Frontiers in Nonlinear Waves The Effect of Nonlinear Fluxes on Spectral Shape and the Generation

1. Frontiers in Nonlinear Waves The Effect of Nonlinear Fluxes on Spectral Shape and the Generation of Wind Waves Don Resio, Senior Scientist ERDC-CHL

2. Waves in the ocean play a critical role in coastal risk “A model should be as simple as possible … but no simpler” A. Einstein (Resio and Westerink: 2007, Physics Today)

3. Recent Wave Model Developments • 1970’s – models begin to incorporate nonlinear interaction source terms (parameterized) • 1980’s – Hasselmann et al. (1985) develops 3G models using a “detailed-balance” approach • Unfortunately, their representation for Snl is extremely flawed • This makes it difficult to get a correct balance of all 3 source terms • 1980’s – Phillips (1985) postulates that all 3 source terms are of comparable magnitude • This opens the door for any combination of “possible” source terms in wave models to be combined with Snl • The new source terms soon dominate Snl – which is in part needed due to the problems with the Discrete Interaction Approximation (DIA) • 2000’s – Today’s models have progressed very little since the 1980’s

4. Full Integral Solution Discrete Interaction Approximation (DIA) Comparison of Full Integral Solution and DIA estimates of Snl for a measured spectrum

5. Examples of different dissipation terms show how “plug and play” has become an accepted way to build models. 4 free parameters 7 free parameters no free parameters

6. Idealized processes in wave generation: APPROACH: “entia non sunt multiplicanda praeter necesstatem” Occam’s Razor Pumping [Sds] Relative Peakedness E(f)f4 Dissipation Region Nonlinear Fluxes: Action, Energy, & Momentum

7. Criticisms of KZ form of wave generation • Directional distribution is not isotropic • Separation between pumping and dissipation regions is not sufficiently large for constant fluxes to produce region of constant fluxes • Interactions are too weak relative to wind input and dissipation • Observational support is lacking (lots of Phillips spectra out there)

8. Chuck Long and I have been on a quest to fill the “missing observation” gap Linear array about 150 m north and east of pier end Baylor gage at end of pier Waverider about 5 km off coast Sled in the Sound Wave Measurements at Duck, NC

10. Characteristics of directional distributions of energy: 1. “Young” waves are very bimodal 2. “Old” waves approach unimodal 3. Both distributions are similar to Hasselmann et al. (1980) Interesting comparison: Bimodal directional spreading characteristics have similar gross amount of spreading. Low inverse wave age (old waves) – almost unimodal High inverse wave age (young waves) very bimoda Variation in “n” obtained when fitting a cos2n function compared to the data from Hasselmann et al (1980)

11. Long & Resio (2007 – JGR) Directionally Integrated spectra Note f-5 form here

12. Analyses From L&R Directional spectra

13. Toba, Belcher and others have postulated that β is linearly proportional to wind speed. This clearly does not work for multiple data sets. β x 1000 General slope Of waverider Data Currituck Sound & Lake George Data This graph shows the importance of multiple data sets in testing theories! ua /g1/2 (m1/2)

14. We found that only by allowing the phase velocity of the spectral peak to enter into the scale for β could we get all the data sets to behave in a consistent manner

15. Observed relationship between spectral peakedness and inverse wave age (Long and Resio, 2007). Relationship is not as chaotic as JONSWAP data indicated.

16. But this is part of a larger pattern showing that both k-5/2 and k-3 ranges co-exist in spectra around the world. What does this mean in terms of the dominant source terms in these regions of the spectrum?

17. (b) (a) (c) (d) Now we need to get into the details of the observations in a direct numerical manner: Constant angular spreading. No dissipation region KZ form extends much higher than upper limit of integration Figure 5. Net energy fluxes through directionally integrated spectra for varying values of n in cos2n directional form: panels a, b, c, and d have values of n equal to 1, 2, 4, and 8, respectively.

18. Results for 3 peakedness values and a Phillips spectrum starting at 4fp Since most transitions to Phillips range began in the 2fp range, this suggests that “real world” spectra should have slopes slightly steeper than f-4

19. But what about momentum?? Transition to Phillips spectrum No transition to Phillips spectrum It is strongly divergent for constant angular spreading and no transition to a Phillips spectrum and even more divergent for the case with a transition.

20. How about for a directional spectrum with angular spreading of the type actually observed in nature? Transition to Phillips spectrum No transition to Phillips spectrum

21. And the momentum fluxes are quite well-behaved, too. No transition to Phillips spectrum Transition to Phillips spectrum

22. But how does this compare to some quantitative estimates of momentum flux from the atmosphere (Note: very tough to measure in situ!!!!!), but Hasselmann inferred it from wave growth rates. Hasselmann estimate: But this is for a JONSWAP spectrum with constant angular spreading. Revised estimate for entire range of peakedness simulated.

23. As explained quite nicely by Lighthill (1962) and Kinsman (1965) for the Miles’ mechanism we have Which is consistent with our data for the equilibrium range coefficient!

24. At 0.8 fp the angular spread is quite broad. From 0.8 fp to 1.0 fp the angular spread becomes much narrower. Since Snl tends to broaden the angular distribution, this must be due to the action of the wind. From 1.0 fp to 1.8 fp the spectrum broadens at almost exactly the rate needed to create a constant momentum flux through the equilibrium range, while maintaining the (almost) constant energy flux through this region of the spectrum. Wind input dominates directional distribution Snl dominates directional distribution Our analyses do not support the existence of large external source terms in the region dominated by Snl

25. Which is similar to the pattern found in observations. Our analyses do not support the existence of large external source terms in the region dominated by Snl

26. Conclusions • This work is attempting to extend the directionally integrated concepts for nonlinear fluxes to include directional properties • Spectra with constant angular spreading produce near-constant energy fluxes through the equilibrium range, but create highly divergent momentum fluxes in this same region of the spectrum • Spectra with distributions of angular spreading based on observations produce near-constant fluxes of both energy and momentum through the equilibrium range • The calculated momentum flux moving through the equilibrium range is in good agreement with estimates of the amount of momentum entering the spectral peak region • It appears that Snl controls spectral densities in the equilibrium range and is able to balance a “Miles” input source function for the wind – producing self-similar growth • Current “external” source terms are inconsistent with obs. • We need a different approach to model validation