On Oceanic Rogue Waves

0.61 MB·English

by  Francesco Fedele

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JOURNAL OF PHYSICAL OCEANOGRAPHY
OnOceanicRogueWaves
FRANCESCOFEDELE∗
SchoolofCivilandEnvironmentalEngineering,SchoolofElectricalandComputerEngineering,GeorgiaInstituteofTechnology,
5
Atlanta,GA,USA.
1
0
2

l
u ABSTRACT
J
Weproposeanewconceptualframeworkforthepredictionofroguewavesandthird-orderspace-time
9
extremesofwindseasthatreliesontheTayfun(1980)andJanssen(2009)modelscoupledwithAdler-Taylor

(2009)theoryontheEulercharacteristicsofrandomfields. Asaspecificapplicationofthisframework,ex-
]
h tremestatisticsofthe2007AndrearoguewaveeventareexaminedandverifiedwithestimatesfromEuropean
Reanalysis(ERA)-interimdata.Inparticular,theeffectsofnonlinearwave-waveinteractionsandspace-time
p
variabilityofthewavefieldareconsidered.ArefinementofJanssen’s(2003)theorysuggeststhatinrealistic
-
o oceanicseascharacterizedbyshort-crestedmultidirectionalwaves,homogeneousandGaussianinitialcondi-
a tionsbecomeirrelevantasthewavefieldadjuststoanon-Gaussianstatedominatedbyboundnonlinearities
s. overtimescalest(cid:29)tc≈0.13T0/νσθ,whereT0,νandσθ denotemeanwaveperiod,spectralbandwidthand
c angularspreadingofdominantwaves. FortheAndreastorm, ERA-interimpredictionsyieldtc/T0∼O(1)
indicatingthatquasi-resonantinteractionsarenegligible. Further,themeanmaximumseasurfaceheightex-
i
s pectedovertheEkofiskplatform’sareaishigherthanthatexpectedatafixedpointwithinthesamearea.
y However,bothofthesestatisticsunderestimatetheactualcrestheighthobs∼1.63Hsobservedatapointnear
h theEkofisksite,whereHsisthesignificantwaveheight. Toexplainthenatureofsuchextreme,weaccount
p forbothskewnessandkurtosiseffectsandconsiderthethresholdhqexceededwithprobabilityqbythemax-
[ imumsurfaceheightofaseastateoveranareaintime.Wefindthathobsnearlycoincideswiththethreshold
h1/1000∼1.62Hsestimatedatapointforatypical3-hourseastate,suggestingthattheAndrearoguewaveis
5 likelytobearareoccurrenceinquasi-Gaussianseas.
v
0
7 1. Introduction Thecrestheightish=18.5m(h/H =1.55)andthecrest-
s
3 to-troughheightH=25.6m(H/H =2.15)(Haver2004;
3 Rogue waves are unusually large-amplitude surface s
Magnusson and Donelan 2013). In the last decade, the
0 wavesthatappearfromnowwhereintheopenocean. Ev-
properties of the Draupner and Andrea waves have been
. idences that such extremes can occur in nature are pro-
1 extensively studied (Dysthe et al. 2008; Osborne 2010;
vided by the Draupner and Andrea events. In particular,
0
Magnusson and Donelan 2013; Bitner-Gregersen et al.
5 the Andrea rogue wave was measured just past 00 UTC
2014; Dias et al. 2015) and references therein). Several
1 on November 9 2007 by a LASAR system mounted on
: theEkofiskplatformintheNorthSeainawaterdepthof physical mechanisms have been proposed to explain the
v
i d =74 m (Magnusson and Donelan 2013). The Andrea occurrence of such giant waves (Kharif and Pelinovsky
X wavehassimilarfeaturesoftheDraupnerfreakwavemea- 2003),includingthetwocompetinghypothesesofnonlin-
r suredbyStatoilatanearbyplatform(d=70m)inJanuary earfocusingduetothird-orderquasi-resonantwave-wave
a
1995 (Haver 2001). Denoting the standard deviation of interactions (Janssen (2003)), and purely dispersive fo-
surfaceelevationsbyσ,theAndreawaveoccurredduring cusing of second-order waves (Fedele and Tayfun 2009;
aseastatewithsignificantwaveheightH =4σ =9.2m, Fedele2008).
s
meanperiodT =13.2sandwavelengthL =220m.The Forinstance,recentstudiesproposethehypothesisthat
0 0
crest height is h=15 m (h/Hs =1.63) and the crest-to- the Draupner wave occurred in crossing seas (see e.g.
troughheightH=21.1m(H/Hs=2.3)(Magnussonand Onorato et al. (2010)). These suggest that angles lying
Donelan2013). TheDraupnerwaveoccurredduringa5- intherange∼10°−30◦betweentwodominantseadirec-
hourseastatewithsignificantwaveheightHs=4σ=11.9 tionsarelikelytoleadtorogue-waveoccurrencesinduced
m,meanperiodT0=13.1sandwavelengthL0=250m. byquasi-resonantwave-waveinteractions. However,Ad-
cocketal.(2011)reportedthatthehindcastfromtheEuro-
peanCentreforMedium-RangeWeatherForecastsshows
∗Corresponding author address: Georgia Institute of Technology swellwavespropagatingatapproximately80°tothewind
Atlanta,GA30332,USA.
