Variational Method for Computing Ray Trajectories and Fronts of Tsunami Waves Generated by a Localized Source
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ATICAL PHYSICS
Variational Method for Computing Ray Trajectories and Fronts of Tsunami Waves Generated by a Localized Source S. Yu. Dobrokhotova,c,*, M. V. Klimenkob,**, I. A. Nosikovb,***, and A. A. Tolchennikova,c,**** a
b
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 117526 Russia Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, West Department, Russian Academy of Sciences, Kaliningrad, 236035 Russia c Moscow Institute of Physics and Technology (National Research University), Dolgoprudnyi, Moscow oblast, 141700 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] ****e-mail: [email protected] Received November 11, 2019; revised February 15, 2020; accepted April 9, 2020
Abstract—A variational approach for solving the boundary value problem of computing ray trajectories and fronts of ocean waves is presented. The solution method is based on Fermat’s principle (of stationary time). A distinctive feature of the proposed approach is that the Fermat functional is optimized directly without solving the Euler–Lagrange equation; moreover, the locations of the wave source and receiver are fixed. Multipath propagation in the boundary value problem is addressed by finding various types of stationary points of the Fermat functional. The technique is numerically tested by applying the method of bicharacteristics with the use of analytical seabed models. The advantages of the variational approach and the prospects of its further development as applied to ocean wave computation are described. The relations between various types of stationary points of the travel time functional, caustics, and foci are discussed. Keywords: ocean waves, tsunami, rays, fronts, Fermat’s principle, functional, method of bicharacteristics DOI: 10.1134/S0965542520080072
1. INTRODUCTION AND FORMULATION OF THE PROBLEM The simulation of propagation of ocean waves, specifically, tsunamis, is an important task for applications [1, 2]. Of greatest interest are the features of ocean wave propagation connected with the formation of caustics, near which tsunami waves can reach enormous destructive power [3]. There are numerous publications in which tsunami parameters in the ocean and nearshore zones are evaluated [4, 5], but a task of current interest is the search for efficient and fast methods for computing and predicting tsunami aftermath. A fast analytical-numerical algorithm for modeling propagation of long waves (e.g., tsunamis) and vortices in the ocean was developed in [6–8]. More specifically, asymptotic formulas were derived for waves and vortices generated by localized sources and described by linearized shallow water equations with a variable depth or by a potential model (with allowance for dispersion effects). These asymptotic formulas were obtained by applying the modified Maslov canonical operator adapted for the construction of solutions localized near points and curves. The formulas are fairly simple and us
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