On the Hamiltonian and geometric structure of Langmuir circulation

Cheng Yang; Cheng Yang

doi:10.3934/cam.2023004

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 58-69. doi: 10.3934/cam.2023004

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On the Hamiltonian and geometric structure of Langmuir circulation

Cheng Yang ^,

Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore

Received: 30 December 2022 Revised: 27 February 2023 Accepted: 08 March 2023 Published: 16 March 2023
76M60, 76E30, 37K30, 37K45, 37K65

The Craik-Leibovich equation (CL) serves as the theoretical model for Langmuir circulation. We show that the CL equation can be reduced to the dual space of a certain Lie algebra central extension. On this space, the CL equation can be rewritten as a Hamiltonian equation corresponding to the kinetic energy. Additionally, we provide an explanation of the appearance of this central extension structure through an averaging theory for Langmuir circulation. Lastly, we prove a stability theorem for two-dimensional steady flows of the CL equation. The paper also contains two examples of stable steady CL flows.
- Langmuir circulation,
- Craik-Leibovich equation,
- Euler equation,
- central extension,
- Hamiltonian structure,
- stability
Citation: Cheng Yang. On the Hamiltonian and geometric structure of Langmuir circulation[J]. Communications in Analysis and Mechanics, 2023, 15(2): 58-69. doi: 10.3934/cam.2023004

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Abstract

The Craik-Leibovich equation (CL) serves as the theoretical model for Langmuir circulation. We show that the CL equation can be reduced to the dual space of a certain Lie algebra central extension. On this space, the CL equation can be rewritten as a Hamiltonian equation corresponding to the kinetic energy. Additionally, we provide an explanation of the appearance of this central extension structure through an averaging theory for Langmuir circulation. Lastly, we prove a stability theorem for two-dimensional steady flows of the CL equation. The paper also contains two examples of stable steady CL flows.

References

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