Numerical investigation of internal solitary waves from the Luzon Strait: Generation process, mechanism and three-dimensional effects

Ocean Modelling 38 (2011) 203–216

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Numerical investigation of internal solitary waves from the Luzon Strait: Generation process, mechanism and three-dimensional effects C. Guo a, X. Chen a,⇑, V. Vlasenko b, N. Stashchuk b a b

Ocean University of China, 238 Songling Road, Qingdao 266100, China School of Marine Science and Engineering, University of Plymouth, Drake Circus, Plymouth PL8 4AA, UK

a r t i c l e

i n f o

Article history: Received 25 September 2010 Received in revised form 9 March 2011 Accepted 11 March 2011 Available online 22 March 2011 Keywords: Internal solitary waves Numerical modeling Three-dimensional Luzon Strait

a b s t r a c t A fully nonlinear, non-hydrostatic model, MITgcm, is used to investigate internal solitary waves (ISWs) from the Luzon Strait (LS). As the ISWs in the South China Sea (SCS) have drawn more and more attention in recent years, they are studied in various ways, i.e., via remote sensing images, in situ measurements, and numerical simulations. The inspiration of this paper derived from the potential flaws of different numerical models that were employed to examine ISWs. In this study, we performed three-dimensional (3D) experiments with realistic topography and stratification, as well as with fully non-hydrostatic terms in the model, which was rather important for investigating the ISWs. Modeling results showed that baroclinic tides in the LS were essentially three-dimensional (3D), and that wave structures around two ridges in the strait were complicated with interesting internal oceanic phenomena. Several zonal cross-sections were chosen to illustrate vertical structures of zonal velocity field, and to show their meridional variances together with surface horizontal velocity gradients in order to highlight the advantages of 3D modeling with fully nonlinear, non-hydrostatic terms. Following Vlasenko et al. (2005), analysis of two parameters (Froude number and slope parameter that is defined as the ratio of inclination of topography to slope of radiated rays) that govern generation regime indicated that internal waves produced in the LS were subject to a mixed lee wave regime rather than baroclinic tide regime or unsteady lee wave regime. The propagation of ISWs beyond the generation area showed that manifestation of 3D effects was not very obvious, which, through further analysis, was mainly attributed to homogeneity of topography, inaccuracy of barotropic forcing, and Kuroshio intrusion in the LS. To better understand the necessity of 3D modeling, we chose several zonal cross-sections and performed various sensitivity experiments to show discrepancies between 2D and 3D cases. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Brief introduction The internal solitary waves (ISWs) in the South China Sea (SCS) have drawn growing attention over the past decade. They originate from the Luzon Strait (LS) and propagate westward or northwestward to deep water, and then onto the slope-shelf areas where wave disintegration or breaking takes place (Zhao et al., 2004; Zhao and Alford, 2006). During their shoaling process on the shelf of northern SCS, they gradually become deformed and eventually dissipate. According to Klymak et al. (2006), waves with amplitudes

up to 170 m and phase speeds of 2.9 ± 0.1 m/s have even been recorded in the deep basin of northern SCS. The Luzon Strait (LS) is a gateway that connects the SCS and the western Pacific Ocean. There are two ridges across the strait in the longitudinal direction over which many small islands and underwater banks are unevenly distributed (Fig. 1). The eastern ridge is relatively taller except in the northern part of the LS where the western ridge connects to the continental shelf. Normally, the eastern ridge is considered to be much more effective in generating the ISWs while the western ridge has comparatively much smaller influences due to its greater depth below the surface (Du et al., 2008; Warn-Varnas et al., 2009). 1.2. Overview of ISWs in the northern SCS

⇑ Corresponding author. Address: College of Physical and Environmental Oceanography, Ocean University of China, 238 Songling Road, Qingdao, Shandong, China. Tel.: +86 13969718201. E-mail address: [email protected] (X. Chen). 1463-5003/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2011.03.002

The ISWs in the northern SCS are amongst the largest ones that have been recorded in world’s oceans. They are frequently seen in remote sensing images (Hsu and Liu, 2000; Zhao et al., 2004; Du

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Fig. 1. Bathymetry of the area around the Luzon Strait. Grey rectangle is the model domain, and two black rectangles are two other potential generation sites for the ISWs that will be discussed later. Green dots in the LS (labeled a–d) are four representative sites for prediction of barotropic tides in the LS with TPXO 7.1 (see Fig. 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

