The run-up of random long-wave ensemble (swell, storm surge, and tsunami) on the constant-slope beach is studied in the framework of the nonlinear shallow-water theory in the approximation of non-breaking waves. If the incident wave approaches the shore from the deepest water, run-up characteristics can be found in two stages: in the first stage, linear equations are solved and the wave characteristics at the fixed (undisturbed) shoreline are found, and in the second stage the nonlinear dynamics of the moving shoreline is studied by means of the Riemann (nonlinear) transformation of linear solutions. In this paper, detailed results are obtained for quasi-harmonic (narrow-band) waves with random amplitude and phase. It is shown that the probabilistic characteristics of the run-up extremes can be found from the linear theory, while the same ones of the moving shoreline are from the nonlinear theory. The role of wave-breaking due to large-amplitude outliers is discussed, so that it becomes necessary to consider wave ensembles with non-Gaussian statistics within the framework of the analytical theory of non-breaking waves. The basic formulas for calculating the probabilistic characteristics of the moving shoreline and its velocity through the incident wave characteristics are given. They can be used for estimates of the flooding zone characteristics in marine natural hazards.

The flooded area size, the water flow depth, and its speed on the coast, and the coastal topography characteristics determine the consequences of marine natural disasters on the coast. The catastrophic events of recent years are well known, when tsunami waves and storm surges caused significant damage on the coast and many deaths. It is worth saying that only in 2018 two catastrophic tsunamis occurred in Indonesia, leading to the deaths of several thousand people (on Sulawesi in September and in the Sunda Strait in December). The calculations of the coastal flooding due to tsunamis and storm surges are mainly carried out within the framework of nonlinear shallow-water equations, taking into account the variable roughness coefficient for various areas of the coastal zone (Kaiser et al., 2011; Choi et al., 2012). The characteristics of the coastal destruction are determined either by using fragility curves (Macabuag et al., 2016; Park et al., 2017) or by using a direct calculation of the tsunami forces (Qi et al., 2014; Ozer et al., 2015a, b; Kian et al., 2016; Xiong et al., 2019).

The computational accuracy was tested on a series of benchmarks, including the idealized problem of the wave run-up onto the impenetrable slope of a constant gradient without friction (Synolakis et al., 2008). The nonlinear shallow-water equations for the bottom geometry of this kind are linearized by using the hodograph (Legendre) transformations. This step makes it possible to obtain a number of exact solutions describing the run-up on the coast. This approach, first suggested by Carrier and Greenspan (1958), was later on used to analyze the run-up of single and periodic waves of various shapes (Synolakis, 1987; Pelinovsky and Mazova, 1992; Carrier, 1995; Carrier et al., 2003; Tinti and Tonini, 2005; Madsen and Fuhrman, 2008; Madsen and Schaffer, 2010; Antuano and Brocchini, 2008, 2010; Didenkulova, 2009; Dobrokhotov et al., 2015; Aydin and Kanoglu, 2017). Moreover, such an approach made it possible to determine the conditions for wave-breaking. The latter means the presence of steep fronts (the gradient catastrophe) within the hyperbolic shallow-water equation framework. The Carrier–Greenspan transformation was further generalized for the case of waves in an inclined channel of an arbitrary variable cross section (Rybkin et al., 2014; Pedersen, 2016; Shimozono, 2016; Anderson et al., 2017; Raz et al., 2018). In a number of practical cases, its use proves to be more efficient than the direct numerical computation within the 2-D shallow-water equation framework (Harris et al., 2015, 2016).

Due to the bathymetry variability and shoreline complexity, diffraction and scattering effects lead to an irregular shape of the waves approaching the coast. Moreover, very often the leading wave is not the maximum one. Such typical tsunami wave records on tide-gauges are well known and are not shown here. It is applied even more often to swell waves, which in some cases approach the coast without breaking (Huntley et al., 1977; Hughes et al., 2010). As a result, the statistical wave theory can be applied to such records and nonlinear shallow-water equations in the random function class can be solved. This approach was used to describe the statistical moments of the long-wave run-up characteristics in Didenkulova et al. (2008, 2010, 2011). Special laboratory experiments were also conducted on irregular-wave run-up on a flat slope, the results of which are not very well described by theoretical dependencies (Denissenko et al., 2011, 2013). As for the field data, we are acquainted with two papers: Huntley et al. (1977) and Hughes et al. (2010), where the statistical characteristics of the moving shoreline were calculated on two Canadian beaches and one Australian beach. They confirmed the fact that the wave process on the coast is not Gaussian. In our opinion, the main problem in the theoretical model of describing the irregular-wave run-up on the shore is associated with the use of two hypotheses: (1) that the small-amplitude wave field (in the linear problem) is Gaussian and that (2) waves run-up on the shore without breaking. It is obvious, however, that in the nonlinear wave field some broken waves can always be present. They affect the distribution function tails and thus the statistical moments of the run-up characteristics.

