Monthly variation on the propagation and evolution of internal solitary waves in the northern South China Sea

Continental Shelf Research 171 (2018) 21–29

Contents lists available at ScienceDirect

Continental Shelf Research journal homepage: www.elsevier.com/locate/csr

Monthly variation on the propagation and evolution of internal solitary waves in the northern South China Sea

T

⁎

Shanwu Zhang , Fuwen Qiu, Junpeng Zhang, Junqiang Shen, Jing Cha Third institute of Oceanography, State Oceanic Administration, Xiamen, China, 361005

A R T I C LE I N FO

A B S T R A C T

Keywords: Internal solitary waves Monthly variation Polarity conversion Environmental parameters Korteweg-de Vries models Northern South China Sea

In this paper, internal solitary waves (ISWs) in the northern South China Sea (NSCS) are investigated with respect to their propagation and evolution processes in different months. To achieve that, environmental parameters associated with ISW are first calculated using climatological datasets derived from the Simple Ocean Data Assimilation 3.3.1 (SODA3.3.1) reanalysis. The environmental parameters are significantly distinct from month to month, especially the quadratic nonlinearity coefficient α. It is negative in the deep basin and turns positive on the shallow continental shelf. A clear dividing line that α equals to zero can be found along the shelf from northeast to southwest from January to February and from October to December. The dividing line is interrupted starting from March and is mostly constrained on the northeast shelf, and from May to September only limited areas can positive α be found. Further simulations based on the variable extended Korteweg-de Vires (veKdV) model indicate that the polarity conversion process of ISWs therefore exhibits significant seasonality. It can be concluded from the simulation results that near the 200-m isobath on the continental shelf the elevation waves are more likely to appear from November to March owing to the positive values of α. The large-scale background current exhibits little effect on the deformation and polarity conversion of ISW in the NSCS although it prominently affects the environmental parameters around the shelf currents from Luzon strait.

1. Introduction For over a decade, a considerable number of in situ observations and numerical simulations have been conducted to study internal solitary waves (ISWs) in the northern South China Sea (NSCS) (Ramp et al., 2004, 2010; Liu et al., 2004; Zhang et al., 2011; Simmons et al., 2011; Cai et al., 2012; Guo and Chen, 2014; Alford and Co-authors, 2015). The ISWs have been found among the world's most powerful ones with amplitude over 200 m (Huang et al., 2016). These waves are believed originating from the interactions between barotropic tide and rapidly changed topography in the Luzon strait (Zhao and Alford, 2006; Buijsman et al., 2010). It has been argued that the waves begin as baroclinic internal tides and propagate into the basin of the NSCS (Alford and Co-authors, 2015). During their propagation to the west, the original internal tides steepen and disintegrate into packets of ISWs (Grimshaw et al., 2014). Though the formation and fate of internal waves in the NSCS have been well illustrated, the long-term variability of ISWs are not well known. The year-round observation has resolved the seasonal variability of the nonlinear internal wave field in the NSCS, and it has been found that there are distinct features of the properties of ISWs in different seasons (Ramp et al., 2010). Using remote sensing

⁎

images, the monthly statistics of the wave occurrence frequency has been reported (Zheng et al., 2008). Long-term mooring efforts have also been carried out, which make it possible to quantify the variability of ISWs on seasonal and annual time scales (Huang et al., 2016, 2017). However, the in-situ observations and remote sensing techniques are quite limited by their spatial and temporal coverage which has been found important in the detective of ISWs under various oceanic background fields, such as the Kuroshio intrusion in the NSCS (Park and Farmer, 2013; Alford and Co-authors, 2015; Li et al., 2016). ISWs in the NSCS have been heavily studied with respect to their propagation and evolution processes both in theoretical and numerical models (Grimshaw et al., 2010, 2014; Li and Farmer, 2011; Zhang et al., 2011). These models introduce the effects of weakly/fully nonlinearity, weakly/fully nonhydrostatic and 2-D/3-D configuration (Simmons et al., 2011). Among them, the canonical Korteweg-de Vries (KdV)-type models are the most commonly used ones. The KdV-type models capture the basic dynamic balance of ISW between weakly nonlinearity and nonhydrostatic dispersion (Lee and Beardsley, 1974). They are modified by some authors to incorporate the effects of variable depth, continuous stratification and background shear flow (Liu et al., 1985; Grimshaw et al., 2004), and are applied to simulate the propagation

Corresponding author. E-mail address: [email protected] (S. Zhang).

https://doi.org/10.1016/j.csr.2018.10.014 Received 2 April 2018; Received in revised form 22 October 2018; Accepted 24 October 2018 Available online 27 October 2018 0278-4343/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

geographic distributions of the environmental parameters, which is of great value to the climatological study of ISWs in the NSCS (Cai et al., 2014). Our purpose is to clearly illustrate the monthly or seasonal characteristics of the environmental parameters in detail in the NSCS, especially in the continental slope and shelf regions where ISWs suffer drastic deformation of their wave profiles. Our contribution in this study is to further investigate the monthly variation of propagation and evolution processes of the ISWs when given the spatial and temporal varying environmental parameters along their propagation path. Much attention is paid on the discussion of deformation and polarity conversion of ISWs. This work, to some extent, is missed in previous studies and deserved more discussions. The rest of this manuscript is arranged as follows. In Section 2, the data and methods are briefly described. The main results are presented in Section 3, including the discussion. We summarize the whole paper and give the main conclusions in Section 4.

