Observation-based dissipation and input terms forWAVEWATCH IIITM: implementation and simple simulations

Stefan Zieger∗‡, Alexander V. Babanin∗, W. Erick Rogers† and Ian R. Young§

∗ Swinburne University of Technology, Melbourne, Victoria, Australia† Naval Research Laboratory, Stennis Space Center, Mississippi§ Australian National University, Canberra, A.C.T., Australia

Abstract

Observational data collected at Lake George, Australia resulted in new insights on the processes of wind waveinteraction and white-capping dissipation and consequently new parameterisations of these source terms.The new wind input source term (Donelan et al., 2006; Babanin et al., 2007a) accounts for the dependenceof growth increment on wave steepness, and on airflow separation with relative reduction of the growthunder extreme wind conditions. The new white-capping dissipation source term (Babanin and Young, 2005;Young and Babanin, 2006) consists of two terms, the inherent breaking term and dissipation induced bylonger waves. The present implementation follows Rogers et al. (submitted). Two novel parameterisationsare validated against existing source terms and buoy measurements for windsea-dominated conditions induration-limited simulations and hindcast.

1 Introduction

Numerical simulation of the evolution of wind-waveenergy density spectrum is routinely conducted inwave forecast and hindcast. In third-generation (3G)models, wind-wave evolution is described by the waveenergy balance equation (1) which totals all energyfluxes, source terms Stot, represented by all physicalprocesses that contribute to wind-wave evolution.The wave energy balance equation can be writtenas (Young, 1999):

∂F

∂t+∇ · cg F = Stot . (1)

The first term of the left hand side of (1) representsthe rate of net change of wave spectral energyF = F (ω, θ, t,x), a function of frequency ω = 2πf ,direction θ, time t and space x. The second termrepresents the advection of wave energy at groupvelocity cg. It is generally accepted (e.g., Komenet al., 1994; Young, 1999; Tolman, 2009) that fordeep water, the total source term (2) that contributesto wind-wave evolution is based on three physicalprocesses, all of which are spectral functions S(ω, θ).These processes are atmospheric input Sin, wavedissipation Sds, and nonlinear interactions betweenwave components of different frequencies Snl:

Stot = Sin + Sds + Snl (2)

Babanin and Westhuysen (2008), and Babanin(2011) argued that these individual terms haveto be further subdivided. For example, Sds isa sum of inherent and cumulative wave-breakingdissipation, of dissipation due to interaction with

‡Corresponding author address: szieger@swin.edu.auProc. of the 12th Int. Workshop on Wave Hindcasting andForecasting, 30 October – 4 November 2011, Kohala Coast, HI

turbulence in the water and the air, with theadverse wind and so on. Provision shouldbe made to let individual dissipation mechanismease and still let swell dissipation continue. Intransitional water and water of finite depthadditional processes, for example wave-bottominteractions and depth-induced breaking, becomesignificant in (2) and have to be considered in thetotal source term Stot (Tolman, 2009). This paperonly focusses on wind input Sin and dissipation Sds.Third-generation wave models such asWAVEWATCH IIITM, use the wave action spectrumN directly to calculate the source terms. Thewave action spectrum is a wavenumber-directionspectrum N(k, θ) which can be converted to the waveenergy spectrum using Jacobian transformation:F (ω, θ) = N(k, θ)ω/cg (Tolman, 2009). Theobservation-based source terms are given in form ofthe more traditional frequency-direction spectrumfor wind input and dissipation hereafter. The presentversion of WAVEWATCH IIITM (3.14) features threesource term packages: WAM3 physics (Komen et al.,1984), WAM4 physics (Ardhuin et al., 2010), andTolman and Chalikov (1996) physics.

The objective here is to test observation-basedsource terms as implemented in WAVEWATCH IIITM

for wind-dominated conditions in duration-limitedsimulations and hindcast. The paper is structuredas follows: Section 2 outlines novel features of theobservation-based source terms including the sourcefunctions; section 3 describes the model setup usedin both simulations. The duration-limited simulationis a purely academic test while the hindcast is basedon the Naval Research Laboratory (NRL) test bedfor Lake Michigan. Results obtained from thosetwo simulations are shown in subsections. Section 4provides a summary and directions for future work.

2 October 2011

2 Observation-based source terms

Spectral parametrisations of wind input and wavedissipation are based on field experiments carried outduring the Australian Shallow Water Experiment(AUSWEX) at Lake George, New South Wales,Australia. For the wind input, the boundary layerstudy is described in detail by Donelan et al. (2005)which was later parameterised by Donelan et al.(2006) as spectral functions to be used in wavemodels. For wave breaking and dissipation, fielddata unveiled novel features of the wave breakingprocess (Banner et al., 2000; Babanin et al., 2001)and spectral dissipation (Babanin and Young, 2005;Young and Babanin, 2006).

So far, the new source terms have been implementedand calibrated for two 3G wave models: (1) theone-dimensional research model WAVETIME1

(Tsagareli et al., 2010) and (2) the U.S. Navyoperational forecast model SWAN2(Rogers et al.,submitted). Wind input and dissipation functionsdescribed in section 2 follow the implementation ofthe NRL operational model as proposed by Rogerset al. (submitted).

