UNIFOB - Uni Computing

BCCS
TECHNICAL REPORT SERIES
Numerical studies of internal solitary wave
trains generated at edges in the topography.
Berntsen, J., Mathisen, J.-P. and Furnes, G.
REPORT No. 21 April 13, 2007
Report on Contract C06127 between NDP (Norwegian Deepwater
Programme) and Fugro OCEANOR AS.
(Subcontract to BCCS)
UNIFOB
the University of Bergen research company
BERGEN, NORWAY

BCCS Technical Report Series is available at http://www.bccs.no/publications/
Requests for paper copies of this report can be sent to:
Bergen Center for Computational Science, Høyteknologisenteret,
Thormøhlensgate 55, N-5008 Bergen, Norway

1 Executive summary
In this report some results from numerical studies of internal wave trains are given. The
parameters of the experiments are chosen to be relevant for the Ormen Lange area. Many of
the characteristics of the measured wave events are reproduced. However, the wave periods
are still too short. In the model results the periods between consecutive waves in a wave
train are typically 20 min whereas they are 45 min in the measurements we have focused
on. A theoretical explanation for this discrepancy is given. Basically the periods between
consecutive waves in a wave train depend on the distance from the generation point, and
it would require more computer resources to follow the waves far enough to reproduce the
45 min periods in numerical experiments. The amplitudes of the waves are reduced with
distance frome the shelf edge. This means that closer to the generation point, the wave
amplitudes and corresponding velocities may be significantly larger than those measured.

2 Description of the numerical experiments
The σ-coordinate ocean model applied in the present studies is a two dimensional, (x,z),
version of the model described in [Berntsen(2000)] where x and z are the horizontal and vertical
Cartesian coordinates respectively. The model is available from www.math.uib.no/BOM/.
The variables are discretized on a C-grid. In the vertical, the standard σ-transformation,
σ = z−η
H+η , where η is the surface elevation, and H the bottom depth, is applied. For advection
of momentum and density a Total Variance Diminishing (TVD)-scheme with a
superbee limiter described in [Yang and Przekwas(1992)] is applied in the present studies.
The standard second order Princeton Ocean Model (POM) method is applied to estimate
the internal pressure gradients ([Blumberg and Mellor(1987), Mellor(1996)]). The model
is mode split with a method similar to the splitting described in
[Berntsen et al.(1981)Berntsen, Kowalik, Sælid, and Sørli] and [Kowalik and Murty(1993)].
Even if the model is two dimensional, flow in the across domain direction is allowed. However,
there will be no across domain variability in this flow. The effects of the earths rotation
are taken into account in the present studies and the Coriolis parameter is set to f =
1.2 × 10−4 s−1 .
It is known that internal waves may be formed behind topographic features in a stratified
ocean, see for instance [Gill(1982), Baines(1995), Kundu and Cohen(2004), Thorpe(2005)].
This is also studied recently in [Roth(2000)] and [Lamb(2007)]. In these papers it is shown
that dispersive wave trains may be generated at a shelf edge in tidally forced systems. Inspired
by the findings in [Roth(2000)] and [Lamb(2007)], a two-dimensional cross shelf
model system has been set up with parameters of topography, forcing, and stratification
that are relevant for the Ormen Lange area. The model area is 18000 m long. At x = 0 m a
flow is forced into the system with a periodic forcing u = U0 sin(ω f t) where u is the velocity
component in x direction, U0 the amplitude of the forced velocity. Generally the tidal
signal is weak near the shelf edge in the Ormen Lange area. The wave trains were observed
during periods with relatively strong winds. The wave trains are therefore assumed to be
driven by inertial oscillations. The inertial frequency ω f is approximately 0.00013s−1 at
our latitude, giving a period of 13.42 hours.
In one set of experiments the focus is on solitary wave propagation outside the shelf,
but generated at the shelf edge. In these experiments the depth profile H(x) in m, see Figure
1, is specified according to
−280 , x < 1000
H(x) =
120
−400+
, x > 1000.
1+((x−1000)/W slope)∗∗4
The slope width Wslope is taken to be 500 m in the present experiments. The studies of
wave propagation off the shelf edge are performed with U0 equal to 0.3 m s −1 and 0.6 m s −1 .
In another set of experiments the focus is on solitary wave propagation on the shelf.
The depth profile in these studies H(x) in m, see Figure 11, is specified according to
−280 , x > 1000
H(x) =
120
−400+
, x < 1000,
1+((x−1000)/W slope)∗∗4
where again the sill width is taken to be 500 m. The studies of wave propagation on the
shelf are performed with U0 equal to 0.3 m s −1 and 0.4 m s −1 .
Based on the measurements provided from this area an idealised initial stratification is
assumed. The density at the surface is 1025.7 kg m 3 , the density at 50 m depth is 1027.3 kg m 3 ,
and at 400 m depth the density is taken to be 1027.6 kg m 3 . Linear interpolation is used to
compute the full density profile. With this density profile and theory about internal wave
propagation the wave speed may be estimated to be in the range from 0.63 to 0.88 m s −1 .
The model grid resolution is 25 m horizontally and 100 equidistant layers are used
vertically. The model is run for two inertial periods. The results from the four experiments
are summarised in the following sections.
2

