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Analysis of kinematic parameters of Internal Solitary Waves in the Northern South China Sea Guanghong Liao a,n, Xiao Hua Xu a, Chujin Liang a, Changming Dong b,c, Beifeng Zhou a, Tao Ding a, Weigen Huang a, Dongfeng Xu a a

State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China College of Marine Science, Nanjing University of Information Science and Technology, Nanjing, 210044, China c Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095, United States b

art ic l e i nf o

a b s t r a c t

Article history: Received 11 April 2014 Received in revised form 1 October 2014 Accepted 4 October 2014 Available online 14 October 2014

The monthly climatology of observed temperature and salinity from the U.S. Navy Generalized Digital Environment Model (GDEM-Version 3.0) is used to derive the geographical and seasonal distribution of kinematic parameters of nonlinear internal waves in the Northern South China Sea (NSCS). Coefﬁcients of the Generalized Extended Korteweg-de Vries Equation (GEKdV) with a background current are investigated (phase speed, dispersion, quadratic and cubic nonlinearity parameters, normalizing factor). These parameters are used to evaluate the possible polarities, shapes of internal solitary waves, their limiting amplitudes and propagation speed. We show that the long wave phase speed and dispersion parameters mainly depend on topography characteristics and have no obvious seasonal variation. The nonlinear parameters and normalizing factor are sensitive to variations in the density stratiﬁcation and topography. Background current also exerts the distinct effects on the kinematic parameters; especially the nonlinear parameter can change by an order of magnitude. The nonlinear parameters take on larger values in the summer (July), and linear internal waves are prone to become steeper and develop into large-amplitude internal solitary waves under such circumstances. This explains why nonlinear internal solitary waves occur more frequently in summer. From the kinematic viewpoint, the dispersion parameter takes on larger values in the Paciﬁc Ocean (PO) due to deeper water depth when compared with that in the NSCS. The stronger dispersion effect in the PO hinders the formation of large amplitude internal solitary waves, explaining why nonlinear internal solitary waves are rarely found to the east of the Luzon Strait. Large near-bottom velocities dominate the shallow area and tend to increase in the warm season. The largest values are induced by internal solitary waves, indicating that internal waves are the major drivers of sediment re-suspension and erosion processes. & 2014 Elsevier Ltd. All rights reserved.

KeyWords: Internal Solitary Wave Background currents Generalized Extended KdV (GEKdV) Kinematic parameters Northern South China Sea

1. Introduction The northern South China Sea (NSCS) is featured by large Internal Solitary Waves (ISWs), which have been extensively observed (Cai et al. 2012; Ramp et al. 2004; Liu and Zhao, 2008; Zhao and Alford 2006; Alford et al., 2010, Liu et al. 2013; Lien et al., 2014) and numerically simulated (Buijsman et al. 2010a, b; Cai and Xie 2010; Zhang et al. 2011; Shaw et al. 2009; Guo et al. 2011). ISWs travel in a near-westward direction over several hundreds of kilometers with wave speeds up to 3 ms 1, vertical isopycnal displacements of about 100 m, and crest lengths of over 100 km in the horizontal plane (Liu and Zhao, 2008). ISWs in the NSCS occur throughout the year (Liu et al., 1998; Liu and Zhao, 2008; Zheng

n

Corresponding author. E-mail address: [email protected] (G. Liao).

http://dx.doi.org/10.1016/j.dsr.2014.10.002 0967-0637/& 2014 Elsevier Ltd. All rights reserved.

et al., 2007), but mainly during the spring and summer (April– July) with the highest frequency of occurrence in July, lower in winter (December–February), and lowest in February (Zheng et al., 2007). This indicates that the seasonal background environmental characteristics exert an impact on the generation and propagation of internal solitary waves. Satellite images also show remarkable differences in the spatial distribution of ISWs to the west and east of the Luzon Strait (LS) (Zheng et al., 2007; Buijsman et al., 2010b). According to satellite synthetic aperture radar (SAR) images from 1995 to 2001 (Zheng et al., 2007), almost all ISWs packets occur west of 120.51. Buijsman et al. (2010b) report that to date, only 17 satellite images have been identiﬁed that show clear evidence of ISWs east of LS, in the region bounded by longitudes 123.51E and 1301E and latitudes 181N–241N. The ﬁne scale of the waves and their sometimes poor organization make their identiﬁcation difﬁcult in low resolution images. The model results by Buijsman et al. (2010b) show that ISWs east of LS are 45% smaller than those to the