and evolution of ISWs in oceanographic practice (Liu et al., 1998, 2017; Holloway et al., 1997, 1999; Cai and Xie, 2010; Grimshaw et al., 2010, 2014, 2016; Li et al., 2015). These modifications are called variablecoefficient KdV (vKdV)-type models, in which the coefficients are variable since the ocean environments are changing accompanied with the forward propagation of ISWs. The derivation of vKdV model requires that the oceanic background field should vary on a length scale which is greater than that of the solitary waves, but is comparable to the length scale over which the wave field evolves. The modal function then depends parametrically on the spatially varied coordinate, and so do the associated coefficients (Grimshaw et al., 2010). The vKdV model itself assumes weak nonlinearity and weak dispersion while it is used to simulate ISWs with amplitudes well beyond the weakly nonlinear limitation (Helfrich and Melville, 2006). For many coastal areas where the coefficient of nonlinear term in vKdV model is small, the high-order nonlinear terms should be considered (Grimshaw et al., 2004). Then, another modification called variable-coefficient extended KdV (veKdV) model is derived when the quadratic nonlinearity coefficients of the vKdV-type models approach zero near the turning depth, i.e. a position where the thickness of the upper layer is equal to that of the lower layer in a two-layer fluid. The veKdV model exists different branches of solutions depending on the sign of quadratic and cubic nonlinearity coefficients (Grimshaw et al., 2004). An ISW converts its polarity from depression to elevation if the quadratic nonlinearity coefficient changes its sign from negative to positive (Liu et al., 1998; Grimshaw et al., 2016). Polarity conversion of the ISW has been verified by previous observations on the continental shelf regions in the NSCS (Orr and Mignerey, 2003; Fu et al., 2012). Recent observation using seismic techniques by Bai et al. (2017) has suggested that depression ISWs in the NSCS begin to convert their polarities at the depth of about 200 m and turn into elevation waves near the depth of 100 m. To determine the conversion depth, however, is really complicated since it depends not only on the oceanic environment but also on the amplitude of the ISW (Bai et al., 2017). By checking the quadratic nonlinearity coefficient of the veKdV model, Liao et al. (2014) has suggested a way to further investigate the seasonal variation of the conversion depth. In their contribution, however, this issue has not been discussed in detail. Therefore, our knowledge regarding the seasonal variation of the polarity conversion depth is not completed yet. In general form, the veKdV model can be written as:

∂3η ∂η ∂η ∂η ∂η c dQ = 0. +β 3 + + α1 η2 + αη +c 2Q dx ∂x ∂x ∂x ∂x ∂t

2. Data and methods 2.1. Datasets The monthly climatological dataset employed in this study is extracted from the SODA3.3.1 reanalysis which provides monthly data of temperature, salinity and current profiles from 1980 to 2015, mapped onto the regular 1/2 × 1/2° Mercator horizontal grid with 50 vertical levels (Carton et al., 2018). The 36-year monthly data are averaged to get the monthly climatological data. The derived monthly data provides 3-D stratification and background current fields for the ISW parametrical study. In addition to the temperature, salinity and current dataset, the bathymetry data is needed to calculate the parameters of ISWs. Here, the gridded bathymetry data is extracted from the NOAA/National Centers for Earth Information 5′ grid Earth topography (ETOPO5) dataset. It is further matched with the former monthly climatological data to ensure the accuracy of the depths at the temperature and salinity grid points. Thus, a full set of temperature, salinity, background current and depth data with a resolution of 0.5° is constructed and prepared for the following calculations. 2.2. Theoretical model The general form of the veKdV model is presented by Eq. (1) where the coefficients are variable and needed to be solved prior to the simulation. First, the linear wave phase speed c is determined from the eigenvalue problem

(1)

where η (x, t) represents the vertical displacement of the pycnocline; x is the horizontal coordinate; t is time; c is the linear wave speed, and the parameters α, α1, β and Q are the quadratic and cubic nonlinearity coefficients, dispersion coefficient and amplification factor, respectively (Grimshaw et al., 2004). These parameters are known as “environmental parameters” as they incorporate the environmental conditions such as bottom topography, continuous stratification and background flow (Liu et al., 1985; Yang et al., 2009; Cai et al., 2014; Chen et al., 2014; Liao et al., 2014). They have been calculated globally (Grimshaw et al., 2007), in the South China Sea (Grimshaw et al., 2010; Liao et al., 2014) and recently in the East China Sea (Cho et al., 2016). Seasonal distribution of the environmental parameters has been given using stratification from monthly climatology in the South China Sea (Cai et al., 2014; Liao et al., 2014; Kurkina et al., 2017). To our knowledge, however, the corresponding deformation of ISWs on monthly or seasonal scale has not been well investigated. In this paper, a monthly climatological dataset of stratification and background current derived from the Simple Ocean Data Assimilation 3.3.1 (SODA3.3.1) reanalysis is used to evaluate the environmental parameters of ISWs. Unlike the spatial and temporal limitations of in situ observations, the climatological dataset is available to give

dϕ d ⎡ (c − U )2 ⎤ + N 2ϕ = 0 dz ⎦ dz ⎣

(2)

with homogenous boundary conditions

ϕ (0) = ϕ (−H ) = 0.

(3)

Here, ϕ is the modal structure function of the vertical displacement, N is the buoyancy frequency, U is the background shear current and H is the undisturbed water depth. Noting that the solution of the boundary problem (2) and (3) is not unique, it is further assumed that the modal function ϕ is normalized so that ϕmax = 1. The quadratic nonlinearity coefficient α and dispersion parameter β in (1) are then given by 0

α=

3 ∫−H (c − U )2ϕz3 dz (4)

I

and 0

β=

∫−H (c − U )2ϕ2dz I

respectively, where 22

,

(5)

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

I=2

0

∫−H (c − U ) ϕz2 dz.

continuous at the boundaries of the upper and lower layers. This procedure continues throughout the water column, and finally an equation constrained by the zero boundary conditions is formed. By solving the equation using iteration method the linear wave speed c is obtained. The corresponding modal function is then derived, and the coefficients of the veKdV model are obtained by the integrals given above. The propagation and evolution processes of ISWs are simulated by solving Eq. (13) with an initial condition given at a fixed point. The “initial condition” here is a single solitary wave with KdV soliton shape or wave packets consisted of KdV solitons with different amplitudes. We use a two-level, five-point Crank-Nicholson scheme to differentiate Eq. (13):

(6)

