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Optimal estimations of directional wave conditions for nearshore field studies
Journal Pre-proof Optimal estimations of directional wave conditions for nearshore field studies R.L. de Swart, F. Ribas, D. Calvete, A. Kroon, A. Orfila PII:
S0278-4343(20)30027-3
DOI:
https://doi.org/10.1016/j.csr.2020.104071
Reference:
CSR 104071
To appear in:
Continental Shelf Research
Received Date: 4 October 2019 Revised Date:
13 January 2020
Accepted Date: 27 January 2020
Please cite this article as: de Swart, R.L., Ribas, F., Calvete, D., Kroon, A., Orfila, A., Optimal estimations of directional wave conditions for nearshore field studies, Continental Shelf Research, https://doi.org/10.1016/j.csr.2020.104071. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Optimal estimations of directional wave conditions for nearshore field studies R.L. de Swarta,∗, F. Ribasa , D. Calvetea , A. Kroonb , A. Orfilac a
Department of Physics, Universitat Polit`ecnica de Catalunya (UPC), Jordi Girona, 1-3, 08034, Barcelona, Spain b Department of Geosciences and Natural Resource Management, University of Copenhagen, Øster Voldgade, 10, 1350, Copenhagen, Denmark c IMEDEA (CSIC-UIB), Mediterranean Institute for Advanced Studies, Miquel Marqu`es, 21, Esporles (Illes Balear), Spain
Abstract Accurate directional wave conditions at shallow water are crucial for nearshore field studies and necessary as boundary conditions for morphodynamic models. However, obtaining reliable results for all wave parameters can be challenging, particularly regarding wave direction. Here, the accuracy of two global hindcast models and propagation of measured wave conditions using linear wave theory or the SWAN wave model (forced by integrated wave parameters or 2D spectra) is assessed to obtain directional wave conditions at shallow water for Castelldefels beach, Northwest Mediterranean Sea. Results are analyzed using different statistical error parameters and for different wave climates (shore-normal, shore-oblique and bimodal). The analysis shows that global hindcast models correctly predict the trends in wave height ∗ Corresponding author at: Department of Physics, Universitat Polit`ecnica de Catalunya (UPC), Jordi Girona, 1-3, Edificio B5, 08034, Barcelona, Spain. Tel: +34 934016087 Email addresses: [email protected] (R.L. de Swart), [email protected] (F. Ribas), [email protected] (D. Calvete), [email protected] (A. Kroon), [email protected] (A. Orfila)
Preprint submitted to Continental Shelf Research
January 29, 2020
and mean wave period but predictions for mean wave direction are only accurate for shore-normal waves. Linear wave theory provides good results for wave height but underestimates refraction, resulting in significant errors in mean wave direction for shore-oblique waves. Finally, SWAN forced with 2D spectra results in the most accurate predictions for all wave parameters. When using integrated wave parameters as boundary conditions, the results for wave height and mean period stay the same whilst the errors in peak period and mean direction worsen for shore-oblique and bimodal wave climates. The reason is that for these wave conditions the directional spectrum constructed out of integrated wave parameters does not resemble the actual directional spectrum. Keywords: Wave propagation, Mediterranean Sea, Global hindcast models, Linear wave theory, SWAN modeling, Directional spectra
1
1. Introduction
2
Many shorelines around the world are highly dynamic and change due
3
to wave action. Understanding this behavior is important for present day
4
coastal zone management and for estimating the vulnerability to climate
5
change (Vitousek et al., 2017). Incoming waves are the most important forc-
6
ing for coastal evolution, and wave characteristics in shallow water depend on
7
the local coastal setting (wave climate, bathymetry, shoreline orientation).
8
Unfortunately, obtaining reliable wave conditions in coastal and inner seas
9
is still a difficult task, especially in semi-enclosed seas like the northwest-
10
ern Mediterranean Sea (Bola˜ nos-Sanchez et al., 2007; Cavaleri et al., 2018).
11
The reason is that this region is characterized by limited fetches, a complex
2
12
bathymetry with deep canyons, significant changes in shoreline orientation
13
and strong variations in the wind climate (S´anchez-Arcilla et al., 2008).
14
Wave conditions can be obtained from a variety of sources. The preferred
15
source for wave conditions are devices that provide real-time measurements.
16
Different types of wave measurement instruments exist (e.g. Allender et al.,
17
1989; Pettersson et al., 2003), but wave buoys are the most common world-
18
wide. However, wave buoys are scarce because they are quite expensive to
19
deploy and maintain. As a result, they are mostly deployed in areas where
20
there is a great need for real-time wave measurements (typically near harbors
21
or populated coastlines). For studying other locations, the measured offshore
22
waves need to be transformed to nearshore study sites using a so called ’wave
23
propagation model’.
24
In addition to buoy measurements, wave conditions can also be obtained
25
from global hindcast models, based on codes such as WAM (The WADMI
26
Group, 1988) and WAVEWATCH-III (Tolman, 2009). Hindcast models typ-
27
ically also assimilate observations and have the advantage that they do not
28
suffer from breakdowns. Furthermore, results can theoretically be acquired
29
for any worldwide location, including those where no other data is available.
30
However, these models may not be accurate in shallow waters (Cavaleri et al.,
31
2018), and proper wave propagation might also be needed.
32
There are different ways to propagate wave conditions to shallow waters.
33
The simplest method is to use linear wave theory based on ray approximation
34
(e.g. Holthuijsen, 2007). Despite its crude assumptions, this method is often
35
applied in nearshore studies to quickly obtain reasonable estimates of the
36
wave conditions. A more thorough method is to use sophisticated wave
3
37
models like SWAN (Booij et al., 1999). Such models include more physical
38
processes and have been extensively used in the past in nearshore regions with
39
gradually variable bathymetry (e.g. Ris et al., 1999). The SWAN model has
40
the additional advantage that it is a spectral model, meaning that it solves
41
the spectral action balance equation without any a priori restrictions on the
42
spectrum for the evolution of wave growth. Spectral models can either be
43
forced by integrated wave parameters or using full 2D spectra.
44
Good predictions of wave height and wave period are important for, e.g.,
45
management of harbors, navigation or fisheries. However, wave direction is
46
crucial in coastal engineering because even small variations in wave direction
47
can already substantially affect estimates of longshore sediment transport
48
(e.g. Soomere and Viˇska, 2014), along with the related dynamics of embayed
49
beaches (Harley et al., 2015) and beach nourishments (Arriaga et al., 2017).
50
On the other hand, nearshore studies in the past typically focused on relat-
51
ing beach morphodynamics to wave height and wave period only (e.g. Van
52
Enckevort et al., 2004; G´omez-Pujol et al., 2007), whereas wave direction was
53
often neglected. However, recent studies indicate that wave direction plays
54
a major role in the evolution of morphodynamic patterns, such as crescentic
55
bars (Calvete et al., 2005; Price and Ruessink, 2011), transverse finger bars
56
(Ribas and Kroon, 2007; Ribas et al., 2012), high angle wave instability and
57
km-scale shoreline sand waves (Ashton et al., 2001; Arriaga et al., 2018).
58
Wave conditions used in nearshore studies are typically obtained by prop-
59
agation of offshore buoy measurements or hindcast model results. This can
60
lead to errors, so that verification of the propagated wave conditions against
61
local measurements is essential. However, many nearshore studies show only
4
62
limited or no verification of wave conditions (e.g. Splinter et al., 2011; Ar-
63
riaga et al., 2018). On the other hand, studies that focus on modeling wave
64
fields in coastal regions with large-scale hindcast models (sometimes coupled
65
to SWAN to increase resolution in the nearshore) mostly verify wave height
66
and period with typical errors of 0.25 m and 1.5 s in both wind-sea and swell
67
conditions (e.g. Pallares et al., 2014; Amrutha et al., 2016), whilst wave di-
68
rection is often not included (e.g. Bola˜ nos-Sanchez et al., 2007; Perez et al.,
69
2017). In the few studies with hindcast models where wave direction is also
70
validated (e.g. Pallares et al., 2014; Amrutha et al., 2016), the corresponding
71
errors are large (> 40◦ ). The errors in wave direction can be substantially
72
reduced by propagating measured wave conditions (Gorrell et al., 2011). Un-
73
fortunately, there is a lack of studies that compare the reliability of different
74
methods to obtain directional wave conditions for nearshore studies, espe-
75
cially in semi-enclosed and coastal seas.
76
The aim of this study is to establish the accuracy of different methods to
77
obtain wave conditions in shallow water for nearshore studies, with a special
78
focus on the wave direction. This is done for the field site of Castelldefels
79
beach, which is located on a limited-fetch, complex-geometry sea (Northwest
80
Mediterranean Sea; Section 2). Three long-term sources of wave conditions
81
are available near this site: a buoy and output points of two hindcast models.
82
An instrument was deployed to measure wave conditions in front of the field
83
site at 21 m depth during a 9 days experiment. This allows to compare five
84
different methods to obtain wave conditions at the field site from the long-
85
term sources (Section 3). Three methods consist of propagating measured
86
wave conditions from the offshore buoy using models with different degrees
5
87
of complexity: the simple wave ray model, SWAN forced by integrated wave
88
parameters and SWAN forced by 2D spectra. The two last methods use the
89
results of the two hindcast models directly. Despite the short duration of
90
the field campaign, distinct wave climates occurred, which allows to charac-
91
terize the accuracy of each method under different wave conditions (Section
92
4). The significance of the findings of this study and the advantages and
93
disadvantages of the different methods are discussed in Section 5, and the
94
conclusions are listed in Section 6.
95
2. Study site and datasets
96
2.1. Study area
97
The study site is Castelldefels beach, located on the north-western Mediter-
98
ranean Sea along the Spanish Catalan coast, approximately 20 km southwest
99
of Barcelona (Figure 1). It is an open beach with an east-west orientation and
100
has a length of approximately 4.5 km, whilst more to the east the shoreline
101
orientation changes rapidly towards the north. Castelldefels beach is part of
102
a continuous stretch of beaches of the Llobregat delta, extending from the
103
Garraf Mountain chain in the west to the Llobregat river outfall in the east.
104
It is mainly composed of sand with a median grain size of 270 µm. Tidal
105
action in this part of the Mediterranean Sea is small, with a range of ap-
106
proximately 20 (10) cm during spring tide (neap tide; Simarro et al., 2015).
107
The bathymetry in the study area (Figure 1) shows a relatively wide shelf in
108
front of the study site, whilst more to the east the shelf is narrower and the
109
water depth quickly reaches more than 100 m.
6
¯
France
Legend
! ( ! (
Spain
# * # * $ +
Portugal
" )
Barcelona
Barcelona wave buoy AWAC location GOW2/SIMAR model point SIMAR deep water GOW2 deep water Wind measurements Castelldefels beach
Bathymetry [m]
El Prat de Llobregat
High : 0 Low : -400
Castelldefels
0 1 2
4
6
Kilometers 8
Figure 1: Overview map showing the Castelldefels study site, the nearshore bathymetry (source Emodnet) and the locations of the different wave data sources used in this study. c 2019 Microsoft Corporation The aerial photography is part of Microsoft Bing Maps ( Earthstar Geographics SIO).
110
The winds in the study area are strongly influenced by orographic bar-
111
riers, which leads to a variable wind climate and the formation of intense
112
north and northwestern winds during December and January. The winds are
113
weaker during the rest of the year and the maximum wind velocities occur
114
during easterly storms that affect the entire Catalan coastline (Pallares et al.,
115
2014). The wind patterns directly influence the local wave climate, which
116
is characterized by calm wave conditions with sudden high energetic wave
117
events (wave height above 1.5 m; Puertos del Estado, 1994). On average,
118
calm conditions prevail during the summer period and energetic conditions
119
occur mostly from October to May (S´anchez-Arcilla et al., 2008). Storm
7
120
waves along the Catalan coast are limited due to the short fetches and an
121
average storm duration of less than 24 hours. As a result, mixed sea states
122
composed of wave trains with more than one peak frequency and mean di-
123
rection occur frequently (S´anchez-Arcilla et al., 2008). Near the study site,
124
the wave climate is dominated by waves from the east and south (Figure 2a).
125
The largest waves come from the east, due to the longest available fetches and
126
the stronger winds that generally blow from this direction (S´anchez-Arcilla
127
et al., 2008; Bola˜ nos et al., 2009).
Figure 2: Wave roses of the long-term wave climate at the Barcelona buoy (a), the SIMAR output point next to the Barcelona buoy (b) and the GOW2 output point nearest to the Barcelona buoy (c) for the period September 2012–May 2018 in terms of spectral wave height Hm0 and mean direction θm .
128
2.2. Short-term data source (AWAC)
129
A short-term dataset of wave conditions was measured from 13 March to
130
22 March 2018 using an AWAC sensor deployed at 21 m depth in front of
131
Castelldefels (Figure 1). This sensor is both a current profiler and a direc-
132
tional wave system that was mounted in a stationary frame at the bottom.
133
During the measurement period, the AWAC provided half-hourly values
134
of the spectral wave height Hm0 , mean zero-crossing period Tm02 , peak period 8
135
Tp , mean direction θm , peak direction θp and mean directional spreading σ θ ,
136
as well as half-hourly 1D frequency and 2D frequency-direction spectra. A
137
detailed description of how these quantities are computed is presented in
138
Section 3.1 and Appendix A. The AWAC used a frequency range of 0.02–
139
0.49 Hz and a frequency resolution of 0.01 Hz to obtain all wave parameters
140
and spectra. The full 2D frequency-direction spectra was estimated using the
141
Maximum Likelihood Method (MLM; Krogstad et al., 1988). All important
142
AWAC settings are summarized in Table 1.
143
The data obtained by the AWAC is used as ground truth in the rest of
144
this study. When comparing the errors of the different methods with the
145
ground truth, the root mean square error (RMSE), BIAS and normalised
146
root mean square error (NRMSE) are used, defined as
s
PN
i=1 (Si
RM SE =
N PN
BIAS =
− Ai )2
i=1 (Si
N RM SE N RM SE = σA
− Ai ) ,
,
(1)
,
(2) (3)
147
in which S denotes the simulated data from the various propagation methods
148
and A denotes the measured data from the AWAC (ground truth). The
149
number of data points is denoted by N , whilst σA denotes the standard
150
deviation of the AWAC measurements. Angles with respect to North are
151
used as input when computing these quantities for angular values. This is
152
appropriate, because wave directions in the study area are such that no jumps
153
from 360 to 0 degrees occur. 9
Table 1: Overview of instrument and deployment settings for the Barcelona buoy and AWAC. Device
AWAC
Barcelona buoy
Water depth
21 m
68 m
Sensor altitude above bottom
0.5 m
Tide-dependent
Burst interval
154
0.5 h (start 5 min
1 h (start at top of
past every half hour)
the hour)
Sampling rate
1 Hz
4 Hz
Length of processed time series
1200 points
5760 points
Start frequency
0.02 Hz
0.03 Hz
End frequency
0.49 Hz
0.64 Hz
Number of frequency bands
48
123
Number of directional bins
90
121
Directional spectrum method
Maximum Likelihood
Maximum Entropy
Method
Method
Start of deployment
13 March 2018 15:05
8 March 2004
End of deployment
22 March 2018 09:05
–
2.3. Long-term available wave datasets
155
Although no long-term wave buoy or other measurement instrument is
156
located directly in front of the study area, there are different long-term wave
157
datasets available around Castelldefels. They comprise both measurements
158
and large-scale global hindcast models.
159
2.3.1. Barcelona wave buoy
160
The Barcelona buoy is the only permanent instrument in the area. It is
161
moored in front of Barcelona harbor (41.32 N, 2.20 E) at a depth of 68 m
162
(Figure 1), managed by Puertos del Estado (Spanish Ports Authority), and
163
it is operational since March 2004. The buoy is located relatively close to the
164
study site (the distance between the buoy and the study site is approximately
10
165
20 km), but there is a significant change in the orientation of the shoreline
166
and the bathymetric contours (Figure 1).
167
The buoy obtains the heave time-series using standard Fourier analysis
168
(Kuik et al., 1988), which are then processed using both spectral analysis
169
and zero-crossing analysis to determine non-directional wave parameters and
170
1D frequency spectra. The frequency range is 0.03–0.64 Hz with a resolution
171
of 0.005 Hz. Finally, the Maximum Entropy Method (MEM; Lygre and
172
Krogstad, 1986) is used to estimate the 2D frequency-direction spectra using
173
the 1D frequency spectra and the available cross-spectra.
174
For this study, hourly values of the spectral wave height Hm0 , mean zero-
175
crossing period Tm02 , peak period Tp , mean direction θm and mean directional
176
spreading σ θ were obtained from the buoy, as well as the hourly 1D frequency
177
and 2D frequency-direction spectra. All important settings of the buoy are
178
summarized in Table 1.