et al., 2008) (also shown in Figs. 2 and 3) and from in situ measurements (Duda et al., 2004; Ramp et al., 2004; Klymak et al., 2006; Farmer et al., 2009; Alford et al., 2010). In recent years, numerical models have also been employed to investigate these wave activities in the northern SCS (Du et al., 2008; Shaw et al., 2009; WarnVarnas et al., 2009; Buijsman et al., 2010; Song et al., 2010). SAR (Synthetic Aperture Radar) images, field observations, and numerical simulations are the three main approaches that are routinely used to study the ISWs in this area. 1.2.1. Remote sensing images From remote sensing images we can see that the ISWs are quite active in the northern SCS (Ebbesmeyer et al., 1991; Hsu and Liu, 2000; Zhao et al., 2004; Du et al., 2008). A good-quality Envisat Advanced Synthetic Aperture Radar (ASAR) image from the European Space Agency (ESA) clearly shows the existence of ISWs in this area from deep water near the LS to the shelf (Fig. 2). There are four forms of ISWs visible from this image: a well developed packet with two strong waves near the generation area whose wave fronts are relatively short (part 1); a much wider yet weaker packet whose northern end is strongly curved under the effect of nonlinear dispersion due to sharp change of water depth (part 2); a packet being refracted by Dongsha Island, in which nonlinear dispersion is also pronounced (part 3); and the final fate of these waves, i.e., they re-encounter with each other after refraction, but in this stage, they are hardly visible due to their evolution towards the shelf where disintegration and dissipation usually take place (part 4). The fact that the four ISW packets appear in one image strongly demonstrates that the ISWs are ubiquitous in the northern SCS. Zhao et al. (2004) compiled a spatial distribution map of solitary wave packets from 1995 to 2001. In this map, the westward and northwestward propagations of the ISWs can be observed at a number of locations from the LS all the way to the shelf of the northern SCS, which suggests that the LS is the source of these waves. Moreover, two types of solitary waves are identified in

the map: Multiple-wave ISW packets with rank-ordered ISWs and single-wave ISW packets containing only one ISW without tails. These two types of ISWs were also illustrated by Du et al. (2008), who also showed several SAR images. Furthermore, with the help of SAR images, these authors speculated different generation sites in the LS according to the propagating directions of the ISWs. Two areas were identified: islands in the middle (Batan Island, Sabtang Island, etc.) and southern (Babuyan Island, etc.) reaches of the LS corresponding to wave propagating directions towards westward and northwestward, respectively. Both multiplewave and single-wave ISW packets are observed in these two directions. Another ASAR image best justified the speculation of Du et al. (2008) (shown in Fig. 3). As can be seen from this figure, two types of waves, a single ISW and a packet, are clearly visible and spread across the northern and central parts of the LS. According to their propagating directions, one source area can be identified (as is shown by the solid white arrow), i.e., a group of islands in the southern LS, including Babuyan Island. Another direction, as is indicated by the dashed white arrow, seems to point to the islands located in the central LS. That the arrow emanating from this area is perpendicular to the southern ends of these two waves shows that very likely the generated waves are combined from both southern and central parts of the LS where some islands are located (i.e., Batan Island, Babuyan Island, etc.). 1.2.2. In situ measurements There are few in situ measurements on the ISWs in the northern SCS. In September 1990, an ADCP was deployed in Lufeng, which is located on the shelf of the northern SCS (Ebbesmeyer et al., 1991). The current meter data clearly showed the passage of ISW packets with the leading soliton inducing the fastest fluid motion. Ebbesmeyer et al. (1991) correlated the solitons in Lufeng with tidal forcing in the channel between Batan and Sabtang islands, and several days’ lag time between barotropic tidal signal in the LS and the solitons’ arrival was consistent with the propagation of first mode solitons. A very large field experiment, Asian Seas

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Fig. 2. An Envisat ASAR image from the ESA taken at 02:07:17 on 24 August 2008. Dashed white rectangles indicate four types of ISWs. Note that the map departs slightly from northerly orientation.

International Acoustic Experiment (ASIAEX), was carried out in the SCS during 2000 and 2001. Some moorings were anchored on the shelf of the northern SCS and recorded many ISWs that passed by. A number of research and analyses were based on the ASIAEX (Duda et al., 2004; Ramp et al., 2004; Zhao and Alford, 2006), which provided more insight into the ISWs in the SCS. Two types of ISWs were also observed during the ASIAEX and very rare second mode ISWs were detected (Ramp et al., 2004). Moreover, in recent years, more moorings were deployed in the northern SCS from the LS all the way to the shelf area (Farmer et al., 2009; Alford et al., 2010). Wave properties, especially in deep water, are investigated, which is quite useful for proper interpretation of the ISW-related phenomena in the SCS since these long time mooring data have fine temporal and spatial (in the vertical direction) resolution. 1.2.3. Numerical simulation As for the numerical simulation of ISWs in the northern SCS, which is the main focus of this paper, a number of models have been developed to study these waves (Cai et al., 2002; Du et al., 2008; Farmer et al., 2009; Shaw et al., 2009; Warn-Varnas et al., 2009; Buijsman et al., 2010). These studies mainly focus on how the ISWs

