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  • View in gallery

    Mode-1 linear internal wave phase speed as a function of depth for the northern South China Sea. The data are derived from temperature and salinity profiles acquired during ship surveys in the region between April 2005 and July 2007. Data are distinguished for quarter and year by symbol and shading, respectively. The dashed line is the nonlinear internal wave phase speed from the empirical model solution presented in Table 1.

  • View in gallery

    Internal wave signatures in the South China Sea from satellite imagery (gray) with the wave front locations from the empirical model (black). There are 141 internal wave signatures from 73 satellite images acquired on 61 days between April 2003 and December 2006. The shape and orientation of the model wave fronts agree well with the satellite signatures.

  • View in gallery

    PIES data from the northern portion of the South China Sea with the east–west TPXO6 model currents from Luzon (20.56°N, 121.24°E). In the PIES data, the nonlinear internal waves are the large spikelike excursions in the direction of shorter delay. Shifting the current +0.677 days (16.24 h) aligns the internal wave occurrences with the westward tidal current maximums (east/west current minimums).

  • View in gallery

    Propagation time residuals (observations − model) as a function of depth for the internal wave signatures presented in Fig. 2. Vertical gray lines are wave observations from moorings B1 (dark gray) and P1 (light gray).

  • View in gallery

    Monthly breakdown for (a) January–June and (b) July–December of the propagation time residuals from the satellite observations presented in Fig. 2. The figures show that the variance can be significant from month to month and be large even in months where the internal wave observations come from a single year. Gray markers are the data from all months and wave events (cf. Fig. 4).

  • View in gallery

    Residual times (observations minus model) of internal waves observed at mooring locations B1 (black) and P1 (gray). The residuals show variations in the propagation time over both the course of the year and within the fortnightly cycle.

  • View in gallery

    Envisat synthetic aperture radar image acquired at 0208 UTC 22 Apr 2007 with the prediction of the leading internal wave fronts (black lines) from the empirical model. The prediction shows good agreement with the shape, orientation, location, and spacing of the wave signatures in the image, even capturing the location of the notch where the waves have recombined after refracting around the Dongsha coral reef.

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An Empirical Model for Estimating the Geographic Location of Nonlinear Internal Solitary Waves

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Abstract

This paper describes the development and performance of a new approach for estimating the geographic location of high-frequency nonlinear internal waves. The approach makes use of the internal wave signatures recorded in satellite imagery to determine the parameters of a model function that relates the internal wave phase speed to depth. The phase speed map produced by the parameterized model function establishes a propagation time and propagation path between an origin and any location in the region of interest. Contours of propagation time represent the internal wave locations for a particular time since generation. The parameters of the model function are found by minimizing the difference between the calculated propagation times and the observed propagation times. This approach is applied to the internal solitary waves observed in the northern portion of the South China Sea west of the Luzon Strait. Propagation time estimates based on this new approach have errors of ±1.32 h (1σ) for depths greater than 1000 m and ±2.55 h (1σ) for all depths over which the waves are observed.

Corresponding author address: Christopher R. Jackson, Global Ocean Associates, 6220 Jean Louise Way, Alexandria, VA 22310. Email: goa@internalwaveatlas.com

Abstract

This paper describes the development and performance of a new approach for estimating the geographic location of high-frequency nonlinear internal waves. The approach makes use of the internal wave signatures recorded in satellite imagery to determine the parameters of a model function that relates the internal wave phase speed to depth. The phase speed map produced by the parameterized model function establishes a propagation time and propagation path between an origin and any location in the region of interest. Contours of propagation time represent the internal wave locations for a particular time since generation. The parameters of the model function are found by minimizing the difference between the calculated propagation times and the observed propagation times. This approach is applied to the internal solitary waves observed in the northern portion of the South China Sea west of the Luzon Strait. Propagation time estimates based on this new approach have errors of ±1.32 h (1σ) for depths greater than 1000 m and ±2.55 h (1σ) for all depths over which the waves are observed.

