A Generalized Parametric Wind Wave Model

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J. Great Lakes Res. 20(4):613-624Internat. Assoc. Great Lakes Res., 1994

A Generalized Parametric Wind Wave Model

Stephen ClodmanAtmospheric Environment Service

Downsview (Toronto), Ontario M3 H 5T4

ABSTRACT. I have re fined an d generalized an existing parametric lake wind wave model to clearlyrelate the model parameters to the wave spectrum an d momentum balance. By solving the model analytically fo r a simple wave growth case, I show that the generalized model can be easily adapted to advancesin the description o f the wind wave spectrum an d the form o f the wave growth function. Case studies onthe Great Lakes demonstrate that systematic errors in the output wave period can be corrected by changing output dependent parameters an d the wave decay adjustment. I also ad d a simple, economical, shallow water parameterization. Use o f the generalized model form will make it easier fo r the non-specialistto simulate wave conditions on the Great Lakes.INDE X WORDS: Wind wave model, wave forecasting.

INTRODUCTIONA wind wave model simulates the physicalprocesses of wave development so that the impor

tant features of the wave field can be calculatedfrom the known wind and other conditions. Such amodel is useful for forecasting, marine design, andother purposes. In general, the final eventual outputof the wave model is the total wave energy E,which is determined from the energy spectrum Efe=Ef.e(f,e) where f is frequency and e is wave direction. In spectral models, this calculation is made formany different values of (f,e), and the interactionsbetween these different components must also befound. It is often simpler and cheaper to do a singlecalculation of E for the whole wave field instead.This can be done accurately if a good assumption,or parameterization, of the spectrum Ef e is made.Models that use such assumptions are referred to asparametric models; they are discussed in more detail in U.S. Army (1984).Fortunately, experience shows that if there is asimple growing wind wave built up by a constantwind velocity, the energy spectrum has a consistentform suitable for parametric models (Hasselmann etat. 1973). The main requirement is that the waterbody not be too large, 500-1,000 km). Withinthis l imit, space and time changes in wind velocitywill usually not dis tort Ef e too much. Therefore ,parametric models are usually suitable for lakes

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such as the Laurentian Great Lakes and small seassuch as the Beaufort Sea.Parametric models come in different types. Simple models (sometimes called Sverdrup-MunkBretschneider models) compute a single value of Efor the wave using only wind speed and fetch(Bretschneider 1973, U.S. Army 1984). By contrast,the type of parametric model described here usesphysical theory more completely, permits variablewind velocity in location and time, gives control ofinitial wave conditions, and provides output valuesat every grid position in a two-dimensional area.Parametric models consist of two main elements: amodel wave growth function and a spectrum parameterization. The growth function resembles that used inspectral models, so that this class of parametric models is somewhat like spectral models with only onespectral component. The growth function relates thetotal wave energy (E) to fetch or time, and is normalized by total wind stress. Hasselmann et at. (1976) reviewed experiments to find the evolution of the waveenergy and frequency, showing that this depends onlocal conditions. Various wind wave models give various growth functions, as shown in Figure 1(SWAMP Group 1985, Khandekar 1989).

It should be noted tha t the growth function includes only that portion of the wind stress that actually contributes to building or d iss ipat ing thewaves. I t is necessary to define these processes inthe wave model as accurately as possible. Modelers


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614 Stephen Clodman

x*

x[m), 04 " .--0_4-,---r----"0_5---,-_----r_,-ro_--,-_---.---, through his choice of 13. I will include various possible growth functions as options in a single model toallow the user freedom to choose the most accurateform; the method will be explained in the model derivation below.

The other principal part of the model is the spectrum of energy as a function of wave frequency anddirection. This controls the relat ion between thephase velocity, momentum, and energy of the wavetrain. Several different forms of the spectrum arepossible (Khandekar 1989), and this choice is stillbeing studied. The commonly used JONSWAP energy-frequency spectrum (Hasselmann et ai. 1973)is shown schematically in Figure 2. Ewans and Kibblewhite (1990) determined how the various parameters of the spectrum vary, showing, for example,that the spectrum becomes less peaked as the wavedevelops. Phil lips (1985) revised the spectrum forhigh frequency, showing that the energy variation athigh frequency f should be be proportional to f-4rather than [-5. Hasselmann et al. (1980) studied theenergy-direction spectrum, finding that the spreadangle of the wave field depends on wave age and onfrequency. Again, I will show in the derivationbelow a model which enables the user to choosethese different factors as needed.

