Se ille, Spain
ARTICLE INFO
Da ase link:Del 3D - FLOW and ba hyme y
iles (O iginal da a)
Keywo ds:
D edging
Tidal p opaga ion
Sal in usion
Basin managemen
Es ua y
ABSTRACT
The need o e icien ma i ime anspo a ion in es ua ies has led o he de elopmen o di e se d edging
s a egies o accommoda e essels wi h deep d a s. Mos ecen s udies assessing he en i onmen al impac s o
channel deepening use ad anced, ailo ed models o simula e he long- e m esponse o his o ical ba hyme ic
changes in es ua ies wo ldwide. Howe e , hese models a e o en ime-consuming and highly speci ic o
local condi ions, limi ing he b oade applicabili y o hei esul s. In addi ion, a common limi a ion is he
signi ican ime gap be ween he ba hyme ic da a used, o en exceeding 100 yea s. This makes i challenging
o quan i y he e ec s o isola ed deepening ope a ions, which is essen ial o unde s anding he in luence o
human in e en ion on es ua ine dynamics. To o e come his limi a ion while ensu ing e icien and adap able
modelling, his pape p esen s a h ee-dimensional idealised model (Del 3D) o quan i y he sho - e m, e.g.,
weeks, hyd odynamic and salini y esponse o d edging ope a ions in mic o-meso idal, well-mixed es ua ies.
Implica ions on channel ope a i i y a e also discussed. The nume ical expe imen s examine a ia ions in bo h
channel dep h and d edging leng h. Key indings sugges ha d edging leng h is c i ical in he es ua ine
esponse. Speci ically, d edging leng h has a g ea e in luence on idal ampli ica ion han channel dep h.
Changes in he low s uc u e a e p ima ily d i en by changes in he ba o opic p essu e g adien and bed
shea o ces, which a y spa ially along he es ua y, de ining h ee dis inc egions o beha iou . In addi ion,
sal in usion inc eases linea ly wi h channel dep h and becomes pa icula ly sensi i e o d edging leng h in
sho e ope a ions. Rega ding basin managemen , esul s e eal ha landwa d ope a i i y is comp omised by
d edging in he lowe i e .
1. In oduc ion
Es ua ies cons i u e e ile and densely popula ed a eas (Sy i ski
and Sai o,2007) ha p o ide mul iple social, economic and en i-
onmen al bene i s (O on e al.,2015). In ecen decades, es ua ies
ha e been inc easingly a ec ed by mul iple p essu es om clima e
change and human ac i i ies, such as i e egula ion and changes
in i e mo phology (Mi chell e al.,2015;Siemes e al.,2024). In
pa icula , he expansion o ma i ime anspo a ion has accele a ed he
de elopmen o channel deepening p ojec s o imp o e na iga ional e -
iciency. These modi ica ions ha e nega i ely al e ed sal and sedimen
anspo (Rals on and Geye ,2019;Reid e al.,2022), impac ing wa e
quali y and ecosys em composi ion. In esponse o hese challenges,
ex ensi e esea ch has ocused on h ee key a eas: (1) analysing he
cu en consequences o his o ical ba hyme ic changes (Rals on e al.,
2019;Van Ma en e al.,2015), (2) assessing he ulne abili y and
en i onmen al isks associa ed wi h u u e d edging ope a ions (Bai
e al.,2003;Alba e al.,2014;Gómez e al.,2014;Paa lbe g e al.,
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (G. Ma ín-Llanes), [email p o ec ed] (A. López-Ruiz).
2015;Ál a ez e al.,2017;Za zuelo e al.,2019;He e al.,2024) and
(3) de eloping mi iga ion and adap a ion s a egies (O on e al.,2015;
Li e al.,2016;Hoagland e al.,2020;Hend ickx e al.,2024).
Rega ding he i s ques ion, changes in es ua ine dynamics due o
channel deepening a ise om he esponse o he ide and he i e low
o changes in he channel dep h. Channel deepening al e s he s uc u e
o ides, ypically esul ing in idal ampli ica ion (DiLo enzo e al.,
1993;Rals on e al.,2019), inc eased cu en ampli udes (Si ien e
e al.,2023), and highe wa e cele i y (Zhang e al.,2021), while also
in luencing idal asymme y (Win e we p and Wang,2013;Win e we p
e al.,2013). Addi ionally, d edging can modi y he es ua ine esponse
o i e loods and s o m su ges, educing peak wa e le els du ing i e
loods bu acili a ing he u he p opaga ion o su ge wa es (Rals on
and Geye ,2019;Bao e al.,2022). Fu he esea ch in Tampa Bay (Zhu
e al.,2015;Meye s e al.,2017), he Seine Es ua y (G asso and
Le Hi ,2019), and he San os Es ua y Sys em (Reid e al.,2022)
sugges ha changes in he i e - ide in e ac ions lead o an inc ease in
h ps://doi.o g/10.1016/j.ocemod.2025.102552
Recei ed 31 Decembe 2024; Recei ed in e ised o m 14 Ma ch 2025; Accep ed 12 Ap il 2025
Ocean Modelling 196 (2025) 102552
A ailable online 25 Ap il 2025
1463-5003/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license ( h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/ ).
G. Ma ín-Llanes and A. López-Ruiz
es ua ine ci cula ion. Addi ional insigh s a e p o ided by Chan e al.
(2018), whose esea ch on Newa k Bay highligh s ha he impac o
channel deepening on exchange low is de e mined by he sensi i i y
o ho izon al salini y g adien s o channel dep h 𝐻, which, in u n,
depends on he d edging leng h. In pa icula , when he d edging
leng h is sho ela i e o he idal wa e leng h, he ho izon al salini y
g adien is insensi i e o he change in channel dep h, and he exchange
low is p opo ional o 𝐻3, consis en wi h he heo y o Hansen
and Ra ay J . (1966). An inc ease in exchange low enhances he
landwa d anspo o salini y and sedimen , leading o g ea e sal
in usion (Chen e al.,2019;Rals on and Geye ,2019;Reid e al.,2022;
Zhao e al.,2022), ele a ed suspended sedimen concen a ions, and
inc eased u bidi y (Talke e al.,2009;Rals on e al.,2012;De Jonge
e al.,2014;Van Ma en e al.,2015;Eidam e al.,2021,2022).