are generated and how they evolve in deep water under the effects of ambient influencing factors. Their conclusions are basically similar and supportive of each other. The simulation results qualitatively coincide with field measurements, i.e., energy from barotropic waves is transferred to internal waves and depressions are formed in the LS. After that they propagate towards the shelf area and later transform into a series of ISW packets due to nonlinear effects. Some impact factors, which include rotation, stratification, bathymetry, tidal forcing, Kuroshio intrusion, among others, are also considered in order to better understand this whole process. Nonetheless, most of these numerical models have some kind of deficiency which renders them unlikely to simulate these distinct phenomena accurately. For instance, two-dimensional (2D) modeling (or even two layers), idealized bathymetry and stratification, and hydrostatic assumption are commonly applied in these models for simplicity. As we can see from Fig. 1, bathymetry in the LS is complicated and essentially 3D. As a consequence, 2D models are not sufficient to delineate the properties of the ISWs in this area because wave refraction and diffraction may take place (shown by the curvature of the ISWs in the SAR images of Figs. 2 and 3; also proved by Vlasenko and Stashchuk (2007)).

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Fig. 3. An Envisat ASAR image from the ESA taken at 14:04:12 on 16 August 2008. Solid and dashed arrows show the two propagating directions in the LS. Note that the map departs slightly from northerly orientation.

In this study, we addressed most of the above-mentioned shortcomings of numerical models. To be more specific, we used realistic bottom topography, realistic stratification, fully nonlinear, non-hydrostatic governing equations, and most importantly 3D simulation. Here, we emphasized the structure of tidal beams, the generation process and mechanism, and 3D effects, while detailed analyses of multimodal structure of the ISWs, some small-scale waves, and the effects of the two ridges in the LS were formulated in another paper (Vlasenko et al., 2010). Moreover, to have a comprehensive understanding of baroclinic tides in the northern SCS, one should consider the whole process including generation, evolution, disintegration, shoaling, and dissipation. However, amongst these processes investigation of the generation process is the most difficult and important and thus is the focus of the present work. The paper is structured as follows. The numerical model set-up is described in Section 2. Analyses of the generation process and mechanism are conducted in Section 3. The 3D effects are shown in Section 4, followed by conclusions in Section 5. 2. Model setup A 3D primitive equation model, MIT general circulation model (MITgcm), with fully nonlinear, non-hydrostatic terms was used to conduct these experiments. The non-hydrostatic term in the

vertical momentum equation is essential for the generation of large amplitude ISWs because it balances nonlinearity when leading waves become steep and narrow; in such a case, Coriolis dispersion can usually be ruled out (nonlinearity is balanced by rotation before non-hydrostatic dispersion takes effect). Many ocean models apply hydrostatic approximations and can only be used to investigate long waves (like internal tides) but are not appropriate for studying ISWs (e.g., Princeton Ocean Model (POM), see Niwa and Hibiya (2004); Regional Ocean Modeling System (ROMS) has just been upgraded to its non-hydrostatic version; see Buijsman et al. (2010)). For detailed description of the structure and algorithm of MITgcm, please refer to Marshall et al. (1997). Our model domain was restricted to the central part of the LS because this area is a major site for the generation of the ISWs (shown as the grey rectangle in Fig. 1). The reason why we chose such an area was primarily based on several preceding works (Zhao et al., 2004; Zheng et al., 2007), which studied the ISWs using SAR images. Of course, there exist other areas in the LS, which can also excite baroclinic tides, for example, in the northern part of the western ridge where it links to the continental shelf and in the southern part of the LS where several islands are scattered (shown by the two black rectangles in Fig. 1 outside our model domain). The bathymetry we used in the model was from ETOPO1 Global Relief Model (2009), with the spatial resolution of 10 . In addition,

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we modified the bathymetry between Batan Island and Sabtang Island manually following Ebbesmeyer et al. (1991). Since we only focused on the generation process of the ISWs, water depth beyond the deep water area was set to 3000 m. Horizontal grid resolution in the x-direction (east–west direction) was set to 250 m, which was fine enough to scrutinize detailed wave structures. (A sensitivity test with the resolution of 500 m was conducted, and the final wave field was almost the same as that from the 250-m grid, indicating that 500 m is also acceptable.) In the y-direction (north– south direction), spatial resolution was 1 km. Along the four open boundaries, we added sponge areas in which the horizontal grid dx (dy) increased exponentially from 250 m (1000 m) to 107 m in order to avoid both barotropic and baroclinic wave reflections. There were 90 layers in the vertical z-direction: 50 layers with 10-m resolution, another 20 layers with 50-m resolution, and the bottom 20 layers with 150-m resolution. Time step was set to 12.5 s, which well satisfied the Courant–Friedrichs–Lewy (CFL) condition. To better understand barotropic tidal properties in this area, a global inverse tidal model of TPXO7.1 (Egbert and Erofeeva, 2002) was employed to evaluate barotropic tides there. Four representative sites marked by green dots in Fig. 1 were chosen to illustrate zonal barotropic currents (see Fig. 4). As can be seen from the figure, both diurnal and semidiurnal tidal components with a fortnightly modulation were clearly visible in the upper three panels,

