Pr

1
Wave-current interactions in three dimensions
Ocean Sciences 2008
Fabrice Ardhuin1 and Nicolas Rascle1,2, Email
ardhuin_at_shom.fr, 1Service Hydrographique et
Océanographique de la Marine, Brest,
France 2now at Ifremer, Brest, France
Momentum conservation wave, mean flow, and total
momentum
The 3D primitive equations (PE) used for
numerical ocean modelling can be extended to
account for wave-current interactions. The most
general form of these equations uses the
Generalized Lagrangian Mean (Andrews and McIntyre
1978). The resulting equations (Ardhuin et al.
2008) is
Extending the approximate GLM
Because the present equations is approximated
from exact equations, one can parameterize the
errors to correct for biases drift velocity,
radiation stresses Here we look at the drift
velocity under incipient breaking waves. Example
of 14 s waves in 3 m water depth For any
water depth, one can compare the drift velocity
(solid) to linear waves with the same energy
(dash), and use the difference to parameterize
the non-linearity (left Rascle and Ardhuin, in
preparation). We have also
re-visited Miches breaking limit (right). More
interesting, is the correction of the linear
radiation stresses to be continued !

• These equations have a few terms on top of the
  usual PE
• The horizontal vortex force
• This is the vector product of the current
  vertical vorticity with the horizontal wave
  pseudo-momentum (Stokes drift). In the radiation
  stress form, this is part of the term
• The Vertical vortex force , product
  of the horizontal current vorticity and the
  vertical wave pseudo-momentum. This has no
  equivalent in 2D theory.
• A wave-modified mean pressure SJ in which one
  can isolate the contribution Sshear of the
  vertical shear of the current.
• A contribution to the Reynolds stresses
  including the momentum lost by breaking waves T
  wc.
• A possible parameterization is
• Contrary to the original GLM equations, the flow
  in these glm2z equations is non-divergent.
• Below is an example with linear waves shoaling on
  a slope, the divergence in wave-induced momentum
  flux is balanced by a set-down in the shallow
  area
• except in the bottom boundary layer, not modelled
  here (figures from Ardhuin et al. 2008b).

For the experts
Assuming a uniform current, Phillips (1977)
obtained the classical total momentum equation

• Interactions of waves and mean flows require a
  special mathematical treatment because wave
  motions are associated with energy and momentum
  fluxes, like turbulence, but also with
  (pseudo)-momentum and mean pressures. Depth
  integration yields equations for the mean
  horizontal momentum M (Mx , My) Mm Mw, the
  sum of mean flow and wave momenta. For a constant
  water density ?w we have (Smith 2006, with the
  simple addition of the Coriolis force),
• These equations use special assumptions and are
  not easily extended to 3D.
• For 3D, two approaches can be followed
• analytical continuation across the surface (e.g.
  McWilliams al. 2004)
• -gt interpretation ??
• surface-following coordinates (e.g. Andrews
  McIntyre 1978, Mellor 2003).
• Mellors (2003) approach is particularly seducing
  because it involves a change of only the vertical
  coordinate. This ? - coordinate is defined by
  following the local wave-induced vertical
  motions, so that wave motions are only along the
  horizontal in the new coordinate. Mellors s is
  the wave-induced vertical displacement ?3. This
  is simple for linear waves over a flat bottom,
  but not so simple when slopes, wind forcing, and
  wave field gradients are introduced (Ardhuin et
  al. 2008b).
• In particular Mellors equation have a 3D
  vertical radiation stress term ,
• that must include a correction of the order of
  the bottom slope.
• This correction is of the same order as the other
  radiation stress terms retained by Mellor.
  Including these correction raises the question of
  the wave potential over complex bottom
  topography which can be solved numerically (e.g.
  Athanassoulis Belibassakis 1999), but for which
  there is no simple expansions in
• powers of the local bottom slope. Asymptotic
  consistency is hard to achieve.

