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Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
409
DOI: 10.4408/IJEGE.2013-06.B-39
LANDSLIDE-TRIGGERED TSUNAMI MODELLING IN ALPINE LAKES
M
artin
FRANZ
(*)
, Y
urY
PODLADCHIKOV
(**)
,
M
ichel
JABOYEDOFF
(*)
& M
arc
-h
enri
DERRON
(*)
(*)
University of Lausanne - CRET - Lausanne, Switzerland,
(**)
University of Lausanne - ISTE - Lausanne, Switzerland
INTRODUCTION
Landslide triggered tsunamis are responsible of a
lot of damages worldwide. [NOAA (http://www.ngdc.
noaa.gov/nndc/struts/form?t=101650&s=70&d=7
);
S
lingerland
& V
oight
, 1979; S
eMenza
& g
hirotti
,
2000]. Mountainous regions are particularly subject
to this phenomenon due to steep topography and as-
sociated slope instabilities, as well as large amount
of water body such as lakes, fjords or reservoirs.
Moreover, the fact that the valleys are highly popu-
lated and that they concentrate the water flow, give
to this type of region a particularly high potential of
catastrophic events.
In Switzerland only, ten tsunamis generated by
landslides are documented since 1923. The slides
volumes range from 5’000 to 300’000 m
3
and the
wave height from 1.5 to 15 m (h
uber
, 1982; l
ouiS
i
ngenieurgeologie
, 2010). In AD 563, a submarine
landslide triggered a tsunami in the Geneva Lake
leading to an 8 m run-up in Geneva, more than 70
km away from the source (K
reMer
et alii, 2012).
In the recent history, the most catastrophic events
that occurred in the Alps is the one of Vajont which
involved a 260 x 106 m
3
slide that triggered a dam
overflow causing over 2000 casualties (S
lingerland
& V
oight
, 1979; W
ard
& d
aY
, 2011; b
oSSa
& P
etti
,
2011). The fact that none of them are listed in the
NOAA Tsunami Events catalogue shows how little it
is known or studied l
Ynett
& l
iu
, 2004; z
iJleMa
&
S
telling
, 2008; W
ard
& d
aY
, 2011).
ABSTRACT
The Alps are the location of potentially cata-
strophic landslide-generated tsunami. Numerical
modelling based on shallow water equations repre-
sents a valuable tool able to provide prediction in
order to assess this threat. However, some inherent
numerical problems (e.g. artefacts development when
applied to real cases, difficulty for wet to dry bed
transition) are well known, but not resolved yet. Our
main objective is to find a method that is relevant for
landslide-triggered tsunami modelling, as accurate as
possible in order to use it as predictive tool.
Thus, we investigated numerical models, based on
shallow water equations, which could run on high reso-
lution bathymetry. Accordingly, four Godunov-type
solvers are confronted to the exact solution. Interesting-
ly, at high resolution, which is needed for bathymetry,
the difference between monotonous first-order meth-
ods gradually disappears. The simplest Lax-Friedrichs
scheme is suggested as the method of choice. The
experiment of a 2D circular dam break, solved using
Lax-Friedrichs scheme, is presented and the resolution
at which results are converging is reported. Perform-
ance of high resolution 2D runs on CPU and GPU are
reported, documenting 50 fold speedup on GPU. As
result of this speedup, the high resolution run take less
than an hour on GPU card that cost less than 1000 SFr.
K
ey
words
: landslide triggered generated tsunami dam lake Alps
modelling shallow water equations Godunov Lax Friedrichs GPU
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M. FRANZ, Y. PODLADCHIKOV, M. JABOYEDOFF & M-H. DERRON
410
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
F
ig. 1 - The left wave and the right wave (S
l
and S
r
as ve-
locity) that can be rarefaction or shock waves. The
middle is a contact wave (S
m
as velocity)
The conservative discrete form is
where F
i+1/2
is the intercell numerical flux correspond-
ing to the intercell boundary at x
=
x
i+1/2
between cells
i and i+1.
The four numerical schemes are only different in
definition of the intercell flux F
i+1/2
.
