Study on Energy Spectrum Instability in the Processes of Propagating and Breaking of Focusing Waves

Based on phase focusing theory, focusing waves with different spectral types and breaking severities were generated in a wave flume. The time series of surface elevation fluctuation along the flume were obtained by utilizing 22 wave probes mounted along the mid-stream of the flume. Based on the wave spectrum obtained using fast Fourier transform (FFT), the instability characteristics of the energy spectrum were reported in this paper. By analyzing the variation of total spectral energy, the total spectral energy after wave breaking was found to clearly decrease, and the loss value and ratio gradually increased and tended to stabilize with the enhancement of breaking severity for different spectral types. When wave breaking occurred, the energy loss was primarily in a high-frequency range of f/fp>1.0, and energy gain was primarily in a low-frequency range of f/fp<1.0. As the breaking severity increased, the energy gain-loss ratio decreased gradually, which demonstrates that the energy was mostly dissipated. For plunging waves, the energy gain-loss ratio reached 24% for the constant wave steepness (CWS) spectrum, and was slightly larger at approximately 30% for the constant wave amplitude (CWA) spectrum, and was the largest at approximately 42% for the Pierson-Moskowitz (PM) spectrum.


Introduction
Wave breaking is a common phenomenon and an important element in oceanographic and off-shore engineering (She et al., 1994). The breaking of surface waves in deep water plays a significant role in the air-sea interaction and is an important fluid dynamics process affecting wave growth, surface current generation and turbulence distribution (Banner and Peregrine, 1993;Duncan, 2001;Longuet-Higgins, 1969;Melville, 1996). The phenomena of wave instability, including wave surface, wave parameters and wave energy, are clearly associated with wave breaking.
Over the past several decades, much research on deepwater wave breaking has been carried out experimentally (Drazen et al., 2008;Duncan et al., 1994;Lin and Rockwell, 1995;Liu et al., 2016;Melville et al., 2002;Perlin et al., 1996;Rapp and Melville, 1990;Tian et al., 2010;Veron and Melville, 1999). Veron and Melville (1999) measured the turbulence generated by breaking waves in laboratory. Melville et al. (2002) used DPIV to investigate the post-breaking velocity field under unsteady breaking waves in the laboratory. Tian et al. (2010) used energy focusing technology to generate breaking waves and gave the definition of characteristic wave parameters. The instability of the wave surface and wave parameters has been studied in detail by Liang et al. (2017), and the evolution of wave energy and the wave spectrum during the wave breaking process has been the focus of many studies. The wave energy transfer from lower to higher frequencies has been observed when waves propagated towards the energy focal zone, and the energy loss in the high-frequency end of the first harmonic band and in the second harmonic band was considerable when wave breaking occurred (Kway et al., 1998;Rapp and Melville, 1990). Tulin and Waseda (1999) analyzed the nonlinear evolution and spectrum evolution of deep-water wave groups initiated by unstable three-wave systems and found that wave breaking enhanced energy transfer from the higher to lower sideband, resulting in an effective downshifting of the spectral energy. Liu et al. (2015) analyzed weakly three-dimensional non-breaking and breaking waves in the laboratory. In addition, wave instability and breaking have also been studied by numerical simulation (Dimas, 2007;Drimer and Agnon, 2006;Seiffert and Ducrozet, 2018;Sullivan et al., 2004).
The analysis of wave energy and of the spectrum due to wave breaking has been primarily focused on the shape change of the wave spectrum, dissipation rate, and so on (Tian et al., 2011;Perlin et al., 2013). However, detailed quantitative analysis of energy loss and evolution among frequencies is very important, which is analyzed less and needs to be studied further. In this study, deep-water focusing waves with different breaking severity and spectral types were generated in the laboratory flume. Energy dissipation and transfer among frequencies were analyzed in detail in this paper, which could provide guidelines for understanding the dissipation mechanism of the energy spectrum due to wave breaking and further improved the dissipation formula.
Following the introduction in Section 1, the remainder of this paper is organized as follows. In Section 2, the experimental apparatus and setup are described, and the theory for deep-water focusing wave generation is introduced. The data analysis and parameters definition in the process of experimental analysis are given in Section 3. In Section 4, the instability of the energy spectrum was analyzed in detail. Lastly, conclusions are drawn in Section 5.

