Tsunami Squares Approach to Landslide-Generated Waves ...ward/papers/Tsunami Squares-final.pdf ·...
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Tsunami Squares Approach to Landslide-Generated Waves: Application to Gongjiafang
Landslide, Three Gorges Reservoir, China
LILI XIAO,1 STEVEN N. WARD,2 and JIAJIA WANG1
Abstract—We have developed a new method, named ‘‘T-
sunami Squares’’, for modeling of landslides and landslide-
generated waves. The approach has the advantages of the previous
‘‘Tsunami Ball’’ method, for example, separate, special treatment
for dry and wet cells is not needed, but obviates the use of millions
of individual particles. Simulations now can be expanded to spatial
scales not previously possible. The new method accelerates and
transports ‘‘squares’’ of material that are fractured into new squares
in such a way as to conserve volume and linear momentum. The
simulation first generates landslide motion as constrained by direct
observation. It then computes induced water waves, given as-
sumptions about energy and momentum transfer. We demonstrated
and validated the Tsunami Squares method by modeling the 2008
Three Gorges Reservoir Gongjiafang landslide and river tsunami.
The landslide’s progressive failure, the wave generated, and its
subsequent propagation and run-up are well reproduced. On a
laptop computer Tsunami Square simulations flexibly handle a
wide variety of waves and flows, and are excellent techniques for
risk estimation.
Key words: Tsunami squares, numerical modeling, landslide,
waves in reservoirs.
1. Introduction
Large, high-velocity landslides entering confined
water bodies (reservoirs, lakes, fiords, and rivers) may
cause potentially extreme tsunami run-up that in-
creases the variety, range, and severity of effects
attributable to the landslide alone. Alaska’s 1958 Li-
tuya Bay tsunami caused by a subaerial landslide has
attracted the attention of scholars, and a series of ex-
perimental and numerical modeling research projects
has been reported (BASU et al. 2010; FRITZ et al. 2009;
WEISS et al. 2009). Two-dimensional cross-sections of
the landslide and the proximal wave were modeled
numerically by MADER and GITTINGS (2002) and QUE-
CEDO et al. (2004) with full Navier–Stokes
hydrodynamic codes. WARD and DAY (2010) devel-
oped a novel ‘‘Tsunami Ball’’ method, which they
demonstrated and validated by simulating the land-
slide and the propagation of the tsunami to the
seaward end of Lituya Bay. WARD (2014) (https://
www.youtube.com/watch?v=6COeNRToYqU) revis-
ited this case by using the Tsunami Squares method
described herein.
Other examples of well-documented rockslide
tsunami include the 1934 Tafjord event in Norway, for
which HARBITZ et al. (2014a, b) recently applied nu-
merical models (denoted GloBouss and DpWaves)
based on Boussinesq equations. A similar Boussinesq
model (Geowave) has also been applied to several
examples of landslide-generated waves, mainly for
open-ocean submarine landslides (WATTS et al. 2003;
POISSON and PEDREROS 2010; WATTS and TAPPIN 2012).
We have developed a new method, named ‘‘T-
sunami Squares’’, for modeling of landslides and
landslide-generated waves. Tsunami Squares can
simulate landslide evolution, wave propagation, the
moving mass of impacting water, and the triggering
wave process. Tsunami Squares is suited both to long-
term hazard assessment in regions where many thou-
sands of possible landslides might need to be modeled,
for example China’s Three Gorges Reservoir, and to
emergencies in which accelerating creep of an unsta-
ble slope has been detected and emergency managers
require a rapid assessment of the tsunami hazard.
2. Overview of the Gongjiafang Landslide
On the afternoon of November 23, 2008, 120 km
up-river of China’s Three Gorges Dam and 4 km
1 Engineering Faculty, China University of Geosciences
(Wuhan), Wuhan, China. E-mail: [email protected] Institute of Geophysics and Planetary Physics, University
of California, Santa Cruz, USA.
Pure Appl. Geophys.
� 2015 Springer Basel
DOI 10.1007/s00024-015-1045-6 Pure and Applied Geophysics
downstream of Wushan County (Figs. 1, 2), the
Gongjiafang landslide slipped from the north shore of
the Yangtze and generated a wave that crossed the
river and washed the far bank to a height of 13 m.
The tsunami damaged the Wuxia town docks 3.5 km
upstream, some navigation aids, roads, orange trees,
and a few small boats. Luckily, no ships were passing
through the busy river corridor, and no one was hurt.
The direct economic loss was limited to 800,000
USD (HUANG et al. 2012).
The Gongjiafang landslide happened after a test
impoundment brought the Three Gorges Reservoir to
a new high water level of 172.8 m. The landslide was
located on a scarp slope on the west of Hengshixi
anticline and the east of Wushan syncline (WU et al.
