Tsunami runup in U and V shaped bays

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Author
Garayshin, Viacheslav Valer'evich
Гарайшин, Вячеслав Валерьевич
Chair
Rybkin, Alexei
Committee
Rhodes, John
Nicolsky, Dmitry
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URI
http://hdl.handle.net/11122/4608
Abstract
Tsunami runup can be effectively modeled using the shallow water wave equations. In 1958 Carrier and Greenspan in their work "Water waves of finite amplitude on a sloping beach" used this system to model tsunami runup on a uniformly sloping plane beach. They linearized this problem using a hodograph type transformation and obtained the Klein-Gordon equation which could be explicitly solved by using the Fourier-Bessel transform. In 2011, Efim Pelinovsky and Ira Didenkulova in their work "Runup of Tsunami Waves in U-Shaped Bays" used a similar hodograph type transformation and linearized the tsunami problem for a sloping bay with parabolic cross-section. They solved the linear system by using the D'Alembert formula. This method was generalized to sloping bays with cross-sections parameterized by power functions. However, an explicit solution was obtained only for the case of a bay with a quadratic cross-section. In this paper we will show that the Klein-Gordon equation can be solved by a spectral method for any inclined bathymetry with power function for any positive power. The result can be used to estimate tsunami runup in such bays with minimal numerical computations. This fact is very important because in many cases our numerical model can be substituted for fullscale numerical models which are computationally expensive, and time consuming, and not feasible to investigate tsunami behavior in the Alaskan coastal zone, due to the low population density in this area
Description
Thesis (M.S.) University of Alaska Fairbanks, 2014.
Table of Contents
Introduction -- Chapter 1. Description of the problem -- Chapter 2. Linearization of the system of shallow water equations -- 2.1. Method of characteristics -- 2.2. Change of variables -- 2.3. Boundary conditions, initial conditions and domain of Φ, the linearized shallow water equation -- 2.4. The limits of applicability of the hodograph transformation -- Chapter 3. Solution of the linearized shallow water equation (the equation (2.19)) -- 3.1. Laplace transformation -- 3.2. Solving the transformed equation -- 3.3. Inverse Laplace transformation -- 3.4. Obtaining the formula for the solution of the linearized shallow water equation -- 3.5. The formula for Φ with a different order of integration -- Chapter 4. Verification of the solution of the linearized equation, obtained by the Laplace transform, for particular cases -- 4.1. Case 1. Plane beach case -- 4.1.1. Method 1. Solving by Laplace transform -- 4.1.2. Method 2. Solving by Fourier-Bessel transform after Carrier-Greenspan -- 4.2. Case 2. An inclined bay with the parabolic cross-section -- 4.2.1. Method 1. Solving by Laplace transform -- 4.2.2. Method 2. Solving by D'Alembert method after Didenkulova and Pelinovsky -- Chapter 5. Relation of the shallow water problem to the wave equation in Rn space -- 5.1. Solution of the wave equation and its spherical mean -- 5.2. Shallow water problems related to the waves in odd-dimensional spaces -- Chapter 6. Sloping bay with the cross-section parameterized by z = cIy ²/₃, c > 0 -- 6.1. Derivation of Φ (λ, σ), the solution of the linearized shallow water equation (6.1) with given initial and boundary conditions -- 6.2. Comparison of the obtained solution Φ for the bay described by z = cIy ²/₃ with the solution obtained through the Laplace transform -- 6.3. Physical characteristics of a wave in the bay and partial derivatives of Φ (λ,σ) -- Chapter 7. Practical experiments -- Conclusion -- References -- Appendix -- Chapter A. Modified Bessel functions and their asymptotic formulas -- A.1. Modified Bessel functions -- A.2. The asymptotic behavior of the modified Bessel functions.
Date
2013-08
Type
Thesis
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