Internal wave simulation for different angles
and shapes of continental shelf
Himansu K. Pradhan*, A.D. Rao, K.K.G. Reddy and S. Mohanty
Centre for Atmospheric Sciences,
Indian Institute of Technology Delhi,
New Delhi, India
[email protected]
Abstract- The average slope of the continental shelf in the
world ocean is 0.5o and its width varies considerably. This
paper illustrates experimental studies describing the internal
wave run-up on different gradients of continental shelf
varying from 0.2o to 0.5o. MIT general circulation model is
configured with a variable grid, tidal information in the
momentum equations and background stratification of
density as initial fields to simulate internal waves. The model
simulated density and temperature time-series is subjected
to Fast Fourier Transform to compute the energy spectra of
internal waves. The results reveal that the peak of internal
wave activity varies spatially for different angles of the
continental shelf. The experiments are further continued for
concave coastline geometry to look at the internal wave
energy distribution over the shelf. The results show that in a
concave coastline the energy is large compared to a straight
coastline inferring convergence of internal wave energy.
Keywords—Continental Shelf, Internal Waves; MITgcm;
Energy spectra.
I. INTRODUCTION
Internal waves (IWs) are common phenomena over
the continental shelves and internal tides are IWs at a tidal
frequency that are generated when barotropic
tides/surface tides interact with shallow topographic
features [1]. Occurring beneath the free surface of a
density stratified water body; they have utmost
importance in submarine acoustics, underwater
navigation,
offshore
structures,
ocean
mixing,
biogeochemical processes, etc. over the shelf-slope
region. The geometry of the shelf-slope region and local
hydrographic conditions determine the effect of this
interaction. Wave trains formed after the interaction
propagate both inward over the continental shelf and
outward into the open ocean [2].
The conversion of barotropic tidal energy into
baroclinic energy in the form of internal tides takes place
at topographic boundaries. Earlier studies of Legg and
Adcroft [3] examined mixing generated by internal wave
reflection from variable topographic slopes of convex and
concave shapes. The upper part of the continental slope
plays an important role in non-linear effects and mixing
as found by Lien and Gregg [4] in their in-situ
observations.
978-1-4799-4918-2/14/$31.00 ©2014 IEEE
Wallace and Wilkinson [5] described the internal
wave run up and its dissipative phase on uniform slopes
of 0.030 and 0.054 radian from a series of laboratory
experiments. Periodic wave trains propagate onto the
slope, steepens forms solitary wave as they travel
shoreward into the continental shelf. Numerical study
conducted by Gerkema et al [6] focused on the non-linear
evolution of the internal tide and showed that the main
region of generation of IWs in upper part of the slope
using two- dimensional MITgcm. Studies by Munroe and
Lamb [7] and Holloway and Merrifield [8] have shown
that there are changes in conversion rates in threedimensional studies as compared to two-dimensional.
Earlier studies considered simulation of IWs over
Gaussian topography, laboratory experiments, etc. with
two-layered and multi-layered stratifications. Present
numerical study considers the real aspect ratio of the
shelf-slope topography found in the world ocean with a
stratified water column of a typical tropical ocean. The
continental shelf across world ocean varies and the
average gradient of the continental shelf is 0.5o . Similarly
the average gradient of continental slope is 3-4o. Here,
we at first reproduce the generation of IWs on different
gradients of continental shelf and compute the spectral
energy distribution. Secondly, a comparison is made for
two different stratification of May and February. Finally,
we demonstrate the crucial role of the coastal geometry.
For this a comparison of spectral energy distribution is
made between a straight and a concave coastline in a
particular stratification.
