Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 239

Implicit simulation of the dissipation in one-phase-medium models is handled by means
of what Zakharov et al. ( 2007 ) call pseudo-viscosity. Techniques can be different, but
the idea is to take the extra energy from the system by artificial means which have the
same overall and long-term effect as would the dissipation due to breaking and other
effects.
Such a technique is described in detail in Chalikov & Sheinin ( 2005 ) and Chalikov &
Rainchik ( 2011 ). The authors employ two types of dissipation, what they call 'tail dissi-
pation' and 'breaking dissipation'. The tail dissipation in a way mimics viscosity which
absorbs the energy fluxes from large to small scales (see e.g. Zakharov & Zaslavskii , 1982 ,
and Section 7.1.2 ). If not attended to, this energy can accumulate at large wavenumbers,
close to the truncation number M , and corrupt the numerical solution.
In order to prevent this, in Chalikov & Rainchik ( 2011 ) 'tail dissipation' was added to
the right-hand sides of (4.7) and (4.8) / (4.14) in Fourier space:

∂η k
∂τ

=− μ k η k , (7.2)

∂φ k
∂τ

=− μ k φ k . (7.3)

Here, μ k is a damping coefficient such that

⎧
⎨

rM |

2

k |− k d

if | k | > k d ,

μ k =

M − k d (7.4)

⎩

0 if | k |≤ k d .

Tuning parameters k d 25 were chosen in Chalikov & Rainchik ( 2011 ),

and little sensitivity of the outcomes to reasonable variations of these parameters was
found. Such a model of tail dissipation is quite straightforward and physical; an increase
of the truncation number M shifts the dissipation towards higher wavenumbers, and in any
case modes at

= M / 2 and r = 0 .

k d are not affected.
Modelling explicitly the dissipation due to breaking, however, represents a considerable
challenge. As discussed in Section 4.1 , there are both mathematical and physical reasons
why potential models cannot go far beyond the breaking-onset point in time or space.
But the breaking is a regular occurrence at the scale of tens of wave periods/lengths or
less in a wave train/field with background steepness ak

| k | <

1, and this is the typical
steepness of natural wave trains. Therefore, some realistic breaking-dissipation means have
to be developed in order to model phase-resolvent evolution of wave trains at the scale of
hundreds/thousands of periods, which is the typical scale for such evolution in deep water
in the field.
Chalikov & Sheinin ( 2005 ) and Chalikov & Rainchik ( 2011 ) used the concept of pre-
venting instability, borrowed from atmospheric models of free convection, and developed
an algorithm of breaking parameterisation based on smoothing the interface when indi-
cations are that it is becoming too steep and the instability can develop. The differential

≥ 0 .