EVALUATING TSUNAMI HAZARDS FROM DEBRIS FLOWS
J. S. WALDER
US Geological Survey, Cascades Volcano Observatory, 1300 Southeast Cardinal Court,
Building 10, Suite 100, Vancouver, Washington, USA 98683
P. WATTS
Applied Fluids Engineering, Inc., 5710 East 7th Street, Private Mail Box #237, Long
Beach, California, USA 90803
Abstract
Characteristics of water waves caused by subaerially generated debris flows vary with
distance from the debris-flow entry point. Three hydrodynamically distinct regions
(splash zone, near field, and far field) may be identified. Experiments demonstrate that
characteristics of the near-field water wave--the only coherent wave to emerge from the
splash zone--depend primarily on submerged volume and submerged travel time of the
debris flow and on water depth where debris-flow motion stops. Near-field wave
characteristics commonly may be used as a proxy source for computational tsunami
propagation. An example application explores hazards associated with potential debris
flows entering a reservoir.
Keywords: Debris flow, tsunami, modeling, scaling relations
1. Introduction
Geophysical mass flows of all types can generate hazardous water waves. Tsunamis
generated by submarine mass flows have recently received increased attention (e.g.,
Tappin et al., 1999, 2001). Destructive water waves generated by mass failure also
strike mountainous and alpine environments (e.g., Plafker & Eyzaguirre, 1979; Fritz et
al., 2001). For brevity, we will refer to all such waves as tsunamis and to all wave
sources as debris flows or “wavemakers”, the terminology of the coastal-engineering
literature (e.g., Dean & Dalrymple, 1991). We restrict our attention to tsunamis
generated by debris flows of subaerial origin.
A variety of methods have been applied to investigate debris-flow generated tsunamis.
Physical scale models have been constructed for a few case studies (e.g., Pugh & Harris,
1982), but this approach is expensive and requires special laboratory facilities, making it
impractical as a general method; moreover, this approach does not elucidate general
physical principles. Theoretical studies have generally been of two sorts. A number of
investigators have considered the tsunami source only to the extent that it acts to
displace water, with coupling between the source and the water ignored. Various
idealized sources have been considered in this way (e.g., Das & Wiegel, 1972; Noda,
1971; Hunt, 1988). Other investigators have considered coupling between wavemaker
and water, with the wavemaker having an assumed rheology (e.g., Norem et al., 1990;
Imamura & Imteaz, 1995; Assier Rzadkiewicz et al., 1997). These rheological
assumptions are, however, called into question by recent work on debris-flow
mechanics (Iverson & Denlinger, 2001; Iverson & Vallance, 2001). There have also
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Walder and Watts
been attempts to combine scaling analysis with experimental results to develop
predictive equations (e.g., Kamphuis & Bowering, 1972; Huber, 1980; Slingerland &
Voight, 1982. We have built upon recent work by Walder et al. (in press), who showed
that experiments with solid-block wavemakers can properly scale the key physics.
Walder et al. developed curve fits for tsunami amplitude and wavelength as functions of
wavemaker volume, wavemaker travel time, and water depth, and showed that these
curve fits do a good job of fitting data for experiments with diverse wavemaker shapes
and materials.
2. Spatial Domains In Tsunami Generation And Propagation
We consider a two-dimensional geometry (Fig. 1). The splash zone, where motion of
the debris flow and water are coupled, extends as far as the debris flow travels. The near
field is defined as the region beyond the splash zone yet before the kinetic- and potential
energy of the wave train approach asymptotic values. The extent of the near field
remains somewhat uncertain; Watts (2000) found that the far field began at a distance of
about three wavelengths beyond the splash zone for linear (small amplitude) water
waves generated by subaqueous block landslides. The near field is also the domain
where a coherent water wave can first be recognized, yet close enough to shore that
propagation effects have not yet altered the waveform substantially. Walder et al. (in
press) showed that a tsunami has well-defined characteristics in the near field, which
can therefore serve as a proxy source for computational wave-propagation purposes. In
other words, as long as the splash zone is small compared to the overall region of
interest, tsunami propagation can be computationally simulated without explicitly
computing splash-zone dynamics. This is a powerful methodological conclusion that
greatly simplifies computational modeling of wave propagation and inundation.
