Semi-Empirical Dissipation Source Functions for Ocean Waves: Part I, Definition, Calibration and Validation

A Generated using V3.0 of the official AMS L TEX template–journal page layout FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! Semi-empirical dissipation source functions for ocean waves: Part I, definition, calibration and validation. Fabrice Ardhuin ∗, Jean-François Filipot and Rudy Magne Service Hydrographique et Océanographique de la Marine, Brest, France Erick Rogers Oceanography Division, Naval Research Laboratory, Stennis Space Center, MS, USA Alexander Babanin Swinburne University, Hawthorn, VA, Australia Pierre Queffeulou Ifremer, Laboratoire d’Océanographie Spatiale, Plouzané, France Lotfi Aouf and Jean-Michel Lefevre UMR GAME, Météo-France - CNRS, Toulouse, France Aron Roland Technological University of Darmstadt, Germany Andre van der Westhuysen Deltares, Delft, The Netherlands Fabrice Collard CLS, Division Radar, Plouzané, France ABSTRACT New parameterizations for the spectral dissipation of wind-generated waves are proposed. The rates of dissipation have no predetermined spectral shapes and are functions of the wave spectrum, in a way consistent with observation of wave breaking and swell dissipation properties. Namely, swell dissipation is nonlinear and proportional to the swell steepness, and wave breaking only affects spectral components such that the non-dimensional spectrum exceeds the threshold at which waves are observed to start breaking. An additional source of short wave dissipation due to long wave breaking is introduced, together with a reduction of wind-wave generation term for short waves, otherwise taken from Janssen (J. Phys. Oceanogr. 1991). These parameterizations are combined and calibrated with the Discrete Interaction Approximation of Hasselmann et al. (J. Phys. Oceangr. 1985) for the nonlinear interactions. Parameters are adjusted to reproduce observed shapes of directional wave spectra, and the variability of spectral moments with wind speed and wave height. The wave energy balance is verified in a wide range of conditions and scales, from the global ocean arXiv:0907.4240v3 [physics.ao-ph] 2 Jan 2010 to coastal settings. Wave height, peak and mean periods, and spectral data are validated using in situ and remote sensing data. Some systematic defects are still present, but the parameterizations probably yield the most accurate overall estimate of wave parameters to date. Perspectives for further improvement are also given. 1. Introduction sities of the surface elevation variance F distributed over a. On phase-averaged models frequencies f and directions θ can be put in the form Spectral wave modelling has been performed for the last dF (f,θ) = Satm(f,θ)+Snl(f,θ)+Soc(f,θ)+Sbt(f,θ), (1) 50 years, using the wave energy balance equation (Gelci dt et al. 1957). This model for the evolution of spectral den- 1 where the Lagrangian derivative is the rate of change of pirical knowledge on the breaking of random waves (Ban- the spectral density when following a wave packet at its ner et al. 2000; Babanin et al. 2001) and the dissipation group speed in physical and spectral space. This spectral of swells over long distances (Ardhuin et al. 2009b). The advection particularly includes changes in direction due to present formulations are still semi-empirical, in the sense the Earth sphericity and refraction over varying topogra- that they are not based on a detailed physical model of dis- phy (e.g. Munk and Traylor 1947; Magne et al. 2007) and sipation processes, but they demonstrate that progress is currents, and changes in wavelength or period in similar possible. This effort opens the way for completely physical conditions (Barber 1949). parameterizations (e.g. Filipot et al. 2010) that will even- The source functions on the right hand side are sepa- tually provide new applications for wave models, such as rated into an atmospheric source function Satm, a nonlin- the estimation of statistical parameters for breaking waves, ear scattering term Snl, an ocean source Soc, and a bottom including whitecap coverage and foam thickness. Other ef- source Sbt. forts, less empirical in nature, are also under way to arrive This separation, like any other, is largely arbitrary. For at better parameterizations (e.g. Banner and Morison 2006; example, waves that break are highly nonlinear and thus Babanin et al. 2007; Tsagareli 2008), but they yet have to the effect of breaking waves that is contained in Soc is in- produce a practical alternative for wave forecasting and trisically related to the non-linear evolution term contained hindcasting. in Snl. Yet, compared to the usual separation of deep-water evolution into wind input, non-linear interactions, and dis- b. Shortcomings of existing parameterizations sipation, it has the benefit of identifying where the energy All wave dissipation parameterizations up to the work and momentum is going to or coming from, which is a nec- of