The Generation and Evolution of Nonlinear Internal Waves in the Deep Basin of the South China Sea [Journal of Physical Oceanography]
(Journal of Physical Oceanography Via Acquire Media NewsEdge) ABSTRACT
Time series observations of nonlinear internal waves in the deep basin of the South China Sea are used to evaluate mechanisms for their generation and evolution. Internal tides are generated by tidal currents over ridges in Luzon Strait and steepen as they travel west, subsequently generating highfrequency nonlinear waves. Although nonlinear internal waves appear repeatedly on the western slopes of the South China Sea, their appearance in the deep basin is intermittent and more closely related to the amplitude of the semidiurnal than the predominant diurnal tidal current in Luzon Strait. As the internal tide propagates westward, it evolves under the influence of nonlinearity, rotation, and nonhydrostatic dispersion. The interaction between nonlinearity and rotation transforms the internal tide into a parabolic or corner shape. A fully nonlinear twolayer internal wave model explains the observed characteristics of internal tide evolution in the deep basin for different representative forcing conditions and allows assessment of differences between the fully and weakly nonlinear descriptions. Matching this model to a wave generation solution for representative topography in Luzon Strait leads to predictions in the deep basin consistent with observations. Separation of the eastern and western ridges is close to the internal semidiurnal tidal wavelength, contributing to intensification of the westward propagating semidiurnal component. Doppler effects of internal tide generation, when combined with a steady background flow, suggest an explanation for the apparent suppression of nonlinear wave generation during periods of westward intrusion of the Kuroshio.
(ProQuest: ... denotes formulae omitted.)
1. Introduction
Largeamplitude nonlinear internal waves are frequently observed in the northern South China Sea, where they propagate westward fromLuzon Strait. Remotely sensed images show thewaves first appearing at longitude 120.58E (Jackson 2009), subsequently traveling across the deep basin before slowing and dissipating on the continental shelf (Fig. 1). These waves have been the subject of intensive study in previous observational programs (especially the U.S. Office of Naval Research Asian Seas International Acoustics Experiment and the Nonlinear Internal Waves Initiative); however, aside from some important moored observations in the deep basin (Ramp et al. 2010) the focus of these studies was primarily on the continental slope on the western side of the deep basin, where topography plays a dominant role in their development and wave dissipation is prominent (see, e.g., Liu et al. 1998; Ramp et al. 2004; Duda et al. 2004).More recently, Alford et al. (2010) summarized extensive observations, including some of those discussed here, of wave shape, speed, and structure as they interacted with the western slope and provided detailed analysis of their arrival times and shape as they traveled across the South China Sea.
A remarkable feature of remotely sensed images of these waves is the clean sweep of their arcs as they propagate across the deep basin. Notwithstanding the highly irregular topography of the source region and consequent threedimensionality of the initial response, which is clearly apparent and extensively analyzed in recent3D simulations (Zhang et al. 2011), the nonlinearity effectively enhances and smoothes out the largeamplitude interfacial depression waves, converting them into the familiar patterns observed from space. This simplified picture adds motivation to our analysis in which we seek a physical understanding based on simplified twolayer, twodimensional models. Our approach is therefore complementary to Alford et al.'s precise analysis of arrival times and wave speeds and to the Zhang et al. (2011) 3D simulations, and focuses on the interpretation of our observations in terms of dynamical processes responsible for wave generation and evolution within the deep basin.
Our observations were acquired with inverted echo sounders for a period of three months in 2005 and six months in 2007 at locations shown in Fig. 1 (Table 1). A preliminary report (Farmer et al. 2009) describes a brief section of the data and uses weakly nonlinear theory to explain some features of the internal tide evolution. Here we draw on the full dataset to reveal the occurrence of nonlinear internal waves in the deep basin and use a fully nonlinear twolayer simulation coupled to an internal tide generation model to explain the observed wave evolution under different tidal conditions.The inverted echo sounder observations, being vertically integrated, can only yield information on the first internal mode, thus limiting our analysis to the firstmode response; moreover, the use of a 2D representation cannot be expected to reproduce the detailed structure and precise arrival times of a comprehensive 3Dsimulation. Notwithstanding these limitations, we find the simplified model helpful in explaining several key features of the time series observations.
Highfrequency nonlinear internal waves in the South China Sea evolve from the nonlinear transformation of internal tides generated in Luzon Strait. These waves propagate ;300 km across the deep basin over periods comparable to the inertial period, ensuring that rotation plays a significant role in their evolution. Following the Ostrovsky (1978) pioneering study, there has been extensive theoretical analysis of nonlinear waves in a rotating frame but few comparisons with observations. Gerkema (1996) compared his twolayer weakly nonlinear predictionswith observations in the Celtic Sea and Massachusetts Bay, but without direct comparison of wave evolution. The observations discussed here provide an opportunity for comparison of observed wave evolution with theoretical models, including the fully nonlinear implementation of Helfrich (2007).
The paper is organized as follows. Section 2 includes a discussion of the observational method, in which inverted echo sounders are deployed together with pressure sensors. Time series observations at three locations over several months in 2007 are summarized so as to illustrate the intermittent occurrence of the highfrequency nonlinear internal waves in the deep basin, in particular its relationship to the strength of semidiurnal tidal currents in Luzon Strait. The subsequent analysis follows a hierarchical approach. In section 3, the nonlinear evolution of a monochromatic internal wave is explored in terms of weakly nonlinear theory so as to illustrate the interaction between nonlinearity and rotation using the OstrovskyHunter equation. The analysis is extended with the Helfrich fully nonlinear treatment in section 4 and comparison made with results of the weakly nonlinear prediction. In section 5, the generation of internal tides over representative topography in Luzon Strait, including both of the prominent ridges, is analyzedwith the Hibiya (1986) twolayermodel. The results are compared with our observations and also used to explore an effect of quasisteady background flows such as the Kuroshio intrusion on internal tide generation. In section 6, output from the same model is used as initial conditions to drive the Helfrich fully nonlinear wave evolution model. The combined model is then used to analyze idealized examples of tidal forcing representative of different tidal forcing conditions encountered in Luzon Strait and the resulting internal tide evolution in the deep basin of the South China Sea. In section 7, we follow the approach of section 6, but with TOPEX/Poseidon global tide model (TPXO; Egbert and Erofeeva 2002) tidal current predictions for Luzon Strait to generate time series for the measurement sites, allowing direct comparison with our observed time series. The paper concludes with a summary in section 8.
2. Description of observations
The South China Sea, a semienclosed marginal sea, is connected to theNorth PacificOcean through Luzon Strait. The strait possesses two major northsouth ridges, LanYu andHengChun (Fig. 1). Interaction between tidal flow and topography in the strait generates internal tides that propagate westward into the South China Sea Basin; eastward propagation into the Pacific Ocean is not considered here. The westward propagating internal tides steepen and form one or more solitarylike internal waves. Prominent signatures of nonlinear internalwaves are readily observed in remotely sensed images (Jackson 2009) and have stimulated numerous theoretical and modeling studies (i.e., Liu et al. 1998).
