An application of one-sided Jacobi polynomials for spectral modeling of vector fields in polar coordinates

A spectral tau-method is proposed for solving vector field equations defined in polar coordinates. The method employs one-sided Jacobi polynomials as radial expansion functions and Fourier exponentials as azimuthal expansion functions. All the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau-coefficients of the radial expansion polynomials of the highest order. The proposed method is applied to linear and nonlinear-dispersive time evolution equations of hyperbolic-type describing internal Kelvin and Poincaré waves in a shallow, stratified lake on a rotating plane.