Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
WAVE PERIOD FORECASTING AND HINDCASTING
– INVESTIGATIONS FOR THE IMPROVEMENT OF
NUMERICAL MODELS –
Christian Schlamkow and Peter Fröhle
University of Rostock/Coastal Engineering Group, Rostock
Abstract:
This paper describes comparative investigations which have been performed using the numerical
wave model SWAN [1]. The comparisons bases on wave some hindcast simulation runs in the area of
the western Baltic Sea, an area which is predominated by fetch-limited waves. The results of these
simulations are compared to wave measurements that have been performed by directional wave
buoys off the coast in water depths of approximately 12m.
The comparisons are mainly focused on the wave periods and the shape of the directional wave
energy spectra. The basis for the simulations are high-resoluted bathymetry informations and highresoluted wind informations. Different simulation variants with different formulations of physical
effects (mainly wind-input terms) are used and the results are compared. As a result some
recommendations for the application of these formulations are given.
I.
INTRODUCTION
Waves are often the most important input
parameter for design, dimensioning and construction
of coastal structures. To get the necessary wave
parameters, wave measurements, analytic wave
forecast methods or numerical wave simulations are
used. The state of the art wave forecast and hindcast
method is the application of numerical wave models.
The numerical model SWAN [1] is one of these
models. SWAN is a so called community-model and
widely used and accepted. SWAN is optimized for
the simulation of nearshore waves and the
simulations include some special physical effect in
shallow water areas [2], e.g. triad interacting.
In comparison to other wave hindcast methods the
results of SWAN simulations show a good
agreement to measured values [5,6]. Especially the
wave hight and wave directions parameters are
reproduced reasonably good. Unfortunately, the
wave periods are often too short in comparison to
measured wave periods. It is the aim of the
presented investigations, to determine the reason for
this effect. Different simulation runs which have
considered different physical formulations have been
performed. In a first approach this has been done by
comparisons of measured and simulated wave
parameters. In a second step, the shape of simulated directional wave energy spectra have also been
compared to the measured wave spectra.
identification of the effects of different physical
formulations on the results, it is important that the
other influences to the model accuracy are as small
as possible. In many cases, the problem for the
accuracy of wave models is the input data. In these
investigations, high-resoluted bathymetry and wind
input data are combined to a high-resoluted
numerical model. The best available input data are
used to ensure the quality of this part.
A. Bathymetry
The simulations area is located in the western part
of the Baltic Sea. The used bathymetry data are a
combination of different surveys and investigations
[4]. The resolution of the bathymetry is ∆x=1' and
∆y=0.5' which is approximately 1km x 1km.
Bathymetry and simulation area are shown in Figure
1.
B. Wind
The wind data is provided by the German Weather
Service (DWD). The DWD operates a model system
for weather forecast and hindcast purposes. The
wind data for the wave model is computed by the so
called “Local Model” (LM). The data bases on
observations which are used for Re-analysing runs.
The spatial resolution of the wind data is
∆x=∆y≈0.0625° what is approximately 7km x 7km.
The time resolution of the wind input fields is one
hour.
II. DESCRIPTION OF MODEL SETUP
Problems in the accuracy of numerical wave
simulations can result from different reasons. In this
investigation the problems of the SWAN model and
the used physical formulations have been analysed.
For the comparison of the results and the
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
Figure 1. Bathymetry and simulation area
III. SIMULATION PARAMETERS
The spatial resolution of the wave simulations is
the same as the spatial resolution of the bathymetry,
approximately 1km x 1km. Tests with higher spatial
resolutions showed no significant differences for the
wave parameters, especially of the waves calculated
outside the breaker zone. For the calculation of wave
parameters in very shallow water and, especially,
close to a very divided coastline, it is obviously
necessary to use higher resoluted bathymetries.
