A propagation model for the internal solitary waves in the northern South China Sea

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2010jc006341, 2010 A propagation model for the internal solitary waves in the northern South China Sea Shuqun Cai 1 and Jieshuo Xie 1 Received 14 April 2010; revised 1 September 2010; accepted 1 October 2010; published 31 December [1] A two dimensional, regularized long wave equation model is developed to study the dynamic mechanisms of the propagation and evolution of the internal solitary waves (ISWs) in the northern South China Sea (SCS). It is shown that the bottom topography would cause the polarity reversal of ISWs, the change of the local wave crestline shape, and some diminution in wave amplitude; even if the ISWs are induced at the small sill channel along the Luzon Strait, they could propagate westward with their crestlines covering a large area in the latitudinal direction in the northern SCS. When there are two trains of ISWs propagating from the same source site with a time lag but different amplitudes of initial solitons, the latter train of ISWs with a larger amplitude may catch then swallow the former one with a smaller amplitude, and the wave amplitude of the merged ISW train decreases while the wave number increases. When there are two trains of ISWs propagating from the different source sites at the same time with the same amplitude of initial solitons, the crestlines of the two ISW trains may meet and a new leading soliton is induced at the connection point. Once the ISW trains collide with the island, before the island, a weak ISW train is reflected; behind the island, the former crestlines of the ISW train are torn by the island into two new trains, which may reconnect after passing around the island. The propagation direction, the wave amplitude, and the reconnection point of the new merged ISW train behind the island depend on the relative orientation of the original soliton source site to the island. Citation: Cai, S., and J. Xie (2010), A propagation model for the internal solitary waves in the northern South China Sea, J. Geophys. Res., 115,, doi: /2010jc Introduction [2] There is an active zone of internal solitary waves (ISWs) in the northern South China Sea (SCS). According to the analyses of satellite photographs and in situ observational data [e.g., Fett and Rabe, 1977; Ebbesmeyer et al., 1991; Liu et al., 1998; Orr and Mignerey, 2003; Ramp et al., 2004; Zhao et al., 2004; Zhao and Alford, 2006; Zheng et al., 2007], it is shown that the ISWs may be generated between the sill channels along the Luzon Strait (which connects the SCS and the western Pacific). The internal wave distribution maps in the SCS have been compiled from hundreds of ERS 1/2, RADARSAT, and Space Shuttle SAR images from 1993 to 1999 by Hsu and Liu [2000], and it is shown that most of the ISWs in the northeastern SCS propagate westward. Traveling to the west, the ISWs encounter nearly no shallow water until reaching the continental shelf of China or near the Dongsha Islands. As the 1 Key Laboratory of Tropical Marine Environmental Dynamics, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China. Copyright 2010 by the American Geophysical Union /10/2010JC ISWs travel westward and approach the Dongsha Islands, shoaling, reflection, refraction, and diffraction become dominant processes, e.g., Figure 1 by Hsu and Liu [2000] shows the wave wave and wave island interactions near the Dongsha Islands. A lot of numerical simulation models are employed to study the generation and evolution of the ISWs. Among them, the one dimensional Korteweg de Vries (KdV) equation numerical model, or the regularized long wave (RLW) equation model, is commonly used to study the effects of nonlinearity, dissipation, and shoaling topography and so forth on the evolution of the ISWs [e.g., Liu et al., 1998; Caietal., 2002a; Helfrich and Melville, 2006; Grimshaw et al., 2007]. Two dimensional [e.g., Lynett and Liu, 2002; Cai et al., 2002b; Du et al., 2008; Warn Varnas et al., 2010] or three dimensional [e.g., Chao et al., 2006; Shaw et al., 2009; Buijsman et al., 2010] model studies are relatively fewer. On the basis of a two layer model on the horizontal plane, Lynett and Liu [2002] simulated the internal wave propagation in the vicinity of the Dongsha Islands, and their numerical results showed strong similarities to the satellite images taken over the same locations. Chao et al. [2006] investigated the reflection and diffraction of the ISWs by an island on the basis of a three dimensional nonhydrostatic numerical model and found that in addition to reflected waves, two wave branches passed around the island 1of14

