On nonlinearity implications and wind forcing in Hasselmann equation
aa r X i v : . [ phy s i c s . a o - ph ] A ug On nonlinearity implications and wind forcing inHasselmann equation
Pushkarev Andrei b,c,d,1 , Zakharov Vladimir a,b,c,d a University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ 85721, USA b LPI, Leninsky Pr. 53, Moscow, 119991 Russia c Pirogova 2, Novosibirsk State University, Novosibirsk, 630090 Russia d Waves and Solitons LLC, 1719 W. Marlette Ave., Phoenix, AZ 85015 USA
Abstract
We discuss several experimental and theoretical techniques historically used forHasselmann equation wind input terms derivation. We show that recently de-veloped
ZRP technique in conjunction with high-frequency damping withoutspectral peak dissipation allows to reproduce more than a dozen of fetch-limitedfield experiments. Numerical simulation of the same Cauchy problem for differ-ent wind input terms has been performed to discuss nonlinearity implicationsas well as correspondence to theoretical predictions.
Keywords:
Hasselmann equation, wind-wave interaction,wave-breaking dissipation, nonlinear interaction, self-similarsolutions, Kolmogorov-Zakharov spectra
1. Introduction
It is generally accepted nowadays that ocean surface wave turbulence is de-scribed by Hasselmann equation (hereafter HE ) ∂ε∂t + ∂ω k ∂~k ∂ε∂~r = S nl + S in + S diss (1)for spectral energy density ε = ε ( ~k, ~r, t ), wave dispersion ω = ω ( k ) and nonlin-ear, wind input and wave-breaking dissipation terms S nl , S in and S diss corre- Corresponding author,
E-mail : [email protected] HE is some sort of mathemat-ical reduction of primordial Euler equations for incompressible fluid with freesurface, it is formally true, in fact, only for advection ∂ω k ∂~k ∂ε∂~r and four-waveinteraction S nl terms.As far as concerns S in and S diss terms, there is no consensus in the worldwideoceanographic community about their parameterization. To our belief, it is oneof the reasons, indeed, for “tuning knobs” (adjusting coefficients) necessity inoperational models for their adjustment to different ocean situations set-ups.Another reason for using “tuning knobs” is the underestimation of the leadingrole of S nl . In other words, the role of the “tuning knobs” also consists in“undoing” the deformation incurred to the primordial equations model throughsubstitution of the exact nonlinear term S nl with DIA -like simplifications.We believe that such currently widely accepted approach of using ”tuning knobs”is conceptually misleading, and continuing efforts on improving the source termsare fruitless because the nonlinear term S nl is the leading term of HE [1], [2].All other source terms are, in a sense, relatively small corrections in significantranges of frequency.Dominations of the nonlinearity exhibit itself in HE in the form of self-similarsolutions, observed in the field and numerical experiments [3, 4, 5, 6, 7, 8].Self-similarity analysis in conjunction with the field and numerical experimentsanalysis allowed to build new ZRP wind input term [9] – analytical solution of HE .In the current paper we use alternative approach to HE simulation, which inaddition to new ZRP wind input term uses another physically based assump-tions – absence of the spectral peak dissipation and implicit wave damping dueto wave-breaking. We justify this new approach through comparison with morethan the dozen of field measurements.Another result of current paper is the development of the set of tests, basedon self-similarity analysis, allowing to make the judgment relative to correct-ness of arbitrary numerical simulation performed in the frame of HE through2omparison of the observed characteristics of wave ensemble with theoreticalprediction.
