A propagation model for the internal solitary wavesin the northern South China Sea

Shuqun Cai1 and Jieshuo Xie1

Received 14 April 2010; revised 1 September 2010; accepted 1 October 2010; published 31 December 2010.

[1] A two‐dimensional, regularized long‐wave equation model is developed to study thedynamic 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 topographywould 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 sillchannel along the Luzon Strait, they could propagate westward with their crestlinescovering a large area in the latitudinal direction in the northern SCS. When there are twotrains of ISWs propagating from the same source site with a time lag but differentamplitudes of initial solitons, the latter train of ISWs with a larger amplitude may catchthen swallow the former one with a smaller amplitude, and the wave amplitude of themerged ISW train decreases while the wave number increases. When there are two trainsof ISWs propagating from the different source sites at the same time with the sameamplitude of initial solitons, the crestlines of the two ISW trains may meet and a newleading soliton is induced at the connection point. Once the ISW trains collide with theisland, before the island, a weak ISW train is reflected; behind the island, the formercrestlines of the ISW train are torn by the island into two new trains, which may reconnectafter passing around the island. The propagation direction, the wave amplitude, and thereconnection point of the new merged ISW train behind the island depend on the relativeorientation 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, C12074, doi:10.1029/2010JC006341.

1. Introduction

[2] There is an active zone of internal solitary waves(ISWs) in the northern South China Sea (SCS). According tothe analyses of satellite photographs and in situ observationaldata [e.g., Fett and Rabe, 1977; Ebbesmeyer et al., 1991; Liuet 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 generatedbetween the sill channels along the Luzon Strait (whichconnects the SCS and the western Pacific). The internal wavedistribution maps in the SCS have been compiled fromhundreds of ERS‐1/2, RADARSAT, and Space Shuttle SARimages from 1993 to 1999 by Hsu and Liu [2000], and it isshown that most of the ISWs in the northeastern SCSpropagate westward. Traveling to the west, the ISWsencounter nearly no shallow water until reaching the conti-nental shelf of China or near the Dongsha Islands. As the

ISWs travel westward and approach the Dongsha Islands,shoaling, reflection, refraction, and diffraction becomedominant processes, e.g., Figure 1 by Hsu and Liu [2000]shows the wave‐wave and wave‐island interactions nearthe Dongsha Islands. A lot of numerical simulation modelsare employed to study the generation and evolution of theISWs. Among them, the one‐dimensional Korteweg‐deVries (KdV) equation numerical model, or the regularizedlong wave (RLW) equation model, is commonly used tostudy the effects of nonlinearity, dissipation, and shoalingtopography and so forth on the evolution of the ISWs [e.g.,Liu et al., 1998; Cai et al., 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., Chaoet al., 2006; Shaw et al., 2009; Buijsman et al., 2010]model studies are relatively fewer. On the basis of a two‐layermodel on the horizontal plane, Lynett and Liu [2002] simulatedthe internal wave propagation in the vicinity of the DongshaIslands, and their numerical results showed strong similaritiesto the satellite images taken over the same locations. Chaoet al. [2006] investigated the reflection and diffraction of theISWs by an island on the basis of a three‐dimensional non-hydrostatic numerical model and found that in addition toreflected waves, two wave branches passed around the island

1Key Laboratory of Tropical Marine Environmental Dynamics, SouthChina Sea Institute of Oceanology, Chinese Academy of Sciences,Guangzhou, China.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2010JC006341

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, C12074, doi:10.1029/2010JC006341, 2010

C12074 1 of 14

and reconnected behind it. For a westward‐propagatingincoming wave, the Coriolis deflection favored northwardwave propagation in the region between the crossover pointand the island, shifting the wave reconnection point behindthe island northward. However, some questions remainunsolved, e.g., could the crestlines of the ISWs generated atthe small source site (i.e., in the sill channels along theLuzon Strait) propagate westward and cover a large domainin the northern SCS? What happens if two trains of ISWsinteract with each other? What happens if the ISWs indifferent propagation directions interact with the DongshaIslands? Above all, what role, if any, does the topographyplay?[3] Pierini [1989] employed a “weakly” two‐dimensional

RLW equation model to simulate the propagation of theISWs in the Alboran Sea successfully. In this paper, weadapt his model and develop it to simulate the horizontalevolution of the ISWs in the northern SCS, major at thenumerical study of the above questions, i.e., the dynamicmechanisms of the wave‐wave and wave‐island inter-actions. In sections 2–4, the setup of the model is givenin section 2; in section 3, the numerical experimental

results and discussion are presented; and section 4 is theconclusion.

2. Model Description, Numerical Scheme,and Choice of Parameters

[4] A one‐dimensional propagation numerical modelbased on the RLW equation developed by Hollowayet 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 cQx

2Q� þ CDc ��j j�

3� ��xxt

c¼ 0 ð1Þ

Here, h is the displacement of interfacial pycnocline, the first

baroclinic modal phase speed c = [g �2��1ð Þh1h2�2h1þ�1h2

]1/2, the non-

linear parameter a = 3c2h1h2

�2h21��1h22�2h1þ�1h2

, the dispersion parameter

b = ch1h26

�1h1þ�2h2�2h1þ�1h2

, and the high‐order nonlinear coefficient

� = 3c2h21h

22[78(

�2h21��1h22�2h1þ�1h2

)2 − �2h31þ�1h32�2h1þ�1h2

]; r1 and r2 are the upperand lower densities; h1 and h2 are the undisturbed upperand lower layer thicknesses, respectively; CD is the drag

Figure 1. RADARSAT ScanSAR image north of the South China Sea (SCS) on 26 April, 1998, show-ing at least four packets of internal waves. The internal wave packets propagated toward the DongshaIsland, the internal wave packets diffracted by Dongsha Islands into two packets of internal waves andthen interacted with each other and remerged as a single wave packet (adapted from Hsu and Liu, 2000).