Experimental methods and apparatus
2.1 Focusing wave generation η In this study, the breaking waves in deep water were generated by the phase focusing method initially proposed by Longuet-Higgins (1974) and further improved by Rapp and Melville (1990) and others (Drazen, 2006;Kway et al., 1998;Loewen and Melville, 1991). In this method, a packet of waves of varying frequencies are generated in a wave tank so that they are in phase with each other at the desired focal point and generate a breaking event. The mechanism of the method is due to the fast spreading of long waves and slow propagation of short waves. For a two-dimensional case, according to linear theory, the free surface displacement, (x, t), can be described as: where N is the number of frequency components and a n , k n , , are the amplitude, the wave number, the angular frequency and the phase of the n-th frequency, respectively. The dispersion relationship is where h is the still water depth and g is the gravitational acceleration. Since we wish the waves to focus at a location (x b , t b ), the phases can be adjusted to ensure: (3) Then, the initial phase of each component is: η By substituting Eq. (4) into Eq. (1) and setting m=0, (x, t) becomes: Thus, we can know that for a given location x b and a time t b of focusing, the wave focusing primarily depends on the amplitude of each component a n .
In Eq. (5), the amplitude of each wave component a n depends on the wave frequency spectrum distribution form. The focusing wave amplitude A is determined by the input spectrum parameters, and we assume that A is the sum of the wave amplitudes at the focal point; thus, The amplitude spectrum can exist in a variety of forms, one of which is the constant wave amplitude (CWA) distribution, assuming each wave component has equal amplitude, namely, a n is constant, i.e., a n = A/N.
Another form is called the constant wave steepness (CWS) distribution, assuming that each wave component has equal steepness and s=k n a n is constant, i.e., Spectral forms can also be found in the Pierson-Moskowitz (P-M) spectrum (Yu, 2000): ω m where is the peak frequency. For a P-M spectrum, the wave component amplitude can be calculated by In addition, we assume that the discrete frequency f n is uniformly distributed in [f 1 , f 2 ], so the frequency width and center frequency can be defined by: . (12)

Experimental flume
The experiments were carried out in a large wave-current flume in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The flume is 69 m in length, 4 m in width and 2.5 m in depth.
The maximum operating depth is 2.0 m. It is equipped with piston wavemaker and the system of wave making is hydraulic servo irregular, which is controlled by computer. In this study, the water depth was fixed at 1.5 m in all the experiments. The flume is equipped with an irregular wave maker, a computer-controlled data acquisition system and bi-directional current generating system of two pumps each with a discharge capacity of 0.8 m 3 /s. With a working depth of h=1.5 m, the value of kh was calculated using the dispersion relationship and satisfied the depth wave conditions of kh>>1 and tan(kh)≈1. To reduce the influence of wave reflections, absorbing devices were arranged at the end of the tank.

Layout of wave probes
To record wave surface elevation, 22 resistance wire wave gauges were used along the wave propagation direction of the wave tank. Specific locations of wave gauges are shown in Fig. 1. To ensure the accuracy of the measurement, strict calibrations were carried out before the experiments. Wave gauges recorded data at a rate of 100 Hz, and the duration of the data acquisition was 163.84 s. The coordinate system is set at the intersection of the balance position of the wave maker and the still water surface, with the x-axis in the direction of the wave propagation and the y-axis pointing upward.

Experimental setup
In this paper, three spectral types were chosen to generate the focusing wave. The details of these tests are presented in Table 1.
Waves in Cases 1-8 were generated using the CWS spectrum. Cases 9-15 and Cases 16-22 were set up to analyze the instability characteristics of wave energy for the CWA spectrum and P-M spectrum, respectively. The parameters in Table 1 were calculated from the amplitude spectrum obtained from the fast Fourier transform (FFT) analysis of the surface elevation, which was measured at the first wave probe station (x=3.65 m). The calculation of initial incident wave steepness (S input ) is described in Section 3.2.