2 Figure 1. RADARSAT ScanSAR image north of the South China Sea (SCS) on 26 April, 1998, showing at least four packets of internal waves. The internal wave packets propagated toward the Dongsha Island, the internal wave packets diffracted by Dongsha Islands into two packets of internal waves and then interacted with each other and remerged as a single wave packet (adapted from Hsu and Liu, 2000). and reconnected behind it. For a westward propagating incoming wave, the Coriolis deflection favored northward wave propagation in the region between the crossover point and the island, shifting the wave reconnection point behind the island northward. However, some questions remain unsolved, e.g., could the crestlines of the ISWs generated at the small source site (i.e., in the sill channels along the Luzon Strait) propagate westward and cover a large domain in the northern SCS? What happens if two trains of ISWs interact with each other? What happens if the ISWs in different propagation directions interact with the Dongsha Islands? Above all, what role, if any, does the topography play? [3] Pierini [1989] employed a weakly two dimensional RLW equation model to simulate the propagation of the ISWs in the Alboran Sea successfully. In this paper, we adapt his model and develop it to simulate the horizontal evolution of the ISWs in the northern SCS, major at the numerical study of the above questions, i.e., the dynamic mechanisms of the wave wave and wave island interactions. In sections 2 4, the setup of the model is given in section 2; in section 3, the numerical experimental results and discussion are presented; and section 4 is the conclusion. 2. Model Description, Numerical Scheme, and Choice of Parameters [4] A one dimensional propagation numerical model based on the RLW equation developed by Holloway et al. [1997], Grimshaw et al. [2001] and Cai et al. [2002a] is adapted here, i.e., t þ c x þ x þ 2 x þ cq x 2Q þ C Dcjj 3 xxt c ¼ 0 Here, h is the displacement of interfacial pycnocline, the first baroclinic modal phase speed c =[ g ð 2 1 Þh 1 h 2 2 h 1 þ 1 h 2 ] 1/2, the nonlinear parameter a = 3c 2 h 2 1 1h 2 2 2h 1 h 2 2 h 1 þ 1 h 2, the dispersion parameter b = ch 1h 2 1 h 1 þ 2 h h 1 þ 1 h 2, and the high order nonlinear coefficient = 3c [ 7 2h 2 1 h2 8 ( 2h 2 1 1h h 1 þ 1 h 2 ) 2 2h 3 1 þ 1h h 1 þ 1 h 2 ]; r 1 and r 2 are the upper and lower densities; h 1 and h 2 are the undisturbed upper and lower layer thicknesses, respectively; C D is the drag ð1þ 2of14

3 Figure 2. Variations of the dispersion parameter b, the high order nonlinear coefficient, the nonlinear parameter a, the first baroclinic modal phase speed c, and the idealized bottom topography of the computational domain in the x direction. coefficient of bottom friction; and Q = Q(x) is a known factor needed to ensure conservation of wave action flux Qh 2. For interfacial waves, here we set Q =2g(r 2 r 1 )c as suggested by Grimshaw et al. [2007]. [5] On the basis of the above one dimensional RLW equation (1), we adapt the two dimensional RLW equation for the waves predominantly traveling along x and whose spatial variation along y is small compared with x (in this sense, it applies to weakly two dimensional waves), as did Pierini [1989], t þ c x þ x þ 2 x þ c 2 þ C Dcj 3 þ c 2 yy ¼ 0 j xxt c [6] The two level, three point Crank Nicholson scheme is used to differentiate equation (2), i.e., 2 x t 1 c 2 x i;j n þ c þ nþ1 i;j þ i;j nþ1 i;j nþ1 x i;j nþ1 0 þ c þ n i;j þ n i;j n i;j x i;j n þ x c C D c n n i;j 2 þ A5 3 þ c 2 2 y n i;j ¼ 0 ð2þ ð3þ Here, each identity is defined as d 2 x h n n i,j =(h i+1,j 2h n n i,j + h i 1,j )/ Dx 2, d 2 y h n n i,j =(h i,j+1 2h n n i,j + h i,j 1 )/Dy 2, d t h n i,j =(h n+1 i,j h n i,j )/ Dt, d x h n n n i,j =(h i+1,j h i 1,j )/(2Dx), i/j and n are the indexes of space along x/y and time, respectively. Equation (3) can be solved by the iteration method. It can be proven that this scheme is unconditionally stable. [7] Ano flux boundary condition [Pierini, 1989], N ¼ 0 is imposed along the rigid boundaries, where N is the direction (x or y) perpendicular to the wall. In our additional sensitive experiment, a radiation boundary condition is substituted and it is found that there is only slight variation near the boundary for both experimental results. As done by Pierini [1989] for the Strait of Gibraltar, the interface displacement at the east open boundary is prescribed along a section from y B to y A (e.g., in Figure 3 (top), where a small channel is 1200 m long in the x direction and 420 m wide in the y direction, and the section from y B to y A represents the source sites of the internal solitons), and the initial incident soliton is given as the solitary wave solution of the extended KdV equation [Grimshaw et al., 2007], ðx; y; tþ ¼ ð4þ ð1 þ BÞ 0 1 þ B cosh½kx ð VtÞŠ ; y B < y < y A ð5þ 3of14

4 Table 1. Model s Running Cases and Their Experimental Parameters Experiment E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 Explanation Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), no island, idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), no island, idealized shoaling bottom topography with the depth ranging from 200 m in the east to 100 m in the west linearly Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), no island, flat bottom topography with a depth of 140 m Incident solitons with h 0 = 36 m at (t =0,x = 0, km y km), no island, flat bottom topography with a depth of 140 m Two trains of incident solitons with the same amplitude of h 0 = 20 m but a time lag of 12.4 h (a period of M 2 tide), one at (t =0,x = 0, N y N), the other at (t = 12.4 h, x = 0, N y N), respectively, real bottom topography in the northern SCS Two trains of incident solitons with a time lag of 15 time steps (i.e., s) but different amplitudes, one at (t =0,x = 0, km y km) with h 0 = 6 m, the other at (t = s, x = 0, km y km) with h 0 = 20 m, respectively, no island, idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly Two trains of incident solitons with the same amplitude of h 0 = 20 m but at different source sites, one at (t =0,x = 0, km y km), the other at (t =0,x = 0, km y km), respectively, no island, idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centered at (x = km, y = 24 km) with a radius of 1.38 km Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centered at (x = km, y = km) with a radius of 1.38 km Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), idealized shoaling bottom topography with the depth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centered at (x = km, y = km) with a radius of 1.38 km Two trains of incident solitons with a time lag of 15 time steps (i.e., s) but different amplitudes, one at (t =0,x = 0, km y km) with h 0 = 6 m, the other at (t = s, x = 0, km y km) with h 0 = 20 m, respectively, no island, flat bottom topography with a depth of 140 m Two trains of incident solitons with the same amplitude of h 0 = 20 m but at different source sites, one at (t =0,x = 0, km y km), the other at (t =0,x = 0, km y km), respectively, no island, flat bottom topography with a depth of 140 m E13 Incident solitons with h 0 = 20 m at (t =0,x = 0, km y km), flat bottom topography with a depth of 140 m, and a circular island centered at (x = km, y = 24 km) with a radius of 1.38 km Here, the nonlinear phase speed is, V ¼ c þ ð1 þ BÞ 0 =6; ð6þ and a(1 + B)h 0 /6 = bk 2, B 2 =1+6bK 2 /a 2. To smooth the initial condition, the initial incident soliton at both ends of the source sites is given as ðx; y; tþ ¼ 0:5ð1þ BÞ 0 1 þ B cosh½kx ð VtÞŠ h tanh y y A W tanh y y B W i ; y ¼ y A ; y B ð7þ where W y B y A. In the computation, we choose W = 0.05(y B y A ). [8] In the computation, we set the relative density difference value between the upper and lower layers at Dr = (r 2 r 1 )/r 1 = , which is obtained by the in situ observation [Cai et al., 2002a]. In our two additional sensitive experiments with the following idealized bottom topography as shown in Figure 2, the other conditions are the same, except that C D values are set 0 and [Grimshaw, 2001], respectively. We find that both experimental results are almost the same, except that the wave amplitude with C D = is slightly weakened. In fact, if we look at the ratio of the friction term to the nonlinear term with parameter a in Equation (2), i.e., C Dcjj 3 x ¼ C DcDx ; ð8þ 3 where Dx is the spatial step, we could find that this dimensionless ratio has an order of about O (0.05) in our computation, which shows that the friction term is unimportant. Thus we set C D = 0 in the following experiments. 3. Numerical Experimental Results and Discussion [9] In all of the experiments, the undisturbed initial upper layer thickness is h 1 = 60 m, and h 2 = H h 1, where H the water depth. We first validate the numerical model in a smaller simulation domain with an idealized bottom topography (Figure 2). For convenience, this computational domain is set within a square domain with each side of 48 km in both x and y directions. The idealized shoaling bottom topography with a mildly constant slope is similar to that in the continental shelf of the northern SCS, but the deepest depth is only 140 m at the eastern boundary and the shallowest depth is only 100 m at the western boundary, so that at x = 24 km away from both the eastern and western boundaries, the water depth is 120 m and the initial upper layer thickness is equal to the lower layer one, i.e., the critical depth for the change of a depression ISW into the elevation one [e.g., Hsu and Liu, 2000]. Thus the critical depth phenomenon can be reflected in the model. Figure 2 also shows the variations of the dispersion parameter b, the high order nonlinear coefficient, the nonlinear parameter a, and the first baroclinic modal phase speed c in the x direction. Thirteen experiments are designed to test whether the possible key 4of14

5 Figure 3. Numerical simulated internal solitary waves (ISWs) in Experiment E1. (top) Horizontal variation of the wave amplitude (unit in m, here and subsequently) at t = 16,425 s. (bottom) Comparison of the wave amplitudes along the section at y =24kmatt = 16,425 s (solid line), t = 21,900 s (dashed line), and t = 32,850 s (thick solid line), respectively. dynamic factors, including the bottom topography, the interactions of wave wave, wave island, and so forth affect the evolution of the ISWs or not. See Table 1 for the numerical experimental cases Validation of the Numerical Experiments With an Idealized Bottom Topography (E1 E4) [10] At first, four experiments are designed to testify to the effects of the topography slope and the amplitudes of the 5of14

