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Natural Hazards
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Nonlinear Processes
in Geophysics
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tural Hazards
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Nonlin. Processes Geophys., 20, 179–188, 2013
Advances
in
Annales
www.nonlin-processes-geophys.net/20/179/2013/
doi:10.5194/npg-20-179-2013
Geosciences
Geophysicae
© Author(s) 2013. CC Attribution 3.0 License.
Chemistry
I. Didenkulova1,2 and A. Rodin1,2
and Physics
1 Laboratory
Chemistry
and Physics
Open Access
Open Access
Discussions
A typical wave wake from high-speed
vessels: its group structure
Atmospheric
Atmospheric
and run-up
Measurement
Measurement
Correspondence to: I. Didenkulova ([email protected])
Techniques
Techniques
Open Access
Open Access
of Wave Engineering, Institute of Cybernetics
at Tallinn University of Technology, Akadeemia tee 21,
Discussions
12618 Tallinn, Estonia
2 Nizhny Novgorod State Technical University,Atmospheric
Atmospheric
Nizhny Novgorod, Russia
Received: 2 September 2012 – Revised: 4 February 2013
– Accepted: 4 February 2013 – Published: 26 February 2013
Discussions
Open Access
Open Access
1 Introduction
Abstract. High-amplitude water wavesBiogeosciences
induced by highare regularly observed in Tallinn Bay,
the
Discussions
Baltic Sea, causing intense beach erosion and disturbing maWaves induced by high-speed vessels, the depth Froude numrine habitants in the coastal zone. Such a strong impact on
ber (the ratio of the ship’s speed and the maximum phase
the coast may be a result of a certain group structure of
of linear water waves for the given depth) of which
the wave wake. In order to understand it, here we present
Climatespeed
during the regular sailing regime exceeds 0.6, have become
Climate
an experimental study of the group structure of these wakes
of the
Pasta subject of intensive study in the last ten years after the
Pikakari beach, Tallinn Bay. The most energetic
vessel
of theatPast
new generation of large and powerful ships operating at
waves at this location (100 m from the coast at the Discussions
water
cruise speeds up to 30 knots has been introduced (Parnell and
depth 2.7 m) have amplitudes of about 1 m and periods of
Kofoed-Hansen, 2001; Parnell et al., 2007, 2008; Soomere,
8–10 s and cause maximum run-up heights on a beach up to
Kurennoy et al., 2009, 2011; Torsvik et al., 2009; Ra1.4 m. These waves represent frequency modulated
packets
Earth System2007;
paglia et al., 2011). It has been demostrated that these waves
Earth System
where the largest and longest waves propagate ahead of other
Dynamics
can be a major contributor of energy to sections of coasts
smaller amplitude and period waves. Sometimes the
groups
Dynamics
Discussionsthat are exposed to significant natural hydrodynamic loads
of different heights and periods can be separated even within
(Soomere, 2005a). The actual effect depends upon the feaone wave wake event. The wave heights within a wake are
tures of the coastal environment and the existing hydrodywell described by the Weibull distribution, which has differGeoscientific
namic loads. In this context, specific types of disturbances,
ent parameters for wakes from different vessels.Geoscientific
Wave run-up
such as high leading waves, monochromatic packets of relanstrumentation
Instrumentation
heights can also be described by Weibull distribution
and its
parameters
of the distri-andtively short waves, solitary and cnoidal wave trains ahead of
Methods
and can be connected to the parametersMethods
the vessel and associated depression areas, all qualitatively
bution of wave heights 100 m from the coast. Finally, the runData Systems
Data
Systems
different from the usual wind waves or constituents of the
up of individual waves within a packet is studied. It is shown
Discussions
linear Kelvin wake, are extremely important (Brown et al.,
that the specific structure of frequency modulated wave pack1989; Neuman et al., 2001; Garel et al., 2008). These specific
ets, induced by high-speed vessels, leads to Geoscientific
a sequence of
wave disturbances have been observed, for example, in the
Geoscientific
high wave run-ups at the coast, even when the original wave
Venice Lagoon, Italy (Rapaglia et al., 2011), Savannah River,
Model
heights are rather moderate. This feature
canDevelopment
be a key to unDevelopment
Georgia (Houser, 2011), New Zealand and Denmark (Parnell
derstanding the significant impact on coasts caused by
fast
Discussions
and Kofoed-Hansen, 2001) and Tallinn Bay, Estonia (Parnell
vessels.
et al., 2008).
Hydrology and All of these cases are characterized by the seriHydrology and
ous damage of coastal environment. In the Marlborough
Earth SystemSounds, New Zealand, the introduction of high-speed ferEarth System
Sciencesries was accompanied by rapid and significant accretion
Sciences
ogeosciences
speed vessels
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Discussions
Ocean Science
Discussions
Open Acces
cean Science
Open Acces
Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
level oscillations were recorded.
180
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
Fig. 1. Location scheme of the Baltic Sea and the study site.
Fig. 1. Location scheme of the Baltic Sea and the study site.
