Internal waves in a two‐layer system using fully nonlinear internal‐wave equations

Abstract

SUMMARYIn order to understand the nonlinear effect in a two-layer system, fully nonlinear strongly dispersive internal-wave equations, based on a variational principle, were proposed in this study. A simple iteration method was used to solve the internal-wave equations in order to solve the equations stably. The applicability of the proposed numerical computation scheme was confirmed to agree with linear dispersion relation theoretically obtained from variational principle. The proposed computational scheme was also shown to reproduce internal waves including higher-order nonlinear effect from the analysis of internal solitary waves in a two-layer system. Furthermore, for the second-order numerical analysis, the balance of nonlinearity and dispersion was found to be similar to the balance assumed in the KdV theory and the Boussinesq-type equations.

Large Amplitude Solitary Waves Due to Solitary Resonance

“…where  1 and  2 indicate the density in the upper and lower layers, H and h 0 indicate the total and lower layer thickness,  is the interfacial displacement, p 1 and p 2 indicate the pressure in the upper and lower layer, a is the amplitude of internal solitary wave, h is the interfacial level, K is the coefficient regarding spatial scale, A ij (i, j = 1~3) indicates the coefficient for interfacial displacement and C i (i = 1~3) indicates the coefficient of wave speed. The upper and lower layer thicknesses were given as 0.20 m and 0.8 m, respectively, the density ratio of the two layers was 1.000/2.000, and the amplitude was given 0.8x0.05 m. Nakayama and Kakinuma [5] revealed that the 3rd order solutions provide a more expanded internal solitary wave, which was also confirmed in the present study ( Figure 2).…”

“…Thus, we derived and approved third-order internal solitary wave equations based on the ninth-order solitary wave equations [3] by using a fully nonlinear and strongly dispersive internal wave model (FSI model; Kakinuma [4], Nakayama and Kakinuma [5]). This study aimed to reveal the mechanisms by which freak waves may occur due to solitary resonance by using the FSI model.…”

Numerical Analyses on Propagation of Nonlinear Internal Waves

“…(7), (8), (9), and (10) or the set of Eqs. (11), (13), (14), and (16) is rewritten to a set of finite difference equations and the time development is carried out by applying implicit schemes similar to that of Nakayama and Kakinuma (2010).…”

Shoaling of Nonlinear Internal Waves on a Uniformly Sloping Beach

“…However, the interface gradient through the present model is milder than that of the BO soliton as shown in Fig. 7, which means that the present model considers wave nonlinearity better than the BO equation since the interface gradient of strongly nonlinear wave is milder than that through the theories for weakly nonlinear waves as the KdV theory (Nakayama and Kakinuma, 2010). The profiles of BO solitons are much different from the calculation results, where larger number of terms for the velocity potential are required to represent the wave dispersion more accurately.…”

“…Nonlinear internal wave eqations based on the variational principle are as follows (Yamashita et al, 2011): (6), (7), and (8) are rewritten to finite difference equations, after which the time development is carried out by applying implicit schemes similar to that of Nakayama and Kakinuma (2010).…”