E*- -SCN/M- - OOGP81'10- ' - ' -DNS.......-- ... EXACT NL----- GOND- - H YPA--MRI- - - - - N:lWAMO--SAIL_ ._ . - TOKlKU- VEN ICE

FIG. 1. Nondimensional growth curves fo r energyE against fetch x for various wind wave models.The diagram is from MacLaren Plansearch Ltd.(1985). Similar diagrams, with model descriptions,are in SWAMP Group (1985) and Khandekar(1989). The wind velocity is assumed to be 20 m s-1at 10 m height. Here x* = g u.-2 x, E' = g2 u.-4 Ewhere u. is friction wind velocity.

have used several significantly different equationsto do this, giving different shapes of wave growthcurves. Assume the wave energy growth equat ionin time dE/dt = 13 E , where 13 is a growth functionto be determined. Now, define the wave age Cn tobe the ratio C/U of wave phase speed to windspeed. Set the scaled wave age S =CiCnf where Cnfis C/U at full wave development, so that S increasesfrom 0 to 1 as the wave field grows under constantwind. Note that the wind speed may be a measuredwind speed (Barnet t 1968), a fric tion wind speed(Plant 1982), or a measured wind speed adjusted forstatic stability (Schwab et ai. 1984).Ursell (1956) used a form drag of the wind pushingon the moving wave shape, so that 13 ex U2 (1 - S)2.Schwab et ai. (1984), following Donelan (1978), builtthe waves entirely by form drag, although considering form drag to be only part of the total stress. Barnett (1968), on the other hand, used a stress of form13 ex U2 (1 - S). The wave-building stress decreaseswith wave age (Donelan 1979, Janssen 1989, Hsu etai. 1982). The modeler can control stress variations

PastWork on this ModelOur model is derived from Schwab et ai. (1984)as revised by Clodman (1989a); Schwab et ai. wasin turn based on Donelan (1978). Donelan chose tostate his wave growth dM/dt in terms of momentumM rather than energy E; since M can be convertedto E, either could have been used. (The HYPA spectral model (SWAMP Group 1985) also uses a mo

mentum balance.) The wind is adjusted to anequivalent neutral wind at 10 m height, and can ifdesired be a function of location and time. TheJONSWAP spectrum (Hasselmann et ai. 1973) isused. Schwab et ai. did a case study test on LakeErie, finding accuracy to be reasonably good. Theirresults, however, may be affected by shallow waterconditions on Lake Erie.

Clodman (1989a) tested the model on severalseasons of U.S. National Data Buoy Center(NDBC) type 191 buoy data (National Oceano-graphic Data Center 1984) on Lakes Michigan(buoy 45007), Superior (buoy 45004), and Huron(buoy 45003), which are shown on Figure 3. Hourlybuoy measured wind velocity, air temperature, andwater temperature values were used to obtain theequivalent neutral stabil ity input wind, which was


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A Generalized Parametric Wind Wave Model 615

t EMAXEf

EMAX' )"- PMEMAXE PMMAX'

,-5fp

FIG. 2. Schematic diagram of JONSWAP spectrum (modifiedfrom Hasselmann et al.1973). Herea =A cn-a, EMAX is peak of Ef , and PM refers toPierson-Moskowitz (y= I)form of spectrum.

tI

assumed to be cons tant with location on the lake.The model was run to get a wave height and meanperiod to compare to the buoy measurements. Themodel accurately computed the wave height, with abias near zero, consistently over different seasons,s tatic stabili ties (within a moderate range), windspeeds, and wind directions. However, Clodmanrecommended an increase in drag coefficient overthat of Schwab et aZ. (1984), based on case studytesting. The model wave height closely followedthe measured wave height as the wave increasedand decreased from day to day. Computations ofwave period, however, were less accurate, with atendency to swing too far between short and longperiods.Clodman (1990) applied this model to a study ofNDBC buoy wind accuracy. Model wave heightscalculated from buoy winds were compared to thebuoy measured heights . A consistent bias in thewind speed , varying between years, was found.Clodman and Eid (1988) successfully tes ted themodel on the Beaufort Sea, showing that it is usablefor that location. Their results were satisfactory, al-

KilometersI I Io 100 200Figure 3. Great Lakes ofNorth America with locations of u.s. NDBC buoys.


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616 Stephen Clodman

(1)

(3)

though limited somewhat by data quality. Clodman(1989b) discussed the model systematic errors. Although wave heights generally had litt le bias, certain situations could cause error. Wave height for avery stable atmospheric boundary layer was overestimated. Decaying waves were somewhat underestimated and so were rapidly turning waves. Waveperiod biases were substantial. The computed period of long period waves tended to be too high andthat of decaying waves too low.In the remainder of the paper, a new, more general form of the model will be demonstrated. First,differential equations of the wave momentum balance will be given, and a parametric wave energyspectrum and the wind stress presented. The equations will be simplified to express an analytic solution for a growing wave using new parameters.Decaying and shallow water waves will be brieflydiscussed. Finally, there is some illustrative testingof the finite difference model to show how some ofits features may be used.

The following improvements will be demonstrated: The wave spectrum can be modified by theuser to accomodate changes in wave modelling theory. The wave frequency to energy relation can bemade output dependent to correct systematic errorsin wave period. Different shapes of wave growthcurve can be used to match different models andconcepts of wave growth. A provision for decayingwave condi tions prevents wave periods from becoming too short. A simple shallow water adjustment is included in the model.

