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A comparison of the measured North Sea Andrea rogue wave with numerical simulations

01 Sep 2013-Vol. 1, Iss: 5, pp 5033-5056

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Topics: Rogue wave (60%)

Summary (2 min read)

1 Introduction

  • A number of studies addressing rogue waves have been conducted theoretically, numerically, experimentally and based on the field data in the last decade.
  • Thus, sampling variability will also be present in 17–30 min wave records extracted from the continuous measurements and in statistical properties derived from them.
  • A spectral wave model (e.g., the WAM model) provides sea state description only in a form of the two-dimensional wave spectrum but does not give any information about the instantaneous position of the sea surface in a given sea state.

3 Hindcast data

  • The hindcast data used in this study were retrieved from the European Centre for Medium-Range Weather Forecast archive.
  • Figure 2 shows a history of the total significant wave height, spectral period and related sea state steepness during the Andrea storm.
  • It is interesting to note that the output from a wave spectral model can be utilized when indicating a mechanism responsible for the occurrence of a rogue wave (e.g. Tamura et al., 2009), even though it may not always allow reaching the firm conclusions.
  • Wind sea clearly dominates the total sea during the growth, peak and decay of the storm.

4 Numerical simulations

  • Numerical simulations have been carried out to get further insight into the Andrea storm characteristics.
  • A comparison of these two approaches (Clamond et al., 2006) has shown that the formulation proposed by Dommermuth and Yue (1987) is less consistent than the one proposed by West et al. (1987) as it does not converge when the amplitude is very small; the latter, therefore, has been applied herein.
  • Nonetheless, comparisons of sta- tistical properties of the surface elevation from HOSM simulations initialized with linear surface (thus no adjustments) and laboratory experiments in directional wave basins (see, e.g., Toffoli et al., 2010a, 2013, for infinite and finite water depth, respectively) showed a very good agreement both in terms of spatial/temporal evolution and maximum values of statistical moments.
  • 6–9 point out that coupling of the wave spectral model and the nonlinear phase-resolving model could predict the occurrence of the Andrea wave.

5 Conclusions

  • The study shows how the wave spectral WAM model and the HOSM model can be coupled to forecast/hindcast the occur- rence of extreme and rogue waves.
  • A spectral model coupled with the HOSM model provides statistical information about waves based on the actual hindcast/forecast spectrum, whether this is bimodal or unimodal.
  • The analysis shows that when the Andrea storm is passing the North Sea rogue waves can be expected in several locations, not only at Ekofisk where the Andrea wave was www.nat-hazards-earth-syst-sci.net/14/1407/2014/ Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014 recorded.
  • The great advance in enhancing computer power has made the coupling between them feasible.
  • Edited by: A. Slunyaev Reviewed by: two anonymous referees.