3 Solitary wave propagation off the shelf (U0 = 0.3 m s −1 )
Below are some results for the case with solitary waves off the shelf, using a maximum
inflow velocity of 0.3 m s −1 . A section of the density field after 11.74 hours and the depth
profile is given in Figure 1. To get a better impression of the wave propagation, the density
fields and two components of the velocity field at three consecutive times are given in
Figures 2, 3, and 4. In these figures the focus is on the internal waves. At time equal to
11.74 hours only one wave of suppression is seen in Figure 2. This wave is also seen after
13.42 hours and 15.09 hours. However, at these later times a group of solitary waves is seen
trailing the leading wave, In Figure 5 the u components at x = 5000 m and x = 15000 m are
plotted as functions of time and depth, again with focus on the passing train of solitary
waves.
The speed of the front of the wave train in the period from 13.42 hours to 15.09 hours is
approximately 0.56 m s −1 . The wave train changes form, and develops the typical solitary
wave train characteristics as it propagates away from the edge. From Figure 5 the period
between consecutive waves in the group may be estimated to be 24 minutes. It is also
clearly seen that the first wave in the train is strongest, and the amplitudes of the waves
behind are gradually reduced.
depth [m]
0
−50
−100
−150
−200
−250
−300
−350
25.8 25.9 25.8 25.9 26 25.8 25.9 26
26
26.1 26.1 26.1
27.5
26.7
27.2
26.2
26.3
26.5
26.8
26.6
27.3
27.4
26.9
26.5 26.3 26.2
26.726.6
26.9 26.8 27
27.1 27.2 27.1
27.3
27.1
26.4
RHO (ci=0.1 )
27
27.5
−400
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
26.4
distance [m]
(a)
26.3 26.2
26.726.6
26.5
27.227.3
27.427.4
Figure 1: Topography and density field after 11.74 hours (7/8 of an inertial periods).
3
27.5
26.8
26.9
26.4
27

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
−40
27.2
−50
−60
−70
−80
−90
26.8
26.9
RHO (ci=0.1 )
−100
4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−30
−20
U (ci=10 cm/s )
−30
distance [m]
(a)
−20
−100
4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
0
W (ci=1 cm/s )
distance [m]
(b)
2
−100
4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000
distance [m]
(c)
Figure 2: The train of solitary waves after 11.74 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
4
0
01
−1
0
−20
−1
−1
−30
−20
0

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
26.6
−40
−50
−60
−70
−80
−90
26.9
25.8
25.9
26
RHO (ci=0.1 )
26.5
26.7
26.326.4
27.3
−100
2000 3000 4000 5000 6000 7000 8000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−10
0
0
−10
0
0
U (ci=10 cm/s )
0
0
0
0
0
0
distance [m]
(a)
0
0
0
26.8
27 27.1
−100
2000 3000 4000 5000 6000 7000 8000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
0
0
0
0
0
0
0
−10
0
0
0 −1
−1
0 −1
0
1
0
2 1
0
W (ci=1 cm/s )
0
−1
0
0
−1
0
0
−2 −1
−1
0
2
−1
−2
−2
0
−1
0
0
1
−2
0
−1−1
0
−10
distance [m]
(b)
00
2
−1
−2
0
−2
−1
−2
−100
2000 3000 4000 5000 6000 7000 8000
0
0
distance [m]
(c)
Figure 3: The train of solitary waves after 13.42 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
5
−1
−10
0
0
0
−20
0
0
0
27.2
−10
26.1
26.2
0
0
0
0
0
0
−1
−1
26.6
−10
−1