2010). The slide mass consisted of thinly-bedded soft
marl stones intersected with thick limestone and
dolomite limestone. Because of strong weathering
and joint damage, the mass had been fractured into
small rocks and vegetated soils. The high shale con-
tent, large developed fissures, and strong weathering
resulted in numerous discontinuities in the rock. In
November, the lower part of the slide mass was
submerged as the reservoir level rose to its new high
level. The clays in the marlstone and shale softened
easily, aggravating a decrease of rock strength. After
being submerged for a long period, the shallow rock
structure at the slope toe began to fail along tension
joints and the upper dry part soon followed.
3. Tsunami Squares Theory
The two-dimensional Tsunami Squares approach
evolved from the ‘‘Tsunami Ball’’ method which
proved to be effective for simulating flow and flow-
like movements (WARD and DAY 2005, 2008,
2010). Tsunami Squares inherits advantages of the
Figure 1Location of Gongjifang Landslide. The lowest map shows the area 6 km up and down river from the landslide in which the tsunami was
modeled. Populated Wushan County, on the left, is the area affected
L. Xiao et al. Pure Appl. Geophys.
Tsunami Ball method, for example not needing
special treatment for dry versus wet cells. Moving
materials in the Tsunami Squares method are,
however, made from divisible squares, which ob-
viates the need for millions of individual particles.
Simulations can be extended to spatial scales not
previously possible.
3.1. Mass Flow and Wave Propagation
Typical tsunami calculations for earthquake and
landslide-generated tsunamis (SATAKE and TANIOKA
1995; LIU et al. 2003; TITOV and GONZALEZ 1997)
solve non-linear, long wave continuity, and momen-
tum equations for variations in water column
thickness H(r,t) and depth averaged horizontal water
velocity v(r,t) at points r = (x,y)
oHðr; tÞot
¼ �rh � vðr; tÞHðr; tÞ½ � ð1Þ
and
oHðr; tÞvðr; tÞot
¼ �rh � vðr; tÞvðr; tÞHðr; tÞ½ �� gHðr; tÞrhfðr; tÞ ð2Þ
or
ovðr; tÞot
¼ �vðr; tÞ � rhvðr; tÞ � grhfðr; tÞ
where g is the acceleration of gravity, rh is the
horizontal gradient, f(r,t) is the elevation of the wa-
ter, and t is time. (To relax the long wave assumption
and include wave dispersion, replace Eq. (2) by
Eq. (22) in Appendix 1. For our purposes here, water
wave dispersion is unnecessary.)
For small steps dt, Eqs. (1) and (2) become:
Hðr; t þ dtÞ ¼ Hðr; tÞ � rh � vðr; tÞHðr; tÞ½ �dt ð3Þ
and
Hðr; t þ dtÞvðr; t þ dtÞ ¼ Hðr; tÞvðr; tÞ� rh � vðr; tÞvðr; tÞHðr; tÞ½ �dt
� gHðr; tÞrhfðr; tÞdt
ð4Þ
Tsunami Squares solves equations that are
equivalent to these, but by use of a new approach.
It considers a regular set of N square cells with
dimension Dc (the length of the uniform squares) and
center points ri = (xi,yi). At time t, each cell holds
material (water or landslide mass) of thickness
Hi(t) = H(ri,t), with mean horizontal velocity
vi(t) = v(ri,t) and mean horizontal acceleration
ai(t) = a(ri,t) (Fig. 3, top left). For locations outside
the flow, Hi(t) would be zero. The entire concept of
wave propagation or mass flow involves updating
those conditions to time t ? dt, where dt is some
small time interval.
Pick one cell, for example i = 10 (red square,
Fig. 3, top right). With its known velocity and
acceleration, displace the cell material to a new
center point:
~ri ¼ riðtÞ þ viðtÞdt þ 0:5aiðtÞdt2
¼ riðtÞ þ 0:5 ½viðtÞ þ ~vi�dt
¼ ½~xiðtÞ; ~yiðtÞ�ð5Þ
and give it a new mean velocity:
~vi ¼ viðtÞ þ aiðtÞdt
¼ ½~vxiðtÞ; ~vyiðtÞ�ð6Þ
we wish to partition the volume and linear momen-
tum of the material in the displaced cell among the
N original cells. It is logical that the partitioned
volume of the displaced ith cell into the jth original
cell is:
Figure 2Landforms at Guongjiafang after the slide (Yichang Center of
China Geological Survey). The slide stopped funnel-shaped on the
riverbed, mainly distributed below 200 m but with thin deposits
above 220 m. The numbers in the red boxes show the positions
where slide velocities were tracked during modeling
Tsunami Squares Approach to Landslide
dVji ¼ ðHiD2cÞ 1�
~xi � xj
��
��
Dc
� �
1�~yi � yj
��
��
Dc
� �
if~xi � xj
��
��
Dc
\1 and~yi � yj
��
��
Dc
\1; otherwise dVji ¼ 0
ð7Þ
The product of the two terms on the right of Eq. (7)
is simply a statement of the fractional area overlap of
the ith displaced cell with the jth fixed cell. Clearly
there is no need to run the partitioning through all
j = N cells because, at most, only four fixed cells
overlap the displaced cell (Fig. 3, bottom left). More-
over because ~ri is known and the cells are square, it is
simple to determine which four cells overlap.
