DTIC ADA591467: Frequency Width in Predictions of Windsea Spectra and the Role of the Nonlinear Solver

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OceanModellingxxx(2013)xxx–xxx
ContentslistsavailableatSciVerseScienceDirect
Ocean Modelling
journal homepage: www.elsevier.com/locate/ocemod
Frequency width in predictions of windsea spectra and the role of the
nonlinear solver
W. Erick Rogersa,⇑, Gerbrant Ph. Van Vledderb,c
aOceanographyDivision,NavalResearchLaboratory,StennisSpaceCenter,MS,USA
bDelftUniversityofTechnology,CivilEngineeringandGeosciences,EnvironmentalFluidMechanics,TheNetherlands
cBMTArgoss,Vollenhove,TheNetherlands
a r t i c l e i n f o a b s t r a c t
Articlehistory: Inthispaper,theaccuracyofpredictionsofthenarrownessinfrequencyspaceofelevationspectrafor
Availableonlinexxxx wind-generatedsurfacegravitywavesisevaluatedwiththespecificobjectiveofdeterminingtheimpact
ofthemethodforcomputingquadrupletinteractions,S .Alternatemetricsarepresentedforconcise
nl4
Keywords: quantificationofthisnarrownessandappliedtoacasestudy:a10-daydurationhindcastforLakeMich-
Windwaves iganduring2002conductedusingtheDiscreteInteractionApproximation(DIA)forS .Under-prediction
nl4
Windsea offrequencynarrownessrelativetoobservationaldataisclearlyidentifiableusingnon-concisemethods.
Spectralnarrowness
Twooffourconcisemethodsforquantifyingspectralnarrownessarefoundtoadequatelyregisterthis
Spectralwidth
bias. By comparing with a hindcast that uses an expensive, exact solution for four-wave nonlinear
Nonlinearinteractions
interactions, it is determined that much of the bias can be attributed to the approximation used for
DIscreteinteractionapproximation
thesolutionoftheseinteractionsinthefirsthindcast,whichcorrespondstotheDIA,whichisthesolution
methodusedtodayinnearlyallroutine,phase-averagedwavemodeling.
PublishedbyElsevierLtd.
1.Introduction important role in design formulas (Goda, 1985; Saulnier et al.,
2011).Particulareffortismadetoexaminetheimpactofapartic-
Theso-called‘‘thirdgeneration’’(3G)modelforwind-generated ular approximation—the Discrete Interaction Approximation—on
surface ocean gravity waves (Komen et al., 1984,1994; WAMDI theaccuracyofthesepredictionsoffrequencywidth.Thisapprox-
Group, 1988; Tolman, 1991; Booij et al., 1999; WISE Group, imationismadeinthephysicscalculationsofnearlyallroutine3G
2007)iscommonlyusedinengineeringandoperationalforecast- wave model applications today, and is introduced below, subse-
ingapplications(e.g.Jensenetal.,2002;Bidlotetal.,2002).Insuch quenttotheintroductionofthemodelphysicsingeneral.
applications, the most common and lowest order quantity pre- The 3G wave models utilize a phase-averaged (spectral)
dictediswaveheight,whichisdirectlyrelatedtothetotalenergy, description of wave conditions; the dependent variable being
m ,inawavespectrum,E(f,h),whereEisthespectraldensityofsea solvedfor in these models is most oftenthe wave action density
0
surfaceelevationvariance(energy),fisthewavefrequency,andh N=E/r(e.g.BrethertonandGarrett,1968).Here,ristheintrinsic
the wave direction: m ¼RREðf;hÞdfdh. Increasingly, attention is angular frequency where ‘‘angular’’ refers to the relation r=2pf.
0
being given to higher order features of the wave spectrum. This Wave action N is solved in five dimensions, e.g. N=N(f,h; x,t),
ismotivatedbysteadyprogressinaccuracyofpredictionsofsim- where x denotes position and t time. The evolution of the wave
plewaveheight,bytheresultingdesiretoexamineingreaterdetail spectrumisdescribedbymeansoftheradiativetransferequation
theaccuracyofE(f,h)predictedbythemodels,and—perhapsmost (Gelcietal.,1956;Hasselmann,1960),whichcanbewrittenas:
importantly—bytheinherentutilityofhigherorderquantitiesfor
@N S S þS þS þS
specific applications. The present paper is specifically concerned þr(cid:2)~cN¼ tot¼ in ds nl4 other ð1Þ
@t r r
withtheaccuracyofpredictionsofthewidth,infrequencyspace,
ofthenon-directionalspectrumEðfÞ¼REðf;hÞdh.Accuratepredic- where c is the energy propagation velocity of the waves in each
tionsofthespectralnarrownessarenotonlyofscientificinterest dimension(cx,cy,cr,ch).TheLHSofthisequationaccountsforkine-
but also for engineering practice, where this property plays an matics,whichareconservative,whereasontheRHS,Stotrepresents
dynamics. In deep water, it is generally accepted that wind-wave
⇑ growth is primarily a result of three physical processes: atmo-
Correspondingauthor.Address:NavalResearchLaboratory,Code7322,Bldg.
spheric input from the wind to the waves, S , wave dissipation
1009,StennisSpaceCenter,MS39529,USA.Tel.:[phone];fax:[phone] in
4759. due to breaking Sds, and nonlinear energy transfer between the
E-mailaddress:[email protected](W.E.Rogers). wave components Snl4. In finite depths, additional terms due to
1463-5003/$-seefrontmatterPublishedbyElsevierLtd.
http://dx.doi.org/10.1016/j.ocemod.2012.11.010
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Frequency width in predictions of windsea spectra and the role of the
5b. GRANT NUMBER
nonlinear solver
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6. AUTHOR(S) 5d. PROJECT NUMBER
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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Naval Research Laboratory,Oceanography Division,Stennis Space REPORT NUMBER
Center,MS,39529-5004
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
In this paper, the accuracy of predictions of the narrowness in frequency space of elevation spectra for
wind-generated surface gravity waves is evaluated with the specific objective of determining the impact of
the method for computing quadruplet interactions, Snl4. Alternate metrics are presented for concise
quantification of this narrowness and applied to a case study: a 10-day duration hindcast for Lake
Michigan during 2002 conducted using the Discrete Interaction Approximation (DIA) for Snl4.