E-mail:[email protected]
Generatedusingv4.3.2oftheAMSLATEXtemplate 1
2 JOURNAL OF PHYSICAL OCEANOGRAPHY
sea. Adcock et al. (2011) also argued that the Draupner (2009);Toffolietal.(2010);Fedeleetal.(2010)).Therel-
waveoccurredduetothecrossingoftwoalmostorthogo- ative importance of such nonlinearities and the increased
nalwavegroupsinaccordwithsecond-ordertheory. This occurrenceoflargewavescanbemeasuredbytheexcess
would explain the set-up observed under the large wave kurtosis
(Walkeretal.2004)insteadofthesecond-orderset-down (cid:10)η4(cid:11)
λ = −3, (1)
normally expected (Longuet-Higgins and Stewart 1964). 40 (cid:104)η2(cid:105)2
However, there is no evidence of significant swell com-
ofthemean-zerosurfaceelevationη, wherebracketsde-
ponents nearby the platform as clearly seen from Fig. 2
notestatisticalaverage.Thisintegralstatisticisdefinedby
in Adcock et al. (2011) and Fig. 1 here, which shows
Janssen (2003) asC =λ /3. In general,C =Cd+Cb
theERA-interimwavedirectionalspectrumattheDraup- 4 40 4 4 4
anditcomprisesadynamiccomponentCd duetononlin-
ner site. Further, in accord with Boccotti’s (2000) quasi- 4
earwave-waveinteractions(Janssen(2003))andabound
determinism (QD) theory the probability that two differ-
contributionCb inducedbythecharacteristiccrest-trough
ent wave groups cross at the same point at the apex of 4
asymmetry of ocean waves (Tayfun (1980); Tayfun and
theirdevelopmentismuchsmallerthantheprobabilitythat
Lo (1990); Tayfun and Fedele (2007); Fedele and Tay-
oneofthetwogroupsfocusesatthesamepoint. Onecan
fun (2009); Fedele (2008); Janssen and Bidlot (2009)).
alsoarguethatreflectionanddiffractionfromtheplatform
Fordeep-waterlong-crestedseaswithaGaussian-shaped
may cause the observed set-up. However, the Draupner
spectrum and within the framework of the higher order
measurements were made from a bridge connecting two
compact Zakharov (cDZ) equation (Dyachenko and Za-
space frame structures. The structural members are rel-
kharov (2011)), Fedele (2014) showed that, correct to
atively small, likely a meter in diameter. The preceding
O(ν2) in spectral bandwidth, the dynamic excess kurto-
greatlylessensthechancesforplatforminterferencefrom
sismonotonicallyincreasesintimetowardtheasymptotic
spray, reflections or diffractions. Clearly, the Draupner
value
wave appears fundamentally different from a typical ex-
(cid:32) √ (cid:33)
pectedextremewavebecauseoftheobservedset-upofthe Cd =Cd 1−4 3+πν2 ≈Cd (cid:0)1−0.40ν2(cid:1),
meansealevel(MSL)belowthelargecrest. However,the 4,cDZ 4,NLS 8π 4,NLS
estimation of the MSL from measurements is ill-defined
andthusmaybenotrobust. Indeed, wenotethatWalker where
π
etal.(2004)estimatedthemeansealevelbylow-passfil- Cd =BFI2 √ (2)
4,NLS
tering the measured time series of the wave surface with 3 3
frequencycutoff f ∼ f /2,where f isthedominantfre- is the dynamic excess kurtosis of unidirectional nar-
c p p
quency. Clearly, the mean surface and wave fluctuations rowband waves in deep water in accord with the one-
are nonlinearly coupled and feed energetically into each dimensional(1-D)nonlinearSchro¨dinger(NLS)equation
other. Asaresult, thelow-passfilteredmeansurfaceisa (MoriandJanssen(2006)). NotethatEq.(2)isalsovalid
mixofthetwocomponents. Ifthetimeseriesisnotlong for the Dysthe (1979) equation as the associated asym-
enough for a statistically significant estimation of wave- metricspectralbroadeningisnotcapturedbytheassumed
waveinteractions,theobservedset-upcouldbethemani- symmet√ric Gaussian spectrum. The Benjamin-Feir index
festationofthelargecrestsegmentthatextendsabovethe BFI = 2µ/ν, with µ denoting an integral measure of
adjacent lowercrests. In thiswork, wewill not dwellon wave steepness defined later in Eq. (20) and the spectral
theDraupnerwave. Thiscallsforfurtherstudiesandnu- bandwidthνisgiveninEq.(21).Clearly,Cd issmaller
4,cDZ
mericalsimulationsoftheseastateinwhichitoccurredto thanCd , especially as the spectral bandwidth widens.
4,NLS
clarify the physics and robustness of the observed set-up ThisisconsistentwiththeresultthatinaccordwithcDZ
belowthelargecrest. Theseissuesaresecondaryandare the linear growth rate of a subharmonic perturbation re-
beyondthescopeofthispaper. Indeed,wewillcapitalize duceswithrespecttotheNLScounterpartforwaveswith
onrecentnumericalsimulationsoftheAndrearoguewave broader spectra. Indeed, for fixed wave steepness, the
(Bitner-Gregersenetal.2014;Diasetal.2015)inorderto initial-stage growth of instabilities away from a Stokes
studythestatisticalpropertiesandspace-timeextremesof waveisattenuatedasthespectralbandwidthincreases(Al-
theseastateinwhichitoccurred. ber(1978);Crawfordetal.(1981)). Thelate-stageevolu-
Other studies propose third-order quasi-resonant inter- tion of modulation instability leads to breathers that can
actionsandassociatedmodulationalinstabilitiesasmech- cause large waves (Peregrine 1983; Osborne et al. 2000;
anisms for rogue wave formation (e.g. Janssen (2003); Ankiewiczetal.2009),especiallyinunidirectionalwaves.
Osborne(2010)). Suchnonlineareffectscausethestatis- Indeed,inthiscaseenergyis”trapped”asinalongwave-
ticsofweaklynonlineargravitywavestosignificantlydif- guide. For small wave steepness and negligible dissipa-
fer from the Gaussian structure of linear seas, especially tion,quasi-resonantinteractionsareeffectiveinreshaping
in long-crested, or unidirectional seas (Janssen (2003); the wave spectrum, inducing larger breathers via nonlin-
Fedele(2008);Onoratoetal.(2009);ShemerandSergeeva earfocusingbeforebreakingoccurs(ShemerandSergeeva
JOURNAL OF PHYSICAL OCEANOGRAPHY 3
6 1
3
)
p
f
5 2 S( 0.5
)/
1 (f
S
4 0
0
d] −1 0 0.1 0.2 0.3 0.4
a f [Hz]
θ [r3 −2
−3
2
−4 0.1
)
θ
−5 D(
1 0.05
−6
0
0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6
f [Hz] θ [rad]
FIG.1. Draupnerstorm: ERA-interim(left)directionalspectrum(logscale)attheDraupnersite(58.2°N,2.5°E)atthetimeofmaximum
developmentofthestorm(Jan2nd199500:00UTC)and(top-right)wavefrequencyspectrumS(f)/S(fp)and(bottom-right)angulardispersion
σ2D(θ)=(cid:82)S(f,θ)df,whereσisthestandarddeviationofsurfaceelevationsand fpthedominantfrequency.Directionθ=0meansgoingtothe
northandθ=π/2totheeast(Oceanographicconvention).