Data analysis and parameters definition
3.1 Data analysis The wave energy spectrum was obtained by FFT of the time series of surface elevation at different locations of the flume. In this study, by taking the location of the maximum wave height as the center, the wave surface elevations with a duration of 41.92 s were chosen for computing the spectrum. Careful observation of the wave surface reveals that this period of time was sufficient for all meaningful wave components and overcame the influence of wave reflection. Based on the spectrum analysis in the present study, the energy of focusing waves mainly concentrated in the frequency range of [0.6 Hz, 1.5 Hz]. To adequately include all meaningful wave components, the frequency range for computing the parameters of wave characteristics was in the range of [0.3 Hz, 4.0 Hz].
3.2 Global wave steepness Rapp and Melville (1990) first proposed a global wave steepness (S c =k c a, k c is a wave number associated with the center frequency, where a is the amplitude at the focal location), which represents the breaking severity. Subsequently,  (Drazen et al., 2008;Kendall Melville, 1994;Loewen and Melville, 1991) provided a definition, S kn = k n a n , and drew similar conclusions as Rapp and Melville (1990). In this paper, the same definition of the global wave steepness in Tian et al. (2010) was adopted as: where a n is the amplitude of the n-th component of the wave train and k s is a spectrally weighted wave number (Liang et al., 2017). The global steepness at the first wave probe station was selected as the incident wave steepness, S input .

Analysis of energy instability
4.1 Variation of total spectral energy along the flume The total spectral energy E sp is the entire energy spectrum in the frequency range [0.3 Hz, 4.0 Hz]. Fig. 2 shows the variation of non-dimensional total spectral energy, E sp k s 2 , along the flume for non-breaking waves, spilling waves and plunging waves under the conditions of different spectral types. Here, the symbols "a" and "b" represent the control position before and after the wave breaking zone. The total energy remained stable along the flume for nonbreaking waves. In this experiment, the working water depth of the flume was 1.5 m, and the width was 4.0 m, so the energy dissipation due to friction from the flume bottom and sidewall was negligible. As the breaking severity increased, the energy loss caused by wave breaking clearly intensified, and the energy decreased significantly downstream of the wave breaking zone compared with the energy value of the upstream region. Compared with the value at the reference position, the changing relationship of the energy loss rate with incident wave steepness due to wave breaking is shown in Fig. 3. The loss value and ratio gradually became larger and tended to stabilize with the increase in breaking severity for different spectral types. The maximum loss ratio was approximately 35% for the CWS spectrum, 40% for the CWA spectrum, and 25% for the PM spectrum.

Changes of spectral energy in the control frequency range
From the evolution of the energy spectrum analyzed in the literature (Liang et al., 2017), there is energy loss in high-frequency parts of the control frequency range, and energy is transferred to lower frequencies. To further analyze the evolution of spectral energy characteristics in the control frequency range, the energy spectrum at different location was subtracted from the reference energy spectrum at x=7.75 m in the flume, and then the differences of energy spectra were normalized by the peak spectrum S p0 at the reference location, as presented in Fig. 4 for plunging wave (Cases 8,15,and 21). The results at x=12.13 m and 13.19 m in Fig. 4 show that as the waves approached the wave breaking zone, the energy spectrum in higher-frequency parts of the control frequency range slightly widened, and the energy increased slightly in the high-frequency range of f/f p >1.5 for the CWS spectrum and CWA spectrum and was at f/f p >1.3 for the PM spectrum. In addition, the energy increased slightly in the low-frequency range of f/f p <1.0 for the three spectral types. For the CWS spectrum and PM spectrum, the increase in the low-frequency range of f/f p <1.0 was more obvious than that in the high-frequency range, but for the CWA spectrum, the tendency was the opposite. The energy decrease primarily occurred in the frequency range of f/f p =1.0-1.5 for the CWS spectrum and CWA spectrum and in the frequency range of f/f p =1.0-1.3 for the PM spectrum. The energy decrease near the peak frequency f p for the CWS spectrum and the PM spectrum and near 1.5f p for the CWA spectrum was most obvious. By comparing the differences at x=18.89 m and 20.63 m with those at x=12.13 m and 13.19 m, the energy loss due to  wave breaking was produced in the high-frequency range of f/f p >1.0, while loss was small in the low-frequency range of f/f p <1.0. After wave breaking, the wave propagated downstream, and the energy redistributed among frequencies again. With the spectrum tending to stabilize, the energy increase in the low-frequency range of f/f p <1.0 was the more obvious, and the energy decrease in the high-frequency range of f/f p >1.0 was more notable compared with the spectrum before wave breaking.

