The estimate of an individual wave run-up is especially important for tsunami warning and risk assessment, as it allows for evaluating the inundation area. Here, as a model of tsunamis, we use the long single wave of positive polarity. The period of such a wave is rather long, which makes it different from the famous Korteweg–de Vries soliton. This wave nonlinearly deforms during its propagation in the ocean, which results in a steep wave front formation. Situations in which waves approach the coast with a steep front are often observed during large tsunamis, e.g. the 2004 Indian Ocean and 2011 Tohoku tsunamis. Here we study the nonlinear deformation and run-up of long single waves of positive polarity in the conjoined water basin, which consists of the constant depth section and a plane beach. The work is performed numerically and analytically in the framework of the nonlinear shallow-water theory. Analytically, wave propagation along the constant depth section and its run up on a beach are considered independently without taking into account wave interaction with the toe of the bottom slope. The propagation along the bottom of constant depth is described by the Riemann wave, while the wave run-up on a plane beach is calculated using rigorous analytical solutions of the nonlinear shallow-water theory following the Carrier–Greenspan approach. Numerically, we use the finite-volume method with the second-order UNO2 reconstruction in space and the third-order Runge–Kutta scheme with locally adaptive time steps. During wave propagation along the constant depth section, the wave becomes asymmetric with a steep wave front. It is shown that the maximum run-up height depends on the front steepness of the incoming wave approaching the toe of the bottom slope. The corresponding formula for maximum run-up height, which takes into account the wave front steepness, is proposed.

Evaluation of wave run-up characteristics is one of the most important tasks in coastal oceanography, especially when estimating tsunami hazard. This knowledge is required for planning coastal structures and protection works as well as for short-term tsunami forecasts and tsunami warning. Its importance is also confirmed by a number of scientific papers (see recent works, e.g. Tang et al., 2017; Touhami and Khellaf, 2017; Zainali et al., 2017; Raz et al., 2018; Yao et al., 2018).

The general solution of the nonlinear shallow-water equations on a plane beach was found by Carrier and Greenspan (1958) using the hodograph transformation. Later on, many other authors found specific solutions for different types of waves climbing the beach (see, for instance, Pedersen and Gjevik, 1983; Synolakis, 1987; Synolakis et al., 1988; Mazova et al., 1991; Pelinovsky and Mazova, 1992; Tadepalli and Synolakis, 1994; Brocchini and Gentile, 2001; Carrier et al., 2003; Kânoğlu, 2004; Tinti and Tonini, 2005; Kânoğlu and Synolakis 2006; Madsen and Fuhrman, 2008; Didenkulova et al., 2007; Didenkulova, 2009; Madsen and Schäffer, 2010).

Many of these analytical formulas have been validated experimentally in laboratory tanks (Synolakis, 1987; Li and Raichlen, 2002; Lin et al., 1999; Didenkulova et al., 2013). For most of them, the solitary waves have been used. The soliton is rather easy to generate in the flume; therefore, laboratory studies of run-up of solitons are the most popular. However, (Madsen et al., 2008) pointed out that the solitons are inappropriate for describing the real tsunami and proposed to use waves of longer duration than solitons and downscaled records of real tsunami. Schimmels et al. (2016) and Sriram et al. (2016) generated such long waves in the Large Wave Flume of Hanover (GWK FZK) using the piston type of wave maker, while McGovern et al. (2018) did it using the pneumatic wave generator.

It should be mentioned that the shape of tsunami varies a lot depending on
its origin and the propagation path. One of the best examples of tsunami
wave shape variability is given in Shuto (1985) for the 1983 Sea of Japan
tsunami, where the same tsunami event resulted in very different tsunami
approaches in different locations along the Japanese coast. These wave shapes
included the following: single positive pulses, undergoing both surging and spilling
breaking scenarios; breaking bores; periodic wave trains, surging as well as
breaking; and a sequence of two or three waves and undular bores. This
is why there is no “typical tsunami wave shape”, and
therefore in the papers on wave run-up cited above, many different wave
shapes, such as single pulses,

A similar study was performed for periodic sine waves (Didenkulova et al., 2007; Didenkulova, 2009). It was shown that the run-up height increases with an increase in the wave asymmetry (wave front steepness), which is a result of nonlinear wave deformation during its propagation in a basin of constant depth. It was found analytically that the run-up height of this nonlinearly deformed sine wave is proportional to the square root of the wave front steepness. Later on, this result was also confirmed experimentally (Didenkulova et al., 2013).

