1. Introduction
Nonlinear internal waves (NLIWs) are often generated as a result of tidal current interaction with topography in stratified water and are ubiquitous in straits and the coastal ocean. These waves may transport energy over hundreds of kilometers, crossing basins and shelf regions, and dissipate far from their generation sites. Vertical isopycnal displacement η, propagation direction θ, and propagation speed C are fundamental NLIW properties that are useful for identifying their generation and dissipation sites and for estimating their intensity, energy, and energy flux.
Shipboard and moored acoustic Doppler current profiler (ADCP) measurements capture velocity fluctuations associated with NLIWs (Chang et al. 2008; Lien et al. 2010, manuscript submitted to J. Phys. Oceanogr.; Alford et al. 2010; Ramp et al. 2010). NLIWs may have vertical currents as large as O(0.1 m s−1) and are often distinct in ADCP vertical velocity records. ADCP velocity measurements have been used to estimate NLIW vertical displacements (Lien et al. 2005; Moum and Smyth 2006), propagation direction (Scotti et al. 2005; Mirshak and Kelley 2009), and propagation speed (Scotti et al. 2005; Moum and Smyth 2006). Conventionally, vertical displacement is estimated by integrating ADCP vertical velocity in time, and propagation direction is estimated as the principal direction of the wave-induced horizontal velocity (Lien et al. 2005; Moum and Smyth 2006). However, these conventional methods do not consider the advection effect of the background flow by NLIWs.
Moum and Smyth (2006) estimate the wave speed C by C = −∂tu′/∂zw′, where u′ and w′ are the horizontal and vertical velocity components associated with NLIWs and ∂t and ∂z are the time and vertical derivatives, respectively. Scotti et al. (2005) indicate that the ADCP beam-spreading effect could seriously distort the observed beam-averaged velocity for nonhydrostatic NLIWs: that is, the large aspect ratio. They proposed a phase-lagged beam-to-earth transformation to correct the beam-spreading effect, and this method is applied to velocity measurements taken from an ADCP platform fixed to the ocean floor that is not disturbed by NLIW passage. NLIW propagation direction and speed are required for the transformation and are estimated using the temporal differences of echo intensity signals arriving at four different ADCP beams.
Here, we propose a scheme to estimate the vertical displacement, propagation direction, and propagation speed of NLIWs using velocity measurements taken from a single moored ADCP platform. The estimation process is iterative and the beam-spreading correction, depending on the propagation speed and direction, is performed at each iteration step. The method is verified using simulated NLIW data generated by a fully nonlinear Dubreil–Jacotin–Long (DJL) model. Propagation speed estimates are also compared with those computed from in situ NLIW arrival time measurements taken by three successive moorings on the Dongsha Plateau in the South China Sea (SCS; Fig. 1).
The SCS: Gray curves (solid and dashed) are surface signatures of nonlinear internal waves from satellite images (Zhao et al. 2004). Black contours are isobaths of 200, 500, 1000, 2000, and 3000 m. Three moorings, LR1, LR2, and LR3, were deployed along the line of 21°05′N near Dongsha Island.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
The paper is organized as follows: In section 2, we present the methods used to estimate the vertical displacement η, the propagation direction θ, and the propagation speed C. In section 3, we use simulation experiments to gain insight into the effects of background flow, beam spreading, and random noise of measurements on our estimates of NLIW properties and to validate our iteration scheme. In section 4, we apply our iteration scheme to in situ ADCP measurements taken in the SCS and confirm our estimates of NLIW propagation speed by comparing them with measurements from other independent methods. A summary is presented in section 5.
2. Data and methods
a. Oceanic data
Three bottom-mounted, upward-looking, 75-kHz Long Ranger ADCPs (LR1, LR2, and LR3) were deployed on the continental slope in the northern SCS in 2006 and recovered in 2007. The moorings were aligned in an east–west direction along 21°05′N on the eastern slope of the Dongsha Plateau, roughly along the prevailing westward propagation direction of NLIWs (Fig. 1). These locations were chosen to catch the evolution of NLIWs on the upper flank of the continental slope (Lien et al. 2010, manuscript submitted to J. Phys. Oceanogr.). The total measurement period was ~11 months at LR1 and LR2 and ~5 months at LR3 (Table 1 and Fig. 2). From east to west, the distance was ~8.5 nautical miles between LR1 and LR2, and ~17 nautical miles between LR2 and LR3. The water depths at LR1, LR2, and LR3 were ~600, ~430, and ~350 m, respectively. Mooring locations, water depths, and ADCP setup are summarized in Table 1.