Figure 1. Sketch illustrating separation of splash zone, near field and far field. The water surface in the splash
zone is highly irregular. In the near field, water displaced by the debris flow has organized itself into a
coherent waveform, with the leading wave commonly being a broad hump of width [x] and amplitude
η ′ relative to the ambient water surface. In the far field, dispersive effects can become important.
The size of the splash zone as well as the tsunami amplitude and wavelength depend on
the submerged debris-flow motion. A full treatment of this problem should arguably
implement a description of the wavemaker as a deforming two-phase granular mass
(Iverson & Denlinger, 2001). We can get some insight into factors that control
wavemaker motion by considering center-of-gravity motion of a block landslide moving
Evaluating tsunami hazards from debris flows
157
down a plane sloped at an angle 2. An approximate equation of motion is (cf. Watts et
al., 2000; Walder et al., in press)
C s  d 2s
s   C  ds 

s

 γ + m  2 = (γ − 1) gχ   + γχg 1 −  −  d  
L  dt
 L
 L   2 L  dt 

2
(1)
where s(t) is distance (measured along the slope) traveled from shore; • is the specific
gravity of the block; L is block length; g is acceleration due to gravity;
χ = sin θ − cos θ tan ϕ , with • being the angle of bed friction; Cm is the added-mass
coefficient, and Cd is the form-drag coefficient (Batchelor, 1967). The various
coefficients may be considered constants as a first approximation. Terms on the righthand side of Equation (1) represents, respectively, friction on the submerged portion of
the block, friction on the subaerial portion, and hydrodynamic drag. The submergence
coefficient s/L becomes equal to 1 once the block is fully submerged. We will denote
total distance traveled from the shoreline by s*, and the time for the block to stop after
hitting the water as t*. Equation (1) is usefully recast in dimensionless form by scaling
distance with [s ] = L and time with [t ] = L / g (cf. Savage and Hutter, 198x), where
the square brackets denote characteristic values. We find, after some algebra,
(γ + C m s )
d 2s
dt
2
= χ (γ − s ) −
C d  ds 
 
2  dt 
2
(2)
where s and t should now be interpreted as dimensionless variables. Any dimensionless
measure of wavemaker motion depends in general on the dimensionless coefficients in
Equation (3), so t*/[t] and s*/[s] are in general functions of &m, and Cd.. The
definitions of [s] and [t] moreover tell us that submerged block motion depends on the
length scale L . Because &m, and Cd are all of order unity, the characteristics of
motion are fairly insensitive to their numerical values (Watts, 1998, 2000). Friction
DQJOH YDULHVZLWKLQUHODWLYHO\QDUURZERXQGVDQG LVERWK VLWH-specific and unlikely
to vary greatly. Thus we might expect that t* ≈ k L / g , where k is a constant of order
unity. This is in fact the result found empirically by Walder et al. (in press).
As runout distance and duration of motion both depend on wavemaker length, data
collected from case studies must be carefully applied. Runout distance and time of
motion determined from any given case study apply only for the corresponding value of
L. A different value of L for some other, hypothetical event will induce changes in
runout distance and duration of motion. If the relative change in length is known, then
the definitions of [s] and [t] show how runout distance and time of motion should
change. Obviously, if L varies by an order of magnitude or more, the effect on time of
motion and runout can be quite pronounced.
3. Tsunami Features in the Near Field
Walder et al. (in press) conducted flume experiments with solid-block wavemakers
either released from rest at the shoreline or entering the water with non-zero velocity.
They described the near-field wave form by the function
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Walder and Watts
(
η (x ) ≈ η ′ sech 2 x λo
)
(3)
where η ′ and λ 0 should be understood simply as fitting parameters. The water wave
represented by Equation (3) is the single significant and only coherent water wave to
emerge from the splash zone. The “wavelength” was found to be
λo ≅ 0.27 t *
g h.