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
−40
26.8
−50
−60
−70
−80
−90
RHO (ci=0.1 )
25.9
26
26.3 26.4
−100
5000 6000 7000 8000 9000 10000 11000 12000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
10
10
20
U (ci=10 cm/s )
26.1
26.2
distance [m]
(a)
25.8
26.526.6 26.7
27 27.1
20
20
−100
5000 6000 7000 8000 9000 10000 11000 12000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
0 0
0
0
0 0
0
1
0 0
0
0
0
1
1
1
0
0
0
0
27.2
distance [m]
(b)
0
0
0
−100
5000 6000 7000 8000 9000 10000 11000 12000
0
−1
−1
0 00 0
0
−1
0
−1
−1
−1
W (ci=1 cm/s )
0 00 0
−1
0
−1
0
0
−1
0
−1
2
0
0
−1
−2
0
−2
−1
0
−1
1
0
2
0
−2
−1−1
0
00
−2
−1
0
distance [m]
(c)
Figure 4: The train of solitary waves after 15.09 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
6
−1
−1
−1
0
0
26.9
10
26.8
27.3
1
20
0
0
0
−1
−1
10
−1
0
0

Depth [m]
Depth [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−100
11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−20
−30
−20
U (ci=10 cm/s )
20
30
U (ci=10 cm/s )
−10−10
−10
−20
20
20
−10
0
Time
(a)
40
30
−100
16 16.5 17 17.5 18 18.5 19 19.5 20
Time
(b)
Figure 5: Time development of the vertical profile of along section velocities at x = 5000 m
(top) and x = 15000 m with focus on the waves. (The time is in hours.)
7
30
30
0
0
30
10
10
20
20
20
20
20
30
10

4 Solitary wave propagation off the shelf (U0 = 0.6 m s −1 )
Below are some results for the case with solitary waves off the shelf, using a maximum
inflow velocity of 0.6 m s −1 . A section of the density field after 13.42 hours and the depth
profile is given in Figure 6. The solitary wave trains generally become much stronger when
the inflow velocity is increased. The density fields and two components of the velocity field
at three consecutive times are given in Figures 7, 8, and 9. In these figures the focus is on
the solitary wave train. In Figure 10 the u components at x = 5000 m and x = 15000 m are
plotted.
The speed of the front of the wave train in the period from 13.42 hour to 15.09 hours is
approximately 0.57 m s −1 and in the period from 15.09 hours to 16.78 hours approximately
1.09 m s −1 . The wave group also changes form, and develops the typical solitary wave train
characteristics as it propagates away from the edge. From Figure 10 the period between
consecutive waves in the group may be estimated to be 15 minutes. It is also clearly seen
that the first wave in the group is strongest, and the amplitudes of the waves behind are
gradually reduced. The speed in the firts wave exceeds 1.00 m s −1 .
depth [m]
0
−50
−100
−150
−200
−250
−300
−350
26
26.7
27
25.8
26.5
25.9
26.1
27.1
27.2
26.6
26.2 26.4
26.3
26.8
26.9
27.3
26.8 26.7
26.9
27
27.4
25.8
−400
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
distance [m]
(a)
25.8
27.3
26
25.9
26 25.9
26.6
26.3
27.3
26.3
27.2
27.1
26.1 26.1
26.2
26.4 26.4
26.526.5
27.5
RHO (ci=0.1 )
27.5
27.4
26.8 26.7
26.9 27
27.1
27.2
27.5
26.6
26.2
26.7
26.8
26.9
26.5
25.8
26.126.2
26.6
26
26.4
2727
Figure 6: Topography and density field after 13.42 hours (one inertial period).
8
27.4
27.5
27.1
27.2
27.3
27.5
27.4
26.3
26.9
26.8
27.1
27.2
27.3
27.5

depth [m]
depth [m]
depth [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
26.7
26.8
26.9
27.1
27
27.2
27.3
RHO (ci=0.1 )
27.4
26.5
26.6
26.1 26.2
26
25.8
26.326.4
26.8 26.7
−200
0 500 1000 1500 2000 2500 3000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
0
0
0
0
0
0
0
0
0
0
0
U (ci=10 cm/s )
10
20
0
0
0
10
0
26.9
27
distance [m]
(a)
0
30
20
0
−200
0 500 1000 1500 2000 2500 3000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
0
0
0
0
0
0
0
0
0
0
0
0
5
−10
0 0
0
−5
−20
0
−5
0
5
−10
−10 −10
10
5
W (ci=5 cm/s )
0
0
−20
0
0
0
−10
−5
−30
−20
−40
0
−10
0
5
10
−10
−20
−30
−10
0
−30
−40
−20
0
distance [m]
(b)
0
27.1
−60
−50
27.2
−10
0
27.3
0
−40
−30
−20
−200
0 500 1000 1500 2000 2500 3000
−5
−15
−10
−5
0
0
5
15
10
25
20
0
−5
0
−5
−20
−30
−25
−15
−10
−15
−5
distance [m]
(c)
Figure 7: The train of solitary waves after 13.42 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
−20
9
−15
−25
−10−10
−5
0
0
0
0
0
0 0
0
0
26.6
−50