In order to demonstrate calculations with new sourceterms for wind input Sin and wave dissipationSds a “transitional spectrum” is used followinga parametric Pierson-Moskowitz spectrum used byTsagareli et al. (2010) and Babanin et al. (2010). The“transitional spectrum” (3) features the transitionfrom ω−4 slope to ω−5 at the transition frequency.Here, a transition frequency of 3ωPM was selectedwhere ωPM (= 2π 0.13 g/U10) refers to the peakfrequency of the Pierson-Moskowitz spectrum for agiven wind speed 10 m above the surface U10 (seePierson and Moskowitz, 1964).

F (ω) =

{α g2

ω4 ωpexp(−[ωPM/ω]4) for ω ≤ 3ωPM

F (3ωPM) [(3ωPM)/ω]5 for ω > 3ωPM

(3)

The “transitional spectrum” is given in (3), whereα = 0.0054 is the Philips constant for thePierson-Moskowitz spectrum, ωp represents the peakfrequency, and g is the gravitational acceleration.An example of the “transitional spectrum” for windspeed of U10 = 8 m s−1 is shown in the top panel ofFigure 1 where the transition from ω−4 slope to ω−5

is marked with the bold dotted line.

2.1 Wind input

The wind input function represents the energy fluxtransferred from wind to waves. It is widelyaccepted

that wind-wave development is due to wave-inducedpressure acting on the slopes of the waves (Donelanet al., 2006). AUSWEX data analysis and the windinput parameterisation reported by Donelan et al.(2006) dependencies that have not been reported inprevious experiments.

Measurements of wave growth were available fora range of wind-forcing conditions including veryyoung waves U10/cp = 5.1 − 7.6 (cp is the phasespeed at the spectral peak) of varying steepness.This unique dataset unveiled a few novel features:(i) full air-flow separation with relative reductionfor strong winds, (ii) non-linear relationship betweenphase speed, crest curvature and the slope of thewaves, and (iii) the dependence of wave growth onwave steepness (Donelan et al., 2006; Babanin et al.,2007a). The resulting parametrisation of the windinput affects the momentum flux parameterisationin opposing ways; enhancement for moderate windsand reduction for strong wind forcing.

2.1.1 Novel featuresFull air-flow separation means that the winddetaches from the flow essentially skipping the wavetroughs before it re-attaches on the windward side ofthe wave crest. Compared to the non-separated flow,the imposed wind input pressure is relatively weakerunder full air-flow separation (Donelan et al., 2006).Full air-flow separation was also part of laboratoryexperiments conducted by Reul et al. (1999).

The wind input Sin = γωF is a function of thewave energy spectrum F and the growth rate ofthe wind-waves γ, hence it is linear dependent(Komen et al., 1984). In terms of friction velocityu?, for example, the relative growth rate γ/ω is afunction of the ratio of wind speed over phase speed.Empirical fitting of the growth rate typically followsthe form γ/ω = ρ a

wβ(u?/c

)n, where ρ a

wis the ratio

of densities of air and water, β is a non-dimensionalconstant, and n is an exponent of either n = 1 as inWAM3 or n = 2 as in WAM4 (Komen et al., 1984;Tolman, 2009).

AUSWEX field data also revealed that the wind-wavegrowth rate γ depends on wave steepness ak(a being the wave amplitude), which has previouslybeen considered as unrelated in potential theory forgravity waves. Donelan et al. (2006) showed thatwave steepness is connected to the phase shift andthe normalised induced pressure amplitude, and thatpotential flow is only valid as ak → 0. In the finalparameterisation the wave steepness ak is replaced bythe spectral saturation Bn (Phillips, 1984). Thus, thegrowth rate γ obtained from the Lake George data

1 developed by G. van Vledder (van Vledder, 2002)2 “Simulating WAves Nearshore” (Booij et al., 1999)

Zieger et al. 3

is also a function of the wave action spectrum, whichin turn makes the wind input non-linear dependenton the action spectrum.

2.1.2 ImplementationThe parameterisation of the wind input Sin asproposed by Donelan et al. (2006) is designed towork from young wind-waves to mature seas, i.e.for conditions of light, moderate, and strong windforcing. The observation-based wind input Sin wasestimated from the quadrature spectrum based onmeasurements of elevation and pressure at the seasurface (Donelan et al., 2006). The proposed windinput is given in (4)–(8) (Donelan et al., 2006;Tsagareli et al., 2010; Rogers et al., submitted).

Sin(ω, θ) = ρ awω γ(ω, θ)F (ω, θ) (4)

γ(ω, θ) = G√BnW (5)

G = 2.8− [1 + tanh(10√BnW − 11)] (6)

Bn = A(ω)F (ω) k3 cg (7)

W = (U10/c − 1)2 (8)

Donelan et al. (2006) calibrated the growth rate(5) based on winds 10 m above mean surface U10.However, 3G wave models typically scale growthrate with friction velocity u? and therefore a scalingof Cd = (U10/u?)

2 was selected in Rogers et al.,where Cd is the drag coefficient given in (13). Thevalue for the drag coefficient depends on variousphysical parameters and the scatter of experimentaldata is usually large (for a review see Babanin andMakin, 2008). Nevertheless, the flux computation istypically parameterised as a function of wind speedand is customisable in WAVEWATCH IIITM. Thedirectional distribution of (8) is implemented as in(9) with c being the phase speed:

W (ω, θ) = max{

0,

√Cdu?c

cos(θ − θw)− 1}2

. (9)

Spectral separation (7), as a measure of wavesteepness, is a function of wave energy densityF (ω) and narrowness A(ω) of the directionaldistribution at a frequency (Babanin and Soloviev,1998b) in which F (ω) =

∫F (ω, θ)dθ being the

non-directional energy density spectrum. Thedirectional narrowness (10) is inversely givenby integration of the normalised energy densityspectrum over all directions, where the normalisationis based on the maximum value in the dominantwave direction F(ω) = max

{F (ω, θ)

}, for all

directions θ ∈ [0, 2π] (Babanin and Soloviev, 1998b).

