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.
Introduction
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, N waves, and periodic symmetric and asymmetric
wave trains, are considered. In this paper, we focus on the nonlinear
deformation and run-up of long single pulses of positive polarity on a plane
beach.
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.
Analytical solution
We solve the nonlinear shallow-water equations for the bathymetry shown in
Fig. 1:
1∂u∂t+u∂u∂x+g∂η∂x=0,2∂η∂t+∂∂xh(x)+ηu=0.
Here η(x,t) is the vertical displacement of the water surface with
respect to the still water level, u(x,t) is the depth-averaged water flow, h(x) is the
unperturbed water depth, g is the gravitational acceleration, x is the
coordinate directed onshore and t is time. The system of Eqs. (1) and (2) is
solved independently for the two bathymetries shown in Fig. 1: a basin of
constant depth h0 and length X0 and a plane beach, where the water
depth h(x)=-x tan α.
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 h0, the wave transforms as a Riemann wave (Zahibo et al., 2008):
3ηx,t=η0t-x+X0+LVx,t,4Vx,t=3gh0+ηx,t-2gh0,
where η0(x=-L-X0,t) is the water displacement at the left
boundary. After the propagation over the section of constant depth
h0, the incident wave has the following shape:
ηX0t=η0t-X0Vx,t,VX0t=3gh0+ηX0t-2gh0.
Following the methodology developed in Didenkulova (2008), we let this
nonlinearly deformed wave described by Eq. (5) run up on a plane beach,
characterized by the water depth h(x)= – x tan α. This approach does
not take into account the merging point of the two bathymetries and,
therefore, does not account for reflection from the toe of the slope and
wave interaction with the reflected wave.
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 ηX0 as a Fourier integral:
ηX0=∫-∞+∞Bωexpiωtdω.
Its complex spectrum B(ω) can be found in an explicit form in terms
of the inverse Fourier transform:
Bω=12π∫-∞+∞ηX0texp-iωtdt.
Equation (7) can be rewritten in terms of the water displacement, produced by
the wave maker at the left boundary (Zahibo et al., 2008):
Bω=12πiω∫-∞+∞dη0dzexp-iωz+x+X0+LVη0,dz,z=t-x+X0+LVη0.
In this study we consider long single pulses of positive polarity:
η0t=Asech2tT,
where A is the input wave height and T is the effective wave period at the
location with the water depth h0. The wave described by Eq. (9) has an
arbitrary height and period and, therefore, does not satisfy properties of
the soliton but just has a sech2 shape. Substituting Eq. (9) into
Eq. (8), we can calculate the complex spectrum B(ω).
Wave run-up oscillations at the coast r(t) and the velocity of the moving
shoreline u(t) can be found from Didenkulova et al. (2008):
10rt=Rt+ugtanα-u22g,11ut=Ut+utgtanα,12Rt=2πτL∫-∞+∞|ω|Hωexpiωt-τL+π4signωdω,13Ut=1tanαdRdt,
where τ=2L/gh0 is the travel time to 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 Rmax of such a wave (Eq. 9) can be found from Didenkulova et
al. (2008) and Sriram et al. (2016):
RmaxA=2.8312cotα1gh02h03T21/4.
If the initial wave is a soliton, Eq. (14) coincides with the famous Synolakis
formula (Synolakis, 1987).
Numerical model
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
h0=3.5 m,
A=0.1 m,
T=20 s and
tan α=1:6. Results of
numerical simulations are shown by the red line, and experimental data are
shown by the blue line.