(2009); Onorato et al. (2009); Chabchoub et al. (2011, (2009)). Thisisconfirmedbyarecentdataqualitycontrol
2012)). Shemer and Alperovich (2013) pointed out that and statistical analysis by Christou and Ewans (2014) of
wavebreakingisinevitableforµ>0.1,andbreatherscan single-pointmeasurementsfromfixedsensorsmountedon
beobservedexperimentallyonlyatsufficientlysmallval- offshoreplatforms,themajorityofwhichwererecordedin
ues of wave steepness (∼0.01−0.09) (Chabchoub et al. theNorthSea.Theanalysisofanensembleof122million
(2011,2012),seealsoShemerandLiberzon(2014)). Fur- individualwavesrevealed3649rogueevents, concluding
ther,theyalsonotedthat’breatherdoesnotbreath’andits that rogue waves observed at a point in time are merely
amplification is smaller than that predicted by the NLS, rareeventsinducedbydispersivefocusing.
in accord with the numerical studies of the Euler equa- More recent studies on the statistics of extreme ocean
tions(SlunyaevandShrira(2013);Slunyaevetal.(2013)). waves provide both theoretical and experimental evi-
However,typicaloceanicwindseasarenot1-Dbutshort- dences that the expected maximum sea surface height
crestedmultidirectionalwavefields. Hence,nonlinearfo- over an area in time (space-time extreme) is larger than
cusingduetomodulationaleffectsisdiminishedsinceen- that expected at a fixed point (time extreme), especially
ergycanspreaddirectionally(Onoratoetal.(2009);Tof- in short-crested multidirectional seas (Forristall (2011,
folietal.(2010)). 2015);Fedele(2012);Fedeleetal.(2013);Barbarioletal.
TheseastateoftheAndreawavewasshort-crested(see (2014)). TheoccurrenceofanextremeinGaussianfields
Fig. 2), and modulation instabilities may have played an is analogous to that of a big wave that a surfer searches
insignificantroleinthewavegrowth(Alber1978;Craw- and eventually finds (Baxevani and Rychlik (2006)). If
fordetal.1981). Further,theactualwaterdepthtowave- thesurfersearchesalargearea,hischancesofencounter-
lengthratio,namelyd/L ∼0.3,suggeststhatwaveswere ing the largest crest obviously increase. Indeed, a large
0
in transitional regime where modulation instabilities, if number of radar signatures of likely rogue waves have
any,arefurtherattenuated(see,e.g. Toffolietal.(2009)). been identified from satellite data in the context of the
Recently, Tayfun (2008) arrived at similar conclusions MaxWaveproject(RosenthalandLehner2008).However,
based on the analysis of data from the North Sea. His therelationbetweenspace-timeextremesandroguewaves
resultsindicatethatlargetimewaves(measuredatagiven isstillunclear.
point)resultfromtheconstructiveinterference(focusing) The preceding review provides the principal motiva-
ofelementarywaveswithrandomamplitudesandphases tion for introducing the theory of stochastic space-time
enhancedbysecond-ordernon-resonantinteractions. Fur- extremes and applying it to study the space-time prop-
ther,thesurfacestatisticsfollowtheTayfun(1980)distri- erties of sea states in which rogue waves occur. In this
butioninagreementwithobservations(TayfunandFedele work, we only focus on the Andrea rogue event as the
(2007);Fedele(2008);Tayfun(2008);FedeleandTayfun Draupner wave requires further studies on the observed
4 JOURNAL OF PHYSICAL OCEANOGRAPHY
6 1
1.5
)
5 1 (fp
S 0.5
0.5)/
f
(
S
4 0
0
d] −0.5 0 0.1 0.2 0.3 0.4
a f [Hz]
r3
[
θ −1
0.6
2 −1.5
−2 θ)0.4
(
1 D
−2.5 0.2
0
0.1 0.2 0.3 0.4 0.5 0 2 4 6
f [Hz] θ [rad]
FIG.2. Andreastorm: ERA-interim(left)directionalspectrum(logscale)attheEkofisksite(56.3°N,3°E)ataboutthetimeoftheAndrea
waveevent(Nov9th2007,00:00UTC)and(top-right)wavefrequencyspectrumS(f)/S(fp)and(bottom-right)angulardispersionσ2D(θ)=
(cid:82)S(f,θ)df,whereσisthestandarddeviationofsurfaceelevationsand fpthedominantfrequency.Directionθ=0meansgoingtothenorthand
θ=π/2totheeast(Oceanographicconvention).
set-up (Walker et al. 2004). We aim at presenting a new servedatapointintime(TayfunandFedele2007)). Sim-
conceptual framework for the prediction of large waves ilarly, h generalizes the crest height average c of the
q 1/n
based on the Tayfun (1980) and Janssen (2009) models 1/nfractionoflargestwaves(TayfunandFedele2007)).
coupled with Adler-Taylor (2009) theory on the Euler- The remainder of the paper is organized as follows.
characteristicsofrandomfields. Theframeworksodevel- First, theessentialelementsofJanssen’s(2003)formula-
opedcanbeappliedtostudybothsecondandthird-order tionfortheexcesskurtosisofdirectionalorshort-crested
nonlinear effects and space-time properties of sea states seas are presented (Fedele 2015). This is followed by a
where rogue waves are expected to occur. In particular, review of the theory of Euler characteristics of random
thenewstochasticmodeldescribesthestatisticalstructure fields (Adler (1981)), space-time extremes (Fedele 2012)
of the wave surface using a Gram-Charlier type distribu- andassociatedstochasticwavegroups(FedeleandTayfun
tion for crest heights formulated in (Tayfun and Fedele 2009). Wethenintroduceanewstochasticmodelforthe
2007). This is able to capture the vertical crest-trough prediction of space-time extremes that accounts for both
asymmetry induced by second-order nonlinearities mea- second and third-order nonlinearities. As a specific ap-
suredbyskewness(Tayfun1980)aswellasintermittency plicationhere,capitalizingontheERA-interimreanalysis
effectsduetothird-ordernonlinearitiesmeasuredbykur-
(Dee et al. (2011)) and numerical simulations of the An-
tosis (Janssen 2009). Moreover, the statistical properties
drea sea state (Bitner-Gregersen et al. 2014; Dias et al.
