The 1958 Lituya Bay landslide-generated mega-tsunami is simulated using the Landslide-HySEA model, a recently developed finite-volume Savage–Hutter shallow water coupled numerical model. Two factors are crucial if the main objective of the numerical simulation is to reproduce the maximal run-up with an accurate simulation of the inundated area and a precise recreation of the known trimline of the 1958 mega-tsunami of Lituya Bay: first, the accurate reconstruction of the initial slide and then the choice of a suitable coupled landslide–fluid model able to reproduce how the energy released by the landslide is transmitted to the water and then propagated. Given the numerical model, the choice of parameters appears to be a point of major importance, which leads us to perform a sensitivity analysis. Based on public domain topo-bathymetric data, and on information extracted from the work of Miller (1960), an approximation of Gilbert Inlet topo-bathymetry was set up and used for the numerical simulation of the mega-event. Once optimal model parameters were set, comparisons with observational data were performed in order to validate the numerical results. In the present work, we demonstrate that a shallow water type of model is able to accurately reproduce such an extreme event as the Lituya Bay mega-tsunami. The resulting numerical simulation is one of the first successful attempts (if not the first) at numerically reproducing, in detail, the main features of this event in a realistic 3-D basin geometry, where no smoothing or other stabilizing factors in the bathymetric data are applied.

Tsunamis are most often generated by bottom displacements due to earthquakes.
However, landslides, either submarine or subaerial, can also trigger
devastating tsunami waves. In addition, they are, on some occasions, extremely
destructive as they form near the coast or in the same coastline if they are
aerial. Sometimes landslides may generate so-called mega-tsunamis, which are
characterized by localized extreme run-up heights (Lituya Bay, 1958,

For seismic tsunami simulations, in general, the most critical phases are generation and arrival at a coast, including inundation. Propagation over deep basins can be modeled using the nonlinear shallow water (NLSW) equations or more typically using a non-diffusive linear approximation. With landslide-generated tsunamis, however, matters become more complicated. The generation phase itself becomes critical and complex effects between the landslide and the water body must be taken into account. Most notably, the case of subaerial-landslide-generated tsunamis is where modeling and numerical implementation becomes most critical, owing to these events producing more complex flow configurations, larger vertical velocities and accelerations, cavitation phenomena, dissipation, dispersion, and complex coupled interaction between landslide and water flow. It is evident that shallow water models cannot take into account and reproduce all of these phenomena, particularly vertical velocities or cavitation. We do, however, demonstrate here that such models can, despite limitations, be useful for hazard assessment in which the main features (from a hazard assessment point of view) of these complex events, such as run-up and main leading wave, are reproduced. The overall aim is not to accurately reproduce the evolution of the displaced solid material or the dispersive nature of the trailing waves as the perturbation propagates, but, alternatively, to accurately reproduce the impact of tsunami waves to coasts in terms of run-up and flood area. Comparison of the numerical results with the observed trimline presented here is shown to support our statement that a fully coupled, vertically integrated shallow water Savage–Hutter model can, effectively and accurately, reproduce the run-up and coastal inundation resulting from aerial landslides generated in fjords and enclosed basins. This study further supports this assessment by comparing model results with observed data for a paradigmatic example of extreme run-up produced by an aerial landslide in an enclosed bay, a simulation that has not been successfully undertaken previously with more comprehensive numerical models.

Full 3-D numerical modeling of landslide-generated tsunamis

In the case of tsunamigenic aerial landslides in fjords, bays, or any long
and narrow water body, confinement and reflection (a process that also makes
propagation and interaction more complex) are relatively more important
considerations than dispersion, which becomes less important. This is
particularly true for the leading wave

Despite this, and independent of the eventual confinement of the flow, many
authors continue to claim the absolute need for dispersive, or even full
Navier–Stokes, models

Among all examples of subaerial-landslide-generated tsunamis, the Lituya Bay
1958 event occupies a paradigmatic place in the records, standing alone as
the largest tsunami ever recorded and representing a scientific challenge of
accurate numerical simulation. Based on generalized Froude similarity,

At 06:16 UTC on 10 July 1958, a magnitude

In order to understand the evolution of the giant Lituya Bay wave, a rough
model at a 1 : 1000 scale was constructed at the University of California,
Berkeley (Robert L. Wiegel in

The landslide was triggered by fault movement and intense earthquake
vibrations

Topographic map of Lituya
Bay (US Geologic Survey, 1961) showing the settings and trimline of the 1958
mega-tsunami (based on data from

Based on the experimental work of Fritz et al. (2009), in the present
numerical study we will follow the same approach: an initial slide speed
(analogous to the impulse of the pneumatic landslide generator in the lab
experiment) will be imposed in order to obtain the 110 m s

Lituya Bay (Fig.