It should be noted that these analytical findings also match tsunami observations. Steep tsunami waves are often witnessed and reported during large tsunami events, such as 2004 Indian Ocean and 2011 Tohoku tsunamis. Sometimes the wave, which approaches the coast, represents a “wall of water” or a bore, which is demonstrated by numerous photos and videos of these events.

The nonlinear steepening of the long single waves of positive polarity has also been observed experimentally in Sriram et al. (2016), but its effect on wave run-up has not been studied yet. In this paper, we study this effect both analytically and numerically. Analytically, we apply the methodology developed in Didenkulova (2009) and Didenkulova et al. (2014), where we consider the processes of wave propagation in the basin of constant depth and the following wave run-up on a plane beach independently, not taking into account the point of merging of these two bathymetries. Numerically, we solve the nonlinear shallow-water equations.

The paper is organized as follows. In Sect. 2, we give the main formulas and briefly describe the analytical solution. The numerical model is described and validated in Sect. 3. The nonlinear deformation and run-up of the long single wave of positive polarity are described in Sect. 4. The main results are summarized in Sect. 5.

We solve the nonlinear shallow-water equations for the bathymetry shown in
Fig. 1:

Equations (1) and (2) can be solved exactly for a few specific cases. In the case of constant depth, the solution is described by the Riemann wave (Stoker, 1957). Its adaptation for the boundary problem can be found in Zahibo et al. (2008). In the case of a plane beach, the corresponding solution was found by Carrier and Greenspan (1958). Both solutions are well-known and widely used, and we do not reproduce them here but just provide some key formulas.

As already mentioned, during its propagation along the basin of constant
depth

Bathymetry sketch. The wavy curve at the toe of the slope regards analytical solution, which does not take into account merging between the constant depth and sloping beach sections.

To do this, we represent the input wave

Wave run-up oscillations at the coast

We also compare this solution with the run-up of a single wave of positive
polarity described by Eq. (9) (without nonlinear deformation). The maximum
run-up height

Numerically, we solve the nonlinear shallow-water equations Eqs. (1) and (2), written in a conservative form for a total water depth. We include the effect of the varying bathymetry (in space) and neglect all friction effects. However, the resulting numerical model will be taken into account for some dissipation thanks to the numerical scheme dissipation, which is necessary for the stability of the scheme and should not influence many run-up characteristics. Namely, we employ the natural numerical method, which was developed especially for conservation laws – the finite-volume schemes.

The numerical scheme is based on the second order in space UNO2 reconstruction, which is briefly described in Dutykh et al. (2011b). In time we employ the third-order Runge–Kutta scheme with locally adaptive time steps in order to satisfy the Courant–Friedrichs–Lewy stability condition along with the local error estimator to bound the error term to the prescribed tolerance parameter. The numerical technique to simulate the wave run-up was described previously in Dutykh et al. (2011a). The bathymetry source term is discretized using the hydrostatic reconstruction technique, which implies the well-balanced property of the numerical scheme (Gosse, 2013).

Water elevations along the 251 m long constant depth
section of the Large Wave Flume (GWK), where

The numerical scheme is validated against experimental data of wave
propagation and run-up in the Large Wave Flume (GWK) in Hanover, Germany. The
experiments were set with a flat bottom, with a constant depth of

Run-up height of the long single wave with

It is reported in Didenkulova et al. (2007) and Didenkulova (2009), for a periodic
sine wave, that the extreme run-up height increases proportionally with the
square root of the wave front steepness. In this section, we study the
nonlinear deformation and steepening of waves described by Eq. (9) and their
effect on the extreme wave run-up height. The corresponding bathymetry used
in analytical and numerical calculations is normalized on the water depth in
the section of constant depth

We underline that in order to have analytical solution, the criterion of no wave breaking should be satisfied. Therefore, all analytical and numerical calculations below are chosen for non-breaking waves.

Maximum run-up height,

Figure 4 shows the dimensionless maximum run-up height,

It is worth mentioning that for small initial wave amplitudes, all run-up heights are close to each other and are close to the thick black line, which corresponds to Eq. (14) for wave run-up on a beach without constant depth section. This means that the effects we are talking about are important only for nonlinear waves and irrelevant for weakly nonlinear or almost linear waves.

