The direct coordinate transformation method,
which only transforms independent variables
and retains Cartesian dependent variables, may not be an appropriate method
for the curvilinear parabolic approximation of the vector form of
the wave-current equation given by
Kirby (1986). In this study, the covariant-contravariant 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 (1986). The derived
curvilinear equation presents the same form as the Cartesian model
equation.
Several examples of wave simulations in curvilinear coordinate systems,
including a case
with wave-current interaction,
are shown with comparisons to theoretical solutions or measurement
data.
Details can be found
in:
Shi, F. and Kirby, J. T.,2005,
Curvilinear parabolic approximation for surface
wave transformation with wave-current interaction,
Journal of Computational Physics , 204, 562-586
Curv-REF/DIF results in simulations of waves in a circular channel
Curv-REF/DIF results of wave-current interaction in Glourlay's case
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