6 Figure 4. Comparison of the wave amplitudes along the section at y = 24 km (a) between Experiment E1 at t = 16,425 s (solid line) and Experiment E2 at t = 14,782.5 s (dashed line), and (b) among Experiments E1 (solid line), E3 (thick solid line), and E4 (dashed line) at t = 16,425 s. incident ISWs on the evolution of the ISWs. Experiment E1 is a standard experiment in which the incident solitons with amplitudes of h 0 = 20 m are prescribed at (t =0,x =0,y B = km y km = y A ). Experiments E2 E3 are designed to show the effect of the topography slope. Experiment E2 is almost the same as Experiment E1, except that the deepest depth of the idealized shoaling bottom topography is 200 m, so that the slope in experiment E2 is a little larger than that in experiment E1. Experiment E3 is almost the same as Experiment E1, except that the shoaling bottom topography is replaced by a flat bottom with a constant depth of 140 m. Experiment E4 is designed to show the effect of the amplitudes of the incident ISWs, which is almost the same as Experiment E3, except that the amplitudes of the incident ISWs are 36 m. The spatial step is chosen as Dx = Dy = 60 m, while the temporal step is Dt = s in experiments E1, E3 E4 and Dt = s in Experiment E2 (with a larger linear phase speed, since the water depth is deeper), respectively, so that the dimensionless linear phase speed is also about 1. [11] The horizontal snapshot of the numerical simulated ISWs at t = s in Experiment E1 is shown in Figure 3 (top), the simulated ISWs propagate westward, with its crestline curvature looking like a smooth arc; the leading soliton in its westernmost point is still along the section at about y = 24 km. Figure 3 (bottom) shows the comparison of the wave amplitudes along the section at y = 24 km (within 0 x 48 km) at t = s, 21,900 s and 32,850 s, respectively, the amplitudes of the leading depression solitons at these three moments, with a value of 15.5 m, 14 m, and 12 m, respectively, decrease gradually with time. The initial incident depression wave is gradually replaced by a train of elevation waves riding on a negative pedestal, which agrees with the one dimensional model results by Grimshaw et al. [2004]. At t = 21,900 s, the leading soliton approaches near the critical depth at x = 24 km, however, the ISWs do not change from depression wave into elevation wave at once; instead, after the leading depression wave passes the critical depth, since the nonlinear parameter a changes from negative to positive, the amplitude of the elevation wave following the leading depression one gets larger and larger, while the amplitude of the leading depression wave gets smaller and smaller; finally, e.g., at about t = 32,850 s, the ISWs seem to be led by the elevation wave with a positive amplitude of 12 m. The case in this experiment is similar to those revealed by the onedimensional models [e.g., Liu et al., 1998; Caietal., 2002a], which shows that the above numerical simulation result by the RLW model is basically reasonable. [12] The simulated results in Experiments E2 E4 are very similar to that in Experiment E1; however, there exists some difference in the polarity reversal, the wave amplitude, and the wave number in the ISW train. Figure 4a shows the comparison of the wave amplitudes along the section at y = 24 km between Experiment E1 at t = 16,425 s and E2 at t = 14,782.5 s. It is found that the wave amplitude ( 19.6 m) in Experiment E2 is larger than that in Experiment E1 ( 15.5 m); meanwhile, the wave number in the ISW train and the elevation wave amplitude following the leading depression soliton are much less than those in Experiment E1, which seems to suggest that when the ISWs propagate in deeper water, they may behave in the mode of KdV type rather than extended Koteweg de Vries (ekdv) type solitons. Moreover, the leading depression soliton in Experiment E2 catches up with that in Experiment E1 in less time, which shows that the propagation speed in Experiment E2 is distinctly faster (since its water depth at the source site is deeper) than that in Experiment E1. This demonstrates that in case 6of14

7 Figure 5. (a) Bottom topography of the computational domain in the northern South China Sea (unit in m, note that the white circular area denotes the Dongsha Islands). (b) Horizontal variation of the wave amplitude simulated in Experiment E5 at t = h. the internal solitons are induced at the source site with a deeper water depth, the leading soliton propagates faster, with its amplitude decreasing slowly with time when compared with that induced at the source site with a shallower water depth. [13] Owing to the flat bottom topography in Experiments E3 and E4, the dispersion parameter b, the high order nonlinear coefficient, the nonlinear parameter a, and the first baroclinic modal phase speed c stay unchanged in the x direction. Figure 4b shows the comparison of the wave amplitudes along the section at y = 24 km among Experiments E1, E3, and E4 at t = 16,425 s. It is found that, when compared with that in Experiment E1, the wave amplitude in Experiment E1 is a little less than that in Experiment E3, which suggests that the bottom topography might lead to some diminution in wave amplitude. Although the amplitudes of the incident ISWs are much larger in Experiment E4 than those in Experiment E3, the amplitude of the leading depression soliton decreases faster than that in Experiment E3; moreover, it is found that the elevation wave train in Experiment E4 lasts from the source site during its propagation, with its wave number much larger than that in Experiment E3. If we look at Equation (2), we can see that this is because the contribution of the high order nonlinear term with coefficient becomes more important when the amplitudes of the incident ISWs get larger. Moreover, since nonlinear parameter a does not change its sign in the x direction, the polarity reversal (i.e., the elevation wave following the leading depression soliton changes into the largest leading