The Boundary Layer Structure Beneath a Gravity Wave Traveling on a Turbulent
Current and the Proper Decomposition of the Wave-Induced Velocity Field
Edwin A. Cowen1 & Stephen G. Monismith2
The understanding of many air-sea processes, including the transport of mass, momentum
and heat at the air-water interface, is confounded by the lack of highly resolved
measurements, in both space and time, of the local near-interface turbulence. Such
measurements are necessary to fully understand transport events at an air-water interface and
deduce the physical processes that govern these events. Further, without highly resolved,
accurate velocity field measurements during momentum transfer events it is difficult to
properly parameterize the physics of momentum transfer for incorporation into models of the
marine boundary layer. Additional measurements, either in the field or in the laboratory (both
of which present unique measurement challenges), are only part of the issue. A major
question needs to be addressed – how should the turbulent velocity field at a moving airwater interface be separated from the unsteady but essentially irrotational motions?
In this paper we will present measurements of the aqueous momentum boundary layer
structure made beneath an essentially monochromatic gravity wave in the laboratory using a
digital particle tracking velocimetry technique (DPTV) developed specifically for wavy freesurface flows (Cowen & Monismith, 1997). The momentum boundary layer has been
resolved to better than 150 micron at eight phases of a gravity wave (wave number, k, 6.21
rad/m, wave amplitude, a, 23.6 mm, water depth, h, 0.354 m). Careful attention has been
paid to the question of velocity field decomposition to accurately separate the turbulence
from the wave-induced orbital motions as well as the inherent orbital velocity variability due
to the wave group structure as well as experimental repeatability considerations.
A free-surface based coordinate system that maps a second-order wavy free-surface profile
onto an appropriately stretched/compressed rectangular domain (Norris & Reynolds, 1975;
McDonald, 1994) is used to follow the boundary layer as a function of phase. Extending the
wave decomposition technique of Thais & Magnaudet (1995) to spatially resolved data fields,
the velocity field is decomposed as follows
u(x, t ) = u (x) + u P (x, t ) + u R (x, t ) + u′(x, t )
where u is the instantaneous velocity vector at spatial location x and time t, u is the temporal
mean velocity vector, u P is the irrotational wave-induced velocity vector, u R is the
rotational wave-induced velocity vector, and u′ is the turbulent velocity perturbation vector.
Importantly, u P is used to separate the effects of amplitude and phase-induced variability in
the wave-induced motions from the turbulence. We will describe the issues and details of the
above wave-turbulence decomposition.
The data set, upon the proper decomposition of the velocity field, reveals four distinct
boundary layer structures. These boundary layers are: the phase dependent oscillatory free1
Asst. Professor and Director, DeFrees Hydraulics Laboratory, School of Civil and Environmental Engineering,
Cornell University, Ithaca, NY 14853-3501, eac20@cornell.edu
2
Professor and Director, Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental
Engineering, Stanford University, Stanford, CA 94305-4020, monismith@ce.stanford.edu
surface Stokes’ layer (prelimary data shown in figure 1), a wave energy decay induced outer
layer (preliminary data shown in figure 2), a turbulent boundary layer, and a capillary wave
influenced layer. The Stokes layer (figure 1) scales similarly to a traditional oscillatory flatplate Stokes layer where β = [σ /(2ν )]1/ 2 is the viscous length scale, σ the radian based
wave frequency and ν the kinematic viscosity. Figure 2 shows the departure of the measured
Eulerian mean velocity profile from the core flow Eulerian mean profile (below kz = 0.12).
This departure is the result of the wave energy decay induced stress which drives a boundary
layer that scales as a turbulent diffusional process (note the considerably longer length scale
of this boundary layer). We will describe and discuss the four boundary layer structures,
including typical length scales and growth rates for each. In addition to all quantities required
for the wave-turbulence decomposition algorithm, the DPTV measurement technique allows
the direct determination of the rms vorticity field, turbulent dissipation, and turbulent kinetic
energy (TKE) field. The rms vorticity field proves to be useful in separating capillary waveinduced velocity fields from the turbulence while the dissipation and TKE allow the turbulent
energy budget to be investigated.
0.000
0.00
-0.005
1.62
0.00
-0.02
3.24
-0.015
4.86
kz
-0.010
βz*
kz*
-0.04
-0.06
-0.08
-0.020
6.48
-15
-10
-5
0
2
<ωE >/(σε )
5
10
15
Figure 1: Phase dependent near-surface
vorticity profiles.
-0.10
-0.12
-1.00
-0.95
-0.90
2
uE wave-induced/(cε )
-0.85
-0.80
Figure 2: Wave-induced Eulerian mean
velocity profile. Dashed line is a
negative stokes drift profile.
References
Cowen, E.A.; Monismith, S.G. (1997). A hybrid digital particle tracking velocimetry
technique. Exp. Fluids 22, 199 –211.
McDonald, B.K. (1994). Modeling Laminar Flow Beneath a Prescribed Small-Amplitude
Wavy Surface. Eng. Dissertation, Dept. of Civil Eng., Stanford University.
Norris, H.L.; Reynolds, W.C. (1975). Turbulent channel flow with a moving wavy boundary.
Dept. of Mech. Eng. Technical Report TF-7, Stanford University.
Thais, L.; Magnaudet, J. (1995). A triple decomposition of the fluctuating motion below
laboratory wind water-waves. J. Geophys. Res. (Oceans) 100, 741-755.