Wind-Generated Waves

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Chapter 1

Wind-Generated Waves

Ian Young

University of Melbourne, Parkville VIC 3010, Australia

1.1. Introduction

As this book considers the dynamics of water waves on the interface between air and water (the ocean in this context), it is necessary to define how one can describe these waves. The simplest approach is to consider that such waves can be approximated by a two-dimensional sinusoidal form. That is, waves which have infinitely long crests with the surface elevation, η is defined by

η = a sin(kx− ωt), (1.1)

where a is the wave amplitude, k = 2π/L is the wave number, L is the wave length, ω = 2π/T is the frequency, T is the wave period, and x and t are space and time, respectively. Small amplitude or linear wave theory [Airy, 1945] can be used to deﬁne relationships between quantities such as the phase speed of the waves, C, orbital motions beneath the surface and k and ω. A full description of linear wave theory is not contained here and the reader is referred to one of the many texts which cover this subject [e.g., Holthuijsen, 2007; Young, 1999].

1.2. Frequency or Omni-Directional Spectrum

Although linear wave theory provides a useful mathematical con-text to consider ocean surface waves, its applicability to real ocean

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Fig. 1.1. Example of a wave record, recorded at a location in the Southern Ocean.

situations is questionable. Figure 1.1 shows a time series of water surface elevation measured at a wave buoy in the Southern Ocean. Although this is a case of extremely large waves, it is typical of wind-generated ocean waves. Although there is clearly a dominant wave period, there is marked variability in both the amplitude and period of individual waves. That is, the wave record does not conform to the sinusoidal form described by (1.1).

It is common within many areas of physics to approximate wave records such as that shown in Fig. 1.1 by the use of a spectral or Fourier model. Under this model, the water surface is represented by the linear summation of sinusoidal components of the form described by (1.1): η(t) = N i=1 aisin(ωit + φi), (1.2)

where ai, ωi and φi are the amplitude, frequency and phase of

com-ponent i, respectively. Provided enough comcom-ponents are included in the summation in (1.2), almost any water surface can be approxi-mated by the summation. The power of this approach is that each of

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the components in the summation will satisfy the properties of linear wave theory.

It is often more convenient to consider the energy of the wave per unit crest length E, rather than the amplitude a. From linear wave theory, this energy is as follows:

E = 1

8ρwgH

2_L, _(1.3)

where H = 2a is the wave height and ρw is the density of water. The

energy per unit area or speciﬁc energy, ¯E is as follows:

¯

E = 1

8ρwgH

2_. _(1.4)

From (1.4) and (1.2), the average energy of the wave proﬁle is as follows: ¯ E = ρwg 8N N i=1 H_i2 (1.5) or ¯ E ρwg = 1 2N N i=1 a2_i = σ2, (1.6)

where σ2 is the variance of the water surface elevation record. It fol-lows from (1.6) that the amplitude components a2_i are related to the energy of the wave record and that one could display the distribution of this energy with frequency by plotting a2_i versus ωi. Such a plot

is termed a discrete amplitude spectrum, being deﬁned only at the discrete values of the Fourier summation.

In the limit, as N → ∞, the amplitude spectrum can be trans-formed into a continuous spectrum, F (f ), where

F (f )Δf = a

2

i

2 . (1.7)

The spectrum F (f ) is called the frequency spectrum or omni-directional spectrum (as no direction is associated with the spectrum)

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or variance [as the integral of the spectrum is the variance from (1.6)] spectrum: σ2 = _∞ 0 F (f )df . (1.8) 1.3. Directional Spectrum

The spectral form F (f ) acknowledges that the water surface is com-posed of components across a range of frequencies; however, it does not consider that these components could propagate in a variety of directions. An extension of (1.2) to account for direction would take the form

η(x, y, t) =

N

i=1

aisin[ki(x cos θi+ y sin θi)− ωit + φi], (1.9)

where θiis the angle between the x-axis and the direction of

propaga-tion of component i. In a similar manner to the frequency spectrum

F (f ), Eq. (1.9) allows a directional spectrum F (f, θ) to be deﬁned as σ2= _2π 0 _∞ 0 F (f, θ)dfdθ. (1.10)