Under-prediction of frequency narrowness relative to observational data is clearly identifiable using
non-concise methods. Two of four concise methods for quantifying spectral narrowness are found to
adequately register this bias. By comparing with a hindcast that uses an expensive, exact solution for
four-wave nonlinear interactions, it is determined that much of the bias can be attributed to the
approximation used for the solution of these interactions in the first hindcast, which corresponds to the
DIA, which is the solution method used today in nearly all routine, phase-averaged wave modeling.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 10
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
2 W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx
wave-bottomfriction,depth-limitedbreakingandtriadinteractions obvious, since the accuracy of the ground truth cannot be ques-
becomesignificant,andmoretermscanbeformulatedwhichmay tioned. However, these comparisons do not answer a particular,
become important in particular circumstances, e.g. non-breaking interestingquestion,whichis,‘‘canevidenceoftheseknownshort-
dissipationorscatteringfrominteractionwithirregularbathymetry comingsoftheDIAbenoted,isolated,andprovenincomparisons
or with sea ice. Every 3G model includes the first three source to observational data?’’. This is the fundamental question of the
terms,whereastheremainingsourcetermscanvaryfromonemod- presentpaper.
eltoanother.Allofthesesourcetermsarespectralfunctions.The Rogers and Wang (2007), henceforth denoted ‘‘RW07’’ use a
readerisreferredtoWISEGroup(2007)foramorecompleteover- multi-monthLakeMichiganhindcastwiththeSWANmodel(Booij
viewofthistechniqueofwavemodeling. etal.,1999)toinvestigatepossiblebiasinpredictionsofdirectional
Ofthesesourceterms,thepresentpaperisprimarilyconcerned spreadingrelativetobuoyobservations.Theirstudywasprimarily
with the term for four-wave nonlinear interactions, S (e.g. motivatedbypreviousclaimsthatuseoftheDIAleadstooverpre-
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Hasselmann, 1962). In a developing sea, this term moves energy dictionofdirectionalspreading.RW07demonstrateoverprediction
from frequencies just above the spectral peak to both lower and of directional spreading above the peak in idealized simulations
higherfrequencies.Thetransfertolowerfrequenciesisaprimary (using the ‘exact’ computations as ground truth) (their Figures 2
mechanismforfrequencydownshifting.Thetransfertohigherfre- and3),butdidnotfindanyoverpredictionofdirectionalspreading
quenciesisbalancedbywindinputandwhite-cappingdissipation incomparisontothebuoymeasurements(theirFigures8and9).A
resultinginpowerlaw-decayofwaveactionwithfrequency.This morepositivebutlessemphasizedfeatureofRW07wastheconsis-
term has also been found to have a primary role in determining tencyoftheresultswithregardtofrequencywidth(thetopicofthe
and stabilizing the spectral shape in general (Hasselmann et al. present paper). Specifically, in both the idealized comparisons
1973;YoungandVanVledder,1993).Unliketheothertwoprimary againstthe‘exact’model(theirFigure2)andthebuoycomparisons
sourceterms,theformofthistermisknownandcanbesolvedfor (theirFigures6and7),thereisanoverpredictionofenergybelow
exactly(e.g.EXACT-NL,HasselmannandHasselmann,1985)butis thespectralpeak,andsothespectrumistoobroad.
fartooexpensiveforroutineuse,necessitatingshortcuts,themost RW07 could only indirectly attribute overprediction of fre-
commonlyusedmethodbeingtheDiscreteInteractionApproxima- quencyspreadingto the DIA,since theydid notapply the ‘exact’
tion (DIA) introduced in the seminal work by Hasselmann et al. S computations in their hindcast due to computational costs.
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(1985),whichusesarelativelysmallsubsetofinteractionsbutpre- This is, however, done by Ardhuin et al. (2007) in a hindcast for
servesthelowestorderfeaturesoftheexactapproach.Sincethis Duck, North Carolina. This was made possible (from a practical
periodofrapidprogressinthe1980s,muchefforthasbeenmade pointofview)throughtheuseofafiniteelementgridwithrela-
in the wave model development community to alternately im- tively few grid points (CREST model, Ardhuin et al., 2001). These
provetheaccuracyoftheDIAorimprovetheefficiencyoftheexact authors find that, in comparison to the ‘exact’ S -based model,
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algorithms(WISEGroup,2007). theDIA-based modelalwayshas higherdirectional spreading(as
The qualitative impact of the DIA ‘‘shortcut’’ on model results expected),butthisdoesnotalwaysleadtoalargererrorincom-
hasmostoftenbeenpresentedforidealizedscenarios(e.g.fetch- parisontobuoydata.Incaseswheredirectionalspreadingpredic-
limitedorturningwinds),asthisobviouslyfacilitatescomprehen- tionsaremadeworsewith‘exact’S ,thisisreasonablyattributed
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sionof themost fundamental features.Hasselmannetal.(1985), toinaccuraciesintheothersourceterms.Ardhuinetal.(2007)also
for example, evaluate fetch-limited growth curves using the DIA mentioninpassingthattheDIAwouldresultinanoverestimation
andnotethatagreementofthemoreimportantscaleparameters, of the growth of low-frequency waves if not for cancellation of
total energy and peak frequency, were excellent, while two less errors via the other source terms, thus leading to overly broad
importantscaleparameterswerepredictedlesswell;theonerelat- frequencyspectra.
ing to spectral peakedness was reported as ‘‘somewhat smaller’’ Ardhuinetal.(2010)conductedLakeMichigansimulationssim-
than the ground truth. In subsequent years, considerable success ilartoRW07,usingtheWAVEWATCHIII(cid:2)model(‘‘WW3’’,Tolman
has been achieved with 3G-models, especially in the context of etal.,2002;Tolman,2009).Thesesimulationsindicateoverpredic-
those‘‘moreimportantscaleparameters’’,whileatthesametime tionof energybelowthespectral peaksimilartothoseshownin
the shortcomings of the DIA are receiving greater scrutiny (e.g. RW07 Figs. 6 and 7. Again, this was not positively attributed to
Van Vledder et al., 2000). These shortcomings are compensated theDIA.However,in thecontextoftheLakeMichiganhindcasts,
bytuningofothersourceterms,thushamperingthedevelopment it suggests a robust feature, independent of modeling platform
of3G-models.Inthesimplestexamples,thelimitationsoftheDIA (SWAN, WW3) and S +S parameterizations (Komen et al.,
in ds
areillustratedbycomputingS usingDIAforafixedwavespec- 1984;Bidlotetal.,2005;Ardhuinetal.,2010,or,aswewillshow,
nl4
trumandcomparingthiswithexactcomputationsofS .Thisexer- Rogersetal.,2012).Theobjectiveofthepresentstudyistodeter-
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cise reveals basic characteristics but is insufficient to anticipate minewhetherthisoverpredictionoffrequencywidthispositively
practical outcomes in a time-stepping solution, because of the attributabletotheDIA.
highly non-linear character of S . Instead, full model exercises Direct, side-by-side comparisons of one-dimensional wave
nl4
areneededinwhichS isusedinconjunctionwithothersource spectra can be a useful method of presenting differences, or a
nl4
terms.Somerecentefforthasbeenmadetocharacterizethepracti- meansofcarefulmodelevaluation.However,inthecontextofrou-
cal implications of the shortcut on realistic applications: Tolman tineorrepetitivemodelvalidationexercises,itisoftenimpractical.