Mooring information and ADCP configuration.
Time series of 2-h, high-pass-filtered zonal velocity at 100-m depth at (a) LR1, (b) LR2, and (c) LR3. (d) Full-depth contour plot of the zonal velocity as one NLIW passed LR3. The white curve represents the vertical displacement beginning at 100-m depth.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
The ADCP was mounted inside a syntactic float at a depth of ~10 m above the bottom to take near-full-depth velocity measurements. The blanking distance of the ADCP velocity measurements from the transducer head was about 7 m. ADCP measurements at 10% of the total water depth beneath the sea surface were contaminated by sea surface reflection and were excluded from analysis.
Satellite images frequently capture surface signatures of westward-propagating NLIWs in the northern SCS (Fig. 1). In situ observations (Yang et al. 2004; Ramp et al. 2004; Lien et al. 2005; Alford et al. 2010) suggest that NLIWs appear at tidal periodicity with amplitudes modulated at a fortnightly tidal cycle. Several possible generation mechanisms have been proposed. NLIWs might be generated within the Luzon Strait by currents flowing over rough topography (Bole et al. 1994; Liu et al. 1998; Ramp et al. 2004). Alternatively, NLIWs might evolve from internal tides (Zhao et al. 2004, Lien et al. 2005; Chao et al. 2007). Strong internal tides are generated in Luzon Strait, propagate into the SCS, are amplified by the shoaling continental slope near Dongsha Island, and then evolve into high-frequency NLIWs (Lien et al. 2005). However, further discussion of the generation mechanism of NLIWs is beyond the scope of the present study.
Zonal velocity fluctuations at 100-m depth at the three mooring sites exhibit clear fortnightly modulation (Fig. 2). Packets of large-amplitude NLIWs arrive predominantly during the semidiurnal period (Fig. 2), with amplitudes modulated fortnightly. The leading wave has a typical vertical displacement of >100 m and a maximum horizontal current speed of >1 m s−1, westward in the upper 200 m and eastward below 200 m (Fig. 2d). The maximum speed of the background flow is 0.3–0.5 m s−1, much weaker than that of the NLIW current. NLIWs were weakest between mid-January and mid-February 2007, presumably because of the mesoscale modulation of internal tidal generation and propagation. This topic is currently under investigation.
b. Separating NLIW velocities from background flow
Both earth- and wave-oriented coordinates are used in the following analysis. In the earth coordinate system, the x and y components represent the zonal (east–west) and meridional (north–south) components, respectively. In the wave coordinate system, x′ and y′ represent the along-wave and across-wave components of the NLIW propagation direction, respectively. The z component is vertical in both coordinate systems. Velocities in earth and wave coordinates are expressed as (u, υ, w) and (u′, υ′, w), respectively. The background velocities in earth and wave coordinates are (U, V, W) and (U′, V′, W), respectively.



Equations (3)–(5) express the total particle velocity in terms of the vertical displacement and propagation velocity of NLIWs and of the background flow. Our purpose here is to describe the method to extract NLIW vertical displacement, propagation speed, and propagation direction using observed velocity measurements from ADCP mooring and to investigate the importance of the modulation by the background flow.
c. Calculating NLIW properties from moored ADCP data



In the absence of the background flow, the expression of η is reduced to the generally used form
Conventionally, the NLIW propagation direction is estimated as
3. NLIW simulations

In the following analysis, we present an idealized fully nonlinear, steady-state DJL NLIW that satisfies the typical density profile and background horizontal current at LR1 (Fig. 3). At LR1, the NLIW propagates at a mean direction θ = 150° (0° at true east, counterclockwise from east), and the typical background current is at 175°, with a maximum current speed of 0.35–0.4 m s−1 in the upper layer (Fig. 3a). The buoyancy frequency squared at LR1 usually has a maximum value of 4 × 10−4 s−2 at ~50 m (Fig. 3b). The background current is projected onto the along-wave and across-wave components. The along-wave component of the background current and the buoyancy frequency are used in (9) to obtain DJL solutions. The total current velocity along the NLIW propagation direction u′ and perpendicular to the NLIW propagation path υ′ are obtained through (3) and (5), respectively. The vertical current velocity w is obtained through (4). The current velocity field (u′, υ′, w) on the wave coordinate can be transformed to the earth coordinate (u, v, w) using the known propagation direction θ.
