Effects of high-order nonlinearity and rotation on the fission of internal solitary waves in the South China Sea*

ZHANG Shan-wu (張善武), FAN Zhi-song (范植松)

College of Physical and Environmental Oceanography, Ocean University of China, 266100 Qingdao, China, E-mail: zhang.shanwu@gmail.com

Effects of high-order nonlinearity and rotation on the fission of internal solitary waves in the South China Sea*

ZHANG Shan-wu (張善武), FAN Zhi-song (范植松)

College of Physical and Environmental Oceanography, Ocean University of China, 266100 Qingdao, China, E-mail: zhang.shanwu@gmail.com

(Received April 5, 2012, Revised July 17, 2012)

A variable coefficient, rotation-modified extended Kortweg-deVries (vReKdV) model is applied to the study of the South China Sea (SCS), with focus on the effects of the high-order (cubic) nonlinearity and the rotation on the disintegration process of large-amplitude (170 m) Internal Solitary Waves (ISWs) and the semi-diurnal internal tide propagating from the deep basin station to the slope and shelf regions in a continuously stratified system. The numerical solutions show that the high-order nonlinearity significantly affects the wave profile by increasing the wave amplitude and the phase speed in the simulated area. It is shown that the initial KdV-type ISW will decay faster when the rotation dispersion is considered, however the wave profile does not change significantly and the rotation effect is not important. The simulations of the semi-diurnal internal tide indicate that the phase of the wave profile is shifted earlier when the rotation effect is included. A solitary wave packet emerges on the shelf, and the wave speed is also greater when considering the rotation dispersion. In addition, the effects of the background currents are discussed further in this paper. It is found that the background currents generally change the magnitude and occasionally change the sign of the nonlinear coefficients in the northern SCS.

Internal Solitary Waves (ISWs), internal tide, high-order nonlinearity, rotation

Introduction

Nonlinear internal waves are frequently observed in the South China Sea (SCS), where both the in-situ measurements and the remote sensing images show that highly nonlinear Internal Solitary Waves (ISWs) propagate a long distance from the generation site. During the Asian Seas International Acoustics Experiment (ASIAEX), ISWs with amplitude greater than 140 m were observed on the continent slope of the northern SCS, and the estimated propagation distance from the generation region of these waves reached about 485 km[1]. In the deep basin of the northern SCS, the amplitude of ISWs is approximately 170 m[2]. These nonlinear waves are often considered to be generated in the Luzon Strait and then to propagate westward across the whole deep basin[1,3-5]. However, the generation mechanism of these ISWs remains to be an issue of study[6].

Weakly nonlinear Kortweg-deVries (KdV) type theories play a primary role in elucidating the essential features of the observations of ISWs despite the oceanic observations show that the mode-one internal waves are often highly nonlinear[7]. In the northeast of the SCS, Ramp et al.[1]found that the two-layer extended KdV (eKdV) model gives results in much better agreement with their observations of both the width and the phase speed than the two-layer KdV model, but the agreement is broken down completely for very large waves (greater than 88 m), which means that it is difficult simulating the large-amplitude ISWs by using the conventional approaches.

The numerical results of the two-dimensional eKdV model are in a reasonably good agreement with the laboratory experiments and also the observed large-amplitude internal waves. This model was applied by Holloway et al.[8]to investigate the evolution of a shoreward propagating internal tide on the Australian North West Shelf (ANWS) by taking account of the effects of the variable depth, the dissi-pation, the spatially variable coefficients and the influence of the background shear.

The earth’s rotation which has a significant effect on the long internal wave is also an important factor. On the modeling of the long shoreward propagation process of ISWs from the Luzon Strait to the coastal zone in the northern SCS, the rotation should be considered[9]. In a two-layer system, the numerical results indicate that the rotation dispersion may inhibit the disintegration of ISWs from the initial internal tide[7]. Based on the rotation modified eKdV (ReKdV) equation in a continuously stratified system, a series of numerical experiments were performed by Holloway et al.[10]on the coast region of the ANWS, and a similar conclusion was reached. Recently, Helfrich[11]used the Miyata and Choi and Camassa (MCC-f) rotation equation in a two-layer system to examine the effect of the rotation on the propagating of ISWs, and the numerical solutions show that the initial ISW will decay by the radiation of longer inertia-gravity waves that may subsequently steepen, leading to the formation of a secondary solitary-like wave. The two-layer MCC-f model was applied to illustrate the observed characteristics of the internal tide evolution in the deep basin of the SCS and the model predictions are consistent with the observations if this model is matched to a wave generation solution, however, the choice of the interface is limited due to the two-layer assumption. When the interface depth is reduced from 500 m to 250 m, both the amplitude and the width of the ISWs are reduced by ~50% though the wave shape of the response remains essentially the same[12]. Moreover, as a modeling tool, the MCC equations have a potential limitation that solitary waves of sufficient amplitude could be unstable at high wave numbers in the form of the Kelvin-Helmholtz instability[7].

In this paper, under the continuously stratified assumption, the fission processes of the large-amplitude ISWs and a linear internal tide propagating from the deep basin to the continent slope and shelf regions in the northeast of the SCS are simulated in the framework of the variable-coefficient ReKdV (vReKdV) model with variable depth, and with the considerations of the spatial change of the background stratification, the effects of the background currents and the dissipation terms. This model was first adopted by Holloway et al.[10]to examine the propagation and the evolution of an initial internal tide represented by a harmonic wave formation, and achieved a good agreement with the observations in the ANWS regions. Furthermore, without considering the rotation and dissipation effects, Grimshaw et al.[13]analyzed this model theoretically and applied it to several oceanic cases, which demonstrate the applicability of the asymptotic theory. In a recent paper by Grimshaw et al.[14], the veKdV model was reviewed and applied to the SCS to study the physical process of the propagation, the deformation and the disintegration of the large-amplitude ISWs, however, the initial amplitude chosen by the authors was far less than the real amplitude of the observed large-amplitude ISWs in this area, and the influence of the background current was not discussed. The goal of this paper is to explore further the roles of the high-order nonlinearity and the rotation in the propagation and the evolution of the large-amplitude ISWs and the internal tide in the northern SCS with the vReKdV model. In addition, the effects of the background currents are discussed further in this paper on the basis of the previous work[9].

The outline of this paper is as follows. In Section 2, the numerical model and the data used are introduced. In Section 3, the simulation results of the largeamplitude ISWs and the semi-diurnal internal tide by the vReKdV model are presented. The effects of the background currents are discussed in Section 4. In Section 5, the conclusions of this paper are given.

1. Model and data

The vReKdV model used in this paper is similar to that presented by Holloway et al.[10]and Grimshaw et al.[13], however, the earth’s rotation was not included in the latter paper. The nonlinear equation is

N( z )and U( z)in Eq.(2) are the buoyancy frequency and the background current, respectively. Themodal function Φ(z)is normalized by its maximal value, and thereforeη(x, t)will be the isopycnal surface with the maximum displacement. Considering the effect of the background current, the expressions of the coefficients in Eq.(1) are as follows:

The variables with subscript “0” in the above equations are those at a fixed point x0, which may be marked as the initial position of a solitary wave. Here, Tin Eq.(5) is the nonlinear correction of the modal structure, and it is determined by the inhomogeneous eigenvalue problem with zero boundary conditions at the sea bottom and the free surface

The function T( z )is normalized so that T( zmax)=0 wherezmaxis the root of Φ(zmax)=1. Considering the nonlinear correction to the modal structure, the vertical displacement of the isopycnal surfaceξ(x, z, t )is given by

If we take accounts of the effects of the friction(κ) of the bottom boundary layer and the horizontal eddy viscosity(ν), a general equation can be obtained as follows

Equation (11) represents a basic model of this paper to investigate the effects of the high-order nonlinearity and the earth’s rotation on the transformation of the large-amplitude ISWs and the internal tide in the continental slope regions of the northern SCS.

