Supplementary information for ”Significant decay of steep swells across ocean basins” Fabrice Ardhuin1∗ , Fabrice Collard3 , and Bertrand Chapron2 , 1

Service Hydrographique et Ocanographique de la Marine, 29609 Brest, France 2 Laboratoire d’Oc´eanographie Spatiale, Ifremer, Centre de Brest, 29280 Plouzan´e, France 3

∗

1

BOOST-Technologies, 29280 Plouzan´e, France

To whom correspondence should be addressed; E-mail: [email protected].

Introduction

In the manuscript titled ”Significant decay of steep swells across ocean basins” we present measurements of swell evolution, including estimates of swell energy dissipation across ocean basins, and propose that swell dissipation is caused by friction at the air-sea interface. In this document we provide supplementary material regarding the data processing and analysis. We further describe numerical modelling experiments and their results, that put in perspective the proposed swell dissipation mechanism, giving an idea of its magnitude for the entire sea state. Previous direct swell analysis used in situ data, and were limited to point measurements[1, 2, 3] or a single swell track[4]. In the latter case, the analysis of swell evolution required the source to be aligned with the instruments, and few storms could be analyzed. Further, the logistics of early measuring devices required islands in the vicinity of most measuring stations, leading to a considerable difficulty in accounting for wave energy dissipated on the shores of all the small islands in the swell propagation path. Attempts to quantify the dissipation of swell energy were thus inconclusive for wave periods larger than 13 s. Using Synthetic Aperture Radar (SAR) data from the ENVISAT satellite mission provides a much larger data set in which favorable measurement conditions can be carefully selected, providing an analysis of swell evolution with unprecedented accuracy.

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Discussion 1. Swell dissipation in existing numerical wave models Another approach to the estimation of swell evolution is indirect, and uses numerical wave models. Until the present work, the overall best parameterization for wind wave generation and dissipation was the one proposed by [5], hereinafter termed ”BAJ”, resulting in very good short term forecasts of the waves in the vicinity of the most severe storms, with errors on 3-hour average wave heights and periods within 14% of observed values in the open ocean (this value is the global average of the normalized root mean square error). Yet, this parameterization, unlike the one by [6], does not specifically account for swell dissipation and invariably leads to overestimations of wave heights in the tropics, unless these wave observations are assimilated. Typical biases reach 45 cm or 25% of the mean observed wave height the East equatorial Pacific (Fig. 1). More importantly the peak periods in the western Pacific are overestimated by 0.6 to 2 s on average, because too much swell energy is present.

Figure 1: Mean difference between modelled and observed wave heights for the year 2007. Observations combine of data from JASON, ENVISAT and GEOSAT-Follow On (GFO) altimeters, with a method described in [7]. Results are provided for (a) the BAJ parameterization, (b) the new parameterization described below with fe given by eq. (13). 2

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Figure 2: Bias between modelled and in situ observed wave periods averaged over 5 hours for the year 2007. Results are provided for (a) the BAJ parameterization, (b) the fitted new parameterization described below. Periods are peak periods except for the UK and French European buoys for which it is the mean period Tm02 . Symbols ∇, 4, ◦, , ♦ and ? correspond to values in the ranges x < −1, −1 ≤ x < −0.5, −0.5 ≤ x < 0, 0 ≤ x < 0.5 , 0.5 ≤ x < 1 , 1 ≤ x, respectively. However, even the BAJ parameterization dissipates some swell energy through the term designed to represent the loss of energy due to wave breaking, inspired from the random pressure pulse model by [8]. This effect was not intended. Indeed, if the whitecaps that cause the surface pressure pulses are not correlated with the swells orbital velocities, there should be no dissipation of the swell since the work of the pressure is the mean product of the pressure times the velocity normal to the surface. Such a correlation could indeed exist, but it is not represented explicitly in the dissipation parameterization, and the best estimates of its effect show that it should be negligible [9, 10]. The residual swell dissipation is thus an unintended benefit of the form of the parameterization. Following Tolman [11], and using recent observations of wave breaking statistics, one can get a quantitative estimate of the expected swell dissipation by removing the dissipation for all the wave components that are not expected to be directly affected by breaking. Based on the 3

observations of [12] and [13], one can use the saturation spectrum Z 2π B (f ) = 2π k 3 F (f, θ)/Cg dθ,