2. Current state of wind input source terms
Nowadays, the number of existing models of S in is large, but neither of themhave firm theoretical justification. Different theoretical approaches argue witheach other. Detailed description of this discussion can be found in the mono-graphs [10], [11] and the papers [12], [13], [14], [15], [16].The development of wind excited waves models has begun as far back as 20-iesof the last century in the well-known works of Jeffreys [17], [18]. His model issemi-empiric and includes unknown ”sheltering coefficient”. All other existingtheoretical models are also semi-empiric, with one exclusion – famous Milesmodel [16]. This model is rigorous, but is related to idealized situation – initialstage of waves excitation by laminar wind with specific wind profile U ( z ).Miles theory application is hampered by two circumstances. First is the factthat atmospheric boundary layer is the turbulent one, and creation of rigorousanalytical theory of such turbulence is nowadays unsolvable problem.There is the opinion, however, that wind speed turbulent pulsations are smallwith respect to horizontal velocity U ( z ), and they can be neglected. This doesn’tmean that turbulence is not taken into account at all. It is suggested that therole of the turbulence consists in formation of the averaged horizontal velocityprofile.This widely spread opinion is that horizontal velocity profile is distributed bylogarithmic law U ( z ) = 2 . u ∗ ln zz (2)Here u ∗ is friction velocity and z – the roughness parameter z = C ch u ∗ g (3)where C ch ≃ − is experimental dimensionless Charnok constant.3ne should note that appearance of anomalously small constants, not having”formal justification”, is extremely rare phenomenon in physics. Eq. (2), (3)mean that roughness parameter is very small: for typical ocean conditions –wind speed 10 m/sec on the height z = 10 m we get z ≃ − m . Suchroughness is only twice the size of viscid layer, defined from multiple experimentson turbulent wind flow over smooth metal plates.Usage of Eqs.(2), (3) assumes therefore that ocean behaves as smooth metalsurface. This is not correct. Horizontal momentum is transferred to the smoothplate on it’s surface itself, while in the ocean this process happens differently.Momentum offtake from atmospheric boundary layer is smoothly distributedover the whole width of the boundary layer and begins from the highest ”con-currence layer”, i.e. from the height where phase speed of the fastest wavematches the horizontal velocity.Momentum offtake leads to horizontal velocity distribution U ( z ) dependence ontime, waves development level and energy spectrum. Meanwhile, Miles insta-bility increment is extremely sensitive to the horizontal velocity profile (there isno waves excitation for linear profile U ( z ) for Miles theory, for example). Thevelocity profile is especially important for slight elevations of the order of severalcentimeters over the water surface, which is almost unknown and difficult forexperimental measurements. However, there are some advances in this direction[19, 20].The necessity of taking into account of waves feedback on the horizontal velocityprofile has been understood long time ago in the works of Fabrikant [21] andNikolaeva, Zyrling [22] later continued in the works of Jannsen [23] and explainedin details in the monograph [11] in the form of ”quasi-laminar” theory. Thistheory is not accomplished.To consider the theory as self-consistent even in the approximation of turbulenceabsence, it is necessary to solve equations describing horizontal velocity profile U ( z ) together with Hasselmann equation, describing energy spectrum evolution.This is not done yet.Aside that fact, many theoreticians do not share share the opinion about tur-4ulent pulsations insignificance, and consider them as the leading factor. Cor-responding T BH theory by Townsend, Belcher and Hunt [12] is alternative toquasi-laminar theory. Both theories are discussed in [24].There is another approach, not connected with experimental analysis – numeri-cal simulation of boundary atmospheric layer in the frame of empiric theories ofturbulence. It was developed in the works [13, 14, 15, 25]. Since those theoriesare insufficiently substantiated, the same relates to derived wind input terms.For all the variety of theoretical approaches of S in definitions, all of them areare ”quasi-linear”, assuming: S in = γ ( ω, φ ) ε ( ω, φ )where standard relation γ ( ω, φ ) = ρ a ρ w ωβ ( ωω p , φ )is being used. Here ω p = gu , where u is the wind speed, defined differently inindividual models. Function β is dimensionless and is growing with the growthof ωω p .However, even for the most ”aggressive” models of wind input terms the valueof β does not exceed several units, but usually 0 < β <
1. In some models(see,for example [15]) β becomes negative for the waves propagating faster thanthe wind, or under large angle with respect to the wind.Looking at multiple attempts of S in experimental definition, one should notethat all of them should be carefully critically analyzed. That criticism is notabout the integrity of measurements itself, but about the used methodology anddata interpretation correctness, and the possibility of transfer of the conclusionsmade in artificially created environment to real ocean conditions.Another significant amount of experiments, belonging to so-called ”fractionalgrowth method” category, has been performed through energy spectrum mea-surement in time and calculation of the corresponding γ through5 ( ω, φ ) = 1 ε ( ω, φ ) ∂ε ( ω, φ ) ∂t (4)Eq.(4) is, in fact, the linear part, or just two terms of the HE Eq.(1). Thismethod is intrinsically wrong, since it assumes that either advection ∂ω k ∂~k ∂ε∂~r andnonlinear S nl terms of Eq.