Martin J. Fengler

A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier--Stokes Equation on the Sphere

This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-

Multiscale Geopotential Solutions from CHAMP Orbits and Accelerometry

3j

Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96

coefficients. To improve the numerical efficiency and economy we introduce an FFT-based pseudospectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with

Regional gravity modeling in terms of spherical base functions

O(N^3)

Harmonic spline-wavelets on the 3-dimensional ball and their application to the reconstruction of the Earth's density distribution from gravitational data at arbitrarily shaped satellite orbits

if

Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE

N

Wavelet Modelling of Regional and Temporal Variations of the Earth´s Gravitational Potential

denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.

The Kaiserslautern multiscale geopotential model SWITCH-03 from orbit perturbations of the satellite CHAMP and its comparison to the models EGM96, UCPH2002_02_0.5, EIGEN-1s and EIGEN-2

CHAMP orbits and accelerometer data are used to recover the long- to medium-wavelength features of the Earths gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earths gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s.

The Spherical Bernstein Wavelet

Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earths gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CuP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.