Modelling Tsunamis Generated by Earthquakes and Submarine Slumps
Using Mike 21
Stephen Anton LUGER, Rhydar Lee HARRIS
Prestedge Retief Dresner Wijnberg, 5th Floor, Safmarine Quay, Clock Tower
Precinct, Victoria and Alfred Waterfront, Cape Town, South Africa, e-mails:
[email protected], [email protected]
Keywords
MIKE 21, tsunami, earthquake, submarine slump
Abstract
This paper describes the modelling of tsunamis generated by earthquakes and
submarine slumps using Mike 21. This includes the calculation of the initial
conditions, the numerical settings applied for the propagation model and three
case studies. A comparison between the MIKE 21 Classic and MIKE 21 Flexible
Mesh models is also described. Given appropriate initial conditions, the MIKE 21
Classic can be used to efficiently simulate the propagation of tsunamis generated
by both earthquakes and submarine slumps. The lack of geographical
coordinates in MIKE 21 Classic can however result in a distorted grid when
simulating larger domains. Although the MIKE 21 Flexible Mesh model includes
geographical coordinates, the numerical scheme (including the higher order
options) appears to be too dispersive to accurately simulate tsunami propagation
over large distances, and the Flexible Mesh version presently does not feature a
landslide option. MIKE 21 Classic is thus the preferred DHI model for tsunami
modelling at present.
INTRODUCTION
A tsunami is a series of water waves generated by rapid, large scale disturbance
of a water body. Only geophysical events that release a large amount of energy
in a very short time into a water body generate tsunamis. Earthquakes are the
most frequent causes, but submarine and subaerial landslides, pyroclastic flows
and caldera collapses during volcanic eruptions, meteorite impacts and ice falls
may also generate tsunamis. Tsunami waves travel over long distances across
the water body with little loss of energy towards the shore, where they shoal as
they reach shallow water. The waves have the potential to inundate significant
areas onshore, depending on local bathymetry and topography, and the
hydrostatic and hydrodynamic forces associated with these waves can be
severely destructive (U.S. NRC, 2009).
This paper describes the modelling of tsunamis generated by earthquakes and
submarine slumps using Mike 21. This includes the calculation of the initial
conditions, the numerical settings applied for the propagation model and three
case studies. A comparison between the MIKE 21 Classic and MIKE 21 Flexible
Mesh models is also described.
INITIAL CONDITIONS FOR TSUNAMIS GENERATED BY EARTHQUAKES
A tsunamigenic earthquake results in a displacement of the water surface, which
is applied as the initial condition in the model. For earthquake-generated
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tsunamis the duration of the source dynamics is generally much smaller than the
period of the tsunami waves, so the coupling between the source and the initial
waveform can be neglected without significantly affecting the properties of the
tsunami waves (U.S. NRC, 2009). The formulation of Okada (1985), which is
based on an elastic earth crust, may be used to calculate the displacement of
the seabed due to the earthquake. A complex rupture pattern can be simulated
by dividing the fault plane into a series of smaller segments, each of these
individually described by an Okada formulation. The vertical displacement of the
seabed induces a corresponding displacement of the water surface, which is
applied as the initial condition for the hydrodynamic model.
An example of the fault parameters for the 26 December 2004 Sumatra tsunami
and the corresponding initial water surface elevation are shown in Table 1 and
Figure 1, respectively.
Table 1: Fault parameters for the 26 December 2004 Sumatra magnitude 9.2
earthquake (Grilli et al, 2007).
Parameter
1
2
3
4
5
Longitude [degrees]
94.10
93.33
92.71
92.17
92.44
Latitude [degrees]
3.48
5.10
7.21
9.68
11.78
Origin
(1)
Origin
(1)
(2)
[degrees]
Strike
Fault Segment Number
323
348
338
356
10
Depth(3) [km]
25
25
25
25
25
Length [km]
220
150
390
150
350
Width [km]
130
130
120
95
95
Dip [degrees]
12
12
12
12
12
Mean Dislocation [m]
18
23
12
12
12
(1) The origin is defined as the mid-point of the upper border of the fault.
(2) An observer facing along strike will see the fault dip to the right (degrees clockwise
from north).