4
Fig. 2. High-speed vessel wakes at the Pikakari beach, Tallinn Bay, Baltic Sea.
Fig. 2. High-speed vessel wakes at the Pikakari beach, Tallinn Bay, Baltic Sea.
The time of video-recording was synchronized with the time of the echosounder, so it
(Parnell and Kofoed-Hansen, 2001). In contrast, in Tallinn
cific shape of the coast, which allows anomalous “nonreflectwasand
possible
to linkgroups
the wave
height measured
the echosounder
and(Didenkulova
run-up at theetcoast
for
Bay, Estonia, long
energetic
of vessel
waves ating”
wave behavior
al., 2009b).
resulted in dramatic
beach
at 86 wakes
Even
though
the significant
effect“Star”,
of waves from higheacherosion
wave in of
theinitially
wake. Inaccreting
total, run-ups
from
were
measured:
36 of them from
Aegna Island (Parnell
et
al.,
2008;
Soomere
et
al.,
2009).
speed
vessels
on
low-energy
coasts
has
been demonstrated
31 from “SuperStar” and 19 from “Viking XPRS”.
These groups could easily smooth out the emerging berm and
for different basins and environments, the reason of its such
sometimes exert very large run-up events (Didenkulova et al.,
strong impact still remains unclear. It can be partially con2009a; Soomere et al., 2009). The highest and longest ship
nected to the net transport of water, excited by ships sailStatistics of waves and corresponding run‐ups in the wake waves from the first 3.
group
reached over 1 m above still waing at transcritical speeds, which may produce water level
ter level with several examples going over 1.5 m above still
set-up under groups of high vessel waves and result in a
water level (DidenkulovaThe
et al.,
2009a;
Torsvik
et
al.,
2009).
of thestudy
coastof(Soomere
et al.,
data set at Pikakari beach, Tallinn Bayrapid
allowsreaction
for a detailed
the statistics
of 2011). This efHowever, on a few days there was evidence of overwash defect may also be reinforced by the specific group structure
vessel-induced waves and corresponding run-ups. The most energetic vessel waves at this
posits at heights about 2 m above water level (Soomere et
of vessel-induced waves, when the largest, the longest and
location
have amplitudes
of one
aboutvessel
1 m and
periods the
of 8-10
with maximum
up more symmetric
al., 2009). The loss
of sediments
caused by
wake
mostsec
asymmetric
wavesrun-up
comeheights
first and
was up to 1 m3 per
meter
of
the
coastline
(Soomere
et
al.,
waves
of
smaller
amplitude
and
period
come
after. The charto 1.4 m.
2009). At Pikakari beach in Tallinn Bay, wake-induced sedacteristic properties of these groups are experimentally studAll measured wave wakes have a specific structure of frequency modulated packets,
iment transport produced a stable convex nearshore beach
ied in this paper for Pikakari beach in Tallinn Bay, the Baltic
whereand
theSoomere,
largest and2011)
longest
wavestocome
first andSea.
waves of smaller amplitude and period after
profile (Didenkulova
similar
a spe(Fig. 3). Sometimes the groups of different heights and periods can be separated even within one
wake, as it is shown in Fig. 3a, where three groups of waves can be clearly identified. The
Nonlin. Processes Geophys., 20, 179–188, 2013
www.nonlin-processes-geophys.net/20/179/2013/
highest waves (with a height of up to 1 m) usually belong to the first group and occur after the
passage of several long, high-amplitude waves.
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
Water displacement, m
0.4
a
181
b
0.4
0.2
0.2
0
0
-0.2
-0.4
0
-0.2
2
4
6
8
10
12
-0.4
0
2
4
6
8
10
12
Time, min
Time, min
Fig.wakes
3. Vessel
wakes
from(18a)June
“SuperStar”
(18(20
June
Fig. 3. Vessel
from (a)
SuperStar
2009), (b) Star
June2009,
2009).16:15),
b) “Star” (20 June 2009, 12:10).
from Star,
23 fromwake
SuperStar
and 14by
from
ExperimentalThe
set-up
averaged distributions of wave heights29within
the vessel
computed
theViking
up- XPRS.
Maximum wave heights (up to 1 m) occurred exclusively for
crossing method are demonstrated in Figs. 4 andthe
5. longest
The averaging
was
performed
over alltypical
66 wind
waves with
periods
∼ 10 s whereas
Tallinn Bay is a semi-sheltered bay in the almost tideless
wave periods were 2–3 s.
events (Fig.by
4)the
and
for each
has crossed the bay several times
Baltic Sea,recorded
which is characterized
relatively
mildspecific
wind shipInwhich
addition, maximum run-ups of all wake waves were
wave climate.
Thethe
peak
periods
wind waves are
usually
measureddistribution
at the coastgathering
manually,the
by data
following
during
time
of theofobservations
(Fig.
5). The average
from the
all inundawell below 3 s, reaching 4–6 s in strong storms and only in
tion of each wake wave and using video recording from 12
ships
might
be reasonable,
since all
studied
vessels
aretill
characterized
similar
lengths,between
widths,the echo
exceptional
cases
exceeding
7–8 s (Soomere,
2005b).