Corresponding author address: Christopher R. Jackson, Global Ocean Associates, 6220 Jean Louise Way, Alexandria, VA 22310. Email: goa@internalwaveatlas.com

1. Introduction

High-frequency nonlinear internal waves are ubiquitous features across the world’s oceans (Jackson 2004, 2007). They represent an important energy transfer mechanism between the tides and small-scale turbulence and mixing, they can affect biological processes through the redistribution of nutrients, and they can cause significant effects on the transmission of acoustic signals (Zhou et al. 1991; Apel et al. 2007). As a result, there is an interest in having a simple and accurate method to determine the location of these waves, particularly those of large amplitude and with solitary-like characteristics.

Internal wave evolution has traditionally been described through the use of the Korteweg–de Vries (KdV) equation and its variants that relate the internal wave’s amplitude to time and a single spatial variable. The KdV equation adequately captures the dynamics of nonlinear internal waves in many situations of interest but is most appropriate for weakly nonlinear gravity waves where the ratios of wave amplitude to upper-layer depth and upper-layer depth to wavelength are both small. Examples of oceanic internal solitary wave train evolution modeled with the KdV equation can be found in Liu et al. (1985), Liu (1988), and Apel (2003). Higher-order nonlinear theories are available when the assumptions mentioned earlier are no longer valid. Descriptions of these may be found in the review papers of Ostrovsky and Stepanyants (1989), Helfrich and Melville (2006), and Apel et al. (2007).

Full three-dimensional simulations of internal wave generation are described by Niwa and Hibiya (2004), Simmons et al. (2004), Jachec et al. (2006), and Legg and Huijts (2006). These models begin with the nonlinear primitive equations to simulate internal waves, which is subject to tidal forcing utilizing either hydrostatic (Niwa and Hibiya 2004; Simmons et al. 2004) or nonhydrostatic (Jachec et al. 2006; Legg and Huijts 2006) conditions. The calculations proceed from first principles with no assimilative constraints; however, they rely on subgrid-scale parameters for their dissipative physics. Exact details of these model configurations are given in the cited publications, but these models generally use a 1-h time step, an O(3 km) grid spacing in the horizontal, and an O(30 m) scale in the vertical, and typically require supercomputer resources for basin-level investigations.

What is currently missing in these modeling approaches is a simple and accurate method to determine the location of an internal wavefront at a particular time in two spatial dimensions (latitude–longitude). This type of two-dimensional location determination can be accomplished with an empirical approach that uses the two-dimensional snapshots of the internal waves recorded in satellite imagery in connection with a parameterized model function that gives the phase speed of the internal wave as a function of geographic location. The “phase speed map” produced by the parameterized model function establishes a propagation time and propagation path between an origin and any location in the region of interest. The parameters of the model function are found by minimizing the difference between the calculated propagation times and the observed propagation times for the waves in the georeferenced satellite imagery.

In section 2, the factors that affect the propagation speed of nonlinear internal waves are briefly reviewed. Section 3 presents a description of the empirical model and its development. The data from the South China Sea used with the model are described in section 4, with the results from the South China Sea presented in section 5. A discussion of the empirical model’s use as a predictive tool is given in section 6.

2. Internal wave dynamics

To understand the factors that determine the phase speed of a high-frequency nonlinear internal wave, it is useful to examine the quadratic one-dimensional KdV equation:
i1520-0426-26-10-2243-e1
where η(x, t) gives the vertical displacement of an isopycnal surface; c0 is the linear long-wave phase speed; and α and β are the coefficients of nonlinearity and dispersion, which also depend on the value of c0. The “classic” soliton solution to Eq. (1), where the nonlinear and dispersive terms are balanced, is the hyperbolic secant pulse given by
i1520-0426-26-10-2243-e2
where η0 is the wave amplitude, C is the soliton propagation speed, and L is a measure of the soliton width (determined by η0, α, and β). The soliton’s propagation speed is a function of the long-wave linear speed, its amplitude, and the nonlinearity coefficient via
i1520-0426-26-10-2243-e3
The coefficients in the KdV equation are determined by the total water depth and the density profile of the water column. For a two-layer fluid, c0 and α are given by
i1520-0426-26-10-2243-e4
i1520-0426-26-10-2243-e5
where h and ρ are the thickness and density of the upper (subscript 1) and lower (subscript 2) fluid layers, respectively.
For a continuously stratified fluid, c0 is the eigenvalue of the Strum–Liouville equation arising from the Navier–Stokes equations:
i1520-0426-26-10-2243-e6
where ϕ = 0 at z = 0 and −H. The Brunt–Väisälä or buoyancy frequency N2 is related to the density profile via
i1520-0426-26-10-2243-e7
and α is obtained from the derivative of the eigenfunction ϕ through
i1520-0426-26-10-2243-e8
More complete discussions on the Strum–Liouville equation and its application and interpretation for stratified fluids can be found in Thorpe (1969), Phillips (1977), and Gill (1982).

