THE HIGH-FREQUENCY PORTION OF WIND WAVE SPECTRA AND ITS IMPLICATIONS Don Resio & Chuck Long ERDC-CHL Vicksburg, MS

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1 THE HIGH-FREQUENCY PORTION OF WIND WAVE SPECTRA AND ITS IMPLICATIONS Don Resio & Chuck Long ERDC-CHL Vicksburg, MS

2 MOTIVATION Spectral shape provides critical information for understanding source term balance in wind wave spectra Example isequilibrium Range forms: Phillips, Kitaigorodksii ( s) E(f)~ f -5 Toba, Donelan, Belcher,etc. ( s) E(f)~uf -4 Resio, Long & Vincent ( ) E(f)~(u 2 c p ) 1/3 f -4 Forristall; Long & Resio ( ) F(f)~f -4 f -5 Impacts: Air-sea interaction Model source terms Wave set up

3 APPROACH Use data from a wide range of scales Develop physical scaling relationships Compare expected scaling behavior with data sets 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 Interpret results via individual wave kinematics Examine impacts on other area Three of the data sets use are wave Measurements at Duck, NC

4 CONCLUSIONS Spectra consistently transition from equilibrium range to an f -5 form at high frequency in all data sets Transition location is consistent with a balance between nonlinear fluxes and high-frequency dissipation Kinematic constraints suggest little or no breaking in the spectral peak region High-frequency breaking primarily due to local accelerations at high frequencies or orbital velocities exceeding c/2 This has very significant implications for wave model source terms, air-sea interactions, and wave set-up at coast

5 Phillips, 1958 Toba, E( f)~ α ugf where α u 4 E( f)~ α g f is the equilibrium range coefficient and is term with units of velocity. Resio, Long, & Vincent 2001 E( f ): α ( u c ) f 4 2 1/3 4 p Where C p = phase velocity of spectral peak Forristall, 1981 E( f ) ~ α ugf ~ α g f fˆ = ufg > const 1 for ( ).

6 Switching to wavenumber spectral basis Fk ( ) = β g k 5/2 where β is the equilibrium range coeff. k is wavenumber. In deep water: β = 1 2 α ug 4 1/2 Equilibrium form appears to transition to different form at high frequencies

7 Simple idea of energy and momentum balance (similar to Hasselmann et al 1973) 4. In equilibrium range: Γ E = ( Sin( f) + Sds( f)) 0 f where Γ E is the net flux of energy through the spectrum S ( f) is the wind input at frequency f, in S ( f) is the dissipation sink at frequency f. ds

8 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 This graph shows the importance of multiple data sets! Currituck Sound & Lake George Data u a /g 1/2 (m 1/2 )

9 Resio, Long & Vincent (2004) Fk ( ) : ( uc) k 2 1/3 5/2 p But where does the transition to a high-frequency form occur?

10 Hypothesis: If wave breaking is confined primarily to a high-frequency range then the energy loss rate in this high-frequency zone must balance the Flux of energy toward high frequencies in equilibrium range given by 3 Λβ Γ E = g 6. where Λ is a nearly constant dimensionless coefficient. Data Sources to examine this hypothesis: 1. Currituck Sound capacitance wave array; 2. Field Research Facility (FRF) Baylor gauge; 3. Field Research Facility (FRF) Waverider buoy; 4. WACSIS Baylor gauge; and 5. WACSIS Waverider buoy.

11 Figure 2. Plot of Plot of alpha v. beta for Currituck Sound data only

12 FRF Baylor gage data Plot of alpha versus beta for three different sets of wave spectra. There is an obvious scaling difference between the ocean-scale spectra and the Currituck spectra.

