Spectral wave evolution and spectral dissipation based on observations: a global validation of new source functions

Fabrice Ardhuin a , Mathieu Hamon b , Fabrice Collard c , Bertrand Chapron and Pierre Queffeulou d a Service

Hydrographique et Ocanographique de la Marine, 29609 Brest, France

b Laboratoire

de Physique des Oc´eans, Universit´e de Bretagne Occidentale, 29000 Brest, France c BOOST-Technologies,

d Laboratoire

29280 Plouzan´e, France

d’Oc´eanographie Spatiale, Ifremer, 29280 Plouzan´e, France

Abstract Existing parameterizations of wave dissipation used in spectral wave models have provided excellent results in most of the world ocean but lead to significant and persisting errors. Here a new parameterization is proposed that simply combines the observed swell dissipation and a saturation-based dissipation compatible with observed wave breaking probabilities. This parameterization is adjusted to provide accurate hindcasts of the global wave field as observed by in situ buoys, and a preliminary validation is presented. The resulting global model is shown to outperform all existing operational models to date in terms of significant wave height, and peak and mean period. The model further provides a better rendering of the high frequency part of the wave spectrum, as validated with C-band radar altimeter cross sections, with important applications for remote sensing. Improvement and adjustment of the model is in progress, with a view to improving coastal sea states.

Key words: surface gravity waves, air-sea fluxes, surface drift, ocean mixing

Proceedings, 4th Chinese-German joint symposium on Coastal and Ocean Engineering

August 2008

1

Introduction

Spectral wave modelling has been performed for the last 50 years, using the wave energy balance equation (Gelci et al., 1957), which describes the radiation of the spectral density of the surface elevation variance F distributed over frequencies f and directions θ, dF (f, θ) = Satm (f, θ) + Snl (f, θ) + Soc (f, θ) + Sbt (f, θ), dt

(1)

where the Lagrangian derivative is the rate of change of the spectral density when following a wave packet at its group speed in physical and spectral space. The source functions on the right hand side are separated into an atmospheric source function Satm (f, θ), a nonlinear scattering term Snl (f, θ), an ocean source Soc (f, θ), and a bottom source Sbt . This separation is somewhat arbitrary, but, compared to the usual separation of deep-water evolution in input, non-linear interactions, and dissipation, it has the benefit of identifying where the energy is going to or coming from. Satm (f, θ), which gives the flux of energy from the atmospheric non-wave motion to the wave motion, is the sum of a wave generation term Sin and a wind-generation term (often referred to as negative wind input, i.e. a wind output) Sout . The nonlinear scattering term Snl (f, θ) represents all processes that lead to an exchange of wave energy between the different spectral components. In deep and intermediate water depth, this is dominated by cubic interactions between quadruplets of wave trains (see Hasselman 1962, Herterich and Hasselmann 1980, Janssen and Onorato 2005), while quadratic nonlinearities play an important role in shallow water (see e.g. WISE Group (2007)). The ocean source Soc (f, θ) may accomodate wave-current interactions and interactions of surface and internal waves, but it will be here restricted to wave breaking and wave-turbulence interactions, and the dissipation of wave energy in the ocean bottom boundary layer 1 . Finally, interactions with the bottom, that will not be considered here, are discussed by Ardhuin et al. (2003) and Elgar and Raubenheimer (2008). The basic principle underlying that equation is that waves essentially propagate as a superposition of almost linear wave groups with a weak-in-the-mean evolution due the processes listed above. Recent review have questioned the possibility of further improving numerical wave models without changing these basic principles (Cavaleri, 2006). Although this may be true in the long term, we demonstrate here that it is still possible to improve model results significantly by including more physical constraints in the source term parameterizations. The main advance proposed in the present paper is the adjustment of a shape-free dissipation function based on 1

Technically this is where the energy is dissipated, although the associated momentum generally ends up in the bottom (Longuet-Higgins, 2005).

2

today’s knowledge on the breaking of random waves (Banner et al. 2000, Babanin et al. 2002) and the dissipation of swells over long distances (Ardhuin et al. 2008).

