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

Ahmed Abdalazeez (1) , Ira Didenkulova (1, 2) , Denys Dutykh (3, 4, 5)

1 TTÜ - Tallinn University of Technology
2 Lobachevsky State University [Nizhni Novgorod]
3 LAMA - Laboratoire de Mathématiques
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
5 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc

The estimate of individual wave run-up is especially important for tsunami warning and risk assessment as it allows to evaluate the inundation area. Here as a model of tsunami we use the long single wave of positive polarity. The period of such 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, what results in a steep wave front formation. Situations, when waves approach the coast with a steep front are often observed during large tsunamis, e.g. 2004 Indian Ocean and 2011 15 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 Riemann wave, while the wave 20 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. 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 25 run-up height, which takes into account the wave front steepness, is proposed.

Domaines: Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Analyse numérique [cs.NA]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Mécanique des fluides [physics.class-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]; Physique Classique [physics.class-ph]; Milieux fluides et réactifs; Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Mathématiques générales [math.GM]; Analyse numérique [math.NA]

Ahmed Abdalazeez, Ira Didenkulova, Denys Dutykh. Nonlinear deformation and run-up of single tsunami waves of positive polarity: numerical simulations and analytical predictions. Natural Hazards and Earth System Sciences, 2019, 19 (12), pp.2905-2913. ⟨10.5194/nhess-19-2905-2019⟩. ⟨hal-02422302⟩