13 3 For a k spectral tail, the energy lost when a wave at wavenumber k breaks: b b 2 λα b 5kb Δ EL = 2 where the amount of energy lost when a wave breaks can approximated as ΔEL is the energy 9. lost when a wave at wavenumber kb breaks, and λ is the proportion of the energy that is lost in a single breaking event. 3 ~ 1/2 5/2 5 x x α k βg k Γ ~ Δ E < f > where < ds L b f b α 5 ~ > is the average frequency of breaking. β c x

14 For transition frequency in relatively deep water: Γ ds ~ α k 5 3/2 x α 17. β 14. ~ 5 cx 3 α 1/2 1/2 1/2 β f β u u 1/3 ~ p ~ ~ 0 5 g c c p p

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16 Some algebra shows that we should have the transition relative to the spectral peak frequency at a location that varies with the square root of cp : β18. ˆ ft ft = ~ f where p p fˆ is the relative frequency t c β of the transition. Note that Forristall (1981) implies f t c ~ p β

17 Traditional spectral random phase simulation in time domain: # angles # frequencies η() t = a cos( ω t + φ ) k j= 1 jk, j jk, # angles # frequencies u () t = ω a cos( ω t + φ )cos( θ ) x j j, k j j, k k k j= 1 # angles # frequencies 2 () = ωj j, k cos( ωj + φj, k) k j= 1 vt & a t with a = E( f, θδ ) fδθ jk, where η( t) is the wave surface elevation at time t, a u jk, x th th is the amplitude of the j frequency and k angle region of the spectrum ( t) is the horizontal velocity in the x direction at time t, vt &( ) is the vertical acceleration of the water surface at time t, and δ f δθ is the discrete bands of frequency and angle used in the simulation.

18 Three constraints from "individual" wave kinematics: 1. wave steepness μ = H / L< 1/ 7 2. acceleration μ = w& /( g/ 2) ratio of orbital velocity to phase velocity μ = u /(2 c) 3 max Frame Oct 2007 Frame Oct p(μ) Breaking Limit for f=f p Breaking Limit for f=2f p Breaking Limit for f=3f p p(μ 1 ) 0.02 p( μ) μ 1 u max /c p

19 Statistical framework for acceleration breaking limit: feq f f t cap b 1 φ p 2 3 φ p < a >= Q f f ( f / f ) df + Q f f df + Q f f ( f / f ) df 0 f eq f t < a > φ( f / f ) Q = Q + Q f + f f f b p t t t t From this representation we see that the f -4 tail cannot extend too high or the accelerations will become very large.

20 1 γ r =1.0 1 γ r = F(μ 2 ) F(μ 2 ) μ 2 =acceleration/(g/2) Frame Nov μ 2 =acceleration/(g/2) 1 F(μ 2 ) γ r =2.6 Cumulative Distribution Function for normalized acceleration as a function of relative peakedness for a range of upper frequencies from f p μ 2 =acceleration/(g/2)

21 Some Implications Waves propagating into a coast Wave Set-up Wave model source terms

22 Problem of transformation from waverider to Baylor gage viewed in frequency space. Dissipation at peak suggests that we need a good source term in that region of the spectrum = breaking at the spectral peak??

23 Problem of transformation from waverider to Baylor gage viewed in wavenumber space. Wavenumber similarity (?) suggests that we need a good source term in the high frequency region of the spectrum?? Note shift to k -3 form.

24 The contribution of the momentum flux into the water column to from the wave field, in the absence of winds, creates a (steady-state) slope which is dependent on the depth of water. η ΓM ΓE ~ ~ y h hc where η is the water surface elevation, y is the direction normal to a straight coast, ΓM is the rate of flux of momentum from the wave field into the water column, h is the water depth. Thus, if wave breaking or other source terms removes energy in deeper water, the setup can be considerable reduced over the case of dissipation which removes energy in shallower water.

25 3 Effect of this breaking form on wave set-up could be very large. Ratio of Momentum Loss Rates T p = 8 seconds T p =12 seconds T p =16 seconds Depth (meters) Calculated ratios of momentum lost from wave field given transition frequency equal to 3 time spectral peak

26 Possible new source term balance with improved Snl and high frequency breaking Major problem in existing models: DIA Other source terms have to be tuned to compensate

27 Jonswap Gamma = 1 Jonswap Gamma = 3.3 Jonswap Gamma = 7 DIA does not give a consistent estimate of Snl for different peakednesses (i.e. it varies with wave age)

28 Full Boltzmann Integral Comparison of DIA to WRT estimate of Snl for a measure spectrum DIA DIA directional characteristics become very garbled when directional characteristics are examined

29 WAY AHEAD: New Model TSA Replaces DIA High Frequency Breaking Replaces Distributed Breaking Solve for Wind Source for Closure

30 Questions? QUESTIONS??