Non dimensional energy E* (wind sea)

Models that use the dissipation parameterizations of the form proposed by Komen et al. (1984) have been refined over the last 25 years, culminating in the form given by Bidlot et al. (2005), or that of Alves and Banner (2003) which have started introducing new features. In spite it successful use for estimating Hs and Tp , the original fixed-shape dissipation functions have terrible built-defects, like the spurious effect of swell on wind sea growth, with stronger growth modelled with higher swells, as shown on figure 1, and discussed by Ardhuin et al. (2007). Associated with that defect also comes a strong reduction of high frequency energy in the pres-

10

-3

Kahma Calkoen (unst.) SHOWEX obs. TEST 304 BAJ no swell TEST 332 BAJ+swell -4

10

3

4

10

Non dimensional fetch X*

10

Fig. 1. Fetch-limited growth of for the SHOWEX case discussed in Ardhuin et al. (2007). The BAJ parameterization is particularly sensitive to swell at short fetch.

ence of swell, which leads to very poor surface slope statistics (Hamon 2008), making these wave models ill-suited for remote sensing studies (figure 2). Another widely used alternative formulation has been proposed by Tolman and Chalikov (1996). This formulation contains important features that we borrow in the present formulation. Namely, it has a dissipation parameterization that is essentially a function of the local spectral density of the frequency spectrum, and it also includes a negative wind input which plays a major role in the swell evolution. However, the magnitude of the source terms appear too weak, as the parameterization yields important biases in wave growth and wave directions at short fetch 3

bouee 46001 (6m NOMAD)

WW3 test 332

0.03

0.03 8

0.02

6

0.015 4 0.01 2

0.005 0

0

8

0.025 pseudo mss

pseudo mss

0.025

0.02

6

0.015 4 0.01 2

0.005 0

10 20 U10 [m/s]

0

WW3 BAJ

WW3 test 321

0.03

0.03 8

0.02

6

0.015 4 0.01 2

0.005 0

8

0.025 pseudo mss

pseudo mss

0.025

0

10 20 U10 [m/s]

0.02

6

0.015 4 0.01 2

0.005 0

10 20 U10 [m/s]

0

10 20 U10 [m/s]

Fig. 2. Typical diagrams of pseudo mean square slopes of the sea surface defined from the frequency R 0.5Hz (2 ∗ pif )4 E(f )/g 2 df from a buoy (here buoy 46001 in the gulf of Alaska) and the spectrum as 0 model with the BAJ parameterization or two tests with saturation-based dissipations. For each data set covering 2006 and 2007 (buoy or model), the data was averaged over 3 hours and binned into wind speed U10 (x axis) and wave height (colors, in meters) classes. For each class the average mss is shown. The BAJ parameterization gives a mss which is essentially a function of U10 , and even decreases with Hs at the high wind speeds, which is contrary to observations. The wind used here for the model forcing are 6 hourly 0.5 degree resolution analyses from ECMWF.

(Ardhuin et al. 2007). Finally, the separation of the two types of dissipation at low and high frequency could have corresponded to the separation between wave breaking (high frequency) and other processes (low frequency). However, it is now well established that this separation cannot be set at twice the peak frequency, since even dominant wave break in young seas. A complete redesign of the source terms is thus necessary. 4

Using observations that no dominant waves break when the saturation spectrum B (f ) = 2π

Z2π

k 3 F (f, θ)/Cg dθ,

(2)

0

is below 1.2 × 10−3 Banner et al. (2000); Babanin and Young (2005), van der Westhuysen et al. (2007) have proposed a dissipation form which uses a similar threshold. Giving encouraging results, their parameterization was verified in coastal conditions only, because they did not know how to treat swell dissipation over large distances. The recent measurement of the swell decay by Ardhuin et al. (2008) now allows an adjustment of the source function for the global scales. This work is presented here, together with a preliminary evaluation of the source terms for coastal areas, which will be used to further adjust the parameterization.

2

Parameterizations

Because we are aiming to produce a practical parameterization, we shall focus only on the dissipation and generation terms. Obviously the poor parameterization of the nonlinear interaction Snl that will be used here, based on the Discrete Interaction Approximation or DIA (Hasselmann et al., 1985), probably causes a significant error, that is partially corrected by the other source functions (e.g. Banner and Young, 1994; Ardhuin et al., 2007).

