Fully nonlinear analytical theories for surface waves and numerical simulations of wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 139

was found to be small and only relevant at very strong wind forcing U / c > 10 (see also

Babanin et al. , 2010a ).
To finalise this section, based on the numerical simulations of initially monochromatic
steep two-dimensional irrotational waves by a fully nonlinear model, it can thus be sum-
marised that there are breaking-onset criteria in terms of free surface. First, there is a critical
initial monochromatic steepness for the wave train. If IMS is greater than this value,

IMS ≈ 0 . 1 , (4.16)

then according to simulations based on the CS model a breaking will always occur. Even
if the wave is initially sinusoidal and linear, the nonlinear evolution of the wave will ulti-
mately lead to breaking. The distance to breaking will be a function of this initial steepness
( Babanin et al. , 2007a , see also Chapter 5 ). The second criterion is the steepness of an
individual wave at the breaking onset. As seen in Figures 4.2 and 4.9 , there appears to be a
limiting value of such a steepness of

2 ∼ 0 . 8 . (4.17)

Nonlinear characteristics of the wave shape, i.e. steepness (1.1) , skewness (1.2) and
asymmetry (1.3) all oscillate, that is the profile of a nonlinear wave does not remain self-
similar and, strictly speaking, is not even that of a Stokes wave: asymmetry is present
and oscillates in quadrature to steepness/skewness. Oscillations, as well as the critical
steepness, are only marginally affected by the wind unless the wind forcing is very strong.

4.2 Lagrangian nonlinear models

Within Lagrangian methodology, the motion of individual particles, rather than fluid dynam-
ics with respect to a fixed coordinate system, is modelled. One can argue that for surface
waves Lagrangian governing equations are more complicated, but on the other hand the
boundary conditions are simpler. One way or another, however, the Lagrangian approach
in fluid mechanics in general and in wave modelling in particular has enjoyed much less
attention compared to Eulerian methods.
A few available Lagrangian nonlinear models, however, demonstrate the impressive
capacity of such approaches. Fouques et al. ( 2006 ) and Fougues & Stansberg ( 2009 )showed
that even first- and second-order Lagrangian simulations are capable of reproducing many
two-dimensional and three-dimensional features of surface waves, which are such compli-
cated and delicate issues in Eulerian models as discussed in Section 4.1 . Fully nonlinear
Lagrangian models can possibly go much further than that, although their potential diffi-
culties are due to problems with conservation of fundamental physical properties. These
need special attention and treatment. For example, in order to satisfy the conservation of
mass, Fougues & Stansberg ( 2009 ) used a second-order residual in the continuity equation
to shift the fluid particles vertically.
A few Lagrangian models have been suggested and applied to wave-breaking studies
by Landrini et al. ( 1998 ), Tulin & Landrini ( 2001 ), Dalrymple & Rogers ( 2006 ) and