Spatial distributions of the main properties of the mode function and kinematic and non-linear parameters of internal waves of the second mode are derived for the South China Sea for typical summer conditions in July. The calculations are based on the Generalized Digital Environmental Model (GDEM) climatology of hydrological variables, from which the local stratification is evaluated. The focus is on the phase speed of long internal waves and the coefficients at the dispersive, quadratic and cubic terms of the weakly non-linear Gardner model. Spatial distributions of these parameters, except for the coefficient at the cubic term, are qualitatively similar for waves of both modes. The dispersive term of Gardner's equation and phase speed for internal waves of the second mode are about a quarter and half, respectively, of those for waves of the first mode. Similarly to the waves of the first mode, the coefficients at the quadratic and cubic terms of Gardner's equation are practically independent of water depth. In contrast to the waves of the first mode, for waves of the second mode the quadratic term is mostly negative. The results can serve as a basis for expressing estimates of the expected parameters of internal waves for the South China Sea.

The South China Sea is an example of shelf seas where highly energetic internal solitary waves often generate up to 100–200 m vertical displacements of water masses. These powerful disturbances are usually excited by tide–topography interaction in the Luzon Strait where the Kuroshio serves as a background current that may greatly modify the generating conditions but does not affect the coefficients of the Gardner model in the South China Sea. The resulting internal waves are further modified by numerous islands, seamounts and other bathymetric features in the Luzon Strait (Liu et al., 1998, 2004, 2006; Cai et al., 2002; Ramp et al., 2004, 2015). Many such structures with amplitudes up to 100 m propagate as solitary waves. Their impact on water masses has been observed in the vicinity of two underwater elevations in an area of the Luzon Strait where the water depth is only about 300 m (Liu et al., 2006). These waves propagate into deeper regions of the South China Sea, cross this water body and exert conspicuous transformations along the continental shelf at depths of 400–200 m. The associated displacements of isopycnals may reach 100 m. The appearance of such waves often matches well theoretical shapes of internal solitons (Klymak et al., 2006).

While most such waves represent internal waves of the first mode, numerous recordings suggest that internal waves of the second and sometimes even third mode are regularly present in the South China Sea (Fig. 1; see also Guo et al., 2006; Yang et al., 2009; Vlasenko et al., 2010; Liu et al., 2013). It is likely that higher modes of long internal waves are often generated in the world ocean. They are frequently excited, for example, as a result of interaction of non-linear waves of the first mode with topography (Ramp et al., 2010, 2015; Shroyer et al., 2010; Vlasenko et al., 2010, 2014; Vlasenko and Stashchuk, 2015). Alternatively, internal waves of any mode may be created in micro-tidal stratified semi-sheltered basins by an intense outflow or inflow (Vlasenko et al., 2009), strong atmospheric disturbances (Ivanov et al., 1987), indirect and/or delayed impact of such disturbances (e.g. release of storm surges) and other phenomena.

Internal waves representing the first, second and third modes (Guo et al., 2006). Numbers at the boxes in the panel indicate the number of the relevant mode.

Most studies of internal waves focus on waves of the first mode. This approach apparently mirrors the abundance of records of various properties of water masses in the uppermost layers of the ocean compared to profiles of the entire water column, namely, profiles of motions and other hydrophysical properties of the upper sections of water masses usually do not provide enough information about the full vertical structure of internal waves. Separation of the internal wave field into components representing different modes is extremely complicated in the vicinity of their generation regions and in areas where the waves interact with one another and with the bottom. Such analysis, however, is feasible in regions remote from the generation and interaction areas because small-amplitude internal waves of different modes propagate with different velocities and become separated after some time.

Solitary internal waves of the first mode may be waves of elevation or waves of depression. The structure of higher-mode internal solitary waves is more complicated (Fig. 2). For example, vertical displacements of the upper and lower jump layers created by an internal solitary wave of the second mode have different polarities. For this reason the notion of internal waves is based on certain topologic features of the instantaneous appearance of the intermediate layer. Waves that create convex modifications of this layer are said to have positive polarity. Such waves are called convex or positive waves in what follows. Waves that create a concave shape of the intermediate layer are said to have negative polarity and are called concave or negative waves.

Scheme of internal solitary waves of the second mode with positive (left) and negative (right) polarity.