(2011)presentsapplicationsinasynthetichurricaneandastorm Thisisespeciallytrueinlongerhindcastswithmultiplelocationsof
eventinLakeMichigan.Onefeaturereportedoftenisbroaderdirec- spectral observational data. In these cases, it is useful to utilize
tional spreading with the DIA than with the full solution (Young bulkparametersthateffectivelyidentifymodelerror,andstatistics
etal.,1987;VanVledder,1990;ForristallandEwans,1998;Ardhuin can be calculated from these. For example, zero-moment wave
etal.,2007;Tolman,2011),broaderfrequencyspectra(alreadyvis- height,H isusedtoquantifytotalenergy,andthespectralmean
m0
ibleinHasselmannetal.,1985,theirFigure9)andanunderestima- period(e.g.T ,T ,T ,seeAppendix)isusedtoquantifya
m,(cid:3)1,0 m,0,1 m,0,2
tionoftheenergylevelatthespectralpeakhasalsobeenreported center(orcentroid)ofthefrequencyspectrum,variouslydefinedto
(Tolman,2011). givemoreorlessweighttofrequenciesfurtherfromthatcenter.In
Tolman(2011)andothersusetheresultsfrom‘exact’calcula- thispaper,existingmethodsforquantifyingthenarrownessinfre-
tions of S as ground truth. The soundness of this approach is quency space of non-directional frequency spectra (as opposed to
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Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx 3
spectralwidth)areevaluated.Wefindthatsomemethodsarenot Lake Michigan depths (m)
sufficientlysensitivetoproblemswithnarrownessthatarereadily 100
visiblebyeye.
450 90
Thispaperisorganizedasfollows.Thewavemodelisdescribed
inSection2,andtheutilizedhindcastisdescribedinSection3.The 400 buoy 45002 80
methods of quantifying frequency narrowness are described in
Section4.InSection5,overpredictionofspectralwidthisdemon- 350 70
stratedinahindcastusingDIA.InSection6,an‘exact’S -based
nl4 300 60
hindcast is analyzed and it is found that the bias in frequency
w7iadntdh8is.reduced.DiscussionandConclusionsaregiveninSections y (km)250 C−MAN SGNW3 50
200 40
2.Modeldescription 150 buoy 45010 30
buoy 45007
ThemodelplatformisSWAN(Booijetal.,1999;SWAN,2010) 100 20
andthewindinput(S ),whitecapping(S ),andnon-breakingdis-
in ds 50 10
sipation (S ) parameterizations are taken from Rogers et al.
swell
(2012). We refer the readerto that paper for details, but the key 0 0
0 100 200
features are briefly given here. The wind input term S is based
in x (km)
onDonelanetal.(2006)andTsagarelietal.(2010),developedfrom
directmeasurementsofwindinputatLakeGeorge,Australia,with Fig. 1. Bathymetry of Lake Michigan (depths in meters). NOAA NDBC station
the wind drag coefficient Cd based on Hwang (2011). Dissipation locationsareindicated.Buoy45007isusedformodel/datacomparisonsherein.
from breaking (whitecapping), S is based on Babanin et al.
ds
(2010) which is developed from Young and Babanin (2006),
used,asecondorderschemewiththirdorderdiffusion(seeSWAN,
Tsagareli(2009),andBanneretal.(2000).S istwo-phase—insofar
ds
2010). Physics for S , S , and S are used as described in Sec-
asitaccountsforbreakingofwavesduetoinherentinstabilityand in ds swell
tion2.Theinitialconditionisfroma‘hotstart’filecreatedbythe
dissipationinducedbythebreakingoflongerwaves—andemploys
modelusing DIAfor S , 0000UTC 12Oct. 2002to 0000 UTC 13
abreakingthreshold,basedonthePhillips(1958)saturationspec- nl4
Oct. 2002. The simulations used for comparisons are from 0000
trum(thesameconceptwasusedearlierbyothers,suchasCollins
UTC 13 Oct. 2002 to 0000 UTC 23 Oct. 2002 (10-day duration),
(1972)and Alves and Banner (2003)). InRogerset al.(2012), the
usingatimestepof6min.
utilizedformofS isdenotedasL4M4.Non-breakingdissipation,
ds
S , is included to account for the slow attenuation of swell by
swell
non-breaking processes, and utilizes work by Ardhuin et al. 4.Methods
(2009,2010).InthenotationofRogersetal.(2012),f =0.006(con-
e
trollingthestrengthofswelldissipation)isusedinsimulationsde- AsmentionedinSection1,theobjectiveofthepresentstudyis
scribed below. For four-wave nonlinear interactions S , we use to determine whether overprediction of frequency width is posi-
nl4
methodsavailableinSWAN(2010)withoutmodification.Theterm tivelyattributable to the DIA. A necessary step is to evaluate the
iscomputedwitheitherDIAor‘exact’computations.Thelatteris utilityofbulkparameterswhichcanbeusedtoquantifythenar-
denoted‘‘XNL’’hereinandisspecificallythe‘‘Webb–Resio–Tracy’’ rowness in frequency space of the non-directional spectrum. In
method(WRT)asimplementedbyvanVledderinSWAN,seeVan thissection,fourmethodsaredescribedtoquantifythisproperty.
Vledder(2002,2006)andSWAN(2010). Allfourmethodsarebasedonsimple,directcalculation,mostlyvia
integration,whichispreferredoverindirectmethods,e.g.fittingto
parametricspectra;thisisasubjectivedecisionandwedonotdis-
3.Hindcastdescription:LakeMichigan2002,10-daysimulation
putethemeritsofmethodsofthelattertype.
MethodA:UsesthemethodofSWAN(2010),asdefinedbyBatt-
LakeMichiganisausefulareaforstudyingtheimpactofmodel
jesandVanVledder(1984).Thecalculationis:
physics,asitismostlydeepwater(minimizingcomplicationsdue
(cid:2) (cid:2)
tofinitedepthsourceterms),largeenoughtoallowgenerationof Q (cid:4)(cid:2)(cid:2)Z fmaxEðfÞei2pfsdf(cid:2)(cid:2)=m ð2Þ
dominant waves measurable using National Data Buoy Center A (cid:2) (cid:2) 0
(NDBC) buoys (e.g. T =6s), and enclosed, minimizing complica- (cid:2) fmin (cid:2)
p
tions due to interaction with swell (older swells being non-exis- where m ¼RfmaxEðfÞdf and s=T (defined in Appendix). The
0 fmin m,0,2
tent). The latter feature is pertinent to the scope of this study, lowerandupperboundsoftheintegrationaredenotedasf and
min
whichisspecifictowindsea. f .Aswithotherintegralquantities,foranygivencomparison,f
max min
TheLakeMichiganmodelsimulationsaresetupasfollows.The and f should be applied consistently: when comparing model
max
computational grid is approximately 252km (east–west) by andobservations,thiswilloftenimplythatthehighestmodelfre-
496km(north–south),with4kmgridresolution.Directionalreso- quenciesareexcludedfromthecalculation.InBattjesandVanVled-
lution is 10(cid:3) (36 bins), and a logarithmic frequency grid is used, der (1984), Q is denoted as parameter j and is referred to as a
A
with35frequenciesfrom0.07to1.97Hz.Thebathymetry,shown ‘‘shapeparameter’’forthepurposeofpredictingwavegroupstatis-
inFig.1,isprovidedat2kmresolutionbytheNOAAGreatLakes ticsfromanon-directionalspectrum.TheBattjesandVanVledder
EnvironmentalResearchLaboratory.Windforcingiscreatedfrom (1984) paper is given as the source of the equation in SWAN
two NOAA NDBC buoys as described in Rogers et al. (2003) and (2010), although the actual source is Rice (1944). However, in
RW07;themethodassumeshomogeneityinlongitudeandspatial SWAN (2010), the quantity is incorrectly referred to as ‘‘FSPR’’,
variation in latitude determined by the buoys 45002 and 45007. ‘‘the normalized frequency width of the spectrum (frequency
JONSWAP (Hasselmann et al. 1973) bottom friction formula is spreading)’’.Liketheothermethodsdescribedhere,thisparameter
used,thoughthisisnotexpectedtoaffectresults,sincethishind- actually quantifies the narrowness of the spectrum and it ranges
cast is predominately in deep water. The default numerics are from0to1.
