Towards Reliable Breaking Wave Forecasts at Sea M. Banner, E. Kriezi and R. Morison Abstract This paper summarizes our recent progress on the goal of computing the spectral distribution of wave breaking from a numerical wave model. Our approach is bas ed on interpreting wave breaking onset as a nonlinear wave group hydrodynamic process. This has led to a significant improvement in parameterising wave breaking probability for sea waves of different scales. These developments allow the wave model to produce initial results that agree closely with recently observed breaking wave spectra. Requirements for using these findings in operational forecast models are discussed

2. BACKGROUND

1. INTRODUCTION

2.1 Wave groups and wave breaking

The occurrence and consequences of large breaking waves at sea has been a persistent concern for centuries of seafarers and coastal dwellers. These waves are responsible for the largest dynamic loading on ships and coastal structures, and can present a significant challenge to human safety on smaller vessels at sea. Also, air-sea interaction scientists have long sought to understand which environmental processes and variables control the relative occurrence rate (probability) and strength of breaking of the dominant sea waves. This applies not only to the dominant sea waves, but also to the shorter breaking waves in the spectrum. Motivation for the inclusion of shorter breaking waves includes improved modelling for the following key processes at the air-sea interface: the aerodynamic consequences (air -flow separation) for the wind input source function and sea surface drag coefficient; enhanced upper ocean mixing processes; increased air-sea fluxes of low-solubility gases. Our recent results from a number of diverse studies provide strong encouragement on which processes need to be included for a reliable parameterisation of wave breaking at sea. This paper is aimed at highlighting recent progress on our ongoing efforts to reliably forecast wave breaking. Applications of this work include routine sea state forecasts of dangerous breaking wave conditions, as well as improved coupled modelling of air-sea fluxes and upper ocean mixing processes, including foam cover.

A significant association of wave breaking with ocean wave groupiness was first noted in the literature in [3] and investigated in detail by [2], who found a remarkably strong correlation between wave breaking and wave group structure. These findings suggest we should take a more global view of breaking than just considering local criteria, and look more closely at group behaviour.

_________________ The authors are with the Centre for Environmental Modelling and Prediction in the School of Mathematics at the University of New South Wales, Sydney 2052, Australia. Email: [email protected]

Recent observational insight from dedicated studies of wave breaking at sea indicates that neither wind speed nor spectral peak inverse wave age U/c p correlate breaking probability successfully [1]. It was also observed [2] that local wave properties such as local wave steepness are not able to separate breaking and non-breaking waves from observed wave height versus wave period distributions. In any event, such local indicators provide no dynamical basis for diagnosing wave breaking onset.

2.2 Insight from modelling of 2-D nonlinear wavetrains In [4], [5], we used two -dimensional ‘exact’ Euler equation boundary element codes (periodic domain and numerical wave tank) to track the evolution of wave group maximum and the associated local depthintegrated energy density, E. It was observed that, travelling with the group, there is a significant flux of energy towards the centre of the group - not a steady flux, but an oscillatory flux, due to the asymmetry of the waveform. Thus from this viewpoint, there are intrinsically two timescales involved in the process. We proposed that the onset of breaking is linked to a threshold in the slower flux, i.e. the mean convergence rate of wave-coherent normalised energy Ek 2 (or E/) at the envelope maximum. Here k is the local carrier wavenumber and is the mean energy of the wave group. From our results, for the typical wave groups studied, there appears to be a threshold nondimensional growth rate for the local non-dimensional energy density, that can distinguish wave groups that evolve to break from those that relax without breaking, i.e. undergo ‘recurrence’.

2.3 Recent observations It is very well known that the dominant sea waves occur routinely in wave groups, so are the ideas arising from 2-D modelling helpful. In particular, is there a parametric threshold for breaking and if so, how can we parameterise the underlying nonlinearity most simply? The mean wave steepness is the traditional parameter used in classical wave train perturbation analysis, so this should provide an initial parameter to investigate. Following up on this possibility for the dominant sea waves, we found [6] a strong c orrelation and clear threshold behaviour when we correlated breaking probability with the significant mean steepness of the dominant waves. It was noted that averages over sufficiently long records were needed to include enough wave groups to gather stable statistics. Also, the breaking probability was defined as relative passage rate past a fixed point of breaking crests to total crests in spectral peak enhancement region.

2.4 Wave breaking probability in the spectrum This is defined as the ratio of the passage rate past a fixed point of breaking crests with velocities in (c, c+dc) to the passage rate past a fixed point of all wave crests with velocities in (c, c+dc ). The breaking probability for wave scale c is then quantified as:

P( c ) =

∫ cΛ( c ) dc ∫ cΠ ( c ) dc

where Λ (c) = spectral density of breaking wave crest length per unit area with velocities in the range (c, c+dc) and Π (c) = spectral density of the total wave crest length per unit area with velocities in the range (c, c+dc ) To quantify the mean nonlinearity of the waves, including the higher wavenumber components above the spectral peak, following [7], we u sed the azimuth-integrated spectral saturation B(k ) = k 4 ∫Φ?(k)dθ = (2π)4 f 5 F(f) 2g2 instead of significant steepness, which is only applicable at the spectral peak. The use of azimuth-integrated B(k ) is complicated by broader ‘directional spreading’ as the wavenumber increases above the spectral peak, but the same qualitative threshold behaviour is evident once the spectral saturation B(k ) is normalized by the mean directional spreading width at wavenumber k. Application of the observed rate of spectral directional spreading with distance from the spectral peak, as reported recently in [7], provides the normalised spectral saturationσ~ (k ). The observed breaking probabilities for different centre frequencies relative to the spectral peak were constructed [8] and found to have a well-defined threshold behaviour, with a common breaking threshold value σ~T ~ 0.0045, as seen in Figure 1.