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Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014
www.nat-hazards-earth-syst-sci.net/14/1407/2014/
doi:10.5194/nhess-14-1407-2014
© Author(s) 2014. CC Attribution 3.0 License.
The North Sea Andrea storm and numerical simulations
E. M. Bitner-Gregersen
1
, L. Fernandez
2
, J. M. Lefèvre
3
, J. Monbaliu
2
, and A. Toffoli
4
1
DNV GL Strategic Research and Innovation, Høvik, Norway
2
Department of Civil Engineering, KU Leuven, Heverlee, Belgium
3
Division Marine et Oceanographie, Meteo-France,Toulouse, France
4
Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn,
VIC. 3122, Australia
Correspondence to: E. M. Bitner-Gregersen (elzbieta.bitner-gregersen@dnvgl.com)
Received: 25 August 2013 Published in Nat. Hazards Earth Syst. Sci. Discuss.: 25 September 2013
Revised: 8 April 2014 Accepted: 15 April 2014 Published: 5 June 2014
Abstract. A coupling of a spectral wave model with a non-
linear phase-resolving model is used to reconstruct the evo-
lution of wave statistics during a storm crossing the North
Sea on 8–9 November 2007. During this storm a rogue wave
(named the Andrea wave) was recorded at the Ekofisk field.
The wave has characteristics comparable to the well-known
New Year wave measured by Statoil at the Draupner plat-
form 1 January 1995. Hindcast data of the storm at the near-
est grid point to the Ekofisk field are here applied as input
to calculate the evolution of random realizations of the sea
surface and its statistical properties. Numerical simulations
are carried out using the Euler equations with a higher-order
spectral method (HOSM). Results are compared with some
characteristics of the Andrea wave record measured by the
down-looking lasers at Ekofisk.
1 Introduction
A number of studies addressing rogue waves have been con-
ducted theoretically, numerically, experimentally and based
on the field data in the last decade. The occurrence of rogue
waves, their generation mechanism, and detailed dynamic
properties are now becoming clear. The state-of-the-art re-
view on extreme and rogue waves can be found in recent re-
view papers and books such as Socquet-Juglard et al., (2005),
Dysthe et al., (2008), Kharif et al., (2009), Osborne (2010),
Slunyaev (2010) and Onorato et al., (2013).
Predictions given by theoretical and numerical wave mod-
els accounting for nonlinearities beyond the second order in
deep water such as HOSM (higher-order spectral method),
nonlinear Schrödinger equations (NLS), the Dysthe model
and the conformal method, compare well with experimental
results (e.g., Onorato et al., 2006a; Galchenko et al., 2010;
Shemer et al., 2010; Toffoli et al., 2010a; Slunyaev et al.,
2012; Oberhagemann et al., 2012).
Unfortunately, there are few studies available based on
field data, partly due to the limited number of rogue waves
recorded in the ocean. Investigations of meteorological and
oceanographic (met-ocean) conditions, in which extreme and
rogue waves occur, together with field analyses of wave time
series are of importance for a getting better insight of the
mechanisms generating these abnormal waves.
It should be noted, however, that field data are usually
recorded in 17–30 min periods every third hour and there-
fore are affected by sampling variability (uncertainty due to
limited number of observations) making it more difficult to
drawn firm conclusions from a field data analysis (see Bitner-
Gregersen and Hagen, 1990, 2003). Further, met-ocean con-
ditions between each 3h measurement are assumed to be
stationary due to lack of information about their variability
within the 3h time period. There are some locations where
continuous measurements of sea surface elevation are taken;
however, they are spare. It is important to note that sea states
recorded continuously every 17–30 min are usually not re-
maining stationary; therefore, not justifying the combination
of 17–30min wave records in one longer wave record. Thus,
sampling variability will also be present in 17–30min wave
records extracted from the continuous measurements and in
statistical properties derived from them.
A long wave time series is needed to obtain reli-
able estimates of extreme values of sea surface elevation
Published by Copernicus Publications on behalf of the European Geosciences Union.

1408 E. M. Bitner-Gregersen et al.: The North Sea Andrea storm and numerical simulations
Figure 1. Location of the Ekofisk field.
characteristics and of their probability of occurrence, which
are of importance for applications. The uncertainty due to
sampling variability will particularly affect the higher-order
statistical moments like skewness and kurtosis, which are
more unstable than the estimates of significant wave height
and spectral/zero-crossing wave period. To obtain reliable es-
timates of these higher-order statistical moments, 250–350
repetitions of a 17 min wave record will often be required,
as demonstrated by, for example, Bitner-Gregersen and Ha-
gen (2003). Therefore numerical wave models remain an im-
portant supporting tool in an analysis of field data.
As pointed out by Tomita (2009), both the numerical non-
linear wave models and the wave spectral models can be uti-
lized in research of extreme and rogue waves and their use
is encouraged. The complimentary nature of these models
is clear; they give different information about a sea state.
A spectral wave model (e.g., the WAM model) provides
sea state description only in a form of the two-dimensional
wave spectrum but does not give any information about
the instantaneous position of the sea surface in a given sea
state. Note also that it accounts for wind forcing and reso-
nant wave interactions but not for quasi-resonance interac-
tions, which are responsible for occurrence of modulational
instability and hence rogue waves (Onorato et al., 2013).
Phase-resolving wave models, however, provide the water
surface elevation from which statistical properties of indi-
vidual waves can be extracted and include quasi-resonance
interactions. Further, these nonlinear wave models allow sim-
ulating a wave record for the required time duration and, by
repeating the 17–30min simulations, significantly reducing
the uncertainty due to sampling variability in estimated sea
surface characteristics and their probability of occurrence.
Table 1. Characteristics of the Andrea and New Year waves.
Wave parameters Andrea wave Draupner wave
H
s
9.2 m 11.9m
T
p
13.2 s 14.4 s
C
max
15.0 m 18.5m
CF= C
max
/H
s
1.63 1.55
H
max
21.1 m 25.0m
HF = H
max
/H
s
2.3 2.1
Although the spectral wave model as well as the nonlinear
numerical wave model are computationally intense, the great
advance in enhancing computer power has made the coupling
between these models feasible.
In the present study we demonstrate the complementary
nature of the wave spectral model WAM and the numer-
ical nonlinear wave model based on the Euler equations
and solved with the HOSM proposed in West et al. (1987).
The coupling is applied to investigate statistical properties
of surface oscillations during the particularly severe Andrea
storm, which crossed the central part of the North Sea on 8–
9 November 2007. During this storm, on 9 November 2007 a
rogue wave called Andrea was recorded at the Ekofisk field
(Magnusson and Donelan, 2013). This wave is comparable in
characteristics, both with respect to the wave height and wave
crest criterion, to the well-known New Year wave (called also
the Draupner wave) measured by Statoil at the Draupner plat-
form on 1 January 1995 (Haver and Anderson, 2000).
The paper is organized as follows: Sect. 2 describes the
Andrea storm and the Andrea wave. Section 3 addresses
hindcast data used in the analysis while Sect. 4 is dedicated
to characteristics of the Andrea storm and comparison of nu-
merical results to some characteristics of the Andrea wave
recorded at the Ekofisk field. Conclusions are summarized in
Sect. 5.
2 The Andrea storm and the Andrea wave
A low pressure area entered the northern North Sea on
8 November 2007. It covered southern Norway and moved
in the morning of 9 November towards southern Sweden.
Strong westerly winds (50–55 knots) followed the low pres-
sure area and a high wave field (significant wave height of
10–11 m) was built up in the north area of the Ekofisk field
(see Fig. 1) in the afternoon of 8 November 2007. The wind
slightly decreased at Ekofisk around 18:00–21:00 UTC (uni-
versal time coordinated); from 22 to 19 m s
1
. The strongest
wind field passed the northeast and east of Ekofisk on
9 November around 06:00 UTC and generated waves of up
to 11–12 m (for details see Magnusson and Donelan, 2013).
Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014 www.nat-hazards-earth-syst-sci.net/14/1407/2014/