Partitioning of vector linear momentum follows in
the same way:
dMji ¼ ðqwHiD2c ~viÞ 1�
~xi � xj
��
��
Dc
� �
1�~yi � yj
��
��
Dc
� �
if~xi � xj
��
��
Dc
\1 and~yi � yj
��
��
Dc
\1; otherwise dMji ¼ 0
ð8Þ
The N updated thickness Hj(t ? dt) and velocity
vj(t ? dt) values in the jth fixed cell comes from
summing and normalizing the volume (Eq. 7) and
momentum (Eq. 8) contributions from all i displaced
cells:
Hjðt þ dtÞ ¼PN
i¼1 dVji
D2c
ð9Þ
vjðt þ dtÞ ¼PN
i¼1 dMji
qwD2cHjðt þ dtÞ ð10Þ
Figure 3Tsunami Squares wave propagation and/or flow simulation concept
L. Xiao et al. Pure Appl. Geophys.
Because only four of dVji and dMji are non-zero
for each i, the sums in Eqs. (9) and (10) involve
4N terms (not N2). Actually, there may be fewer than
4N terms because there is no need to displace and
partition cells that are dry.
This process has time-stepped a wave propagation
and/or flow simulation on a fixed set of cells while:
1. Conserving material volume exactly. It is possible
to verify that the sum of the four non-zero
partitioned volumes dVji in Eq. (7) is equal to
(HiDc2), the volume of the displaced cell. Equa-
tion (9) simply replaces the ‘‘continuity equation’’
common to most numerical approaches (SATAKE
AND TANIOKA 1995; LIU et al. 2003; HEINRICH 1992;
WALDER et al. 2003) by tracking material from one
cell to another.
2. Conserving linear momentum exactly. dMji ¼qwdVji ~vi sums to the final momentum of the
material in the ith cell. By tracking momentum
from one cell to another, Eq. (10) replaces the
advected component �rh � vðr; tÞvðr; tÞHðr; tÞ½ �in the momentum equation (Eq. 2). For high-speed
landslides and flows, conservation of momentum
is critical for constructing realistic simulations.
For deep water wave propagation, however, fluid
velocities are small and advected momentum is
negligible. This aspect of Tsunami Squares can be
linearized by replacing Eq. (10) by vjðt þ dtÞ ¼ ~vj
where ~vj is determined by use of Eq. (6)
3. Requiring no special treatment of dry cells or any
mention of topography.
4. Reducing a N2 summation to a 4N summation.
5. Obviating the need for a single numerical
derivative.
To show that Eqs. (9) and (10) are equivalent to
Eqs. (3) and (4), Eq. (9) is evaluated with dt very
small (dt � 1) so that vx(r,t)dt � Dc and
vy(r,t)dt � Dc for all cells, and terms with dt2 are
-0
H(rj;t þ dt) = H(rj;t)� H(rj;t)vxðrj;tÞ��
��
Dc
+vyðrj;tÞ��
��
Dc
� �
dt
þXN¼j
i¼1
H(rj,t)vxðri;tÞj j
Dc
� �
dt +XN¼j
i¼1
H(rj,t)vyðri;tÞ��
��
Dc
� �
dt
ð11Þ
The first RHS term in Eq. (11) is the original
material in cell j. The second RHS term in Eq. (11) is
the material originally in cell j that has moved into
adjacent cells. The sums in Eq. (11) only include
those cells in the x and y directions that overlap the
jth cell after their displacements (Eq. 5). They
account for the material originally in adjacent cells
that has moved into cell j. Another statement of
(Eq. 11) is:
Hðrj; t þ dtÞ ¼ Hðrj; tÞ � r � vðrj; tÞHðrj; tÞ� �
dt
ð12Þ
Hence, Eq. (11) is exactly equivalent to Eq. (3)
for vanishingly small dt. We can evaluate Eq. (10) in
the same way:
H(rj;t þ dt)v(rj;t) = H(rj;t)~v(rj;t)� H(rj;t)v(rj;t)
�vxðrj;tÞ��
��
Dc
+vyðrj;tÞ��
��
Dc
� �
dt
+XN 6¼j
i¼1
H(rj,t)v(rj,t)vxðri;tÞj j
Dc
� �
dt
+XN 6¼j
i¼1
H(rj,t)v(rj,t)vyðri;tÞ��
��
Dc
� �
dt
ð13Þ
The first RHS term in Eq. (13) is the original
momentum of cell j. The second RHS term in
Eq. (13) is momentum originally of cell j that was
transferred to adjacent cells. The sums in Eq. (13)
account for momentum originally in adjacent cells
transferred to cell j. Equation (13) is equivalent to the
first three terms in Eq. (4) for vanishingly small
dt. Tsunami Squares updates flow velocities through
the slope of the surface (third RHS term in Eq. (4)) as
a separate step.