Hence, p ecisely quan i ying he e ec s o channel deepening on
es ua ine hyd odynamics is essen ial o assessing i s impac on salini y
and sedimen concen a ions, bo h o which play a c i ical ole in
de e mining ecological heal h and wa e quali y. Howe e , much o he
exis ing li e a u e elies on analy ical o semi-analy ical models (Talke
e al.,2009;Cai e al.,2012a,b;Di Risio e al.,2017) and obse -
a ional da a s udies (Chan e al.,2018;Bao e al.,2022), which
o en lack p ecision (Zhang e al.,2011) and a e unable o accu a ely
ep esen complex es ua ine sys ems (Win e we p and Wang,2013).
In con as , nume ical modelling eme ges as a powe ul al e na i e o
assessing he e ec s o channel deepening (Bai e al.,2003;Alba e al.,
2014;Gómez e al.,2014;Paa lbe g e al.,2015;Van Ma en e al.,
2015;Chen e al.,2019;Zhao e al.,2022;Yi e al.,2024). Among
hei ad an ages, nume ical models enable he e alua ion o spa ially
a iable isks associa ed wi h human in e en ions (Familkhalili and
Talke,2016;Meye s e al.,2017;Familkhalili e al.,2020) and o
o e come linea isa ion assump ions ha a e no accep able o he
analysis o s ong non-linea p ocesses, such as idal asymme y (Guo
e al.,2014). Howe e , hese s udies o en ace h ee key limi a ions:
(1) he signi ican compu a ional ime equi ed o high-de ail models,
(2) hei s ong dependence on local condi ions, which es ic s he
gene alisa ion o esul s o o he es ua ine sys ems, and (3) he ex en-
si e ime gap in he ba hyme ic da a used o simula e he es ua ine
esponse o channel deepening (Meye s e al.,2017;Rals on e al.,
2019;Familkhalili e al.,2020;Eidam e al.,2021,2022). This ime
gap, o en exceeding 100–150 yea s, makes i challenging o isola e
he speci ic e ec s o channel deepening om o he en i onmen al
and an h opogenic changes ha may ha e occu ed o e he same
pe iod (Reid e al.,2022). Fo ins ance, Eidam e al. (2021) sugges ed
ha se e al mo phological modi ica ions, such as land eclama ion
and he disposal o d edged ma e ial wi hin he es ua y, ook place
du ing he analysed ime ame, u he complica ing he a ibu ion o
obse ed changes o d edging alone. Addi ionally, while many s udies
ha e ocused on long- e m ends, he sho - e m esponse o indi idual
d edging ope a ions ha e ecei ed ela i ely li le a en ion (Vellinga
e al.,2014).
In his con ex , his pape de elops a h ee-dimensional idealised
model (Del 3D) o isola e he sho - e m hyd odynamic and salin-
i y esponse o d edging ope a ions. Nume ical expe imen s assess
he in luence o changes in channel dep h and d edging leng h, a
ac o ha has ecei ed limi ed a en ion (Chan e al.,2018). Speci -
ically, 15 d edging scena ios a e simula ed in a mic o-meso idal,
well-mixed es ua y (Ma ín-Llanes and López-Ruiz,2024) o quan i y
changes in idal s uc u e and esidual low, which a e u he analysed
h ough momen um balance a ia ions. The e ec s o hese hyd ody-
namic al e a ions on sal in usion and channel ope a i i y (i.e., he
ime ac ion du ing which wa e le el is highe han a ce ain essel
d a ) a e also discussed. By using an idealised modelling app oach, he
p oposed me hodology elimina es ba hyme ic i egula i ies and local
e ec s, he eby imp o ing he applicabili y o indings while educing
compu a ional ime. This makes i pa icula ly ele an o es ua ine
po sys ems subjec ed o mic o- and meso- idal es ua ies, ypically
ound in Eu opean egions (Ga el and D’Alimon e,2017;Díez-Mingui o
e al.,2012), p o iding aluable insigh s o po managemen and he
po en ial en i onmen al isks associa ed wi h d edging ac i i ies.
The con en o he pape is o ganised as ollows. Sec ion 2desc ibes
he physical domain and he nume ical amewo k. The nume ical ex-
pe imen s ep esen ing he baseline and d edging scena ios a e de ined
in Sec ion 3. Sec ion 4analyses he impac o d edging in es ua ine
hyd odynamics and sal in usion. The implica ions o he ob ained
esul s on he managemen o na iga ional channels a e discussed in
Sec ion 5. Finally, he main conclusions a e p esen ed in Sec ion 6.
2. Ma e ials and me hods
2.1. Physical domain
The physical domain in which he nume ical model is implemen ed
ollows he idealised geome y o mula ion (Sa enije,2012), which is
ep esen ed by a i e wid h (𝐵) ha a ies exponen ially along he
longi udinal, linea axis as:
𝐵(𝑥) =𝐵0exp (−𝑥
𝑏)(1)
whe e 𝐵0is he mou h wid h, 𝑥 ep esen s he axial (along-channel)
dis ance (posi i e landwa d) om he mou h o he ups eam sec ion
o he es ua y a 𝑥=𝐿, and 𝑏is he con e gence leng h.
Acco ding o Eq. (1), he geome y o he es ua y is ob ained using
𝐵0= 1000 m,𝑏= 157 k mand 𝐿= 80 k m(Fig. 1a). This geome y
ep esen s an es ua y wi h mode a e con e gence acco ding o Ma ín-
Llanes and López-Ruiz (2024), whe e mo e de ails on he geome ic
cha ac e is ics can be ound. The es ua y halweg 𝐻𝑚𝑎𝑥(𝑥)and he
coas al shel a e de ined wi h cons an slopes o 𝑆= 2.5 × 10−5 and 𝑆𝑛=
2.5 × 10−3, espec i ely. Hence, conside ing he a e age wa e dep h a
he mou h 𝐻0= 10 m, a e age dep hs o 𝐻𝑟= 8 mand 𝐻𝑛= 25 ma e
ob ained in he ups eam and o sho e bounda ies, espec i ely. The
c oss-channel geome y is de ined ollowing a Gaussian shape:
𝐻(𝑥, 𝑦) =𝐻𝑚𝑎𝑥(𝑥) exp (−𝑦2
2𝑐2)(2)
whe e 𝑦is he c oss-channel coo dina e, which a ies om −𝐵(𝑥)∕2 o
𝐵(𝑥)∕2. The shape o he Gaussian is de ined using a s anda d de ia ion
o 𝑐= 152 m, which esul s in a c oss-channel a e age slope o 0.04.
