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The Dispersion Relation of Short Wind Waves from Space–Time Wave Measurements*

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The Dispersion Relation of Short Wind Waves from Space–Time Wave Measurements* 1936 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 The Dispersion Relation of Short Wind Waves from Space±Time Wave Measurements* DAVID W. W ANG AND PAUL A. HWANG Oceanography Division, Naval Research Laboratory, Stennis Space Center, Mississippi (Manuscript received 17 November 2003, in ®nal form 13 March 2004) ABSTRACT To study the dispersion relation of short wind waves, a linear wave gauge array (WGA) is con®gured and mounted on a wave-following buoy to conduct in situ space±time measurements of short gravity waves. Results from two ®eld deployments of the WGA buoy in growing seas are presented. The two-dimensional (2D) wave- number±frequency spectra derived from the space±time measurements provide a direct examination on the relation of wave frequency and wavenumber of short waves in the along-wind direction. Both wavenumber-based and frequency-based phase velocities are extracted from the 2D spectra. The effect of higher harmonics resulting from the Fourier decomposition of nonlinear wave pro®les is more prominent to the frequency-based phase velocity than the wavenumber-based phase velocity. The wavenumber-based phase velocity is consistent with that according to the linear dispersion relation, while the frequency-based phase velocity becomes larger due to the higher harmonics. 1. Introduction g 0.5 CL(k) 5 or (2) The wave dispersion relation serves as a key discrim- 12k inator of the dynamical properties of the surface wave g ®eld. For deep water gravity waves, the linear dispersion CL(v) 5 . (3) relation between the magnitude of wavenumber k and v frequency v can be expressed as Studies based on wave measurements from laboratory and ®eld experiments have shown that the short wave v 2 5 gk, (1) components do not follow the linear dispersion relation, and the phase velocity C(v) is signi®cantly larger than where g is the gravitational acceleration. This relation that according to the linear dispersion relation (e.g., provides an avenue to connect wave characteristics ob- Ye®mov et al. 1972; Mitsuyasu et al. 1979; Ramamon- served between spatial and temporal domains. The dis- jiarisoa and Mollo-Christensen 1979; Rebuffat and Ra- persion relation is crucial to remote sensing applica- mamonjiarisoa 1981). Effects of nonlinear steep dom- tions, such as scattering of radar waves by ocean surface inant waves on the dispersion relation and phase ve- roughness. One quanti®able variable that can be ex- locity of short waves have been studied (Longuet-Hig- tracted from the dispersion relation between the wave- gins and Phillips 1962; Huang and Tung 1976; Masuda number and frequency is the phase velocity C 5 v/k. et al. 1979; Laing 1986). It is generally believed that In practice, the phase velocity can be derived either as waves in open oceans are less steep and do not affect a wavenumber-based variable C(k) from spatial mea- the dispersion relation of short waves. Due to the dif- surements or a frequency-based variable C(v) from tem- ®culties of conducting reliable in situ measurements of poral measurements. According to the linear dispersion short waves in open oceans, there is a lack of solid relation (1), we can have evidence to validate this assumption. Most ®eld studies on the dispersion relation focus on either long gravity waves [up to 0.5 Hz in Donelan et al. (1985), up to 1 Hz in Hara and Karachintsev (2003), and up to 12.6 rad * U.S. Naval Research Laboratory Contribution Number NRL/JA/ 21 7330-03-21. m in Gotwols and Irani (1980)] or gravity-capillary waves (Hara et al. 1998; wavenumbers ranging from 50 to 800 rad m21). It is of great interest and importance Corresponding author address: Dr. David W. Wang, Meso- and to have reliable ®eld measurements for short gravity Finescale Ocean Physics Section, Department of the Navy, Naval Research Laboratory, Code 7330, Stennis Space Center, MS 29529- waves to bridge the data gap and improve our under- 5004. standing. E-mail: [email protected] Methods used to study the dispersion relation are Unauthenticated | Downloaded 09/30/21 02:46 AM UTC DECEMBER 2004 WANG AND HWANG 1937 based on the assumption that the wave ®eld can be range of 1 m, based on the design by Chapman and treated by a linear superposition of sinusoidal wave Monaldo (1991). The wires are installed at a spacing of components. One commonly used method is the cross- 0.0508 m, which provides a spatial measurement for spectral method, which estimates frequency-based wavenumbers ranging from 6.18 to 61.8 rad m21. The wavenumber k(v) and phase velocity C(v) from the temporal sampling rates for the deployment of St. An- cross spectra of temporal wave measurements