The averaged temperature, salinity, density, and buoyancy frequency derived from CTD data acquired during each of three deploymentrecovery cruises inApril, July, and October 2007 are shown in Fig. 2. The stratification and sound speed are dominated by temperature. Normal mode analysis calculated from profiles acquired on each cruise, using 3000m water depth, yields first internal mode wave speeds of 2.64, 2.72, and 2.73 m s21, respectively, and corresponding depths for the vertical velocity eigenfunction maximum were 850 m, 800 m and 800 m. Seasonal variability during the period of our measurements was not dominant. Table 2 shows representative amplitude and phase of eight primary tidal current constituents derived from the TPXO prediction, for a location (218N, 1228E) near the Batan Islands in Luzon Strait. The properties of this tidal flow are generally consistent with the Zhang et al. (2011) numerical results and Ramp et al. (2010) field observations; we use the tidal flow at this location to derive the tidal flow at our model boundary under the assumption of mass conservation. The tides have mixed characteristics with the diurnal constituents K1 and O1 both exceeding the largest semidiurnal constituentM2. We use the envelope of the sum of four semidiurnal tidal currents (M2 1 S2 1 N2 1 K2) and four diurnal tidal currents (K11O11P11Q1) to indicate the magnitude of semidiurnal and diurnal forcing, respectively. The inertial period in this region is ;33 h.
Our observations make use of pressure sensor equipped inverted echo sounders (PIES), the performance of which has been analyzed by Li et al. (2009). Although a nonlinear inversion procedure can be applied to inverted echo sounder data, a linear inversion is appropriate for treating long time series. For the present study, a 1ms acoustic travel time change is equivalent to ;24 m of isotherm vertical displacement at the depth of the vertical velocity eigenfunction maximumbased on a normalmode analysis, and this transformation is used in what follows. Two inverted echo sounders (P1 and P2) were deployed in 2005 for a pilot study and three (A1, A2, and A3) were deployed in 2007 (Fig. 1). A1 was placed between the two ridges in Luzon Strait, and the other four deployments were in the deep basin west of HengChun Ridge.
The acoustic travel times were measured every 6 s and filtered, as described by Li et al. (2009), into 1min samples. This data processing method limits effects due to sea surface roughness and underwater ambient noise so that internal tides and internal solitary waves can be properly interpreted. Representative 4day segments are shown in Fig. 3, illustrating the internal wave response in the South China Sea for different tidal forcing conditions in Luzon Strait. They show the progressive distortion of the internal tide leading, under the right conditions, to formation of highfrequency nonlinear internal waves. It is common to refer to these high frequency internal waves as solitary, solitary like, or solitons. For consistency we use the term internal solitary wave to describe these features, recognizing that they are part of a connected evolution of the internal tide and are neither strictly independent of the surrounding wave field nor the permanent form.
Internal tides are typically associated with vertical isopycnal displacements up to 100 m, and internal solitary waves with corresponding displacements of 150 m(Klymak et al. 2006 reports wave ranges of 170 m). A representative half width of the solitary waves discussed here is 35 km. Using a combination of several datasets such as inverted echo sounders, thermistor chains, and remote sensing, the average wave speed in the deep basin is found to be ;3 m s21 (Zhao and Alford 2006; Klymak et al. 2006; Farmer et al. 2009), although the wave speed varies depending onwave properties (Alford et al. 2010) or local water depth and stratification.
Solitary waves do not appear on every tidal period or at every location. Between the two ridges (A1, see Fig. 1), there is little evidence of solitary waves either in our measurements or in remotely sensed images. In the middle of the South China Sea (P1, P2, A2, and A3), the appearance and temporal distribution of solitary waves exhibit different and distinctive patterns related to the diurnal inequality of tidal forcing in Luzon Strait.
(i) Figure 3a: When the semidiurnal component of tidal currents in Luzon Strait is dominant, solitary waves appear in the middle of the basin. Usually only a single internal solitary wave appears every ;12 h at our westernmost deployment (A3 or P2).
(ii) Figure 3b:When the magnitudes of semidiurnal and diurnal tidal currents are comparable, the amplitude of the combined tidal flow alternates due to the phase difference between semidiurnal and diurnal forcing. Flow to the east is stronger and of shorter duration than flow to the west. Solitary waves appearing in the middle of the basin (P2 or A3) alternate between single and multiple waves on successive semidiurnal tides. These two types of waves have been called ''a'' and ''b'' waves according to their local arrival time and are discussed in several papers (i.e., Ramp et al. 2004; Zhao and Alford 2006; Farmer et al. 2009; Ramp et al. 2010; Alford et al. 2010).
(iii) Figure 3c: When the tidal current in Luzon Strait is predominantly diurnal, our measurements show that the semidiurnal internal tide is prominent in the deep basin (A2 and A3), although not between the ridges at station A1. The observations show that internal tides evolve into cuspshaped waves with smooth crests and sharp troughs, referred to as ''corner waves'' (Helfrich and Grimshaw 2008). In this case, solitary waves are not observed at our westernmost measurement sites (P2 and A3).
Six months of data from 2007 are summarized in Fig. 4. The TPXO tidal current prediction in Luzon Strait is shown in Fig. 4a, with the semidiurnal component in black and the diurnal component in gray. The dominance of diurnal forcing is immediately apparent. Figure 4b shows the observed first internal mode response in Luzon Strait (A1). While there is some similarity to the tidal current signal, there is considerable variability in response, possibly indicative of sensitivity to the energy distribution between first and higher mode constituents, as well as effects of wave generation at the two separate ridges. An indication of background changes in stratification, possibly associated with mesoscale eddies or variability in the Kuroshio, is provided by a 28day moving average of the A1 signal (heavy black line).
Figures 4c,d show vertical excursions associated with solitary waves when they occur atA2 and A3, respectively. Only the leading wave is shown for a wave packet and the vertical excursion is referenced to the lowpass filtered internal tide from which it evolved, which accounts for variability in the upper bound of themeasurement. Comparing Fig. 4d with 4a, it is apparent that the intermittent occurrence of solitary waves at A3 is correlated with maxima of the envelope of semidiurnal currents in Luzon Strait. This is particularly clear, for example, at the beginning of September when the peaks in the maxima of the envelope of diurnal and semidiurnal currents are well separated. Figure 5 shows the relationship between the magnitude of the tidal current in Luzon Strait and the corresponding height of the leading solitary wave at A2 andA3. There is little apparent relationship to the diurnal forcing (Figs. 5a,b) but a clear correlation between (negative) wave heights and the semidiurnal tidal currents (Figs. 5c,d).