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
4 W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx
MethodBis: Table1
ExamplecalculationsoffrequencynarrownessparametersonJONSWAPspectra.In
Q (cid:4)ðT =T Þ2 ð3Þ eachcase,twovaluesforQaregiven.Thefirstvaluecorrespondstoupperboundof
B m;0;2 m;(cid:3)1;0
integrationfmax=1.0Hzandthesecondvalue,inroundbrackets(),correspondsto
whereTm,0,2andTm,(cid:3)1,0aredefinedintheAppendix.Thus,itisthe fmax=0.4Hz.
squareoftheratiobetweenTm,0,2,whichisoften70–90%ofthepeak JONSWAPc Narrowness
period,andT ,whichisusuallymuchclosertothepeakperiod.
Thecloserthims,(cid:3)p1a,0rameteristoone,themorepeaked,ornarrow,the QA QB QC QD
frequencyspectrum. 1.0 0.42(0.37) 0.72(0.83) 1.67(1.72) 2.01(2.25)
2.0 0.50(0.47) 0.74(0.85) 2.60(2.66) 2.49(2.74)
Asimilarquantity,T T wasusedbyVanVledderetal.
m,0,1/ m,(cid:3)1,0 3.3 0.57(0.55) 0.77(0.87) 3.43(3.49) 3.15(3.41)
(2008), in studies of the Dutch Waddensea to assist in detection 6.0 0.65(0.65) 0.80(0.89) 4.54(4.60) 4.32(4.59)
ofareaswherelongerwavesweredominantandwherebi-model 10.0 0.72(0.72) 0.84(0.91) 5.54(5.59) 5.62(5.87)
spectraoccurred.InthenumeratorofQ ,T isusedhereinpref- 100.0 0.92(0.92) 0.96(0.98) 9.56(9.58) 12.15(12.25)
B m,0,2
erence to T as it provides more separation from T , and
m,0,1 m,(cid:3)1,0
thusitisexpectedtohavegreatersensitivitytochangesinshape.
QCM(cid:4)emthaoxdðECfniÞs=:Tm;(cid:3)1;0 ð4Þ m2/Hz)23 13−Oct−2002 0Sb7uW:0o0Ay:0N0 m2/Hz)468 13−Oct−2002 13:00:00
where: E(f) (01 E(f) (02
E ðfÞ(cid:4)EðfÞ=m ð5Þ 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
fn 0 frequency (Hz) frequency (Hz)
aZnfmfdimna:xEfnðfÞdf ¼1 ð6Þ E(f) (m2/Hz)0123 13−Oct−2002 19:00:00 E(f) (m2/Hz)012 15−Oct−2002 04:00:00
ThusQCisthepeakofthefunctionE(f)afterithasbeennormalized 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
by the area under the same function.1 The division by T is frequency (Hz) frequency (Hz)
m,(cid:3)1,0
idnecvliusdeeddintothmisaskteudtyhefolqlouwanintigtythdeimfoernmsuiolantlieosns.oTfhtihsemAeptahroadmwetaesr 2/Hz)2 19−Oct−2002 01:00:00 2/Hz)4 19−Oct−2002 14:00:00
usedbyBabaninandSoloviev(1987,1998a)toquantifynarrowness m m
oizfastpioenc,trwaeinfinddirethctaitonthaelsfpoarmceu.lHaoiswseivmeirl,awrittohtthhoesenounse-ddimbyenLesiBolnoanld- E(f) (01 0.1 0.2 0.3 0.4 E(f) (02 0.1 0.2 0.3 0.4
etal.(1982),Belberovetal.(1983),MansardandFunke(1990),and frequency (Hz) frequency (Hz)
BabaninandSoloviev(1998b).Ofthefourmethodsusedhere,only
thismethodincludesanon-integraloperation,the‘‘max(E )’’opera- Fig.2. Comparisonofobservedandcomputednon-directionalspectrausingthe
tionin(4).ThisimpliesthatQ respondstospectralpeakfendness. simulationwithDIAforSnl4forsixtimeperiods.
C
MethodDis:
Z fmax spondstoyoungerseas, whichtendtobe relatively narrowinfre-
Q (cid:4)2=m2 fE2ðfÞdf ð7Þ quency space. Values of c>6 are not necessarily realistic for wind
D 0
fmin seas, but are included here for illustration.3 Also note that peak
enhancementisaspecificquantityassociatedwith‘‘overshoot’’near
ThisistakenfromGoda(1970,1985).Ithasbeenreferencedinthe
thespectralpeak(e.g.Young(1999),Figure5.20),whereasspectral
freakwaveliterature(Janssen,2003)as‘‘Goda’speakednessfactor
Q ’’withtheBenjaminFeirIndexbeingBFI¼k Q pffimffiffiffiffiffiffi2ffiffiffipffiffiffi.Thehigh- narrowness is a more general quantity that might be affected by
p o p 0
other features of the spectrum, e.g. a bimodal frequency spectrum
erQ ,themorepeakedthespectrum.
p
willtendtohavealowspectralnarrowness.
As it is less useful to evaluate higher order moments (in this
Allfourquantitiesaredimensionless.Separatecalculationswith
case,spreading)incaseswherethelowerordermomentsaredis-
the JONSWAP spectra (not shown) indicate that none of the Q
similar,weonlyincludepointsinthetimeseriesforwhiche <0.2
H
parametersexhibitasignificantdependenceonthepeakfrequency
ande <0.2,where:
T
f .Withfrequencydistributionsshapedlikenormaldistributions,
(cid:2) (cid:2) p
eeHT ¼¼(cid:2)(cid:2)(cid:2)THmm;(cid:3)0:1o;b0s;o(cid:3)bsH(cid:3)mT0m;m;(cid:3)od1e;0l(cid:2);m=Hodeml(cid:2)(cid:2)0=;oTbsm;(cid:3)1;0;obs ð8Þ QprCoapnodrtiQonDaelixthyi)b,iat slotrgoicnaglacnodnsiedqeunetniccaelodfeptheendfeonrmceoofnthfpe(leiqnueaa-r
tions.Q andQ alsoexhibitsomedependenceonf usingnormal
A B p
Tofamiliarizewiththesequantities,wecompareinTable1values distributions.Thedifferingbehaviorassociatedwithuseofnormal
calculated using the JONSWAP spectrum(see for example, Young, distributionsversusJONSWAPspectraiscausedbythedependence
1999,pg.112)fordifferentvaluesoftheJONSWAPspectralpeaked- oftheJONSWAPspectrumonf beyondsimplyshiftingthespec-
p
nessparameterc,whichmanyreaderswillbefamiliarwith.2JON- truminfrequencyspace.
