TY - JOUR

T1 - On regularizing the strongly nonlinear model for two-dimensional internal waves

AU - Barros, Ricardo

AU - Choi, Wooyoung

N1 - Funding Information:
WC gratefully acknowledges support from the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology through the World Class University (WCU) program with grant no. R31-2008-000-10045-0 .

PY - 2013

Y1 - 2013

N2 - To study the evolution of two-dimensional large amplitude internal waves in a two-layer system with variable bottom topography, a new asymptotic model is derived. The model can be obtained from the original Euler equations for weakly rotational flows under the long-wave approximation, without making any smallness assumption on the wave amplitude, and it is asymptotically equivalent to the strongly nonlinear model proposed by Choi and Camassa (1999) [3]. This new set of equations extends the regularized model for one-dimensional waves proposed by Choi et al. (2009) [30], known to be free from shear instability for a wide range of physical parameters. The two-dimensional generalization exhibits new terms in the equations, related to rotational effects of the flow, and possesses a conservation law for the vertical vorticity. Furthermore, it is proved that if this vorticity is initially zero everywhere in space, then it will remain so for all time. This property-in clear contrast with the original strongly nonlinear model formulated in terms of depth-averaged velocity fields-allows us to simplify the model by focusing on the case when the velocity fields involved by large amplitude waves are irrotational. Weakly two-dimensional and weakly nonlinear limits are then discussed. Finally, after investigating the shear stability of the regularized model for flat bottom, the effect of slowly-varying bottom topography is included in the model.

AB - To study the evolution of two-dimensional large amplitude internal waves in a two-layer system with variable bottom topography, a new asymptotic model is derived. The model can be obtained from the original Euler equations for weakly rotational flows under the long-wave approximation, without making any smallness assumption on the wave amplitude, and it is asymptotically equivalent to the strongly nonlinear model proposed by Choi and Camassa (1999) [3]. This new set of equations extends the regularized model for one-dimensional waves proposed by Choi et al. (2009) [30], known to be free from shear instability for a wide range of physical parameters. The two-dimensional generalization exhibits new terms in the equations, related to rotational effects of the flow, and possesses a conservation law for the vertical vorticity. Furthermore, it is proved that if this vorticity is initially zero everywhere in space, then it will remain so for all time. This property-in clear contrast with the original strongly nonlinear model formulated in terms of depth-averaged velocity fields-allows us to simplify the model by focusing on the case when the velocity fields involved by large amplitude waves are irrotational. Weakly two-dimensional and weakly nonlinear limits are then discussed. Finally, after investigating the shear stability of the regularized model for flat bottom, the effect of slowly-varying bottom topography is included in the model.

KW - Internal waves

KW - Regularization

KW - Strongly nonlinear model

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U2 - 10.1016/j.physd.2013.08.010

DO - 10.1016/j.physd.2013.08.010

M3 - Article

AN - SCOPUS:84884773357

VL - 264

SP - 27

EP - 34

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -