Ja-Rin Park

Fourier-series representation and projection of spherical harmonic functions

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10−15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.

Geopotentinl Field in Nonlinear Balance with the Sectoral Mode of Rossby-Haurwitz Wave on the Inclined Rotation Axis

Analytical geopotential field in balance with the sectoral mode (the first symmetric mode with respect to the equator) of the Rossby-Haurwitz wave on the inclined rotation axis was derived in presence of superrotation background flow. The balanced field was obtained by inverting the divergence equation with the time derivative being zero. The inversion consists of two steps, i.e., the evaluation of nonlinear forcing terms and the finding of analytical solutions based on the Poisson`s equation. In the second step, the forcing terms in the from of Legendre function were readily inverted due to the fact that Legendre function is the eigenfunction of the spherical Laplacian operator, while other terms were solved either by introducing a trial function or by integrating the Legendre equation. The balanced field was found to be expressed with six zonal wavenumber components, and shown to be of asymmetric structure about the equator. In association with asymmetricity, the advantageous point of the balanced field as a validation method for the numerical model was addressed. In special cases where the strength of the background flow is a half of or exactly the same as the rotation rate of the Earth it was revealed that one of the zonal wavenumber components vanishes. The analytical balanced field was compared with the geopotential field which was obtained using a spherical harmonics spectral model. It was found that the normalized difference lied in the order of machine rounding, indicating the reliability of the analytical results. The stability of the sectoral mode of Rossby-Haurwitz wave and the associated balanced field was discussed, comparing with the flrst antisymmetric mode.

High-Order Spectral Filter for the Spherical-Surface Limited Area

AbstractA high-order spectral filter for the spherical-surface limited-area domain, either window or sector type, is presented, where the window domain is finite both in longitude and latitude and the sector domain is finite in longitude, but is ranged from Pole to Pole in latitude. The data given in the physical domain are extended to either extended window or sector domain by padding artificial data that are appropriate for spectral decomposition with half-ranged Fourier series. The high-order filter equation of Laplacian operator type was split into first- or second-order spherical elliptic equations as in the global domain high-order spectral filter. Each low-order elliptic equation is discretized using half-ranged Fourier series both in longitudinal and latitudinal direction. Since the domain is of the spherical geometry, the window domain spectral filter consists of full matrices for each zonal wavenumber and thus performs filtering with O(N3) operation for N × N grids. On the other hand, the sector...

Spectral analysis method for a limited area using the eigenmodes of the spherical Laplacian operator

We propose a spectral analysis method using the eigenmodes of the spherical Laplacian operator on the limited area domain. Two numerical methods are considered for the horizontal discretization: One uses the half-ranged Fourier series for both longitudinal and latitudinal directions, and the other uses the Fourier finite-element method with piecewise linear basis functions for the latitudinal direction. The field variable for the two numerical algorithms is represented as linear combinations of the eigenvectors of the Laplacian operator on the limited area domain; we define the one-dimensional spectrum with the eigenvector coefficients as a function of the indices equivalent to the total wavenumbers of the Laplacian operator on the global domain. The spatial robustness of this method was verified through the self-consistency test comparing the spectra of isotropic Gaussian bells on the sphere. We used the method in the kinetic energy spectral analysis for a limited area with global atmospheric data, and compared the results for different seasons. The kinetic energy spectra represented the well-known characteristics with scale and different powers with season.