Internal solitary waves of the second mode with both polarities have been regularly observed on the north-western continental shelf of the South China Sea (Yang et al., 2010; Ramp et al., 2015). Such waves may be generated during interactions of solitary waves of the first mode with various bathymetric features (Vlasenko and Hutter, 2001). Positive (convex) solitary waves of this kind appear substantially (by about 20 times) more frequently in the records than negative (concave) waves (Yang et al., 2010).

The dynamics of long internal waves of the second mode can be described with reasonable accuracy using weakly non-linear evolution equations of the Korteweg–de Vries (KdV) family. In particular, Gardner's equation is commonly used as the classic model of internal waves of the first mode (Holloway et al., 1999; Grimshaw et al., 2004; Talipova et al., 2014, 2015). This model has been applied inter alia to explain and replicate a polarity switch of internal solitons propagating along the north-eastern continental shelf in the South China Sea (Liu et al., 1998; Orr and Mignerey, 2003; Zhao et al., 2004; Grimshaw et al., 2010).

The models of this kind are correct only asymptotically. Their core advantage is that a small set of parameters governs the appearance and properties of internal solitary waves. This feature makes it possible to use these models to isolate and identify principally new features of the dynamics of internal waves even if some details of the system are not reproduced. For example, a new kind of quasi-steady non-linear internal wave (a so-called breather) has been predicted using the framework of Gardner's equation. The possibility of generation of such phenomena by solitary waves of the second mode and the basic properties of its long-term propagation have been obtained in a numerical “wave tank” using Euler's equations (Lamb et al., 2007; Terletska et al., 2016). Several features of the process of generation of table-top solitary waves were also extracted based on Gardner's equation (Kurkina et al., 2016). The effect of a change in the polarity of solitary waves predicted by the asymptotic theory has been repeatedly observed in various areas, including the South China Sea (see above). It is however inevitable that many specific features and details (e.g. radiation of short waves, properties of strongly non-linear disturbances or breaking of solitonic structures) cannot be reproduced using equations for weakly non-linear waves and specific configurations of stratification may require the use of higher-order analysis and equations.

The parameters selected for the model have a significant effect on many features of internal solitary waves. In other words, the appearance and core qualities of the propagation and transformation of such waves are governed by spatial variations in the coefficients of Gardner's equation along the propagation path of the waves in question. The associated variations have been thoroughly studied for internal solitary waves of the first mode using common databases of the vertical structure of temperature and salinity (Levitus, 1982; Carnes, 2009). This approach made it possible to construct climatologically valid maps of spatio-temporal variations in various coefficients of Gardner's equation for internal waves of the first mode in different regions of the world ocean. These maps depict the values of phase speed of long waves (also called wave speed because for long waves it is also equal to group speed) and coefficients of various terms (linear, quadratic and cubic terms) in the relevant Gardner equation (Pelinovsky et al., 1995; Talipova and Polukhin, 2001; Polukhin et al., 2003, 2004; Kurkina et al., 2011, 2017). Similar maps have also been calculated for the South China Sea (Grimshaw et al., 2010; Liao et al., 2014).

As many regions of the world ocean support propagation of internal waves of higher modes, it is important to expand this kind of “climatology” of internal wave propagation regimes to cover, to a first approximation, the properties of large-amplitude internal waves of the second mode. Such maps of the kinematic parameters (wave speed and the coefficient at the linear term) and coefficients at the non-linear terms of the relevant evolution equation make it possible to rapidly evaluate several core properties of the dynamics and impact of internal waves, build pathways of the propagation of waves from their typical areas of generation and identify which regions are possibly affected by hydrodynamic loads created by large internal waves.

For example, the polarity of solitons is governed by the sign of the coefficient at the quadratic term of Gardner's equation (Grimshaw et al., 2007). The values of this coefficient as well as other kinematic parameters of waves can be calculated in a straightforward manner from the so-called mode function and its derivatives. The lines where the coefficient at the quadratic term vanishes or changes its sign mark the regions of a switch of the polarity of internal solitons. This switch may be accompanied by radical changes in the further behaviour of waves or the region may even be a location of the onset of wave breaking. This feature is valid for solitons of the first and second modes. Similar maps of the values of the coefficient at the cubic term specify e.g. the regions where modulational instability of internal wave trains may modify wave properties or where a specific type of soliton – internal breather – may exist (Talipova et al., 2011).

