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Yang, Cheng

On the Hamiltonian and geometric structure of Langmuir circulation. (English) Zbl 1517.76015

Commun. Anal. Mech. 15, No. 2, 58-69 (2023).
Summary: 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.

Cited in 1 Document

MSC:

76B07 Free-surface potential flows for incompressible inviscid fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76E30 Nonlinear effects in hydrodynamic stability
35Q31 Euler equations
53Z05 Applications of differential geometry to physics

Keywords:

Craik-Leibovich equation; Euler equation; Lie algebra central extension; stability; averaged Langmuir circulation
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References:

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