Frontiers in Nonlinear Waves

The Effect of Nonlinear Fluxes on Spectral Shape and the Generation

of Wind Waves

Don Resio, Senior ScientistERDC-CHL

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)

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

in ds in ds nlS S S S S

f(Hz)

angl

e(d

egre

es)

0.4 0.6 0.8 10

50

100

150

3.1E-072.7E-072.3E-071.9E-071.5E-071.1E-077.1E-083.1E-08

-9.3E-09-5.0E-08-9.0E-08-1.3E-07-1.7E-07

f(Hz)

angl

e(d

egre

es)

0.4 0.6 0.8 10

50

100

150

2.4E-061.9E-061.4E-069.3E-074.2E-07

-8.6E-08-5.9E-07-1.1E-06

Full Integral Solution

Discrete Interaction Approximation(DIA)

Comparison of Full Integral Solution and DIA estimates of Snl for a measured spectrum

3

4

673

4

12

21 5

2

ds

"Quasi-linear" form

,( )

Alves-Banner "saturation" form

( , )

Flux form

S ( )

a

ads

PM

z zz

zds

m

nl

S a k ka

zS z z F k

z

S

4 free parameters

7 free parameters

no free parameters

Examples of different dissipation terms show how “plug and play” has become an accepted way to build models.

43/ 2

44 43

4

We want the compensated spectrum to look something like:

2( ) exp( ) 1

(2 )

where

is the equilibrium range constant as defined in Resio et al. (2004)

z is a constant =

p

g fE f f z

f

r r

4

2

4

for ; 1 for

is the relative peakedness as defined in Long and Resio (2007)

is a peakedness factor given by

( )

2

with values the same as for the JONSWAP sp

p p

r

p

p

f f f f

f f

f

ectrum,

up to the point where wave breaking becomes dominant.E(f)f4

Pumping[Sds]

Nonlinear Fluxes:Action, Energy,& Momentum

DissipationRegion

Idealized processes in wave generation:

APPROACH: “entia non sunt multiplicanda praeter necesstatem”Occam’s Razor

Relative Peakedness

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)

Linear array about 150 m north and east of pier end Baylor gage at end of pier

Waverider about5 km off coast

Sled in the Sound

Wave Measurements at Duck, NC

Chuck Long and I have been on a quest to fill the “missing observation” gap

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)

Characteristics of directional distributions of energy: 1. “Young” waves are very bimodal2. “Old” waves approach unimodal3. Both distributions are similar to Hasselmann et al. (1980)

Interesting comparison:Bimodal directional spreading characteristics have similar gross amount of spreading.

Long & Resio(2007 – JGR)Directionally

Integratedspectra

Note f-5 form here

AnalysesFrom L&RDirectional

spectra

ua /g1/2 (m1/2)

β x 1000

Toba, Belcher and others have postulated that β is linearly proportional to wind speed. This clearly does not work for multiple data sets.

Currituck Sound &Lake George

Data

General slopeOf waverider

Data

This graph shows the importance of multiple data sets in testing theories!

4( )E f f 4( ) ~E f f

4( )E f f

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

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

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?

Now we need to get into the details of the observations in a direct numerical manner:

2

2 1

1

2

0

[ ( , ) ( , )]f

net netE E in ds

f

S f S f dfd

(a) (b)

(d)(c)

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.

Constant angular spreading.

No dissipation region KZ form extends much higher than upper limit of integration

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

But what about momentum??

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.

No transition to Phillips spectrum Transition to Phillips spectrum

How about for a directional spectrum with angular spreading of the type actually observed in nature?

No transition to Phillips spectrum Transition to Phillips spectrum

And the momentum fluxes are quite well-behaved, too.

No transition to Phillips spectrum Transition to Phillips spectrum

2*ret a

eq retw

u

g

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.

2*3 ret a

eq retw

u

g

Hasselmann estimate:

But this is for a JONSWAP spectrum with constant angular spreading.

Revised estimate for entire range of peakedness simulated.

+ 6eq

+ 6atm

For a typical Currituck spectrum:

1.1 10 ( )

For a Garrett drag law and 13 m/s wind speed:

1.3 10 ( )

m

m

As explained quite nicely by Lighthill (1962) and Kinsman (1965) for the Miles’ mechanism we have 2

*ret a p netin p in E

w

u cE c

g

2 23* *ret a p pnet

E nl nlw

u c u c

g g g

Which is consistent with our data for the equilibrium range coefficient!

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

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

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