2.2. Model desc ip ion and se up
2.2.1. Model desc ip ion
Hyd odynamics and sal anspo in he es ua y a e ob ained using
he Del 3D model (Lesse e al.,2004). This modelling amewo k has
been widely applied in ecen s udies on mo phodynamics, hyd ody-
namics, and sal anspo (Ma y -Kolle e al.,2017;Ruiz-Reina and
López-Ruiz,2021;Za zuelo e al.,2021), pa icula ly in esponse o
clima e change and human in e en ions (Mulligan e al.,2019;Yin
e al.,2019;Wu e al.,2021;A e alo e al.,2022), including channel
deepening (Alba e al.,2014;Guo e al.,2014;Paa lbe g e al.,2015;
Van Ma en e al.,2015;Za zuelo e al.,2015,2019;Reid e al.,2022;
Zhao e al.,2022). The hyd odynamic FLOW-module o he Del 3D
model is used o sol e he uns eady shallow wa e equa ions in 3D
and he ad ec ion–di usion equa ion o sal anspo coupled o a
u bulence closu e model. Fu he de ails o hese equa ions a e ound
in Ma ín-Llanes and López-Ruiz (2024). A sigma-laye scheme wi h a
cons an numbe o laye s is used o he e ical disc e isa ion.
Ocean Modelling 196 (2025) 102552
2
G. Ma ín-Llanes and A. López-Ruiz
Fig. 1. Model se up and nume ical expe imen s. (a1) Es ua y ba hyme y. (a2) De ail o he ed a ea highligh ed in (a1). (b) C oss-channel geome y o he di e en dep h
inc eases. (c) Thalweg p o iles a e d edging ope a ions ex ending 15 km (𝛥𝐻 = 15,30,40%). (d) Thalweg p o iles a e d edging ope a ions wi h 𝛥𝐻 = 40% (d =5, 10, 15, 20,
30 km). Black dashed line in (c, d) indica es mean wa e le el.
2.2.2. Model se up
The model se up consis s o a egula g id co e ing bo h he es ua y
and he shel . G id esolu ion a ies om 854 ×328 m2nea he
o sho e and c oss-sho e bounda ies o 109 ×24 m2nea he mou h
and a he ups eam sec ions o he i e . To ensu e adequa e e ical
esolu ion, he ini ial 𝜎-laye scheme (Ma ín-Llanes and López-Ruiz,
2024), consis ing o 10 e ical laye s wi h hickness a ying acco ding
o local dep h, was assessed agains a 20-laye scheme. In pa icula ,
ha monic analysis o wa e le els and eloci ies was conduc ed along
he channel. Addi ionally, axial salini y p o iles a neap and sp ing
ides and he empo al e olu ion o sal in usion we e analysed o
bo h e ical esolu ions. The 10-laye scheme demons a ed su icien
accu acy in all sensi i i y es s and was hus adop ed o his s udy.
Fu he de ails on he laye sensi i i y analysis a e p o ided in Sec ion
1 o he Supplemen a y Ma e ial. The ime s ep, which ensu es s abili y
and accu acy, gi en he spa ial g id esolu ion and dep hs, is 6 s.
Bounda y condi ions a e imposed in ou open bounda ies. The
o sho e bounda y p esc ibes an as onomical wa e le el o cing wi h
wo semi-diu nal componen s: (1) 𝑀2(𝐴= 1.00 m;𝜙= 180◦) and (2)
𝑆2(𝐴= 0.25 m;𝜙= 90◦). Hence, he es ua y is conside ed mic o-
meso idal. A he c oss-sho e bounda ies, a Neumann- ype condi ion
is imposed wi h a ze o longsho e wa e le el g adien (Roel ink e al.,
2004). Finally, a o al discha ge o 𝑄𝑟= 150 m3s−1 wi h uni o m
e ical dis ibu ion o eloci ies is conside ed in he ups eam sec ion
o he es ua y. This alue is es ablished acco ding o he mean idal
p ism (6.47 × 107m3) and he pe iod o he main idal ha monic o
ep oduce well-mixed condi ions, i.e., wi h a Can e -C eme s Numbe
below 0.1, acco ding o Dye (1973). The de ini ion o he physical pa-
ame e s used in he model can be ound in Sec ion 2 o Supplemen a y
Ma e ial. The model is se a he equa o and hen he Co iolis e ec s
a e neglec ed.
3. Nume ical expe imen s
The nume ical expe imen s comp ise 15 simula ions whe e he es-
ua y desc ibed in Sec ion 2and Fig. 1a (Scena io 0) is modi ied by
a ying he channel dep h (see Table 1) while p ese ing he c oss-
sec ional shape. Speci ically, h ee di e en dep h inc eases o 𝛥𝐻 =
15,30,40% (Fig. 1b,c) a e conside ed, ela i e o he o iginal mou h
dep h. These alues con o m a ep esen a i e sample o common dep h
inc eases obse ed in di e en es ua ies a ec ed by d edging ope a-
ions, which usually ange be ween 10 − 40% (Chan e al.,2018;Rals on
e al.,2019;Rals on and Geye ,2019;Amo im e al.,2023). The in-
c eased dep h ex ends o e i e di e en leng hs: 𝑑= 5,10,15,20,30 k m
(Fig. 1d), which co espond o a ac ion o he maximum sal in usion
in Scena io 0 (0.35, 0.7, 1, 1.35 and 2, espec i ely). F om his poin ,
a cons an ansi ion slope o 1∕200 is applied o connec he d edged
plane wi h he o iginal channel ba hyme y. Since his slope emains
ixed ac oss all 15 ope a ions, he ex en o he ansi ion zone a ies
depending on each d edging scena io.
Simula ions a e se conside ing he bounda y condi ions desc ibed
in Sec ion 2.2.2. Ini ial condi ions o each scena io we e ob ained
om p io spin-up simula ions wi h iden ical ex e nal o cing, ensu ing
s eady-s a e hyd odynamic and anspo condi ions. The ime ame
o he nume ical expe imen s is 15 days, so ha di e ences in he low
s uc u e due o neap-sp ing cycles a e cap u ed.
Ocean Modelling 196 (2025) 102552
3
G. Ma ín-Llanes and A. López-Ruiz
Table 1
Summa y o nume ical expe imen s.