The Lax-Friedrichs (LF) scheme uses (t
oro
,
2001, page 163):
The 2 step Lax-Wendroff (LW) scheme uses
(t
oro
, 2001, page 164):
The Godunov centred scheme (GC) uses (t
oro
,
2001, page 164):
GODUNOV UPWIND
The exact Riemann solution deals with tree type
of waves (Fig. 1). The region between left and right
waves is the star region. In this region h and u are
characterized by h* u*, for which a nonlinear algebra-
ic equation is solved by an iterative method. The flux
of the Godunov upwind scheme (Gup) is computed
for the exact local Riemann solution U
i+1/2
(0)
at every
contact between the cells (t
oro
, 2001):
We concentrate on the development of model
based on shallow water equations, since some limita-
tions were encountered. When applied to real cases,
development of numerical artefacts were reported
(W
ieczoreK
et alii, 2007; F
ranz
et alii, 2012). S
iMPSon
& c
aStelltort
(2006) describe a model formulation
based on the shallow water equations and numerical
algorithm to solve it based on the approximate Rie-
mann solvers allowing for dry bed transition. In K
re
-
Mer
et alii (2012) this approach was applied for real
bathymetry but without the wet to dry bed transition
(i.e. run-up). t
oro
(2001) is the most comprehensive
presentation of the numerical techniques for the shal-
low water equations. More recently, t
oro
& g
arcia
-
n
aVarro
(2007) review the remaining numerical
problems, in particular the treatment of the wet/dry
front propagation. z
iJleMa
& S
telling
(2008) states
that, in case of variable topography, the solutions of
nonlinear shallow water equations are confronted to
numerical difficulties, even with exact Riemann solv-
er. To resolve these numerical difficulties, we reiniti-
ated testing of the various methods described in t
oro
(2001) with emphasis on higher resolution amendable
by modern computers. We introduced additional tests
relevant for tsunami imitated by landslides and tests
for treating of variable bathymetry.
METHODS
The four numerical schemes tested here are: the
Lax-Friedrichs, the 2 step Lax-Wendroff (or Richtmy-
er), the Godunov centred, and the Godunov upwind
(t
oro
, 2001). Those are all based on the two-dimen-
sional shallow water equations
where U the solution vector, F and G the flux vectors
defined as
where h is the water depth, u and v are the components
of the depth-averaged velocity vector, g is the gravity
acceleration. Dimensional splitting is used to reduce it
to the augmented one-dimensional problem
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LANDSLIDE-TRIGGERED TSUNAMI MODELLING IN ALPINE LAKES
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
411
Generation of a dry bed (test 5)
This test is similar to the second test, which simu-
late a sudden aperture in the water body. In this test the
aperture reaches the bed and spontaneously generates
a dry bed. It tests the capacity of the codes to generate
a dry bed condition from a wet bed. It is considered as
the most difficult of the five tests (Tab. 2, test 5).
All methods, including LF, work well with high res-
olution. The LF scheme, which is known to be too diffu-
sive to be used in practice (t
oro
, 2001), becomes accept-
able when the resolution is increased over thousand grid
points. Moreover it is not oscillating and generally fits
correctly the exact solution. The LF scheme, due to its
simplicity, can be the best choice in order to model wave
generation and propagation over complex bathymetry.
GROUP 2: ADDITIONAL TESTS FOR LANDSLI-
DE-GENERATED TSUNAMI
The first five tests do not explore interplay of initial
water level jump (like in tests 1, 3 and 4) and divergent
velocity jump potentially leading to the spontaneous
formation of the dry state (like in tests 2 and 5). Simi-
larly to the tests 2 and 5, we introduce tests 6 and 7 in
such a way that test 6 does not lead to the spontaneous
dry state while test 7 does. Tests 6 and 7 are different
from tests 2 and 5 by allowing for a water level jump
in initial conditions. These additional tests are relevant
for the landslide-generated tsunami. Indeed, sliding
into shallow water landslide can be accommodated by
water level elevation on the top of the landslide and
divergent horizontal velocity laterally displacing the
water and creating the space for the landslide. The lat-
ter possibility may potentially lead to the spontaneous
generation of the dry state. Newly suggested tests 6
and 7 are proposed as a validation for the ability of a
numerical scheme to handle and to correctly predict
the spontaneous dry state around landslide.