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west due to asymmetric modulated barotropic tides at the east ridge, 39% smaller than those due to a deeper Paciﬁc Ocean (PO), 28% smaller than those due to westward thermocline shoaling, and 23% smaller than those due to internal tide resonance in a double ridge conﬁguration. ISWs in the NSCS normally originate from the interaction between barotropic tidal currents and submarine topography, especially the LS is main source area (Ramp et al., 2004). This is also conﬁrmed in model experiments, for example by Cai et al. (2002) and Chao et al. (2007). Several generation mechanisms have been identiﬁed, including internal tide onset (Lee and Beardsley, 1974; Zhao and Alford, 2006), Lee-waves (Maxworthy, 1979; Cai et al., 2002), combination of the internal tide onset mechanism and Lee-wave release (Shaw et al., 2009; Buijsman et al., 2010a), collapse of the internal mixed region (Maxworthy, 1979; Du et al., 2008), and the scattering of an internal tide beam at the surface thermocline (Gerkema, 2001). Which of these mechanisms occurs depends on the details of the barotropic ﬂow, density stratiﬁcation, and topographical features. Generally speaking, nonlinearity and dispersion play an important role in the production and propagation of ISWs: nonlinearity leads to steepening, breaking is prevented by dispersion, and the balance between nonlinearity and dispersion results in the generation of a rank-ordered packet of internal solitary waves. Comparisons between observations and theoretical models indicate that the two-dimensional nonlinear internal wave structure in the NSCS can be well described by solitary wave solutions of the Kortewegde Veries (KdV) family of equations. Many characteristics of the internal wave ﬁeld, e.g., the dispersion relation, depend primarily on the structure of the density stratiﬁcation. The horizontal variability of hydrological ﬁelds is evident in weakly nonlinear internal wave theory (Talipova et al., 1998; Pelinovsky et al., 1995). A world maps of KdV equation coefﬁcients, inferred from the long-term mean annual hydrologic data with one-degree latitude-longitude resolution is ﬁrstly given by Grimshaw et al. (2007). Then Grimshaw et al. (2010) studied the distribution of KdV equation coefﬁcients in January over the South China Sea (SCS) in the absence of a background current; Cai et al. (2014) further investigated the monthly variation in some parameters of ISWs in the SCS, similarly, the background currents are not considered. However, some authors have demonstrated that the background current ﬁeld plays an important role in the propagation and breakup of ISWs (Polukhin et al., 2004; Cai et al. 2008; Fan et al., 2013). The purpose of this paper is to examine the geographical distribution and seasonal variation of the kinematic parameters of long ISWs in the NSCS using the U.S Navy Generalized Digital Environment Model (GDEM-Version 3.0) and a weakly nonlinear internal wave propagation and evolution model. Because GDEM represents the long-term mean density stratiﬁcation, our approach allows a climatological estimate of the expected internal wave parameters for different regions in the NSCS. We try to explain the east-west asymmetric distribution of ISWs about the LS based on the zonal difference of kinematic parameters. Another key outcome is the near-bottom velocities induced by internal waves in the NSCS, where ISWs are the major driver of sediment resuspension and erosion processes.

2. Data The hydrography data for the present study are from GDEM‐ Version 3.0, which was derived from the temperature and salinity proﬁles extracted from the Mater Oceanographic Observational Data Set edited at the Naval Research Laboratory (NRL). The monthly climatology dataset has a horizontal resolution of

Fig. 1. Topographic map of the study sea area and distribution of data stations, the symbols star (★) is selected points for analysis in Section 3.3 (see Fig. 2 and Tables 1 and 2).

0.251 0.251, and 78 standard depths from surface to 6600 m, with a vertical resolution varying from 2 m at the surface to 200 m below the 1600 m. In order to remove erroneous proﬁles, the NRL manually examines all proﬁles within groups covering small geographic regions and short seasonal or monthly time periods. Carnes (2009) reported the data evaluation for global oceans, and Li et al. (2002), Qu et al. (2006) and Wang et al. (2011) demonstrated that the GDEM captures the detailed temperature and salinity distribution in the SCS by comparison with previous water mass analysis. The topography data comes from the Smith and Sandwell (1997) dataset with a resolution of (1/601). The topography map and data stations for the study area are shown in Fig. 1. The investigated area focusses on the NSCS (1101–1251E, 181–231N), where a high frequency of ISWs is known to occur. The Luzon Strait, (18.51 to 22.01 N and 1201 to 1221 E), which lies between the islands of Taiwan and Luzon, is the most probable source region of internal waves. Its bottom topography features two parallel north– south-oriented submarine ridges. The deeper Hengchun Ridge (HCR) lies 100 km to the west of the shallower Lanyu Ridge (LYR), and a trench deeper than 4000 m exists between them. The interaction between ridges and high tidal energy, together with the year-round strong stratiﬁcation that characterize the LS, are appropriate conditions for the generation of internal waves.

3. Weakly nonlinear internal wave propagation model 3.1. Generalized Extended Korteweg-de Vries equation Many observations show that ISWs in the NSCS are well described by the KdV equation. Based on the weakly nonlinear theory of long ISWs (Grimshaw et al., 2004, Holloway et al., 1999), the vertical isopycnal displacement ζðz; x; tÞ of internal solitary wave can be expressed as (up to the second in nonlinearity) ζðz; x; tÞ ¼ ηðx; tÞΦðzÞ þ η2 ðx; tÞTðzÞ;

ð1Þ

where t is time, x is the horizontal coordinate along the propagation direction of ISWs, z the vertical directed upward, Φ(z) describes the vertical structure of the long internal wave, and T(z) is the ﬁrst nonlinear correction to Φ(z). Under the Boussinesq approximation, the function η(x,t) describes the transformation of a wave along the axis of propagation and its evolution in time, and satisﬁes the following equation (Generalized Extended KdV equation, i.e. GEKdV): ∂η ∂η ∂η ∂3 η c dQ þ ðc þ αηÞ þα1 η2 þ β 3 þ η ¼ 0; ∂t ∂x ∂x ∂ x 2Q dx