The cubic nonlinearity coefficient α1 is given by

α1 =

1 I

0

∫−H [3(c − U )2 (3Tz − 2ϕz2) ϕz2 − α2ϕz2 + αc (5ϕz2 − 4Tz ) ϕz] dz, (7)

where T is the nonlinear correction to the modal function ϕ and is obtained by solving the eigenvalue problem

dϕ ⎤ d ⎡ dT d (c − U )2 ⎤ + N 2T = −α ⎡ (c − U ) dz ⎦ dz ⎣ dz ⎦ dz ⎣ +

dϕ 2 3 d ⎡ (c − U )2 ⎛ ⎞ ⎤ ⎢ 2 dz ⎣ ⎝ dz ⎠ ⎥ ⎦

ζi j + 1 − ζi j

(8)

Δx

with zero boundary conditions that T(-H) = T(0) = 0. The solution of the inhomogeneous eigenvalue problem (8) with homogeneous boundary conditions is also not unique and has the form of Tp + pϕ where Tp is a particular solution of (8), and p is an arbitrary constant. The normalization of T follows the step that proposed by Grimshaw et al. (2004) in which the value of p is specified so that T(zmax) = 0, where zmax is found from ϕ(zmax) = 1. The additional term Q in Eq. (1) represents the amplification factor of wave amplitude and is given by

Q=

0 (c −H 2 0 c0 −H (c0

c2

∫

∫

−

U )(dϕ/ dz )2dz

− U0)(dϕ0 / dz )2dz

−

(11)

Eq. (1) can be reduced to (Holloway et al., 1999; Grimshaw et al., 2004)

β ∂3ζ ∂ζ ∂ζ αQ α Q2 + ⎛ 2 ζ + 1 2 ζ 2⎞ + 4 3 = 0. s c c c ∂s ∂x ∂ ⎝ ⎠ ⎜

⎟

(12)

The dissipation effects on the ISW evolution are also considered in the simulation processes since the ISW breaking and turbulent mixing cannot be ignored. It can be achieved by introducing the eddy viscosity ν and bottom friction coefficient κ in the equation above (Holloway et al., 1997; Liu et al., 1998), and the new model using for the simulation is

β ∂3ζ ∂ζ ∂ζ κcQ ν ∂ 2ζ αQ α Q2 ζ ζ − 3 2 = 0. + ⎛ 2 ζ + 1 2 ζ 2⎞ + 4 3 + s c s β c c c ∂s ∂ ∂x ∂ ⎝ ⎠ ⎜

1 ν Ltt (ζi j + ζi j + 1) = 0 2 c3

(14)

Lt (ζi j ) = (ζi +j 1 − ζi −j 1)/(2Δt ),

(15)

Ltt (ζi j ) = (ζi +j 1 − 2ζi j + ζi −j 1)/(Δt )2 ,

(16)

(17)

respectively. The scheme is semi-implicit and can be proved unconditionally stable (Cai and Xie, 2010). Radiation boundary conditions, ζ1 = ζ2 and ζn-1 = ζn, are applied on both sides. The application of the boundary conditions ensures low reflection of wave energy and reduces false fluctuations which may spread out and contaminate the model results. The eddy viscosity υ and bottom friction coefficient κ are neglected until the waves have propagated for 100 km to near the shelf break (Fig. 1), and set to 1.2 m2/s and 0.0026 respectively after 100 km. The including of eddy viscosity and quadratic bottom friction efficiently smooths the horizontal gradients and damps the nonphysical large amplitude of ISW in shallow water. Eq. (14) together with its boundary conditions constitute linear equations with a pentadiagonal matrix on the left side and can be solved by iteration method.

(10)

− t, x = x,

⎟

Lttt (ζi j ) = (ζi +j 2 − 2ζi +j 1 + 2ζi −j 1 − ζi −j 2)/2(Δt )3,

and coordinate

∫ cdx(x )

⎜

and (9)

The subscript “0” denotes a reference location at a fixed point x0 and can be chosen as the initial location where the ISWs originate. Introducing the change in variable (Holloway et al., 1997)

s=

1 ⎛ αQ 1 β α Q2 Lttt (ζi j + ζi j + 1) + 1 2 ζi j ⎞ Lt (ζi j ζi j + 1) + 2 2⎝ c 2 c4 c ⎠

Here, the operators Lt, Ltt, and Lttt are defined as

.

ζ = η Q (x ) ,

+

⎟

(13)

In the following section, this spatial version of veKdV model with dissipation will be chosen as the basic model to investigate the propagation and evolution of ISWs under different oceanic background stratifications. 2.3. Numerical methods The eigenvalue problems (2) and (8) with their homogenous boundary conditions can be solved by various numerical methods, such as the shooting method (Alford et al., 2010) and the Thomson-Haskell (T-H) method (Fliegel and Hunkins, 1975). It is suitable in this paper to use the T-H method because of the relatively coarse vertical resolution of the climatological datasets. This method assumes constant buoyancy frequencies between adjacent layers of the climatological profiles. Within the adjacent layers the differential equation is solved analytically, and it is assumed that the solution and its first derivative are

Fig. 1. Map of study region of the NSCS (top) and the bathymetry along the cross section (bottom). The red solid line in the top panel indicates the simulation cross section and the green lines show the 200-m and 100-m isobaths. B1 and S7 (black dots) are two stations of the mooring sites of Ramp et al. (2010). 23

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

Fig. 2. Monthly distribution of the first mode linear phase speed (color) on which the background current (quiver) is superimposed. The black (gray) solid line in each panel gives the contour line of 1.5 m/s in the case of with (without) background current. The background current is an averaged result of the SODA3.3.1 reanalysis in the upper 100 m. Shallow water area (depth lower than 50 m) is masked with blank.

Fig. 3. Same as Fig. 2, but for the monthly distribution of the quadratic nonlinearity coefficient α.

and as a result, from May to September only limited areas can the positive α been found. The dividing line becomes continuous again extending from northeast to southwest in October. This distribution feature of α may result in the wave polarity difference as we know that positive α can give rise to an elevation wave while negative α can only support a depression wave. The absolute values of α in most deep areas are in O (10−3 s−1) or lower and becomes larger as the water depth turns shallower. Approaching the turning zone where α vanishes, the absolute values of α can reach to O (10−2 s−1). They quickly decrease to zero in the turning zone and gradually increase in shallow waters. Unlike the quadratic nonlinearity coefficient, there is a clear dividing line in the distribution of the cubic nonlinearity coefficient α1 throughout the year, which is visible in Fig. 4. The dividing line that α1 = 0, however, is in the water deeper than that of α = 0. In the deep water, α1 is almost positive and smaller than O (10−4 m−1 s−1). In the shallow water, however, the sign of α1 is hard to determine. As shown in Fig. 4, α1 is mostly negative but sometimes may have positive values in the continental shelf areas. Owing to the much smaller magnitude of α1, the cubic nonlinearity may have little effect on the ISW evolution unless the quadratic nonlinearity vanishes or the wave amplitude is

3. Results and discussion 3.1. Environmental parameters distribution The bathymetry of NSCS and the cross section used in this paper are shown in Fig. 1. The initial ISW is given at station B1 and assumed to propagate northwestward along the cross section which is about 288° due north consistent with the propagation directions (282–288°) observed by Ramp et al. (2010). The shallow end of the cross section is located on the continental shelf between the water depth of 100–200 m. When crossing the 200-m isobath the shoaling ISW is expected having a drastic deformation (Bai et al., 2017), which will be simulated in the following section. Given the bathymetry data together with the stratification and current profiles, the environmental parameters associated with ISW are computed. As done by Liao et al. (2014), we choose zonal component of the SODA3.3.1 velocity output as the background current in the following calculation due to the small direction difference from the propagation path. Fig. 2 shows the monthly distribution of the linear wave phase speed c. Clearly, the spatial distribution of c roughly follows that of the bathymetry. It decreases from 3 m/s in the deep basin to lower than 0.5 m/s on the shelf. The linear phase speed has larger values in summer and autumn (redder colors in c from June to October in Fig. 2), which may be attributed to the stronger stratification in the two seasons. The monthly background current extracted from SODA3.3.1 product is averaged in the upper 100 m and overlaid on the fill plots in Fig. 2. The significant characteristic captured by the products is the Kuroshio intrusion in winter and the persistent along continental slope currents. The background current does affect the wave speed; As seen from the contour line of c = 1.5 m/s, there is a slight increase in c along the continental slope where the background current has a weak westward component (lower than 0.1 m/s). Monthly distribution of the quadratic nonlinearity coefficient α is shown in Fig. 3. The coefficient is negative in the deep basin throughout the year, while it turns to positive on the continental shelf. A clear dividing line along the whole shelf from northeast to southwest can be found from January to February and from October to December. The dividing line is interrupted starting from March and is mostly constrained on the northeast shelf. It is further interrupted and suppressed,

Fig. 4. Same as Fig. 2, but for the monthly distribution of the cubic nonlinearity coefficient t α1. 24

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

Fig. 6. Averaged depth of (a) c = 1.5 m/s, (b) α = 0, and (c) α1 = 0. Both cases of including (with) and excluding (without) background current (BC) are shown.

Fig. 5. Distribution of the difference (values including background current minus those excluding background current) of environmental parameters in the NSCS: (first column) linear phase speed, (second column) quadratic and (third column) cubic nonlinearity coefficients.

difference can arise from the sign change of α1 in shallow water in winter months (Fig. 4).

large enough. Since we perform the simulation of the ISW propagating from deep basin to continental shelf region in which the quadratic nonlinearity coefficient may change its sign and the wave may experience drastic amplification in amplitude, the cubic nonlinearity could not be excluded from the model. Another parameter in model (12) is the dispersion coefficient β. The magnitude of β experiences a multi-order change decreasing from O (106 m3/s) in deep water to O (102 m3/s) in shallow water. The distribution of β shows similar feature with that of the phase speed c as revealed by previous studies (Grimshaw et al., 2010; Liao et al., 2014). It is also mainly determined by the water depth, and there is no significant seasonal variation. Thus, it will be not shown in this paper again. To further discuss the effect of large-scale background current on the environmental parameters, we calculate the monthly difference of each parameter in the case of including (excluding) background current and the results are shown in Fig. 5. Distinct variation of phase speed and quadratic nonlinearity coefficient can be found near the Kuroshio intrusion region and along the continental slope. A westward shift of 0.1 m/s of phase speed is apparent and accordingly evident change in the modal function can be expected. As a result, the quadratic nonlinearity coefficient has larger absolute value (second column in Fig. 5). Another feature is the relatively weak variation in July compared with other three months, which can be attributed to the weak along-slope current during the summer monsoon season. In additional, the quadratic and cubic nonlinearity coefficients have large difference in the shallow water indicating the changes of sign of both parameters on the continental shelf when considered the effect of background current. Fig. 6 shows the averaged depth where the linear phase speed equals to 1.5 m/s, quadratic nonlinearity turns to zero and cubic nonlinearity coefficient turns to zero, respectively. Both cases of including and excluding background current are presented. When considered the large-scale background current, there is an obvious shoreward shift for the depth of wave speed contour line due to the westward component of the current. However, no significant change can be found in the depth difference of α = 0. The depth referred to as the turning depth for shoaling ISW has its maximum in winter months (~300 m) and minimum in summer months (~100 m), and the difference is not prominent when considered the effect of current. The background current has more prominent effect on the cubic nonlinearity coefficient. The depth difference can be 100 m in winter as shown in Fig. 6c. The