II. NUMERICAL MODEL CONFIGURATION
MITgcm [9],[10] is an open source code available to
the community, (see http://mitgcm.org/) designed to study
both oceanic and atmospheric phenomena. This finite
volume, z-coordinate model solves the incompressible
Navier-Stroke equations with Boussineq approximation
on an Arakawa-C grid. The lopped cell representing the
topography [11] is essential for accurate representation of
the interaction of barotropic tide with topography. For the
current study the model uses polynomial equation of state
[12] for the computation of density field. The main
physical parameters governing the tidal flow are
shelf break
o
0.2 - RED
0.3o - GREEN
0.4o - PINK
0.5o - BLUE
-1000
-1200
-1400
0
5
10
15
20
continental slope
( avg. angle ~ 3 o )
25
30
35
distance ( km )
40
45
50
55
60
Fig. 1: Schematic representation of bottom topography at a
particular cross-section.
The monthly climatological temperature and salinity
fields are derived from World Ocean Atlas (2009) of
National Oceanographic Data Centre that are taken as
initial density fields representing the background density
[16],[17]. The profiles are taken for the month of May and
February bearing different stratifications found in the
tropical waters of the Bay of Bengal in the North Indian
Ocean.
The simulation is carried out for 18 days of which the
first 10 days are considered for spin-up and the next 8
days for analysis. A 3.5days time span is chosen
containing a spring tide for computing the spectral
estimates. This is obtained by direct method using the
Fast Fourier Transform algorithm of Cooley and Tukey
[18]. Spectral estimates of IWs is computed from the time
>
<
>
>
<
pycnocline
depth ( m )
-200
<
thermocline
0
-400
-600
(a)
-800
4
(b)
8 12 16 20 24 28
33
temperature ( oC )
(c)
34
35
36 1020
salinity ( psu )
1023
1026
density ( kg/m3 )
Fig. 2: Vertical profile for May (a) Temperature (b) Salinity (c)
Density.
The experiment details are tabulated below:
open ocean
-400 (angle varies between 0.2o to 0.5o )
-600
-800
The first set of experiments are carried out for
different continental shelf angles 0.2o, 0.3o, 0.4o and 0.5o
for a shelf width of 17km. Correspondingly, the shelf
break falls at a water depth of 70m, 110m, 145m and
175m respectively. The cross-section at 90th grid point
that falls at the centre of the domain along the y-axis is
taken into account for study. The area averaged
temperature and salinity for the month of May were given
as initial conditions as shown in the Fig. 2a,b,c. This
shows that the water is stratified continuously from the
surface bearing a small mixed layer of depth 20-25m. The
Coriolis frequency is fixed at 4.3E-5 and the beta plane
approximation is at 2.18E-11.
(a)
(b)
(c)
(d)
spectral estimate ( oC )2
depth ( m )
-200
III. RESULTS AND DISCUSSION
Shelf
angle
Shelf
width
Depth at
shelf break
Slope
angle
Stratification
0.20
0.30
0.40
0.50
17km
17km
17km
17km
70m
110m
145m
175m
30
30
30
30
May
May
May
May
300
0.2 o - RED
0.3o - GREEN
0.4o - PINK
0.5o - BLUE
250
200
150
100
50
0
(a)
10
15
20
0
25
30
35
40
distance ( km )
45
open ocean
continental
shelf
-200
50
55
70 m
110 m
145 m
175 m
-50
-100
-150
-200
-250
continental slope
-300
-300
impact points
at shelf break
-350
for different
angles of
-400
continental shelf
-400
-500
60
0
shelf break
-100
depth ( m )
coast
continental shelf
>
0
series of density and temperature that describes the
distribution of signal power over wave frequencies.
halocline
(a)frequency of the barotropic tide, ωo (b) amplitude of
the barotropic tide, Uo (c) coriolis frequency fo.
The model is forced with real-time tides by adding
tidal components in the momentum equations [13],[14] of
the model. An oscillating barotropic flow is imposed
uniformly throughout the domain as a surface forcing.
Four tidal components M2, S2, K1 and O1 were
considered for tidal forcing. The respective time periods
are 12.4hr, 12hr, 23.93hr and 25.82hr and corresponding
velocity amplitudes are 28.98cm/s, 30cm/s, 15.041cm/s
and 13.943 cm/sec for the components.