Dep h inc ease
𝛥𝐻 [%]
D edging leng h
𝑑[km]
Scena io 0 0 0
Scena io 1 15 5
Scena io 2 15 10
Scena io 3 15 15
Scena io 4 15 20
Scena io 5 15 30
Scena io 6 30 5
Scena io 7 30 10
Scena io 8 30 15
Scena io 9 30 20
Scena io 10 30 30
Scena io 11 40 5
Scena io 12 40 10
Scena io 13 40 15
Scena io 14 40 20
Scena io 15 40 30
4. Resul s: impac o channel deepening on hyd odynamics and
salini y dis ibu ion
4.1. Hyd odynamics and salini y dis ibu ion in he unal e ed es ua y
This sec ion examines hyd odynamics and salini y dis ibu ion a
he unal e ed scena io (Scena io 0). The ide is i s cha ac e ised
h ough he ha monic analysis o wa e le el and eloci y a 10 poin s
along he channel, using he _ ide MATLAB package (Pawlowicz e al.,
2002). Resul s a e shown in Fig. 2. Rega ding wa e le el ampli udes
(Fig. 2a), he damping o he semi-diu nal componen 𝑀2is signi ican
up o hal o he channel (𝑥= 40 k m); a e his sec ion, he highe
in luence o he i e discha ge compensa es he damping o he idal
wa e and leads o a synch onous beha iou wi h wa e le el ampli udes
app oaching 0.4 m. Reduced ex e nal o e ide 𝑀4ampli udes a e
ob ained along he channel, wi h a maximum o 0.1 m in he ups eam
sec ion. Tidal cu en s a he mou h o he es ua y a e shown in Fig. 2b.
Maximum alues du ing he lood (posi i e landwa d) each 1.1 and
0.6 ms−1 du ing sp ing and neap ide cycles, espec i ely. On he o he
hand, maximum cu en s du ing he ebb (nega i e seawa d) each 1.6
and 1.2 ms−1 in sp ing and neap ide cycles, espec i ely. These alues
dec ease mono onically along he channel (Fig. 2c); in he ups eam
sec ion, he ampli ude o he semidiu nal componen 𝑀2is less han
one i h o he alue a he mou h.
Tidal in e ac ions wi h ba hyme y and i e discha ge lead o
idal asymme y (LeBlond,1991;Pa ke ,1991), which is e lec ed in
an imbalance be ween lood and ebb du a ions and maximum low
eloci ies (Guo e al.,2014). The s eng h o he asymme y is measu ed
by he ampli ude a io be ween he o e ide 𝑀4and he semidiu nal
componen 𝑀2while he di ec ion o he esul ing ne anspo is a
unc ion o he phase lag be ween hese wo componen s. The esul s
in Fig. 2d, show an inc easing ebb idal asymme y along he channel
(2𝜙𝑀2−𝜙𝑀4≈ 350◦), enhanced by he landwa d dep h educ ion, which
is pa icula ly no o ious du ing neap ides (Fig. 2b), when maximum
ebb eloci ies a e wice hose achie ed a he lood. This ebb dominan
beha iou is common in s ongly dissipa i e ide domina ed es ua ies
a ec ed by i e discha ge (Lanzoni and Semina a,1998). Finally, he
phase lag be ween wa e le el and eloci ies (Fig. 2e) inc eases along
he channel om 0.93 h a he mou h o 2.79 h a he ups eam sec ion.
Consequences o he anspo o conse a i e subs ances (e.g. sal
anspo ) de i ed om his esul will be signi ican when he ole
o he i e - ide ela ion is ele an , i.e. when i e discha ge changes
wi h a ime scale close o he idal pe iod a e analysed.
Ho izon al salini y dis ibu ion in he unal e ed es ua y is quan i ied
in e ms o sal in usion, i.e., he landwa d spa ial bounda y whe e he
salini y o he bo om laye is educed o 1 psu. This magni ude changes
pe iodically in ela ion o he ide, eaching a maximum alue o 15
km, app oxima ely, a sp ing ide. Fo he idal and i e discha ge
condi ions de ined in Sec ion 2.2.2, he wa e column is well-mixed.
The esul s indica e ha he es ua y is ide domina ed, exhibi ing
p onounced ebb idal asymme y. In consequence, he hyd odynamic
esponse and he anspo changes due o d edging will be mainly
de e mined by changes in he idal p ope ies. The ide can be modi ied
by changes in i e discha ge, con e gence o i e ma gins, channel
dep h and ic ional e ec s. D edging will di ec ly modi y he channel
dep h and ic ional e ec s, bu will also induce changes in he i e -
ide ela ionship and he ole o con e gence, all o which de e mine
es ua ine dynamics in he new con igu a ion.
4.2. Tidal p opaga ion and asymme y
The impac o channel deepening on hyd odynamics is i s as-
sessed by analysing he a io o he 𝑀2ampli ude (𝜂) and phase (𝜖)
be ween he d edged scena ios and he unal e ed es ua y along he
channel (Fig. 3). The educ ion o ic ion leads o idal ampli ica ion
(Fig. 3a,c,e) and a gene al dec ease in he idal phase (Fig. 3b,d, ).
Changes a e p opo ional o he d edging leng h and dep h inc ease,
wi h he la e ha ing a g ea e in luence on idal ampli ica ion. As
shown in Fig. 3e, he maximum ampli ude is eached o a 40%
dep h inc ease and a d edging leng h o 30 km, esul ing in a idal
ampli ica ion o 60%. In e ms o idal phase, he educ ion in bo om
ic ion due o d edging inc eases he idal wa e cele i y. The minimum
phase a io is hus ob ained o scena io 15 and esul s in 93.5% (i.e. a
educ ion in a el ime o 0.178 h a e d edging), as shown in Fig. 3 .
In con as , when analysing idal p opaga ion ac oss di e en d edg-
ing leng hs o a gi en dep h inc ease (e.g. o 𝛥𝐻 = 30% in Fig. 3c,d),
wo di e en esul s a e obse ed. Fi s , idal ampli ica ion and phase
educ ion inc ease along he d edged sec ion o he channel, eaching
hei peak a 𝑥=𝑑. Howe e , he d edging leng h ha p oduces he
highes idal ampli ica ion a a gi en loca ion a ies depending on ha
loca ion posi ion. Fo example, a 𝑥= 17 k m he g ea es ampli ica ion
occu s o 𝑑= 20 k m, whe eas a 𝑥= 7 k m, he maximum is obse ed
o 𝑑= 10 k m. On he o he hand, in he unal e ed sec ion o he
channel, i.e. o 𝑥 > 𝑑, idal ampli ica ion and phase educ ion dec ease,
app oaching an asymp o ic alue ha di e s om he co esponding
alue in he unal e ed es ua y. Asymp o ic idal ampli ude inc eases
wi h d edging leng h and dep h inc ease, while he opposi e is obse ed
o he idal phase. Consequen ly, e en when d edging is limi ed o a
small ac ion o he channel, i induces signi ican al e a ions in he
idal s uc u e h oughou he en i e es ua y: wi h he maximum dep h
inc ease and he longes d edging leng h, idal ampli ica ion eaches
nea ly 40% in he ups eam sec ion o he channel. This beha iou was
obse ed by Rals on and Geye (2019) in he Hudson Ri e es ua y,
whe e a local change in channel dep h a he mou h a ec ed 100 km
ups eam.
