There is a general lack of understanding of tsunami wave
interaction with complex geographies, especially the process of inundation.
Numerical simulations are performed to understand the effects of several
factors on tsunami wave impact and run-up in the presence of gentle submarine
slopes and coastal cliffs, using an in-house code, a constrained interpolation profile (CIP)-based model. The model employs a high-order
finite difference method, the CIP method, as the flow solver; utilizes a
VOF-type method, the tangent of hyperbola for interface capturing/slope
weighting (THINC/SW) scheme, to capture the free surface; and treats the
solid boundary by an immersed boundary method. A series of incident waves are
arranged to interact with varying coastal geographies. Numerical results are
compared with experimental data and good agreement is obtained. The
influences of gentle submarine slope, coastal cliff and incident wave height
are discussed. It is found that the tsunami amplification factor varying with
incident wave is affected by gradient of cliff slope, and the critical
value is about 45

A tsunami is one of the most disastrous coastal hazards in the world and can be caused by earthquakes, volcanic eruptions and submarine landslides. The 2004 tsunami in South-East Asia was one of the most destructive tsunami events in human history. There were over 100 000 victims in 11 countries during the tsunami period (Liu et al., 2005). In the recent Great East Japan Earthquake and resulting tsunami in 2001, over 24 000 people were killed or went missing and 300 000 buildings were damaged (Mimura et al., 2011). A serious nuclear disaster at the Fukushima Daiichi Nuclear Power Plant was caused by the powerful run-up and destructive force of the tsunami wave. All these events and lessons from the previous tsunami wave disasters indicate that the actual tsunami wave run-up and the corresponding destructive forces were underestimated (Dao et al., 2013).

Investigation of tsunami wave transformation in the near-shore area is a feasible approach to learning the action mechanism and inverting the tsunami source. Post-disaster studies were mostly done through field observations and numerical simulations. The sediment in the near-shore area is frequently regarded as traces of a tsunami. Monecke et al. (2008) analysed sand sheet deposited by the 2004 tsunami and extended tsunami history 1000 years into the Aceh past, pointing out the recurrence frequency of damage-causing tsunamis. However, due to the strong backflow of tsunamis, sediment can be brought back to sea. Hence, there may be an underestimation of the run-up using sediment information to study palaeotsunamis. Dawson (1994) pointed out that the upper limit of sediment deposition lay well below the upper limit of wave run-up, which was marked by a well-defined zone of stripped vegetation and soil. Goto et al. (2011) found that previous estimates of palaeotsunamis were probably underestimated by considering their newly acquired data on the 2011 Japanese tsunami event.

However, limited by the complexity in field observations, numerical simulation is another effective approach to investigate tsunamis. The computational domain should include a large area in which a tsunami generates, propagates and inundates (Sim and Huang, 2015). Therefore, shallow water equations (SWE) were popularly used due to their high efficiency. However, when there is interaction between wave and complex geography, SWE is unable to capture flow structures in detail and often underestimates the result (Liu et al., 1991). An amendment is needed to improve the result of SWE by using data from physical experiments or field observation.

Since the available time-series data of tsunami waveform and flow field in the near-shore area are scarce, there is a general lack of understanding of tsunami interaction with complex geography. A more accurate method is required to reproduce the process of tsunami evolution in the coastal area. Due to the development of supercomputing technology and a precise numerical algorithm, computational fluid dynamics (CFD) with viscous flow theory and fluid–solid coupling mechanics are capable of dealing with the complex flow problems when geographies exist. The finite difference method is widely used in various CFD models as a flow solver. Hitherto, the accuracy of the finite difference method has been a great challenge. In this paper, we introduce a CFD model based on the constrained interpolation profile (CIP) algorithm. The CIP method was first introduced by Takewaki et al. (1985) as a high-order method to solve the hyperbolic partial differential equation. Tanaka et al. (2000) proposed a new version of the CIP-CSL4 that overcomes the difficulty of conservative property. Hu et al. (2009) simulated strongly non-linear wave–body interactions used a CIP-based Cartesian grid method, and the results were in good agreement with experiment data. Kawasaki and Suzuki (2015) developed a tsunami run-up and inundation model based on the CIP method, and a highly accurate water surface profile was observed by using slip conditions on the wet–dry boundary. Fu et al. (2017) simulated the flow past an in-line forced oscillating square cylinder using a CIP-based model. The CIP method can be applied in CFD and shows good performance in other areas. Sonobe et al. (2016) employed the CIP method to simulate sound propagation involving the Doppler effect.

