The form of the asymptotic depth-limited wind-wave spectrum: Part III — Directional spreading

doi:10.1016/j.coastaleng.2009.09.001

Coastal Engineering

Volume 57, Issue 1, January 2010, Pages 30-40

https://doi.org/10.1016/j.coastaleng.2009.09.001 Get rights and content

Abstract

The directional spreading of both the wavenumber and frequency spectra of finite-depth wind generated waves at the asymptotic depth limit are examined. The analysis uses the Wavelet Directional Method, removing the need to assume a form for the dispersion relationship. The paper shows that both the wavenumber and frequency forms are narrowest at the spectral peak and broaden at wavenumbers (frequencies) both above and below the peak. The directional spreading of the wavenumber spectrum is bi-modal above the spectral peak. In contrast, the frequency spectrum is uni-modal. This difference is shown to be the result of energy in the wind direction being displaced from the linear dispersion shell. A full parametric relationship for the directional spreading of the wavenumber spectrum is developed. The analysis clearly shows that typical dispersion relationships are questionable at high frequencies and that such effects can be significant. This result supports greater attention being focussed on the routine recording of wavenumber spectra, rather than frequency spectra.

Introduction

The asymptotic depth-limited spectrum is the limiting form which results when a constant wind blows over a large body of uniformly finite-depth water. The resulting spectrum is hence a balance between atmospheric input, bottom friction, breaking dissipation and various nonlinear processes (3-wave and 4-wave). In essence, this spectral form is the finite-depth analogy of the deep-water asymptote represented by the Pierson–Moskowitz form (Pierson and Moskowitz, 1964). The finite-depth asymptotic form is of importance for a number of reasons. As the asymptotic form, it is the limiting design condition in finite-depth situations and is hence of engineering significance. As, by definition, all source terms (input, dissipation, bottom friction and nonlinear interaction) are in balance, this spectral form provides valuable indirect information on the physics of wind-wave generation in finite-depth conditions. Finally, as will become obvious in this paper, nonlinear processes become important in these situations and hence the spectral form gives insight to the importance of such processes in intermediate-depth conditions.

There are numerous observations of finite-depth spectra reported in the literature. The vast majority of these represent shoaling conditions, where deep-water waves have propagated into areas of finite depth. Observations of locally-generated waves in finite-depth conditions are much rarer. The classic data sets from Lake Okeechobee, USA (Thijsse, 1949, U.S. Army Corps of Engineers, 1955, Bretschneider, 1958) considered the asymptotic limits of integral parameters such as total energy (wave height) and period. The more recent studies have largely considered the data sets taken in the mid 1990s at Lake George, Australia (Young and Verhagen, 1996a, Young and Verhagen, 1996b, Young et al., 1996, Resio et al., 2004). These studies, together with data from Lake Ijssel, The Netherlands (Bottema, 2007, Bottema and van Vledder, 2009) have considered the form of the full frequency spectrum, including some analysis of directional spreading.

This paper is the third in a series which analyses data taken with a high resolution spatial array in Lake George in 1997 and 1998. Young and Babanin (2006) (henceforth called Part I) considered the asymptotic limits for total energy (significant wave height) and peak frequency, as well as developing a parametric form for the one-dimensional frequency spectrum. This result was extended in Young and Babanin (2009) (henceforth called Part II) where they used the Wavelet Directional Method to study the one-dimensional wavenumber spectrum. This paper extends the analysis of Part II to consider the full directional spectrum in both wavenumber and frequency forms.

The arrangement of the paper is as follows. Section 2 provides a brief overview of observations of the directional wave spectrum, with particular reference to finite-depth conditions. Section 3 summarises the present experimental configuration and the data used in the analysis. This is followed in Section 4 by a description of the Wavelet Directional Method (MDM) which is used to analyse the present data set. Initial observations of the directional spreading are made in Section 5, which is followed by a detailed analysis in Section 6. The results are discussed in detail in Section 7 and conclusions are drawn in Section 8.

Section snippets

Directional spreading function

It is common practice to consider the directional frequency spectrum, E(f, θ) or directional wavenumber spectrum, F(k, θ), where f is frequency, k = |k̲| is the modulus of the wavenumber vector and θ is direction (Longuet-Higgins et al., 1963) in the form: $E (f, θ) = E (f) D (f, θ)$ $F (k, θ) = F (k) D (k, θ)$ where E(f) is the one-dimensional frequency spectrum, F(k) is the one-dimensional wavenumber spectrum and D is a directional spreading function, defined such that $\int_{0}^{2 π} D (k, θ) d θ = \int_{0}^{2 π} D (f, θ) d θ = 1$

An alternative

Experimental setup and available data

Wave data were collected in Lake George in south-eastern Australia. This site has been well documented in the previous studies of Young and Verhagen, 1996a, Young and Verhagen, 1996b, Young et al., 1996. The measurements were made from a platform on the eastern shore of the lake. This experimental site and the instrumentation and data recorded for this experiment have been reported in detail in Young et al. (2005), Part I and Part II. A broad range of environmental parameters were recorded as

The wavelet directional method

The most obvious way to measure the directional wavenumber spectrum, F(k̲), where k̲ is the two-dimensional wavenumber vector is to digitize the two-dimensional spatial domain (x, y). This can be done by stereo-photography (Holthuijsen, 1983, Banner et al., 1989) or indirectly through a remote-sensing technique (e.g. Alpers et al., 1981, Young et al., 1985, Hasselmann et al., 1985, Walsh et al., 1989). These approaches have either logistical difficulties (stereo-photography) or rely on still

Observations of the directional spreading function

As outlined in Parts I and II, the data considered in this paper were recorded over a period of approximately 1 year from September 1997 to October 1998. A total of 55 records were analysed using the WDM. All of these records could be classified as intermediate water depth (k_pd between 0.7 and 1.99) with approximately half at the asymptotic limit to growth defined in Part I.