The gain and loss characteristics of the energy spectra
For further analysis of the energy transfer and loss due to wave propagating and breaking, according to the variation of total spectral energy along the flume in Section 4.2, the locations a and b were chosen as the control positions before and after the wave breaking zone for different spectral types (Fig. 5). Near the control position, the shape of the energy spectrum remained stable, so it can be compared. The differences normalized by peak spectrum (S ap ) at position a before (a) and after (b) the breaking zone are shown as a function of the frequency normalized by the peak spectral frequency in Fig. 5.
When no wave breaking is occurring, the change tendency of spectral differences among frequencies was similar for the CWA and CWS spectra, but it was different for the PM spectrum in the low-frequency range. Additionally, the frequency ranges of energy increase and decrease were diverse for different spectral types. For the CWS spectrum, the energy increase occurred in two frequency bands which were in high-frequency ranges of (1.48-1.6)f p and low-frequency ranges of (0.9-1.18)f p , where the decrease was the most obvious at peak frequency and the energy increase distinctly between 1.18f p and 1.48f p . For the CWA spectrum, the decrease of energy spectrum occurred in two frequency bands, at (0.84-1.12)f p and at (1.34-1.48)f p . The increase primarily occurred between 1.12f p and 1.34f p . For the PM spectrum, the energy increase occurred in low-frequency ranges of (0.8-0.98)f p whose upper boundary will be further away from 0.98f p if the incident wave steepness is smaller and in high-frequency ranges of (1.15-1.45)f p , where the increase was more notable and the width was larger in high-frequency ranges than that in low-frequency ranges. The energy decrease primarily occurred in frequency ranges of (0.98-1.15)f p , and some decrease oc- curred in high-frequency ranges of (1.45-1.65)f p .
Wave breaking occurred as the incident wave steepness increased. When the wave breaking severity was weak (e.g., spilling wave), both the gain and loss occurred in two frequency bands, with the frequency range and width varying at different wave breaking severity. When the wave breaking severity increased (e.g., plunging wave), the extent of energy gain in the high-frequency range larger than f p became weaker and even negative (for the CWS spectrum), and the extent of energy gain in the low-frequency range smaller than f p became stronger. For the CWS spectrum, as the wave breaking severity increased (e.g., spilling wave), the energy gain occurred in two frequency bands, which were in low-frequency ranges of (0.8-1.0)f p and high-frequency ranges of (1.18-1.48)f p where the gain amount was smaller compared with non-breaking waves. Loss also occurred in two frequency bands in frequency ranges of (1.0-1.2)f p and frequency ranges of (1.48-1.8)f p . When the wave breaking was stronger (e.g., plunging wave), the energy gain in low-frequency in ranges of (0.8-1.02)f p was increasingly obvious, and the frequency bands of energy gradually became wider, between 1.02f p and 1.8f p . When the wave breaking severity was weak, energy gains occurred in frequency ranges of (1.18-1.48)f p , and as the wave breaking severity increased, the energy was lost and the loss was increasingly notable. For the CWA spectrum, with wave breaking occurring and the wave breaking severity increasing, the energy gain occurred between 0.8f p and 0.95f p and gradually became more notable. The gain was also present but gradually weakened in the frequency ranges of (1.12-1.34)f p , in which the gain occurred for non-breaking waves. In addition, its width narrowed and its upper boundary moved towards a higher frequency (e.g., Case 14, where the frequency ranges were between 1.27f p and 1.32f p ). The loss mainly occurred in the frequency range of (0.95-1.16)f p , whose lower boundary moved towards a higher frequency (e.g., approximately 1.27f p for Case 14) as the wave breaking severity increased, and the frequency range of (1.32-1.6)f p . For the PM spectrum, as the incident wave steepness increased and the wave breaking severity increased accordingly, the gain in low-frequency ranges of (0.8-0.98)f p became more obvious. For the frequency ranges of (1.15-1.45)f p , the gain for non-breaking waves gradually narrowed ((1.18-1.35)f p for spilling wave and (1.2-1.32)f p for plunging wave), and the value of energy gain gradually decreased. Energy loss occurred in the frequency ranges of (0.98-1.20)f p and in higher frequency ranges where the loss became more obvious, compared with only a small amount of loss produced for non-breaking waves, whose width increased (from (1.45-1.65)f p to (1.32-1.65)f p ) with the increase in wave breaking severity.
To further calculate the energy gain (increase) and loss (decrease) after wave breaking, the energy gain and loss were obtained by integrating the positive and negative parts of the spectral density difference along the frequency, respectively, and the energy gain-loss ratio for different cases was analyzed, as shown in Table 2. The energy gain-loss ratio was larger than 85% for the CWS spectrum, 99% for the CWA spectrum and 73% for the PM spectrum, which demonstrates that most energy decrease was obtained again by other frequencies for non-breaking waves. With incident wave steepness and wave breaking severity increasing accordingly, the energy loss became larger and larger, but the corresponding energy gain did not increase proportionally, which accounts for the loss due to wave breaking. Under different spectral types, the energy gain-loss ratio partially decreased and then gradually became stable, as shown in Fig. 6. For plunging waves, the energy gain-loss ratio reached 24% for the CWS spectrum, was slightly larger at approximately 30% for the CWA spectrum, and was the largest at approximately 42% for the PM spectrum. As previously analyzed (Liang et al., 2017) for the PM spectrum, the energy spectrum is most stable and the energy loss due to wave breaking is the lowest under the same incident wave steepness.

