Role of wave scattering in instability-induced Langmuir circulation
Abstract: We consider a classical problem about dynamic instability that leads to the Langmuir circulation. The problem statement assumes that there is initially a wind-driven shear flow and a plane surface wave propagating in the direction of the flow. The unstable mode is a superposition of i) shear flow and ii) surface waves both modulated in the horizontal spanwise direction and iii) circulation that is made up with vortices forming near-surface rolls whose axis are coaligned along the shear flow streamlines and whose transverse size corresponds to the modulation period. Usually, the Langmuir circulation is understood as the vortical part of the mode slowly varying in time, which is the combination of the first and the last flows. The novelty of our approach is that we, firstly, take into account the scattering of the initial surface wave on the slow current. Second, we find the interference of the scattered and the initial waves generating a Stokes drift modulated in the same direction. Third, we establish the subsequent affect of the circulation by the vortex force created by the nonlinear interaction of the initial shear flow and the modulated part of the Stokes drift. S. Leibovich & A.D.D. Craik previously showed that the third part of the mechanism could maintain the Langmuir circulation. We calculate the growth rate which is approximately twice smaller than that obtained by A.D.D. Craik. The vertical structure of the circulation in the mode consists of two vortices, that corresponds to the next mode in Craik's model. Considering the wave scattering, we describe the fast-wave motion as a potential flow with relatively weak vortical correction. Application of the technique can be expanded on other flows where fast oscillating surface waves coexist with a slow current.
- S. A. Thorpe, ‘‘Langmuir circulation,’’ Annual Review of Fluid Mechanics 36, 55–79 (2004).
- M. A. C. Teixeira, ‘‘Langmuir circulation and instability,’’ in Encyclopedia of Ocean Sciences (Third Edition), edited by J. K. Cochran, H. J. Bokuniewicz, and P. L. Yager (Academic Press, Oxford, 2019) pp. 92–106.
- R. A. Weller and J. F. Price, ‘‘Langmuir circulation within the oceanic mixed layer,’’ Deep Sea Research Part A. Oceanographic Research Papers 35, 711–747 (1988).
- A. D. D. Craik, ‘‘The generation of Langmuir circulations by an instability mechanism,’’ Journal of Fluid Mechanics 81, 209–223 (1977).
- S. Leibovich, ‘‘Langmuir circulation and instability,’’ Elements of Physical Oceanography: A derivative of the Encyclopedia of Ocean Sciences 124, 288 (2009).
- A. J. Faller and E. A. Caponi, ‘‘Laboratory studies of wind-driven Langmuir circulations,’’ Journal of Geophysical Research: Oceans 83, 3617–3633 (1978).
- P. P. Sullivan and J. C. McWilliams, ‘‘Langmuir turbulence and filament frontogenesis in the oceanic surface boundary layer,’’ Journal of Fluid Mechanics 879, 512–553 (2019).
- A. D. D. Craik and S. Leibovich, ‘‘A rational model for Langmuir circulations,’’ Journal of Fluid Mechanics 73, 401–426 (1976).
- A. D. D. Craik, ‘‘Wave-induced longitudinal-vortex instability in shear flows,’’ Journal of Fluid Mechanics 125, 37–52 (1982).
- T. Kawamura, ‘‘Numerical investigation of turbulence near a sheared air–water interface. part 2: Interaction of turbulent shear flow with surface waves,’’ Journal of marine science and technology 5, 161–175 (2000).
- F. Veron and W. K. Melville, ‘‘Experiments on the stability and transition of wind-driven water surfaces,’’ Journal of Fluid Mechanics 446, 25–65 (2001).
- Y. Fujiwara and Y. Yoshikawa, ‘‘Mutual interaction between surface waves and Langmuir circulations observed in wave-resolving numerical simulations,’’ Journal of Physical Oceanography 50, 2323–2339 (2020).
- N. Suzuki, ‘‘On the physical mechanisms of the two-way coupling between a surface wave field and a circulation consisting of a roll and streak,’’ Journal of Fluid Mechanics 881, 906–950 (2019).
- S. Leibovich, ‘‘On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and averaged current,’’ Journal of Fluid Mechanics 79, 715–743 (1977).
- L.-A. Couston, M. A. Jalali, and M.-R. Alam, ‘‘Shore protection by oblique seabed bars,’’ Journal of Fluid Mechanics 815, 481–510 (2017).
- H. Lamb, Hydrodynamics (University Press, 1932).
- D. D. Holm, ‘‘The ideal Craik-Leibovich equations,’’ Physica D: Nonlinear Phenomena 98, 415–441 (1996).
- R. H. Stewart and J. W. Joy, ‘‘HF radio measurements of surface currents,’’ in Deep sea research and oceanographic abstracts, Vol. 21 (Elsevier, 1974) pp. 1039–1049.
- M. S. Longuet-Higgins, ‘‘Mass transport in water waves,’’ Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 245, 535–581 (1953).
- J. A. Nicolás and J. M. Vega, ‘‘Three-dimensional streaming flows driven by oscillatory boundary layers,’’ Fluid Dynamics Research 32, 119–139 (2003).
- C. Garrett, ‘‘Generation of Langmuir circulations by surface waves — a feedback mechanism,’’ Journal of Marine Research 34, 117 (1976).
- A. A. Abrashkin and E. N. Pelinovsky, ‘‘Gerstner waves and their generalizations in hydrodynamics and geophysics,’’ Physics–Uspekhi 65, 453–467 (2022).
- S. Leibovich, ‘‘The form and dynamics of Langmuir circulations,’’ Annual Review of Fluid Mechanics 15, 391–427 (1983).
- A. D. D. Craik, ‘‘A wave-interaction model for the generation of windrows,’’ Journal of Fluid Mechanics 41, 801–821 (1970).
- S. Manna and A. Dhar, ‘‘Modulational instability of obliquely interacting capillary-gravity waves over infinite depth,’’ Archives of Mechanics 73, 583–598 (2021).
- H. Yin, Q. Pan, and K. Chow, ‘‘Modeling “crossing sea state” wave patterns in layered and stratified fluids,’’ Physical Review Fluids 8, 014802 (2023).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.