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and these three sites had very similar tidal structure, indicating tidal coherence along the meridional direction in our model domain. Velocity amplitude in Fig. 4d was much smaller due to the much larger water depth there; and tidal harmonics, although not pronounced, also comprised both diurnal and semidiurnal components. Such information on barotropic tides is very important since it acts as a yardstick to justify our choice of magnitude of tidal potential. In our model, baroclinic tides were not forced by open boundary forcing as was done in many other works (e.g., Warn-Varnas et al., 2009; Buijsman et al., 2010). Instead, we added extra terms in the momentum equations to drive the MITgcm. These two terms are solved from simplified barotropic equations and have the following forms: Fx = UH0/H(x, y)rcos(rt) and Fy = UH0/H(x, y)f sin(rt) in the zonal and meridional directions, respectively. In these two terms, H0 is the depth where barotropic tide has velocity U, H(x, y) is local water depth, and f is the Coriolis parameter. We need to choose an appropriate value for U so that amplitudes of the barotropic tides generated at these four sites comply with those predicted by TPXO 7.1 in order to make our results realistically representative. We want to emphasize here that, although the barotropic tides in this area feature both diurnal and semidiurnal characters, we investigate barotropic M2 and K1 tidal forcing separately. As is well known, the critical latitude for K1 internal tides in the northern hemisphere is about 30°N, which is very close to 20.5°N where

Fig. 4. Zonal velocities (u) over a period of one month derived from TPXO 7.1 at four different sites as shown in Fig. 1: (a) 20°N, 121.85°E; (b) 20.5°N, 121.85°E; (c) 21°N, 121.85°E; and (d) 20.5°N, 121.5°E.

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the centre of our model domain is located. Actually, the effect of rotation will suppress the generation of internal waves (Gerkema, 1996; Vlasenko et al., 2005; Helfrich and Grimshaw, 2008), especially at high latitudes. The generated waves will be weaker if the local latitude is closer to the critical latitude (Vlasenko et al., 2005). As is shown by Vlasenko et al. (2010), the generated waves do not reveal any intensive signals carrying energy from the bottom features in the LS, which justifies our assumption that K1 barotropic tidal forcing is not effective for the generation of internal tides in the LS. Numerical experiments (Niwa and Hibiya, 2004) have shown that M2 internal tides are generated effectively above the two ridges in the LS. Consequently, in this paper we only focus on the generation of the ISWs with M2 tidal forcing. Five M2 periods are performed and the last period output is used for analysis. Background temperature and salinity profiles were monthly averaged data taken from World Ocean Atlas (2005). We chose data in January and July to represent winter and summer, respectively. Density and buoyancy frequency were computed through   the linear equation of state and formula N 2 ¼  qg @@zq , respectively (see Fig. 5). From Fig. 5 we can see that in summer the stratification is much stronger than that in winter, and that the pycnocline in summer resides at the depth of about 80 m, while in winter the stratification is relatively uniform and the pycnocline is a little deeper.

To reduce mixing and to ensure that the model runs smoothly, Richardson number-dependent parameterizations (Pacanowski and Philander, 1981) and the scheme by Leith (1996) are employed in our model.

3. Generation process and mechanism In this section, we describe the model output that is driven by M2 tidal forcing with its magnitude close to that predicted by TPXO 7.1 and the background temperature/salinity is from the summer profiles. We show a plane view of surface horizontal velocity gradients (Fig. 6), which is a measure of sea surface roughness and represents the signal captured by SAR images, and a vertical cross-section in the zonal direction illustrating the internal temperature field (Fig. 7) after five M2 tidal periods. The leading edge of a first mode depression ISW is associated with the convergence of surface current, and the trailing edge corresponds with the divergence. According to the classic theory by Alpers (1985), in SAR images the front section of such a wave is bright and the rear section is dark, due to the scattering of signals emitted by satellite sensors at the sea surface. Such sea surface roughness caused by the convergence/divergence of sea water is reproduced in Fig. 6. As is shown in the figure, wave energy is

Fig. 5. Computed background density (left panel) and buoyancy frequency (right panel) based on monthly averaged temperature and salinity data (from WOA05). Solid lines are for summer, and dashed lines are for winter.

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Fig. 6. Surface horizontal velocity gradient from the model after five M2 tidal periods overlapped on bathymetry near the LS. Numbers 1–5 correspond to different underwater vertical wave structures, as will be shown in Fig. 7.

