Radar Investigation of the Structure of Wind Waves

Transcription

1 Journal of Oceanography, Vol. 59, pp. 49 to 63, 2003 Radar Investigation of the Structure of Wind Waves TIMOTHY LAMONT-SMITH 1 *, JOERN FUCHS 2 and MARSHALL P. TULIN 2 1 QinetiQ, St. Andrew s Road, Malvern, Worcestershire, WR14 3PS, England 2 Ocean Engineering Laboratory, University of California Santa Barbara, 6740 Cortona Drive, Goleta, CA 93117, U.S.A. (Received 6 November 2001; in revised form 8 May 2002; accepted 9 May 2002) Radar data from three experiments are analysed. Scatter characteristics of 50 cm wind-generated waves have been investigated with a C-band radar in a large windwave tank. Evidence of wave groups in sea clutter from the west coast of Scotland in the Sound of Sleat is also presented. The spectrum of the waves in the sound is narrow-banded and the waves are young, like the wind-wave spectrum in the laboratory. Clutter measurements, collected on the English south coast at Portland, of more oceanlike waves with broad band spectra also suggest the presence of wave groups. Evidence of the presence of wave groups is demonstrated in range-time images, as well as in the Fourier domain. Some ad hoc processing schemes, the normalised variance and binary threshold techniques, were successfully applied to enhance the appearance of the wave groups. The wind waves change frequency with fetch in the wave tank and the downshifting process is investigated using range-frequency maps of the radar data. The waves appear to change frequency in discrete steps that are associated with wave breaking events. The difference in wave period before and after breaking could be measured, and a wave crest was shown to be lost to compensate for the change in period, as expected. Some downshifting could also be measured in the Sound of Sleat. The ratio of wave group frequency and wave frequency is in accordance with Benjamin-Fier sideband instability theory, as it is for the data measured at Portland. Keywords: Radar, wind waves, wave groups, frequency downshifting. 1. Introduction A common feature of radar studies of sea backscatter is that in the frequency-wavenumber domain (ω-k space), there is a non-dispersive feature, e.g. Ivanov and Gershenzon (1993), Werle (1995, 1998), Smith et al. (1996), and Stevens et al. (1999). Typically, this non-dispersive feature is a straight line through the origin whose gradient corresponds to the group velocity of the dominant waves. Consequently, it is usually referred to as the group line. For gravity waves in deep water, the group velocity is half the phase velocity. If wave groups were present, then within some region in space-time they would repeat at fixed space and time intervals. In the 2D Fourier domain, this should be apparent as a point (δω, δk) somewhere along the group line, possibly with harmonics present at higher frequencies. δω is the frequency of the periodic modulation envelope and δk is the wavenumber of the group. Data from Ivanov and Gershenzon (1993) of radar data collected in * Corresponding author. tlsmith@qinetiq.com Copyright The Oceanographic Society of Japan. the open ocean show these distinctive spots along the group line, and an unambiguously periodical wind wave group structure. At this stage, it is important to make a distinction between a peaked envelope caused by random focussing of waves and a non-linear wave group, periodic in at least a few group lengths. The term wave group used in this paper applies only to the latter, although elsewhere in the literature a peaked envelope is also described as a wave group. The definition used here is that a wave group has a finite lifetime, characterised by a modulation envelope periodic in at least a few cycles, which propagates with the group velocity over a trajectory of many group lengths. Repeating or multiple wave groups also occur wherein a wave group appears locally in space-time, loses coherence to be followed by a separately identifiable wave group. In essence, a group of groups appears. The existence of wave groups is the result of the nonlinear instability of finite waves, which gives rise to their modulation. As waves pass through the group they tend to break near the centre of the group where the amplitude is highest. Breaking gives rise to high radar cross-section, which ultimately allows the crest of the wave group 49

