TY - JOUR

T1 - High-order strongly nonlinear long wave approximation and solitary wave solution

AU - Choi, Wooyoung

N1 - Funding Information:
This research was supported by the US National Science Foundation through Grant No. DMS-2108524, and the Brain Pool Program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (Grant No. 2021H1D3A2A01082312).
Publisher Copyright:
© 2022 The Author(s). Published by Cambridge University Press.

PY - 2022/8/25

Y1 - 2022/8/25

N2 - We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.

AB - We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.

KW - solitary waves

KW - surface gravity waves

UR - http://www.scopus.com/inward/record.url?scp=85134876547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85134876547&partnerID=8YFLogxK

U2 - 10.1017/jfm.2022.544

DO - 10.1017/jfm.2022.544

M3 - Article

AN - SCOPUS:85134876547

VL - 945

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

M1 - A15

ER -