The time resolution of the simulations is one hour,
like the resolution of the wind input fields and
measurements used for the comparison. The
maximum number of iterations is limited to 20 (which
had never been reached in our simulations) in order
to get the same convergence to the measured data
as for shorter time steps (e.g. 10 minutes)
The wave energy spectrum (or, in SWAN “action
density spectrum”) is discretized from 0.02 Hz to
1 Hz, corresponding to the recommendations in the
SWAN user manual [2] and to theoretical
considerations about the wave periods appearing in
the simulation area. The wave directions are
discretized in 36 steps (10°). The resolution in
frequency-domain is calculated by SWAN (normally
41 meshes) in order to meet numerical demands [2].
All relevant physical effects (triads, friction,
quadruplets if available) are activated for the
simulations. Depth induced breaking was not
activated for these simulations, because it does not
occur in the considered water depths in the
simulation area.
A. Simulated variants
SWAN can be used as a first, second or third
generation wave model. This setting activates and/or
deactivates different physical formulations in the
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simulation and can be combined with special
switches for some special physical formulations.
In general, the wave growth is described by two
mechanisms: First: the waves grow linear in time
caused by pressure waves in the wind field. Second:
the waves grow exponential in time by transfer of
wind energy in dependence to the wave energy,
because higher waves cause bigger pressure
differences in the wind field [3]. Based on these
mechanisms the wind input energy term can be
described as:
Sin (σ , θ ) = A + BE (σ , θ )
(1)
A and B are depending on the wave frequency,
the wave direction and wind speed and -direction.
The expression for A uses a formulation of Cavaleri
and Malanotte-Rizzoli [7]. The term for B can be
chosen according to Snyder et. al [8] or to Janssen
[9]. A new non-linear saturation based whitecapping
term combined with an exponential wind growth term
by Yan [10] is provided by Westhuysen [11].
In GEN1-mode, the linear wind growth following a
modified Cavaleri and Malanotte-Rizzoli approach [4]
can be combined with the exponential wind growth
description following a modified formulation
developed by Snyder et al [4]. The whitecapping
formulation of Holthuijsen and De Boer [12] is used
in this mode. The consideration of quadruplet wavewave interaction is not possible.
The GEN2-mode uses the same formulations as
the GEN1-mode with the modification that the
Phillips “constant” of the formulation is variable in
this mode.
The GEN3-mode is the state-of-the-art mode. In
this mode, the (optional) linear wind growth term
uses the Cavaleri and Malanotte-Rizzoli formulation [7]. The exponential wind growth uses the
Snyder et al formulation [8]. Alternative the Janssen
Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
[9] or Westhuysen [11] formulation can be used. The
whitecapping formulations use the basic approach of
Komen et al. [13] or alternatively Janssen [9]. In this
mode, it is possible to consider quadruplet wavewave interactions, following the Hasselmann et al.
[14] approach.
The simulation variants and the corresponding
switches and modes are complied in Table 1.
TABLE I.
DESCRIPTION OF SIMULATION VARIANTS
V0
default GEN1-mode
V1
default GEN2-mode
V2
default GEN3-mode, linear wind growth term is not
activated
V3
GEN3-mode with “Janssen” wind growth term, linear
wind growth term is not activated
V4
GEN3-mode with “Westhuysen” wind growth term,
linear wind growth term is not activated
V5
default GEN3-mode, linear wind growth term is
activated with (default) proportionality coefficient of
a=0.0015
V6
GEN3-mode with “Janssen” wind growth term, linear
wind growth term is activated with (default)
proportionality coefficient of a=0.0015
V7
GEN3-mode with “Westhuysen” wind growth term,
linear wind growth term is activated with (default)
proportionality coefficient of a=0.0015
IV.
WAVE MEASUREMENTS
Figure 3. Directional waverider buoy
V. RESULTS
For a first assessment of the accuracy of the
simulations in comparison to the measured values,
the reduced wave parameters were compared.
Bases on these results the shapes of the directional
wave energy spectra have been compared.
