We study the Gaussian random elds indexed by R d whose covariance is de ned in all generality as the parametrix of an elliptic pseudo-di erential operator with minimal regularity asumption on the symbol. We construct new wavelet bases adapted to these operators the decomposition of the eld on this corresponding basis yields its iterated logarithm law and its uniform modulus of continuity. We also characterize the local scalings of the eld in term of the properties of the principal symbol of the pseudodi erential operator. Similar results are obtained for the Multi-Fractional Brownian Motion.

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The behavior of planetary (Rossby) waves in finite-element numerical models is investigated by using a quasi-geostrophic approximation to the midlatitude, β-plane, two-dimensional linear shallow-water equations. A dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive/dissipative nature of the finiteelement Galerkin mixed formulation. Three finite-element pairs are considered by using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow Rossby waves, due to the coupling between the momentum and continuity equations. Numerical solutions of two test problems to simulate slow Rossby waves are in good agreement with the analytical results.

SUMMARYAn analysis of the discrete shallow-water equations using the Raviart-Thomas and Brezzi-Douglas-Marini finite elements is presented. For inertia-gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with the analytical results.