In the ocean, the nonlinear and dispersive terms are rarely completely balanced (or balanced only over short periods of time) and the amplitude of the soliton varies in time. The nonlinearity coefficient α disappears entirely when the upper and lower fluid layers become equal in thickness, requiring a cubic nonlinearity term to be included in the equation for the KdV description to remain valid. The addition of this term produces the extended KdV (eKdV) equation that also better captures the broadening and flattening of large solitary pulses (Lee and Beardsley 1974; Djordjevic and Redekopp 1978; Ostrovsky and Stepanyants 1989; Grimshaw et al. 2002). The recent review papers by Helfrich and Melville (2006) and Apel et al. (2007) also discuss the properties of the KdV and eKdV equations and their solutions as well as providing a description of the fully nonlinear theories.

3. Empirical model development and description

High-frequency nonlinear internal waves are distinctive features in satellite imagery of the sea surface. The waves manifest themselves as alternating bands of light and dark quasi-linear strips that remain coherent over tens to hundreds of kilometers away from their generation region. Each satellite image containing an internal wave signature is a two-dimensional snapshot of the internal wave at a particular time from generation. Repeated satellite observations over a region, when combined, map the extent of the waves both geographically and temporally (in propagation time). The internal wave positions and propagation times are related through a two-dimensional (longitude–latitude) map of the internal wave’s phase speed. By establishing the phase speed of the internal waves over a region of interest, the propagation time and propagation path between an origin and any point in the region can be calculated by using either ray tracing or through an eikonal differential equation.

The empirical model approach will use the internal wave signatures observed in satellite images (two-dimensional position/propagation time sampling) to estimate the parameters of a model function that describes the phase speed of the internal wave as a function of geographic location. The model function will provide the phase speed map that establishes the propagation time and propagation path between an origin and any location in the region of interest. With both the observed and model propagation times available for each wave front, the model parameters for creating the phase speed map can be found though a numerical minimization method using the difference between the observed and model propagation times as its measure of fit.

The first step is to create a parameterized function that can be used to relate the internal wave phase speed to the wave’s geographic location. Equation (3) shows that the nonlinear phase speed is the long linear phase speed plus a correction for amplitude and nonlinearity, which is on the order of 5%–10% of the linear phase speed. By examining the linear long-wave phase speed derived from either in situ observations or historical climatological data, it is possible to get an idea for a form of the parameterized function. Figure 1 shows the linear long-wave speed as a function of depth derived from the Strum–Liouville Eq. (6), using temperature, salinity and pressure measurements acquired during ship-based surveys in the South China Sea between 2005 and 2007. The arrangement of the data points on Fig. 1 suggests a phase speed versus depth function of the form
i1520-0426-26-10-2243-e9
where H(x, y) is the total water depth (specified by latitude y and longitude x), CMax is the maximum wave phase speed, and B1 and B2 are terms that determine how rapidly variations in depth will affect the slowing and refracting of the waves. This form has the advantage of being well defined over all values of H and having only three parameters to estimate from the measurements. Equation (9) is not unique, because there are a number of possible expressions for a phase speed versus depth function. Alternatively, a third-order polynomial could have been used or Eq. (3) could be rewritten using the expressions for c0 and α for a two-layer fluid [Eqs. (4) and (5)] and expressing h2 as (Hh1). This latter formulation produces an equation with three terms (h1, η0, and Δρ/ρ) that are directly connected to the physical condition of the internal wave and the water column, but it then requires each of these terms to be parameterized for their variation with geographic location. Preliminary work using these approaches has not yielded results that are any better than those achieved using Eq. (9).
Having specified the speed of the wave at all points in the region of interest, the remaining problem is to determine the propagation time and propagation path for a wave whose speed varies depending on its location along the propagation path. The problem is familiar in geometric optics and both oceanic and solid earth acoustics. If the speed of the wave is only a function of position (and not a property of the wave front itself), then the problem can be solved using the two-dimensional eikonal differential equation,
i1520-0426-26-10-2243-e10
where T(x, y) represents the travel time between an origin (x0, y0) and location (x, y) and s is the wave slowness (1/velocity) at that location.