(1) are absent together, or relation ∂ω∂~k ∂ε∂~r = S nl (5)is fulfilled.First assumption is simply not correct, since neglected terms are defining inocean conditions. The second assumption is also wrong since Eq.(5) contradictsEq.(4). Therefore, in the relation to ”fractional growth method” we are justciting the single relevant publication by Plant [26] where, it seems, author wellunderstood the scarcity of the ”fractional growth method”.As a matter of fact, the real interest present the experiments, which used mea-surements of the correlation between the speed of the surface growth and thepressure to the surface: Q ( ω ) = Re < η ( ω ) P ∗ ( ω ) > (6)Unfortunately, the number of such experiments is limited, and not all of themhave significant value for ocean phenomena description. Also, one should takeout of consideration the experiments performed in laboratory conditions.Consider, for example, the set of experiments described in [27]. These exper-imenst were performed in the wave tank of 40 m length and 1 m depth. Ex-perimentators created the wind blowing at the speed up to 16 m/sec , but theystudied only short waves no longer than 3 m , moving no faster than 1 . m/sec .Therefore, they studied the very short-wave tail of the function β in the condi-tions far from the ocean ones. The value of these measurements is not significant.6he same arguments relate to multiple and precisely performed measurements inthe Lake George, Australia [28]. The depth of this lake, in average, is about 1 m .That is why on its surface can propagate the waves not faster than 3 . m/sec .The typical wind speed, corresponding these measurements, was 8 − m/sec .Therefore, while the results of these measurements are quite interesting andcorrespond to theoretical predictions [29], obtained expression for S in is quitearguable, not only because is non-improvement to ”quasi-linear” theory, butalso being in complete contradiction with it.After critical analysis of experiments on S in measurements, only three of thosedeserve an attention. Those are the experiments by Snyder et al. [30], Hsiao,Shemdin [31] and Hasselmann, Bosenberg [32]. The experiments were performedin the open ocean and measured direct correlations of surface speed change andthe pressure.Those experiments were performed fairly long time ago, their accuracy is notquite high and scatter of data is significant. Therefore, their interpretation isquite ambiguous. Anyhow, these experiments produced two well-known formulafor β . For Snyder and Hasselmann, Bosenberg experiments: β = 0 . ξ − , ξ = ωω cos φ (7) β = 0 , ξ < β = 0 . . ξ − (9) β = 0 otherwise (10)The difference between these S in formula is significant. Comparison of windforcing performed on measured spectra [5] shows that Snyder-Hasselmann-Bosenbergform gives 5 ÷ S i n than Hsiao-Shemdin one.Furthermore, the Hsiao-Shemdin form agrees with Jeffreys theoretical model,while Snyder-Hasselmann-Bosenberg one is in disagreement with any known7heoretical models.Summing up, we can conclude that at the moment there is no solid parameter-ization of S in accepted by worldwide oceanographic community. Keeping thatfact in a mind, we decided to go our own way – not to build new theoreticalmodels and not to reconsider both old and completely new measurements of S in .For almost seventy year, counting from works of Garrett and Munk [33], physi-cal oceanography assimilated tremendous amount of experimental facts on basicwind-wave characteristics – wave energy and spectral peak frequency as a func-tions of limited fetch. Such experiments are described in works [5, 6, 7, 8].From the other side, numerical methods of Hasselmann equation 1 solution forexact term S nl and specified in advance terms S in and S diss have been improvedsignificantly. This can be done not only for duration-limited domain, but forfetch-limited domain too.Therefore, we proposed purely new pragmatic approach to definition of S in . Wehave chosen S in function in such a way that numerical solution of Hasselmannequation explains maximum amount of known field experiments.The result wasthe S in function described in details in [4] and named thereafter ZRP function.It is important to emphasize that work [4] assumed localization of energy dis-sipation in short waves. This assumption contradicts widely accepted concept,but we explain the difference in the following chapter.
3. Two scenarios of wave-breaking dissipation term: spectral peak orhigh-frequency domination?
In current section we explain why there is no need to use dissipation in thespectral peak area.The spectral peak frequency damping is widely accepted practice, and is in-cluded as an option in the operational models
W AM , SW AN and
W W
3. His-torically, it was done apparently by need.The necessity was caused by wind input function S in in Snyder form. Fast wave8nergy growth was observed in no-dissipation calculations, which didn’t matchresults of field measurements. Despite the result was obtained with the help of DIA model of S nl , it is qualitatively correct, because is also confirmed by ournumerical calculations using exact nonlinear interaction term S nl .It is shown below that Snyder wind input without dissipation gives 5 − ZRP , Tolman-Chalikov and Hsiao-Shemdin). This doesn’t mean, however, that long-wavedissipation exist, indeed. From our viewpoint, the necessity of its introduc-tion is explained by Snyder model imperfection, based on not quite accurateexperiments.We don’t see the physical reasons for energy-containing long waves breakings.Their steepness in the conditions of typically developed wave turbulence is notbig: µ = < ∇ η > / ∼ .
1, or even smaller. Because this value is very far fromlimiting steepness of Stokes wave µ S ≃ .