At open boundaries, Orlanski radiation boundary
conditions [15] are prescribed which allow any
disturbance generated in the domain to pass through
without any significant distortion. No-slip boundary
condition is applied at the bottom and a free-slip at the
lateral boundary. It uses implicit free surface and no rigid
lid for surface pressure.
Four experiments with gradients of continental shelf
varying from 0.2o to 0.5o and gradient of continental slope
is 3o as shown in Fig. 1. The shelf break/shelf edge point
is smoothened to avoid abrupt fall from continental shelf
to continental slope. An orthogonal curvilinear grid
bearing 390*180*23 grid points in x, y and z direction
respectively. From the 23 vertical levels, the first 12
levels are from the sea surface with 10m interval each and
next 4 levels are kept at 20m interval. However, the
maximum depth at this cross-section of the domain is
2500m (not shown in figure).
(b)
-450
-500
10
15
20
25
30
35
40
distance ( km )
45
50 10
15
55 20
25
60 30
o
temperature ( C )
Fig. 3: (a) Spectral estimate of temperature for all continental
shelf angles. (b) Bottom topography normal to the shelf for all
angles (left) and vertical temperature profile (right).
0.2 o - RED
0.3o - GREEN
0.4o - PINK
0.5o - BLUE
40
30
10
>
pycnocline
<
>
<
>
6 10 14 18 22 26 30 33
o
(c)
34
35
salinity ( psu )
36 1020
1023
1026
density ( kg/m3 )
Fig. 5: Vertical profile for February (a) Temperature (b) Salinity
(c) Density.
The detail of the experiments are tabulated below:
Shelf
angle
Shelf
width
Depth
at
shelf break
Slope
angle
Stratificatio
n
0.20
0.30
0.40
0.50
17km
17km
17km
17km
70m
110m
145m
175m
30
30
30
30
Feb
Feb
Feb
Feb
(a)
10
15
20
25
30
35
40
distance ( km )
45
50
shelf break
open ocean
continental
shelf
-200
(b)
15
20
25
30
35
40
distance ( km )
70 m
110 m
145 m
175 m
impact points
at shelf break
for different
angles of
continental shelf
-400
10
60
-50
-100
-150
-200
-250
continental slope
-300
55
0
-100
-500
(b)
temperature ( C )
45
50
1022
55
-300
-350
-400
-450
spectral estimate ( oC )2
0
<
(a)
-800
(a)
(b)
(c)
(d)
20
thermocline
depth ( m )
-400
-600
-500
102560 1028
3
density (kg/m )
Fig. 4: (a) Spectral estimate of density across all cross-sections.
(b) Bottom topography normal to the shelf for all angles (left)
and vertical density profile (right).
The spectral estimate of density representing spectral
energy distribution is also shown in Fig. 4. On
comparison of Fig. 3 and 4, the behaviour of the profile
for spectral estimate of both temperature and density is
similar except its magnitude. Since the variables are
different, it is expected the magnitudes of the spectral
estimate to be different.
The above simulations are compared with an
experiment using a different stratification for February
using February temperature while retaining May salinity
to see the effect of temperature alone. The vertical profile
of temperature, salinity and density calculated from both
temperature and salinity are shown in Fig. 5.
300
0.2 o - RED
0.3o - GREEN
0.4o - PINK
0.5o - BLUE
250
200
150
100
50
0
(a)
10
15
20
0
25
30
35
40
distance ( km )
45
50
open ocean
continental
shelf
-200
55
70 m
110 m
145 m
175 m
-50
-100
-150
-200
-250
continental slope
-300
-300
impact points
at shelf break
-350
for different
angles of
-400
continental shelf
-400
-500
60
0
shelf break
-100
depth ( m )
spectral estimate ( d )2
-200
50
0
depth ( m )
0
halocline
Fig. 3a shows the computed spectral estimate of model
simulated temperature for different angles (0.2o, 0.3o, 0.4o
and 0.5o in Fig. 3b) of the shelf represented in different
colours. The activity is predominant over the shelf-slope
region as the wave excitement tends to increase with the
height and the steepness of the shelf-slope. As the angle
increases, the internal wave activity is seen more towards
inner shelf. Particularly for 0.2o shelf, the activity is just
over the shelf break and the estimate is less since the
water column is limited to the upper part of the
thermocline at the shelf break, whereas depth at the shelf
break increases for all other angles [see Fig. 3b ]. In
general, the spectral estimate in the open ocean is less due
to less or no interaction with the bottom topography.