MODEL EQUATIONSMomentum Balance Equation and SpectrumI will now describe the detailed formulation ofthe model. This procedure, while based on Schwabet ai. (1984) and Clodman (1989a), is more general.

Assume linear waves and a single (unimodal) wavetrain, in deep water, with no water current, on an x,y surface. The wave energy, as stated above, is afunction of frequency and direction so that 0-2 = E =I I Efe df d8 over all f, 8. Potential and kinetic energy of the wave are equal in this case. Assume thespectrum is separable: E fe =Ef G . Here G =G(8)is the normalized (f G d8 = 1) direction spectrum,and G is assumed symmetric about the mean wavedirection 80 (as in Hasselmann et ai. 1980), with areasonably narrow angular spread. Ef is the frequency spectrum, with I Ef df =E .

Define the followingH == 40- significant wave heightRs == I G cos(8-8o) d8 wave angular spreadparameterfp frequency at peak of spectrum Erfs == I Eet df / E mean frequencyRr == f /fp wave spectrum skewness parameterg gravityPw ' Pa water and air densityc(f) =g / (2 1t f) wave phase speed at givenfrequencyC =c(fp) wave train phase speed at peak frequency

State components of wave momentum M j and fluxtensor T jj (Schwab et ai. 1984):Mx =g I I (Er,s/c) cos 8 d8 dfM =g I I (Er s/c) sin 8 d8 dfy ,Txx =1/2 g I I Er,s cos2 8 d8 dfTyy =1/2 g I I Er,s sin2 8 d8 dfTxy = Tyx = 1/2 g I I Er,s cos 8 sin 8 d8 dfThen, with 1: = (' tx' 't y) the wave-building windstress, Schwab et ai. (1984) give the wave momentum balance:aMx (fJ'xx (fJ'xy r x--+--+--=-at ax dy PwaMy (fJ'yx (fJ'yy ry--+--+--=-at ax dy Pw

Now simplify M by separating E f e and using thedefinitions for Rs' Rf' fs ' c(f): 'M = (Mx,My)=RfRsgC1a2 (cos 00 , sin ( 0 ) (2)

Now Txx = ~ g 0-2 I G cos2 8 d8 . Integrate a suitablestep or cosine function for G meeting the conditionsabove, finding I G cos2(8-80) d8 == Rs2 for narrowspread. Similarly simplify Txy =Tyx and Tyy and get

Txx = ,7iga2 [(2R; -1)cos2 80 + (1 - R;)]Txy = TyX = ,7iga2 (2R; -1)cos80 sin 80Tyy = ,7iga2 [(2R; -1)sin2 00 + (1 - R;)]Now consider the frequency spectrum in the form

(4)


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A Generalized Parametric Wind Wave Model 617where A and a are constants and Sh(f) is the spectrum shape. Then define the spectrum energy ratioRp' the nondimensional energy of Sh(f), and use (4):

Ef( f) = A C ~ a i ( 2 1 t ) - - 4 r 5 exp[-1.25(fr)--4Hy expo]where 1)

Frequency and energy can be found from the otherquanti tie s using (2) and (5). Once M is found ateach grid point and time step from (1) and (3), thewave height and period are computed, with wavedirection eo along M.To complete the evaluation of (2), Rf and Rp arefound from Sh(f). Various spectra can be usedfor Sh(f), but we will assume the commonly usedJONSWAP formula (Hasselmann et al. 1973):

(9)

(8)

coefficient D =D crbs , and assumed dissipationws sCnf-2 C C , skin drag is

More generally, the drag can be stated as a mixtureof skin and form drag. For simplicity, and becausethe result is not much affected, assume bf =bs =b.Defining combined effective drag coefficient D ==D f + Ds and form drag fraction F == DID we have! = PaDab[FIQ - ~ I Cnfl(Q - ~ I Cnf )

+ (1- F)(UQ - C , ~ i C ~ ) ]The model can now be solved numerically. If waveheight and direction, and wind velocity, are known ateach grid point at initial time, (1) is solved in finite

difference form to get the momentum at the next timestep. The new wave height, period, and direction canthen be derived. The calculation can be carried as farforward in time as needed, provided the wind velocityis known at each time and place.

(5)Rp = f p4JSh(f)df= a 2 I(Ag-2U aC4 - a )

Figure 2 illustrates this shape. Here Hasselmann etal. use the values A = 0.0097, a = 2/3, 'Y = 3.3,13a =0.07, 13b =0.09.

