Propagation regimes of interfacial solitary waves in a three-layer fluid

Abstract

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

Kinematic parameters of internal waves of the second mode in the South China Sea

“…Similarly, the typical value of the coefficient at the cubic non-linear term is α 1 = 0.0005 m −1 s −1 and the magnitude of this term is α 1 A 2 = 0.2 m s −1 . The case when both non-linear terms are small is discussed in detail by Pelinovsky et al (2007) and Kurkina et al (2015). The typical magnitude of the phase speed of linear long internal waves of the second mode is c ∼ 0.4 m s −1 .…”

“…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).…”

Kinematic parameters of internal waves of the second mode in the South China Sea

“…The impact of stratification and mode correction on the value and sign of the coefficient at the cubic term in Gardner's equation is analysed in detail in Grimshaw et al (1997) and Kurkina et al (2015). Consequently, to specify the coefficients of Gardner's equation, it is necessary to evaluate the mode function from Eq.…”

“…Similarly, the typical value of the coefficient at the cubic non-linear term is α 1 = 0.0005 m −1 s −1 and the magnitude of this term is α 1 A 2 = 0.2 m s −1 . The case when both non-linear terms are small is discussed in detail by Pelinovsky et al (2007) and Kurkina et al (2015). The typical magnitude of the phase speed of linear long internal waves of the second mode is c ∼ 0.4 m s −1 .…”

Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

“…The Korteweg-de Vries (KdV) equation and its generalisations such as the Gardner (or more generally, higher-order KdV), Ostrovsky and Kadomtsev-Petviashvili (KP) equations are well known as good weakly-nonlinear models describing long surface and internal waves that are commonly observed in the oceans, e.g. Grimshaw, Pelinovsky & Talipova (1997); Grimshaw et al (1998); Helfrich & Melville (2006); Apel et al (2007); Grimshaw et al (2010); Ablowitz & Baldwin (2012); Grimshaw, Helfrich & Johnson (2013); Chakravarty & Kodama (2014); Liao et al (2014); Alias, Grimshaw & Khusnutdinova (2014); Grue (2015); Kurkina et al (2015); Khusnutdinova, Stepanyants & Tranter (2018) and references therein. These models apply to the waves with plane or nearly-plane fronts.…”