Observation on modulation of short wind waves Xin Zhang Scripps Institution of Oceanography Univ. of Califor., San Diego, MS-0213 La Jolla, California 92093, USA
[email protected] Abstract—Modulation of the propagation of short wind waves by the dominant long wind wave components is investigated in a laboratory experiment. The theoretical predictions are confirmed by the observed three dimensional frequencywavenumber spectra of short waves. (Abstract) Keywords: short waves, free waves, bond waves, modulation, microwaves, Doppler backscatter
I.
INTRODUCTION
Short wind-waves, rough surface elements of the sea, are of great interest in understanding air-sea interchange and microwave remote sensing. Techniques based on microwave backscattering, such as scatterometer determination of ocean wind stress and ocean wave imaging by synthetic aperture radar, have demonstrated the capabilities for monitoring the dominant ocean directional wave spectrum, the ocean wind stress, internal waves, fronts, currents and shallow water bathymetry under a wide range of ambient conditions. It is the spatial distribution of modulated very short waves that manifests the above large scale phenomena in the radar images. The surface processes that modulate short surface waves are the keys to understanding of electromagnetic remote sensing. Here, we consider in particular the interactions between dominant wind waves and short wind wave components. Long waves modulate not only the energy but also the wavelength, frequency and propagation speed of short waves. In the first order of long wave slope AK [1][2]:
σ f a = σ 1 + AK cos(ψ ) (1 + AK cos(ψ )) F
(1)
k = k (1 + AK cos(ψ ))
(2)
Here, f a is the short wave apparent frequency, σ and k are the values of short wave intrinsic frequency, σ , and wavenumber, k , at mean surface level, F , K , A , and ψ are the frequency, wavenumber, amplitude, and phase of the underlying long waves respectively. The apparent speed of short gravity waves, ca is: ca =
fa C = c p 1 + −1 AK cos(ψ ) + U d c p k
(3)
where c p is the value of short wave phase speed, c p , at the mean water level, and C is long wave phase speed. U d is the
surface wind drift. As dominant long waves become steeper, the modulation at the long wave crests (compress) is much more effective than at the troughs (stretching) [3]. Here, we report some observations of frequency modulation from our laboratory wind wave experiments, in which three dimensional short wave spectra are obtained. II.
EXPERIMENTAL SET UP
Our experiment is carried out at the wind wave facility at Scripps Institution of Oceanography. The wind-wave tank is about 44 m long with a cross section of 2.4 m by 2.4 m. Water level was set to 1 m for the experiment. Wind-generated short gravity waves were measured with a refraction-based optical imaging technique [4]. Due to a low camera frame rate (30 Hz) as compared to the apparent frequency of short surface waves, surface imaging methods are often limited to measure two-dimensional wavenumber spectrum only. In order to measure three dimensional frequency-wavenumber spectrum, our surface imaging system was towed along the wind direction following the propagation of waves. The slope image data were taken at the fetches from 15 m to 19 m. In addition, near surface flows were also observed at the same time with particle image velocimetry and infrared surface imaging. Wave height is also measured by wave wires. III.
PROPAGATION MODULATION OF SHORT WAVES
Due to the long wave orbital advection, the propagation speeds of short waves are unevenly distributed on the different phases of the underlying long waves. They are faster near the long wave crests and slower near the troughs. The modulation of short wave speed is depended on the both of steepness and wavelength of the long waves. The modulation of propagation speed and amplitude of short waves by the orbital velocity of the underlying long waves can actually be observed directly from our image sequences. Figure 1 shows composed surface slope image time sequences. Each sequence consists of slices of surface gradient image (2.2 cm by 23.2 cm). Each slice was taken out in the wind direction from a surface image. The image slices are placed side by side in the order of time. Images are captured 1/30 second apart. The image intensity represents the magnitude of surface gradient. The brightist features (high in surface slope) are usually associated with short wave crests where parasitic capillaries reside. The orientation of gray value structures indicates the propagation speed of the wave structures. The long wave crests propagate at the most fast speed as marked in the figure by white straight lines. The underlying long wave in the top image is less steep
than that in the bottom image, and the modulation is stronger in the case of the bottom image than the top image. The short waves become steeper (higher intensity) and faster (higher orientation) near the long wave crests. The second long wave crest in the lower sequence is involved in a small-scale breaking, sweeping all the short waves.
Figure 1 Surface slope image sequences illustrate the orbital modulations of short wave speed and slope. The size of each slice of surface gradient image is 2.2 cm by 23.2 cm. Wind speed is 9.4 m/s and in upward direction. The time interval between the subsequent images is 1/30 second. The image intensity is proportional to the surface gradient and the scale is shown at the bottom of the figure.
The averaged propagation properties of short wind waves can be best presented by wavenumber-frequency spectrum. In figure 2, along-wind-direction sections of three dimensional wave spectra (degree of saturation) at wind speeds of 4.0 m/s, 5.3 m/s, and 7.7 m/s are shown respectively. The spectra are calculated from an average of 60 runs slope image sequences. For each run, 5 seconds of 150 images are captured started
immediately after the cart carrying our image apparatus began to move. The cart accelerates smoothly and reaches a uniform speed within a half second. The images within the first half second are not used here in calculating spectrum. In three dimensional frequency-wavenumber space, the energy of surface waves are concentrated only in the domain restricted by free or force modes. The aliasing of wave components that over Niquist frequency is not as bad as one dimensional case and some of these high frequency components can be recovered by periodic extension of spectral values as show in figure 2. Linear dispersion relation is plotted as solid curves. When we calculate these curves, the cart speed and surface drift from separate measurements are taken into consideration. The dashed straight lines are for bound waves, the wave components that propagate at the phase speeds of dominant long waves. Our cart is moving at a speed faster than the speeds of short waves but slower than the speeds of dominant long waves, so that free propagating short waves and bound waves can be maximally separated in spectral domain. At low wind of 4.0 m/s, the harmonics of dominant waves are much stronger as compared to the higher wind speed cases. As wind speed increases, the bound waves components are quickly diminished as compared to the free wave components. The bound waves are not as strong as observed in shorter fetch laboratory wind wave experiments [5]. This may due to two facts. First, it is well known [6] that young wave age waves are much steeper, and naturally have stronger harmonics. Second, the harmonics decay more rapidly than free wave spectrum. The wave length of dominant wave increases as wind and the corresponding higher order harmonics have less weight in the short wave range. The modulation of short wave propagation is also evident from the frequency-wavenumber spectra. In figure 3, each curve is a spectral profile of figure 2 with a fixed wavenumber. The spectral profile series are plotted from top to bottom with an increasing wavenumber from 0.7 rad/cm to 3 rad/cm, representing short waves from about 9 cm to 1 cm. The
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Figure 2 Three dimensional wave spectra of degree of saturation. The contour plots are the sections along wavenumber in the wind direction. Solid curves are for free propagating linear waves and dashed straight lines are for bound waves.
triangles mark the corresponding wave frequencies for bound waves travel at the speed of dominant waves, and squares for free waves travel at linear wave speeds. The modulation increases the breath of free wave peaks. As the theory suggested suggested (Eq. 1), the modulation increases with the ratio of σ / F . Indeed, the peak of free waves broadens as decreasing in short wave wavelength ( σ ↑ ) or increasing wind speed ( F ↓ ). As wind speed increases, the wavelength of the dominant wind wave increases, which corresponds to decreasing of frequency of dominant waves. Also, this modulation is much more skewed towards higher frequency, corresponding a stronger modulation at the longwave crests than the troughs. This is due to the non-linearity of longwaves as shown theoretically by Longeut-Higgins [3]. IV IMPRECATIONS ON DOPLLER MICROWAVE RETURNS In papers published over a 30 year period, a difference was noted in the mean Doppler shift of horizontally and vertically polarized microwave sea return obtained at large incidence angles. The sea return depends theoretically on surface waves of a certain wavelength that resonate with microwave wavelength according to the Bragg resonant mechanism. However, the different Doppler shifts of polarized sea returns imply that the polarized microwaves are resonant with different waves that propagate at different speeds (fast and slow scatters). The physics of scattering at grazing incidence has been disputed over the years. The propagation modulation has been over looked in the past. Although, at high wind speeds, we did not find the existence of pronounced bound waves which propagate at high speeds, freely travelling short waves do propagate fast near long wave crests, and slowly near the troughs. We therefore wind speed = 4.0 m/s
log(F)
suggest that speed modulation can also be a substantial contributor to the Doppler microwave return from not very young wind waves in the laboratory tanks. In these cases, the dominant waves are not as pronounced as wind waves of young wave age, so that the harmonics of the dominant waves are not strong in comparison to the free waves. Based on the basic idea of composite surface scattering theory, the HH cross section is weighted towards the short waves near the crests and it has a larger Doppler frequency. In contrast, the VV cross section is weighted slightly towards the short waves near the trough and has a lesser Doppler frequency. ACKNOWLEDGMENT This work was supported by the National Science Foundation. (OCE0118028). REFERENCES [1]
[2] [3] [4]
[5]
[6]
Longuet-Higgins, M. S. and R. W. Stewart, “Changes in the form of short gravity waves on long waves and tidal currents”, J. Fluid Mech. 1960, v8, pp565-583. Phillips O. M “The dispersion of short wavelets in the presence of a dominant long wave”. J. Fluid Mech. 1981, v107, pp465-485. Longuet-Higgins, M. S. “The propagation of short surface waves on longer gravity waves”. J. Fluid Mech. 1987, v177, pp293-306 Zhang, X and C.S.Cox, “Measuring the Two Dimensional Structure of a Wavy Water Surface Optically: A Surface Gradient Detector”, Exp. Fluids, 1994, v 7, pp225-237. Plant, W.J., Keller, W. C., Hesany, V., Hara, T., Bock, E., and Donelan, M. A., “Bound waves and Bragg scattering in a wind-wave tank,” J. Geophys. Res. 1999,v104, No. C2, 3243-3263Y. Donelan, M. A., Hamilton, J. and Hui, W. H. “Directional spectra of wind-generated waves” Phil. Trans. R. Soc. Lond., 1985 A 513, pp509562.
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Figure 3. Spectral profiles with fixed wavenumbers. The spectral profile series are plotted from top to bottom with an increasing wavenumber for 0.7 rad/cm to 3 rad/cm. The squares mark the corresponding wave frequencies for free waves travel at linear wave speeds, and triangles for bound waves travel at the speed of dominant waves.