Conclusions
In this study, based on the phase focusing theory, focusing waves with different breaking severities for the CWS spectrum, CWA spectrum and PM spectrum were generated in the laboratory flume. The time series of surface elevation fluctuation along the flume were obtained utilizing 22 wave probes along the mid-stream of the flume. The in-stability of the energy spectrum due to focusing wave propagating and breaking was analyzed.
The total spectral energy after wave breaking decreased remarkably with increasing breaking severity. The loss value and ratio gradually increased and tended to stabilize with breaking severity enhancement for different spectral types. The maximum loss ratio was approximately 35% for the CWS spectrum, 40% for the CWA spectrum, and 25% for the PM spectrum.
Energy loss primarily occurred in the high-frequency range of f/f p >1.0, and energy gain primarily in the low-frequency range of f/f p <1.0 when wave breaking occurred. When the wave breaking severity was weak, there was a high-frequency range of f/f p >1.0 with energy gain, and the frequency band became narrow and even disappeared, accompanied by decreasing energy gain as the wave breaking severity increased. For non-breaking waves, the energy gain-loss ratio was larger, which demonstrates that energy transfers among frequencies. As the wave breaking severity increased, the energy gain-loss ratio decreased gradually,  . 6. Relationship between the energy gain-loss ratio and incident wave steepness for the CWS, CWA, and PM spectra.
which shows that the energy loss accounted for wave breaking. For plunging waves, the energy gain-loss ratio reached 24% for the CWS spectrum, was slightly larger at approximately 30% for the CWA spectrum, and was the largest at approximately 42% for the PM spectrum.