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 h0=3.5 m, length of
[a, b] =251 m and a plane beach with a slope of tan α=1:6 (see
Fig. 1). The flume had 16 wave gauges along the constant depth section and a
run-up gauge on the slope. The incident wave had an amplitude of A=0.1 m and
period of T=20 s. The detailed description of the experiments can be found
in Didenkulova et al. (2013). The results of numerical simulations are in
good agreement with the laboratory experiments along the constant depth
section (see Fig. 2) as well as on the beach (Fig. 3). The comparison of the run-up
height is calculated numerically and analytically using the approach described
in Sect. 2 and with the experimental record shown in Fig. 3. It can be seen
that the experimentally recorded wave is slightly smaller, which may be
caused by the bottom friction, especially on the slope. Both numerical
and analytical models describe the first wave of positive polarity rather
well. The numerical prediction of run-up height is slightly higher than the
analytical one. This additional increase in the run-up height in the numerical
model may be explained by the nonlinear interaction with the reflected wave,
which is not taken into account in the analytical model. The wave of
negative polarity is much more sensitive to all the effects mentioned above
than the wave of positive polarity and, therefore, looks different for all
three lines in Fig. 3. By introducing additional dissipation in the numerical
model, one can easily reach perfect agreement between the numerical
simulations and experimental data. However, we do not do so, since below we
focus on the analysis of analytical results and for clarity would
like to avoid additional parameters in the numerical model. Also, we focus
on the maximum run-up height and, therefore, expect small differences
between the results of analytical and numerical models. The data used for all figures of this paper are available at 10.13140/rg.2.2.27658.41922 (Abdalazeez et al., 2019).
Run-up height of the long single wave with
A=0.1 m and
T=20 s on a beach slope, where tan
α=1:6. The numerical
solution is shown by the red dotted line, the analytical solution is shown by
the blue dashed line and the experimental record is shown by the black solid
line.
Results of numerical and analytical calculations
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 h0 and is shown in Fig. 1. The input wave
parameters such as wave amplitude, A/h0, and effective wave length,
λ/X0, where λ=Tgh0, are changed. The beach
slope is taken as tan α=1:20 for all simulations.
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,
Rmax/A, as a function of initial wave amplitude,
A/h0,
for different distances to the slope,
X0/λ. Analytical solution described in Sect. 2 is
shown by lines, and numerical solution described in Sect. 3 is shown by
symbols (diamonds, triangles, squares and circles) with matching colours.
The thick black line corresponds to Eq. (14) for wave run-up on a beach
without constant depth section, where kh0=0.38.
Figure 4 shows the dimensionless maximum run-up height, Rmax/A, as a
function of the initial wave amplitude, A/h0. The incident wave propagates
over different distances to the bottom slope, X0/λ=1.7,
3.4, 5.1 and 6.8, where kh0=0.38. The analytical solution described in Sect. 2
is shown with lines, and the numerical solution described in Sect. 3 is shown
with symbols (diamonds, triangles, squares and circles). It can be seen that
in most cases and especially for small values of X0/λ=1.7
and 3.4, numerical simulations give larger run-up heights than analytical
predictions. These differences can be explained by the effects of wave
interaction with the toe of the underwater beach slope, which are not taken
into account in the analytical solution. For larger distances
X0/λ=6.8, both analytical and numerical solutions give
similar results, supported by the numerical scheme dissipation described in
Sect. 3, which can be considered a “numerical error”. It should be
mentioned that we use a physical dissipation rate of zero for these simulations;
however, a small dissipation for stability of the numerical scheme is still
needed, and this may become noticeable at large distances. For the
sech2-shaped wave (A/h0=0.03, λ/X0=0.12)
propagation, the reduction of initial wave amplitude constitutes
∼2 %.
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,
Rmax/A,
as a function of distance to the slope,
X0/λ, for
different amplitudes of the initial wave,
A/h0.
Analytical solution described in Sect. 2 is shown by lines, and numerical
solution described in Sect. 3 is shown by symbols (triangles, squares and
circles) with matching colours, where kh0=0.38.
The same effects can be seen in Fig. 5, which shows the maximum run-up
height, Rmax/A, as a function of distance to the slope,
X0/λ, for different amplitudes of the initial wave,
A/h0. The distance X0/λ changes from 0.8 to 9.4, where
kh0=0.38. The analytical solution is shown with lines, while the
numerical solution is shown with symbols (triangles, squares and circles).
It can be seen in Fig. 5 that for smaller values of X0/λ<6, numerical predictions provide relatively larger run-up values compared
with analytical predictions, while for higher values of X0/λ>6, the differences are significantly reduced. A relevant change of this
behaviour is given for A/h0=0.06. We can observe that numerical
predictions for this amplitude become smaller than analytical predictions
for X0/λ>8. As stated above, we believe that this can be a result
of interplay of two effects: interaction with the underwater bottom slope,
which is not taken into account in the analytical prediction, and the
numerical scheme dissipation (“numerical error”), which affects the
numerical results.