Previous studies proposed a link between solitary wave generation in the South China Sea and the maximum westward (Zhao and Alford 2006) or eastward (Buijsman et al. 2010a) tidal flow in Luzon Strait, concepts that were further explored by Alford et al. (2010). Figures 4 and 5 suggest the association of solitary waves in the deep basin with the magnitude of semidiurnal tidal currents over the ridges in Luzon Strait, but it is less supportive of a direct relationship with the magnitude of diurnal tidal currents in Luzon Strait. The link between tidal forcing and the farfield signal is taken up with a dynamical model in section 5.
3. Interaction between nonlinearity and rotation
Rotation plays an important role in nonlinear internal wave propagation across the South China Sea. Ostrovsky (1978) described the interaction between nonlinearity, rotation, and nonhydrostatic dispersion under the weakly nonlinear, weakly nonhydrostatic, and slowly rotational approximation:
... (1)
where ? represents the interface displacement, c is the linear phase speed of the firstmode baroclinic internal wave, f is the Coriolis parameter, and a and b are coefficients of nonlinearity and nonhydrostatic dispersion, respectively.
The Ostrovsky equation is not completely integrable and possesses many integrals of motion. Stationary solutions, corresponding to different shapes of internal solitary waves and internal tides, have been analyzed by Ostrovsky (1978) and Stepanyants (2006). However, the ''antisoliton theorem'' states that solitary waves in the form of a stationary pulse cannot exist in the presence of rotation owing to negative values (i.e., b > 0) of nonhydrostatic dispersion (Leonov 1981; Galkin and Stepanyants 1991). Resonance between the solitary wave and linear perturbations always exists, leading to energy leakage from the rear of the wave. During its decay, a new solitary wave is generated upstream, sharing common properties with the initial wave, similar to the Fermi PastaUlam recurrence phenomenon (Fermi et al. 1974). As time progresses, the background isopycnal tends toward a parabolic shape (Gilman et al. 1996). Stationary solitary wave regimes have also been analyzed by Boyd and Chen (2002) and Stepanyants (2006). In contrast to the antisoliton theorem, the KdV solitary wave can coexist with a smooth periodic wave of small amplitude and long wavelength (Gilman et al. 1996). It moves slightly faster than the background wave with varying amplitude depending on its position on the background wave. This property is explained by an approximate adiabatic theory of the interaction between solitary waves and the external disturbance (Gorshkov and Ostrovsky 1981). The observations discussed here involve the coexistence of solitary waves and large amplitude internal tides.
Two implicit nondimensional numbers are readily derived from a scaling analysis: the Ursell number (Ur) defining the ratio between nonlinearity and nonhydrostatic dispersion (Turner 1973) and the Ostrovsky number (Os) defining the ratio between nonlinearity and rotation (Farmer et al. 2009). For a twolayer fluid, the Ursell number is
... (2)
where L is the horizontal length scale, A is the wave amplitude, h1 and h2 are the upper and lower layer thicknesses, a=3c0(h1 h2)/2h1h2, and b=c0h1h2/6. In the following analysis, h1 = 500 m and h2 = 2500 m in the deep water of the South China Sea.
Given typical amplitudes of 50 m for semidiurnal (L = 134 km) and 25 mfor diurnal (L=259 km) internal tides in the South China Sea, the corresponding Ursell numbers are 1.1 × 103 and 2.2 × 103, respectively. Thus, nonhydrostatic effects are initially small. However, the Ursell number of the highfrequency nonlinear internal waves drops to ;4 because of their reduced width (L = 35 km). Thus nonhydrostatic effects become important when the interface steepens sufficiently that internal solitary waves are formed under the balance between nonlinearity and nonhydrostatic dispersion: that is, when Ur ; O(1). Neglect of nonhydrostatic effects is justified prior to the interface becoming steep.
The corresponding definition of the Ostrovsky number for a twolayer flow (Farmer et al. 2009) is
... (3)
where g9 is the reduced gravity and ? is the wavelength. Based on the Boyd (2005) weakly nonlinear stability analysis, for a simple harmonic wave for which Os < 1, rotation inhibits breaking, whereas for Os > 1 nonlinearity dominates and the wave will break. Breaking in this context refers to progressive steepening leading to a nonhydrostatic response.
Dropping the nonhydrostatic term, the Ostrovsky equation reduces to the rotationmodified shock wave, or OstrovskyHunter equation (Hunter 1990):
... (4)
The breaking time for harmonic internal waves with rotation may be numerically evaluated from (4) with periodic boundary conditions, using a Fourier pseudospectral method (Boyd 2005). Breaking in this context implies steepening up to the point at which nonlinear effects broaden the internal wave spectrum in the high band to a given level at which nonhydrostatic effects comes into play. We set the level of the energy in the highest band (normalized>632) to be such that the ratio between the spectrum at breaking and the initial spectrum is greater than 100. This choice ensures that the time to breaking tb is not sensitive to the precise choice of this criterion and is also consistent with the shock wave solution without rotation,
... (5)
where k is the maximum slope of the initial interface.
Solutions of the OstrovskyHunter equation are only determined by the Ostrovsky number (see appendix). The relationship between the breaking time and the Ostrovsky number is shown in Fig. 6, superimposed on the nonrotating shock wave solution. Three regimes are distinguished in Fig. 6, with corresponding wave shapes shown in Fig. 7.
1) Figure 7a: Regime I, Os < 1. In contrast to the nonrotating shockwave solution, no breaking occurs (Boyd 2005). Given the latitude and stratification in the South China Sea, this regime would be limited to small amplitude waves for which linear theory applies. If the wave amplitude is large (i.e., A*h1), the Ostrovsky number can also be <1 for a small Rossby number or special stratification, but the weakly nonlinear assumption would not hold in this case.
2) Figure 7b: Regime II, 1 < Os < ;2. The interface slope caused by nonlinear steepening is reduced. The shock position is shifted toward the wave center. The wave shape approximates that of the corner wave with smooth crest and sharp troughs. Rotation causes breaking to occur later than predicted by the shock wave solution.
3) Figure 7c: Regime III, Os>;2. A shock forms first and the breaking time approaches that of the shock wave solution. Nonlinear steepening dominates and rotation plays a minor role. The asymmetry of internal tides is more apparent.
In summary, analysis of the OstrovskyHunter equation illustrates rotational inhibition of nonlinear steepening. The Coriolis force is proportional to the fluid velocity, which depends on thermocline slope: nonlinear steepening leads to an asymmetric wave shape, but rotation converts internal wave energy into inertial motion, thus offsetting nonlinear steepening and tending to retain a symmetric wave shape. Wave shapes corresponding to these three regimes can all be found in the inverted echo sounder observations (see Fig. 3).