Wind ForcingTo solve (1), completing the model, the stress 1

forcing the wave is needed. We will assume thestress is a combination of form and skin drag (seeabove). Schwab et al. (1984) and Clodman (1989a)used a form drag, with stress proportional to thesquare of the velocity difference between the windand the moving wave. They state form drag W f DTf'where DTf is the total form drag coefficient and W fis the proportion of the drag which goes to buildthe wave. Schwab et al. used DTf = [0.4l1n(50/cr)]2,W f = 0.028 while Clodman (1989a) recommendedW f = 0.08 with the same DTf . We here combineW f DTf == Dwf ' the effective form drag coefficient,and express it in the more general DWf = Df crbf Clodman's DWf is approximately equivalent to Df =0.0008, bf= )0 . The form drag equation is(7)

where the air density term Pa is a correction toSchwab et al.Skin drag is assumed to be tangential stress onthe water proportional to the square of the wind velocity, less a wave dissipation term. With skin drag

Parameter ListThe model parameters and their typical valuesare listed in Table 1. The following parameters givethe energy-frequency spectrum: spectrum amplitudeA, nondimensional energy of the spectrum Rp ' waveage amplitude variation a, and frequency skewnessR f . The simplified energy-direction spectrum is represented by the angular spread ratio Rs ' The generalwind stress parameters are the effective drag coeffi

cient D, the form drag fraction F, and the waveheight adjustment b. Together these provide thevariable read-in or output-dependent specificationsfor the model. Variations in drag coefficient and angula r sp read ratio were d iscussed in Clodman(1989a), but we will here consider the effect ofvarying all of the parameters.We will show that the parameters have specific applications to improving the wave modeling. Rf' A,R , and R can be used to take into account newp sknowledge of the wave spectrum. For example, inte-grating the JONSWAP equation (6) with Hasselmannet al. 's (1973) parameter values gives R f = 1.19,R = 0.305, but if an f-4 term replaces the f5 term(Phillips 1985), then R f = 1.38, R =0.365. Reducing'Y0r 13 in (6) would reduce R ana increase Rr- Making Rf output dependent cari be used to correct thesystematic biases in the wave period (demonstratedin model testing below). The stress parameters F andb can be used to control the shape of the wavegrowth curve as discussed in the next section.


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618 Stephen ClodmanTABLE 1. Summary of model parameters with definitions and typicalrange of values. Bracketed numbers are equation numbers defining theparameters. Note that the growth curve parameter m =a + 2b - %ab and thetotal spectrum amplitude AR = A Rp"Spectrum parametersA (4)Rp (5)a (4)R f f I fRs 1Gcos(S-So) dSStress parametersCnf CIU at S = 1D D f +D sF DIDb (9)

0.008-0.0120.3-0.40.0-1.51.15-1.450.85-1.0

1.0-1.20.0003-0.0010.0-1.00.0-0.5

spectral nominal amplitudenondimen. energy of spectrum shapeamplitude wave age dependencefrequency skewness ratioangular spread parameterwave age CIU at full developmenteffective drag coefficientform drag fractiondrag variation with wave height

GROWINGWAVE SIMPLIFIED SOLUTIONS

Then the spec ial cases discussed above-skin,mixed, and form drag-can be stated:

(15)

(14)

where T=_I_Kr-IVm-3U2b-It3 -a2-a I

'h K - 4-aR RA4-a - I D- Iwi t r = g f s R Pa Pw

dX = [1 - 2FS + (2F -1)S2r l S3-mdSh X - 2 K-1Vm-4U2b-2were =-- x4 -a x

. h K R2 - I D- Iwi t x =g sPa PwA time-dependent equation in scaled time T can befound similarly

dT = [1- 2FS+ (2F -1)S2r l S2-mdS

Fetch and Duration Equations and SolutionsAnalytic solutions to certain forms of the model

equations will now be stated and displayed (Table2, Figs. 4-6). These solutions, although highly simplified, relate the model to its parameters to showhow the model can be tuned by considering the effects of the various parameters on the model results. To state the fetch-dependent ODE in terms ofS, proceed from (13) with aE/at =0 . Use (10) for"t, and s ta te crb and dE in terms of S using (12).Then set m == a + 2b - Yzab and convert to scaledfetch X:

The combination of (10), (12), and (13) gives a partial differential equation in one variable which canbe solved numerically. Here (13) could have beenstated with S or M instead of E. I f aE/at =0 , (13)becomes an ordinary differential equation (ODE)for fetch dependent growth, and if aE/ax =0 , (13)is an ODE for time-dependent growth.

(13)

(12)

(11)

(10)

r = PaDcybU2(l- S2) F =0r = PaDcybU2(1- S) F =12r = PaDcybU2 (1- S)2 F =1

Note that the mixed drag equation in (10) is similar tothat of Barnett (1968). Now from (5), since C =Cnf US , S can be stated in terms of energy (AR == A4 IS = V-IU - 4-a E4-a

Iwhere V = Cnf (g-2 AR )4-a

Without loss of generality, assume eo =O. Then (1)becomes, using (2) and (5) for the a/at term, and (3)for the a/ax term:I 13-a R R ( 2-aA ua)4-aE-4-a JE4 -a f s g R at

1/ R2 JE._ r+/2g ---ax Pw

Differential EquationsNow we derive a simple growing wave to permita scale analysis and show how to choose the values

of the parameters. Set the wind direction equal tothe wave direction eo and recall that scaled waveage S increases from 0 to I as the wave develops,so the stress is


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A Generalized Parametric Wind Wave ModelTABLE 2. Solutions of (14) and (15) for nondimensionalfetch X and timeT in terms of scaled wave age S, for different m and F, with S =0 at X =0or T= O.