(3) Depth from the seabed to the upper border of the fault.
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Figure 1: Initial water surface elevation calculated for the 26 December 2004
Sumatra tsunami using the fault parameters in Table 1.
INITIAL CONDITIONS FOR TSUNAMIS GENERATED BY SUBMARINE
SLUMPS
Submarine mass failures can be categorised as either slip events, which are
typically large translations in landslide masses, or rotational failure leading to a
slump event (see Figure 2). Only slumps will be considered here, although a
similar methodology can be applied for slips. Unlike tsunami generation by
earthquakes, which can be accurately modelled using the instantaneous coseismic displacement of the water surface as an initial condition, submarine
slumps or slides typically take place over a number of minutes. The velocity of
landslides can often be comparable to the phase velocity of the tsunami waves
generated by it and an explicit landslide model should thus be employed in order
to initiate the tsunami (U.S. NRC, 2009).
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Figure 2 The main morphological and structural features of submarine slumps
(Dingle, 1977).
Subaerial or submarine slumps and slides can be simulated explicitly in the
MIKE 21 Classic model using the landslide option. The effect of the landslide is
modelled by forcing terms representing the temporal dynamic vertical
deformation of the bathymetry. The main task in preparing the input data for the
model is to generate a time varying bathymetry file.
We have developed a numerical routine to define the dynamic changes in seabed
level arising from a slump. The submarine slump is simulated as a rigid body
moving down a slope. The body has a Gaussian shape as specified in Grilli and
Watts (2005). The equation describing the slump motion follows Watts et al
(2003), where the slump motion is modelled as a rigid body undergoing a
rotation around a point described as the centre of rotation of a circle prescribed
by the arc of the circular failure plane (see Figure 3). The rigid body is subject to
external moments due to gravity, added mass and shear stress summed over
the failure plane. The slump motion is described with a cosine function and as
such experiences an initial angular acceleration, relatively constant maximum
angular velocity and a subsequent deceleration before coming to rest in a
position such that the centre of mass of the slump is vertically under the axis of
rotation.
Figure 3: Parameters defining the slump model of Watts et al (2003).
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The magnitude of the tsunami generated by a slump depends on a number of
parameters including slump volume, water depth, slump thickness, initial
acceleration and the maximum velocity of the slump. An example of the
application of this methodology is described later in this paper.
PROPAGATION MODELLING USING MIKE 21 CLASSIC
The MIKE 21 Classic hydrodynamic model is used to simulate the propagation of
the tsunami wave from the source to the shore. The model solves the non-linear
two-dimensional shallow water equations (conservation of mass and verticallyintegrated momentum) on a series of dynamically-nested rectangular grids using
an implicit time scheme. Processes simulated include spatially-varying Coriolis
force, bottom shear stress, momentum dispersion and flooding / drying.
Our tests indicate that the grid spacing should be selected to ensure at least 20
to 30 grid points per tsunami wavelength. Although a Courant number of 5 to 20
may be acceptable for the stability of the implicit solver, our tests indicate that a
Courant number of approximately 1 is required for the accurate propagation of
the tsunami wave over large distances. Given the grid spacing and model time
step, the maximum and minimum allowable water depths in a particular grid
area can be calculated as follows:
dmin imum =
dmax imum =
where
∆x
=
k
=
T
=
C
=
∆t
=
(∆x k)2
9.81 T 2
(C ∆x)2
9.81 ∆t 2
grid spacing [m]
grid points per tsunami wavelength, 20 to 30 are recommended
tsunami period [s], typically 5 to 40 minutes
Courant number, 1 is recommended
model time step [s]
The nested grid facility allows progressively finer grid sizes to be used as the
tsunami propagates into shallower water. For smaller model domains in the
order of 500 km, the default map projection such as UTM is used. In the case of
large model domains, e.g. modelling tsunami propagation across an ocean, a
limitation of MIKE 21 Classic is the lack of geographical coordinates, which will
result in a distorted grid and associated errors in the tsunami propagation. In
this case the best option is to use the Map Projection Editor to set up a custom
map projection designed to minimise the distortion in the area of interest, e.g.
using the projection type: Lambert conformal conic with two standard parallels.