Changes in idal asymme y a e d edging along he es ua y a e
quan i ied in Fig. 4. This pa ame e is ep esen ed as he a io be ween
he ampli ude o he o e ide ha monic 𝑀4and ha o he p incipal
idal cons i uen 𝑀2. In e ms o wa e le el, channel deepening esul s
in a educ ion o he 𝑀4ampli ude (while inc easing he 𝑀2ampli ude,
as seen in Fig. 3), leading o an o e all dec ease in idal asymme y
along he channel compa ed o he unal e ed es ua y (Fig. 4a,d,g).
This e ec becomes mo e p onounced wi h bo h he inc ease in dep h
and he d edging leng h, achie ing a maximum educ ion o 45% a
𝑥= 33 k m o 𝛥𝐻 = 40% and 𝑑= 30 k m(scena io 15, Fig. 4g).
Howe e , d edging does no al e he o e all end in he 𝑀4∕𝑀2
a io along he channel, which inc eases om he es ua y mou h up
o 𝑥= 33 k m, dec eases un il 𝑥= 49.5 k mand inc eases again in
he emaining sec ion o he es ua y. The inc easing asymme y up o
𝑥= 33 k mag ees he hyposynch onous beha iou (dec easing idal
ampli ude) o he es ua y seen in Fig. 2a. In his egion, wo dis inc
linea slopes a e obse ed. The i s slope, which is smalle , co esponds
Ocean Modelling 196 (2025) 102552
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G. Ma ín-Llanes and A. López-Ruiz
Fig. 2. Hyd odynamics in he unal e ed es ua y. (a) 𝑀2and 𝑀4wa e le el ampli udes along he channel. (b) Su ace cu en s a he es ua y mou h a neap (NT) and sp ing (ST)
ide. (c) 𝑀2and 𝑀4cu en ampli udes along he channel. (d) Tidal asymme y (s eng h). (e) Phase lag be ween cu en and ele a ion. ( ) Tidal asymme y (di ec ion).
Fig. 3. Ampli ude and phase a ios along he es ua y. (a),(c),(e) Tidal ampli ude a io. (b),(d),( ) Tidal phase a io. Each ow co esponds o a di e en dep h inc ease. Each cu e
co esponds o a di e en d edging leng h. Subindex 0 indica es magni udes a he unal e ed es ua y.
o he d edged s e ch. Since he 𝑀2ampli ude is highe nea he
mou h and he o e ide is small, idal asymme y inc eases sligh ly in
he d edged a ea. In addi ion, mo e signi ican di e ences be ween he
unal e ed and d edged scena ios a e obse ed in he hyposynch onous
ame compa ed wi h he emainde o he channel. Since bo h idal
ampli udes and idal ampli ica ion a e smalle in he uppe es ua y,
changes in idal asymme y a e mainly a ibu ed o he ha monic 𝑀2,
i.e. d edging leads o mo e symme ic cycles (compa ed o he unal-
e ed scena io) when idal ampli udes a e highe and less symme ic
cycles when idal ange is smalle .
Tidal cu en asymme y is quan i ied by he a io 𝑈𝑀4∕𝑈𝑀2. As
shown in Fig. 4b,e,h, he magni ude o idal asymme y is educed
h oughou he es ua y p opo ionally o bo h he inc ease in dep h
and he d edging leng h. In addi ion, conside ing he phase lag be-
ween bo h ha monics, Fig. 4c, ,i shows ebb dominance in all scena ios
(2𝜙𝑀2−𝜙𝑀4≈ 350◦). Bo h esul s imply ha d edging leads o a
weake ebb asymme y, compa ed o he unal e ed es ua y. This is
consis en wi h he esul s om Win e we p and Wang (2013), who
obse ed highe lood dominance a e d edging. As he ebb dom-
inance is educed, he idally a e aged seawa d anspo o wa e
Ocean Modelling 196 (2025) 102552
5

G. Ma ín-Llanes and A. López-Ruiz
Fig. 4. Tidal asymme y along he channel. (a),(d),(g) S eng h o wa e le el asymme y. (b),(e),(h) S eng h o cu en s asymme y. (c),( ),(i) Di ec ion o cu en s asymme y.
Each ow co esponds o a di e en dep h inc ease. Each cu e co esponds o a di e en d edging leng h, as indica ed in Fig. 3. Ligh blue dashed line indica es magni udes in
he unal e ed es ua y.
will be weake in he new con igu a ions, consequen ly enhancing he
landwa d anspo o sal (Chen e al.,2019;Zhao e al.,2022) and
suspended sedimen (Van Ma en e al.,2015). This beha iou is u he
analysed in he ollowing sec ions.
4.3. Along-channel esidual low s uc u e
A u he analysis o desc ibe he wa e anspo in ol es he
e alua ion o he esidual ( ide a e aged) uni wid h wa e lux (RUWF)
(Chen e al.,2019):
RUWF =1
𝑇∫𝑇
0

𝑄 𝑑 𝑡(3)
whe e 
𝑄is he ins an aneous a e o wa e anspo pe uni wid h
h ough he wa e column, 𝑇is he 𝑀2 idal pe iod, and 𝑡is ime. The
di e ence in he RUWF be ween he d edged scena ios wi h 𝛥𝐻 = 40%
and he unal e ed es ua y is shown in Fig. 5.
D edging behind he sal in usion limi inc eases he exchange
low, de ined as he di e ence be ween he ne bo om in low and
ne su ace ou low, in he d edged a ea, and he ne ou low in he
unal e ed pa o he es ua y (Fig. 5a,b,c), leading o an inc ease in
sal in usion. In con as , when he d edging leng h is g ea e han
he sal in usion leng h (Fig. 5d,e), a di e en beha iou is ob ained
be ween he esidual sal in usion limi and he ansi ion slope, whe e
negligible esidual lux di e ence is obse ed. Since esidual cu en s
a e di e en om ze o in bo h he unal e ed and d edged scena ios,
esul s indica e equi alen idally a e aged hyd odynamic condi ions
in bo h cases. Inc easing he d edging leng h esul s in g ea e sal
in usion, a sligh educ ion in he exchange low a iance wi hin he
d edged egion, and an enhanced ou low in he unal e ed sec ion o
he channel. On he o he hand, he magni ude o he exchange low
and he sal leng h inc ease p opo ionally wi h he dep h inc emen .
