Curvilinear parabolic approximation for surface wave transformation with wavecurrent interaction
The direct coordinate transformation method, which only transforms independent variables and retains Cartesian dependent variables, may not be an appropriate method for the purpose of simplifying the curvilinear parabolic approximation of the vector form of the wavecurrent equation given by Kirby [Higher-order approximations in the parabolic equation method for water waves, J. Geophys. Res. 91 (1986) 933-952]. In this paper, the covariantcontravariant tensor method is used for the curvilinear parabolic approximation. We use the covariant components of the wave number vector and contravariant components of the current velocity vector so that the derivation of the curvilinear equation closely follows the higher-order approximation in rectangular Cartesian coordinates in Kirby [Higher-order approximations in the parabolic equation method for water waves, J. Geophys. Res. 91 (1986) 933-952]. The resulting curvilinear equation can be easily implemented using the existing model structure and numerical schemes adopted in the Cartesian parabolic wave model [J.T. Kirby, R.A. Dalrymple, F. Shi, Combined Refraction/Diffraction Model REF/DIF 1, Version 2.6. Documentation and Users Manual, Research Report, Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware, Newark, 2004]. Several examples of wave simulations in curvilinear coordinate systems, including a case with wavecurrent interaction, are shown with comparisons to theoretical solutions or measurement data.
parabolic approximation wavecurrent interaction curvilinear coordinates wave equations surface wave simulation