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StabilityofAlternateBarsandObliqueDunes
AlternatebarsintheTokachi River,JapanImagecourtesyofV.Langlois
MarcoColom
bini–Università
diGen
ova
–Ita
ly
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;Ø PartIII– linearstabilityofalternatebars&obliquedunes;
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;Ø PartIII– linearstabilityofalternatebars&obliquedunes;
Ø PartIV– weaklynonlinearanalysis
Alternateba
rsinstraightch
anne
ls
Ingeniería deríos,JorgeAbad2011
𝛽 =𝑊$
∗
𝐷∗
2𝑊$∗
𝐿)∗
Thena
megame:alte
rnateba
rs
Thena
megame:alte
rnateba
rsAlternatebarsinaflume.
ImagecourtesyofS.Ikeda.
Thena
megame:alte
rnateba
rs
Thena
megame:alte
rnateba
rs"A small dune sings only the few days in which there is no wind and no clouds”
(Andreotti, B. - LiveScience, 2005)
Thena
megame:alte
rnateba
rsDiagonalbarsinaflume.
FromEinstein&Shen,JGR1964
Thena
megame:alte
rnatevs.d
iagona
lbars
Alternatebarsinaflume.ImagecourtesyofS.Ikeda.
Diagonalbarsinaflume.FromEinstein&Shen,JGR1964
Diagonalbarsinaflume.FromEinstein&Shen,JGR1964
Ø “...itisprobable thatdiagonalbarshavesometimesbeenclassifiedasalternatebarsbysomeauthors.”
Ø “...experimentsseemtoindicatethatagrouping ofthree-dimensionalmesoforms[i.e.scalingwithflowdepth],inwhichthefrontsof themesoforms werediagonallyalligned overthechannelwidth,wasresponsible forthesefeatures.”
FromJaeggi,JHE1984
Thena
megame:alte
rnatevs.d
iagona
lbars
(Roughregime)
JSM:Jaeggi (1984),Sukegawa (1971),Muramoto &Fujita(1978)GSR:Guy,Simons&Richardson(1966)
Longitu
dina
lscalin
g:bars&
dun
es
𝐶 =𝑈∗
𝑢-∗=
8𝑓 =
1𝜅 ln
11.092.5𝑑
𝐿)∗ 𝑊$∗⁄
𝐿)∗ 𝐷∗⁄
JSM:Jaeggi (1984),Sukegawa (1971),Muramoto &Fujita(1978)GSR:Guy,Simons&Richardson(1966)
≈ 20
≈ 8
Long
itudi
nalscalin
g: b
ars
& du
nes
(Roughregime)𝐶 =𝑈∗
𝑢-∗=
8𝑓 =
1𝜅 ln
11.092.5𝑑
𝐿)∗ 𝑊$∗⁄
𝐿)∗ 𝐷∗⁄
Longitu
dina
l&tran
sversewaven
umbe
rs Ø LongitudinalwavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘) =2𝜋𝐷∗
𝐿)
𝑘)
Longitu
dina
l&tran
sversewaven
umbe
rs Ø LongitudinalwavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘) =2𝜋𝐷∗
𝐿)
Ø TransversewavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘< =2𝜋𝐷∗
𝐿<=2𝜋𝐷∗
4𝑊$∗=
𝜋2𝛽
𝑘<
𝑘)
Ø LinearØ 2DFlowmodelØ FreeBars
Ø LinearØ 2DFlowmodelØ FreeandForcedBars
Ø WeaklyNonLinearØ 2DFlowmodelØ FreeBars
Ø LinearØ 3DFlowmodelØ FreeBars
Stab
ilityofa
lternateba
rs
• 3DROTATIONALFLOWMODEL(infinitelywidechannel)
• BOUSSINESQ’SCLOSURE(algebraicmixinglength)
• COORDINATETRANSFORMATION(rectangulardomain)
• EQUILIBRIUMMODEL(Exner)
• BEDLOADONLY(MPMbedloadfunction)
• CORRECTIONSFORSEDIMENTWEIGHT(x– Fredsøe,y– Engelund)
• CORRECTIONFORBEDLOADLAYERTHICKNESSFlow
and
sedimen
ttranspo
rtmod
els
FLOWMODEL
SEDIMENTTRANSPORTMODEL
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
Ω = Ω 𝑘), 𝑘<; 𝐹𝑟, 𝐶
DUNEFLAVOUR
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
Ω = Ω 𝑘), 𝑘<; 𝐹𝑟, 𝐶 Ω = Ω 𝜆,𝛽; 𝜗, 𝑑
DUNEFLAVOUR BARFLAVOUR
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
𝜆 = 𝑘) 𝛽 𝛽 =𝜋2𝑘<
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Ω = Ω 𝑘), 𝑘<; 𝐹𝑟, 𝐶 Ω = Ω 𝜆,𝛽; 𝜗, 𝑑
DUNEFLAVOUR BARFLAVOUR
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
Ω = Ω 𝜆, 𝛽; 𝜗, 𝑑 𝜆 = 𝑘) 𝛽 𝛽 =𝜋2𝑘<
𝜆
𝛽
𝜗𝜗[
= 3 𝑑 = 0.025
𝐹𝑟 = 1 𝐶 = 13
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
𝜆
𝛽
Ω = Ω 𝜆, 𝛽; 𝜗, 𝑑 𝜆 = 𝑘) 𝛽 𝛽 =𝜋2𝑘<
Resonant conditions- 𝜆], 𝛽]
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
𝜆
𝛽
Ω = Ω 𝜆, 𝛽; 𝜗, 𝑑 𝜆 = 𝑘) 𝛽 𝛽 =𝜋2𝑘<
𝛽[
𝜆[
Criticalconditions- 𝜆[, 𝛽[
Resonant conditions- 𝜆], 𝛽]
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
𝜆
𝛽
Criticalconditions- 𝜆[, 𝛽[
Resonant conditions- 𝜆], 𝛽]
Ω = Ω 𝜆, 𝛽; 𝜗, 𝑑 𝜆 = 𝑘) 𝛽 𝛽 =𝜋2𝑘<
𝛽[
𝜆[
𝛽 > 𝛽[ 𝜗,𝑑
AlternateBarsdonotforminanarrowchannel
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
Criticalconditions- 𝜆[, 𝛽[ Resonant conditions- 𝜆], 𝛽]
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
Bar instability
Antidune instability
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
𝜔 > 0 𝜔 > 0 𝜔 < 0
𝛽 =𝜆𝑘)
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
AlternateBa
rs:3D
line
arstab
ility
𝛽 =𝜆𝑘)
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝑘)
𝛽
𝜆
𝛽
𝑘)
𝛽
AlternateBa
rs:3D
line
arstab
ility
𝛽
𝑘)
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
𝑘< =𝜋2𝛽
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝛽
𝑘)
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
Bar instability Antidune instability
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
Criticalconditions- 𝜆[, 𝛽[
Resonant conditions- 𝜆], 𝛽]
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
2D disturbances are stable!