1

A(ω)=

2π∫0

F (ω, θ)

F(ω)dθ (10)

Note that the spectral saturation Bn is then non-directional and was applied to all directions in orderto calculate the directional growth rate γ(ω, θ).

2.1.3 Wind stressThe exchange of momentum between the wind, theatmosphere, and the water is determined by thewind stress, a form of action exerted by the windon the surface and is an indicator of the strengthof air-sea interaction (Tsagareli et al., 2010). Closeto the surface, the contribution to the total stressτ is due to wave-induced stress τw, turbulent stressτt, and viscous stress τv. At the surface, theturbulent momentum flux in the boundary layerapproaches zero and therefore turbulence vanishes(Tsagareli et al., 2010). As a result, the total stressat the surface can be written as:

τ = τv + τw (11)

Calculation of the wave-induced stress requiresknowledge on the drag coefficient Cd and the viscousdrag coefficient Cv in order to calculate τ and τv,respectively. The drag coefficient is used to translatewinds in the boundary layer to the wind stress at thesurface. Let ρa be the density of the air then thetotal stress can be computed with τ = ρaCdU

210 =

ρau2? and the viscous stress with τv = ρaCvU

210.

Substituting both expressions into (11) yields thevalue for the wave-induced stress:

τw = ρaU210 (Cd − Cv). (12)

For the drag coefficient, parameterisation (13)was selected as proposed by Hwang (2010)which accounts for saturation, and even declinefor extreme winds, of the sea drag at windspeeds in excess of 30 m s−1. To prevent u?from dropping to zero at very strong windsU10 ≥ 50.33 m s−1, expression (13) was modified toyield u? = 2.026 m s−1 (Rogers et al., submitted).

Cd × 105 = 80.58 + 9.67U10 − 0.16U210 (13)

The viscose drag coefficient was parameterised as afunction of wind speed from data by Banner andPeirson (1998):

Cv × 103 = 1.1− 0.05U10. (14)

The wave-induced stress (12) is used as a principalconstraint for the wind input. The constraint is thatthe normal stress τn obtained from integration overthe wind-momentum-input function should be equalto wave-induced stress: τn = τw. Let ρw be thedensity of water and Sin(ω) be the non-directionalform of the wind input in radian frequency space:

Sin(ω) =∫ 2π

0Sin(ω, θ)dθ, then the normal stress τn

can be calculated from the wind-momentum-input:

τn = ρwg

∫ ω1

ω0

Sin(ω)

c(ω)dω. (15)

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Fig. 1: Wind source term computation on (top panel) “transitional spectrum” (3) for wind speed of 8 m s−1 withtransition from f−4 to f−5 at f = 0.48 Hz (dotted line). The lower panel shows the non-directional wind input (4)based on the prescribed spectrum with (red solid line) and without (gray solid line) reduction (16) applied to matchthe constraint for the wind input. The highest discrete frequency of the spectrum is f = 1.07 Hz with extrapolationup to 10 Hz shown (grey solid and black dashed lines).

Integral limits in (15) range from the first discretefrequency of the spectrum ω0 up to the highfrequency tail ω1 = 20π (i.e. 10 Hz). If thehighest discrete frequency of the spectrum is lessthan ω1, a diagnostic tail up to 20π is attachedto the wind input using an approximation for thespectral slope: Sin(ω) ∝ ω−2. Note that, forF (ω) ∝ ω−5 the Donelan et al. (2006) windinput has exactly that slope at the highest discretefrequency of the spectrum. In order to satisfy theconstraint and in case of τn > τw, a frequencydependent factor L (16) is applied to reduce energyfrom the high frequency part of the spectrum:Sin(ω) = L(ω)Sin(ω):

L(ω) = min{

1, exp(µ [1− U10/c]

)}. (16)

The reduction (16) follows an exponential formdesigned to disproportionally reduce energyfrom the discrete part of the spectrum. Thestrength of reduction is controlled by parameter

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Fig. 2: Same as Fig. 1 but for “transitional spectrum”with U10 = 10 m s−1 .

µ where greater energy is reduced at high frequencieswith only little impact on the energy-dominant partof the spectrum, where the wind input was actuallymeasured by Donelan et al. (2006). The value ofµ is dynamically calculated by iteration at eachintegration time step.

An illustrative example for the implemented windinput source term is given in Figure 2 where the windinput is shown as a function of frequency f based on(Fig. 1) the “transitional spectrum” (3) for a windspeed of 8 m s−1. Also shown is the disproportionalreduction in energy of the spectrum to match theimposed constraint on the wind stress. In orderto calculate the normal stress (15) the wind inputspectrum was extrapolated to ω1 = 20π. Note,that the disproportional reduction of energy onlyapplies to the discrete part of the spectrum witha constant reduction of energy in the diagnostictail to maintain the spectral slope of the tail. Asubstantial part of the reduction of energy occursat the higher frequencies of the discrete spectrumand in the diagnostic tail, however, as illustratedin Figure 2, such reduction is less significant whenthe wind exceeds 10 m s−1 partially due to air-flowseparation, with relative reduction of energy forstrong winds.