Tsunami Squares satisfies the same non-linear
continuity and momentum equations used in tradi-
tional methods, but does so in a way that is more
intuitive (by just moving cell mass and momentum by
appropriate partitioning).
3.2. Gravitational Acceleration
To complete the time step, it is necessary to
update mean cell accelerations ai(t). As is customary
Tsunami Squares Approach to Landslide
in ‘‘long wave’’ theory, the mean acceleration of
material in the cell is proportional to the slope of
upper surface f(ri,t):
aiðtÞ ¼ agðri; tÞ ¼ �grhfðri; tÞ¼ �grh TðriÞ þ Hðri; tÞ½ �
ð14Þ
H(ri,t) = Hi(t) is material thickness found above,
T(ri) is the fixed topography over which it moves, g is
the acceleration of gravity, and rh is the horizontal
gradient (Fig. 4).
The rhf(ri,t) in Eq. (14) is the only step in which
a numerical derivative must be evaluated. Even this
sole differentiation can, however, be avoided by
fitting a plane to f(ri,t) and its eight adjacent
neighbors, then fixing the horizontal gradient from
the slope of that surface. This plane-fitting approach
helps stabilize the calculation by estimating the
gradient across a two-dimensional region as a func-
tion of adjacent points alone. Another advantage of
the plane-fitting approach is the ability to ‘‘punch
out’’ specific locations near ri by excluding them
from the fit. Where wet cells are near dry ones, the
dry sites would normally be ‘‘punched out’’ during
calculation of the slope of the surface f(ri,t): For
example, if a steep dry cliff was adjacent to a wet
area, the cliff surface would not be included when
computing the slope of the fluid.
In most cases the computation grid is made
sufficiently large that flows or waves do not reach the
ends of the domain in the time period of interest. If
need be, an ‘‘absorbing buffer zone’’ can be included
near domain walls to minimize reflections.
4. Landslide Simulation
A difficulty in simulating landslides is that they
behave partly like a solid, because of a cohesive
fraction, and partly like a fluid, because they flow into
valleys and channels (COUSSOT and MEUNIER 1996).
Landslides have solid characteristics when they ini-
tially fail and when they near a stop (HUNGR and
MCDOUGALL 2009). In between, after a few seconds of
acceleration and before the final seconds of de-ac-
celeration, slide material loses cohesion and takes on
fluid characteristics. This is why Tsunami Squares
does not differentiate fluids from moving slide ma-
terial. For landslide material considered solid during
the first few seconds, the driving force on the square
depends on the slope of the bottom surface. For
landslide material regarded as fluid-like, the driving
force on the square depends upon the slope of the top
surface (Eq. 14).
4.1. Frictional Acceleration
Two types of frictional acceleration are added to
Eq. (14) to oppose sliding, one due to basal friction
(ab) and one due to dynamic friction (ad):
aiðtÞ ¼ agðri; tÞ þ adðri; tÞ þ abðri; tÞ ð15Þ
Basal friction is the simple static resistance of the
sliding surface because of interactions between
moving materials and the rough bed. It depends on
material type, solid fraction, bed roughness, and
normal stress. The acceleration of a cell of material as
a result of basal friction is:
Figure 4Geometry of slide mass and water flow. Tsunami Squares considers moving masses more fluid-like than solid, so the upper surface horizontal
gradient drives the flow for both the slide and the water
L. Xiao et al. Pure Appl. Geophys.
abðri; tÞ ¼ �lbgv̂slideðri; tÞ ð16Þ
Here, v̂slide is the unit velocity vector and lb is the
basal friction coefficient. Basal friction is treated as a
solid-like moving resistance.
Dynamic friction originates from resistance en-
countered by the moving bulk material through air or
water. Dynamic resistance is intrinsically velocity-
dependent. ‘‘V-square’’ friction, as it is called,
originates from pressure or viscous-like forces acting
on the top and bottom surfaces of moving slides of
thickness H(ri,t):
adðri; tÞ ¼ �ldvðri; tÞ vðri; tÞj j=Hðri; tÞ ð17Þ
Here, ld is the dynamic friction coefficient that
expresses all velocity-dependent properties of particle
interaction. Because dynamic friction increases as
|v2|, it imparts a terminal velocity to motion, with
thicker slides attaining a higher speed. It is weak
during landslide initiation and stoppage, but is
dominant for high-speed moving masses.
