GUIDE WAVE ANALYSIS AND FORECASTING - WMO

86
different. For very shallow water the result is an f –3
frequency tail. This assumption has lead Bouws et al.
(1985) to propose a universal shape of the spectrum in
shallow water that is very similar to the JONSWAP spectrum
in deep water (with the f –5 tail replaced with the
transformed k –3 tail). It is called the TMA spectrum.
The evolution of the significant wave height and the
significant wave period in the described idealized situation
is parameterized from observations in deep and
shallow water with the following formulations:
⎡ * m1
* * m
X
H = Atanh( h ) tanh ⎢
κ
3
1
κ3
*
⎢tanh
κ h
⎣ 4
m4
( )
⎡ * m
* * m
X
T = Btanh( h ) tanh ⎢
κ
4
2
2π κ4
⎢tanh
κ h
⎣ 4
2
* m4
( )
⎤
⎥,
(7.12)
⎥
⎦
where dimensionless parameters for significant wave
height, H * , significant wave period, T * , fetch, X * , and
depth, h * , are, respectively: H * = gH s/u 2 , T * = gT s/u,
Figure 7.4 —
Shallow water growth curves for
dimensionless significant wave
height (upper) and period
(lower) as functions of fetch,
plotted for a range of depths
Dimensionless significant wave height H*
Dimensionless period T*
GUIDE TO WAVE ANALYSIS AND FORECASTING
0.3
0.2
3
2
1
0.1
0.02
0.01
8
10 0
10 0
⎤
⎥
⎥
⎦
10 1
10 1
X * = gX/u2 , h * = gh/u2 (fetch is the distance to the upwind
shore). The values of the coefficients have been estimated
by many investigators. Those of CERC (1973) are:
A = 0.283 B = 1.2
κ1 = 0.0125 κ2 = 0.077 κ3 = 0.520 κ4 = 0.833
m1 = 0.42 m2 = 0.25 m3 = 0.75 m4 = 0.375
Corresponding growth curves are plotted in Figure 7.4.
7.6 Bottom friction
The bottom may induce wave energy dissipation in various
ways, e.g., friction, percolation (water penetrating
the bottom) wave induced bottom motion and breaking
(see below). Outside the surf zone, bottom friction is
usually the most relevant. It is essentially nothing but the
effort of the waves to maintain a turbulent boundary
layer just above the bottom.
Several formulations have been suggested for the
bottom friction. A fairly simple expression, in terms of
the energy balance, is due to Hasselmann et al. (1973) in
the JONSWAP project:
10 2
10 2
10 3
Dimensionless fetch X*
10 3
10 4
10 4
Dimensionless fetch X*
10 5
10 5
10 6
10 6
h* = ∞
5
1
0.5
0.2
0.1
0.05
0.02
h* = ∞
10 7
5
1
0.5
0.2
0.1
0.05
0.02
10 7