elevation one) that happens in Experiment E1 does not occur all the way in both Experiments E3 and E4 (figure omitted) Real Bottom Experiment in the Northeastern SCS (E5) [14] One question put forward in section 1 is, Could the crestlines of the ISWs generated at the small source site propagate westward and cover a large domain in the northern SCS? Now, we employ the above numerical model to simulate the propagation of the ISWs in the northeastern SCS with a real bottom topography (Figure 5a). In the computation, owing to the limitations of the computer, the spatial and temporal steps are chosen as Dx = m, Dy = m and Dt = s, respectively, so that the dimensionless linear phase speed in the x direction is also near 1. The depth is linearly interpolated based on the ETOPO5 Global Earth Topography with a resolution of 5. It is supposed that there are two trains of initial incident solitons from y B = Ntoy A = N at the east 7of14

8 Figure 6. Numerical simulated ISWs in Experiment E6, with the horizontal variation of the wave amplitude (a) at t = 5475 s, (b) at t = 12,045 s and (c) at t = 21,900 s, and (d) comparison of the wave amplitudes along the section at y =24kmatt = s (solid line), t = 5475 s (dashed line), and t = 21,900 s (thick solid line), respectively. open boundary with the same amplitude of 20 m but a time lag of 12.4 h (i.e., about the period of M 2 tide) in the experiment. [15] The horizontal snapshot of the numerical simulated ISWs at t = h when the waves have propagated far into the basin in Experiment E5 is shown in Figure 5b, two trains of the simulated ISWs propagate westward, each train has a crestline like a bending bow, and the leading soliton in its westernmost point is within the sections from y B = N toy A = N. Each crestline of the ISWs could cover a large area in the latitudinal direction. The simulated case is basically similar to that shown in Figure 1. It is shown that even if the internal solitons are induced at the small sill channel along the Luzon Strait, they could propagate westward with their crestlines covering a large area in the latitudinal direction in the northern SCS. [16] The above numerical experiment run costs a lot of computer time; meanwhile, it should be stressed that the northern and southern boundaries might be touched by the wave in the later simulation, which would cause some reflected waves that we do not intend to discuss, since in this study we just want to understand some dynamic mechanisms affecting the propagation and evolution of the ISWs. Therefore, in the following, we still use the smaller simulation domain with an idealized bottom topography (Figure 2) to carry out the numerical experiments Wave Wave and Wave Island Interactions Experiments With an Idealized Bottom Topography (E6 E10) [17] What happens if the ISWs interact with each other? Experiments E6 E7 are designed to answer this question. Experiment E6 is designed to show the collision of two ISW trains which propagate from the same source site with a time lag but different amplitudes of incident solitons. It is the same as Experiment E1, except that now there are two trains of incident solitons at (t =0,x =0,y B = km y km = y A ) and at (t = s, x =0,y B = km y km = y A ), respectively, and the amplitudes of the 8of14

9 Figure 7. Numerical simulated ISWs in Experiment E7, with the horizontal variation of the wave amplitude (a) at t = s, (b) at t = 12,045 s, and (c) at t = 21,900 s, and comparison of the wave amplitudes along the section at y = 24 km (dashed line) with those at y = km (solid line) at (d) t = 12,045 s and (e) t = 21,900 s. first incident solitons are 6 m while the second ones are 20 m, respectively. Experiment E7 is designed to show the collision of two ISW trains which propagate from the different source sites at the same time, but with the same amplitude of incident solitons, i.e., two trains of incident solitons have the same amplitudes of h 0 = 20 m with one source site at (x =0,y B = km y km = y A ) and the other at (x =0,y D = km y km = y C ), 9of14

10 Figure 8. Numerical simulated ISWs in Experiment E8, with the horizontal variation of the wave amplitude (a) at t = 12,045 s, (b) at t = 16,425 s, and (c) at t = 21,900 s, and (d) comparison of the wave amplitudes along the section at y =24kmatt = 16,425 s (solid line) with those at t = 21,900 s (dashed line), respectively. respectively. The numerical simulated result in Experiment E6 is shown in Figure 6. At first, the solitons with smaller amplitudes are leading (e.g., at t = 5475 s in Figure 6a or at t = s in Figure 6d), then, since the second train of ISWs with larger amplitudes has a faster propagation speed according to Equation (6), gradually, the latter train of ISWs catches the former one (e.g., at t = 12,045 s in Figure 6b). Finally, the latter train of ISWs seems to swallow the former one (e.g., at t = 21,900 s in Figures 6c and 6d), the two trains of ISWs merge, and the merged ISW train keeps propagating westward. By comparison of Figure 6d with Figure 3 (bottom) at t = 21,900 s, it is found that the amplitude of the leading depression wave (with a value of 11 m) in Experiment E6 is much smaller than that (with a value of 14 m) in Experiment E1, which demonstrates that owing to the interaction of the two trains of ISWs, both trains may lose their energies so