(1)

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to define a threshold B(f ) = 0.0012 [14] below which no wave with a similar frequency is observed to break. Allowing for a margin of error, we have re-ran the model (fully described in Discussion 4) with the dissipation set to zero at all wave frequencies for which B < 0.001. Because the wave heights are underestimated in the strong swell generation areas at mid-latitudes, the result in figure 3 can be interpreted as a lower bound for the wave overestimation in the absence of swell dissipation. In the new parameterization used below, the result without swell dissipation is more easily obtained by setting the dissipation factor fe to zero. In that case, because the the wave heights are overestimated in the strong swell generation areas at midlatitudes, we get an upper bound for the bias in the model without swell dissipation. In such model calculations the excess of energy in the swell band is limited by nonlinear interactions that are stronger for these steep non-dissipated swells than for the real swells. Such a model study for the year 2007 show that in the absence of swell dissipation, wave heights are overestimated by 46 to 83 cm when averaged over the world ocean, which amounts to 24 to 44% of the average observed wave height. The large uncertainty in these estimates arises from the uncertainty in the initial wave generation. Any attempt to quantify swell dissipation with this type of indirect modelling method, even when using complex data assimilation methods[15], is thus fundamentally limited by the many assumptions made in numerical wave models about other processes.

Discussion 2. Data quality and error model for the SAR data All SAR data used are level 2 (L2) products, provided by the European Space Agency (ESA) and obtained with the L2 processor version operational at ESA since November 2007[16]. Before this date we used reprocessed archive data with the same processor version, because previous processor versions insufficient filtered non-wave signatures in the radar images, often causing low wavenumber artefacts[16]. This filtering is necessary to remove the contributions of atmospheric patterns or other surface phenomena, with spectral signatures that can overlap the swell spectra like ships, slicks, sea ice, or islands. The wave measurements used here are directional SAR wave spectra in which the 180◦ directional ambiguity has been removed, using the co-spectra of separate image looks[17]. Additionally, the backscatter-derived wind speed using wind direction analyses from the European Center for Medium Range Weather Forecasting (ECMWF), is used in our analysis. Other information are also provided such as the radar image normalized variance, generally used to flag images with land, ships, slicks or other causes of backscatter variations not related to the wave field, SAR azimuth cut-off, which is used to assess the minimum swell wavelength properly imaged by the SAR, typically in the range 100 to 200 m. ENVISAT ASAR Wave Mode imagettes are 5 by 10 km radar scenes acquired every 4

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Figure 3: Mean difference between modelled and observed wave heights for the year 2007. Same as figure 1, except that in each model the swell dissipation was de-activated. 100 km looking to the right of the flight direction, 23 deg from nadir. The complete orbital cycle of ENVISAT around the earth is 35 days with 3 day subcycles. Here we consider errors on wave height estimates, which are the most important for the following estimation of swell energy decay. In previous work[16] the truncated significant R 1/2 1/12 wave height Hs,12 = 4 0 E(f )df was compared to buoy data, and the SAR showed an r.m.s. error of 0.5 m, which includes a bias of 0.2 m. In our study we use the heights of swell partitions, which are expected to be more accurate because • Hs,12 is sometimes poorly estimated due the azimuthal cut-off • we have excluded low and high wind conditions in the data used for the decay estimation (see below) • we use only data with peak periods greater or equal to 13 s, which is further away from 5