van der Westhuysen et al. (2007) had no quantitative re- essary feature when ocean waves are used to drive or are lationship with observed features of wave dissipation, and coupled with atmospheric or ocean circulation models (e.g. the parameterizations were generally used as set of tuning Janssen et al. 2004; Ardhuin et al. 2008b). knobs to close the wave energy balance. The parameteri- Satm, which gives the flux of energy from the atmo- zation of the form proposed by Komen et al. (1984) have spheric non-wave motion to the wave motion, is the sum produced a family loosely justified by the so-called ‘random of a wave generation term Sin and a wind generation term pulse’ theory of (Hasselmann 1974). These take a generic Sout (often referred to as “negative wind input”, i.e. a form wind output). The nonlinear scattering term Snl repre- sents all processes that lead to an exchange of wave energy 2 0.5 4.5 4 k k and momentum between the different spectral components. Soc (f,θ)= Cdsg kr Hs δ1 + δ2 , (2) kr kr In deep and intermediate water depth, this is dominated " # by cubic interactions between quadruplets of wave trains, in which Cds is a negative constant, and kr is an energy- while quadratic nonlinearities play an important role in weighted mean wavenumber defined from the entire spec- shallow water (e.g. WISE Group 2007). The ocean source 1 trum, and Hs is the significant wave height. In the early Soc may accomodate wave-current interactions and inter- and latest parameterizations, the following definition was actions of surface and internal waves, but it will be here used restricted to wave breaking and wave-turbulence interac- 1/r tions. fmax 2π 16 r The basic principle underlying equation (1) is that waves kr = k E (f,θ) dfdθ , (3) H2 essentially propagate as a superposition of almost linear " s Z0 Z0 # wave groups that evolve on longer time scales as a result of where r is a chosen real constant, typically r = 0.5 or weak-in-the-mean processes (e.g. Komen et al. 1994). Re- r =0.5. − cent reviews have questioned the possibility of further im- These parameterizations are still widely used in spite of proving numerical wave models without changing this basic inconsistencies in the underlying theory. Indeed, if white- principle (Cavaleri 2006). Although this may be true in the caps do act as random pressure pulses, their average work long term, we demonstrate here that it is possible to im- on the underlying waves only occurs because of a phase cor- prove model results significantly by including more physical relation between the vertical orbital velocity field and the features in the source term parameterizations. The main moving whitecap position, which travels with the breaking advance proposed in the present paper is the adjustment wave. In reality the horizontal shear is likely the dominant of a shape-free dissipation function based on today’s em- mechanism (Longuet-Higgins and Turner 1974), but the 1In the presence of variable current, the source of energy for the question of correlation remains the same. For any given wave field, i.e. the work of the radiation stresses, is generally hidden whitecap, such a correlation cannot exist for all spectral when the energy balance is written as an action balance (e.g. Komen wave components: a whitecap that travels with one wave et al. 1994). leads to the dissipation of spectral wave components that 2 propagate in similar directions, with comparable phase ve- An alternative and widely used formulation has been locities. However, whitecaps moving in one direction will proposed by Tolman and Chalikov (1996), and some of its give (on average) a zero correlation for waves propagating features are worth noting. It combines two distinct dissipa- in the opposite direction because the position of the crests tion formulations for high and low frequencies, with a tran- of these opposing waves are completely random with re- sition at two times the wind sea peak frequency. Whereas spect to the whitecap position. As a result, not all wave Janssen et al. (1994) introduced the use of two terms, k and components are dissipated by a given whitecap (others k2 in eq. (2), in order to match the very different balances should even be generated), and the dissipation function in high and low frequency parts of the spectrum, they still 0.5 4.5 4 cannot take the spectral form given by Komen et al. (1984). had a common fixed coefficient, Cdsg kr Hs . In Tolman A strict interpretation of the pressure pulse model gives and Chalikov (1996) these two dissipation terms are com- a zero dissipation for swells in the open ocean because the pletely distinct, the low frequency part being linear in the swell wave phases are uncorrelated to those of the shorter spectrum and proportional to wind friction velocity u⋆, the 2 breaking waves. There is only a negligible dissipation due high frequency part is also linear and proportional to u⋆.

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