D EPTH - INDUCED WAVE BREAKING D EFAULT PARAMETERIZATION

Most parameterizations for the depth-induced breaking in spectral wave models are based on the work ofBattjes and Janssen(1978) who coupled a Rayleigh distribution to represent the wave heights in a random wave field with the dissipation of a single break-ing wave, assumed analogous to the dissipation of a 1D bore (per unit time and per unit bottom area;Lamb,1932andLe Méhauté,1962). The bulk dissipation can then be given as:

εB J= −1

4αB Jf¯ρgQbH_m² (4.2)

whereαB J= O(1) is a tunable coefficient, ¯f is the mean wave frequency (based on the first and zero-th order moment of the wave variance density spectrum i.e., ¯f = fm01; see Section2.4),ρ is the density of water, g is the gravitational acceleration, Qbis the fraction of breakers and H_mis the maximum possible wave height in the local water depth.

Since the details of the distribution function for the wave heights is not required to estimate overall properties,Battjes and Janssen(1978) use an approximate distribution.

They assume a Rayleigh distribution for the non-breaking waves and truncate this at Hm

with a delta function at this limit to represent the breaking waves. This yields an implicit expression for the fraction of breakers dependent on the root-mean square wave height H_{r ms}: To estimate Hm,Battjes and Janssen(1978) used an estimate based onMiche(1944):

H_m= 0.88k⁻¹t anh¡

γkd/0.88¢ (4.4)

where k = 2π/L is the wave number and L is the wavelength, γ is introduced as an ad-justable coefficient and d is the local water depth. This expression has two limits: in deep water as kd → ∞, the expression reduces to Hm= 0.88k⁻¹and in shallow wa-ter as kd → 0, the expression reduces to Hm= γd. The first limit represents a limiting steepness where waves begin to break in deep water which is often referred to as white capping whereas the second limit represents depth-induced breaking. As most spectral wave models use a separate source terms for white capping, Eq. (4.4) is reduced in SWAN to H_m= γB Jd . Battjes and Janssen(1978) showed thatγB J= 0.80 provided reasonable results over their observations, however most operational third-generation wave models useγB J = 0.73 which is taken as the averaged value over a larger data set (Battjes and Stive,1985, their Table 1). This has been shown to provide reasonable results over a large

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Figure 4.2: Calibratedγ_β−kdas a function of the local bottom slope n = t an⁻¹β and local normalized wave number ˜kd withγB J= 0.73 for reference (fromSalmon et al.,2015).

range of bathymetries and wave conditions, particularly for slopes and non-locally gen-erated waves (e.g.Salmon et al.,2015). We subsequently refer to theBattjes and Janssen (1978) dissipation model as the BJ78 model and its use withγB J= 0.73 as the ‘default’.

ALTERNATIVE PARAMETERIZATIONS

Beta-kd (β − kd) parameterization Many studies that have addressed improving the modeling of depth-induced wave breaking have focused on re-scaling the dissipation in terms of either local topography i.e., bottom slope (e.g.Madsen,1976) or wave param-eters either offshore or local i.e., wave number or wave steepness (e.g.Ruessink et al., 2003;Apotsos et al.,2008). However previous studies (e.g.Rattanapitikon,2007) have shown that these scalings do not provide substantially better results than using the de-fault parameterization.Salmon et al.(2015) suggest an alternative approach of combin-ing both local bottom slope (β) and normalized wave number (kd) in a joint scaling and show this to provide similar results to usingγB J= 0.73 in laboratory cases with sloping bathymetries and significantly improved results over horizontal laboratory cases. This β − kd scaling is shown in Figure4.2.

In their scaling, they assume that in shallow water (kd < 1) waves behave more like solitary waves so that wave number becomes irrelevant for the kinematics and dynamics of such waves. Therefore, under such conditions, the waves are only dependent on the

4.3.WAVE MODEL FOR DEPTH-INDUCED BREAKING

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local bottom slope,β. However, in deeper water (kd > 1),van der Westhuysen(2010) suggests an equilibrium state where the wave growth limit of locally generated waves is due to a balance between the local wind-wave growth and dissipative sources i.e., depth-induced breaking, white capping and bottom friction. Over (near-) horizontal bottoms, the influence of depth-induced breaking was shown to be small in this balance and not well represented withγB J= 0.73. To account for this, under these conditions, a posi-tive kd -dependency onγ_β−kdto represent a reduction in Q_b(as for exampleTing,2001;

Ruessink et al.,2003) is introduced so that waves become dependent on bothβ and kd.