(b)
-450
-500
10
15
20
25
30
35
40
distance ( km )
45
50
10
15
55 20
25
60 30
o
temperature ( C )
Fig. 6: (a) Spectral estimate of temperature across all crosssections. (b) Bottom topography normal to the shelf (left) for all
angles and vertical temperature profile (right).
20
(a)
40
45
50
open ocean
-100
continental
shelf
impact points
at shelf break
for different
angles of
continental shelf
-400
-500
(b)
10
15
20
25
30
35
40
distance ( km )
70 m
110 m
145 m
175 m
-50
30
17
-100
-150
39
16
0
-200
po
-250
continental slope
-300
60
0
shelf break
-200
55
45
50
1022
55
-300
15
82
-350
-400
-450
-500
102560 1028
3
density (kg/m )
Fig. 7: (a) Spectral estimate of density across all cross-sections.
(b) Bottom topography normal to the shelf (left) for all angles
and vertical density profile (right).
in
(a)
ts
83
84
85
longitude
200
b(i)
150
30
100
50
0
0
80 100 120 140 160 180
>
shelf break
( 145 m )
-400
-600
0
80
90
50
60
0
80 100 120 140 160 180
0
0
-200
-200
-400
-400
-600
40
60
80 100 120 140 160
-600
-800
-1000
80 & 100
-1200
c(ii)
-1400
60
80 100 120 140 160 180
b(iii)
100
-800
c(i)
80 100 120 140 160 180
150
-1000
-1400
60
200
-800
-1200
100
50
-1000
30 & 150
b(iv)
100
50
0
80 100 120 140 160 180
150
b(ii)
100
50
150
100
150
-200
depth ( m )
200
b(v)
150
87
200
0
90
-1200
80 100 120 140 160 180
40
grid points
grid points
c(iii)
-1400
60
80 100 120 140 160
grid points
Fig. 8: (a) The domain of the concave coastline. b(i-v) The
spectral energy distribution at different cross sections. c(i-iii)
The bottom topography at the respective cross sections.
150
>
20
depth (m)
0
-10
-50
-100
-150
-200
-300
-400
-500
-600
-800
-1000
-1500
-2000
-2500
-3000
100
90
80
latitude
19
18
30
spectral estimate ( d ) 2
15
82
po
po
0
0
in
18
39
16
in
ts
17
(a)
ts
83
84
200
b(i)
150
100
30
depth ( m )
-400
-600
87
200
100
b(ii)
80 100 120 140 160 180
shelf break
( 145 m )
0
200
200
50
0
0
80
90
50
80 100 120 140 160 180
0
0
0
-200
-400
-400
-600
80 100 120 140 160 180
-600
-800
grid points
b(iii)
100
-1000
80 100 120 140 160 180
80 100 120 140 160 180
200
-200
-1200
100
50
-800
c(i)
b(iv)
100
-1000
-1400
80 100 120 140 160 180
150
-800
-1200
150
100
-1000
30 & 150
b(v)
150
150
50
0
-200
86
150
50
0
85
longitude
>
Fig. 6 shows spectral estimate of temperature for all
continental shelf angles for February stratification. It also
shows the propagation of IWs towards the coast enhances
with the larger gradient of continental shelf as in the case
of May. However, the overall spectral estimate are less
compared to that of simulated for May. This fact is
attributed to prevailing deeper mixed layer (extending up
to 60m) in Feb. The peak spectral estimate at 0.2o lies
outside the continental shelf region as observed in May
due to small portion of stratified water column interacts at
the shelf break (the depth of the shelf break is 70m as
shown in figure).
