Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays
Abstract—We present an analytical study of the propagation and
run-up of long waves in piecewise sloping, U-shaped bays using the
cross-sectionally averaged shallow water equations. The nonlinear
equations are transformed into a linear equation by utilizing the gen-
eralized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech
748:416–432, 2014). The solution of the linear wave propagation is
taken as the boundary condition at the toe of the last sloping segment,
as in Synolakis (J Fluid Mech185:523–545, 1987). We then consider a
piecewise sloping bathymetry, and as in Kanoglu and Synolakis (J
Fluid Mech 374:1–28, 1998), ﬁnd the linear solution in the near shore
region, which can be used as the boundary condition for the nonlinear
problem. Our primary results are an analytical run-up law for narrow
channels and breaking criteria for both monochromatic waves and
solitary waves. The derived analytical solutions reduce to well-known
solutions for parabolic bays and plane beaches. Our analytical pre-
dictions are veriﬁed in narrow bays via a comparison to direct
numerical simulation of the 2-D shallow water equations.
Key words: Long-wave run-up, shallow water wave equa-
tions, carrier–greenspan transformation, piecewise sloping bays.
Tsunamis originating in the open ocean pose a
signiﬁcant threat to coastal communities (Dunbar and
Weaver 2015). Understanding of the tsunami propa-
gation and run-up is of a paramount importance for
saving lives and mitigating the potential impacts on
coastal infrastructure (Synolakis and Bernard 2006;
Kanoglu et al. 2015). In Alaska, British Columbia,
and the North Western states of the U.S. tsunami
waves, before reaching coastal communities, have to
propagate through long and narrow channels and
bays. The communities are commonly located at the
bay head areas—a potential worst case location for
the community, since long waves can greatly amplify
at the head of the bays, as shown in Fig. 1.
To assess the potential tsunami inundation areas,
2-D shallow water equations (SWEs) are commonly
used to model the propagation and run-up of tsuna-
mis. The 2-D SWEs for general bathymetry/
topography proﬁles are usually hard to analyze ana-
lytically and are hence solved numerically. An
interested reader is referred to NTHMP (2012) for a
list of models employed to assess the tsunami inun-
dation in the U.S. states and territories. However, in
many physically relevant circumstances—along the
plane beaches (Carrier and Greenspan 1957; Syno-
lakis 1987) or in long and narrow bays (Zahibo et al.
2006; Choi et al. 2008; Rybkin et al. 2014; Diden-
kulova and Pelinovsky 2011), the SWEs can be
greatly simpliﬁed into a 1-D system of equations. We
recall that the vast majority of results have been
derived by use of Carrier–Greenspan (CG) transform
in the context of plane beaches (Carrier and Green-
span 1957). In particular, a wide variety of wave
proﬁles run-up behaviors have been investigated, and
expressions for maximum run-up and wave ampliﬁ-
cation factors were derived for monochromatic waves
(Synolakis 1991; Madsen et al. 2008; Carrier and
Greenspan 1957; Keller and Keller 1964), Solitary
waves (Synolakis 1987), Gaussian pulse (Carrier
Department of Applied Mathematics, University of Color-
ado Boulder, Colorado 80309, USA.
Department of Applied Mathematics, University of
Waterloo, Ontario N2L 3G1, Canada.
Department of Mathematics and Statistics, University of
Alaska Fairbanks, Alaska 99775, USA. E-mail: email@example.com
Physics Department, University of Alaska Fairbanks,
Alaska 99775, USA.
Geophysical Institute, University of Alaska Fairbanks,
Alaska 99775, USA.
Institute of Applied Physics, Nizhny Novgorod 603155,
Nizhny Novgorod State Technical University n.a. R.E.
Alekseev, Nizhny Novgorod 603155, Russia.
Special Research Bureau for Automation of Marine
Researches, Yuzhno-Sakhalinsk, Russia.
Pure Appl. Geophys. 174 (2017), 3185–3207
Ó 2017 Springer International Publishing
Pure and Applied Geophysics