The weakly nonlinear analysis discussed above, as well as the unified generationpropagation model (Gerkema 1996), reveal the importance of rotation in nonlinear internal wave generation. However, internal waves in the SouthChina Sea are among the largest reported, motivating use of a finite amplitude wave evolution model to investigate internal wave evolution in a fully nonlinear and rotational frame.
4. Fully nonlinear evolution
The Helfrich (2007) fully nonlinear twolayer model provides a framework for analyzing nonlinear evolution with rotation. It is derived from the continuity equation and the Euler equations in a rotating twolayer fluid. This nonlinear system includes two dimensionless parameters:
... (6)
where L is the horizontal length scale and H is the total depth. The deformation radius is LR 5 (g9H)1/2/f, where g9 is reduced gravity. The term bH is small (bH 1/2 5 0.02 in the deep basin of the South China Sea if L 5LR). The nonhydrostatic effect is relatively weak within the rotational frame, consistent with the previous analysis.
A monochromatic internal tide A coskx is chosen as the initial condition for a basic illustration in Fig. 8. We choose model parameters consistent with our observations in the South China Sea (Table 3). The initial wave amplitudes A are 50 and 25 m for the semidiurnal (M2: 12.42 h) and diurnal (K1: 24 h) internal tides, corresponding to Ostrovsky numbers 5.21 and 0.64, respectively. After 30 h, which is the approximate travel time fromthe LanYu Ridge to station P2 or A3, three solitary waves evolve from the semidiurnal internal tide, but the diurnal internal tide evolves into a corner wave. Examples of both shapes are also found in our observations (Fig. 3).
We compare fully nonlinear with weakly nonlinear solutions for the South China Sea for each of the three regimes discussed above. Figures 9ac show the internal tide shape at the time the OstrovskyHunter equation predicts breaking.
(i) For Os < 1 (Fig. 9a, corresponding to regime I in Fig. 7), no breaking occurs. Comparison between weakly and fully nonlinear evolution (Figs. 9a,d) shows little difference. The interface retains its initial harmonic shape and the weakly nonlinear assumption is adequate.
(ii) For 1<Os<2 (Fig. 9b, corresponding to regime II in Fig. 7), breaking is postponed by rotation. The diurnal internal tide shape is closer to the hydrostatic weakly nonlinear prediction. The cusp shape is obvious for the semidiurnal, rather than the diurnal internal tide. The steep interface at the trough may also lead to nonhydrostatic effects so as to form internal solitary waves.
(iii) For Os > 2 (Fig. 9c, corresponding to regime III in Fig. 7), the fully nonlinear model also exhibits shock formation, but the shock position differs from that of the weakly nonlinear theory. Rotational effects can also be recognized in regime III unless the Ostrovsky number is so large that nonlinearity predominates.
Figures 9df show the same calculation one nondimensional time later. The most noticeable effect is the single welldefined solitary wave in Fig. 9e. This feature also appears in the observed time series.
Two types of stationary finiteamplitude inertiagravity waves are predicted in a twolayer rotational hydrostatic fluid: corner and lobate waves (Plougonven and Zeitlin 2003; Helfrich and Grimshaw 2008). The corner waves are found in our observations as well as our numerical models. However, lobate waves are not observed.
Comparison between Helfrich's model and solutions of the OstrovskyHunter equation support the need for a fully nonlinear analysis in the South China Sea. The solution of the fully nonlinear system is determined by multiple parameters (Helfrich and Grimshaw2008), rather than just theOstrovsky number.A composite method was proposed by Cai et al. (2002), including a linear internal tide generation and a KdVtype weakly nonlinear evolution model. Their weakly nonlinear model may underestimate nonlinear effects and also excludes rotational effects. In what follows, we use a composite model so as to incorporate both fully nonlinear and rotational effects in nonlinear internal wave development.
5. Internal tide generation
a. Application of Hibiya's internal tide generation model
Strong tidal currents over topography in Luzon Strait radiate internal tides into the South China Sea.Consistent with the simplified twolayer representation discussed above, we use the Hibiya (1986) 2D linear hydrostatic solution to explore internal tide generation in Luzon Strait. The TPXO prediction reveals that the Froude number over the ridges of Luzon Strait is generally less than 1. Unsteady lee waves are not apparent in our observations at A1, implying that, at least in this location, the hydrostatic assumption is appropriate. However, the spatial spectrum h(k) of the topography h(x) also contributes to the generation of internal tide harmonics.
The internal tides generated over a ridge depend on the strength of tidal current, the degree of stratification, and the ridge geometry. Hibiya solved the linearized Boussinesq equation for tidal flow U = Ub sinvt over a ridge under the assumption that the topographic perturbation of barotropic tidal flow Ub and the bottom topography are both relatively small. This assumption can be relaxed to requiring that the aspect ratio between ridge height and width is small (;0.1 in the South China Sea). The interface displacement of the internal tides is then given by
... (7)
Here the superscript 7 indicates the wave direction and c, A1, and Z1 are the wave speed, amplitude, and eigenfunction of the first baroclinic mode, respectively. This model assumes no lateral variation in topography and neglects rotation.
Before applying Eq. (7) to the doubleridge topography of Luzon Strait, we explore the relationship between barotropic tides and internal tides in a nondimensional space for the simpler case of an isolated ridge, represented here by a Gaussian function. The horizontal distance, vertical distance, and velocity are normalized by ridge half width b, ridge heightH0, and first internal mode wave speed c. Time t is normalized by b/c. The monochromatic tidal forcing is indicated by the normalized maximum barotropic tidal flow Ub/c and period Tc/b. In Luzon Strait, the normalized tidal periods for M2 and K1 are 5.4 and 10.4, respectively.
Internal tide generation encompasses three regimes depending on the value of the densimetric Froude number over the ridge (Vlasenko et al. 2005).TheTPXOprediction shows that the Froude number along the Luzon Strait ridges is generally subcritical (F2 r <1) so that internal tide generation is expected to be of the mixed tidallee wave type, sharing common characteristics of harmonic oscillation and unsteady lee wave generation.
Figure 10 shows three key parameters as a function of harmonic forcing evaluated fromHibiya's solution: internal wave height, maximum leading internal wave slope, and theOstrovsky number.Wave height refers to the range of the vertical thermocline displacement. For harmonic M2 orK1 tidal forcing, the wave height andmaximumslope of leading waves increase with increasing tidal forcing. The leading slope of the internal tide becomes steeper when tidal forcing increases due to the topographic gradient.