(4)
The same functional form was proposed by Watts (1998, 2000) in connection with
tsunamis generated by submarine landslides. Wave amplitude 00 is well described by
 *
t
g h3
η ′ ≈ 1.32 h 
Vw





−0.68
(5)
for (t * / Vw ) gh 3 varying from about 2 to 100. The quantity (t * / Vw ) gh 3 may be
interpreted as a dimensionless measure of the wavemaker travel time per unit volume,
where Vw is the wavemaker volume per unit width along the shoreline and h is the water
depth near the end of debris-flow motion. The influence of t* on η’has been recognized
with regards to tsunamis generated by earthquakes (Hammack, 1973) and submarine
mass flows (Watts, 1997, 1998, 2000), but has not previously been considered in
discussions of tsunamis generated by subaerial mass flows. Equation (5) also does a
good job of fitting data for previous experiments for which t* may be inferred
(Bowering, 1970; Huber, 1980). For (t * / Vw ) gh 3 less than about 2, the asymptotic
limit η ’≈ 0.85 h is reached for the given depth h (Dean and Dalrymple, 1991). An
intriguing consequence of Equation (5) is that η ’ is practically independent of h, making
wave generation very nearly a function only of debris-flow volume and the duration of
submerged debris-flow motion.
4. Computing tsunami effects: modeling approach and some results
The experimental work of Walder et al. (in press) was done in a flume, which is rarely a
reasonable representation of actual water bodies. One must generally account for lateral
spreading of the wavefront as the debris flow submerges. We have done this by
incorporating the results presented in Equations (3) through (5) into a software package
called the Tsunami Open and Progressive Initial Conditions System (TOPICS). The
output of TOPICS is the free-surface profile of the near-field wave corrected for
geometrical spreading. This profile is used as the initial condition in a tsunamipropagation model. In other words, the initial condition for wave-propagation purposes
corresponds not to the moment at which the debris flow impacts the water, but rather to
the moment at which debris-flow motion stops.
Evaluating tsunami hazards from debris flows
159
All pertinent experimental studies show that water waves generated by debris flows
commonly have a wavelength of about 5 to 10 times h in the near field. Such waves are
dispersive and moderately to strongly nonlinear, as indicated by values of the Ursell
parameter η ′λ20 / h 3 commonly in the range 1 to 100 (cf. Dean and Dalrymple, 1991). A
Boussinesq model, rather than a shallow-water model (in which horizontal velocity is
assumed uniform over depth), is an appropriate tool for simulating wave propagation
and inundation. We have used the Boussinesq model Geowave to illustrate the
significance and hazards of debris-flow generated tsunamis for a specific scenario of a
debris flow entering a lake. Geowave is based on the public-domain software
FUNWAVE (Wei et al., 1995; Wei & Kirby, 1995). The code is fully nonlinear and
handles dispersion in a manner that correctly simulates deep-water waves. Geowave
takes the surface elevation from TOPICS and inputs this as an initial condition into
FUNWAVE at the characteristic time t* after debris-flow impact.
We have modeled a hypothetical tsunami generated by a debris flow entering Baker
Lake, a reservoir on the flanks of Mount Baker, which is an active volcano in northern
Washington, USA, that last erupted about 150 years ago. Baker Lake is an important
resource for both hydropower generation and recreation. We used Geowave to model
wave inundation at the shoreline for a range of debris-flow volumes. We used
topographic- and bathymetric data from the U.S. Geological Survey, and established a
simulation grid (with a grid spacing of 15 m) using Surfer software. Results for one
simulation, for a debris-flow volume of 107 m3, are shown in Fig. 2. The model does not
account for bathymetric changes owing to sedimentation. As the modeled debris-flow
volume is only about 2% of the capacity of Baker Lake, assuming constant bathymetry
is unlikely to introduce significant error. However, modeling the effect of, say, a 108 m3
debris flow—about the largest plausible flow, based on geologic evidence (Gardner et
al., 1995)—would entail accounting for bathymetric changes as the debris flow enters
the lake. Sufficiently large debris flows would likely create a blockage splitting Baker
Lake in half, in which case the simulation process, which does not account for
bathymetric changes caused by sedimentation, would not be meaningful. We suggest
that the Fig. 2 simulation provides a reasonable, effective benchmark for assessing
effects of tsunamis generated by volcanogenic debris flows from Mount Baker. One
important result from this simulation is that the wave height at the dam would be just
enough to overtop the dam if the water in the reservoir were at the normal operating
level.