TESTS
The tests used here are those proposed in t
oro
(2001) (tests 1 to 5, Tab. 2). These tests are relevant to
the dam break. As we are interested in landslide gener-
ated waves, we add three additional tests (6, 7 and 8 in
Tab. 2) that better capture the generation of the wave
by the penetration of a landslide in the water body.
GROUP 1: THE TORO (2001) TESTS
Left critical Rarefaction and Right Shock (test 1)
This test simulates a dam break over a wet bed in
order to test the speed of propagation, the absence of
oscillations, and the steepness of the front. Initial data
(Tab. 2, test 1) produce shock wave to the right and
rarefaction to the left.
Two rarefaction and nearly dry bed (test 2)
It simulates a sudden aperture in the water body.
The initial data (Tab. 2, test 2) generate two rarefac-
tion waves travelling in opposite directions. The water
depth at the created aperture is close to zero. The test
investigates the symmetry of propagation, the ability
of the schemes to sustain the near zero water depth
situation and smoothness of the velocities profiles.
Right dry bed Riemann problem (test 3)
Simulate a dam break with a dry bed on the right.
The initial conditions (Tab. 2, test 3) produce a rarefac-
tion wave. The test evaluates the capacity of a numeri-
cal scheme to handle the transition from dry to wet bed.
Left dry bed Riemann problem (test 4)
Simulate a dam break with a dry bed on the right.
The initial conditions (Tab. 2, test 4) produce a rar-
efaction wave. The test evaluates the capacity of a
numerical scheme to handle the transition from dry to
wet bed and negative velocities.
Tab 1 - GPU time for the runs presented at
Figure 7 (same physics and model
tested but different resolution)
Tab 2 - The initial parameters of the
tests, where hL, and hR are
the water level, u
L
and u
R
the
velocity, respectively on the
left and the right, x
0
the posi-
tion of the wall and Tout the
time of the simulation end
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M. FRANZ, Y. PODLADCHIKOV, M. JABOYEDOFF & M-H. DERRON
412
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
the left and a shock wave that propagates to the right.
The Figure 4 illustrates this test. The cross section is
assumed perpendicular to the slide direction. For this
test, as for the next two tests, there are only Gup and
the LF schemes are shown.
The moderate test (Fig. 4) is successfully
simulated by the both schemes (Gup & LF). The
main relevant aspect is that an increased resolu-
Test 6: Moderate test of landslide penetration
(100m depth, 1m/s velocity)
This test simulates the penetration of a landslide
into a deep water body at a relatively slow velocity.
The initial conditions (Table 2, test 6), as the low ini-
tial velocities, make this test similar to the test n° 1,
where the elevation of water due to the landslide be-
come affected by a rarefaction wave that propagates to
Fig. 2 - Four methods are com-
pared with the exact Rie-
mann solution, respec-
tively from tests 1 to test 5
(from top to bottom) at a
resolution of 2000 numer-
ical grid points. At such a
resolution, all the meth-
ods fit quite well to exact
solution. The LF is the
most numerically diffused
solution (velocity & eleva-
tion). There are only two,
LF and Gup, that show no
numerical oscillations
Fig. 3 - Close-ups on the shock waves (elevation and velocity) of the test 1.Left: grid resolution is 500; Right: grid resolution is 2000.
Only Gup and LF schemes are not oscillatory. The LF is the most diffusive solution, but it captures the wave location well
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LANDSLIDE-TRIGGERED TSUNAMI MODELLING IN ALPINE LAKES
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
413
Test 8: a rough bed
This case tests the capacity of the methods to
simulate the propagation of the waves over a rough
bed (Fig. 6). This is a crucial point for a reliable nu-
merical model for landslide-generated tsunami over
real bathymetry. The Gup and the LF methods suc-
cessfully passed this rough bed test (Fig. 6). Again
the solutions at higher resolution are less diffusive.