2.2 Wave dissipation

Wave dissipation Sds in this paper is attributed toenergy loss due to wave breaking and is implementedas a negative contribution of energy in the totalsource term equation (2) in WAVEWATCH IIITM

(Tolman, 2009). Wave breaking and dissipation hasbeen poorly understood and served in operationalwave models as tuning parameterisation tooutbalance residual energy of the wind input,but lately observations and numerical modellingmade understanding of this process availablefor parameterisations based on physics of thephenomenon (Babanin, 2011).

Zieger et al. 5

2.2.1 Novel featuresAUSWEX data analysis yield three novel features ofprocesses of wave breaking and spectral dissipationnamely: (i) the threshold behaviour of wave breaking(initially observed by Banner et al., 2002; Babaninet al., 2001), (ii) the cumulative dissipative effectdue to breaking and dissipation of short wavesaffected by longer waves (Babanin et al., 2010),and (iii) direct dependence of the dissipation on thewind input when the wind forcing is very strong(Babanin, 2011). The threshold behaviour postulatesthat waves will not break unless they exceed ageneric steepness in which case the wave breakingprobability depends on the level of exceedence abovethe threshold (Babanin et al., 2010). For spectralwave models, Babanin and Young (2005), Young andBabanin (2006) and Babanin et al. (2007b) suggestedquantitative formulations of the new dissipationterm that accommodate threshold behaviour. Thecumulative dissipation effect is a previously knownphysical feature (e.g. Phillips, 1961; Longuet-Higginsand Stewart, 1961), but until recently was notaccounted for in the dissipation term of spectralwave models. Therefore, the wave dissipation termSds consists of two distinct terms (“two-phase”behaviour): an inherent breaking component T1 andforced dissipation term T2 (Babanin et al., 2010;Rogers et al., submitted). The third feature isimportant at the spectrum tail or at very strongwind forcing. It signifies a condition when the windinput is so strong that the nonlinear interactions arenot able to transfer exclusive energy flux, and it isdissipated locally through excessive breaking.

2.2.2 ImplementationThe “two-phase” behaviour of the wave breaking anddissipation term is implemented as:

Sds(ω, θ) =(T1 + T2

)F (ω, θ), (17)

where T1 is the inherent breaking term and T2accounts for the cumulative effect of short-wavebreaking due to longer waves. The threshold spectraldensity FT is calculated as in (18), where k isthe wavenumber and with εT = 0.0352 being theempirical constant for the wave breaking probability(Babanin et al., 2007b):

FT(ω) =εT

A(ω) cg k3. (18)

Furthermore, let the level of exceedence above thecritical threshold spectral density at which stagewave breaking is prominent being defined as ∆(ω) =F (ω) − FT(ω), and F(ω) being a generic spectraldensity (see later this section), then the inherentbreaking component can be calculated as:

T1(ω) = a1A(ω)ω

2π

[∆(ω)

F(ω)

]L. (19)

The cumulative dissipation term, is not local infrequency space and is based on an integral thatgrows towards higher frequencies dominating atsmaller scales:

T2(ω) = a2

ω∫0

A(w)

[∆(w)

F(w)

]Mdw. (20)

Since the cumulative term (20) is based onintegration with altering cut-off frequency, it willnot destabilize longer waves and force them to break(Babanin et al., 2010; Rogers et al., submitted).

The dissipation terms (19) and (20) depend onfive selected parameters: a generic spectral densityF(ω) used for normalisation, and four coefficientsa1, a2, L, and M . In previous studies, Babaninet al. (2010) and Tsagareli et al. (2010) selectedthe spectral density F (ω) as generic spectrumfor normalisation, whereas Ardhuin et al. (2010)selected the threshold spectral density FT(ω). Thecoefficients L and M control the strength of thenormalised threshold spectral density ∆(ω)/F(ω) ofthe dissipation terms. Rogers et al. (submitted)recently calibrated the dissipation terms based onduration-limited academic tests and proposed foursets of coefficients listed in Table 1. Banner et al.(2002) introduced the directional narrowness A(ω)as correction for the directional spread to reconcileobserved values of the wave-breaking threshold acrossdifferent bands. Proceeding studies by Babaninet al. (2007b) and Babanin (2009) showed thatsuch correction does not affect the magnitude ofthe threshold value used in the dissipation terms.Consequently, the directional narrowness parameteris set to unity A(ω) ≈ 1 in all calculations.

Tab. 1: Generic spectral density F(ω) used fornormalisation, and four coefficients for wave dissipationterms (19) and (20) after Rogers et al. (submitted).