Typical vertical profiles of (a) background velocity U′ and (b) buoyancy frequency squared N2 at LR1.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
a. Background flow effect
We chose the DJL solution with C = −2.21 m s−1 as our test wave. The horizontal and vertical velocities of this test wave are shown in Figs. 4a,d. The along-wave component of velocity in (3) includes the strain term Cηz associated with the NLIW (Fig. 4b) and the effect of the background flow
Contour of (a) the total horizontal velocity u′, (b) the horizontal velocity induced by an NLIW without the background flow
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
This test wave is used to validate the method and the iteration scheme for extracting the NLIW propagation speed and direction and vertical displacements using velocity observations (section 2c). The propagation direction and speed of the wave are computed using (7) and (8) (Figs. 5a,b, respectively). The estimates of C = −2.21 m s−1 and θ = 150° are identical to those of the test wave, by construction. In estimating the propagation direction, ignoring the effect of the vertical displacement of the background current, i.e.
(a) Scatterplot between ∂zw and ∂tu′ for the estimate of the propagation speed, (b) scatterplot between u − U(z − η) and υ − V(z − η), (c) scatterplot between u − U(z) and υ − V(z), and (d) scatterplot between u and υ for estimates of the propagation direction. Dot colors represent measurements at different water depths. The blue line in (a) represents the Deming regression fit for the estimate of the propagation speed. Black lines in (b),(c), and (d) represent the Deming regression fit of NLIW propagation direction using (7). Gray lines in (c) and (d) represent the Deming regression fit for the propagation direction using equations
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
b. ADCP beam-spreading effect and correction
The 75-kHz Long Ranger ADCP used on moorings LR1, LR2, and LR3 has four beams of angle φ that are 20° from the vertical and separated 90° horizontally. The horizontal separation between beams (i.e., the beam spreading) increases with the distance from the transducer heads. Vertical profiles of horizontal and vertical velocity components are computed using the along-beam velocities observed by the four beams. Therefore, the average horizontal spatial scale of ADCP velocity measurements increases with the distance from the transducer head, referred to as the “beam-spreading effect.”
Comparison between Figs. 6a,d shows three distorted features due to the beam-spreading effect: 1) the nodal point depth is raised from ~260 to ~220 m, 2) a maximum eastward velocity core exists at 300–400 m, and 3) an M-shaped westward velocity feature is present at depths between 200 and 300 m in the distorted velocity field (Fig. 6a). The vertical displacement, estimated as
Contours of (a) simulated zonal velocity, (b) simulated meridional velocity, (c) simulated vertical velocity, (d) corrected zonal velocity, (e) corrected meridional velocity, and (f) corrected vertical velocity. The simulated velocities (a)–(c) illustrate the ADCP beam-spreading effect. The contour line interval in (a),(b),(d), and (e) is 0.2 m s−1. The contour line interval in (f) is 0.1 m s−1. White curves in (c) represent the vertical displacements of the DJL test wave. Black curves in (c) represent the vertical displacement computed as
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1






We compute the propagation speed of the test wave using beam-spreading distorted velocities according to (8). The beam spreading introduces noise into the estimate (cf. Figs. 5a, 7a) and results in an estimate of propagation speed C = 3.22 m s−1, which is 46% faster than that of the test wave. The scatterplot between ∂zw and ∂tu′ shows significant deviations from the regression line in the upper 250 m, where the beam-spreading effect is strong (Fig. 7a). The beam-spreading effect does not introduce significant error into the estimate of propagation direction for the test wave (cf. Figs. 5b–d with Figs. 7b–d).