TheOstrovsky number (Fig. 10c) depends on wave slope and height. The potential for nonlinear effects increases with greater forcing but the rotational effect increases with a forcing period closer to the inertial period.The thick solid line indicates the criticalOstrovsky number (Os=1). For wave generation over a single ridge, Ub/c must exceed 0.03 for anM2 internal tide to break andmust exceed 0.13 for a K1 tide to break.
b. The Doppler effect on internal tide generation
The internal diurnal tide is influenced to a much greater extent by rotational dispersion than the semidiurnal component. However, semidiurnal forcing is effectively modulated by the large diurnal signal (Fig. 11), thus influencing initial conditions for nonlinear evolution in the deep basin. In a related context, initial conditions can also be influenced by the Kuroshio intrusion into Luzon Strait resulting in asymmetry of the tidal forcing, thus modifying the internal tide evolution.
The internal wave response for asymmetric forcing by a steady flow, such as the Kuroshio inflow, is illustrated in Fig. 12. The superimposed flow U0 increases the amplitude of internal tides propagating against the steady current but reduces the amplitude of the internal tide propagating with the current. This result is explained by the Doppler effect during internal tide generation. Given a tidal flow U(t) superimposed on a constant barotropic flow U0, Eq. (7) can be written as
... (8)
Or, in the wavenumber domain,
... (9)
where the caret indicates the wavenumber spectrum. The factor c/(c 2 U0) in Eqs. (8) and (9) is not dominant if the background flow is weak, that is, U0/c is small, which is also the case in the South China Sea. The Doppler shift induced by the superimposed flow changes the equivalent Froude number FrDoppler (1or2indicates the direction):
... (10)
As shown in Fig. 10a, the superimposed flow increases the amplitude of internal waves propagating against the superimposed flow (i.e., 2U0, hence increasing Os and thus the corresponding potential for nonlinear effects in Fig. 10c).
The Kuroshio intrusion in winter across much of the LanYu Ridge must be associated with a corresponding westward component on which the tidal currents are superimposed. The Doppler effect may therefore provide a basic explanation for the occurrence of reduced nonlinear internal wave generation during winter in the South China Sea, consistent with historical satellite images (Zheng et al. 2007). Other oceanographic consequences of the Kuroshio, such as vertical shear or horizontal stratification gradients across the ridge, may also contribute to this effect (Shaw et al. 2009; Buijsman et al. 2010b).
c. Internal tide generation from two ridges in Luzon Strait
We now apply Hibiya's model solution (7) to Luzon Strait to obtain initial conditions for the subsequent nonlinear analysis. We represent the LanYu and Heng Chun ridges in Luzon Strait by two Gaussian curves,
... (11)
where hs1 and hs2 are the heights of the two ridges and b1 and b2 indicate their half widths (Table 3). Although the topography between 208 and 218N is often cited as a primary region for generation of internal waves propagating into the South China Sea (Ebbesmeyer et al. 1991; Ramp et al. 2004; Zhao and Alford 2006), lack of highresolution bathymetry leads to uncertainty in the detailed generation characteristics. To derive the farfield internal tides, we use the meridionally averaged topography between 208 and 218N(Fig. 13) derived from 2min gridded elevations/bathymetry for the world (ETOPO2v2) (NGDC 2006). West of the HengChun Ridge, the topography is assumed flat in our simplified representation; thus, predicted linear internal tides maintain their shape and speed in the far field. Tidal currents in Luzon Strait are predicted from the global tidal model TPXO with eight primary constituents (Table 2). Comparisons between Hibiya's linear model and our observations are shown in Fig. 14 for station A1 in August 2005 and Fig. 15 for station P1 in April 2007, respectively. The internal tide observations at A1 (Fig. 14) are composed of westward propagating waves generated from the eastern ridge and eastward propagating waves generated from the western ridge. The latter will propagate across the eastern ridge into the Pacific Ocean and will not significantly influence wave evolution in the South China Sea.
The eastern ridge is the dominant source of internal tide generation at this latitude, but the western ridge contribution cannot be ignored. The distance between the two ridges is ;110 km, which is moderately close to the semidiurnal internal tide wavelength (134 km). If the two ridges are similar in shape, superposition of the westward propagating internal tide radiating from the western ridge with the westward propagating internal tide originating at the eastern ridge will lead to a corresponding increase in amplitude of the inphase components in the deep basin with a ;2p delay (we neglect wave scattering and reflection here). However, the western ridge is deeper than the eastern ridge in the middle of Luzon Strait, so the modulation from the western ridge becomes more complicated depending on the phase difference between the internal tides generated from each ridge. For example, during the ''alternating'' tidal forcing (at 2005 UTC on 48 August, Fig. 15) the smaller internal tides generated from the eastern ridge are superimposed on the larger internal tides radiating from the western ridge; on the following tide, a larger internal tide generated at the eastern ridge is superimposed on the smaller wave radiated from the western ridge.
Figure 16 shows Ostrovsky numbers approximated for the combined diurnal and semidiurnal signal, as described by Farmer et al. (2009), from the farfield model predictions for the period covered in Fig. 15. Their magnitudes illustrate an alternating pattern, consistent with the single multiplesingle alternating pattern of nonlinear internal waves observed in the middle of the basin (Fig. 3b).
The internal tide amplitude increases with the strength of tidal forcing. Given a moderate Froude number and the specified topography and stratification, lee waves are generated during the ebb tide when the wave height increases with increasing tidal flow. Subsequently, the accumulated energy is released during the deceleration stage, but the wave height also increases continuously with time as the tidal flow slackens due to formation of a deceleration wave. The amplitude of acceleration and deceleration waves is comparable (Hibiya 1986). Because of the tidal inequality, we do not directly relate generation of nonlinear internal waves with either themaximum westward or maximum eastward flow: the full tidal inequality is an essential aspect of our interpretation.
Attenuation of internal tides propagating across the western ridge (Johnston and Merrifield 2003; Chao et al. 2007) is excluded due to the small topographic perturbation assumption. The leading slope predicted from the model is smaller than observed, especially in Fig. 15, because of the absence of nonlinear steepening. Ray tracing shows a merging of rays generated from the two ridges (Farmer et al. 2009). Resonant amplification between ridges is not considered in this simplified linear model, although superposition of waves generated at both ridges is included and does enhance the semidiurnal signal. Despite the simplifications involved, Hibiya's model provides a consistent prediction of internal tide generation, which we now use to drive Helfrich's (2007) fully nonlinear model of wave evolution.
6. Relationship between the tidal current in Luzon Strait and nonlinear internal waves in the South China Sea
Although we use a linear model to generate the internal tide, nonlinear evolution begins immediately. The Hibiya (1986) model is used to generate the internal tide in the far field, which is then back propagated to the ridge. We use the internal tide generated at the eastern ridge as input to theHelfrich (2007) fully nonlinear evolution model with a starting point at the eastern ridge. When this wave reaches the western ridge, we superimpose the internal tide generated at the western ridge. This second source is also calculated with Hibiya's model for the far field and back propagated to the source. This superposition of the internal tide (nonlinearly evolved) that has arrived from the east, with the internal tide generated over the western ridge, now becomes a revised input to Helfrich's fully nonlinear model with a new starting point at the location of the western ridge. This procedure allows us to account for nonlinear propagation of the waves generated at both eastern and western ridges while consistently incorporating output from Hibiya's generation model.