E. M. Bitner-Gregersen et al.: The North Sea Andrea storm and numerical simulations 1409
Figure 2. History of significant wave height, spectral wave period
and sea state steepness for the total sea during the Andrea storm.
The Andrea wave was recorded at Ekofisk by the down-
looking lasers just past 00:00UTC on 9 November 2007.
This wave is comparable in characteristics to the well-known
New Year wave (the Draupner wave) measured by Statoil at
the Draupner platform on 1 January 1995 (Haver and An-
derson, 2000). The characteristics of the Andrea wave, as
reported by Magnusson and Donelan (2013), are compared
to the New Year wave ones in Table 1. H s denotes the sig-
nificant wave height, T
p
the spectral peak period, C
max
the
maximum crest height in the wave record, H
max
the max-
imum zero-downcrossing wave height with the crest C
max
,
CF is the maximum crest factor (crest criterion), HF is the
maximum height factor (height criterion).
CF> 1.3 (or > 1.2) and HF> 2 within a 20 min wave
record represent simplified definitions of a rogue wave (see
e.g. Bitner-Gregersen and Toffoli, 2012). If both criteria are
fulfilled a rogue wave can be classified as a double rogue
wave (Krogstad et al., 2008). As seen in Table 1 both the
New Year wave as well as the Andrea wave can be called a
double rogue wave. Note that both waves are recorded in the
North Sea from the platforms located over a water depth of
ca. 75 m.
3 Hindcast data
The hindcast data used in this study were retrieved from
the European Centre for Medium-Range Weather Forecast
(ECMWF) archive. Wave parameters were acquired at the
nearest grid point to Ekofisk. The data cover the period of
the Andrea storm history from 00:00UTC 8 November 2007
to 00:00 UTC 11 November 2011 and are stored every 6h.
The selected grid point is at a water depth of 74m and within
a distance of ca. 50 km from the Ekofisk field. The data in-
clude the wind speed as well as the significant wave height
and spectral peak period for the total sea, wind sea and swell.
The components of the wind sea are those that are still under
the influence of the local wind forcing and are detected as the
part of wave spectrum where the wind input source term is
positive. The remaining part of the wave spectrum is consid-
ered as swell (see, e.g., Hauser et al., 2005, for details).
Figure 2 shows a history of the total significant wave
height, spectral period and related sea state steepness dur-
ing the Andrea storm. The significant wave height reaches its
maximum at 06:00UTC on 9 November 2007. This is con-
sistent with the findings of Magnusson and Donelan (2013)
based on the NORA10 (Norwegian 10 km Reanalysis
Archive) hindcast data developed at the Norwegian Meteo-
rological Institute with major support from a consortium of
oil companies (see e.g., Aarnes et al., 2011). The maximum
wave height of 9.8 m is associated with the largest spectral
peak period and the highest sea state steepness of 0.14 dur-
ing the storm. It should be noted that the same high steep-
ness is observed before the significant wave height reaches
its maximum. Because the hindcast data are sampled every
6 h it is not possible to detect the steepness at 00:40 UTC on
9 November 2007, when the Andrea wave was recorded at
the Ekofisk field.
The probability of occurrence of rogue waves is related
to mechanisms generating them. It is interesting to note that
the output from a wave spectral model can be utilized when
indicating a mechanism responsible for the occurrence of a
rogue wave (e.g. Tamura et al., 2009), even though it may
not always allow reaching the firm conclusions. We illustrate
this below for the Andrea wave recorded at Ekofisk during
the Andrea storm.
The recognized mechanisms responsible for the occur-
rence of rogue waves can be classified as follows (Onorato et
al., 2006a, b, 2010, 2013; Toffoli et al., 2011; Didenkulova,
2010; Didenkulov and Pelinovsky, 2011; Sergeeva et al.,
2011, 2013):
linear Fourier superposition (frequency or angular linear
focussing)
wave–current interactions
crossing seas
quasi-resonance nonlinear interactions (modulational
instability)
shallow water effects.
These mechanisms have also been considered in order to in-
dicate a possible phenomenon responsible for generating the
Andrea wave.
The linear focusing is occurring very seldom and be-
cause the present study is addressing nonlinear waves the
linear focusing has been eliminated from further consider-
ations. Further, no strong current has been reported in the
Ekofisk area representing the intermediate water depth ocean
zone. Therefore wave–current interactions and shallow wa-
ter effects seem not to be responsible for the occurrence
of the Andrea wave.
www.nat-hazards-earth-syst-sci.net/14/1407/2014/ Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014