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

2 of 14

coefficient of bottom friction; and Q = Q(x) is a knownfactor needed to ensure conservation of wave action fluxQh2. For interfacial waves, here we set Q = 2g(r2 − r1)c assuggested by Grimshaw et al. [2007].[5] On the basis of the above one‐dimensional RLW

equation (1), we adapt the two‐dimensional RLW equationfor the waves predominantly traveling along x and whosespatial variation along y is small compared with x (in thissense, it applies to “weakly” two‐dimensional waves), as didPierini [1989], i.e.,

@

@x�t þ c�x þ ���x þ ��2�x þ cx

2� þ CDc ��j j�

3� ��xxt

c

� �

þ c

2�yy ¼ 0 ð2Þ

[6] The two‐level, three‐point Crank‐Nicholson schemeis used to differentiate equation (2), i.e.,

�x �t 1� �

c�2x

� ��ni;j þ

1

2cþ ��nþ1

i;j þ ��nþ1i;j �nþ1

i;j

� ��x�

nþ1i;j

24

þ 1

2cþ ��ni;j þ ��ni;j�

ni;j

� ��x�

ni;j þ �ni;j

�xc

2þCDc ��ni;j

��� ���3

0@

1A35

þ c

2�2y�

ni;j ¼ 0 ð3Þ

Here, each identity is defined as dx2hi,j

n = (hi+1,jn − 2hi,j

n + hi−1,jn )/

Dx2, dy2hi,j

n = (hi,j+1n − 2hi,j

n + hi,j−1n )/Dy2, dthi,j

n = (hi,jn+1 − hi,j

n )/Dt, dxhi,j

n = (hi+1,jn − hi−1,j

n )/(2Dx), i/j and n are the indexesof space along x/y and time, respectively. Equation (3) canbe solved by the iteration method. It can be proven thatthis scheme is unconditionally stable.[7] A no‐flux boundary condition [Pierini, 1989],

�N ¼ 0 ð4Þ

is imposed along the rigid boundaries, where N is thedirection (x or y) perpendicular to the wall. In our additionalsensitive experiment, a radiation boundary condition issubstituted and it is found that there is only slight variationnear the boundary for both experimental results. As done byPierini [1989] for the Strait of Gibraltar, the interface dis-placement at the east open boundary is prescribed along asection from yB to yA (e.g., in Figure 3 (top), where a smallchannel is 1200 m long in the x‐direction and 420 m wide inthe y‐direction, and the section from yB to yA represents thesource sites of the internal solitons), and the initial incidentsoliton is given as the solitary wave solution of the extendedKdV equation [Grimshaw et al., 2007],

� x; y; tð Þ ¼ 1þ Bð Þ�01þ B cosh K x� Vtð Þ½ � ; yB < y < yA ð5Þ

Figure 2. Variations of the dispersion parameter b, the high‐order nonlinear coefficient �, the nonlinearparameter a, the first baroclinic modal phase speed c, and the idealized bottom topography of the com-putational domain in the x‐direction.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

3 of 14

Here, the nonlinear phase speed is,

V ¼ cþ � 1þ Bð Þ�0=6; ð6Þand a(1 + B)h0/6 = bK2, B2 = 1 + 6b�K2/a2. To smooth theinitial condition, the initial incident soliton at both ends ofthe source sites is given as

� x; y; tð Þ ¼ 0:5 1þ Bð Þ�01þ B cosh K x� Vtð Þ½ �� tanh

y� yAW

� �� tanh

y� yBW

� �h i; y ¼ yA; yB ð7Þ

where W � yB − yA. In the computation, we chooseW = 0.05(yB − yA).[8] In the computation, we set the relative density dif-

ference value between the upper and lower layers at Dr =(r2 − r1)/r1 = 0.0036, which is obtained by the in situobservation [Cai et al., 2002a]. In our two additional sen-sitive experiments with the following idealized bottomtopography as shown in Figure 2, the other conditions arethe same, except that CD values are set 0 and 0.0025[Grimshaw, 2001], respectively. We find that both experi-mental results are almost the same, except that the waveamplitude with CD = 0.0025 is slightly weakened. In fact, ifwe look at the ratio of the friction term to the nonlinear termwith parameter a in Equation (2), i.e.,

CDc ��j j�3

���x

���������� ¼

CDcDx

3; ð8Þ

where Dx is the spatial step, we could find that thisdimensionless ratio has an order of about O (0.05) in ourcomputation, which shows that the friction term is unim-portant. Thus we set CD = 0 in the following experiments.