To summarize, we include westward radiation and nonlinear evolution of the internal tides generated at both east and west ridges. We neglect nonlinear interaction inside the Luzon Basin between opposing waves radiating from the two ridges. Although such interaction undoubtedly occurs, the internal tide propagating eastward from the western ridge is relatively small compared to the tide propagating westward across the Luzon Basin, and the nonlinear interaction is consequently neglected. We neglect nonlinear interactions at both ridges and also neglect reflections of the internal tides at the ridges and associated dissipation mechanisms. Comprehensive numerical models can dispense with these and other simplifications, but the results of this simplified model provide insight on the underlying physical mechanisms and facilitate investigation of wave evolution for different tidal forcing patterns.
We first consider three idealized patterns similar to those observed in Fig. 3. Consistent with the tidal analysis (Table 2), a harmonic tidal forcing in water of 3000m depth is used to drive the model for semidiurnal or diurnal tides. The model results are shown in Fig. 17 in three columns, corresponding to three characteristic tidal forcing patterns: semidiurnal, diurnal, and mixed. For each pattern, the internal tide used as an initial condition at the eastern ridge is shown as well as the nonlinear wave calculated with Helfrich's model at location A3.
For the semidiurnal tidal flow U= Ub sinv1t (Fig. 17a), quasiharmonic semidiurnal internal tides are generated (Fig. 17b). The internal tides generated from the eastern ridge are superimposed on those generated by the western ridge. The internal tide has a dimensional amplitude of 36 m in the far field, corresponding to Os = 4.0. After ;30 h, corresponding to the travel time from the eastern ridge to our sites A3 and P2, four internal solitary waves are generated.
With diurnal forcing U = Ub sinv2t (Fig. 17d), the internal tides are reduced by the outofphase contribution from the western ridge (Fig. 17e). The amplitude of the internal tide is 14 m and Os = 0.38. Rotational dispersion inhibits breaking of these waves in the deep basin. Cornershaped internal tides appear after 30 h of evolution (Fig. 17f).
Finally, instead of using a combination of three frequencies (Buijsman et al. 2010a), we add the semidiurnal and diurnal forcingU=Ub(sinv1t1sinv2t) (Fig. 17g). The amplitude of internal tides now alternates from 36 to 30 m every 12 h (Fig. 17h), corresponding to Os= 4.0 and Os= 3.3, respectively, owing to diurnal modulation. The resulting internal tide alternately exhibits five and then one internal solitary wave.
This composite twolayer model predicts the three distinctive patterns observed with inverted echo sounders. The appearance of highfrequency nonlinear internal waves is related to the magnitude of semidiurnal tidal forcing, even though the diurnal tides are stronger than the semidiurnal tides in Luzon Strait (Table 2). Rotational dispersion plays an important role at the diurnal frequency, modifying internal tides into a corner shape and inhibiting highfrequency nonlinear internal wave generation. The alternating pattern results from diurnal modulation of the semidiurnal forcing.
7. Time series model comparison with observations
Owing to the topographic effect, the internal tides are not exactly harmonic. We therefore set up the initial conditions to drive Helfrich's evolution model through Fourier decomposition with linear hydrostatic solutions of the nonlinear model,
... (12)
where h0 represents the thickness of the top layer and s and y are the longitudinal and transverse vertical shears, respectively. The term gH represents the ratio between the length scale and the radius of deformation. The initial transverse barotropic transport is set to zero. The dispersion relation for each Fourier component is given by
... (13)
The Fourier decomposition can be used to facilitate control of numerical instability due to high frequency harmonics in the initial condition.
The model result with the single eastern ridge deviates from our observations as expected (Li 2010), and we therefore incorporate internal tide generation from both ridges following the method used in section 6. The predictions at P1 and P2 are compared with observations in Fig. 18. Remaining disparities between model results and observations are likely related to the various simplifications that we have made, including the twolayer assumption, our selection of interface depth and layer density difference to represent a continuous density profile, and especially the simplified twodimensional topographic representation with meridionally averaged topography used to predict the internal tides in the far field and the neglect of depth changes away from the ridges. The bathymetry differs from north to south in Luzon Strait, so internal tide generation along the strait must vary with latitude.
In Fig. 19, the influence of rotational dispersion is illustrated by eliminating rotation from the prediction at P2. The shape of predicted internal tides becomes progressively more asymmetric without rotation, differing significantly from the observations. Instead of single nonlinear internal waves, shocks are formed and multiple nonlinear waves emerge. The amplitudes of the multiple nonlinear internal waves are greater than the rotational results. Based on the dispersion relationship (13), the phase speed is greater with rotation. However, rotation also shifts the shock position backward and toward the wave center, so the joint effect causes highfrequency nonlinear internal waves to arrive at P2 slightly earlier without rotation.
The three patterns of nonlinear internal waves in 2007 illustrated in Fig. 3 are shown in Fig. 20. In each case the primary characteristics of the observations are reconstructed. Small time disparities exist between the observations andmodel predictions. These aremost likely related to the simplified representation of stratification and bathymetry in our model calculations. The modulation in wave shape due to the western ridge is apparent through comparison between the model results with the western ridge included (black solid) and without the western ridge (black dashed) in the bottom three panels of Fig. 20.
''Corner waves'' are apparent in Figs. 20f,i. These waves are generated not only by diurnal tides but also by smallamplitude semidiurnal tides. Diurnal internal tides are suppressed by the rotation, but semidiurnal internal tides are intensified due to the doubleridge structure. Nevertheless, theOstrovsky number of internal tides every 12 h is between 0.78 and 2.17; corner waves are formed due to the competing effects of nonlinearity and rotation.
8. Summary and discussion
The deep basin of the South China Sea provides an excellent environment for examining the dynamics of nonlinear internal wave evolution. Time series of inverted echo sounder observations are analyzed using twolayer models so as to gain a general understanding of the generation and evolution of nonlinear internal waves. The role of rotation is emphasized.
Highfrequency nonlinear internal waves in the South China Sea arise from internal tides generated by tidal flow over topography in Luzon Strait, accompanied by nonlinear steepening. The Ostrovsky number based on weakly nonlinear analysis suggests three possible outcomes as a result of internal tide evolution. The study also shows the importance of the fully nonlinear description (i.e., Fig. 9), whichmay be compared with previous weakly nonlinear studies (i.e., Cai et al. 2002; Liu et al. 1998).