179
2.3.2. SIMAR database
180
The SIMAR database, maintained by Puertos del Estado, is a hindcast
181
of wind, sea level and wave parameters for the entire Spanish coast that
182
spans from 01/01/1958 until today. Since 2001, the wind field at 10 m above
183
the sea surface is obtained using the regional HIRLAM model provided by
184
the Spanish State Meteorological Agency (AEMET; Und´en et al., 2002). The
185
obtained wind fields are used in both WAM and WAVEWATCH III to obtain
186
the wave fields along the Spanish coast. The resolution of the wind model is
187
3 km since 2012 and that of the wave model is 5 km since 2006. Data output
188
is available each hour since 2012 (Puertos del Estado, 2015).
189
The SIMAR database has multiple data points in front of the Barcelona 11
190
coast. One of these data points is located directly in front of the study site
191
at the location where the AWAC was deployed at a depth of approximately
192
21 m and another is located close to the Barcelona wave buoy (41.32 N, 2.21
193
E) at a depth of approximately 75 m (Figure 1). For both output points,
194
the integrated wave parameters provided by the SIMAR database are the
195
spectral wave height Hm0 , mean zero-crossing period Tm02 , peak period Tp
196
and mean direction θm .
197
2.3.3. GOW2 database
198
The GOW2 database, which is developed and maintained by the hydraulic
199
institute IH Cantabria (Perez et al., 2017), is based on the numerical model
200
WAVEWATCH III and provides wave data for the period 01/01/1979 until
201
today. Since 2011, GOW2 is forced with the CFSv2 model (Saha et al.,
202
2010). The grid used in GOW2 for coastal continental areas has a resolution
203
of 0.25◦ × 0.25◦ which is approximately 21 km × 28 km (lon × lat) in the
204
Barcelona area.
205
Although GOW2 has a coarser resolution compared to SIMAR, it also
206
has a data point directly in front of Castelldefels (at 21 m depth, see Fig-
207
ure 1). Analogous to SIMAR, a second data point was selected (at deeper
208
water) located closest to the Barcelona buoy (41.25 N 2.25 E) at a depth
209
of approximately 250 m. From both points, hourly time series of significant
210
wave height Hm0 , mean zero-crossing period Tm02 , peak period Tp and mean
211
direction θm have been obtained.
12
212
3. Methods
213
3.1. Definition of wave parameters
214
215
216
217
218
The 2D frequency-direction spectrum, E(f, θ), represents the frequencydirection density spectrum of the sea-surface elevation variance E, defined R fhigh R 2π E(f, θ)df dθ. The 1D frequency spectrum E(f ) is related as E = flow 0 R 2π to E(f, θ) by the expression E(f ) = 0 E(f, θ)dθ and in a similar way it is possible to define a 1D direction spectrum as Z
fhigh
E(f, θ)df
E(θ) =
.
(4)
flow 219
Here, the symbol E denotes variance regardless of the unit. The units are
220
m2 /Hz for E(f ), m2 /deg for E(θ) and m2 /Hz/deg for E(f, θ). Notice that
221
the surface elevation variance multiplied by 21 ρg is equal to the total energy
222
Etot of the waves per unit surface area. Lastly, by changing flow and fhigh , it is
223
possible to apply high- and low-pass filters, respectively. Various integrated
224
wave parameters can be computed from the obtained spectra (Table 2). Ad-
225
ditional information about obtaining 2D spectra and computing directional
226
wave parameters is given in Appendix A. It is important to note that the
227
peak frequency fp and thus the peak period Tp obtained from buoy measure-
228
ments can have a large variability. This is especially true for multi-modal
229
spectra, where the observed maximum can easily switch between multiple
230
frequency peaks. The mean period generally does not have this problem,
231
which is why it is common to define a synthetic peak period, Tp∗ as a func-
232
tion of the mean zero-crossing period Tm02 (Rogers and Wang, 2007). The
233
relation used here is Tp∗ = 1.33Tm02 , which is the average of the relations be-
234
tween Tp and Tm02 given for the Pierson-Moskowitz and JONSWAP spectral 13
235
shapes (see Soulsby, 1998). In this study, the synthetic peak period is used
236
as input in both the simple wave ray propagation model (Section 3.2) and
237
for the SWAN model forced with integrated wave parameters (Section 3.3).
238
The randomness in fp also directly affects the measurements of θp , meaning
239
that mean direction θm is a more stable parameter.
240
To force the simple wave ray and SWAN models, all integrated wave
241
parameters (except Tp∗ ) are obtained directly from the buoy. For the hours
242
that no buoy data is available, the integrated wave parameters are computed
243
from the available 1D and 2D spectra using the expressions in Table 2. Table 2: Definitions of integrated wave parameters computed from 1D frequency and 2D frequency-direction spectra, where the Fourier coefficient a1 (f ), b1 (f ), a1 and b1 are given in Appendix A or computed from the buoy measurements. Parameter nth moment of 1D frequency spectrum
Definition R mn = 0∞ f n E(f )df
Spectral wave height
√ Hm0 = 4 m0
Root-mean square wave height
Hrms =
Mean wave period
Tm01 =
Mean zero-crossing period
Tm02 =
Wave period at peak of spectrum
Tp =
Mean wave direction per frequency Mean direction for entire spectrum
H√ m0 2 m0 m1
q
m0 m2
1 , fp
E(fp ) = max E(f ) b (f ) θm (f ) = tan−1 a1 (f ) 1 b1 −1 θm = tan a 1
Mean directional spreading for full spectrum
p σ θ = q 2[1 − r1 ], 2 r1 = a21 + b1
Peak wave direction
θp = θm (fp )
14
244
3.2. Simple wave ray model
245
The first model used to propagate waves from deep water to Castelldefels
246
is a simple wave propagation model that is based on ray approximation (linear
247
wave theory) and assumes monochromatic waves and parallel depth contours
248
(i.e. alongshore uniform bathymetry). Due to this latter assumption, the real
249
bathymetry is not used and the model inputs are simply the wave height,
250
period and angle in a certain deep water depth, together with the shallow
251
water depth where wave conditions are required.
252
First, the dispersion relationship is solved using a Newton’s numerical
253
scheme to compute the wave numbers at deep and shallow water (T is as-
254
sumed to be constant), after which Snell’s law is applied to obtain the wave
255
angle at shallow water. In the dispersion relation, the current contribution
256
(Doppler shift) is neglected, because at the AWAC it is smaller than 1% for
257
96% of the time (maximum contribution is 1.8%). Finally, the model ap-
258
plies wave energy conservation, i.e. a constant cross-shore wave energy flux
259
Fx (again assuming alongshore uniformity) to compute the wave height at
260
shallow water. The model uses Hrms for the wave height (characteristic wave
261
height for wave energy), a synthetic peak period Tp∗ and θm for the angle of
262
incidence.
263
3.3. SWAN wave propagation model
264
3.3.1. Model description
265
SWAN (Simulating Waves Nearshore) is a third-generation spectral wave
266
model that describes the evolution of the 2D frequency-direction spectrum in
267
coastal regions and inland waters by accounting for many relevant physical
15
268
processes (Booij et al., 1999). The SWAN model, as WAM and WAVE-
269
WATCH III, is based on the spectral action balance equation with sources
270
and sinks, but SWAN is specifically designed for coastal areas. Usually, wave
271
models use the action density N (defined as N = E/ωr where ωr is the
272
relative radian frequency), because it is conserved when propagating in the
273
presence of an ambient current (Whitham, 1974). Note that in our case ωr is
274
equal to the absolute radian frequency ω because ambient currents are negli-
275
gible (as explained in Section 3.2). The action balance equation in Cartesian
276
coordinates reads (e.g. Komen et al., 1994)
277
∂N ∂cx N ∂cy N ∂cωr N Stot ∂cθ N + + + = + , (5) ∂t ∂x ∂y ∂ωr ∂θ ωr where cx , cy , cωr and cθ are the group velocities in x, y, ωr and θ space. The
278
right hand side of equation (5) contains Stot , which is the source/sink term
279
and includes six physical processes that are important in generating, dissipat-
280
ing, or redistributing wave energy: the wave growth by wind, the nonlinear
281
transfer of wave energy through three-wave and four-wave interactions, the
282
wave decay due to whitecapping, bottom friction and depth-induced wave
283
breaking. More information about these terms can be found in the SWAN
284
Scientific and technical documentation (SWAN Team, 2018a).
285
3.3.2. Model setup
286
The SWAN model Cycle III version 41.20A is used in this study with
287
spherical coordinates and nautical convention. The model domain consists
288
of a curvilinear grid that stretches approximately 60 km alongshore. The
289
grid follows the coastline and the bathymetric line of the Barcelona buoy
290
(68 m; Figure 3). The spatial resolution varies throughout the grid and is 16
Figure 3:
SWAN domain including detailed bathymetry that is used to propagate the
waves to the study site.
291
approximately 200 m around the AWAC location. The grid bathymetry is
292
obtained from the EMODnet database and has a resolution of 120 m (Figure
293
3). The landward boundary of the SWAN model is set at approximately 10
294
m depth, because the EMODnet database becomes unreliable for shallower
295
depths. The output point in front of Castelldefels beach (the AWAC location
296
and the GOW2/SIMAR output point) is located at 21 m depth.
297
In this study, SWAN is used in 2D non-stationary mode, and station-
298
ary computations (recommended for domains smaller than 1 deg) with a
299
maximum of 15 iterations per computation. The frequency grid contains
300
38 logarithmically spaced values in the range 0.03–1 Hz, with the recom-
301
mended frequency resolution of df /f = 0.1 (SWAN Team, 2018b) and the
302
directional resolution used is 5◦ . The default JONSWAP formulation for
303
bottom friction is used with a coefficient value of 0.038 m2 s-3 . Many differ-
304
ent sensitivity tests have been conducted to determine the default settings
305
(Section 4.4.2). Following the results of these sensitivity tests, the default
306
third-generation physics formulation of Komen et al. (1984) is used, whilst
307
whitecapping, quadruplets, depth-induced breaking, triad wave-wave inter17
308
actions, wind growth are switched off. The integrated wave parameters ob-
309
tained with SWAN at the AWAC location are integrated over the frequency
310
range of the AWAC (0.02–0.49 Hz), following the recommendations of Pal-
311
lares et al. (2014).
312
Two types of offshore wave conditions are used in this study. The first
313
(and most used) type of boundary conditions is to force SWAN with a single
314
set of integrated wave parameters from which SWAN computes an artificial
315
single-component JONSWAP spectrum with a default peak enhancement
316
factor γ = 3.3. The integrated wave parameters used are Hm0 , synthetic
317
Tp∗ , θm and σ θ (see also Section 3.1). The SWAN manual (SWAN Team,
318
2018b) states that the wave direction θp should be used for forcing but in
319
this study the mean direction θm is used instead. This choice was made
320
because θp directly depends on fp , which is not a reliable parameter in buoys
321
(see Section 3.1). The second way to apply the offshore wave conditions is
322
to use the full 2D frequency-direction spectra, which are the most detailed
323
measurements of the wave climate that can be obtained from the buoy. Note
324
that when using SWAN forced with 2D frequency-direction spectra only the
325
energy that propagates into the domain is taken into account. Furthermore,
326
the Barcelona buoy has a fine linear frequency resolution of 0.005 Hz and
327
a directional resolution of 3◦ , whilst in SWAN the frequency resolution is
328
coarser and logarithmic and the directional resolution is 5◦ . Thus, the buoy
329
spectra are interpolated to the frequencies and directions used by SWAN.
330
The boundary conditions are imposed along the entire south boundary
331
of the grid but not in the lateral boundaries. However, since the region of
332
interest is in the center of the domain and the domain is sufficiently wide,
18
333
the absence of lateral boundaries does not influence the results. The off-
334
shore waves are assumed to be alongshore uniform and equal to those at the
335
Barcelona buoy.
336
4. Results
337
After a description of the wave conditions during the field campaign,
338
the performance of the different methods will be validated hourly using the
339
AWAC data. A total of 211 values are available for the SIMAR and GOW2
340
hindcast models, 188 for the simple wave ray model and 207 for the SWAN
341
simulations. It is stressed that the results for Tp shown in this section re-
342
fer to the actual measured or modeled peak period and not the previously
343
mentioned synthetic Tp∗ .
Figure 4: Wave roses showing the wave climate during the field campaign at the Barcelona buoy (a) and AWAC (b) in terms of spectral wave height Hm0 and mean direction θm .
344
345
346
4.1. Wave conditions during field campaign The wave climate during the 9-days field campaign was dominated by south-southeasterly waves during the first 6 days, after which east-southeasterly 19
347
waves were present (Figure 4a and 5). During the majority of the campaign,
348
the wave conditions were quite energetic (Hm0 > 0.5 m). Only on 14 March,
349
Hm0 dropped below 0.5 m, whilst the largest Hm0 was registered on 18 March
350
(1.8 m at the buoy and 1.9 m at the AWAC).
351
The various datasets of the campaign have been separated into three
352
different wave climates: southerly, easterly and bimodal. This separation is
353
made based on the measured 2D frequency-direction spectra at the Barcelona
354
wave buoy. The criteria for classifying a wave climate as easterly or southerly
355
are that at least 70% of the wave energy comes from that direction. If this
356
is not the case, the wave climate is classified as bimodal. The threshold
357
angle (147◦ ) to discriminate between easterly and southerly wave energy
358
is set by computing the average mean direction for both the easterly and
359
southerly wave conditions and subsequently taking the mean. This results
360
in southerly wave climates being present during 68% of the field campaign,
361
11% for easterly wave climates and 21% for bimodal wave climates (Figure
362
5).
363
Table 3 shows the statistics of 5 integrated wave parameters during the
364
complete field campaign and for the different wave climates. The average
365
wave height during southerly and bimodal wave climates was quite similar,
366
but it increased during easterly wave climates. During easterly and bimodal
367
wave conditions, the wave direction at the AWAC changes with respect to
368
the Barcelona buoy due to refraction and the average wave height decreases.
369
As shown in Figure 2a, easterly waves are important in the overall wave
370
climate of the Barcelona buoy, which is why the easterly waves that occurred
371
during the end of the field campaign have been analyzed in detail. More-
20
Table 3: Statistical values (mean, standard deviation, minimum and maximum) of integrated wave parameters measured at the Barcelona buoy and AWAC during the field campaign for the entire period and the different wave climates. Wave
Method
climate
Hm0 [m] Mean,
Tm02 [s] stdev,
Mean,
Tp [s] stdev,
Mean,
θm [deg] stdev,
Mean,
θp [deg] stdev,
Mean,
stdev,
min, max
min, max
min, max
min, max
min, max
Buoy
1.0, 0.32, 0.40, 1.8
4.8, 0.84, 2.9, 7.7
7.0, 1.4, 3.9, 10
169, 43, 74, 213
−
AWAC
0.88, 0.26, 0.34, 1.8
5.0, 0.82, 3.4, 7.7
7.2, 1.9, 3.2, 11
180, 20, 130, 225
180, 24, 32, 244
Buoy
0.97, 0.27, 0.40, 1.7
4.7, 0.88, 2.9, 7.7
6.6, 1.3, 3.9, 10
196, 11, 171, 213
−
AWAC
0.91, 0.27, 0.34, 1.8
4.8, 0.83, 3.4, 7.7
6.5, 1.6, 3.2, 10
191, 11, 164, 225
191, 20, 32, 244
Buoy
1.5, 0.26, 0.80, 1.8
5.6, 0.49, 4.9, 6.6
8.2, 1.2, 6.3, 10
86, 7, 74, 99
−
AWAC
0.91, 0.14, 0.55, 1.1
5.6, 0.64, 4.1, 6.7
9.6, 0.81, 6.8, 11
145, 11, 130, 174
150, 9, 135, 175
Buoy
0.94, 0.31, 0.40, 1.3
4.8, 0.58, 3.3, 6.1
7.4, 1.3, 5.4, 10
128, 27, 75, 173
−
AWAC
0.77, 0.25, 0.34, 1.1
5.1, 0.68, 3.4, 6.4
7.7, 1.8, 4.4, 10
162, 11, 131, 199
161, 16, 130, 197
− −
−
Full period − −
−
Southerly −
−
−
Easterly − −
−
Bimodal
372
over, propagating easterly waves is more challenging because they experience
373
strong refraction before reaching Castelldefels due to the change in shoreline
374
orientation, whilst southerly waves only experience limited changes (see data
375
at AWAC in Figure 4b and Figure 5).
376
4.2. Results of large-scale hindcast models
377
Results of the SIMAR and GOW2 database during the field campaign
378
at the AWAC location are shown in Figure 6, in yellow and purple colors,
379
respectively. The modeled trends in Hm0 and Tm02 agree quite well with the
380
measurements. This is also clear from Table 4, which contains the statisti-
381
cal errors for the various wave parameters and the different wave climates.
382
On average, both models underpredict Tm02 but the general pattern in the
383
measured data is still captured. Regarding θm , both models clearly show
384
large deviations. During southerly waves, the errors in direction are still 21
Figure 5:
Time series of integrated wave parameters and 1D directional spectra during
the field campaign. The three top plots show, respectively, the spectral wave height Hm0 , mean zero-crossing period Tm02 and mean direction θm both for the Barcelona buoy and the AWAC. The last two plots shows the 1D directional spectrum E(θ) measured by the Barcelona buoy and the AWAC respectively. The background shading in the three tops plots indicates the different wave climates; white for southerly, light-gray for bimodal and dark-gray areas for easterly wave climates.