The dependence of maximum run-up height, Rmax/A, on kh0 is shown in Fig. 6 for A/h0=0.03. It can be seen that the difference between numerical
and analytical results decreases with an increase in kh0. We relate this
effect with the wave interaction with the slope, which is not properly
accounted in our analytical approach. As one can see in Fig. 7, this
difference for a milder beach slope tan α=1:50 is reduced.
Maximum run-up height,
Rmax/A, as a function of kh0 for
different distances to the slope,
X0/λ. Analytical solution described in Sect. 2 is shown by lines, and
numerical solution described in Sect. 3 is shown by symbols (diamonds,
triangles, squares and circles) with matching colours. The thick black line
corresponds to Eq. (14) for wave run-up on a beach without constant depth
section
(A/h0=0.03).
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 kh0=0.38 for X0/λ > 8.
Maximum run-up height,
Rmax/A,
as a function of initial effective wave length, λ/X0 (blue axes) and kh0 (black
axes). Analytical solutions for tan
α=1:20 and
tan α=1:50 are
shown by dotted and dashed lines, respectively, while numerical simulations
for tan α=1:20 and
tan α=1:50 are
shown by circles and crosses, respectively (A/h0=0.03).
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 X0/λ. This result is in agreement with the conclusions
of Didenkulova et al. (2007) and Didenkulova (2009) for sinusoidal waves. In
order to be consistent with the results of Didenkulova et al. (2007) and Didenkulova (2009), we connect the distance X0/λ with the
incident wave front steepness in the beginning of the bottom slope. The wave
front steepness s is defined as the maximum of the time derivative of water
displacement, d(η/A)/d(t/T), and is studied in
relation with the initial wave front steepness, s0, where
s(x)=maxdη(x,t)/dtA/T,s0=maxdη(x=a,t)/dtA/T.
In order to calculate the incident wave front steepness in the beginning of
the bottom slope from results of numerical simulations, we should separate
the incident wave and the wave reflected from the bottom slope. At the same
time, the wave steepening along the basin of constant depth is very well
described analytically, as demonstrated in Fig. 9.
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 A/h0 and kh0 and can be
approximated well by the power fit (coefficient of determination
R2=0.99):
Rmax/R0=s/s00.42,
where Rmax/A is the maximum run-up height in the conjoined basin (with a
section of constant depth); R0/A is the corresponding maximum run-up
height on a plane beach (without a section of constant depth).
Maximum run-up height,
Rmax/A,
as a function of the distance to the slope,
X0/λ , for different
values of kh0. Analytical solution described
in Sect. 2 is shown by lines, and numerical solution described in Sect. 3
is shown by symbols (triangles, squares and circles) with matching colours
(A/h0=0.03).
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 N waves in Tadepalli and Synolakis
(1994).
Wave evolution at different locations,
x/λ=0, 0.85, 1.71, 2.56,
3.41, 4.27 and 5.12, along the section of constant depth for a basin with
X0/λ=5.12 and tan α=1:20. Numerical results are shown by solid lines,
while the analytical predictions are given by the black dotted lines. The
parameters of the wave are
A/h0=0.03 and
kh0=0.19.
The ratio of maximum run-up height in the conjoined
basin,
Rmax/A, and the maximum run-up height on a plane beach,
R0/A, versus the wave front steepness,
s/s0, for A/h0=0.057,
kh0=0.38 (brown points); A/h0=0.086,
kh0=0.38 (red plus signs); A/h0=0.057,
kh0=0.29 (blue points); A/h0=0.086,
kh0=0.29 (turquoise
plus signs); A/h0=0.057, kh0=0.22 (violet
points); A/h0=0.086, kh0=0.22 (pink plus
signs); A/h0=0.057, kh0=0.19 (dark-green
points); and A/h0=0.086,
kh0=0.19 (light-green
plus signs). All markers correspond to the results of numerical simulations,
while the asymptotic analytical predictions are given by the red solid line.
Black dashed line corresponds to the power fit of the analytical results of
Eq. (16).