Some further remarks about inundation effects are in order. In water of constant depth, a
wave front will spread and the wave amplitude will decrease with distance from the
source even in the absence of dissipation (Mei, 1983). Owing to wave refraction over a
sloping bottom, wave fronts converge at bathymetric highs, causing an increase in wave
amplitude. Conversely, relatively deep water near the shoreline will tend to reduce the
impact of a tsunami owing to wave-front divergence.
Many of the patterns that appear in Fig. 2 can therefore be attributed solely to
bathymetry. Water entering an embayment may be confined if the embayment narrows,
thereby forcing the entering water to build in amplitude.
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Walder and Watts
Figure 2. Maximum water-surface elevation of Baker Lake, relative to ambient water level, in meters,
following entry of a 107 m3 debris flow at local coordinates easting ≈ 3000 m, northing ≈ 8000 m. The local
origin is at UTM zone 10 coordinates (594950, 5388000). We assumed that the debris flow would overtop its
subaerial banks and enter the lake over a broad front of about 2 km. The duration of submerged motion was
estimated to be 44 s, using the empirical scaling relation of Walder et al. (in press). Characteristic near-field
water depth was chosen as 20 m based on measured bathymetry. Inferred values for “wavelength” and
maximum amplitude of the near-field wave are 154 m and 12.7 m, respectively The solid line is the shoreline
at an ambient water level of 220 m a.s.l. The elliptical mound of high water in the center of the lake opposite
the point of debris-flow entry reflects the initial condition for wave propagation given by TOPICS.
This effect may be responsible for the increased modeled wave amplitude near the dam.
There are two significant wave fronts in our simulations: the first is due to the debris
flow entering the lake, the second to reflections from shore. The second wave front is
nearly as strong as the first wave front. Any enclosed or semi-enclosed body of water
will have resonance characteristics determined by the shape of the shoreline and the
bathymetry (Mei, 1983). One result of our computations not shown here is that debrisflow generated tsunamis in Baker Lake appear to set several inlets into resonance, as
indicated by raising and lowering of much of the water in a consistent fashion. Although
we have not modeled long-term behavior in this work, it would not be surprising for
resonance to set in and to persist for hours.
Evaluating tsunami hazards from debris flows
161
Dark stripes that parallel the shoreline in Fig. 2 represent the effect of edge waves,
which are highly dispersive waves formed when energy is trapped along a sloping shore
and travels parallel to the shoreline (Liu et al., 1998). Edge waves in fact produce many
of the largest values of inundation in the simulations, and can be highly nonlinear.
5. Conclusions
Experimental studies and scaling analyses lead to the important conclusion that
numerical models of tsunami propagation, for the case of debris-flow generated
tsunamis, can be carried out independently of the complicated exercise of computing
splash-zone dynamics, as long as the splash zone is much smaller than the overall
region of interest. This greatly simplifies computational modeling of wave propagation
and inundation. The effective initial condition for computational wave propagation is
defined by the “near-field” water wave, the only coherent water wave that emerges from
the splash zone. Amplitude and wavelength of this near-field wave depend primarily on
debris-flow volume, debris-flow time of motion, and the water depth at the point where
debris-flow motion stops.
6. Acknowledgments
Mention of trade names is for identification only does not constitute endorsement by the
U.S. Geological Survey.
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