F(ω) L M a1 a2

DL1M1 FUL2M2 FT

UL1M4 FT

UL4M4 FT

1214

1244

2.0 10-4

8.8 10-6

5.7 10-5

5.7 10-7

1.6 10-3

1.1 10-4

3.2 10-6

8.0 10-6

Based on the “transitional spectrum” describedearlier, Figure 3 gives an illustrative example for twoof the models which shows first-order dependence(DL1M1) and fourth-order dependence (UL4M4)of the dissipation terms. With higher-order,the majority of dissipated energy shifts fromlower frequencies towards higher frequencies inthe spectrum. Figure 3 also shows the relativecontribution of the inherent breaking term T1and the cumulative dissipation term T2. This isconsistent with observations from the real ocean

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ra E

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Fig. 3: Dissipation source term computation on “transitional spectrum” (Fig. 1 top panel) for wind speed of 8 m s−1 .Source terms Sds = T1 +T2 are shown as a function of frequency f for coefficients (top) DL1M1 and (bottom) UL4M4(see Tab. 1).

in which for fully developed waves the dissipationis dominated by the cumulative wave breakingT2 (Babanin et al., 2010). Hence, parameters a1and a2 from Table 1 were calibrated such that thecumulative contribution T2 accounts for 75-80% ofthe total dissipation after 12 hours of simulation.

The calibration procedure carried out by Rogerset al. (submitted) that yields to the four dissipationmodels listed in Table 1 matched the totalenergy in academic duration-limited simulations topre-existing physics in the 3G wave model SWANwhich however did not consider any other metricassociated with the spectral distribution of energy.

3 Simulations

In order to evaluate the performance of theobservation-based input and dissipation in WAVE-WATCH IIITM (labeled STX hereafter), simulationsfor wind-dominated conditions of low to mediumrange wind forcing were carried out for each ofthe four dissipation models listed in Table 1. Theevolution covers cover an idealised duration-limitedacademic test and comparisons of integral metricsbetween NODC3 buoys and Lake Michigan hindcast.

3.1 Academic testThe academic test is an idealised duration-limitedsimulation of the wave evolution for a single pointin the infinite ocean and under homogeneouswind-forcing with a wind speed of 12 m s−1 andconstant wind direction commencing from calmconditions. Wind-wave evolution for new wind-inputand breaking functions was investigated by twomeans of nonlinear interactions: with DIA (discreteinteractive approximation, Hasselmann et al., 1985)parameterisation and with exact computations(XNL for exact nonlinear hereafter) by meansof WRT4 (Tracy and Resio, 1982). The latter

provides accurate estimates of energy fluxes withinthe wave system, in addition to those due to windand breaking, but is computational expensive. Theformer is a fast approximate and thus routinelyemployed in operational forecast. The durationof this test is limited to 12 hours of simulationwith a global time stepping of 30 seconds (15seconds for source term integration). Spectralproperties feature 24 directions and 40 frequencies,logarithmically spaced between f = 0.042 . . . 1.7 Hz.The performance of the academic test is comparedagainst existing physics: TC96 (Tolman andChalikov, 1996), WAM3 (Komen et al., 1984), andWAM4 (Ardhuin et al., 2010).

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STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

10-1 100

frequency f

10-3

wave s

pect

ra E

(f)

f5

Date: 1968-06-06 06:00:00WAM3 DIA

TC96 DIA

WAM4 DIA

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

Fig. 4: Level of the spectral tail versus frequency after6 hours of simulation. Shaded areas represent theobservational parameterisation of Babanin and Soloviev(1998a) with 95% confidence limits.

3 U.S. National Oceanographic Data Center,available online: http://www.nodc.noaa.gov/BUOY/

4 Webb-Resio-Tracy method

Zieger et al. 7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

TC96 DIA

WAM3 DIA

WAM4 DIA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

TC96 DIA

WAM3 DIA

WAM4 DIA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX XNL DL1M1

STX XNL UL1M4

STX XNL UL2M2

STX XNL UL4M4

TC96 XNL

WAM3 XNL

WAM4 XNL

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX XNL DL1M1

STX XNL UL1M4

STX XNL UL2M2

STX XNL UL4M4

TC96 XNL

WAM3 XNL

WAM4 XNL

Fig. 5: Non-dimensional wave evolution by means oftotal wave energy versus peak frequency for selectedsource functions with (top) DIA and (bottom) XNL.The dependences are normalised by the observationalparameterisation of Babanin and Soloviev (1998a) withshaded ares showing 95% confidence limits.

Integral growth curves shown in Figure 5 areusually subject to main scrutiny and tuning andall the source functions perform reasonably wellwith respect to the experimental dependence. InFigure 5, results are shown for (top) approximatenonlinear interactions and (bottom) exact nonlinearinteractions. Since tuning is typically done withthe use of DIA in evolution test, it is noticeablethat the performance actually deteriorates oncethe exact nonlinear term is employed. Whennormalised by the observational parameterisationof Babanin and Soloviev (1998a) with respect tothe 95% confidence intervals (shaded gray), thenon-dimensional evolution of dissipation modelsUL1M4 and UL4M4 are in good agreement to theexisting physics TC96, WAM3 and WAM4.

For the level of the spectral tail, all four newdissipation parameterisations overestimate the tailin a similar fashion to WAM3 as illustrated in

Figure 4 which shows the spectrum after 6 hoursof simulation by means of approximate nonlinearinteraction computation (DIA). Figure 4 also showsthe observational parameterisation of Babanin andSoloviev (1998a) with 95% confidence limits inrespect to the peak frequency fWAM4

p = 0.175 Hzas estimated by WAM4 simulation. Estimatesof peak frequency for all other source termsare: fWAM3

p = 0.187 Hz, fTC96p = 0.232 Hz, and

fSTXp ≈ 0.19 Hz, for WAM3, TC96, and the

observation-based physics, respectively. Of allfour dissipation models, parameterisation UL4M4features the lowest level of the spectral tail, however,still significantly exceeding observed levels and levelsof TC96 and WAM4. As the waves develop the levelof the spectral tail increases (not shown). Due tothe high level of the spectral tail metrics of highermoments in the hindcast of Lake Michigan areexpectedly biased (see later section 3.2).