SWAP c=1 corresponds to fully developed seas, which tend to be ThenarrownessvaluesinTable1arecalculatedwithtwoalter-
relatively broad in frequency space, and JONSWAP c=3.3 corre- nate values for the upper limit of integration, f =1.0Hz and
max
f =0.4Hz.Thisrevealssomedependencyoftheparameterson
max
1 Apartfromthedivisionbythespectralwaveperiod,onecanrecoverthesame fmax.MethodBinparticularissensitive,whichisnotsurprisinggi-
valueusingthereciprocalvalueoftheareaofthefunctionthatisnormalizedbythe venthatitutilizesahigherordermomentofthespectrum.4Meth-
peakvalue.
2 Though it is not expected to have much impact on the calculations, for
completeness, the other JONSWAP parameters used are: fp=0.19 and a=0.02, 3 Using c>6 is actually a technique for providing SWAN with parameterized
ra=0.07, rb=0.09 (see e.g., Young, 1999 for definition) with integration of boundaryforcingthatisnarrow,i.e.anunrealisticallypeakedwindseacanbeusedto
frequenciesfromfmin=0.03HzwithgridspacingDf=0.001Hzandanf(cid:3)5spectral approximateolderswells,sincethelattertendtobenarrowintherealocean.
tail. 4 ThesensitivitywouldpresumablybereducedbyusingTm,0,1insteadofTm,0,2.
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx 5
/v55UL4M4H fe 0.0060 DIA 2Hz frequency narrowness Q frequency narrowness Q
1 3 A B
0.5f to 0.8f 0.8f to 1.2f
p p p p n. bias = −0.16 0.95 n. bias = −0.03
0.8 2.4 CC = 0.43 CC = 0.72
0.6 slope = 0.86 slope = 0.97
RMSE = 0.13 0.9 RMSE = 0.04
00..46 nB i=a s1 =9 70.14 m 11..28 nB i=a s1 =9 9−0.03 m model0.4 SI = 0.19 model0.85 SI = 0.04
RMSE = 0.16 m RMSE = 0.14 m
0.2 NSIR =M 0S.4E8 = 0.93 0.6 NSIR =M 0S.1E5 = 0.14 0.2 0.8
C.C. = 0.61 C.C. = 0.94
0 0 0 0.75
0 0.2 0.4 0.6 0.8 1 0 0.6 1.2 1.8 2.4 3 0 0.2 0.4 0.6 0.75 0.8 0.85 0.9 0.95
buoy buoy
frequency narrowness Q frequency narrowness Q
2 1.2f to 2f 1 2f to 3f C D
p p p p
n. bias = −0.28 n. bias = −0.20
1.6 0.8 5 CslCop =e =0. 502.71 5 CsloCp =e =0. 605.79
RMSE = 1.03 RMSE = 0.82
1.2 0.6 el4 SI = 0.19 el 4 SI = 0.15
n = 199 n = 98 d d
0.8 BRiMasS E= −=0 0.0.120 m m 0.4 BRiMasS E= 0=. 000.0 m4 m mo3 mo 3
NRMSE = 0.14 NRMSE = 0.14 2
0.4 SI = 0.15 0.2 SI = 0.16
C.C. = 0.95 C.C. = 0.97 1 2
0 0
0 0.4 0.8 1.2 1.6 2 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 2 3 4 5
buoy buoy
Ffriegq.u3e.nEcnyebrgayndpsr:eDdiIcAt-iobnaseedstimmoadteeslv(4sp.offimffiffibffiffi0sffiffi,eirnvamtioentesr.sS),tadtiivstidiceadlqinutaontfoituiersreshlaotwivne Fig.4. ValidationoffrequencynarrownessparametersQA,QB,QC,andQD.Allmodel
resultsshownarefromtheDIAsimulation.Theredlineshowsthelinearbest-fit
are in order: number of observations, bias, root-mean-square error, normalized
relation.StatisticalquantitiesshownaredescribedintheAppendix.(Forinterpre-
root-mean-squareerror(calculationgivenintheAppendix),scatterindex(standard
tationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe
deviationoferrorsdividedbythemeanofobservations),andcorrelationcoefficient
webversionofthisarticle.)
CCcalculatedasshownintheAppendix.
wherefrequenciesf andf areindicatedineachpanel.Thebiasin
1 2
thefrequencyrangeof0.5f <f<0.8f is+14cm.5
od C appears to be the least sensitive. For example, the relative p p
ThefrequencynarrownessparametersintroducedinSection4
change from f =1.0Hz to f =0.4Hz with JONSWAP c=1.0 is
max max areappliedtoevaluatethesensitivityofthesefourparametersto
(cid:3)12%, +15%, +3%, and +12%, for Q , Q , Q , and Q , respectively. At
A B C D the problems visible in comparisons such as Fig. 2. This is done
c=6.0,itis0%,+11%,+1%,and+6%.
fortheDIA-basedsimulationresultsinFig.4.Inallcases,theupper
bound on integration, f , is 0.4Hz, corresponding to the buoy’s
max
5.ResultsusingDIA upper limit. We find that parameters Q and Q do not show (or
A B
only weakly show) a bias that is apparent in the non-directional
Inthissection,resultsfromthemodelwhichemploystheDIA spectral comparisons. Parameters Q and Q do demonstrate this
C D
for Snl4 are compared to observations from NOAA NDBC buoy sensitivity. QC indicates the largest normalized bias, and it was
45007.Traditionalbulkparameterssuchaswaveheightandwave demonstratedinSection4thatthisparameterhaslesssensitivity
periods are only of peripheral concern in this study, but for sake totheselectionoff ,soitisselectedforallsubsequentcompar-
max
ofcompleteness,someareevaluatedintheAppendix. isons,thoughitisnotedthatQ isalsosuitable.
D
As a first step, the most direct method of comparison is used.