The eikonal differential equation is the basic mathematical model for describing the travel time for a given velocity model. The major advantages of the eikonal method compared with ray tracing techniques are the ability to implement it on a regular grid and a fair numerical robustness (Fomel 1997). The eikonal Eq. (10) gives only the first arrival time for a propagating wave front and does not account for subsequent arrivals resulting from wave reflection. For the case where the velocity is always greater than zero (a continually expanding wave front), Eq. (10) can be solved for in a computationally efficient way by using the fast marching methods described by Sethian (1999). Fast Marching Methods are numerical techniques for computing interface motions and belong to the family of upwind finite-difference schemes that has been recognized as one of the most efficient means of travel-time computations (Popovici 1991; Fomel 1997). In the case where s = 1, Eq. (10) give the distance between the origin and the location.

To calculate the propagation time from Eq. (10), it is necessary to specify a geographic origin for the waves. The location, or several locations, can be explicitly specified or can be additional model parameters that are derived from the observations. The origin’s location has a geometric effect on the shape of the resulting model wavefronts (e.g., determining the radius of curvature at a particular point or establishing the principal propagation direction).

With the model propagation time obtained from Eq. (10), the observed propagation times must be obtained from the satellite imagery (in situ internal wave observations are also suitable but are usually much more limited in their range of geographic diversity). To determine the observed propagation time, each individual internal wave needs to be associated with a generation time. The observed propagation time is then simply the image acquisition time minus the generation time. Knowledge about the internal wave generation time relative to the tidal phase allows the generation times to be obtained from a tidal model (e.g., Oregon State University’s TPXO6; Egbert and Erofeeva 2002). For the approach as implemented, the exact phase of the tide at generation need not be known, but the relationship between generation and phase must be applied consistently when tagging the waves with a generation time. The relationship can be either constant or via some other function (e.g., a smooth function of maximum tide amplitude). Any mismatch between the selected generation phase and the actual generation phase will be visible in the residuals as a bias or trend.

With the parameterized form of the function relating internal wave phase speed to geographic position, the positions of the internal waves from the satellite imagery, each internal observation associated with a generation time, and a way to calculate propagation times from the parameterized phase speed function, it is now possible to find the parameters that will provide the best fit between the observed and modeled internal wavefront propagation times. Determining the best-fit parameters is now a multidimensional minimization problem over parameters CMax, B1, B2, and origin latitude and longitude. There are a variety of methods that can be used to solve problems of this type. The results presented in section 5 were found by using a downhill simplex method (via the Matlab fminsearch function), with the sum of the squares of the difference between the observed and model propagation times as the measure of fit.

