03 Application study of shallow water wave model (SWAN)

Yang dezhou
Yin boshu
Institute of Oceanology
Chinese Academy of Sciences

1: Abstract
SWAN (Simulating Wave Nearshore) wave model has been put into use to
examine its applicability in Bohai Sea.
There is generally poor agreement between model results and
measurements, when we use default parameters of the formulas used by
SWAN to implement the simulations in Bohai Sea.
Also, it is surprising that for the same wind process simulated results from
two wind generation and three white-capping dissipation expressions (Komen
and Janssen, Alkyon) demonstrated large difference.
Further study showed that the proportionality coefficient in linear growth
term of wind growth source function plays an unperceived role in the process
of wave development.

Based on experiments and analysis, we found default constant
coefficient should change rather than a constant. Therefore, a new
changing coefficient with the variation of friction velocity u* was
introduced into the linear growth term of wind growth source function in
SWAN model.
Then, we experimented with three weather processes (two cold waves
and a south wind) to validate the improvements in the linear growth term.
It is found that the results from the improved SWAN wave model agree
much better with the measurements than those from the previous SWAN
model.
Furthermore, the large differences of results from two wind generation
and three whitecapping dissipation expressions were eliminated.
The improved SWAN wave model makes a solid foundation for
setting up a proper model of wave numerical forecasting in Bohai Sea.

2:Background
(1)The experiments of Becagra Toulany et al.
Becagra Toulany et al indicated that for 20m/s winds and constant
depths, Komen expression always gives higher Hs (significant
wave heights) than Jassen expression.
For long duration (>24hr) and large fetch, wave heights from
Janssen is as much as 50% of Komen for depths in excess of 50m.
By comparison, for 30m/s winds, Komen gives Hs estimates that
are 30% in excess of Hs estimates resulted from Janssen
expression. Study of simulation of a meteorological “bomb” for the
NW Atlantic [1] suggested that the difference between Hs from
Komen and Janssen expression is as much as 2m.

(2)Our experiments in different sea areas.
When we use the SWAN model in Fushanwan
bay ,we get results that always are not higher enough
to match with the measurements.

3: Physical processes of SWAN
We only take the expressions of wind and whitecapping into
consideration in our experiments.
(1) Input by wind
Wind energy to waves is commonly described as the sum of linear
and exponential growth. There are two kinds of wind growth
models in SWAN that are available for us. Both expressions of
wind growth model of them share the following form (1) and the
same linear growth (2), while the exponential growth term is
different.
S in
( σ , θ ) = A + B × E(
σ , θ )
A describes linear growth and B×E describes exponential growth.
(1)

Linear growth by wind
A
=
α
[ U
2 *
max(0,cos( θ −θ
w))]
g 2π
4
H
H
= exp( −(
σ / σ
PM
)
* −4
)
*
σ = 0.13g
PM
2π
28U
*
We find the value is unsuitable to the shallow water applications.

Exponential growth
a. Expression due to Komen et al. (1984) [2]:
ρa
U*
B = max[ 0,0.25 (28 cos( θ −θw)
−1)]
× σ
ρ C
U*
ρa
w
is friction velocity,
ρw
ph
θ wind direction , C
w
ph
and are the density of air and water respectively.
b. Expression due to Janssen (1991) [4]:
is the phase speed,
B
ρa
U*
2
2
= β ( ) (max( 0,cos( θ −θw)))
× σ
ρ C
w
ph
β is the Miles “constant”.

(2) Two expressions of white-capping
a: White-capping is primarily controlled by the steepness of the
waves. In presently operating third generation wave models
(including SWAN) the white-capping formulations are based on a
pulse-based model (Hasselmann, 1974) [5], as adapted by the
WAMDI group (1988) :
S
ds
Γ = Γ
~ k
, w
( σ , θ ) = −Γσ
~ E(
σ , θ )
k
KJ
k ~ s
= C
ds
(( 1−δ ) + δ ~)( )
k
~ s
Γ is a steepness dependent coefficient, k is wave number ,
σ ~ denote a mean frequency and k ~ a mean wave number, respectively.
PM
p

Komen et al. (1984) estimated the value of Γ by closing the energy
balance of the waves in fully developed conditions. This implies that this
value depends on the wind input formulation that is used.. Since two
different wind input formulations are used in SWAN model, two sets of
coefficients are used. For the wind input of Komen et al: C =2.36×10 -5
ds
, p=4, and δ = 0 which are the default value in SWAN model . In our
computation we set the value of to 0.5. because its value
represent the state of wave and 0.5 is suitable to the wave state of
Bohai Sea, while for Janssen et al, we don’t adapt the values of ,
p .
δ
C ds

: An alternative formulation for whitecapping is based on the
Cumulative Steepness Method as described in Alkyon et al. (2002)
[6]. With this method, dissipation due to white-capping depends on
the steepness of the wave spectrum at and below a particular
frequency. It is defined as (directionally dependent):
σ 2π
2
' m
Sst ( σ , θ ) k | cos( θ −θ
) | E(
σ , θ ) dθdσ
= ∫∫
0
0
In this expression the coefficient m controls the directional
dependence. The new white-capping source term is given by
S
st
wc
( σ , θ
st
) = −C
S ( σ , θ ) E(
σ , θ )
wc
st

st
with C wc
=4.0 is a tunable coefficient. We find this value is so big
that much of wave energy is dissipated. In our experiments , we
st
amended this value C wc =0.15.

