Modelling the influence of
wind waves on the turbulent boundary layers above and below the air-water
interface
Alastair D. Jenkins, University of Bergen & Bjerknes Centre for Climate Research, Bergen, Norway
Øyvind Sætra, Norwegian Meteorological Institute, Oslo, Norway Yuliya Troitskaya, Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia
Acknowledgements
National science funding authorities in
Norway (Research Council of Norway) – projects Atmosphere- Ocean Interaction and IPY/ArcChange
Russian Federation
Introduction
The exchange of momentum, mechanical and thermal energy, and mass, between the atmosphere and the ocean is mediated by laminar and turbulent boundary-layer processes
above and below the air-water interface, and at the interface itself.
Wind-driven waves on the interface interact with the mean flow in both boundary layers, via several mechanisms:
Stokes drift due to non-closed fluid particle paths
critical layers where the mean flow speed is equal to the phase speed of wave Fourier components,
injection of momentum into the mean flow in association with wave energy dissipation by wave breaking and other processes.
Wave breaking also provides a source of turbulent kinetic energy for mixing.
The processes mentioned above depend nonlinearly on the wave amplitude.
They may are slowly-varying "mean" quantities of second order in a small parameter such as wave steepness,
May be computed using the wave Fourier spectrum and the wave energy input and dissipation terms employed in spectral wave prediction models.
Since the flow variables change rapidly in the direction normal to the interface, one should employ a suitable surface-following coordinate system.
Example of surface-following coordinates
Example of surface-following coordinates:
In this case, orthogonal above surface and approximately Lagrangian below surface
Strong nonlinearities and wave breaking
Steeper, (micro)breaking wave effects may be regarded as “weak in the mean” (K. Hasselmann 1974).
Even if perturbative methods cannot be applied to the breaking processes, the net effect may be regarded as more well-behaved
Breakup of the surface (spray, bubbles) may or may not need special treatment
Newell & Zakharov, energy of surface formation
Phillips' “b” parameter, representing strength of breaking events, may be related to geometrical scale of breaking crest structure
(e.g. Jenkins, JFM 1994)
Geometrical scale of breaking crest, proportional to 2/3 power of flux of fluid through jet (Jenkins 1994)
Modelling the effects of surface waves
Within the water column, the effect of the waves on the
mean flow may be computed directly, via an analytical description of the wave orbital motion.
Above the surface it is necessary to invoke an iterative procedure, since equations of the Orr-Sommerfeld type must be integrated
numerically for each wave component.
Wind- and wave-induced near-surface current
(Jenkins, JPO 1986, 1987; Dt. Hydrogr. Z. 1989) Driven using 1-point version of WAM model
Hodograph, squares mark every 3 pendulum hours
Wave-mean flow-turbulence interaction being incorporated into GOTM turbulence closure model
Turbulence model with pressure transport terms calculated by the wave model
∂ e
∂ t = A
w{ ∂ ∂ U z 2 ∂ ∂ V z
2} − 1 ρ ∂ ∂ z p'w ' ∂ ∂ z Az ∂ ∂ e z − 2 Bl qe
∂ ∂ e z − 2 Bl qe
− 1 ρ
∂
∂ z p'w' = 2
ρ ∫ kS dis e −2k∣z ∣ dk
Relation between pressure transport term and dissipation of
waves:
Boundary condition at Z=0
A w ∂ e
∂ z = 0
A w ∂ e
∂ z = α u ∗ 3
When pressure term is not included:
z
0= 1.8 Hs
z
0= 0.1ν / u
wWhen pressure term is included:
• Compare with observations of dissipation
reported by Terray et al. (1996) and Drennan et al (1996).
• The observed wind is used in a one-grid WAM model till the observed wave height is obtained
• Then the energy flux from WAM is then used to
force the turbulent Ekman model
U=12.05m/s, Hs=2.9m
Black lines no pressure transport terms Red lines – with pressure transport terms
Run with all obs: U=11.00m/s, Hs=2.5m
Black lines no pressure transport terms Red lines – with pressure transport terms
Quasi-linear model of Quasi-linear model of
turbulent wind over waved turbulent wind over waved
water surface
water surface
Vertical profile of
momentum flux terms [Jenkins 1992 JPO]
1 = mean (turbulent) shear stress
2 = pressure-slope covariance
. . . .
Model of the wind flow
Wind ≡ The turbulent air boundary layer over water surface
ν - the eddy viscosity coefficient is a given function of z
The first order semi-empirical model of turbulence
Boundary conditions at the air-sea interface
z= ξ (x,y,t)
The quasi-linear approximation
The set of linear equations for the
disturbances induced in the air by a single harmonic wave at the water surface
Boundary conditions
The mean velocity components
x
y
The mean wind velocity
The quasi-linear approximation
Momentum balance in the air turbulent boundary layer
, ,
F k
C a lc u l a t i o n s f o r Ω = 4 0 s m o o t h r e g i m e
r o u h r e g i m e
( D o n e l a n e t a l , G R L , 2 0 0 4 ) ( D o n e l a n e t a l , G R L , 2 0 0 4 )
( D o n e l a n e t a l , G R L , 2 0 0 4 )
M. D. Powell, P. J. Vickery & T. A. Reinhold, Nature, 2003
Friction velocity
M. D. Powell, P. J. Vickery & T. A. Reinhold, Nature, 2003
Roughness height
Concluding remarks
The effect of surface waves on the adjacent atmospheric and oceanic boundary layers can in many cases be modelled
successfully by quasilinear perturbation methods.
Such methods require us, formally or informally, to employ surface-following coordinate systems
Successful applications include:
The prediction of near-surface (and near-bottom) drift currents
Estimation of near-surface ocean turbulent energy dissipation
The computation of air-sea momentum flux (drag coefficient, effective surface roughness)
In particular, it is not necessary to assume any dominating effect of spray in order to reproduce observed drag coefficient reductions at very high wind speeds
Air-sea mass (and heat) flux may be investigated suitable parameterizations of the effect of breaking wave crests.