2 HYDRAULICS TEST SECTION RADAR ANTENNA CROSS SECTION INLET TUNNEL BEACH 4.3m 1.5m 2.4m HONEYCOMB 1.5m 2.1m WAVEMAKER 30.5m 53.3m Fig. 1. Large wind-wave tank at the Ocean Engineering Laboratory, University of California, Santa Barbara. to be imaged by radar. The details of the radar backscatter from breaking waves, including polarisation effects and the Doppler spectra from the breaking process, are described in detail by Fuchs et al. (1999). Most modelling of ocean waves has been done by treating the ocean as a linear stochastic system. Longuet- Higgins (1984), for instance, has investigated the properties of the envelopes that arise in a Gaussian random sea state. The basis for this model is that the waves at any point in the sea are due to the linear superposition of waves that were generated by wind at many different points of the sea surface, so that the wave sources are uncorrelated from each other. Lake and Yuen (1978) made the earliest serious proposal to model the ocean surface as a non-linear system in which non-linear hydrodynamic phenomena play a dominant role. The possibility of wave groups originating through the instability of wind waves was specifically investigated by them. As time has passed, wave groups and other non-linear aspects of wave phenomena have been increasingly studied, and knowledge about them grows. It is natural to expect the appearance of wave groups in steady wind wave systems approximating finite waves, or a system of finite waves, such as might be expected in sheltered environments that produce young waves with narrow banded spectra. The laboratory data are narrowbanded, as are the data from one of the field sites. As waves grow due to the action of wind, the frequency of the most energetic wave decreases with fetch, a process called downshifting (Tulin, 1996). It has been observed in wave tank studies, see Hatori (1984) for instance, that the wind wave spectrum is discrete; this is almost certainly related to non-linear wave interactions. Observations of wave group evolution by Tulin and Waseda (1999) using mechanical waves in the wave tank have shown that the wave group modulation, which originates in the unstable growth of sidebands, is eventually accompanied by a highly discretized energy spectrum. There are peaks in the spectrum, equally separated, with a constant increment of frequency as the result of the transfer of energy through non-linear wave interactions. They showed that in the presence of strong wave breaking, significant energy is irreversibly transferred from the peak to the lower sideband, such that the new peak in the spectrum occurs at the former sideband position. Ocean waves may be thought about in two complementary ways: deterministic and statistical. The presence of strong non-linear structure in ocean wave systems suggests that both a deterministic, and a statistical viewpoint are necessary for analysis, as will be shown. The objectives of this study were to investigate the presence of wave groups and the nature of wind wave growth at a number of different sites that mimic the conditions of the open ocean with progressively greater realism, as discussed next. 1.1 Experimental set-up The radar backscatter from wind-generated waves has been investigated in the wind-wave tank of the Ocean Engineering Laboratory at the University California Santa Barbara. A wind speed of 8 m s 1, at a height of 1 m above the water surface, was used and the fetch was around m. The wave tank has dimensions 4.3 m wide, by 2.1 m deep, and by 53.3 m long, as shown in the sketch of the laboratory layout in Fig. 1. The Frequency Modulated Continuous Wave (FMCW) radar used in the laboratory has a frequency chirp from 4 8 GHz, which gives a range resolution of 3.77 cm. Details of the radar system and its calibration are given by Fuchs et al. (1996, 1999). The radar is polarimetric with a 1 khz PRF, 500 Hz for each polarisation. It is operated at short range and at a low grazing angle (LGA) of around 6. The data presented here are only from the vertical polarisation (VV), although horizontal (HH), as well as the cross-polar (HV/VH) return were also collected. The 3 db beamwidth is 3, which corresponds to around 50 cm in the centre of the swath for the centre frequency. An array of wave wires measured the wave height of the wind waves, spaced along the length of the tank attached to one sidewall, and outside the beamwidth of the radar. Data collected from field experiments are also presented. Experiments were conducted on the west coast of Scotland in 1991 in the Sound of Sleat and at Portland on the South coast of England in The Maritime Clifftop Radar (MCR), built and operated by the defence company Racal (now called Thales), is a pulse Doppler radar 50 T. Lamont-Smith et al.

3 Table 1. Main radar parameters. Radar UCSB (C-band) Maritime clifftop radar (MCR) Centre frequency 6 GHz 10 GHz Bandwidth 4 GHz 100 MHz Range resolution 4 cm 1.5 m Swath length 8.5 m 384 m/1536 m Polarisation Dual (VV/HH/VH) Dual (VV/HH/VH) PRF 1 khz 1 khz Beamwidth Grazing angle 6 variable (2 /5 ) that operates at 10 GHz with a 100 MHz analogue chirp that gives a range resolution of 1.5 m. The radar is typically operated at a range of around 2 3 km. The azimuthal beamwidth of the radar is 1.2 ; this translates to a crossrange extent of the order of m for both the sites used. The grazing angle of the radar to the water surface was 5 at the west coast of Scotland experiment, and 1 2 at Portland. Table 1 gives a summary of the main specifications of the radars used in this paper. Low grazing angle staring radars give clear contrast between the wave peaks and troughs, allowing the wave to be clearly distinguished. This is true even for nonbreaking waves in vertical polarisation. A non-breaking wave peak and trough cannot be distinguished in horizontal polarisation as readily as in vertical. In horizontal polarisation the backscatter is much spikier and the dominant return comes almost exclusively from the crest of breaking waves. The radar imaging mechanisms for breaking waves are discussed by Fuchs and Tulin (2000), using the same 4 8 GHz wide bandwidth radar. Caponi et al. (1999) show simulations of range-time intensity images of radar backscatter from wind waves and discuss the imaging mechanisms for VV and HH at different grazing angles. Narrow beamwidth and high-resolution radars exhibit good spatial selectivity, such that waves travelling towards or away from the radar are particularly strongly imaged. These two capabilities together make the radars used in this study extremely useful for investigating oceanographic features such as wave groups. It is possible to make an approximation that waves of a particular wavelength are not resolvable if there are many peaks and troughs of those waves in each radar range cell. If all the waves are long-crested and there is only one wave crest of wavelength λ in a range cell at any time, then for the wave to be resolvable at some angle θ to the radar line-of-sight, λ Rφsinθ + rcosθ () 1 2 R is the range and φ is the beamwidth, which together give the azimuthal extent of the radar (Rφ), and r is the range resolution. Radars have long been established as remote sensing tools, and are shown here to be particularly effective in studying the structure of individual waves and wave groups in range and time. For example, Walsh et al. (1989) used an airborne radar system to observe the evolution of fetch limited directional wave spectra. In comparison, wave buoys and single wave wires are omni-directional sensors that measure a superposition of all the waves present at a single point. One disadvantage of this is that the sensor must be deployed at the point required, which may not be convenient, and even an array cannot collect data over any great spatial extent, compared to a radar. Typically, a wave buoy or wire is better suited to collecting time series data for the spectral information than an analysis of the behaviour of individual waves in range and time. 2. Wave Groups and Wave Evolution Figure 2 shows a range-time amplitude image of one minute of vertically polarised data, collected in the laboratory wave tank with the 4 8 GHz FMCW radar. The data have been block averaged to 40 ms per pixel. The range swath extends from 6 14 m from the radar, which corresponds to a fetch of around m from the wind generator, as shown in the sketch in Fig. 1. The grazing angle is 6, and the antenna beam pattern causes the fall-off in the backscatter that is visible in the image at the near and far ranges. In this experiment the fan speed is held constant and generates a wind speed of approximately 8 m s 1 at a height of 1 m above the mean water level. The phase speed of the wind waves is an order of magnitude smaller, 80 cm s 1, which means that the waves are still young, and at 50 cm the waves are an order of magnitude shorter than the width of the tank, at 4.3 m. The crests of the individual waves are the dominant features in Fig. 2. They are travelling at the phase veloc- Radar Investigation of the Structure of Wind Waves 51