3. Numerical Experimental Resultsand Discussion

[9] In all of the experiments, the undisturbed initial upperlayer thickness is h1 = 60 m, and h2 = H − h1, where H thewater depth. We first validate the numerical model in asmaller simulation domain with an idealized bottomtopography (Figure 2). For convenience, this computationaldomain is set within a square domain with each side of48 km in both x and y directions. The idealized shoalingbottom topography with a mildly constant slope is similar tothat in the continental shelf of the northern SCS, but thedeepest depth is only 140 m at the eastern boundary and theshallowest depth is only 100 m at the western boundary, sothat at x = 24 km away from both the eastern and westernboundaries, the water depth is 120 m and the initial upperlayer thickness is equal to the lower layer one, i.e., the criticaldepth for the change of a depression ISW into the elevationone [e.g., Hsu and Liu, 2000]. Thus the critical depth phe-nomenon can be reflected in the model. Figure 2 also showsthe variations of the dispersion parameter b, the high‐ordernonlinear coefficient �, the nonlinear parameter a, and thefirst baroclinic modal phase speed c in the x‐direction. Thir-teen experiments are designed to test whether the possible key

Table 1. Model’s Running Cases and Their Experimental Parameters

Experiment Explanation

E1 Incident solitons with h0 = −20 m at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km), no island, idealized shoaling bottom topography with thedepth ranging from 140 m in the east to 100 m in the west linearly

E2 Incident solitons with h0 = −20 m at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km), no island, idealized shoaling bottom topography with thedepth ranging from 200 m in the east to 100 m in the west linearly

E3 Incident solitons with h0 = −20 m at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km), no island, flat bottom topography with a depth of 140 mE4 Incident solitons with h0 = −36 m at (t = 0, x = 0, 23.82 km≤ y ≤ 24.18 km), no island, flat bottom topography with a depth of 140 mE5 Two trains of incident solitons with the same amplitude of h0 = −20 m but a time lag of 12.4 h (a period of M2 tide),

one at (t = 0, x = 0, 20°10.83′N ≤ y ≤ 20°15.41′N), the other at (t = 12.4 h, x = 0, 20°10.83′N ≤ y ≤ 20°15.41′N), respectively,real bottom topography in the northern SCS

E6 Two trains of incident solitons with a time lag of 15 time steps (i.e., 821.25 s) but different amplitudes,one at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km) with h0 = −6 m, the other at (t = 821.25 s, x = 0, 23.82 km ≤ y ≤ 24.18 km)with h0 = −20 m, respectively, no island, idealized shoaling bottom topography with the depth ranging from 140 min the east to 100 m in the west linearly

E7 Two trains of incident solitons with the same amplitude of h0 = −20 m but at different source sites,one at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km), the other at (t = 0, x = 0, 21.06 km ≤ y ≤ 21.42 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

E8 Incident solitons with h0 = −20 m at (t = 0, x = 0, 22.32 km ≤ y ≤ 25.68 km), idealized shoaling bottom topography with thedepth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centered at(x = 14.28 km, y = 24 km) with a radius of 1.38 km

E9 Incident solitons with h0 = −20 m at (t = 0, x = 0, 22.32 km ≤ y ≤ 25.68 km), idealized shoaling bottom topographywith the depth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centeredat (x = 14.28 km, y = 25.38 km) with a radius of 1.38 km

E10 Incident solitons with h0 = −20 m at (t = 0, x = 0, 22.32 km ≤ y ≤ 25.68 km), idealized shoaling bottom topographywith the depth ranging from 140 m in the east to 100 m in the west linearly, and a circular island centered at(x = 14.28 km, y = 22.62 km) with a radius of 1.38 km

E11 Two trains of incident solitons with a time lag of 15 time steps (i.e., 821.25 s) but different amplitudes,one at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km) with h0 = −6 m, the other at (t = 821.25 s, x = 0, 23.82 km ≤ y ≤ 24.18 km)with h0 = −20 m, respectively, no island, flat bottom topography with a depth of 140 m

E12 Two trains of incident solitons with the same amplitude of h0 = −20 m but at different source sites,one at (t = 0, x = 0, 23.82 km ≤ y ≤ 24.18 km), the other at (t = 0, x = 0, 21.06 km ≤ y ≤ 21.42 km),respectively, no island, flat bottom topography with a depth of 140 m

E13 Incident solitons with h0 = −20 m at (t = 0, x = 0, 22.32 km ≤ y ≤ 25.68 km), flat bottom topography with a depth of 140 m,and a circular island centered at (x = 14.28 km, y = 24 km) with a radius of 1.38 km

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

4 of 14

dynamic factors, including the bottom topography, the in-teractions of wave‐wave, wave‐island, and so forth affect theevolution of the ISWs or not. See Table 1 for the numericalexperimental cases.

3.1. Validation of the Numerical Experiments Withan Idealized Bottom Topography (E1 ∼ E4)

[10] At first, four experiments are designed to testify tothe effects of the topography slope and the amplitudes of the

Figure 3. Numerical simulated internal solitary waves (ISWs) in Experiment E1. (top) Horizontal var-iation of the wave amplitude (unit in m, here and subsequently) at t = 16,425 s. (bottom) Comparison ofthe wave amplitudes along the section at y = 24 km at t = 16,425 s (solid line), t = 21,900 s (dashed line),and t = 32,850 s (thick solid line), respectively.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

5 of 14

incident ISWs on the evolution of the ISWs. Experiment E1is a standard experiment in which the incident solitons withamplitudes of h0 = −20 m are prescribed at (t = 0, x = 0, yB =23.82 km ≤ y ≤ 24.18 km = yA). Experiments E2 ∼ E3 aredesigned to show the effect of the topography slope.Experiment E2 is almost the same as Experiment E1, exceptthat the deepest depth of the idealized shoaling bottomtopography is 200 m, so that the slope in experiment E2 is alittle larger than that in experiment E1. Experiment E3 isalmost the same as Experiment E1, except that the shoalingbottom topography is replaced by a flat bottom with a con-stant depth of 140 m. Experiment E4 is designed to show theeffect of the amplitudes of the incident ISWs, which is almostthe same as Experiment E3, except that the amplitudes of theincident ISWs are −36 m. The spatial step is chosen as Dx =Dy = 60 m, while the temporal step is Dt = 54.75 s inexperiments E1, E3 ∼ E4 and Dt = 49.47 s in ExperimentE2 (with a larger linear phase speed, since the water depthis deeper), respectively, so that the dimensionless linearphase speed is also about 1.[11] The horizontal snapshot of the numerical simulated