3, thise waves are essentially weakly-nonlinear. Besides those waves, more shorter waves inevitably develop, havingthe steepness approaching to the critical one, and those waves break. There isnevertheless no reason to expect that these waves have the same phase velocityas the energy-containing ones.Unfortunately, the theory of ”wave-breakers” is not developed yet. In our view,which we don’t consider based enough, one of the possible variants of such theorycould be the following.The primordial Euler equations for potential flow of deep fluid with free surfacehas the self-similar solution η ( x, t ) = gt F (cid:18) xgt (cid:19) (11)This solution was studied numerically in the framework of simplified M M T ( M aida − M cLaughlin − T abak ) model of Euler equations [34].In Fourier space this solution describes the propagation to high wave-numbersand returning back to dominant wave spectral peak of fat spectral energy tail,corresponding in real space to sharp wedge formation at time t = 0 and space9oint x = 0. This solution describes formation of the ”breaker”.In the absence of dissipation, this event is invertible in time. Presence of high-frequency dissipation chops off the end of the tail, just like “cigar cutter”, andviolates the tail invertability. Low and high harmonics, however, are stronglycoupled in this event due to strong nonlinear non-local interaction, and deformedhigh wave-numbers tail is almost immediately returns to the area of spectralpeak. As soon as fat spectral tail return to the area of the spectral peak, totalenergy in the spectrum diminishes, which causes settling of the spectral peakat lower level of energy. This process of ”shooting” of the spectral tail towardhigh wave-numbers, and its returning back due to wave breaking is the realreason of ”sagging down” of the energy profile in the spectral peak area, butwas erroneously associated with the presence of the damping in the area ofspectral peak.This explanation shows that individual wave-breakings studies [29], [35] are notthe proof of spectral peak damping presence.Also there is another, direct proof of the fact that the damping is localizedin the area of short waves. It is the measurements of quasi-one-dimensional”breakers” speed propagation – strips of foam, which accompany any developedwave turbulence. Those airplane experiments, recently performed by P.Hwangand his team [36, 37, 38, 39], show that wave breakers propagate 4-5 times slowerthan crests of leading waves.Based on the above discussion, we propose to use only high-frequency dampingas a basis of alternative framework of HE simulation. One can “implicitly”insert this damping very easily without knowing its analytic form via spectraltail continuation by Phillips law ∼ ω − .Replacement of high-frequency spectrum part by Phillips law is not our inven-tion. It is the standard tool offered as an option in operational wave forecastingmodels, known as the ”parametric tail”, and corresponds to high-frequency dis-sipation, indeed. For the practical definition of Phillips tail it’s necessary toknow two more parameters: coefficient in front of it and starting frequency.The coefficient in front of ω − is not exactly known, but is unnecessary to be10efined in the explicit form – it is dynamically determined from the continuitycondition of the spectrum. As far as concerns another unknown parameter –the frequency where Phillips spectrum starts – we define it as f = ω π = 1 . HE . We think that the question of finerdetails of high-frequency damping structure is of secondary importance at cur-rent stage of alternative framework development.
4. Checking of the new modeling framework against theoretical pre-dictions and field measurements
To check alternative framework for HE simulation, we performed numericaltests for waves excitation in limited fetch conditions. As it was already men-tioned, alternative framework is based on exact nonlinear term S nl in W RT form and
ZRP new wind input term: S in = γε (12) γ = 0 . ρ air ρ water ω (cid:18) ωω (cid:19) / f ( θ ) (13) f ( θ ) = cos θ for − π/ ≤ θ ≤ π/
20 otherwise (14) ω = gu , ρ air ρ water = 1 . · − (15)The coefficient 0 .
05 in front of Eq.(13) was found through carefully performednumerical experiments with different coefficient values to get the best corre-spondence with experimental data.Thus β = 0 . ω (cid:18) ωω (cid:19) / f ( θ ) (16)We have chosen β in the form of power function of frequency for the followingreasons. It is well known from various field experiments [6] that wave energy and11pectral frequency maximum dependencies on the fetch are the power functionsof fetch: ε = ε χ p (17) ω = ω χ − q (18)This observation is in excellent agreement with the fact that conservative sta-tionary Hasselmann equation ∂ω∂k ∂ε∂x = S nl (19)has two-parameter family of self-similar solutions [3, 4, 5, 6, 7, 8]: ε = χ p + q F ( ωχ q ) (20)which lead to dependencies Eqs.(17), (18).One of the p, q is free, but they are connected by relation10 p − q = 1 (21)called thereafter the ”magic law”.Analysis of field experiments [6] shows that ”magic law” is fulfilled with highaccuracy in many of them. HE Eq.(1) has self-similar solutions only if it is supplied by wind input term inpower form: β = ω s f ( θ ) (22)and self-similar substitution gives in this case q = s . In the most knownexperiments p = 1 and q = 3 /
10. That gives us the value s = 4 / β ( ωω p ) is not the power function, the Eq.(1)solution in the absence of long-wave dissipation is ”quazi-self-similar” in typicalcases. In this case p and q are slow functions of the fetch, but the ”magiclaw” Eq.(21) is still fulfilled [8]. Strictly speaking, an existence of the universal12 ZRP forcing , wind speed U=10 m/sec xg/U^20.0000.0020.0040.0060.008 E g ^ / U ^ (a) Solid line - numerical experiment,dashed line – fit by 2 . · − · xgU ZRP forcing , wind speed U=10 m/sec xg/U^2-0.50.00.51.01.52.0 d l n ( E ) / d l n ( x ) (b) Exponent p of the energy growthas the function of fetch x Figure 1for any ocean conditions expression of β ( ω, θ ) is not proved, because turbulentboundary layer is different for typical oceans, passats (trade winds) or mountaincoastal line. Therefore, the fact of explanation by ZRP expression for S in of atleast half of known field experiments can be considered as big success.Fig.1 shows that total energy is growing along the fetch by power law in accor-dance with Eq.(17) with p = 1 . q = 0 . ω −