It is a significant research project to deal with the free-surface problem
in CFD. Hirt et al. (1981) put forward a mass conservation method named
“volume of fluid” (VOF), which is flexible and efficient for treating
complicated free boundary configurations. Based on the principle of VOF,
several improved methods were developed: PLIC (Youngs, 1982), THINC (Xiao et
al., 2005), WLIC (Yokoi, 2007) and tangent of hyperbola for interface
capturing with slope weighting (THINC/SW; Xiao et al., 2011). Yokoi et al. (2013) proposed a numerical framework consisting of the CLSVOF method,
multi-moment methods and density-scaled CSF model. The framework can
capture free-surface flows with complex interface geometries well. More recently,
conventional VOF has been widely used by combining it with various additional
schemes. Malgarinos et al. (2015) proposed an interface sharpening scheme on
the basis of the standard VOF method, which effectively restrained interface
numerical diffusion. Gupta et al. (2016) used a coupled VOF and pseudo-transient method to solve free-surface flow problems, and the numerical
solution compared well with analytical or experimental data. Quiyoom et al. (2017) simulated the process of gas-induced liquid mixing in a shallow
vessel and found that the mixing time predicted by EL

When coastal geographies are included, special handling of solid boundary is required. Peskin (1973) proposed an immersed boundary method (IBM) to treat the blood flow patterns of the human heart, which was later introduced to simulate the interactions between solid objects and incompressible fluid flows (Ha et al., 2014; Lin et al., 2016).

CFD is more convenient and economical than laboratorial experiments and field observation. The most attractive feature is that it can provide time-series data of waveform and flow field, which are helpful for a better understanding of the inundation mechanism of waves in near-shore areas. Markus et al. (2014) introduced a virtual free-surface (VFS) model, which enabled the simulation of fully submerged structures subjected to pure waves and combined wave–current scenarios. Vicinanza et al. (2015) proposed new equations to predict the magnitude of forces exerted by the wave on its front face. The equations were added to five random-wave CFD models and good agreement was obtained when compared with empirical predictions. Oliveira et al. (2017) utilized PFEM to simulate complex solid–fluid interaction and free surface, so that a piston numerical wavemaker was implemented in a numerical wave flumes. Regular long waves were successfully generated in the numerical wave flumes.

When the efficient numerical algorithm is adopted, CFD numerical simulation can be applied to study the tsunami inundation in coastal areas. However, limited by the computational efficiency and numerical dissipation of finite difference method, the quantity and slenderness ratio of computational grids should be moderate. Because of the high proportion of spatial span in horizontal and vertical coordinates in most cases, reasonable abnormal model scale in horizontal and vertical is necessary, similarly to laboratory experiments.

The purpose of the present work is to understand the characteristics of a tsunami, help invert the generation mechanisms and provide reference for tsunami forecasting and post-disaster treatment. In this study, tsunami wave impact and run-up on coastal cliffs are simulated using an in-house code, a CIP-based model. Considerable attention is paid to the influence of different coastal topographies, steep cliffs on the beach and submerged gentle slopes. Coastal cliffs are one of the most common coastal landforms, representing approximately 75 % of the world's coastline (Rosser et al., 2005), such as the coast of Banda Aceh in Indonesia and the steep slope at San Martin (Baptista et al., 1993). The existence of a cliff can influence not only the impact and run-up of a tsunami wave but also the erosion deposition. Different layers provide variations in resistance to erosion (Stephenson and Naylor, 2011). Particularly, some coastal cliffs consisting of soft rocks are eroded at the toe (Yasuhara et al., 2002), which causes them to be more easily destroyed. This is indispensable to understanding the function of coastal cliffs and submerged gentle slopes when the tsunami wave approaches the shore. The submerged gentle slope, such as the continental shelf, affects the waveform evolution and wave celerity before tsunami waves reach the shoreline. The tsunami amplification factor (Satake, 1994), relative wave height, run-up on the cliff and impact pressure will be analysed in this work.

In this paper, Sect. 2 describes the governing equations and the numerical methods, Sect. 3 provides the initial condition and numerical wavemaker and Sect. 4 presents model validation. Dimensionless analysis is then used to examine the effect of front slope length, depth ratio and cliff angles on the run-up, and impact pressure. Finally, the discussion and conclusion are in Sects. 5 and 6, respectively.