3.2. Simulation results To simulate the wave evolution across the basin to the continental slope and shelf regions, a typical cross section in the NSCS is selected (Fig. 1). The cross section is roughly along the propagation path of ISW as suggested by previous numerous studies in this area (Klymak et al., 2006; Ramp et al., 2010). It is nearly 160 km long, and the water depth changes from 2000 m in the basin to 400 m on the continental slope. Further onto the shelf, the cross section is approaching depth of 200 m and cross the 200-m isobath to about 150 m of the end station (Fig. 1b). The buoyancy frequency and current profiles representing the monthly stratification and background shear flow along the cross section are calculated by the SODA3.3.1 monthly climatology, which is shown in Fig. 7. The depth of the maximum buoyancy frequency is deeper in winter months (December, January and February) and turns shallower in summer (June, July and August) indicating significant monthly variation of pycnocline depth. The pycnocline depth is located at the depth approximately 100 m in winter while it is located at the depth ranging from 50 m to 100 m in summer months. Spring (March, April and May) and autumn (September, October and November) are the two transitional seasons with relatively weaker strength and shallower depth of the pycnocline in the former. The pycnocline associated with the maximum buoyancy frequency separates the water column into roughly two layers. In summer months, the upper layer is obviously shallower than that in winter months, which may lead to significant changes in the environmental parameters closely related with the variation of the water stratification. The zonal component of the background current is persistently westward at the deep stations throughout the year while it is eastward in the upper layers for the shallow water stations with depth lower than 300 m from March to September. The magnitude of background current decreases from 0.2 m/s at deep stations to lower than 0.1 m/s at shallow stations. Each profile shows significant current shear in the vertical and the shear effect as well as the stratification is included in the calculations of modal function and environmental parameters. As shown in Fig. 8, there is no obvious change in the magnitude of c until the wave propagates to the station at 50 km. The phase speed gradually decreases shoreward and is approximately 1 m/s near 150 km. The distributions of c along the cross section in different months are similar, and the background current seems to have little 25

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

Fig. 9. Same as Fig. 8, but for the quadratic nonlinearity coefficient. The gray dashed line gives the location where the quadratic nonlinearity coefficient turns to zero in the case of excluding background current. The corresponding turning depth is also shown with gray text.

close to zero. In addition, the absolute values (approaching to 10−2 s−1) in summer (June to August) are obviously larger than in other seasons. The background current has little effect on α except for those stations further than 100 km in summer months. However, the difference is only in order of 10−3 s−1. The monthly variation of α may induce a prominent deformation of ISW during its shoaling, which will be shown in the following simulation results. Since the ISWs propagate for a long distance from the deep basin to the continental shelf where they cross a transition zone in which the quadratic nonlinearity vanishes, the cubic nonlinearity cannot be ignored, and the coefficient also needs to be calculated prior to the simulation. The results are shown in Fig. 10. Along the cross section the cubic nonlinearity coefficient α1 is in O (10−5 m−1 s−1). From October to March, α1 is positive in the deep water before 100 km. The coefficient changes its sign from positive to negative at much shallower locations further than 100 km from April to September. The annual mean depth where α1 changes its sign is 391 m, much deeper than the turning depth of α. The largest absolute value of α1 for the cross section is approximately 3.0 × 10−4 m−1 s−1 with little monthly variation. The background current also has little effect on a1. Noticeable difference can only be found at those stations further than 100 km from May to August. As shown above, the ISW will undergo a variable environmental

Fig. 7. Monthly distribution of buoyancy frequency and background current profiles along the cross section shown in Fig. 1. Only the values in the upper 500 m are presented to clearly show the pycnocline and the topography of the continental slope (gray area of the lower left corner).

Fig. 8. Monthly distribution of the first mode linear phase speed along the cross section. Both cases of including (with) and excluding (without) background current (BC) are shown.

effect. In fact, magnitude of the background current (0.1–0.2 m/s) is much smaller than the wave speed. Therefore, it may not significantly affect the wave propagation and evolution. We then check the nonlinearity coefficients. The quadratic nonlinearity coefficient along the cross section is first calculated and shown in Fig. 9. The coefficient varies between −5 and −10 (in 10−3 s−1) and exhibits little difference before the waves arrive at 50 km. There is a noticeable increase in the absolute values of α after 50 km followed by a quick decrease approaching to zero. A distinct feature is that α changes its sign to positive from November to March while it remains negative in other months. The averaged depth of α crossing zero is 229 m for this cross section. Noting that in April and October, α is negative but very

Fig. 10. Same as Fig. 8, but for the cubic nonlinearity coefficient. 26

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

Fig. 12. Simulation results at 150 km from the origin. The control (sensitivity) run with amplitude of 50 m (80 m) is presented by thick (thin) solid line.

those in January to March. Propagation and evolution of the wave after 100 km, however, is totally different. The ISW disintegrates into a wave train of depression with several small-amplitude waves following the leading wave, and no elevation wave can be found. In October, the polarity conversion process of ISW seems to emerge again, and the wave plots in November and December are much the same as those in January and February. The simulation results including the effect of background current are also shown in Fig. 11. Clearly, there is no obvious difference between the results of including and excluding background current. The large-scale background current in some extent only affects the wave speed and results in a phase shift of the simulated wave. Neither the wave disintegration nor polarity conversion process is significantly affected when considered the background current. The simulation results at 150 km from the origin are shown in Fig. 12. A sensitivity run has been performed using an initial wave with amplitude of 80 m. As shown in Fig. 12, the polarity conversion process is similar with the control run with amplitude of 50 m. The wave profile of the former run has faster speed than that of the latter one due to the larger amplitude contribution to the nonlinear phase speed. Thus, the amplitude of the initial wave may take effect on the phase shift but the polarity conversion process is robust since it is controlled by the predefined parameters derived from the background stratification and current. For the simulation cross section, the quadratic nonlinearity coefficient changes its sign from negative to positive in November to March as shown in Fig. 9. Therefore, the simulated waves become flatten in the front, and elevation waves emerge at the rear. This polarity conversion process can be clearly found from the simulation results using the monthly climatological data, which has not been reported by previous studies.

Fig. 11. Simulation results for an initial wave of KdV soliton with an amplitude of 50 m. The results are shown in multi-wave stacked plots in a s-x axis moving with the first mode linear phase speed. The distance interval between the adjacent wave profiles is 20 km. Red thick line in the last panel gives the scale of wave amplitude. Both cases of including (black solid line) and excluding (gray solid line) background current are shown. Gray dashed line gives the location where the quadratic nonlinearity coefficient turns to zero in the case of excluding background current.

condition in which the parameters of model (12) are changing not only the magnitude but also the sign, and the large-scale background current only takes significant effect after the wave propagates into continental slope region. Other parameters of model (12) including the dispersion coefficient and the amplification parameter along the cross section are also calculated. Their monthly distributions are similar with that of the phase speed c. Thus, we do not present and discuss them in this section due to space limitation. With the environmental parameters calculated above, an initial wave profile is input at the origin of the cross section. Fig. 11 presents the simulation results for an initial wave with amplitude of 50 m. The initial wave is subjected to the typical KdV soliton solution (in sech2 form) with wave parameters given at the initial station of the cross section. Previous in-situ observation by Ramp et al. (2010) has revealed that the annual mean amplitude of type-a ISW at B1 is 53 ± 27 m. Therefore, the amplitude of the initial wave is a typical value for ISW in the deep basin. From Fig. 11, apparent deformation of the initial ISWs cannot be found before the waves arrive at 80 km from the origin, which agrees with the little variation of the associated environmental parameters along the cross section. The waves become shaper and narrower as they propagate further onto the continental slope and shelf after 80 km. This deformation process is clearly different from month to month as shown in Fig. 11. From January to March the initial ISW is deformed and amplified at 80 km, and the amplitude is largest at 100 km. Approaching the turning point the front of the wave profile is flattened, and apparent elevation waves can be found after that. There is significant difference in the number of waves in the wave packets, which is decreased from January to March. The simulated wave profiles of April to September before 100 km have similar deformation features with

3.3. Discussion The polarity conversion of ISW from depression in deep water to elevation in shallow water is commonly found in real ocean (Liu et al., 1998; Orr and Mignerey, 2003; Fu et al., 2012; Bai et al., 2017). In a two-layer system, this process occurs at the position where the thickness of the upper layer is equal to that of the lower layer (Liu et al., 1998). For KdV-type theories, it means that the quadratic nonlinearity coefficient vanishes (Grimshaw et al., 2004). This process has been verified by the simulation above. In this paper, we focus on the monthly variation of this process, and as shown above, polarity conversion of ISW in the NSCS does reveal distinct scenarios in different months. This feature is attributed to the quadratic nonlinearity coefficient that varies from month to month. 27

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

process of the ISW. Thus, neither the parametric study nor simulation of moed-2 wave is discussed in the paper. The reader can refer to Kurkina et al. (2017) the parametric study of mode-2 wave in the SCS. Another aspect of the background current effect on ISW can be the influence from the mesoscale eddy which is also a frequently observed feature in the NSCS. Previous studies have shown that mesoscale eddy can change the propagation path of ISW and induce the wave energy focusing and stretching (Alford and Co-authors, 2015; Xie et al., 2015). In this paper, however, we mainly focus on the large-scale background current and have not simulated the propagation and evolution of ISW when encountered the eddies, which means the results are meaningful in climatological sense since we use the monthly climatological datasets of stratification and background current. In fact, the parametric study has shown that the environmental parameters with the background current are significantly distinct along the continental slope although the magnitude of the difference is not sufficient to change the deformation and polarity conversion features of ISW.

Table 1 The first layer depth derived from the maximum first modal function depth and its ratio to the total water depth at 150 km. Month

D1 (m)

D (m)

D1/D

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

111 109 104 97 90 87 86 87 90 97 103 110

200 200 200 200 200 200 200 200 200 200 200 200

0.56 0.55 0.52 0.49 0.45 0.44 0.43 0.44 0.45 0.49 0.52 0.55

The quadratic nonlinearity coefficient in a two-layer system can be given by

3c h − h2 α= 0 1 , 2 h1 h2

4. Summary and conclusion ISW, as a high-frequency fluctuation in the ocean pycnocline, is sensitive to the variation of background fields, such as stratification and background current. Multi-scale variation in the stratification and background current can modulate the properties of ISW, including its polarity. In this paper, using the climatological datasets derived from SODA3.3.1 reanalysis, we have investigated the conversion depth where the ISW in the NSCS start to change polarity from depression to elevation and presented its seasonality. The main conclusions are drawn as follows:

(18)

where h1 and h2 are the thickness of the upper and lower layer, respectively. The sign of α therefore depends on the selection of h1 and h2 determined from the water stratification and background current. A preferable choice is to let h1 be the depth where the maximum buoyancy frequency is located (Bai et al., 2017). In this paper we choose the depth where the first modal function reaches its maximum. The depth in our study is derived based on a continuously stratified assumption of the water column and including the effect of background current, which makes it more reasonable for the oceanic practice. Table 1 gives the upper layer depth derived from the maximum fist modal function and its ratio to the total depth. The ratio is larger than 0.5 from November to March when noticeable elevation waves exist and is very close to 0.5 in April (0.49) and October (0.49). The depth ratio is significantly less than 0.5 and reaches its minimum in July (0.43), which means only depression waves are supported by the background fields when they cross the isobath near 200 m. Climatologically, the polarity conversion process appears on the continental shelf in the NSCS mainly in other three seasons rather than in summer. However, it does not mean that there is no polarity conversion of ISW near 200-m depth in summer months. In their contribution, Bai et al. (2017) has reported that the polarity conversion of ISW starts from the 200-m isobath by using seismic data collected in July 2009. The waves observed are interpreted as “transition waves” before they finally developed to “elevation waves”. By calculating the quadratic nonlinearity coefficients based on continuous layer model at four corresponded locations, Bai et al. (2017) found that only one location with positive α value can support the observed results. However, their results are not so reliable since they use the temperature and salinity profiles spatially and temporally different with the observed seismic cross sections. As shown in Fig. 9, the α values vary rapidly near the continental slope and shelf regions. Therefore, the sign of α is quite sensitive to the water depth especially in the transition zone. Another result of our calculation of environment parameters supports the observation results by Bai et al. (2017). As shown in Fig. 6b, the quadratic nonlinearity coefficient has an averaged turning depth near 100 m in July, which is consistent with the depth reported by Bai et al. (2017). Though the in-situ observations of ISW in the NSCS have shown the existence of mode-2 waves especially in winter months, our calculation of environmental parameters and simulation of wave propagation and evolution do not resolve these waves. As reported by Liu et al. (2013), only 65% of mode-1 wave energy has been transferred from deep water to shallow water, and the mode-1 wave disintegrates into wider mode-2 wave of which the wave energy can be 34% of the former. However, our primary purpose in this paper is to discuss the polarity conversion

1) The parametric study of ISW in the NSCS exhibits significant month to month variation especially for the quadratic nonlinearity coefficient. Apparent dividing line of α equals to zero can be found along the continental shelf from northeast to southwest from January to February and from November to December, which indicates the polarity conversion of ISW is easier to appear during these months. The averaged turning depth for α = 0 is found near 200 m (annual mean) with its maximum (~300 m) in January and minimum (~100 m) in July. 2) The simulation of ISW propagation and evolution reveals that elevation waves are more likely to appear from November to March when the quadratic nonlinearity coefficient changes its sign from negative to positive. 3) Although the large-scale background current prominently affects the environmental parameters of ISW around the shelf current from Luzon strait, it exhibits little effect on the deformation and polarity conversion of ISWs which propagate from the deep basin to the continental shelf of water depth between 200 m and 100 m. By presenting the distribution of the environmental parameters especially the quadratic nonlinearity coefficient in the NSCS, it is shown that the properties of ISW exhibit distinct variations on seasonal scales. It can be also expected that the variation of ISW properties on annual or longer time scales could be probably illustrated if given the appropriate datasets as the background fields. This could be done by calculating the environmental parameters, demonstrating their variations on different time scales and further simulating the propagation and evolution of ISW under varied background fields. Acknowledgements The present study is supported by the National Key Research and Development Program of China (2016YFC1401403), SOA Program on Global Change and Air-Sea Interactions (GASI-IPOVAI-02), the Scientific Research Foundation of Third Institute of Oceanography, SOA (2014029, 2017011), and the National Natural Science Foundation of 28

Continental Shelf Research 171 (2018) 21–29

S. Zhang et al.

China (41506014, 41505041).

Huang, X., Zhang, Z., Zhang, X., Qian, H., Zhao, W., Tian, J., 2017. Impacts of a mesoscale eddy pair on internal solitary waves in the northern South China Sea revealed by mooring array observations. J. Phys. Oceanogr. 47 (7), 1539–1554. Klymak, J.M., Robert, P., Liu, C.-T., Liu, A.K., Laura, D., 2006. Prototypical solitons in the South China Sea. Geophys. Res. Lett. 33 (11), L11607. Kurkina, O., Talipova, T., Soomere, T., Giniyatullin, A., Kurkin, A., 2017. Kinematic parameters of internal waves of the second mode in the South China Sea. Nonlinear Prog. Geophys. 24 (4), 1–26. Lee, C.Y., Beardsley, R.C., 1974. The generation of long nonlinear internal waves in a weakly stratified shear flow. J. Geophys. Res. 79 (3), 453–462. Li, L., Wang, C., Grimshaw, R., 2015. Observation of internal wave polarity conversion generated by a rising tide. Geophys. Res. Lett. 42 (10), 4007–4013. Li, Q., Farmer, D.M., 2011. The generation and evolution of nonlinear internal waves in the deep basin of the South China Sea. J. Phys. Oceanogr. 41 (7), 1345–1363. Li, Q., Wang, B., Chen, X., Chen, X., Park, J.H., 2016. Variability of nonlinear internal waves in the South China Sea affected by the Kuroshio and mesoscale eddies. J. Geophys. Res. Oceans 121 (4), 2098–2118. Liao, G., Xu, X.H., Liang, C., Dong, C., Zhou, B., Ding, T., Huang, W., Xu, D., 2014. Analysis of kinematic parameters of Internal Solitary Waves in the Northern South China Sea. Deep Sea Res. 94, 159–172. Liu, A.K., Holbrook, J.R., Apel, J.R., 1985. Nonlinear internal wave evolution in the Sulu Sea. J. Phys. Oceanogr. 15 (12), 1613–1624. Liu, A.K., Chang, Y.S., Hsu, M.K., Liang, N.K., 1998. Evolution of nonlinear internal waves in the East and South China Seas. J. Geophys. Res. 103 (C4), 7995–8008. Liu, A.K., Ramp, S.R., Zhao, Y., Tang, T.Y., 2004. A case study of internal solitary waves during ASIAEX-2001. IEEE J. Ocean. Eng. 29 (4), 1144–1156. Liu, A.K., Su, F.-C., Hsu, M.K., Kuo, N.-J., Ho, C.-R., 2013. Generation and evolution of mode-two internal waves in the ocean. Cont. Shelf Res. 59, 18–27. Liu, Z., Grimshaw, R., Johnson, E., 2017. Internal solitary waves propagating through variable background hydrology and currents. Ocean Model. 116, 134–145. Orr, M.H., Mignerey, P.C., 2003. Nonlinear internal waves in the South China Sea: observation of the conversion of depression internal waves to elevation internal waves. J. Geophys. Res.: Oceans 108 (C3). https://doi.org/10.1029/2001JC001163. Park, J.H., Farmer, D., 2013. Effects of Kuroshio intrusions on nonlinear internal waves in the South China Sea during winter. J. Geophys. Res.: Oceans 118 (12), 7081–7094. Ramp, S.R., Tang, T.Y., Duda, T.F., Lynch, J.F., Liu, A.K., Chiu, C.S., Bahr, F.L., Kim, H.R., Yang, Y.J., 2004. Internal solitons in the Northeastern South China Sea Part I: sources and deep water propagation. IEEE J. Ocean. Eng. 29, 1157–1181. Ramp, S.R., Yang, Y.J., Bahr, F.L., 2010. Characterizing the nonlinear internal wave climate in the northeastern South China Sea. Nonlinear Process. Geophys. 17 (5), 481–498. Simmons, H., Chang, M.-H., Chang, Y.-T., Chao, S.-Y., Fringer, O., Jackson, C.R., Ko, D.S., 2011. Modeling and prediction of internal waves in the South China Sea. Oceanography 24 (4), 88–99. Xie, J., He, Y., Chen, Z., Xu, J., Cai, S., 2015. Simulations of internal solitary wave interactions with mesoscale eddies in the northeastern South China Sea. J. Phys. Oceanogr. 45, 2959–2978. Yang, Y.J., Fang, Y.C., Chang, M.H., Ramp, S.R., Kao, C.C., Tang, T.Y., 2009. Observations of second baroclinic mode internal solitary waves on the continental slope of the northern South China Sea. J. Geophys. Res.: Oceans 114 (C10). https://doi.org/10. 1029/2009JC005318. Zhang, Z., Fringer, O.B., Ramp, S.R., 2011. Three-dimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the South China Sea. J. Geophys. Res. 116 (C5). https://doi.org/10.1029/2010JC006424. Zhao, Z.X., Alford, M.H., 2006. Source and propagation of internal solitary waves in the northeastern South China Sea. J. Geophys. Res. 111 (C11). https://doi.org/10.1029/ 2006JC003644. Zheng, Q., Song, T., Ho, C., Lin, H., 2008. Statistics of internal waves in the SCS. In: Liu, A.K., Ho, C., Liu, C.-T. (Eds.), Satellite Remote Sensing in the SCS. Tingmao Publishing Co., Taipei, pp. 67–80.