Moreover, both schemes have passed the consist-
ency test: rough bathymetry does not cause wave
motion if the water level is flat and initial velocity
is zero. We conclude that both schemes are admis-
sible while having similar computational cost. LF
is preferred due to considerable simplicity of its
implementation.
tion reduces numerical diffusion for both numeri-
cal schemes and reduces the difference between the
two numerical solutions.
Test 7: Extreme test of landslide penetration (0.1 m
depth, 6 m/s velocity)
This test, presented in Figure 5, simulates, as the
test 6, the penetration of a landslide in a water body,
except that the water is very shallow (0.1 m) and the
initial velocity is faster (6 m/s). The initial data (Table
2, test 7) induce lateral displacement of the water lead-
ing to a spontaneous dry state like test 5.
This extreme test is simulated well by both meth-
ods (Fig. 5). Again we observe a better fit and less
diffusion with a higher resolution (here 2000).
Fig. 5 - Results of the extreme test 7 for Gup and
LF methods. Top: grid resolution is 200;
Bottom: grid resolution is 2000
Fig. 4 - Results of the moderate test 6 for Gup
and LF methods. Top: grid resolution is
200; Bottom: grid resolution is 2000
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M. FRANZ, Y. PODLADCHIKOV, M. JABOYEDOFF & M-H. DERRON
414
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
GROUP 3: RESOLUTION TESTS IN 2D
Four runs of growing resolution are computed in
2D in order to confirm the positive effect of the reso-
lution on the numerical diffusivity of the simplest LF
scheme. They are performed on a regular worksta-
tion to check if needed high resolution is practically
achievable. The Figure 7 presents the results of the
exactly the same experimental setup (circular dam
break) conducted at different grid resolutions.
The Figure 7 illustrates that the diffusive problem
decreases with the increase of the resolution. The run
time rises with the resolution (see Table 1). We ob-
serve that GPU computing is practical already
SUMMARY
We have conducted three different groups of
tests. The group 1 consisting of five tests from t
oro
(2001) have been successfully passed by all four nu-
merical schemes considered. Despite the fact that the
LF method is commonly regarded as too diffusive
and not practical (e.g. t
oro
, 2001), it appears that the
LF method, together with the Gup method, are the
best solutions, considering that they are not oscillato-
ry. The groups 2 and 3 continued the testing with only
these two methods. In our second testing suit (group
2) consisting of tests 6, 7 and 8, we have considered
more difficult numerical scenarios related to land-
slides-triggered tsunami problems. Running these
tests with different resolutions, it appears clearly that
a higher resolution solves the diffusive problems. The
test 8 demonstrates robustness of both LF and Gup
over rough bed. Finally, the tests of the group 3 shows
that very high resolution is affordable, even with 2D
models, when computed on GPU.
With ever growing computer power, the choice of
the method can be reconsidered. Indeed the LF meth-
od can be a good choice because: it is not oscillatory,
its diffusive problems disappear with high resolution,
it withstands rough beds, and its simplicity.
CONCLUSION
Landslide-triggered tsunami in alpine lakes or
reservoirs can potentially lead to catastrophic events.
Modelling this phenomenon is necessary in order to
assess its consequences, but it is known to be prob-
Fig. 6 - Results of the rough bed test 8 for Gup
and LF methods. Top: grid resolution is
200; Bottom: grid resolution is 2000
Fig. 7 - Results of four resolution tests in 2D
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LANDSLIDE-TRIGGERED TSUNAMI MODELLING IN ALPINE LAKES
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
415
This study suggests modern reassessment of the Lax-
Friedrichs method. Indeed, with modern high resolu-
tion computing, this method is no more too diffusive,
captures correct propagation velocities, non-oscilla-
tory and easy to implement.
lematic. This paper compares four numerical meth-
ods with the exact Riemann solution, proposes three
new tests directly linked to the landslide-generated
tsunami and investigates the effects and the feasi-
bility of (very) high resolution 2D modelling tests.
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M. FRANZ, Y. PODLADCHIKOV, M. JABOYEDOFF & M-H. DERRON
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