Our model is established in a two-dimensional Cartesian coordinate system,
based on viscous fluid theory with incompressible hypothesis. The governing
equations are continuity equation and Navier–Stokes equations written as
follows:

Multiphase flow theory is employed to solve the problem of solid–liquid–gas
interaction. A volume function,

Physical properties, such as the density and viscosity in a mesh, can be
calculated by

A fractional step approach is applied to solve the time integration of the
governing Eqs. (1) and (2). The first step is to calculate the
advection term, neglecting the diffusion term and pressure term, as Eq. (5)
shows.

The principle of CIP method:

A CIP method is employed to solve Eq. (5).
The second step is to solve the diffusion term by a central difference
scheme:

The final step is the coupling of the pressure and velocity by considering
Eq. (1):

The free surface is captured by a THINC/SW scheme, which is based on the principles of the VOF method. The solid boundary is treated by an IBM (Peskin, 1973).

The basic principle of CIP is that when computing the advection of a
variable

To explain the steps of the CIP method, we take the following 1-D advection
equation as an example.

CIP scheme as a kind of semi-Lagrangian method.

The four unknown coefficients in Eq. (11) can be determined by using known
quantities

The THINC/SW scheme, first put forward by Xiao et al. (2011), is used for
free-surface capturing of incompressible flows. Some test examples have
indicated that the scheme has the features we need, such as mass conservation and a
lack of oscillation (Ji et al., 2013). The basic idea of THINC/SW is that, for the
profile of a volume function

In this section, 2-D numerical wave tanks, including incident waves and geographies, are introduced. Simulation cases are divided into two categories according to different parameters and purposes.

The first simulations are completed in Tank 1, as shown in Fig. 3. This wave
tank is 10.0 m in length and 1.0 m in height. Four slopes compose the
topography profile, representing continental slope, continental shelf, beach
and cliff. The still-water depth in front of the
topography profile is fixed at 0.35 m, so that the still-water shoreline is
located at the starting point of the beach. This point is regarded as the
original point of this tank to determine other positions mentioned in this
paper. Tank 1 is used for verifying the accuracy of our model and
investigating cliff slope gradient and incident wave height, which may
influence the tsunami amplification factor. Five cliff slopes are tested:

Schematic diagram of Tank 1. tan

Schematic diagram of Tank 2. tan

The second simulations utilize Tank 2, similar to Tank 1 except for
slight differences, as shown in Fig. 4a. In this tank, the still-water
shoreline also lies on the starting point of the beach, which is the
original point of this tank. Four gentle submarine slopes (representing the
continental shelf) of different lengths are used:

In this work, considerable attention will be paid to Tank 2. It is necessary to number the simulated cases in Tank 2 to avoid confusion, as shown in Table 1.

Summary of basic parameters calculated in Tank 2.

By declaring a velocity of water particle in the left-most grid and assigning it a value from laboratory wave-paddle velocity, a numerical paddle wavemaker is set at the left side of wave tank (Figs. 3 and 4a).

For a solitary wave, the approximate solution of wave profile near the wave
paddle can be described as follows (Boussinesq, 1872):

The wave-paddle velocity can be calculated as

Using Eqs. (21), (23) and (24), wave-paddle trajectory can be derived as an
implicit expression:

To verify the accuracy of our model, numerical results from one of the cases
in Tank 1 are compared with available experimental data (Sim, 2017). The
incident wave height and the cliff slope gradient of this case are

Parameter of three sets of grids (Unit: m).

Figure 5 concerns the predicted time series of water elevations at
locations S1–3, and the physical measurements (Sim, 2017) are
also presented for comparison. Figure 5a illustrates the comparison results
at S1. It can be observed that wave has not reached the topography, so that
the waveform has not transformed excessively and is similar to the original
waveform. Good general agreement is found for all computations. The relative
wave height at S1 is 1.0 m, which reveals the accuracy of the target incident
wave. Figure 5b shows the results at S2,

Time series of experimental data and predicted water elevations using
different grids:

Figure 6 describes the results of the tsunami wave amplification factor in
Tank 1. The tsunami amplification factor is defined as a ratio of the local
tsunami height to the tsunami height at a reference location. The vertical
coordinates are

Wave amplification factors,

Time series of relative wave elevation in Tank
2:

Figure 7 depicts the time series of relative wave elevation in Tank 2 for
cases 1–12. Four gentle submarine slope lengths and three incident wave
heights are considered. The predicted results at

Maximum relative wave height in front of the cliff in Tank 2:

Wave run-up on the cliff.