ISWs at t = 16425 s in Experiment E1 is shown in Figure 3(top), the simulated ISWs propagate westward, with itscrestline curvature looking like a smooth arc; the leadingsoliton in its westernmost point is still along the section atabout y = 24 km. Figure 3 (bottom) shows the comparison ofthe wave amplitudes along the section at y = 24 km (within0 ≤ x ≤ 48 km) at t = 16425 s, 21,900 s and 32,850 s,respectively, the amplitudes of the leading depression soli-tons at these three moments, with a value of −15.5 m, −14 m,and −12 m, respectively, decrease gradually with time. Theinitial incident depression wave is gradually replaced by atrain of elevation waves riding on a negative pedestal, whichagrees with the one‐dimensional model results by Grimshaw

et al. [2004]. At t = 21,900 s, the leading soliton approachesnear the critical depth at x = 24 km, however, the ISWs donot change from depression wave into elevation wave atonce; instead, after the leading depression wave passes thecritical depth, since the nonlinear parameter a changes fromnegative to positive, the amplitude of the elevation wavefollowing the leading depression one gets larger andlarger, while the amplitude of the leading depressionwave gets smaller and smaller; finally, e.g., at about t =32,850 s, the ISWs seem to be led by the elevation wavewith a positive amplitude of 12 m. The case in thisexperiment is similar to those revealed by the one‐dimensional models [e.g., Liu et al., 1998; Cai et al.,2002a], which shows that the above numerical simulationresult 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 existssome difference in the polarity reversal, the wave amplitude,and the wave number in the ISW train. Figure 4a shows thecomparison of the wave amplitudes along the section aty = 24 km between Experiment E1 at t = 16,425 s and E2at t = 14,782.5 s. It is found that the wave amplitude(−19.6 m) in Experiment E2 is larger than that in Experi-ment E1 (−15.5 m); meanwhile, the wave number in the ISWtrain and the elevation wave amplitude following the leadingdepression soliton are much less than those in ExperimentE1, which seems to suggest that when the ISWs propagate indeeper water, they may behave in the mode of KdV typerather than extended Koteweg‐de Vries (eKdV) type soli-tons. Moreover, the leading depression soliton in ExperimentE2 catches up with that in Experiment E1 in less time, whichshows that the propagation speed in Experiment E2 is dis-tinctly faster (since its water depth at the source site is deeper)than that in Experiment E1. This demonstrates that in case

Figure 4. Comparison of the wave amplitudes along the section at y = 24 km (a) between Experiment E1at t = 16,425 s (solid line) and Experiment E2 at t = 14,782.5 s (dashed line), and (b) among ExperimentsE1 (solid line), E3 (thick solid line), and E4 (dashed line) at t = 16,425 s.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

6 of 14

the internal solitons are induced at the source site with adeeper water depth, the leading soliton propagates faster,with its amplitude decreasing slowly with time when com-pared with that induced at the source site with a shallowerwater depth.[13] Owing to the flat bottom topography in Experiments

E3 and E4, the dispersion parameter b, the high‐ordernonlinear coefficient �, the nonlinear parameter a, and thefirst baroclinic modal phase speed c stay unchanged in thex‐direction. Figure 4b shows the comparison of the waveamplitudes along the section at y = 24 km among Ex-periments E1, E3, and E4 at t = 16,425 s. It is found that,when compared with that in Experiment E1, the waveamplitude in Experiment E1 is a little less than that inExperiment E3, which suggests that the bottom topographymight lead to some diminution in wave amplitude.Although the amplitudes of the incident ISWs are muchlarger in Experiment E4 than those in Experiment E3, theamplitude of the leading depression soliton decreases fasterthan that in Experiment E3; moreover, it is found that theelevation wave train in Experiment E4 lasts from thesource site during its propagation, with its wave numbermuch larger than that in Experiment E3. If we look atEquation (2), we can see that this is because the contri-bution of the high‐order nonlinear term with coefficient �

becomes more important when the amplitudes of theincident ISWs get larger. Moreover, since nonlinearparameter a does not change its sign in the x‐direction, thepolarity reversal (i.e., the elevation wave following theleading depression soliton changes into the largest leadingelevation one) that happens in Experiment E1 does notoccur all the way in both Experiments E3 and E4 (figureomitted).

3.2. Real Bottom Experiment in the NortheasternSCS (E5)

[14] One question put forward in section 1 is, Could thecrestlines of the ISWs generated at the small source sitepropagate westward and cover a large domain in thenorthern SCS? Now, we employ the above numerical modelto simulate the propagation of the ISWs in the northeasternSCS with a real bottom topography (Figure 5a). In thecomputation, owing to the limitations of the computer, thespatial and temporal steps are chosen as Dx = 102.89 m,Dy = 771.67 m and Dt = 72.83 s, respectively, so that thedimensionless linear phase speed in the x‐direction is alsonear 1. The depth is linearly interpolated based on theETOPO5 Global Earth Topography with a resolution of 5′.It is supposed that there are two trains of initial incidentsolitons from yB = 20°10.83′N to yA = 20°15.41′N at the east

Figure 5. (a) Bottom topography of the computational domain in the northern South China Sea (unit inm, note that the white circular area denotes the Dongsha Islands). (b) Horizontal variation of the waveamplitude simulated in Experiment E5 at t = 56.63 h.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