4 B C A Fig. 2. Range-time amplitude image of vertically polarised backscatter from 8 m s 1 wind waves in the OEL wind-wave tank. The lines at the side of the image indicate the crests of the individual waves that are propagating at the phase velocity. Fig. 3. Normalised variance of the intensity of the same data as from Fig. 2. The lines at the side of the image at A, B and C indicate wave groups. ity of gravity waves. Some lines have been sketched at the side of the image that indicate the typical spacing and gradient of the individual wave crests. There are within the image, places where there are short stripes of high intensity as a result of wave breaking. The slope of the stripes is again just the phase speed of the breaking waves, but these stripes are also organised in bands, whose slope is the corresponding group speed. One such band is visible ending at 6 m range and time T = 20 s. The waves break as they pass through the crest of a group, and the breaking results in a high radar cross-section. It is the encounter speed between the phase speed of the waves and the group speed of the modulation that is responsible for the orientation of the steeper dark bands in the image. In the case of deep water, the gradient of the steeper dark bands in the range-time diagram is also the speed of the group. Figure 3 shows an image formed from the normalised variance of the intensity. The same data as Fig. 2 are used except with an integration period of 160 ms for each pixel, rather than 40 ms. The normalised variance of the intensity, I, is defined as 2 σ 2 µ I 2 2 I I. ( ) 2 2 The normalising factor µ 2 is just the square of the mean intensity. The normalised variance is frequently used as a texture estimator in SAR imagery (Blacknell, 1994) because it gives a useful measure of the spikiness of the data. For instance, radar data are frequently represented by the K-distribution, amongst others, in which case the normalised variance is equal to 1 + 2/ν where ν is the shape parameter of the K-distribution. The compound model of sea clutter (Ward et al., 1990) suggests that the normalised variance of the intensity should be equal to unity everywhere in the image, given the sufficiently short time scales used to process the data; thus, we might expect some speckle in Fig. 3, but no structure. However structure is visible there and in particular there are high intensity lines, with values ~2, caused by high variability in backscatter intensity at those points. The number of effective scatterers in a range cell is likely to be very small because the area illuminated by the radar is extremely small, just 4 cm in range 52 T. Lamont-Smith et al.

5 the mean spectrum, and where all values below the mean are shown as white pixels. The gravity wave dispersion relation and its harmonics are visible (the curved lines), as is the group line, which is straight. In general, for Stokes waves of small steepness and in the absence of current, the harmonics take the form, 2 ω n = nk g () 3 n Fig. 4. ω-k diagram of 8 m s 1 wind waves in the OEL windwave tank. 2D Fourier transform of Fig. 2. Dashed lines show the dispersion relations and the group line. and 50 cm cross-range, consequently the statistics do not obey the central limit theorem, and thus the compound model is not readily applicable. There may in fact only be a single dominant scatterer in the regions that have high variability. In certain portions of the image, for instance the first 30 seconds, marked A, the lines appear to be periodic. The side of the image has lines sketched that indicate the spacing and velocity of the group. The periodicity may be seen most easily, if the image is held up to the reader s eye and examined in the direction of the lines. Other groups have been sketched at B and C which have the same group velocity but with different crest spacing, corresponding to both smaller and larger δω and δk, B and C respectively. The group marked at A will be considered in detail from now on. A comparison with Fig. 2 shows that the lines visible in Fig. 3 correspond to the crest of the modulations where wave breaking occurs and which results in the high radar cross-section. The velocity of the features may be estimated from the gradient of the lines in the image. The group velocity of a gravity wave in deep water is half that of the phase velocity (ignoring drift currents), as may be seen by comparing the relative gradients of the lines from Figs. 2 and 3. Figure 4 shows the 2D Fourier transform of the first 33 seconds taken from the amplitude data in Fig. 2. The absolute values of the Fourier transformed data are shown in the image with a colour scale in db that is relative to where, n = 1, the first harmonic, corresponds to the equation for the deep-water gravity wave dispersion relation in still water, and n = 2,3,4,... are the higher harmonics. Stokes law for the harmonic amplitudes show that the higher harmonics, although always smaller than the first harmonic, grow rapidly relative to it with increasing steepness. Philips and Banner (1974), for instance, show experimental evidence for this. A current represents a transport of the water mass, and may be thought of as a translation of the co-ordinate frame of the waves. If there is a current V then there will be an additional shift k V applied to the still-water frequency ω 0 seen in the ω-k spectrum, such that, ω = ω + k V. ( ) 0 4 Thus, the current may be measured from the distortion of the dispersion curves. Overlaid on Fig. 4 are dashed lines showing the estimated positions of the dispersion curves, based on Eqs. (3) and (4), corrected for a deduced current of around 8 cm s 1, found by fitting the curve to the data. This method is a well-established remote sensing technique for measuring ocean currents, and can also be applied in three dimensions (k x, k y and ω) by using a scanning radar (Young et al., 1985). The third and even fourth harmonic of the dispersion relation are also faintly visible at higher frequencies, but without any lines overlaid. Figure 5 shows the 2D Fourier transform of the first 33 seconds taken from Fig. 3, which shows range-time data processed using the normalised variance. The scale of the image is magnified with respect to Fig. 4, and a linear colour scale is used. The group line is again visible and there is a high intensity black point corresponding to the frequency δω and wavenumber δk of the finite wave group sketched in Fig. 3. The frequency of this brightest point in the image is 0.35 Hz and it has an inverse wavelength of about 0.7 m 1. There are other frequencies present in the image as well, but these are less strong. The wavelength of the dominant waves could be estimated from the range-time amplitude image in Fig. 2, however a better method is to measure the spectral amplitude of the pixels along the dispersion curve that is visible in Fig. 4. The asterisks shown in Fig. 6 come from measurements of the spectral amplitude along the disper- Radar Investigation of the Structure of Wind Waves 53