4. Improvement of proportionality
coefficient α
As we all know, transfer of wind energy to the waves is commonly
described in SWAN with a resonance mechanism and a feed-back
mechanism. In SWAN, the value of the proportionality coefficient is
constant in a stationary or non-stationary wind process, although it is
variable with different wind process. We find that the proportionality
coefficient α in linear growth term of wind growth model plays an
unperceived role in the process of wave developing. From the results
of computation, we find that it is unseemly that the proportionality
coefficient is a constant value.

α
a: The variation of α almost hasn’t effect on the wave developing,
when we experiment with wind speed below 7.5m/s(at 10m
elevation).
α
b: However, the variation of has much effect on the wave
developing when wind speed is between 7.5m/s and 15m/s,
c: while it has little effect when wind speed is higher than 15m/s .

In view of the relation between Philips coefficient and friction velocity
in equilibrium range of high frequency, based on the experiments of
different wind field, we introduce a parabolic function concerning the
friction velocity into the linear growth term of wind growth model in
place of constant proportionality coefficient (in default, it is constant
in SWAN model):
α =
⎧0.0015
⎪
⎨−
6.1834×
U
⎪
⎩0.0015
2
*
+ 5.8989×
U
*
−1.2568
U
U
10
≤ 7.5 m / s
7.5 m / s ≤ U
10
10
≥15
m / s
≤15
m / s
U 10
is wind speed at 10m elevation, is friction velocity.
2
U∗
= CD ×
U
2
10
, C D is drag coefficient.
U *

When wind speed is weaker or stronger, the value of is
equivalent to the default value in SWAN ,otherwise it is variant with
wind speed. We adapted the source code of SWAN in order to
apply the new formula to SWAN model and the amended SWAN
model is as convenient as the default SWAN model for us to use.
If we want to use the amended SWAN model, what we need to
do just is to set a special value to the in the ‘input’ file of SWAN.
α
α

5. Numerical implementation and result
analysis
The designs of experiment
To validate the new formula, we assume different wind input
expressions and white-capping expressions to compute the
significant wave heights using amended and default models.
As follow, it is described in detail.
a. Assume Komen wind input expression and Alkyon expression of
white-capping to compute the significant wave heights, which is
abbreviated to ‘default Komen-Komen’ when default value is used
and ‘amended Komen-Komen’ when new formula is implemented.

. Assume Komen wind input expression and corresponding
Komen coefficient of white-capping to compute the significant wave
heights, which is abbreviated to ‘default Komen-Komen’ when
default value is set and ‘amended Komen-Komen’ when new
formula is implemented.
c. Assume Janssen wind input expression and corresponding
Jassen coefficient of white-capping to compute the significant wave
heights, which is abbreviated to ‘default Janssen-Janssen’ when
default value is set and ‘amended Janssen-Janssen’ when new
formula is implemented.
d. Assume Janssen wind input expression and Alkyon expression
of white-capping to compute the significant wave heights, which is
abbreviated to ‘default Komen-Komen’ when default value is set
and ‘amended Komen-Komen’ when new formula is implemented.

Computational domain
All our experiment are implemented in Bohai Sea: N37 o -41 o ,
E117.5 o -122.5 o . The resolution of computational grid is 5’× 5’, that
ο
ο
⎛ 1 ⎞ ⎛ 1 ⎞
is to say, ⎜ × ⎜ . As follow, the computational grid is
⎝12
⎟
⎠
⎝12
described in figure 1 in detail.
⎟
⎠

41 o
8805
8710
40 o E
N
39 o
9804
38 o
118 o 119 o 120 o 121 o 122 o
Figure 1 computational grid in Bohai Sea and locations of observation .

Wind processes
we select four typical wind processes from historical data as
following:
Serial
number
Process
Computing time
Locations
of stations
Depth
(m)
1
8710
1987.10.29 02:00—1987.11.01 02:00
120.75 o E
39.90 o N
31
2
8805
1988.05.26 08:00—1988.05.29 08:00
121.12 o E
40.208 o N
25
3
9804
1998.04.22 20:00—1998.04.25 02:00
118.817 o E
38.217 o N
10
4
9904
1999.04.18 20:00—1999.04.20 14:00
118.817 o E
38.217 o N
10
Wind fields used in this study were prepared by Ocean
University of China.