oflocalwavesinspaceandtimearerelatedtohigherorder
2015), the extreme statistics of the Andrea rogue-wave
momentsofthewavedirectionalspectrumcapitalizingon
eventisexaminedindetail. Inconcluding,wediscussthe
Adler-Taylor(2009)theory.Asaresult,weareabletode-
implicationsoftheseresultsonrogue-wavepredictions.
finetheprobabilitystructureoftherandomvariableη
max
representing the maximum third-order nonlinear surface
waveheightoveranareaintime. Inaccordwiththenew
2. Excesskurtosisofshort-crestedseas
model,wewillshowthatstatisticalaveragesofthemaxi-
mumη donotexplainrareoccurrencesoflargeoceanic Fedele (2015) revisited Janssen’s (2003) formulation
max
waves, which instead need be interpreted using quantile- forthetotalexcesskurtosisC4ofweaklynonlineargravity
typeapproaches.Inparticular,wewillconsiderthethresh- waves in deep water. This comprises a dynamic compo-
oldh exceededbyη withprobability0≤q≤1andthe nentCd duetononlinearwave-waveinteractions(Janssen
q max 4
conditionalmeanh =(cid:10)η |η >h (cid:11),wherebrackets and Bidlot (2009)) and a bound contributionCb (Janssen
q max max q 4
denotestatisticalaverage. Thestatistich isageneraliza- (2009)). Forwavesthatarenarrowbandandcharacterized
q
tion to space-time maxima of the threshold c exceeded byaGaussiantypedirectionalspectrum,Cd isexpressed
n 4
bythecrestheightofthe1/nfractionoflargestwavesob- asasix-foldintegralthatdependsontimet,BFI andthe
JOURNAL OF PHYSICAL OCEANOGRAPHY 5
parameter inspaceandtime. IntheleftpanelofFig.3, thepreced-
σ2 ingapproximationinEq.(5)iscomparedagainstthethe-
R= 2νθ2, (3) oreticalC4d,max fornarrowbandwaves(Fedele(2015), see
also Appendix A). Evidently, the latter is slightly larger
whichisadimensionlessmeasureofshort-crestednessof
thanthemaximumexcesskurtosisderivedbyJanssenand
dominant waves, with ν and σ denoting spectral band-
θ Bidlot (2009), who have also used the fit in Eq. (5) but
width and angular spreading (Janssen and Bidlot (2009);
with b=1. Their maximum follows by first taking the
Morietal.(2011)). Theassociatedexcesskurtosisgrowth
limitoftheexcesskurtosisatlargetimesandthensolving
rate can be solved analytically for narrowband waves
the associated six-fold integral (Fedele (2015)). Clearly,
(Fedele (2015), see also Appendix A). It is found that in
thedynamicexcesskurtosisshouldvanishatlargetimes.
thefocusingregime(0<R<1)thedynamicexcesskur-
Janssen (personal communication, 2014) confirmed that
tosisinitiallygrowsattainingamaximumC atthein-
4,max Eq.(A3)holdsandprovidedanalternativeproofthatthe
trinsictimescale
large-timeCd tendstozerousingcomplexanalysis.
4 √
τ =2πν2tc = √1 , or tc ∼ 0.13 (4) Further, in the focusing regime (R < 1,τc < 1/ 3),
c T0 3R T0 νσθ fromEq. (5)
givenbytheleast-squaresfit C4d,max(τc) ≈ b −1+3τc2 . (7)
BFI2 (2π)21+3bR τ2
0 c
C4d,max(R) b 1−R Clearly,themaximumkurtosisbecomeslargerforlonger
≈ , 0≤R≤1, (5)
BFI2 (2π)2R+bR0 time scales τc, as illustrated in the righ√t panel of Fig. 3.
Inthedefocusingregime(R>1,τ >1/ 3),thedynamic
√ c
where R0 = 3π3 and b = 2.48. Eventually the excess excess kurtosis is negative and its minimum valueC4d,min
dynamic kurtosis tends monotonically to zero as energy canbecomputedfromEq.(6). Thisresultholdsfordeep-
spreads directionally, as in the numerical simulations of waterwaves.DrawingonJanssenandOnorato(2007)and
Annenkov and Shrira (2009). In the defocusing regime Janssen and Bidlot (2009), the extension to intermediate
(R>1), the dynamic excess kurtosis is always negative. watersofdepthdfollowsbyredefiningtheBenjamin-Feir
It attains a minimum at tc given by (Janssen and Bidlot Index as BFIS2 =αSBFI2, where the depth factor αS de-
(2009)) pends upon the dimensionless depth k0d, where k0 is the
dominantwavenumber(seeAppendixA).Inthisworkwe
(cid:18) (cid:19)
1 choose k as the mean wavenumber k corresponding to
Cd =−RCd (R), 0≤R≤1. (6) 0 m
4,min R 4,max themeanspectralfrequencyωm=m001/m000,wheremijk
arespectralmoments(seeAppendixB).Inthedeep-water
andthentendstozerointhelongtime. Thus,thepresent limitα becomes1,andBFI reducestotheusualdefini-
S S
theoretical predictions indicate a decaying trend for the tionofBFI (Janssen(2003)). Asthedimensionlessdepth
dynamicexcesskurtosisoverlargetimes. k d decreases, BFI2 reduces and it becomes negative for
0 S
For time scales t 10t , a cold start with initial ho- k d<1.363andsothedynamicexcesskurtosis.