In pa icula , o 𝑑= 5 k m he ne in low is inc eased by 0.08 m3s−1,
0.15 m3s−1 and 0.2 m3s−1 o 𝛥𝐻 = 15,30 and 40%, espec i ely. Fu he
de ails on he impac o sal in usion a e p esen ed in Sec ion 4.5.
To be e unde s and he unde lying mechanisms d i ing changes
in he esidual low s uc u e, a de ailed analysis o idal cu en s
is conduc ed. Speci ically, a ep esen a i e c oss-sec ion om each
o he h ee egions iden i ied is selec ed, and he di e ence in he
e ical dis ibu ion o he axial cu en ( halweg) be ween Scena io 14
and he unal e ed es ua y is examined o e wo idal cycles (Fig. 6).
Rep esen a i e sec ions a e loca ed a 𝑥= 7.5 k m,𝑥= 18 k m,𝑥=
22.5 k m, espec i ely. In he es ua ine ci cula ion egion (Fig. 6a),
which co esponds o he d edged a ea below he sal in usion limi ,
he cu en di e ence shi s sign be ween he lood and ebb ides. On
he one hand, a nega i e di e ence is obse ed a lood ide. Since
hese cu en alues a e posi i e, his esul indica es a educ ion in
he lood cu en h oughou he en i e wa e column a e d edging.
The maximum nega i e di e ence (−0.25 ms−1) is obse ed be ween he
maximum lood and he high wa e slack (HWS) a he su ace, and i
dec eases g adually owa ds he bo om. On he o he hand, a posi i e
di e ence is obse ed du ing he ebb ide, indica ing a dec ease in
he ebb cu en h oughou he wa e column, excep a he su ace
du ing he maximum ebb. Maximum posi i e alues (+0.20 ms−1) a e
ob ained be ween maximum ebb and low wa e slack (LWS) a he
bo om, dec easing owa ds he su ace. As a esul , ime a e aging
leads o a ne ou low a he su ace and a ne in low a he bo om,
as shown in Fig. 5.
Conside ing he d edged a ea be ween he sal in usion limi and
he ansi ion slope (Fig. 6b), he same beha iou is obse ed, wi h
nega i e alues mainly du ing he lood, and posi i e alues mainly
du ing he ebb. Howe e , he e ical dis ibu ion o he axial cu en
di e ence is uni o m, and he maximum posi i e and nega i e alues
a e equi alen . This means ha lood and ebb cu en s dec ease by
he same amoun h oughou he wa e column, wi h he magni ude
o he change a ying depending on he idal phase. The maximum
nega i e di e ence occu s nea high wa e slack (HWS), while he
maximum posi i e di e ence is obse ed nea low wa e slack (LWS).
Symme ic changes in idal cu en s lead o equi alen esidual cu en
con igu a ions in bo h he d edged and unal e ed es ua ies, esul ing in
a negligible di e ence in he RUWF ac oss he en i e wa e column in
bo h scena ios (Fig. 5).
Finally, in he unal e ed s e ch o he es ua y (Fig. 6c), he sign
shi is e e sed, esul ing in la ge lood and ebb cu en s. This esul is
consis en wi h he ampli ica ion o he 𝑀2cu en wi hin his egion,
leading o idal cycles wi h highe ampli udes o bo h lood and ebb
Ocean Modelling 196 (2025) 102552
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G. Ma ín-Llanes and A. López-Ruiz
Fig. 5. Di e ence in he esidual uni wid h wa e lux be ween he d edged scena ios wi h a dep h inc ease o 𝛥𝐻 = 40% and he unal e ed es ua y. Each panel co esponds o
a di e en d edging leng h. Solid black line ep esen s he d edged ba hyme y; dashed black line ep esen s he unal e ed es ua y. Solid ed line indica es esidual sal in usion
a e d edging; dashed ed line indica es esidual sal in usion in he unal e ed es ua y. Posi i e alues indica e in low.
cu en s. This ampli ica ion is uni o m h oughou he wa e column.
In addi ion, since ebb cu en s inc ease mo e han lood cu en s and
pe sis o a g ea e ac ion o he idal cycle, a ne ou low (nega i e
RUWF di e ence) is obse ed beyond he ansi ion slope in Fig. 5. This
esul aligns wi h he ebb dominance shown in Fig. 4c, ,i.
4.4. Momen um balance
This sec ion examines a ia ions in he momen um balance o iden-
i y he p ima y ac o s d i ing hyd odynamic impac s in d edged es-
ua ies. The Del 3D momen um equa ion in he ho izon al axial di ec-
ion, neglec ing he accele a ion due o Co iolis o ce, eads (Del a es,
2016):
𝜕 𝑢
𝜕 𝑡+𝑢𝜕 𝑢
𝜕 𝑥+𝑣𝜕 𝑢
𝜕 𝑦+𝜔
𝐻
𝜕 𝑢
𝜕 𝑧+𝑔𝜕 𝜂
𝜕 𝑥+𝑔𝐻
𝜌0
𝜕 𝜌
𝜕 𝑥−𝐹𝑥−1
𝐻2
𝜕
𝜕 𝑧(𝜈𝜕 𝑢
𝜕 𝑧)−𝑀𝑥= 0(4)
whe e 𝑢, 𝑣, 𝜔a e he along-channel, c oss-channel and e ical cu en s,
𝜂is he wa e le el, 𝑔is he g a i a ional accele a ion cons an , 𝐻is
he wa e dep h measu ed along he z-coo dina e, 𝜌0is he seawa e
densi y, 𝜌is he wa e densi y, and 𝜈is he eddy iscosi y. These e ms
in Eq. (4) ep esen he local accele a ion, he idal s ess, he accele -
a ion due o la e al anspo , he e ical ad ec ion o momen um, he
ba o opic p essu e g adien , he ba oclinic g adien , he accele a ion
due o iscosi y (𝐹𝑥), he e ical di usion o momen um and he bed
shea o ce (𝑀𝑥), espec i ely.