4. South China Sea observations

Nonlinear internal solitary waves in the northern portion of the South China Sea are generated at (or near) the Luzon Strait, propagate westward across the deep basin, refract around the Dongsha coral reef, and dissipate on the continental shelf after persisting for more than four days and traveling over 500 km. The internal waves occur year round (regularly between March and November and intermittently from December to February) and in situ measurements in the area west of Luzon have found waves with amplitudes in excess of 100 m and propagation speeds close to 3 m s−1 (Klymak et al. 2006). It is currently believed that these solitary waves are produced through a steepening of the internal tide as it travels westward from the Luzon Strait into the northern South China Sea (Zhao and Alford 2006; Helfrich and Grimshaw 2008).

a. Phase speed relationship to depth

The empirical model relies on a parameterized function that gives the internal wave phase speed as a function of geographic location. The form of this function was selected by examining the relationship between the linear long-wave phase speed and depth for the northern portion of the South China Sea (Fig. 1). The phase speeds presented in Fig. 1 were derived using Eqs. (6) and (7), with temperature and salinity profiles of the water column acquired between April 2005 and July 2007 as part of ship-based surveys. The data show that the phase speeds remain within a relatively narrow range of values (<20% variance), even though the profiles were collected in every season over the 27-month time frame. As previously described in section 3, the distribution of points in the figure suggests that a square root of the hyperbolic tangent can be used as a suitable function to relate the phase speed to depth (and subsequently geographic location).

b. Internal wave satellite observations

From satellite, internal waves near the Dongsha coral reef were first noted in Defense Meteorological Satellite Program (DMSP) imagery from 1973 (Fett et al. 1979). Since that time, a large amount of satellite imagery (both synthetic aperture radar and optical sun glint) containing internal wave signatures has been acquired over the northeastern portion of the South China Sea and several authors (Hsu and Liu 2000; Zhao et al. 2004) have used to data create composite maps showing the spatial distribution of waves between the Luzon Strait and the continental coast.

Figure 2 shows the spatial distribution of 141 wave fronts extracted from 72 georeferenced images acquired on 61 distinct dates from April 2003 to December 2006. Each gray curve in Fig. 2 represents either an isolated solitary wavefront or the leading wave front of an internal wave packet. The majority of the images (46) were acquired by the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor onboard the National Aeronautics and Space Administration (NASA) satellites Terra and Aqua. To be visible in the MODIS imagery, the internal waves must be present in the sun-glint region of the collection swath. A total of 26 images were acquired by the synthetic aperture radar on either Envisat or on Radarsat-1, which operates independent of both the lighting conditions and cloud coverage.

These satellite data show that the eastern extent of the internal wave signatures is near 120.5°E, and the waves remain visible until they are inside the 100-m isobath. The phase speed dependence on depth causes the waves to refract as they reach close to Dongsha. The number of wave fronts appears to increase from east to west, but this is actually an artifact of sampling. With the waves immediately west of Luzon moving approximately twice as fast as the waves on the shelf, they persist for only half as long over the same horizontal scale and so the number of waves appears reduced in the deep basin.

c. Tidal relationship to generation

Zhao and Alford (2006) have shown a strong correlation between internal wave occurrences near the Dongsha coral reef and the maximum of the westward tidal current at the Luzon Strait using mooring data acquired in April and May 2001 as part of the Asian Seas International Acoustics Experiment (ASIAEX; Ramp et al. 2004). This same strong relationship between the maximum westward current at Luzon and the occurrence of internal solitary waves has also been found in the central basin at 21.31°N, 118.64°E in 2514 m of water by measurements obtained from a pressure-sensor-equipped inverted echo sounder (PIES) mooring deployed for 75 days during August and September 2005.

The PIES instrument transmits an acoustic signal every 6 s to produces one useful estimate per minute of the signal’s round-trip travel time (Farmer et al. 2006; Li et al. 2009). Figure 3 shows the acoustic echo delay from PIES along with the normalized amplitude of the east(+)/west component of current at Luzon from the TPXO6; tidal model for 16–30 August 2005. The black dots represent the 1-min PIES measurements, and the black line is an 18-min average of the PIES measurements using a Savitzky–Golay smoothing filter. Nonlinear internal waves manifest themselves as large spikelike excursions in the direction of shorter delay visible on 17–26 August and marked by the triangles under the largest internal wave for a particular occurrence.

