Introduction
Due to its semi-enclosed morphology and shoaling bathymetry, the North Sea
experiences extreme wave conditions, in particular during winter periods
. When combined with sea-level surges such events
can lead to damaging inundation of low-lying coastal regions, due to
wave overtopping of sea defences. Development of a modelling and predictive
capability for high-resolution wave conditions in the North Sea is therefore
a high priority. However, the task is complicated due to interaction between
locally generated waves and incoming swell from outside the region, and
especially due to interactions between waves and tidal currents.
Crossing, or bimodal, sea states occur between 5 and 40 % of the time in
the North Sea . These are generated when
swell waves propagating into the region from distant storm events interact
with locally generated waves which may be of very different direction,
period, and height. Swell waves from the North Atlantic and the Norwegian Sea propagate
into the North Sea, interacting with local wind-sea-generated waves, modifying
the main spectral parameters. The interaction between differing wave trains
is not fully understood, but crossing seas have been statistically associated
with freak wave incidence, and shipping accidents
.
The North Sea is particularly prone to rogue wave events, such as the famous
Draupner wave recorded in 1994 , the first ever
recorded rogue wave event, that occurred in crossing sea conditions
. In the paper, the effect of the enhancement of the
wind-sea waves due to swell is assessed during storms.
In addition to crossing seas, wave–current interactions are a well-known
cause of wave height amplification or attenuation. Wave–current interactions
(WCIs) are depth- and current-induced modification of wave features. A
seminal study carried out by highlighted that
WCIs are significant in the North Sea, changing the
significant wave height (Hs) and the mean wave period
(Tm) by 5 and 10 %, respectively, during storm periods.
However,
the model that was used by to assess this effect
was at very course resolution and broad scale. In particular, the effect of
the WCI in the coastal shallow areas
was not considered.
showed that in the absence of wave breaking, local wave
amplitude is given by
AA0=c0c(c+2U),
where A is the resulting wave amplitude, A0 is the unperturbed
amplitude of the wave field, c is the wave phase speed, and U is the
current that interacts with the wave train. It is important to notice that
the sign of the current is determinant on the effect of the WCI: if the
current travels in opposite direction with the wave train, there will be an
enhancement of the significant wave height. Conversely, if current and waves
are in the same direction, the Hs decreases. For deep water
waves, the phase speed c depends only on the period of the wave, while in
shallow water c depends only on the depth. Equation (1) shows that the
waves travelling in a direction opposing the current (U<0) have a positive
ratio A/A0 and consequently, an enhancement of the wave amplitude.
The area studied in the present paper.
![]()
The computational grid generated with MIKE Zero software.
![]()
WCIs could also lead to the breaking of the wave: if the
current is strong enough to block the wave train ,
these waves can break and lose energy before arriving to the coastline
.
WCIs are particularly difficult to quantify empirically, and computationally
intensive to model. In addition, the WCIs are a
well-known mechanism for the formation of rogue waves in the ocean: the
Agulhas current, that flows near the coastline of South Africa, was one
of the first places in which this mechanism was identified
. Recently,
many studies
highlighted that the interaction between a train of waves with an adverse
current could increase its wave steepness, and cause rogue waves due to the
modulational instability . However, it is clear that
shallow coastal waters, embayments, and headlands are particular foci for
interactions . Model studies have
concentrated on comparing wave height and period in coupled and uncoupled
model versions showing, for example, 3 % difference in wave height and
20 % in wave period in the Dutch and German coastal waters of the North Sea
. Similar results have been obtained for coastal waters
of the Adriatic during bora conditions , finding a
maximum reduction for the Hs of 0.6 m in the central Adriatic
and a simultaneous increase up to 0.5 m in the gulfs of Trieste and Venice.
WCIs were also studied during hurricane conditions off the eastern seaboard of the USA
.
Sea defences of coastal settlements along the
northeast coast of Scotland have suffered several damaging events during the
period 2009–2014 as a result of surges and waves. The coastal waters are
dominated by strong tidal currents and wind-driven residuals, are exposed
to wave trains entering the North Sea from the north, and generated by storm
events in the central and southern North Sea. Although the oceanography of
the North Sea as a whole has been intensively studied since the 1830s
, and the region was one of the
earliest to be subjected to computational hydrodynamic modelling
, high-resolution modelling
activity has been largely concentrated in areas with potential for wave and
tidal energy extraction
.
However, there are no such models for the northeast coast of mainland
Scotland, and none which include coupled WCIs. Our
objective here was to develop and test such a model for the stretch of
coastline between the Firth of Tay and Peterhead, centred on the
strategically important port city of Aberdeen and the town of Stonehaven
(Fig. 1). The latter is the base for a governmentally supported marine
monitoring site with a >15-year time series of high-resolution data on a
wide range of environmental parameters .
Materials and methods
The MIKE by DHI model was used to simulate the tidal- and wind-driven
circulation, and the wave propagation. The MIKE software is composed of
different modules, for the creation of a model grid
and input files and to simulate different hydrodynamical features at the same time
or separately. The following modules were used:
The MIKE Zero modules for generating the computational grid and the input
files.
The MIKE 3 FM module for simulating the tidal and the wind-driven
circulation.
The MIKE 21 SW module for modelling the wave propagation.
For the simulation of the WCIs, a one-way coupling
between MIKE 3 FM and MIKE 21 SW was set up. The depth average flow fields and
the water level output from MIKE 3 FM was provided as input to the MIKE 21 SW
model.