References Alford, M.H., Lien, R.-C., Simmons, H., Klymak, J., Ramp, S., Yang, Y.J., Tang, D., Chang, M.-H., 2010. Speed and evolution of nonlinear internal waves transiting the South China Sea. J. Phys. Oceanogr. 40 (6), 1338–1355. Alford, M.H., Co-authors, 2015. The formation and fate of internal waves in the South China Sea. Nature 521 (7550), 65–69. Bai, Y., Song, H., Guan, Y., Yang, S., 2017. Estimating depth of polarity conversion of shoaling internal solitary waves in the northeastern South China Sea. Cont. Shelf Res. 143, 9–17. Buijsman, M.C., Kanarska, Y., McWilliams, J.C., 2010. On the generation and evolution of nonlinear internal waves in the South China Sea. J. Geophys. Res. 115 (C2), C02012. Cai, S.Q., Xie, J.S., 2010. A propagation model for the internal solitary waves in the northern South China Sea. J. Geophys. Res. 115 (C12), C12074. Cai, S.Q., Xie, J.S., He, J.L., 2012. An overview of internal solitary waves in the South China Sea. Surv. Geophys. 33 (5), 927–943. Cai, S.Q., Xie, J.S., Xu, J.X., Wang, D.X., Chen, Z.W., Deng, X.D., Long, X.M., 2014. Monthly variation of some parameters about internal solitary waves in the South China sea. Deep Sea Res. 84, 73–85. Carton, J.A., Chepurin, G.A., Chen, L., 2018. SODA3: a new ocean climate reanalysis. J. Clim. 31, 6967–6983. https://doi.org/10.1175/JCLI-D-18-0149.1. Chen, Z., Xie, J., Wang, D., Zhan, J., Xu, J., Cai, S., 2014. Density stratification influences on generation of different modes internal solitary waves. J. Geophys. Res.: Oceans 119 (10), 7029–7046. Cho, C., Nam, S.H., Song, H., 2016. Seasonal variation of speed and width from kinematic parameters of mode‐1 nonlinear internal waves in the northeastern East China Sea. J. Geophys. Res. Oceans 106 (1), 499–504. Fliegel, M., Hunkins, K., 1975. Internal wave dispersion calculated Using the ThomsonHaskell method. J. Phys. Oceanogr. 5 (3), 541–548. Fu, K.H., Wang, Y.H., Laurent, L.S., Simmons, H., Wang, D.P., 2012. Shoaling of largeamplitude nonlinear internal waves at Dongsha Atoll in the northern South China Sea. Cont. Shelf Res. 37, 1–7. Grimshaw, R., Pelinovsky, E., Talipova, T., 2007. Modelling internal solitary waves in the coastal ocean. Surv. Geophys. 28, 273–298. Grimshaw, R., Wang, C., Li, L., 2016. Modelling of polarity change in a nonlinear internal wave train in Laoshan Bay. J. Phys. Oceanogr. 46 (3), 965–974. Grimshaw, R., Pelinovsky, E., Talipova, T., Kurkin, A., 2004. Simulation of the transformation of internal solitary waves on oceanic shelves. J. Phys. Oceanogr. 34 (12), 2774–2791. Grimshaw, R., Pelinovsky, E., Talipova, T., Kurkina, O., 2010. Internal solitary waves: propagation, deformation and disintegration. Nonlinear Process. Geophys. 17 (6), 633–649. Grimshaw, R., Guo, C., Helfrich, K., Vlasenko, V., 2014. Combined effect of rotation and topography on shoaling oceanic internal solitary waves. J. Phys. Oceanogr. 44 (4), 1116–1132. Guo, C., Chen, X., 2014. A review of internal solitary wave dynamics in the northern South China Sea. Prog. Oceanogr. 121, 7–23. Helfrich, K.R., Melville, W., 2006. Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425. Holloway, P.E., Pelinovsky, E., Talipova, T., 1999. A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res. 104 (C8), 18333–18350. Holloway, P.E., Pelinovsky, E., Talipova, T., Barnes, B., 1997. A nonlinear model of internal tide transformation on the Australian North West Shelf. J. Phys. Oceanogr. 27 (6), 871–896. Huang, X., Chen, Z., Wei, Z., Zhang, Z., Zhou, C., Yang, Q., Tian, J., 2016. An extreme internal solitary wave event observed in the northern South China Sea. Sci. Rep. 6. https://doi.org/10.1038/srep30041.

29