The directional frequency spectrum F (f, θ) deﬁnes the distribu-tion of energy as a funcdistribu-tion of frequency and direcdistribu-tion. It follows from (1.9) that noting that wave number is a vector, the energy could also have been described in terms of a wave number spectrum Q(kx, ky) or

Q(k, θ). Then the variance in a similar fashion to (1.10) is as follows: σ2=

Q(kx, ky)dkxdky =

F (f, θ)dfdθ. (1.11) Noting that dkxdky =|k|dkdθ, the spectral forms can be related as

F (f, θ) = Q(kx, ky)|k|

dk

df, (1.12)

and the quantity dk/df can be determined from linear wave theory.

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1.4. Statistical Properties of Ocean Wave Heights

In Eq. (1.2), the statistical variability of the water surface elevation is modeled using a Fourier series. Assuming the Fourier amplitudes

ai (or spectrum) are narrow banded, it can be shown that the

prob-ability density function (pdf) for individual wave heights follows a Rayleigh distribution [Rice, 1954].

p(H) = H

4σ2e

−H2_/8σ2

, (1.13)

where the pdf must satisfy the requirement p(H)dH = 1.

Longuet-Higgins [1952] derived relationships for a number of char-acteristic wave heights based on (1.13):

Mean wave height: ¯

H =

Hp(H)dH =√2πσ2_, _(1.14) Root mean square (rms):

H_rms2 =

H2p(H)dH = 8σ2. (1.15) From (1.15) and (1.13), the Rayleigh distributions can be written as

p(H) = 2H H2

rms

e−H2/Hrms2 _. _(1.16)

The cumulative form of (1.16), i.e., the probability that H is greater than ˆH, is given by P (H > ˆH) = _∞ ˆ H p(H)dH = e−H2/Hrms2 _. _(1.17)

The average height of all waves greater than ˆH is given by

¯ H( ˆH ) = _∞ ˆ H H2e−(H/Hrms) 2 dH _∞ ˆ H He−(H/Hrms) 2 dH . (1.18)

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Table 1.1. Representative waves calculated from the Rayleigh distribution [after Young, 1999]. N H1/N/σ H1/N/ ¯H H1/N/Hrms Comments 100 6.67 2.66 2.36 50 6.24 2.49 2.21 20 5.62 2.24 1.99 10 5.09 2.03 1.80 Highest 1/10 wave 5 4.50 1.80 1.59 3 4.00 1.60 1.42 Significant wave 2 3.55 1.42 1.26 1 2.51 1.00 0.87 Mean wave

Rather than wanting to know the average height of all waves greater than a speciﬁed value, it is more common to want to know the average height of the highest 1/N waves. This can also be determined from the Rayleigh distribution [Goda, 1985]. Following Young [1999], typical values are given in Table 1.1.

From Table 1.1, it follows that the wave whose height is equal to the average of the highest 1/3 of the waves, is also equal to 4 times the variance of the record. This value is called the signiﬁcant wave height Hs and is often used as a representative wave height for a

record:

Hs= 4σ≈ H1/3. (1.19) Similarly, it follows that the highest 1% of waves is given by H1 ≈ 1.66Hs and it can also be shown that the maximum wave height is

Hmax≈ 2Hs. Therefore, the signiﬁcant wave height can be calculated

either from a time-domain analysis (e.g., H_1/3) or from the integral of the spectrum (1.10) (e.g., Hs).

1.5. Spectral Analysis of Recoded Data

A number of instruments can be used to measure the water sur-face elevation at a discrete sampling interval Δt. A good summary of approaches can be found in Tucker [1991]. Assuming the water

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surface elevation has been measured, following (1.2) it can be repre-sented as η(t) = _∞ −∞X(f )e iωt_dt, _(1.20)

where i =√−1 and X(f) is the Fourier transform of η. Noting that

eiθ = cos θ + i sin θ, Eq. (1.20) can be expressed as

η(t) =

_∞

−∞X(f )[cos ωt + i sin ωt]dt, (1.21)

which is the same form as in (1.9). As η(t) is a real quantity, the integral in (1.21) must also be a real quantity. Hence, the Fourier transform X(f ) must, in general, be a complex quantity. It follows that X(f ) = _∞ −∞η(t)e −iωt_dt _(1.22) or X(f ) = _∞ −∞η(t)[cos ωt− i sin ωt]dt. (1.23)