Figure 1. Breaking probability fRWR against ~ s , the azimuth-integrated saturation normalized by the local spectral spreading width θ(fc) for the range of nondimensional centre frequencies fc/fp investigated: (a) fc/fp = 1.0 (b) fc/fp = 1.16 (c) fc/fp = 1.35 (d) fc/fp = 1.57 (e) fc/fp = 1.83 (f) fc/fp = 2.13 (g) fc/fp = 2.48. Each data point is based on a one-hour data record from three North Pacific storms, as described in [8].

3. WAVE MODEL IMPLEMENTATION 3.1 Radiative transfer equation The radiative transfer equation (deep water, no currents) for describing the evolution of the waveheight spectrum F(k) is given by

∂F + c ⋅∇ F = S tot g ∂t

where F = F(k,θ) is the directional wave spectrum, c g is the group velocity, S tot = Sin + Snl + Sds is the total source term, in which Sin is the atmospheric input spectral source term, S nl is the nonlinear spectral transfer source term representing nonlinear wave-wave interactions within the spectrum and Sds is the spectral dissipation rate due primarily to wave breaking

3.2 New spectral dissipation rate term S ds Based on the new insight on breaking onset, we developed a refined form of S ds (k) based on the strong saturation threshold behaviour described above in section 2.4. It is based on treating waves in different directional spectral bands as nonlinear wave groups. In this form of Sds , shown below, we used a power law function of the spectral saturation ratio to reflect the observed threshold behaviour, refining the form proposed in [9]: S ds (k , θ ) = C [(σ~ − σ~T ) / σ~T )] n ( k / k m ) ω F(k,θ )

where

km

is

the

mean

wavenumber

given

by

∫ k F (k) dk / ∫ F (k ) dk ). The exponent n was taken as 2

to 1.5 Hz, for a forcing wind speed of U10 = 10 m/s. For this calculation, we used a standard parametric form of Sin [11],

based on matching to the expected high wavenumber form of S in(k ) and the tuning constant C was chosen to provide the optimal match to observed fetch evolution of the spectral peak energy.

3.3 Exact NL computation of fetch-limited wind wave evolution Our new S ds is based on local saturation ratio in contrast to the integral wave steepness used in the quasi-linear form of S ds presently used in most operational wave forecasting models. As evidenced by the excellent reproduction of the observed growth curves compiled by [10], this new form has exc ellent flexibility to model S ds within the spectrum over the range of sea states from young to old.

3.4 Spectral breaking wave computations Two key properties of the data are assumed in the modelling: (i) a common breaking probability threshold in terms of the normalised saturationσ~ (k ) for all k in the computational domain. This is shown in Figure 2. (ii) the normalised crest passage rate density cΠ(f??) as a function of distance from the spectral peak, as shown in Figure 3. This was assumed to be applicable for all wave ages:

Figure 3. Assumed empirically-based relationship between actual to nominal wave crest passage rate past a fixed point against distance from the spectral peak frequency. a form of the ‘exact’ form of the nonlinear transfer source term Snl due to [12] and the form of S ds described in 3.2 above. The one-dimensional transect wavenumber spectrum calculated for the old wind sea conditions reported in the observations of [13] agreed very closely in level and shape with their measurements. This close correspondence assures that the levels of spectral saturation are also in close agreement. Further, our resulting computed Λ(c) was found to be in very close agreement with the measured spectral distribution for this case.

Present status Further calculations at other wind speeds and wave ages are proceeding. Our goal of operational forecasts of dominant breaking wave occurrence probability will need to rely on the availability of a suitable parametric form of Snl that can calculate the spectral saturation as for the ‘exact’ form. Extensive validation will be essential before this feature can be added to routine sea state forecasts.

4. CONCLUDING REMARKS Figure 2. Assumed model threshold function for breaking probability at any scale against the local normalised spectral saturation at that scale. The symbols show the computed breaking probabilities for a young and an old wind sea during a run for U10 =10 m/s. Initial computations of the directional wave spectrum were made for the full spectral bandwidth covering 0.06 Hz

Our approach identifies nonlinear wave group dynamics as the primary mechanism involved in the breaking onset of 2-D deep water wave. In this framework, there appears to be a common threshold for a dimensionless mean growth rate reflecting the mean convergence rate of energy at the envelope maximum, that separates breaking from recurrent evolution, applicable in the presence of strong forcing

by wave-slope-coherent surface pressure and surface layer shear. For the dominant wind sea, a threshold significant wave steepness relationship appears to be a good first approximation for correlating breaking probability. For different wave scales, observations indicate that if the normalised spectral saturation is used to quantify the nonlinearity, breaking probability curves show self-similar threshold behaviour at different scales. A refined form for Sds based on these observations performs well in reproducing observed behaviour both at the spectral peak (integral fetch growth curves) and at the spectral tail (level, directional spreading). Reliable predictions of the normalized spectral saturation σΝ and observed distributions of total wave crest passage rates allow computations of the breaking crest length spectral density Λ(c). Comparison of our initial computational results with recent field data is very encouraging, and has the potential to provide spectral breaking wave forecasts as part of routine wave model computations. Extending the computations to address severe sea state conditions is in progress.

Acknowledgements