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 kh0. In our opinion, this dependence on kh0 is a result
of merging a plane beach with a flat bottom. This effect can be parameterized
with the factor (L/λ)1/4. The result of this parameterization
is shown in Fig. 11. Here we can see that for smaller face front wave
steepness, s/s0<1.5, the run-up height is proportional to the
analytically estimated curve shown by Eq. (16), while for larger face front
wave steepness, s/s0>1.5, the dependence on s/s0 is
weaker. This dependence for all numerical run-up height data, presented in
Fig. 11, can be approximated by the power fit (coefficient of determination
R2=0.85):
Rmax/R0=1.17λ/L1/4s/s01/4.
The normalized maximum run-up height,
Rmax/R0 (L/λ)1/4, calculated numerically versus
the wave front steepness,
s/s0, for
the same values of A/h0 and kh0 as in Fig. 10. Red solid line
is proportional to the “analytically estimated” Eq. (16), while black
solid line corresponds to Eq. (17).
Conclusions and discussion
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: Rmax/R0=s/s00.42.
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 N waves
(Tadepalli and Synolakis, 1994).
Numerical simulations in general support this analytical finding. For
smaller face front wave steepness (s/s0<1.5), numerical curves
of the maximum tsunami run-up height are parallel to the analytical ones, while
for larger face front wave steepness (s/s0>1.5), this
dependence is milder. The latter may be a result of numerical dissipation
(error), which is larger for a longer wave propagation and, consequently,
larger wave steepness. The suggested formula, which gives the best fit with
the data of numerical simulations in general, is Rmax/R0=1.17λ/L1/4s/s01/4.
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 nonlinear shallow-water equations, which are used in this study and
commonly utilized for tsunami modelling, are also known to neglect
dispersive effects. In this context, it is important to mention the recent
work of Larsen and Fuhrman (2019). They used Reynolds-averaged Navier–Stokes (RANS) equations and k–ω
model for turbulence closure to simulate the propagation and run-up of positive
single waves, including full resolution of dispersive short waves (and their
breaking) that can develop near a positive tsunami front. They similarly
showed that this effect depends on the propagation distance prior to the
slope if a simple toe with a slope type of bathymetry is utilized. This
work shows that these short waves have little effect on the overall run-up
and hence give additional credence to the use of shallow-water equations.
These results largely confirm what was previously hypothesized by Madsen et
al. (2008), namely that these short waves would have little effect on the overall
run-up and inundation of tsunamis (though they found that they could
significantly increase the maximum flow velocities).
Data availability
The data used for all figures of this paper are available at 10.13140/rg.2.2.27658.41922 (Abdalazeez et al., 2019). The source code (in MATLAB) used to generate
these data may be shared upon request.
Author contributions
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.
Competing interests
The author declares that there is no conflict of interest.
Acknowledgements
The authors are very grateful to Professor Efim Pelinovsky, who came up with the idea
for this study a few years ago.
Financial support
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.
Review statement
This paper was edited by Mauricio Gonzalez and reviewed by two anonymous referees.
ReferencesAbdalazeez, A. A., Didenkulova, I., and Dutykh, D.: Data_Nonlinear deformation and run-up of single tsunami waves of positive polarity numerical simulations and analytical predictions.zip, 10.13140/rg.2.2.27658.41922, 2019.Brocchini, M. and Gentile, R.: Modelling the run-up of significant wave
groups, Cont. Shelf Res., 21, 1533–1550,
10.1016/S0278-4343(01)00015-2, 2001.Carrier, G. F. and Greenspan, H. P.: Water waves of finite amplitude on a
sloping beach, J. Fluid Mech., 4, 97–109,
10.1017/S0022112058000331, 1958.Carrier, G. F., Wu, T. T., and Yeh, H.: Tsunami run-up and draw-down on a
plane beach, J. Fluid Mech., 475, 79–99,
10.1017/S0022112002002653, 2003.Didenkulova, I.: New trends in the analytical theory of long sea wave runup,
in: Applied Wave Mathematics, edited by: Quak, E. and Soomere, T., Springer,
Berlin, Heidelberg, Germany, 265–296,
10.1007/978-3-642-00585-5_14, 2009.Didenkulova, I., Pelinovsky, E., Soomere, T., and Zahibo, N.: Run-up of
nonlinear asymmetric waves on a plane beach, in: Tsunami and nonlinear
waves, edited by: Kundu, A., Springer, Berlin, Heidelberg, Germany, 175–190,
10.1007/978-3-540-71256-5_8, 2007.Didenkulova, I., Pelinovsky, E., and Soomere, T.: Runup characteristics of
symmetrical solitary tsunami waves of “unknown” shapes, Pure Appl.