Fig. 1 shows a typical example of the resulting directional spectra. The case shown is for c061323.oc8, as defined in Part

Data analysis and parametric representation

Noting that the wavenumber directional spreading factor does not conform to a common parametric form (e.g. cos² or sech²), the shape independent measure of spreading proposed by Babanin and Soloviev (1998) $A^{- 1} (k) = \int_{0}^{2 π} K (k, θ) d θ$ was applied to the full data set. Fig. 5 shows both A(k) and A(f) as functions of k/k_p and f/f_p, respectively. Noting that a large value of A represents narrow spreading, Fig. 5 confirms that both spectra are narrowest at the spectral peak and increase in width for

Discussion of results

The results presented above raise a number of interesting questions, which need to be considered. The first of these is how is it possible for the directional wavenumber spectrum to be bi-modal, whilst the directional frequency spectrum is uni-modal? Clearly, if there is a linear dispersion relationship relating wavenumber and frequency, the shapes of the spreading should be similar. If, however, some process results in a more complex mapping between wavenumber and frequency, then it is

Conclusions

The results of this paper have provided a description of the directional spreading of both the wavenumber and frequency spectra of the asymptotic depth-limited wind-wave spectrum. There are significant differences in the spreading of these two forms. The wavenumber spectrum is bi-modal, consistent with previous measurements. In contrast, the frequency spectrum is uni-modal. This apparent conflict is explained by energy in the wind direction at high frequencies being displaced from the linear

Acknowledgements

The authors gratefully acknowledge the financial support of the U.S. Office of Naval Research (grants N00014-97-1-0234, N00014-97-1-0277 and N0014-97-1-0233) and the Australian Research Council (grant A00102965). We also express our gratitude to the staff and students of the School of Civil Engineering of the Australian Defence Force Academy: Jim Baxter, Karl Shaw, Ian Shephard and Michael Wilson who offered highly professional and prompt responses to all urgent demands during the experiments.

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Directional spreading is an active field of research and there have been many advances since the classical measurements of Mitsuyasu et al. (1975), Hasselmann et al. (1980), and Donelan et al. (1985). There are now a variety of techniques (Fourier Expansion Method, Maximum Entropy Method, Maximum Likelihood Method, and Wavelet Directional Method) to extract the directional spectra and accuracy will depend upon the method chosen (Young, 1994; Alves and Melo, 1999; Young, 2010). In addition, there is now strong evidence that fetch-limited waves in deep water are bimodal for frequencies higher than the peak frequency (Young et al., 1995; Ewans, 1998; Alves and Melo, 1999; Hwang et al., 2000).
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View full text

The form of the asymptotic depth-limited wind-wave spectrum: Part III — Directional spreading

Abstract

Introduction

Section snippets

Directional spreading function

Experimental setup and available data

The wavelet directional method

Observations of the directional spreading function

Data analysis and parametric representation

Discussion of results

Conclusions

Acknowledgements

Coast. Eng.

Appl. Ocean Res.

Coast. Eng.

Coast. Eng.

On the detectability of ocean surface waves by real and synthetic aperture radar

J. Geophys. Res.

Variability of directional spectra of wind-generated waves, studied by means of wave staff arrays

Mar. Freshw. Res.

Equilibrium spectra of wind waves

J. Phys. Oceanogr.

Modeling spectral dissipation in the evolution of wind waves — part 1. Assessment of existing model performance

J. Phys. Oceanogr.

Wavenumber spectra of short gravity waves

J. Geophys. Res.

Measured wind-wave climatology Lake IJssel (NL), report RWS RIZA 2007.020

Similarity of the wind wave spectrum in finite depth water, 1. Spectral form

J. Geophys. Res.

Similarity of the wind wave spectrum in finite depth water, 2. Statistical relationships between shape and growth stage parameters

Dtsch. Hydrogh. Z.

Revisions in wave forecasting: deep and shallow water

Directional spectra of wind-generated waves

Philos. Trans. R. Soc. Lond., A

Nonstationary analysis of the directional properties of propagating waves

J. Phys. Oceanogr.

Observations of the directional spectrum of fetch-limited waves

J. Phys. Oceangr.

On the non-linear energy transfer in a gravity-wave spectrum, part 1. General theory

J. Fluid Mech.

Directional wave spectra observed during JONSWAP 1973

J. Phys. Oceanogr.

Theory of synthetic aperture radar ocean imaging — a MARSEN view

J. Geophys. Res.

Observations of the directional distribution of ocean-wave energy in fetch-limited conditions

J. Phys. Oceanogr.