A. Wave parameters
Figure 4 shows the time-series of measured and
calculated significant wave heights (Hm0). In this kind
of diagram it is difficult to identify the problems of the
accuracy of the simulations or e.g. systematic errors.
Therefore scattering graphs are used. Figure 5
shows the comparison of measured and simulated
(V2) wave heights. In this and following graphs, the
measured values are plotted in the x-axis and the
calculated values in the y-axis.
Figure 2. Locations of wave measurements
The wave measurements have been performed at
different locations in the Baltic Sea (Figure 2). The
measurements are covering different periods. For
this paper, only one typical location is selected for
the comparisons (Warnemünde, Figure 2). The
water depth at the location of the buoy is d=12 m. It
is planned to expand the analyses to the other
showed locations indicated in figure 2.
The used measurement-device is a Datawell
directional waverider-buoy (Figure 3). The measurement device works on basis of acceleration
measurements. The analyses of the raw data, wave
energy spectrum, directional spectrum and wave
parameters are available.
Figure 4. Time-series of wave height at Warnemünde
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
Figure 5. Comparison of measured and calculated (V2) wave
heights
1) Variants V0, V1 and V2
Figure 6 shows the graphs for the wave heights of
the simulation-variants V0 (GEN1-mode), V1 (GEN2mode) and V2 (default GEN3-mode). It is noticeable
that in all three variants the most values with low and
moderate wave heights show a good agreement to
the measurements. Some outliers with calculated
wave height of 0.1 m to 0.2 m and measured wave
heights of 0.7 m to 1.3 m are caused by special
conditions at the measurement location which was
near the port of Rostock. In calm conditions
sometimes ship waves (especially from ferry boats)
are measured but obviously not simulated.
In case of storm conditions and corresponding
high waves, the results are differing. V0
underestimates these waves clearly. In variant V1
these values are also underestimated, but the
deviation is not so high (Table 2). In variant V2 a
small overestimation was observed, but these
variant shows the highest accuracy of wave heights
of all variants (Figure 6).
The scattering graphs for the mean wave periods
are shown in Figure 7. The calculated wave periods
of the variants V0 and V1 are nearly the same. The
values are underestimated in most cases. In variant
V2 (default GEN3-mode) the wave periods are
always clearly underestimated and not useable for
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many practical cases. The reason for this behaviour
is explained in section 4.2 and is related to limited
accuracy of the prognosis of the shape of the wave
energy spectrum.
The wave directions are compared in the
scattering plot in Figure 8, where only wave
directions of waves with a height exceeding
Hm0=0.5m are accounted. In general, the simulated
wave directions show a good agreement to the
measured directions for all simulations.
2) Variants V2, V3 and V4
The simulation variants V2, V3 and V4 contain the
three different wind input terms of the GEN3-mode in
SWAN. Variant V2 uses the default formulation, V3
the Janssen formulation and V4 the Westhuysen
formulation. Figure 9 shows scattering diagrams for
the wave height of these three variants. The graph of
variant V2 shown at Figure 6 and gives a good
agreement to the measured values. The variant V3
(Janssen formulation) shows complete implausible
wave heights. It seems that these problems depend
on a bug in the actual implementtation of SWAN
(version 40.51AB). The calculated wave heights of
variant V4 are overestimated in many cases,
especially in case of high waves and the accuracy of
the wave heights is worse than in V2.
The wave periods of V2, V3 and V4 are compared
in the scattering graphs in Figure 10. Figure 10a is
also known from Figure 7c. The wave periods,
calculated in variant V3 are as implausible as the
wave heights. The wave periods, calculated by
variant V4 are the best of all evaluated simulations,
but also to low in most cases.
3) Variants V5. V6 and V7
In the variants V5, V6 and V7 the exponential
resp. non-linear growth wind input energy terms of
Janssen, Komen and Westhuysen are combined
with the linear growth terms of Cavalerie and
Malanotte with the default proportionality term of
0.0015. Figure 11 shows the scatter graphs of
simulated and measured wave heights for these
variants.
Variant V6 is again completely implausible (see
above). Variant V5 and V7 show very similar results
compared to the graphs in Figure 9. To analyze this,
some further comparisons are made. Figure 12
shows results compared the wave heights of variant
V2 to V5 and variant V4 compared to V7. It is
obvious that these variants are identical in most
cases. Hence, the variants V5, V6 and V7 are
ignored
for
the
further
analyses
a) V0
b) V1
c) V2
Figure 6: Comparison of measured and calculated (V2) wave heights
a) V0
b) V1
c) V2
Figure 7: Comparison of wave periods of V0, V1 and V2
a) V0
b) V1
c) V2
Figure 8: Comparison of wave directions of V0, V1 and V2
Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
a) V2
b) V3
c) V4
Figure 9: Comparison of wave heights of V2, V3 and V4
a) V2
b) V3
c) V4
Figure 10: Comparison of wave periods of V2, V3 and V4
a) V5
b) V6
c) V7
Figure 11: Comparison of wave heights of V5, V6 and V7
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
a) V2 to V5
b) V4 to V7
Figure 12. Comparison of wave heights of V2 to V5 and V4 to V7
In Table 1 the slope of the best fit linear regression line H m 0, calculated = a ⋅ H m 0 , measured and the standard
deviation of the deviations between calculated and measured values is printed.
TABLE II.
Variant
SLOPE OF BEST FIT STRAIGHT LINE AND STANDARD DEVIATIONS
Wave height
Wave period
slope of best fit straight line
standard deviation of
deviations [m]
slope of best fit straight line
standard deviation of
deviations [m]
V0
0.86
0.19
0.79
0.58
V1
0.96
0.17
0.81
0.60
V2
1.08
0.17
0.60
0.45
V4
1.16
0.19
0.78
0.56
Table 1 shows that the calculated wave heights
are clearly underestimated in variant V0 and V1.
Variant V2 overestimates the wave heights slightly,
variant V4 overestimates the wave heights more. In
all variants, the spread of the deviations between
calculated and measured wave heights in nearly the
same.
The calculated wave periods are too short in every
analysed variant. The smallest mean deviations
were observed in variant V1 (but with the largest
spread of deviations), the highest deviations shows
V2. In summary, the experience of underestimated
wave periods in SWAN is also verified in this study.
B. Spectral wave informations
The SWAN wave model is based on the
calculation of the changes of the wave energy
spectra. The above compared parameters are
derived from the wave energy spectra. This means,
these problems in the accuracy of these parameters
has their reasons in problems of the simulation of the
energy density spectra. In some of the investigated
variants, the wave height is comparatively good
reproduced, but the wave period is underestimated
significantly. This indicates that the shape of the
simulated spectra is not identical to the measured
spectra.
This is investigated in the next chapter. In the first
part, the spectra of some selected time steps are
compared. After that, average wave spectra are
compared.
1) Comparison of results at selected time steps
December, 4th 1999, 00:00.
On December, 4th 1999, the highest wave height
in the considered period occured. Table 3 shows the
measured and calculated wave parameters for this
time step. The maximum wind velocity was
U=21 m/s (average over one hour) from West for the
location at Warnemünde.
TABLE III.
TH
WAVE PARAMETER AT 4 DECEMBER 1999, 00:00
Variant
Hm0 [m]
T02 [s]
Measured
2.96
5.76
V0
1.91
4.55
V1
2.50
4.84
V2
3.05
3.87
V4
3.25
4.94
Figure 13 shows the comparison of the measured
and the calculated wave energy spectra and
illustrates possible problems of each variant:
• V0 underestimates the energy peak of the
spectrum and also the high-frequency-tail
(f>0.2 Hz). This results in a too low calculated
wave height and in a too short wave period.