the azimuthal cut-off, compared to the 12 s threshold in that previous work. • the buoys used in the comparison are not located exactly at the same place as the SAR image, and are in areas with relatively large wave field gradients, compared to the middle of ocean basins that we sample here. Indeed, another work on swell partition heights[18], used the same processing as for the level2 ESA wave mode product, but applied to 4 by 4 km tiles from narrow swath images. That study found a 0.37 m r.m.s. error, which includes a 0.17 m bias. This smaller error was obtained in spite of a much smaller image size (a 4 times smaller area) and a wide range of radar incidence angle for which the modulation functions used to relate the radar image spectrum to the ocean elevation spectrum are not so well known as for the 23◦ incidence of the wave mode data. This analysis has now been repeated for the present paper on wave mode data using buoys located within 100 km and 1 hour of the SAR observation. By selecting buoys with good quality spectra, this yields 1100 data points with wind speeds between 1 and 10 m s−1 . Overall the bias is 0.24 m and the standard deviation of the errors is 0.5 m. The bias is found to be primarily a function of the swell height and wind speed, increasing with height and decreasing with wind speed. The standard deviation is instead dominated by the swell height and peak period, with the most accurate estimations for large periods (figure 4). When the distance between the buoy and SAR data is reduced to 50 km, much less data points are available (100), but the overall standard deviation decreases to 0.4 m. A significant part of the errors reported in figure 4 are thus due to the distance between the SAR and buoy measurement, which should not be included in our errors. Based on these result, a conservative error model for the swell system heights is a gamma distribution with a bias given by Hss − µ = 0.1Hss − 0.15 max{0, U10 − 7}

(2)

where µ is the expected value in meters and the wind speed U10 is in m s−1 , and a standard deviation given by σ = max {0.15, min {0.25Hss , max {0.4, 0.4 − 0.05(T − 13)}}}

(3)

where σ is in meters and the period T is in seconds. Early calculations were also done with a more simple error model Hss − µ = 0.15 m and σ = max{0.3, 0.15Hss } based on previously published analyses[19], and gave similar results with slightly larger error bars on the estimates of α.

Methods. SAR database generation Virtual SAR buoys A first type SAR wave mode data analysis was performed to verify the capability of SAR to track swell systems. A region of the ocean covering 2 by 2 degrees in latitude and longitude is 6

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Figure 4: Bias and standard deviation of swell partition heights Hss over an ensemble of 1200 buoy measurements located within 100 km of the SAR image, as a function of Hss (a and b), wind speed (c and d) and peak period (e and f) selected. For each wave spectrum observed in that square area, swell partitions are extracted using standard procedures, and a time history of these partitions is formed. In order to increase the quantity of data, swell partitions from a wider region are propagated in space and time following their theoretical great circle track in the wave direction, at the group speed corresponding to the peak wavelength. In fig. 2a, each horizontal colored segment thus corresponds to one swell partition that crosses the spatial window. The segment length corresponds to the time during which the propagated swell system is present in the spatial window. Some segments are very short, corresponding to trajectories that barely cut one corner of the square. Clearly the SAR detects the directions of the most energetic part of the wave spectrum which is also measured by the buoys (fig. 2b).

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Figure 5: Storm identification using swells arrivals observed with buoys and SAR. (a) Peak frequency fp = 1/Tp of and direction of swell partitions at a SAR virtual buoy located around the Christmas Island buoy (WMO number 51028). The sloping straight line fitted to the observed SAR ridge corresponds to the buoy observation.(b) Energy and mean direction spectrum measured in situ: contours, equally spaced from 0.1 to 1.4 indicate the natural logarithm of the spectral energy density E(f ) with, f = 1/T the wave frequency and T the wave period. Colors indicate the mean arrival direction at each frequency.

Analysis of SWAO tracks and estimation of swell dissipation A database of energetic storms was compiled by tracking back each observed long swell partition from SAR wave spectra along great circle trajectories at its group velocity. The location and date of storms was defined as the spatial and temporal center of the convergence area and time of the trajectories. Wind fields from the satellite scatterometer QuikSCAT provided storm date and location that always agreed within a few hundreds of kilometers and 6 hours. From these detected storm centers, great circle tracks were traced in all directions, and tracks away from islands were chosen based on a global shoreline database[20]. Each track was followed at a fixed group speed (discrete speeds corresponding to integer values of the peak wave period from 11 to 19 s). Along each track, defined by its origin in space and time, its outgoing direction θ0 and the wave period T , all ENVISAT ASAR wave spectra and significant wave height Hs from ENVISAT and JASON altimeters were gathered with an acceptable time window of plus or minus 3 hours and 100 km from the theoretical position. In a first filtering