ð2Þ

G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

where β, α, and α1 are called the dispersion, quadratic and cubic nonlinear parameters, respectively. The function Q(x) characterizes the ampliﬁcation of the linear long internal wave, reﬂecting slow change in depth and horizontal change of background density ﬁeld. The phase speed c of linear long internal waves can be found from the eigenvalue problem for Ф(z) d dΦ ½ðc UðzÞÞ2 þ N 2 ðzÞΦ ¼ 0; dz dz Φð0Þ ¼ Φð HÞ ¼ 0:

ð5Þ

The buoyancy frequency N(z) is determined by the following expression: g dρ ; N 2 ðzÞ ¼ ρ dz

ð6Þ

where g is the acceleration due to gravity and ρ is undisturbed density proﬁle. Considering the effect of the background current, the coefﬁcients of the GEKdV Eq. (1) are determined through Ф(z) and its correction term T(z) R0 ðc UÞ2 ðdΦ=dzÞ3 dz 3 α ¼ R0H ; ð7Þ 2 ðc UÞðdΦ=dzÞ2 dz H

R0

ðc UÞ2 Φ2 dz 1 ; β ¼ R 0 H 2 2 H ðc UÞðdΦ=dzÞ dz

ð8Þ

R0

c2

2 H ðc0 U 0 ÞðdΦ0 =dzÞ dz ; R0 2 H ðc UÞðdΦ=dzÞ dz

ð9Þ

α1 ¼ R 0

2 H ðc UÞðdΦ=dzÞ dz

I ¼ 9ðc UÞ2

;

ð10Þ

dT dΦ 2 dΦ ð Þ 4αðc UÞdT ; dz dz dz dz

dΦ dΦ dΦ II ¼ α2 ð Þ2 þ 5αðc UÞð Þ3 6ðc UÞ2 ð Þ4 : dz dz dz The ﬁrst nonlinear correction T(z) is a solution of the following ordinary differential equation: d dT d dΦ 3 d dΦ ½ðc UÞ þ N 2 T ¼ α ½ðc UÞ þ ½ððc UÞ2 Þ2 ; dz dz dz dz 2 dz dz

ð11Þ

with the boundary conditions Tð0Þ ¼ Tð HÞ ¼ 0;

ð12Þ

and with the normalization Фmax ¼ 1. T ðzmax Þ ¼ 0;

whereФðzmax Þ ¼ 1:

Δρ h1 h2 1=2 Þ ; ρ h1 þ h2

3c h1 h2 ð Þ; 2 h1 h2 3c

2

2 2

8h1 h2

ð14Þ ð15Þ

2

ðh1 þ h2 þ 6h1 h2 Þ

β ¼ 6c h1 h2

ð16Þ ð17Þ

where Δρ=ρ⪡1 is the relative layer density difference, h1 is the upper-layer thickness (which is given by the maximal depth for buoyancy frequency), and h2 ¼H h1 is the lower-layer thickness (H is the total water depth). The density of the two-layer ﬂuid are determined by the formulae: Z 1 h1 ρupper ¼ ρðzÞdz; ð18Þ h1 0 ρlower ¼

1 h2

Z

h2 0

ρðzÞdz;

ð19Þ

3.2. Shapes amplitudes of nonlinear internal waves After substituting the change of variables Z dx ηðx; sÞ t and ςðx; sÞ ¼ ; s¼ cðxÞ Q ðxÞ

ð20Þ

In Eq. (2) (Grimshaw et al., 2004), the steady-state internal solitary wave solution for constant coefﬁcients is described in general form ς¼

A ; 1 þ B cosh ½ðγðs kxÞ

ð21Þ

where cosh( ) is hyperbolic cosine function. In Eq. (21) only one parameter is arbitrary; the parameters A, B, k, and γ are given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6βγ 2 α2 c2 2 βγ 2 A¼ 2 ; γ¼ ðB 1Þ; k ¼ 4 ; ð22Þ 6α1 β αc Q c The amplitude of the solitary wave is

R0

H ðI þ IIÞdz

α¼

ð4Þ

ζðzmax ; x; tÞ ¼ ηðx; tÞ

c20

c ¼ ðg

α1 ¼

where N(z) is the buoyancy frequency, H is the total water depth, and U(z) is background current along the wave propagation direction. Cai et al.(2008) point out that the phase speed and wave structure are modiﬁed by background currents, although the current shear has little effect on vertical structure function, the current curvature could have a strong impact on it. Based on measured current proﬁles, Polukhin et al. (2004) also demonstrated background current have the distinct effects on kinematic parameters of ISWs, It is convenient to use the normalization Фmax ¼ Ф(zmax)¼1, and in the case where the leading solution ηðx; tÞ coincides with the isopycnal surface displacement at zmax

Q¼

For a two-layer model (interfacial wave) with a rigid lid approximation, the environmental parameters are simply:

ð3Þ

with the boundary conditions

161

a¼

A α ¼ ðB 1Þ ; 1þB Q α1

The signs of nonlinear coefﬁcients α and α1 vary. When α1 o0 and 0o Bo 1, the solitary wave polarity also depends on the sign of quadratic nonlinear coefﬁcient α; when α 40, the solitary wave is positive, otherwise it is negative. Moreover, the solitary wave amplitude ranges between 0 and a limiting value alim ¼ α=Q α1 :

The subscript 0 in Eq. (9) indicates values at some arbitrary point x0; almost all internal waves in the NSCS propagate westward, so the points at 1201E will be chosen as reference points in the following computation (Fig. 1).

