Hyun-Gyu Kang

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.

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...

An efficient implementation of a high-order filter for a cubed-sphere spectral element model

Abstract A parallel-scalable, isotropic, scale-selective spatial filter was developed for the cubed-sphere spectral element model on the sphere. The filter equation is a high-order elliptic (Helmholtz) equation based on the spherical Laplacian operator, which is transformed into cubed-sphere local coordinates. The Laplacian operator is discretized on the computational domain, i.e., on each cell, by the spectral element method with Gauss–Lobatto Lagrange interpolating polynomials (GLLIPs) as the orthogonal basis functions. On the global domain, the discrete filter equation yielded a linear system represented by a highly sparse matrix. The density of this matrix increases quadratically (linearly) with the order of GLLIP (order of the filter), and the linear system is solved in only O ( N g ) operations, where N g is the total number of grid points. The solution, obtained by a row reduction method, demonstrated the typical accuracy and convergence rate of the cubed-sphere spectral element method. To achieve computational efficiency on parallel computers, the linear system was treated by an inverse matrix method (a sparse matrix–vector multiplication). The density of the inverse matrix was lowered to only a few times of the original sparse matrix without degrading the accuracy of the solution. For better computational efficiency, a local-domain high-order filter was introduced: The filter equation is applied to multiple cells, and then the central cell was only used to reconstruct the filtered field. The parallel efficiency of applying the inverse matrix method to the global- and local-domain filter was evaluated by the scalability on a distributed-memory parallel computer. The scale-selective performance of the filter was demonstrated on Earth topography. The usefulness of the filter as a hyper-viscosity for the vorticity equation was also demonstrated.

Fourier Finite-Element Method with Linear Basis Functions on a Sphere: Application to Elliptic and Transport Equations

AbstractThe Fourier finite-element method (FFEM) on the sphere, which performs with an operation count of O(N2 log2N) for 2N × N grids in spherical coordinates, was developed using linear basis functions. Dependent field variables are expanded with the Fourier series in the longitude, and the Fourier coefficients are represented with a series of first-order finite elements. Different types of pole conditions were incorporated into the Fourier coefficients of the scalar and vector variables in order to avoid discontinuity at the poles. For the Laplacian operator, the linear element was defined as a function of the sine of latitude instead of the latitude. The FFEM was applied to the derivatives of the first- and second-order elliptic equations and the transport equations. The scale-selective high-order Laplacian-type filter was implemented as a hyperviscosity. For the first-order derivative the fourth-order convergence rate of the accuracy, as is expected from the theoretical analysis, was achieved. Ellipt...

Effect of a High-Order Filter on a Cubed-Sphere Spectral Element Dynamical Core

AbstractA high-order filter for a cubed-sphere spectral element model was implemented in a three-dimensional spectral element dry hydrostatic dynamical core. The dynamical core incorporated hybrid ...