385
moderate (they can be as much as 40◦ ), but they increase significantly when
386
easterly waves are present (up to 80◦ or even more). Both models predict
22
387
waves at Castelldefels coming from the east, whilst in reality these waves
388
have refracted substantially and come more from the south-southeast.
Figure 6:
Time series of integrated wave parameters during the field campaign at the
AWAC location. From top to bottom, the spectral wave height Hm0 , mean zero-crossing period Tm02 and mean direction θm are shown for different data sources (AWAC measurements or one of the models). The background shading indicates the different wave climates; white for southerly, light-gray for bimodal and dark-gray areas for easterly wave climates.
389
4.3. Results of simple wave ray model
390
The results of the simple wave ray model are also shown in Figure 6 (in
391
green) and in Table 4. Please note again that the wave period remains un-
392
changed when using this model. The modeled Hm0 and Tm02 agree quite well
393
with the measurements, although errors in modeled Hm0 can be significant
394
during easterly wave conditions. Regarding Tp and θm , the model shows good 23
395
agreement for southerly waves. For easterly waves, θm improves compared
396
with those of the GOW2 and SIMAR databases, but errors can still be quite
397
high (up to 40◦ ). The performance of θm for the different wave climates
398
is shown in Figure 7a. The gaps in the time series of the simple wave ray
399
model during easterly wave conditions are because the model can only prop-
400
agate waves in Barcelona buoy that have a direction ±90◦ with respect to
401
the shore-normal at Castelldefels beach.
402
4.4. Results of SWAN model
403
4.4.1. SWAN default settings
404
The results of the default SWAN simulation forced with integrated wave
405
parameters are shown in Figure 6 and in Table 4. The modeled results of
406
Hm0 and Tm02 agree well with the measurements for all wave climates, but
407
Tp , θm and θp are only reproduced well for southerly waves. The modeled
408
θm shows an improved performance for easterly and bimodal wave climates
409
when compared to the results of the simple wave ray model, but the error in
410
θm is still substantial (Figure 7b). Moreover, although not shown in previous
411
comparisons, notice that the results of θp worsen compared to θm (particularly
412
the RMSE).
413
When forcing SWAN with 2D spectra, not only are Hm0 and Tm02 well
414
reproduced, but also Tp now shows much more similarity with the mea-
415
surements (Figure 6 and Table 4). Most importantly, θm now shows good
416
agreement with the measurements for all wave climates (Figure 7c). Finally,
417
the RMSE in θp becomes larger compared to the default SWAN simulation
418
forced with integrated wave parameters (although the RMSE was already
419
large), but the BIAS improves. 24
Table 4: Summary of the statistical errors for the different methods during the full period and for the three wave climates. Entries in italics indicate the cases for which NRMSE > 1. Wave
Method
climate
Hm0 [m]
Tm02 [s]
RMSE,
BIAS,
NRMSE
RMSE,
Tp [s] BIAS,
RMSE,
BIAS,
θm [deg]
θp [deg]
RMSE, BIAS,
RMSE, BIAS,
NRMSE
NRMSE
NRMSE
NRMSE
0.98
1 .0 , −0 .77 , 1 .2
1.7,
0.28,
0.94
44 , −11 , 2 .2
−
−
−
GOW2
0.26, −0.14,
SIMAR
0.24,
0.004, 0.92
0 .95 , −0 .32 , 1 .2
1.6,
0.64,
0.86
37 , −9 .7 , 1 .8
−
−
−
Wave ray model
0.14, −0.001, 0.53
0.42, −0.13, 0.51
1.4, −0.57,
0.74
18, −3.9, 0.87
−
−
−
SWAN parameters* 0.11, −0.036, 0.41
0.41,
0.11, 0.50
1.5, −0.82,
0.82
14, −7.3, 0.66
22, −6.6, 0.90
SWAN 2D spectra* 0.12, −0.07,
0.47
0.46,
0.08, 0.56
0.82,
0.006, 0.44
8.1,
0.94, 0.40
26 ,
0 .92 , 1 .1
1.0
0 .96 , −0 .74 , 1 .2
1 .7 ,
0 .41 , 1 .1
13 ,
6 .7 , 1 .2
−
−
−
1.0
12 ,
8 .5 , 1 .1
−
−
−
3.7, 0.74
−
−
−
Full period
GOW2
0.28, −0.18,
SIMAR
0.24, −0.041, 0.87
0 .88 , −0 .22 , 1 .1
1.6,
0.10,
0.43, −0.12, 0.51
1.0, −0.31,
0.67
8.3,
SWAN parameters* 0.12, −0.047, 0.44
0.43,
0.18, 0.52
1.1, −0.39,
0.69
7.5, −2.5, 0.67
17,
2.4, 0.83
SWAN 2D spectra* 0.12, −0.056, 0.44
0.43,
0.13, 0.52
0.69, −0.002, 0.44
7.2,
18,
8.2, 0.93
GOW2
0 .18 ,
0 .06 , 1 .3
0 .92 , −0 .67 , 1 .4
0 .9 , −0 .42 , 1 .1
72 , −70 , 6 .8
−
−
−
SIMAR
0 .31 ,
0 .28 , 2 .3
0 .68 ,
0 .22 , 1 .1
1 .4 ,
0 .59 , 1 .7
60 , −59 , 5 .7
−
−
−
0 .46 , −0 .14 , 3 .4
0.34,
0.089, 0.52
2 .5 , −0 .80 , 3 .0
38 , −13 , 3 .6
−
−
−
SWAN parameters* 0.06, −0.005, 0.44
0.47,
0.053, 0.73
2 .5 , −2 .3 ,
21 , −19 , 2 .0
31 , −29 , 3 .5
SWAN 2D spectra* 0.094, −0.076, 0.70
0.62, −0.18, 0.96
1 .0 , −0 .18 , 1 .3
10, −7.5, 0.95
43 , −40 , 4 .9
GOW2
0.21, −0.083, 0.85
1 .2 , −0 .94 , 1 .8
2 .1 ,
77 , −37 , 7 .1
−
−
−
SIMAR
0.20, −0.070, 0.81
1 .2 , −0 .92 , 1 .8
1.8,
0.97
65 , −44 , 5 .9
−
−
−
0.096,
0.040, 0.39
0.39, −0.27, 0.57
2 .0 , −1 .3 ,
1 .1
31 , −24 , 2 .8
−
−
−
SWAN parameters* 0.076, −0.015, 0.31
0.31, −0.069, 0.46
2 .0 , −1 .4 ,
1 .1
21 , −17 , 1 .9
29 , −24 , 1 .9
SWAN 2D spectra* 0.14, −0.11,
0.46,
1.0,
0.57
9.6,
32 , −0 .80 , 2 .1
Southerly Wave ray model
Easterly Wave ray model**
Bimodal Wave ray model
0.009, 0.38
0.57
0.047, 0.67
0.59,
3 .0
0 .22 , 1 .1 0.84,
0.13,
2.3, 0.65
0.95, 0.88
*Results are shown for default SWAN settings **Only 8 datapoints available due to most wave directions < 90 deg with respect to North
25
420
4.4.2. Sensitivity to changing SWAN settings
421
Several sensitivity tests have been conducted to find the default SWAN
422
settings (see Section 3.3) for both forcing types. These tests can be sepa-
423
rated into numerical tests (grid refinement and increasing frequency resolu-
424
tion), tests of physical processes (including quadruplets and whitecapping
425
with and without wind) and methodological tests (default integration range,
426
choice of SWAN input parameters, changing spectral shape and directional
427
spreading value). Refining the grid by a factor 3 or increasing the frequency
428
resolution by a factor 2 does not improve the results. This proves that the
429
chosen numerical resolution is sufficient. The results also remain the same
430
when including quadruplets and whitecapping as physical processes, although
431
Battjes (1994) indicates that these processes are important in coastal waters.
432
The probable reason is that the domain used in this study is fairly small (less
433
than 10 km cross-shore) and measured wave conditions are imposed at the
434
boundary instead of using wind data for wave generation. When wind is also
435
included in the computations (using measurements from Barcelona Airport),
436
the results for θm worsen when forcing with 2D spectra (RMSE increases
437
from 8 to 10◦ and BIAS from 1 to 3.5◦ ). On the contrary, the results for
438
θm improve slightly when forcing with integrated wave parameters (RMSE
439
decreases from 14 to 13◦ and BIAS improves from −7 to −5◦ ). The rest of
440
the parameters stay approximately the same. It is also possible to change
441
the frequency range that is used in SWAN to compute the integrated wave
442
parameters. When applying no cutoff frequency (as done in SWAN default
443
settings) instead of using the frequency range of the AWAC (Section 3.3.2),
444
the results for Tm02 slightly worsen (RMSE increases by at most 0.1 s whilst
26
445
the BIAS can worsen by 0.2–0.3 s). Again, the rest of the parameters do not
446
display any changes.
Figure 7: Scatterplots showing θm modeled using the simple wave ray model (a), SWAN forced with integrated wave parameters (b) and SWAN forced with 2D spectra (c) versus AWAC measurements. The colors denote the different wave climates.
447
A choice has to be made regarding the input wave period (Tm01 or Tp ) and
448
wave angle (θm or θp ) when forcing SWAN with integrated wave parameters.
449
As explained in Section 3.1, a synthetic peak period is used in this study
450
that is computed from the Tm02 measured by the buoy. When using Tm01
451
instead (computed from the buoy 1D frequency spectra), no changes are
452
observed in the results. When using θp instead of θm , the results for Hm0
453
and Tm02 are similar, but errors increase substantially for Tp , θm and θp . The
454
results remain the same when changing the spectral shape from JONSWAP
455
for fetch-limited seas (Hasselmann et al., 1973) to Pierson-Moskowitz for
456
fully developed seas (Pierson and Moskowitz, 1964). Finally, when forcing
457
with integrated wave parameters the directional spreading σ θ can be set to a
458
constant value instead of using the time-variable measured data. Tests have
459
been done for constant σ θ ranging from 10◦ to 60◦ with a step size of 10◦ .
460
The SWAN user manual advises a σ θ of 30◦ and the results indicate that this 27
461
value gives the best results. For smaller values of σ θ , the errors in θm and
462
θp increase substantially (RMSE increases from 14 to 18–21◦ and from 22 to
463
24–26◦ respectively), whilst the errors remain the same for higher values of
464
σ θ . Changing σ θ does not lead to changes in the rest of the parameters.
465
5. Discussion
466
5.1. Performance of wave propagation methods
467
In this work, the results of different wave modeling approaches with mea-
468
sured data are analyzed. Despite the short duration of the field campaign,
469
the different wave climates that occurred allow to distinguish some clear
470
characteristics of the different methods. The assessment is done based on
471
the statistical errors defined in Section 2.2 and the results are discussed for
472
the three different wave climates: southerly (∼5◦ from shore-normal at 21 m
473
depth), easterly (∼30◦ from shore-normal at 21 m depth) and bimodal wave
474
climates. An error is deemed acceptable when NRMSE ≤ 1, i.e., when the
475
RMSE < standard deviation. The cases when this does not occur are marked
476
in italics in Table 4. The confidence limit of θm is estimated by Kuik et al.
477
(1988) to be 5◦ –10◦ in terms of RMSE.
478
For southerly waves, the three propagation methods give good results for
479
all wave parameters (Table 4). The main reason for this is that propagating
480
southerly waves is straightforward, since they are nearly shore-normal. As
481
a result, the simplifications of the simple wave ray model (monochromatic
482
waves and parallel depth contours) do not affect the propagation of these
483
waves. When forcing SWAN with 2D spectra, the only clear improvement is
484
obtained for Tp . This is a result of differences in the 2D spectra between the 28
485
simulations. Figure 8 shows the measured 2D spectra in the buoy and in the
486
AWAC together with the modeled spectra for SWAN forced with a single set
487
of integrated wave parameters (used to build a single-component JONSWAP
488
spectrum) and SWAN forced with the measured 2D spectrum. Using 2D
489
spectra as forcing leads to modeled 2D spectra at the buoy and AWAC loca-
490
tions that are very similar to the measured 2D spectra. When forcing with
491
integrated wave parameters, the modeled 2D spectra are slightly wider in di-
492
rection and the frequency peak is smaller. However, the energy distribution
493
is generally correct in the latter case indicating that the single-component
494
JONSWAP spectrum generally works well in southerly wave climates.
495
For easterly waves, the simple wave ray model performs badly (N RM SE >
496
1 for all parameters except Tm02 , see Table 4), and in particular refraction
497
is strongly underestimated. Using SWAN forced with integrated wave pa-
498
rameters improves the results for Hm0 and θm , although the errors in Tp and
499
θm are large and refraction is still underestimated. Forcing SWAN with 2D
500
spectra leads to a big improvement in Tp and θm and most parameters show
501
a N RM SE < 1. In particular, refraction is well captured and the RM SE
502
of θm is only 10◦ . The reason is that forcing SWAN with 2D spectra leads
503
to modeled 2D spectra similar to the observed ones (Figure 9), also at the
504
AWAC. When forcing SWAN with integrated wave parameters, the modeled
505
spectrum at the Barcelona buoy is much wider in direction compared to the
506
observed spectrum and only displays one peak in frequency (in reality there
507
are several). At the AWAC, the energy in the modeled spectrum is much
508
more concentrated in direction (hence the higher energy values), whilst in
509
reality there is much more spread in direction. For these easterly waves,
29
Figure 8: Measured and modeled 2D frequency-direction spectra E(f, θ) for a southerly wave climate (15 March 2018 20:00 UTC). From left to right the results are shown for the measured data, SWAN forced with integrated wave parameters and SWAN forced with 2D frequency-direction spectra (data in both cases taken from the Barcelona buoy). The top row shows the results for the Barcelona buoy location and the bottom row the results for the AWAC location at Castelldefels. Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.3/1.3 m, Tm02 = 5.0/5.1 s, θm = 202/193 deg.
510
the single-component JONSWAP spectrum does not work well because it
511
cannot represent multiple frequency peaks and it is too wide in direction.
512
The performance of the wave ray model is worse than that of SWAN with
513
integrated wave parameters. This is a result of the crude simplifications, in
514
particular the assumption of an alongshore-uniform bathymetry. As a result,
515
the refraction of easterly waves over the alongshore-variable bathymetry is
516
not correctly reproduced. Furthermore, as mentioned before, the waves that
517
arrive at the Barcelona buoy with a direction < 90◦ with respect to North
30
Figure 9:
Identical to Figure 8, but now for a easterly wave climate (22 March 2018
03:00 UTC). Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.8/1.0 m, Tm02 = 5.6/6.7 s, θm = 80/136 deg.
518
cannot be considered with the simple wave ray model (hence the gaps in
519
Figure 6).
520
For bimodal wave climates, the only tested method to accurately prop-
521
agate waves is forcing SWAN with 2D spectra (N RM SE < 1 for all wave
522
parameters except θp ). SWAN forced with integrated wave parameters leads
523
to modeled spectra that differ greatly from the measurements (Figure 10).
524
The single-component JONSWAP spectrum is not valid for bimodal wave
525
climates because the directional distribution used only allows for one direc-
526
tional peak in the 2D spectrum and the distribution on both sides of the
527
directional peak is symmetrical. The result is that the peak is located in an
528
area where in reality no energy is present (the mean of the two wave fields).
529
Consequently, the resulting errors in Tp and θm are large, although Hm0 and 31
Figure 10: Identical to Figures 8 and 9, but now for a bimodal wave climate (20 March 2018 03:00 UTC). Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.3/0.9 m, Tm02 = 4.6/5.0 s, θm = 107/164 deg.
530
Tm02 are still relatively well reproduced. The simple wave ray model also
531
performs well for Hm0 and Tm02 but the errors in Tp and θm are large. The
532
causes for these errors are twofold. First, the simple wave ray model assumes
533
monochromatic waves, meaning that all wave energy is concentrated in a
534
single set of integrated wave parameters, which is not sufficient to accurately
535
describe bimodal wave climates. Second, the model assumption of along-
536
shore uniform bathymetry forbids to accurately reproduce wave refraction
537
(as explained before).