Figure 6 compares the ratios of existing physicsrelative to the observation-based parameterisationUL4M4 (U10/cp = 1.5) after 6 hours of simulation(legend and time as in Fig. 4). When comparedto the observation-based parameterisation ofDonelan et al. (2006), WAM3 and WAM4 windinput is consistently overestimating up to a factorof 10 between 1.0 and 2.0 peak frequencies. Incontrast, differences to TC96 wind input are fromoverestimating to underestimating by a factor of 5.This higher wind input is essentially compensatedby the dissipation of similar magnitude, except forTC96 dissipation where between 1.4 and 2.0 peakfrequencies the dissipation is underestimating by afactor of 2 (Fig. 6, bottom subplot).

10-1

100

101

norm

alis

ed w

ind input Date: 1968-06-06 06:00:00

U10/cp =1.5WAM3 DIA

TC96 DIA

WAM4 DIA

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

1.0 1.2 1.4 1.6 1.8 2.0 2.2

dimensionless frequency f/fp

10-1

100

101

norm

alis

ed d

issi

pati

on

Fig. 6: (Top) wind input and (bottom) dissipation after 6hours as a function of relative peak frequency normalisedby observation-based model UL4M4 (legend as in Fig. 4).

8 October 2011

0 1 2 3 40

1

2

3

4

wave height [m]n=1765ρ=0.955ε=0.19b=0.07SI=0.23

2 4 6 8

2

4

6

8

mean wave period [s]n=1662ρ=0.753ε=0.4b=-0.05SI=0.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350mean wave dir. [°N]n=1662ρ=0.589ε=101.67b=2.04SI=0.56

2 4 6 8

2

4

6

8

true peak period [s]n=1662ρ=0.798ε=0.72b=0.29SI=0.17

2 4 6 8

2

4

6

8

T0,2

n=1662ρ=0.775ε=0.04b=-0.02SI=0.17

2 4 6 8

2

4

6

8

T−1,0

n=1662ρ=0.544ε=0.39b=-0.16SI=0.07

Lake Michigan - stx4 DIANDBC directional wave spectra

NDBC 45007

WA

VEW

ATC

H STX

4

5

25

50

75

100

150

200

250

300

Fig. 7: Scatter density comparisons between observation-based source terms (UL4M4) with approximate nonlinearinteractions (DIA) and NODC buoy 45007 located 42.67˚N and 87.02˚W. Each of the legends indicates scatterstatistics: number of scatter points, correlation coefficient, root-mean-square error, bias, and scatter index.

1 2 3 4 5 6

inverse wave age (U10c−1p )

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

nor

mal

stre

ss(τ

n)

wave-induced stress (*_dia_*.mat)

WAM3 DIA

TC96 DIA

WAM4 DIA

Donelan et al. (2006)

Fig. 8: The normal stress τn as a function of inverse waveage U10/cp. The normal stress was calculated from thewind-input (15).

The apparent overestimate of the wind input shownin Figure 6 is partially due to constraint appliedto the wind input, that is the wave induced stressshould equal the normal stress exercised by the wind.Figure 8 shows the normal stress τn as calculatedfrom (15) as a function of inverse wave age U10/cpfor all existing source functions. At inverse waveage of U10/cp = 1.5 the normal stress exceeds thewave induced stress by 2.4 times for WAM3 and1.6 times for WAM4 while it accounts only for 80%for TC96 physics. Note that, none of the existingphysics applies a constraint on the normal stress τn.However, WAM4 computes the value of τn from thewind-input in order to estimate the friction velocity

u? (τ = τv + τn = ρau2?) and the roughness surface

length, where as all other source term packages usevarious sea-drag parameterisations to estimate dragcoefficient Cd and friction velocity u?.

3.2 Lake MichiganThe Lake Michigan hindcast is a real test casethat simulates windsea-dominated conditions. Thegeneration of swell is essentially prevented due to thelimited fetch and the topography of the lake. LakeMichigan is instrumented with two NODC buoyslocated in the southern and the northern center ofthe late respectively. The duration of the hindcastcovers a period of 45 days, from 1 Septemberto 14 November 2002, with a time stepping of7.5 minutes (15 seconds for source term integration).Spectral properties are set to 36 directions and29 frequencies logarithmically scaled betweenf = 0.07 . . . 0.9 Hz. At every full hour of thehindcast, the performance of observation-baseddissipation model UL4M4 is compared against insitu instruments: NODC buoys 45002 and 45007.Note that, only for the latter station directionaldata is available for the period of the hindcast.

Since, for the new dissipation model UL4M4, thegrowth curves of total wave energy are reproducedwell in the academic test (see Fig. 5), so arethe integral wave properties as shown in Figure 6(i.e. wave height, true peak period, and meanwave direction). However, the overestimate of the

Zieger et al. 9

spectral tail yields to an apparent bias in thehigher-moment periods: mean wave period (T0 1),T0 2, and T−1 0. Higher-moment periods depend onthe higher-moment spectra in which case the periodis proportional to the ratio of two spectral momentsfollowing the form: Ti j ∝ mi/mj , where for n = i, jmn is the nth-order-moment of the spectrum definedas mn =

∫ωnF (ω)dω (Holthuijsen, 2007).