The computational grid
Both MIKE 3 FM and MIKE 21 SW use an unstructured grid approach, with
triangular elements . Unstructured grids can
represent complex coastlines better than a rectangular grid and potentially
provide more realistic flows, enabling the geography of the coastline to
affect the propagation of tidal and surface waves in a realistic manner. In
addition, triangular grid elements allow smoothly changing cell sizes across
a region, with the highest resolution concentrated in an area of particular
interest. The mesh for the area of study is shown in Fig. 2. An enhanced-resolution
area was created near Stonehaven, because this work is part of a
wider study focusing on the resuspension of the sediments in this part of the
domain. This high-resolution area also covered the Firth of Forth and the
Aberdeenshire coastline, where previous studies have shown enhanced
currents due to interaction between the tidal wave and the Scottish coastline
.
The MIKE 3 FM hydrodynamic model
MIKE 3 FM (flow model) is based on the numerical solution of the 3-D
incompressible Reynolds-averaged Navier–Stokes equations, under the
Boussinesq and the hydrostatic pressure approximations . The
spatial discretization of the primitive equations is performed using a
cell-centred finite-volume method. In the finite-volume method the volume
integrals in the partial differential equations with a divergence are
converted to surface integrals using the Gauss–Ostrogradsky theorem
.
MIKE 3 FM has a flexible approach for simulating the
flow in the water column. It is possible to choose between sigma layers
, z layers, and coupled sigma and z layers. For our
purpose we decided to use the equidistant sigma layers approach, because the
bathymetry of the area was not sufficiently complex to require a more accurate
description with a coupled sigma and z layer that would be extremely
computationally expensive. Sigma layers are also useful for resolving the
water column well throughout the tidal cycle, given the large tidal range.
The coupled sigma and z layers were tested, but there was no significant
improvement for simulating the flow.
For the horizontal eddy viscosity the
formulation proposed by was used, in which the
sub-grid-scale transport is expressed by an effective eddy viscosity related
to a characteristic length scale rather than a constant eddy viscosity. This
sub-grid-scale viscosity is given by
A=cs2l22SijSij,
where cs is the Smagorinsky constant, l is the characteristic length of
the grid size, and the deformation rate is given by
Sij=12∂ui∂xj+∂uj∂xi.
The bed resistance was parameterized using a constant quadratic drag
coefficient cf. The average bottom stress is determined by a
quadratic friction law:
τ¯b=cfρ0u¯bu¯b,
where u¯b is the average flow velocity above the bottom and
ρ0 is the density of the water. The value of the above parameters
chosen for the model are reported in Sect. 3.1.
The model was forced with a time series of
tidal elevations at the open boundaries from the open-source OSU (Oregon
State University) Tidal Prediction Software (OTPS)
, based on TOPEX satellite observation of the
water-level observations interpolated with tide gauge data from the European
shelf region. In order to take account of the wind-driven circulation and
surge in the model, meteorological forcing was applied across the model
domain, using the ERA-Interim reanalysis for wind velocity and mean sea-level
pressure .
The MIKE 21 SW wave model
The MIKE 21 SW (spectral wave) is an unstructured grid model for wave
prediction and analysis . The MIKE 21 SW is based on the wave
action conservation equation , where
the dependent variable is the frequency-directional wave action
spectrum. This is given by
∂N∂t+∇⋅(cN)=Sσ,
where S is the energy source term, defined as
S=Sin+Snl+Sds+Sbot+Ssurf
that depends on the energy transfer from the wind to the wave field
Sin, on the nonlinear wave–wave interaction Snl, on the
dissipation due to depth-induced wave breaking Ssurf, on the
dissipation due to bottom friction Sbot,
and on the dissipation caused by the white-capping Sds.
The wave action density spectrum N(σ,θ) is defined as
N(σ,θ)=E(σ,θ)σ,
where E is the wave energy density spectrum, σ=2πf is the
angular frequency (where f is the frequency), and θ is the direction
of wave propagation. The momentum transfer from the wind to the waves follows
the formulation in . The momentum transfer and the
drag depend not only on the strength of the wind but also on the wave state
itself.
For the physics of the propagation and breaking of the waves we
choose the following parameters:
The depth-induced wave breaking is based on the formulation of , in which the gamma parameter is a constant 0.6 across the domain.
The formulation of the depth-induced wave breaking can be written as
Ssurf(σ,θ)=-αQbσ¯Hm28πE(σ,θ)Etot,
where α ≈ 1.0 is a calibration constant, Qb is the
fraction of breaking waves, σ¯ is the spectrum average frequency,
Etot is the total wave energy that is linked to the wave action density
spectrum, and Hm is the estimated maximum wave height, that is
defined as Hm=γd , in which d
is the depth and γ is the free breaking parameter
.
The bottom friction is specified in the model as the Nikuradze roughness (kN) .
The white-capping formulation described in in order to consider the dissipation of waves, based on the theory of
. For the fully spectral formulation, the white capping assumes a form that
is dependent on the mean frequency σ¯ and on the wavenumber k:Sds(σ,θ)=-Cdsk¯2m021-δkk¯+δkk¯2σ¯N(σ,θ).
Here the two parameters, Cds and δ, are the two dissipation
coefficients that control the overall dissipation rate and the strength of
dissipation in the energy/action spectrum, respectively, and m0 is the
zeroth moment of the overall spectrum.
The values of the above parameters chosen for the model are reported in
Sect. 3.2.
The model also included nonlinear energy transfer such as
the quadruplet wave interaction and the triad wave
interaction which is the dominant nonlinear interaction in shallow water
.