Fig. 7. Temperature profile along zonal cross-section (b) (marked in Fig. 6) after 5 M2 tidal periods. Arrows 1 to 5 correspond to five different wave structures whose positions are also shown in Fig. 6 (see the text for further analysis) (Vlasenko et al., 2010).

radiated out from the LS and after certain evolution in deep water near the two ridges, an ISW packet with several waves emerges west of the LS. It is obvious that the internal signals in this area are essentially 3D, especially in the near field of the two ridges east of 120°E, while in the far field, the 3D effect is not so pronounced. On the whole, the signal in the northern part of the domain is stronger than that in the south. We will come back to this 3D issue in Section 4. In Fig. 7, underwater wave motions are clearly delineated. There is a wave packet containing two waves at about 360 km (amplitude up to 120 m; see Arrow 1), followed by a second mode ISW (amplitude up to 80 m; see Arrow 2) on which some small scale waves are riding. Behind them is another smooth second mode ISW (Arrow 3), followed by a single first mode ISW (Arrow 4). At about 190 km, there exists a complex structure, which looks much like a second mode but actually is the superposition of high modes and will disintegrate if we continue this run a little further. Here we only roughly describe these wave structures of different modes. Detailed analysis and explanations are given in another paper (Vlasenko et al., 2010).

As for the generation mechanism of the internal waves in the LS, different conclusions have been reached in previous studies. Using a simplified two-layer model, Cai et al. (2002) suggested a mechanism that belonged to a lee wave regime. Zhao and Alford (2006) made a comparison between observed ISWs on the shelf and the corresponding westward propagating tidal current peaks in the LS, and found that they were correlated very well. Since the eastward and westward tidal currents are asymmetrical and there is no obvious correlation between the ISWs and eastward tidal currents, it is suggested that the ISW packets are generated by nonlinear steepening of internal tides rather than by lee wave mechanism. Du et al. (2008) employed a 2D numerical model with idealized topography to investigate the generation process of the ISWs in the LS; they suggested two mechanisms: the evolution of depression and the internal mixing disturbance. The former is close to the lee wave mechanism, while the latter much resembles the theory of Maxworthy (1979). Following Gerkema (2001), Shaw et al. (2009) proposed that generation of the ISWs was a two-step process: the production of internal wave beams and their nonlinear interaction with a sharp thermocline. By using a simplified

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linear analytical model, Yuan et al. (2006b) suggested a western boundary current instability generation mechanism for these internal waves, which was quite different from the above-mentioned mechanisms. It is understandable that different assertions are concluded, since different people use different approaches to investigate the ISWs in this area and some of them are not mutually exclusive. Take numerical simulations for example, various conditions, from type of topography and value of tidal forcing to the model itself (i.e., algorithm, hydrostatic or non-hydrostatic, how motions are forced, 2D or 3D, etc.) are applied in different models, which may easily lead to the diversity of conclusions. Next, we will discuss ISWs’ generation mechanism in the LS, with the control of two non-dimensional parameters that classify generated baroclinic tides over variable bottom topography: (1) Froude number

Fr ¼ kU 0 =r

ð1Þ

where k is the horizontal wavenumber of the generated waves, U0 is the amplitude of barotropic tidal velocity and r is the tidal frequency here and (2) slope parameter

dh=dx

a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr2  f 2 Þ=ðN2  r2 Þ

ð2Þ

which defines the ratio of topography inclination (numerator) to ray slope of generated waves (denominator). According to Nakamura et al. (2000), total time variation of tidal forcing can be expressed as a sum of local time variation and advection, i.e.,

dF @F @F ¼ þu dt @t @x

ð3Þ

where u represents barotropic flow. The first term on the right-hand side of the above equation is responsible for the generation of baroclinic tides with tidal frequency, while the second term stands for the advection effect and is responsible for the formation of lee waves. When the first term is much larger than the second term, i.e., Fr  1, baroclinic tides are excited; when the second term is dominant, i.e., Fr  1, unsteady lee waves are produced. In the intermediate case when these two terms are comparable, i.e.,

Fr  1, it corresponds to a mixed lee wave regime where both baroclinic tides and lee waves are present. After a rough estimation of the Froude number in our model domain, the area around the islands where barotropic velocity is larger has the value in the order of unity, while in the deeper channels between and beyond the islands the Froude number is much smaller than unity, indicating a tendency for a baroclinic tides regime. As for the slope parameter, an isopleth map is drawn to examine its distribution in our domain (Fig. 8). It is known that steep topography is favorable for the conversion of energy from surface tides to internal tides (Vlasenko et al., 2005). On supercritical slopes, well defined beams will radiate obliquely upward and downward, respectively. From Fig. 8 we can see that critical condition for semidiurnal tide is not broadly distributed in this area and critical and supercritical slopes concentrate in the band between 121.5°E and 122°E. Following Vlasenko et al. (2005), for steep topography (i.e., slope parameter is of the order of unity in some regions) below the critical latitude, when Fr  1, baroclinic tidal beams and multimodal solution exist, while for Fr  1, multimodal baroclinic tides and their further evolution to first and second mode wave trains and ISWs together with mixed unsteady lee waves are produced. In our case, in the areas with shallow water, especially over the eastern ridge, the Froude number is mostly in the order of unity, while at other sites it is much smaller. As for the slope parameter, it is basically critical or supercritical along the eastern ridge and partly along the western ridge. Thus, we can expect a number of wave phenomena around this area according to the classification of Vlasenko et al. (2005). To examine its vertical wave structure, zonal velocity field along cross-section (b) is contoured to illustrate the above-mentioned wave phenomena (Fig. 9). In Fig. 9, tidal beams calculated from initial stratification are superimposed on zonal velocity field at the end of the fifth M2 period when westward barotropic tide slackens and starts to propagate eastward. Scrutinizing Fig. 9, we can see that wave motions around the two ridges are quite active and strongly nonlinear. Both the eastern ridge and western ridge are able to radiate beams, which are further reflected at the thermocline or bottom and are finally scattered in the upper layer due to their interactions with the thermocline and the effects of nonlinearity. The distance between two adjacent intersections on the free surface is about