7 of 14

open boundary with the same amplitude of 20 m but a timelag of 12.4 h (i.e., about the period of M2 tide) in theexperiment.[15] The horizontal snapshot of the numerical simulated

ISWs at t = 56.63 h when the waves have propagated farinto 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 theleading soliton in its westernmost point is within thesections from yB = 20°10.83′N to yA = 20°15.41′N. Eachcrestline of the ISWs could cover a large area in thelatitudinal direction. The simulated case is basically sim-ilar to that shown in Figure 1. It is shown that even if theinternal solitons are induced at the small sill channelalong the Luzon Strait, they could propagate westwardwith their crestlines covering a large area in the latitudinaldirection in the northern SCS.[16] The above numerical experiment run costs a lot of

computer time; meanwhile, it should be stressed that thenorthern and southern boundaries might be “touched” by the

wave in the later simulation, which would cause some re-flected waves that we do not intend to discuss, since in thisstudy we just want to understand some dynamic mechanismsaffecting the propagation and evolution of the ISWs.Therefore, in the following, we still use the smaller simula-tion domain with an idealized bottom topography (Figure 2)to carry out the numerical experiments.

3.3. Wave‐Wave and Wave‐Island InteractionsExperiments 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 ISWtrains which propagate from the same source site with a timelag but different amplitudes of incident solitons. It is thesame as Experiment E1, except that now there are two trainsof incident solitons at (t = 0, x = 0, yB = 23.82 km ≤ y ≤24.18 km = yA) and at (t = 821.25 s, x = 0, yB = 23.82 km ≤y ≤ 24.18 km = yA), respectively, and the amplitudes of the

Figure 6. Numerical simulated ISWs in Experiment E6, with the horizontal variation of the wave ampli-tude (a) at t = 5475 s, (b) at t = 12,045 s and (c) at t = 21,900 s, and (d) comparison of the wave amplitudesalong the section at y = 24 km at t = 766.5 s (solid line), t = 5475 s (dashed line), and t = 21,900 s (thicksolid line), respectively.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

8 of 14

first incident solitons are ‐6 m while the second ones are−20 m, respectively. Experiment E7 is designed to show thecollision of two ISW trains which propagate from the dif-ferent source sites at the same time, but with the same

amplitude of incident solitons, i.e., two trains of incidentsolitons have the same amplitudes of h0 = −20 m with onesource site at (x = 0, yB = 23.82 km ≤ y ≤ 24.18 km = yA)and the other at (x = 0, yD = 21.06 km ≤ y ≤ 21.42 km = yC),

Figure 7. Numerical simulated ISWs in Experiment E7, with the horizontal variation of the wave ampli-tude (a) at t = 2737.5 s, (b) at t = 12,045 s, and (c) at t = 21,900 s, and comparison of the wave amplitudesalong the section at y = 24 km (dashed line) with those at y = 22.56 km (solid line) at (d) t = 12,045 s and(e) t = 21,900 s.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

9 of 14

respectively. The numerical simulated result in ExperimentE6 is shown in Figure 6. At first, the solitons with smalleramplitudes are leading (e.g., at t = 5475 s in Figure 6a orat t = 766.5 s in Figure 6d), then, since the second train ofISWs with larger amplitudes has a faster propagation speedaccording to Equation (6), gradually, the latter train ofISWs catches the former one (e.g., at t = 12,045 s inFigure 6b). Finally, the latter train of ISWs seems toswallow the former one (e.g., at t = 21,900 s in Figures 6cand 6d), the two trains of ISWs merge, and the mergedISW train keeps propagating westward. By comparison ofFigure 6d with Figure 3 (bottom) at t = 21,900 s, it isfound that the amplitude of the leading depression wave(with a value of −11 m) in Experiment E6 is much smallerthan that (with a value of −14 m) in Experiment E1, whichdemonstrates that owing to the interaction of the two trains ofISWs, both trains may lose their energies so that the ampli-tude of the new merged leading soliton decreases largely, andaccording to Equation (6), the merged ISW train propagateswestward slower owing to the diminution in wave amplitude.Meanwhile, Figure 6d shows that the wave number in theISW train following the leading soliton increases, which maybe due to the nonlinear interaction of the two trains of ISWs.The evolution process of Experiment E6 might also suggestthat in case one train of ISWs with smaller amplitudes is

induced prior to the other train of ISWswith larger amplitudesat the same source site, then the largest wave appearing in themiddle of the ISW packets may be a temporary phenomenonduring their propagations (e.g., in Figures 6a and 6d). Rampet al. [2004] suggested that the reason why there existso‐called B‐type ISW packets (i.e., the wave packets gener-ally had the largest wave in the middle rather than at theleading edge) in the northern SCS is that they are generated indifferent places at different tidal periods. However, accordingto Experiment E6, it seems possible that the B‐type ISWs aregenerated in the same place but at different times, for the tidein 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 ISWtrains, each train of ISWs would propagate westward aloneas does that in Experiment E1; but once the crestlines ofboth ISW trains meet (Figure 7a), the crestlines of the twoISW trains look like a flying bird stretching its two wings, asthe wave packet northwest of the Dongsha coral reefs shownin Figure 1. Owing to the nonlinear wave‐wave interaction,the wave amplitude at the connection point (which is just atthe middle of the two incident solitons) increases gradually,and a new soliton‐like wave with a comparative largeamplitude such as those of the former two leading solitons isinduced (Figures 7b and 7d); finally, the three leading so-