The complex nature of X(f ) signiﬁes that the sinusoidal spectral component has both an amplitude ai and a phase φi. As the water

surface is recorded as discrete values over a ﬁnite period of time, it is convenient to convert the integral in (1.23) to a discrete summation:

X (n/Tr) =f = Tr N =dt N−1 j=0 η(jTr/N )[cos(2πjn/N )− i sin(2πjn/N)], (1.24) where N is the number of points in the discrete time series, Tr is

the length of the time series and n is a counter which ranges from 0 to N − 1. With the Fourier transform (1.24) deﬁned, the frequency spectrum is as follows:

F (f ) = 2

Tr|X(f)|

2_. _(1.25)

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The modulus of the Fourier transform in (1.25) means that all phase information in the Fourier approximation to the water surface is lost. Therefore, the spectrum contains no phase information. As will be outlined in Chapter 6, this means that models which pre-dict the spectrum are often termed phase-averaged models, whereas models which represent the actual water surface elevation record are commonly termed phase-resolving models (Chapter 7).

Figure 1.2 shows the spectrum obtained from Fourier analysis of the time series shown in Fig. 1.1. Based on the integral of the spectrum, the signiﬁcant wave height is given by Hs= 12.18 m. It is

interesting to note that the highest crest to trough height in the time series is approximately 21 m. Noting that H₁ ≈ 1.66Hs, this yields a

value of 21.2 m in good agreement with the observed largest wave in the record.

Fig. 1.2. The frequency spectrum of the time series as shown in Fig. 1.1.

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1.6. Fetch-Limited Growth

A basic understanding of the important factors governing wind wave growth can be obtained from the study of one of the simplest wave evolution cases: fetch-limited growth. In fetch-limited growth, a con-stant wind U10 blows for a long period (such that it reaches steady state) perpendicular to an infinitely long straight coastline. The evo-lution of the wave spectrum is then studied with distance offshore: the fetch x. Based on a suggestion by Olson [1943], Sverdrup and Munk [1947] and Kitaigorodskii [1962, 1973] assumed that the fol-lowing variables should define the situation: the fetch length x; the wind speed U₁₀; gravitational acceleration g; the variance of the water surface elevation σ2 = H_s2/16 (1.19); and the peak frequency of

the spectrum fp. Dimensional analysis yields three non-dimensional

groupings of these quantities:

ε = σ 2_g2 U₁₀4 (non-dimensional energy), (1.26) ν = fpU10 g (non-dimensional frequency), (1.27) χ = gx U2 10 (non-dimensional fetch). (1.28) There have been numerous ﬁeld and laboratory studies aimed at determining ε = f₁(χ) and ν = f₂(χ), where f₁ and f₂ denote the functional dependence.

Young [1999] considers the results from a number of these studies, which include:

• the pioneering work of Sverdrup and Munk [1947] and

Bretschnei-der [1952, 1958], who developed the so-called SMB curves;

• the studies of fully-developed or long fetch asymptotic limits by

Pierson and Moskowitz [1964];

• the JONSWAP studies of Hasselmann et al. [1973]; • measurements in the Bothnian Sea by Kahma [1981];

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Fig. 1.3. Data from a number of fetch-limited studies showing the development of the non-dimensional energy ε as a function of the non-dimensional fetch χ. After Young [1999].

• Lake Ontario measurements by Donelan et al. [1985];

• measurements in the North Atlantic by Dobson et al. [1989].

Young [1999] combined these results and a variety of other datasets into composite diagrams relating the non-dimensional vari-ables. Figure 1.3 shows the combined datasets for ε versus χ and Fig. 1.4 shows the corresponding results for ν versus χ.

These results show that the functions f₁ and f₂are well described by power laws, ﬁnally asymptoting to a constant value at large

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Fig. 1.4. Data from a number of fetch-limited studies showing the development of the non-dimensional peak frequency ν as a function of the non-dimensional fetchχ. After Young [1999].

non-dimensional fetch, χ. Young [1999] approximated the composite dataset by ε = max (7.5± 2.0) × 10−7χ0.8, (3.6± 0.9) × 10−3, (1.29) ν = max (2.0± 0.3)χ−0.25, (0.13± 0.02). (1.30)

Equations (1.29) and (1.30) are shown in Fig. 1.5.

