Abstract. 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.
Nonlinear deformation and run-up of single tsunami waves of positive polarity: numerical simulations and analytical predictions