2.1

Wave breaking

We have chosen to use the simplest possible dissipation term formulated in terms of the directionintegrated spectral saturation B (f ), defined above, with a realistic threshold Br = 1.2 × 10−3 corresponding to the onset of wave breakingBabanin and Young (2005). This isotropic dissipation term takes the form " iso Sds (f, θ) = σCds

(

)#2

B (f ) − 1, 0 max Br

F (f, θ).

(3)

We then adjusted the dissipation parameter Cds = −2.4 × 10−5 . However, the resulting directional spectra appeared generally narrower than observed spectra. Because Banner et al. (2002) showed that the threshold Br increased with the width of the directional spectrum, and because one may expect wave groups to exhibit larger orbital velocities 5

and wave heights when the directional spreading is small, we modify the isotropic form in eq. (3) by introducing a partially-integrated saturation, 0

B (f, θ) = 2π

θ+∆θ Z

k 3 F (f, θ)/Cg dθ,

(4)

θ−∆θ

to give,  " 

Sds (f, θ) = σCds δ max

(

)#2

B (f ) − 1, 0 Br

"

+ (1 − δ) max

(

0

)#2  

B (f, θ) − 1, 0 Br

F (f, θ).



(5)

2.2

Swell dissipation

Another very important part of the parameterization, at least when considering oceanic scales, is the dissipation of swells. Historically, swell has been thought to dissipate due to oceanic turbulence (Groen and Dorrestein, 1950), a process which is now regarded as negligible (Ardhuin and Jenkins, 2006). Recent theories have instead focused on air-sea interactions, and, in particular the correlation of pressures and surface slopes over swells (e.g. Kudryavtsev and Makin, 2004). Progress in this area was hampered by the lack of any reliable estimate of the swell dissipation. This has now changed with the analysis of swell propagation using synthetic aperture radar (SAR) (Ardhuin et al., 2008). The main finding of this investigation is that swell decay is non-linear, with a relative stronger decay of steeper swells, and there are indications that there is a threshold steepness below which dissipation is negligible. Taken together, these facts support that swell dissipation is dominated by friction in the air-side boundary layer which is either smooth or with very low roughness. Because this process yields a net momentum flux to the atmosphere, which generates wind (Harris, 1966), the corresponding source term will be called here ”wind output”. The oscillatory boundary layer is expected to be smooth for a Reynolds number Re= 2uorb,s Hs /νa where νa is the air viscosity, and the significant orbital velocity amplitude is ZZ

uorb,s = 2

2

σ F (k, θ) dkdθ

1/2

.

(6)

In this case the dissipation source term for swells is given by Dore (1978)’s viscous theory, o ρa n √ Sout (f, θ) = −s5 2k 2νσ F (f, θ) . (7) ρw 6

0

E(f) (m 2/Hz)

10

1

10

2

10

0.1

0.2

0.3

f (Hz)

150

0.5

0.6

Data TEST 304 ( D q = 180°) BAJ, no swell TEST 332 ( D q = 70°) BAJ, with swell

120 mean dir. (deg)

0.4

90 60 30 0 30 60 90 0.1

0.2

0.3

0.4

0.5

0.6

dir. spread sq (f) (deg)

80 60 40 20 0

Fig. 3. Model-data comparison at buoys X3 during the SHOWEX event discussed in Ardhuin et al. (2007). (a) Frequency spectra, (b) mean directions, (c) directional spreads. Using a non-isotropic dissipation function (TEST332, compared to the isotropic TEST304) allows the reproduction of both the energy level and the directional spreading. Otherwise an increase in dissipation yields a lower and narrower spectrum. A proper balance may be found by adjusting the wind input, but this typically does not work at all frequencies.

