the nonlinear schrödinger equaon, dissipaon and ocean swell · 2013. 5. 8. · overall objecve:...
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ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell
WorkshoponOceanWaveDynamics‐FieldsDianeHenderson,HarveySegurPennStateU UofColorado
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!
!
!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)
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Overallobjec2ve:
Findagood(approximate)model,topredictaccuratelytheevolu2onofoceanswellasitpropagatesoverlongdistancesintheocean.Candidate#1:nonlinearSchrödingereq’nCandidate#2:dampednonlinearSchrödingereq’nCandidate#3:???
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Chapter1:NonlinearSchrödingerequa2on
(Zakharov,1968)Anapproximatemodelforwavesondeepwater:
i!"A+#!x2A+$!y
2A+% | A |2 A = 0
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Chapter1:NonlinearSchrödingerequa2on
(Zakharov,1968)Anapproximatemodelforwavesondeepwater:
surface slowmodula2on fastoscilla2onseleva2on
i!"A+#!x2A+$!y
2A+% | A |2 A = 0
!(X,Y,T;") ~ "[A("(X ! cgT ),"Y,"2X) "e
i#+ A
*e!i#]+O(" 2 )
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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),…
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Modula2onalinstability
• Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds
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Modula2onalinstability
• Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds
• Inadispersivemediumwithoutdissipa2on,auniformtrainofplanewavesoffiniteamplitudeislikelytobeunstable
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Modula2onalinstability
• Dispersivemedium:wavesatdifferentfrequenciestravelatdifferentspeeds
• Inadispersivemediumwithoutdissipa2on,auniformtrainofplanewavesoffiniteamplitudeislikelytobeunstable
• Maximumgrowthrateof(nonlinear)instability:
=amplitudeofcarrierwave
! = K A0
2
A0
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Experimentalevidenceofmodula2onalinstabilityindeepwater‐Benjamin(1967)
nearthewavemaker 60mdownstream “uniform” wavetrain “disintegrated”mess
frequency=0.85Hz,wavelength=2.2m, waterdepth=7.6m
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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
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Ques2ons:mapfrom
Snodgrassetal,1966
StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.
Q1:Ifoceanswellisunstable,howdo
wavestravelcoherentlyover13,000km?
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Ques2on2:
Lakeetal(1977)soughtexperimentalevidenceofFPUrecurrenceondeepwaterIni2alfrequency:ω =3.6Hzλ=12cm
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Lake,Yuen,Rungaldier,Ferguson(1977)
Frequencydownshiling,whichisimpossibleinNLS
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AlbertEinstein
“Everythingshouldbemadeassimpleaspossible,butnotsimpler.”
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Chapter2:DampednonlinearSchrödingerequa2on
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.•Foranyδ≥0,thereisnodownshiling,accordingtodampedNLS.=>Thismathema2calmodelhasthepoten2altoansweroneofthetwoques2ons.
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Mathema2calresults(Seguretal,2005)•Auniformtrainofoscillatoryplanewavesoffiniteamplitudeondeepwaterisunstableifδ=0.•Butthesamewavetrainisstableforanyδ>0.Q:Whatmakesthisinstabilitysounusual?
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Q:Whatmakesthisinstabilitysounusual?
Standardsitua2on:Anon‐dissipa2vemodelpredictsaninstabilitywithgrowthrateΩ.
Withphysicaldissipa2on(notinmodel),expect:
Observedgrowthrate=Predictedgrowthrate–physicaldecayrate
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Q:Whatmakesthisinstabilitysounusual?
NLS:Predictedgrowthrate
i!"A+#!x2A+$!y
2A+% | A |2 A = 0
! = K A0
2
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Q:Whatmakesthisinstabilitysounusual?
NLS:Predictedgrowthrate
DampedNLS:
Observedgrowthrate=
Predictedgrowthrate–physicaldecayrate
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
! = K A0
2
! = K A0
2
"e#2 A
0
2!
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Experimentalverifica2onoftheory
(former)1‐DtankatPennState
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Experimentalwaverecords
x1
x66
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Amplitudesofseededsidebands(dampingfactoredoutofdata)
(withoveralldecayfactoredout) ___dampedNLStheory ‐‐‐Benjamin‐Feirgrowthrate experimentaldata
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Q:Whatifnonlinearity>>dissipa2on?
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
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Q:Whatifnonlinearity>>dissipa2on?A:Frequencydownshiling
notpredictedbyeitherNLS(δ=0orδ>0)
x1 x12
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Recallthe2tleoftalk:
ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell
Q:Dothetheoryandthelaboratoryexperimentsactuallypredictwhathappenstooceanswell?
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RecallSnodgrassetal,1966
StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.
Q:Howmuchdissipa2ondidtheswelltrackedby
Snodgrassetalexperience?
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RecallSnodgrassetal,1966
StormsnearAntarc2cageneratedoceanswellthatpropagated13,000kmacrossthePacific.
Q:Howmuchdissipa2ondidtheswelltrackedby
Snodgrassetalexperience?
Snodgrass,p.432:“negligibleatenua2on”
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Figure20ofSnodgrassetal(1966)Wavespectra,measuredat12‐hourintervalsat4sequen2almeasuringsta2ons,arenarrowatTutuila,andbecomenarroweratsubsequentsta2ons..
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DatafromSnodgrassetal(1966)August1.9storm
Energydecayrate:Δ=0.43x10‐3km‐1
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DatafromSnodgrassetal(1966)August13.7storm
Energydecayrate:
Δ=0.25x10‐3km‐1
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SARdatafromCollardetal.(2009)
Sta2s2calaveragefor15‐secondwaves,over35swelltracks:
Energydecayrate:Δ =0.37x10‐3km‐1Uncertainty:0.31x10‐3<Δ<0.40x10‐3km‐1
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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
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HowtorelateΔtoδ ? Recalldissipa2veNLS:Derivedusingasmallparameter: ε=2|A0|k0<<1, τ=ε2k0X
=>tonondimensionalizeΔ:
i!"A+#!x2A+$!y
2A+% | A |2 A+ i&A = 0
! =!
2" 2k0
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Dimensionlessdecayratesoffreelypropaga2ngwaves
Event k0(m‐1) Δ(m‐1) ε δ
Aug1.9(S) 0.017 0.43x10‐6 0.011 0.105Aug13.7(S) 0.016 0.25x10‐6 0.011 0.065Jul23.2(S) 0.014 0.23x10‐6 0.0046 0.39
Collard0.018 0.37x10‐6 0.029 0.012PSUlab 44.1 0.22 0.10 0.25
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Conclusions
• Dissipa2onisimportantintheevolu2onofsurfacewaves,inthelabandintheocean
• Dissipa2oncanactonthesamedistance‐scaleasnonlinearityanddispersion
• Frequencydownshilingoccursinthelabandintheocean
• Openques2on:whatcausesthedissipa2on?• Openques2on:whatcausesdownshiling?
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Thankyouforyouraten2on
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Downshilingofwavetrains
Define:
Show:
i!"A+#!x2A+$!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
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DownshilinginSnodgrassdata?
Recall:dissipa2veNLS=>P(τ)/M(τ)=constant
Jul23.2 Aug1.9 Aug.13.7
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Theroleofdissipa2onintheevolu2onofoceanswell
IMACSConference–2013DianeHenderson,HarveySegurPennStateU UofColorado
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ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell
AMSsec2onalmee2ng–Boulder,2013
DianeHenderson,HarveySegurPennStateU UofColorado
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Conclusions1. Thedampingrateforoceanswellisvastlysmallerthan
thatforlaboratorywaterwaves.Butoceanswellisalsolessnonlinearthantypicallaboratorywaves.Theimportantparameterisδ,whichcomparesdistance‐scalesofdampingandnonlinearity.
2.Therangeofvaluesofδforoceanswelloverlapstherangeofvaluesforlabwaves.
3.Foroceanswellwithsmallenoughnonlinearity,dissipa2onimpedesandcanstopthemodula2onalinstability.Frequencydownshilingoccursforlabwavesandforoceanswell.ItisnotpredictedbyNLS,withorwithoutdamping.
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ThenonlinearSchrödingerequa2on,dissipa2onandoceanswell
WorkshoponOceanWaveDynamicsDianeHenderson,HarveySegurPennStateU UofColorado