c 0
mogeneousandGaus(cid:63)sianconditionsbecomeirrelevantas Thus, we expect that third-order quasi-resonant inter-
thewavefieldtendstoanon-Gaussianstatedominatedby actions do not play any role in the formation of large
bound nonlinearities as the total kurtosis of surface ele- waves in realistic oceanic seas. However, the effects of
vations asymptotically approaches the value represented bound nonlinearities on skewness and kurtosis should be
by the bound component (Annenkov and Shrira (2013, accounted for in the extreme value analysis. In this re-
2014)). Intypicaloceanicstormswheredominantwaves gard,Janssen(2009)deeplyinvestigatedthepropertiesof
are characterizedwith ν ∼0.2−0.4 andσ ∼0.2−0.4, weaklynonlinearwaterwaveswithintheHamiltonianfor-
θ
thisadjustmentisrapidsincethetimescalet /T ∼O(1) malism developed by Krasitskii (1994). Further, he de-
c 0
withT ∼10−14sandthedynamickurtosispeakisneg- rivedgeneralexpressionsforskewnessandkurtosisofthe
0
ligiblecomparedtotheboundcounterpart.Fortimescales wavesurfacevalidforfinitedepthsandarbitraryspectra.
oftheorderoforlessthant ,thedynamiccomponentcan Inparticular,thecanonicaltransformationforathird-order
c
dominate and the wave field may experience rogue wave narrowband (NB) wavetrain can be evaluated explicitly
behavior induced by quasi-resonant interactions (Janssen and Janssen (2009) gives a simple expression for the NB
(2003)).However,onecanarguethatthelargeexcesskur- boundkurtosisgiveninEq. (A4)ofAppendixA.
tosis transient observed during the initial stage of evolu- Bitner-Gregersen et al. (2014) and Dias et al. (2015)
tion is a result of the unrealistic assumption that the ini- performedMonteCarlosimulationsofthehindcastedsea
tialwavefieldishomogeneousGaussianwhereasoceanic stateinwhichtheAndreawaveoccurredusingtheirrota-
wave fields are usually statistically inhomogeneous both tionalandinviscidEulerequations.Theestimatedkurtosis
6 JOURNAL OF PHYSICAL OCEANOGRAPHY
0.7
NB theory NB theory
0.6 Least−squares fit 0.6 Least−squares fit
Janssen−Bidlot fit (2009)
0.5 0.5
2 2
I I
F F
B 0.4 B 0.4
/ x / x
a a
m0.3 m0.3
d4, d4,
C C
0.2 0.2
0.1 0.1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2
R 1/τ
c
FIG.3.MaximumdynamicexcesskurtosisC4d,maxasafunctionof(left)Rand(right)1/τc:(boldcurve)presenttheoreticalprediction,(thincurve)
least-squaresfitfromEq.(5)(b=2.48)and(dashedcurve)Janssen-Bidlot(2009)fit(b=1).
ismainlyduetonon-resonantwave-waveinteractionsand resultthedynamickurtosisisroughlyhalfthecorrespond-
C =Cb ∼0.033, or λ =λb =3C ∼0.1. This is in ing value in deep waters, which is already negligible as
4 4 40 40 4
fair agreement with Janssen’s NB estimate λb,NB =0.09 theseastateisbroadbanded. Clearly,theGaussianadjust-
40
fromEq. (A4)usingERA-interimdata. Bitner-Gregersen ment timetc ∼O(T0)∼15 seconds, indicating that non-
et al. (2014) also reported a larger kurtosis (∼0.35) sev- linear wave-wave interactions are negligible. Indeed,Cd
4
eralhoursbeforetheAndreawaveoccurrence. Theactual is slightly negative, implying a defocusing wave regime
lasermeasurementsatEkofiskgiveinsteadoscillatingval- due to the short-crestedness of the sea state whereas the
ueswithintherange0−0.3indicatingunstableestimates non-zeroandpositiveCb componentindicatesthatbound
4
of kurtosis as they are based on short 20-min time series nonlinearitiesarenotnegligible.
((MagnussonandDonelan2013)). Clearly, to date numerical simulations of the Euler
Drawing on the ERA-interim reanalysis data, we now equations are computationally expensive and unfeasible
considerthehindcastedseastateduringwhichtheAndrea for an operational forecasting of extreme waves. One
waveoccurred(UTC00onNov9th,2007). Thetoppanel should consider Janssen’s general third-order expression
on the left of Fig. 4 shows the spatial distribution of sig- for kurtosis, which is valid for any depth and arbitrary
nificant wave height. The maximum H is about 8.3 m, spectra. Thisisacomplexformulawhoseimplementation
s
which is smaller than 9.2 m actually observed (Magnus- requirescareandeffortbeyondthescopeofthispaper.
son and Donelan (2013)). It is well known that ERA- In this work we propose that statistical predictions of
interim underestimates peak values and predicts broader extreme waves of realistic oceanic seas can be based on
directionalspectrabecauseofthelowspaceandtimeres- Adler-Taylor’s (2009) theory of random fields coupled
olution of the data. Indeed, wave parameters are solved withtheTayfun(1980)andJanssen(2009)modelstoac-
every 6 hours and the grid cell areal size is A ∼1002 countforbothsecondandthird-ordernonlinearities.Inthe
0
km2 with60verticallevels(Deeetal.(2011)). Neverthe- following, we will first present the theory of space-time
less, such predictions provide leading order estimates of extremesforGaussianseas(Fedele(2012)). Adler-Taylor
the sea-state parameters that can be refined further in fu- theory is then extended to third-order non-Gaussian seas
turestudies,usingforecastmodelswithhigherresolution and apply it to study the statistical properties of the An-
(PoncedeLeo´nandGuedesSoares(2014)). Thetoppan- drearoguewaveevent.