The
June
1 July 2009. by
Since
the distance
significant wave height in the bay exceeds 0.5 m with a probsounder wave
and coastal
measurements
was rather
draughts and operational speeds and induce similar
wakes,
and, therefore,
canshort
be(slightly
ability of 10 % (Soomere, 2005b).
more than 2 wavelengths), it was easy to follow the propagaIn contrast
to this low
wave activity,
Tallinn
Bayheight
reg- distribution.
considered
aswind
an averaged
wake
wave
It can bewave
seenupthat
waveshoaling
tion of each particular
to themost
coast.ofA wake
ularly hosts fast vessel traffic, with fairways located close
on the beach at the experiment site is shown in Fig. 2. On the
heightsand
arevessels
distributed
within
the interval
and the number of particularly high waves in
to the shoreline
operating
at cruise
speeds up<30
to cmcoast,
one can see a pole for the run-up measurements and
30 knots (Parnell
et al.,
et al.,example,
2009; Soomere
et
farther heights
offshore are
a tripod,
levelconstitute
oscillations were
the wake
is 2008;
ratherTorsvik
low. For
the waves
whose
largerwhere
than sea
40 cm
al., 2011). Fast vessel generated waves considerably exceed
recorded.
just 2%
the wave
heights
one wake.
fromofdifferent
vessels itwas
varies
from 1%with the
typical periods
andof
heights
of wind
waveswithin
(Soomere,
2005a; For wakes
The time
video-recording
synchronized
Parnell et al., 2008). The periods of the highest waves are of
time of the echo sounder, so it was possible to link the wave
for “Viking XPRS”, 2% for “Star” and 3% for “SuperStar”.
8–15 s and their heights are up to 2 m depending on location
height measured at the echo sounder and run-up at the coast
(Parnell et al., 2008; Kurennoy et al., 2009, 2011).
for each wave in the wake. In total, run-ups from 86 wakes
An experiment focusing on the properties of vessel wakes
were measured: 36 of them from Star, 31 from SuperStar and
6
and their impact on coasts was performed
at the Pikakari
19 from Viking XPRS.
beach, Tallinn Bay (Fig. 1) during the high traffic period
5
from 12 June till 1 July 2009. The environmental conditions
in the period of the experiment were
4 rather low, with the
3 Statistics of waves and corresponding run-ups in
wind wave background mostly of about 10 cm and reaching
3
the wake
40 cm during three days. The vessels of our interest were the
high-speed ferries Star, SuperStar and Viking XPRS, follow2
The data set at Pikakari beach, Tallinn Bay, allows for a deing routes Helsinki–Tallinn. All three vessels are large contailed study of the statistics of vessel-induced waves and corventional ships with lengths of about1200 m and operational
speeds of 30 knots, which are capable to generate largeresponding run-ups. The most energetic vessel waves at this
0 in Tallinn bay. Howhave
of about 1 m and periods of 8–10 s
amplitude, long and long-crested waves
0
0.1
0.2
0.3 location
0.4
0.5 amplitudes
0.6
0.7
with
maximum
run-up
heights
up to 1.4 m.
ever, due to the variability in ship track and its speed along
Wave height, m
the track, the parameters of ship generated waves may vary,
All measured wave wakes have a specific structure of freFig. values
4. Averaged
distribution
of wave
the wave
wake;packets,
red dashed
linethecorresponds
quency
modulated
where
largest and longest
their averaged
are described
in Kurennoy
et al.heights
(2009). within
Vessel generated
waves
were
recorded
using
a
downwardwaves
come
first
and
waves
of
smaller
amplitude
and period
to the Rayleigh distribution (  =0.12 m), black solid line corresponds to the Weibull distribution
after (Fig. 3). Sometimes the groups of different heights and
looking ultrasonic echo sounder (LOG aLevel®) with a sam(  =0.16onm,
q =1.53)
fitted
using the
maximum
estimate.
pling frequency of 5 Hz, mounted
a stable
tripod
approxperiods
can belikelihood
separated even
within one wake, as it is shown
in Fig. 3a, where three groups of waves can be clearly idenimately 100 m from the shoreline, 2.4 km from the sailing
tified. The highest waves (with a height of up to 1 m) usually
line at a water depth of about 2.7 m from 17 June till 1 July
belong to the first group and occur after the passage of sev2009, and were studied overall and with respect to the type of
6
the vessel. The record contains 66 clearly identifiable wakes:
eral long, high-amplitude waves.
pdf, m-1
2
www.nonlin-processes-geophys.net/20/179/2013/
Nonlin. Processes Geophys., 20, 179–188, 2013
f the wave heights within one wake. For wakes from different vessels it varies from 1%
ng XPRS”, 2% for “Star” and 3% for “SuperStar”.