3. Phillips high frequency tail ω − Fig.3b shows log-log derivative of the spectral curve from Fig.3a figure, whichcorresponds to the exponent of the local power law. Again, one can see the areas13
ZRP forcing , wind speed U=10 m/sec xg/U^20.00.20.40.60.81.01.2 < o m ega > U / g (a) Solid line - numerical experiment,dashed line – fit by 3 . · (cid:0) xgU (cid:1) − . . ZRP forcing , wind speed U=10 m/sec xg/U^2-0.5-0.4-0.3-0.2-0.10.0 d l n ( < o m ega > ) / d l n ( x ) (b) Exponent q of the mean frequencydependence on fetch x Figure 2corresponding to Kolmogorov-Zakharov index − −
5. Thevalue of the index to the left side from − − / ε ≃ ω − asymptotics. It was observed in all our numer-ical experiments. It is Zakharov-Filonenko spectrum, which is the solution ofequation S nl = 0 (23)It is predicted by weak-turbulent wind-wave turbulence theory and appearsroutinely in numerical experiments [2, 3, 4, 5, 6, 9], see also [42, 43, 44, 45, 46].This spectrum is confirmed by multiple ocean field [47, 48, 49, 50], wave tanks[51] and Lake George [29] measurements.The ”inverse cascade” spectrum ε ω ≃ ω − / was also predicted by weak-turbulent theory [29], [44, 45] and observed in numerical experiments [5]. Itsfield measurements, however, are less confident.In reality, nonlinear S nl term is the leading term in the ocean energy balance14 irectional energy spectrum for limited fetch -8 -6 -4 -2 l n e Fetch = 20081.08 m (a) Logarithm of spectral energy den-sity as a function of logarithm of fre-quency f = ω π - solid line. Dashedline - fit ∼ ω − , dash-dotted line - fit ∼ ω − . P o w e r i nde x Time = 20081.08 m (b) Local exponent of ω calculatedfrom the solid line Fig.3a. Figure 3[1], [2]. It consists of two parts: S nl = F k − Γ k ε k (24)which almost compensate each other. Otherwise, one can not explain persistentpresence of Zakharov – Filonenko asymptotics ε ω ≃ ω − .Fig.4 presents relation ”(10 q − p ) as a function of fetch x . It is in perfectaccordance with self-similar prediction Eq.(21).We conclude that alternative framework for HE simulation reproduces the fol-lowing analytical features of HE :1. Self-similar solutions with correct exponents2. Kolmogorov-Zakharov spectra ∼ ω − Table 1 presents results of calculation of exponents p and q (see Eqs.(17)-(18))for 14 different experimental observations with the last row corresponding tolimited fetch growth numerical experiment within alternative framework. One15 ZRP forcing , wind speed U=10 m/sec xg/U^2-2-1012 - Figure 4: Relation (10q-2p) as a function of the fetch x .can see good correspondence between theoretical, experimental and numericalvalues of p and q .
5. Tests for separation of trustworthy wind input terms from non-physical ones
As it was already discussed, there are plenty of historically developed param-eterizations of wind input terms. Analysis of nonlinear properties of HE inthe form of specific self-similar solutions and Kolmogorov-Zakharov law for di-rect energy cascade allows us to propose the set of tests, which would allowseparation of physically justified wind-input terms S in from non-physical ones.As such, we propose:1. Checking powers of observed energy and mean frequency dependenciesalong the fetch versus predicted by self-similar solutions.2. Checking the “Magic relations” Eq.(21) between exponents p and q forobserved energy and frequency dependencies along the fetch.3. Checking exponents of directional spectral energy dependencies versusKolmogorov-Zakharov exponent − HE simulations which used the followingpopular wind input terms within alternative framework:16 xperiment p q Babanin, Soloviev 1998 0.89 0.28Walsh et al. (1989) US coast 1.0 0.29Kahma, Calkoen (1992) unstable 0.94 0.28Kahma, Pettersson (1994) 0.93 0.28JONSWAP by Davidan (1980) 1.0 0.28JONSWAP by Phillips (1977) 1.0 0.25Kahma, Calkoen (1992) composite 0.9 0.27Kahma (1981, 1986) rapid growth 1.0 0.33Kahma (1986) average growth 1.0 0.33Donelan et al. (1992) St Claire 1.0 0.33Ross (1978), Atlantic, stable 1.1 0.27Liu, Ross (1980), Lake Michigan, unstable 1.1 0.27JONSWAP by Hasselmann et al. (1973) 1.0 0.33Mitsuyasu et al. (1971) 1.0 0.33ZRP numerics 1.0 0.3Table 117 E g ^ / U ^ (a) xg/U^2-0.50.00.51.01.52.0 d l n ( E ) / d l n ( x ) (b) Figure 5: Same as Fig.1, but for Chalikov S in
1. Chalikov S in term [25, 15]2. Snyder S in term [30]3. Hsiao-Shemdin S in term [31]4. W AM S in term [52]
6. Test of Chalikov wind input term
Fig.5 shows that total energy growth along the fetch significantly exceeds ob-served in
ZRP simulation, and value of the corresponding exponent significantlydeviates from theoretical value p = 1 . ZRP numerical results and corresponding self-similar exponent q = 0 . ZRP case we observe:1. Spectral maximum area2. Kolmogorov-Zakharov segment ∼ ω −
3. Phillips high frequency tail ∼ ω − Fig.7b shows log-log derivative of the energy curve from Fig.7a, corresponding tothe exponent of the local power law. Again, one can see the areas corresponding18 < o m ega > U / g (a) d l n ( < o m ega > ) / d l n ( x ) (b) Figure 6: Same as Fig.2, but for Chalikov S in to Kolmogorov-Zakharov index − −
5. The value of theindex to the left side of − − / q − p ) as a function of fetch distance x . It issurprising that it is in perfect accordance with the relation Eq.(21). It mean thatdespite incorrect values p and q along the fetch, their combination (10 q − p )still holds in complete accordance with theoretical prediction, i.e. self-similarityis fulfilled locally.