ð24Þ

When the solitary wave amplitude approaches this value, its width tends to inﬁnity (B-0), and such solitary waves are known as “top-table” solitons. When α1 40 and 1oBo 0 and solitary waves are known as “top-table” solitons width tend (B 1). These are called “algebraic” solitons aa lg ¼ 2α=Q α1 :

ð13Þ

ð23Þ

ð25Þ

In this paper, we construct and discuss the geographic distribution of parameters c, β, α, α1 , Q, alim, and aalg and their seasonal variations in the GEKdV equation for the NSCS. We also analyze and discuss the geographical features of the near-bottom velocity induced by the solitary waves.

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G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

Fig. 2. Horizontal distribution of velocity vectors (unit cm/s) at 200 m level (a) January, (b) August, and (c) proﬁles of eastern velocity component at the selected points (labeled by symbols star (★) in Fig. 1). Table 1 Kinematic parameters of ISWs at the selected points in January, the parameters are calculated without (superscript “0”) and with regard to the current (notice that the quadratic nonlinear parameter, cubic nonlinear parameter and dispersion parameter is scaled by factor 10 3, 10 4, and 106 respectively).

c0 cn α0 αn

Point A (1231E, 201N)

Point B (121.751E, 201N)

Point C (120.51E, 201N)

Point D (119.251E, 201N)

Point E (1181E, 201N)

Point F (116.751E, 201N)

Point G (115.51E, 201N)

Point H (114.251E, 201N)

3.38

2.55

2.74

2.54

2.42

2.18

2.02

1.25

3.27 (0.97c0 ) 3.59

2.38 (0.93c0 ) 4.22

2.74 (1.0c0 ) 2.96

2.50 (0.98c0 ) 2.68

2.36 (0.98c0 ) 3.29

2.14 (0.98c0 ) 4.62

2.02 (1.0c0 ) 5.26

1.23 (0.98c0 ) 4.84

1.57(0.44α0 ) α1 0 1.09 α1 n 1.09(1.0α1 0 ) β0 2.16

3.49(0.83α0 ) 1.61

2.28(0.77α0 ) 1.07

1.87(0.70α0 ) 0.95

2.33(0.71α0 ) 1.11

2.43(0.53α0 ) 1.74

4.48(0.85α0 ) 1.97

0.43

0.76(0.48α1 0 ) 0.26

0.68(0.63α1 0 ) 0.88

0.51(0.53α1 0 ) 0.68

0.59(0.53α1 0 ) 0.50

1.27(0.72α1 0 ) 0.21

1.98(1.0α1 0 ) 0.12

0.91(2.1α1 0 ) 0.01

βn

0.23(0.91β0 )

0.87(1.0β0 )

0.66(0.97β0 )

0.49(0.97β0 )

0.20(0.94β0 )

0.11(0.97β0 )

0.01(1.0β0 )

1.91(0.89β0 )

G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

3.3. Solution method for coefﬁcients of the GEKdV equation Propagation and evolution of internal waves mainly depend on topographic features and oceanic background environment conditions, which are represented in the coefﬁcients of the GEKdv equation. The buoyancy frequency N(z) and background velocity U(z) are needed to solve the boundary value problems (3) and (4). The density stratiﬁcation N(z) can be calculated from long-term mean temperature and salinity proﬁles provide by GDEM-3.0.

163

Generally speaking, the geostrophic current is a main contribution to the total background current in the ocean though the barotropic tidal current is also important in some continental area. In present study we only consider the geostrophic current. In order to obtain the background velocity ﬁeld, we employed a 3-D ocean current diagnostic model. The model has successfully calculated the velocity ﬁeld from temperature and salinity data; the model is described in detail in Liao et al. (2007, 2008) and Yuan et al. (2009). The Luzon Strait is a major generation source

Table 2 Kinematic parameters of ISWs at the selected points in July. The parameters are calculated without (superscript “0”) and with regard to the current (notice that the quadratic nonlinear parameter, cubic nonlinear parameter and dispersion parameter is scaled by factor 10 3, 10 4, and 106 respectively). Point A (1231E, 201N)

Point B (121.751E, 201N)

Point C (120.51E, 201N)

Point D (119.251E, 201N)

Point E (1181E, 201N)

Point F (116.751E, 201N)

Point G (115.51E, 201N)

Point H (114.251E, 201N)

3.47

2.68

2.87

2.70

2.59

2.23

2.00

1.38

3.41(0.98c0 ) 3.87

2.48(0.93c0 ) 5.09

2.89(1.0c0 ) 4.01

2.77(1.02c0 ) 3.88

2.62(1.01c0 ) 4.59

2.28(1.02c0 ) 5.91

2.03(1.01c0 ) 8.29

1.38(0.99c0 ) 14.43

1.89(0.49α0 ) α1 0 1.49 α1 n 1.79(1.20α1 0 ) β0 2.16

4.40(0.86α0 ) 2.63

3.26(0.81α0 ) 2.3

3.33(0.86α0 ) 2.3

3.71(0.81α0 ) 2.57

4.5(0.76α0 ) 3.68

6.98(0.84α0 ) 5.49

13.14 (0.91α0 ) 9.73

1.68(0.64α1 0 ) 0.26

1.9(0.82α1 0 ) 0.82

1.93(0.84α1 0 ) 0.63

1.89 (0.74α1 0 ) 0.48

2.78(0.76α1 0 ) 0.2

4.2(0.77α1 0 ) 0.10

9.06 (0.93α1 0 ) 0.01

βn

0.23(0.90β0 )

