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

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The Effect of Nonlinear Fluxes on

Spectral Shape and the Generation

of Wind Waves

Don Resio, Senior Scientist

ERDC-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

Discrete Interaction Approximation

(DIA)

Comparison of Full Integral Solution

and DIA estimates of Snl for a

measured spectrum

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

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

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)

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

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)

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)

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.

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?

(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.

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

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.

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

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

No transition to Phillips spectrum

Transition to Phillips spectrum

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.

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!

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

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