To represent these dependenciesSalmon et al.(2015) suggest:

γ_β−kd= γ_β/t anh£ γ_β/γkd˜

¤ (4.5)

and their extensive calibration findsγβ= 0.5 + 7.59 |∇d| ≥ 0 for the bottom slope, β-dependency where the bottom slope is estimated as the magnitude of the bottom gra-dient i.e., t anβ = |∇d| (as taken from the computational grid;Salmon et al.,2015) and γkd˜ = −8.06+8.09 ˜kd ≥ 0 for the ˜kd-dependency where ˜k = k_−1/2is a characteristic wave number (to be determined from the local spectrum;WAMDI Group,1988; see Section 2.4).

To account for inherent differences between 1D long-crested (directionally narrow) and 2D short-crested (directionally spread) wave conditions, Eqs. (4.2) and (4.3) are modified bySalmon et al.(2015) to:

ε^θ_{B J}= −1 be sufficiently long-crested so that the 1D bore assumption ofBattjes and Janssen(1978) is applicable to that partition,σ_θis the directional spreading of the spectrum (Kuik et al., 1988) andσ^∗_θis the upper limit for directional spreading where the 1D bore assumption holds. This was shown to provide the best results withσ^∗_θ= 15^◦(Salmon et al.,2015). We subsequently refer to this parameterization, i.e., the BJ78 dissipation model with both bottom slope and wave number dependency (Eq.4.5) and modified with the directional partitioning (Eqs.4.6and4.7) as the beta-kd orβ − kd parameterization.

Bi-phase (ϕ) parameterization A development of theBattjes and Janssen(1978) model was proposed byThornton and Guza(1983) who suggested, based on their field obser-vations, using a Rayleigh distribution for the breaking waves shifted toward the higher waves. This is achieved by the use of a weighting function (their Eq. 21) to give the bulk dissipation as:

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whereαT Gis a calibration coefficient (αT G≈ 0.5 for laboratory cases and αT G≈ 3.4 for field cases in their study) andγT G= Hr ms,max/d , the ratio of the maximum rms wave height to local depth. This dissipation model is subsequently referred to as the TG83 model.

An alternative weighting function was suggested byvan der Westhuysen(2009,2010) who argued that the weighting function should be based on wave nonlinearity which is introduced through the bi-phase parameterization ofEldeberky(1996). The bulk dissi-pation is then given as:

For which van der Westhuysen (2009) calibrated αT G ≈ 0.95; ϕ = 0.5 π[tanh (0.2/Ur ) − 1] represents the parameterization of the bi-phase of the self-self triad interaction at the peak of the spectrum as a function of Ur , the Ursell number (see Sec-tion2.4);ϕr e f = −4π/9 and n = 4©1 − π⁻¹ar c t an [s − 0.038]ª with the local wave steep-ness s as defined byWAMDI Group(1988, see Section2.4). In contrast to theβ − kd parameterization, the relaxation of the depth-induced breaking criterion for locally gen-erated waves is provided by a reduction of nonlinearity (ϕ) and therefore dissipation. We subsequently refer to the TG83 model with the weighting function ofvan der Westhuysen (2009,2010) as the bi-phase orϕ parameterization.

4.4. R

To assess the performance of the three parameterizations introduced in Section4.3.3, we calculate two commonly applied performance metrics which have been used in pre-vious studies: the scatter index (s.i .) and the relative bias (r.b.) for the field data sets as described in Section4.2(e.g.Janssen et al.,1984;Komen et al.,1994). These performance metrics are defined in Section3.3.3. Following previous studies of depth-induced wave breaking over field data sets (e.g.Rattanapitikon,2007;Salmon et al.,2015), scatter in-dices of < 10% are considered very good; between 10% and 20% as reasonable and > 20%

as poor.

4.4.1. D