The subcritical Froude number over the ridges is consistentwithmixed tidalleewave generation. Although the eastern ridge plays a dominant role in internal tide generation at the latitude (218N) of our measurements, wave generation over the western ridge cannot be ignored and both ridges must be included to reproduce essential features. The effect of the western ridge is sensitive to the phase of internal tides and the distance between the two ridges approximates the internal semidiurnal wavelength, leading to enhancement of the semidiurnal signal.
A mean background flow intensifies internal tide generation, if themean flowand internal tide propagation are in opposite directions, but reduces it if they are in the same direction. This effect helps explain the influence of the Kuroshio intrusion on internal tide generation and is also relevant to the semidiurnal internal tide generation, which is modulated by the diurnal current.
For the parameters used here, rotation is sufficient to inhibit formation of an internal hydraulic jump. Nonhydrostatic dispersion becomes significant as the wave steepens, leading to formation of internal solitary waves. Rotation tends to disperse energy into internal inertial gravity waves, either decreasing the number of internal solitary waves or inhibiting their formation. Rotational dispersion results in a close relationship between the amplitude of semidiurnal tidal forcing and the occurrence of internal solitary waves in the South China Sea. Its relative importance for semidiurnal and diurnal internal tides results in an alternating pattern of internal solitary waves near spring tides.
The Ostrovsky number, which is determined by the wave amplitude, stratification, and Coriolis parameter, indicates the relative importance of nonlinearity and rotation. Observations and weakly nonlinear analysis suggest three distinct regimes for harmonic internal wave evolution: solitary wave generation associated with shock formation, solitary waves generated in the trough of corner waves, and the complete suppression of breaking. The separation of these three regimes differs from that described by Boyd and Chen (2002) in which there are five overlapping regimes in the steady periodic solution of the Ostrovsky equation. The fully nonlinear analysis illustrates the limitations of the weakly nonlinear theory and is determined by multiple parameters rather than the Ostrovsky number alone.
A composite approach using the Hibiya (1986) generation model coupled to the Helfrich (2007) fully nonlinear evolution model shows how the appearance of highfrequency nonlinear internal waves is related most closely to the magnitude of semidiurnal tidal flow in Luzon Strait, rather than diurnal tidal flow. Calculations of the sensitivity to the different parameters have been carried out. The selected interface depth used here is 500 m. If the depth is reduced to 250 mand the stratification adjusted to preserve the observed wave speed, the shape of the response remains essentially the same but the amplitude and width of internal solitary waves are both reduced by;50%. Slightly varying the interface depth near 500 m does not cause obvious changes in wave response.However, the twolayer assumption limits our attempt to set the interface depth below the ridge height. The choice of interface depth and ridge height seems reasonable, given the natural variability and approximations required.
This analysis is based on a twolayer, twodimensional approximation. However, twolayer models have some limitations. Higher mode internal waves and intermodal interactions are excluded; higher mode effects may be significant near the ridges. The twodimensional approximation does not account for the possibility of localized forcing of internal waves (i.e., WarnVarnas et al. 2010) or localized conditions of controlled flow leading to depression waves (i.e., Cummins et al. 2006). Improved bathymetric data, detailed flow measurements, and finescale modeling within the strait are required to resolve the full implications of topographic variability. Notwithstanding the simplifications of our approach, a twolayer, twodimensional coupled wave generation and wave evolution analysis reveals physical mechanisms underlying nonlinear internal wave properties and behavior in the South China Sea and provides a useful complement to more complex models.
Acknowledgments. We are grateful toDr. Steve Ramp (MontereyBayAquariumResearch Institute) for leading cruises during which our instruments were deployed and recovered; to Prof. David Tang (National Taiwan University) for his wide ranging support of the program; and to Erran Sousa and Gerald Chaplin for expert assistance in instrument preparation, deployment, and recovery.We thank Dr. Karl Helfrich (Woods Hole Oceanographic Institute) for stimulating our approach to this problem and acknowledge the helpful comments of our referees. This work was supported by the U.S. Office of Naval Research under the Nonlinear Internal Wave Initiative.
REFERENCES
Alford, M. H., R.C. Lien, H. Simmons, J. Klymak, S. Ramp, Y. J. Yang, D. Tang, and M.H. Chang, 2010: Speed and evolution of nonlinear internal waves transiting the South China Sea. J. Phys. Oceanogr., 40, 13381355.
Boyd, J. P., 2005: Microbreaking and polycnoidal waves in the OstrovskyHunter equation. Phys. Lett., 338, 3643.
_____,and G. Chen, 2002: Five regimes of the quasicnoidal, steadily translating waves of the rotationmodified Kortewegde Vries (''Ostrovsky'') equation. Wave Motion, 35, 141155.
Buijsman, M. C., Y. Kanarska, and J. C. McWilliams, 2010a: On the generation and evolution of nonlinear internal waves in the South China Sea. J. Geophys. Res., 115, C02012, doi:10.1029/ 2009JC005275.
_____, J. C. McWilliams, and C. R. Jackson, 2010b: Eastwest asymmetry in nonlinear internal waves from Luzon Strait. J. Geophys. Res., 115, C10057, doi:10.1029/2009JC006004.
Cai, S., X. Long, and Z. Gan, 2002: A numerical study of the generation and propagation of internal solitary waves in the Luzon Strait. Oceanol. Acta, 25, 5160.
Chao, S., D. Ko, R. Lien, and P. Shaw, 2007: Assessing the west ridge of Luzon Strait as an internal wave mediator. J. Oceanogr., 63, 897911.
Cummins, P. F., L. Armi, and S. Vagle, 2006: Upstream internal hydraulic jumps. J. Phys. Oceanogr., 36, 753769.
Duda, T. F., J. F. Lynch, J. D. Irish, R. C. Beardsley, S. R. Ramp, C. S. Chiu, T. Y. Tang, and Y. J. Yang, 2004: Internal tide and nonlinear wave behavior in the continental slope in the northern South China Sea. IEEE J. Oceanic Eng., 29, 11051130.
Ebbesmeyer, C. C., C. A. Coomes, R. C. Hamilton, K. A. Kurrus, T. C. Sullivan, B. L. Salem, R. D. Romea, and R. J. Bauer, 1991: New observations on internal waves (solitons) in the South China Sea using an acoustic doppler current profiler. Proc. Marine Technology Society, New Orleans, LA, Marine Technology Society, 165175.
Egbert,G.D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183204.
Farmer, D., Q. Li, and J.H. Park, 2009: Internal wave observations in the South China Sea: The role of rotation and nonlinearity. Atmos.Ocean, 47, 267280.
Fermi, E., J. Pasta, and S. Ulam, 1974: Studies of nonlinear problems, I. Nonlinear Wave Motion, A. C. Newell, Ed., Lectures in Applied Mathematics, Vol. 15, American Mathematics Society, 143156.