538
The above discussion highlights the importance of using high quality
539
boundary conditions when propagating waves over alongshore-variable bathyme-
540
tries. The simple wave ray model uses straightforward boundary conditions
541
(wave height, wave period and wave direction), that work well for shore32
542
normal waves but are not accurate enough for shore-oblique waves and bi-
543
modal wave climates. When applying a spectral wave model like SWAN, the
544
boundary conditions are more detailed because either integrated wave pa-
545
rameters (including directional spreading) are used to build the 2D spectra
546
or the measured 2D spectra are applied explicitly. However, a 2D spectrum
547
constructed out of a single set of integrated wave parameters can be signifi-
548
cantly different from the measured spectrum because a certain combination
549
of Hm0 , Tp∗ , θm and σ θ can fit to many different spectral shapes. As noted by
550
Portilla-Yand´ un et al. (2015) and Cavaleri et al. (2018), a lot of detailed infor-
551
mation that is available in the full 2D spectrum is blurred or lost when only
552
using the integrated wave parameters. The present study underlines that,
553
in the Catalan coast, this is critical for easterly and bimodal wave climates,
554
where a single set of integrated wave parameters is unable to accurately de-
555
scribe the complex wave spectrum. This mostly affects the results of Tp and
556
θm , whilst for Hm0 and Tm02 no changes are noticed. When using the full 2D
557
spectrum as boundary conditions, it is possible to obtain accurate results for
558
Tp and θm in all wave climates, which is in agreement with previous research
559
on the western American coast (Gorrell et al., 2011). Furthermore, notice
560
that for both forcing types the errors in θp are much larger compared to θm
561
(for the three wave climates). The results using integrated wave parameters
562
might improve in case the 2D spectrum could be reconstructed out of mul-
563
tiple sets of wave parameters (e.g. Portilla-Yand´ un et al., 2015) instead of
564
using a single set.
33
565
5.2. Performance of hindcast models
566
The two hindcast models (SIMAR and GOW2) correctly predict the
567
trends in wave height and wave period (Table 4) for all wave climates, whilst
568
the errors for θm are very large for shore-oblique waves and bimodal wave
569
climates. There are several reasons for these mismatches. Global hindcast
570
models typically have coarse spatial resolution (5 km for SIMAR and 0.25◦
571
for GOW2), which is not sufficient to take into account the necessary details
572
in changes in the orientation of the shoreline, bathymetry and orography
573
(Cavaleri et al., 2018). Apart from that, global hindcast models are often
574
used without shallow water physics and without incorporating the effects of
575
bottom friction. Furthermore, the output points of the two hindcast models
576
in front of the study site are located at a shallow depth of 21 m. Using
577
data from an output point at deeper water and propagating this data with
578
SWAN might yield better results. To test this, data from SIMAR and GOW2
579
output points located closest to the Barcelona buoy at deeper water (75 m
580
and 250 m respectively; Figure 1) have been compared to the measured data
581
from the Barcelona buoy (at 68 m depth). Again substantial differences are
582
seen between the modeled and measured wave data (Figure 2). The results
583
for Hm0 (RMSE of 0.2 m for both models), Tm02 (RMSE of 1.0 and 0.6 s
584
respectively) and Tp (RMSE of 1.5 and 1.4 s respectively) are good for both
585
models, but the errors in θm (RMSE of 37◦ for both models) remain large.
586
The match between the modeled and measured wave climates at deeper water
587
is not deemed sufficient (particularly for wave direction) to use deep water
588
hindcast data as input for the wave propagation models. Furthermore, prop-
589
agating this data with SWAN would lead to the same problems that occur
34
590
when forcing SWAN with integrated wave parameters from the buoy.
591
Another reason that hindcast models display large errors in wave direction
592
is that they are forced by wind fields from atmospheric models that can
593
contain important errors (Bola˜ nos-Sanchez et al., 2007). Unfortunately, it is
594
difficult to obtain reliable wind fields in coastal and semi-enclosed seas (like
595
the Mediterranean Sea), due to the strong spatial and temporal gradients in
596
wind speed and direction (Cavaleri et al., 2018). Various studies (e.g. Ardhuin
597
et al., 2007; Mentaschi et al., 2015) demonstrated that choosing a different
598
wind model or changing the resolution of the wind model can strongly affect
599
the results of wave models in the Western Mediterranean Sea. Specifically
600
for the Catalan coast, Bertotti et al. (2014) tested four different atmospheric
601
models as input for the same wave model. The results showed that during
602
non-extreme but complicated wave conditions (consisting of multiple wave
603
systems), none of the atmospheric models was able to correctly reproduce
604
all the observed wind patterns (each model missed at least one of the wind
605
systems), which in turn led to an incorrect representation of the wave field.
606
Other studies that used large-scale wave models forced by wind fields
607
and focused on the Catalan coast (e.g. Bola˜ nos-Sanchez et al., 2007; Alomar
608
et al., 2014; Pallares et al., 2014) also indicated that errors in the modeled
609
wave fields were mainly related to inaccuracies in the wind fields. Although
610
typically only wave height and wave period were validated, Bola˜ nos-Sanchez
611
et al. (2007) also compared measured and modeled 1D frequency spectra
612
in which they found large differences, particularly for cases with multiple
613
frequency peaks. Pallares et al. (2014) obtained good results for wave height
614
and wave period but large errors in mean direction (RMSE above 40◦ –60◦
35
615
depending on the study period). Amrutha et al. (2016), a similar study in the
616
Eastern Arabian Sea that also used a large-scale wave model forced by wind,
617
found good agreement for mean direction during monsoon periods (RMSE
618
of 13◦ ), but errors increased substantially outside this period because the
619
wind patterns were not correctly simulated. Finally, the study by Rogers
620
and Wang (2007) focused on modeling directional wave properties in Lake
621
Michigan using SWAN forced by wind measurements from two buoys. They
622
found better results for θm , with an RMSE of 17◦ .
623
5.3. Recommendations
624
Based on the results discussed above, the following recommendations
625
can be given to obtain directional wave conditions for coastal studies in
626
semi-enclosed and coastal seas that are surrounded by a relevant orogra-
627
phy. Coarse-resolution hindcast models (or fine-resolution models forced by
628
coarse atmospheric models) usually reproduce the trends in wave height and
629
mean period correctly, meaning that in most cases they can be safely used
630
after applying some simple calibration methods (e.g. bias correction). In
631
particular, both SIMAR and GOW2 results are nowadays used extensively
632
to evaluate global changes in wave energy (e.g. Rodriguez-Delgado et al.,
633
2018) and can be very useful for engineering applications such as predict-
634
ing beach inundation and harbor management or for ecological studies (e.g.
635
De la Hoz et al., 2018). However, the large errors in wave direction make
636
global hindcast models not suited for nearshore morphodynamic studies in
637
semi-enclosed and coastal seas.
638
In such environments, the alternative is to propagate measured offshore
639
wave conditions. In case the wave parameters of interest are wave height 36
640
and mean period (but not direction), any of the tested propagation methods
641
can be safely used. However, the propagation method must be chosen care-
642
fully if accurate results for wave direction are needed, especially when the
643
bathymetry is alongshore-variable. The most reliable way to propagate the
644
wave conditions is by forcing a spectral model (such as SWAN) with mea-
645
sured 2D spectra as boundary conditions. Although it can be complicated to
646
obtain and handle the 2D spectra, this is the only method providing errors
647
below 10◦ for all the wave climates. This accuracy in wave direction can
648
be of crucial importance to determine if a site is prone to suffer from high
649
angle wave instability and the formation of shoreline sand waves (Ashton
650
et al., 2001; Arriaga et al., 2018) or whether crescentic or transverse bars
651
will be formed or destroyed (Calvete et al., 2005; Price and Ruessink, 2011;
652
Ribas et al., 2012). The only parameter that is not well reproduced (by any
653
method) is the peak direction θp , so it is discouraged to use this parameter
654
in nearshore studies.
655
When 2D spectra are not available, the most ideal situation would be to
656
construct a 2D spectrum out of multiple sets of integrated wave parameters
657
(e.g. separate wind-sea and swell components). Unfortunately, this is often
658
not possible because integrated wave parameters are typically only provided
659
for the entire spectrum. The next option is to force a spectral wave model
660
with 2D spectra constructed out of a single set of integrated wave param-
661
eters or use the integrated wave parameters directly in a simple wave ray
662
model. Compared with forcing with the full 2D spectra, these methods yield
663
the same accuracy for wave height and mean period, but wave direction is
664
only well-reproduced for cases characterized by limited refraction and par-
37
665
allel depth contours. As soon as the wave climate is bimodal or the waves
666
refract strongly, the errors in θm increase substantially. Although these two
667
methods have the advantage to be much easier to use, the errors in wave
668
angle can be too large to correctly characterize the dominant processes. The
669
underestimation of wave refraction is stronger in the simple wave ray model,
670
which sometimes cannot even account for all waves (in case of changes in the
671
orientation of the shoreline). In general, it is highly recommended to validate
672
the propagation results for wave direction with measurements before using
673
such simpler methods. Nevertheless, it should be pointed out that for the
674
case studied here, the effect of incorporating physical processes (like white-
675
capping and quadruplets) in spectral models is negligible (see Section 4.4.2).
676
This indicates that conditions are favorable for applying the simple wave ray
677
model and its shortcomings compared to spectral models are expected to
678
become even more evident in more complex phenomenological frameworks.
679
Results for wave ray tracing could be improved by using more advanced wave
680
ray models that can be forced with multiple sets of integrated wave param-
681
eters, take into account alongshore-variable bathymetry and include some
682
representation of shallow water processes.
683
Finally, the location of wave buoys can also cause measurement problems.
684
Wave buoys are often deployed near harbor entrances, meaning that they are
685
often located in water depths of less than 30 m. As a result, the measurements
686
are affected by shallow water processes (shoaling, wave breaking and friction),
687
whilst also the close proximity of harbor constructions like breakwaters can
688
render the data useless for accurately describing the offshore wave climate.
689
Furthermore, wave buoys can be sheltered from certain wave directions due
38
690
to changes in the orientation of the shoreline. For example, the Barcelona
691
wave buoy used in this study is sheltered for the rare waves coming from the
692
west-southwest, whilst they can freely arrive at Castelldefels. In this sense,
693
having information from a second buoy located in the southern part of the
694
domain would be ideal.
695
6. Conclusions
696
The accuracy of different methodologies to obtain directional wave char-
697
acteristics for nearshore field studies has been tested for various wave climates
698
at a beach located on a limited-fetch, complex-geometry sea. Such verifica-
699
tion of wave propagation methods at the location of interest turns out to
700
be crucial regarding wave direction. Global hindcast models with spatial
701
resolutions of 10 km or more cannot be expected to represent the coastal
702
zone and this also shows up in this study, meaning that their results must be
703
interpreted with care. The trends in wave height and mean period are cor-
704
rectly predicted, but the errors in wave direction for shore-oblique waves are
705
large (RMSE above 60◦ ), so that hindcast models are not suited for studies
706
of nearshore processes that depend on wave direction.
707
Better accuracy can be obtained by propagating wave conditions mea-
708
sured at offshore buoys, although the propagation method must be chosen
709
carefully. A simple wave ray model based on linear wave theory that as-
710
sumes monochromatic waves as boundary conditions and alongshore-uniform
711
bathymetry provides good results regarding wave height and period. How-
712
ever, the crude model assumptions cause the model to underestimate refrac-
713
tion of oblique waves over an alongshore-variable bathymetry, which leads to 39
714
large errors in wave direction (RMSE above 30◦ ) for easterly and bimodal
715
wave climates.
716
The results for these wave climates can be improved by propagating waves
717
using a properly scaled, third-generation wave model like SWAN. As offshore
718
boundary condition, such models require a full 2D spectrum, and one option
719
is to reconstruct it out of a single set of integrated wave parameters. When
720
applying this method, the results for wave direction during shore-oblique
721
waves and bimodal wave climates are improved, although the RMSE are
722
still above 20◦ . The other option is to directly prescribe a measured 2D
723
spectrum, which gives the best results for wave direction and reduces the
724
RMSE to values below 10◦ for all wave climates. Such accuracy in wave
725
direction is essential in many nearshore studies, particularly when studying
726
the evolution of crescentic bars, transverse bars and shoreline sand waves.
727
During the short sampling period of this study, the SWAN results are
728
robust to changes in model settings. Using simple settings suffice and includ-
729
ing physical processes like wind, quadruplets or whitecapping do not improve
730
the results. This confirms that the dominant process modifying the offshore
731
waves over the alongshore-variable bathymetry to the nearshore study site
732
is wave refraction. Under such circumstances, the use of spectral boundary
733
conditions is highly important and observed offshore 2D wave spectra can be
734
extremely useful to obtain accurate results of the nearshore wave field using
735
wave propagation models.
40
736
Acknowledgments
737
The authors would like to thank Benjam´ın Casas of IMEDEA and the
738
technicians from SOCIB (Coastal Ocean Observing and Forecasting System
739
of the Balearic Islands) for their help and assistance during the field cam-
740
paign. Special thanks to Pilar Gil from Puertos del Estado for providing
741
the SIMAR and wave buoy data and IH Cantabria for providing the GOW2
742
data used in this study. The bathymetric data used in this study has been
743
obtained from the European Marine Observation and Data Network (EMOD-
744
net, http://www.emodnet-bathymetry.eu/) and the wind data of Barcelona
745
airport has been provided by the Spanish meteorological agency (Agencia
746
Estatal de Meteorolog´ıa), which is part of the Ministry for the Ecological
747
Transition (MITECO). This work has been funded by the Spanish govern-
748
ment through the research projects CTM2015-66225-C2-1-P and CTM2015-
749
66225-C2-2-P (MINECO/FEDER). Finally, the constructive comments of
750
two anonymous reviewers helped to improve the quality of this paper.
751
Declarations of interest
752
753
754
755
None. Appendix A. Measurement of directional wave parameters Directional wave parameters are defined using the 2D frequency-direction spectrum, E(f, θ), which is often written as
E(f, θ) = E(f )D(f, θ) ,
41
(A.1)
756
757
where D(f, θ) is called the directional distribution function that has units R 2π of 1/deg. It is defined in such a way that 0 D(f, θ)dθ = 1. A lot of
758
different methods have been developed to obtain D(f, θ) out of wave buoy
759
measurements. The most common method starts by decomposing D(f, θ)
760
conceptually into a Fourier series using (Longuet-Higgins et al., 1963) ∞ 1 1 X D(f, θ) = + [an cos(nθ) + bn sin(nθ)] , π 2 n=1
(A.2)
761
where an (f ) and bn (f ) are the standard Fourier coefficient that are defined
762
as
Z
2π
an (f ) =
D(f, θ) cos(nθ)dθ
and
D(f, θ) sin(nθ)dθ
.
Z 02π bn (f ) =
(A.3)
0 763
Only the first four Fourier coefficients (a1 , b1 , a2 , b2 ) of D(f, θ) can be ob-
764
tained from buoy measurements (Hoekstra et al., 1994). As a result, direc-
765
tional spreading is generally overestimated, whereas D(f, θ) is mostly too
766
broad and can even become negative (Young, 1994). To overcome this prob-
767
lem, parametric models like the cos2s method (Longuet-Higgins et al., 1963)
768
and data-adaptive methods like the MLM (Krogstad et al., 1988) and MEM
769
(Lygre and Krogstad, 1986) approximate the complete Fourier series to give
770
a full estimate of D(f, θ). However, the drawback of these models is that they
771
give details of D(f, θ) that are not determinable from buoy measurements
772
and sometimes even generate spurious data (Benoit et al., 1997; Rogers and
773
Wang, 2007). Taking these problems into account, Kuik et al. (1988) defined
774
expressions for mean wave direction, θm (f ), and directional spreading, σθ (f ), 42
775
per frequency, as a function of the first two determinable Fourier coefficients.
776
Their expression for θm (f ) reads " θm (f ) = tan−1
b1 (f ) a1 (f )
# .
(A.4)
777
However, the frequency range is always finite so the way to compute the
778
overall mean direction averaged over the flow -fhigh frequency band is " θm = tan−1
b1 a1
# ,
(A.5)
779
and the peak wave direction θp is computed by taking the mean direction
780
at the frequency peak (θp = θm (fp )). Furthermore, the directional spreading
781
averaged over the frequency range can be computed using p σ θ = 2[1 − r1 ] ,
782
q 2 where r1 = a21 + b1
,
(A.6)
in which a1 and b1 are the first 2 Fourier coefficients that are defined as R fhigh a1 =
a1 (f )E(f )df R fhigh E(f )df flow
flow
and
R fhigh b1 =
b1 (f )E(f )df flow R fhigh E(f )df flow
.
(A.7)
783
Wave buoys normally obtain the Fourier coefficients from measurements as
784
described by Hoekstra et al. (1994).
785
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786
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Pettersson, H., Graber, H.C., Hauser, D., Quentin, C., Kahma, K.K., Dren-
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Ek, M., Meng, J., Wei, H., Yang, R., Lord, S., van den Dool, H., Kumar,
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A., Wang, W., Long, C., Chelliah, M., Xue, Y., Huang, B., Schemm,
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J.K., Ebisuzaki, W., Lin, R., Xie, P., Chen, M., Zhou, S., Higgins, W.,
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Reanalysis. Bulletin of the American Meteorological Society 91, 1015–1058.
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S´anchez-Arcilla, A., Gonz´alez-Marco, D., Bola˜ nos, R., 2008.