1410 E. M. Bitner-Gregersen et al.: The North Sea Andrea storm and numerical simulations
Figure 3. History of significant wave height for wind sea and swell
during the Andrea storm.
The two remaining rogue wave generation mechanisms,
namely crossing seas and quasi-resonance nonlinear interac-
tions (modulational instability), could be regarded as the only
possible candidates that generated the Andrea wave. In order
to select one of them the time history of wind sea and swell
during the Andrea storm has been studied.
Figure 3 shows evolution of significant wave height for
wind sea and swell during the Andrea storm at the nearest
grid point to Ekofisk. Wind sea clearly dominates the total
sea during the growth, peak and decay of the storm. Most of
the time the significant wave height of swell is only slightly
above 1m, much lower than the wind sea significant wave
height, which mostly reaches 10 m at the peak of the storm.
Therefore the total sea of the Andrea storm is dominated by
wind sea. The wind sea and swell have approximately the
same energy (the significant wave height around 1m) only at
the beginning and end of the storm.
It is well established that two wave trains with similar en-
ergy and frequencies traveling at particular angles can trigger
modulational instability and be responsible for the formation
of rogue waves (Onorato et al., 2006a, 2010). Such results
have been confirmed through recent numerical simulations
of the Euler equations and experimental work performed in
the MARINTEK Laboratories (Toffoli et al., 2011). The in-
vestigations have showed that the kurtosis, a measure of the
probability of occurrence of extreme waves, depends on an
angle β between the crossing wave systems. The maximum
value of kurtosis is achieved for 40 < β < 60. No such con-
ditions where wind sea and swell have the same energy and
spectral peak frequency and are crossing each other under the
angle 40 < β < 60 have been identified in the hindcast data
from the Andrea storm. The ECMWF hindcast data seems to
point out that the Andrea wave might have occurred in the
sea state more prone to extreme waves as a result of modula-
tional instability (i.e., the sea state with relatively high steep-
ness and not a particularly broad spectrum).
This conclusion is supported by Figs. 4 and 5 present-
ing evolution of the directional wave spectrum during the
Andrea storm at the location considered. One wave system
is seen in the period from 18:00UTC 8 November 2007 to
06:00 UTC 9 November 2007 within which the Andrea wave
was recorded.
4 Numerical simulations
Numerical simulations have been carried out to get further in-
sight into the Andrea storm characteristics. Short-term wave
records at the sampling interval of 6 h have been generated
by solving the Euler equations with the HOSM as proposed
by West et al. (1987).
In the case of constant water depth (h = 74 m in this
study), the velocity potential 8(x,z,t) of an irrotational,
inviscid, and incompressible liquid satisfies the Laplace’s
equation everywhere in the fluid. The boundary conditions
are such that the vertical velocity at the bottom (z = 74) is
zero, and the kinematic and dynamic boundary conditions
are satisfied for the velocity potential 9(x,y,t)= 8(x,y,
η(x,y,t),t) on the free surface; that is, z = η(x,y,t) (see Za-
kharov, 1968). The expressions of the kinematic and dynamic
boundary conditions are as follows:
9
t
+ gη +
1
2
(9
2
x
+ 9
2
y
)
1
2
W
2
(1+ η
2
x
+ η
2
y
) = 0, (1)
η
t
+ 9
x
η
x
+ 9
y
η
y
W (1+ η
2
x
+ η
2
y
) = 0, (2)
where the subscripts denote the partial derivatives, and
W (x, y, t) = 8
z
|η represents the vertical velocity evaluated
at the free surface.
The time evolution of an initial surface elevation can be
calculated from Eqs. (1) and (2). For this study, we have
used the HOSM, which was independently proposed by West
et al. (1987) and Dommermuth and Yue (1987). A compar-
ison of these two approaches (Clamond et al., 2006) has
shown that the formulation proposed by Dommermuth and
Yue (1987) is less consistent than the one proposed by West
et al. (1987) as it does not converge when the amplitude is
very small; the latter, therefore, has been applied herein. The
advantage of HOSM in comparison to other methods is that
it allows simulating a large number of random realizations
of the surface elevation, within a reasonable computational
time, without limitations in terms of the spectral bandwidth.
HOSM uses a series expansion in the wave slope of the
vertical velocity W (x, y,t) about the free surface. In the
present study we have considered a third-order expansion so
that the four-wave interaction is included (see Tanaka, 2001,
2007). Under these circumstances, the solution presented
herein is not fully nonlinear. The expansion is then used to
evaluate the velocity potential 9(x,y,t) and the surface el-
evation η(x,y,t ) from Eqs. (1) and (2) at each instant of
time. The time integration is performed by means of a fourth-
order Runge–Kutta method with a time step 1t = T
p
/200
(T
p
is the spectral peak period). All aliasing errors generated
in the nonlinear terms are removed (see West et al., 1987,
and Tanaka, 2001, for details). A small time step is used to
Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014 www.nat-hazards-earth-syst-sci.net/14/1407/2014/