Non-directionalspectraE(f)arecomparedforsixtimeperiodsdur-
6.ResultsusingXNL
ingthe10-daysimulationsinFig.2.Thesesixtimesareselectedas
periodsinwhichtheabsoluteerrorofQ,
c The10-daytimesimulationisrepeatedusingXNLforS .For
nl4
E ¼jQ (cid:3)Q j ð9Þ theothersourceterms,nochangesaremade,e.g.thereisnore-cal-
Qc c;model c;obs
ibrationofthecoefficientsusedinS .Fig.5showsresultsfrom6h
ds
matchesthemedianE forthe10-daysimulation.Thus,thissmall duringthe10-daysimulations,forcomparisonwithFig.2.Qualita-
Qc
subsetofthelargersimulationcanberegardedasatypicalrepre- tively,theXNL-basedsimulationappearstoprovideabettermatch
sentationof the whole, in the context of the narrowness quantity totheobservations—especiallyinthecontextoftheenergylevelat
Q. Features of the DIA-based simulation are noticeably similar to thepeak,andamountofenergybelowthepeak—inatleastfourof
c
thosenotedbyRW07,Ardhuinetal.(2007)andTolman(2011).Rel- the six examples. A comparison of Hm0;partial, similar to Fig. 3, is
ativetotheobservations:(1)theenergylevelatthepeakislowin made in Fig. 6. Whereas the bias in the frequency range of
fiveofthesixplots,and(2)thereistoomuchenergybelowthepeak 0:5f <f <0:8f is +14cm using the DIA-based model, corre-
p p
in at least three plots. However, as mentioned in Section 1, such spondingresultswiththeXNL-basedmodelindicatemuchsmaller
qualitativecomparisonscannotbedirectlyusedtogeneratestatis- overallbiasinenergybelowthepeak,+5cm.Thus,ourdataclearly
ticsforlongtimeseries.Anotherformofcomparisonistolookat supporttheinterpretationthatinaccuracieswiththeDIAcontrib-
energylevel,separatedaccordingtofrequencyrelativetothepeak, utetotheoverpredictionofenergybelowthepeak.
suchasdoneinRW07(theirFigure6).Wecantherebyverifythat Toprovidefurthersupporttothisinterpretation,thefrequency
the DIA-based model exhibits the same overprediction of energy narrownessparametersintroducedinSection4areapplied.Figs.7
below the peak, as observed by that earlier study. This is shown and8compareresultsusingXNLversusthoseusingDIAintermsof
inFig.3;thequantitycomparedis: frequency narrowness parameters QC and QD. As shown in Fig. 7,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z f2 5 UsingasimilarsimulationwiththeKomenetal.,1984physics,wefindabiasof
H ¼4 EðfÞdf ð10Þ
m0;partial +11cm;usingthe74-daysimulationwithKomenetal.,1984physics,RW07reporta
f1 biasof+9cm.
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
6 W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx
E(f) (m2/Hz)0123 0.1 freq0u.2en13c−yO c(t−H02.0z30)2 07:0XDb0uNI:oA0L0y0.4 E(f) (m2/Hz)02468 0.1 freq0u.2en13c−yO c(t−H02.0z30)2 13:00:000.4 M4H fe 0.0060 w/XNL46 fnCsRSrl.eICoM bq=p iS=e ua0 E s=0e. 2 =. =n306 c80.09.y.087 18narrowness QC M4H fe 0.0060 w/DIA46 fnCsRSrl.eICoM bq=p iS=e ua0 E s=0e. 1 =. =n509 c2−.17y0.10. 23n8arrowness QC
E(f) (m2/Hz)0123 13−Oct−2002 19:00:00 E(f) (m2/Hz)0123 15−Oct−2002 04:00:00 model = v55UL4020 2 buo4y XN6L model = v55UL4020 2 buo4y DIA6
0.1 freq0u.2ency (H0.z3) 0.4 0.1 freq0u.2ency (H0.z3) 0.4 NL frequency narrowness QD A frequency narrowness QD
FiE(f) (m2/Hz)g.5024. Co0m.1parifsroenq0u.2oefn19oc−byOs c(et−H0r2v.0z3e0)2d 01a:0n0d:000c.4ompE(f) (m2/Hz)uted024non0.-1direfcrteioq0nu.2aeln19sc−pyOe c(ct−Ht02r.0za30)2u 1s4:i0n0g:000t.4he model = v55UL4M4H fe 0.0060 w/X02460nCsRSl.ICoM b=p iS=e a0 E s=0. 1 =. =2508 8−.090.55.0b93uo4y XN6L model = v55UL4M4H fe 0.0060 w/DI02460nCsRSl.ICoM b=p iS=e a0 E s=0. 1 =. =2605 5−.070.98.2b20uo4y DIA6
simulationswithDIAandXNLforSnl4forsixtimeperiods.
Fig.7. ValidationoffrequencynarrownessusingtheDIAandXNLforevaluatingthe
quadrupletinteractions.Thefrequencynarrownessisquantifiedusingparameter
1 /v55UL4M4H fe 0.0060 XNL 2Hz 3 QC(upperpanels)andQD(lowerpanels).Statisticalquantitiesshownaredescribed
0.5f to 0.8f 0.8f to 1.2f intheAppendix.
p p p p
0.8 2.4
0.6 1.8 model = v55UL4M4H fe 0.0060
n = 194 n = 194 5
0.4 Bias = 0.05 m 1.2 Bias = −0.03 m buoy
RMSE = 0.10 m RMSE = 0.16 m
NRMSE = 0.59 NRMSE = 0.15 XNL
0.2 SI = 0.57 0.6 SI = 0.17 4.5 DIA
C.C. = 0.56 C.C. = 0.92
0 0
0 0.2 0.4 0.6 0.8 1 0 0.6 1.2 1.8 2.4 3 C 4
Q
s
s
2 1.2f to 2f 1 2f to 3f wne 3.5
p p p p o
1.6 0.8 arr 3
n
1.2 0.6 cy
n
0.8 nB i=a s1 =9 4−0.03 m 0.4 nB i=a s9 =8 0.00 m que 2.5
0.4 RNSIMR =MS 0ES. 1E=8 =0 .01.21 6m 0.2 RNSIMR =MS 0ES. 2E=0 =0 .00.51 7m fre 2
C.C. = 0.93 C.C. = 0.95
0 0
0 0.4 0.8 1.2 1.6 2 0 0.2 0.4 0.6 0.8 1 1.5
1

Fig.6. SimilartoFig.3,exceptforXNL-basedmodel. 10/13 10/14 10/15 10/16 10/17 10/18 10/19 10/20 10/21 10/22
MM/DD
thebiasinQC(upperpanelsofFig.7)isalmostcompletelyremoved Fig.8. AsinupperpanelsofFig.7,exceptastimeseriescomparison.
usingXNLandthebest-fitslopeisnearunity.However,randomer-
ror,as quantified bythescatter indexand correlationcoefficient,
actuallyincreases;thecauseofthisisnotknown.Boththepositive studies,suchas thespectralperiodT ,themeanwavedirec-
m,(cid:3)1,0
andnegativeconsequencesofreplacingDIAwithXNLareseenalso tion h and the circular RMS directional spreading r. Together
m h
intheQDcomparison(lowerpanelsofFig.7),buttheconsequences theseparametersformacomprehensiveset toassessmodelper-
are notaspronouncedas withQC. Thetimeseriescomparisonof formance, but their error behavior can also be used to pinpoint
QC,showninFig.8,clearlyillustratestheimprovedresultsbyusing modeldeficiencies.
XNLinsteadoftheDIA. Inthisstudyfrequencynarrownessisaddedtothislisttoinves-
tigatetheeffectofcomputationalmethodsfornonlinearquadru-
7.Discussion plet interactions on the narrowness of the frequency spectrum.