depth [m]
depth [m]
depth [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
26.4
25.8
26.9
26.1
26.3
26.2
26.5
26.6
26.8
−200
2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
0
−20
−40
−60
−80
−100
40
−120
−140
−160
−180
30
27
RHO (ci=0.1 )
27.4
26.7
27.3
27.1
26.9
27.2
distance [m]
(a)
−200
2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
0
0
0
30
0
0
50
U (ci=10 cm/s )
0
0
0
0
0
0
0
0
0
0
W (ci=5 cm/s )
00
0
0
50
0
00
0
−5
60
20
0
0
5
−5
−5
−5
−10
30
40
60
20
50
distance [m]
(b)
−200
2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
0
0
0
0
5
0
0
20
10
40
−10
0
5
0
27
30
40
30
10
26.4
50
70
60
20
distance [m]
(c)
10
20
10
0
0
30
−10
0
25
−15−5
−10 −15
Figure 8: The train of solitary waves after 15.09 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
0
−5
−5
−10
−10
10
−20
−15
−5
−15
−10
0
−5
0
15
0
−10
−20
−25
0
−5
−15
−20
25.9
26
26.3
−20
10
20
0
−5
−5
10
0 0
10
5
5
8070
60
40
25.8
26.2
26.5
27.1
26.7
26.1
26.6
26.8
10
0
20
20
0
30
27.3
20
50
0
−25
0
20
15
25
−5
5
−5
−30
−15
−20
−5
−35
−10
30
40
−10
0
−20
−25
−10
−15
−10
−5
0
0
27.2
27.4
0
26.9
−10
0
0
0
0
0
27
0
0

depth [m]
depth [m]
depth [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
25.8
26
26.1
26.5
26.2
26.3
26.4
26.8
26.6
26.7
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
x 10 4
−200
distance [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
27.4
RHO (ci=0.1 )
26.9
27.1
27
27.2
27.3
(a)
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
x 10 4
−200
distance [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
40
0
40
0
0
40
0
0
−5
40
70
50
U (ci=10 cm/s )
0
0
5
0
0
W (ci=5 cm/s )
−5
60
40
−5
0
0
0
70
60
10
30
80
0
5
0
5
−5
−10
0
50
−5
40
−10
−5
40
(b)
0
0
0
70
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
x 10 4
−200
distance [m]
(c)
Figure 9: The train of solitary waves after 16.78 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
5
60
10
11
100
30
90
0
80
50
−5
0
5
−10
0
−5
40
−15
−10
0
0
−5
0
50
10
70
40
100
30
80
26.1
26.9 27
27.1
90
0
0
5
5
−5
−15
0
60
−5
−10
−10
25.9
26.5
−5
−5
0
26
27.2
27.3
40
0
5
26.2
50
26.4
26.8
40
26.3
26.6
26.7
15
27.4
100
80
30
110
70
90
0
10
0
5
−5
−10
−5
−15
60
−10
−5
−5
25.8

Depth [m]
Depth [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
20
10
−200
14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
40
−20
−10
U (ci=10 cm/s )
50
−30
80
100
40
30
−40
70
110
90
60
40
80
0
30
70
100
90
U (ci=10 cm/s )
60
50
70
90
80
60
30
−30
−20
30
70
80
60
−10
90
−10
30
0
Time
(a)
50
30
20
30
70
40 40
10
20
10
0
20
50
60
30
0
40
40
−200
17 17.5 18 18.5 19 19.5 20
20
Time
(b)
Figure 10: Time development of the vertical profile of along section velocities at x = 5000 m
(top) and x = 15000 m with focus on the waves.
12
50
70
80
20
60
60
80
50
30
40
7060
10
20 20
30
20
40
20
40
80
50
7060
10
50
20
40
30
50
60
70
30
40
50
70
30
0
0

