Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 240

parameterisation employs diffusion-type terms which are added to the right-hand sides of
(4.9) and (4.10) as

∂
∂ξ

B ∂η J − 1 η τ =

∂ξ ,

(7.5)

∂
∂ξ

B ∂φ J − 1

φ τ =

∂ξ .

(7.6)

Their diffusion coefficient B depends on the second derivative of the surface:

⎧
⎨

2

2 2 z

∂ξ

ξ ∂

η
∂ξ

if ∂ C b = > s ,

2 2

B = (7.7)

⎩

2 z
∂ξ

if ∂ 0 ≤ s

2

where the tunable parameters were chosen as coefficient C b ≈ 0 . 1 and the second deriva-

tive as s 300.

The algorithm (7.5) - (7.7) obviously does not affect the solution in the absence of break-
ing. When active, it does not change the volume of the water, but reduces the momentum
and energy of the waves which are assumed to be passed on to the motions beyond the
wave model, i.e. mean current and turbulence. Such a transfer can be attended to separately
outside the potential approach (e.g. Chalikov & Belevich , 1993 ).
Thus, on the macroscopic level, the differential diffusion-type parameterisation plays
the role of the wave breaking in the wave system, and the authors showed this behaviour
to be quite realistic. Within the potential model, the parameterisation of the rotational phe-
nomenon prevents development of instability if the second derivative s is becoming too
large and thus prevents development of the breaking as such. Therefore, this is not in itself
a physical model of the breaking, it is simulating the physical consequences of the break-
ing necessary for modelling longer-term wave evolution. Like any parameterisation, it has
to be applied with caution, and the authors outline the limits of such applications - i.e. the
scheme may fail to prevent the breaking occurrence within the model if the initial steepness
or the wind energy input are too high.
In this regard, explicit modelling of the breaking in progress, from the start to end of
breaking, is an interesting task for fluid mechanics. Such models, capable of handling
two-phase flows, have existed for over a decade now (e.g. Abadie et al. , 1998 ; Zhao & Tani-
moto , 1998 ; Chen et al. , 1999 ; Watanabe & Saeki , 1999 ; Mutsuda & Yasuda , 2000 , among
the earlier ones), and these days they can even include the effects of viscosity, surface
tension, air and water compressibility (i.e. acoustic effects), and three-dimensionality.
These models reproduce the behaviour and multiple observed features of the breaking
in progress very realistically, including the formation of turbulence, bubble injection, pro-
duction and break down, spray emission - key features of the dissipation, i.e. dynamical
processes on which the wave spends its lost energy and momentum. In the past, the serious
limitation for this kind of model was the extremely high computational demands, and this

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