619

0. 5

0.1

i l lo Fo1/21o1/21

x- ~ l n ( I - S 2 )

-S3/3 - - S - In(l-S)+ 2S + 3 In(l-S) + S/(I-S)

-S + ~ l n ( l +S) - ~ l n ( l - S )_ ~ S 2 - S - In(l-S)-(l-S) + 2 In(l-S) + ( l - S ) ~ l

1. 0

0. 5

I52 0. 20.1

T-S + ~ l n ( l + S ) - ~ l n ( l - S )

_ ~ S 2 - S - In(I-S)-( l-S) + 2 In(l-S) + (l-S)-l

- ~ l n ( l - S 2 )(l-S) - In(l-S) - 1In(l-S) + S/(l-S)

0.05 ---- .... . L _ ~ - - . L _ _ , _ ~ .._..1 J. J - L ._0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10

x -FIG. 4. Nondimensional fetch growth curves forsolutions of (14) shown in Table 2, with S2 vs Xshown for various m and R Lines are m = 0 (solid)and m = I (dashed), with F = 0 (no dot), F =(small dot), and F = I (large dot). The result of thenumerical model run for m = I, F = 0, to confirmthat the model matches the analytical solution, isrepresented by X's (compare to the dashed, no dotline). Numerical model results fo r other F, m,match similarly to the corresponding solutions.

0. 5 .

0. 2

0.02 ~ _ l _ . . . I. .._ ~ . L _0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10T -

FIG. 5. Nondimensional time growth curves fo rsolutions of (15) shown in Table 2, with S2 vs. Tshown for various m and F, and with numericalmodel run. See Figure 4.

O.05 ~ . L . - - . - " - - - - . ' - ' - - - - - - ' . - - - - - ' - - - - .0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10x-

FIG. 6. Difference of shape of solution curvesillustrated. Curves are for fetch, S2 vs. x (x in arbi-trary units), for m = 1 with F = 0, 1. These arethe same curves as Figure 4, but shifted left andright to pass through a common solution point.This shows schematically that the drag coefficientcan be adjusted to fit a known solution point(intersection of lines) and a known fully developedwave height (H at S = 1), but the growth curve isstill a function ofR

Assuming the parameters and the wind velocityare constant in space and time, (14) and (15) can besolved analytically for m an integer and general F.Table 2 gives solutions for various m, F. Figure 4(fetch-dependent) and Figure 5 (time-dependent)show the log-log curves of these solutions graphically for S2. Here S2 is approximately a heightgrowth curve (actually S2 DC H4/(4.a. Figure 6 further shows the difference in the shape of the solution curves by setting them to pass through anarbitrari ly chosen solution point, with a change inthe factor Kx shifting the curve left or right.Thus F and m control the shape of the growthcurves of S. As F increases from 0 to 1, the wavegrowth near full development becomes slower, thiseffec t becoming greatest near F = 1. In otherwords, drag coefficient D must increase as F in-


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620 Stephen Clodmancreases to give the same rate of growth. Increasingm reduces development for small fetch or time butgives slightly more rapid growth near full development. Put another way, changing F and mallowsfor var iat ions in rate of wave development withwave age (e.g., Janssen 1989, Donelan 1979). Although m is a function of both a and b, we suggestthat m would normally be adjusted to control thegrowth by varying b, since b is a stress parameterand since the H-S relation is independent of b, butdependent on a.The growth curves for spectral models, illustratedin Figure 1, can be qualitatively compared withthose of the present model (Fig. 4). The growthcurves of this model can be adjusted to resemblethose of any of the spectral models by changing Fand m as discussed above. For example, some models that approach full development quickly resemble our model with skin drag (F = 0), while thosewhich take longer to reach full development, aremore like form drag (F = 1). The growth curves ofsimple empirical models (e.g., Bretschneider 1973)can also be matched to our model by a similar strategy, but we will not give the details here.

The model can now be tuned to existing wavegrowth and spectral data. The directional spectrumgives Rs and Sh(f) in (4) gives Rp' Rf I f a and Cnfare now set, and wave height H for full development (S = 1) is known for given wind speed V, thenA can be computed from (12). Then adjust band Fto control the shape of the growth curve. I f S isknown for a particular scaled fetch X or time T, thedrag coefficient D follows from (14) or (15).The numerical model also computes the wave behavior for turning waves (wind at an angle to thewave train). Equat ions in dS/dt and d8/dt can beformulated, using (9) in (1), and differentiating (2)and solving; we will not show the details. I f F =0,d8/dt oc sin 8 which matches the result of Guntheret al. (1981).