Figure 8. Numerical simulated ISWs in Experiment E8, with the horizontal variation of the wave ampli-tude (a) at t = 12,045 s, (b) at t = 16,425 s, and (c) at t = 21,900 s, and (d) comparison of the waveamplitudes along the section at y = 24 km at t = 16,425 s (solid line) with those at t = 21,900 s (dashedline), respectively.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

10 of 14

litons seem to merge together as a single wave packet, andthe ISW train propagates westward (Figure 7c). Figure 7dshows that at t = 12,045 s, the wave amplitude of the newinduced soliton along the section of the connection point aty = 24 km is about −17 m, which is almost equal to thewave amplitude along the section at y = 22.56 km (i.e.,along the center of one source site of the incident solitons);meanwhile, there is only one depression wave followingthe leading depression soliton along the section of theconnection point. However, Figure 7e shows that at t =

21,900 s, the wave amplitude along the section at y =22.56 km is still almost the same as that in Experiment E1;it is larger than the wave amplitude of the new inducedsoliton along the section of the connection point. Thisseems to suggest that, first, the collision of the two ISWtrains from different source sites has no effect on the waveamplitude and the propagation speed of each initial inci-dent ISWs train; second, the new induced soliton along thesection of the connection point would weaken morequickly during its propagation.[19] What happens if the ISWs interact with an island?

The following three experiments E8 ∼ E10 are designed toinvestigate this question. The circular island has a sameradius of 1.38 km, but with its center at (x = 14.28 km,y = 22.62 km) in Experiment E8, at (x = 14.28 km, y =25.38 km) in Experiment E9 and at (x = 14.28 km, y =22.62 km) in Experiment E10, respectively. A train of theinitial incident solitons at the eastern boundary is prescribedwithin (x = 0, yB = 22.32 km ≤ y ≤ 25.68 km = yA), so that thewidth of soliton source sites (3.36 km) is greater than thediameter of the island (2.76 km) in the y‐direction. In Exper-iment E8, the source sites of the incident solitons are sym-metric relative to the island,while they are situated southeast ofthe island in Experiment E9 and situated northeast of the islandin Experiment E10, respectively. In these three experiments,before the crestlines of ISW trains get to the island, they alsopropagate westward alone as those in Experiment E1 (figureomitted). Once the wave crestlines collide with the island, asudden large wave amplitude may appear at the collision pointnear the island, e.g., Figure 8a shows that the wave amplitudesnear the northeast and southeast rims of the island can reachabout −30 m at the colliding moment and reduce quickly later,and owing to the nonlinear wave‐island interaction, the formercrestlines are torn into two parts by the island. Since the sourcesites of the incident solitons are in different orientations rela-tive to the island in the three experiments, the correspondingresults are also different.[20] In Experiment E8, since the incident solitons are

symmetric relative to the island and thus the two torntrains of ISWs are also symmetric, each train is led by aleading soliton with the same amplitude (Figure 8b).Finally (Figure 8c), after passing around the island, the twotorn trains of ISWs meet again and remerge as a singlewave packet propagating westward as shown in Figure 1,and the reconnection point of the two torn ISW trains isalong the center of the island. It is also found that, east ofthe island, owing to the reflection of the island, there is anew weak train of ISWs propagating eastward, e.g., Figure8d shows that along the section at y = 24 km, this reflectedwave 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. Theresult of the reflection and diffraction of ISWs by an islandagrees with the conclusion by Chao et al. [2006]. Fur-thermore, it is interesting to note that, west of the island,the leading depression wave amplitude at t = 16,425 s isonly about −5 m due to the collision with the island, butafter the two torn trains of ISWs meet again, the leadingdepression wave amplitude at t = 21,900 s reaches −17 m,which is larger than that in the no island Experiment E1 inFigure 3 (bottom); meanwhile, the amplitude of the leadingelevation wave following the depression wave is not thelargest, which is also different from the ranked‐order ele-

Figure 9. Horizontal variation of the simulated waveamplitude in Experiment E9 (a) at t = 12,045 s, (b) att = 16,425 s, and (c) at t = 21,900 s, respectively.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

11 of 14

vation waves in the no island Experiment E1, and the wavenumber in the elevation waves train also decreases. Thissuggests that the potential energy of the elevation wavesmight be converted into the potential energy of the depres-sion ones. Thus, the wave‐island interaction would cause theamplitude of the leading depression wave to increase whilethe amplitude and the wave number in the subsequent ele-vation wave train to decrease.[21] In Experiments E9 and E10, since the incident soli-

tons are not symmetric relative to the island, the two trains

of ISWs torn by the island are also not symmetric. The re-sults shown in Figures 9 and 10 are also different from thatin Experiment E8. When the wave crestlines collide withthe island (Figures 9a, 9b, 10a, and 10b), east of the island,the wave amplitudes near the southeast and northeast rimsof the island can reach about −30 m at the collidingmoment and reduce quickly later, and owing to the

Figure 10. Same as Figure 9 but for Experiment E10.

Figure 11. Horizontal variation of the simulated waveamplitude at t = 21,900 s in (a) Experiment E11, (b) Exper-iment E12, and (c) Experiment E13.

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

12 of 14

reflection of the island, there is a new weak reflected orrefracted train of ISWs propagating northeastward andsoutheastward (Figures 9c and 10c). West of the island, theformer crestlines are also torn into two trains by the island,but one train is weaker with smaller wave amplitudes andthe other train is stronger with larger wave amplitudes. Theamplitude and the later major westward propagationdirection of the leading depression soliton are related to therelative orientation of the original soliton source sites tothe island, i.e., if the incident solitons are situated south-east or northeast of the island (in Experiments E9 andE10), then, after the ISWs’ collision with the island, theweaker or stronger train of ISWs with smaller or largerwave amplitudes is situated north or south of the island,and the later major westward propagation direction of thetorn train of ISWs is still south or north of the island alongthe initial propagation direction. Finally, the weaker andstronger trains of ISWs may meet again during theirwestward propagation and remerge as a single wave packet(Figures 9c and 10c), but the wave crestlines near thereconnection point are very irregular, and the waveamplitude at the reconnection point in Experiments E9 andE10 is distinctly weaker that in Experiment E8.