0.8(0.98β0 )

0.62 (0.99β0 )

0.47 (0.99β0 )

0.2(1.0β0 )

0.10(1.0β0 )

0.01(1.0β0 )

c0 cn α0 αn

1.86(0.86β0 )

Fig. 3. Geographic distribution of phase speed c (m s 1) in January and July.

164

G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

for ISWs in the northern SCS, and most of them are found to propagate westward from the Luzon Strait to the Dongsha Islands or the continental shelf of China based on the analysis of more than 500 synthetic aperture radar (SAR) and optical satellite images (Wang et al., 2012). On the other hand, the in-situ observed ISWs travel in directions rotated clockwise from 1951 to 3451 in the northeast of the Dongsha Islands (Liu et al., 2004), while from the Luzon Strait to northeast of the Dongsha Islands, the ISWs travel in directions of 282–2881 (Klymak et al., 2006). So it is reasonable to merely consider zonal components of velocities as background current U(z) in the following calculation. As an example, we select the velocity ﬁeld at the 200 m level from the diagnostic model for January and July (see Fig. 2a and b). The circulation pattern from the diagnostic model is basically consistent with previous investigation. Firstly, it is clearly shown that the strong Kuroshio currents pass through the Luzon Strait and ﬂows northward. A branch of Kuroshio intrudes into the northeast of the SCS in January, while there are no distinct intrusions in August. In January, a larger cyclonic circulation with two cores is present in the northwest of Luzon Island (Fig. 2a ), and the core centers are located at (181N, 1191E) and (18.51N, 116.751E), respectively. In August, the current is weaker than that in January, and multi-eddy structures are identiﬁed in the north of the SCS. Fig. 2(c) shows current proﬁle of eastern component at the

selected points (labeled by symbol star ★ in Fig. 1), the main difference between January and August occurs at upper 1000 m depth. Based on buoyancy frequency proﬁles and background velocity ﬁeld, the Thompson–Haskell method (Fliegel and Hunkins, 1975; Shi and Fan, 2009) is used to solve the boundary value problems (3) and (4), and to calculate the internal-wave vertical mode function Фn(z) and the corresponding long-wave phase speed cn. The superposition of all modes constitutes the complete solution to boundary value problem, but the ﬁrst several terms are the most important; for simpliﬁcation, the ﬁrst three modes will be considered in this paper. After obtaining the phase speed cn, the boundary value problems (11) and (12) can be solved numerically. Finally, the coefﬁcients can be calculated from formulae (7)–(10). For the preliminary analysis, the kinematic parameters of ISWs at the selected points are listed in Tables 1 and 2 for January and August, respectively. In order to make comparison, the tables contain the parameters calculated without (superscript “0”) and with the current. As it can be seen, the variations of the phase speed and dispersions parameters are not sensitive to background currents, and the presence of the current yields differences around 5%. While, the variations of the nonlinear parameters caused by the background current are distinct, for example, the values of α vary by more than 50% at points Point B, and even vary by more

Fig. 4. Geographic distribution of dispersion parameter β ( 103 m6 s 1) in January and July.

G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

than a factor of two at points Point H (see Table 1 for January). Similar variations are also present in August (Table 2).

4. Analysis and discussion 4.1. Kinematic characteristics of the internal solitary waves in the NSCS The coefﬁcients of the GEKdV, phase speed (c), normalizing factor (Q), dispersive (β) and both nonlinear (α,α1 ) parameters, are obtained as functions of position. Their seasonal variations (January and July) will also be discussed in the section. As Figs. 3 and 4 show, the distribution of linear parameters c and β in the NSCS mainly has a close relationship with the water depth, whereas its seasonal variation is insigniﬁcant; this is especially obvious for the dispersion parameter (β). The phase speed (c) increases as the water depth increases and its contours basically follow the isobaths in both seasons, and the pattern of c is very similar with the results from Cai et al. (2008), where background current are not considered. According to two layer model (see formula (14)), the depth of thermocline (h1) and water depth (H) play key role. By contrast, variation of h1 is less than that of H in the NSCS, so variation in phase speed c mainly depend on water depth