Fig. 8. Slope parameters (i.e., the ratio between topography inclination and ray slope of generated waves) are calculated for M2 tides in the whole domain. Values above five are all set to five, and those below unity are all set to unity, in order to highlight the features of interest.

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Fig. 9. Contoured zonal velocity along cross-section (b). Positive (red) and negative values (blue) correspond to eastward and westward velocity, respectively. Superimposed are several tidal beams (calculated from background stratification) radiated from the two ridges. Dashed and solid lines are beams emanating from western ridge and eastern ridge, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

125 km, which is exactly the wavelength of first mode internal tide. The location where the eastern ridge emits beams is at the depth of about 150 m, which is slightly below the pycnocline. Farther beyond the two ridges, first and second mode ISWs gradually detach from tidal beams and evolve westward. To highlight our 3D modeling, another two cross-sections that lie north and south to cross-section (b), cross-sections (a) and (d) indicated by dashed lines in Fig. 6, are chosen to illustrate both coherence and variance of the wave field in meridional direction (shown in Figs. 10 and 11). Topography of cross-section (a) is much more regular compared to cross-section (b), and clearer and stronger rays are visible around both ridges. On either side of the eastern ridge, two beams propagating in different directions are perceived; one propagates upward and then is reflected back at the thermocline, and the other goes downward and turns back at the bottom. Again, nonlinear superposition of tidal beams generated by the two ridges makes the final signals much more significant, which eventually leads to the complexity of ray structures and multiple modes. On the other hand, cross-section (d), which lies southernmost within the three sections, possesses basically similar structure with cross-sections (a) and (b). Nonetheless, signals in cross-section (d) are the weakest and the generated ISWs have the smallest amplitudes (actually the second mode ISW is hardly visible from the pure zonal velocity field) due to the reduction of nonlinearity. This signal reduction from north to south is closely related to topography of different cross-sections and is in accordance with the distribution of surface horizontal velocity gradients shown in Fig. 6. It is beyond our expectation that the western ridge can also produce such evident beams, taking into account its much greater depth and relatively gentler topography. To check this out, the same numerical experiment was repeated along cross-section (d) with the eastern ridge truncated, i.e., the depth where the eastern ridge is located was set to 3000 m (Fig. 12). The reason why we

chose cross-section (d), rather than cross-section (b), for the truncation experiment is just because that the western ridge in crosssection (b) is relatively irregular (as we can see the three peaks in Fig. 3), thus the generated velocity field in such a case is less illustrative. In this case with truncated topography, the signals produced by the western ridge alone are not as strong as those when the eastern ridge exists. In fact, linear M2 internal tide wavelength derives from background stratification and depth is about 117 km, which is very close to the distance between the two ridges. Thus, wave motion generated above the western ridge is reinforced by the signal produced from the eastern ridge due to resonant characteristics. Moreover, the production of second mode ISWs is also owing to the nonlinear superposition of the signals generated by both ridges, and either of them alone cannot produce evident second mode signals (Vlasenko et al., 2010). Returning to Fig. 12, the Froude number above the bottom feature is much less than the unity, and the slope parameter is close to critical status, indicating that in such a case it is much more like the linear regime and that regular baroclinic tidal beams are expected (Vlasenko et al., 2005). This is verified by wave velocity field in Fig. 12. In this figure almost symmetrical tidal beams are radiated on both sides of the ridge and nonlinear ISWs are absent in the deep water west of 300 km, showing the near linearity of internal waves in this case. Furthermore, when we plotted another very similar graph with the existence of only the eastern ridge, a quite different scenario arose (not shown), i.e., asymmetrical beams with strong nonlinearity were present, which featured characteristics of mixed lee waves. In fact, whether baroclinic tides in the LS belong to a mixed lee wave regime, or not, can also be illustrated by the wave generation process in the first tidal period. In fact, wave fields around the ridges are blurred in cycles 3–5 (shown in Fig. 7 at around x = 0 km), due to the strong barotropic tides and the induced baroclinic wave motions in the previous tidal cycles, so it would not be appropriate to show the generation process in tidal

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Fig. 10. The same as Fig. 9, except for cross-section (a) without superimposing the tidal beams.