• In V1 is the energy at the peak-frequency
also underestimated. The high- and low
frequency-parts shows a fair agreement to
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
the measured values. The calculated wave
height is too low. The wave period has the
best fit in this time, but is still too low.
• The variant V2 shows the result of the default
SWAN GEN3-mode. This variant results in a
good agreement of the wave height.
Unfortunately the wave period is the worst of
all variants. The energy at the peak of the
spectrum is too low, and the energy at the
high-frequency-tail is overestimated. This
results in small deviations of the wave
heights, but strong underestimation of the
wave periods. The basic shape of the
spectrum shows the worst agreement to the
measured values.
• In variant V4, the calculated wave height is
too high. The reason for that is an
overestimation in the low-frequency-tail of the
spectrum, which results in to much wave
energy in this part of the spectra. The other
parts of the spectra show a comparatively
good agreement with the measured values,
also the peak of the spectrum.
It must be pointed out, that the internal calculation
runtime of SWAN computes wave periods shorter
than an external program which uses the SWAN
output spectra, because SWAN adds a analytical
(diagnostic) high-frequency tail to the discrete
spectrum. To assess the influence of this effect, the
wave periods are computed using an external
program. Table 4 shows a comparison of both wave
periods (computed by SWAN and by external
program). The analyses of the wave measurements
are performed also without such a diagnostic tail, so
that the parameter “wave period” has a different
basis. This effect changes the wave periods about
12% (in variant V2).
TABLE IV.
COMPARISON OF WAVE PERIODS, COMPUTED BY
SWAN AND BY EXTERNAL PROGRAM
Variant
T02 [s],
computed by
SWAN
T02 [s],
computed by
external
program
V0
4.55
4.63
V1
4.84
4.91
V2
3.87
4.34
4.94
5.28
V4
th
January, 30 2000, 09:00
The event of January, 30th 2000, 09:00 is
characterized by wind from WNW with a maximum
wind velocity of U=21.7 m/s (average over one hour).
Table 5 shows the wave parameters measured and
simulated for this time step.
TABLE V.
TH
WAVE PARAMETER AT 30 JANUARY 2000, 09:00
Variant
Hm0 [m]
T02 [s]
Measured
2.85
5.60
V0
2.05
4.67
V1
2.36
4.70
V2
2.96
3.93
V4
3.13
4.86
The comparison of measured and calculated
spectra is shown in Figure 14. The shape of the
spectra of variants V0, V1 and V2 is similar to the
event at the 4th December. The spectrum, computed
by V4 shows the problem of this variant very clear:
There is to much energy in the low-frequency-tail
and also in the high-frequency-tail of the spectrum.
This produce to high waves and to short wave
periods.
th
Figure 13. Wave energy spectra at 4 December 1999, 00:00
th
Figure 14. Wave energy spectra at 30 January 2000, 09:00
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
December, 26th 1999, 04:00
At the December, 26th the wind blows from South
with maximum wind speed of U=6.3 m/s. At 04:00
was the local maximum in wind speed and wave
height at location Warnemünde. Table 2 shows the
measured and the calculated wave parameters for
this time step. In this example the wave height is
relatively low, but very good calculated by variant V2.
Variant
Hm0 [m]
T02 [s]
Measured
0.88
3.92
V0
0.66
2.82
V1
0.73
2.86
V2
0.89
2.19
V4
0.94
2.94
January, 24th 2000, 02:00
Variant
Hm0 [m]
T02 [s]
SWAN
T02 [s]
external
program
Measured
0.85
3.35
3.35
V0
0.79
2.98
3.08
V1
0.91
3.17
3.26
V2
0.88
2.39
3.82
V4
0.91
2.84
3.16
th
Table 7: Wave parameter at January, 24 2000, 02:00
th
Table 6: Wave parameter at January, 26 2000, 04:00
The comparison of the spectra is shown in Figure
15. It is obvious that all calculated spectra
overestimate the wave energy at the high frequencytail in this case. This results in too short wave
periods. Furthermore, it is shown again, that the
shape of the spectrum, calculated by variant V2,
does not match the measured spectra. The wave
energy is clearly overestimated at the high
frequency-tail and clearly underestimated at the
peak of the spectrum. Anyhow, the calculated wave
high is nearly identical with the measured value. This
is typical for the results of variant V2. The variant V4
shows the best match in the peak of the spectrum,
but overestimates the wave energy at the low- and
high-frequency-tail .This results in too high waves.