3.4. Flat Bottom Experiments (E11 ∼ E13)

[22] What role, if any, does the topography play in thewave‐wave and wave‐island interactions? Three flat bottomexperiments, E11 ∼ E13, based on the correspondingexperiments E6 ∼ E8 with idealized bottom topography(e.g., Experiment E13 is almost the same as ExperimentE6, except it is a flat bottom experiment) are designed toanswer this question. Figure 11 shows the distribution ofthe simulated wave amplitude at t = 21,900 s in the threeexperiments. When comparing Figures 11a, 11b, and 11cwith those in Figures 6c, 7c, and 8c, respectively, in theexperiments with bottom topography, it is found that, near

the leading depression soliton, the local wave crestlinesshape is different and the bottom topography effectcauses some diminution in wave amplitude of the leadingsoliton. Comparison of the wave amplitudes along thesection at y = 24 km at t = 21,900 s between the experimentswith and without idealized bottom topography is also shownin Figure 12. In fact, as revealed by Experiments E1 and E3,the bottom topography leads to some diminution in waveamplitude, then the diminution in wave amplitude causes thewave propagation speed to reduce and the associated wave‐crestlines shape changes accordingly.

4. Conclusions

[23] In this paper, a two‐dimensional RLW numericalmodel is set up to simulate the evolution of ISWs duringtheir propagation to the continental sea area in the northernSCS. According to the above experimental results, someconclusions can be drawn as follows.[24] First, according to the Experiments E1 ∼ E4 and

E11 ∼ E13, during the propagation of the ISW train, thebottom topography effect may change the sign of the non-linear parameter near the critical depth, cause the polarityreversal of ISWs, lead to some diminution in wave ampli-tude, and change of the local wave crestlines shape. More-over, even if an ISW train is induced at the small sill channelalong the Luzon Strait, it could propagate westward, with itswave crestline covering a large area in latitudinal direction inthe northern SCS, which is shown by the real bottomExperiment E5.[25] Second, two cases of wave‐wave interaction are

studied. In Experiment E6, when there are two trains ofISWs propagating from the same source site with a time lagbut different amplitudes of incident solitons, the latter trainof ISWs with a larger amplitude of the leading soliton and afaster propagation speed may catch and then swallow theformer one with a smaller amplitude of the leading soliton

Figure 12. Comparison of the wave amplitudes along the section at y = 24 km at t = 21,900 s between(a) Experiments E6 (solid line) and E11 (dashed line), (b) Experiments E7 (solid line) and E12 (dashedline), and (c) Experiments E8 (solid line) and E13 (dashed line).

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

13 of 14

and a slower propagation speed, and subsequently, owing tothe nonlinear interaction, the propagation speed and thewave amplitude of the merged ISWs train decrease while thewave number in the train increases. This suggests that incase one train of ISWs with smaller amplitudes is inducedprior to the other train with larger amplitudes at the samesource site, then the largest wave appearing in the middle ofthe ISW packet may be a temporary phenomenon due to thenonlinear wave‐wave interaction. In Experiment E7, whenthere are two trains of ISWs propagating from the differentsource sites at the same time but with the same amplitude ofincident solitons, once the crestlines of the two ISWs trainsmeet, a new leading soliton with a large amplitude isinduced at the connection point. The nonlinear collision ofthe two ISW trains has no effect on the wave amplitude andthe propagation speed of each initial incident ISWs train,and the induced soliton at the connection point will weakenquickly during its propagation.[26] Third, as for the wave‐island interaction shown by

Experiments E8 ∼ E10, once the ISW train collides with theisland, before the island, there is one weak ISW train re-flected by the island, while behind the island, the formercrestlines of the ISW train are torn by the island into twonew ISW trains. If the incident solitons are symmetric rel-ative to the island, then the reflected and torn ISW trains arealso symmetrical relative to the island, and after passingaround the island, the two torn ISWs trains reconnect,propagating westward with the reconnection point and themajor propagation direction overlapped along the center ofthe island, which agrees with the model results withoutearth’s rotation by Chao et al. [2006]. It is interesting thatjust after the collision with the island, the leading elevationwave following the depression wave is not the largestamplitude wave, and the collision would cause the ampli-tude of the leading depression wave to increase while theamplitude and the wave number in the subsequent elevationwaves train to decrease, which suggests the conversion ofthe potential energy of the elevation waves into the potentialenergy of the depression ones. If the incident solitons are notsymmetric relative to the island, then the reflected and tornISW trains are also not symmetrical relative to the island. Itdepends on the relative orientation of the original solitonsource site to the island. When the original source site ofsolitons is situated south or north of the island, then beforethe island, the weak ISW train reflected by the island wouldpropagate northeastward or southeastward later, whilebehind the island, the larger wave amplitude and the majorpropagation direction of the two torn ISW trains appear tothe south or north of the island. Although the two torn ISWtrains reconnect behind the island and propagate westward,their reconnection point and the major propagation directiondo not overlap.[27] The propagation model study of the ISWs here is still

very robust, since only the first baroclinic mode wave isconsidered, and some other factors such as the earth’srotation are not taken into account, which requires furthernumerical model studies.