In theory s5 = 1 for pure air and water. Here we shall take s5 = 1.2. The transition to turbulence was found for Rec ' 100000, which is consistent with smooth boundary layers over fixed bottoms (Jensen et al., 1989). Above this threshold, the turbulent dissipation can be parameterized as Sout (f, θ) = −

o ρa n 16fe σ 2 uorb,s /g F (f, θ) . ρw

7

(8)

Using a wave model to estimate the full orbital velocity (i.e. including the wind sea), the dissipation factor fe may be estimated form the SAR observations.

dissipation parameter (dE/dx) /E

Fig. 4. Dissipation factor fe as a function of the Reynolds number for swells 4000 km away from their generating storm. The colors show the corresponding spatial decay parameter that was inferred by fitting theoretical dissipation curves to the observed variation in swell wave height. Each dot correspond to one swell system from a storm, which was followed from 4000 to 8000 or more kilometers away from the storm(Ardhuin et al., 2008). Most of the values of fe fall in the range 0.005 to 0.01, slightly higher than the smooth boundary layer data of Jensen et al. (1989), which, for similar Reynolds numbers, are in the range 0.004 to 0.007. The larger values of fe are probably overstimated due to a general underestimation of Hs , and thus uorb,s , in the biggest storms.

It was found that the model tended to underestimate large swells and overestimate small swells, with regional biasses. This defect is likely due, in part, to errors in the generation or non-linear evolution of theses swells. However, it was chosen to adjust fe as a function of the wind speed and direction, fe = 0.7fe,GM + [0.015 − 0.018 cos(θ − θu )] u? /uorb ,

(9)

where fe,GM is the friction factor given by Grant and Madsen (1979)’s theory for rough oscillatory boundary layers without a mean flow, using a roughness length adjusted to 0.04 times the roughness for the wind. This gives a stronger dissipation for swells opposed to winds. 8

Thus the full wind input source term reads ρa βmax Z 4 u? + zα e Z ρw κ2 C 

Sin (f, θ) =

2

up cosp (θ − θu )σF (f, θ) + Sin (f, θ) ,

(10)

where βmax is (constant) a non-dimensional growth parameter, κ is von K´arm´an’s constant, and p is a constant power, here taken at 1.7 instead of the more usual p = 2. In the present implementation the air/water density ratio is constant. We define Z = log(µ) where µ is given by Janssen (1991, eq. 16), and corrected for intermediate water depths, so that Z = log(kz1 ) + κ/ [cos (θ − θu ) (u0? /C + zα )] ,

(11)

where z1 is a roughness length modified by the wave-supported stress τw , and zα is a wave age tuning parameter. The effective roughness z1 is implicitly defined by u? 10 m U10 = log κ z1 ( ) u2? z0 = max α0 , 0.0020 g z0 . z1 = q 1 − τw /τ 



(12) (13) (14)

The maximum value of z0 was added to reduce the unrealistic stresses at high winds that are otherwise given by the standard parameterization. This is equivalent to setting a maximum wind drag coefficient of 2.8 × 10−3 . This, together with the different value of p and the use of an effective friction velocity u0? (f ) instead of u? in (11) are the only changes to the general form of Janssen (1991)’s wind input. That friction velocity is defined by 2

(u0? (f )) =

Zf Z2π Sin (f 0 , θ0 ) 2 u? (cos θu , sin θu ) − |su | C 0 0

0 0 (cos θ, sin θ) df dθ , .

(15)

This correction is also applied to the high frequency tail if su > 0. The dissipation parameters Cds = −2.4 × 10−5 , ∆θ = 70◦ and δ = 0.25 have been adjusted to the directional short fetch measurements of Ardhuin et al. (2007), together with the wind input parameters βmax = 1.75, and the wave age correction factor zα = 0.005, instead of the values 9

1.2 and 0.011 typically usedBidlot et al. (2005). Results with these parameters are presented below.