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procedure, only SAR swell partitions which peak wavelength and direction are within 50 m and 20 degrees of expected were retained. A list of 12 storms was then selected that has significant number of measurements along trajectories without island sheltering. For each track of each of these storms, a short ASCII file was created containing one line for each SAR observation indicating distance from storm, time, position, swell partition observed wave height, wavelength and direction, direction mismatch from trajectories and local SAR derived wind speed. These files are available via anonymous ftp at the address ftp.ifremer.fr under the directory ifremer/cersat/products/gridded/wavewatch3/HINDCAST/SWAO/. All the analysis is based on these files only, with further filtering described below.

Swell track combination and SAR data selection A typical track file contains 3 to 20 SAR data. In order to define a reliable attenuation of the swell energy, tracks with neighbouring values of the outgoing direction θ0 are merged in relatively narrow direction bands, of the order of 5 to 10◦ (table 1), so that the combined wave properties are similar enough. In order to have enough SAR data at large distance from the storm to allow for a reliable estimation of the swell attenuation, we defined 32 such track ensembles. These ensembles include a total of 149 tracks out of the 1245 in the original database. In each ensemble of swell tracks, some SAR data was filtered out based on the following criteria • the distance from the source should be more than 4000 km, in order to satisfy the point source hypothesis which gives a reference wave height decay to which observed decay is compared to estimate the attenuation, and also to minimize errors due to the source localization. • the wind speed should be more than 3 m s−1 and less than 13 m s−1 : this filters out weak wind conditions in which the waves are poorly imaged by the SAR, and high wind conditions in which the azimuthal cut-off may contaminate the swell height estimation, and generally increases the SAR error (figure 4). • the significant swell height Hss , after bias correction based on eq. (5), should be more that 0.5 m. This makes sure that the signal to noise ratio in the image is large enough so that the wave height estimation is accurate enough. • The selected data should span a range of distances from the source larger than 2000 km, in order to be representative of various locations along the track. • The selected data should contain 6 or more SAR measurements, in order to provide a reliable swell dissipation estimate. Eventually, 3 of the 32 track ensembles gave no reliable estimate of the swell attenuation because no SAR data matched all four criteria, thus table 1 and 2 only gives information on 29 track ensembles. 9

number Storm time 1 20040216 00 2 20040216 00 3 20040216 00 4 20040418 18 5 20040418 18 6 20040418 18 7 20040418 18 8 20040630 23 9 20040709 18 10 20051021 00 11 20051021 00 12 20051114 03 13 20051113 12 14 20051113 12 15 20060310 00 16 20060310 12 17 20060310 23 18 20060427 00 19 20060427 06 20 20060427 06 21 20060427 06 22 20070212 18 23 20070324 00 24 20070324 00 25 20070324 00 26 20070812 00 27 20070812 00 28 20070812 00 29 20071030 00 Units date and hour UTC

Latitude 160 E 160 E 160 E 165 E 165 E 165 E 165 E 145 E 177 W 155 W 155 W 160 E 160 E 160 E 137 E 136 E 136 E 155 E 150 E 143 E 140 E 168 E 165 W 165 W 165 W 100 W 100 W 100 W 155 W deg.

Longitude 37 N 37 N 37 N 52 S 52 S 52 S 52 S 25 N 55 S 50 N 50 N 40 N 40 N 40 N 45 N 45 N 45 N 54 S 58 S 53 S 53 S 38 N 53 S 53 S 53 S 55 S 55 S 55 S 47 S deg.

T 14 15 16 14 15 16 17 13 14 15 17 13 15 17 16 14 13 15 14 16 17 15 15 17 18 15 17 18 15 s

θmin θmax 76 85 85 95 75 85 63 94 85 90 77 88 75 85 75 80 32 37 120 130 135 150 90 100 85 95 80 90 140 150 145 155 130 140 65 75 65 75 35 45 35 45 74 90 85 90 77 82 71 73 -30 -24 -27 -17 -27 -17 75 90 deg. deg.