Tothatendvariousmetricsquantifyingspectralnarrownesswere
Modelperformanceisoftenexpressedintermsofbulkstatistics investigatedandoneparameterwasselectedthatisabletoshowa
ofsignificantwaveheightH andpeakperiodT orameanspec- specific deficiency of the Discrete Interaction Approximation on
m0 p
tralperiod,e.g.T .Formanyapplicationsthesefirstordermet- the narrowness (or peakedness) of the frequency spectrum, and
m,0,2
rics are sufficient to judge model performance. However, with the over-prediction of spectral density below the peak frequency
increasing requirements on model skill, other integral spectral in particular. Based on simulations in Lake Michigan using the
parameters are gradually included in more model verification SWANmodelandDIAandXNLparameterizationofthequadruplet
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx 7
Table2
Statisticsforsimulationsofduration10days.Thenumberofspectrausedisindicatedasn.Columns2and4correspondtothesimulationsdiscussedinthemaintext,andthe
physicsdescribedinSection2aredenotedasRBW2012.
XNLRBW2012 DIARBW2012 DIAKHHnk=2
Fullset Subset Fullset Subset Fullset Subset
n 240 156 240 166 240 172
Hm,0(m)
Bias (cid:3)0.05 (cid:3)0.01 (cid:3)0.03 (cid:3)0.00 0.00 0.03
RMSE 0.21 0.13 0.18 0.12 0.17 0.12
CC 0.93 0.97 0.95 0.98 0.96 0.98
Tm,0,1(s)
Bias (cid:3)0.18 (cid:3)0.15 (cid:3)0.10 (cid:3)0.06 0.08 0.15
RMSE 0.37 0.28 0.33 0.23 0.32 0.26
CC 0.85 0.92 0.86 0.94 0.89 0.95
Tm,0,2(s)
Bias (cid:3)0.18 (cid:3)0.15 (cid:3)0.11 (cid:3)0.07 0.06 0.12
RMSE 0.35 0.27 0.31 0.21 0.29 0.23
CC 0.85 0.91 0.86 0.94 0.89 0.95
Tm,(cid:3)1,0(s)
Bias (cid:3)0.18 (cid:3)0.14 (cid:3)0.07 (cid:3)0.01 0.12 0.20
RMSE 0.41 0.31 0.37 0.26 0.38 0.33
CC 0.85 0.92 0.86 0.93 0.88 0.94
QA
Norm.bias (cid:3)0.04 (cid:3)0.06 (cid:3)0.14 (cid:3)0.16 (cid:3)0.09 (cid:3)0.11
CC 0.43 0.51 0.42 0.43 0.46 0.44
Slope 0.94 0.94 0.86 0.86 0.91 0.90
QB
Norm.bias (cid:3)0.00 (cid:3)0.01 (cid:3)0.02 (cid:3)0.03 (cid:3)0.02 (cid:3)0.03
CC 0.74 0.79 0.69 0.72 0.69 0.72
Slope 1.00 0.99 0.98 0.97 0.98 0.97
QC
Norm.bias (cid:3)0.01 0.01 (cid:3)0.27 (cid:3)0.28 (cid:3)0.26 (cid:3)0.26
CC 0.34 0.38 0.47 0.52 0.49 0.49
Slope 0.96 0.98 0.71 0.71 0.73 0.73
QD
Norm.bias (cid:3)0.01 (cid:3)0.03 (cid:3)0.17 (cid:3)0.20 (cid:3)0.16 (cid:3)0.18
CC 0.53 0.59 0.56 0.66 0.59 0.65
Slope 0.95 0.95 0.80 0.79 0.82 0.81
interactions,weareabletoshowtheapplicabilityofthespectral applytomethodCasitmayaffectthepeakvaluemax(E ).Assuch,
fn
narrowness parameter Q to illustrate a potential weak point of validation with these metrics (such as our Fig. 7) must be inter-
c
theDIA. pretedinthecontextofthedatasetused.Further,disparateobser-
Othermetricsforfrequencywidth,notshownhere,havebeen vationaldatasetsshouldnotbecombinedwhencreatingstatistics
used with success. For example, in cases of uni-modal spectra, forvalidation:thestatisticsshouldbetreatedseparately.Thisisin
the distance (in Hz) between f and f can be used, where contrasttotreatmentoftraditionalquantitiessuchaswaveheight.
1 2
Eðf Þ¼Eðf Þ¼0:5maxðEðfÞÞ. Or, the spectrum can be normalized Thoughmuchmoreusefulthansubjectivevisualcomparisonof
1 2
and treated as a pseudo-probability distribution function, and non-directionalspectra,comparisonssuchasFig.3stillhavelimita-
standard deviation-like parameter can then be calculated. Such tions.Themostsignificantlimitationistherequirementtoidentify
parameters might in fact be preferred to the Q parameters since thespectralpeak,andthatallenergyisevaluatedaccordingtoitspo-
theunits(Hz)aremoretangible.Saulnieretal.(2011)recentlypro- sitionrelativetothissinglepeak.Thisiseasilyjustifiedinenviron-
videdanexcellentoverviewofanumberofmetrics,bothdimen- ments such as Lake Michigan, which is typically dominated by
sional and non-dimensional. As mentioned already, fitting a c windsea.6However,intheopenocean,itismuchmorecommonto
parametertotheJONSWAPhasbeenusedsuccessfullyinthepast, havemultiplepeaksduetoswell,andsothesecomparisonsareimpos-
e.g. by Hasselmann et al. (1985). Experiments with 16 alternate siblewithoutfirstseparatingthewindseafromswell.Thefournar-
metricshavebeenperformedandwillbedocumentedseparately rownessparametersappliedhere,bycontrast,stillhavemeaningin
inareportfocusedonmetrics. thecaseofbimodalspectra,justaswaveheightH hasmeaning.Even
m0
Validation of higher order moments of directional and non- so,itisfeltthatapplicationofawindsea/swellseparationalgorithm
directional spectra introduces new challenges. Not only are the (e.g.Gerling,1992;HansonandPhillips,2001)priortocalculations
models normally less reliable for prediction of these moments, ofnarrownesswilloftenbeworthwhile,anditobviouslynecessary
butsuchapplicationsalsorequiremorefidelityfromourobserva- ifwindsea(e.g.impactofsourcetermsonwindsea)isbeingstudied.
tionaldatasets,perhapspushingtotheirlimitsinsomecases.Van InSection5,theoverpredictionofwaveenergybelowthespec-
Vledder and Battjes (1992) raise concerns about the statistical tralpeakusingaDIA-basedmodelisdocumented,usingthephys-
properties of narrowness metric Q , which is based on Goda
D
(1970, 1985). In short, though the metric can be quite useful for
application to model spectra, application to measured spectra is 6 Thisdominanceisnotabsolute,however:intheGreatLakes,becauseoftheirsize,
buoymeasurementssometimesindicatetwopeaksduringrapidlyrotatingwinds,
less reliable due to sensitivity to the amount of smoothing used
implying a new windsea and old windsea (alternately named young swell)
increatingthespectra.Weanticipatethatthiscriticismwouldalso component.
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
8 W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx
Table3
Statisticsforsimulationsofduration74daysusingDIA.Thethirdcolumn(KHHnk=2,fullset)isessentiallythesamesimulationasusedbyRW07.