7. Test of Snyder wind input term
Fig.9 shows that total energy growth along the fetch significantly exceeds
ZRP case, but has the value of growth exponent close to p = 1 . x .Dependence of mean frequency against the fetch shown on Fig.10 is lower than ZRP numerical results, but has fairly close value to self-similar solution index q = 0 . .1 1.0ln f10 -8 -6 -4 -2 l n e Fetch = 20096.52 m (a) P o w e r i nde x Time = 20096.52 sec (b)
Figure 7: Same as Fig.3, but for Chalikov S in - Figure 8: ”Magic number” 10 q − p as a function of the fetch x for Chalikovwind input term. 20 E g ^ / U ^ (a) xg/U^2-2-1012 d l n ( E ) / d l n ( x ) (b) Figure 9: Same as Fig.1, but for Snyder S in < o m ega > U / g (a) d l n ( < o m ega > ) / d l n ( x ) (b) Figure 10: Same as Fig.2, but for Snyder S in .1 1.0ln f10 -8 -6 -4 -2 l n e Fetch = 20027.48 m (a) P o w e r i nde x Time = 20027.48 sec (b)
Figure 11: Same as Fig.3 but for Snyder S in
2. Kolmogorov-Zakharov segment ∼ ω −
3. Phillips high frequency tail ∼ ω − Fig.11b shows log-log derivative of the energy curve from Fig.11a, which cor-responds to the exponent of the local power law. Again, one can see the areascorresponding to Kolmogorov-Zakharov index − −
5. Thevalue of the index to the left side of − − / q − p ) as the function of the fetch. Again,it is in perfect accordance with the theoretical relation Eq.(21). As in Chalikovcase it means that despite not perfect values of p and q and wrong energy growthalong the fetch, their combination (10 q − p ) still holds in complete accordancewith theoretical prediction, i.e. self-similarity is also fulfilled locally in Snydercase.
8. Test of
Hsiao-Shemdin wind input term
Fig.13 shows that total energy growth along the fetch strongly underestimates
ZRP simulation, and has the asymptotic value of exponent p ≈ . ZRP results and asymptotic value of index q ≈ . - Figure 12: Relation (10 q − p ) as a function of the fetch x for Snyder wind inputterm. xg/U^20.0000.0010.0020.0030.004 E g ^ / U ^ (a) xg/U^2-2-1012 d l n ( E ) / d l n ( x ) (b) Figure 13: Same as Fig.1, but for
Hsiao − Shemdin S in xg/U^20.00.20.40.60.81.0 < o m ega > U / g (a) xg/U^2-0.5-0.4-0.3-0.2-0.10.0 d l n ( < o m ega > ) / d l n ( x ) (b) Figure 14: Same as Fig.2, but for
Hsiao − Shemdin S in Fig.15 presents directional spectrum as a function of frequency in logarithmiccoordinates. One can see:1. Spectral maximum area2. Kolmogorov-Zakharov segment ∼ ω −
3. Phillips high frequency tail ∼ ω − Fig.15b shows log-log derivative of the energy curve from Fig.15a, correspondingto the exponent of the local power law. Again, one can see the areas correspond-ing Kolmogorov-Zakharov index − −
5. The value of theindex to the left side of − − / q − p ) as the function of the fetch coordinate x .It is in total agreement with the theoretical predictions Eq.(21), which meansthat self-similarity is fulfilled locally in Hsiao − Shemdin case.