The climatic data of the temperature and the salinity with a high resolution of 0.25o×0.25ogrid are obtained based on the database World Ocean Atlas 2001 (WOA01), and only the data in May are used in this paper. The monthly mean data of WOA01 contain only 24 layers (upper 1 500 m) in the vertical direction, and they are made up of 33 layers (5 500 m) according to the seasonal averaged data from the same database. Then the buoyancy frequency N( z)is evaluated according to the equation of state of the seawater EOS80. The determination of the background currentsU( z)is a difficult problem in the study of the propagation process of ISWs. In this paper, we only consider those composed of the monthly mean baroclinic circulation and the barotropic tidal current, which is a simple approximation of the background currentsU( z). The monthly mean baroclinic circulation data come from the model results ofLICOM1.0[15], which is a quasi-global (75oS-65oN) model and has a resolution of a uniform 0.5o×0.5ogrid in the horizontal direction and with 30 layers in the vertical direction. A 950 year spin-up run is conducted from the initially motionless ocean, and then the modeled results are interpolated to the 0.25o×0.25ogrid. The barotropic tidal current data come from the results in the SCS (99oE-122oE, 2oN-25oN) using the barotropic POM model without considering the effect of the wind. In the modeling of the barotropic tide, a 30 d spin-up run is conducted with a horizontal resolution of 1/3o×1/3ogrid, with the same boundary conditions as the Fes95 model, and with only the semidiurnal M2 and S2, the diurnal K1 and O1 being taken into account. The model results are divided into the surface layer, the middle layer and the bottom layer in the vertical direction according to theσcoordinate (the value ofσis 0, –0.3, and –1.0, respectively). Because the difference among the three layers is very small, we will adopt the surface result as a representative value of the barotropic tidal current in the following numerical simulation. By interpolating the barotropic tidal current from the 1/3o×1/3ogrid to 0.25o× 0.25ogrid, we can obtain the background currents U( z).

Fig.1 The sea floor map and the model stations (S1–S7) along the latitudinal zone of 21oN in the northern SCS. The buoyancy frequency and the background baroclinic current profiles at these stations

The 1'×1' bathymetry data are provided by the famous Smith and Sandwell sea floor topography, and only seven stations along the 21oN in the northern SCS are used in this paper. The distribution of the cross-sections S1-S7 studied in this paper is shown in Fig.1. The profiles of the stratification and the background baroclinic currents at the corresponding stations are plotted in Fig.1(a) and Fig.1(b), respectively. The initial station S1 is located at (21oN, 118.5oE), and the station S7 (21oN, 117oE) is in the continental slope region close to the observation station (21o2.8'N, 117o13.2'E) of Yang et al.[4].The background current is dominantly westward with a magnitude of about 0.1 m/s in the upper layer.

Fig.2 The first mode of the modal function and its nonlinear correction at S7

Based on the Thomson-Haskell method[16], the long wave phase speedc is determined by solving the eigenvalue problem (2) and (3). Then the first mode of the linear modal function Φ(z)and its nonlinear correction T( z )are obtained. The vertical profiles of Φ(z)and T( z )at S7 are shown in Fig.2. The depth of the maximum value of Φ(z)is around 120 m, corresponding to the point that the first mode internal wave reaches the maximal amplitude according to the KdV theories. There are two turning points for T( z )around 50 m and 200 m, respectively.

The distributions of the parametersc,α,α1and βin the SCS along two particular sections were given by Grimshaw et al.[14], however the effects of the background currents were not discussed in their work. In this paper, the distributions of these parameters along sections S1-S7 are calculated for cases with and without the background currents (Fig.3). The existence of the background current only slightly shifts the phase speed c , and the effect of the background currents on the nonlinear coefficients is more prominent as shown in Fig.3. The variations of the magnitu-des of the coefficientsαand α1are nearly the same, both increasing from S1 to S5, and then decreasing from S5 to S7. The values ofβandQare nearly unchanged, which means that they are not sensitive to the background current.

Fig.3 Coefficients of Eq.(7) calculated for the bathymetry and stratification in the northern SCS along 21oN cross section

2. Simulation results

Equation (11) is solved by using the numerical method developed for the vKdV equation by Holloway et al.[8], where the numerical stability criterion for the differential scheme without the dissipative terms is also given. To run this model, an initial wave form should be determined. In our study, a large-amplitude KdV-type solitary wave representing the initial observed ISW and a sinusoidal long wave representing an internal tide are given to investigate the propagation and the deformation of the ISWs and the internal tide, respectively.