N 35 26 6 11 17 35 32 23 11 23 30 21 49 24 19 13 10 47 44 16 19 48 15 11 17 19 14 8 62

Table 1: Ensembles of swell tracks selected for swell attenuation analysis. Each ensemble is defined by the source storm, the minimum and maximum outgoing directions θmin and θmax . The number of SAR data that was retained for the estimation of the attenuation is N . All storms are located in the Pacific Ocean.

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number α 1 26.3 2 22.1 3 9.4 4 17.8 5 6.1 6 12.0 7 8.7 8 11.5 9 13.3 10 12.8 11 7.0 12 16.3 13 11.9 14 1.4 15 3.9 16 12.1 17 11.8 18 10.1 19 15.4 20 9.9 21 6.2 22 37.4 23 21.6 24 20.7 25 6.0 26 0.6 27 0.4 28 -5.9 29 18.6 Units 10−8 m−1

α1 22.3 16.1 -0.1 3.6 -8.0 5.3 2.7 2.5 -6.5 7.5 2.6 -1.2 4.8 -5.8 0.0 4.1 -1.4 5.8 9.2 -1.2 -12.0 31.5 10.7 -5.2 -11.1 -6.7 -9.7 -13.0 13.4 −8 10 m−1

α2 29.3 25.9 20.3 36.6 16.8 18.3 15.1 18.6 24.4 15.1 10.2 39.8 13.8 6.2 9.6 16.2 19.3 13.3 22.2 18.5 30.5 40.2 30.4 41.1 13.5 2.4 3.5 3.0 25.9 10−8 m−1

H 5.6 4.0 3.8 3.3 2.4 3.0 2.8 2.0 2.2 3.0 2.6 2.3 2.4 2.3 2.3 2.0 1.8 3.5 3.5 2.9 2.2 4.6 3.0 2.5 1.6 1.5 1.3 1.0 2.5 m

ε1 8 17 7 14 9 14 11 25 10 7 21 13 17 12 9 7 9 10 8 6 13 10 14 11 8 11 12 20 13 %

Res fe,s 10.1 0.0161 4.8 0.0201 4.1 0.0082 3.5 0.0136 1.7 0.0052 2.5 0.0136 2.1 0.0146 1.4 0.0197 1.6 0.0039 2.7 0.0134 1.8 0.0631 1.8 0.0217 1.7 0.0152 1.4 0.0021 1.5 0.0045 1.3 0.0293 1.1 -0.0022 3.7 0.0065 3.9 0.0091 2.4 0.0096 1.3 0.0327 6.3 0.0334 2.7 0.0284 1.6 0.0641 0.6 N.A. 0.7 N.A. 0.4 N.A. 0.2 N.A. 1.9 0.0380 105

fe 0.0129 0.0137 0.0082 0.0091 0.0035 0.0092 0.0098 0.0064 0.0066 0.0110 0.0178 0.0088 0.0146 0.0013 0.0029 0.0083 0.0076 0.0045 0.0060 0.0064 0.0109 0.0238 0.0156 0.0215 N.A. N.A. N.A. N.A. 0.0152

ε2 U10 13 6.2 17 6.4 7 7.0 13 6.0 9 6.5 13 7.6 11 7.4 25 5.1 10 5.7 9 7.7 19 9.2 13 7.9 17 6.5 12 6.2 9 6.5 7 7.3 9 7.5 10 6.7 8 7.1 6 5.7 13 5.8 11 7.0 14 9.1 11 9.8 8 8.6 11 6.1 12 6.8 19 7.5 12 7.3 % m s−1

Table 2: Swell dissipation estimates. The fitted wave height at 4000 km from the source and constant linear decay coefficients are H and α, with ε1 the mismatch of the linear attenuation to the observed wave heights, normalized by the r.m.s. observed height. The analysis was repeated 400 times using a Monte Carlo simulation of observation errors. The 16% and 84% levels in the estimation of α are given by α1 and α2 . The fitted swell dissipation factor and total dissipation factor are fe,s and fe , the latter with a relative error ε2 . Finally the mean wind speed over the SAR images used in the fit is also given by U10 , and the significant swell Reynolds number Res is estimated at 4000 km based on the linear fit.

