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);