Gilbert Inlet shoreline and glacier, bathymetries, and shorelines
before and after 1958.

The shores around the main part of the bay are composed mainly of rocky
beaches that rise steeply away from the shoreline. There are two adjoining
land masses that rise away from the beach, ranging in elevations from less
than 30 m at a horizontal distance of 2 km, around Fish Lake, to 170 m at
a horizontal distance of 370 m at The Paps (see Fig.

According to

According to

The bathymetric data used for the modeling work presented here were obtained
from the US National Ocean Service: Hydrographic Surveys with Digital
Sounding. Data from survey ID H08492, 1959, were used as reference
bathymetry since this survey is the nearest in time to the data of the
tsunami and there were enough data collected to provide a good representation
of the entire Lituya Bay seafloor. Data from survey ID H04608, 1926, were used
to reconstruct Gilbert Inlet bathymetry as these were the closest pre-event
data available. Unfortunately, data from this survey are not sufficient in
resolution to provide an acceptable bathymetric grid for our study of the
entire bay. Nevertheless, the survey provides both enough data and detailed
information of pre-tsunami bathymetry in the Gilbert Inlet. In
Fig.

The dimensions of the landslide on the northeast wall of Gilbert Inlet were
determined with reasonable accuracy by

To locate and reconstruct the volume of the slide mass, the following
procedure was implemented: first, based on aerial photos and data provided by

The three criteria we tried to fulfill in order to reconstruct the slide geometry were (1) to place it in its exact location (projected area), (2) keep an approximate location for the centroid, also in height, and (3) to recover an accurate volume for the numerical slide.

In order to reproduce the main features of the slide impact,
Hermann M. Fritz and
collaborators (Fritz et al., 2009) designed a pneumatic landslide generator.
They intend to model the transition from rigid to granular slide motion.
Thus, at the beginning, the granular material is impulsed until the landslide
achieves 110 m s

In this work, we have followed the same idea: assuming that 110 m s

From a detailed analysis of the bathymetry surveys available for this study,
an unexpected shifted location of the slide deposit on the floor of the
Gilbert Inlet (see Fig.

Coulomb-type models for granular-driven flows have been intensively
investigated in the last decade, following the pioneering work of
Savage and Hutter

HySEA models have been fully validated for tsunami modeling using all of the
benchmark problems posted on the NOAA NTHMP web site for propagation and
inundation

Initial conditions for the slide: slide location and initial velocity vector direction.

This section describes the system of partial differential equations modeling
landslide-generated tsunamis based on layered average models. The 2-D
Landslide-HySEA model is a two-dimensional version of the model proposed in

Let us consider a layered medium composed by a layer of inviscid fluid with
constant homogeneous density

Details about the reconstructed slide. The isolines into the yellow
area have been modified to reconstruct the location of the slide previous to
the event.

In these equations, the subscript

The terms

The term

Note that

Finally,

System Eq. (

To discretize system Eq. (

We denote by

Let us suppose that

To check the precise definition for the numerical fluxes,

The numerical scheme described above, when applied to wet–dry situations, may
produce incorrect results: the gradient of the bottom topography generates
spurious pressure forces and the fluid can artificially climb up slopes. In

The implementation of the wet–dry front treatment in the numerical scheme results in not having to impose boundary conditions at the coasts. The coastline becomes a moving boundary, computed by the numerical scheme. Depending on the impact wave characteristics or the water backflow movement, the computational cells are filled with water or they run dry. Consequently, no specific stabilization model technique is required.

In this section the numerical scheme when the friction terms

Sketch of the two-layer model. Relation among

The resulting scheme is exactly well balanced

Landslide initial conditions have been described in
Sect.

The computational grid considered for the numerical simulation is a
4 m

The numerical scheme described in this section has been implemented in the CUDA programming language in order to be able to run the model in GPUs. A highly efficient GPU-based implementation of the numerical scheme allows us to compute highly accurate simulations in very reasonable computational time.

