stability of alternate bars and y oblique dunes a à i · stability of alternate bars and oblique...
TRANSCRIPT
![Page 1: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/1.jpg)
StabilityofAlternateBarsandObliqueDunes
AlternatebarsintheTokachi River,JapanImagecourtesyofV.Langlois
MarcoColom
bini–Università
diGen
ova
–Ita
ly
![Page 2: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/2.jpg)
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;
![Page 3: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/3.jpg)
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;Ø PartIII– linearstabilityofalternatebars&obliquedunes;
![Page 4: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/4.jpg)
StabilityofRiverBedFormsPartIII
MarcoColom
bini–Università
diGen
ova
–Ita
ly
Ø PartI– linearstabilityofroll-waves,dunes&antidunes;Ø PartII– linearstabilityofripplesanddunes;Ø PartIII– linearstabilityofalternatebars&obliquedunes;
Ø PartIV– weaklynonlinearanalysis
![Page 5: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/5.jpg)
Alternateba
rsinstraightch
anne
ls
Ingeniería deríos,JorgeAbad2011
𝛽 =𝑊$
∗
𝐷∗
2𝑊$∗
𝐿)∗
![Page 6: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/6.jpg)
Thena
megame:alte
rnateba
rs
![Page 7: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/7.jpg)
Thena
megame:alte
rnateba
rsAlternatebarsinaflume.
ImagecourtesyofS.Ikeda.
![Page 8: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/8.jpg)
Thena
megame:alte
rnateba
rs
![Page 9: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/9.jpg)
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)
![Page 10: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/10.jpg)
Thena
megame:alte
rnateba
rsDiagonalbarsinaflume.
FromEinstein&Shen,JGR1964
![Page 11: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/11.jpg)
Thena
megame:alte
rnatevs.d
iagona
lbars
Alternatebarsinaflume.ImagecourtesyofS.Ikeda.
Diagonalbarsinaflume.FromEinstein&Shen,JGR1964
![Page 12: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/12.jpg)
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
![Page 13: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/13.jpg)
(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∗⁄
![Page 14: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/14.jpg)
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$∗⁄
𝐿)∗ 𝐷∗⁄
![Page 15: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/15.jpg)
Longitu
dina
l&tran
sversewaven
umbe
rs Ø LongitudinalwavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘) =2𝜋𝐷∗
𝐿)
𝑘)
![Page 16: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/16.jpg)
Longitu
dina
l&tran
sversewaven
umbe
rs Ø LongitudinalwavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘) =2𝜋𝐷∗
𝐿)
Ø TransversewavenumbersforDiagonalBarsarelargerthanforAlternateBars
𝑘< =2𝜋𝐷∗
𝐿<=2𝜋𝐷∗
4𝑊$∗=
𝜋2𝛽
𝑘<
𝑘)
![Page 17: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/17.jpg)
Ø LinearØ 2DFlowmodelØ FreeBars
Ø LinearØ 2DFlowmodelØ FreeandForcedBars
Ø WeaklyNonLinearØ 2DFlowmodelØ FreeBars
Ø LinearØ 3DFlowmodelØ FreeBars
Stab
ilityofa
lternateba
rs
![Page 18: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/18.jpg)
• 3DROTATIONALFLOWMODEL(infinitelywidechannel)
• BOUSSINESQ’SCLOSURE(algebraicmixinglength)
• COORDINATETRANSFORMATION(rectangulardomain)
• EQUILIBRIUMMODEL(Exner)
• BEDLOADONLY(MPMbedloadfunction)
• CORRECTIONSFORSEDIMENTWEIGHT(x– Fredsøe,y– Engelund)
• CORRECTIONFORBEDLOADLAYERTHICKNESSFlow
and
sedimen
ttranspo
rtmod
els
FLOWMODEL
SEDIMENTTRANSPORTMODEL
![Page 19: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/19.jpg)
• LINEARLEVEL:differentialeigenvalueproblem
AlternateBa
rs:3Dlin
earstability
𝐺 𝑥, 𝑦, 𝑧, 𝑡 = 𝐺D 𝑧 + 𝜀𝑔H 𝑧 exp 𝑖𝑘) 𝑥 − 𝜔𝑡 + 𝑖𝑘<𝑦 + Ω𝑡 + 𝑐. 𝑐.
