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

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

TL;DR: This research presents a probabilistic procedure to assess whether the presence of non-volatile substance such as phosphorous, nitrogen, or phosphorous in the response of the immune system to earthquake-triggered diarrhoea and finds that the response is positive.

Abstract: Introduction Conclusions References Tables Figures

## 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(ω, 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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TL;DR: In this paper, the authors developed a robust numerical method for modeling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness.

Abstract: We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

616 citations

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12 Jan 2009

TL;DR: In this paper, the authors present deterministic and statistical approaches for studying the behavior of Rogue Waves in Waters of Infinite and Finite Depths and Shallow-Water Rogue Waves, respectively.

Abstract: Observation of Rogue Waves.- Deterministic and Statistical Approaches for Studying Rogue Waves.- Quasi-Linear Wave Focusing.- Rogue Waves in Waters of Infinite and Finite Depths.- Shallow-Water Rogue Waves.- Conclusion.

550 citations

### "A comparison of the measured North ..." refers result in this paper

...(2008), Kharif et al. (2009), Osborne (2010), Slunayev (2010) and Onorato et al....

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...(2008), Kharif et al. (2009), Osborne (2010), Slunayev (2010) and Onorato et al. (2013). Predictions given by theoretical and numerical wave models accounting for nonlin20 earities beyond the second order in deep water such as: HOSM, Nonlinear Schrödinger Equations (NLS), the Dysthe model and the Conformal Method, compare well with experimental results (e....

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