5 Solitary wave propagation on the shelf (U0 = 0.3 m s −1 )
Below are some results for the case with solitary waves on the shelf, using a maximum
inflow velocity of 0.3 m s −1 . A section of the density field after 11.74 hours and the depth
profile is given in Figure 11. The density fields and two components of the velocity field at
three consecutive times are given in Figures 12, 13, and 14. In Figure 15 the u components
at x = 10000 m and x = 15000 m are plotted.
The speed of the front of the wave train in the period from 11.74 hour to 13.42 hours is
approximately 0.53 m s −1 and in the period from 13.42 hours to 15.09 hours approximately
0.68 m s −1 . The wave group also changes form, and develops the typical solitary wave train
characteristics as it propagates away from the edge. From Figure 15 the period between
consecutive waves in the group may be estimated to be 24 minutes.
depth [m]
0
−50
−100
−150
−200
−250
−300
−350
27.2
27.3
27.4
26.2
26.3
26
25.9
26.5 26.7
27.127.1
26
25.8 25.9 25.8 25.9 25.8
26.3 26.1
26.5
26.7
26.7
26.9
26.8
27.1 26.9
26.9
27.227.2
27.3
26.126.1
26.6
27.5
26.4
26.8
27.4
27.3
27
RHO (ci=0.1 )
27.5
26.2
26.3
26.5
26.6
−400
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
distance [m]
(a)
26.2
26
26.6
26.4 26.4
Figure 11: Topography and density field after 11.74 hours (7/8 of an inertial periods).
13
27.4
26.8 27
27.3
27.5
27.4
27

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
RHO (ci=0.1 )
27
27.1
−100
6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−10
distance [m]
(a)
−100
6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
−30
0
0
0
−20
U (ci=10 cm/s )
0
0
0
00
U (ci=2 cm/s )
−10
0
0
0
−30
−20
0
0
−10
distance [m]
(b)
2
0
−100
6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000
0
distance [m]
(c)
Figure 12: The train of solitary waves after 11.74 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
0
14
0
−2
0
−2
−30
26.3
−2
25.9
−30
0
27.2
26.5
0
26.4
0

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
26.8
26.1
26.2
RHO (ci=0.1 )
26.6
26.7
25.825.9
27.2
−100
8000 8500 9000 9500 10000 10500 11000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
U (ci=10 cm/s )
distance [m]
(a)
0
0
26.5
27.1
26.927
−100
8000 8500 9000 9500 10000 10500 11000
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0 0
0
0
0
0
0
0
0
0
U (ci=2 cm/s )
−2
0
0
0
distance [m]
(b)
0
0
0
−2
−100
8000 8500 9000 9500 10000 10500 11000
distance [m]
(c)
Figure 13: The train of solitary waves after 13.42 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
−2
15
−2
0
0
2
20
4
30
0
0
0
−4
−2
10
−2
−4
−2
0
0
0
0
0

depth [m]
depth [m]
depth [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
25.8
26.3 26.4
27.1
27
26.6
26.9
RHO (ci=0.1 )
26.5
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7
x 10 4
−100
distance [m]
0
−10
−20
−30
−40
−50
30
−60
−70
−80
−90
(a)
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7
x 10 4
−100
distance [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
0
0
50
U (ci=10 cm/s )
0
0
0
00
2
U (ci=2 cm/s )
0 0
0
−2
50
−2
−2
0
0
(b)
0
40
1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7
x 10 4
−100
distance [m]
(c)
Figure 14: The train of solitary waves after 15.09 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
16
30
4
26
0 0
60
0
2
−2
50
−2
−4
−4
−2
0 0
0
26.1
26.2
0
0
0
30
0
0
27.3
00
25.9
0
30
0
0
0
00
0

Depth [m]
Depth [m]
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
10
30
40
0
20
−100
13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 15
0
−10
−20
−30
−40
−50
−60
−70
−80
−90
0
−10
30
U (ci=10 cm/s )
U (ci=10 cm/s )
0
0
40
0
50
0
70
60
Time
(a)
−100
14.5 15 15.5 16 16.5
Time
(b)
Figure 15: Time development of the vertical profile of along section velocities at x =
10000 m (top) and x = 15000 m with focus on the waves.
17
30
50
40
60
40
40
50
40
40
50