ADCP beam-spreading effect on a DJL test wave. Scatterplots for (a) the NLIW propagation speed, (b) the propagation direction using (7), (c) the propagation direction using
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
c. Random noise effect in ADCP measurements
The random noise intrinsic to the ADCP velocity measurements may affect the estimate of propagation direction [i.e., the regression slope between u − U(z − η) and υ − V(z − η)] and the estimate of the propagation speed [i.e., the regression slope between ∂tu′ and ∂zw′]. To examine the effects, we added a random noise value of 0.05 m s−1 to the test wave’s horizontal velocity and 0.025 m s−1 its vertical velocity. Using the standard linear regression method, our estimates of the NLIW’s propagation direction and wave speed have errors of ~3° and ~0.5 m s−1, respectively.
The standard linear regression analysis is inappropriate when both fitted (y axis) and reference (x axis) variables are subject to random errors. On the other hand, Deming regression analysis, also called the errors-in-variables regression, accounts for measurement errors in both fitted and reference variables (Deming 1943). Using this method, we recovered the true propagation speed and direction of the test wave.
d. An iterative method
An iterative method is used to estimate η, θ, and C and to correct the ADCP beam-spreading effect from the velocity measurements. The iteration begins from the estimate of the vertical displacement (6) with the initial estimate
Iterative analysis estimates of NLIW properties using (6)–(8). Each iteration is plotted to estimate (a) propagation speed; (b) propagation direction; (c) root-mean-squares of the difference between corrected zonal velocity u and zonal velocity uo, between the corrected meridional velocity υ and meridional velocity of the test wave υo, and between the corrected vertical velocity w and vertical velocity of test wave wo; and (d) root-mean-squares of the difference between the corrected vertical displacement and vertical displacement of the test wave ηo.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
This test wave analysis shows that 1) the background flow may lead to errors in the estimates of η and θ by ~15 m and ~5° for the test wave; 2) the beam-spreading effect leads to the distortion of the measured velocities of the test wave and produces serious errors in estimates of η and C (i.e., ~30 m and ~2 m s−1); and 3) the iteration scheme to estimate η, C, and θ using (6), (7), (8), and (11) is effective.
e. Temporal sampling effect in ADCP measurements
As the NLIW passes the mooring location, the temporal sampling number Ns could have an impact on the estimate of properties because the sparse measurements are expected to distort the flow field. The sampling interval of ADCP is fixed (Table 1), so Ns is primarily determined by the propagation speeds and horizontal scales of NLIW. The speed of NLIW propagation correlates negatively to Ns. To assess the effect, the iterative method mentioned above was applied to the temporally resampled test wave to estimate C and θ. We found that the scarcity of Ns results in the propagation speed of NLIW being underestimated, possibly because u′(t) is oversmoothing because of sparse sampling. The estimate error decreases when Ns increases, and the significant estimate errors of C (e.g., >25% of the true values) occur when Ns ≤ 3. The estimate errors of C are <8% of the true values when Ns ≥ 6 and then decrease below 4% of the true values when Ns ≥ 7. However, the effect of temporal sampling does not have a significant impact on the estimate of θ. The estimate errors of θ are <3% of the true values if Ns ≥ 3. We will further discuss the effect of Ns in the next section.
4. NLIW observations in the South China Sea
a. Iterative method applied to ADCP velocity measurements
NLIWs are active in the northern SCS, with horizontal velocities as strong as ~2 m s−1 and amplitudes > 100 m. The background flow is primarily associated with internal tides with a maximum current speed of 0.3–0.5 m s−1. At LR1, LR2, and LR3, the separation between ADCP beams is ~1 m near the transducer head and reaches to ~400, ~300, and ~200 m near the surface, respectively. The typical NLIW horizontal scale is ~1 km (Chang et al. 2008). The assumption of a spatially homogenous velocity field, as in the standard ADCP data processing algorithm, is clearly inappropriate for these small-scale NLIWs. The beam-spreading effect is significant for our velocity measurements.
We compute spectra of u, υ, and w velocities and determine the spectral noise floor. The random noise for u and υ is 0.05–0.06 m s−1 and for w is 0.01–0.02 m s−1. These random noise estimates are used for the Deming regression analysis.
Scotti et al. (2005) proposed a method to correct the beam-spreading effect in ADCP measurements. This method is straightforward if the ADCP platform is fixed. During NLIW events in the SCS, the passing waves jolted our ADCP platform, mounted in a buoyant float. The ADCP pitch and roll remained within 4° and may be ignored. However, the ADCP platform often experienced strong rotational forces with heading changes of nearly 90° during the NLIW passage.