182
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
wave fields, a generalization of Rayleigh distribution – twoparametric Weibull distribution is used. Since Rayleigh dis5
tribution fails to describe ship-generated waves, we follow the same scheme as for wind waves and apply two4
parametric Weibull distribution for a better fit. Both approximated distributions are plotted in Figs. 4 and 5. It can be
3
seen that Rayleigh pdf does not work well for description
2
of the waves in the vessel wake. The best observed fit is for
the Viking XPRS. However, Weibull pdf seems to be a fairly
1
good model describing distribution of wave heights within a
single vessel wake. Even though it underestimates the height
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
of the distribution for waves of small (∼ 10 cm) amplitude,
Wave height, m
it describes well the distribution of all larger wave heights
veraged distribution
of wave
heights within
wavewithin
wake;theredwave
dashed>line
corresponds
Fig. 4. Averaged
distribution
of wavethe
heights
15 cm.
The mentioned discrepancy for small amplitude
wake; red dashed line corresponds to the Rayleigh distribution
waves can be related to the existence of wind wave backyleigh distribution
 =0.12
m), black
solid lineto corresponds
to the Weibull distribution
(σ = 0.12(m),
black solid
line corresponds
the Weibull distribuground, whose amplitudes were < 15 cm during most of the
tion m,
(σ =q0.16
m, qfitted
= 1.53)
fittedthe
using
the maximum
likelihood
time of the experiment except a few days.
(  =0.16
=1.53)
using
maximum
likelihood
estimate.
estimate.
Approximated scale parameter σ remains the same for
overall wave distribution (Fig. 4) and for distributions found
for particular vessel types (Fig. 5) and is of the same order
The averaged distributions6of wave heights within the
for both Rayleigh and Weibull pdfs. It changes from 0.1 m till
vessel wake, computed by the up-crossing method, are
0.13 m for Rayleigh distribution and from 0.15 m to 0.16 m
demonstrated in Figs. 4 and 5. The averaging was performed
for Weibull distribution.
over all 66 recorded events (Fig. 4) and for each specific
The shape parameter q in Weibull distribution varies more
ship that crossed the bay several times during the time of
significantly. It is equal to 1.53 for overall distribution and
the observations (Fig. 5). The average distribution gatherreaches the value of 1.45 for Star, 1.52 for SuperStar and
ing the data from all ships might be reasonable, since all
1.71 for Viking XPRS vessels. This variety can be explained
studied vessels are characterized by similar lengths, widths,
if we consider a physical meaning of the parameter q comdraughts and operational speeds and induce similar wave
paring Weibull distribution with Glukhovsky distribution obwakes, and, therefore, can be considered as an averaged wake
tained for description of sea wave statistics at intermediwave height distribution. It can be seen that most of wave
ate water depth (Glukhovsky, 1966). Glukhovsky distribuheights are distributed within the interval < 30 cm and the
tion has been obtained semi-empirically for wind waves in
number of particularly high waves in the wake is rather low.
the coastal zone of the Caspian Sea by including an addiFor example, the waves whose heights are larger than 40 cm
tional parameter, the water depth. Since both Weibull and
constitute just 2 % of the wave heights within one wake. For
Glukhovsky distributions represent a generalization of the
wakes from different vessels it varies from 1 % for Viking
Rayleigh distribution, we use this analogy and apply it to
XPRS, 2 % for Star and 3 % for SuperStar.
ship-generated waves in shallow water. For this purpose it
The obtained distributions of wave heights H within a
is more convenient to consider cumulative distribution funcwave wake are approximated by the mostly used probability
tions (cdf). For Weibull distributions it has the following
density functions (pdf) in wave statistics: the one-parametric
form
Rayleigh pdf
q "
#
H
H
H2
.
(3)
FW (H, σ, q) = 1 − exp −
fR (H, σ ) = 2 exp − 2 ,
(1)
σ
σ
2σ
The Glukhovsky cdf is described by the following expression
described by the only one scale parameter σ , measured in m,
(Glukhovsky, 1966)
and two-parametric Weibull pdf


q 2
1−H̄ / h
q H q−1
H
H
π
,
fW (H, σ, q) =
exp −
,
(2)
FG H, H̄, h = 1 − exp − σ σ
σ
H̄
4 1 + √H̄
pdf, m-1
6
2π h
described by a scale parameter σ and a shape parameter q, using a nonlinear least squares method. Rayleigh
distribution describes amplitudes of narrow-band Gaussian
processes. However, sea wave measurements often deviate
from Rayleigh. This is why, in oceanology, e.g. for wind
Nonlin. Processes Geophys., 20, 179–188, 2013
(4)
where H̄ is the mean wave height and h is the water depth.