Both basal and dynamic friction act to slow the
slide, but are not allowed to reverse the sliding
direction. ld and lb can be functions of time and
space and made as complicated as necessary. For
example, ld of subaerial slides might be less than ld
for subaqueous slides; lb might change from a low
value to a high value as the velocity of the slide falls
below a critical value (WARD and DAY 2006).
Alternatively, the transition could be determined by
an ‘‘angle of repose hrepose’’ of the solidifying
material. When the slope of the top surface drops
below a specific angle, basal friction grows.
Transition between two types of friction enables
Tsunami Squares to embrace the dual behavior of
landslides. When slides move at high speeds, fluid
behavior and dynamic friction dominate and the slope
of the upper surface drives the motion. When slow or
stopped, solid behavior and basal friction dominate.
4.2. Observational Constraints on the Landslide
The landslide model incorporates several obser-
vational constraints: the topography of the bed, the
initial and final landslide shape, and the initial and
final landslide thickness. The Yangtze River, at an
elevation of 173 m in this region, has a generally
parabolic cross-section 470 m wide and 122 m deep.
Freshly exposed slip surfaces on the north bank
reveal the isosceles triangular outline of the landslide
(Fig. 2). The slide was 40 m wide at the top and
160 m wide at the water surface. It formed an upslope
from 120 to 400 m above sea level (350 m above the
deepest part of the river bed) with a horizontal
distance of 290 m from the shoreline (Fig. 5), and
dipped steeply with upper and lower slope angles of
63� and 44�. From measured centerline values, we
interpolated landslide thicknesses across the slide
face, arriving at a total volume of 380,000 m3 and an
average thickness of 17 m. Slide deposits were
distributed at elevations from 210 to 50 m above
sea level, with the thickest part at approximately
90 m; the toe of the landslide deposit is at the floor of
the river bed.
We are also fortunate to have an eyewitness video
of the landslide. This shows that the lower part of the
slide slipped first, followed by the upper part three
seconds later, and that the whole sliding event lasted
approximately 31 s. According to field investigation
the landslide is divided into two parts, below and
above 240 m. The program imposes sequential fail-
ure that keeps the upper part fixed for the first 3 s and
then releases it.
4.3. Landslide Simulation Results
4.3.1 Model Setup
We ran the simulations on 10 m digital topography
that defines the dimension of the cells, Dc = 10. In
general, to maintain reasonable resolution in
Eqs. (5)–(10), it is best to select a time step, dt, such
that vrep 9 dt \ Dc where vrep is some representative
peak velocity of the flow. Because vrep is unknown
before the fact, experimentation is needed to select
dt. In these calculations we took dt = 0.5 s. For
values of friction we set dynamic friction ld equal to
0.05 and 0.2 for landslide material moving on dry
land and under water, respectively. For basal friction
lb gradually changes from 0.2 to 6.0 when the slope
angle becomes less than the 31� angle of repose
measured from the final landslide shape. These values
of friction were derived after several runs to best fit
the observed field data.
Tsunami Squares Approach to Landslide
4.3.2 Model Results
Figure 6 shows instantaneous profiles of the middle
cross-section and plane view of the slide at four times
from start to finish (a Quicktime animation of the
landslide is available at http://es.ucsc.edu/*ward/
Yangtze1.mov). Rainbow colors indicate sliding ve-
locity. At t = 4 s, the speed of the middle part
exceeded that of the upper part, a consequence of the
sequential failure assumption. At t = 10 s, the upper
part, now moving faster than the lower parts, helps to
push the whole mass deeper into the river. The model
slide moved downward for 26 s, stopping on the
riverbed with a uniform angle of repose and only a
thin layer depositing above water (Fig. 6, lower right
and left), consistent with the behavior of the real
landslide (Fig. 5).
Impact velocity and landslide shape are critical
variables for calculation of wave heights in the
laboratory (FRITZ et al. 2003). We tracked slide-
impact velocities near the river surface and on the
centerline above and below water (five squares in
Fig. 2 (right) and five circles in Fig. 6 (left)).
Figure 7 shows the velocity curves of the tracked
spots. The orange line that follows the above-water
portion at 240 m elevation (#4) has the highest peak
speed of 21 m/s. For the initial four seconds,
however, speeds there are actually slower than in
other places before increasing sharply to peak at
approximately nine seconds. The initial delay reflects
the applied sequential bottom-to-top failure (Fig. 8).
Figure 5Engineering geological profile of the Guongjiafang landslide. The initial and final surfaces provide the thickness of the central section and
residuals
Figure 6Moving landslide profiles at t = 0, 4, 10 and 30 s. For the plane
views (left), background colors brown and red indicate land above
and underwater. The blue to white shading indicates the thickness
of the landslide, thick to thin. The arrows indicate slide direction
and speed. Arrow colors follow the rainbow legend. The red spots
mark the positions where velocities are being tracked in Fig. 7.
Points 1, 2, and 3 are at the water surface (173 m). Points 4 and 5
are above and under water, at 240 m and 140 m. For the cross
sections (right), the rainbow colors represent landslide speeds from
0 to greater than 15 m/s. The lower part of the slide has a higher
speed during first few seconds. The upper part moves fastest later
because of higher gravitational potential energy. A Quicktime
animation of the landslide is available at http://es.ucsc.edu/*ward/
Yangtze1.mov
c
L. Xiao et al. Pure Appl. Geophys.
Tsunami Squares Approach to Landslide
At the water surface (locations #1, 2, and 3),
velocity on the left (#2) is much lower and falls to
zero faster than the other two. This is because of
differences in slide thickness and upslope slide
extent. V-square friction (18) offers resistance in-
versely proportional to slide thickness. As apparent in
the left column of Fig. 6, thin bits near slide edges
move slower than thicker bits near the middle.
Upslope extent also affects slide speed and duration.
The more material upslope, the longer it takes to pass
by a given place. The higher on the slope it originated
the higher velocity it acquires in transit. After the
slide mass has completely passed by at the left,
material continues to pass at the middle and right. As
a result, more deposits pile on the middle and right
sides, which agrees with the observed residual
(Fig. 2).
With regard to peak speeds on the centerline (#1,
4, and 5), the above-water position (#4) ranks highest
and the underwater position (#5) ranks lowest. This is
expected, because greater frictional resistance was
applied underwater. At t = 20 s, velocities at posi-
tions #4 and #5 vanish, but masses still pass by
position #1. The last few moving masses slip into the
water there but they stop just below the surface.
Previous analysis of the witness video (HUANG
et al. 2012) inferred a peak slide speed of 11.65 m/s,
somewhat lower than our value. HUANG et al. (2012)
quantified the velocity of the uppermost part of the
landslide, but we know the landslide deformed as it
moved with different velocity profiles at different
locations (e.g. Fig. 7). Measured velocity at the slide
top does not necessarily reflect the slide velocity at
the water level, where waves are produced.
5. Wave Generation, Propagation, and Inundation
5.1. Wave Sources
Given an initial distribution of still water H(ri,t)
and the landslide model computed above, there are
Figure 7Slide velocities at five positions above, below, and at the water level. The blue, purple, red, orange, and green lines indicate velocities at
positions 1–5 in Fig. 2
Figure 8Landslide tsunami-generating mechanisms. ‘‘Lift-up’’ (NMT) and
‘‘drag-along’’ (DA) are the main sources of wave generation
L. Xiao et al. Pure Appl. Geophys.
several ways to introduce waves. The classical
approach simply lifts or drops the water by an
amount equal to the passing slide thickness. Gravita-
tional energy is imparted to the water in this way, but
there is no direct transfer of momentum from the
slide to the water. We call this approach ‘‘no
momentum transfer (NMT)’’. NMT may be adequate
for some tsunami sources, for example submarine
earthquakes, but for high speed slides into water it
does not suitably describe wave generation.
Here, in addition to NMT, we introduce ‘‘drag-
along (DA)’’. DA assumes that extra forces exist at
the water–landslide interface. These forces act to
slow the slide but also accelerate the water, much like
anti-friction. For a submarine landside moving at
velocity vs(ri,t), drag-along acceleration of an over-
lying water layer of thickness H(ri,t) would be:
adaðri; tÞ ¼ cdavsðri; tÞ vsðri; tÞj j=Hðri; tÞ ð18Þ
The drag-along coefficient, cda, may or may not
equal the dynamic friction coefficient ld. Unlike
friction (Eq. 17) that acts only in the direction
opposite to fluid flow, DA (Eq. 18) can accelerate
the flow in any direction that the slide is moving. DA
transfers momentum from the slide to the water and
enhances wave production beyond that of NMT
alone.
5.2. Observational Constraints on the Tsunami
Field workers record tsunami heights at the shore
in two ways: measurement of the wave trail on trees,
docks, and structures, or measurement of the trace of
the dry land–wet land transition. The former records
wave height as it reaches those objects (this is
denoted ‘‘flow depth’’). The latter reveals the highest
elevation the wave reaches on shore (run-up height).
The two types of measurement can lead to sig-
nificantly different results, especially where tsunamis
inundate complex terrain, for example that around the
Gongjiafang site.