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Whitham, G.B., 1974. Linear and Nonlinear Waves. John Wiley & Sons, Inc., Hoboken, NJ, USA. doi:10.1002/9781118032954. Young, I.R., 1994. On the measurement of directional wave spectra. Applied Ocean Research 16, 283–294. doi:10.1016/0141-1187(94)90017-5. 52
¯
France
Legend
! ( ! (
Spain
# * # * $ +
Portugal
" )
Barcelona wave buoy AWAC location GOW2/SIMAR model point SIMAR deep water GOW2 deep water Wind measurements Castelldefels beach
Bathymetry [m]
El Prat de Llobregat
High : 0 Low : -400
Castelldefels
0 1 2
4
6
Kilometers 8
Barcelona
Highlights • • • •
Five methods to obtain directional wave conditions at shallow waters are tested. Hindcast models provide proper wave height and period but unreliable wave angle. Propagating idealized buoy-derived spectra still leads to inaccurate direction. Only propagating buoy-measured directional spectra produces accurate wave direction.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0278-4343(20)30027-3
DOI:
https://doi.org/10.1016/j.csr.2020.104071
Reference:
CSR 104071
To appear in:
Continental Shelf Research
Received Date: 4 October 2019 Revised Date:
13 January 2020
Accepted Date: 27 January 2020
Please cite this article as: de Swart, R.L., Ribas, F., Calvete, D., Kroon, A., Orfila, A., Optimal estimations of directional wave conditions for nearshore field studies, Continental Shelf Research, https://doi.org/10.1016/j.csr.2020.104071. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Optimal estimations of directional wave conditions for nearshore field studies R.L. de Swarta,∗, F. Ribasa , D. Calvetea , A. Kroonb , A. Orfilac a
Department of Physics, Universitat Polit`ecnica de Catalunya (UPC), Jordi Girona, 1-3, 08034, Barcelona, Spain b Department of Geosciences and Natural Resource Management, University of Copenhagen, Øster Voldgade, 10, 1350, Copenhagen, Denmark c IMEDEA (CSIC-UIB), Mediterranean Institute for Advanced Studies, Miquel Marqu`es, 21, Esporles (Illes Balear), Spain
Abstract Accurate directional wave conditions at shallow water are crucial for nearshore field studies and necessary as boundary conditions for morphodynamic models. However, obtaining reliable results for all wave parameters can be challenging, particularly regarding wave direction. Here, the accuracy of two global hindcast models and propagation of measured wave conditions using linear wave theory or the SWAN wave model (forced by integrated wave parameters or 2D spectra) is assessed to obtain directional wave conditions at shallow water for Castelldefels beach, Northwest Mediterranean Sea. Results are analyzed using different statistical error parameters and for different wave climates (shore-normal, shore-oblique and bimodal). The analysis shows that global hindcast models correctly predict the trends in wave height ∗ Corresponding author at: Department of Physics, Universitat Polit`ecnica de Catalunya (UPC), Jordi Girona, 1-3, Edificio B5, 08034, Barcelona, Spain. Tel: +34 934016087 Email addresses: [email protected] (R.L. de Swart), [email protected] (F. Ribas), [email protected] (D. Calvete), [email protected] (A. Kroon), [email protected] (A. Orfila)
Preprint submitted to Continental Shelf Research
January 29, 2020
and mean wave period but predictions for mean wave direction are only accurate for shore-normal waves. Linear wave theory provides good results for wave height but underestimates refraction, resulting in significant errors in mean wave direction for shore-oblique waves. Finally, SWAN forced with 2D spectra results in the most accurate predictions for all wave parameters. When using integrated wave parameters as boundary conditions, the results for wave height and mean period stay the same whilst the errors in peak period and mean direction worsen for shore-oblique and bimodal wave climates. The reason is that for these wave conditions the directional spectrum constructed out of integrated wave parameters does not resemble the actual directional spectrum. Keywords: Wave propagation, Mediterranean Sea, Global hindcast models, Linear wave theory, SWAN modeling, Directional spectra
1
1. Introduction
2
Many shorelines around the world are highly dynamic and change due
3
to wave action. Understanding this behavior is important for present day
4
coastal zone management and for estimating the vulnerability to climate
5
change (Vitousek et al., 2017). Incoming waves are the most important forc-
6
ing for coastal evolution, and wave characteristics in shallow water depend on
7
the local coastal setting (wave climate, bathymetry, shoreline orientation).
8
Unfortunately, obtaining reliable wave conditions in coastal and inner seas
9
is still a difficult task, especially in semi-enclosed seas like the northwest-
10
ern Mediterranean Sea (Bola˜ nos-Sanchez et al., 2007; Cavaleri et al., 2018).
11
The reason is that this region is characterized by limited fetches, a complex
2
12
bathymetry with deep canyons, significant changes in shoreline orientation
13
and strong variations in the wind climate (S´anchez-Arcilla et al., 2008).
14
Wave conditions can be obtained from a variety of sources. The preferred
15
source for wave conditions are devices that provide real-time measurements.
16
Different types of wave measurement instruments exist (e.g. Allender et al.,
17
1989; Pettersson et al., 2003), but wave buoys are the most common world-
18
wide. However, wave buoys are scarce because they are quite expensive to
19
deploy and maintain. As a result, they are mostly deployed in areas where
20
there is a great need for real-time wave measurements (typically near harbors
21
or populated coastlines). For studying other locations, the measured offshore
22
waves need to be transformed to nearshore study sites using a so called ’wave
23
propagation model’.
24
In addition to buoy measurements, wave conditions can also be obtained
25
from global hindcast models, based on codes such as WAM (The WADMI
26
Group, 1988) and WAVEWATCH-III (Tolman, 2009). Hindcast models typ-
27
ically also assimilate observations and have the advantage that they do not
28
suffer from breakdowns. Furthermore, results can theoretically be acquired
29
for any worldwide location, including those where no other data is available.
30
However, these models may not be accurate in shallow waters (Cavaleri et al.,
31
2018), and proper wave propagation might also be needed.
32
There are different ways to propagate wave conditions to shallow waters.
33
The simplest method is to use linear wave theory based on ray approximation
34
(e.g. Holthuijsen, 2007). Despite its crude assumptions, this method is often
35
applied in nearshore studies to quickly obtain reasonable estimates of the
36
wave conditions. A more thorough method is to use sophisticated wave
3
37
models like SWAN (Booij et al., 1999). Such models include more physical
38
processes and have been extensively used in the past in nearshore regions with
39
gradually variable bathymetry (e.g. Ris et al., 1999). The SWAN model has
40
the additional advantage that it is a spectral model, meaning that it solves
41
the spectral action balance equation without any a priori restrictions on the
42
spectrum for the evolution of wave growth. Spectral models can either be
43
forced by integrated wave parameters or using full 2D spectra.
44
Good predictions of wave height and wave period are important for, e.g.,
45
management of harbors, navigation or fisheries. However, wave direction is
46
crucial in coastal engineering because even small variations in wave direction
47
can already substantially affect estimates of longshore sediment transport
48
(e.g. Soomere and Viˇska, 2014), along with the related dynamics of embayed
49
beaches (Harley et al., 2015) and beach nourishments (Arriaga et al., 2017).
50
On the other hand, nearshore studies in the past typically focused on relat-
51
ing beach morphodynamics to wave height and wave period only (e.g. Van
52
Enckevort et al., 2004; G´omez-Pujol et al., 2007), whereas wave direction was
53
often neglected. However, recent studies indicate that wave direction plays
54
a major role in the evolution of morphodynamic patterns, such as crescentic
55
bars (Calvete et al., 2005; Price and Ruessink, 2011), transverse finger bars
56
(Ribas and Kroon, 2007; Ribas et al., 2012), high angle wave instability and
57
km-scale shoreline sand waves (Ashton et al., 2001; Arriaga et al., 2018).
58
Wave conditions used in nearshore studies are typically obtained by prop-
59
agation of offshore buoy measurements or hindcast model results. This can
60
lead to errors, so that verification of the propagated wave conditions against
61
local measurements is essential. However, many nearshore studies show only
4
62
limited or no verification of wave conditions (e.g. Splinter et al., 2011; Ar-
63
riaga et al., 2018). On the other hand, studies that focus on modeling wave
64
fields in coastal regions with large-scale hindcast models (sometimes coupled
65
to SWAN to increase resolution in the nearshore) mostly verify wave height
66
and period with typical errors of 0.25 m and 1.5 s in both wind-sea and swell
67
conditions (e.g. Pallares et al., 2014; Amrutha et al., 2016), whilst wave di-
68
rection is often not included (e.g. Bola˜ nos-Sanchez et al., 2007; Perez et al.,
69
2017). In the few studies with hindcast models where wave direction is also
70
validated (e.g. Pallares et al., 2014; Amrutha et al., 2016), the corresponding
71
errors are large (> 40◦ ). The errors in wave direction can be substantially
72
reduced by propagating measured wave conditions (Gorrell et al., 2011). Un-
73
fortunately, there is a lack of studies that compare the reliability of different
74
methods to obtain directional wave conditions for nearshore studies, espe-
75
cially in semi-enclosed and coastal seas.
76
The aim of this study is to establish the accuracy of different methods to
77
obtain wave conditions in shallow water for nearshore studies, with a special
78
focus on the wave direction. This is done for the field site of Castelldefels
79
beach, which is located on a limited-fetch, complex-geometry sea (Northwest
80
Mediterranean Sea; Section 2). Three long-term sources of wave conditions
81
are available near this site: a buoy and output points of two hindcast models.
82
An instrument was deployed to measure wave conditions in front of the field
83
site at 21 m depth during a 9 days experiment. This allows to compare five
84
different methods to obtain wave conditions at the field site from the long-
85
term sources (Section 3). Three methods consist of propagating measured
86
wave conditions from the offshore buoy using models with different degrees
5
87
of complexity: the simple wave ray model, SWAN forced by integrated wave
88
parameters and SWAN forced by 2D spectra. The two last methods use the
89
results of the two hindcast models directly. Despite the short duration of
90
the field campaign, distinct wave climates occurred, which allows to charac-
91
terize the accuracy of each method under different wave conditions (Section
92
4). The significance of the findings of this study and the advantages and
93
disadvantages of the different methods are discussed in Section 5, and the
94
conclusions are listed in Section 6.
95
2. Study site and datasets
96
2.1. Study area
97
The study site is Castelldefels beach, located on the north-western Mediter-
98
ranean Sea along the Spanish Catalan coast, approximately 20 km southwest
99
of Barcelona (Figure 1). It is an open beach with an east-west orientation and
100
has a length of approximately 4.5 km, whilst more to the east the shoreline
101
orientation changes rapidly towards the north. Castelldefels beach is part of
102
a continuous stretch of beaches of the Llobregat delta, extending from the
103
Garraf Mountain chain in the west to the Llobregat river outfall in the east.
104
It is mainly composed of sand with a median grain size of 270 µm. Tidal
105
action in this part of the Mediterranean Sea is small, with a range of ap-
106
proximately 20 (10) cm during spring tide (neap tide; Simarro et al., 2015).
107
The bathymetry in the study area (Figure 1) shows a relatively wide shelf in
108
front of the study site, whilst more to the east the shelf is narrower and the
109
water depth quickly reaches more than 100 m.
6
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France
Legend
! ( ! (
Spain
# * # * $ +
Portugal
" )
Barcelona
Barcelona wave buoy AWAC location GOW2/SIMAR model point SIMAR deep water GOW2 deep water Wind measurements Castelldefels beach
Bathymetry [m]
El Prat de Llobregat
High : 0 Low : -400
Castelldefels
0 1 2
4
6
Kilometers 8
Figure 1: Overview map showing the Castelldefels study site, the nearshore bathymetry (source Emodnet) and the locations of the different wave data sources used in this study. c 2019 Microsoft Corporation The aerial photography is part of Microsoft Bing Maps ( Earthstar Geographics SIO).
110
The winds in the study area are strongly influenced by orographic bar-
111
riers, which leads to a variable wind climate and the formation of intense
112
north and northwestern winds during December and January. The winds are
113
weaker during the rest of the year and the maximum wind velocities occur
114
during easterly storms that affect the entire Catalan coastline (Pallares et al.,
115
2014). The wind patterns directly influence the local wave climate, which
116
is characterized by calm wave conditions with sudden high energetic wave
117
events (wave height above 1.5 m; Puertos del Estado, 1994). On average,
118
calm conditions prevail during the summer period and energetic conditions
119
occur mostly from October to May (S´anchez-Arcilla et al., 2008). Storm
7
120
waves along the Catalan coast are limited due to the short fetches and an
121
average storm duration of less than 24 hours. As a result, mixed sea states
122
composed of wave trains with more than one peak frequency and mean di-
123
rection occur frequently (S´anchez-Arcilla et al., 2008). Near the study site,
124
the wave climate is dominated by waves from the east and south (Figure 2a).
125
The largest waves come from the east, due to the longest available fetches and
126
the stronger winds that generally blow from this direction (S´anchez-Arcilla
127
et al., 2008; Bola˜ nos et al., 2009).
Figure 2: Wave roses of the long-term wave climate at the Barcelona buoy (a), the SIMAR output point next to the Barcelona buoy (b) and the GOW2 output point nearest to the Barcelona buoy (c) for the period September 2012–May 2018 in terms of spectral wave height Hm0 and mean direction θm .
128
2.2. Short-term data source (AWAC)
129
A short-term dataset of wave conditions was measured from 13 March to
130
22 March 2018 using an AWAC sensor deployed at 21 m depth in front of
131
Castelldefels (Figure 1). This sensor is both a current profiler and a direc-
132
tional wave system that was mounted in a stationary frame at the bottom.
133
During the measurement period, the AWAC provided half-hourly values
134
of the spectral wave height Hm0 , mean zero-crossing period Tm02 , peak period 8
135
Tp , mean direction θm , peak direction θp and mean directional spreading σ θ ,
136
as well as half-hourly 1D frequency and 2D frequency-direction spectra. A
137
detailed description of how these quantities are computed is presented in
138
Section 3.1 and Appendix A. The AWAC used a frequency range of 0.02–
139
0.49 Hz and a frequency resolution of 0.01 Hz to obtain all wave parameters
140
and spectra. The full 2D frequency-direction spectra was estimated using the
141
Maximum Likelihood Method (MLM; Krogstad et al., 1988). All important
142
AWAC settings are summarized in Table 1.
143
The data obtained by the AWAC is used as ground truth in the rest of
144
this study. When comparing the errors of the different methods with the
145
ground truth, the root mean square error (RMSE), BIAS and normalised
146
root mean square error (NRMSE) are used, defined as
s
PN
i=1 (Si
RM SE =
N PN
BIAS =
− Ai )2
i=1 (Si
N RM SE N RM SE = σA
− Ai ) ,
,
(1)
,
(2) (3)
147
in which S denotes the simulated data from the various propagation methods
148
and A denotes the measured data from the AWAC (ground truth). The
149
number of data points is denoted by N , whilst σA denotes the standard
150
deviation of the AWAC measurements. Angles with respect to North are
151
used as input when computing these quantities for angular values. This is
152
appropriate, because wave directions in the study area are such that no jumps
153
from 360 to 0 degrees occur. 9
Table 1: Overview of instrument and deployment settings for the Barcelona buoy and AWAC. Device
AWAC
Barcelona buoy
Water depth
21 m
68 m
Sensor altitude above bottom
0.5 m
Tide-dependent
Burst interval
154
0.5 h (start 5 min
1 h (start at top of
past every half hour)
the hour)
Sampling rate
1 Hz
4 Hz
Length of processed time series
1200 points
5760 points
Start frequency
0.02 Hz
0.03 Hz
End frequency
0.49 Hz
0.64 Hz
Number of frequency bands
48
123
Number of directional bins
90
121
Directional spectrum method
Maximum Likelihood
Maximum Entropy
Method
Method
Start of deployment
13 March 2018 15:05
8 March 2004
End of deployment
22 March 2018 09:05
–
2.3. Long-term available wave datasets
155
Although no long-term wave buoy or other measurement instrument is
156
located directly in front of the study area, there are different long-term wave
157
datasets available around Castelldefels. They comprise both measurements
158
and large-scale global hindcast models.
159
2.3.1. Barcelona wave buoy
160
The Barcelona buoy is the only permanent instrument in the area. It is
161
moored in front of Barcelona harbor (41.32 N, 2.20 E) at a depth of 68 m
162
(Figure 1), managed by Puertos del Estado (Spanish Ports Authority), and
163
it is operational since March 2004. The buoy is located relatively close to the
164
study site (the distance between the buoy and the study site is approximately
10
165
20 km), but there is a significant change in the orientation of the shoreline
166
and the bathymetric contours (Figure 1).
167
The buoy obtains the heave time-series using standard Fourier analysis
168
(Kuik et al., 1988), which are then processed using both spectral analysis
169
and zero-crossing analysis to determine non-directional wave parameters and
170
1D frequency spectra. The frequency range is 0.03–0.64 Hz with a resolution
171
of 0.005 Hz. Finally, the Maximum Entropy Method (MEM; Lygre and
172
Krogstad, 1986) is used to estimate the 2D frequency-direction spectra using
173
the 1D frequency spectra and the available cross-spectra.
174
For this study, hourly values of the spectral wave height Hm0 , mean zero-
175
crossing period Tm02 , peak period Tp , mean direction θm and mean directional
176
spreading σ θ were obtained from the buoy, as well as the hourly 1D frequency
177
and 2D frequency-direction spectra. All important settings of the buoy are
178
summarized in Table 1.