6 Fig. 6. Measurement along the dispersion curve from Fig. 4. Asterisk symbols show the relative amplitude of the dispersion curve and diamond symbols show measurements from the harmonic curve with the inverse wavelength scaled down by a factor 2. The solid vertical line shows an estimate of the position of the dominant wavelength. Fig. 5. Zoomed in portion of the ω-k diagram formed from the normalised variance of the intensity from Fig. 3. sion curve and the diamond symbols come from measurements along the second harmonic curve. The spectral measurements of the second harmonic have been scaled by a factor 2 to allow a comparison to be made more conveniently on the one graph. The spectral amplitude of the second harmonic is a relatively consistent 6 db less than the first harmonic for the wavelengths that are present. The peak amplitudes occur for an inverse wavelength of 2.1 m 1 corresponding to a wavelength of around 50 cm. A graph of inverse wavelength is plotted, rather than frequency, because the uniform drift current has no effect on the wavelength. The inverse wavelength of the wave group is around 0.7 m 1 for the highest amplitude point in the ω-k plot of the normalized variance in Fig. 5. This suggests that in the spatial domain there are 3 waves in each wave group. A visual inspection of Figs. 2 and 3 suggests that this is a reasonable estimate. The number of waves in a wave group depends on whether it is examined in the spatial or time domain. The situation of three waves in the spatial domain, δk/k = 1/3, corresponds to six waves in the time domain, δω/ω = 1/6, since the individual waves travel at twice the velocity of the group envelope. Observations made from a wave wire or other point measurement that collected time series data would therefore see six waves in the group. To avoid confusion, only the number of waves in the spatial domain will be discussed. Benjamin and Fier (1967) suggested a simple relationship between the steepness of the waves, ak 0, and the regime of the most unstable sidebands, given by the inequality 0 < δω/ω 0 2ak 0. The most unstable sidebands correspond to a particular value of δω, either side of the primary frequency, ω 0. Benjamin and Fier (1967) found there was maximum wave growth when δω 1 δk = = ak0 ( 5) ω 2 k 0 0 where the frequency and wavenumber of the wave is ω 0 and k 0 and for the group is δω and δk. Melville (1982) showed a comparison between some laboratory results and various theoretical relationships, which suggested that Eq. (5) gives a small underestimate of the true wave steepness. Equation (5) suggests that the steepness of the wind waves in the laboratory should be around ak 0 = In fact the r.m.s. slope for these wind waves has been measured by the wave wire array at the side of the tank as Figure 7 shows a 1-D Fourier transform in the time direction only of the range-time amplitude data from Fig. 2, with no spectral averaging. The length of time used was a minute (thus giving a spectral resolution of 1/60 Hz), and it had already been incoherently integrated in time over 40 ms samples before the Fourier transform was applied. The image has been normalised by the beam weighting, so that the intensity is more or less uniform across the image, and it is displayed with a linear colour scale that shows the relative amplitude. There is a broad strip across the image at around 2 Hz that changes frequency with range (fetch). The fetch increases from right to left across the image. The nearrange at 6 m corresponds to the furthest fetch of about 54 T. Lamont-Smith et al.

7 Fig. 8. Range collapsed spectrum formed from Fig. 7. Fig. 7. Range-frequency image of 8 ms 1 wind waves in the OEL wave tank m, and the furthest range displayed of 14 m corresponds to a fetch of 29.5 m. The frequency of the broad strip in the image corresponds to the phase frequency of the waves. The wind waves grow with the fetch, they get longer and faster and the frequency decreases, this process is known as downshifting, and is governed for fetchlimited ocean waves by the well-known fetch laws (Hasselmann et al., 1973). The strip across Fig. 7 is made up of lines of constant frequency. At different ranges, exactly the same frequencies are dominant, with amplitudes well above the background. It is significant that there are lines making the strip, rather than a more random spread of energy. This suggests that the waves may have discrete frequencies, and the frequency of a particular wave does not change continuously with the fetch but changes in discrete steps. If the wave spectrum were continuous then there would not be horizontal lines. With the small amount of data used here the range-frequency map might still be noisy, but there would be no reason to expect the same frequency components to be present at different ranges. Figure 8 shows the ensemble average of the 212 spectra from the different ranges in Fig. 7 that were collapsed in the range direction to show the discreteness of the spectrum more clearly. The harmonic frequency at around 4 Hz and the group frequency at around 0.3 Hz may now be seen more easily. Fig. 9. Zoomed in range-frequency map of laboratory wind waves formed from 7 hours of data showing spectral downshifting with fetch (decreasing range). The sampling frequency of the radar was reduced from 500 Hz to Hz, and 7 hours of data were collected and processed. The 7 hour period time series was split into approximately 375 chunks of data, each 67 seconds long and Fourier transformed and then incoherently integrated, in order to reduce the overall spectral variance. Figure 9 now shows the spectral downshifting more clearly than Fig. 7, which had no spectral averaging applied. The peak spectrum changes by around 0.3 Hz across Radar Investigation of the Structure of Wind Waves 55

8 Fig. 10. Range-time amplitude image of the first 6 seconds from Fig. 2 of the range-time radar data collected in the wind-wave tank. the 8 m of fetch. Figure 10 shows an expanded range-time image with the first 6 seconds of the data from Fig. 2 plotted, where the wave breaking associated with a group envelope may be identified. Figure 11 shows a schematic of some of the waves breaking. Successive waves break one wavelength down wave, producing white water and high radar cross section, and after a period corresponding to two wave periods, T, as a consequence of the encounter speed between the individual waves and the peak of the modulation envelope. The encounter speed between the individual waves and the peak of the envelope is just v p v g, which is v g, the group velocity, as the phase velocity v p is twice the group velocity in this case. Donelan et al. (1972) observed exactly this kind of periodicity from airborne and shipborne observations of whitecaps in the ocean. The actual track of the waves in Fig. 10 is not by any means as straight as the schematic in Fig. 11 suggests. Overlaid lines in Fig. 10 identify the tracks of four individual wave crests before breaking. If the wave at a range of 12.5 m at t = 0 seconds is followed, it can be seen that its track is not straight. The wave begins to break at around 11.5 m and t = 1.5 s where it almost seems to catch up with the previous wave in front of it. The adjacent wave tracks appear pinched together. Immediately after the Fig. 11. Sketch of the wave pattern for 50 cm wavelength waves as they encounter the crest of a modulation envelope travelling at the group velocity, and break. The interval between breaking is twice the wave period T, and the interval in range is one wavelength. wave has finished breaking, the wave appears to slow down, and then it is almost caught by the wave behind when it too breaks, forming another pinched looking region. The reasons for this are related to how a wave modulates, deforms and breaks (Tulin and Waseda, 1999). When breaking, larger waves develop steep front faces that plunge ahead of the wave, for example see the radar images in Fuchs et al. (1999). For shorter waves, as in Duncan et al. (1999), a bulge is formed rather than a jet. In either case a feature appears that has a high radar crosssection that moves forward ahead of the wave crest. The wave then forms broken white water when the jet splashes into the front face, or decays into turbulent flow in the case of the spilling wave. This broken or turbulent water still has a relatively high radar cross-section, but is left behind as the broken crest moves forward. As the white water decays away, the ongoing wave crest strengthens and becomes re-established as the dominant scatterer. In Fig. 10 there are four parallel lines overlaid on the image at the bottom of the figure around 12 m which mark four wave crests, which have not yet broken. The period of these waves as plotted on the bottom of the figure is T 1 = 0.48 s. The period of the three lines plotted at the top of the page around 9 m is T 2 = 0.56 s. This corresponds to a difference in frequencies f of 0.3 Hz, as might be expected from the group frequency. Using the figure 56 T. Lamont-Smith et al.

9 Isle of Skye Sound of Sleat + Knoydart Fig. 12. Map drawn from the NOAA NGDC World Vector Shoreline database of the site of the Sound of Sleat radar experiment in July symbol shows the radar site. Shaded box shows the approximate area illuminated by the radar. of f = 0.3 Hz, δω/ω is about 1/6, just as was found from estimating the period of the wave groups in earlier figures. Frequency downshifting implies by definition a reduction of the wave crest density. From the difference in wave periods pre- and post-breaking, it should be expected that seven wave crests become six as one wave crest is lost to compensate for the increased period. This can actually be seen occurring in Fig. 10. The tracks of four wave crests have been highlighted at the bottom of the page, but only three at the top, and the first and last waves may be easily followed across the page from bottom to top. Tulin and Waseda (1999) give a discussion of wave loss during downshifting with clear examples from wave wire data collected in the wave tank. It is their view, supported by measurements, that breaking is necessary for permanent downshifting. The process of wave downshifting through local discrete wave fusion events is also discussed by Huang et al. (1996); they suggest that wave fusion occurs at the local amplitude minima and need not necessarily depend on wave breaking. Fuchs and Tulin (2000) have also shown a linear relationship between wave wire spectral densities and the radar power spectral density for a number of different wind speeds within the wave tank, with further evidence that the spectra are discrete in both the radar and wave wire data. 2.1 Comparison with sea clutter Figure 12 shows the site of a radar experiment conducted off the west coast of Scotland in the Sound of Sleat, on 23rd July A number of groups participated in this experiment. For instance, Lee et al. (1995) examined Doppler spectra to investigate Bragg and non-bragg scatter. A spar buoy deployed by the Applied Physics Laboratory of Johns Hopkins University (Chapman, 1992) Fig. 13. Range-time amplitude image of low grazing angle radar backscatter from the Sound of Sleat. The lines at the side of the image indicate a finite wave group. collected meteorological data during the experiment. The wind speed was between 2 and 6 m s 1 and the wind was blowing down the sound, while the radar data shown in Fig. 13 were collected with the Racal Maritime Cliff-top Radar (MCR) looking crosswind, across the sound towards the Isle of Skye. The + symbol in Fig. 12 shows the radar site and the grey patch in the middle of the sound shows the approximate area illuminated by the radar. The grazing angle from the cliff-top site to the water was 5. The predominant wind direction was in general either up or down the sound. It should be understood that the sheltered conditions in the sound are not necessarily entirely typical of the open ocean. Figure 14 is the corresponding 2-D Fourier transform of Fig. 13. The ω-k spectrum is dominated by wind-waves with wavelengths in the range 4 10 m. It may be noted that there are no higher harmonics visible. This may be a result of the low wave steepness, compared to the wave tank, and it may also be a consequence of the low clutter to noise ratio of the radar system. Wave groups, just as in the laboratory, modulate the wind waves and produce high radar cross-sections at the crests of the groups, as might be expected from energetic wave breaking. The peak in the ω-k spectrum in Fig. 14 occurs around 0.15 m 1, corresponding to 6.6 m wave- Radar Investigation of the Structure of Wind Waves 57

10 Fig D Fourier transform of Fig. 13. length. Wave groups are visible in certain portions of Fig. 13, for instance at t = 150 s at least four equally spaced crests are visible. A series of equally spaced lines has been drawn by the side of the image to indicate the crests of the finite wave group. The periodicity of the crests of the group may be seen most easily if the image is held up to the eye and examined in the direction of the lines. The spacing between the lines marked on Fig. 13 is around 40 m, thus δk/k = 0.165, which indicates that there are approximately six waves within each group envelope. This implies from Eq. (5) a steepness of 0.08, which is realistic for ocean waves, and is half the steepness of the waves in the wave tank. Range-frequency analysis may also be performed on the data from the Sound of Sleat. The range-time data from Fig. 13 are Fourier transformed just in the time direction in Fig. 15. The noise floor appears to be much higher than from the data collected in the laboratory, but the spectrum again shows some evidence of downshifting. At a range of 3400 m the dominant peak is around 0.53 Hz. At 3600 m the peak frequency appears to be around 0.46 Hz. Although the data are again very noisy, the dominant frequency appears to remain the same over substantial ranges, of the order of at least 100 m, before changing. The dominant wind flow in the Sound of Sleat is up and down the sound, which is almost perpendicular to the radar line-of-sight, so perhaps strong spectral downshifting should not be expected. An experiment was carried out at Portland on the south coast of England, on 26th October A sketch of the site is given in Fig. 16, and because it is more open, Fig. 15. Range-frequency map of 250 seconds of radar data collected from the Sound of Sleat. the site is in certain respects more like the open ocean than the Sound of Sleat, consequently the waves are much longer than at the west coast of Scotland site. The height of the radar was 60 m above sea level, which gave a 1 2 grazing angle at the ranges used, and the beamwidth of the radar was 1.2. The wind as measured at the radar was blowing at a speed of around 10 m s 1 away from the radar. There was quite a strong tidal current with a component in the radar direction that changed significantly with the tidal cycle and range to the radar (Lamont-Smith, 1997), but the current is near enough constant over the short time and length scales used here. Both the wind and current are likely to be highly localised near-shore effects, and not typical of the open ocean. Figure 17 shows the range-time amplitude image of some of the radar data. There are long wavelength swell waves present in the image that predominate and that have a wavelength of around 50 m, which is illustrated by the lines sketched at the side of the image. There are also some shorter wind waves only faintly visible as streaks within the image. The swell waves visible in Fig. 17 are weakly modulated. Unlike the previous data sets from the laboratory and the Sound of Sleat where the waves were almost certainly breaking and had short crests, the modulation in the backscatter is comparatively broad and probably unrelated to breaking. There is enhanced backscatter from 58 T. Lamont-Smith et al.

11 Portland + The English Channel Fig. 16. Map drawn from the NOAA NGDC World Vector Shoreline database of the site of the 1994 Portland radar experiment. + symbol shows the radar site. Shaded box shows the approximate area illuminated by the radar. Fig. 18. Binary threshold applied to Fig. 17. The black pixels correspond to the lowest backscatter power, all other values above the threshold are white pixels. Fig. 17. Range-time amplitude image of low grazing angle radar backscatter from Portland on the south coast of England. the crest, and significantly reduced backscatter from the troughs of the waves, which may well be due to shadowing. The modulations are made more obvious in Fig. 18 by using a binary threshold applied to the lowest values in the original image. The threshold is shown by the dotted line on the colour scale beneath Fig. 17. The white pixels in Fig. 18 come from all the data values above the threshold. The black pixels come from the lowest backscatter regions, which correspond to particularly deep troughs in the waves that probably are shadowed, and which occur when the envelope of the waves is at a maximum. The lines sketched at the side of the image in Fig. 18 indicate the possibility of a finite wave group. The spacing between the lines is 400 m, which suggests there are spatially eight waves within each group envelope. A simple simulation in Fig. 19 shows the tracks of the individual waves, which have a wavelength of 50 m. The speed of the waves is estimated from the dispersion relation, taking into account the actual water depth of 40 m and the estimated tidal current of 0.26 m s 1 (discussed later). The thicker, darker lines in Fig. 19 represent what would occur for a finite wave group that had eight waves within each envelope. The darker lines show reduced backscatter from the troughs of the waves. If the simulation in Fig. 19 is compared to the data shown in Figs. 17 and 18, then relatively close agreement may be observed. If the radar had been operating at higher grazing than the 1 2 that it was, then it is possible the deepest troughs would not have been shadowed and the group modulation would have been almost invisible. The Institute of Oceanographic Sciences Deacon Laboratory (IOSDL) deployed a wave rider buoy during the experiment at Portland and collected significant wave height data, ξ 1/3, as well as current data. The average of Radar Investigation of the Structure of Wind Waves 59

12 Fig. 19. Sketch of 50 m waves travelling in 40 m depth of water in the presence of a 26 cm s 1 current, which are modulated by a finite wave group with 8 waves in each group envelope. the top 1/3 highest peak-to-trough wave heights was 1.8 m. This quantity is approximately equivalent to four times the r.m.s. wave height (Apel, 1990). ξ13 / 4σ. ( 6) Equation (6) suggests that the r.m.s. slope for the 50 m waves is around Using ak 0 = 0.057, Eq. (5) implies that there should be around nine waves in the wave group envelope. Given all the uncertainties in the measurements, and that Eqs. (5) and (6) are only approximations, the agreement between the measured group length of eight waves and the length predicted here from the wave steepness is excellent, and gives some confidence in the result. The ω-k spectrum seen in Fig. 20 is comparatively broad, and it shows that there are wavelengths present from 4 m to 75 m. The period of integration used whilst collecting the data was unfortunately too large to avoid any temporal aliasing. As a result the negative half of the frequency spectrum has been included in the image and wrapped around, so that all of the spectrum may be seen. There are waves travelling both towards and away from the radar, as indicated by the positive and negative arms of the spectrum respectively. The higher harmonics of the positive spectrum may also be seen. Figure 21 shows a sketch of the deep-water gravity Fig D Fourier transform image of Fig. 17. The negative portion of the spectrum is shown wrapped around into the positive spectrum, because of aliased frequencies. wave dispersion relation in the presence of a current of 26 cm s 1 moving towards the radar, which was found by curve-fitting the dispersion curve to the actual data. The solid lines show what would be expected for a continuous spectrum of waves of wavelengths from 75 m down to 4 m. The asymmetry between the positive and negative arms of the dispersion relation is due to the current. The positive arm is pushed up by the positive Doppler shift of the current (k V) and the negative arm is moved down. The two dashed lines show the first two harmonics of the positive arm of the dispersion curve. In much of the published literature that shows ω-k diagrams, there is typically a straight line feature through the origin that is non-dispersive, and the gradient is close to the group velocity of the dominant waves, or at least the group velocity of the waves that are breaking. For this reason, it is commonly referred to as the group line. Stevens et al. (1999) related the group line directly to the breaking wave spectrum, and they suggested that a welldefined straight group line implies that the breaking wave spectrum for their data is narrow. The group line in the ω-k diagram of the Portland data in Fig. 20 is not a straight line, but a curved fan that occupies a spread of values. The simulation of the group line in Fig. 21 shown by the two dotted lines is in fact a beat line generated by assuming the presence of wave beating. Two wave trains of different frequencies, ω 1 and ω 2, will form a modulated wave when they are superposed, in a phenomenon called beating. The periodicity of the beating is governed by the differences in their respective temporal and spa- 60 T. Lamont-Smith et al.

13 group line that is visible in most published ω-k radar imagery, rather than the kind of wave group phenomenon discussed in this paper. To establish the presence of wave groups it seems necessary to look in range-time diagrams for modulations in the radar cross section to identify the loci of wave group peaks or troughs. The modulation should be periodic in at least a few cycles, and should propagate at the group velocity over a trajectory of many group lengths, in order to be able to reasonably claim the presence of a wave group. Fig. 21. Sketch of the dispersion curves (solid lines) and the harmonics (dashed lines) with the effect of current, for a spectrum of wavelengths spanning 75 m down to 4 m. The dotted lines show the periodicity of wave beating for a 50 m wavelength wave with the other waves in the spectrum (upper line), as well as for a 10 m wavelength wave (lower line). tial frequencies ω 1 -ω 2 and k 1 -k 2. The upper of the two dotted lines is for a 50 m wavelength wave (ω 1, k 1 ) that is beating with all the other possible waves in the spectrum down to 4 m. The lower line is for a 10 m wavelength wave, similarly. Only two wavelengths are shown here for simplicity, but obviously a continuous fan would be expected for all the possible combinations of waves. It is this beating phenomenon within a broad spectrum of waves that may be responsible for the wide spread of values seen in the curved fan visible in the ω-k plot of Fig. 20. A narrow banded spectrum such as the Sound of Sleat data, shown in Fig. 14, would have a narrower range of beating waves and thus would have a narrower fan, with a straighter beat line, exactly as observed in the radar imagery. The wave group identified in the thresholded rangetime plot of Fig. 18 and modelled in the sketch in Fig. 19 was visible because it repeated at fixed space and time intervals. In the 2D Fourier domain, this should be apparent as a point (δω, δk) in the ω-k domain, and this is represented by a + symbol on the schematic plot in Fig. 21. In general, the presence of a discrete bright spot in ω-k imagery in the data analysed here, has not been obvious without the application of ad hoc processing schemes; such as the normalised variance technique that was applied to the laboratory data, or the binary threshold applied to the Portland data. Where the spectrum is close to being discrete with closely spaced discrete frequencies, then a wave group description may be most appropriate. However a stochastic model of the sea surface with wave beating is probably adequate to explain the so-called 3. Conclusions Wave phenomena were examined using radar data collected from three different sites with a view to understanding the nature of wind wave growth. The first case examined showed radar backscatter from 50 cm wind waves generated in a large (4.3 m wide) laboratory wind wave tank. The waves were young and the spectrum was narrow, as observed in the ω-k plots by looking at the dispersion curve and its harmonics. The other data sets were collected from coastal sites that were progressively more like the open ocean, with broader spectra. In the laboratory, wave groups were found by identifying and measuring detailed structure in range-time diagrams of the radar backscatter. Three waves per group were seen in the spatial domain, equivalent to six waves in the temporal domain. The size of the group was entirely consistent with the expected relationship (Benjamin and Fier, 1967) between the measured wave steepness (ak = 0.16), and the unstable sideband frequencies responsible for the wave growth and the appearance of wave groups. The nature of the wind wave spectral growth was investigated in the wave tank using range-frequency maps produced from the radar data. It was found that the dominant frequency of the waves changed significantly, even over the short fetch examined. Frequency downshifting implies a reduction in the wave crest density, which could also be seen in the laboratory data. The change in period of the waves and hence frequency, as well as the loss of a wave crest, could be seen to occur during the wave breaking phase. The change in frequency ( f = 0.3 Hz) of the waves as a consequence of the wave loss matched with the wave group frequency δω and the separation of the sideband from the spectral peak, exactly as it should do. Radar data collected at the Sound of Sleat on the west coast of Scotland showed that the wave spectrum was narrow banded like the laboratory data. The waves in the sheltered environment of the sound were likely to be young, again very like the waves in the laboratory. A wave group was readily identifiable in the range-time radar data without any additional processing. There were around six waves in each group envelope, which implied a steepness of 0.08 that is realistic of ocean waves, and which Radar Investigation of the Structure of Wind Waves 61

14 was half the steepness of the waves in the tank. A rangefrequency map of the radar data was also able to show that some downshifting was occurring. A longer time data set with a greater range swath, perhaps looking down the sound, would have been better to study spectral downshifting in more detail. The data from Portland were collected at a coastal site that was much more open and maybe more representative of the open ocean than the Sound of Sleat. The waves at Portland were older and there was a much wider band spectrum in the ω-k plot. There was some evidence of a wave group with eight waves in the group envelope, which was visible in the range-time diagram after some additional processing. The length of the group was again consistent with what would be expected for the frequency of the unstable sideband given the wave steepness (ak = 0.057) that was measured by a wave buoy. Investigation of the so-called group line visible in the ω-k imagery suggested that the feature could probably best be described in terms of wave beating between all the waves present, rather than the kind of group phenomenon discussed in this paper. Radar systems are already widely used to remotely sense the oceans to measure current, wind, and even sea state. The ω-k plots and range-frequency maps described in this paper are radar techniques that could be useful in studying a variety of wave phenomena. It has been shown here that radar has the potential to investigate the fetch laws and the details of the mechanisms of wind wave growth in the ocean, as a considerable amount of spectral information on the spatial wave field is readily available by using these kinds of processing schemes. Radar, for instance, could collect thousands of range samples over many kilometres from any direction over the ocean (depending on the site) to produce range-frequency maps of the type used here to investigate spectral downshifting. A similar experiment using wave buoys would be extremely difficult to arrange, and would not have the same kind of flexibility or capability. References Apel, J. (1990): Principles of Ocean Physics. International Geophysics Series, Vol. 38, Academic Press. Benjamin, T. B. and J. E. Fier (1967): The disintegration of wave trains in deep water. Part 1. Theory. J. Fluid Mech., 27, Blacknell, D. (1994): Comparison of parameter estimators for K-distribution. IEE Proc.-Radar, Sonar Navig., 141(1), Caponi, E. A., B. M. Lake and H. C. Yuen (1999): Hydrodynamic effects in low-grazing angle backscattering from the ocean. IEEE Trans Antennas & Propagat., 47(2), Chapman, R. D. (1992): Meteorological data from the 1991 west coast Scotland experiment. JHU/APL Tech. 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