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Estimation of dissipation coefficients 0 (φ) was fitted. Three fits were performed, one with a For each set of SAR data, a function Hss constant linear decay α, the others with constant dissipation factors fe,s or fe . In each case the parameter α, fe or fe,s was fitted together with the height Hss (φ0 ) at a distance x0 = Rφ0 = 4000 km from the storm source. In practice the possible values of Hss (φ= π/5), from 1 to 12 m, and α or fe /fe,s (from −2.0 × 10−7 to 1 × 10−6 m−1 and -0.1 to 0.4, respectively) were scanned and the pair (Hss ,α 00 0 (φ), fe ) that gave the minimum root mean square difference with (φ), fe,s ), and (Hss ), (Hss observations Hss (φi ) were retained (table 2). In order to perform this fit, the function Hss (φ), was obtained from a numerical integration of dF (f, θ, φ) = αRF ((f, θ), φ) (4) dφ or

ρa 32π 4 p dF (f, θ, φ) = 16 (fe,s or γfe ) R Es F (f, θ, φ) dφ ρw gT 4 if Res > 28000 or Re > 100000 dF (f, θ, φ) = αv RF (f, θ, φ) otherwise, dφ

(5)

where γ is max{1.5, uorb /uorb,s }, in which the minimum value of 1.5 is meant to correct for the systematic underestimation of the large swells wave heights by the numerical model. Numerical integrations were performed from x = 4000 to x = 15000 km, for each pair, e.g. (H, α), using a simple first order Euler scheme that was found to converge fast enough. Here the swell Reynolds number is defined as Res = 4uorb,s aorb,s /ν. The wind sea and other swell systems is taken into account in the fe fits via the γ factor. The error function was computed by linearly interpolating the discretized Hs0 (φj ) at the positions φi where selected observations were made. In order to take into account the uncertainty of the SAR-derived wave heights, the estimation of α was repeated 400 times using uncorrelated random values of each SAR measurement, with a gamma distribution with a mean given by the observed value and a standard deviation of 30 cm or 15% of the measured value, whichever is greater. This Monte-Carlo estimation gave 400 values of α and H. The values corresponding to the 16 and 84 percentiles (this would correspond to one standard deviation if the values were Gaussian), are given in table 2 and shown as error bars on figure 1 of the paper. The estimated swell dissipation coefficient α was found to be weakly sensitive on the exact choice of the distance x0 and the minimum and maximum values for the wind speed and wave height. The variability of values of α for any range of wave slope is limited, and the confidence intervals of most of the estimates are relatively narrow. This suggests that our analysis is more sensitive than previous studies, in which attenuations less than 1.0 × 10−7 m−1 were not reliable 12

(this value corresponds to 0.05 dB/degree in [4]). This was likely due to the misalignement of swell tracks with fixed measuring stations, and errors introduced by corrections for islands, problems that are absent in our dataset. On the contrary, the estimation of fe is limited by the known biasses of the model described below, used to estimate the significant surface orbital velocity amplitude uorb . Indeed, uorb = 2πHs /Tm02 , and although both Hs and Tm02 are accurately estimated for average sea states, up to Hs = 8 m, there is a strong negative bias on wave heights in big storms, (for Hs > 10 m the bias is of the order of 10 to 15% of the value), which is typically the type of conditions found in some cases here. We have thus corrected uorb values from the model to be at least 1.5 times the SAR-derived swell orbital velocity uorb,s . The values of fe are thus indicative, and are not expected to be accurate to better than 50%.