Correlating the PIES time series observations and TPXO6; current over the time of the PIES deployment at 1-day intervals and using a 14-day correlation window shows that the PIES observations lag the westward current maximum by 0.677 ± 0.007 days (16.26 ± 0.17 h) with a correlation coefficient of 0.717 ± 0.032. The TPXO6; currents shifted forward by this amount of time align the westward current extrema with the internal wave occurrences. When aligned in this manner, the PIES data show that internal wave generation does not always occur at the exact westward maximum of the current, but it can either lead or lag current extrema. The PIES data also show that the internal waves manifest themselves as both solitary pulses and as packets of up to 3 waves.

5. Model results for the South China Sea

Once each internal wave in the satellite imagery is associated with a generation time (based on the tidal relationship discussed above), establishing the wave’s observed propagation time, the empirical model parameters (CMax, B, B2, and origin latitude and longitude) for the South China Sea were found through simplex minimization by using the sum of the squares of the weighted difference between the observed and model propagation times as its measure of fit. The minimization was done by using approximately 300 data points (selected from the locations presented in Fig. 2) to achieve a roughly uniform geographic distribution over the region and a balance over the number of points from each month of the year.

The estimated model parameters for the South China Sea are presented in Table 1. The point source origin was found to be at 20.53°N, 124.9°E, well to the east of the South China Sea. This is clearly not the true origin location of the waves but a “virtual origin” that allows the model to produce wave fronts with a large radius of curvature that matches the wave signatures from the satellite imagery in the area east of 118°E (if the internal waves originate from a steepening of the internal tide, there is no single point source origin). The longitude of the origin introduces a bias in the residuals of almost two days, which when subtracted from the model propagation time map places the appearance of the waves at the time of maximum westward current near 120.0°E (cf. Figure 2). Because the internal waves observed in satellite imagery first appear closer to 120.5°E, indicating that, although each wave can be associated with maximum westward current, the generation (or initial manifestation) time is on average around 4.5 h before the maxima.

The CMax parameter was estimated to be 2.97 m s−1 in the deep basin, which agrees with the speeds observed in the mooring record and from ship-based observations. The phase speed versus depth curve corresponding to the derived model parameters is shown as the dashed line in Fig. 1. The curve sits roughly 10% above the mean long linear wave speed for any given depth. This is consistent with Eq. (3), which shows the nonlinear wave speed is the long-wave linear speed plus a correction for internal wave amplitude and nonlinearity, which is on the order of 5%–10%.

Figure 2 shows the model wavefronts at half-day intervals along with the satellite observations. The shape and orientation of the model wavefronts agree well with the satellite signatures. The model wavefronts show that it only takes a little over a day for the internal waves to go from their first appearance near 120.5°E to the 1000-m isobath of the continental shelf and another 1.5 days for the waves to reach inside the 200-m isobath.

a. Residual analysis

The residuals (the difference between the observed and model propagation times) for the South China Sea are shown in Fig. 4. The plot contains 7090 points from satellite images acquired between April 2003 and December 2006 (and shown in Fig. 2) along with 304 data points from the B1 (see below) and PIES moorings. The plot shows that, for depths greater than 1000 m, the residuals have a standard deviation of ±1.32 h and stay within with a constant extent of approximately ±4 h. The extent of the residuals increases for depths less than 1000 m and continues to grow as depths become shallower (and propagation times become longer), indicating that other factors, not included in the model, are having an effect on the internal wave phase speed. For all data, the standard deviation is ±2.55 h.

Figure 5 shows monthly breakdowns of the propagation time residuals. April and July stand out as the two months with the largest number of internal waves (and points) yet show some of the smallest variance although the mean of the April data is at 1.51 h. The August results show both a large mean and variance. Figure 5 shows that the variance can vary significantly from month to month and be large even in months where the internal wave observations come from a single year, usually where a majority of the wave observations are at depths of less then 1000 m. At shallow depths, even individual wave fronts have residuals that vary over several hours highlighting the uncompensated effects along specific propagation paths as the individual wave front becomes distributed over a greater geographic area. No clear seasonal pattern is observed in the satellite residuals, most likely because the satellite observations cover such a wide variety of generation dates and conditions and the seasonal effects may be small. Figure 1 showed the mode-1 long-wave linear phase speeds in the deep basin covering a variety of seasons remained within a relatively narrow range of values.