OTHERMODIFICATIONSDecaying wave adjustment

Since the model is intended mainly for short tomedium fetch (less than 1000 km), decaying wavescan be assumed to become small or reach shorequickly. Therefore a roughly approximate solutionis sufficient for such waves; complete modeling ofswell is not needed. A decaying wave simplified solution analogous to (15) could be found. This would

show a model period Cmod decreasing as the wavedecays, which is unrealistic. However, we havehere added a simple decay wave adjustment. In thisversion, phase speed (and wave period) is held constant by treating it as a conservative advected quantity Cadvec ' when Cadvec > Cmod . Testing of thisfeature is discussed below. (An adjustment decreasing the drag coefficient during decay was alsotested on the Great Lakes, but was found to haverelatively little effect.)

ShallowWater AdjustmentThe model described above in this paper is designed for deep water situations. However, certainlakes (Lake Erie, Lake St. Clair), and nearshore regions of other lakes, are shallow enough to makethe wind waves depth-dependent. I have implemented a simple, economical procedure for suchsituations. This method is not, of course, equivalentto complete shallow water models having detailedbottom effects and wave refraction. However, thecomplexity and computational burden of completeshallow water models makes it difficult to use themroutinely. For moderate depth and gentle bottomslope, a simpler method might give adequate performance, so that the model error is at most a littlegreater than for deep water.Consider the empirical wave height equation of

V.S. Army (1984), based on their modeling resultsfor general shallow water conditions:H sh = O . 0 2 8 9 U ~ F d t a n h ( O . 0 0 5 6 5 X ~ . 5 FiJI)

where FD == tanh(0.53Z?75) (16)and Xe ==gU;/x, Ze ==gU"A2Zd

with x the fetch, zd the water depth, and VA =0.71V1.23 where V is wind speed adjusted for stability.Since tanh y < 1 , tanh y


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A Generalized Parametric Wind Wave Model 621lated. To prevent occasional unrealistic low heightsfor rapidly decreasing winds, Hsh 1.5 m is imposed (done for large lakes only) . To actually use itin the model, Hsh is converted to shallow water momentum by (2) and (5). The minimum of shallowwater and deep water momentum is then taken, andthe other model outputs are then computed. Sincethe new wave momentum is inserted in the modelequations (1) to be used in the next time step, theshallow water wave adjustment is an integral part ofthe model, not just a correction. This helps ensurethat the wave varies in a realist ic way when thewind velocity or water depth varies.

MODEL TESTINGThis section briefly discusses testing of the finitedifference wind wave model to compare the numerical results to the analytic solution above, and toshow the use of the model innovations in specif iccases. The purpose is not so much to discuss precise model accuracy, since this will depend on thelocation and on the data sets used, but rather to illustrate the method. The mathematical equations ofthe model are given above by (1)-(3), (6), and (9).Testing with data from Lake St. Clair and LakeErie (buoy 45005-see Fig. 3) shows the effect ofthe shallow water adjustment to be qualitativelyreasonable and beneficial - the height error is usually a few tenths of a meter, and is usually smallerthan with the deep water model alone. However, a

detailed quantitative verification is quite difficultand beyond the scope of this paper. Note that thismethod adds only about 5% to the model computational time. The shal low water adjustment can beshut off by the user if not needed. In the remainderof this section, the shallow water adjustment wasnot used, since the testing was done in a deep waterarea.The model was run for simple constant wind andgrowing wave situations for various values of a, b,F, and U to show that its behavior is accurately represented by the solutions to (14) and (15). It wasfound for various F, m, that in fact these solutionsdo match the model result closely. To illustrate this,model solutions for m = 1 and F =0 are plotted inFigures 4 and 5, with S and X or T scaled by (12)and either (14) or (15).

Case StudiesCase study modeling was then done, based on theprocedures of Clodman (1989a). The testing used

u.S. National Data Buoy Center (NDBC) type 191buoy data (National Oceanographic Data Center1984). For the cases discussed here the continuousperiod 22 March - 8 December 1982 on LakeMichigan (NDBC buoy 45007) was used (Fig. 3).The gridsize was 15 km, with verification of waveheight and mean period every 3 h. (A complete runtook 150 s on a 24 MIPS computer.) The buoy windspeed was increased 10% to correct the systematicbias found by Clodman (1990), and adjusted to anequivalent neutral wind at 10 m height using thesurface air to water temperature difference.