This paper focuses on the construction of maps of phase speed and coefficients at various terms of Gardner's equation. These quantities are often called kinematic and non-linear parameters of long internal waves of the second mode. The target area is the South China Sea where such maps are useful to better evaluate the core properties of internal waves and their propagation. We start with a short description of the set-up of the problem of internal wave propagation. An asymptotic solution to this problem can be provided by an evolution equation for such internal waves – Gardner's equation. To properly evaluate the values of its coefficients that govern the appearance and dynamics of internal waves of the second mode, it is necessary to adequately describe the structure of the mode function. A relevant non-linear correction to this function is derived using an asymptotic procedure, which is discussed in Sect. 2 together with the main features of the appearance of internal solitary waves of the second mode. Section 3 describes the resulting maps of phase speed and various coefficients at the non-linear terms of the Gardner model for internal waves of the second mode and the applicability of the entire model for the conditions of the continental shelf of the South China Sea. The main conclusions of the study are formulated in Sect. 4.

Similarly to the treatment of internal waves of the first mode, the dynamics
of long internal waves of the second mode in the ocean can be adequately
described using a classic evolution equation – Gardner's equation (Holloway
et al., 1999; Grimshaw et al., 2004, 2007). This model equation, presented
here in the nondimensional form

The mode function

The phase speed

We follow the tradition of selecting

The vertical structure of internal waves of the second mode is more complicated than the same structure for the classic internal waves of the first mode. The core difference between these structures can be illustrated with the example of a simple model of quasi-two-layer stratification (Fig. 3a). The water masses described by such a model have one jump layer of density. The Brunt–Väisälä frequency (Fig. 3b) has one maximum along each vertical cross section. The maximum is located in the region of the fastest variation in density. Similarly, the mode function for the waves of the first mode has one extremum (maximum or minimum depending on the normalisation) along each vertical cross section. Importantly, the mode function for the waves of the first mode exhibits no sign change within the entire water column.

An example of the vertical profile of the density

In contrast, the mode function for the waves of the second mode changes its
sign. It has the maximum positive value near the upper boundary of the jump
layer and the minimum (negative) value near the lower boundary of the jump
layer (Fig. 4a). This means that two more intrinsic quantities are present
in the system: the locations of zero-crossing

There are different approaches in the literature. Liu and Wang (2012) rely
on the values of the mode function at its minimum

Other recent studies of internal solitons of the second mode in the South
China Sea address the situation on the continental shelf where the largest
absolute values of

The vertical structure of the mode function

In such environments it is convenient to adjust Gardner's equation so that it
describes the deviations of the isopycnals that correspond to

Importantly, with this choice of normalisation the polarity of the internal
solitons of the second mode matches the sign of the coefficient at the
quadratic term of Eq. (1). In other words, positive (convex) solitons
correspond to positive values of the coefficient

This feature may lead to certain problems in the analysis of the dynamics of internal waves of this kind. However, the normalisation is, in essence, arbitrary and the vertical structure of motions is independent of the chosen normalisation. Therefore, the relevant issues are purely technical and do not impact on the results of the analysis. Thus, we decided to meet the possible technical implications but still follow the more logical and straightforward normalisation using the maximum of the absolute values of the mode function at the upper boundary of the jump layer.

The South China Sea is a large (surface area about 3.5 million km

To construct spatial maps of these parameters, we followed the approach
implemented for the calculation of similar parameters of internal waves of
the first mode and the coefficients of Gardner's equation for this basin
(Grimshaw et al., 2010; Liao et al., 2014). We employed generalised
climatologic information about long-term mean temperature and salinity
profiles. This information was extracted from the Generalized Digital
Environment Model (GDEM) (Teague et al., 1990; Carnes, 2009). The GDEM
(

The maps in this paper were calculated for the stratification that is characteristic in July. Seasonal variations in these maps are discussed shortly at the end of Sect. 3.2. The field of large-scale currents was ignored because there is no detailed information about the currents in this area in the existing databases. We stress that both large-scale currents and mesoscale structures may considerably affect the local stratification and greatly impact the values of coefficients of Gardner's equation. However, their impact is highly variable in space and time and it is likely that it will become evident via limited variations of the seasonal values of the coefficients in question.