179
2.3.2. SIMAR database
180
The SIMAR database, maintained by Puertos del Estado, is a hindcast
181
of wind, sea level and wave parameters for the entire Spanish coast that
182
spans from 01/01/1958 until today. Since 2001, the wind field at 10 m above
183
the sea surface is obtained using the regional HIRLAM model provided by
184
the Spanish State Meteorological Agency (AEMET; Und´en et al., 2002). The
185
obtained wind fields are used in both WAM and WAVEWATCH III to obtain
186
the wave fields along the Spanish coast. The resolution of the wind model is
187
3 km since 2012 and that of the wave model is 5 km since 2006. Data output
188
is available each hour since 2012 (Puertos del Estado, 2015).
189
The SIMAR database has multiple data points in front of the Barcelona 11
190
coast. One of these data points is located directly in front of the study site
191
at the location where the AWAC was deployed at a depth of approximately
192
21 m and another is located close to the Barcelona wave buoy (41.32 N, 2.21
193
E) at a depth of approximately 75 m (Figure 1). For both output points,
194
the integrated wave parameters provided by the SIMAR database are the
195
spectral wave height Hm0 , mean zero-crossing period Tm02 , peak period Tp
196
and mean direction θm .
197
2.3.3. GOW2 database
198
The GOW2 database, which is developed and maintained by the hydraulic
199
institute IH Cantabria (Perez et al., 2017), is based on the numerical model
200
WAVEWATCH III and provides wave data for the period 01/01/1979 until
201
today. Since 2011, GOW2 is forced with the CFSv2 model (Saha et al.,
202
2010). The grid used in GOW2 for coastal continental areas has a resolution
203
of 0.25◦ × 0.25◦ which is approximately 21 km × 28 km (lon × lat) in the
204
Barcelona area.
205
Although GOW2 has a coarser resolution compared to SIMAR, it also
206
has a data point directly in front of Castelldefels (at 21 m depth, see Fig-
207
ure 1). Analogous to SIMAR, a second data point was selected (at deeper
208
water) located closest to the Barcelona buoy (41.25 N 2.25 E) at a depth
209
of approximately 250 m. From both points, hourly time series of significant
210
wave height Hm0 , mean zero-crossing period Tm02 , peak period Tp and mean
211
direction θm have been obtained.
12
212
3. Methods
213
3.1. Definition of wave parameters
214
215
216
217
218
The 2D frequency-direction spectrum, E(f, θ), represents the frequencydirection density spectrum of the sea-surface elevation variance E, defined R fhigh R 2π E(f, θ)df dθ. The 1D frequency spectrum E(f ) is related as E = flow 0 R 2π to E(f, θ) by the expression E(f ) = 0 E(f, θ)dθ and in a similar way it is possible to define a 1D direction spectrum as Z
fhigh
E(f, θ)df
E(θ) =
.
(4)
flow 219
Here, the symbol E denotes variance regardless of the unit. The units are
220
m2 /Hz for E(f ), m2 /deg for E(θ) and m2 /Hz/deg for E(f, θ). Notice that
221
the surface elevation variance multiplied by 21 ρg is equal to the total energy
222
Etot of the waves per unit surface area. Lastly, by changing flow and fhigh , it is
223
possible to apply high- and low-pass filters, respectively. Various integrated
224
wave parameters can be computed from the obtained spectra (Table 2). Ad-
225
ditional information about obtaining 2D spectra and computing directional
226
wave parameters is given in Appendix A. It is important to note that the
227
peak frequency fp and thus the peak period Tp obtained from buoy measure-
228
ments can have a large variability. This is especially true for multi-modal
229
spectra, where the observed maximum can easily switch between multiple
230
frequency peaks. The mean period generally does not have this problem,
231
which is why it is common to define a synthetic peak period, Tp∗ as a func-
232
tion of the mean zero-crossing period Tm02 (Rogers and Wang, 2007). The
233
relation used here is Tp∗ = 1.33Tm02 , which is the average of the relations be-
234
tween Tp and Tm02 given for the Pierson-Moskowitz and JONSWAP spectral 13
235
shapes (see Soulsby, 1998). In this study, the synthetic peak period is used
236
as input in both the simple wave ray propagation model (Section 3.2) and
237
for the SWAN model forced with integrated wave parameters (Section 3.3).
238
The randomness in fp also directly affects the measurements of θp , meaning
239
that mean direction θm is a more stable parameter.
240
To force the simple wave ray and SWAN models, all integrated wave
241
parameters (except Tp∗ ) are obtained directly from the buoy. For the hours
242
that no buoy data is available, the integrated wave parameters are computed
243
from the available 1D and 2D spectra using the expressions in Table 2. Table 2: Definitions of integrated wave parameters computed from 1D frequency and 2D frequency-direction spectra, where the Fourier coefficient a1 (f ), b1 (f ), a1 and b1 are given in Appendix A or computed from the buoy measurements. Parameter nth moment of 1D frequency spectrum
Definition R mn = 0∞ f n E(f )df
Spectral wave height
√ Hm0 = 4 m0
Root-mean square wave height
Hrms =
Mean wave period
Tm01 =
Mean zero-crossing period
Tm02 =
Wave period at peak of spectrum
Tp =
Mean wave direction per frequency Mean direction for entire spectrum
H√ m0 2 m0 m1
q
m0 m2
1 , fp
E(fp ) = max E(f ) b (f ) θm (f ) = tan−1 a1 (f ) 1 b1 −1 θm = tan a 1
Mean directional spreading for full spectrum
p σ θ = q 2[1 − r1 ], 2 r1 = a21 + b1
Peak wave direction
θp = θm (fp )
14
244
3.2. Simple wave ray model
245
The first model used to propagate waves from deep water to Castelldefels
246
is a simple wave propagation model that is based on ray approximation (linear
247
wave theory) and assumes monochromatic waves and parallel depth contours
248
(i.e. alongshore uniform bathymetry). Due to this latter assumption, the real
249
bathymetry is not used and the model inputs are simply the wave height,
250
period and angle in a certain deep water depth, together with the shallow
251
water depth where wave conditions are required.
252
First, the dispersion relationship is solved using a Newton’s numerical
253
scheme to compute the wave numbers at deep and shallow water (T is as-
254
sumed to be constant), after which Snell’s law is applied to obtain the wave
255
angle at shallow water. In the dispersion relation, the current contribution
256
(Doppler shift) is neglected, because at the AWAC it is smaller than 1% for
257
96% of the time (maximum contribution is 1.8%). Finally, the model ap-
258
plies wave energy conservation, i.e. a constant cross-shore wave energy flux
259
Fx (again assuming alongshore uniformity) to compute the wave height at
260
shallow water. The model uses Hrms for the wave height (characteristic wave
261
height for wave energy), a synthetic peak period Tp∗ and θm for the angle of
262
incidence.
263
3.3. SWAN wave propagation model
264
3.3.1. Model description
265
SWAN (Simulating Waves Nearshore) is a third-generation spectral wave
266
model that describes the evolution of the 2D frequency-direction spectrum in
267
coastal regions and inland waters by accounting for many relevant physical
15
268
processes (Booij et al., 1999). The SWAN model, as WAM and WAVE-
269
WATCH III, is based on the spectral action balance equation with sources
270
and sinks, but SWAN is specifically designed for coastal areas. Usually, wave
271
models use the action density N (defined as N = E/ωr where ωr is the
272
relative radian frequency), because it is conserved when propagating in the
273
presence of an ambient current (Whitham, 1974). Note that in our case ωr is
274
equal to the absolute radian frequency ω because ambient currents are negli-
275
gible (as explained in Section 3.2). The action balance equation in Cartesian
276
coordinates reads (e.g. Komen et al., 1994)
277
∂N ∂cx N ∂cy N ∂cωr N Stot ∂cθ N + + + = + , (5) ∂t ∂x ∂y ∂ωr ∂θ ωr where cx , cy , cωr and cθ are the group velocities in x, y, ωr and θ space. The
278
right hand side of equation (5) contains Stot , which is the source/sink term
279
and includes six physical processes that are important in generating, dissipat-
280
ing, or redistributing wave energy: the wave growth by wind, the nonlinear
281
transfer of wave energy through three-wave and four-wave interactions, the
282
wave decay due to whitecapping, bottom friction and depth-induced wave
283
breaking. More information about these terms can be found in the SWAN
284
Scientific and technical documentation (SWAN Team, 2018a).
285
3.3.2. Model setup
286
The SWAN model Cycle III version 41.20A is used in this study with
287
spherical coordinates and nautical convention. The model domain consists
288
of a curvilinear grid that stretches approximately 60 km alongshore. The
289
grid follows the coastline and the bathymetric line of the Barcelona buoy
290
(68 m; Figure 3). The spatial resolution varies throughout the grid and is 16
Figure 3:
SWAN domain including detailed bathymetry that is used to propagate the
waves to the study site.
291
approximately 200 m around the AWAC location. The grid bathymetry is
292
obtained from the EMODnet database and has a resolution of 120 m (Figure
293
3). The landward boundary of the SWAN model is set at approximately 10
294
m depth, because the EMODnet database becomes unreliable for shallower
295
depths. The output point in front of Castelldefels beach (the AWAC location
296
and the GOW2/SIMAR output point) is located at 21 m depth.
297
In this study, SWAN is used in 2D non-stationary mode, and station-
298
ary computations (recommended for domains smaller than 1 deg) with a
299
maximum of 15 iterations per computation. The frequency grid contains
300
38 logarithmically spaced values in the range 0.03–1 Hz, with the recom-
301
mended frequency resolution of df /f = 0.1 (SWAN Team, 2018b) and the
302
directional resolution used is 5◦ . The default JONSWAP formulation for
303
bottom friction is used with a coefficient value of 0.038 m2 s-3 . Many differ-
304
ent sensitivity tests have been conducted to determine the default settings
305
(Section 4.4.2). Following the results of these sensitivity tests, the default
306
third-generation physics formulation of Komen et al. (1984) is used, whilst
307
whitecapping, quadruplets, depth-induced breaking, triad wave-wave inter17
308
actions, wind growth are switched off. The integrated wave parameters ob-
309
tained with SWAN at the AWAC location are integrated over the frequency
310
range of the AWAC (0.02–0.49 Hz), following the recommendations of Pal-
311
lares et al. (2014).
312
Two types of offshore wave conditions are used in this study. The first
313
(and most used) type of boundary conditions is to force SWAN with a single
314
set of integrated wave parameters from which SWAN computes an artificial
315
single-component JONSWAP spectrum with a default peak enhancement
316
factor γ = 3.3. The integrated wave parameters used are Hm0 , synthetic
317
Tp∗ , θm and σ θ (see also Section 3.1). The SWAN manual (SWAN Team,
318
2018b) states that the wave direction θp should be used for forcing but in
319
this study the mean direction θm is used instead. This choice was made
320
because θp directly depends on fp , which is not a reliable parameter in buoys
321
(see Section 3.1). The second way to apply the offshore wave conditions is
322
to use the full 2D frequency-direction spectra, which are the most detailed
323
measurements of the wave climate that can be obtained from the buoy. Note
324
that when using SWAN forced with 2D frequency-direction spectra only the
325
energy that propagates into the domain is taken into account. Furthermore,
326
the Barcelona buoy has a fine linear frequency resolution of 0.005 Hz and
327
a directional resolution of 3◦ , whilst in SWAN the frequency resolution is
328
coarser and logarithmic and the directional resolution is 5◦ . Thus, the buoy
329
spectra are interpolated to the frequencies and directions used by SWAN.
330
The boundary conditions are imposed along the entire south boundary
331
of the grid but not in the lateral boundaries. However, since the region of
332
interest is in the center of the domain and the domain is sufficiently wide,
18
333
the absence of lateral boundaries does not influence the results. The off-
334
shore waves are assumed to be alongshore uniform and equal to those at the
335
Barcelona buoy.
336
4. Results
337
After a description of the wave conditions during the field campaign,
338
the performance of the different methods will be validated hourly using the
339
AWAC data. A total of 211 values are available for the SIMAR and GOW2
340
hindcast models, 188 for the simple wave ray model and 207 for the SWAN
341
simulations. It is stressed that the results for Tp shown in this section re-
342
fer to the actual measured or modeled peak period and not the previously
343
mentioned synthetic Tp∗ .
Figure 4: Wave roses showing the wave climate during the field campaign at the Barcelona buoy (a) and AWAC (b) in terms of spectral wave height Hm0 and mean direction θm .
344
345
346
4.1. Wave conditions during field campaign The wave climate during the 9-days field campaign was dominated by south-southeasterly waves during the first 6 days, after which east-southeasterly 19
347
waves were present (Figure 4a and 5). During the majority of the campaign,
348
the wave conditions were quite energetic (Hm0 > 0.5 m). Only on 14 March,
349
Hm0 dropped below 0.5 m, whilst the largest Hm0 was registered on 18 March
350
(1.8 m at the buoy and 1.9 m at the AWAC).
351
The various datasets of the campaign have been separated into three
352
different wave climates: southerly, easterly and bimodal. This separation is
353
made based on the measured 2D frequency-direction spectra at the Barcelona
354
wave buoy. The criteria for classifying a wave climate as easterly or southerly
355
are that at least 70% of the wave energy comes from that direction. If this
356
is not the case, the wave climate is classified as bimodal. The threshold
357
angle (147◦ ) to discriminate between easterly and southerly wave energy
358
is set by computing the average mean direction for both the easterly and
359
southerly wave conditions and subsequently taking the mean. This results
360
in southerly wave climates being present during 68% of the field campaign,
361
11% for easterly wave climates and 21% for bimodal wave climates (Figure
362
5).
363
Table 3 shows the statistics of 5 integrated wave parameters during the
364
complete field campaign and for the different wave climates. The average
365
wave height during southerly and bimodal wave climates was quite similar,
366
but it increased during easterly wave climates. During easterly and bimodal
367
wave conditions, the wave direction at the AWAC changes with respect to
368
the Barcelona buoy due to refraction and the average wave height decreases.
369
As shown in Figure 2a, easterly waves are important in the overall wave
370
climate of the Barcelona buoy, which is why the easterly waves that occurred
371
during the end of the field campaign have been analyzed in detail. More-
20
Table 3: Statistical values (mean, standard deviation, minimum and maximum) of integrated wave parameters measured at the Barcelona buoy and AWAC during the field campaign for the entire period and the different wave climates. Wave
Method
climate
Hm0 [m] Mean,
Tm02 [s] stdev,
Mean,
Tp [s] stdev,
Mean,
θm [deg] stdev,
Mean,
θp [deg] stdev,
Mean,
stdev,
min, max
min, max
min, max
min, max
min, max
Buoy
1.0, 0.32, 0.40, 1.8
4.8, 0.84, 2.9, 7.7
7.0, 1.4, 3.9, 10
169, 43, 74, 213
−
AWAC
0.88, 0.26, 0.34, 1.8
5.0, 0.82, 3.4, 7.7
7.2, 1.9, 3.2, 11
180, 20, 130, 225
180, 24, 32, 244
Buoy
0.97, 0.27, 0.40, 1.7
4.7, 0.88, 2.9, 7.7
6.6, 1.3, 3.9, 10
196, 11, 171, 213
−
AWAC
0.91, 0.27, 0.34, 1.8
4.8, 0.83, 3.4, 7.7
6.5, 1.6, 3.2, 10
191, 11, 164, 225
191, 20, 32, 244
Buoy
1.5, 0.26, 0.80, 1.8
5.6, 0.49, 4.9, 6.6
8.2, 1.2, 6.3, 10
86, 7, 74, 99
−
AWAC
0.91, 0.14, 0.55, 1.1
5.6, 0.64, 4.1, 6.7
9.6, 0.81, 6.8, 11
145, 11, 130, 174
150, 9, 135, 175
Buoy
0.94, 0.31, 0.40, 1.3
4.8, 0.58, 3.3, 6.1
7.4, 1.3, 5.4, 10
128, 27, 75, 173
−
AWAC
0.77, 0.25, 0.34, 1.1
5.1, 0.68, 3.4, 6.4
7.7, 1.8, 4.4, 10
162, 11, 131, 199
161, 16, 130, 197
− −
−
Full period − −
−
Southerly −
−
−
Easterly − −
−
Bimodal
372
over, propagating easterly waves is more challenging because they experience
373
strong refraction before reaching Castelldefels due to the change in shoreline
374
orientation, whilst southerly waves only experience limited changes (see data
375
at AWAC in Figure 4b and Figure 5).
376
4.2. Results of large-scale hindcast models
377
Results of the SIMAR and GOW2 database during the field campaign
378
at the AWAC location are shown in Figure 6, in yellow and purple colors,
379
respectively. The modeled trends in Hm0 and Tm02 agree quite well with the
380
measurements. This is also clear from Table 4, which contains the statisti-
381
cal errors for the various wave parameters and the different wave climates.