6 Solitary wave propagation on the shelf with stronger inflow
(U0 = 0.4 m s −1 )
Below are some results for the case with solitary waves on the shelf, using a stronger forcing.
Generally the solitary waves develops earlier and are seen much more clearly, see
below a section of the density field after 11.74 hours. The density fields and two components
of the velocity field at three consecutive times are given in Figures 17, 18, and 19.
In these figures the focus is on the solitary wave train. In Figure 20 the u components at
x = 5000 m and x = 15000 m are plotted, again with focus on the passing group of solitary
waves.
The speed of the front of the wave train in the period from 11.74 hour to 13.42 hours is
approximately 0.63 m s −1 and in the period from 13.42 hours to 15.09 hours approximately
1.0 m s −1 . Thus as in the more weakly forced case in the previous section, the wave group
gains speed away from the edge. From Figure 20 the period between consecutive waves in
the group may be estimated to be 20 minutes. It is again clearly seen that the first wave in
the group is strongest, and the amplitudes of the waves behind are gradually reduced.
depth [m]
0
−50
−100
−150
−200
−250
−300
−350
27
26.5
26.7
27.3
27.4
27.5
25.9
26
25.8
26.1 26.1
26.3
26.2
26.4
26.5
26.6
26.2
26.3
26
26.6
26.8 26.8
26.9
27.1
27.2
26.7
27
27.3
27.2
RHO (ci=0.1 )
26.4
25.9
27.1 26.9
27.4
27.5
27
25.8
26.7 26.7
27.3
26.1
−400
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
distance [m]
(a)
25.8
25.9
25.9 26.1
27.1
26
26.2 26.2
26.3
26.3
26.8
26.5 26.5
27.5
26.8
26.6
26.9
27.2 27.1
26
26.426.4
Figure 16: Topography and density field after 11.74 hours (7/8 of an inertial periods).
18
27.4
27.5
27
27.3
26.6
27.2
26.9
27.4

depth [m]
depth [m]
depth [m]
0
−20
−40
−60
−80
−100
27.4
−120
−140
−160
−180
26.7
26.3
26.4
25.9
26.1
26
26.2 26.2
26.5
26.6
−200
1000 1500 2000 2500 3000 3500 4000 4500 5000
−20
10
0
−40
−60
−80
−100
−120
−140
−160
−180
−10
0
27.1
RHO (ci=0.1 )
26.9
distance [m]
(a)
26.3
26.7
25.8
26.1
26.4
26.8 26.8
27.3
0
−200
1000 1500 2000 2500 3000 3500 4000 4500 5000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
−20
0
0
0
0
0
0
−10
−50
0
10
0
U (ci=10 cm/s )
0 0
0
0
0
−30
−40
−50
00
0
U (ci=5 cm/s )
−5
0
0 0
0
−10
−50
0
0
0
−20
0
0
5
20
10
0
−50
−10
−40
20
0
27
10
−10
0
−30
−50 −50
distance [m]
(b)
0
10
0
−5
−10
−5
0
0
0
−200
1000 1500 2000 2500 3000 3500 4000 4500 5000
0
−40
distance [m]
(c)
Figure 17: The train of solitary waves after 11.74 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
19
0
10
5
−10
−5
0
−5
0
0
5
20
−20
−50
5
15
10
30
10
−60
27.2
−10
0
0
0
0
0
−50
−15
−5
−10
−5
−10
0
0
−5
0
26.5
27.1
5
0
26.6
5
30 20
10
0
−20
−50
10
15
25.9
27.3
−10
−30
−40
−60
0 0
0
−5
−10
−5
−50
−15 −20
−10
−5
26
00
0
−40

depth [m]
depth [m]
depth [m]
0
−20
−40
−60
−80
−100
−120
−140
27.4
−160
−180
25.8 25.9
26.1
26.5 26.4
26.5
26.9
26.8
26.6
26.7
26.2
27.1 27.1
27
26.3
−200
3000 4000 5000 6000 7000 8000 9000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
RHO (ci=0.1 )
26
27.2
27.3
distance [m]
(a)
−200
3000 4000 5000 6000 7000 8000 9000
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
20
20
0
30
U (ci=10 cm/s )
0
30
40 50
−10
−10
10
40
30
0
50
20
−10
−10
40
30
20
distance [m]
(b)
−200
3000 4000 5000 6000 7000 8000 9000
00
0
0
0
0
0
0
00
0
0
0
0 0
0
0
0
U (ci=5 cm/s )
0
−5
0
0
0
5
−5 −5
0
5
0
0
0
0
−5
−10
−5
−10
0
−10
distance [m]
(c)
Figure 18: The train of solitary waves after 13.42 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
0
20
0
0
10
5
−10
−5
0
−5
0
0
26.4
26.9
26.8
40
20
60
−10
10
27.4
50
30
−10
0
0
5
0
−5
0
−5
−10
−5
0
0
5
27
40
20
−10
−20
26.7 26.6
60
10 15
5
50
70
30
−20
0
−5
0
0
10
−10
−15
−10
−5
−10
26.2
26.3
0
−5
0
0
0
0
00
00
0
26.1
0
0
25.9
0 0
0

depth [m]
depth [m]
depth [m]
0
26.3
−20
−40
−60
−80
−100
−120
−140
−160
−180
26.4
27.2
27.1
26.6
26.7
26.8
27.3
25.8
27
25.9
26
26.9
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
x 10 4
−200
distance [m]
0
−20
50
−40
−60
−80
−100
−120
−140
−160
−180
40
40
RHO (ci=0.1 )
40
30
(a)
26.5
26.2
27.1
27.2
27.4
26.1
26.3
26.4
40 40
26.6
26.8
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
x 10 4
−200
distance [m]
0
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
00
0 0
0
50
40
0
0
0
0
0
0
60
U (ci=10 cm/s )
0
0
−5
U (ci=5 cm/s )
0
50
0
0
0
50
60
0
0
00
70
60
50
5
(b)
0
−5
0
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
x 10 4
−200
distance [m]
(c)
Figure 19: The train of solitary waves after 15.09 hours. The density field is given in a), the
u component of the velocity field is given in b), and the vertical velocities in c).
21
00
80
50
30
0
70
60
0
5
0
−5
−5
0 0
0
0
0
26.7
90
80
30
5
70
60
30
0
0
−5
0
−5
−10
27.3
50
40
40
0
0
0
0
60
30
100
80
27
90
00
10
70
−5
−15
5
5
−10
0
−5
50
40
−10
00
−5
26.9
40
0
27.4
40
00
0
26.5
0
0
0
0
0
0

Depth [m]
Depth [m]
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
200
10
10
20
20
0
10
00
−200
12 12.5 13 13.5 14 14.5 15
0
−20
−40
−60
−80
−100
−120
−140
−160
−180
−30
40 40
−30
60
40
30
50
U (ci=10 cm/s )
50
−20
50
60
90
80
−10
100
40
70
0
−30
40
−40
−30 −30
−20 −20
50
50
50
U (ci=10 cm/s )
80
90
70
60
40
50
40
30
−30
50
0
−10
−20
−20
70
80
50
−10
60
50
40
60
−20
50
−10−10
30
−10
0
30
Time
(a)
70
80
60
−200
15 15.5 16 16.5 17 17.5 18
Time
(b)
Figure 20: Time development of the vertical profile of along section velocities at x = 5000 m
(top) and x = 15000 m with focus on the waves.
22
40
−10
50
60
0
70
0
60
30
40
50
0
0
50
0
10
60
50
50
40
60
10
10
20
30
30
40
60
20
40
30
30
50
40