Comparing Eqs. (3) and (4) we find a connection between parameter σ and q in Eq. (3) and characteristics of wave field:
www.nonlin-processes-geophys.net/20/179/2013/
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
6
183
a
pdf, m-1
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wave height, m
pdf, m-1
6
6
b
5
5
4
4
3
3
2
2
1
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0
c
0.1
0.2
0.3
0.4
0.5
Wave height, m
Wave height, m
Fig. 5. Averaged distribution of wave heights within the wave wake of a) “Star”, b) “SuperStar”
Fig. 5. Averaged distribution of wave heights within the wave wake of (a) Star, (b) SuperStar and (c) Viking XPRS; red dashed line corresponds to the fitted Rayleigh
distribution
with corresponding
parameter
σ (a)to
0.12
(b) 0.13
m, (c) distribution
0.11 m; blackwith
solid line corresponds to
and c) “Viking
XPRS”;
red dashed line
corresponds
them,
fitted
Rayleigh
the fitted Weibull distribution with corresponding parameters (σ , q) (a) (0.15 m, 1.45), (b) (0.16 m, 1.52), (c) (0.15 m, 1.71) fitted using the
corresponding
parameter  a) 0.12 m, b) 0.13 m, c) 0.11 m; black solid line corresponds to the
maximum likelihood
estimate.
fitted Weibull distribution with corresponding parameters (  , q ) a) (0.15 m, 1.45), b) (0.16 m,
1.52), c) (0.15 m, 1.71) fitted using the maximum
likelihood estimate.
2.5
pdf, m-1
2
3/2 ! 1−H̄2 / h
H̄
4
2 distributions
2
The obtained
of wave heights H within a wave wake are approximated by
, σ = H̄
+
.
(5)
q=
π
h density functions (pdf) 1.5
1 − H̄ / h the mostly πused probability
in wave statistics: the one-parametric
pdf, m-1
Rayleigh pdf
1
Taking into account the water depth at the location of conducted measurements (h = 2.7 m) and applying the mean
0.5
wave height (H̄ = 0.14 m), parameters of the Weibull dis-H
 H2 
H ,which
(1)
   2 exp  2  ,
tribution can be estimated as σ ≈ 0.16 m and q ≈f R2.1,

200 

0.2
0.4
0.6
0.8
1
1.2
1.4
are close to the values calculated from the original data.
Runup
height,
m
The averaged distributions of corresponding run-up
heights on a beach, caused by a single wave wake are shown
Fig. 6.
Averaged
distribution ofWeibull
run-up heights
within the wave
by the only one scale parameter
, measured
in m,
and two-parametric
pdf the
Averaged
heights
wave
wake; red dashed
in Figs. 6 and 7described
for overall
run-up distribution andFig.
with6.rewake;distribution
red dashed of
linerun-up
corresponds
to within
the Rayleigh
distribution
spect to vessels of different types. Similar to wave heights,
(σ = 0.38 m),
black solid (line
corresponds
to thesolid
Weibull
distribucorresponds to the Rayleigh
distribution
 =0.38
m), black
line
corresponds to th
these distributions can also be approximated by Rayleigh and q tion
(σ = 0.56q m, q = 2.77) fitted using the maximum likelihood
1


q  H  estimate.
H 
Weibull pdfs, which are marked by the corresponding
(  =0.56
maximum likelihood estimat
lines
  m, , q =2.77) fitted using the(2)
fW H , , q distribution
  exp
  
in Figs. 6 and 7, and it can be seen that the Weibull distribu    
tion works quite well even for description of run-up heights.
2.5
all distribution,
2.75 for Star, 2.93 for SuperStar and 2.70
a for
However, the shape parameters are more diversed for run-up 7
Viking XPRS vessels.
heights. For the Rayleigh distribution, σ = 0.38 m for over2
Knowing parameters of distributions, we can calculate the
all distribution and for Star, 0.40 m for SuperStar and 0.33 m
corresponding
mean wave and run-up heights straight from
1.5
for Viking XPRS vessels. Similar to wave heights, scale pathe distribution:
rameter in Weibull distribution for run-ups is slightly larger
1
Z∞
than the one for the Rayleigh pdf: σ = 0.56 m for overall distribution and for Star, 0.59 m for SuperStar and 0.49 m for
H̄ = Hf (H ) dH ,
(6)
0.5
Viking XPRS vessels. The shape parameter q = 2.77 for over0
0
0
www.nonlin-processes-geophys.net/20/179/2013/
2.5
0.2
0.4
0.6
0.8
1
1.2
1.4
RunupGeophys.,
height, m 20, 179–188, 2013
Nonlin. Processes
b
3
Fig. 6. Averaged distribution of run-up heights within the wave wake; red dashed line
corresponds to the Rayleigh distribution (  =0.38 m), black solid line corresponds to the Weibull
184
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
distribution (  =0.56 m, q =2.77) fitted using the maximum likelihood estimate.