Mean density profiles were computed for each horizontal pixel of the GDEM
database from temperature and salinity profiles using the International
Equation for State of Seawater (Fofonoff and Millard Jr., 1983). With these
density profiles, we evaluated the mode function

As described above, the calculations of various parameters and coefficients
rely on the values of maxima

A large part of the values of

The locations of zero-crossings of the mode function largely follow the
relevant locations of the maximum. Therefore, both zero-crossings and maxima
of the mode function roughly reflect the core variations in the water depth.
In contrast, the minima of the mode function are only weakly, if at all,
correlated with

Another view of the nature of the distributions of the quantities

Scatter plot of normalised values of

As discussed above, it is debatable whether the maximum or minimum values of
the mode function should be used for normalising this function. The spatial
distribution of the values of

Spatial distributions of phase speeds of long linear internal waves of the
first and second modes are very similar to each other in the South China Sea
(Fig. 11). The phase speeds for internal waves of the second mode are mostly
below 1.5 m s

Map of phase speeds of long linear internal waves of the second

Even though water depth is one of the most important factors governing the
propagation speed of internal waves, stratification of water masses equally
contributes to the properties of the propagation of internal waves. Its
impact is apparently complemented by variations in the amount of incoming
radiation from the Sun. These variations may be one of the reasons for the
presence of the meridional pattern of the phase speed of internal waves of
the second mode. This meridional pattern is well known for internal waves of
the first mode (Talipova and Polukhin, 2001). Its presence is a likely reason
why the dependence of the phase speed on water depth shows substantial
scatter in the study area (Fig. 12a). The level of scatter is, however,
fairly moderate and the relationship between the water depth and phase speed
can be reasonably approximated using a power function:

Map of coefficients at the dispersive term of the Gardner equation
for internal waves of the second mode

Spatial distributions of the coefficient at the dispersive term of Gardner's
equation (1) for internal waves of the second (Fig. 13a) and first (Fig. 13b)
modes are also qualitatively similar. However, the numerical values of this
coefficient differ substantially. This coefficient (and, consequently, the
impact of linear dispersion on the wave propagation and dynamics) for waves
of the second mode is about 3–4 times smaller than the similar coefficient
for the waves of the first mode. The relationship between this coefficient
and water depth (Fig. 13c) is remarkably different from a similar
relationship (8) for phase speed. Figure 13c clearly represents a quadratic
relationship that graphically can be presented as a parabola:

Differently from the coefficient at the dispersive term, the values of coefficients at the non-linear terms of Gardner's equation are mostly governed by properties of stratification and only insignificantly depend on the water depth (Talipova and Polukhin, 2001). It is therefore not surprising that the maps of these coefficients for waves of the second (Fig. 14a) and first (Fig. 14b) modes are qualitatively similar to each other and that the numerical values of these coefficients for the two modes are comparable.

Map of coefficients at the quadratic term of the Gardner equation
for internal waves of the second mode

The histogram of the values of the coefficient at the quadratic term in Eq. (1) indicates that, differently from several other quantities addressed
above, this coefficient has a clearly skewed but unimodal distribution. The
values with both signs are more or less equally represented (Fig. 14c). The
range of values is from

Interestingly, a smaller peak exists for zero values of this coefficient. Gardner's equation transforms into the modified KdV equation in locations where the coefficient at the quadratic term vanishes and one has to use this equation in order to properly describe weakly non-linear dynamics of internal waves in such regions. Importantly, the study area contains regions characterised by large gradients and changes in the sign of this coefficient. In general, the signs of this coefficient are different in deeper-water and shallower regions of the South China Sea. Interestingly, the signs of this coefficient are also different in the north-western and south-western regions of the continental shelf for both modes.

The coefficient in question is positive in most of the northern part of the shelf; consequently, the situation is favourable for the existence of positive internal solitons of the second mode. This feature explains why convex solitons are predominantly recorded in the north-eastern segments of the continental shelf (Yang et al., 2010). In contrast, this coefficient is generally negative for internal waves of the first mode, whereas its positive values are found only in a few small areas of the South China Sea. Further, this coefficient for waves of the second mode is predominantly positive in the southern relatively shallow part of the South China Sea, whereas for waves of the first mode the sign of this coefficient is highly variable.