Geophys., 165, 2249–2264,
10.1007/978-3-0346-0057-6_13, 2008.Didenkulova, I., Denissenko, P., Rodin, A., and Pelinovsky, E.: Effect of
asymmetry of incident wave on the maximum runup height, J. Coastal
Res., 65, 207–212, 10.2112/SI65-036.1, 2013.Didenkulova, I., Pelinovsky, E. N., and Didenkulova, O. I.: Run-up of long
solitary waves of different polarities on a plane beach. Izvestiya,
Atmos. Ocean. Phys., 50, 532–538,
10.1134/S000143381405003X, 2014.Dutykh, D., Poncet, R., and Dias, F.: The VOLNA code for the numerical
modeling of tsunami waves: Generation, propagation and inundation, Eur.
J. Mech. B-Fluid., 30, 598–615,
10.1016/j.euromechflu.2011.05.005, 2011a.Dutykh, D., Katsaounis, T., and Mitsotakis, D.: Finite volume schemes for
dispersive wave propagation and runup, J. Comput. Phys.,
230, 3035–3061, 10.1016/j.jcp.2011.01.003, 2011b.Glimsdal, S., Løvholt, F., Harbitz, C. B., Romano, F., Lorito, S., Orefice, S., Brizuela, B., Selva, J., Hoechner, A., Volpe, M., Babeyko, A., Tonini, R., Wronna, M., and Omira, R.: A new approximate method for quantifying tsunami maximum inundation height probability, Pure Appl. Geophys., 176, 3227–3246, 10.1007/s00024-019-02091-w, 2019.
Gosse, L.: Computing qualitatively correct approximations of balance laws:
exponential-fit, well-balanced and asymptotic-preserving, Springer Milan,
Italy, 2013.Kânoğlu, U.: Nonlinear evolution and runup–rundown of long waves
over a sloping beach, J. Fluid Mech., 513, 363–372,
10.1017/S002211200400970X, 2004.Kânoğlu, U. and Synolakis, C.: Initial value problem solution of
nonlinear shallow water-wave equations, Phys. Rev. Lett., 97,
148501, 10.1103/PhysRevLett.97.148501, 2006.Larsen, B. E. and Fuhrman, D. R.: Full-scale CFD simulation of tsunamis. Part
1: Model validation and run-up, Coast. Eng., 151, 22–41,
10.1016/j.coastaleng.2019.04.012, 2019.Li, Y. and Raichlen, F.: Non-breaking and breaking solitary wave run-up,
J. Fluid Mech., 456, 295–318,
10.1017/S0022112001007625, 2002.Lin, P., Chang, K. A., and Liu, P. L. F.: Runup and rundown of solitary waves
on sloping beaches, J. Waterw. Port C., 125, 247–255,
10.1061/(ASCE)0733-950X(1999)125:5(247), 1999.Løvholt, F., Glimsdal, S., Harbitz, C. B., Zamora, N., Nadim, F.,
Peduzzi, P., Dao, H., and Smebye, H.: Tsunami hazard and exposure on the global
scale, Earth-Sci. Rev., 110, 58–73,
10.1016/j.earscirev.2011.10.002, 2012.Madsen, P. A. and Fuhrman, D. R.: Run-up of tsunamis and long waves in terms
of surf-similarity, Coast. Eng., 55, 209–223,
10.1016/j.coastaleng.2007.09.007, 2008.Madsen, P. A. and Schäffer, H. A.: Analytical solutions for tsunami run-up
on a plane beach: single waves, N-waves and transient waves, J.
Fluid Mech., 645, 27–57, 10.1017/S0022112009992485,
2010.Madsen, P. A., Fuhrman, D. R., and Schäffer, H. A.: On the solitary wave
paradigm for tsunamis, J. Geophys. Res.-Oceans, 113,
1–22, 10.1029/2008JC004932, 2008.