Fig. 11. The same as Fig. 9, except for cross-section (d) without superimposing the tidal beams.

cycles 3–5. Considering that our model needs some spin-up time before it comes to a stable and regular state, waves appearing in the very beginning of the model integration are irrelevant, and have smaller amplitudes, especially in the first period. Neverthe-

less, we can still qualitatively see the evolution process in the beginning of the model (figures not shown). When barotropic tides at the very beginning flow eastward and gradually reach their maximum value (i.e., in the first 1/4 T), downstream depressions

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Fig. 12. Contoured zonal velocity along cross-section (d).The eastern ridge has been truncated in this figure (see the text).

and lee waves on the east side of the ridge are formed, but they are trapped there and gradually grow in amplitude by absorbing energy from the background flow. When eastward tide slackens and turns westward, these waves are released and start to evolve westward. These processes occur above the bottom feature and possess the character of lee waves. At the same time, beyond the ridge depression waves are formed on both sides of the ridge and propagate out of the source region efficiently with specific phase speed, indicating properties of baroclinic tides as well. In sum, waves generated in this domain possess characters of both baroclinic tides and lee waves, and are indeed of the intermediate case – mixed lee waves. 4. Discussion of 3D effects From the above analysis of sea surface signal (horizontal velocity gradients) and three different cross-sections, we can see that 3D simulation is crucial to understanding baroclinc tides in the northern SCS correctly due to its complex topography. However, in the far field beyond the generation area, 3D effects are not as pronounced, which can be seen by the relatively straight wave fronts at about 121°E in Fig. 6. Although from SAR images some waves do have the feature of straightness, most waves have curved fronts (Hsu and Liu, 2000; Zhao et al., 2004). In this section we will discuss some issues on 3D effects. Possible reasons for straight fronts in the far field that should be curved are listed as follows. First, water depth in the far field has been arbitrarily set to 3000 m, which means that waves experience the same environmental conditions there (i.e., topography, stratification). As a consequence, large waves could adjust easily to be uniform without the refraction that is induced by non-constant latitudinal and longitudinal water depth. Another reason could be due to the idealized tidal forcing in the LS. In our model, barotropic tides are elliptic with major axis pointing

in the east–west direction. Although this is qualitatively similar to what is predicted by TPXO 7.1, there do exist some discrepancies between these two models. Moreover, predictions from this barotropic model may also be somewhat inaccurate. Instantaneous tidal currents in the LS between Batan Island and Sabtang Island can be as large as 283 cm/s (Ebbesmeyer et al., 1991), while in TPXO 7.1 the largest velocity is at least 4 times smaller than 283 cm/s. Since the area around Batan and Sabtang islands is considered to be a major generation site and also lies right in the middle of our model domain, larger tidal currents there may induce larger internal waves which may further result in the curvature of generated waves in the far field. Furthermore, the Kuroshio could also be an important player in this area with active oceanic phenomena. Considering the multiple intrusion styles in the LS and its large velocity (about 1 m/s) (Yuan et al., 2006a), the Kuroshio can be an uncertain factor in reality and may cause a larger driving force in the middle of our model domain, which may finally lead to the curvature of wave fronts as well. Some numerical experiments have been performed to examine the impact of this strong western boundary flow (see Warn-Varnas et al., 2009; Buijsman et al., 2010), and their results show that the Kuroshio intrusion does have, though not dominant, impacts on the resulting fields west of the LS. However, our model is driven by tidal potential rather than open boundary forcing as discussed in Section 2, it is impossible to include the Kuroshio in our study. The above-mentioned factors, i.e., variant water depth, realistic tidal forcing, and Kuroshio intrusion, which can result in 3D effects of the wave field, are actually neglected in our model, due to the reasons stated above. To illustrate the advantage of our 3D modeling over the 2D one, we chose some zonal cross-sections and performed several 2D tests, i.e., everything was the same except that we selected several representative 2D transects to examine the difference between 2D and 3D modeling; in this way, we can see clearly the importance of 3D performance.

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Fig. 13. Comparison of vertical temperature field for 2D (dashed line) and 3D (solid line) cases after 5 M2 tidal periods: panels (a) and (b) correspond to cross-sections (c) and (d), respectively.

First of all, cross-sections (c) and (d), which intersect with the eastern ridge at relatively deep water rather than on islands, were chosen to display wave discrepancies between the 2D and 3D cases (Fig. 13). Quite evident differences are seen along cross-section (c) in Fig. 13a. Wave structures in the near field of the two ridges are more or less the same, but in the far end where the ISWs have already formed, wave amplitudes in the 2D case are obviously larger. Surprisingly, larger waves in the 2D case unexpectedly propagate more slowly than the waves in the 3D case, against the classic KdV (Korteweg–de Vries) theory, which states that waves with larger amplitude propagate faster due to nonlinear dispersion. Additionally, in cross-section (d) discrepancies are quite small, and the two wave profiles coincide well, although the same 3D effects exist as well (i.e., larger yet slower waves in the 2D case, though it is not as pronounced along this cross-section). The reason for this should be that the complex 3D bathymetry in the LS makes the generation sites and processes different in the 2D and 3D cases. For instance, the meridional flanks of an undersea mount or island in the 3D simulation are able to affect the resulting wave fields, while in the 2D case such flanks do not exist. To be more specific, the processes of wave refractions developed near the local bottom elevations or depressions in the LS result in such different behaviors during generation and subsequent evolution and interaction processes (Vlasenko and Stashchuk, 2007). By performing 3D numerical experiments in the Andaman Sea, Vlasenko and Stashchuk (2007) found that wave refractions can produce concave and convex fragments of the wave fronts, which can lead to a transverse redistribution of energy perpendic-