![Page 20: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/20.jpg)
• 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
![Page 21: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/21.jpg)
• 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
![Page 22: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/22.jpg)
• 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
![Page 23: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/23.jpg)
• 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
![Page 24: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/24.jpg)
• 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]
![Page 25: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/25.jpg)
• 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]
![Page 26: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/26.jpg)
• 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
![Page 27: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/27.jpg)
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
Criticalconditions- 𝜆[, 𝛽[ Resonant conditions- 𝜆], 𝛽]
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 28: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/28.jpg)
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 29: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/29.jpg)
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 30: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/30.jpg)
𝜆
𝛽
𝜆
𝛽
AlternateBa
rs:linearstability
SWMODEL 3DMODEL
Bar instability
Antidune instability
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 31: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/31.jpg)
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
![Page 32: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/32.jpg)
AlternateBa
rs:3D
line
arstab
ility
𝛽 =𝜆𝑘)
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝑘)
𝛽
𝜆
𝛽
![Page 33: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/33.jpg)
𝑘)
𝛽
AlternateBa
rs:3D
line
arstab
ility
𝛽
𝑘)
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 34: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/34.jpg)
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
𝑘< =𝜋2𝛽
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
𝛽
𝑘)
![Page 35: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/35.jpg)
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 36: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/36.jpg)
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
Bar instability Antidune instability
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 37: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/37.jpg)
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 38: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/38.jpg)
𝑘)
𝑘<
AlternateBa
rs:3D
line
arstab
ility
Criticalconditions- 𝜆[, 𝛽[
Resonant conditions- 𝜆], 𝛽]
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 39: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/39.jpg)
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
2D disturbances are stable!
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 40: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/40.jpg)
AlternateBa
rs:3D
line
arstab
ility
𝑘)
𝑘<
2D disturbances are the most unstable!
𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗[
= 3 𝐶 = 13
![Page 41: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/41.jpg)
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)
![Page 42: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/42.jpg)
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
![Page 43: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/43.jpg)
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
![Page 44: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/44.jpg)
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
![Page 45: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/45.jpg)
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
![Page 46: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/46.jpg)
𝑘<
𝑘)
𝜗𝜗[
= 2
Movingup
:C=10
As𝜗 increases:
![Page 47: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/47.jpg)
𝑘<
𝑘)
𝑘<
𝑘)
𝜗𝜗[
= 2𝜗𝜗[= 2.5
Movingup
:C=10
As𝜗 increases:Ø Anewregionofinstabilityappears:2Dantidunes;Ø Theregionsofinstabilityforbarsandantidunes aredistinct;Ø Barsandantidunes linearlycoexist:barsarelessunstable;
![Page 48: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/48.jpg)
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;
![Page 49: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/49.jpg)
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;
![Page 50: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/50.jpg)
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
![Page 51: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/51.jpg)
Regimediagram:bars&
dun
es
![Page 52: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/52.jpg)
Regimediagram:bars&
dun
es
![Page 53: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/53.jpg)
Regimediagram:bars&
dun
es
![Page 54: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/54.jpg)
Regimediagram:bars&
dun
es
![Page 55: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/55.jpg)
Regimediagram:bars&
dun
es
![Page 56: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/56.jpg)
Regimediagram:bars&
dun
es
![Page 57: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/57.jpg)
Regimediagram:bars&
dun
es
![Page 58: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/58.jpg)
Dune
-rippletran
sitio
n
AsC increases:
Ø thelongitudinalwavenumberofmaximumgrowthrateincreases:barsbecomeshorter;Ø 2Ddisturbances becomeunstablebutarelessunstable than3Ddisturbances;Ø 2Ddisturbances becomethemostunstable;Ø Thisresultsinatransitionfrom3Dbarsto2Ddunes viadiagonalbars(3Ddunes);
![Page 59: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/59.jpg)
Dune
-rippletran
sitio
n
AsC increases:
Ø thelongitudinalwavenumberofmaximumgrowthrateincreases:barsbecomeshorter;Ø 2Ddisturbances becomeunstablebutarelessunstable than3Ddisturbances;Ø 2Ddisturbances becomethemostunstable;Ø Thisresultsinatransitionfrom3Dbarsto2Ddunes viadiagonalbars(3Ddunes);
![Page 60: Stability of Alternate Bars and y Oblique Dunes a à i · Stability of Alternate Bars and Oblique Dunes Alternate bars in the Tokachi River, Japan Image courtesy of V. Langlois i](https://reader031.vdocuments.net/reader031/viewer/2022030902/5b4054407f8b9a5e528d2f2e/html5/thumbnails/60.jpg)