y Weakly nonlinear analysis of river bed forms a à i · Colombini , Seminara & Tubino (1987)...

MarcoColom

bini–Università

diGen

ova–Ita

ly Weaklynonlinearanalysisofriverbedforms

Weaklynonlinearanalysisofriverbedforms

Colombini, Seminara &Tubino (1987)Finite-amplitude alternatebarsJournal ofFluidMechanics181

Colombini, Seminara &Tubino (1987)Finite-amplitude alternatebarsJournal ofFluidMechanics181

Schielen,Doelman &deSwart(1993)Onthenonlinear dynamics offreebarsinstraightchannelsJournal ofFluidMechanics252

Stab

ilityand

Bifu

rcationTh

eory

StabilityandBifurcations

StabilityandBifurcationsStab

ilityand

Bifu

rcationTh

eory Ø Bedformsaretheresultoftheinstabilityofthesystem

composedbytheflowandbythebed(thecontainer).Ø Avarietyofflow-bedconfigurationsareobserved,which

correspondtodifferentregimesinthespaceofphysicalparameters.

Ø BifurcationTheoryprovidesamathematicaltooltodeterminetheregionsoftheparameterspacecharacteristicofeachregimeandthecorrespondingshapeofthebedform.

Ø ThestabilityofaBaseStateisstudiedwithrespecttoperturbationsoftheflowandthebed.Here,theBaseStateisrepresentedbyasteadyuniformflowinaninfinitelywidechannelwithactivesedimenttransport.

Ø Bifurcationsistheprocesswherebyanewsolutiontakesover(bifurcates)astheboundaryofastableregionintheparameterspaceiscrossed.


ilityand

Bifu

rcationTh





Ø Thestabilityofa BaseStateisstudiedwithrespecttoperturbationsoftheflowandthebed.Here,theBaseStateisrepresentedbyasteadyuniformflowinaninfinitelywidechannelwithactivesedimenttransport.



ilityand

Bifu

rcationTh








ilityand

Bifu

rcationTh





Ø Thestabilityofa BaseStateisstudiedwithrespecttoperturbationsoftheflowandthebed.Here,theBaseStateisrepresentedbyasteadyuniformflowinaninfinitelywidechannelwithactivesedimenttransport.



ilityand

Bifu

rcationTh







FlowandSedimentTransportModels

• 2DSHALLOWWATER FLOWMODEL

• EMPIRICALCLOSUREFORBEDSHEARSTRESS(Chézy conductancecoefficient)

• EQUILIBRIUMMODEL(Exner)

• BEDLOADONLY(MPMbedloadfunction)

• CORRECTIONSFORSEDIMENTWEIGHT(x– Fredsøe,y– Engelund)

FLOWMODEL

SEDIMENTTRANSPORTMODEL

Flow

and

sedimen

ttranspo

rtmod

el

2DSWEQUATIONS+CONTINUITY(dimensionlesswith𝝆,𝑼𝟎∗ ,𝑫𝟎∗)𝐷𝑈,* + 𝐷𝑈𝑈,, + 𝐷𝑉𝑈,. =

𝑆𝐷𝐹𝑟3

−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.

𝐷𝑉,* + 𝐷𝑈𝑉,, +𝐷𝑉𝑉,. = −𝐷𝐹𝑟3

𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

DEPTH-AVERAGINGPROCEDURE

𝑈 = ; 𝑢𝑑𝜁?

@𝑉 = ; 𝑣𝑑𝜁

?

@

𝜁 =𝑧 − 𝐵𝐷

𝑇,,8 = ; 𝜏,,𝑑𝜁?

@𝑇,.8 = ; 𝜏,.𝑑𝜁

?

@𝑇..8 = ; 𝜏..𝑑𝜁

?

@

𝑇,7 = 𝜏,D EF@𝑇.7 = 𝜏.D EF@

𝑇,,9 = ; 𝑢− 𝑈 3𝑑𝜁?

@𝑇,.9 = ; 𝑢 −𝑈 𝑣 −𝑉 𝑑𝜁

?

@𝑇..9 = ; 𝑣− 𝑉 3𝑑𝜁

?