382
On average, both models underpredict Tm02 but the general pattern in the
383
measured data is still captured. Regarding θm , both models clearly show
384
large deviations. During southerly waves, the errors in direction are still 21
Figure 5:
Time series of integrated wave parameters and 1D directional spectra during
the field campaign. The three top plots show, respectively, the spectral wave height Hm0 , mean zero-crossing period Tm02 and mean direction θm both for the Barcelona buoy and the AWAC. The last two plots shows the 1D directional spectrum E(θ) measured by the Barcelona buoy and the AWAC respectively. The background shading in the three tops plots indicates the different wave climates; white for southerly, light-gray for bimodal and dark-gray areas for easterly wave climates.
385
moderate (they can be as much as 40◦ ), but they increase significantly when
386
easterly waves are present (up to 80◦ or even more). Both models predict
22
387
waves at Castelldefels coming from the east, whilst in reality these waves
388
have refracted substantially and come more from the south-southeast.
Figure 6:
Time series of integrated wave parameters during the field campaign at the
AWAC location. From top to bottom, the spectral wave height Hm0 , mean zero-crossing period Tm02 and mean direction θm are shown for different data sources (AWAC measurements or one of the models). The background shading indicates the different wave climates; white for southerly, light-gray for bimodal and dark-gray areas for easterly wave climates.
389
4.3. Results of simple wave ray model
390
The results of the simple wave ray model are also shown in Figure 6 (in
391
green) and in Table 4. Please note again that the wave period remains un-
392
changed when using this model. The modeled Hm0 and Tm02 agree quite well
393
with the measurements, although errors in modeled Hm0 can be significant
394
during easterly wave conditions. Regarding Tp and θm , the model shows good 23
395
agreement for southerly waves. For easterly waves, θm improves compared
396
with those of the GOW2 and SIMAR databases, but errors can still be quite
397
high (up to 40◦ ). The performance of θm for the different wave climates
398
is shown in Figure 7a. The gaps in the time series of the simple wave ray
399
model during easterly wave conditions are because the model can only prop-
400
agate waves in Barcelona buoy that have a direction ±90◦ with respect to
401
the shore-normal at Castelldefels beach.
402
4.4. Results of SWAN model
403
4.4.1. SWAN default settings
404
The results of the default SWAN simulation forced with integrated wave
405
parameters are shown in Figure 6 and in Table 4. The modeled results of
406
Hm0 and Tm02 agree well with the measurements for all wave climates, but
407
Tp , θm and θp are only reproduced well for southerly waves. The modeled
408
θm shows an improved performance for easterly and bimodal wave climates
409
when compared to the results of the simple wave ray model, but the error in
410
θm is still substantial (Figure 7b). Moreover, although not shown in previous
411
comparisons, notice that the results of θp worsen compared to θm (particularly
412
the RMSE).
413
When forcing SWAN with 2D spectra, not only are Hm0 and Tm02 well
414
reproduced, but also Tp now shows much more similarity with the mea-
415
surements (Figure 6 and Table 4). Most importantly, θm now shows good
416
agreement with the measurements for all wave climates (Figure 7c). Finally,
417
the RMSE in θp becomes larger compared to the default SWAN simulation
418
forced with integrated wave parameters (although the RMSE was already
419
large), but the BIAS improves. 24
Table 4: Summary of the statistical errors for the different methods during the full period and for the three wave climates. Entries in italics indicate the cases for which NRMSE > 1. Wave
Method
climate
Hm0 [m]
Tm02 [s]
RMSE,
BIAS,
NRMSE
RMSE,
Tp [s] BIAS,
RMSE,
BIAS,
θm [deg]
θp [deg]
RMSE, BIAS,
RMSE, BIAS,
NRMSE
NRMSE
NRMSE
NRMSE
0.98
1 .0 , −0 .77 , 1 .2
1.7,
0.28,
0.94
44 , −11 , 2 .2
−
−
−
GOW2
0.26, −0.14,
SIMAR
0.24,
0.004, 0.92
0 .95 , −0 .32 , 1 .2
1.6,
0.64,
0.86
37 , −9 .7 , 1 .8
−
−
−
Wave ray model
0.14, −0.001, 0.53
0.42, −0.13, 0.51
1.4, −0.57,
0.74
18, −3.9, 0.87
−
−
−
SWAN parameters* 0.11, −0.036, 0.41
0.41,
0.11, 0.50
1.5, −0.82,
0.82
14, −7.3, 0.66
22, −6.6, 0.90
SWAN 2D spectra* 0.12, −0.07,
0.47
0.46,
0.08, 0.56
0.82,
0.006, 0.44
8.1,
0.94, 0.40
26 ,
0 .92 , 1 .1
1.0
0 .96 , −0 .74 , 1 .2
1 .7 ,
0 .41 , 1 .1
13 ,
6 .7 , 1 .2
−
−
−
1.0
12 ,
8 .5 , 1 .1
−
−
−
3.7, 0.74
−
−
−
Full period
GOW2
0.28, −0.18,
SIMAR
0.24, −0.041, 0.87
0 .88 , −0 .22 , 1 .1
1.6,
0.10,
0.43, −0.12, 0.51
1.0, −0.31,
0.67
8.3,
SWAN parameters* 0.12, −0.047, 0.44
0.43,
0.18, 0.52
1.1, −0.39,
0.69
7.5, −2.5, 0.67
17,
2.4, 0.83
SWAN 2D spectra* 0.12, −0.056, 0.44
0.43,
0.13, 0.52
0.69, −0.002, 0.44
7.2,
18,
8.2, 0.93
GOW2
0 .18 ,
0 .06 , 1 .3
0 .92 , −0 .67 , 1 .4
0 .9 , −0 .42 , 1 .1
72 , −70 , 6 .8
−
−
−
SIMAR
0 .31 ,
0 .28 , 2 .3
0 .68 ,
0 .22 , 1 .1
1 .4 ,
0 .59 , 1 .7
60 , −59 , 5 .7
−
−
−
0 .46 , −0 .14 , 3 .4
0.34,
0.089, 0.52
2 .5 , −0 .80 , 3 .0
38 , −13 , 3 .6
−
−
−
SWAN parameters* 0.06, −0.005, 0.44
0.47,
0.053, 0.73
2 .5 , −2 .3 ,
21 , −19 , 2 .0
31 , −29 , 3 .5
SWAN 2D spectra* 0.094, −0.076, 0.70
0.62, −0.18, 0.96
1 .0 , −0 .18 , 1 .3
10, −7.5, 0.95
43 , −40 , 4 .9
GOW2
0.21, −0.083, 0.85
1 .2 , −0 .94 , 1 .8
2 .1 ,
77 , −37 , 7 .1
−
−
−
SIMAR
0.20, −0.070, 0.81
1 .2 , −0 .92 , 1 .8
1.8,
0.97
65 , −44 , 5 .9
−
−
−
0.096,
0.040, 0.39
0.39, −0.27, 0.57
2 .0 , −1 .3 ,
1 .1
31 , −24 , 2 .8
−
−
−
SWAN parameters* 0.076, −0.015, 0.31
0.31, −0.069, 0.46
2 .0 , −1 .4 ,
1 .1
21 , −17 , 1 .9
29 , −24 , 1 .9
SWAN 2D spectra* 0.14, −0.11,
0.46,
1.0,
0.57
9.6,
32 , −0 .80 , 2 .1
Southerly Wave ray model
Easterly Wave ray model**
Bimodal Wave ray model
0.009, 0.38
0.57
0.047, 0.67
0.59,
3 .0
0 .22 , 1 .1 0.84,
0.13,
2.3, 0.65
0.95, 0.88
*Results are shown for default SWAN settings **Only 8 datapoints available due to most wave directions < 90 deg with respect to North
25
420
4.4.2. Sensitivity to changing SWAN settings
421
Several sensitivity tests have been conducted to find the default SWAN
422
settings (see Section 3.3) for both forcing types. These tests can be sepa-
423
rated into numerical tests (grid refinement and increasing frequency resolu-
424
tion), tests of physical processes (including quadruplets and whitecapping
425
with and without wind) and methodological tests (default integration range,
426
choice of SWAN input parameters, changing spectral shape and directional
427
spreading value). Refining the grid by a factor 3 or increasing the frequency
428
resolution by a factor 2 does not improve the results. This proves that the
429
chosen numerical resolution is sufficient. The results also remain the same
430
when including quadruplets and whitecapping as physical processes, although
431
Battjes (1994) indicates that these processes are important in coastal waters.
432
The probable reason is that the domain used in this study is fairly small (less
433
than 10 km cross-shore) and measured wave conditions are imposed at the
434
boundary instead of using wind data for wave generation. When wind is also
435
included in the computations (using measurements from Barcelona Airport),
436
the results for θm worsen when forcing with 2D spectra (RMSE increases
437
from 8 to 10◦ and BIAS from 1 to 3.5◦ ). On the contrary, the results for
438
θm improve slightly when forcing with integrated wave parameters (RMSE
439
decreases from 14 to 13◦ and BIAS improves from −7 to −5◦ ). The rest of
440
the parameters stay approximately the same. It is also possible to change
441
the frequency range that is used in SWAN to compute the integrated wave
442
parameters. When applying no cutoff frequency (as done in SWAN default
443
settings) instead of using the frequency range of the AWAC (Section 3.3.2),
444
the results for Tm02 slightly worsen (RMSE increases by at most 0.1 s whilst
26
445
the BIAS can worsen by 0.2–0.3 s). Again, the rest of the parameters do not
446
display any changes.
Figure 7: Scatterplots showing θm modeled using the simple wave ray model (a), SWAN forced with integrated wave parameters (b) and SWAN forced with 2D spectra (c) versus AWAC measurements. The colors denote the different wave climates.
447
A choice has to be made regarding the input wave period (Tm01 or Tp ) and
448
wave angle (θm or θp ) when forcing SWAN with integrated wave parameters.
449
As explained in Section 3.1, a synthetic peak period is used in this study
450
that is computed from the Tm02 measured by the buoy. When using Tm01
451
instead (computed from the buoy 1D frequency spectra), no changes are
452
observed in the results. When using θp instead of θm , the results for Hm0
453
and Tm02 are similar, but errors increase substantially for Tp , θm and θp . The
454
results remain the same when changing the spectral shape from JONSWAP
455
for fetch-limited seas (Hasselmann et al., 1973) to Pierson-Moskowitz for
456
fully developed seas (Pierson and Moskowitz, 1964). Finally, when forcing
457
with integrated wave parameters the directional spreading σ θ can be set to a
458
constant value instead of using the time-variable measured data. Tests have
459
been done for constant σ θ ranging from 10◦ to 60◦ with a step size of 10◦ .
460
The SWAN user manual advises a σ θ of 30◦ and the results indicate that this 27
461
value gives the best results. For smaller values of σ θ , the errors in θm and
462
θp increase substantially (RMSE increases from 14 to 18–21◦ and from 22 to
463
24–26◦ respectively), whilst the errors remain the same for higher values of
464
σ θ . Changing σ θ does not lead to changes in the rest of the parameters.
465
5. Discussion
466
5.1. Performance of wave propagation methods
467
In this work, the results of different wave modeling approaches with mea-
468
sured data are analyzed. Despite the short duration of the field campaign,
469
the different wave climates that occurred allow to distinguish some clear
470
characteristics of the different methods. The assessment is done based on
471
the statistical errors defined in Section 2.2 and the results are discussed for
472
the three different wave climates: southerly (∼5◦ from shore-normal at 21 m
473
depth), easterly (∼30◦ from shore-normal at 21 m depth) and bimodal wave
474
climates. An error is deemed acceptable when NRMSE ≤ 1, i.e., when the
475
RMSE < standard deviation. The cases when this does not occur are marked
476
in italics in Table 4. The confidence limit of θm is estimated by Kuik et al.
477
(1988) to be 5◦ –10◦ in terms of RMSE.
478
For southerly waves, the three propagation methods give good results for
479
all wave parameters (Table 4). The main reason for this is that propagating
480
southerly waves is straightforward, since they are nearly shore-normal. As
481
a result, the simplifications of the simple wave ray model (monochromatic
482
waves and parallel depth contours) do not affect the propagation of these
483
waves. When forcing SWAN with 2D spectra, the only clear improvement is
484
obtained for Tp . This is a result of differences in the 2D spectra between the 28
485
simulations. Figure 8 shows the measured 2D spectra in the buoy and in the
486
AWAC together with the modeled spectra for SWAN forced with a single set
487
of integrated wave parameters (used to build a single-component JONSWAP
488
spectrum) and SWAN forced with the measured 2D spectrum. Using 2D
489
spectra as forcing leads to modeled 2D spectra at the buoy and AWAC loca-
490
tions that are very similar to the measured 2D spectra. When forcing with
491
integrated wave parameters, the modeled 2D spectra are slightly wider in di-
492
rection and the frequency peak is smaller. However, the energy distribution
493
is generally correct in the latter case indicating that the single-component
494
JONSWAP spectrum generally works well in southerly wave climates.
495
For easterly waves, the simple wave ray model performs badly (N RM SE >
496
1 for all parameters except Tm02 , see Table 4), and in particular refraction
497
is strongly underestimated. Using SWAN forced with integrated wave pa-
498
rameters improves the results for Hm0 and θm , although the errors in Tp and
499
θm are large and refraction is still underestimated. Forcing SWAN with 2D
500
spectra leads to a big improvement in Tp and θm and most parameters show
501
a N RM SE < 1. In particular, refraction is well captured and the RM SE
502
of θm is only 10◦ . The reason is that forcing SWAN with 2D spectra leads
503
to modeled 2D spectra similar to the observed ones (Figure 9), also at the
504
AWAC. When forcing SWAN with integrated wave parameters, the modeled
505
spectrum at the Barcelona buoy is much wider in direction compared to the
506
observed spectrum and only displays one peak in frequency (in reality there
507
are several). At the AWAC, the energy in the modeled spectrum is much
508
more concentrated in direction (hence the higher energy values), whilst in
509
reality there is much more spread in direction. For these easterly waves,
29
Figure 8: Measured and modeled 2D frequency-direction spectra E(f, θ) for a southerly wave climate (15 March 2018 20:00 UTC). From left to right the results are shown for the measured data, SWAN forced with integrated wave parameters and SWAN forced with 2D frequency-direction spectra (data in both cases taken from the Barcelona buoy). The top row shows the results for the Barcelona buoy location and the bottom row the results for the AWAC location at Castelldefels. Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.3/1.3 m, Tm02 = 5.0/5.1 s, θm = 202/193 deg.
510
the single-component JONSWAP spectrum does not work well because it
511
cannot represent multiple frequency peaks and it is too wide in direction.
512
The performance of the wave ray model is worse than that of SWAN with
513
integrated wave parameters. This is a result of the crude simplifications, in
514
particular the assumption of an alongshore-uniform bathymetry. As a result,
515
the refraction of easterly waves over the alongshore-variable bathymetry is
516
not correctly reproduced. Furthermore, as mentioned before, the waves that
517
arrive at the Barcelona buoy with a direction < 90◦ with respect to North
30
Figure 9:
Identical to Figure 8, but now for a easterly wave climate (22 March 2018
03:00 UTC). Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.8/1.0 m, Tm02 = 5.6/6.7 s, θm = 80/136 deg.
518
cannot be considered with the simple wave ray model (hence the gaps in
519
Figure 6).
520
For bimodal wave climates, the only tested method to accurately prop-
521
agate waves is forcing SWAN with 2D spectra (N RM SE < 1 for all wave
522
parameters except θp ). SWAN forced with integrated wave parameters leads
523
to modeled spectra that differ greatly from the measurements (Figure 10).
524
The single-component JONSWAP spectrum is not valid for bimodal wave
525
climates because the directional distribution used only allows for one direc-
526
tional peak in the 2D spectrum and the distribution on both sides of the
527
directional peak is symmetrical. The result is that the peak is located in an
528
area where in reality no energy is present (the mean of the two wave fields).
529
Consequently, the resulting errors in Tp and θm are large, although Hm0 and 31
Figure 10: Identical to Figures 8 and 9, but now for a bimodal wave climate (20 March 2018 03:00 UTC). Wave conditions at the Barcelona buoy/AWAC: Hm0 = 1.3/0.9 m, Tm02 = 4.6/5.0 s, θm = 107/164 deg.
530
Tm02 are still relatively well reproduced. The simple wave ray model also
531
performs well for Hm0 and Tm02 but the errors in Tp and θm are large. The
532
causes for these errors are twofold. First, the simple wave ray model assumes
533
monochromatic waves, meaning that all wave energy is concentrated in a
534
single set of integrated wave parameters, which is not sufficient to accurately
535
describe bimodal wave climates. Second, the model assumption of along-
536
shore uniform bathymetry forbids to accurately reproduce wave refraction
537
(as explained before).