2.5
a
pdf, m-1
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Runup height, m
2.5
3
2
2.5
Knowing parameters of distributions, we can calculate the correspondingbmean wave and
pdf, m-1
up heights straight from the distribution:
c
2
1.5
1.5
1

H   Hf H dH ,
0.5
0
0
1
0.2
0.4
(6)
0.5
0
0.6
0.8
1
1.2
1.4
0
0
0.2
0.4
0.6
0.8
1
Runup height, m
Runup height, m
consider an average wave amplification on a beach. The calculated values of mean wave and
Fig. 7. Averaged distribution of run-up heights within the wave wake of a) “Star”, b)
Fig. 7. Averaged
distribution
run-up
within the
wave
of (a)Rayleigh
Star, (b) SuperStar and (c) Viking XPRS; red dashed line
up heights, calculated
from the
originalofdata
andheights
reconstructed
from
thewake
obtained
“SuperStar”
and c) “Viking
XPRS”; redparameters
dashed line
corresponds
to 0.40
the Rayleigh
distribution
corresponds to the Rayleigh
distribution
with corresponding
σ (a)
0.38 m, (b)
m, (c) 0.33
m; black solid line corresponds
Weibull distributions
for distribution
overall statistics
and for each
vessel separately
in (0.59 m, 2.93), (c) (0.49 m, 2.70) fitted using the
to the Weibull
with corresponding
parameters
(σ , q) (a) are
(0.56presented
m, 2.75), (b)
with corresponding parameters  a) 0.38 m, b) 0.40 m, c) 0.33 m; black solid line corresponds
maximum likelihood estimate.
8.
to the Weibull distribution with corresponding parameters (  , q ) a) (0.56 m, 2.75), b) (0.59 m,
2.93), c) (0.49 m, 2.70) fitted using thebution
maximum
likelihood
are so
close toestimate.
each other that they practically coinRunup height, m
0.55
0.5
0.45
0.4
0.13
0.14
0.15
Wave height, m
0.16
0.17
cide, which strongly indicates that Weibull distribution is an
10 appropriate model for description of waves within a vessel
wake. The heights reconstructed from Rayleigh distributions
underestimate run-up observations by 10–20 %. Based on the
calculated data, the corresponding regression curves of wave
amplification at the coast can be plotted (Fig. 8). Though regression lines, based on four points only, are not satisfactory,
they can still indicate some trend. In the case of the Rayleigh
and Weibull distributions they can be described by the following expressions
= 3.1
H̄,
8. Mean run-up and wave height in the wake, calculated from the original dataR̄(red
squares)
(7)
Fig. 8. Mean run-up and wave height in the wake, calculated from
the from
original
data (red
squares)
and
reconstructed
fromtriangles)
Rayleigh distributions; blue
d reconstructed
Rayleigh
(blue
circles)
and
Weibull (black
R̄ = 3.5H̄.
(8)
(blue circles) and Weibull (black triangles) distributions; blue solid
olid and black
dashed lines corresponds to the regression curve fitted the heights calculated
and black dashed lines correspond to the regression curve fitted
heights
calculated
from the
Weibull distributions,
reAs it has been noticed before, the Rayleigh approximation
from
the Rayleigh
andRayleigh
Weibulland
distributions
respectively.
spectively.
gives smaller amplification at the coast and underestimates
run-up heights.
It is remarkable that original values of mean wave and run-up heightsFor
andcomparison,
those
we estimate the amplification of nonbreaking
sinusoidal
waves
with a period T = 10 s, which corand
consider
an
average
wave
amplification
on
a
beach.
The
nstructed from the Weibull distribution are so close to each other that practically coinside,
responds to the period of the largest vessel waves, by the
calculated values of mean wave and run-up heights, calcuch strongly indicates that Weibull distribution is an appropriate model for shallow
description
offormula for long wave run-up on a plane beach
water
lated from the original data and reconstructed from the obes within tained
a vessel
wake.andThe
heights
reconstructed
fromstatisRayleigh(Didenkulova
distributionset al., 2007)
Rayleigh
Weibull
distributions
for overall
s
tics
and
for
each
vessel
separately
are
presented
in
Fig.
8.
erestimate run-up observations by 10-20%. Based on the calculated data, the corresponding
R
2L
It is remarkable that original values of mean wave and run=π √
≈ 6.
(9)
ession curves
wave amplification
at the coastfrom
can be
8). Though
upofheights
and those reconstructed
theplotted
Weibull(Fig.
distriH regression
ghT
s based on four points only are not satisfactory, still they can indicate some trend. In the case
Nonlin.
Processes
Geophys.,
20, 179–188,
2013
he Rayleigh and
Weibull
distributions
they can
be described
by the following expressionswww.nonlin-processes-geophys.net/20/179/2013/
wave height. This is clearly seen in Figs. 9b and 9c, where the absolute maximum wave heights
are not so pronounced as in Fig. 9a and are just slightly larger than for other waves.
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
185
Wave/runup heights, m
1
a
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
Time, min
Wave/runup heights, m
0.8
1.2
c
b
1
0.6
0.8
0.4
0.6
0.4
0.2
0.2
0
0
2
4
6
8
10
12
0
0
1
2
Time, min
3
4
5
6
7
8
9
Time, min
Fig. 9. Wave heights (blue circles) and corresponding run-up heights (red triangles) within the
Fig. 9. Wave heights (blue circles) and corresponding run-up heights (red triangles) within the wake from (a) SuperStar (19 June 2009),
(b) Star (20 June 2009)
(c)a)Viking
XPRS (21
2009).
wake and
from
“SuperStar”
(19June
June
2009, 19:15), b) “Star” (20 June 2009, 12:10) and c) “Viking
XPRS” (21 June 2009, 13:50).