3.3 Calibration and fine tuningAs a result of the academic test and the LakeMichigan hindcast, additional calibration and tuningis required to match the level of the spectral tailand higher-moment periods. It is now knownexperimentally that at strong forcing at a particularscale the extra wind energy flux cannot be effectivelytransferred by nonlinear interactions and a partof it is dissipated locally (Babanin and Young,2005; Babanin et al., 2007b). This effect shouldreduce the tail level and is now parameterisedin order to optimise the tail and higher-momentmetrics. This was done by introducing a frequencydependent non-dimensional correction factor R ina way that the total dissipation (17) corrects to:Sds = RSds. In fact, the correction postulatedhere disproportionally enhances the dissipation atthe high-frequency tail of the spectrum to matchthe observational parameterisation of Babanin andSoloviev (1998a) with 95% confidence limits (as inFig. 4) with only little impact on the integral growthcurves (Fig. 5). The dissipation enhancement factorR is given in (21).

R = max{

1, exp(a3 exp

(a4

U10

cp

)[U10

c − a5])}

(21)

In (21), coefficients a3 accommodate for themagnitude of the tail correction while a4 providesadditional control as a function of wave development(i.e. inverse wave age U10/cp). Since the additionalcontrol function exp(a4U10/cp) grows with wavedevelopment, values of coefficient a4 are negative.Coefficient a5 serves as a threshold value for which,when exceeded, the tail correction kicks in. For eachof the four dissipations models the coefficients a3,a4, and a5 are listed in Table 2, in extension todissipation models in Table 1.

Tab. 2: The four dissipation models (see Tab. 1) extendedby coefficients a3, a4, and a5 for the correction (21) of thelevel of the high-frequency tail of the spectrum.

a3 a4 a5

DL1M1UL2M2UL1M4UL4M4

3.826.127.964.1

-0.72-1.93-1.87-2.48

2.42.42.42.4

10-1 100

frequency f

10-3

wave s

pect

ra E

(f)

f5

Date: 1968-06-06 06:00:00WAM3 DIA

TC96 DIA

WAM4 DIA

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

10-1 100

frequency f

10-3w

ave s

pect

ra E

(f)

f5

Date: 1968-06-06 12:00:00WAM3 DIA

TC96 DIA

WAM4 DIA

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

10-1 100

frequency f

10-3

wave s

pect

ra E

(f)

f5

Date: 1968-06-06 12:00:00WAM3 DIA

TC96 DIA

WAM4 DIA

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

Fig. 9: Level of the spectral tail versus frequency withenhancement factor R (21) applied after (top) 6 hoursand (bottom) 12 hours of simulation. Shaded areasrepresent the observational parameterisation of Babaninand Soloviev (1998a) with 95% confidence limits.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

TC96 DIA

WAM3 DIA

WAM4 DIA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

non-dimensional frequency (f=fp U10 g−1 )

100

non-d

imensi

onal energ

y (

E=σ

2g2

U−

410

)

STX DIA DL1M1

STX DIA UL1M4

STX DIA UL2M2

STX DIA UL4M4

TC96 DIA

WAM3 DIA

WAM4 DIA

Fig. 10: Non-dimensional wave evolution by means oftotal wave energy versus peak frequency for selectedsource functions. The dependences are normalised by theobservational parameterisation of Babanin and Soloviev(1998a) with shaded ares showing 95% confidence limits.

10 October 2011

The performance of the tail correction in theacademic test with approximate nonlinearinteractions (DIA) is shown in Figure 9 andFigure 10. The latter shows the growth curves oftotal energy in a similar fashion to those plotted inFigure 5. The former shows the level of the spectraltail which is when compared to Figure 4 now ingood agreement with observational parameterisationof Babanin and Soloviev (1998a) for frequenciesgreater than 0.3 Hz. Results are shown for 6 hoursand 12 hours of simulation.

The Lake Michigan hindcast with the tail correctionapplied yield almost identical scatter density plots(not shown) as in Figure 7. As expected, thehigher-moment periods perform slightly better asillustrated in Figure 11, but improvements aremarginal. The energy in the high frequency part ofthe spectrum seem to have little impact on the totalwave energy higher-order moments of the spectrum.

0 1 2 3 40

1

2

3

4

wave height [m]n=1765ρ=0.956ε=0.2b=0.1SI=0.26

2 4 6 8

2

4

6

8

mean wave period [s]n=1662ρ=0.752ε=0.4b=-0.05SI=0.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350mean wave dir. [°N]n=1662ρ=0.595ε=100.83b=2.23SI=0.55

2 4 6 8

2

4

6

8

true peak period [s]n=1662ρ=0.795ε=0.74b=0.32SI=0.18

2 4 6 8

2

4

6

8

T0,2

n=1662ρ=0.777ε=0.04b=-0.02SI=0.17

2 4 6 8

2

4

6

8

T−1,0

n=1662ρ=0.538ε=0.39b=-0.15SI=0.07

Lake Michigan - STX4C6 DIANDBC directional wave spectra

NDBC 45007

WA

VEW

ATC

H STX

4C

6

5

25

50

75

100

150

200

250

300

0 1 2 3 40

1

2

3

4

wave height [m]n=1765ρ=0.956ε=0.2b=0.1SI=0.26

2 4 6 8

2

4

6

8

mean wave period [s]n=1662ρ=0.752ε=0.4b=-0.05SI=0.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350mean wave dir. [°N]n=1662ρ=0.595ε=100.83b=2.23SI=0.55

