This research aims to study the numerical solution of the non-divergent barotropic model in spherical coordinates, linearized around a resting background state. For this purpose, we shall use the spectral method that consists of expanding the stream function in terms of the Eigen solutions of the Laplace operator, the so-called spherical harmonics. As demonstrated in several studies in the literature, the mutual orthogonality between the spherical harmonics leads to a diagonalized representation of the linear operator. Consequently, the time evolution of the expansion coefficients in the linearized version of the model is described by a linear harmonic oscillator equation whose characteristic oscillation frequency satisfies the dispersion relation of the Rossby-Haurwitz waves. In this context, different discretization schemes of these ordinary differential equations will be analyzed by comparing with the exact solution represented by the rotation in the complex plane around the circle centered at the origin whose radius is given by the initial amplitude of the expansion coefficient and the angular frequency given by the respective Eigen frequency of the corresponding spherical harmonic. Through the spectral expansion of the relative vorticity and stream function fields, the impact of the time discretization of the numerical solution in the energy dispersion shall be analyzed.
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