### Rotational Solitary Wave Interactions Over an Obstacle

#### Abstract

In this work we investigate the propagation of rotational solitary waves over a submerged obstacle in a vertically sheared shallow water channel with constant vorticity. In the weakly nonlinear regime, the problem is formulated in the forced Korteweg-de Vries framework. The initial value problem for this equation is solved numerically using a Fourier pseudospectral method with integrating factor. Solitary waves are taken as initial data, and the interaction wave-current-topography is analysed. We identify three types of regimes according to the intensity of the vorticity. A rotational solitary wave can bounce back and forth over the obstacle remaining trapped for large times, it can pass over the obstacle without reversing its direction or the wave can be blocked, i.e., it bounces back and forth above the obstacle until reaching a steady state. Such behaviour resembles the classical damped spring-mass system.

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Akylas, TR. On the excitation of long nonlinear water waves by a moving pressure distributions. J Fluid Mech. 1984; 141:455-466. DOI: 10.1017/S0022112084000926.

Baines, P. Topographic effects in stratified flows. Cambridge: Cambridge University Press; 1995.

Flamarion, MV. Rotational flows over obstacles in the forced Korteweg- de Vries framework. Selecciones Matemáticas. 2021; 8:(1) 125-130. DOI: 10.17268/sel.mat.2021.01.12.

Flamarion, MV; Milewski, PA; Nachbin, A. Rotational waves generated by current-topography interaction. Stud Appl Math. 2019; 142: 433-464. DOI: 10.1111/sapm.12253.

Flamarion, MV; Ribeiro-Jr, R. Trapped solitary-wave interaction for Euler equations with low-pressure region. Comp Appl Math. 2021; 40:(20) 1-11. DOI: 10.1007/s40314-020-01407-0.

Flamarion, MV; Ribeiro-Jr, R. Gravity-capillary flows over obstacles for the fifth-order forced Korteweg-de Vries equation. J Eng Math. 2021; 129:17 DOI: 10.1007/s10665-021-10153-z.

Grimshaw, R; Smyth N. Resonant flow of a stratified fluid over topogra- phy in water of finite depth. J. Fluid Mech. 1986; 169: 235-276. DOI: 10.1017/S002211208600071X.

Grimshaw. R; Pelinovsky, E; Tian, X. Interaction of a solitary wave with an external force. Physica D. 1994; 77: 405-433. DOI: 10.1002/sapm1996973235.7

Johnson, RS; Freeman, NC. Shallow water waves on shear flows. J. Fluid Mech. 1970; 42: 401-409. DOI: 10.1017/S0022112070001349.

Johnson, RS. Models for the formation of a critical layer in water wave propaga- tion. Phil. Trans. R. Soc. A. 2012; 370:1638-1660. DOI: 10.1098/rsta.2011.0456.

Kim, H; Choi, H. A study of wave trapping between two obstacles in the forced Korteweg-de Vries equation. J Eng Math. 2018; 108:197-208. DOI: 10.1007/s10665-017-9919-5.

Lee, S. Dynamics of trapped solitary waves for the forced KdV equation. Sym- metry. 2018; 10(5):129. DOI: 10.3390/sym10050129.

Lee, S; Whang, S. Trapped supercritical waves for the forced KdV equation with two bumps. Appl Math Model. 2015; 39:2649-2660. DOI: 10.1016/j.apm.2014.11.007.

Milewski, PA. The Forced Korteweg-de Vries equation as a model for waves generated by topography. CUBO A mathematical Journal. 2004; 6:33-51.

Pratt, LJ. On nonlinear flow with multiple obstructions. J. Atmos. Sci. 1984; 41:1214-1225. DOI: 10.1175/1520-0469.

Trefethen, LN. Spectral Methods in MATLAB. Philadelphia: SIAM; 2001.

Wu, TY. Generation of upstream advancing solitons by moving disturbances. J Fluid Mech. 1987; 184: 75-99. DOI: 10.1017/S0022112087002817.

Wu, DM; Wu, TY. Three-dimensional nonlinear long waves due to moving surface pressure. In: Proc. 14th. Symp. on Naval Hydrodynamics. Nat. Acad. Sci., Washington, DC. 1982; 103-25.

Trefethen LN. Spectral Methods in MATLAB. Philadelphia: SIAM; 2001.

DOI: https://doi.org/10.5540/tcam.2022.023.03.00531

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