The default drying depth of 0.2 m and flooding depth of 0.3 m are applied. Eddy
viscosity is generally found to have an insignificant influence on these
simulations and is often set to zero, although it can be used to suppress high
frequency disturbances, particularly when modelling landslides. Bed resistance is
specified by a Manning number of 32 m1/3/s.
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Where open boundaries exist, it is important that the tsunami waves are not
reflected back into the model domain. In the absence of an absorbing boundary
condition in MIKE 21 Classic, we have applied an artificial cubic-shaped beach in
front of the open boundaries. The beach is typically 100 grid cells wide. This is
easily created in the MIKE Grid Editor by deleting the existing depths near the
boundary, setting the depth at the boundary to zero, and then using the “inverse
distance ^3 weighted” interpolation tool. In addition, the bed resistance on the
artificial beach slope is increased by specifying a Manning number of 0.1 m1/3/s.
CASE STUDY: PROPAGATION OF 2004 SUMATRA TSUNAMI TO SOUTH
AFRICA
The methodology described above has been applied to the Sumatra tsunami of
26 December 2004. The fault parameters and the corresponding initial water
surface elevation are shown in Table 1 and Figure 1, respectively. The tsunami is
propagated across the Indian Ocean to South Africa.
Nine nested grids are used, with the grid spacing varying from 120 m at Port
Elizabeth to 9720 m at the model boundaries. The model time step is 6 s, which
ensures a Courant Number of less than 1.0. The model results are shown in
Figure 4. The bathymetry and orientation of Port Elizabeth Bay is seen to amplify
the tsunami, particularly in the western corner of the bay and inside the two
ports.
The 26 December 2004 event was measured at a number of tidal stations along
the South African coastline, with the largest water level variation measured in
the Port of Port Elizabeth. The measured tide has been subtracted from
predicted tide and then adjusted for the average storm surge of 0.18 m
measured during the tsunami. The resulting tsunami signal is shown in Figure 5.
It should be noted that the maximum crest of the tsunami was not recorded due
to an instrument problem. Hartnady and Okal (2007) estimate the maximum
crest level to have been approximately 2.11 m above the predicted tidal level. If
the 0.18 m average storm surge is taken into account the maximum tsunami
crest level reduces to 1.93 m.
The modelled tsunami-induced water levels inside the Port of Port Elizabeth
compare well to the measurements (see Figure 5). The model slightly underpredicts the maximum water level (model: 1.7 m, measured: approximately
1.9 m) while over-predicting the minimum water level (model: -2.0 m,
measured: -1.5 m). Both measured and modelled tsunamis have periods of
between 30 and 40 minutes.
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Figure 4: Modelled propagation of the Sumatra tsunami of 26 December 2004.
(The maximum water surface elevations in the large plot are calculated from the
model solution saved at 10 minute intervals, which allows the propagation of the
wave crests to be visualised. The small plot uses the model solution saved at
1 minute intervals in order to obtain an accurate estimate of the maximum
elevations near the coastline.)
Figure 5: Calibration of tsunami model against tide gauge measurements in the
Port of Port Elizabeth
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CASE STUDY: TSUNAMI GENERATED BY A HYPOTHETICAL SUBMARINE
SLUMP OFFSHORE OF CAPE TOWN
The methodology described above is applied to model the tsunami generated by
a hypothetical 80 km3 slump on the continental shelf edge offshore of Cape
Town. The geometry of the hypothetical slump described here is based on the
measured geometry of the proximal portion of the much larger historical Agulhas
Slump, located on the continental shelf edge of South Africa (Dingle, 1977). The
slump parameters are given in Table 2 and the resulting tsunami is shown in
Figure 6.
Table 2: Parameters describing a hypothetical 80 km3 slump on the continental
shelf edge offshore of Cape Town.
Volume(1) [km3]
80
(2)
18
Length
“b” [km]
Width(3) [km]
18
Thickness “T” [km]
0.3
Rotation “φ” [degrees]
0.4
Radius “R” [km]
135
Displacement “S” [km]
1.0
Centroid Longitude [degrees]
17.18
Centroid Latitude [degrees]
-34.37
(4)
Strike
[degrees]
10
Water Depth [m]
2000
Initial Acceleration [m/s2]
0.011
Maximum Velocity [m/s]
2.3
Duration of Slump Movement [minutes]
11.3
(1) Since the slump is elliptic, the volume = π/4 x length x width x thickness
(2) Items in “inverted commas” are defined in Figure 3
(3) Width of the slump is measured transversely across the slope.