that the amplitude of the new merged leading soliton decreases largely, and according to Equation (6), the merged ISW train propagates westward slower owing to the diminution in wave amplitude. Meanwhile, Figure 6d shows that the wave number in the ISW train following the leading soliton increases, which may be due to the nonlinear interaction of the two trains of ISWs. The evolution process of Experiment E6 might also suggest that in case one train of ISWs with smaller amplitudes is 10 of 14 induced prior to the other train of ISWs with larger amplitudes at the same source site, then the largest wave appearing in the middle of the ISW packets may be a temporary phenomenon during their propagations (e.g., in Figures 6a and 6d). Ramp et al. [2004] suggested that the reason why there exist so called B type ISW packets (i.e., the wave packets generally had the largest wave in the middle rather than at the leading edge) in the northern SCS is that they are generated in different places at different tidal periods. However, according to Experiment E6, it seems possible that the B type ISWs are generated in the same place but at different times, for the tide in the Luzon Strait is a complicated, irregular, mixed one. [18] The numerical simulated result in Experiment E7 is shown in Figure 7. Before the meeting of the two ISW trains, each train of ISWs would propagate westward alone as does that in Experiment E1; but once the crestlines of both ISW trains meet (Figure 7a), the crestlines of the two ISW trains look like a flying bird stretching its two wings, as the wave packet northwest of the Dongsha coral reefs shown in Figure 1. Owing to the nonlinear wave wave interaction, the wave amplitude at the connection point (which is just at the middle of the two incident solitons) increases gradually, and a new soliton like wave with a comparative large amplitude such as those of the former two leading solitons is induced (Figures 7b and 7d); finally, the three leading so-

11 Figure 9. Horizontal variation of the simulated wave amplitude in Experiment E9 (a) at t = 12,045 s, (b) at t = 16,425 s, and (c) at t = 21,900 s, respectively. litons seem to merge together as a single wave packet, and the ISW train propagates westward (Figure 7c). Figure 7d shows that at t = 12,045 s, the wave amplitude of the new induced soliton along the section of the connection point at y = 24 km is about 17 m, which is almost equal to the wave amplitude along the section at y = km (i.e., along the center of one source site of the incident solitons); meanwhile, there is only one depression wave following the leading depression soliton along the section of the connection point. However, Figure 7e shows that at t = 21,900 s, the wave amplitude along the section at y = km is still almost the same as that in Experiment E1; it is larger than the wave amplitude of the new induced soliton along the section of the connection point. This seems to suggest that, first, the collision of the two ISW trains from different source sites has no effect on the wave amplitude and the propagation speed of each initial incident ISWs train; second, the new induced soliton along the section of the connection point would weaken more quickly during its propagation. [19] What happens if the ISWs interact with an island? The following three experiments E8 E10 are designed to investigate this question. The circular island has a same radius of 1.38 km, but with its center at (x = km, y = km) in Experiment E8, at (x = km, y = km) in Experiment E9 and at (x = km, y = km) in Experiment E10, respectively. A train of the initial incident solitons at the eastern boundary is prescribed within (x =0,y B = km y km = y A ), so that the width of soliton source sites (3.36 km) is greater than the diameter of the island (2.76 km) in the y direction. In Experiment E8, the source sites of the incident solitons are symmetric relative to the island, while they are situated southeast of the island in Experiment E9 and situated northeast of the island in Experiment E10, respectively. In these three experiments, before the crestlines of ISW trains get to the island, they also propagate westward alone as those in Experiment E1 (figure omitted). Once the wave crestlines collide with the island, a sudden large wave amplitude may appear at the collision point near the island, e.g., Figure 8a shows that the wave amplitudes near the northeast and southeast rims of the island can reach about 30 m at the colliding moment and reduce quickly later, and owing to the nonlinear wave island interaction, the former crestlines are torn into two parts by the island. Since the source sites of the incident solitons are in different orientations relative to the island in the three experiments, the corresponding results are also different. [20] In Experiment E8, since the incident solitons are symmetric relative to the island and thus the two torn trains of ISWs are also symmetric, each train is led by a leading soliton with the same amplitude (Figure 8b). Finally (Figure 8c), after passing around the island, the two torn trains of ISWs meet again and remerge as a single wave packet propagating westward as shown in Figure 1, and the reconnection point of the two torn ISW trains is along the center of the island. It is also found that, east of the island, owing to the