165

(topography). The smallest phase speed is less than 2 ms 1, which is found in depths shallower than 1500 m. According to results from Asian Seas International Acoustics Experiments (ASIAEX), the typical propagation speed in the shelf decreased from 1.8 ms 1 to 0.72 ms 1 with the water depth from 800 m to 70 m (Liu and Zhao, 2008). The largest phase speeds are greater than 3 ms 1 and are found in the deep-water sea basin region. The propagation speed of ISWs in the deep-water area of the NSCS amounts to 2.85 ms 1 or so as revealed in ﬁeld observations (Klymak et al., 2006). The spatial distribution of the dispersion parameter β is mainly related to topographic features, as its isoline distribution trend is consistent with that of topographic depth (Fig. 1). In the deep water area β is largest, with maximum value more than 2 106 m3 s 1. As water depth decreases, β gradually decreases to zero in the coastal sea area where the breaking of ISWs frequently occurs. The variation of β is relatively large at the shelf break, as can be seen from the thick isoline of β. In short, the linear parameters c andβ appear to be mostly determined by the bathymetry features in the NSCS. However the nonlinear parameters (α,α1 ) are sensitive to the ﬁne structure of the density stratiﬁcation and background current (Figs. 5 and 6), with signiﬁcant seasonal variation. Sign change may arise from season to season for the nonlinear coefﬁcient. When the sign of the

Fig. 5. Geographic distribution of quadratic nonlinear parameter α ( 10 3 s 1) in January and July.

166

G. Liao et al. / Deep-Sea Research I 94 (2014) 159–172

Fig. 6. Geographic distribution of cubic nonlinear parameter α1 ( 10 4 m 1 s 1) in January and July.

quadratic nonlinear parameter α changes from negative to positive, the nonlinear internal wave will deform from a depression into an elevation, i.e. polarity conversion occurs. The position of the depth for the maximum buoyancy frequency plays an important role in the sign change of the quadratic nonlinear coefﬁcient according to two layer model. In January, α is mainly negative in the deeper 500 m area and the maximum value of about 0.01 s 1 presents at the continental slope. In July, the distribution of α is somewhat similar to that in January, except holding stronger nonlinearity with a larger negative maximum at continental slope. Along 100 m isobaths on the continent shelf, α change its sign from negative to positive, and this can lead to polarity conversation of ISWs. Depression and elevation ISWs can be recognized in SAR images by different signatures, i.e. elevation ISWs have signatures in alternating dark stripes leading bright stripes while depression ISWs wave signatures in alternating bright stripes leading dark strips. Based on ERS-1 SAR images, Liu et al. (1998) has shown that the depression waves may be converted to elevation waves as they propagate from deep water onto a shallow shelf. The polarity conversions on the southern continental shelf are also conﬁrmed by shipboard acoustic ﬂow visualization data in the ASLAEX (Orr and Mignerey, 2003). Fig. 5 also overlay the locations of polarity conversions inquired from a

collection of about 300 oceanic internal wave SAR images including ERS-1/2 SAR, ENVISAT ASAR and RADARSAT-1 images from 1998 to 2008, respectively. It can be seen that the elevation ISWs mainly occur in the areas between 1141E and 1151E longitude and 201 and 221N latitude, where the depths are shoaling from over 200 m to nearly 50 m. Obviously it is basically consistent with the positive α area, especially for winter season. When α is zero, the high-order nonlinear terms, i.e. cubic nonlinear terms, should be taken into account in order to balance the dispersion term, which also primarily depends on variations of density stratiﬁcation with distinct seasonal variation (Fig. 6). In deep sea, α1 changes little with a value of about 1.0 10 4 m 1 s 1 in January; the isolines are crowed with a steeper gradient near the continental shelf, where α1 also may change its sign from positive to negative. In July, the values of α1 is larger than that in January, while the pattern is somewhat similar to each other. The variation of nonlinear parameters further depends on the stratiﬁcation and background currents features in the shallow continental area.. Generally speaking, values of the nonlinear parameters (α,α1 ) are larger in July than in January, and larger quadratic nonlinear parameter values mainly occur in the shelf break with water depth from 100 m to 1000 m. Its maximal value is about 1.6 10 2 s 1 in July. The calculated quadratic nonlinear coefﬁcient in the deep-water area is

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Fig. 7. Geographic distribution of normalizing factor Q in January and July.

about 0.008–0.012 s 1 (Fig. 6), so the amplitude of the ISWs can be estimated to be 75–225 m; at present the amplitude of the observed ISWs in the deep-water area of the NSCS is 170 m (Klymak et al., 2006). The nonlinear parameters take on larger values in summer. This background environment is favorable for an internal wave to become steeper and develop into a large-amplitude nonlinear internal wave. This point can be used to explain why ISWs occur more frequently in summer (Zheng et al., 2007; Buijsman et al., 2010b). The gradient of nonlinear parameters is larger at the shelf break, the phenomena is similar for the dispersion parameter. Fig. 7 gives the geographic distribution of the normalizing factor Q and algebraic soliton amplitudes. Obviously, the spatial distribution of Q is related to both the topographic variation and the density stratiﬁcation, and also with seasonal variation. Note that the negative cubic nonlinearity parameters are only present in the small shallow area (see Fig. 6). So “top-table” solitons should be rarely observed in the NSCS. In other words, the “algebraic soliton” dominates. Fig. 8 illustrates the geographical distribution of minimum algebraic soliton amplitudes for points with positive cubic nonlinearity parameters. The range for the minimum amplitudes of the algebraic soliton is between 10 and 90 m. Among these solitons, the larger ones exist in the area shallower than 1000 m water depth.