The forcings
included in the model are the local wind and the swell wave field from
outside the model area and specified at the model boundaries. For the model
boundaries we used boundary conditions from the
North Atlantic model, a larger
wave model that encompass the southern Norwegian Sea and the North Atlantic
Ocean. The ERA-Interim 0.125∘×0.125∘ model was used
to provide a wind field across the model domain with a time resolution of 6 h .
Wave–current interactions
The WCIs are implemented using a one-way coupling
between currents and waves. The model was run without and with currents
implemented, and then the differences between the two runs were studied. The
WCIs in the MIKE model are taken in account in the
dispersion relation for the angular frequency term, since the current due by
tides and wind affect the propagation and changes the wavelength of the
wave train. The MIKE 21 SW dispersion relation in fact is
σ=gktanh(kd)=ω-k‾⋅U‾.
The one-way coupling has, however, some limitations since it is not taking into
account the modification of the current by the wave itself
.
Swell detection
In the present study, the wind-sea and the swell waves and their interaction
are studied. MIKE 21 SW gives the opportunity to separate spectrally the
windsea waves and the swell waves. There are two criteria, based on a dynamic
threshold, available to make this separation.
The first criterion is based on
the difference of the energy between the spectrum and the fully developed sea
condition . In this case, the threshold frequency
is identified as
fthreshold=αfp,PMEPMEModelβ,
where α=0.7, β=0.31, EModel is the total energy at
each node point calculated by the MIKE 21 SW model,
and the Pierson–Moskowitz peak frequency and the energy are estimated as
fPM=0.14gU10EPM=U101.4g4.
The second method is based on the wave-age criterion
from empirical wave measurements in wave tanks and in Lake Ontario field
measurements. From , swell waves are the
components fulfilling the following relation:
U10cpcos(θ-θw)<0.83,
where U10 is the wind speed at 10 m, cp is the phase speed,
θ is the wave propagation direction, and θw is the
direction of the wind. For discriminate swell and windsea waves we used the
second method, since it is the most widely used for this purpose and is the more
reliable method .
Validation data sets
The model was validated using five independent data sets. The hydrodynamic
model was validated using data from the UK National Tide Gauge Network in
Aberdeen and Leith and using the tide gauge data from the Scottish
Environmental Protection Agency (SEPA) in Buckie. Validation was performed
comparing harmonic components extracted from time series of both model and
real data. The harmonic components of the sea level were extracted using the
UTide Matlab function .
Current meter observations
from the British Oceanographic Data Centre (BODC) were used to validate the
modelled currents. For the wave model, we compared recorded data from wave
gauges in the Moray Firth and in the Firth of Forth (obtained from CEFAS) and
data from a wave rider buoy deployed in Aberdeen Bay (data obtained from
University of Aberdeen). In addition, we used significant wave height and
mean wave period from satellite data provided by WaveNet (CEFAS) for June
2008. Especially important in this case is the Aberdeen wave gauge, since
this is the only one in shallow water (the depth of the sea in the mooring
location is 10 m). This allows us to evaluate the ability of the model in
coastal areas, in which the WCIs are strongest.
Table 1 shows details of the observations used for validating and calibrating
the tidal and wave model, while in Fig. S1 in the Supplement we show the
position of the tide and wave gauges used for the validation and the position
of the satellite data.
Location of the validation/calibration instrumentation.
Description
Coordinates
Depth (m)
Use
longitude (∘)
latitude (∘)
Aberdeen tide gauge
-2.0803
57.144
–
water level val/cal
Leith tide gauge
-3.1682
55.9898
–
water level validation
Buckie tide gauge
-2.9667
57.6667
–
water level validation
Firth of Forth buoy
-2.5038
56.1882
–
waves validation
Moray Firth buoy
-3.3331
57.9663
–
waves val/cal
Aberdeen wave rider
-2.0500
57.1608
–
waves validation
BODC 4551 RCM
-2.8000
57.7910
12
current validation
BODC 4561 RCM
-1.9680
57.2320
12
current validation
BODC 4562 RCM
-1.9680
57.2320
27
current validation
BODC 4571 RCM
-1.9020
57.2260
12
current validation
BODC 4572 RCM
-1.9020
57.2260
52
current validation
BODC 4582 RCM
-2.1500
56.9870
23
current validation
BODC 4591 RCM
-2.0980
56.9820
12
current validation
BODC 4592 RCM
-2.0980
56.9820
47
current validation
The validation for waves was carried out
using four statistical indices: the bias, the root mean square error (RMSE),
the correlation coefficient (R), and the scatter index (SI). These indices
are defined below.
Bias=1N∑i=1Nxoi-xmiRMSE=1N∑i=1Nxoi-xmi2R=∑i=1Nxoi-x‾oxmi-x‾m∑i=1Nxoi-x‾o2xmi-x‾m2SI=RMSEx‾o
For tidal current validation we used, instead of the SI, the normalized
root mean square error (NRMSE), that is defined as
NRMSE=RMSEmax(x)-min(x).
The validation was performed for different years. For the hydrodynamic model
the agreement between modelled and observed water level was evaluated for the
entire year 2007, while the currents were validated for 1992, where the rotor
current meter observations were available. The wave model was validated for
2010 and 2008, where observations and boundary inputs were available.
