On the distortion of turbulence by a progressive surface wave

Abstract

A rapid-distortion model is developed to investigate the interaction of weak turbulence with a monochromatic irrotational surface water wave. The model is applicable when the orbital velocity of the wave is larger than the turbulence intensity, and when the slope of the wave is sufficiently high that the straining of the turbulence by the wave dominates over the straining of the turbulence by itself. The turbulence suffers two distortions. Firstly, vorticity in the turbulence is modulated by the wave orbital motions, which leads to the streamwise Reynolds stress attaining maxima at the wave crests and minima at the wave troughs; the Reynolds stress normal to the free surface develops minima at the wave crests and maxima at the troughs. Secondly, over several wave cycles the Stokes drift associated with the wave tilts vertical vorticity into the horizontal direction, subsequently stretching it into elongated streamwise vortices, which come to dominate the flow. These results are shown to be strikingly different from turbulence distorted by a mean shear flow, when ‘streaky structures’ of high and low streamwise velocity fluctuations develop. It is shown that, in the case of distortion by a mean shear flow, the tendency for the mean shear to produce streamwise vortices by distortion of the turbulent vorticity is largely cancelled by a distortion of the mean vorticity by the turbulent fluctuations. This latter process is absent in distortion by Stokes drift, since there is then no mean vorticity.The components of the Reynolds stress and the integral length scales computed from turbulence distorted by Stokes drift show the same behaviour as in the simulations of Langmuir turbulence reported by McWilliams, Sullivan & Moeng (1997). Hence we suggest that turbulent vorticity in the upper ocean, such as produced by breaking waves, may help to provide the initial seeds for Langmuir circulations, thereby complementing the shear-flow instability mechanism developed by Craik & Leibovich (1976).The tilting of the vertical vorticity into the horizontal by the Stokes drift tends also to produce a shear stress that does work against the mean straining associated with the wave orbital motions. The turbulent kinetic energy then increases at the expense of energy in the wave. Hence the wave decays. An expression for the wave attenuation rate is obtained by scaling the equation for the wave energy, and is found to be broadly consistent with available laboratory data.

Attenuation of Gravity Waves by Turbulence

“…Teixeira and Belcher (2002) developed a rapid distortion model to study the interaction of a single irrotational monochromatic wave and weak turbulence. The model is applicable when the orbital velocity is larger than the turbulence and the slope of the wave is sufficiently high that the straining of the turbulence by waves dominates over the straining of the turbulence itself.…”

“…Comparison with Teixeira and Belcher (2002) was carried out using the extrapolated value of u'=0.0033 at z 0 obtained previously. The use of this value requires increasing the original constant α in a factor of 35 in order to match the observations.…”

“…Theoretical estimates of wave attenuation due to turbulence validated with limited field and experimental data (Teixeira andBelcher 2002, Ardhuin andJenkins 2006) have recently been incorporated in wave models, e.g. WAVEWATCH III (Tolman 2009).…”

Wave modelling – The state of the art

“…In order to represent the stretching, rapid-distortion theory was applied on the water-side of the surface by Teixeira and Belcher (2002). The theory assumes that the eddy turn-over time is less than the wave period, or, said differently, that the strain rate of the turbulence by the wave motion is more than that of the turbulence by itself, and that the turbulent velocity is much less than the wave-induced velocity.…”

A Note on the Effect of Wind Waves on Vertical Mixing in Franks Tract, Sacramento–San Joaquin Delta, California

“…Surface waves can modify hydrodynamics near the free surface in three ways (Craig and Banner 1994). First, the interaction of the wave Stokes drift (the time-averaged drift of a particle being moved by a wave) with the wind-driven surface shear current can result in Langmuir circulation formation (as reported by Craik and Leibovich 1976;Skyllingstad and Denbo 1995;Teixeira and Belcher 2002). Second, Reynolds stresses (the mean forces imposed on the mean flow by turbulent fluctuations) can be created when the waves are not perfectly irrotational (e.g., Magnaudet and Thais 1995).…”