The results of experiments
The following figures (figure 2- figure 3) illustrate the difference of
computed significant wave heights between default SWAN model
and amended SWAN model in above two wind processes using
different wind input formula and white-capping expressions.

7
Measurements
Default Komen -Alkyon
Amended Komen -Alkyon
5
Default Komen -Komen
Default Janssen-Janssen
Default Janssen-Alkyon
Amended Komen -Komen
Amended Janssen -Jassen
Amended Janssen -Alkyon
hs/m
3
1
02:00 14:00 14:00 14:00
T/h (1987.10.29.02:00--1987.11.01.02:00)
Figure 2 Measured data are compared with the default SWAN
model and amended SWAN model in 8710 process.. Location of
observation is (120.75 o E 39.90 o N).

6
5
4
Default Komen -Alkyon
Default Komen -Komen
Default Janssen-Janssen
Default Janssen-Alkyon
Measurements
Amended Komen -Alkyon
Amended Komen -Komen
Amended Janssen -Jassen
Amended Janssen -Alkyon
hs/m
3
2
1
0
20:00 08:00 20:00 08:00 20:00
T/h (1998.04.22.20:00-1998.04.25.02:00)
Figure 3 Measured data are compared with the default SWAN
model and amended SWAN model in 9804 process. Location of
observation is (118.817oE 38.217o N).

It is easy to find out that the HS (significant wave height)
that we attained is much less than the HS of
measurements, when we use the default values of
SWAN model. This poor agreement with observational
data indicates that the default setting of SWAN does
not fit to the Bohai Sea. The input of wave energy is not
enough to render the corresponding wave height that
we have observed. For other two wind processes,
similar results have been obtained.

It is seen from above results that not only at the
nearshore observation point (10m), but also at the deep
water observation point (31m) in Liaodong Bay, the
results that we computed from the amended SWAN
model, are more consistent with the observation data
than the results from the default SWAN model in all of
wind processes. Moreover, the difference of HS derived
from the two different wind input formula (Komen and
Janssen) also is diminished. The good agreement with
measurements suggests that the improvement that we
made by introducing a new formula is reasonable and
effective in Bohai Sea.

c: In addition, Cumulative Steepness Method is used to compute
st
the wave height and we find that the default value of C wc , is so
high that the wave energy is consumed excessively. When the
st
value of C wc is set to 0.15 and new formula is applied to SWAN, no
matter which wind energy formula is selected the reasonable
results always are able to be attained.

5. Conclusions
When SWAN is implemented in Bohai Sea and the default value of
model is applied to computation, the HS (significant wave height)
of SWAN is much less than the HS of measurements.
a: By means of analysis of defect of SWAN, we introduce a
parabolic function concerning the friction velocity into the linear
growth term of wind growth model in place of constant
proportionality coefficient (in default, it is constant in SWAN
model) .
b: We conduct four experiments in every wind process to verify the
new formula. The computed results validate our modification in all
of the four wind processes, because the computed results of
amended SWAN model always are more consistent with the
observational data than results of default SWAN model in every
experiment. This improvement maintains good precision in higher
wind speed and improves the poor results when wind is weak.

c: In addition, the amended SWAN model has the same
convenience as the default SWAN model. If we set the value of
to a specified value in ‘input’ file, we can get computed results of
amended SWAN model, otherwise we get the results of default
SWAN model.
From what has been discussed above, by means of SWAN
applicability study in Bohai Sea, we have set up a wave forecast
model for Bohai Sea. It is helpful to improve the level of shallow
sea wave forecast in China.
α

References:
[1] Becgara Toulany, Will Perrie, Peter C.Smith and Baoshu
Yin,2002,A fine-resolution operational wave model for the new
atlantic, The proceedings of 7th International workshop on wave
hindcasting and forecasting, Oct.21-
25,Albert,Canada,Ed.V.Swail,Published by Met. Service of
Canada,pp420-427.
[2]Komen,G.J., S. Hasselmann, and K.Hasselmann,1984:On the
existence of a fully developed windsea
spectrum,J.Phys.Oceanogr.,14,1271-1285.
[3] WAMDI Group,1988.The WAM model-a third generation ocean
wave prediction model.J.Phys.Oceanogr.,18,1775-1810.
[4]Janssen, P.A.E.M., 1991a:Quasi-linear theory of wind-wave
generation applied to wave forecasting,
J.Phys.Oceanogr.,21,1631-1641.

[5] Hasselmann, K., 1974: On the spectral dissipation of ocean
waves due to whitecapping, Bound.-layer Meteor., 6, 1-2, 107-127
[6]Alkyon and Delft Hydraulics, 2002: SWAN fysica plus, report
H3937/A832 (by order of RIKZ/RWS as a part of the project HR-
Ontwikkeling).
[7] Wen Shengchang andYu Zhouwen,1984.Sea wave theory and
computation principles,Science Press,411-416.