Mazova, R. K., Osipenko, N. N., and Pelinovsky, E. N.: Solitary wave climbing a
beach without breaking, Rozprawy Hydrotechniczne, 54, 71–80, 1991.McGovern, D. J., Robinson, T., Chandler, I. D., Allsop, W., and Rossetto, T.:
Pneumatic long-wave generation of tsunami-length waveforms and their runup,
Coast. Eng., 138, 80–97,
10.1016/j.coastaleng.2018.04.006, 2018.Pedersen, G. and Gjevik, B.: Run-up of solitary waves, J. Fluid
Mech., 135, 283–299, 10.1017/S002211208700329X, 1983.Pelinovsky, E. N. and Mazova, R. K.: Exact analytical solutions of nonlinear
problems of tsunami wave run-up on slopes with different profiles, Nat.
Hazards, 6, 227–249, 10.1007/BF00129510, 1992.Raz, A., Nicolsky, D., Rybkin, A., and Pelinovsky E.: Long wave runup in
asymmetric bays and in fjords with two separate heads, J.
Geophys. Res.-Oceans, 123, 2066–2080,
10.1002/2017JC013100, 2018.Schimmels, S., Sriram, V., and Didenkulova, I.: Tsunami generation in a
large scale experimental facility, Coast. Eng., 110, 32–41,
10.1016/j.coastaleng.2015.12.005, 2016.Shuto, N.: The Nihonkai-chuubu earthquake tsunami on the north Akita coast,
Coastal Engineering Japan, JSCE, 28, 255–264, 10.1080/05785634.1985.11924420, 1985.Sriram, V., Didenkulova, I., Sergeeva, A., and Schimmels, S.: Tsunami
evolution and run-up in a large scale experimental facility, Coast.
Eng., 111, 1–12, 10.1016/j.coastaleng.2015.11.006,
2016.
Stoker, J. J.: Water waves, the mathematical theory with applications,
Interscience Publishers Inc., New York, 1957.Synolakis, C. E.: The run-up of solitary waves, J. Fluid Mech.,
185, 523–545, 10.1017/S002211208700329X, 1987.Synolakis, C. E., Deb, M. K., and Skjelbreia, J. E.: The anomalous behavior of
the runup of cnoidal waves, Phys. Fluids, 31, 3–5,
10.1063/1.866575, 1988.Tadepalli, S. and Synolakis, C. E.: The run-up of N-waves, P. Roy. Soc.
Lond. A, 445, 99–112, 10.1098/rspa.1994.0050, 1994.Tang, J., Shen, Y., Causon, D. M., Qian, L., and Mingham, C. G.: Numerical
study of periodic long wave run-up on a rigid vegetation sloping beach,
Coast. Eng., 121, 158–166,
10.1016/j.coastaleng.2016.12.004, 2017.Tinti, S. and Tonini, R.: Analytical evolution of tsunamis induced by
near-shore earthquakes on a constant-slope ocean, J. Fluid
Mech., 535, 33–64, 10.1017/S0022112005004532, 2005.Touhami, H. E. and Khellaf, M. C.: Laboratory study on effects of submerged
obstacles on tsunami wave and run-up, Nat. Hazards, 87, 757–771,
10.1007/s11069-017-2791-9, 2017.Yao, Y., He, F., Tang, Z., and Liu, Z.: A study of tsunami-like solitary wave
transformation and run-up over fringing reefs, Ocean Eng., 149,
142–155, 10.1016/j.oceaneng.2017.12.020, 2018.Zahibo, N., Didenkulova, I., Kurkin, A., and Pelinovsky E.: Steepness and
spectrum of nonlinear deformed shallow water wave, Ocean Eng., 35
47–52, 10.1016/j.oceaneng.2007.07.001, 2008.Zainali, A., Marivela, R., Weiss, R., Yang, Y., and Irish, J. L.: Numerical
simulation of nonlinear long waves in the presence of discontinuous coastal
vegetation, Mar. Geol., 396, 142–149,
10.1016/j.margeo.2017.08.001, 2017.