reflection of the island, there is a new weak train of ISWs propagating eastward, e.g., Figure 8d shows that along the section at y = 24 km, this reflected wave front at t = 16,425 s arrives at about x = 10 km, while at t = 21,900 s, it arrives at about x = 7 km. The result of the reflection and diffraction of ISWs by an island agrees with the conclusion by Chao et al. [2006]. Furthermore, it is interesting to note that, west of the island, the leading depression wave amplitude at t = 16,425 s is only about 5 m due to the collision with the island, but after the two torn trains of ISWs meet again, the leading depression wave amplitude at t = 21,900 s reaches 17 m, which is larger than that in the no island Experiment E1 in Figure 3 (bottom); meanwhile, the amplitude of the leading elevation wave following the depression wave is not the largest, which is also different from the ranked order ele- 11 of 14

12 of ISWs torn by the island are also not symmetric. The results shown in Figures 9 and 10 are also different from that in Experiment E8. When the wave crestlines collide with the island (Figures 9a, 9b, 10a, and 10b), east of the island, the wave amplitudes near the southeast and northeast rims of the island can reach about 30 m at the colliding moment and reduce quickly later, and owing to the Figure 10. Same as Figure 9 but for Experiment E10. vation waves in the no island Experiment E1, and the wave number in the elevation waves train also decreases. This suggests that the potential energy of the elevation waves might be converted into the potential energy of the depression ones. Thus, the wave island interaction would cause the amplitude of the leading depression wave to increase while the amplitude and the wave number in the subsequent elevation wave train to decrease. [21] In Experiments E9 and E10, since the incident solitons are not symmetric relative to the island, the two trains Figure 11. Horizontal variation of the simulated wave amplitude at t = 21,900 s in (a) Experiment E11, (b) Experiment E12, and (c) Experiment E of 14

13 Figure 12. Comparison of the wave amplitudes along the section at y =24kmatt = 21,900 s between (a) Experiments E6 (solid line) and E11 (dashed line), (b) Experiments E7 (solid line) and E12 (dashed line), and (c) Experiments E8 (solid line) and E13 (dashed line). reflection of the island, there is a new weak reflected or refracted train of ISWs propagating northeastward and southeastward (Figures 9c and 10c). West of the island, the former crestlines are also torn into two trains by the island, but one train is weaker with smaller wave amplitudes and the other train is stronger with larger wave amplitudes. The amplitude and the later major westward propagation direction of the leading depression soliton are related to the relative orientation of the original soliton source sites to the island, i.e., if the incident solitons are situated southeast or northeast of the island (in Experiments E9 and E10), then, after the ISWs collision with the island, the weaker or stronger train of ISWs with smaller or larger wave amplitudes is situated north or south of the island, and the later major westward propagation direction of the torn train of ISWs is still south or north of the island along the initial propagation direction. Finally, the weaker and stronger trains of ISWs may meet again during their westward propagation and remerge as a single wave packet (Figures 9c and 10c), but the wave crestlines near the reconnection point are very irregular, and the wave amplitude at the reconnection point in Experiments E9 and E10 is distinctly weaker that in Experiment E Flat Bottom Experiments (E11 E13) [22] What role, if any, does the topography play in the wave wave and wave island interactions? Three flat bottom experiments, E11 E13, based on the corresponding experiments E6 E8 with idealized bottom topography (e.g., Experiment E13 is almost the same as Experiment E6, except it is a flat bottom experiment) are designed to answer this question. Figure 11 shows the distribution of the simulated wave amplitude at t = 21,900 s in the three experiments. When comparing Figures 11a, 11b, and 11c with those in Figures 6c, 7c, and 8c, respectively, in the experiments with bottom topography, it is found that, near the leading depression soliton, the local wave crestlines shape is different and the bottom topography effect causes some diminution in wave amplitude of the leading soliton. Comparison of the wave amplitudes along the section at y =24kmatt = 21,900 s between the experiments with and without idealized bottom topography is also shown in Figure 12. In fact, as revealed by Experiments E1 and E3, the bottom topography leads to some diminution in wave amplitude, then the diminution in wave amplitude causes the wave propagation speed to reduce and the associated wavecrestlines shape changes accordingly. 