  The prediction of the wind wave spectrum of the ocean using numerical models
are an important tool for researchers, engineers, and communities living in
coastal areas. The governing equation of the wind wave models, the Wave Action
Balance Equation, presents unique challenges for implementing reliable
numerical models because it is highly advective, highly nonlinear and high
dimensional. Historically, most operational models have utilized finite
difference methods, others have used finite volume methods but relatively few
attempts at using finite element methods. In this work, we seek to fill this
gap by investigating several different finite element discretizations of the
Wave Action Balance Equation. The methods, which include streamline upwind
Petrov-Galerkin (SUPG), least squares, and discontinuous Galerkin, are
implemented and convergence properties are examined for some simplified 2-D
test cases. Then, a new spectral wind wave model, WAVEx, is formulated and
implemented for the full problem setting. WAVEx uses continuous finite elements
along with SUPG stabilization in geographic/spectral space that allows for
fully unstructured triangular meshes in both geographic and spectral space. For
propagation in time, a second order fully implicit finite difference method is
used. When source terms are active, a second order operator splitting scheme is
used to linearize the problem. In the splitting scheme, propagation is solved
using the implicit method and the nonlinear source terms are treated
explicitly. Several test cases, including analytic tests and laboratory
experiments, are demonstrated and results are compared to analytic solutions,
observations, as well as output from another model that is used operationally.