2 4 6 8

2

4

6

8

true peak period [s]n=1662ρ=0.795ε=0.74b=0.32SI=0.18

2 4 6 8

2

4

6

8

T0,2

n=1662ρ=0.777ε=0.04b=-0.02SI=0.17

2 4 6 8

2

4

6

8

T−1,0

n=1662ρ=0.538ε=0.39b=-0.15SI=0.07

Lake Michigan - STX4C6 DIANDBC directional wave spectra

NDBC 45007

WA

VEW

ATC

H STX

4C

6

5

25

50

75

100

150

200

250

300

0 1 2 3 40

1

2

3

4

wave height [m]n=1765ρ=0.956ε=0.2b=0.1SI=0.26

2 4 6 8

2

4

6

8

mean wave period [s]n=1662ρ=0.752ε=0.4b=-0.05SI=0.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350mean wave dir. [°N]n=1662ρ=0.595ε=100.83b=2.23SI=0.55

2 4 6 8

2

4

6

8

true peak period [s]n=1662ρ=0.795ε=0.74b=0.32SI=0.18

2 4 6 8

2

4

6

8

T0,2

n=1662ρ=0.777ε=0.04b=-0.02SI=0.17

2 4 6 8

2

4

6

8

T−1,0

n=1662ρ=0.538ε=0.39b=-0.15SI=0.07

Lake Michigan - STX4C6 DIANDBC directional wave spectra

NDBC 45007

WA

VEW

ATC

H STX

4C

6

5

25

50

75

100

150

200

250

300

Fig. 11: Scatter density comparison of observation-basedsource terms (UL4M4) with approximate nonlinearinteractions (DIA) in the Lake Michigan hindcast for thesame location as in Fig. 7 but with enhancement factor R(21) applied with only higher-moment periods T0 2 andT−1 0 shown. Other integral parameters performed aswell as before and are therefor omitted.

4 Summary and future work

The observation-based source terms implementedin WAVEWATCH IIITM were tested in a duration-limited academic test and the Lake Michiganhindcast. These tests showed that the totalenergy is in good agreement with observations bymeans of the parameterisation by Babanin andSoloviev (1998a) (see Fig. 9 and Fig. 10) and buoydata (see Fig. 7). However, scatter comparisonsbetween NODC buoy 45007 and the Lake Michiganhindcast also show that the observation-based sourceterms underestimate true peak period, mean waveperiod T0 1 and higher moment periods T0 2 andT−1 0 even with the high level of the spectraltail adjusted (Fig. 9) using the correction factor(21). One possible reason for these results mightbe the distribution of energy along all discretedirections. For example, the constraint applied on

Tab. 3: Metrics used to assess the performance ofthe observation-based source terms. Tested metricsare marked with crosses while dashes indicate metricsthat still need to be verified unless it is not applicable(i.e. n/a). The metrics tabulated cover: dimensionlessenergy E, dimensionless peak frequency fp, fetch x, andtime t, wave height Hs, wave periods Tp and Tn, wavedirection θ, peak frequency fp, the level of the spectraltail α, spectral peakedness γ, transitional frequencyωt, directional narrowness A, directional spreading Dθ,wave induced stress (as scalar τn and vector ~τn), windinput Sin, wave dissipation Sds, the ratio of input anddissipation and swell dissipation Swl.

ACADEMIC TEST HINDCAST

METRICS duration-limited Lake Michigan

growth curves E vs. f pE vs. xE vs. t

xn /a−

n /an /an /a

integral parameters Hs

Tp

Tn

θf p

n/an/an/an/an/a

xxxxx

spectral parameters

αγωt

x−−

−−−

directional parameters

A (f p)A(2f p)Dθ

−−−

−−−

constraints τnτnSi n

Sds

SdsS in−1

Swl

−−−−

−−

x−xx

−−

the normal stress τn and the “two-phase” behaviourof the wave-breaking and dissipation term T1 andT2 are based on non-directional parameterisations.Therefore, additional testing need to be carriedout in order to qualify the directional distributionof energy of the spectrum (see directional metricslisted in Table 3), which also plays a significantrole in situations of turning wind fields (i.e. greaterthan 90˚) and for opposing winds.

At this stage, the implementation of theobservation-based source terms lacks swelldissipation Swl and the white-capping dissipationSds remains zero if the spectral energy density doesnot exceed the threshold spectral density FT. Also,in case of opposing wind forcing, the growth rate γin the wind input parameterisation remains zero, butlaboratory experiment conducted by Donelan (1999)showed that the growth rate becomes negative ifUλ/2/c − 1 < 0, where Uλ/2 is the wind speed ata height of one-half the wave length λ. This plays

Zieger et al. 11

a significant role in order to model open-oceanconditions and conditions of extreme wind forcingsuch as hurricanes or tropical cyclones. The swelldissipation will be implemented after Babanin(2011), in which the dissipation is due to turbulentmixing of the water column as a function of theorbital velocity of the waves at the surface, afterArdhuin et al. (2009), in which it is due to interactionwith atmospheric turbulence, and after Donelan(1999) in which it is due to waves interacting withopposing winds.

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