3

Model performance

All model runs discussed here are performed using version 3.13 of WAVEWATCH, kindly provided by Tolman (2007) for development and beta testing. The model configuration used is global with a 0.5 degree regular resolution in latitude and longitude. The spectral grid uses 32 frequencies from 0.0373 to 0.716 Hz, and 24 directions. The model is forced with analyzed winds and sea ice concentration every 6 hours with the same spatial resolution, provided by the European Center for Medium Range Weather Forecasting (ECMWF). These fields are interpolated in time, preserving the square of the wind modulus. Further, subgrid islands are treated following the method of Tolman (2003). Besides the sensitivity to swells in coastal areas (figure 1), the main motivation for designing a new source term package was the same as the motivation of Tolman and Chalikov (1996), i.e. the overestimation of swell heights and periods in the Pacific, a feature common to all the parameterizations derived from that of Komen et al. (1984). In addition, we also desired a proper variability of the high frequency tail level, for remote sensing applications. That latter aspect will be described in detail elsewhere. For this, an example of improvement is the correlation against mean square slopes derived from C-band satellite altimeter data, which was increased from 0.58 to 0.71 (numbers given by Hamon, 2008) for 10 s averaged satellite data from JASON 1 for December 2007). Here we will focus on parameters that are more widely used for ocean and coastal engineering, namely Hs , the peak frequency fp or peak period Tp = 1/fp and the mean period Tm,02 . The global biasses against altimeters has been dramatically improved, with the positive biasses in the East Pacific and East Indian ocean now completely erased. In fact, it is likely that now swells are slightly underestimated in the areas, as in a good fraction of the South Pacific. From the analysis of the swell propagation using SAR data, it appears that this is rather due to an underestimation of the most severy storms, together with an overestimation of the dissipation of more moderate swells. Work is now under way to further adjust the parameterizations in order to correct these errors. Interestingly the East-West gradients in biases found with the BAJ parameterization is greatly reduced. This was already the case with the parameterization by Tolman and Chalikov (1996), as revealed by Tolman (2003). We believe that this is because there is generally less swell on the Western side of ocean basins. With less swell the BAJ parameterization gives a lower windsea growth, and leads to the previously found bias. The improvement in the swell fields is also revealed by the lower bias on the periods in the 10

Fig. 5. 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 Rascle et al. (2008). Results are provided for (a) the BAJ parameterization, (b) the new parameterization described here.

Pacific (figure 6). The results, however, are most spectacular when random errors are considered. Here we use root mean square (r.m.s.) errors (which include the bias) and normalize these by the local observed r.m.s. values in order to provide a scale for the model errors that may be compared from one location to another (figure 7). The fraction of model grid points where the normalized error (NRMSE) is less than 10%, has expanded from 19% with BAJ to 42% with the new parameterization. The median error with BAJ is reduced from 12% to 10.5%. There is even 11% of the world ocean with errors on Hs under 8%, while only 2% had such a low error with BAJ. 11

60N

(a) Old model

40N 20N 0 20S 40S

150W

120W

-1

90W

-0.8

60W

-0.6

30W

0

-0.4 -0.2

0

30E

0.2

60E

0.4 0.6

90E

0.8

Period bias (Tp or Tm02, in seconds)

60N

120E

150E

1

(b) New model

-2

0

40N 20N 0 20S 40S

Fig. 6. 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, ◦, 2, ♦ 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.

4

Summary and discussion

Previous parameterizations for wind wave generation and evolution have been constantly refined over the last few decades. The family of parameterizations based on an overall mean steepness for the entire spectrum (Komen et al., 1984), culminated in the parameterization (BAJ) used at present at ECMWF (Bidlot et al., 2005)Janssen2007. That parameterization, until recently, provided the best available analysis and forecasts of sea states for most applications. Here we present a new set of parameterizations that combine a wave breaking pa12

Fig. 7. Normalized root mean square difference (NE), in percent, for Hs over the year 2007 between models and altimeter data from satellites JASON, GFO and ENVISAT.

rameterization compatible with the threshold behaviour of breaking statistics (Banner et al., 2000), and a nonlinear swell dissipation consistent with observed swell decays (Ardhuin et al., 2008). Although many details of this parameterization are still to be refined, these two important features remove the spurious swell sensitivity in the BAJ parameterization, leading to a better estimate of common wave parameters at global scales, as summarized in table 1. The present model has been validated by a 4 year hindcast, avaialable via ftp at ftp: //ftp.ifremer.fr/ifremer/cersat/products/gridded/wavewatch3/, and is routinely used for forecasting since May 10, 2008. In enclosed areas where little altimeter data can be used to correct the model by assimilation, the model presented here actually provides the best operation analyses and forecasts so far. This will become clearer in the statistics for the coming months presented in the JCOMM model verification page http://www. jcomm-services.org/Wave-Forecast-Verification-Project.html/. 13