We identified 358 NLIW packets passing the LR1 and LR2 moorings. The LR3 mooring was deployed for a shorter period and only captured 151 NLIW packets. For the following analysis, we identified leading waves with ADCP heading fluctuations within 20° and maximum horizontal speeds > 0.5 m s−1. A total of 205 leading waves passed LR1 and LR2 sequentially and satisfied the criteria, of which 70 also sequentially passed LR2 and LR3 and were also selected.
An example wave event is shown in Fig. 9, with a maximum NLIW horizontal velocity of ~1.7 m s−1 and a maximum vertical displacement of ~130 m. The background flow velocity is ~0.3 m s−1 (Fig. 9a). The structure of the observed zonal velocity field shows features typical of beam-spreading distortion (cf. Fig. 6a). Given estimates of the propagation speed and the direction of the NLIW, the beam-spreading effect is corrected using (11) and the true velocity field is recovered (Fig. 9b).
One NLIW event: (a) Raw ADCP zonal velocity measurements, (b) beam-spreading corrected zonal velocity, and (c) ADCP echo intensity. The white curves in (c) represent estimates of vertical displacements between 0 and 600 m at 50-m intervals computed using (6). The velocity contour interval in (a) and (b) is 0.2 m s−1.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
The iterative method applied to ADCP velocity observations at LR1, LR2, and LR3 yields estimates of η, θ, and C (Fig. 10). The analysis converges after the fourth iteration for estimates of NLIW θ, C, and velocity. The vertical displacements converge after the ninth iteration, and the estimated vertical displacement agrees well with the water scatter fluctuation induced by the NLIW as shown by ADCP echo intensity (Fig. 9c). Propagation speed is estimated at −1.83 m s−1 (Fig. 11c), close to the maximum westward current speed. Without the beam-spreading correction, the propagation speed estimate is −3.91 m s−1 (Fig. 11a) and is much noisier. The propagation direction is 154° using the beam-spreading corrected velocities (Fig. 11d) and is 151° without the correction (Fig. 11b). Again, as seen in the simulated test wave, the beam-spreading has a minimal effect on the estimate of propagation direction.
Iterative analysis to estimate NLIW properties from oceanic ADCP measurements using (6)–(8) with the beam-spreading correction. Each iteration step is plotted to estimate (a) propagation speed; (b) propagation direction; (c) root-mean-squares of the difference between zonal velocity in iterations u and the corrected zonal velocity un, between meridional velocity in iterations υ and the corrected meridional velocity υn, and between vertical velocity in iterations w and the corrected vertical velocity wn; and (d) root-mean-squares of the difference between vertical displacement in iterations and the corrected vertical displacement ηn.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
An example of estimating the propagation speed and the propagation direction from ADCP mooring velocity measurements: scatterplots for estimating (a) the propagation speed, (b) the propagation direction using (7), (c) the propagation speed using beam-spreading corrected velocities, and (d) the propagation direction using beam-spreading corrected velocities. Dot colors represent measurements at different water depths.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
Moored ADCP velocity observations show large variations in the slope between ∂zw and ∂tu′, revealing significant scattering (Figs. 11a,c), most of which appears in the upper 100–200 m, where the beam-spreading effect is stronger. The iteration analysis yields convergent results (Figs. 11b,d); the regression slope between ∂zw and ∂tu′ for propagation speed and between u − U(z − η) and υ − V(z − η) for propagation direction are much better defined. Through the iteration procedure, the correlation between ∂zw and ∂tu′ increases from 0.45 to 0.86 and the correlation between u − U(z − η) and υ − V(z − η) increases from 0.84 to 0.92 (cf. Figs. 11a,c and cf. Figs. 11b,d). The iteration procedure reduces the error of the propagation speed estimate by ~76% and for the of the propagation direction by ~40%.