[28] Acknowledgments. This work is jointly supported by the KeyProgram KZCX1‐YW‐12‐03 from the Chinese Academy of Sciences,China National Funds for Distinguished Young Scientists 41025019,

“863” Hi‐Tech Programs (grants 2008AA09Z112 and 2008AA09A402),and NSFC grant 40676021.

ReferencesBuijsman, M. C., Y. Kanarska, and J. C. McWilliams (2010), On the gen-eration and evolution of nonlinear internal waves in the South China Sea,J. Geophys. Res., 115, C02012, doi:10.1029/2009JC005275.

Cai, S., Z. Gan, and X. Long (2002a), Some characteristics and evolutionof the internal soliton in the northern South China Sea, Chin. Sci. Bull.,47(1), 21–26.

Cai, S., X. Long, and Z. Gan (2002b), A numerical study of the generationand propagation of internal solitary waves in the Luzon Strait, Oceano-logica Acta, 25(2), 51–60.

Chao, S. Y., et al. (2006), Reflection and diffraction of internal solitarywaves by a circular island, J. Oceanogr., 62(6), 811–823.

Du, T., Y. H. Tseng and X. H. Yan (2008), Impacts of tidal currentsand Kuroshio intrusion on the generation of nonlinear internal wavesin Luzon Strait, J. Geophys. Res., 113, C08015, doi:10.1029/2007JC004294.

Ebbesmeyer, C. C., et al. (1991), New observation on internal wave (solitons)in the South China Sea using an acoustic Doppler current profiler, inMarine Technology Society 91 Proceedings, pp. 165–175, New Orleans,LA, USA.

Fett, R. W., and K. Rabe (1977), Satellite observation of internal waverefraction in the South China Sea, Geophys. Res. Lett., 4(5), 189–191,doi:10.1029/GL004i005p00189.

Grimshaw, R. (2001), Internal solitary waves, in Environmental StratifiedFlows, chapter 1, pp. 1–28, R. Grimshaw (ed.), Kluwer, Boston.

Grimshaw, R., E. Pelinovsky, T. Talipova, and A. Kurkin (2004), Simula-tion of the transformation of internal solitary waves on oceanic shelves,J. Phys. Ocean., 34, 2774–2779.

Grimshaw, R., E. Pelinovsky, and T. Talipova (2007), Modeling internalsolitary waves in the coastal ocean, Surv. Geophys., 28, 273–298.

Helfrich, K. R., and W. K. Melville (2006), Long nonlinear internal waves,Annu. Rev. Fluid Mech., 38, 395–425.

Holloway, P. E., E. Pelinovsky, T. Talipova, and B. Barnes (1997), A non-linear model of internal tide transformation on the Australian North WestShelf, J. Phys. Oceanogr., 27, 871–896.

Hsu, M. K., and A. K. Liu (2000), Nonlinear internal waves in the SouthChina Sea, Can. J. Remote Sensing, 26(2), 72–81.

Liu, A., Y. Chang, M.‐K. Hsu, and N. Liang (1998), Evolution of nonlinearinternal waves in the East and South China Seas, J. Geophys. Res., 103(C4), 1613–1624, doi:10.1029/97JC01918.

Lynett, P. J., and P. L.‐F. Liu (2002), A two‐dimensional, depth‐integratedmodel for internal wave propagation over variable bathymetry, WaveMotion, 36(3), 221–240.

Pierini, S. (1989), A model for the Alboran Sea internal solitary waves,J. Phys. Oceanogr., 19(6), 755–772.

Orr, M. H., and P. C. Mignerey (2003), Nonlinear internal waves in theSouth China Sea: Observation of the conversion of depression internalwaves to elevation internal waves, J. Geophys. Res., 108(C3), 3064,doi:10.1029/2001JC001163.

Ramp, S. R., et al. (2004), Internal solitons in the northeastern South ChinaSea Part I: Sources and deep water propagation, IEEE J. Oceanic Eng.,29(4), 1157–1181.

Shaw, P.‐T., D. S. Ko, and S.‐Y. Chao (2009), Internal solitary wavesinduced by flow over a ridge: With applications to the northern SouthChina Sea, J. Geophys. Res., 114, C02019, doi:10.1029/2008JC005007.

Warn‐Varnas, A., et al. (2010), Solitary wave generation dynamics atLuzon Strait, Ocean Modelling, 31(1–2), 9–27.

Zhao, Z., and M. H. Alford (2006), Source and propagation of internalsolitary waves in the northeastern South China Sea, J. Geophys. Res.,111, C11012, doi:10.1029/2006JC003644.

Zhao, Z., V. Klemas, Q. Zheng, and X.‐H. Yan (2004), Remote sensingevidence for baroclinic tide origin of internal solitary waves in thenortheastern South China Sea, Geophys. Res. Lett., 31, L06302,doi:10.1029/2003GL019077.

Zheng, Q., R. D. Susanto, C. R. Ho, Y. T. Song, and Q. Xu (2007), Statis-tical and dynamical analyses of generation mechanisms of solitary inter-nal waves in the northern South China Sea, J. Geophys. Res., 112,C03021, doi:10.1029/2006JC003551.

S. Cai and J. Xie, Key Laboratory of Tropical Marine EnvironmentalDynamics, South China Sea Institute of Oceanology, Chinese Academy ofSciences, 164 West Xingang Rd., Guangzhou 510301, China. (caisq@scsio.ac.cn)

CAI AND XIE: MODEL FOR INTERNAL SOLITARY WAVES C12074C12074

14 of 14