4
Run-up of individual waves in the vessel wake
At the same time Fig. 9a with an outstanding peak for the highest wave demonstrates
In order to better understand the details of individual wave
numerous waves of much smaller height, whichrun-up
result within
in an the
almost
thewesame
(justthe
slightly
wake,
consider
run-up of three packetsdifferent
corresponding
three
different
smaller) run-up. This can be partially explained by
periodstoof
wake
waves,types
whichofisvessels. As it has
been mentioned before, the time at the echo sounder and at
reflected in the time-frequency spectrum in Fig. 10.video recording was synchronized, making possible to identifydifferent
the run-up
of most
of waves
in in
theFig.
wake,
The approach of three different groups with
periods
is clearly
seen
10.except waves of
very small amplitude (< 10 cm at the echo sounder record).
The first group of waves with peak period at 8 sec appears at the time moment of 1 min. Then, 3
For this analysis we select a wave wake from SuperStar,
minutes after the second group with peak period which
5 sec approaches.
finally,
afterat 319:15;
more from Star, ococcurred onAnd
19 June
2009
minutes the third group of waves with 4 sec periodscurred
comes.on 20 June 2009 at 12:10; and from Viking XPRS, occurred on 21 June 2009 at 13:50. Corresponding wave and
So, the structure of the wave packet is such that waves of longer period come first. At the
run-up heights for all of them are shown in Fig. 9.
same time the run-up height increases with decreaseItincan
thebewave
period from
[see Eq.
andthere
this is a fairly good
concluded
Fig.(9)]
9 that
correlation
between
wave
run-up waves
data. Increase in the
may explain, why waves of the second group, which
have smaller
periods
thanand
the largest
wave height generally corresponds to the increase in the runof the first group, produce the same run-up (see, time
min in the
Fig.maximum
9a).
up interval
height. 4-8
However,
run-up in the wake does
not always correspond to the maximum wave height. This is
13
clearly seen in Fig. 9b and c, where the absolute maximum
Fig. 10. Time-frequency spectrum for SuperStar (19 June 2009).
wave heights are not so pronounced as in Fig. 9a and are just
slightly larger than for other waves.
At the same time Fig. 9a, with an outstanding peak for
Here L is the distance to the shore (in our case L = 100 m),
the highest wave, demonstrates numerous waves of much
g is the gravity acceleration and h is the water depth (h =
smaller height, which result in almost the same (just slightly
2.7 m).
smaller) run-up. This can be partially explained by different
Thus, averaged amplification of vessel waves is almost
periods of wake waves, which is reflected in the timetwice as smaller than the one predicted by Eq. (9). This can
frequency spectrum in Fig. 10.
be explained by strong dissipation during wave breaking, as
The approach of three different groups with different perivery often vessel wakes break before reaching the coast (see
ods is clearly seen in Fig. 10. The first group of waves, with
Fig. 2).
peak period at 8 s, appears at the time moment of 1 min.
www.nonlin-processes-geophys.net/20/179/2013/
Nonlin. Processes Geophys., 20, 179–188, 2013
closer to the “breaking” curve, while smaller-amplitude waves are closer to the “non-breaking”
line, what corresponds to our expectations.
186
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
a
Amplification
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wave height, m
8
Amplification
Amplification
6
4
6
4
2
2
0
0
c
b
8
0.1
0.2
0.3
0.4
0.5
Wave height, m
0.6
0.7
0
0
0.1
0.2
0.3
0.4
Wave height, m
Fig. 11.onWave
amplification
on the wave
beachheight
plotted
wave height
forSuperStar
wake waves
Fig. 11. Wave amplification
the beach
plotted against
foragainst
wake waves
from (a)
(19from
June a)
2009), (b) Star (20 June
2009) and (c) Viking XPRS
(21
June
2009).
Dashed
lines
correspond
to
Eq.
(9)
and
dotted
lines
correspond
to
calculation
of breaking run-up
“SuperStar” (19 June 2009), b) “Star” (20 June 2009) and c) “Viking XPRS” (21 June 2009).
with the characteristic period of 6 (blue) and 14 s (black). Black solid line corresponds to the power approximation described by Eq. (10).
Dashed lines correspond to Eq. (9) and dotted lines correspond to calculation of breaking run-up
with the characteristic period of 6 (blue) and 14 (black) sec. Black solid line corresponds to the
Then, 3 min after, the second group with peak period
5s
up on(10).
a plane beach (Eq. 9), calculated for waves with peripoweratapproximation
approaches. And finally, after 3 more minutes, the third group
ods of 6 and 14 s. It can be seen that some smaller (probably,
of waves with 4 s periods comes.
non-breaking) waves with wave heights < 20 cm may have
So, the structure of the wave packet is such that waves of
the same amplification, but the majority of wake waves lie
longer period come
first.
At
the
same
time
the
run-up
height
below
threshold.