ular to the wave propagation direction. They also showed that transects that have relatively abrupt topography were supposed to focus wave energy, which was well applicable to the LS case. To be more exact, since both cross-sections (c) and (d) exhibit smaller waves in the 3D case and considering that these two cross-sections cross the eastern ridge with underwater banks rather than islands (the topography in the peripheral of an island is much steeper), we can see that areas between the islands in the central LS ought to scatter wave energy to the ambient transects. In the meantime, accordingly, the islands (Batan Islands) are the places that ‘‘focus’’ energy. To further corroborate our conclusion, an idealized 2D/3D experiment with regular Gaussian topography (in contrast with the irregular LS topography) was conducted to examine the 3D effects more carefully, and the test results (not shown) presented very similar behavior, i.e., in the 2D case, energy was radiated out of the generation area more slowly but its generated ISWs had slightly larger amplitudes, revealing the generality of such discrepancies between the 2D and 3D cases. Despite the fact that the islands are the energy convergence areas, it is unlikely for us to do 2D experiments if we choose a cross-section that intersects an island; and we tried to ‘‘truncate’’ the island to an undersea bank to see whether the resulting field would be comparable to that in the 3D case. However, this was not the case, and there were even greater discrepancies, showing that this was not a good approach. Nevertheless, we find something novel in this tentative experiment, i.e. a wave packet with non-rank-ordered waves is produced, with its shape and location

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much resemble the waves measured by Farmer et al. (2009). Furthermore, field observations on shelf area during the ASIAEX also reported the existence of such packets (Ramp et al., 2004), which, together with observations by Farmer et al. (2009), confirms the generality of such waves. Nonetheless, such a wave packet is discovered in our model with unrealistic topography, and taking into account the wide and complex topography in the northern SCS, more future works need to be done in order to look into this phenomenon more comprehensively.

5. Conclusions We present in this paper 3D modeling of nonlinear ISWs in the northern SCS, and focus on the generation process around the two ridges in the LS. Previous numerical simulations of the ISWs in this area usually introduce specific simplifications, among which 2D modeling and hydrostatic approximation are usually applied. This could be the reason why these previous simulations did not reproduce internal wave phenomena in the SCS correctly, due to the lack of complexity of topography and the exclusion of non-hydrostatic terms. We aimed to better understand these fascinating wave activities by focusing on their generation process, mechanism, and 3D effects. Satellite images show that the LS is the source region for the generation of large ISWs in the northern SCS (Hsu and Liu, 2000; Zhao et al., 2004). In our model, we chose the central part of the LS as our model domain, since this area was the most effective one among three main generation sites in the region. According to the model output, both first and second mode ISWs with very large amplitudes arise beyond the LS after certain evolution processes. The eastern ridge in the LS can efficiently radiate tidal beams, which then nonlinearly superimpose on beams generated by the western ridge. This leads to very strong signals above both ridges and also the emergence of interesting second mode ISWs. The wave field in our model is essentially 3D, especially in the area east of 120°E, which is closer to the generation sites. Not only can we see the rich activities produced by latitudinal topography variation, as was illustrated by Figs. 7 and 9, the contoured zonal velocity fields also exhibit evident variance induced by topography difference in longitudinal direction. Overall structures in different cross-sections are basically similar, but some detailed wave motions, especially above the eastern ridge, and the intensities of wave signals vary, weakening from north to south. The possible reasons for the homogeneity of wave fronts in the far field are attributed to several factors, including the uniformity of water depth and stratification there, the irregularity of the Kuroshio intrusion, and incomplete setting of barotropic forcing in our model domain. Three additional experiments with 2D transects were performed to understand the differences between the 2D model results and the 3D case. Wave refraction caused by local bottom elevations and depressions can result in wave profile differences. To be more specific, areas between and around the islands respectively scatter and focus energy, due to the steeper topography across the islands. The generation mechanism of baroclinic tides in the LS, which is one of the key points of this paper, is analyzed with the classification of two non-dimensional parameters: Froude number and slope parameter (refer to the text for their definitions). The Froude number is of the order of unity around islands but much smaller in relatively deeper water, and the slope parameter is basically critical or supercritical along the eastern ridge and partly along the western ridge. On the whole, internal waves in the LS belong to the mixed lee wave regime and multiple wave processes like mul-

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