To try and get a better understanding of any seasonal trends, it is more useful to examine the wave events recorded at several deep water moorings deployed during 2005–06. The two vertical lines visible in Fig. 4 near 2500-m depth are the data recorded at the PIES mooring (light gray) and at mooring B1 (dark gray). B1 was deployed at 21.365°N, 118.59°E in 2465 m of water from May 2005 through May 2006 as part of the Office of Naval Research’s Windy Islands Soliton Experiment. The B1 mooring had, as part of its instrument suite between approximately 100- and 1300-m depth, two Sea Bird Electronics (SBE) 37-SM MicroCATs and six SBE-39 recorders for measuring temperature and pressure. All eight instruments sampled both temperature and pressure at 1-min intervals. To identify the internal waves in the temperature record, the temperature data from each sensor were high-pass filtered using a 400-min-long Savitzky–Golay smoothing filter and at each 1-min observation time the filtered temperatures from each mooring instrument were added together to create a vertically integrated temperature profile. Internal wave signatures were identified by looking for the largest temperature spike greater than three standard deviations, within ±4 h of the occurrence time predicted by the model. Figure 6 shows that the residuals of propagation time, at both the B1 and PIES mooring, varies by several hours over the fortnightly (14 day) tidal cycle. For the approach as implemented, the internal waves were assumed to be generated at the extrema of the westward current and the propagation time to each mooring was taken as a constant. The variations in the residuals indicate that one (or both) of these assumptions is incorrect. This behavior can be seen in Fig. 3, where the nonlinear internal wave spikes are not perfectly aligned with the model current minimums. Internal wave phase speeds are also affected by variations in internal wave amplitudes, the stratification conditions, and the strength of the tidal forcing over the 14-day tidal cycle. No variable has been found to be highly correlated with the residual, indicating that these factors are all influencing the generation and/or propagation time in a complex way.

Figure 6 shows two additional trends. First, the wave occurrences become more intermittent between late November 2005 (day 330) and early March 2006 (day 60), most likely because of changes in the stratification during the winter or possibly because of an intrusion of the Kuroshio into the basin. Second, the residuals appear to show a change in mean over the course of the year, decreasing between days 150 and 350 in 2005, and it appears to increase again in early 2006, indicating seasonality to the phase speed profiles and propagation times.

6. Application to prediction

With the model parameters established and thus the ability to determine the propagation time to any location in the geographic region of interest, the empirical model, combined with a generation criteria, can serve as the basis for a prediction system. Arrival time predictions can be made for fixed locations or propagation time contours (like those shown in Fig. 2) produced at intervals based on the generation relationship to the tidal cycle can be used to create wave front location maps. The maps can be combined to create animations to understand the wave field evolution (an animation showing the waves in the South China Sea over a 30-day period is available online at http://www.internalwaveatlas.com/SCS).

An example of a two-dimensional comparison between the predicted wave front locations and the wave signatures at 0208 UTC 22 April 2007 is shown in Fig. 7. Figure 7 is an Envisat synthetic aperture radar image containing four internal wave signatures around the Dongsha coral reef overlaid with the predicted positions. Figure 7 shows the prediction to be in good agreement with the shape, orientation, location, and spacing of the wave signatures in the image, even capturing the location of the “notch” where the waves have recombined after refracting around the Dongsha coral reef. The predicted wavefront (TPred = 1.103) leads the easternmost wave as well as the wave to the west of Dongsha (TPred = 2.117) by approximately 8 km (between 1.0 and 1.5 h based on the respective phase speeds of 2 and 1.45 m s−1). There is closer agreement in position with the wave immediately east of Dongsha (TPred = 1.415) and faint remains of the signature farther up on the shelf (TPred = 2.492). The discrepancy between the model and observed positions is most likely due to the waves at prediction times 1.415 and 2.492 being generated closer to the time of the westward current extrema.

7. Summary

This paper describes the development and performance of a new approach for estimating the geographic location of nonlinear internal wavefronts. The approach makes use of the wave signatures recorded in satellite imagery and a parameterized model function to determine a two-dimensional phase speed map and calculate a propagation time to locations over the region of interest. Contours of the propagation times represent the internal wave locations for a particular time from generation.

The model has been generally conceived and can be readily adapted to other regions, requiring only the latitude and longitude positions of internal waves, some understanding of when wave generation occurs, and a parameterized function relating geographic location to internal wave velocity. The parameters can be solved for globally (as for the South China Sea) or for specific time period that represent a particular set of conditions.

The approach was applied to the internal solitary waves observed in the northern portion of the South China Sea west of the Luzon Strait. The model wavefront locations were found to be in good agreement with the shape, orientation, location, and spacing of the wave signatures observed in the satellite imagery. Propagation-time estimates based on the model have errors of ±1.32 h (1σ) for depths greater than 1000 m and ±2.55 h (1σ) for all depths over which the waves are observed.

Acknowledgments

This work was supported by the Office of Naval Research Physical Oceanography Program through Contract N0001405C0190. The author would also like to thank Dr. Louis St. Laurent of FSU for his discussion, Dr. David Farmer and Mr. Qiang Li of URI for providing the PIES data, Dr. Steve Ramp and Mr. Fredrick Bahr of the NPGS for providing access to the B1 mooring data, Drs. Steve Ramp and Y. J. Yang for supplying the CTD data from R/V OR1, and the two reviewers for their useful comments and suggestions. Bathymetry data were obtained from Smith and Sandell (available online at http://topex.ucsd.edu/marine_topo/mar_topo.html), and the maps were generated using Rich Pawlowicz’s M_Map toolbox (available online at http://www2.ocgy.ubc.ca/~rich/map.html).

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Fig. 1.
Fig. 1.

Mode-1 linear internal wave phase speed as a function of depth for the northern South China Sea. The data are derived from temperature and salinity profiles acquired during ship surveys in the region between April 2005 and July 2007. Data are distinguished for quarter and year by symbol and shading, respectively. The dashed line is the nonlinear internal wave phase speed from the empirical model solution presented in Table 1.

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 2.
Fig. 2.

Internal wave signatures in the South China Sea from satellite imagery (gray) with the wave front locations from the empirical model (black). There are 141 internal wave signatures from 73 satellite images acquired on 61 days between April 2003 and December 2006. The shape and orientation of the model wave fronts agree well with the satellite signatures.

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 3.
Fig. 3.

PIES data from the northern portion of the South China Sea with the east–west TPXO6 model currents from Luzon (20.56°N, 121.24°E). In the PIES data, the nonlinear internal waves are the large spikelike excursions in the direction of shorter delay. Shifting the current +0.677 days (16.24 h) aligns the internal wave occurrences with the westward tidal current maximums (east/west current minimums).

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 4.
Fig. 4.

Propagation time residuals (observations − model) as a function of depth for the internal wave signatures presented in Fig. 2. Vertical gray lines are wave observations from moorings B1 (dark gray) and P1 (light gray).

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 5.
Fig. 5.

Monthly breakdown for (a) January–June and (b) July–December of the propagation time residuals from the satellite observations presented in Fig. 2. The figures show that the variance can be significant from month to month and be large even in months where the internal wave observations come from a single year. Gray markers are the data from all months and wave events (cf. Fig. 4).

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 6.
Fig. 6.

Residual times (observations minus model) of internal waves observed at mooring locations B1 (black) and P1 (gray). The residuals show variations in the propagation time over both the course of the year and within the fortnightly cycle.

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Fig. 7.
Fig. 7.

Envisat synthetic aperture radar image acquired at 0208 UTC 22 Apr 2007 with the prediction of the leading internal wave fronts (black lines) from the empirical model. The prediction shows good agreement with the shape, orientation, location, and spacing of the wave signatures in the image, even capturing the location of the notch where the waves have recombined after refracting around the Dongsha coral reef.

Citation: Journal of Atmospheric and Oceanic Technology 26, 10; 10.1175/2009JTECHO638.1

Table 1.

Empirical model parameters for the South China Sea.

Table 1.
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