Momen um e ms we e i s e alua ed du ing wo idal cycles a
he channel halweg o he h ee ep esen a i e sec ions de ined in
Sec ion 4.3. The ob ained esul s, which a e shown in Sec ion 3 o he
Supplemen a y Ma e ial, sugges ha he ba o opic p essu e g adien
and bed shea o ces a e he dominan ac o s, so ha impac s o
channel deepening on hyd odynamics can be assessed in e ms o
changes in hese wo key e ms. Speci ically, al e a ions in he low
s uc u e a he su ace a e closely linked o changes in he ba o opic
p essu e g adien , while modi ica ions in bo om cu en s a e p ima ily
in luenced by changes in bed shea o ces. This is shown in Fig. 7,
whe e he ba o opic g adien and bed shea o ces in bo h he unal-
e ed and d edged (Scena io 14) si ua ions a e ep esen ed along wo
idal cycles. Fi s , as shown in Fig. 7a, he p essu e g adien is educed
in he d edged a ea below he sal in usion limi , and exhibi s a 2-
hou phase lag compa ed o he esul s in he unal e ed es ua y. A
clea co ela ion wi h he axial cu en is obse ed, as he sign o he
di e ence in he ba o opic e m be ween he d edged and unal e ed
con igu a ions is nega i e du ing lood ide and posi i e du ing ebb
Ocean Modelling 196 (2025) 102552
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G. Ma ín-Llanes and A. López-Ruiz
Fig. 6. Di e ence in he e ical dis ibu ion o he axial cu en be ween he d edged
(Scena io 14) and unal e ed scena ios along wo idal cycles. (a) 𝑥= 7.5 k m. (b)
𝑥= 18 k m. (c) 𝑥= 22.5 k m. Posi i e alues indica e in low. Black con ou s indica e
ze o eloci y. The igh axis ep esen s he su ace cu en a he halweg.
ide, which is consis en wi h he e ical dis ibu ion o cu en s a
he su ace (Fig. 6a). Mo eo e , he ime e olu ion o he ba o opic
e m sugges s a clea ebb-domina ed asymme y. Since his e m is
p opo ional o he su ace cu en , his esul implies a nega i e idally
a e aged cu en , i.e., a ne su ace ou low, which is consis en wi h
obse a ions in Fig. 5. Rega ding he ic ional e m in his egion
(Fig. 7b), since bed shea opposes he bo om cu en , maximum posi-
i e alues a e obse ed a ebb ide and ice e sa. D edging leads o a
gene al dec ease in he ic ional e m, which is pa icula ly ele an a
ebb ide. Asymme y changes om ebb-domina ed (posi i e bed shea
accele a ion) o lood-domina ed (nega i e bed shea accele a ion) a e
d edging, leading o a posi i e ne in low a he bo om, as shown
in Fig. 5. In Fig. 6a, s onge posi i e cu en di e ences we e also
obse ed a he bo om. The o e all esul , conside ing changes in bo h
su ace and bo om cu en s, is an inc ease in es ua ine ci cula ion,
which is p opo ional o he dep h inc ease.
In he d edged pa o he channel beyond he sal in usion limi ,
he ba o opic g adien shows a sligh phase lag (∼ 1hou ) compa ed
o he unal e ed es ua y (Fig. 7c), wi h no signi ican changes in ampli-
ude. Inc easing channel dep h educes he wa e su ace slope. How-
e e , as ic ion dec eases due o d edging, idal eloci ies and e ec i e
d ag inc ease, o se ing he e ec o he inc eased dep h (Rals on e al.,
2019). Consequen ly, he p essu e g adien unde goes a ime shi in
his egion, wi hou signi ican changes in i s ampli ude. Rega ding
Fig. 7d, d edging esul s in a lowe and mo e symme ic ime dis ibu-
ion o bed shea . Values a he su ace and bo om sugges equi alence
be ween he idally a e aged hyd odynamic condi ions a he d edged
and unal e ed es ua ies, esul ing in a negligible di e ence in he RUWF
(Fig. 5).
Finally, in he unal e ed pa o he channel, he p essu e g adien
inc eases du ing he ebb ide and shows no signi ican changes du ing
he lood ide. This indica es ha he inc eased dep h compensa es
o he lood ide inc ease, bu i does no balance he inc eased ebb
(Fig. 7e). Bed shea accele a ion also inc eases a he ebb (Fig. 7 )
in esponse o he la ge inc ease in bo om ebb cu en . Bo h esul s
highligh he dominance o he ebb cu en h oughou he en i e wa e
column, leading o a ne ou low, as obse ed in Fig. 5.
4.5. Impac s on sal in usion
In his sec ion, he e ec s o channel deepening on sal in usion a e
add essed. Fig. 8shows he a io o he maximum sal in usion in he
di e en d edged scena ios o he alue ob ained o he unal e ed sce-
na io. The o e all esul s indica e ha sal in usion inc eases wi h bo h
channel dep h and d edging leng h. The maximum a io is obse ed
in Scena io 15, whe e sal in usion inc eases in 15.5%. Addi ionally,
a linea ela ionship be ween sal in usion and dep h inc ease is ob-
se ed, wi h he slope a ying be ween 0.19 and 0.37 depending on
he d edging leng h. A ending o Fig. 8b, a local minimum is obse ed
o a d edging leng h equal o he maximum sal leng h in he unal e ed
es ua y (𝑑=𝑋𝑚𝑎𝑥,0= 15 k m). This beha iou , occu ing o any dep h
inc ease, sugges s a dis inc esponse in sal in usion depending on
whe he deepening is conside ed sho o long, ela i e o he maximum
sal leng h in he unal e ed es ua y. Speci ically, sal in usion exhibi s
highe sensi i i y o he d edging leng h o sho ope a ions, wi h his
e ec becoming mo e p onounced as 𝛥𝐻 inc eases. In con as , his
e ec weakens o long-dis ance ope a ions. Sal in usion as a unc ion
o he d edging leng h can be exp essed by Eq. (5):
𝛥𝑋[%] =⎧
⎪
⎨
⎪
⎩
𝑏𝑑
𝑋𝑚𝑎𝑥,0
+𝑏0𝑑 < 𝑋𝑚𝑎𝑥,0
𝑐𝑑
𝑋𝑚𝑎𝑥,0
+𝑐0𝑑 > 𝑋𝑚𝑎𝑥,0
(5)
whe e 𝑏= [4.392,8.864,11.4] and 𝑐= [3.95,6.281,7.45] o 𝛥𝐻 =
[15,30,40]%. No ably, he di e ence in slope be ween sho (b) and long
(c) ope a ions o 𝛥𝐻 = 40% is en imes g ea e han ha ob ained
o 𝛥𝐻 = 15%. This esul highligh s ha he ex en o d edging plays
a mo e signi ican ole in modula ing sal in usion han he dep h
inc ease alone. Speci ically, he inc ease in sal in usion be ween he
sho es and longes d edging leng hs o 𝛥𝐻 = 40% su passes he
inc ease obse ed be ween any o he dep h inc emen s.