Based on Schwab et aZ. (1984) and Clodman(1989a) we start by setting the spectral parameters toA = 0.0097, a = 2/3, R = 0.305, Rf = 1.19, Rs = 0.96.Set the stress p a r a m e t ~ r s Cnf = 1.2, F = 1, with Wf'DTf as in Clodman (1989a) (equivalent toDf=0.0008, b = ~ - s e e above) . Figure 7 illustratesthe height forecast graphically with a time series fora 2-week period of active weather. Note that themodel wave height closely follows the observedwave height as the wind increases and decreasesfrom day to day. Period computations with Rf= 1.19(Fig. 8) have some systematic biases; the period istoo long for long waves and sometimes too short forshort waves. In other words, the wave height resultsare accurate. However, the wave height and periodcan not be tuned to be simultaneously correct, because one or more of the spectrum parameters are incorrect . We will attempt below to correct this byadjusting the spectrum skewness Rf.Table 3 shows results for mean periods greaterthan 3 s (about 40% of the time) to emphasize thelarger waves (H > 0.7 m approx). The algebraicmean error and the error root mean square of Hareabout 0 .0 m and 0.25 m respectively for thesecases. In Table 3, where a is decreased to 0 and A isincreased to 0.012, the mean height is kept aboutthe same. The effect of this change to A and a depends on the wave age; for the relatively smallwave age in most of these cases, the period is reduced. Varying Rs has almost no effect on mean statistics, but some strong wind and long fetch caseshave higher waves for higher Rs (not shown).We will now show in Table 3 and Figure 8 howvarying the parameter Rf and using the decay adjustment can correct the systematic period errors.Increasing Rf to 1.31 has the effect of reducing thecomputed period, in this case improving accuracy.I t does not affect H for the fe tch-dependent case.For the runs of Table 3, only about 2% of the periods are more than 1 s too short and only about 2%


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622 Stephen Clodman

3

2H

15

10U

5

221098210 13 14 15 16 17OCTOBER 1982 (00 UTC)FIG. 7. Time series plot o f wave heights and wind speeds at buoy 45007 on Lake Michigan (ClodmanI989a). Solid line-model H, long dash l ine-measured H, short dash l ine-measured U (m/s). The dateat 0000 UTC each day is shown and plots are every 3 h.

more than 1 s too long. Table 3 also tests R f increasing with decreasing computed frequency f ,making the spectrum shape an output function 6fperiod. This gives the lowest error variance becauseit corrects the overestimate of mean period for thelong period waves. Figure 8 shows that with Rf =1.19 the period maxima are too high and the minima too low. Using a variable Rf reduces high period values to realistic levels (e.g., 16 October)without reducing the lower period values.The advection adjustment for decaying waves isintended to prevent wave periods from becomingtoo low in decay situations while leaving growingwave situations unchanged. This sometimes workswell, for instance on 17 October (Fig. 8). The slightincreases in error variance and intercept shown inTable 3 suggests the period is sometimes increasedtoo much. Therefore this adjustment is not alwaysbeneficial, but i t is helpful for some cases with significant swell.

CONCLUSIONSThis study extends the parametric lake wind

wave model of Schwab et al. (1984) to make it

more general. More complete forms of the wavespectrum and of the wind stress are introduced.This a llows new developments in understandingwave behavior to be used, based on the sca leanalysis of the parameter effects. Only those features and parameters affecting the user need be employed.The JONSWAP spectrum high frequency tailorpeakedness can be varied by altering the spectrumparameters. Making the frequency ratio output dependent is used to remove systematic bias from thecomputation of the period so that it is more accurate. Wind stress parameters can be used to changethe wave growth curve, for example to make thegrowth of older waves faster or slower in proportion to that of younger waves. This helps keep themodel accurate in different physical si tuations. Inaddition, a decay advection of the wave period prevents excessive decrease of the wave per iod indecay situations. Finally, a simple shallow wateradjustment is introduced, placing a depth-dependentupper bound on wave height.The model computer code is wri tten in FOR-TRAN. The author will provide it for research purposes on request.


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PERIOD(SEC) 7

A Generalized Parametric Wind Wave Model

o-\---.-----r----,--,-----,-----,----,--. ,---,---.----r----,--,--,8 10 11 12 13 14 15 16 17 18 19 20 21 22

OCTOBER 1982 (00 UTC)

623

FIG. 8. Time series plot ofmodel wave periods at buoy 45007 compared to measured (x) values, fo r certain runs from Table 3. Short dash line with + : Rf =1.19 (A =0.0097), long dash l ine with .1: Rf variable,solid line with 0 : Rf variable with decay advection. Note that the "Rf variable" line does not appear whereit is equal to the "Rf variable with decay advection" line.

TABLE 3. Accuracy of hindcasts ofmean period fo r buoy 45007, 22 March-8 December1982. Mean observed height and period for these cases is about 1.45 m and 3.9 s.Rs =0.96, Rp =0.305, F =1, b =1/2. Err is the mean algebraic error, Err* is the rootmeanvariance of the error about the mean error. The slope and intercept (int) are of a l inearregression estimate of model period as a function of the mean period. With Rf varying,Rf =1.31 [1+0.5(0.25-fp )]. DA indicates that the decay advection procedure is usea.