The connection of the run-up parameters at the nonlinear stage with the
linear field at a fixed point is described either in a parametric form or
implicitly in the nonlinear equation (Didenkulova et al., 2010). This does
not allow for using the standard methods of random processes. At the same time,
it is known that this implicit equation is equivalent to a partial
first-order differential equation (PDE), i.e., to the simple (the Riemann
wave) equation (Rudenko and Soluyan, 1977). In statistical problems, this
equation arises in nonlinear acoustics. This equation, or its generalization,
the nonlinear diffusion equation called the Burgers' equation (Burgers,
1974), is the model equation in the hydrodynamic turbulence theory (Frisch, 1995). It should be noted that for the one-dimensional Burgers turbulence, as well as its three-dimensional version, used for the model description of the large-scale universe structure (Gurbatov et al., 2012), it is possible to give an almost comprehensive statistical description for certain initial conditions (Gurbatov et al., 1991, 1997, 2011; Gurbatov and Saichev, 1993; Molchanov et al., 1995; Frisch, 1995; Woyczynski, 1998; Frisch and Bec, 2001; Bec and Khanin, 2007). In particular, the single-point and the two-point probability distributions of the velocity field and even the

This paper is devoted to the analytical study of the probabilistic characteristics of the narrow-band long-wave run-up on the coast. Section 2 gives the basic equations of nonlinear shallow-water theory and the Carrier–Greenspan transformation, with the latter making it possible to linearize the nonlinear equations. Section 3 describes the moving shoreline dynamics when the deterministic sine wave approaches the slope. The probability characteristics of the deformed sine oscillations of the moving shoreline with a random phase are described in Sect. 4. Section 5 contains the probabilistic characteristics of the vertical displacement of the moving shoreline if the incident narrow-band wave has a random amplitude and phase. The discussion of the wave-breaking effects and their influence on the distribution of the run-up characteristics is given in Sect. 6. The results obtained are summarized in Sect. 7.

Here we will consider the classical formulation of the problem of a long-wave run-up on the constant-gradient slope in an ideal fluid (Fig. 1). The
wave is one-dimensional and propagates along the

The problem geometry.

It is important to note that the hodograph transformation is valid if the
Jacobian transformation is nonzero

We will assume that the wave approaches the coast from the area far from the
shoreline (

Detailed calculations of the long-wave run-up on the coast were carried out repeatedly; see, for example, Carrier and Greenspan (1958), Synolakis (1987), Pelinovsky and Mazova (1992), Tinti and Tonini (2005), Madsen and Fuhrman (2008), Madsen and Schaffer (2010), Antuano and Brocchini (2008, 2010), Didenkulova (2009), Dobrokhotov et al. (2015), and Aydin and Kanoglu (2017).

It is worth mentioning that the nonlinear time transformation in Eqs. (11) and (12) leads to the shoreline oscillation distortion in comparison with the
linear theory predictions. So, for large amplitudes the wave shape becomes
multivalued (broken). The first moment of the wave breaking on the
shoreline (the gradient catastrophe) is easily found from Eq. (12) by
calculating the first derivative of the moving shoreline velocity

The similar Carrier–Greenspan transformation is obtained for waves in narrow-inclined channels, fjords, and bays (Rybkin et al., 2014; Pedersen, 2016; Anderson et al., 2017; Raz et al., 2018); only the wave equation (Eq. 4) and relations (Eqs. 5–8) change. However, the moving shoreline dynamics are still described by Eqs. (11) and (12), valid for arbitrary cross section channels.

The monochromatic wave run-up on a flat slope by using the Carrier–Greenspan transformation has been studied in a number of papers cited above. Let us reproduce here the main features of the moving shoreline dynamics necessary for us to draw the statistical description below. Mathematically, the monochromatic wave run-up is described by an elementary solution of Eq. (4)

Figure 2 shows the moving shoreline dynamics at different wave height values
in terms of the breaking parameter up to the limiting value (Br

The moving shoreline dynamics

Let us now consider the probabilistic characteristics of the initially sine
wave run-up with a random phase on the shore, assuming it to be uniformly
distributed over the interval [0–2

The probability density of the moving shoreline vertical displacement for the initially sine wave run-up at Br

The obtained probability density function can be used to calculate the
statistical moments of the shoreline oscillations. Technically, however, it
is easier to use the parametric equations (Eqs. 27 and 28) and calculate
all the moments.