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Fig. 1.5. Growth law relationships given by (1.29) (top) and (1.30) (bottom). The shaded regions show the typical spread of published results and give an indication of the accuracy of these relationships. After Young [1999].

1.7. Spectral Evolution

Although evolution of integral parameters of the spectrum, such as

Hs and fp (ε and ν), provide valuable insight, it is often more

infor-mative to understand how the full spectrum F (f ) evolves as a func-tion of fetch, x.

As can be seen from Fig. 1.2, the high frequency face of the one-dimensional frequency spectrum can be approximated by a power law of the type F (f )∝ f−n. Phillips [1958] assumed that this region was controlled by gravity such that dimensional analysis yields

F (f )∝ g2f−5. (1.31) Toba [1973] assumed that the wind speed as characterized by the friction velocity u_∗ was also important, dimensional analysis

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yielding

F (f )∝ gu_∗f−4. (1.32)

1.7.1. JONSWAP form: f−5

Based on the high-frequency spectral form proposed by Phillips [1958], Hasselmann et al. [1973] considered data obtained from the Joint North Sea Wave Project (JONSWAP), proposing a spectral form F (f ) = αg2(2π)−4f−5 −5 4 f fp ₋₄ Pierson–Moskowitz spectrum · γexp » −(f−fp)2 2σ2f2_p – , (1.33) where σ = σa for f ≤ fp, σb for f > fp. (1.34) Figure 1.6 shows a typical spectral form generated by (1.33). This relationship has ﬁve free parameters. The parameters α and fp are

scale parameters. The parameter α defines the overall scale of the spectrum and was first proposed by Phillips [1958] and fp defines

the peak of the unimodal spectral form. The remaining three param-eters are shape variables. The peak enhancement parameter γ deﬁnes the ratio of the maximum spectral energy to the maximum spec-tral energy of the corresponding Pierson–Moskowitz [Pierson and Moskowitz, 1964] form. The parameters σa and σb are the left and

right spectral width parameters, respectively. These shape param-eters deﬁne the shape of the spectral peak region. They have lit-tle inﬂuence on the spectrum, as one moves away from this region. At high frequency (e.g., f > 3fp), the spectrum reverts to a form

F (f )≈ f−5.

As part of the JONSWAP experiment, Hasselmann et al. [1973] parameterized the spectral parameters α, γ, σ in terms of the non-dimensional peak frequency ν (1.27). Through (1.30), these rela-tionships could also be expressed in terms of the non-dimensional

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Fig. 1.6. Comparison between proposed forms for the one-dimensional frequency spectrum. JONSWAP (1.33) (dashed line) and Donelan et al. [1985] form (1.36) (solid line).

Fig. 1.7. JONSWAP data [Hasselmann et al., 1973] showing the relationship betweenα and non-dimensional peak frequency ν. The solid line shows the form (1.35).

fetch χ (1.28). Figures 1.7–1.9 show the JONSWAP data for these parameters.

Hasselmann et al. [1973] found no consistent relationship within the data scatter for the shape parameters, adopting their mean

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Fig. 1.8. JONSWAP data [Hasselmann et al., 1973] showing the relationship betweenγ and non-dimensional peak frequency ν. The solid line shows the mean valueγ = 3.3.

Fig. 1.9. JONSWAP data [Hasselmann et al., 1973] showing the relationship betweenσ_a,σ_band non-dimensional peak frequencyν. The solid lines show the mean values σa= 0.07 and σb= 0.09.