The numerical simulation presented in this work covers a wall clock time period of 10 min. A total of 14 516 time iterations were necessary to evolve from initial conditions to final state, 10 min later. This required a computational time of 1528.83 s (approx 25.5 min), which means 44 million computational cells were processed per second in a nVidia GTX 480 graphic card.

To overcome the uncertainty inherent in the choice of model parameters and in
order to produce a numerical simulation as close as possible to the real
event, a sensitivity analysis has been performed. To do so, the three key
parameters, (1) Coulomb friction angle,

Location and spatial extension of the four regions considered to compute maximum run-up for the sensitivity analysis.

Previous to searching the optimal set of parameters that will determine the
optimal solution, a sensitivity analysis was carried out. In order to assess
model sensitivity to parameters, four regions have been selected (see
Fig.

For a fixed value of the Coulomb friction angle,

Each panel in Fig.

From this first set of figures, we can conclude that

In Fig.

For a fixed value of the friction coefficient,

In Fig.

For a fixed value of the ratio of densities,

Images showing the four criteria used to select the optimal
numerical solution and comparison of simulated optimal inundated area with
the observed trimline.

Now, in order to find the optimal parameters, according to four criteria that
will be defined below, a more detailed “microscopic” grid of parameters
is used for performing additional numerical experiments. Taking into account
the results presented in Figs.

Giant wave generation and initial evolution in Gilbert
Inlet.

Then, the four criteria considered in order to select the optimal parameters
were as follows.

The run-up on the spur southwest of Gilbert Inlet had to be the closest to the optimal 524 m.

The wave moving southwards to the main stem of Lituya bay had to cause a peak close to 208 m in the vicinity of Mudslide Creek.

The simulated wave had to break through Cenotaph Island, opening a narrow channel through the trees

The trimline maximum distance of 1100 m from the high-tide shoreline at Fish Lake had to be reached.

More than 250 simulations were performed in order to find the optimal values
for the parameters that best verified the four conditions mentioned above.
Finally, the optimal parameters selected were

Setting the three parameters to the values given above, simulation satisfied
the previous four criteria with very good accuracy, more precisely,

the run-up in Gilbert Inlet reached 523.85 m
(Fig.

the run-up peak in the vicinity of Mudslide Creek reached a height close to 200 m
(Fig.

the wave produced a narrow channel crossing through Cenotaph Island (Fig.

the run-up reached more than 1100 m in distance from the high-tide shoreline in the Fish Lake area (Fig.

Wave evolution from

Sensitivity analysis provided the optimal set for the three key parameters considered for this study. In this section, model results corresponding to the simulation performed with these optimal parameters are presented: first the main characteristics of the giant wave generated in Gilbert Inlet, second optimal description of wave evolution through the main stem of the Lituya Bay, and third inundation details resulting from a comparison of numerical simulation run-up with the real trimline observed in several areas of interest. Additional material in the form of a numerical simulation movie is also provided.

Wave evolution from

Following the landslide trigger, the generated wave reaches its maximum
amplitude,

While the maximum run-up on the east side of Gilbert head is reached, the
southern propagating part of the initial wave, with a height of more than

Simulated inundated areas and maximum height all around Lituya Bay. Trimline in pink.

While the initial wave propagates through the main axis of Lituya Bay to
Cenotaph Island, a larger second wave appears as a reflection of the first one
from the south shoreline (Fig.

The first wave reaches Cenotaph Island after

After hitting Cenotaph Island, the wave splits into two parts; one advances
in the shallow channel north of the island and the second travels through the
deeper channel south of the island (Fig.

Large inundated areas were formed both around Fish Lake, on the north
shoreline, and in the flat areas surrounding The Paps, while the main wave
reaches the narrow area near La Chaussee Spit. During this time, the wave
amplitude is larger than 15 m, indeed, over 20 m on the north shoreline. At

In this section, the computed inundated area limit is compared with the real
trimline drawn by Miller. Figure

As it has been shown, on Gilbert Head, the maximum run-up (523.85 m) is
reached on the east slope. Furthermore, the run-up is extended
oblique to slope on the western face of Gilbert Head. Run-up extent and
trimline coincide quite remarkably in this sector
(Fig.

Figure

The agreement between model and observation around the flat areas surrounding
Fish Lake is good (Fig.

The computed run-up underestimates the trimline over steeper slopes on the
eastern third of the south shores (Fig.

Around the central third of the south shoreline, the agreement between model
and observed trimline is less coherent (Fig.

