- Access by Xinjiang University
Surface gravity waves propagating in a rotating frame: The Ekman-Stokes instability
Phys. Rev. Fluids 4, 104802 – Published 30 October, 2019
DOI: https://doi.org/10.1103/PhysRevFluids.4.104802
Abstract
We report on an instability arising when surface gravity waves propagate in a rotating frame. The Stokes drift associated to the uniform wave field, together with global rotation, drives a mean flow in the form of a horizontally invariant Ekman-Stokes spiral. We show that the latter can be subject to an instability that triggers the appearance of an additional horizontally-structured cellular flow. We determine the instability threshold numerically, in terms of the Rossby number Ro associated to the Stokes drift of the waves and the Ekman number E. We confirm the numerical results through asymptotic expansions at both large and low Ekman number. At large E the instability reduces to that of a standard Ekman spiral driven by the wave-induced surface stress instead of a wind stress, while at low E the Stokes-drift profile crucially determines the shape of the unstable mode. In both limits the instability threshold asymptotes to an Ekman-number-independent critical Rossby number, which in both cases also corresponds to a critical Reynolds number associated to the Lagrangian base-flow velocity profile. Parameter values typical of ocean swell fall into the low-E unstable regime: the corresponding “anti-Stokes” flows are unstable, with possible consequences for particle dispersion and mixing.
Physics Subject Headings (PhySH)
Article Text
References (36)
- O. M. Phillips, The scattering of gravity waves by turbulence, J. Fluid Mech. 5, 177 (1959).
- K. E. Kenyon, Wave refraction in ocean currents, Deep-Sea Res. Oceanogr. 18, 1023 (1971).
- K. B. Dysthe, Refraction of gravity waves by weak current gradients, J. Fluid Mech. 442, 157 (2001).
- O. Bühler, Waves and Mean Flows (Cambridge University Press, Cambridge, 2009).
- B. Gallet and W. R. Young, Refraction of swell by surface currents, J. Mar. Res. 72, 105 (2014).
- P. Gutierrez and S. Aumaître, Surface waves propagating on a turbulent flow, Phys. Fluids 28, 025107 (2016).
- A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge University Press, Cambridge, 1985).
- W. R. C. Phillips and Z. Wu, On the instability of wave-catalysed longitudinal vortices in strong shear, J. Fluid Mech. 272, 235 (1994).
- M. A. C. Teixeira and S. E. Belcher, On the distortion of turbulence by a progressive surface wave, J. Fluid Mech. 458, 229 (2002).
- W. R. C. Phillips, On the spacing of Langmuir circulation in strong shear, J. Fluid Mech. 525, 215 (2005).
- T. Humbert, S. Aumaître, and B. Gallet, Wave-induced vortex recoil and nonlinear refraction, Phys. Rev. Fluids 2, 094701 (2017).
- M. C. McWilliams, P. P. Sullivan, and C.-H. Moeng, Langmuir turbulence in the ocean, J. Fluid Mech. 334, 1 (1997).
- J. C. McWilliams and J. M. Restrepo, The wave-driven ocean circulation, J. Phys. Oceanogr. 29, 2523 (1999).
- A. D. D. Craik and S. Leibovich, A rational model for Langmuir circulations, J. Fluid Mech. 73, 401 (1976).
- S. Leibovich, On wave-current interaction theories of Langmuir circulations, J. Fluid Mech. 99, 715 (1980).
- A. D. D. Craik, Wave-induced longitudinal-vortex instability in shear flows, J. Fluid Mech. 125, 37 (1982).
- A. D. D. Craik, The generation of Langmuir circulations by an instability mechanism, J. Fluid Mech. 81, 209 (1977).
- Wu-ting Tsai et al., On the formation of coherent vortices beneath nonbreaking free-propagating surface waves, J. Phys. Oceanogr. 47, 533 (2017).
- Wu-ting Tsai, Shi-ming Chen, and Guan-hung Lu, Numerical evidence of turbulence generated by nonbreaking surface waves, J. Phys. Oceanogr. 45, 174 (2015).
- N. E. Huang, On surface drift currents in the ocean, J. Fluid Mech. 91, 191 (1979).
- M. S. Longuet-Higgins, Mass transport in water waves, Philos. Trans. Royal Soc. A 245, 535 (1953).
- Ü. Ünlüata and C. C. Mei, Mass transport in water waves, J. Geophys. Res. 75, 7611 (1970).
- O. S. Madsen, Mass transport in deep-water waves, J. Phys. Oceanogr. 8, 1009 (1978).
- Z. Xu and A. J. Bowen, Wave- and wind-driven flow in water of finite depth, J. Phys. Oceanogr. 24, 1850 (1994).
- Z. Hui and Y. Xu, The impact of wave-induced Coriolis-Stokes forcing on satellite-derived ocean surface currents, J. Geophys. Res. Oceans 121, 410 (2016).
- N. Suzuki and B. Fox-Kemper, Understanding Stokes forces in the wave-averaged equations, J. Geophys. Res.: Oceans 121, 3579 (2016).
- J. A. Polton, D. M. Lewis, and S. E. Belcher, The role of wave-induced Coriolis–Stokes forcing on the wind-driven mixed layer, J. Phys. Oceanogr. 35, 444 (2005).
- A. Gnanadesikan and R. A. Weller, Structure and instability of the Ekman spiral in the presence of surface gravity waves, J. Phys. Oceanogr. 25, 3148 (1995).
- G. E. Backus, The effect of the Earth's rotation on the propagation of ocean waves over long distances, Deep-Sea Res. 9, 185 (1962).
- F. Ursell, On the theoretical form of ocean swell on a rotating Earth, Monthly Notices Roy. Astron. Soc. 6, 1 (1950).
- R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res. 75, 5895 (1970).
- K. Hasselmann, Wave-driven inertial oscillations, Geophys. Astrophys. Fluid Dyn. 1, 463 (1970).
- A. J. Faller and R. Kaylor, Instability of the Ekman spiral with applications to the planetary boundary layers, Phys. Fluids 10, S212 (1967).
- U. Frisch, Turbulence: The legacy of A. N. Kolmogorov, (Cambridge University Press, Cambridge, 1995).
- B. D. Dore, On mass transport velocity due to progressive waves, Q. J. Mech. Appl. Math. 30, 157 (1977).
- J. E. Weber, Attenuated wave-induced drift in a viscous rotating ocean, J. Fluid Mech. 137, 115 (1983).