RBW2012 KHHnk=2 KHHnk=1
Fullset Subset Fullset Subset Fullset Subset
n 1685 1092 1688 1089 1688 912
Hm,0(m)
Bias (cid:3)0.02 (cid:3)0.01 0.02 0.01 (cid:3)0.12 (cid:3)0.06
RMSE 0.17 0.12 0.17 0.12 0.21 0.14
CC 0.96 0.97 0.96 0.98 0.95 0.98
Tm,0,1(s)
Bias (cid:3)0.07 (cid:3)0.07 0.11 0.12 (cid:3)0.36 (cid:3)0.26
RMSE 0.33 0.24 0.34 0.26 0.51 0.35
CC 0.90 0.94 0.90 0.94 0.88 0.95
Tm,0,2(s)
Bias (cid:3)0.08 (cid:3)0.08 0.09 0.10 (cid:3)0.35 (cid:3)0.26
RMSE 0.31 0.22 0.32 0.24 0.49 0.34
CC 0.90 0.94 0.90 0.94 0.88 0.95
Tm,(cid:3)1,0(s)
Bias (cid:3)0.05 (cid:3)0.03 0.15 0.16 (cid:3)0.37 (cid:3)0.23
RMSE 0.36 0.26 0.40 0.32 0.55 0.35
CC 0.90 0.94 0.91 0.94 0.87 0.95
QA
Norm.bias (cid:3)0.11 (cid:3)0.13 (cid:3)0.08 (cid:3)0.08 (cid:3)0.08 (cid:3)0.14
CC 0.76 0.70 0.76 0.68 0.63 0.61
Slope 0.88 0.87 0.90 0.92 0.88 0.83
QB
Norm.bias (cid:3)0.02 (cid:3)0.02 (cid:3)0.02 (cid:3)0.02 (cid:3)0.00 (cid:3)0.02
CC 0.84 0.85 0.83 0.84 0.74 0.80
Slope 0.98 0.98 0.98 0.98 0.99 0.98
QC
Norm.bias (cid:3)0.29 (cid:3)0.28 (cid:3)0.32 (cid:3)0.25 (cid:3)0.30 (cid:3)0.30
CC 0.53 0.61 0.57 0.63 0.58 0.55
Slope 0.70 0.70 0.71 0.73 0.69 0.68
QD
Norm.bias (cid:3)0.28 (cid:3)0.20 (cid:3)0.34 (cid:3)0.17 (cid:3)0.27 (cid:3)0.19
CC 0.46 0.78 0.52 0.81 0.53 0.74
Slope 0.80 0.80 0.80 0.81 0.83 0.78
icsofRogersetal.(2012)forS andS .Thesamebehaviorisdoc- (cid:5) Using a 10-day hindcast for Lake Michigan here, and in prior
in ds
umented for longer duration simulations and more traditional idealized computations (Hasselmann et al., 1985, RW07), we
forms of S and S in the Appendix. Further, the behavior is ob- findaconsistentoutcome:themodelswhichuseexactcompu-
in ds
servedbyArdhuinetal.(2010)usingathirdsetofphysics,anda tations for S produce more narrow frequency spectra than
nl4
differentmodel,WAVEWATCHIII,appliedinLakeMichigan.Ard- comparablesimulationsthatusetheDIAforS .
nl4
huinetal.(2007)observesimilarbehaviorusingathirdmodel,ap- (cid:5) Fourmethodsarepresentedforquantifyingthenarrownessofa
pliedattheNorthCarolinacontinentalshelf.Therefore,itmaybea spectruminfrequencyspace.ThosedenotedasMethodsCand
robustfeature,notspecifictoaparticularmodel,orphysicspack- Dpresentedherearefoundtobemostuseful,insofarastheyare
age,orhindcast.Herein,thisbehaviorispartiallyattributedtothe found to be suitably sensitive to the narrowness that is clear
DIA by applying the same hindcast with XNL. In this study, only from visual inspection. Method C is a frequency analog of a
onemodel(SWAN),onehindcast(LakeMichigan),andonesetof method proposed by Babanin and Soloviev (1987, 1998a) for
physics are employed. It is not demonstrated that utilization of quantifying directional width. Method D employs the peaked-
XNLinanothermodel,hindcast,orwithotherphysicswouldyield nessfactorusedbyGoda(1970,1985).
the same improvement to model agreement with observations. (cid:5) The narrowness quantity associated with Method C agrees
However, it is noted that narrower or more peaked spectra with well—in the mean—with observed frequency spectra if exact
XNL is a robust characteristic of comparisons between DIA and computationsareusedforS inthe10-dayhindcastpresented.
nl4
XNL (e.g. Hasselmann et al., 1985, RW07, Tolman, 2011), so it is (cid:5) Scatterofthesamenarrownessquantityfor thesamesimula-
reasonable to expect that where this robust bias feature exists, tionis,however,worsewhenexactcomputationsareusedfor
applicationofXNLmayimproveagreementwithobservations. S .
nl4
8.Conclusions
Acknowledgments
Wesummarizetheresultsofthisstudyasfollows:
The authors thank Dr. David Wang and anonymousreviewers
(cid:5) FromanalysisofhindcastsforLakeMichiganpresentedinthis for constructive comments and suggestions. Dr. Wang was espe-
study, as well as prior cited works, overprediction of energy cially helpful in pointing out earlier work with metrics. Results
below the spectral peak—resulting in overly broad non-direc- fromthisstudywerefirstsharedwiththewavemodelingcommu-
tionalspectra—maybeaconsistentfeatureof3Gmodelswhich nitybyProf.AlexBabaninatthe2011WavesinShallowEnviron-
utilizetheDiscreteInteractionApproximation(DIA)forSnl4. mentsmeeting,forwhichtheauthorsaregrateful.Thisworkwas
Pleasecitethisarticleinpressas:Rogers,W.E.,VanVledder,G.P.Frequencywidthinpredictionsofwindseaspectraandtheroleofthenonlinearsolver.
OceanModell.(2013),http://dx.doi.org/10.1016/j.ocemod.2012.11.010
W.E.Rogers,G.Ph.VanVledder/OceanModellingxxx(2013)xxx–xxx 9
supported by the Office of Naval Research under the National ofintegralwaveheightandperiodmeasures)thantheXNLbased
OceanPartnershipProgram. runs.ThisisprobablyduetothefactthattheXNLbasedmodelwas
notcalibrated,butwedeemthisnotessentialforourpurpose.We
alsopresentstatisticsforalongersimulation,thatusedbyRW07.
AppendixA.Additionalerrorstatistics,includingresultswith ThesearegiveninTable3.Forthis74-daysimulation,calculations
alternatephysics wereperformedonlywiththeDIA.Asalreadyconcludedforthere-
sults presented in Table 2, the type of input/dissipation physics
Thoughnotaspecificgoalofthisstudy,itisusefultodocument packagehaspracticallynoeffectontheapplicabilityofthenarrow-
theperformanceofthemodelsintermsofconventionalquantities, nessparameters.
waveheightandmeanperiod.Asmentionedearlier,validationof
higher order moments is more meaningful in cases where lower
order moments are in good agreement. In this appendix, we in-
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