4. Conclusions [23] In this paper, a two dimensional RLW numerical model is set up to simulate the evolution of ISWs during their propagation to the continental sea area in the northern SCS. According to the above experimental results, some conclusions can be drawn as follows. [24] First, according to the Experiments E1 E4 and E11 E13, during the propagation of the ISW train, the bottom topography effect may change the sign of the nonlinear parameter near the critical depth, cause the polarity reversal of ISWs, lead to some diminution in wave amplitude, and change of the local wave crestlines shape. Moreover, even if an ISW train is induced at the small sill channel along the Luzon Strait, it could propagate westward, with its wave crestline covering a large area in latitudinal direction in the northern SCS, which is shown by the real bottom Experiment E5. [25] Second, two cases of wave wave interaction are studied. In Experiment E6, when there are two trains of ISWs propagating from the same source site with a time lag but different amplitudes of incident solitons, the latter train of ISWs with a larger amplitude of the leading soliton and a faster propagation speed may catch and then swallow the former one with a smaller amplitude of the leading soliton 13 of 14

14 and a slower propagation speed, and subsequently, owing to the nonlinear interaction, the propagation speed and the wave amplitude of the merged ISWs train decrease while the wave number in the train increases. This suggests that in case one train of ISWs with smaller amplitudes is induced prior to the other train with larger amplitudes at the same source site, then the largest wave appearing in the middle of the ISW packet may be a temporary phenomenon due to the nonlinear wave wave interaction. In Experiment E7, when there are two trains of ISWs propagating from the different source sites at the same time but with the same amplitude of incident solitons, once the crestlines of the two ISWs trains meet, a new leading soliton with a large amplitude is induced at the connection point. The nonlinear collision of the two ISW trains has no effect on the wave amplitude and the propagation speed of each initial incident ISWs train, and the induced soliton at the connection point will weaken quickly during its propagation. [26] Third, as for the wave island interaction shown by Experiments E8 E10, once the ISW train collides with the island, before the island, there is one weak ISW train reflected by the island, while behind the island, the former crestlines of the ISW train are torn by the island into two new ISW trains. If the incident solitons are symmetric relative to the island, then the reflected and torn ISW trains are also symmetrical relative to the island, and after passing around the island, the two torn ISWs trains reconnect, propagating westward with the reconnection point and the major propagation direction overlapped along the center of the island, which agrees with the model results without earth s rotation by Chao et al. [2006]. It is interesting that just after the collision with the island, the leading elevation wave following the depression wave is not the largest amplitude wave, and the collision would cause the amplitude of the leading depression wave to increase while the amplitude and the wave number in the subsequent elevation waves train to decrease, which suggests the conversion of the potential energy of the elevation waves into the potential energy of the depression ones. If the incident solitons are not symmetric relative to the island, then the reflected and torn ISW trains are also not symmetrical relative to the island. It depends on the relative orientation of the original soliton source site to the island. When the original source site of solitons is situated south or north of the island, then before the island, the weak ISW train reflected by the island would propagate northeastward or southeastward later, while behind the island, the larger wave amplitude and the major propagation direction of the two torn ISW trains appear to the south or north of the island. Although the two torn ISW trains reconnect behind the island and propagate westward, their reconnection point and the major propagation direction do not overlap. [27] The propagation model study of the ISWs here is still very robust, since only the first baroclinic mode wave is considered, and some other factors such as the earth s rotation are not taken into account, which requires further numerical model studies. [28] Acknowledgments. This work is jointly supported by the Key Program KZCX1 YW from the Chinese Academy of Sciences, China National Funds for Distinguished Young Scientists , 863 Hi Tech Programs (grants 2008AA09Z112 and 2008AA09A402), and NSFC grant References Buijsman, M. C., Y. Kanarska, and J. C. McWilliams (2010), On the generation and evolution of nonlinear internal waves in the South China Sea, J. Geophys. Res., 115, C02012, doi: /2009jc Cai, S., Z. Gan, and X. Long (2002a), Some characteristics and evolution of the internal soliton in the northern South China Sea, Chin. Sci. Bull., 47(1), Cai, S., X. Long, and Z. Gan (2002b), A numerical study of the generation and propagation of internal solitary waves in the Luzon Strait, Oceanologica Acta, 25(2), Chao, S. Y., et al. (2006), Reflection and diffraction of internal solitary waves by a circular island, J. Oceanogr., 62(6), Du, T., Y. H. Tseng and X. H. 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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: /2006jc S. Cai and J. Xie, Key Laboratory of Tropical Marine Environmental Dynamics, South China Sea Institute of Oceanology, Chinese Academy of Sciences, 164 West Xingang Rd., Guangzhou , China. (caisq@ scsio.ac.cn) 14 of 14