Model run

ACC

BAJ

ECMWF

Altimeters Hs

12.3

14.0

Buoys Hs

12.4

13.6

11.0

Buoys Tp

20.1

24.1

16.9

Buoys Tm02 6.8 7.6 8.5 Table 1 Averages of normalized root mean square errors against various datasets for the new parameterization (ACC) and the previous best parameterization (BAJ) for the entire year 2007 (buoys observations are averaged over 5 hours). The buoys considered here are those with WMO numbers 41002, 41010, 42001, 42002, 42003, 44004, 44008, 44011, 44137, 44138, 44139, 4414 (U.S. and Canadian East coasts and Gulf of Mexico), 46001, 46004, 46035, 46066, 46184, 46002, 46005, 46036, 46059 (U.S. and Canadian west coasts), 51001, 51002, 51003, 51004 (Hawaii), 62029, 62081, 62163, 64045 (North East Atlantic). The values for Tp correspond to all the American buoys, while the values for Tz correspond only to the 4 European buoys. For reference we also give the results at the buoys for the analyses from the operational ECMWF model that uses the BAJ parameterization are also recalled. Because the ECMWF model assimilates altimeter wave heights its performance is largely improved compared to the BAJ free run (without assimilation) in open ocean areas. In enclosed seas such as the North Sea and the Gulf of Mexico the model with ACC is found to provide better results than ECMWF operational analyses.

The present parameterization, although it gives the best result so far for in terms of bias and random error on Hs against altimeter data for Hs up to 8 m, which is very good for delicate marine operations. However, it also produces important negative biasses (up to 8%) for Hs over 10 m, which is not so good for extreme value analysis and design. Further work will focus on correcting this effect, which can probably be done by reducing the sheltering coefficient su in eq. (15). Further we will also attempt a simplification of the parameterization with a view to validate it also further at coastal scale. For this, the directional distribution of the wave energy is of critical importance. The present parameterization tends to give spectra that are too narrow, in particular at high frequencies (figure 3), and work is under way to correct this bias. Finally, a general improvement of the physical basis of the source functions is also sought. In particular, work on the wave breaking term should eventually separate the term in a breaking probability expressed as a properly weighted convolution integral over the spectrum, and a breaking severity. Because individual waves break, one has to go from the spectrum to the individual waves, for which breaking statistics can be verified, and back to the spectrum. The source term should thus be a deconvolution of this dissipation per breaking scale back to the Fourier modes that contribute to a given scale, as proposed by Filipot et al. (2008). With that approach, the use of both B and B 0 as measures of the saturation level may not be necessary and B 0 may suffice. Preliminary work in that direction shows that a better directional distribution is already obtained by introducing a cos2 weighting function in B 0 , and redefining B(f ) as max{B 0 (f, θ), 0 ≤ θ < 2π}. Such a weighting corresponds to a saturation parameter in terms of the orbital velocity, which may provide a generic wave breaking term for both deep and 14

shallow water. The convolution-deconvolution approach, in both the directional and frequency domains, still has to be implemented and calibrated. As soon as reliable breaking statistics are introduced in the model, these can be in turn used in the wind generation term to reproduced the enhanced generation of breaking waves (e.g. Reul et al., 2008), and for many direct applications such as remote sensing and offshore engineering.

Acknowledgments. This research would not have been possible without the wind and ice fields provided by ECMWF and Meteo-France, the satellite altimeter data provided by ESA and CNES, and the many in situ observations acquired by all contributors to the JCOMM (WMOIOC) exchange program, including NOAA/NDBC, Meteo-France, Puertos del Estado, the U.K. Met. Office, the Australian Weather Bureau and the Irish Marine Institute. Finally, the quality of the present model and feasibility of our research owes much to the very kind help from H.L. Tolman (NOAA/NCEP) and Jean Bidlot and Peter Janssen (ECMWF) and anonymous reviewers. We are indebted to Denis Croiz´e-Fillon and Pierre Queffelou (Ifremer) who performed the comparison of model and altimeter data. This work was partially funded by the ANR project HEXECO.

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