The data from all three vessels on Fig. 11 can be very
wellthis
described
using the power regression
increases with decrease in the wave period (see Eq. 9) and
The dotted lines in Fig. 11 represent the result of nucurve (see, solid line in Figs. 11)
merical simulation of wave amplification along a Pikakari
this may explain why waves of the second group, which have
beach profile. The numerical simulation is performed ussmaller periods than the largest waves of the first group, proing the CLAWPACK (Conservation Laws Package) packduce the same run-up (see, time interval 4–8 min in Fig. 9a).
This effect may also be a key to the understanding of rapid 15 age (LeVeque, 2004), which models shallow water equations
in the form of laws of conservation and allows formation
beach erosion caused by fast-vessel wakes. The group struc ture of the wake works
so that waves with the largest run-up
and propagation of breaking waves parameterized by the
heights come one after another resulting in the anomalous
shock waves. The propagation of such waves along a flat
bottom is described in Pelinovsky and Rodin (2011); and Diimpact on the coast.
denkulova et al. (2011). The calculated amplification curve
Another factor, which influences run-up height, is wave
decreases with an increase in the wave height, which is in a
breaking, which affects waves of large amplitudes reducing
good agreement with the behavior of the experimental data.
their run-up. Observations show that very often the largest
Waves of larger amplitude, which are more subject of wave
vessel induced waves break before they reach the coast (see,
breaking, are closer to the “breaking” curve, while smallerFig. 2). Waves of larger amplitude break farther from the
amplitude waves are closer to the “non-breaking” line, what
coast and dissipate stronger than smaller amplitude waves.
corresponds to our expectations.
This can be seen in Fig. 11 for wave amplification at the
The data from all three vessels in Fig. 11 can be very well
coast (R/H ), where waves of smaller amplitude amplify
described using the power regression curve (see, solid line in
stronger than the larger ones. For comparison, blue and black
dashed horizontal lines correspond to the non-breaking runNonlin. Processes Geophys., 20, 179–188, 2013
www.nonlin-processes-geophys.net/20/179/2013/
I. Didenkulova and A. Rodin: A typical wave wake from high-speed vessels
Figs. 11)
R
= aH −b ,
(10)
H
where the power coefficient varies within 0.75 ≤ b ≤ 0.95
depending on the type of the vessel and dimensional coefficient changes within the range 0.4 ≤ a ≤ 1.0. Obtained
experimentally, Eq. (10) can be used for estimates of run-up
heights from vessel generated waves.
5
Conclusions
The anomalous impact of waves from high-speed vessels on
coasts is considered with respect to their specific group structure. The intensive vessel-induced waves and their run-ups
are studied experimentally at the Pikakari beach in Tallinn
Bay, the Baltic Sea, in summer 2009. In total, 66 wave wakes
were recorded and 86 run-ups were measured from three different high-speed vessels: Star, SuperStar and Viking XPRS.
It is shown that Weibull distribution can be a good model
describing the distribution of wave heights in a single wake
and corresponding run-up heights. This works for the overall distribution, as for wakes from different vessels. Both
parameters of the Weibull distribution for wave heights can
also be estimated from the Glukhovsky distribution for wind
waves in shallow or intermediate water (Glukhovsky, 1966)
and these estimates are in a good agreement with measured
data. The mean wave and run-up heights, reconstructed from
the Weibull distributions, demonstrate a great coincidence
with the measurements. There is also a correspondence between mean wave and run-up heights showing that increase
in one results in the increases in the other. It is found that, on
average, vessel wave heights are amplified 3.5 times during
100 m of propagation to the coast.
The run-up of three particular wake waves, which correspond to three different vessels, is studied in detail. It is
demonstrated that group structure of the wake, where the
largest and the longest waves come first and waves of smaller
amplitude and period come after, works so that waves with
largest run-up heights come one after another resulting in the
anomalous impact on the coast. This effect may be a key to
understanding of the rapid beach erosion caused by fast vessel wakes.
Wave amplification varies significantly within a wake.
Waves of large amplitude have very little amplification,
which is explained by the influence of the wave breaking.
At the same time small amplitude (< 20 cm) waves amplify
significantly at the coast and can be described by the run-up
of long non-breaking waves on a plane beach in the shallow
water framework. Waves of moderate amplitude represent an
intermediate case where wave breaking effects are still important, but they are influenced by wave dispersion and dissipation in the near bottom layer. For the description of these
waves an empirical formula based on experimental data is
provided.
www.nonlin-processes-geophys.net/20/179/2013/
187
Acknowledgements. This research was partially supported by
the Federal Targeted Program “Research and educational personnel of innovation Russia” for 2009-2013, targeted financing
by the Estonian Ministry of Education and Research (grant
SF0140007s11), Estonian Science Foundation (Grant 8870), grants
MK-1440.2012.5 and RFBR-12-05-33087. ID acknowledges the
support provided by the Alexander von Humboldt Foundation and
AR thanks DoRA 4 program. Authors thank Kevin Parnell and the
team of Institute of Cybernetics (Tallinn, Estonia) for their help in
organizing the experiment.
Edited by: E. Pelinovsky
Reviewed by: two anonymous referees
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