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values, γ = 3.3, σa = 0.07 and σb = 0.09. This is not surprising

as these are notoriously difficult parameters to determine accurately from a curve fit to a spectrum defined at only discrete values of frequency. For α, they obtained the following relationship (Fig. 1.7):

α = 0.033ν0.67. (1.35)

1.7.2. The Toba form: f−4

As shown by (1.32), Toba [1973] proposed an alternative high frequency spectral form proportional to f−4. Utilizing this form, Donelan et al. [1985] proposed a spectral formulation similar to JONSWAP F (f ) = βg2(2π)−4f_p−1f−4 − f fp ₋₄ · γexp » −(f−fp)2 2σ2f2_p – d . (1.36)

Based on a combination of both ﬁeld and laboratory data, Donelan

et al. [1985] proposed the relationships for the parameters

ε = 6.365× 10−6ν−3.3, (1.37) β = 0.0165ν0.55, (1.38) γd= 6.489 + 6 log ν, ν≥ 0.159, 1.7, ν < 0.159, (1.39) σ = 0.08 + 1.29× 10−3ν−3. (1.40) The Donelan et al. [1985] data relating β and ν are shown in Fig. 1.10 and relating γd and ν in Fig. 1.11. As shown by (1.37)–

(1.40), Donelan et al. [1985] found consistent relationships for all parameters with much less scatter than Hasselmann et al. [1973]. This may be because (1.36) is a more appropriate spectral form than (1.33) or simply that the data quality is superior for the Donelan

et al. [1985] experiment.

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Fig. 1.10. The data of Donelan et al. [1985] showing the dependence of β on ν. The solid line is (1.38).

Fig. 1.11. The data of Donelan et al. [1985] showing the dependence ofγ on ν. The solid line is (1.39).

1.8. Directional Spreading

The directional spectrum (1.10) is often represented in terms of the one-dimensional spectrum, F (f ) [Longuet-Higgins et al., 1963], as follows:

F (f, θ) = F (f )D(f, θ). (1.41)

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The directional spreading function D(f, θ) is deﬁned by

D(f, θ)dθ = 1. (1.42)

A number of diﬀerent forms have been proposed for D(f, θ). The most detailed data was compiled by Donelan et al. [1985] who pro-posed the form

D(f, θ) = 0.5βssech2 βs[θ− θm(f )], (1.43)

where θm is the mean wave direction at frequency f and βs is

given by βs= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2.61 f fp 1.3 for 0.56 < f /fp < 0.95, 2.28 f fp _−1.3 for 0.95 < f /fp < 1.60, 10{−0.4+0.8393 exp[−0.567 ln(f/fp)2]} _{for f /f} p > 1.6. (1.44) The ﬁrst two branches of the relationship (1.44) were obtained from the Donelan et al. [1985] wave staﬀ data. The relationship for

f /fp > 1.6 was obtained from stereo-photographic data by Banner

[1990].

Equations (1.44) deﬁne directional spreading which is narrowest at frequencies near the spectral peak and broadens at frequencies both above and below this value.

1.9. Spectral Energy Balance

The original studies of spectral shape used dimensional arguments [Phillips, 1958; Toba, 1973]. A major development in understand-ing of the processes responsible for the evolution of the spec-trum was brought about by the work of Hasselmann [1969] and

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Hasselmann et al. [1973]. It can be assumed that the evolution of the directional spectrum can be deﬁned by the total derivative

DF (f, θ)

Dt = Stot, (1.45)

where, for deep water, the total source term Stot is given by

Stot = Sin+ Snl+ Sdis, (1.46) and

S_in≡ atmospheric input from the wind

S_nl≡ nonlinear interaction between spectral components S_dis≡ dissipation due to whitecap wave breaking.

It is the balance between these terms which give rise to a spec-trum of the general form shown in Fig. 1.6. Numerous ﬁeld mea-surements have shown that wind-generated spectra (as opposed to swell) consistently follow a form similar to that shown in Fig. 1.6, i.e., a spectrum which is unimodal with a high frequency face≈ f−n. The peak of the spectrum gradually “downshifts” to lower frequen-cies with increasing non-dimensional fetch (i.e., fetch and/or wind speed increasing). Although, all the source terms in (1.46) play a role in this evolution, it is the nonlinear term which is principally responsible for this consistent spectral form. Young and van Vledder [1993] showed using numerical experiments that S_nl has a “shape stabilizing” role, continually forcing the spectrum back to this form. Even in cases of strongly turning winds, the nonlinear term shapes the spectrum towards this standard spectral form. However, should wave components become separated too far in either frequency or direction space, the nonlinear coupling can breakdown and separate swell systems can develop, largely independent of the wind-generated waves in the sea (spectrum).

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