538
The above discussion highlights the importance of using high quality
539
boundary conditions when propagating waves over alongshore-variable bathyme-
540
tries. The simple wave ray model uses straightforward boundary conditions
541
(wave height, wave period and wave direction), that work well for shore32
542
normal waves but are not accurate enough for shore-oblique waves and bi-
543
modal wave climates. When applying a spectral wave model like SWAN, the
544
boundary conditions are more detailed because either integrated wave pa-
545
rameters (including directional spreading) are used to build the 2D spectra
546
or the measured 2D spectra are applied explicitly. However, a 2D spectrum
547
constructed out of a single set of integrated wave parameters can be signifi-
548
cantly different from the measured spectrum because a certain combination
549
of Hm0 , Tp∗ , θm and σ θ can fit to many different spectral shapes. As noted by
550
Portilla-Yand´ un et al. (2015) and Cavaleri et al. (2018), a lot of detailed infor-
551
mation that is available in the full 2D spectrum is blurred or lost when only
552
using the integrated wave parameters. The present study underlines that,
553
in the Catalan coast, this is critical for easterly and bimodal wave climates,
554
where a single set of integrated wave parameters is unable to accurately de-
555
scribe the complex wave spectrum. This mostly affects the results of Tp and
556
θm , whilst for Hm0 and Tm02 no changes are noticed. When using the full 2D
557
spectrum as boundary conditions, it is possible to obtain accurate results for
558
Tp and θm in all wave climates, which is in agreement with previous research
559
on the western American coast (Gorrell et al., 2011). Furthermore, notice
560
that for both forcing types the errors in θp are much larger compared to θm
561
(for the three wave climates). The results using integrated wave parameters
562
might improve in case the 2D spectrum could be reconstructed out of mul-
563
tiple sets of wave parameters (e.g. Portilla-Yand´ un et al., 2015) instead of
564
using a single set.
33
565
5.2. Performance of hindcast models
566
The two hindcast models (SIMAR and GOW2) correctly predict the
567
trends in wave height and wave period (Table 4) for all wave climates, whilst
568
the errors for θm are very large for shore-oblique waves and bimodal wave
569
climates. There are several reasons for these mismatches. Global hindcast
570
models typically have coarse spatial resolution (5 km for SIMAR and 0.25◦
571
for GOW2), which is not sufficient to take into account the necessary details
572
in changes in the orientation of the shoreline, bathymetry and orography
573
(Cavaleri et al., 2018). Apart from that, global hindcast models are often
574
used without shallow water physics and without incorporating the effects of
575
bottom friction. Furthermore, the output points of the two hindcast models
576
in front of the study site are located at a shallow depth of 21 m. Using
577
data from an output point at deeper water and propagating this data with
578
SWAN might yield better results. To test this, data from SIMAR and GOW2
579
output points located closest to the Barcelona buoy at deeper water (75 m
580
and 250 m respectively; Figure 1) have been compared to the measured data
581
from the Barcelona buoy (at 68 m depth). Again substantial differences are
582
seen between the modeled and measured wave data (Figure 2). The results
583
for Hm0 (RMSE of 0.2 m for both models), Tm02 (RMSE of 1.0 and 0.6 s
584
respectively) and Tp (RMSE of 1.5 and 1.4 s respectively) are good for both
585
models, but the errors in θm (RMSE of 37◦ for both models) remain large.
586
The match between the modeled and measured wave climates at deeper water
587
is not deemed sufficient (particularly for wave direction) to use deep water
588
hindcast data as input for the wave propagation models. Furthermore, prop-
589
agating this data with SWAN would lead to the same problems that occur
34
590
when forcing SWAN with integrated wave parameters from the buoy.
591
Another reason that hindcast models display large errors in wave direction
592
is that they are forced by wind fields from atmospheric models that can
593
contain important errors (Bola˜ nos-Sanchez et al., 2007). Unfortunately, it is
594
difficult to obtain reliable wind fields in coastal and semi-enclosed seas (like
595
the Mediterranean Sea), due to the strong spatial and temporal gradients in
596
wind speed and direction (Cavaleri et al., 2018). Various studies (e.g. Ardhuin
597
et al., 2007; Mentaschi et al., 2015) demonstrated that choosing a different
598
wind model or changing the resolution of the wind model can strongly affect
599
the results of wave models in the Western Mediterranean Sea. Specifically
600
for the Catalan coast, Bertotti et al. (2014) tested four different atmospheric
601
models as input for the same wave model. The results showed that during
602
non-extreme but complicated wave conditions (consisting of multiple wave
603
systems), none of the atmospheric models was able to correctly reproduce
604
all the observed wind patterns (each model missed at least one of the wind
605
systems), which in turn led to an incorrect representation of the wave field.
606
Other studies that used large-scale wave models forced by wind fields
607
and focused on the Catalan coast (e.g. Bola˜ nos-Sanchez et al., 2007; Alomar
608
et al., 2014; Pallares et al., 2014) also indicated that errors in the modeled
609
wave fields were mainly related to inaccuracies in the wind fields. Although
610
typically only wave height and wave period were validated, Bola˜ nos-Sanchez
611
et al. (2007) also compared measured and modeled 1D frequency spectra
612
in which they found large differences, particularly for cases with multiple
613
frequency peaks. Pallares et al. (2014) obtained good results for wave height
614
and wave period but large errors in mean direction (RMSE above 40◦ –60◦
35
615
depending on the study period). Amrutha et al. (2016), a similar study in the
616
Eastern Arabian Sea that also used a large-scale wave model forced by wind,
617
found good agreement for mean direction during monsoon periods (RMSE
618
of 13◦ ), but errors increased substantially outside this period because the
619
wind patterns were not correctly simulated. Finally, the study by Rogers
620
and Wang (2007) focused on modeling directional wave properties in Lake
621
Michigan using SWAN forced by wind measurements from two buoys. They
622
found better results for θm , with an RMSE of 17◦ .
623
5.3. Recommendations
624
Based on the results discussed above, the following recommendations
625
can be given to obtain directional wave conditions for coastal studies in
626
semi-enclosed and coastal seas that are surrounded by a relevant orogra-
627
phy. Coarse-resolution hindcast models (or fine-resolution models forced by
628
coarse atmospheric models) usually reproduce the trends in wave height and
629
mean period correctly, meaning that in most cases they can be safely used
630
after applying some simple calibration methods (e.g. bias correction). In
631
particular, both SIMAR and GOW2 results are nowadays used extensively
632
to evaluate global changes in wave energy (e.g. Rodriguez-Delgado et al.,
633
2018) and can be very useful for engineering applications such as predict-
634
ing beach inundation and harbor management or for ecological studies (e.g.
635
De la Hoz et al., 2018). However, the large errors in wave direction make
636
global hindcast models not suited for nearshore morphodynamic studies in
637
semi-enclosed and coastal seas.
638
In such environments, the alternative is to propagate measured offshore
639
wave conditions. In case the wave parameters of interest are wave height 36
640
and mean period (but not direction), any of the tested propagation methods
641
can be safely used. However, the propagation method must be chosen care-
642
fully if accurate results for wave direction are needed, especially when the
643
bathymetry is alongshore-variable. The most reliable way to propagate the
644
wave conditions is by forcing a spectral model (such as SWAN) with mea-
645
sured 2D spectra as boundary conditions. Although it can be complicated to
646
obtain and handle the 2D spectra, this is the only method providing errors
647
below 10◦ for all the wave climates. This accuracy in wave direction can
648
be of crucial importance to determine if a site is prone to suffer from high
649
angle wave instability and the formation of shoreline sand waves (Ashton
650
et al., 2001; Arriaga et al., 2018) or whether crescentic or transverse bars
651
will be formed or destroyed (Calvete et al., 2005; Price and Ruessink, 2011;
652
Ribas et al., 2012). The only parameter that is not well reproduced (by any
653
method) is the peak direction θp , so it is discouraged to use this parameter
654
in nearshore studies.
655
When 2D spectra are not available, the most ideal situation would be to
656
construct a 2D spectrum out of multiple sets of integrated wave parameters
657
(e.g. separate wind-sea and swell components). Unfortunately, this is often
658
not possible because integrated wave parameters are typically only provided
659
for the entire spectrum. The next option is to force a spectral wave model
660
with 2D spectra constructed out of a single set of integrated wave param-
661
eters or use the integrated wave parameters directly in a simple wave ray
662
model. Compared with forcing with the full 2D spectra, these methods yield
663
the same accuracy for wave height and mean period, but wave direction is
664
only well-reproduced for cases characterized by limited refraction and par-
37
665
allel depth contours. As soon as the wave climate is bimodal or the waves
666
refract strongly, the errors in θm increase substantially. Although these two
667
methods have the advantage to be much easier to use, the errors in wave
668
angle can be too large to correctly characterize the dominant processes. The
669
underestimation of wave refraction is stronger in the simple wave ray model,
670
which sometimes cannot even account for all waves (in case of changes in the
671
orientation of the shoreline). In general, it is highly recommended to validate
672
the propagation results for wave direction with measurements before using
673
such simpler methods. Nevertheless, it should be pointed out that for the
674
case studied here, the effect of incorporating physical processes (like white-
675
capping and quadruplets) in spectral models is negligible (see Section 4.4.2).
676
This indicates that conditions are favorable for applying the simple wave ray
677
model and its shortcomings compared to spectral models are expected to
678
become even more evident in more complex phenomenological frameworks.
679
Results for wave ray tracing could be improved by using more advanced wave
680
ray models that can be forced with multiple sets of integrated wave param-
681
eters, take into account alongshore-variable bathymetry and include some
682
representation of shallow water processes.
683
Finally, the location of wave buoys can also cause measurement problems.
684
Wave buoys are often deployed near harbor entrances, meaning that they are
685
often located in water depths of less than 30 m. As a result, the measurements
686
are affected by shallow water processes (shoaling, wave breaking and friction),
687
whilst also the close proximity of harbor constructions like breakwaters can
688
render the data useless for accurately describing the offshore wave climate.
689
Furthermore, wave buoys can be sheltered from certain wave directions due
38
690
to changes in the orientation of the shoreline. For example, the Barcelona
691
wave buoy used in this study is sheltered for the rare waves coming from the
692
west-southwest, whilst they can freely arrive at Castelldefels. In this sense,
693
having information from a second buoy located in the southern part of the
694
domain would be ideal.
695
6. Conclusions
696
The accuracy of different methodologies to obtain directional wave char-
697
acteristics for nearshore field studies has been tested for various wave climates
698
at a beach located on a limited-fetch, complex-geometry sea. Such verifica-
699
tion of wave propagation methods at the location of interest turns out to
700
be crucial regarding wave direction. Global hindcast models with spatial
701
resolutions of 10 km or more cannot be expected to represent the coastal
702
zone and this also shows up in this study, meaning that their results must be
703
interpreted with care. The trends in wave height and mean period are cor-
704
rectly predicted, but the errors in wave direction for shore-oblique waves are
705
large (RMSE above 60◦ ), so that hindcast models are not suited for studies
706
of nearshore processes that depend on wave direction.
707
Better accuracy can be obtained by propagating wave conditions mea-
708
sured at offshore buoys, although the propagation method must be chosen
709
carefully. A simple wave ray model based on linear wave theory that as-
710
sumes monochromatic waves as boundary conditions and alongshore-uniform
711
bathymetry provides good results regarding wave height and period. How-
712
ever, the crude model assumptions cause the model to underestimate refrac-
713
tion of oblique waves over an alongshore-variable bathymetry, which leads to 39
714
large errors in wave direction (RMSE above 30◦ ) for easterly and bimodal
715
wave climates.
716
The results for these wave climates can be improved by propagating waves
717
using a properly scaled, third-generation wave model like SWAN. As offshore
718
boundary condition, such models require a full 2D spectrum, and one option
719
is to reconstruct it out of a single set of integrated wave parameters. When
720
applying this method, the results for wave direction during shore-oblique
721
waves and bimodal wave climates are improved, although the RMSE are
722
still above 20◦ . The other option is to directly prescribe a measured 2D
723
spectrum, which gives the best results for wave direction and reduces the
724
RMSE to values below 10◦ for all wave climates. Such accuracy in wave
725
direction is essential in many nearshore studies, particularly when studying
726
the evolution of crescentic bars, transverse bars and shoreline sand waves.
727
During the short sampling period of this study, the SWAN results are
728
robust to changes in model settings. Using simple settings suffice and includ-
729
ing physical processes like wind, quadruplets or whitecapping do not improve
730
the results. This confirms that the dominant process modifying the offshore
731
waves over the alongshore-variable bathymetry to the nearshore study site
732
is wave refraction. Under such circumstances, the use of spectral boundary
733
conditions is highly important and observed offshore 2D wave spectra can be
734
extremely useful to obtain accurate results of the nearshore wave field using
735
wave propagation models.
40
736
Acknowledgments
737
The authors would like to thank Benjam´ın Casas of IMEDEA and the
738
technicians from SOCIB (Coastal Ocean Observing and Forecasting System
739
of the Balearic Islands) for their help and assistance during the field cam-
740
paign. Special thanks to Pilar Gil from Puertos del Estado for providing
741
the SIMAR and wave buoy data and IH Cantabria for providing the GOW2
742
data used in this study. The bathymetric data used in this study has been
743
obtained from the European Marine Observation and Data Network (EMOD-
744
net, http://www.emodnet-bathymetry.eu/) and the wind data of Barcelona
745
airport has been provided by the Spanish meteorological agency (Agencia
746
Estatal de Meteorolog´ıa), which is part of the Ministry for the Ecological
747
Transition (MITECO). This work has been funded by the Spanish govern-
748
ment through the research projects CTM2015-66225-C2-1-P and CTM2015-
749
66225-C2-2-P (MINECO/FEDER). Finally, the constructive comments of
750
two anonymous reviewers helped to improve the quality of this paper.
751
Declarations of interest
752
753
754
755
None. Appendix A. Measurement of directional wave parameters Directional wave parameters are defined using the 2D frequency-direction spectrum, E(f, θ), which is often written as
E(f, θ) = E(f )D(f, θ) ,
41
(A.1)
756
757
where D(f, θ) is called the directional distribution function that has units R 2π of 1/deg. It is defined in such a way that 0 D(f, θ)dθ = 1. A lot of
758
different methods have been developed to obtain D(f, θ) out of wave buoy
759
measurements. The most common method starts by decomposing D(f, θ)
760
conceptually into a Fourier series using (Longuet-Higgins et al., 1963) ∞ 1 1 X D(f, θ) = + [an cos(nθ) + bn sin(nθ)] , π 2 n=1
(A.2)
761
where an (f ) and bn (f ) are the standard Fourier coefficient that are defined
762
as
Z
2π
an (f ) =
D(f, θ) cos(nθ)dθ
and
D(f, θ) sin(nθ)dθ
.
Z 02π bn (f ) =
(A.3)
0 763
Only the first four Fourier coefficients (a1 , b1 , a2 , b2 ) of D(f, θ) can be ob-
764
tained from buoy measurements (Hoekstra et al., 1994). As a result, direc-
765
tional spreading is generally overestimated, whereas D(f, θ) is mostly too
766
broad and can even become negative (Young, 1994). To overcome this prob-
767
lem, parametric models like the cos2s method (Longuet-Higgins et al., 1963)
768
and data-adaptive methods like the MLM (Krogstad et al., 1988) and MEM
769
(Lygre and Krogstad, 1986) approximate the complete Fourier series to give
770
a full estimate of D(f, θ). However, the drawback of these models is that they
771
give details of D(f, θ) that are not determinable from buoy measurements
772
and sometimes even generate spurious data (Benoit et al., 1997; Rogers and
773
Wang, 2007). Taking these problems into account, Kuik et al. (1988) defined
774
expressions for mean wave direction, θm (f ), and directional spreading, σθ (f ), 42
775
per frequency, as a function of the first two determinable Fourier coefficients.
776
Their expression for θm (f ) reads " θm (f ) = tan−1
b1 (f ) a1 (f )
# .
(A.4)
777
However, the frequency range is always finite so the way to compute the
778
overall mean direction averaged over the flow -fhigh frequency band is " θm = tan−1
b1 a1
# ,
(A.5)
779
and the peak wave direction θp is computed by taking the mean direction
780
at the frequency peak (θp = θm (fp )). Furthermore, the directional spreading
781
averaged over the frequency range can be computed using p σ θ = 2[1 − r1 ] ,
782
q 2 where r1 = a21 + b1
,
(A.6)
in which a1 and b1 are the first 2 Fourier coefficients that are defined as R fhigh a1 =
a1 (f )E(f )df R fhigh E(f )df flow
flow
and
R fhigh b1 =
b1 (f )E(f )df flow R fhigh E(f )df flow
.
(A.7)
783
Wave buoys normally obtain the Fourier coefficients from measurements as
784
described by Hoekstra et al. (1994).
785
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786
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France
Legend
! ( ! (
Spain
# * # * $ +
Portugal
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Barcelona wave buoy AWAC location GOW2/SIMAR model point SIMAR deep water GOW2 deep water Wind measurements Castelldefels beach
Bathymetry [m]
El Prat de Llobregat
High : 0 Low : -400
Castelldefels
0 1 2
4
6
Kilometers 8
Barcelona
Highlights • • • •
Five methods to obtain directional wave conditions at shallow waters are tested. Hindcast models provide proper wave height and period but unreliable wave angle. Propagating idealized buoy-derived spectra still leads to inaccurate direction. Only propagating buoy-measured directional spectra produces accurate wave direction.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: