WAVEx : a new spectral wind wave model using stabilized finite elements
Creators: name: Loveland, Mark Douglass
name: 0000-0002-2164-2884
Publisher: name: The University of Texas at Austin
Year: 2023
Subjects: Wind waves
Finite elements

description: The focus of this dissertation is the development of mathematical models for wind-generated gravity waves on the ocean surface. Wind-generated gravity waves, commonly referred to as wind waves, are waves on the ocean surface that typically exist with wavelengths of 1-1500 m and periods ranging from 0.25-30 seconds. Spectral wind wave models estimate the transport of energy from these wind waves by numerically solving a form of a partial differential equation (PDE) called the Wave Action Balance Equation. More specifically, these models seek to predict the changes in action density due to forcing from winds on the surface, dissipation, and nonlinear interactions. In turn, these models can be used to generate forecasts of statistical quantities which include significant wave height, wave radiation stress, significant wave period, as well as the wave spectrum itself. This dissertation focuses on two primary objectives related to spectral wind wave models. The first being development of finite element methods (FEM) for the spectral wind wave model and the second being investigation into source term implementations. As it pertains to the latter, the dissertation discusses the history of source terms and those used in contemporary spectral wind wave models. A study is conducted that focuses on the trade-off between run time and accuracy of 1st, 2nd, and 3rd generation source terms. In the study, a coupled wave-circulation model called ADCIRC+SWAN is used to model Hurricane Ike. Water levels from the coupled model are compared to gauge data and significant wave height, peak period, and mean wave direction are compared to buoys. Results show potential for efficacy of reduced order source terms in order to eliminate computational cost while sacrificing minimal accuracy. Several potential FEM discretizations of the Wave Action Balance Equation are discussed. The methods, which include streamline upwind Petrov-Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented using the open source finite element library FEniCS for simplified 2-D cases. Open source finite element libraries, such as FEniCS, typically only support geometries up to dimension of 3. The Wave Action Balance Equation is 4 dimensions in space so this presents difficulties. A method to use a FEM library, such as FEniCS, to solve problems in domains with dimension larger than 4 using the product basis is discussed and demonstrated for some simpler problems such as the Poisson equation in 4 dimensions and the advection-reaction-diffusion problem. Using these ideas, a new spectral wind wave model, WAVEx, is formulated and implemented using the new finite element library FEniCSx. WAVEx is designed to allow for construction of multiple FEM discretizations with relatively small modifications in the code base. An example implementation is then demonstrated with WAVEx using continuous finite elements and SUPG stabilization in geographic/spectral space. For propagation in time, a generalized one step implicit finite difference method is used. When source terms are active, the second order operator splitting scheme known as Strang splitting is used. In the splitting scheme, propagation is solved using the aforementioned implicit method and the nonlinear source terms are treated explicitly using second order Runge-Kutta. Several test cases which are part of the Office for Naval Research Test Bed (ONR Test Bed) are demonstrated both with and without 3rd generation source terms and results are compared to analytic solutions, observations, and SWAN output.