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SURFACE WAVES AND THEIR EFFECT ON THE LARGE-SCALE FLOW 
From the viewpoint of large-scale modelling and compared to turbulent fluctuation, waves 
are understood to be a special type of fluctuation with different properties. Waves are 
considered to be a sub-scale phenomenon, with respect to both time and space. Here 
we are interested in the way in which the (mean) circulation is affected by the presence of 
waves, including sea and swell. 
A wave should be considered not only as a travelling pattern of surface deflection but 
also as an orbiting phase-organised motion of water particles. The resulting net effect on 
the average motion is due to the fact that the particle orbits are not closed. After a 
complete wave cycle, a net displacement is left. Particles move back and forth but these 
two steps do not cancel each other out exactly. When a particle moves forward (in the 
direction of wave propagation) it moves faster than during the backward motion. The 
resulting effect can be illustrated even with a harmonic wave, see LeBlond & Mysak 
(1978). It is also plausible that the non-cancellation of water transport goes with a net 
transport of momentum. The wave-induced extra mass flux is nothing but the Stokes 
flow, the wave-induced extra momentum flux is called radiation stress. 
Although waves are usually perceived as a fluctuation of the water surface they affect the 
entire water column down to the bottom whereby their effect is also felt throughout, at 
least in principle. By way of additional fluxes of mass and momentum the waves affect 
the mean flow not only at surface but at any depth. Inasmuch as we are interested in its 
one-way forcing effect on the filtered (large-scale and time-averaged) flow, it acts as a 
driving agent, like a body force. The wave-induced terms are derived from a harmonic 
oscillation and phase averaging. The procedure gives typical vertical profiles. 
The detailed calculation is left out here. We refer to Dolata & Rosenthal (1984) where 
phase-averaging to second order in wave amplitude is carried out from a partly 
Lagrangian viewpoint (moving internal interfaces). The result of the radiation stress 
calculation remarkably is the same as with ordinary Reynolds averaging. 
In our large-scale circulation model, waves are considered to be ever-present throughout 
the domain. (We suppose that the current status of the wave field is available at any time 
from a simulation of its evolution.) From this point of view, the wave field is a large-scale 
phenomenon which, of course, affects the circulation. As the Stokes drift caused by sea 
waves contributes to real flow we consider it, therefore, as part of the large-scale 
circulation. Our goal is to model, in the presence of waves, the total flow consisting of 
Eulerian average and Stokes drift. (Note that sometimes the term large-scale flow means 
only the Eulerian average as, e.g., in Dolata & Rosenthal). 
Thus, in order to take account of the effect of waves in our model, we have to build the 
forcing of Stokes flow into the momentum equations. What has to be included in the 
momentum budget is the radiation stress term which, in our understanding of circulation, 
makes an additional contribution. 
We adhere to Dolata & Rosenthal, but carry over their calculation to spherical co