Results
Calibration and validation of the hydrodynamic model
The hydrodynamic model was calibrated for the year 2007, based on the
agreement with the recorded water level at the tide gauge in Aberdeen. The
calibration parameters were the time step that was fixed at 1 s after an
analysis of the Courant–Friedrichs–Lewy (CFL) conditions (higher time steps
were investigated but the model was unstable); the Smagorinsky constant that
was set to 0.2 (for values >0.3 the model showed some blow-up); and the
bottom roughness that was parameterized with the drag coefficient
cf=0.0025. After calibration, the MIKE 3 tidal model was
validated against harmonic components extracted from both observed and
modelled data for water level. The agreement between modelled and observed
currents was also investigated. RMSE for the
amplitude of harmonic components was less than 1 % for all the cases, while
the phase of the main semidiurnal component was well modelled. In particular
for the dominant M2 component, the phase error was very low and the
amplitude was well modelled (see Tables 2 and S1 in the Supplement for more
details). The validation results show that the modelled results are in good
agreement with the recorded tidal amplitude and phase. The model was run for
1992 and measurements obtained from BODC from eight locations were used to
validate the currents in the model. The validation of the single components
u and v is reported in Table S2. Table 3 shows
that the model adequately represents the current speeds in the domain. The
validation shows that the model slightly underestimates the current,
however it can be noticed that the bias of the model was very low. The RMSE,
except for one observation, does not exceed 15 % of the maximum speed.
Computed RMSE for the main harmonic components, the validation
for each tide gauge is reported in the Table S1.
Components
RMSE
A (cm)
g (∘)
M2
2.52
0.78
S2
1.64
3.68
N2
1.31
3.02
O1
0.76
5.42
K1
0.44
14.4
Q1
0.42
13.6
Results from the validation of the currents, showing the difference
between the modelled and observed current speeds at the eight locations
reported in Table 1.
RCM no.
Lat
Long
Depth
RMSE
NRMSE
R2
Bias
(m)
(m s-1)
(m s-1)
4551
-2.8
57.791
12
0.094
0.157
0.17
0.001
4561
-1.968
57.232
12
0.111
0.124
0.70
-0.03
4562
-1.968
57.232
27
0.075
0.105
0.75
-0.02
4571
-1.902
57.226
12
0.223
0.147
0.23
-0.05
4572
-1.902
57.226
52
0.087
0.112
0.80
-0.02
4582
-2.15
56.987
23
0.075
0.124
0.80
0.02
4591
-2.098
56.982
12
0.125
0.132
0.73
-0.062
4592
-2.098
56.982
47
0.073
0.121
0.82
-0.05
Calibration and validation of the wave model
The calibration of the wave model was carried out for 3 months in 2008 and
was based on the agreement between the observed and the modelled
Hs of the Firth of Forth wave gauge. There were three calibration parameters:
the wave-breaking parameter γ and the two dissipation
coefficients associated to the wave breaking (Cdis and δ).
During this procedure we noticed how the most sensible parameter was the
γ controlling the wave breaking. We investigated the behaviour of the
γ for the range 0.6–1.0, since most of the observation studies in
literature were reporting such values
,
and we found that γ=0.6 was the value giving better results for the
Hs. The wavebreaking dissipation coefficients were fixed to
Cdis=2.5 and δ=0.8, while the Nikuradze bottom roughness
was fixed to 0.01 m. The results of the wave gauges and satellite validation
are reported in Table 4. We evaluated the performance of the wave model with
WCIs implemented (coupled) and without WCIs (uncoupled).
There was a good agreement between the modelled-with-WCI and measured wave data.
The bias does not exceed 0.15 m for significant wave height. Table 4 and
Fig. 3 shows that the model estimates correctly the significant wave height
in the Firth of Forth and in Aberdeen but underestimates this parameter in
the Moray Firth. However, the agreement with the data is still satisfactory.
In particular, low RMSE values were recorded for the Aberdeen wave gauge,
which is the only coastal shallow-water wave gauge that is available in the area
(the depth of the mooring site is 10 m). The model performance against
satellite data randomly sampled throughout the domain shows good agreement.
Without the WCI included in the model, small or no differences were estimated
for significant wave height, but larger differences were seen for mean wave
period (Tm01=2πm0/m1): the calculated RMSE for the uncoupled
model was 0.97 s in Aberdeen, 1.24 s in the Firth of Forth and 1.83 s in
the Moray Firth. Comparing satellite observations in spring and winter
conditions, it is possible to conclude that, in general,
the model provides accurate predictions for wave heights <1.5–2 m, but slightly underestimates the height of larger waves. On the other hand,
wave periods are better modelled in the winter period when the waves are
higher. No or very small differences were recorded between the coupled and
uncoupled models for satellite validation. This is because the resolution of the
satellite data is low and because the satellite data are often in deep water,
where the WCI are less important.
Comparison between observed and modelled (both with and without
WCIs implemented) wave heights and periods for wave
gauges and satellite observations. The reported validation was carried out
for 2010 (Firth of Forth and Moray Firth) and for 2008 (Aberdeen wave
gauge and satellite observations). Details of the observation data are
reported in Table 1 of the paper and in Table S2 of the Supplement.
Coupled
Uncoupled
Bias
RMSE
R
SI
Bias
RMSE
R
SI
Firth of Forth
Hs
-0.02 m
0.30 m
0.941
0.27
-0.01 m
0.30 m
0.939
0.27
Tm
-0.70 s
1.17 s
0.767
0.25
-0.76 s
1.24 s
0.758
0.27
Moray Firth
Hs
-0.14 m
0.42 m
0.849
0.38
-0.15 m
0.42 m
0.848
0.39
Tm
-1.18 s
1.75 s
0.668
0.39
-1.23 s
1.83 s
0.656
0.41
Aberdeen
Hs
-0.07 m
0.21 m
0.836
0.32
-0.07 m
0.22 m
0.831
0.32
Tm
-0.25 s
0.91 s
0.715
0.20
-0.30 s
0.97 s
0.701
0.21
Satellite
Winter
Hs
-0.2 m
0.4 m
–
0.25
-0.2 m
0.4 m
–
0.25
Tm
+0 s
0.8 s
–
0.15
+0 s
0.8 s
–
0.15
Spring
Hs
-0.1 m
0.3 m
–
0.21
-0.1 m
0.3 m
–
0.21
Tm
+0.1 s
1.2 s
–
0.23
+0.1 s
1.2 s
–
0.23
Comparison between observed and modelled Hs in the Firth
of Forth and the Moray Firth for 2010.