(4) An observer facing along strike will see the slump moving down to the right (degrees
clockwise from north).
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Figure 6: Modelled propagation of a tsunami generated by a hypothetical 80 km3
slump on the continental shelf edge offshore of Cape Town. (The maximum
water surface elevations in the large plot are calculated from the model solution
saved at 10 minute intervals, which allows the propagation of the wave crests to
be visualised. The small plot uses the model solution saved at 1 minute intervals
in order to obtain an accurate estimate of the maximum elevations near the
coastline.)
The modelled 80 km3 slump results in a maximum runup of approximately 2 m
at the shoreline. Although there is clear evidence of historical slumps around the
shelf margin of South Africa, the present-day risk posed by slumps is unclear.
The historical Agulhas Slump located on the edge of the Agulhas Bank
approximately 400 km south-west of Cape Town is one of the largest slumps
identified world-wide with an estimated length of 750 km, width of 106 km and
volume of 20 000 km3. The slump involved Pliocene sediments and may
therefore be Quaternary (1.8 million years to present) in age (Dingle, 1977).
The volume of the historical Agulhas Slump is 250 times larger than the slump
that has been modelled, implying a devastating tsunami should the slump have
occurred as a single unit rather than a number of smaller events at different
times. Preliminary numerical modelling indicates that for this slump region the
duration of the tsunami-induced water level disturbance at the shore is 1 to 2
hours, implying that individual slumps separated by longer than this time are
effectively separate smaller events rather than one large event. The present-day
risk posed by these events to sensitive infrastructure such as power stations is
the subject of ongoing research.
CASE STUDY: TSUNAMI-INDUCED WATER LEVELS AND CURRENTS FOR
PORT DESIGN IN CHILE
Mejillones Bay is located in northern Chile. The subduction zone of the Nazca and
South American plates lies approximately 100 km to the west of the bay. The
existing port infrastructure includes the port terminals of Mejillones and
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Angamos. Currently in design and construction is the Terminal Graneles Del
Norte (TGN) to the north of Angamos Port. Maximum tsunami-induced water
levels and currents are required to check the structural integrity of the port
structures.
The two tsunami events modelled are based on two previous seismic events in
Chile: the 1877 event with a magnitude of 8.7 and the February 2010 event with
a magnitude of 8.8. Although these previous earthquake epicentres were over
land or within 7 km of the shore, respectively, for this study both events are
assumed to occur in deep water offshore of Mejillones. The sensitivity to the
fault position and fault depth were investigated. It was found that a shallow
(depth = 10 km) source located in deep water along the Peru-Chile trench
(longitude = -71.5°) resulted in the worst tsunami runup and velocities in
Mejillones Bay.
Six nested grids were used, with a grid spacing of 6075 m, 2025 m, 675 m,
225 m, 75 m and 25 m. The model bathymetry was obtained from the MIKE CMAP electronic hydrographic charts and local bathymetric surveys. Topographic
data up to +30 m Chart Datum was included in the model. The model time step
was 1 s, which ensured a Courant number of less than 1.3. The grid spacing was
selected to ensure at least 30 grid points per tsunami wavelength. The drying
depth was set at 0.2 m and the flooding depth was 0.4 m. A constant velocitybased eddy viscosity of 0.5 m2/s was applied. Bed resistance was specified by a
Manning number of 32 m1/3/s.
An example of the modelled propagation of the tsunami generated by a
magnitude 8.8 earthquake is shown in Figure 7. The tsunami takes
approximately 12 minutes to reach the coast. The maximum water levels at the
port are approximately 12 m above still water level. For the magnitude 8.7
event, the current speeds are predicted to reach 5.2 m/s on top of the platform
structure at the Angamos terminal, and 3.7 m/s around the seaward caissons
(see Figure 8).
