Ocean internal waves interpreted as oscillation travelling waves in consideration of ocean dissipation

Abstract

Ocean internal waves interpreted as oscillation travelling waves in consideration of ocean dissipation *Jiang Zhu-Hui(姜祝辉) a)b) † , Huang Si-Xun(黄思训) a) , You Xiao-Bao(游小宝) b)c) , and Xiao Yi-Guo(肖义国) b) a)

Variational principles for two kinds of non-linear geophysical KdV equation with fractal derivatives

“…Under such a small scale, the media becomes discontinuous, and the fractal calculus [38][39][40][41][42] has to be adopted. Equations ( 1) and ( 2) are very useful models to describe oceanic internal solitary waves in continuous media [49][50][51][52][53], however an unsmooth boundary will greatly affect the properties of non-linear waves. Therefore, the smooth space (X, T) should be replaced by a fractal space (X β , T α ), where β and α are fractal dimensions in space and time, respectively.…”

“…Ocean internal wave is a common natural phenomenon, which is investigated by researchers using some non-linear internal wave models such as the KdV equation [49][50][51][52][53]. However, in practical application, some internal solitary waves often have strong nonlinearity, which is inconsistent with the weak non-linear assumption of the KdV equation.…”

“…So, it is more practical to use the high-order EKdV eq. ( 10) to simulate the internal solitary wave in the ocean on the free surface [44][45][46][47][48][49][50]. At the same time, the high-order EKdV has the ability to simulate the propagation and fission process of large amplitude and strong non-linear internal solitary waves.…”

“…The KdV-type equations have numerous applications in various branches of science and engineering. Various research works have been reported over the recent years related to KdV like equations by different authors [43][44][45][46][47][48][49][50][51]. Here we aimed to study the results of gKdV equation with Coriolis term, which represents the impact of the Earth's rotation on the fluid, and is admissible for large-scale ocean waves [43].…”

“…In this paper, variational principles are established by the semi-inverse method [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] for the non-linear geophysical KdV (gKdV) equation with Coriolis term, and the high-order extended KdV (EKdV) equation, in fractal space and time derivatives [36][37][38][39][40][41][42], respectively. Although both KdV-type equations in this paper have been extensively studied for a long time by some scientists [43][44][45][46][47][48][49][50][51][52][53], but, up to now, variational principles for them with fractal derivatives have not been dealt with. Therefore, finding variational principles for them is of great value, and might find lots of applications in numerical simulations and scientific researches.…”