Maximum run-up height,

The same effects can be seen in Fig. 5, which shows the maximum run-up
height,

The dependence of maximum run-up height,

Maximum run-up height,

The next figure, Fig. 8, supports all the conclusions drawn above. It also shows that
the difference between analytical and numerical results increases with an
increase in the wave period. As pointed out before for small wave periods, the
numerical solution may coincide with the analytical one or even become
smaller as in

Maximum run-up height,

It is important that both analytical and numerical results in Figs. 5 and 8
demonstrate an increase in maximum run-up height with an increase in the
distance

It can be seen that the wave transformation described by the analytical
model is in a good agreement with numerical simulations. Therefore, below we
make reference to the analytically defined wave front steepness, keeping in mind
that it coincides well with the numerical one. Having said this, we approach
the main result of this paper, which is shown in Fig. 10. The red solid line
gives the analytical prediction. It is universal for single waves of
positive polarity for different amplitudes

Maximum run-up height,

The fit is shown in Fig. 10 by the black dashed line. For comparison, the
dependence of the maximum run-up height on the wave front steepness obtained
using the same method for a sine wave is stronger than for a single wave of
positive polarity (Didenkulova et al., 2007) and is proportional to the
square root of the wave front steepness. This is logical, as the sinusoidal wave
has a sign-variable form and, therefore, excites a higher run-up. For
possible mechanisms, see the discussion on

Wave evolution at different locations,

The ratio of maximum run-up height in the conjoined
basin,

The results of numerical simulations are shown in Fig. 10 with different
markers. It can be seen that numerical data for the same period but
different amplitudes follow the same curve. The run-up is higher for waves
with smaller

The normalized maximum run-up height,

In this paper, we study the nonlinear deformation and run-up of tsunami waves, represented by single waves of positive polarity. We consider the conjoined water basin, which consists of a section of constant depth and a plane beach. While propagating in such basin, the wave shape changes forming a steep front. Tsunamis often approach the coast with a steep wave front, as was observed during large tsunami events, e.g. the 2004 Indian Ocean Tsunami and 2011 Tohoku tsunami.

The study is performed both analytically and numerically in the framework of the nonlinear shallow-water theory. The analytical solution considers nonlinear wave steepening in the constant depth section and wave run-up on a plane beach independently and, therefore, does not take into account wave interaction with the toe of the bottom slope. The propagation along the bottom of constant depth is described by a Riemann wave, while the wave run-up on a plane beach is calculated using rigorous analytical solutions of the nonlinear shallow-water theory following the Carrier–Greenspan approach. The numerical scheme does not have this limitation. It employs the finite-volume method and is based on the second-order UNO2 reconstruction in space and the third-order Runge–Kutta scheme with locally adaptive time steps. The model is validated against experimental data.

The main conclusions of the paper are the following.

It is found analytically that the maximum tsunami run-up height on a beach depends on
the wave front steepness at the toe of the bottom slope. This dependence is
general for single waves of different amplitudes and periods and can be
approximated by the power fit:

This dependence is slightly weaker than the corresponding dependence for a
sine wave, proportional to the square root of the wave front steepness
(Didenkulova et al., 2007). The stronger dependence of a sine wave run-up on
the wave front steepness is consistent with the philosophy of

Numerical simulations in general support this analytical finding. For
smaller face front wave steepness (

These results can also be used in tsunami forecasts. Sometimes, in order to save time for tsunami forecasts, especially for long distance wave propagation, the tsunami run-up height is not simulated directly but estimated using analytical or empirical formulas (Glimsdal et al., 2019; Løvholt et al., 2012). In these cases we recommend using formulas which take into account the face front wave steepness. The face front steepness of the approaching tsunami wave can be estimated from the data of the virtual (computed) or real tide-gauge stations and then be used to estimate the tsunami maximum run-up height on a beach.

The data used for all figures of this paper are available at

AAA ran all the calculations, prepared the data for sharing, discussed the results and wrote the first draft of the manuscript. ID initiated this study, provided the numerical code for analytical solution, discussed the results and contributed to the writing of the manuscript. DD developed and provided numerical solvers for nonlinear shallow-water equations, discussed the results and contributed to the writing of the manuscript. All authors reviewed the final version of the paper.

The author declares that there is no conflict of interest.

The authors are very grateful to Professor Efim Pelinovsky, who came up with the idea for this study a few years ago.

Analytical calculations were performed with the support of Russian Science Foundation grant no. 16-17-00041. Numerical simulations and their comparison with the analytical findings were supported by ETAG grant no. PUT1378. The authors also thank the PHC PARROT project no. 37456YM, which funded the authors' visits to France and Estonia and allowed this collaboration.

This paper was edited by Mauricio Gonzalez and reviewed by two anonymous referees.