4.2. Near-bottom velocity and maximum velocity induced by ISWs Internal waves frequently occur in the NSCS and thus play an important role in driving sediment re-suspension and erosion processes. This section analyzes the near-bottom processes induced by ISWs. With the use of Eq. (2), the horizontal velocity (u) of ﬂuid particles along the propagation direction of ISWs can be expressed as follows: uðx; z; tÞ ¼ cηðx; tÞ

dΦ α dΦ dT þð þ c Þη2 ðx; tÞ; dz 2 dz dz

ð26Þ

The ﬁrst term in Eq. (26) correspond to the leading order of the asymptotic expansion, and the remaining terms reﬂect the nonlinear correction. It should be mentioned that one has to specify the isopycnal displacement ηðx; tÞ, the vertical mode function Ф(z), and its nonlinear correction function T(z). It is difﬁcult to obtain ηðx; tÞ, but it can be inferred from analysis of many SAR images or by a detailed numerical simulation. Here we follow Cai et al. (2014) and the ISW amplitude is prescribed based on previous observation, that is, setting ISW amplitude of 150 m in the deep sea and an amplitude of 90 m on the continental shelf/slope (note that when the water depth H is less than 90 m, the ISW amplitude is set η ¼ H 10). Here we will discuss the near-bottom horizontal

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Fig. 8. Geographic distribution of algebraic solitons amplitudes (m) in January and July.

velocities and maximum horizontal velocities induced by ISWs. As statement in Section 3, we set the horizontal coordinate x as eastwest direction, which is basically consistent with the propagation direction of ISWs in the NSCS. So the calculated horizontal velocities u takes positive values when it directed eastward. The geographical features of the near-bottom horizontal velocities (u) are shown in Fig. 9. The largest values of near-bottom horizontal velocities can be expected along shallow areas with the tendency to increase in the warm season. Fig. 9 shows that the near-bottom horizontal velocities with large values around 0.7 m s 1 are mainly present in the trap between 100 m and 500 m in January. In July, near-bottom horizontal velocities increase, dominate the area shallower than 1000 m depth, and have isolines approximately parallel to the isobaths. An interesting feature is that the near bottom horizontal velocities are mostly negative (westward), i. e., the near bottom currents induced by ISWs nearly head perpendicularly to the continent shelf. Fig. 10 gives the distribution of the maximum horizontal velocities induced by ISWs, and they nearly direct eastward. The larger values of maximum horizontal velocities occur in the continent shelf and slope, especially between 100 m and 1000 m water depth, where the largest horizontal velocities is over 1 m s 1 in January, and even reaches 2 m s 1 in July. Fig. 10 also

shows the maximum horizontal velocity in July is larger than that in January. 4.3. Mechanisms affecting soliton evolution Three parameters govern steepening and breaking of ISWs (Helfrich and Grimshaw 2008; Buijsman et al. 2010b): the nonlinearity parameters, the non-hydrostatic dispersion parameter, and the Coriolis dispersion parameter. Nonlinearity alone will result in steeping and breaking of the ISWs. The non-hydrostatic dispersion alone will lead to the ISWs to disperse and break up into its Fourier components. The combined effects of both terms sustain a solitary wave. Similar to non-hydrostatic dispersion, Coriolis dispersion may also prevent an internal wave from disintegrating into solitons. However, Coriolis dispersion acts on larger length scales, whereas non-hydrostatic dispersion acts on shorter length scales. For typical NSCS conditions, the analysis by Buijsman et al. (2010b) shows Coriolis dispersion is more important than non-hydrostatic dispersion for low frequency internal waves. But when strong nonlinear effects make lower frequency and longer length internal waves to steep (for example, an internal tide: internal wave at tidal frequency), they eventually evolve into ISWs with higher frequencies higher and smaller length scales.

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Fig. 9. Geographic distribution of near bottom horizontal velocity induced by ISWs in January and July.

Then the non-hydrostatic dispersion becomes much more important, eventually, resulting in a balance between the nonlinear and the non-hydrostatic dispersion. In the remainder of this section, the zonal distribution of kinematic parameters controlling formation and development of the ISWs is assessed. Plots of nonlinearity and dispersion coefﬁcients for the GEKdV equation are shown in Fig. 11 as a function of longitude. The lines with squares are averages of values along the meridional direction and indicate results from the Continuous Stratiﬁcation Model (CSM). The lines with circles are results from the Two-Layer Model (TLM). In order to compare the parameter plots, Fig. 9d shows the zonal distribution of the water depths averaged along the meridional direction. It can be seen that the nonlinearity parameters are very similar in the NSCS and the Paciﬁc Ocean (PO) (Fig. 11a). The values are approximately 5 10 3 in January, and they increase slightly around 1171E and west of 1141E, which shows more variation due to the temporal and spatial variability of the density stratiﬁcation in the shallower area. The nonlinearity parameters are larger in July than in January. The nonlinearity coefﬁcients calculated from the two-layer model are overestimated when compared with the continuous stratiﬁcation model. It is obvious that the twolayer model overvalues the nonlinearity coefﬁcients; especially they are two times larger in January. The dispersion parameter is about three times larger in the PO than that in the NSCS (Fig. 11b).