Fig.4 The vertical isopycnal displacements at the depth of the maximum Φ(z)for the first mode ISWs. The results are normalized by the maximum amplitude ηmaxat S1 (170 m), and the displacement is incremented by –1 unit at each station

2.1 Large-amplitude ISWs

The initial ISW at S1 is assumed to be a depression ISW of the first mode, as a solution of the classical KdV equation

The definition of the half width ΔKdVis

where ζ0is the amplitude of the ISW at station S1.

In the first run,ζ0is supposed to be –170 m, according to the observation in the center basin of the northern SCS[2]. The numerical solutions of the vKdV model in the absence of the rotation and the cubic nonlinearity are shown in Fig.4. Before the initial wave arrives at station S4, the amplitude slowly decays due to the dispersion effect, meanwhile the half width turns broader, as predicted by the classical KdV theory. Then, the wave steepens at station S4 due to the topography and nonlinearity effects, and subsequently a solitary like wave with amplitude of nearly 170 m is formed at the front while a relatively smaller wave emerges at the rear, which means the fission of the ISW has occurred. When the wave propagates into the shallow water, the leading wave continues to decay and disintegrate into smaller-amplitude solitary waves. At station S6, the leading soliton is about 115 m in height and in the second wave packet, the largest one is about 50m in height and 3 h behind. The phase speed of the leading solitary wave is estimated as 1.62 m/s at station S6 by Eq.(11), and the value will be reduced to 1.57 m/s if the background currents (about 0.05 m/s) are neglected. The disintegration process will continue after S6, and finally the amplitude of the leading wave reduces to only 73 m at station S7.

Fig.5 As in Fig.4 but for the veKdV model. The results are normalized by the maximum amplitude ηmaxof S4

As shown in Fig.5, the numerical simulation results of the veKdV model in which the cubic nonlinearity is considered and the rotation effect is not included are quite different from those shown in Fig.4. In Fig.5, the initial wave steepens and the leading wave gradually turns narrower from station S1 to station S4, then the leading solitary wave disintegrates into two solitons. The leading wave keeps disintegrating, but the amplitude becomes smaller, being reduced to about 115 m at station S6. The amplitude of the leading solitary wave at S6 is nearly the same as the former vKdV run. However the phase speed of this wave is different, since it is defined by

and α1retains positive during the whole run. The phase speed at station S6 determined by Eq.(15) is about 1.74 m/s, which is notably larger than the estimated value by Eq.(13). A comparison between these two runs is shown in Fig.6. In contrast, the wave amplitude predicted by the veKdV model is larger, and the fission process is more complex. The nonlinear effect is more prominent in the veKdV case because of the higher amplitude. Figure 6 indicates clearly that the veKdV wave is indeed faster than the vKdV wave.

Fig.6 The comparisons of vertical isopycnal displacements at the depth of the maximum Φ(z)for the first mode ISWs at station. The results are normalized by the simulated maximum amplitude ηmaxof S4. The initial amplitude at S1 is 170 m

The rotation effect is considered in the following run. The numerical simulation results of ISWs by using the vKdV and vRKdV models, respectively are shown in Fig.7. The comparison of results between these two models shows that a disintegrated solitary wave packet will gradually occur after 50 km, and the solitary-like waves will travel slightly faster whenconsidering the rotation dispersion. The effect of the rotation is evident after 75 km, and the decay process in the vRKdV model seems to be more rapid than in the vKdV case. A comparison of results between these two models at three stations from S4 to S6 is shown in Fig.8 and the same results may be obtained. The leading solitary wave is about 30 min earlier in the rotation case than that in the non-rotation case. However, the wave profile seems nearly unchanged when including the rotation effect, which means that the rotation effect is not significant in our large-amplitude ISW simulations.

Fig.7 The vertical isopycnal displacements of the vKdV model (upper panel) and the vRKdV model (lower panel) at the depth of the maximum Φ(z)for the first mode ISWs. The displacementηis shown at equal distance intervals in a coordinate moving with the long wave phase speed. The initial amplitude of the KdV-type solitary wave is 170 m

2.2 Internal tide

A first-mode sinusoidal long wave is used to represent the westward propagating semi-diurnal internal tide, and the expression of this wave is

Fig.8 The comparisons of vertical isopycnal displacement at the depth of the maximum Φ(z)for the first mode of ISWs at station. The results are normalized by the simulated maximum amplitudeηmaxof S4.