7 Discussion
Wave phenomena with periods of approximately 45 min are observed near the shelf edge in
the Ormen Lange area. They are characterised by one leading wave with 4 to 5 gradually
weaker waves behind it. The observed waves thus have the characteristics of internal solitary
wave trains in stratified oceans. The tides are generally weak in this area. Therefore,
wind and inertial oscillations are taken as forcing in the numerical experiments reported
here. See the description of the general oceanography in this area
[Mathisen et al.(2007)Mathisen, Berntsen, and Furnes].
It is known that accelerating flow over an edge easily starts off a train of solitary waves
in a stratified ocean. These waves may have wave lengths of approximately 500 to 1000 m
and grid sizes of approximately 25 m are necessary to resolve them in numerical studies.
This means that they are not captured in general studies of the circulation in this area,
and the strong current events, with speeds that my exceed 1 m s −1 , associated with the
waves are not captured. In the preliminary studies reported here current speeds that exceeds
1.10 m s −1 are found without increasing the background inflow velocities towards
the maximum values measured along the shelf edge. With stronger forcing, the wave trains
develop earlier, and they get larger amplitudes. However, the periods between consecutive
waves in a wave train seem to be reduced as the forcing is increased. With stronger forcing,
the speed of the wave train also seems to increase with distance from the generation point.
The waves gradually changes shape as they propagate. In the present studies they are
still fairly strong 17 km away from the generation point, which is as far as they have been
’followed’ using simulations that take approximately 15 hours each.
Basically the waves produced in these numerical studies have many of the characteristics
of the observed waves. The major discrepancy is the period, which is approximately
20 min in the model results and 45 min in the observations. It may be noted that periods of
20 min are also measured over the continental shelf off Oregon,
see [Moum and Smyth(2006)].
The periods between consecutive waves in a wave train generated by a strong pulse
at for instance an edge in the topography may, however, change with travel time or with
distance from the generation point, see [Dysthe(2004), Whitham(1974)].
For surface wave trains generated by a strong initial water elevation, the surface elevations
η(x,t) may be given by
η(x,t) ∼ 1
2Ct 1/3 Ai(x − √ gh
Ct
1/3 ),
where x is the horizontal coordinate, t is time, g is the gravity, h is the water depth, Ai is the
Airy function, and C = ( h2√ gh
2 ) 1/3 , see [Dysthe(2004)]. For an internal solitary wave train,
the speed √ gh must be replaced with √ g ′ h ′ where g ′ = g Δρ
ρ0
and Δρ is the density difference
and ρ0 the background density. Furthermore, h must be replaced with h1h2
h1+h2 where h1 is the
depth of the surface layer and h2 the thickness of the bottom layer in a two layer system.
The solution given above is for a more simple system than what we have in the Ormen
Lange area, and for a more idealised initial state than what we have in the numerical
simulations. However, the solution has many of the characteristics of both the observed
waves and the modelled waves, with one strong leading wave and decaying trailing waves
behind it. The factor t 1/3 in the denominator of the argument to the Airy function also
indicate that the period between consecutive waves in a wave train may be doubled, if t is
increased with a factor 8. Since the waves travel with almost constant speed, this indicate
that if the period is 20 min 10 km from the generation point, or the shelf edge, the period
will be 40 min 80 km from the generation point. Also the amplitudes of the waves will be
reduced as they move. In essence this also means that if the observed periods of the waves
are the same for all events, it is lightly that the waves come from the same generation point,
or from generation points equally far from the observation point. The factor t 1/3 in the
23

denominator of the amplitude of the Airy function means that closer to the shelf edge, the
current velocities may be substantially larger.
We would also like to point at the papers on Waves in Geophysical Fluids in
[Grue and Trulsen(2006)] including [Grue(2006)], [Kharif and Pelinovsky(2006)], and
[Morozov(2006)]. In [Kharif and Pelinovsky(2006)] dispersion enhancement of transient
wave packets is discussed and an Airy function solution of the type given above is presented.
In [Morozov(2006)] also internal waves in the Arctic region are described.
A full explorations of these wave trains, including sensitivity to the parameters involved
(forcing, period forcing, topography, stratification), investigations of possible maximum
speeds, and studies where we follow the waves further is a full project in itself. The waves
are followed 18 km in the present studies, and approximately 100 km is possible using more
computer resources. It could also be mentioned that even if the focus in this report has been
on solitary wave trains outside mid-Norway such waves are observed many places in the
ocean, including elsewhere in Norwegian waters, see [Global Ocean Associates(2004)].
Acknowledgement. Many thanks to Kristian B. Dysthe for pointing us the theory on
wave fronts.
24

References
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[Berntsen et al.(1981)Berntsen, Kowalik, Sælid, and Sørli] H. Berntsen, Z. Kowalik,
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splitting procedure. Modeling, Identification and Control, 2:181–199, 1981.
[Berntsen(2000)] J. Berntsen. USERS GUIDE for a modesplit σ-coordinate numerical
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Bergen, Johs. Bruns gt.12, N-5008 Bergen, Norway, 2000. 48p.
[Blumberg and Mellor(1987)] A.F. Blumberg and G.L. Mellor. A description of a
three-dimensional coastal ocean circulation model. In N.S. Heaps, editor, Three-
Dimensional Coastal Ocean Models, volume 4 of Coastal and Estuarine Series, pages
1–16. American Geophysical Union, 1987.
[Dysthe(2004)] K. Dysthe. Lectures given at the summer school on:Water Waves and
Ocean Currents. Nordflordeid 21-29 June 2004., 2004.
[Gill(1982)] A.E. Gill. Atmosphere-Ocean Dynamics. Academic Press, 1982. ISBN-0-
12-283520-4.
[Global Ocean Associates(2004)] Global Ocean Associates. An Atlas of Oceanic Internal
Solitary Waves, 2004.
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Rogue Waves, Internal Waves and Internal Tides. SpringerWienNewYork, 2006.
[Grue and Trulsen(2006)] J. Grue and K. Trulsen, editors. Waves in Geophysical Fluids:
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NewYork.
[Kharif and Pelinovsky(2006)] C. Kharif and E. Pelinovsky. Freak waves phenomenon:
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