Numerical realization of the generalized Carrier-Greenspan Transform for the shallow water wave equations

Thesis (M.S.) University of Alaska Fairbanks, 2015We study the development of two numerical algorithms for long nonlinear wave runup that utilize the generalized Carrier-Greenspan transform. The Carrier-Greenspan transform is a hodograph transform that allows the Shallow Water Wave equations to be transformed into a linear second order wave equation with nonconstant coefficients. In both numerical algorithms the transform is numerically implemented, the resulting linear system is numerically solved and then the inverse transformation is implemented. The first method we develop is based on an implicit finite difference method and is applicable to constantly sloping bays of arbitrary cross-section. The resulting scheme is extremely fast and shows promise as a fast tsunami runup solver for wave runup in coastal fjords and narrow inlets. For the second scheme, we develop an initial value boundary problem corresponding to an Inclined bay with U or V shaped cross-sections that has a wall some distance from the shore. A spectral method is applied to the resulting linear equation in order to and a series solution. Both methods are verified against an analytical solution in an inclined parabolic bay with positive results and the first scheme is compared to the 3D numerical solver FUNWAVE with positive results

On the Klein-Gordon equation originating on a curve and applications to the tsunami run-up problem

Thesis (M.S.) University of Alaska Fairbanks, 2019Our goal is to study the linear Klein-Gordon equation in matrix form, with initial conditions originating on a curve. This equation has applications to the Cross-Sectionally Averaged Shallow Water equations, i.e. a system of nonlinear partial differential equations used for modeling tsunami waves within narrow bays, because the general Carrier-Greenspan transform can turn the Cross-Sectionally Averaged Shallow Water equations (for shorelines of constant slope) into a particular form of the matrix Klein-Gordon equation. Thus the matrix Klein-Gordon equation governs the run-up of tsunami waves along shorelines of constant slope. If the narrow bay is U-shaped, the Cross-Sectionally Averaged Shallow Water equations have a known general solution via solving the transformed matrix Klein-Gordon equation. However, the initial conditions for our Klein-Gordon equation are given on a curve. Thus our goal is to solve the matrix Klein-Gordon equation with known conditions given along a curve. Therefore we present a method to extrapolate values on a line from conditions on a curve, via the Taylor formula. Finally, to apply our solution to the Cross-Sectionally Averaged Shallow Water equations, our numerical simulations demonstrate how Gaussian and N-wave profiles affect the run-up of tsunami waves within various U-shaped bays.Chapter 1: Introduction and overview -- 1.1 Introduction -- 1.2 Overview -- Chapter 2: Preliminaries -- 2.1 Wave equation -- 2.2 Wave equation initial value problem -- 2.3 Wave equation for spatially-variable speed -- 2.4 The 1D Klein-Gordon Equation -- 2.5 Initial conditions long an arbitrary curve -- Chapter 3: Statement of the problem -- Chapter 4: Solution to the problem -- 4.1 Projection of initial conditions -- 4.2 Solutions to the Klein-Gordon Equation with specific A and B -- Chapter 5: Applications to the shallow water wave equation -- 5.1 Cross-sectionally averaged shallow water equations -- 5.2 U-shaped Bays -- 5.3 Numerical simulations -- Chapter 6: Conclusions -- References

Inverse non-linear problem of the long wave run-up on coast

The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erd\'elyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach.Comment: 15 pages, 8 figure

Modelling tsunami inundation on coastlines with characteristic form

This paper provides an indication of the likely difference in tsunami amplification and dissipation between different characteristic coastal embayments, coastal entrances and estuaries. Numerical modeling is performed with the ANU/Geoscience Australia tsunami inundation model. Characteristic coastal morphology is represented by simpler generic morphological shapes which can be applied easily in the ANUGA model, such that key non-dimensional parameters (e.g. embayment depth/bay width) can be varied. Modeling is performed with a range of bay shapes, seabed gradient and different incident tsunami wave shapes and wave angles, including sine waves, solitary waves and leading depression Nwaves. The results show a complex pattern for both large and small embayments, with wave breaking an important control on the amplification of the wave between the 20m contour and the shore. For large embayments, the wave run-up can be amplified by a factor six in comparison to the amplitude at the model boundary. For small embayments, the amplification is dependent on the location of the ocean water line, or tidal stage

Tsunami runup in U and V shaped bays

Thesis (M.S.) University of Alaska Fairbanks, 2014.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 areaIntroduction -- 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

Electro-impulse de-icing testing analysis and design

Electro-Impulse De-Icing (EIDI) is a method of ice removal by sharp blows delivered by a transient electromagnetic field. Detailed results are given for studies of the electrodynamic phenomena. Structural dynamic tests and computations are described. Also reported are ten sets of tests at NASA's Icing Research Tunnel and flight tests by NASA and Cessna Aircraft Company. Fabrication of system components are described and illustrated. Fatigue and electromagnetic interference tests are reported. Here, the necessary information for the design of an EIDI system for aircraft is provided

Axial wind effects on stratification and longitudinal sediment transport in a convergent estuary during wet season

Author Posting. © American Geophysical Union, 2020. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research-Oceans 125(2), (2020): e2019JC015254, doi:10.1029/2019JC015254.The Coupled Ocean‐Atmosphere‐Wave‐Sediment Transport (COAWST) modeling system was used to examine axial wind effects on vertical stratification and sediment transport in a convergent estuary. The model demonstrated that stratification dynamics in the upper estuary (Kelvin number <1; Ke= fB/√ g'hs) are dominated by longitudinal wind straining, whereas the dominant mechanism governing estuarine stratification in the lower estuary (Kelvin number ~1) is lateral wind straining. Barotropic advection contributes to seaward sediment transport and peaks during spring tides, whereas estuarine circulation causes landward sediment transport with a maximum during neap tides. Down‐estuary winds impose no obvious effects on longitudinal sediment flux, whereas up‐estuary winds contribute to enhanced seaward sediment flux by increasing the tidal oscillatory flux. The model also demonstrates that bottom friction is significantly influenced by vertical stratification over channel regions, which is indirectly affected by axial winds.This research was funded by the National Natural Science Foundation of China (Grants 41576089, 51761135021, and 41890851), the National Key Research and Development Program of China (2016YFC0402603) and the Guangdong Provincial Water Conservancy Science and Technology Innovation Project (Grant 201719). We thank Professor Liangwen Jia at the Sun Yat‐sen University for his kindly providing the surficial sediment samples data in 2011. We also thank graduate students Guang Zhang and Yuren Chen from the Sun Yat‐sen University for their help in data analysis. We are grateful to two anonymous reviewers for their insightful comments to help improve this manuscript. The data related to this article is available online at the Zenodo website (https://zenodo.org/record/3606471).2020-07-1