elsofFig.5showstheGaussianadjustmenttimet /T and
c 0
theJanssenNB total excesskurtosisλ =3C . TheNB
40 4 3. Space-time(ST)extremes
dynamiccomponentλd =3CdfromEq.(5)andthebound
40 4
counterpartλb =3CbfromEq.(A4)areshowninthebot- In accord with ERA-interim reanalysis, we can as-
40 4
tompanelsofthesamefigure. AttheEkofisklocationthe sume that sea states are both stationary in a time inter-
waterdepthisd=74m,sok d∼3andα =0.58. Asa val D≤D =6 hours and homogeneous over an area
0 S ERA
JOURNAL OF PHYSICAL OCEANOGRAPHY 7
H [m] Tayfun NB steepness µ
s
66oN 66oN
4
8 4 0.08
63oN 7 63oN 0.075
6 2 0.07
60oN 60oN 6 0.065
6
5
57oN 57oN 6 8 0.06
4 4 6 0.055
54oN 3 54oN 4 0.05
2 2
51oN 2 51oN 2 0.045

8oW 4oW 0o 4oE 8oE 8oW 4oW 0o 4oE 8oE
*
Wave dimension β Boccotti Ψ
66oN 66oN
4 4 2.9 4 4 0.75
6 6
63oN 63oN 0.7
2.85
60oN 60oN 0.65
6 4 6 4
57oN 6 8 2 2.8 57oN 6 8 2 0.6
64 64 0.55
54oN 2 54oN 2
2 2.75 2 0.5
51oN 2 51oN 2

8oW 4oW 0o 4oE 8oE 8oW 4oW 0o 4oE 8oE
FIG.4.ERA-interimreanalysisatthepeakoftheAndreastorm.Toppanels:(left)significantwaveheightHs=4σand(right)TayfunNBwave
steepnessµS(Eq. (22)). Bottompanels: (left)wavedimensionβ (Eq. (12))and(right)narrowbandednessBoccottiparameterψ∗. Dashedlines
areHscontours.ThetrianglesymbolindicatestheEkofisksiteposition.
A≤A =1002 km2. Then,thefreesurfaceη(x,t)can it, plus the number of hollows inside. Further, the prob-
ERA
be modeled as a three-dimensional (3-D) homogeneous abilityofexceedancethattheglobalmaximumofη over
Gaussianrandomfieldoverthespace-timevolumeΩde- Ω, say ηmax, exceeds z depends on the domain size and
finedbytheareaAandthetimeintervalD,andx=(x,y) it is well approximated by the expected EC of the excur-
denotesthehorizontalcoordinatevector.Thus,theassoci- sionset,providedthatzissufficientlyhigh(Adler(1981,
atedprobabilitydistributionsatanypointofthevolumeis 2000);AdlerandTaylor(2009)).Intuitively,aszincreases
thesameandGaussian.DrawingonAdler(1981),wenext theholesandhollowsintheexcursionsetUΩ(z)disappear
considertheEulercharacteristics(EC)ofexcursionsetsof until each of its connected components includes just one
η defined as follows. Given a threshold z, the excursion local maximum of η, and EC counts the number of lo-
setU (z)isthepartofΩwithinwhichη isabovez: calmaxima. Forverylargethresholds,ECequals1ifthe
Ω
global maximum exceeds the threshold and 0 otherwise.
Thus,ECoflargeexcursionsetsisabinaryrandomvari-
U (z)={(x,t)∈Ω:η(x,t)>z}. (8)
Ω
ablewithstates0and1,and,forlargez,
In1-DGaussianprocesses,theECsimplycountsthenum- Pr{η >z}=Pr{EC(U (z))=1}=(cid:104)EC(U (z))(cid:105),
max Ω Ω
ber of z-upcrossings. Thus, (8) provides the generaliza- (9)
tion of this concept to to higher dimensions. Indeed, for where angled brackets denote expectation. This heuris-
twodimensional(2-D)randomfields,ECcountsthenum- tic identity has been proved rigorously to hold up to an
ber of connected components minus the number of holes errorthatisingeneralexponentiallysmallerthantheex-
of the respective excursion set. In 3-D sets instead, EC pectedECapproximation(AdlerandTaylor(2009);Adler
counts the number of connected volumetric components (2000)). For 3-D random fields, which are of interest in
of the set, minus the number of holes that pass through oceanicapplications,theprobabilityP (ξ;A,D)thatthe
ST
8 JOURNAL OF PHYSICAL OCEANOGRAPHY
Gaussian adjustment time t/T Janssen NB total kurtosis λ
0 40
66oN 66oN
4 4 0.21 4 4 0.12
63oN 0.2 63oN 0.1
0 0
2 2
60oN 6 0.19 60oN 6
6 6 0.08
57oN 6 8 0.18 57oN 6 8
4 0 6 4 0 6 0.06
0.17
54oN 4 54oN 4
51oN 2 2 0 2 0 0.16 51oN 2 2 0 2 0 0.04
8oW 4oW 0o 4oE 8oE 8oW 4oW 0o 4oE 8oE
Janssen NB dynamic kurtosis λd Janssen NB bound kurtosis λb
40 40
x 10−3
66oN 66oN
4 4
4 −1 4 0.12
63oN 63oN
−2
0 0 0.1
2 2
60oN 6 −3 60oN 6
6 6 0.08
57oN 6 8 −4 57oN 6 8
0 6 0 6
4 4 0.06
54oN 4 −5 54oN 4
2 0 2 2 0 2
51oN 2 0 −6 51oN 2 0 0.04
8oW 4oW 0o 4oE 8oE 8oW 4oW 0o 4oE 8oE
FIG.5. ERA-interimreanalysisatthepeakoftheAndreastorm. Toppanels: (left)Gaussianadjustmenttimetc/T0(Eq. (4))and(right)total
NBkurtosisλ40=λ4d0+λ4b0. Bottompanels: (left)NBdynamickurtosisλ4d0(Eq. (5))and(right)NBboundkurtosisλ4b0. DashedlinesareHs
contours[m].ThetrianglesymbolindicatestheEkofisksiteposition.
maximumsurfaceelevationη overtheareaAanddur- whereξ isthemostprobablesurfaceelevation,ormodal
max m
ing a time interval D exceeds the threshold ξH is given value which, according to Gumbel (1958) and Eq. (10),
s
by(AdlerandTaylor(2009)) satisfies
P (ξ;A,D)=Pr{η >ξH }
ST max s
(10) P (ξ ;A,D)=(16M ξ2+4M ξ +M )P (ξ )=1.