@

Flow

and

sedimen

ttranspo

rtmod

el

𝑧

𝑢 =𝑈𝜅𝐶 ln

𝜁 + 𝜁@𝜁@

𝑧 = 𝐵@ = 𝜁@𝑧 = 0

𝜈L = 𝜅𝐷𝑈𝐶 𝜁 + 𝜁@ 1 − 𝜁

𝐶 =1𝜅 ln

11.09𝑘


@𝜁@ = exp −𝜅𝐶 − 1 =

𝑘30 =

2.5𝑑30 =

𝑑12

𝜁 = 0


Flow

and

sedimen

ttranspo

rtmod

el

𝑧

𝑢 =𝑈𝜅𝐶 ln

𝜁 + 𝜁@𝜁@

𝑧 = 𝐵@ = 𝜁@𝑧 = 0

𝜈L = 𝜅𝐷𝑈𝐶 𝜁 + 𝜁@ 1 − 𝜁

𝐶 =1𝜅 ln

11.09𝑘


@𝜁@ = exp −𝜅𝐶 − 1 =

𝑘30 =

2.5𝑑30 =

𝑑12

𝑇,7 = 𝜏,D EF@ = 𝜈L𝑢,D EF@=𝑈3

𝐶3

𝜁 = 0


𝑇,,8 = ; 𝜏,,𝑑𝜁?

@=2𝑁𝐷𝑈𝑈,,

𝐶3 𝑁 =16 𝜅𝐶 +

16

𝑇,,9 = ; 𝑢 −𝑈 3𝑑𝜁?

@=

𝑈3

𝜅3𝐶3

Flow

and

sedimen

ttranspo

rtmod

el

2DSWEQUATIONS+CONTINUITY(dimensionlesswith𝝆,𝑼𝟎∗ ,𝑫𝟎∗)

𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

DEPTH-AVERAGINGPROCEDURE

𝑇,,8 =2𝑁𝐷𝒰𝑈,,

𝐶3𝑇,.8 =

𝑁𝐷𝒰 𝑈,. + 𝑉,,𝐶3

𝑇..8 =2𝑁𝐷𝒰𝑉,.

𝐶3

𝑇,7 =𝒰𝑈𝐶3

𝑇.7 =𝒰𝑉𝐶3

𝒰 = 𝑈3+ 𝑉3

𝑇,,9 =𝑈3

𝜅3𝐶3𝑇,.9 =

𝑈𝑉𝜅3𝐶3

𝑇..9 =𝑉3

𝜅3𝐶3

Flow

and

sedimen

ttranspo

rtmod

el

𝐷𝑈,* + 𝐷𝑈𝑈,, + 𝐷𝑉𝑈,. =𝑆𝐷𝐹𝑟3

−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

2DSWEQUATIONS+CONTINUITY(dimensionlesswith𝝆,𝑼𝟎∗ ,𝑫𝟎∗)

𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

2DSEDIMENTCONTINUITY(Exner)

𝜙,, 𝜙. = Φ cos𝛿 , sin 𝛿 sin 𝛿 =𝑇.7

𝐶3−𝜇.𝜗𝐵,.Φ = 𝐴 𝜗 − 𝜗d

e3

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0𝒬 =𝑠 − 1 𝑔𝑑∗e

𝑈@∗𝐷@∗(1 − 𝑝)=

𝑠 − 1 𝑑e

𝐹𝑟(1 − 𝑝)

CORRECTIONSFORSEDIMENTWEIGHT(x– Fredsøe,y– Engelund)

𝜗 − 𝜗l =𝒰3𝐹𝑟3

𝑠 − 1 𝑑𝐶3− 𝜗dm − 𝜇. 𝑆− 𝐵,,

Flow

and

sedimen

ttranspo

rtmod

el


−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

2DSWEQUATIONS+2DFLUID&SEDIMENTCONTINUITYEQUATIONS

𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0

+ALGEBRAICCLOSURESFORSTRESSES&SEDIMENTTRANSPORT

Flow

and

sedimen

ttranspo

rtmod

el


−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

Expansions,Interactions&Cascadeprocesses

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦Grou

ndfloo

r:basestate

𝐺@


Grou

ndlevel:Ba

sestate


𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0



−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.


𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0



−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

Grou

ndlevel:Ba

sestate


𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0



−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

Grou

ndlevel:Ba

sestate

𝑈@,𝑉@, 𝐷@,𝐵@ = 1, 0, 1, 𝜁@

𝑇,@7 ,𝑇.@7 ,𝑇,,@8 ,𝑇,.@8 ,𝑇..@8 ,𝑇,,@9 ,𝑇,.@9 ,𝑇..@9 =1𝐶3, 0, 0,0, 0,

1𝜅3𝐶3

, 0

𝜗@, 𝜗d@, Φ@ =𝑆

𝑠− 1 𝑑, 𝜗dm − 𝜇,𝑆, 𝐴 𝜗@ − 𝜗d@

e3

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦Firstfloor:linearlevel

𝐺@

𝒜 𝑔??𝐸? 𝒜∗ 𝑔??∗ 𝐸w?


𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0



−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.

Firstfloor:Linearlevel


𝐷,* + 𝑈𝐷,, + 𝑉𝐷,. +𝐷𝑈,, +𝐷𝑈,. = 0

𝐵,* + 𝒬 𝜙,,, + 𝜙.,. = 0



−𝐷𝐹𝑟3

𝐵 + 𝐷 ,, − 𝑇,7 + 𝑇,,8 − 𝑇,,9 𝐷 ,, + 𝑇,.8 − 𝑇,.9 𝐷,.


𝐵 + 𝐷 ,. − 𝑇.7+ 𝑇,.8 − 𝑇,.9 𝐷,,+ 𝑇..8 − 𝑇..9 𝐷

,.


𝐺?? = 𝑔?? exp 𝑖𝑘, 𝑥 − 𝜔𝑡 + 𝑖𝑘.𝑦

𝐷@𝑈??,* +𝐷??𝑈@,* +𝐷@𝑈@𝑈??,, + 𝐷@𝑈??𝑈@,, +𝐷??𝑈@𝑈@,, + 𝐷@𝑉@𝑈??,. + 𝐷@𝑉??𝑈@,. + 𝐷??𝑉@𝑈@,.

−𝑖𝑘,𝜔𝑢??+ 𝑖𝑘,𝑢??

• LINEARLEVEL:algebraiceigenvalueproblem

𝐺ox = 𝑔ox exp 𝑖𝑝𝑘, 𝑥 − 𝜔𝑡 + 𝑖𝑞𝑘.𝑦Firstfloor:Linearlevel

𝑨?? − 𝑘,𝜔𝑰 | 𝒙?? = 0

𝑨?? =

𝑎?? 𝑎?3𝑎3? 𝑎33

𝑎?e 𝑘, 𝐹𝑟3⁄𝑘. 𝐹𝑟3⁄ 𝑘. 𝐹𝑟3⁄

𝑘, 𝑘.𝛾𝑎�? 𝛾𝑘.

𝑘, 00 𝛾𝑎��

𝒙?? =

𝑢??𝑣??𝑑??𝑏??

where:𝛾 = 𝒬Φ@ ≪ 1

and𝑎�� dependonthewavenumbersandonbaseflowquantities



det 𝑨?? − 𝑘,𝜔𝑰 = 0

det

𝑎?? − 𝑘,𝜔 𝑎?3𝑎3? 𝑎33 − 𝑘,𝜔



𝑘,−𝑘,𝜔 00 𝛾𝑎�� − 𝑘,𝜔

= 0

whichprovidesaquarticpolynomial: FOUR eigenvalues,threefortheflow, oneforthebed;



det 𝑨?? − 𝑘,𝜔𝑰 = 0

det

𝑎?? 𝑎?3𝑎3? 𝑎33



𝑘, 00 𝛾𝑎�� − 𝑘,𝜔

= 0

whichprovidesaquarticpolynomial: FOUR eigenvalues,threefortheflow, oneforthebed;ifthequasi-steadyhypothesis ismade(notimederivativesinflowequations):ONE eigenvalueforthebed;

𝑘,𝜔 = 𝛾 𝑎�?𝑢�?? + 𝑘.𝑣�?? + 𝑎��



where𝑢�?? and𝑣�??aresolutions ofthelinearnonhomogeneous reducedalgebraicsystem:

𝑘,𝜔 = 𝛾 𝑎�?𝑢�?? + 𝑘.𝑣�?? + 𝑎��

𝑎?? 𝑎?3 𝑎?e𝑎3? 𝑎33 𝑘. 𝐹𝑟3⁄𝑘, 𝑘. 𝑘,

|𝑢�??𝑣�??𝑑�??

= −𝑘, 𝐹𝑟3⁄𝑘. 𝐹𝑟3⁄

0Obtainedfromthepreviouseigensystem byeliminating thelastrow(Exner equation)andbymoving thelastcolumn (proportional to𝑏??)totherighthandside.Thesolution ofthissystemprovides theflowresponsetoabedperturbation ofunitaryamplitude.


Ω = Ω 𝑘,, 𝑘.; 𝐹𝑟, 𝐶 Ω = Ω 𝜆,𝛽; 𝜗, 𝑑

DUNEFLAVOUR BARFLAVOUR

𝜆 =2𝜋𝑊�

∗

𝐿,∗= 𝑘, 𝛽 𝛽 =

𝑊�∗

𝐷∗ =𝜋2𝑘.

𝐶 =1𝜅 ln

11.092.5𝑑𝜗 ≅ 0.14

𝐹𝑟3𝑒 d

𝐶3


Ω = 𝑘,𝜔� = 𝛾 𝐴@𝑘,𝑇¡,??7�

𝑇,@7− 𝑘,

𝜇,𝜗@

+ 𝑘.𝑇¡.??7�

𝑇,@7− 𝑘.

𝜇.𝜗@

StabilizingDestabilizing

𝜆

𝛽


Criticalconditions- 𝜆d, 𝛽d

𝐺 𝑥, 𝑦, 𝑡 = 𝐺@ + 𝜀𝑔? exp 𝑖𝑘, 𝑥 − 𝜔𝑡 + 𝑖𝑘.𝑦 + Ω𝑡 + 𝑐. 𝑐.

Resonant conditions- 𝜆8, 𝛽8

Ω = Ω 𝜆,𝛽; 𝜗, 𝐶 𝜆 =2𝜋𝑊�

∗

𝐿,∗= 𝑘, 𝛽 𝛽 =

𝑊�∗

𝐷∗ =𝜋2𝑘.

𝛽d

𝜆d

𝛽 > 𝛽d 𝜗,𝐶

AlternateBarsdonotforminanarrowchannel


𝐹𝑟 = 1 𝑑 = 0.025𝜗𝜗d

= 3 𝐶 = 13

𝜆

𝛽


𝐶

𝛽−𝛽 d

𝛽 d⁄

𝑑

Firstfloor:Linearlevel BIFURCATIONPARAMETER

𝜆theory

𝜆experim

ents


WAVELENGTHOFMAXIMUMAMPLIFICATION

𝜆

𝛽

• LINEARLEVEL:algebraiceigenvalueproblemΩ = Ω 𝜆, 𝛽, 𝜗,𝑑 Criticalconditions- 𝑘d, 𝐹𝑟d

Criticalcon

ditio

nsindun

einstab

ility

𝛽d

𝜆d

𝛽 > 𝛽d 𝜗,𝐶

𝛽d representsthebifurcationpoint:AlternateBarsdonotformbelowcriticalconditions

𝐺 𝑥, 𝑦, 𝑡 = 𝐺@ + 𝜀𝑔? exp 𝑖𝑘, 𝑥 − 𝜔𝑡 + 𝑖𝑘.𝑦 + Ω𝑡 + 𝑐. 𝑐.

• LINEARLEVEL:differentialeigenvalueproblem

𝐺 𝑥, 𝑧, 𝑡 = 𝐺@ 𝑧 + 𝜀𝑔? 𝑧 exp 𝑖𝑘 𝑥 − 𝜔𝑡 + Ω𝑡 + 𝑐. 𝑐.

Ω = Ω 𝑘, 𝐹𝑟, 𝐶 Criticalconditions- 𝑘d, 𝐹𝑟d

Criticalcon

ditio

nsindun

einstab

ility

Thereareseveralcriticalvalues𝐹𝑟d thatrepresentdifferentbifurcationpointsforDunes,Antidunes andRollWaves

WNL AnalysisØ WNLanalysisprovidesatooltoinvestigatetheneighbourhood ofthe

criticalpoints;

Ø Theperturbationparameterisexpandedas𝛽 = 𝛽d 1 + 𝜀3 ;

Ø Aslow timescale𝑇 = 𝜀3𝑡isintroduced;Ø Theamplitudeoftheperturbationevolvesontheslowtimescaledueto

thefactthatweslightlyexceedthebifurcationpoint,wherethegrowthrateoftheperturbationvanishes;

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦Firstfloor:linearlevel

𝐺@

𝒜 𝑔??𝐸? 𝒜∗ 𝑔??∗ 𝐸w?

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦Second

floo

r:qua

draticinteractions

𝐺@

𝒜 𝑔??𝐸? 𝒜∗ 𝑔??∗ 𝐸w?

𝒜𝒜∗ 𝑔3@𝐸@ 𝒜3 𝑔33𝐸3 𝒜∗3 𝑔33∗ 𝐸w3

𝐺@

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦

𝒜 𝑔??𝐸? 𝒜∗ 𝑔??∗ 𝐸w?Third

floo

r:cu

bicinteractio

ns

𝒜𝒜∗ 𝑔3@𝐸@ 𝒜3 𝑔33𝐸3 𝒜∗3 𝑔33∗ 𝐸w3

𝒜3𝒜∗ 𝑔e?𝐸? 𝒜e 𝑔ee𝐸e 𝒜∗e 𝑔ee∗ 𝐸we 𝒜𝒜∗3 𝑔e?∗ 𝐸w?

𝐺@

𝐸o = exp 𝑖𝑝𝑘, 𝑥 −𝜔𝑡 + 𝑖𝑝𝑘.𝑦

𝒜 𝑔??𝐸? 𝒜∗ 𝑔??∗ 𝐸w?Third

floo

r:cu

bicinteractio

ns

𝒜𝒜∗ 𝑔3@𝐸@ 𝒜3 𝑔33𝐸3 𝒜∗3 𝑔33∗ 𝐸w3

𝒜3𝒜∗ 𝑔e?𝐸? 𝒜e 𝑔ee𝐸e 𝒜∗e 𝑔ee∗ 𝐸we 𝒜𝒜∗3 𝑔e?∗ 𝐸w?

Secularterms

Nonlinearitygivesrisetointeractionsbetweenthefundamentalanditselfwhichleadtothegenerationofhigherharmonicsboth inthelongitudinalandinthetransversedirections.Following thiscascadeprocessonefindsthatthefundamental isreproducedatthirdorder,whichleadstothegenerationof secularterms.

Inorder topreventtheiroccurrencetheslow timedependenceof theamplitudeofthefundamentalmustalsobeforcedtoproduceacontribution atthirdorder.

Thisprovidesasolvabilitycondition thatyieldstheLandau-Stuartamplitudeequation

WNLAnalysis:Landau-Stuart

𝑑𝒜𝑑𝑇 = 𝛼?𝒜 + 𝛼3𝒜3𝒜∗

WNLAnalysis:Landau-Stuart𝑑𝒜𝑑𝑇 = 𝛼?𝒜 + 𝛼3𝒜3𝒜∗

𝑑 𝒜 3

𝑑𝑇 = 2𝛼?§ 𝒜 3+ 2𝛼3§ 𝒜 �

𝛼?§ isalwayspositive (relatedtothefactthatthegrowth rateincreasesas𝛽 > 𝛽d;If𝛼3§ isnegativethebifurcationissupercritical;If𝛼3§ ispositivethebifurcationissubcritical;

WNLAnalysis:Landau-Stuart𝑑 𝒜 3

𝑑𝑇 = 2𝛼?§ 𝒜 3+ 2𝛼3§ 𝒜 � = 0

𝒜¨3 = −

𝛼?§

𝛼3§𝛼?§ isalwayspositive (relatedtothefactthatthegrowth rateincreasesas𝛽 > 𝛽d;If𝛼3§ isnegativethebifurcationissupercritical;If𝛼3§ ispositivethebifurcationissubcritical;Anequilibrium amplitude isreachedonlyifthebifurcationissupercritical

DUNES: EQUILIBRIUM AMPLITUDE

OBSERVED

ESTIMATED

DUNES: EQUILIBRIUM SOLUTION

BARS: EQUILIBRIUM AMPLITUDE

OBSERVED

ESTIMATED

BARS: EQUILIBRIUM SOLUTION

Finite-amplitude alternate bars 229

3

2

Bar

1

0

0

0 1 2 3 4 5 H B M

FIGURE 8. The maximum scour qM calculated for the values of ( 9 , d , ) corresponding to the experiments referred to in figure 5 is plotted versus the maximum bar height HBM. The average dependence detected by Ikeda (1982) is represented by the solid line.

FIGURE 9. An overall prospectic view of bed topography of alternate bars as predicted by the present model accurate to O(E).

to emerge, namely the formation of diagonal fronts and the increased steepness of the bottom downstream of the fronts. It may help the reader to present the values attained by the amplitudes of each second-order harmonic for the bottom elevation in the case plotted in figure 9: we find

1AeI2 (Ch22-d2.J = (7.66 x lo-', 0.24), IAe12 (e hzo-d20) = (0.15) I A , ~ ( p O 2 - d o 2 ) = (7.81 x 10-4,8.76 x 10-4).

InvitationtoriverstabilityphenomenaGaryParkerSummerschoolonstabilityofriverandcoastalforms– Perugia, ItalySeptember3-14,1990

Leave nature to its devices and it composes its own poetry. Simple rules interact to give rise to a hierarchy of structures, each nevertheless possessing, manifest or hidden, an internal symmetry that reveals itself to that part of the human mind capable of recognizing beauty. To be a scientist is to listen to the song of nature. To experience the instant when a mist of dissonance lifts to reveal the harmony of a heretofore unexplained phenomenon is to watch the sun rise on a clear day from the top of a mountain.

y Weakly nonlinear analysis of river bed forms a à i · Colombini , Seminara & Tubino (1987)...

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