For the dispersion parameter, the results from the two-layer model are consistent with the continuous stratiﬁcation model in the shallow area, i.e. east of 1181 E. However, in the deep area the twolayer model also takes on larger values than the continuous stratiﬁcation model. Fig. 11d gives the ratio of the dispersion parameter to nonlinear parameter (β=α), which shows larger values in the PO. The formation of large-amplitude internal waves by a tidal ﬂow is normally considered to involve the following three stages. First, tide–topography interactions generate internal tides, and the waves are considerably ampliﬁed at the “critical slope” condition, i.e. the characteristic slope of internal tide is equal to that of the topography (Baines 1982). The second stage is the ampliﬁcation of internal tides at their generation regions. This process is thought to be the most important and was investigated by Hibiya (1986); the dimensionless Froude number, deﬁned as the ratio of the barotropic ﬂow speed to phase velocity of the nth mode cn, plays a key role in this stage. The third stage represents the nonlinear evolution of the large-amplitude internal waves as they propagate away from the generation region or alternatively when the wave amplitude is sufﬁciently large, internal tides grow and disintegrate into large-amplitude internal solitary wave trains (Gerkema and Zimmerman, 1995). Eventually, the balance between nonlinearity and dispersion results in the propagation of a rank-ordered packet of internal solitary waves. As known from the distribution of the

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Fig. 10. Geographic distribution of maximum horizontal velocity induced by ISWs in January and July.

nonlinearity and dispersion parameters above (Fig. 11), the dispersion parameter is larger on the west than on the east side of the LS, whereas the nonlinearity parameters have nearly the same values on both sides. In consequence, westward propagating solitons are larger and more distinguished. This is why ISWs are easily observed and occur more frequently on the west of the LS (NSCS), but they are difﬁcult to observe on the east side of the LS (PO). In addition, the nonlinear parameter in summer (July) is stronger than in winter (January), whereas the dispersion parameter has the same value in both seasons. Thus, internal tides more easily transform to ISWs under such conditions.

5. Summary and conclusions Based on the climatological temperature and salinity proﬁles from the Generalized Digital Environment Model (GDEM-Version 3.0), current velocities were computed using a three dimensional ocean current diagnostic model. Then the kinematic parameters for the propagation and evolution of ISWs were evaluated using the GeKdV equation with background current. The geographic distribution and seasonal variations of these parameters were investigated; the main conclusions are summarized as follows:

The linear phase speed and dispersion parameter show little variation with season, depends primarily on the topographic features of the Northern South China Sea, and is weakly related to the density stratiﬁcation. Their spatial distribution is approximately consistent with that of topographic isobaths in the NSCS. The nonlinear parameters and normalizing factor rely not only on water depth but also on density stratiﬁcation, which possesses a seasonal characteristic. Especially in the shallow area, these parameters further depend on the local marine environmental characteristics. The nonlinear parameters hold larger values in summer (July) than in winter (January), and such circumstances are favorable for internal waves (such as internal tides) to steepen and develop into large-amplitude internal solitary waves. This point also can be used to explain why internal solitary waves are observed more frequently in summer. The important existence condition for a stable solitary wave is the balance between nonlinear term and dispersion term. Comparing the quadratic nonlinear and dispersion parameters between the NSCS and the PO, the quadratic nonlinear parameters hold almost the same values, whereas the dispersion parameters in the PO are signiﬁcantly greater than in the NSCS. To date, internal solitary waves are mostly observed on the west side of the Luzon Strait (NSCS), and are rarely found on the east side of the Luzon Strait (PO). The results from this work explain that the

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41376033, 41476022); the scientiﬁc research fund of the Second Institute of Oceanography, State Oceanic Administration (Grant nos. JG1302 and JG1206); the project of global change and interaction between ocean and atmosphere, and Bureau of Ocean Energy Management (M14AC00021). The authors thank Greg King for proofreading the manuscript. All authors appreciate three anonymous reviewers for their comments to help improve the paper.

References

Fig. 11. Zonal distribution of the nonlinear coefﬁcient (a), dispersion coefﬁcient (b) and (c) ratio of dispersion to nonlinear coefﬁcient in January and July, respectively. All values are averaged in intervals of 0.251 along the meridional direction. Red lines are from the Continue Stratiﬁcation Model (CSM), and green lines are results from the Two-Layer Model (TLM). (d) Zonal distribution of the water depths averaged along the meridional direction. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

stronger dispersion effect in the PO due to deeper water depth forms a hindrance, so that the internal tides hardly develop into a larger amplitude internal solitary wave in the PO. The nonlinearity coefﬁcients calculated from the two-layer model are overestimated when compared with the continuous stratiﬁcation model, whereas the dispersion parameters from the two-layer model and the continuous stratiﬁcation model are nearly consistent, aside from the exception in deep water. The largest values of near-bottom velocities induced by internal solitary waves mainly dominate the shallow area with the tendency to increase in the warm season as expected, especially the near bottom currents induced by ISWs nearly head perpendicularly to the continent shelf. This will be important to investigate further due to the impact internal waves have on sediment resuspension and erosion processes. We note that this study has not taken into account the frictional dissipation effect, nor the dispersion effect due to earth rotation. Holloway et al. (1999) demonstrated that the frictional dissipation term attenuates the amplitude of an internal wave and reduces the number of solitons formed; earth rotation will also affect and decrease the number of soliton waves generated. The effects of the above factors will be taken into account in the future study on the evolution of internal waves in the northern South China Sea.

Acknowledgment This study was supported by National Basic Research Program of China (Grant nos. 2011CB403503, 2014CB441501, 2012CB417303), the National Natural Science Foundation of China (Grant nos.

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