![]()
Root mean square difference between wave model output with and
without WCI: (a) significant wave height (m), (b) peak wave
period (s), (c) wave directional spreading (degrees).
![]()
Wave–current interaction
Predicted wave field with and without WCI were compared
during a 7-month period in 2010, covering both winter and summer conditions,
for evaluating the importance of WCI on wave features. The results are shown
in Fig. 4. For the comparison between the coupled and the uncoupled model the
root mean square (rms) between the two runs was computed. Results show some
differences between the two runs. In particular, the largest deviations due
to WCI are found in coastal areas, such as around headlands and bays, and in
estuaries, in which the currents (mostly driven by tides) are strongest. As
expected, the highest differences were seen in the proximity of the coastline
: this was because the strength of the mainly
tidal-driven currents are stronger . During
spring tides, higher values for the current were recorded off northeast
England and near Peterhead and Aberdeen (see Fig. 1). Wave periods are more
affected than wave heights in this coupling, with rms deviations that can be
on average 20 % (absolute value) in shallow-water coastal areas. We also
considered the effect of the WCIs on the wave
directional spreading, as this is an important variable for the stability of
the wave train in deep water and for its evolution . The
results showed that during the 7-month period the significant wave height
was, on average, less affected than directional spreading or wave periods:
the difference was of the order of magnitude of 0.1 m near the coastline and
less offshore, while the difference in peak spectral wave period
(Tp) exceeded 1 s in some of the east coast firths such as the
Moray Firth and the Firth of Forth.
Maximum modelled positive deviation of the Hs (m) due to
the WCIs recorded in the 7-month period run in 2010.
![]()
Maximum positive and negative variation during the
7-month period were also studied (Figs. 5 and 6). The figure is similar to
the RMSE: the larger variation is reported only in the coastal areas, while
in the open sea the maximum variation is limited up to 1 m. Spatially, the
maximum variation of Hs between the coupled and the uncoupled run
was +2.8 and -1.8 m, both occurring during storm events and both
occurring in coastal areas, near the coast of Aberdeen and Peterhead and
south of the Firth of Forth, in which the tidally driven current is stronger
.
Current and swell effect on the windsea wave field
In order to study the importance of the WCIs and the
coupling between swell and windsea waves off the east coast of Scotland,
three storms were considered in the period January–August 2010. Storm events
were identified by examining the time series in the Firth of Forth and the
Moray Firth in which the highest Hs were recorded. These three
storms were selected because they were the three most intense storms during
the considered period and originated from different weather conditions.
Maximum modelled negative deviation of the Hs (m) due to
the WCIs recorded in the 7-month period run in 2010.
![]()
The mean sea level pressure fields (hPa) before and during the
25–26 March 2010 storm.
![]()
The 26–27 February 2010 storm
Between 25 and 27 February 2010, the UK was affected by a low pressure
system, that moved rapidly from west to east. From the afternoon of the 25th
to the 26th, the centre of the storm was over the North Sea (Fig. 7). At the
same time, another low pressure system (not shown in the map) was over the
Norwegian Sea, causing a train of swell moving from north to south. Comparison of
modelled and observation wave heights and wave period conditions for this
storm are reported in Fig. 8. In addition, the modelled conditions in the
Aberdeen wave rider location are reported. The figure shows that the model
reproduces adequately the conditions during that storm, in particular around
the time in which the maximum Hs was reached.
Wave conditions during the 25–26 February 2010 storm:
(a) comparison between coupled and uncoupled modelled Hs
(m) with observed data in Firth of Forth wave gauge; (b) comparison
between coupled and uncoupled modelled Hs (m) in the Aberdeen wave
gauge; no observation data were available from this wave gauge during this
storm; (c) comparison between coupled and uncoupled modelled
Tm (s) with observed data in Firth of Forth wave gauge;
(d) comparison between coupled and uncoupled modelled Tm
(s) in the Aberdeen wave gauge, no observed data were available from this wave
gauge during this storm.
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The modelled Hs on the east coast of Scotland at
12:30 UTC on 26 February 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and uncoupled
models in the Moray Firth area, (e) windsea waves, (f) swell
waves.
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The low pressure over North Sea caused windsea
waves exceeding 4 m. In Fig. 9 the situation in the sea is shown at
12:30 UTC of the 26th: swell waves contributed to enhancing the
Hs in the centre of the storm, while a train of swell waves was
forming from this storm, travelling west to the Moray Firth. Interaction of
the windsea and the swell waves caused high waves along the east coast: the
maximum recorded Hs by the Firth of Forth wave gauge was 4.8 m.
WCI contributed to the enhancement of Hs by up to 1 m in coastal
areas, while in the open sea the contribution was very low, up to 0.1 m. In
the afternoon of the 26th (Fig. 10, at 19:00 UTC) the storm was near the
Firth of Forth. The contribution of the swell waves was significant,
increasing the Hs by up to 1 m: model outputs showed that the
central part of the storm had an Hs>5 m, while without the
swell coming from north the centre of the storm would have been an
Hs < 4.5 m. To our knowledge, no significant damages were
recorded for this storm.