=(16M ξ2+4M ξ+M )P (ξ), ST m 3 m 2 m 1 R m
3 2 1 R (13)
where Drawing on Fedele (2012), the correct average number
P (ξ)=Pr{h>ξH }=exp(−8ξ2) (11) ofspace-timewavesofdimensionβ occurringwithinthe
R s
space-timevolumeΩspannedbyareaAandtimeinterval
istheRayleighexceedanceprobabilityofthecrestheight
Dis
hofatimewaveobservedatasinglepointwithinA.Here,
M1 andM2 aretheaveragenumberof1-Dand2-Dwaves N (A,D)= 16M3ξm2+4M2ξm+M1. (14)
thatcanoccurontheedgesandboundariesofthevolume ST (4ξ )β−1
m
Ω, and M is the average number of 3-D waves that can
3
occur within the volume (Fedele (2012)). These all de- The parameter β represents a scale dimension of waves,
pend on the directional wave spectrum and are given in i.e. therelativescaleofaspace-timewavewithrespectto
AppendixA. thevolume’ssizeand1≤β ≤3.Inparticular,ifwaveex-
Astatisticalindicatorofthegeometryofspace-timeex- tremesare3-D(β >2)theyareexpectedtooccurwithin
tremesinthevolumeΩisthewavedimensionβ defined thevolumeΩawayfromtheboundaries,whereasthelim-
byFedele(2012)as iting case of 1-D time extremes (β ∼1) occur for time
4M ξ +2M waves observed at a single point. Furthermore, Fedele
β =3− 2 m 1 , (12)
16M ξ2+4M ξ +M (2012) showed that space-time extremes are larger than
3 m 2 m 1
JOURNAL OF PHYSICAL OCEANOGRAPHY 9
time extremes in agreement with recent stereo measure- where µ =k σ, whichisanupperboundforC . From
m m 3
mentsofoceanicseastates(Fedeleetal.(2013),seealso thelineardispersionrelationk =ω2/gisthewavenum-
m m
Fig.7). ber corresponding to the mean spectral frequency ω =
m
The bottom-left panel of Fig. 4 shows the map of the m /m ,
001 000
estimatedwavedimensionβ fortheNorthSeaareaatthe
(cid:113)
peak time of the Andrea storm for sea states of duration ν = m m /m2 −1 (21)
000 002 001
D=3 hours over the cell area A . Clearly, sea states
ERA
wereshort-crestedandextremesareroughly3-D,indicat- is the spectral bandwidth and m are spectral moments
ijk
ingthattheareaconsideredislargecomparedtothemean (see Appendix B). In intermediate water depths, the nar-
wavelength. Thus, in accord with Boccotti’s (2000) QD rowbandapproximationsleadto(Tayfun2006)
theory, a space-time extreme most likely coincides with
thecrestofafocusingwavegroupthatpassesthroughthe µ∼µS=µmfS, (22)
areaasdescribedbelow.
where f =D +D ,with
S 1 2
4. Extremenonlinearwavegroups 1 4n−1
D = ,
1 2n2tanh(q )−q
Drawing on Boccotti’s (2000) QD theory, Fedele and m m
Tayfun(2009)showedthatforsecond-orderweaklynon- cosh(qm)[2+cosh(2qm)]
D = ,
linear waves the expected space-time dynamics near a 2 2sinh3(q )
m
largewavecrestisthatofastochasticwavegroupwhose
freesurfaceisdescribedby n=[1+2qm/sinh(2qm)]/2, qm =kmd and d is the wa-
ter depth. The coefficients D and D arise from the
1 2
ζ /H =ξ ζ +ξ2ζ , (15) frequency-difference and frequency-sum terms, i.e. A−
c s 0 1 0 2 12
and A+, in Eq. (18) as the spectral bandwidth ν →0. A
12
whereξ0=h0/Hsisthedimensionlesslinearcrestheight, generalexpressionforskewnessvalidforfinitedepthsand
arbitraryspectraisgivenbyJanssen(2009). TheNBlimit
ζ (X,T)=Ψ(X,T) (16)
1 of Janssen’s skewness yields the same Eq. (22) derived
byTayfun(2006).
isthelinearcomponent,
Further, the wave trough following the large crest oc-
(cid:104)η(x,t)η(x+X,t+T)(cid:105) curs att =T∗, where T∗ is the abscissa of the first min-
Ψ(X,T)=
σ2 imumofthetimecovarianceψ(T)=Ψ(X=0,T)(Boc-
(17)
(cid:90) S cotti2000). Second-ordernonlinearitiesdonotaffectthe
= σ12cos(χ1)dω1dθ1 crest-to-trough heights of large waves since wave crests
andtroughsaredisplacedupwardequally. Thus,themax-
isthespace-timecovarianceofη (Boccotti(2000))and imum second-order nonlinear crest-to-trough height ob-
served at a point in time remains essentially the same
(cid:90) S S (cid:16)
ζ2= σ132 A+12cos(χ1+χ2)+ (18) ξas(t1h+atψo∗f),thwehelirneeaψr∗g=roψup(Tζ∗1) iasndthegiBveonccobtyti’Hs /(2H0s00=)
(cid:17) 0
A− cos(χ −χ ) dω dθ dω dθ narrowbandedness parameter. Note that for narrowband
12 1 2 1 1 2 2
waves ψ∗ →1. The left panels of Fig. 4 show the maps
is the second-order correction. Here, S = S(ω ,θ ) of the Tayfun NB steepness µ (top) from Eq. (22) and
j j j
and χ = k · X − ω T, where X=(X,Y) and Boccottiψ∗(bottom)estimatedfromthehindcastedERA-
j j j
k =(k sinθ,k cosθ) with k tanh(k d) = ω2/g from interim sea state in which the Andrea wave occurred.
lijnear djisperjsiojn, andj the coeffij cientsjA± canjbe found It appears that ψ∗ ∼ 0.75 as the characteristic value of
12 short-crested sea states dominated by wind waves (Boc-
in Sharma and Dean (1979). For generic sea states,
cotti (2000)). At the Ekofisk location, the NB prediction
the largest nonlinear crest amplitude is attained at the
givesµ=C /3∼0.05infairagreementwiththepredicted
focusingpoint(X=0,T =0)andgivenby 3
valuesfromnumericalsimulationsoftheEulerequations
ξ =ξ +2µξ2, (19) (Bitner-Gregersenetal.2014;Diasetal.2015)andactual
0 0
lasermeasurements(MagnussonandDonelan2013)).
where ξ =h/H is the nonlinear crest height. The Tay- Wehaveshownthatthatthird-orderquasi-resonantin-
s
funwavesteepnessµ=C /3relatestothewaveskewness teractions do not play any role in the formation of large
3
C ofsurfaceelevations. Foroceanicapplicationsindeep waves in realistic oceanic seas. However, the effects of
3
waters, Fedele and Tayfun (2009) proposed the approxi- third-orderboundnonlinearitiesonthepredictionofmax-
mation imum crest and wave heights must be accounted for as
µ∼µ =µ (cid:0)1−ν+ν2(cid:1), (20) addressedinthenextsection.
a m
10 JOURNAL OF PHYSICAL OCEANOGRAPHY
5. A new stochastic third-order space-time (FST) the wave surface relying on Janssen’s (2009) third-order
model Hamiltonian formulation, but this is beyond the scope of
thispaper.
DrawingonFedele(2008,2012)andTayfunandFedele
Giventheprobabilitystructureofthewavesurfacede-
(2007), weproposeanewstochasticmodel, hereafterre-
finedbyEq.(23),thenonlinearmeanmaximumsurfaceor
ferred to as FST, which accounts for both second and
crestheighth =ξ H attainedovertheareaAduring
third-order nonlinearities in the prediction of space-time FST FST s
atimeintervalDisgiven,accordingtoGumbel(1958),by
extremes. Inparticular,considera3-Dnon-Gaussianfield
over an area A for a time period of D. Clearly, the area
h
cannot be too large since the wave field may not be ho- ξFST= FST
H
mogeneous. Thedurationshouldbeshortsothatspectral s
γ (1+4µξ )
changesoccurringintimearenotsosignificantandthesea =ξ +2µξ2+ e m ,
state can be assumed as stationary. Then, the third-order m m 16ξ − 32M3ξm+4M2 −ΛG(ξ )
(nl) m 16M3ξm2+4M2ξm+M1 m
nonlinearprobabilityP (ξ;A,D)thatthemaximumsur- (29)
FST
faceelevationη overtheareaAandduringthetimein-
max
tervalDexceedsthegenericthresholdξHs isequaltothe wherethemostprobablesurfaceelevationvalueξm satis-
probabilityofexceedingthethresholdξ0,whichaccounts fiesPFST(ξm;A,D)=1,orequivalentlyfromEq.(23)
forkurtosiseffectsonly,thatis
(16M ξ2+4M ξ +M )P (ξ )(cid:0)1+Λξ2(4ξ2−1)(cid:1)=1,
3 m 2 m 1 R m m m
P(nl)(ξ;A,D)=P (ξ ;A,D)(cid:0)1+Λξ2(4ξ2−1)(cid:1). (23)
FST ST 0 0 0 and
2ξ (8ξ2−1)
The Gaussian probability of exceedance PST is given in G(ξm)= 1+Λmξ2(4mξ2−1).
Eq. (10) and the amplitude ξ, which accounts for both m m
skewnessandkurtosiseffects,relatestoξ0 viatheTayfun The nonlinear mean maximum surface or crest height hT
(1980)quadraticequation expectedatapointduringthetimeintervalDfollowsfrom
Eq.(29)bysettingM =M =0andM =N ,namely
2 3 1 D
ξ =ξ +2µξ2. (24)
0 0
γ (1+4µξ )
ξ =h /H =ξ +2µξ2+ e m , (30)
Further,theparameter T T s m m 16ξ −ΛG(ξ )
m m
Λ=λ40+2λ22+λ04 (25) where,now,ξmsatisfies
isameasureofthird-ordernonlinearitiesanditisafunc- N P (ξ )(cid:0)1+Λξ2(4ξ2−1)(cid:1)=1.
D R m m m
tionofthefourthordercumulantsλ ofthewavesurface
nm
η anditsHilberttransformηˆ (TayfunandFedele2007) Here, ND = D/T¯ denotes the number of wave occur-
ring during D and T¯ is the mean up-crossing period (see
λ =(cid:104)ηnηˆm(cid:105)/σn+m+(−1)m/2(n−1)(m−1), (26) Appendix B). The linear mean counterpart follows from
nm
Eq.(30)bysettingµ=0andΛ=0.
whereσ isthestandarddeviationofthewavesurfaceand Drawing on the Boccotti (2000) distribution for wave
n+m=4. DrawingonMoriandJanssen(2006), weas- heights, the third-order nonlinear mean maximum wave
sumethefollowingrelationsbetweencumulants heightexpectedatapointisgivenby
λ22=λ40/3, λ04=λ40, (27) HT=h(TΛ)(cid:112)2(1+ψ∗), (31)
andEq.(25)simplifiesto (Λ)
where h is the nonlinear mean maximum crest height
T
thataccountsforkurtosiseffectsonly. Thisfollowsfrom
8λ
Λ=Λappr= 40, (28) Eq.(30)bysettingµ=0,namely
3
whichwillbeusedinthiswork. Then,Eq.(23)reducesto h(Λ)/H =ξ + γe , (32)
T s m 16ξ −ΛG(ξ )
amodifiedEdgeworth-Rayleigh(MER)distribution(Mori m m
and Janssen 2006). For realistic oceanic seas the kur-
where
tosis λ ∼λb is only due to bound nonlinearities. To
40 40
date,thevalidityofthecumulantrelationsinEq.(27)has N P (ξ )(cid:0)1+Λξ2(4ξ2−1)(cid:1)=1.
D R m m m
beenproventoholdforsecond-orderNBwavesonly(Tay-
fun and Lo 1990). Further studies are desirable to deter- For narrowband waves, ψ∗ =1 and Eq. (31) reduces to
minethegeneralexpressionsoffourthordercumulantsof the MER model proposed by Mori and Janssen (2006).