Figure 7: Modelled tsunami propagation for the first 16 minutes after a
magnitude 8.8 earthquake offshore of Mejillones.
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Figure 8: Instantaneous current speeds at Angamos terminal during tsunami
drawdown from a magnitude 8.7 earthquake. Maximum current speeds reach
5.2 m/s on top of the platform structure.
APPLICATION OF MIKE 21 FLEXIBLE MESH MODEL FOR TSUNAMI
MODELLING
A brief assessment of the MIKE 21 Flexible Mesh model has been undertaken by
repeating the Chilean case study described above using the Flexible Mesh model
instead of the MIKE 21 Classic model. A triangular mesh was generated to mimic
the refinement achieved with the nested rectangular grids, although the large
number of resulting elements meant that the element areas in the Flexible Mesh
model were approximately 30% larger compared to the Classic model. The
Flexible Mesh model was run with both the lower and higher order time
integration and space discretization schemes. The default critical CFL number of
0.8 was tested, as well as a lower value of 0.4.
The results in Figure 9 show that the higher order scheme improves the model
accuracy, particularly for the current speeds. Reducing the critical CFL number to
0.4 does not improve the results further. Compared to the Classic model, the
Flexible Mesh is seen to underestimate the current speeds by approximately
25% in this particular example.
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Figure 9: Comparison between the MIKE 21 Classic and Flexible Mesh models.
Plots show the surface elevation and velocities in Mejillones Bay due to
magnitude 8.7 earthquake offshore.
The Classic model was found to be more computationally efficient in this case, as
shown by the following runtimes for the four models:
•
•
•
•
HD Classic, CFL = 1.3
FM Lower Order, CFL = 0.8
FM Higher Order, CFL = 0.8
FM Higher Order, CFL = 0.4
:
:
:
:
2.5 hours
4.0 hours
12.2 hours
22.3 hours
In cases where frequency dispersion is important, e.g. where the source is close
to the shoreline, the use of a Boussinesq wave model such as MIKE BW should
be considered in preference to shallow water equation models such as MIKE 21
Classic and Flexible Mesh.
CONCLUSIONS
Given appropriate initial conditions, the MIKE 21 Classic can be used to
efficiently simulate the propagation of tsunamis generated by both earthquakes
and submarine slumps. The lack of geographical coordinates in MIKE 21 Classic
can however result in a distorted grid when simulating larger domains.
Although the MIKE 21 Flexible Mesh model includes geographical coordinates,
the numerical scheme (including the higher order options) appears to be too
dispersive to accurately simulate tsunami propagation over large distances, and
the Flexible Mesh version presently does not feature a landslide option.
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MIKE 21 Classic is thus the preferred DHI model for tsunami modelling at
present.
REFERENCES
Dingle R (1977) The anatomy of a large submarine slump on a sheared
continental margin (SE Africa). Journal of the Geological Society of London 134;
293-310.
Grilli S and Watts P (2005) Tsunami Generation by Submarine Mass Failure, 1:
Modelling, Experimental Validation and Sensitivity Analysis. Journal of Waterway,
Port, Coastal and Ocean Engineering, November/December 2005.
Grilli S, Ioualalen M, Asavanant J, Shi F, Kirby J and Watts P (2007) Source
Constraints and Model Simulation of the December 26, 2005, Indian Ocean
Tsunami. Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 133,
No. 6, November, 2007.
Hartnady C and Okal E (2007) Mentawai tsunami effect at Port Elizabeth, South
Africa on 12-14 September 2007, South Afr. J. Sci.
Okada Y (1985) Surface Deformation to Shear and Tensile Faults in a HalfSpace. Bull. Seism. Soc. Am., 75, [4], 1135-1154.
U.S. NRC (2009) Tsunami Hazard Assessment at Nuclear Power Plant Sites in
the United States of America, Final Report. U.S. Nuclear Regulatory Commission,
Report NUREG/CR-6966, March 2009.
Watts P, Grilli S, Kirby J, Fryer G and Tappin D (2003). Landslide Tsunami Case
Studies Using a Boussinesq Model and a Fully Nonlinear Tsunami Generation
Model. Natural Hazards and Earth System Sciences, Vol. 3, 2003, pp. 391-402.
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