Fig.9 The vertical isopycnal displacements of the vReKdVmodel at the depth of the maximum Φ(z)in the semidiurnal internal tide run. The displacementηis shown at equal distance intervals in a coordinate moving with the long wave phase speed. The initial amplitude is 20 m

A solitary wave appears when the distorted wave passes station S6 (125 km), and a wave packet is gradually formed. The amplitude of the leading wave in this packet is about 75 m at station S7 (150 km), and the speed of this packet is just slightly greater than the local long wave phase speed.

Fig.10 The comparisons of vertical isopycnal displacements at the depth of maximum Φ(z)for the first mode of the semi-diurnal internal tide from stations S1 to S7. The initial amplitude of the internal tide is 20 m

The rotation effect on the evolution of the semidiurnal internal tide is examined. The results of the vKdV and vRKdV models for stations S1 to S7 are shown in Fig.10, which indicates that the vRKdV wave is significantly faster than the vKdV wave. The phase of the wave profile seems to be shifted earlier in the vRKdV case than that in the vKdV case. A solitary wave packet emerges at S7, and the packet speed is also greater when considering the rotation dispersion.

Fig.11(a) Bathymetry of the northern SCS

Fig.11(b) Topography of the cross-section P1-P2

3. Effects of background currents

The effects of the background currents on the propagation and the evolution of the ISWs and the internal tides are discussed further here on the basis of

Fig.12 Coefficients of linear long wave phase speed c, quadratic nonlinearity αand cubic nonlinearity α1determined for bathymetry and stratification in the northern SCS along the cross section P1-P2

the previous work[9,19]. The background currents do not significantly affect the modal structure since the monthly mean baroclinic circulation is less than

0.1 m/s and the barotropic tidal current is not large enough to be dominant. However, when the background current turns sufficiently large, there will be notable changes in the coefficients of the vKdV model and its varied modifications. In order to investigate the influences of the background current on the evolution of ISWs in the SCS, a section P1-P2 across the whole basin and extending to the continent shelf region is selected (Fig.11). The water depth is about 3 617 m at station P1 in the deep basin and 89 m at station P2 on the shelf. The corresponding distributions of c,α and α1are shown in Fig.12. The values of the model parameters at several stations along the section P1-P2are given in Table 1 for the cases with and without the consideration of the background current. In the deep water, when the background current is small(＜0.1 m/s), the influence of the background current on the model parameters is not significant. But, in the continental shelf region, the background current is much larger (＞0.1m/s), and the model parameters are prominently affected by the background current, especially the nonlinear coefficients. The background currents may generally change the magnitude and even the sign of the nonlinear coefficients (Fig.12, Table 1).

Table 1 Model parameters at several stations along the section P1-P2 for cases with and without consideration of the background current

4. Conclusion

The effects of the high-order nonlinearity and the earth’s rotation on the fission of ISWs in the northern SCS are investigated by using the vKdV model and its varied modifications (with/without the cubic nonlinearity, with/without the rotation) in a continuously stratified system. To simulate the propagation and the evolution of the ISWs in the northern SCS, an initial KdV-type ISW with amplitude of 170 m is assumed, and the numerical solutions show that the initial wave will first decay if the high-order nonlinearity is not included whereas it will first steepen with the consideration of the effect of the high-order nonlinearity. The initial wave disintegrates into two solitons in the vKdV model, as well as in the veKdV model at the station with depth about 1 000 m. The numerical solutions show that the high-order nonlinearity significantly affects the wave profile by increasing the amplitude and phase speed in the simulated area. It is shown that the initial KdV-type ISW will decay faster when the rotation dispersion is considered, however the wave profile does not change significantly, which means that the rotation effect is not significant in our simulation. The simulations of the semi-diurnal internal tide indicate that the phase of the wave profile is shifted earlier by considering the rotation effect. A solitary wave packet emerges on the shelf, and the packet speed is also greater when considering the rotation dispersion. In addition, it is found that the background currents generally change the magnitude, and may occasionally change the sign of the nonlinear coefficients in the northern SCS.

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10.1016/S1001-6058(13)60357-1

* Project supported by the National Natural Science Foundation of China (Grant No. 41030855).

Biography: ZHANG Shan-wu (1987-), Male, Ph. D. Candidate

FAN Zhi-song,

E-mail: fanzhs@hotmail.com