Discussion 3. Verification of geometrical optics asymptotes and point source hypothesis The asymptotic energy-conserving solution E ∝ 1/ [φ sin(φ)] was verified using a semi-analytic model. This model uses the conservation of the spectral density along geodesics, which are computed analytically on the spherical Earth. At time t = 0 the initial wave spectra are prescribed to vary in space with a Gaussian storm distribution centered on the equator, with a width σx . At each position, the initial wave spectrum is prescribed to be a JONSWAP-type spectrum with a peak enhancement factor γJ , which is related to a spectral width parameter σf [21]. Finally, the initial spectra are taken isotropic in directions. That latter aspect is not very realistic but simplifies the calculations since the initial spectral density is only given by the frequency and not the direction. A space-time swell track is defined by the successive positions of an idealized wave packet travelling from the storm center at time t = 0 to a distance of 15000 km along the equator. At regular interval along this track, the wave spectrum is estimated by computing the spectral densities at a relative frequency resolution of 2% and a directional resolution of 0.5◦ . The quasianalytic total wave energy is then obtained by summation over the spectrum, and compared to the asymptotic value. The spatial decay of waves from such storms is thus completely specified by σx and γJ . Due to the finite size of the storm source, the asymptotic decay should be attained in the limit x  σx . Further, the dispersive decay requires a finite width of the wave spectrum but it is also affected by the size of the source. Indeed, the dispersive spreading induces travel distance differences of the order of δx = xσf (∂Cg /∂f )/Cg . This corresponds to a difference δx in the initial wave packet position at time t = x/Cg . The asymptotic decay is reached for δx  σx . In practice, beyond 4000 km from the source and for a large storm with σx = 550 km, the error relative to the asymptotic decay is less than 0.4% for a Pierson-Moskowitz[22] spectrum (defined by γJ = 1, which corresponds to a large σf ), and 4.5% for a JONSWAP spectrum (defined by γJ = 3.3, which corresponds to a small σf ). These two spectra correspond the 13

the extremes of broad and narrow spectra in the open ocean, with the JONSWAP form being rather rare and corresponding more to a coastal or enclosed sea situation. For a very compact storm, with σx = 220 km the maximum error is 1.2% for a Pierson-Moskowitz spectrum and up to 9.6% (negative bias) for JONSWAP spectrum. Thus very compact storms with young waves may lead to a significant departure from the generic decay asymptote. however, such an extreme deviation is still several times smaller than the differences between observed decays and the conservative asymptotic decay. Further, errors in source location gives an error proportional to the position mismatch divided by the distance from the source, and this effect is expected to be negligible. These calculations were done for fixed storms. The reader is referred to [3, 4], for a discussion of the effects of storm motion, that are likewise negligible. Thus, beyond 4000 km from the storm center, E(φ) is not expected to deviate by more than 10% from the 1/ [φ sin(φ)] asymptote for realistic storm sizes and spectral widths.

Discussion 4. Boundary layer theory For the sake of simplicity we will consider here the case of monochromatic waves propagating in the x direction only, and we will neglect the curvature of the surface. For the small steepness swells considered here that latter approximation is well founded and a more complete analysis can be found in [23]. Because the boundary layer is expected to be very thin compared to the wavelength, one can consider a local section of that boundary layer, for a given swell phase, in the local frame of reference moving at the sub-surface velocity u− (x, t) = σa cos(kx − σt), where a is the swell amplitude. The free stream velocity above the waves, just outside of the boundary layer is u+ (x, t) = −σa cos(kx−σt) (figure 6). Due to the oscillations that propagate at the phase velocity C, the horizontal advection of any quantity X by the flow velocity u, given by u∂X/∂x, can be neglected compared to its rate of change in time ∂X/∂t since the latter is a factor u/C smaller than the former, which is typically less that 0.1 for the swells considered here. Defining u e(x, z, t) = hu(x, z, t)i − u− (x, t), where the brackets denote an average over the wave phase, the horizontal momentum equation is thus approximated by, 1 ∂p ∂u− ∂e u =− − +G ∂t ρa ∂x ∂t

(6)

where G represents the divergence of the vertical viscous and turbulent fluxes of horizontal momentum, ∂ 2u e ∂ hu0 w0 i G=ν 2 + . (7) ∂z ∂z Because the boundary layer thickness δ is small compared to the wavelength, the pressure gradient in the boundary layer is given by the pressure gradient above the boundary layer, in balance with the horizontal acceleration, which is another way to write Bernoulli’s equation[24],