R f A a Err Er r* slope int1.19 0.0097 0.67 0.35 0.47 1.18 -0.341.19 0.012 0.00 0.19 0.45 1.14 -0.361.31 0.0097 0.67 0.01 0.40 1.02 -0.08vary 0.0097 0.67 -0.10 0.36 0.90 0.29DA vary 0.0097 0.67 -0.05 0.41 0.86 0.49


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624 Stephen ClodmanACKNOWLEDGMENTS

Dr. Madhav Khandekar, of the author's institution, provided a useful internal review of the paper.Dr. Barry Lesht , Argonne National Laboratory,Argonne, I llinois, as an Associate Editor of thisJournal, made helpful suggestions for the benefit ofnon-specialist readers.REFERENCES

Barnett , T.P. 1968. On the generation, dissipation, andprediction of ocean wind waves. J. Geoph. Res.73:513-529.Bretschneider, C.L. 1973: Prediction of waves and currents. Look Lab / Hawaii 3: 1-17, Univ. of Hawaii.

Clodman, S. 1989a . Refinement and testing of a lakewind wave model on seasonal data. Atmos.-Ocean.27: 588-596.___ , 1989b. Evaluation and reduction of systematicerrors in lake wind wave hindcasts. In 2nd Int'l Work-shop on Wave Hindcas ting and Forecas ting ,Preprints, pp. 322-327. Environment Canada, Atmospheric Environment Service.___ , 1990. Estimating the variation of buoy wind andwave data biases. J. Great Lakes Res., 16: 288-298.___ , and Eid, B. 1988. Adapting a lake wind wavemodel to a small enclosed sea. In Fourth Con! onMeteorology and Oceanography of the Coastal Zone,pp. 62-65. Amer. Meteor. Soc.Donelan, M.A. 1978. A simple numerical model for waveand wind stress prediction. Canada Centre for InlandWaters, Burlington, Ont., Canada.____, 1979. On the fra ct ion of wave momentumretained by waves. In Marine forecasting: predictabil-ity and modell ing in ocean hydrodynamics, pp.141-159. Elsevier.Ewans, K.C., and Kibblewhite, A.C. 1990. An examination of fetch-limited wave growth of f the wes t coastof New Zealand by a comparison with the JONSWAPresults. J. Phys. Oceanogr. 20: 1278-1296.Gunther, H., Rosenthal, W., and Dunckel, M. 1981. Theresponse of surface gravity waves to changing winddirection. J. Phys. Oceanogr. 11: 718-728.Hasselmann, K., Barnett , T.P., Bouws, E., Carlson, H.,Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H.,Hasselmann, D.E. , Kruseman, P. , Meerburg, A.,

Muller, P., 0lbers, D.J., Richter, K., Sel l, W., andWalden, H. 1973: Measurement of wind wave growthand swell decay during the JONSWAP. Deut. Hyd.Inst. A(8). 96 pp.___ , Ross, D.B., Muller, P., and Sell, W. 1976. Aparametric wave prediction model. J. Phys. Oceangr.6: 200-228.____ , Dunckel, M., and Ewing, J .A. 1980. Directional wave spectra observed during JONSWAP 1973.J. Phys. Oceanogr. 10: 1264-1280.Hsu, C-T, Wu, H-Y, Hsu, E-Y, and Street, R.L. 1982.Momentum and energy transfer in wind generation ofwaves. J. Phys. Oceanogr. 12: 929-951.Janssen, P.A. 1989. Wave-induced stress and the drag ofair flow over sea waves. J. Phys. Oceanogr. 19: 745754.Khandekar, M.L. 1989. Operational analysis and predic-tion of ocean wind waves. Springer-Verlag, Coastaland Estuarine Studies 33.MacLaren Plansearch Ltd. 1985. Evaluation of the Spec-tral Ocean Wave Model (SOWM) for supporting real-time wave forecasting in the Canadian East Coast off-shore. Report prepared for Atmospheric EnvironmentService of Canada.

National Oceanographic Data Center. 1984. NationalOceanographic Data Center Users Guide. Key toOcean . Records Docum. 14, Nat'l OceanographicData Center, Washington, D.C.Phillips, O.M. 1985. Spectral and statistical properties ofthe equilibrium range in wind-generated gravitywaves. J. Fluid Mech. 156: 505-531.Plant, W.A. 1982. A relationship between wind stressand wave slope. J. Geoph. Res. 87: 1961-1967.Schwab, D.J ., Benne tt , J .R. , Liu, P.c., and Donelan,M.A. 1984. Application of a simple numerical waveprediction model to Lake Erie, J. Geoph. Res. 89:3586-3592.SWAMP Group, The. 1985. Ocean wave modeling.Plenum.Ursell, F. 1956. Wave generation by wind. In Surveys inMechanics, pp. 216-249. Cambridge Univ.U.S. Army. 1984. Shore Protection Manual, 4th edit .,Vol. 1. U.S. Army Coastal Engineering Research Center, Vicksburg, Miss.

Submitted: II January 1993Accepted: 27 June 1994

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A Generalized Parametric Wind Wave Model

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