A field investigation of wave run-up was con-
ducted by two groups soon after the landslide (DAI
et al. 2010; HUANG et al. 2012). They did not reveal
the methods used for measurement nor did they
indicate precision or local variability. We note the ten
surveyed values from the two groups in Figs. 11 and
12. Wave run-up decayed both upstream and down-
stream from the landslide, and the further from the
landslide, the slower was the rate of decrease. Waves
ran up highest on the north shore where the landslide
occurred. Three-hundred and 4,000 m upstream on
the north shore the impulse wave ran up 13.1 and
1.1 m, respectively.
5.3. Tsunami Simulation Results
5.3.1 Model Setup
Using the same 10-m spaced digital topography as for
the landslide simulation, the Yangtze reservoir was
filled to 173 m elevation extending 6 km up and
downstream and the simulated Gongjiafang landslide
was allowed to fall into the River. For water flows,
both dynamic and basal friction, ld and lb, are equal
to 0, except near the shore where, ld increases to
0.02, because of the higher friction there. For this
study we set the drag-along coefficient, cda, at 0.2.
5.3.2 Model Results
Within seconds, the landslide ‘‘pushes up’’ and
‘‘drags along’’ the water to form a tsunami that
spreads out over the river (Figs. 9, 10). The wave
reaches the opposite bank in approximately 15 s, runs
up on the land, then flows back as a reflected wave.
The return wave peaks over the landslide bank at
t = 61 s. The orange curve in each part of Fig. 9
shows the maximum flow depth up to that time along
the river cross-section.
Figure 10 shows a plot of the tsunami waveforms
at three positions (#1, #2, and #3 in Fig. 9). In
general, the first wave crest is higher than the later
ones. The 17-s interval between the first crests of the
blue and green curves corresponds to the cross-river
propagation time from point #1 to point #3. For long
waves, the cross-river transit time Dt can be ex-
pressed as:
Dt ¼Z#3
#1
ds=ffiffiffiffiffiffiffiffiffiffiffi
ghðsÞp
ð19Þ
where h(s) is the water depth, g is the acceleration of
gravity, and s is the along-path position. For known
Tsunami Squares Approach to Landslide
Figure 9Waves propagate and reflect in river cross section at the 4th, 10th, 20th, and 61st seconds. The orange lines depict the highest flow height and
have been exaggerated tenfold for better visualization. The yellow spots 1 (right), 2 (middle), and 3 (left) on the water surface mark positions
where flow height is tracked in Fig. 10. A Quicktime animation is available at http://es.ucsc.edu/*ward/Yangtze2.mov
L. Xiao et al. Pure Appl. Geophys.
water depth, Eq. (19) gives a cross-river transit time
of 17 s, which perfectly matches the modeling result.
The dominant wave period at sites #1 and #3 is
32 s. Wave crests at #1 correlate with troughs at #3,
and vice versa. The dominant period at mid-river site
#2 is 16 s, much less than at the river edge sites.
These features fit the character of standing wave
modes. The period of the nth river mode is:
Tn ¼ 2Dt=n ð20Þ
with Dt calculated by use of Eq. (19). n/2 corresponds
to the number of wavelengths that fit, bank to bank.
Figure 10Wave trains at three positions on a cross-river line. The locations are shown in Fig. 9
Figure 11Extent of the river tsunami after T = 7, 20, and 50 s, and 2:33 and 4:01 min. Numbers in circles show run-up heights in decimeters. Numbers
beside red spots are field data, also in decimeters. A Quicktime animation is available at http://es.ucsc.edu/*ward/Yangtze3.mov
Tsunami Squares Approach to Landslide
The fundamental (‘‘slosh’’) mode (n = 1) has a pe-
riod of T = 32 s with peak amplitudes of opposite
sign at the river banks, as is apparent from the blue
and green curves in Fig. 9. The n = 1 mode has a
node at the river center, so this oscillation is not seen
in the red curve. The n = 2 overtone has a peak in
mid-river and a period of T = 16 s. It is the largest
contributor to the red curve.
Landslide-generated tsunami transit rapidly and
negatively affects Yangtze shores for many kilome-
ters in our model, as in the real event. Figure 11
shows a map of wave propagation at five typical
moments. Within 7 s, slide masses hit the river and
water waves radiate outward in an arc (Fig. 11a).
They cross the river and wash the far bank in
approximately 20 s (Fig. 11b). Within 50 s the waves
have propagated a distance of 2 km (Fig. 11c). In
2.5 min the first wave reaches Wushan County
(Fig. 11d). For several minutes afterwards, the signal
echoes between shores before dying out (Fig. 11e).