9. Test of
WAM3 wind input term
Fig.17 shows that total energy growth along the fetch dramatically underesti-mates
ZRP simulation, and has the value of exponent p asymptotically goingto 0 versus fetch coordinate x . 24 .1 1.0ln f10 -8 -6 -4 -2 l n e Fetch = 20021.75 m (a) P o w e r i nde x Time = 20021.75 m (b)
Figure 15: Same as Fig.3 but for
Hsio − Shemdin S in xg/U^2-2-1012 - Figure 16: Relation (10 q − p ) as a function of the fetch x for Hsiao − Shemdin wind input term. 25 xg/U^20.00000.00050.00100.00150.0020 E g ^ / U ^ (a) xg/U^2-2-1012 d l n ( E ) / d l n ( x ) (b) Figure 17: Same as Fig.1, but for
W AM S in Dependence of the mean frequency against the fetch shown on Fig.18 demon-strates strong discrepancy with
ZRP results and corresponding index q alsogoes to 0 asymtotically.Fig.19a presents directional spectrum as a function of frequency in logarithmiccoordinates. One can see:1. the spectral maximum area2. Kolmogorov-Zakharov segment ∼ ω −
3. Phillips high frequency tail ∼ ω − Fig.19b shows log-log derivative of the energy curve from Fig.19a, correspondingto the exponent of the local power law. Again, one can see the areas correspond-ing Kolmogorov-Zakharov index − −
5. The value of theindex to the left side of − − / q − p ) as the function of the fetch coordinate x . It is in total disagreement with the theoretical predictions. There is no anyindication of “magic relation” Eq.(21) fulfillment.Comparing the results, obtained for Snyder and W AM
W AM xg/U^20.00.20.40.60.81.01.2 < o m ega > U / g (a) xg/U^2-1.0-0.50.00.51.0 d l n ( < o m ega > ) / d l n ( x ) (b) Figure 18: Same as Fig.2, but for
W AM S in -5 -4 -3 -2 -1 l n < e > Fetch = 20142.99 m (a) P o w e r i nde x Fetch = 20142.99 m (b)
Figure 19: Same as Fig.3 but for
W AM S in xg/U^2-2-1012 - Figure 20: Combination (10 q − p ) as a function of the fetch x for W AM
10. Conclusion
We are offering alternative framework for numerical simulation of HE . Beingsupplied with ZRP wind input term, such approach reproduces the results ofmore than a dozen of experimental observations.We also performed numerical simulations of HE for four another historicallywell-known wind input terms within the same alternative framework. Theydemonstrated the results deviating from ZRP simulation.To classify the results of the above simulations we applied the set of nonlineartests to different kinds of wind input terms, and here is the conclusion:1.
ZRP forcing term perfectly corresponds to theoretically predicted resultslike Kolmogorov-Zakharov spectrum ∼ ω − , self-similar solutions for28 xperiment p -test q -test KZ -spectrum Magic relation Energy growth ZRP
YES YES YES YES YES
Chalikov
NO NO YES YES NO
Snyder ≈ ≈
YES YES NO
Hsiao − Shemdin
NO NO YES YES NO
W AM p = 1 and q = 0 . p − q = 1 and reproduces more than a dozen offield experiments. Therefore, it can serve as the benchmark.2. All wind input terms pass the test for presence of Kolmogorov-Zakharovlaw ε ∼ ω − . This means that effects of nonlinearity are so strong, thatpresumably no variation of the wind input term parameterization cansuppress it.3. Chalikov and
Hsiao − Shemdin cases fail p − and q − tests, but pass“Magic relation” (quasi-self-similarity) test.4. Snyder case “approximately” passes p − , q − and “Magic relation” tests.5. W AM KZ − spectrum test.6. None of the wind-input parameterization, except ZRP one, can correctlyreproduce experimentally observed limited fetch growth.In summary, the nonlinearity influence is so robust in the dynamics of HE thatone can’t “spoil” Kolmogorov-Zakharov law ∼ ω − for any tested wind inputterm S in . Self-similarity tests like p − and q − tests are the most sophisticatedbetween suggested ones. And the “magic relation” test is probably somewherein-between versus detection of the “quality” of particular wind input term.The summary of the tests is presented in Table 2.29
1. Acknowledgments
This research was supported by ONR grant N00014-10-1-0991, NSF grant 1130450and program of RAS presidium ”Nonlinear dynamics in mathematical and phys-ical sciencies”. The authors greatfully acknowledge the support of these foun-dations.