Time histories of impact pressure on the
cliff:

Wave heights at five gauges in Tank 2,

Figure 9 displays the wave run-up on the cliff in Tank 2; different lengths of gentle
submarine slope are compared. The predicted run-up is normalized to
the incident wave height,

Figure 10 shows the time histories of impact pressure on the cliff at two
pressure sensors, P3 and P4. The detailed locations of pressure sensors are
shown in Fig. 4b. Figure 10a, c, e, g, i and k give the
predicted results at location P3, and the results at location P4 are shown
in Fig. 10b, d, f, h, j and l. The recorded pressure is
normalized to the hydrostatic pressure due to incident wave amplitude,

The maximum pressure at five pressure sensors of all cases is also shown
in Fig. 11. According to the analysis of Fig. 10, there is no peak when the
maximum relative pressure is smaller than 2.5, and so our attention will be
paid to maximum relative pressure greater than 2.5. The
inclination of a cliff affects the appearance of the pressure peak. Under the
condition of a toe-eroded cliff, the generation of pressure peak is
frequent. As for the normal cliff, the pressure peak is rare, but it can be
significantly large once it appears. In addition, the pressure peak is found to
be related to the length of gentle submarine slope

Maximum impact pressure at five measuring points.

Figure 12 demonstrates the snapshots of the pressure distribution at different
time instances. Two typical cases of a normal cliff

Snapshots of impact pressure distribution in front of the
cliff.

The tsunami amplification factor is essentially a kind of relative wave height that normalized to the height at a reference location. The interesting result in the present work is as follows. In Tank 1, we analyse the tsunami amplification factor near the steepest cliff and find that it increases with the initial wave height (as Fig. 6e and f shown). However, in Tank 2, when the gauge is close to the normal cliff, the relative wave height decreases as the initial wave height increases (seen in Fig. 8a, c and e). It seems that the results of Tank 1 and Tank 2 are contradictory. One of the possible explanations is the influence of the beach. The most significant difference between Tank 1 and 2 is the length of beach. The effect of beach can be simply summed up as follows. The longer the beach is, the more energy is lost before wave impact. The beach is also an area for the mixing of incident and reflected wave; for a large wave, which requires a long area for mixing when the beach is not long enough, the drastic mixing will occur under the coastal line. The details of the beach effect, including process of mixing and energy dissipation, are a meaningful research subject for future work.

As for the run-up in Tank 2, a critical length of gentle submarine slope is
found for some cases. Before the wave gets across the coastal line, gentle
submarine slope facilitates wave deformation and energy focus. A
proper slope helps a wave to adequately prepare before it touches the
cliff. When the slope is too long, as the wave reaches the shore line it may
have broken or be on the verge of breaking, which makes energy dissipate ahead
of time. However, the optimum length is affected by several factors such as
initial wave height and cliff slope; this is why there is no critical value
found in some cases. From the present work, it is reasonable to speculate
that higher initial waves require longer, gentle submarine slopes to achieve
the critical value. This can be connected to the analysis of Fig. 11 that
when

The present work is only a start as the understanding of tsunami inundation needs to be expanded upon and quantified.

In this study, tsunami wave impact and run-up in the presence of submarine gentle
slopes and a coastal cliff are investigated numerically using a
CIP-based model. Numerical results are initially compared with available
experimental data and the good agreement revealed the ability of our model
to solve the complex flow field, such as wave breaking, water–air mixing and
violent impact. The results can be summarized as follows.

The gradient of cliff slope has a critical value about 45

The length of gentle submarine slope influences the tsunami wave run-up
and has a critical value of about

When wave transforms near the cliff, the cases with small incident wave height have a larger relative wave height, which means a devastating tsunami may be caused by a moderate source.

It is easier for tsunami waves to run up on a normal cliff than on a toe-eroded cliff.

There are two opportunities for the appearance of pressure peak during the process of tsunami wave run-up and impact. One is the direct impacting pressure when tsunami waves first hit the coastal cliff, and the other is caused by the backflow from the cliff after run-up with a widely affected area.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This work was financially supported by the National Natural Science Foundation of China (grant nos. 51479175, 51679212 and 51409231) and Zhejiang Provincial Natural Science Foundation of China (grant no. LR16E090002). Edited by: M. Gonzalez Reviewed by: two anonymous referees