Collection of simulated run-up heights along
shore line enables preparation of a wave-decay curve
(Fig. 12). This includes broad variations with dis-
tance especially within the first 4 km up and down
river. These variations probably result from interfer-
ence of many positive and negative waves reflected
from the curved river shores (Fig. 11a, for example).
Some sites are naturally prone to higher or lower run-
up because of local geography. Likewise, waves run-
up higher in such narrowing water channels as valley
and branch stream outlets than at straight shore
locations.
The highest wave run-up in the simulation,
17.2 m, was located near the landslide. Maximum
run-up height on the opposite bank was 14.2 m. The
wave height had dropped to 0.4 m by the time it
reached the docks at Wushan County. Field observa-
tions measured 13 m at the landslide and 12 m at the
opposite bank. For the south shore, the observed data
(red dots) are scattered around the simulation result
(red line). The observed north shore values (blue
spots) are slightly higher than the values calculated.
Considering that the specific run-up quantities mea-
sured in the field are not completely clear nor were
any uncertainties assigned, we cannot make a formal
statement of goodness of fit, but the run-up heights on
both the north and south shores are reasonable fits
with the observations, with correlation coefficient
R2 = 0.88.
6. Conclusions
This paper introduces Tsunami Squares, a new
approach for modeling of landslide-generated waves.
Tsunami Squares has the advantages of the previous
Tsunami Ball method, for example, special separate
treatment for dry and wet cells is not needed, but it
Figure 12Run-up height decay as a function of distance along the north (blue) and south (red) shores. The landslide occurred on the north shore. The
scattered dots are the observed values on the north (blue) and south (red) shores
L. Xiao et al. Pure Appl. Geophys.
obviates the use of millions of individual particles.
Simulations can be expanded to spatial scales not
possible previously. The new method accelerates and
transports squares of material that are fractured into
new squares in such a way as to conserve volume and
linear momentum.
The simulation first computes landslide motions
on dry land. A novel aspect of Tsunami Squares is
that it considers landslides as part solid and part fluid.
Landslides are more fluid-like within a few seconds
of initiation after the material loses cohesion. Solid
landslides are driven by the slope of the bottom
surface. Fluid-like landslides are driven by the slope
of the top surface.
The second step in the simulation introduces
water. The falling landslide generates waves by two
means. One is the simple uplift of the fluid, known as
‘‘no momentum transfer’’. The second mechanism is
drag-along, a force representing the interaction be-
tween the slide and water. Velocity-dependent drag-
along contributes substantially to tsunami wave
generation from high-speed landslides.
We have demonstrated and validated Tsunami
Squares by modeling the 2008 Three Gorges Reser-
voir Gongjiafang landslide and river tsunami. The
landslide’s progressive failure, generated wave, and
subsequent propagation and run-up are well repro-
duced (WARD and XIAO 2013).
On a laptop computer, Tsunami Square simula-
tions flexibly handle a wide variety of waves and
flows, and are excellent techniques for risk estimation,
hazard assessment, and emergency management
applications.
Acknowledgments
L. Xiao was supported by the National Natural
Science Foundation of China (no. 41202247). We
thank Professor K. Yin, from the China University of
Geosciences (Wuhan), for supporting the geology
background research and field investigation. We
thank Dr Simon Day, from University College
London, for careful revision. We also thank Chongq-
ing Three Gorges Reservoir Geological Hazards
Prevention and Control Office for supplying the
10-meter digital elevation map.
Appendix 1: Inclusion of wave dispersion
Long wave theory assumes that the depth-aver-
aged horizontal acceleration of a water column is
proportional to the gradient of the fluid surface as
stated in Eq. (21):
ovðr; tÞot
¼ �vðr; tÞ � rhvðr; tÞ � grhfðr; tÞ ð21Þ
According to linear dispersive wave theory
(WARD AND DAY 2010), the depth-averaged horizontal
acceleration of a water column is proportional to the
gradient of the fluid surface smoothed over a di-
mension comparable with the water depth. Linear
dispersion can be accommodated in tsunami squares
simply by replacing Eq. (21) by Eq. (22)
ovðr; tÞot
¼ �vðr; tÞ � rhvðr; tÞ � grhfsmoothðr; tÞ
ð22Þ
where
fsmoothðr; tÞ ¼ fðr; tÞ � SðrÞ
¼Z
fðr0; tÞSðr� r0Þdr0ð23Þ
and
SðrÞ ¼ Re
Z
k
eik�r tanhðkhÞ4p2 kh
dk ð24Þ
This indicates that short wave (kh � 1) contri-
butions to the surface gradient impart less depth-
averaged acceleration to the water column than do
longer wave (kh � 1) contributions. As a result,
short waves fall behind long waves, as dictated by
linear dispersive theory. For the applications in this
paper, water wave dispersion is not important.
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