References [1] V. E. Zakharov, Energy balances in a wind-driven sea, Physica ScriptaT142 (2010) 014052.[2] V. Zakharov, S.I.Badulin, On energy balance in wind-driven sea, DokladyAkademii Nauk 440 (2011) 691–695.[3] V. E. Zakharov, Theoretical interpretation of fetch-limited wind-driven seaobservations, NPG 13 (2005) 1 – 16.[4] A. Pushkarev, D. Resio, V. Zakharov, Weak turbulent approach to thewind-generated gravity sea waves, Physica D 184 (2003) 29 – 63.[5] S.I.Badulin, A.N.Pushkarev, D.Resio, V.E.Zakharov, Self-similarity ofwind-driven sea, NPG 12 (2005) 891–945.[6] S. I. Badulin, A. Babanin, D. Resio, V. E. Zakharov, Weakly turbulentlaws of wind-wave growth, JFM 591 (2007) 339 – 378.[7] E. Gagnaire-Reno, M. Benoit, S. Badulin, On weakly turbulent scaling ofwind sea in simulation of fetch-limited growth, JFM 669 (2011) 178 – 213.[8] V. E. Zakharov, S. I. Badulin, P. A. Hwang, G. Caulliez, Universality of seawave growth and its physical roots, arXiv:1411.7235 [physics.ao-ph] (2014)34.[9] V. E. Zakharov, D. Resio, A. Pushkarev, New wind input termconsistent with experimental, theoretical and numerical considerations,arXiv:1212.1069 [physics.ao-ph]. 3010] I. R. Young, Wind Generated Ocean Waves, Elsevier, 1999.[11] G. J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann,P. A. E. Janssen, Dynamics and Modeling of Ocean Waves, CambridgeUniversity Press, 1994.[12] S. E. Belcher, J. C. R. Hunt, Turbulent flow over hills and waves, AnnualReview of Fluid Mechanics 30 (1998) 507–538.[13] D. V. Chalikov, V. K. Makin, Models of the wave boundary layer,Boundary-Layer Meteorology 56 (1991) 83 – 99.[14] V. N. Kudryavtsev, V. K. Makin, J. F. Meirink, Simplified model of the airflow above the waves, Boundary-Layer Meteorology 98 (2001) 155 – 171.[15] H. L. Tolman, D. Chalikov, Source terms in a third-generation wind-wavemodel, JPO 26 (1996) 2497–2518.[16] J. Miles, On the generation of surface waves by shear flows, JFM 3 (1957)185–204.[17] H. Jeffreys, On the formation of water waves by wind, Proc. Roy. Soc. 107A(1924) 189 – 206.[18] H. Jeffreys, On the formation of water waves by wind ii, Proc. Roy. Soc.107A (1925) 341 – 347.[19] T. S. Hristov, S. D. Miller, C. A. Friehe, Dynamical coupling of wind andocean waves through wave-induced air flow, Nature 422 (2003) 55 – 58.[20] Y. Troitskaya, D. Sergeev, O. Ermakova, G. Balandina, Statistical param-eters of the air boundary layer over steep water waves measured by pivtechnique, JPO 41 (2011) 1421–1454.[21] A. Fabricant, Quasilinear theory of wind-waves generation, Izv. Atmos.Ocean. Phys. 12 (1976) 858 – 862.3122] Y. L. Nikolaeva, L. S. Zimring, Kinetic model of sea wind waves generationby turbulent wind, Izv. Atmos. Ocean. Phys. 22 (1986) 135 – 142.[23] P. A. E. M. Janssen, Quasilinear approximation for the spectrum of wind-generated water waves, JFM 117 (1982) 493 – 506.[24] M. Donelan, A. V. Babanin, I. R. Young, M. Banner, Waves-follower fieldmeasurements of the wind-input spectral function. part ii: Parameteriza-tion of the wind input, JPO 36 (2006) 1672–1689.[25] D. Chalikov, The parameterization of the wave boundary layer, JPO 25(1995) 1333 – 1349.[26] W. J. Plant, A relationship between wind stress and wave slope, JGR 87(1982) 1961 – 1967.[27] C. Mastenbroek, V. K. Makin, M. H. Garat, J. P. Giovanangeli, Experi-mental evidence of the rapid distortion of turbulence in the air flow overwater waves, JFM 318 (1996) 273–302.[28] M. Donelan, A. V. Babanin, I. R. Young, M. L. Banner, C. McCormick,Wave-follower field measurements of the wind-input spectral function. parti: Measurements and calibrations, J. Atmos. Oceanic Technol. 22 (2005)799 – 813.[29] I. R. Young, A. V. Babanin, Spectral distribution of energy dissipation dueto dominant wave breaking, JPO 36 (2006) 376 – 394.[30] R. L. Snyder, F. W. Dobson, J. A. Elliott, R. B. Long, Array measurementsof atmospheric pressure fluctuations above surface gravity waves, JFM 102(1981) 1 – 59.[31] S. V. Hsiao, O. H. Shemdin, Measurements of wind velocity and pressurewith a wave follower during marsen, JGR 88 (1983) 9841 – 9849.[32] D. Hasselmann, J. Bosenberg, Field measurements of wave-induced pres-sure over wind-sea and swell, JFM 230 (1991) 391 – 428.3233] H. U. Sverdrup, W. H. Munk, Wind, sea and swell: theory of relations forforecasting (1947).[34] A.Pushkarev, V.E.Zakharov, Quasibreathers in the