The good agreement in the calculated wave
periods (variants V1 / V4) is founded by good
agreements of the shape of the spectra for the
measured and calculated values. Figure 16 shows
the comparison of the spectra for this event. It is
obvious that the shape of the spectra from variant V4
shows a relatively good agreement, but the low
frequency-tail is again overestimated. In this
example another problem becomes apparent: The
measured spectrum has no high frequency-tail,
because the buoy only supply informations from
fmin=0.005 Hz to fmax=0.635 Hz. It is possible that in
cases of low waves the maximum frequency of
natural waves is higher. This is not indicated by the
buoy and may cause the problems in the agreement
with simulated values.
th
Figure 16: Wave energy spectra at 24 January 2000, 02:00
th
Figure 15: Wave energy spectra at 30 January 2000, 09:00
2) Comparison of averaged wave energy spectra
In the previous paragraphs, measured and
calculated spectra at discrete points in time are
compared. To compare the spectra in general,
average deviations from measured spectra are
calculated. The measured spectra are approximated
by linear interpolations to the discrete frequency
sampling points of the SWAN calculated spectra and
the energy of the measured spectra is subtracted
from the calculated spectra and averaged over time.
Figure 17 shows the result of the analyses. The
graph shows that in variant V0 (GEN1-mode of
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Chinese-German Joint Symposium on Hydraulic and Ocean Engineering, August 24-30, 2008, Darmstadt
SWAN), the spectral peak is clearly underestimated.
This results in too low wave heights. Variant V1
(GEN2-mode) has the same problem, but not so
significant. The shape of the spectra calculated by
variant V2 (default GEN3-mode), matches good
around the peak of the spectrum, but the wave
energy is overestimated in the high frequency-tail.
That will lead to too short wave periods. The results
of variant V4 (GEN3-mode with Westhuysen wind
input term) clearly overestimates the energy at the
low frequency-tail of the spectrum and also at the
high frequency-tail. This tends to too high wave
heights but relative low differences of measured and
calculated wave periods.
of the calculated wave energy spectra. It seems that
the formulation of Westhuysen resp. Yan is
reproducing the shape of the wave spectra better
than the other approaches.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Figure 17: Average difference of calculated and measured wave
energy spectra
VI. CONCLUSION
The performed investigations show, that wave
forecasting and hindcasting with the SWAN-model
are applicable for questions of coastal engineering.
The results of such simulations have to be evaluated
for the accuracy of the required wave parameters.
SWAN simulations show a comparatively high
accuracy for the wave heights, especially with the
default GEN3-mode. Unfortunately, the wave periods
have poor accuracy in this mode. The GEN3-mode
with the Westhuysen wind energy input term shows
a better agreement of the wave periods to the
measured values in the Baltic Sea, an area where
fetch limited sea-state predominates.
It is recommended after these investigations to
use the default GEN3-mode (Komen formulation) for
questions concentrating to the wave heights and the
GEN3-mode with Westhuysen formulation for
questions concentrating to the wave periods. At this
time it is not possible to give recommendations for
both parameters in one mode.
It is planned to enlarge the investigations
described above to other locations. Also these
investigations are a basis for improvements of the
numerical model SWAN. It seems that these
improvements should be concentrated to the change
of the wind input energy term, because the change
of this term causes significant changes in the shape
502
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