The modelled Hs on the east coast of Scotland at
19:00 UTC on 26 February 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and
uncoupled models in the Moray Firth area, (e) windsea waves, (f) swell
waves.
![]()
The mean sea level pressure fields (hPa) before and during the
30–31 March 2010 storm.
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The 30–31 March 2010 storm
The larger storm in 2010 occurred during the night of 30 March 2010. Between
29 March and 1 April 2010 the southeast coast of Scotland and the north of England
were struck by severe weather and very strong winds. These conditions were
caused by a strong depression that originated from a weak minimum near the
Azores islands, in the North Atlantic, in front of the Portuguese coast. This
low pressure was <990 hPa once over Great Britain and Ireland at midnight
of 30 March 2010 and reached its minimum the day after with a depression
of <980 hPa over the north of England. The evolution of the storm from
surface pressure charts from ECMWF ERA-Interim reanalysis is reported in
Fig. 11 . These figures clearly
show that the depression, at its maximum strength, is just above the south of
Scotland during the night between 30 and 31 March 2010. This depression
generated both very high waves (Hs exceeded 6 m, measured in the
Firth of Forth) and surge waves exceeding 0.5 m (measured both by the Aberdeen
and Leith tide gauges). The waves caused significant damages to the coastal
defences of cities in the southeast of Scotland. In particular, the City of Edinburgh
Council estimated the damages to coastal defences to be about
GBP 23 000. Also, in Berwick, at the southern entrance of the Firth
of Forth, some damages were caused to the harbour infrastructures. To the
east, in Dumbar, waves topped the roofs of two-floor houses.
Damaging conditions associated with this storm were caused by a combination
of simultaneous factors: (1) tides in the spring period, (2) a surge wave of
about 0.5 m generated by local pressure and wind, (3) windsea waves
generated locally that interacted with strong currents, (4) a weak but
significant swell waves field that interacted with the windsea waves.
Figure 12 shows the intensity of the current in the Aberdeen wave gauge location
and the resulting WCI. It can be seen that the current
was strongly enhanced by the wind, and consequently the WCI effect was
stronger.
Modelled currents and waves conditions in the Aberdeen wave gauge
location during the 30–31 March 2010 storm (depth of the mooring location is
10 m).
![]()
The modelled surge wave due to the local wind and pressure at
02:00 UTC on 31 March 2010.
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At about 00:30 UTC on 31 March 2010, the storm was at its maximum,
causing the wave field to hit the coastline at around the same time as high
tide and surge. The different components of the storm were analysed. First,
the surge wave generated by the minimum of pressure above the North Sea was
studied. Figure 13 shows the difference between the total water level and the
water level due to tides at 02:00 UTC on 31 March 2015. The model predicted
a surge wave up to 0.5 m. A comparison between the recorded water level and
the model output showed that the model underestimated the surge wave by about
0.1 m. The reason for this underestimation could be because the boundary
conditions for the model only included tidal water level and did not include
the surge wave from outside the model. The surge wave extended from the Firth
of Forth southwards: the water level in those regions was enhanced by about 0.4–0.5 m. In addition to these surge conditions, the
Hs of the waves at the same time was exceeding 7 m in the same
areas (see Figures 14 and 15). Figures 11 and 12 show the wave field at two
different times in the storm, at 00:30 and at 02:00 UTC, respectively. The
swell wave effect was very low, but contributed to the enhancement of
Hs up to 0.5 m, while on the coastline the contribution of the
WCI was very strong. At 02:00 UTC on 31 March 2010 (Fig. 15), when the storm
reached the coastline, WCI increased Hs by up to 2.5 m in many
locations near the Firth of Forth (see Fig. 15d). Figures 14f and 15f show
high Hs swell waves at the entrance of the Firth of Forth. These
were waves generated by the large storm shown in Figure 14e, but are no
longer influenced by the local wind, but are propagating outside the centre
of the windsea waves to the coastline. Hs recorded by the Firth
of Forth wave gauge measured a peak of significant wave height of 6.46 m at
05:00 UTC on 31 March 2015. The model matched the peak recorded in the wave
gauge reasonably well, predicting higher S values of the Firth of Forth,
where more damages were caused. The wave–wave interactions due to the
interaction between swell and windsea waves was important for the enhancement
of the Hs in the northern part of the Scotland, where the windsea
wave conditions were less intense, while the contribution was low in the
central part of the storm.
The modelled Hs on the east coast of Scotland at
00:30 UTC on 31 March 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and
uncoupled models in the Firth of Forth area, (e) windsea waves, (f) swell
waves.
![]()
The modelled Hs on the east coast of Scotland at
02:00 UTC on 31 March 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and
uncoupled models in the Firth of Forth area, (e) windsea waves, (f) swell
waves.
![]()
The mean sea level pressure fields (hPa) before and during the
19 June 2010 storm.
![]()
The 19 June 2010 storm
The third storm that is considered in this paper was one that generated high
off-shore wave conditions, with swell propagating to the coastline. This is
an example of how the coupling of swell and windsea waves could lead to
extreme wave conditions, with significant wave height exceeding 6 m offshore
and 4–5 m on the coastline. Figure 16 shows the pressure conditions between
18 and 20 June 2010. On 17 June 2010 (not shown) a system of low
pressure was generated between Greenland and Iceland. This minimum moved
quickly to the Scandinavian peninsula, intensifying and remaining in the area
of Sweden and Norway for 72 h. This low pressure caused strong winds in the
northern North Sea and consequently the generation of waves in the area
between the Norway and Scotland. Recorded wave conditions in the Firth of
Forth are compared with the model output (Fig. 17a–c) and model output
from the Aberdeen wave rider location is shown (Fig. 17b–d). The model
demonstrates the wave conditions present during this storm well (both for wave heights
and periods) and the results show the limited effect of the WCI in those
locations. This field of waves arrived at the Scottish coastline at the same
time as the low pressure was generating high waves in the bulk of the North
Sea, causing two trains of waves to be in the same place at the same time.
This condition, known as crossing or bimodal sea, is quite common in the
North Sea . The model hindcasted that the
storm offshore was at its maximum near 16:00 UTC on 19 June 2010
(Fig. 18). At 16:00 UTC on 19 June 2010, the modelled offshore, mid-North
Sea, windsea-generated waves peaked at Hs∼ 5 m (Fig. 14e),
whereas the swell waves were a little smaller with Hs∼3–4 m (Fig. 18f). Further north, in the Moray Firth, the swell waves
dominated with the swell having Hs∼6 m and the windsea
having Hs∼2 m. The resulting predicted wave field had
Hs>6 m (Fig. 18b). In the Moray Firth, an Hs of more
than 5 m was recorded. However, at this time, the coupling between currents
and waves caused a decrease of the significant wave height at the coastline
(Fig. 18c). In some locations Hs was reduced by more than 0.5 m
(see Fig. 18c–d). Three hours later (Fig. 19), the turning tidal currents enhanced
the waves by more than 1.5 m in coastal locations. In this storm, the
WCIs play a role in the enhancement of the wave
conditions: spatially, the effect (Figs. 18–19) is significant on the
coastline. In addition, the windsea wave field is significantly enhanced by
swell waves, and the bimodal sea conditions are effective in changing the
Hs due to the interactions between swell and windsea waves.
Wave conditions during the 19 June 2010 storm:
(a) comparison between coupled and uncoupled modelled Hs
(m) with observed data in Firth of Forth wave gauge; (b) comparison
between coupled and uncoupled modelled Hs (m) in the Aberdeen wave
gauge, no observed data were available from this wave gauge during this
storm; (c) comparison between coupled and uncoupled modelled
Tm (s) with observed data in Firth of Forth wave gauge;
(d) comparison between coupled and uncoupled modelled Tm
(s) in the Aberdeen wave gauge, no observed data were available from this wave
gauge during this storm.
![]()
The modelled Hs on the east coast of Scotland at
16:00 UTC on 19 June 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and
uncoupled models in the Firth of Forth area, (e) windsea waves, (f) swell
waves.
![]()
The modelled Hs on the east coast of Scotland at
19:00 UTC on 19 June 2010: (a) coupled model (mean WCI included),
(b) uncoupled model (mean WCI not included), (c) difference between
coupled and uncoupled models, (d) difference between coupled and
uncoupled models in the Firth of Forth area, (e) windsea waves, (f) swell
waves.
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Effect of WCI on the wave spectra
Considering the second storm (30–31 March 2010) we analysed the effect of
the WCI on the 1-D and 2-D spectra. Modelled
spectra were extracted from the model output in three locations in
correspondence with the wave gauges, and the output with and without
WCIs was analysed (Figs. 20–22). Some significant
variation of the energy density of the spectra (≈ 20 %) were seen
for the considered storm, in particular for the Aberdeen wave gauge, but also
for the Firth of Forth wave gauge, in which high waves were recorded; the
major changes were reported near the spectral peak. The model also predicted
a shift of the spectral peak and variation in swell magnitude. Since large
variations were recorded for the Aberdeen wave gauge and the Firth of Forth
wave gauge, we analysed the modelled directional spectra with and without
WCIs for the considered storm. In Figs. 23–24, we show
the results for the 2-D spectrum, in which not only the distribution of the
energy with the frequency was shown but also the distribution with the angle.
Variation in the magnitude of the spectral energy with the angle along with
small variation in the direction of the wave train were modelled.
Modelled 1-D spectrum in the Firth of Forth wave gauge, the red line is
the coupled model (with WCIs incorporated), while the blue
line is the uncoupled model: (a) 31 March 2010 at 00:30,
(b) 31 March 2010 at 01:15, (c) 31 March 2010 at 02:00,
(d) 31 March 2010 at 04:15, (e) 31 March 2010 at 06:00, and
(f) 31 March 2010 at 08:30 UTC.
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Modelled 1-D spectrum in the Moray Firth wave gauge, the red line is the
coupled model (with WCIs incorporated), while the blue line
is the uncoupled model: (a) 31 March 2010 at 00:30,
(b) 31 March 2010 at 01:15, (c) 31 March 2010 at 02:00,
(d) 31 March 2010 at 04:15, (e) 31 March 2010 at 06:00,
and (f) 31 March 2010 at 08:30 UTC.
![]()
Modelled 1-D spectrum in the Aberdeen wave gauge, the red line is the
coupled model (with WCIs incorporated), while the blue line
is the uncoupled model: (a) 31 March 2010 at 00:30,
(b) 31 March 2010 at 01:15, (c) 31 March 2010 at 02:00,
(d) 31 March 2010 at 04:15, (e) 31 March 2010 at 06:00, and
(f) 31 March 2010 at 08:30 UTC.