Galkin, V. N., and Yu. A. Stepanyants, 1991: On the existence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech., 55, 939943.
Gerkema, T., 1996: A unified model for the generation and fission of internal tides in a rotating ocean. J. Mar. Res.., 54, 421450.
Gilman, O. A., R. Grimshaw, and Yu. A. Stepanyants, 1996: Dynamics of internal solitary waves in a rotating fluid. Dyn. Atmos. Oceans, 23, 403411.
Gorshkov, K. A., and L. A. Ostrovsky, 1981: Interactions of solitons in nonintegrable system: Direct perturbation method and applications. Physica D, 3, 428438.
Helfrich, K. R., 2007: Decay and return of internal solitary waves with rotation. J. Phys. Fluids., 19, 026601, doi:10.1063/1.2472509.
_____, and R. H. J. Grimshaw, 2008: Nonlinear disintegration of the internal tide. J. Phys. Oceanogr., 38, 686701.
Hibiya, T., 1986: Generation mechanism of internal waves by tidal flow over a sill. J. Geophys. Res., 91, 76977708.
Hunter, J. K., 1990: Numerical solutions of some nonlinear dispersive wave equations. Computational Solution of Nonlinear Systems of Equations, E. L. Allgower and K. Georg, Eds., Lectures in Applied Mathematics, Vol. 26, 301316.
Jackson, C., 2009: Internal wave detection using the Moderate Resolution Imaging Spectroradiometer (MODIS). J. Geophys. Res., 112, C11012, doi:10.1029/2007JC004220.
Johnston, T. M. S., and M. A. Merrifield, 2003: Internal tide scattering at seamounts, ridges, and islands. J. Geophys. Res., 108, 3180, doi:10.1029/2002JC001528.
Klymak, J. M., R. Pinkel, C.T. Liu, A. K. Liu, and L. David, 2006: Prototypical solitons in the South China Sea. Geophys. Res. Lett., 33, L11607, doi:10.1029/2006GL025932.
Leonov, A. I., 1981: The effect of Earth rotation on the propagation of weak nonlinear surface and internal long oceanic waves. Ann. N. Y. Acad. Sci., 373, 150159.
Li, Q., 2010: Nonlinear internal waves in the South China Sea. Ph.D. thesis, University of Rhode Island, 307 pp.
_____, D. M. Farmer, T. F. Duda, and S. Ramp, 2009: Acoustical measurement of nonlinear internal waves using the inverted echo sounder. J. Atmos. Oceanic Technol., 26, 22282242.
Liu, A. K., Y. S. Chang, M. K. Hsu, and N. K. Liang, 1998: Evolution of nonlinear internal waves in the East and South China Seas. J. Geophys. Res., 103 (C4), 79958008.
National Geophysical Data Center, cited 2006: 2minute gridded global relief data (ETOPO2v2). National Oceanic and Atmospheric Administration National Geophysical Data Center. [Available online at http://www.ngdc.noaa.gov/mgg/fliers/06mgg01. html.]
Ostrovsky, L. A., 1978: Nonlinear internal waves in a rotating ocean. Oceanology, 18, 119125.
Plougonven, R., and V. Zeitlin, 2003: On periodic inertiagravity waves of finite amplitude propagating without change of form at sharp densitygradient interfaces in the rotating fluid. Phys. Lett., 314, 140149.
Ramp, S. R., and Coauthors, 2004: Internal solitons in the northeastern South China Sea. Part I: Sources and deep water propagation. IEEE J. Oceanic Eng., 29, 11571181.
_____,Y. J. Yang, and F. L. Bahr, 2010: Characterizing the nonlinear internal wave climate in the northeastern South China Sea. Nonlinear Processes Geophys., 17, 481498, doi:10.5194/npg17481 2010.
Shaw, P.T., D. S. Ko, and S.Y. Chao, 2009: Internal solitary waves induced by flow over a ridge: with applications to the northern South China Sea. J. Geophys. Res., 114, C02019, doi:10.1029/ 2008JC005007.
Stepanyants, Yu. A., 2006: On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compaction and compound solitons. Chaos Solitons Fractals, 28, 193204.
Turner, J. S., 1973: Buoyancy Effects in Fluid. Cambridge University Press, 373 pp.
Vlasenko, V., N. Stashchuk, and K. Hutter, 2005: Baroclinic Tides: Theoretical Modeling and Observational Evidence. Cambridge University Press, 351 pp.
WarnVarnas,A., J.Hawkins, K.G. Lamb, S. Piacsek, S. ChinBing, D. King, and G. Burgos, 2010: Solitary wave generation dynamics at Luzon Strait. Ocean Modell., 31, 927, doi:10.1016/ j.ocemod.2009.08.002.
Zhang, Z., O. B. Fringer, and S. R. Ramp, 2011: Threedimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the SouthChina Sea. J.Geophys. Res., 116, C05022, doi:10.1029/2010JC006424.
Zhao, Z., and M. H. Alford, 2006: Source and propagation of internal solitary waves in the northeastern South China Sea. J. Geophys. Res., 111, C11012, doi:10.1029/2006JC003644.
Zheng, Q., R. D. Susanto, C.R. Ho, Y. T. Song, and Q. Xu, 2007: Statistical and dynamical analyses of generation mechanisms of solitary internal waves in the northern South China Sea. J. Geophys. Res., 112, C03021, doi:10.1029/ 2006JC003551.
QIANG LI* AND DAVID M. FARMER
Graduate School of Oceanography, University of Rhode Island, Kingston, Rhode Island
(Manuscript received 21 September 2010, in final form 29 January 2011)
* Current affiliation: Center for Ocean Science and Technology, Graduate School at Shenzhen, Tsinghua University, Shenzhen, China.
Corresponding author address: David M. Farmer, Graduate School of Oceanography, University of Rhode Island, South Ferry Road, Kingston, RI 02884.
Email: dfarmer@gso.uri.edu
APPENDIX
Normalization of the OstrovskyHunter Equation
Given the OstrovskyHunter Eq. (4), a dilation theorem is readily proved (Boyd 2005): If y(x, t) is a solution of the OstrovskyHunter equation, from any positive ?,
... (A1)
is also a solution. Given an initial condition h5A coskx, it has the same solution as
... (A2)
Here g2 = f 2/2c. Therefore, for an arbitrary initial harmonic internal wave h 5 A coskx, the amplitude A, distance x, and time t are normalized by
... (A3)
so h9 = A9 cos(x9) can be generally analyzed as shown in section 3. The corresponding Ostrovsky number is directly related to the wave amplitude,
... (A4)
Therefore, the solutions of the OstrovskyHunter equation are uniquely determined by the Ostrovsky number.
(c) 2011 American Meteorological Society
[ Back To SIP Trunking Home's Homepage ]
