Surface gravity waves propagating in a rotating frame: the Ekman-Stokes instability

We study the stability properties of the Eulerian mean flow generated by monochromatic surface-gravity waves propagating in a rotating frame, see illustration in figure 1a. The wave averaged equations, also known as the Craik-Leibovich equations [1], govern the evolution of the mean flow. For propagating waves in a rotating frame these equations admit a steady depth-dependent base flow sometimes called the Ekman-Stokes spiral [3, 2, 4, 5], because of its resemblance to the standard Ekman spiral. This base flow profile is controlled by two non-dimensional numbers, the Ekman number Ek = f/(ν λ2) and the Rossby number Ro = Us/(f λ). Here λ is the wavelength of the surface waves, f is twice the rotation rate, Us is the Stokes drift velocity associated with the surface waves and ν is the kinematic viscosity.

We show that this steady laminar velocity profile is linearly unstable above a critical Rossby number Roc(Ek). We determine the threshold Rossby number as a function of Ek using a numerical eigenvalue solver, before confirming the numerical results through asymptotic expansions in the large/low Ek limit. We show the instability threshold in figure 1b (data points) along with the asymptotic results (dashed lines). These were also confirmed by nonlinear simulations of the Craik-Leibovich equations. When the system is well above the linear instability threshold, Ro >> Roc , the resulting flow fluctuates chaotically. We will discuss the possible implications in an oceanographic context, as well as for laboratory experiments.

Figure 1: a) Left: Schematic of the system under study. b) Right: Instability threshold Roc as a function of Ek. The symbols indicate the numerical values, while the dashed lines show the asymptotic results.

References

[1] A. D. D. Craik and S. Leibovich. A rational model for langmuir circulations. J. Fluid Mech., 73(3):401–426, 1976.

[2] A. Gnanadesikan and R. A. Weller. Structure and instability of the ekman spiral in the presence of surface gravity waves. J. Phys. Oceanogr., 25(12):3148–3171, 1995.

[3] N. E. Huang. On surface drift currents in the ocean. J. Fluid Mech., 91(1):191–208, 1979.

[4] J. C. McWilliams, E. Huckle, J.-H. Liang, and P. P. Sullivan. The wavy ekman layer: Langmuir circulations, breaking waves, and reynolds stress. J. Phys. Oceanogr., 42(11):1793–1816, 2012.

[5] 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(4):444–457, 2005.