E. M. Bitner-Gregersen et al.: The North Sea Andrea storm and numerical simulations 1411
Figure 4. Evolution of the directional wave spectrum during the Andrea storm, from 00:00 UTC 8 November 2007 to 06:00 UTC 9 Novem-
ber 2007.
Figure 5. Evolution of the directional wave spectrum during the Andrea storm, from 12:00 UTC 9 November 2007 to 11 November 2007
18:00 UTC .
minimize the energy leakage. Throughout the simulations the
variation of total energy remains lower than 0.5%.
The model works under the assumption that the water
depth is uniform. At the Ekofisk area, including the location
considered, the variation of bottom topography is negligible
and hence such an assumption does not affect the end re-
sult of the simulations. It is worth mentioning, however, that
where bottom topography is changing wave dynamics could
be affected and thus a variable bathymetry should be consid-
ered (e.g., the numerical model of Fructus and Grue, 2007).
The HOSM requires as input an initial sea surface and
velocity potential with periodic boundary conditions. For
the purpose of the present study, the initial conditions are
extracted from the hindcast wave spectrum. Initially, the
spectrum E, θ) is converted into a wavenumber spectrum
E(k
x
,k
y
) with the linear dispersive relation. An input surface
www.nat-hazards-earth-syst-sci.net/14/1407/2014/ Nat. Hazards Earth Syst. Sci., 14, 1407–1415, 2014

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Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "The north sea andrea storm and numerical simulations" ?

In this paper, the authors used a coupling of a spectral wave model with a nonlinear phase-resolving model to reconstruct the evolution of wave statistics during a storm crossing the North Sea on 8-9 November 2007.