ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell

WorkshoponOceanWaveDynamics‐FieldsDianeHenderson,HarveySegurPennStateU UofColorado

!

!

!t" +!# "!" =!

z#,

!t# +

1

2|!# |2 +g" =

$

%!" (

!"

1+ |!" |2),

on z = !(x,y,t),

! " = 0 -h(x,y) < z < !(x,y,t),

!

"z# +$# %$h = 0 on z = –h(x,y). !

Preliminaries:Stokes’equa2onsofwaterwaves(1847)

Overallobjec2ve:

Findagood(approximate)model,topredictaccuratelytheevolu2onofoceanswellasitpropagatesoverlongdistancesintheocean.Candidate#1:nonlinearSchrödingereq’nCandidate#2:dampednonlinearSchrödingereq’nCandidate#3:???

Chapter1:NonlinearSchrödingerequa2on

(Zakharov,1968)Anapproximatemodelforwavesondeepwater:

i!"A+#!x2A+$!y

2A+% | A |2 A = 0

Chapter1:NonlinearSchrödingerequa2on

(Zakharov,1968)Anapproximatemodelforwavesondeepwater:

surface slowmodula2on fastoscilla2onseleva2on

i!"A+#!x2A+$!y

2A+% | A |2 A = 0

!(X,Y,T;") ~ "[A("(X ! cgT ),"Y,"2X) "e

i#+ A

*e!i#]+O(" 2 )

BIGdiscoveryinthe1960s:

Themodula2onalinstability(orBenjamin‐Feirinstability)wasdiscoveredbyseveralpeople,indifferentscien2ficdisciplines,indifferentcountries,usingdifferentmethods:

Lighthill(1965),Whitham(1967),Zakharov(1967,1968),Ostrovsky(1967),Benjamin&Feir(1967),Benjamin(1967),Benney&Newell(1967),…

Modula2onalinstability

•  Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds

Modula2onalinstability

•  Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds

•  Inadispersivemediumwithoutdissipa2on,auniformtrainofplanewavesoffiniteamplitudeislikelytobeunstable

Modula2onalinstability

•  Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds

•  Inadispersivemediumwithoutdissipa2on,auniformtrainofplanewavesoffiniteamplitudeislikelytobeunstable

•  Maximumgrowthrateof(nonlinear)instability:

=amplitudeofcarrierwave

! = K A0

2

A0

Experimentalevidenceofmodula2onalinstabilityindeepwater‐Benjamin(1967)

nearthewavemaker 60mdownstream “uniform” wavetrain “disintegrated”mess

frequency=0.85Hz,wavelength=2.2m, waterdepth=7.6m

Experimental evidence of modulational instability of EM waves in an optical fiber

•  Tai, Hasegawa •  & Tomita (1986)

L = 1.3*10-6 m, •  T = 4*10-15 s

•  Recall: Recall:

• 

! = K A0

2

Ques2ons:mapfrom

Snodgrassetal,1966

StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.

Q1:Ifoceanswellisunstable,howdo

wavestravelcoherentlyover13,000km?

Ques2on2:

Lakeetal(1977)soughtexperimentalevidenceofFPUrecurrenceondeepwaterIni2alfrequency:ω =3.6Hzλ=12cm

Lake,Yuen,Rungaldier,Ferguson(1977)

Frequencydownshiling,whichisimpossibleinNLS

AlbertEinstein

“Everythingshouldbemadeassimpleaspossible,butnotsimpler.”

Chapter2:DampednonlinearSchrödingerequa2on

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.•Foranyδ≥0,thereisnodownshiling,accordingtodampedNLS.=>Thismathema2calmodelhasthepoten2altoansweroneofthetwoques2ons.

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.Q:Whatmakesthisinstabilitysounusual?

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Q:Whatmakesthisinstabilitysounusual?

Standardsitua2on:Anon‐dissipa2vemodelpredictsaninstabilitywithgrowthrateΩ.

Withphysicaldissipa2on(notinmodel),expect:

Observedgrowthrate=Predictedgrowthrate–physicaldecayrate

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Q:Whatmakesthisinstabilitysounusual?

NLS:Predictedgrowthrate

i!"A+#!x2A+$!y

2A+% | A |2 A = 0

! = K A0

2

Q:Whatmakesthisinstabilitysounusual?

NLS:Predictedgrowthrate

DampedNLS:

Observedgrowthrate=

Predictedgrowthrate–physicaldecayrate

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

! = K A0

2

! = K A0

2

"e#2 A

0

2!

Experimentalverifica2onoftheory

(former)1‐DtankatPennState

Experimentalwaverecords

x1

x66

Amplitudesofseededsidebands(dampingfactoredoutofdata)

(withoveralldecayfactoredout) ___dampedNLStheory ‐‐‐Benjamin‐Feirgrowthrate experimentaldata

Q:Whatifnonlinearity>>dissipa2on?