It is well known that the values of the coefficient of the quadratic term of
Eq. (2) for internal waves of the first mode are practically independent of
the water depth (Talipova and Polukhin, 2001). This property is also true
for internal waves of the second mode in the South China Sea (Fig. 15). The
largest absolute values of this coefficient (corresponding to both negative
and positive values) occur in relatively shallow areas. The range of its
values in deeper parts (depths

Scatter diagram of coefficients at the quadratic term of the Gardner
equation against water depth for internal waves of the second mode

The coefficient at the cubic non-linear term of Eq. (1) has relatively small
(but positive) values for waves of the first mode in the entire deep-water
region of the South China Sea (Fig. 16b). This coefficient for waves of the
second mode also has small absolute values in this area. It is positive only
on the continental slope and turns negative in the entire eastern part of the
sea. This coefficient for waves of the second mode has large positive values
in selected locations of the Sulu Sea. The north-eastern shelf of the South
China Sea is characterised by intermittent variations in the sign of this
coefficient for both modes of internal waves. This area also shows the
largest absolute values of this coefficient (0.001 m

Map of coefficients at the cubic term of the Gardner equation for
internal waves of the second mode

The histogram of different values of the coefficient at the cubic term of
Eq. (2) is moderately skewed. It covers values from

Scatter diagram of coefficients at the cubic term of the Gardner
equation against water depth for internal waves of the second mode

Even though several properties of water masses of the South China Sea exhibit
extensive seasonal variations, this feature does not necessarily become
evident in terms of kinematic parameters of internal waves of the second
mode. The maps of quantities that express the normalised stratification
conditions

Gardner's equation is, strictly speaking, only an asymptotically valid model
for weakly non-linear long internal waves. Thus, its applicability should be
discussed for each particular environment and set of parameters of internal
waves. The observed amplitudes of internal waves of the second mode in the
shelf region of the South China Sea were in the range of 10–30 m. According
to Yang et al. (2009, 2010), amplitudes of internal solitons of the second
mode are about 20 m. We use the value

The above shows that in this region usually

The derived maps of various parameters of the governing quantities of the underlying model (such as the location of the maxima of the modal function) and the parameters of the weakly non-linear models provide a new insight into qualitative features of the propagation and transformations of internal waves of the second mode in the South China Sea. The presented climatologically valid distributions of the phase speed and coefficients at the non-linear terms of Gardner's equation (1) (or other equations of the family of KdV-type equations) may be used for expressing estimates of various parameters of internal waves of this kind. This includes inter alia evaluation of hydrodynamic loads on the seabed (and on offshore engineering structures) created by the propagation of such waves, forecasting of areas and depths strongly affected by the internal wave activity after intense wave generation events, and identification of regions with a very high probability that such waves will break.

A promising development is the possibility of evaluation of the limiting amplitude of internal solitons that correspond to negative values of the coefficient at the cubic non-linear term (Kurkina et al., 2011, 2017) as well as the amplitude of algebraic solitons that correspond to the positive values of this coefficient.

The main conclusions of the study are the following.

Spatial distributions of all kinematic parameters of internal waves of the second mode in the South China Sea (except for the coefficient at the cubic non-linear term of Gardner's equation) are qualitatively similar to analogous distributions for internal waves of the first mode.

The dispersive term of Gardner's equation for internal waves of the second mode is about 3–4 times smaller than this term for waves of the first mode.

The phase speed for internal waves of the second mode is about half of that for waves of the first mode.

The coefficients at the quadratic and cubic terms of Gardner's equation for internal waves of the second mode mainly depend on the stratification and much less on the total water depth.

In contrast to internal waves of the first mode, the quadratic term of Gardner's equation is mostly negative for waves of the second mode in the South China Sea.

The derived maps of the parameters as well as underlying data for histograms and scatter plots may be obtained from the authors in digital form via requests by e-mail.

The authors declare that they have no conflict of interest.

This study was initiated in the framework of the state task programme in the sphere of scientific activity of the Ministry of Education and Science of the Russian Federation (project nos. 5.4568.2017/6.7 and 5.1246.2017/4.6) and financially supported by this programme, grants of the President of the Russian Federation (NSh-6637.2016.5 and MK-5208.2016.5), the Russian Foundation for Basic Research (grant no. 16-05-00049), and institutional support IUT33-3 from the Estonian Research Council.Edited by: Vasiliy I. Vlasenko Reviewed by: three anonymous referees