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i.e. −∂p/∂x/ρa = −σ 2 a sin(kx − σt) = ∂u+ /∂t. This yields ∂e u ∂u+ =2 +G ∂t ∂t

(8)

with the boundary condition for z  δ, u e goes to 2u+ (x, t). The equation for the horizontal momentum is thus exactly identical to the one for the oscillatory boundary layer over a fixed bottom with wave of the same period but with an amplitude twice as large. In the viscous case, one recovers, after some straightforward algebra, the known viscous result, i.e., for z > ζ, (9) u e(x, z, t) = 2σa [ez+ cos (kx − σt + z+ ) − cos (kx − σt)] + O(ρa /ρw ) p where z+ = (z − ζ)/ 2ν/σ, with the surface elevation ζ(x, t) = a cos(kx − σt). Evaluating the work of the viscous stresses hρa√ νu∂u/∂zi, eq. (9) gives the low frequency asymptote to the viscous decay coefficient, αv = 2k 2νσρa /ρw /Cg . This result was previously obtained using a Lagrangian approach without all the above simplifying assumptions[25]. The full viscous result is obtained by also considering the water viscosity νw , which gives the O(ρa /ρw ) correction for the motion in the air, and the classical dissipation term with a decay αvw = 4k 2 νw /Cg , which dominates for the short gravity waves.

orbital velocity profile across the wave crests, in the air and in the water

Streamli ne

uorb,s

Figure 6: Boundary layer over waves in the absence of wind. Because of the larger inertia of the water compared to the air, most of the adjustment from the sub-surface velocity to the free stream velocity in the air occurs on the air-side of the surface. As a result, for a comparison with fixed bottom boundary layers, the Reynolds number based on the orbital motion should be redefined with a doubled velocity and a doubled displacement, 15

i.e. Re= 4uorb aorb . For monochromatic waves aorb = a and uorb = aσ = 2πa/T . For random waves, investigations of the ocean bottom boundary layer suggest that the boundary layer properties are roughly equivalent to that of a monochromatic boundary layer defined by significant properties[26].

Discussion 5. Improvement in numerical wave modelling based on the present analysis Model description A preliminary validation of a new wave model parameterization has been performed using the present results. Although relatively few tests have been carried out, one of the parameterizations turned out to outperform today’s best wave models by a significant margin. This parameterization was thus implemented in the wave model routinely used at SHOM as part of the Previmer project, providing wave information to a variety of users (http://www.previmer.org). The results discussed below do not constitute any proof of the correctness of the wave attenuation mechanism highlighted here, but rather gives an indication on the usefulness of this result. Further, they provide an order of magnitude of the effect on the entire sea state, beyond the few swell partitions studied above. In order to highlight this magnitude, the model was ran with three constant values of the dissipation factor fe ( 0, 0.0035 and 0.0070). Another run was performed with an adjusted function form for the dependence of fe on the wind speed and direction, as explained below. This adjusted model was used in figures 1, 2, 5 and 6. Wave models are by no means perfect. They predict the wave spectrum based on the wave action balance equation[27], which writes, in deep water and without current, dF (f, θ) = Sin (f, θ) + Snl (f, θ) + Sds (f, θ), dt

(10)

where the spectrum F and source terms S are also functions of the geographical position, omitted here for simplicity. Solving that equation presents a number of challenges. First of all, the wind-wave generation Sin and wave dissipation terms Sds are poorly known. Second, the better known non-linear interaction term Snl (f, θ) requires extensive computer power that make routine wave forecasting barely feasible today. That term is thus usually parameterized with the DIA approximate form Snl [28]. Using that term may compensate for errors in the other two[29, 30] but the source terms are essentially uncertain in active wind wave generation conditions, and their numerical integration in time is not simple either[31]. Finally, the integration of the wave action balance requires accurate numerical schemes when swells are to be propagated across ocean basins[32]. In previous work, it was found that the input term by Janssen[33], used in the WAM-Cycle 4 model, probably has the right order of magnitude[34, 30]. On top of this term, we now add a negative wind input term in order to represent the upward momentum flux associated with the 16

wave attenuation observed here. If Re