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

Q:Whatifnonlinearity>>dissipa2on?A:Frequencydownshiling

notpredictedbyeitherNLS(δ=0orδ>0)

x1 x12

Recallthe2tleoftalk:

ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell

Q:Dothetheoryandthelaboratoryexperimentsactuallypredictwhathappenstooceanswell?

RecallSnodgrassetal,1966

StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.

Q:Howmuchdissipa2ondidtheswelltrackedby

Snodgrassetalexperience?

RecallSnodgrassetal,1966

StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.

Q:Howmuchdissipa2ondidtheswelltrackedby

Snodgrassetalexperience?

Snodgrass,p.432:“negligibleatenua2on”

Figure20ofSnodgrassetal(1966)Wavespectra,measuredat12‐hourintervalsat4sequen2almeasuringsta2ons,arenarrowatTutuila,andbecomenarroweratsubsequentsta2ons..

DatafromSnodgrassetal(1966)August1.9storm

Energydecayrate:Δ=0.43x10‐3km‐1

DatafromSnodgrassetal(1966)August13.7storm

Energydecayrate:

Δ=0.25x10‐3km‐1

SARdatafromCollardetal.(2009)

Sta2s2calaveragefor15‐secondwaves,over35swelltracks:

Energydecayrate:Δ =0.37x10‐3km‐1Uncertainty:0.31x10‐3<Δ<0.40x10‐3km‐1

Measuredenergy‐decayratesoffreelypropaga2ngwaves

Event k0(m‐1) Δ(m‐1)

Aug1.9(S) 0.017 0.43x10‐6Aug13.7(S) 0.016 0.25x10‐6Jul23.2(S) 0.014 0.23x10‐6

Collard0.018 0.37x10‐6PSUlab 44.1 0.22

HowtorelateΔtoδ ? Recalldissipa2veNLS:Derivedusingasmallparameter: ε=2|A0|k0<<1, τ=ε2k0X

=>tonondimensionalizeΔ:

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

! =!

2" 2k0

Dimensionlessdecayratesoffreelypropaga2ngwaves

Event k0(m‐1) Δ(m‐1) ε δ

Aug1.9(S) 0.017 0.43x10‐6 0.011 0.105Aug13.7(S) 0.016 0.25x10‐6 0.011 0.065Jul23.2(S) 0.014 0.23x10‐6 0.0046 0.39

Collard0.018 0.37x10‐6 0.029 0.012PSUlab 44.1 0.22 0.10 0.25

Conclusions

•  Dissipa2onisimportantintheevolu2onofsurfacewaves,inthelabandintheocean

•  Dissipa2oncanactonthesamedistance‐scaleasnonlinearityanddispersion

•  Frequencydownshilingoccursinthelabandintheocean

•  Openques2on:whatcausesthedissipa2on?•  Openques2on:whatcausesdownshiling?

Thankyouforyouraten2on

Downshilingofwavetrains

Define:

Show:

i!"A+#!x2A+$!y

2A+% | A |2 A+ i&A = 0

M (! ) = A(x, y,! )2

D! dxdy, P

1(! ) = i A

*"xA# A"xA*$% &'D

! dxdy

M (! ) =M (0) !e"2"! , P(! ) = P(0) !e"2"! ,

!P(! )

M (! )=P(0)

M (0)= average_ frequency

DownshilinginSnodgrassdata?

Recall:dissipa2veNLS=>P(τ)/M(τ)=constant

Jul23.2 Aug1.9 Aug.13.7

Theroleofdissipa2onintheevolu2onofoceanswell

IMACSConference–2013DianeHenderson,HarveySegurPennStateU UofColorado

ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell

AMSsec2onalmee2ng–Boulder,2013

DianeHenderson,HarveySegurPennStateU UofColorado

Conclusions1.  Thedampingrateforoceanswellisvastlysmallerthan

thatforlaboratorywaterwaves.Butoceanswellisalsolessnonlinearthantypicallaboratorywaves.Theimportantparameterisδ,whichcomparesdistance‐scalesofdampingandnonlinearity.

2.Therangeofvaluesofδforoceanswelloverlapstherangeofvaluesforlabwaves.

3.Foroceanswellwithsmallenoughnonlinearity,dissipa2onimpedesandcanstopthemodula2onalinstability.Frequencydownshilingoccursforlabwavesandforoceanswell.ItisnotpredictedbyNLS,withorwithoutdamping.

ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell

WorkshoponOceanWaveDynamicsDianeHenderson,HarveySegurPennStateU UofColorado