The second moment determines the dispersion

Finally, the total flooding time and its drainage time are easy to find from
Eqs. (27) and (28) by finding from the Eq. (28) the value

The total flooding time (the solid curve) and the drainage time (the dashed curve) depending on the parameter Br.

It is worth noting that, in contrast to the vertical displacement, the
moving shoreline velocity distribution [

Let us consider the run-up of a quasi-harmonic wave with a random amplitude
and phase on a flat slope. To do this, we will first rewrite Eqs. (32) and (37) for them to include the wave amplitude. It is convenient to
enter the maximum height

Let us construct the finite amplitude distribution at which the linear field
distribution is close to the Gaussian form and modify the Rayleigh
distribution for wave heights in the area

The modified Rayleigh distribution (Eq. 43) for different distribution values

When

Figure 6 shows the distribution of the run-up characteristics for different
ratios of

The probabilistic density function of the vertical shoreline
displacement in the nonlinear theory (the solid lines) and in the linear
theory (the dashed lines) for different

The finite (

The probabilistic density function of the shoreline vertical displacement in the linear theory (the dashed line) and nonlinear theory (the solid line).

The theory described above is valid for non-breaking waves. The mentioned
wave ensemble, strictly speaking, cannot be the Gaussian one, as it always
has unlimited tails in the probability density function. Let us briefly
discuss what the formulas obtained for non-breaking waves lead to in the
presence of broken waves. Figure 8 shows the parametric curve (Eqs. 27 and 28) when Br

The parametric curve (Eqs. 27 and 28) with Br

The probability density function at Br

In this paper, we study the run-up of irregular narrow-band waves with a random envelope (swell, storm surges, and tsunami) on a beach of a constant slope. The work was carried out in the framework of the nonlinear wave theory with one important assumption: there should be no breaking waves in the wave ensemble. This restriction is quite strict for field and laboratory conditions, but nevertheless, there are cases when it is performed. For instance, 75 % of historical tsunami waves climbed onto the coast with no breaking (Mazova et al., 1983). In the experiments performed in the Warwick University tank and in the Large Tank in Hanover (Denissenko et al., 2011, 2013) this condition was fulfilled.

The wave nonlinearity at the run-up stage leads to increased deviations from Gaussianity, as might be expected from general considerations. Nevertheless, it is shown that the probability distribution of the moving shoreline velocity does not depend on the wave nonlinearity and can be calculated within the linear theory framework. The same conclusion can be drawn about the distribution of the extreme run-up characteristics (the moving shoreline displacement and speed), which, in fact, has been discussed previously (Didenkulova et al., 2008). However, the probabilistic density function of the moving shoreline displacement differs from that predicted in the linear theory framework. It is described by Eq. (43) by using either the theoretical or the measured distribution of the incident wave amplitudes. The paper gives the calculation results of the probabilistic run-up characteristics with the modified Rayleigh distribution for wave amplitudes.

The wave-breaking leads to the inapplicability of the wave run-up theory based on the Carrier–Greenspan transformation. If, nevertheless, the share of large amplitude waves is small, the breaking occurs mainly at the run-down stage, having little effect on the long-wave coastal flooding characteristics (see Sect. 6). This question, however, requires a special study based on direct numerical solutions of the shallow-water equations or their nonlinear-dispersive generalizations.

Finally, it is worth noting that we considered the narrow-band wave run-up with a random amplitude and phase. As far as the random waves with a wide spectrum are concerned, they may be the problem for further consideration.

The obtained probability density functions of the vertical displacement of the moving shoreline are useful to compute statistical characteristics of flooding time and force on coasts and constructions, which are necessity for the mitigation of natural marine hazards.

Now, in practice, various generalizations of shallow-water equations are used to analyze tsunami run-up including the wave dispersion; see, for instance, Løvholt et al. (2012). The wave dissipation is a quadratic dissipative term that prevents us from getting analytical results, so its influence on statistical characteristics should be investigated in the future.

All data have been obtained using the analytic formulas given in the text.

SG performed the computing of the random Riemann wave characteristics, and EP performed the computations of the probability density function of the moving shoreline.

The authors declare that they have no conflict of interest.

The work is supported by the grants from the Russian Science Foundation: grant nos. 19-12-00256 (for computing the random Riemann wave characteristics) and 16-17-00041 (for the computations of the probability density function of the moving shoreline).

This research has been supported by the Russian Science Foundation (grant nos. 19-12-00256 and 16-17-00041).

This paper was edited by Ira Didenkulova and reviewed by two anonymous referees.