![]()
Polar plot of the modelled 2-D directional spectrum (energy density,
m2 × s / degrees) in the Firth of Forth
wave gauge, red indicates the contour plot of the coupled model spectrum (with
WCIs incorporated), while black indicates the contour plot of
the uncoupled model. Contour lines are plotted every
0.01 m2 s degrees-1: (a) 31 March 2010 at 00:30,
(b) 31 March 2010 at 01:15, (c) 31 March 2010 at 02:00,
(d) 31 March 2010 at 04:15, (e) 31 March 2010 at 06:00, and
(f) 31 March 2010 at 08:30 UTC.
![]()
Polar plot of the modelled 2-D directional spectrum (energy density,
m2 × s / degrees) in the Aberdeen wave gauge, red indicates the
contour plot of the coupled model spectrum (with WCIs
incorporated), while black indicates the contour plot of the uncoupled model.
Contour lines are plotted every 0.01 m2 s degrees-1:
(a) 31 March 2010 at 00:30, (b) 31 March 2010 at 01:15,
(c) 31 March 2010 at 02:00, (d) 31 March 2010 at 04:15,
(e) 31 March 2010 at 06:00, and (f) 31 March 2010 at
08:30 UTC.
![]()
Similar results
for the spectrum variations are reported in for the
WCIs at the mouth of Danube, while similar spectral
changes were identified in laboratory experiments, as in
and .
Conclusions
In this study we presented a model capable of hindcasting surge
and storms on the east coast of Scotland. The combination of spring tide,
strong wind, and high waves can be extremely threatening in coastal areas. The
North Sea is one of the areas most affected by this forcings. Storms in North
Sea can generate extremely high waves as well as rogue waves
.
Results indicate that WCIs play a fundamental role in the wave
propagation during severe storms in the coastal areas, while for the open
sea, the maximum contribution of this interaction is less than 0.5 m of
magnitude. The results are consistent with other studies of WCI in other
parts of the world, such as in the southern North Sea , in
which the difference based on the normalized rms difference of a 1-month period
is about 3 % for the Hs and has an rms of 20 % for
Tm Tables 3 and 4 of. Such is the case in the northern Adriatic Sea
in the shallow areas between the Venetian Lagoon, the Gulf of Trieste, and the
Istrian peninsula, where deviations up to 1 m were modelled during
bora and sirocco conditions .
The validation shows that the model performs reasonably well during both calm
periods and storms for waves, and also performs well for tides and surges.
During severe storms, in particular when the low pressure was over England
and Scotland, it was found that the WCIs are
significant, causing an increase or decrease in Hs that can exceed 2 m
in some coastal areas, depending on the direction of the wave field compared
to the current. A similar result was found for the peak spectral wave period:
Fig. 4 shows that in the time period considered here the largest deviation
of wave periods due to WCI is in the estuarine areas of the east coast, with
rms deviations of more than 1.2 s.
Wave propagation in the Firth
of Forth during storms generated in the mid-North Sea is driven by trains of
swell waves detaching from the open sea storm. During the stormy periods
considered here, the windsea waves in the Firth of Forth did not exceed 3.5 m
in the outer area of the estuary and 1 m in the inner part, while the swell
field exceeded 5 m at the entrance of the Firth of Forth. In the inner
firth, the swell waves have a similar magnitude to the windsea waves. Conversely,
the area of the estuary of the Firth of Forth is mainly driven by locally generated
waves. A similar behaviour was noticed in the other two estuarine areas on the
east coast: the Tay estuary and the Moray Firth.
The northeast coast of
Scotland is more exposed to swell arriving from the North Atlantic and the
Norwegian Sea, while the central and southern parts are more exposed to
local windsea waves and to storms generated in the bulk of the North Sea.
Spectra were also considered in the analysis of the WCIs: spectral variations, in particular in the energy peak, were
significant and exceeding 20 % in some cases. Wave periods are adequately
modelled by the model presented in this paper. Wave models, however, have a
large error for the wave period, since they do not include nonlinear
quasi-resonant interactions
that are also
fundamental for the correct estimation of the Hs when the
spectrum is narrow. In addition, wave periods from satellite data are often very
difficult to estimate . Another limitation
of the study is that no surge boundaries were available, so the water level
and the current fields were only due by tides and the local field of wind and
pressure. This led to an overall underestimation of the strength of the
current and a possible underestimation of the total effect of the WCI.
The model also has forecasting capabilities, in particular when nested with
large-scale models, such as the North Atlantic model
. A limitation of the model is
that the MIKE by DHI software does not allow an online coupling between waves
and tides, slowing the simulation process. In fact, currents and waves are
simulated by different modules and it is not possible to perform a direct
coupling. For this work, the currents were simulated first and then the output
data were saved in order to use them as input for the wave model. Another
limitation of the model, due to the one-way coupling, is that we can not
study the effect of the wave setup and setdown on the surge water level
, and most importantly, the wave radiation
effect on the current field itself
. Previous work on this
interaction shows that the modification of the current field is more
important in very shallow water areas (< 10 m depth). In this paper,
however, we were more interested in the effect of the current field on the
wave. Future work will focus on understanding what effect the
waves have on the current dynamics on the east coast of Scotland. This will be
implemented by first running the wave model, then using the wave radiation in
the hydrodynamic model to estimate the enhancement of the water level due to
waves near the shoreline and to estimate the variation of the current due to
the wave radiation stress.
This research also underlines the importance of high-resolution regional-scale
models for the understanding of sea dynamics and the forecasting of
dangerous sea states: larger models usually have inadequate resolution to
estimate the effect of such processes near the coastline. Future work will be
focused on the hindcasting of freakish wave states based on the estimation of
the kurtosis from the parameters of the model
and on the sediments
resuspension in the area of Stonehaven , which is an
intensive study site for suspended sediment and other biological variables in
the water column .