at two drew Bay and the Gulf of Mexico are 25 and 50 Hz, adjacent locations. The method has its limitations and respectively. The mean measurement resolution of the bias (Huang 1981; Dudis 1981) and is only considered 20 wires is 0.23 mm with a standard deviation of 0.0028 to be valid for phase velocities in the frequency region mm for the St. Andrew Bay deployment and 0.19 mm around the dominant wave component. Nonetheless, it with a standard deviation of 0.0037 mm for the Gulf of has been widely adopted in many studies because of its Mexico deployment. The measurement resolutions are simplicity and availability (e.g., Ye®mov et al. 1972; obtained from static calibrations. The potential impacts Rikiishi 1978; Ramamonjiarisoa and Mollo-Christensen to the resolutions from the capillary effect and self- 1979; Mitsuyasu et al. 1979; Thornton and Guza 1982; disturbance of the capacitance probe are not considered Elgar and Guza 1985; Laing 1986). The relation be- and require further study. This wave gauge array pro- tween wavenumber and frequency of a random wave vides a direct space±time measurement of surface wave ®eld in open oceans should be best studied via the use elevation h(y, t) along a transect direction over the of a wavenumber±frequency spectrum, which maps the gauge array length (1 m). energy distribution in both space and temporal domain. During the two ®eld deployments, the array is mount- Following the energy distribution on the 2D plane, we ed on a surface-following buoy and is oriented along can directly examine the relation between wavenumber the direction of the buoy's bow. The buoy is deployed and frequency and extract the phase velocity. in free drifting mode to reduce the relative velocity be- Using the wavenumber±frequency spectra estimated tween the surface current and the structural members from directional wave measurements, Donelan et al. piercing through the water surface. This operation can (1985) and Hara and Karachintsev (2003) examined the signi®cantly reduce local wave generation by the buoy dispersion relation and phase velocity. Their results are members and minimize ¯ow distortion to the sensing limited by the directional resolutions and the data-adap- region (Hwang et al. 1996). This free-drifting operation tive techniques (Irani et al. 1986). Gotwols and Irani also reduces the potential effect of Doppler frequency (1980) studied the phase velocity of short waves using shifts (see the appendix). The buoy's bow is aligned into 2D spectra derived directly from wave images recorded the local wind direction by a ®n (about 0.4 m2 in area) by a video camera. Their results are limited by the ef- attached on the mast. Under steady wind conditions, the fects of sky radiance and re¯ection in the image (Irani orientation of the wave gauge array (referred to as the et al. 1986). The above studies are all subject to un- along-wind array) is approximately in the direction of certainties and limitations imposed by the measurement dominant wind wave. The locations of the free-drifting systems and data processing schemes. To have an ac- buoy are registered by the GPS recording onboard the curate examination of the dispersion relation, it is highly buoy. The recordings are then used to estimate drifting desirable to use wavenumber±frequency spectra derived speed and direction. Effects from the buoy's transla- from in situ high-resolution space±time measurements tional and angular motions on wave measurements from of short waves. However, the dif®culties of conducting the arrays are corrected using the outputs from motion such space±time data acquisition make this rarely prac- sensors mounted on the buoy following the approach ticed in the ®eld experiments. by Hanson et al. (1997). More details about the buoy Recently, we con®gured a linear wave gauge array designs and its operations can be found in Hwang et al. (WGA) mounted on a wave-following buoy to carry out (1996). An ultrasonic anemometer (Handar 425A) in situ space±time measurements for short wind waves. mounted on the buoy mast at about 1.3 m about mean In this study, we present results from two ®eld deploy- water line provides wind speed and direction measure- ments under active wave generation conditions: one in ment at 1-Hz sampling rate. Average wind speed at 10- a well-sheltered estuary of St. Andrew Bay, Florida, and m elevation U is then estimated from the measured wind another in nearshore waters of the Gulf of Mexico. Sec- speed combined with humidity and air and water tem- tion 2 provides a description of the wave gauge array peratures using the method by Liu et al. (1979). system and environmental conditions of the two de- ployments. Analysis results are presented in section 3. b. St. Andrew Bay deployment Concluding remarks are given in section 4. The ®rst ®eld deployment of the WGA buoy was 2.
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