The spectral estimate of energies for May and Feb at
0.5o is approximately the same, which may be attributed
to less effect of the mixed layer at higher depth (175m) of
shelf break. In both the months, the shelf break at 175m
interacts with lower portion of the pycnocline. Also in
these cases, the extent of spectral estimate of energy
propagation over the shelf is nearly the same however
their peak varies. In all other continental shelf angles
(0.2o, 0.3o and 0.4o ) the spectral estimate in Feb is less
than May, as a deep mixed layer doesn't support growth
of IWs propagation/generation. As discussed previously,
the spectral estimate of density shown in Fig.7 describes
the energy spectra have the same behaviour as spectral
estimate of temperature.
The experiments are further continued to see the effect
of the geometry of the coastline. For this, an experiment
with May stratification is carried out with concave
coastline for a continental shelf angle of 0.4o and
continental slope angle of 3o [see Fig. 8a] . Spectral
energy distribution at the cross-section of interest i.e. at
30, 80, 90, 100 and 150 in solid lines shown in Fig. 8b(iv) is analysed. The 80th, 90th and 100th grid point falls in
the concave portion with the 90th point falling at the
86
200
spectral estimate ( d )2
35
ts
30
distance ( km )
in
25
po
20
0
15
18
18
10
latitude
0
0
-10
-50
-100
-150
-200
-300
-400
-500
-600
-800
-1000
-1500
-2000
-2500
-3000
100
90
80
19
10
depth (m)
spectral estimate ( d ) 2
150
spectral estimate ( d )2
20
spectral estimate ( d ) 2
30
>
40
spectral estimate ( d ) 2
spectral estimate ( d )2
0.2 o - RED
0.3o - GREEN
0.4o - PINK
0.5o - BLUE
0
depth ( m )
centre of curvature. In this experiment, rotation is not
taken into account i.e. coriolis force is kept as zero.
50
80 & 100
-1400
c(ii)
-1200
90
-1400
c(iii)
80 100 120 140 160 180
80 100 120 140 160 180
grid points
grid points
Fig. 9: (a) The domain of a straight coastline. b(i-v) The spectral
energy distribution at different cross sections. c(i-iii) The bottom
topography at the respective cross sections.
The energy is maximum at the 90th cross-section and
decreases on either side indicating the convergence of
energy due to the concave shape of the coastline.
To understand further this energy accumulation due to
the concave shape of the coastline, an experiment with
straight coastline is carried out keeping the same
congfiguration (continental shelf angle of 0.4o and
continental slope angle of 3o) shown in Fig. 9. The
energy distribution shows no change at all the crosssections [see Fig. 9b(i-v)]. The spectral energy in the
concave coastline is almost double at 90th cross-section
as compared to the straight coastline. Thus, the energy
maximum at the curved portion is the due to the geometry
of the coastline.
[8]
[9]
[10]
[11]
[12]
[13]
IV. CONCLUSION
The shelf edge remains the region of maximum
internal wave activity. With the increasing depth of the
shelf break the stratified water column allows more IW
activity over the continental shelf. The peak of spectral
estimate of energy for 0.2o lies outside the shelf region in
both the cases (May & Feb) as the interaction with the
continental shelf region is limited only to the upper part of
the thermocline. Because of the presence of a deeper
mixed layer (60-70m) in Feb, the associated spectral
energy estimates are relatively less compared to that of
continuously stratified waters of May. The peak spectral
estimate of energy for 0.5o shelf angle for both the months
(May & Feb) is nearly the same as the effect of mixed
layer depth of 175m is negligible. The geometry of the
coastline also plays a major role in IW energy distribution
with an increased magnitude seen in the concave part of
the coast.
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