with
the wave-induced
mean pressure N.B. Srad ? SJ Cg?Ek?/?
the radiation stresses re the sum of two very
different terms the mean wave-induced pressure,
and the homogenous wave momentum flux.
??
??
Pressure under the waves
Horizontal Stokes drift
Mean current
Vertical Stokes drift
Application Momentum balance and velocity
profiles in the inner shelf and surf zone
Conclusions A practical GLM-based set of
equations was proposed. This approach has the
benefit of allowing strong shears in the
near-surface drift velocity with strong mixing at
the same time, consistent with observations. It
is consistent with McWilliams et al.s (2004)
Eulerian averages, thus providing an
interpretation of their velocities in the
crest-to-trough region as GLM averages, and
allowing for simple extensions to nonlinear
waves, standing waves... This GLM set is for
the mean flow momentum only. This choice avoids
complications that arise due to wave momentum
vertical advection/radiation in non-homogenous
conditions, which causes inconsistencies in
Mellors (2003) equations (Ardhuin et al. 2008b).
The present equations provide an extension, based
on first principles, of Smiths (2006) equations
to depth-varying currents. Implementation and
tests were performed in the ROMS code and will be
continued using other models. Outstanding
issues Turbulence closure in the presence of
waves We propose to use the GLM of the TKE
equation but this should be validated. Also, does
the mixing length vary as the water depth is
modulated by the wave motion (e.g. Huang and Mei
2003) ? How do I measure a GLM velocity ?
Video-based PIV gives û ½ Us ADCP data can
probably be averaged in a GLM way
not easy with stacked EMs. This, and
more, will be tried in the Truc Vert Beach
experiment of 2008, right now, in France.
Stokes-Coriolis force
Vertical and Horizontal vortex forces
Simple test case with ROMS
Equation (2) was implemented in ROMS, with due
care to the mass conservation. The Stokes drift
divergence is imposed at the surface as if using
Newberger Allen (2007) equations (but note that
the vertical vortex force is missing in
NA07). The shore is supposed infinite in the
y-direction, the bottom profile is linear with a
slope of 1/100. Wave forcing was computed for
monochromatic waves of period 12 s, with the
energy dissipation given by the model of Thornton
Guza (1983) using B1, and ? given by Battjes
Stive (1984). In order to have a significant
undercurrent for these monochromatic waves, and
to simplify the situation, the eddy viscosity was
fixed and uniform (Kz0.03 m2s-1). The Coriolis
parameter is set to f10 -4 s-1.
The horizontal vortex force drives the jet
towards the shore, the vertical vortex force
breaks the jet. The vertical profiles of both
the current and the Stokes drift are important as
wave-current interaction terms are dominant
forcing terms in the surf zone.
Effect of strong wave nonlinearity
Truc Vert Beach from the sky
For incipient breaking waves, the Stokes drift is
computed using a numerical fully non-linear
solution of the potential flow over a flat bottom
(Darlymple 1969). The Stokes drift of such
incipient breaking waves exhibits a strong
near-surface vertical shear.
References Ardhuin, F., N. Rascle, and K.
A. Belibassakis, Explicit waveaveraged primitive
equations using a generalized Lagrangian mean,
Ocean Modelling, vol. 20, pp. 3560,
2008a. Ardhuin, F., A. D. Jenkins, and K.
Belibassakis, Commentary on the
three-dimensional current and surface wave
equations by George Mellor, J. Phys.
Oceanogr., 2008b. accepted, available at
http//arxiv.org/abs/physics/0504097. Huang, Z.,
and C. C. Mei, Effects of surface waves on a
turbulent current over a smooth or rough seabed,
J. Fluid Mech., vol. 497, pp. 253 287, 2003.
McWilliams, J. C., J. M. Restrepo, and E. M.
Lane, An asymptotic theory for the interaction
of waves and currents in coastal waters, J.
Fluid Mech., vol. 511, pp. 135178, 2004. Mellor,
G., The three-dimensional current and surface
wave equations, J. Phys. Oceanogr., vol. 33, pp.
19781989, 2003. Miche, A. (1944), Mouvements
ondulatoires de la mer en profondeur croissante
ou décroissante. forme limite de la houle lors de
son déferlement. application aux digues
maritimes. Troisième partie. Forme et propriétés
des houles limites lors du déferlement.
Croissance des vitesses vers la rive,
Annales des Ponts et Chaussées, Tome 114,
369406. Newberger, P. A., and J. S. Allen,
2007 Forcing a three-dimensional, hydrostatic,
primitive-equation model for application in the
surf zone, Part 1 Part 2, J. Geophys.
Res., vol. 112 N. Rascle, Influence of waves on
the ocean circulation. PhD thesis, Université de
Bretagne Occidentale, 2007.
http//tel.archives-ouvertes.fr/tel-00182250/. Rue
ssink, B. G., D. J. R. Walstra, and H. N.
Southgate (2003), Calibration and verification of
a parametric wave model on barred beaches,
Coastal Eng., 48, 139149. check out our
Waves In Shallow Environments (WISE)
bibliographic database http//surfouest.free.fr/
WISEBIB
Velocity profiles. Dashed no Coriolis Solid
full Coriolis (including Stokes-Coriolis) Here
the bottom slope is 1/1000 and Kz0.01 m2s-1
How does the transition occur between the surf
zone, where the undertow is pushed by the
wave-induced set-up, and deep water with an
under-current pushed by the Stokes-Coriolis
forcing ?
But finite amplitude waves are also associated
with larger momentum fluxes relative to linear
waves. This effect is under investigation for
comparison with set-up measurements (Raubenheimer
et al. 2001)