Introduction - Breaking and Dissipation of Ocean Surface Waves - page 5

0.5

0.4

0.3

a 1

0.2

0.1

b 2 b 1

0

a 2

−0.1

−0.2

−0.3

−0.4

−0.5

0 1 2 3 4 5 6

horizontal distance, x

Figure 1.2 Geometric definition of wave skewness and asymmetry. The wave propagates from
left to right. b 1 and b 2 are horizontal distances from the breaker crests to the zero-upcrossing
and -downcrossing respectively. a 1 and a 2 are the breaker crest height and trough depth respec-
tively. Solid line - numerically simulated incipient breaker (skewness is S k

= .

1 15, asymmetry is

A s =−

.

51). Dashed line - harmonic wave of the same wavelength and wave height ( S k = A s =

0).
Dash-dotted line - nonlinear wave of the same length and height obtained by means of perturbation
theory ( S k =

0

0). Dotted lines - mean (zero) water level (horizontal) and line drawn from
the breaker crest down to the level of its trough (vertical). Figure is reproduced from Babanin et al.
( 2010a ) with permission

0 . 39, A s =

An incipient breaker shown in Figure 1.2 (solid line) is produced numerically by means
of the Chalikov-Sheinin model (hereinafter CS model ( Chalikov & Sheinin , 1998 ; Cha-
likov & Sheinin , 2005 )) which can simulate propagation of two-dimensional waves by
means of solving nonlinear equations of hydrodynamics explicitly. The shape of such
a wave is very asymmetric, with respect to both vertical and horizontal axes, and even
visually the wave looks unstable.
Instability is a key word in the breaking process. The wave that we interpret as the
incipient breaker in Figure 1.2 cannot keep propagating as it is: it will either relax back
to a less steep, skewed and asymmetric shape, or collapse. We will define the steepness,
skewness and asymmetry (with respect to the vertical axis) as

H
λ ,

= ak = π (1.1)

a 1
a 2 −

S k = 1 , (1.2)

b 1
b 2 −

A s = 1 , (1.3)