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
2D disturbances are the most unstable!
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
ExperimentaldatasetsJSM:Jaeggi (1984),Sukegawa (1971),Muramoto &Fujita(1978)GSR:Guy,Simons&Richardson(1966)
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Movingupalongaverticallinemeans(samesediment):
Ø C constant⇒ dconstant⇒ D*constant
Ø θ increases⇒ Fr increases⇒U*increases
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Movingupalongaverticallinemeans(samesediment):
Ø C constant⇒ dconstant⇒ D*constant
Ø θ increases⇒ Fr increases⇒U*increases
a ⇒ 𝑆 ∝ 𝐹𝑟X increases
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Movingupalongaverticallinemeans(samesediment):
Ø C constant⇒ dconstant⇒ D*constant
Ø θ increases⇒ Fr increases⇒U*increases
Ø Flowrateincreaseswithconstantflowdepth
a ⇒ 𝑆 ∝ 𝐹𝑟X increases
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Movingupalongaverticallinemeans(samesediment):
Ø C constant⇒ dconstant⇒ D*constant
Ø θ increases⇒ Fr increases⇒U*increases
Ø Flowrateincreaseswithconstantflowdepth⇒Slopeincreases
Ø FromAlternateBarstoAlternateBars& Antidunes
𝑘<
𝑘)
𝜗𝜗[
= 2
Movingup
:C=10
As𝜗 increases:
𝑘<
𝑘)
𝑘<
𝑘)
𝜗𝜗[
= 2𝜗𝜗[= 2.5
Movingup
:C=10
As𝜗 increases:Ø Anewregionofinstabilityappears:2Dantidunes;Ø Theregionsofinstabilityforbarsandantidunes aredistinct;Ø Barsandantidunes linearlycoexist:barsarelessunstable;
𝑘<
𝑘)
𝑘<
𝑘)
𝑘<
𝑘)
𝜗𝜗[
= 2𝜗𝜗[= 2.5
𝜗𝜗[
= 3
Movingup
:C=10
As𝜗 increases:Ø Anewregionofinstabilityappears:2Dantidunes;Ø Theregionsofinstabilityforbarsandantidunes aredistinct;Ø Barsandantidunes linearlycoexist:barsarelessunstable;Ø 3Dantidunes becomethemostunstable;
𝑘<
𝑘)
𝑘<
𝑘)
𝑘<
𝑘)
𝜗𝜗[
= 2𝜗𝜗[= 2.5
𝜗𝜗[
= 3
Movingup
:C=10
As𝜗 increases:Ø Anewregionofinstabilityappears:2Dantidunes;Ø Theregionsofinstabilityforbarsandantidunes aredistinct;Ø Barsandantidunes linearlycoexist:barsarelessunstable;Ø 3Dantidunes becomethemostunstable;Ø Thisresultsinatransitionfrom2Dto3Dantidunes;
Regimediagram:bars&
dun
es
𝐶 =1𝜅 ln
11.092.5𝑑
𝜗 ≅ 0.14𝐹𝑟X𝑒Z[
𝐶X
Movingrightalongahorizontallinemeans(samesediment):
Ø C increases⇒d decreases⇒ D* increases
Ø θ constant⇒ 𝑈∗ ∝ 𝐶 increases
Ø Flowrateincreasesbut Fr decreases
Ø FromAlternateBarstoDiagonalBarsto2DDunes
a ⇒ 𝑆 ∝ 1/𝐷∗ decreases
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Regimediagram:bars&
dun
es
Dune
-rippletran
sitio
n
AsC increases:
Ø thelongitudinalwavenumberofmaximumgrowthrateincreases:barsbecomeshorter;Ø 2Ddisturbances becomeunstablebutarelessunstable than3Ddisturbances;Ø 2Ddisturbances becomethemostunstable;Ø Thisresultsinatransitionfrom3Dbarsto2Ddunes viadiagonalbars(3Ddunes);
Dune
-rippletran
sitio
n
AsC increases:
Ø thelongitudinalwavenumberofmaximumgrowthrateincreases:barsbecomeshorter;Ø 2Ddisturbances becomeunstablebutarelessunstable than3Ddisturbances;Ø 2Ddisturbances becomethemostunstable;Ø Thisresultsinatransitionfrom3Dbarsto2Ddunes viadiagonalbars(3Ddunes);