The average half-amplitude full width of the NLIWs is ~1 km (Chang et al. 2008), and the full width of NLIWs is ~1.5 km. At LR1, the sampling intervals are 90 s (Table 1). Accordingly, Ns is ~7 and ~8 for NLIWs propagating with typical speeds of 2 and 2.5 m s−1, respectively. The simulation from the previous section suggests that the estimate errors for C can be expected to be <4% of the true value, and the maximum estimate errors of C are ~0.1 m s−1. At LR2 and LR3, the sampling intervals are 60 or 30 s (Table 1). The estimate errors of C are O(0.01 m s−1). For this reason, we did not correct the temporal sampling effect here. Otherwise, the sampling interval effect is also related to the wave shape and the observation points where the wave is sampled. The estimate errors calculated from test wave simulated from the DJL model may not accurately represent individual waves as measured by ADCP.
b. Validation of propagation speed estimates
Arrival time data for NLIWs at the three mooring sites provide another independent estimate of the zonal component of NLIW propagation speed. Using the estimates of propagation direction from the iteration analysis, we compute the meridional component and the average propagation speeds (including both zonal and meridional components) of NLIWs between LR1 and LR2, CLR12, and between LR2 and LR3, CLR23. Using the iteration results, we compare estimates of the propagation speed CLR12, from the arrival time of NLIWs at LR1 and LR2, with (CLR1 + CLR2)/2, which is the average of the propagation speed estimates at LR1 and LR2, and also compare CLR23 with (CLR2 + CLR3)/2. These two independent estimates are in general agreement, albeit with large scatterings (Fig. 12). The linear correlation coefficient is 0.7, and the 95% significance level is 0.21.
Estimates of propagation speeds using the difference of arrival times at three moorings and estimates of propagation speeds using the iterative method applied to velocity measurements at single moorings. Vertical bars represent the 95% confidence interval of the propagation speed using the iteration method.
Citation: Journal of Atmospheric and Oceanic Technology 28, 6; 10.1175/2010JTECHO814.1
Several factors may explain the discrepancies between the two independent propagation speed estimates. Our estimate of C using the iteration analysis might contain errors because 1) the motion of ADCP platform motion during the NLIW event could result in an inaccurate beam-spreading correction, 2) some oceanic processes are excluded in the iteration analysis (6)–(8) (e.g., unsteady internal waves, turbulence mixing, and trapped core), and 3) the NLIW propagation speed has a strong nonlinear dependency on the local depth. The second and third reasons suggest that NLIW propagation speed may not be linear between mooring sites, so the average speed between the two moorings (e.g., CLR12) is different from the average speed at two mooring sites [e.g., (CLR1 + CLR2)/2].
5. Conclusions
Moored ADCPs are used to capture velocity measurements during NLIW events. However, important NLIW parameters, including propagation speed, propagation direction, and vertical displacement, are not directly measured by the moored ADCPs. We present an iterative method to estimate these parameters using velocity measurements from a single moored ADCP. This method also corrects the beam-spreading effect of ADCP measurements.
A simulated NLIW that satisfies the DJL solution and three sets of moored ADCP measurements taken in the SCS validate our approach. NLIW vertical displacement is computed as the integration time of the direct vertical velocity observations, with the effect of the background flow corrected. The propagation direction is estimated as the principal direction of the wave-induced horizontal velocity, with the effect of background current removed. The propagation speed is estimated as ∂tu′/∂zw′ in the NLIW’s propagation direction, assuming that the wave structure and the propagation speed remain constant. We use Deming regression analysis to compute the slope between ∂tu′ and ∂zw′. Our analysis concludes that NLIW vertical displacement, propagation direction, and propagation speed can be estimated from velocity measurements from a single moored ADCP. The background flow and its vertical advection by NLIWs significantly affect the estimates of NLIW properties significantly and cannot be ignored, especially for large-amplitude NLIWs and in strong shear flows.
Acknowledgments
The authors thank the crew of the Taiwanese R/V Ocean Researcher 1, Mr. Her, and graduate students at the Institute of Oceanography of National Taiwan University for the mooring preparation and operation. Discussions with Frank Henyey at the Applied Physics Laboratory, University of Washington, and Stastna Marek at the Department of Applied Mathematics, University of Waterloo, greatly assisted in our analysis and presentation. We also thank Kevin Lamb for providing his DJL model program codes. This work was supported by National Science Council (Grants NSC98-2745-M-019-00, NSC95-2611-M-002-016-MY3, NSC 94-2611-M-012-001, and NSC 95-2611-M-012-003-MY3) of Taiwan and by the U.S. Office of Naval Research (Grant N00014-04-1-0237).
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