5. Discussion: Implica ions on he managemen o na iga ional
channels
Many po s wo ldwide a e loca ed in es ua ies and may conse-
quen ly be a ec ed by nume ous coas al p ocesses, including sedimen
anspo and deposi ion. Sedimen in illing educes he a ailable wa e
dep h, hus sho ening he ime o which essels can na iga e h ough
he es ua y and limi ing po ope a i i y (Ál a ez e al.,2017), i.e., he
ac ion o ime du ing which wa e dep h is highe han he equi ed
d a o a ce ain po ope a ion (Za zuelo e al.,2019). Rega ding
his issue, d edging ope a ions a e pe iodically planned (Sepeh i e al.,
2024).
This sec ion explo es a ia ions in he channel ope a i i y a e
d edging, which is de e mined by changes in he wa e dep h. Wa e
dep h a e d edging esul s om he ensemble conside a ion o : (1)
he inc eased ele a ion o bed le el below he mean sea le el (𝛥𝐻)
and (2) he esul ing changes in he wa e le el due o he al e ed ide.
Al hough he e ec o he i s e m is limi ed o he d edged po ion o
he channel (𝑥≤𝑑), changes in he idal le el a ec he en i e sys em.
As seen in Sec ion 4.2 (Fig. 3), d edging leads o idal ampli ica ion,
esul ing in inc eased high and low wa e le els and di e en du a ions
o he idal cycle. Rega ding he la e , inc eased low le els las ing
o a longe pe iod o ime play an un a ou able ole in espec o
channel ope a i i y. Al hough his e ec is expec ed o be negligible
compa ed o he inc eased bed le el ele a ion in he d edged po ion o
he channel, channel ope a i i y o low a ge d a s in ce ain egions
o he es ua y whe e 𝑥 > 𝑑could be comp omised by he du a ion o
he low ide.
The change in channel ope a i i y (𝛥𝑂 𝑝𝑡𝑣) o a gi en a ge d a
(𝑦𝑜𝑏𝑗 ) is exp essed as he absolu e di e ence be ween ope a i i y in
he d edged es ua y (𝑂 𝑝𝑡𝑣) and ope a i i y in he unal e ed es ua y
(𝑂 𝑝𝑡𝑣0). Bo h quan i ies ep esen he ime ac ion, o e a 15-day
Ocean Modelling 196 (2025) 102552
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G. Ma ín-Llanes and A. López-Ruiz
Fig. 7. Momen um balance a ia ion: Ba o opic p essu e g adien and bed shea o ces along wo idal cycles. (a), (b) 𝑥= 7.5 k m. (c), (d) 𝑥= 18 k m. (e), ( ) 𝑥= 22.5 k m. Blue
solid and dashed lines ep esen he d edged (Scena io 14) and unal e ed es ua ies, espec i ely. Solid black line ep esen s su ace cu en ( igh axis).
Fig. 8. Di e ence in he maximum sal in usion be ween he unal e ed (𝑋𝑚𝑎𝑥,0) and d edged (𝑋𝑚𝑎𝑥) scena ios. Cu es in (a) co espond o each d edging leng h; cu es in (b)
co espond o he di e en dep h inc eases.
pe iod, du ing which wa e dep h exceeds he a ge d a . Hence,
𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 )≤100%; posi i e alues indica e an inc eased ope a i i y
a e d edging; and nega i e alues indica e he opposi e. In addi ion,
𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 ) = 0implies ha he ope a i i y is he same in bo h he
unal e ed and d edged scena ios. This si ua ion will occu , o ins ance,
o low a ge d a s o which he e will always be a g ea e wa e
dep h in bo h he unal e ed and d edged ba hyme ies. In addi ion,
𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 ) = 0will also occu o high a ge d a s o which wa e
dep h is always lowe in bo h he unal e ed and d edged ba hyme ies.
Fig. 9 ep esen s 𝛥𝑂 𝑝𝑡𝑣 ac oss he es ua y in Scena io 5 conside ing
a ange o a ge d a s scaled wi h he d edged channel dep h (𝐻).
As illus a ed, he magni ude and he sign o 𝛥𝑂 𝑝𝑡𝑣 change ab up ly
om he d edged (𝑥≤30 k m) o he unal e ed (𝑥≥30 k m) pa o
he channel. In addi ion, hese wo a iables a e di ec ly ela ed o he
alue o he a ge d a .
Fi s , conside ing he egion a ec ed by channel deepening, he
unc ion 𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 )is posi i e and adop s a pa abolic shape a each
poin o he es ua y. This esul implies ha ope a i i y in his e-
gion inc eases o main ains i s alue be o e d edging, depending on
he alue o he a ge d a . As a o emen ioned, changes in channel
ope a i i y a e de e mined by changes in he wa e dep h. In his pa
o he channel, he con ibu ion o he inc eased bed le el ele a ion
causes he pos -d edge wa e dep h ime se ies (𝑊𝑑(𝑡)) o be highe
han he wa e dep h ime se ies a he unal e ed es ua y (𝑊𝑑 ,0(𝑡))
o each ime ( ) (Fig. 10a). Hence, he unc ion 𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 ) akes a
di e en alue depending on he alue o 𝑦𝑜𝑏𝑗 wi h espec o 𝑊𝑑(𝑡)and
𝑊𝑑 ,0(𝑡), as illus a ed in Fig. 10b. A ending o his igu e, i e di e en
egions wi hin he 𝑊𝑑−𝑡space a e dis inguished: Region (1), whe e
𝑦𝑜𝑏𝑗 < 𝑊𝑑 ,0(𝑡); Region (2), whe e 𝑦𝑜𝑏𝑗 is wi hin he in e al de ined
by 𝑊𝑑 ,0(𝑡); Region (3), whe e 𝑦𝑜𝑏𝑗 > 𝑊𝑑 ,0(𝑡)and 𝑦𝑜𝑏𝑗 < 𝑊𝑑(𝑡); Region
(4), whe e 𝑦𝑜𝑏𝑗 is wi hin he in e al de ined by 𝑊𝑑(𝑡)and Region (5),
whe e 𝑦𝑜𝑏𝑗 > 𝑊𝑑(𝑡). The ex eme egions (1, 5) a e cha ac e ised by a
null change in he channel ope a i i y; in Region (1) a ge d a s a e
su icien ly low o ha e an ope a i i y o 100% in bo h he und edged
and d edged channels, whe eas in Region (5) high d a s lead o a
ope a i i y o ze o in bo h cases. Bo h si ua ions lead o 𝛥𝑂 𝑝𝑡𝑣(𝑦𝑜𝑏𝑗 ) = 0,
which means ha no imp o emen in ope a i i y has been achie ed o
hese d a s. On he o he hand, ope a i i y a e d edging shi s om
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