One of the items to be checked in the sensitivity analysis presented in
Sect.

As has been previously described, La Chaussee Spit was completely covered by
water from minute

Same as in Fig.

The Landslide-HySEA model was tested against analytic solutions and
laboratory measurements of

Nonetheless, as it was previously mentioned, potential sources of model errors are the quality of model initialization parameters, the initial landslide conditions, or the digital elevation model (DEM) due to limitations associated with bathymetric data. Moreover, in real landslides, the material is neither homogeneous nor granular, as assumed here in the present study. Nevertheless, this type of model can be used in practice to provide general information about the generated tsunami and the flooded areas as demonstrated in the presented results.

A high-quality DEM is necessary to properly model tsunami wave dynamics and inundation onshore, especially in areas with complicated bathymetries. In this study there was an additional need for good information on topo-bathymetric data just before the event in order to produce realistic pre-event geometry.

Though we have combined the DEM based on the best available data in the
region (described in Sect.

Due to the choice of optimal parameters in the sense described in
Sect.

In a second study stage, an inundation assessment is performed. A detailed
description of the run-up areas along the shores of the bay is presented. In
general, computed inundation areas are in very good agreement with Miller
observations. Nevertheless, the model provides larger inundation areas than
the

It has been demonstrated that the landslide-triggering mechanism proposed by

It has been shown that the numerical model used can simulate subaerial
scenarios similar to the Lituya Bay case provided that some information is
available to calibrate the model. The main question that remains to be
answered is then obvious: what happens when information to calibrate the
model is not available? In that case, which approach is followed? In other
words, how would an actual risk assessment study would be performed without post-event information? In that case two approaches can be followed. One first
option is a deterministic approach, in which, depending on the
characteristics of the slide, some coefficients are selected as default. In
the case considered in this work, consisting in a slide moving, essentially
as a solid block, the “blind” proposed parameters would be (in parentheses
the optimal values for comparison)

Concerning future work, as uncertainty in the data (initial condition, model
parameters, etc.) is of paramount importance in real applications, a
promising line of research is uncertainty quantification. Therefore, some
information of the main probabilistic moments should be provided. Uncertainty
quantification is currently a very active area of research, with one of the
most efficient techniques utilized being multilevel Monte Carlo methods. To
run such a method, a family of embedded meshes is first considered. Then, a
large enough number of samples of the stochastic terms are chosen and, for
each sample, a deterministic simulation is run. Finally, the probabilistic
moments are then computed by a weighted average of the deterministic
computations

Another improvement of the model envisioned will be carried out by
considering shallow Bingham dense avalanche models, like those introduced in

Our definition of the bathymetry is based on
public domain topo-bathymetric data cited in the paper and on information
extracted from the work of Miller (1960). Data concerning the numerical
results can be obtained by a personal request to the corresponding author. Any material requested from the authors will
be made available in a public repository of our university. A visualization of the numerical
simulation is available as additional material at

JMGV designed, with available topo-bathymetric data and literature, the initial condition and wrote the first version of the paper. JM wrote the subsequent versions of the paper and carried out the entire revision process, performed the sensitivity analysis, and generated the corresponding figures. MJC contributed to the definition of the mathematical model and the design of the numerical scheme. CSL generated the reconstructed topo-bathymetry used in the simulations and developed the discretization of the source terms. SOA performed the numerical simulations and produced some of the figures. MA implemented the CUDA GPU version of the numerical code used. He also performed some numerical tests and provided some figures. DA collected the historical information officially available for this event in the USGS and NOAA archives. He contributed to the writing of the initial version of the article.

The authors declare that they have no conflict of interest.

All numerical experiments required for the development of this research have been performed at the Unit of Numerical Methods of the University of Málaga (Ada Byron Building). This work was partially funded by the NOAA Center for Tsunami Research (NCTR), Pacific Marine Environmental Laboratory (USA) contract no. WE133R12SE0035, by the Junta de Andalucía research project TESELA (P11-RNM7069), by the Spanish Government Research project SIMURISK (MTM2015-70490-C02-01-R), and by the Universidad de Málaga, Campus de Excelencia Andalucía TECH. All data required to perform the numerical simulations presented in this study are described in the text or are publicly available. Diego Arcas was supported by the National Oceanic and Atmospheric Administration with this report being PMEL contribution no. 4665. Edited by: Maria Ana Baptista Reviewed by: two anonymous referees