Patrick Fischer

A reexamination of the QBO period modulation by the solar cycle

[1] Using the updated Singapore wind from 1953 to 2007 for the lower stratosphere 70-10 hPa, courtesy of Barbara Naujokat of Free University of Berlin, we examine the variation of the period of the Quasi-Biennial Oscillation (QBO) as a function of height and its modulation in time by the 11-year solar cycle. The analysis is supplemented by the ERA-40 reanalysis up to 1 hPa. Previously, it was reported that the descent of the easterly shear zone tends to stall near 30 hPa during solar minimum, leading to a lengthened QBO westerly duration near 44-50 hPa and the reported anticorrelation of the westerly duration and the solar cycle. Using an objective method, continuous wavelet transform (CWT), for the determination of local QBO period, we find that the whole QBO period is almost invariant with respect to height, so that the stalling mechanism affects only the partition of the whole period between easterly and westerly durations. Using this longest data set available for equatorial stratospheric wind, which spans five and half solar cycles (six solar minima), we find that in three solar minima, the QBO period is lengthened, while in the remaining almost three solar cycles, the QBO period is lengthened instead at solar maxima. We suggest that the decadal variation of the QBO period originates in the upper stratosphere, where the solar-ozone radiative influence is strong. The solar modulation of the QBO period is found to be nonstationary; the averaged effect cannot be determined unless the data record is much longer. In shorter records, the correlation can change sign, as we have found in segments of the longest record available, with or without lag.

Numerical Solution of the Schrödinger Equation in a Wavelet Basis for Hydrogen-like Atoms

An iterative method is proposed to solve the Schrodinger eigenvalue problem in a wavelet framework. Orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix. This representation, called the nonstandard (NS) form, is obtained by means of the Beylkin--Coifman--Rokhlin (BCR) algorithm and simplifies the numerical calculations. Problems due to the one-dimensional mathematical model and to the discretization process receive special attention.

Intensity of vortices: from soap bubbles to hurricanes

By using a half soap bubble heated from below, we obtain large isolated single vortices whose properties as well as their intensity are measured under different conditions. By studying the effects of rotation of the bubble on the vortex properties, we found that rotation favors vortices near the pole. Rotation also inhibits long life time vortices. The velocity and vorticity profiles of the vortices obtained are well described by a Gaussian vortex. Besides, the intensity of these vortices can be followed over long time spans revealing periods of intensification accompanied by trochoidal motion of the vortex center, features which are reminiscent of the behavior of tropical cyclones. An analysis of this intensification period suggests a simple relation valid for both the vortices observed here and for tropical cyclones.

The structures responsible for the inverse energy and the forward enstrophy cascades in two-dimensional turbulence

Two-dimensional turbulence admits two different ranges: an inverse energy cascade at large scales and a cascade of enstrophy to the small scales. Here we show that two flow structures govern the transfers of either enstrophy or energy. Vortical structures are responsible for the transfers of energy upscale while filamentary structures are responsible for the forward transfer of the enstrophy.

The Wavelet Transform: A New Mathematical Tool for Quantum Chemistry

Since the early years of Quantum Chemistry, many attempts have been made to visualize electronic distributions of atoms, molecules or solids. Generally, the problem has been approached by representing the one-electron charge density. However, this density on its own does not provide a complete insight into electronic distributions, e.g. information on the momenta of the electrons is completely smoothed out. Therefore the possibility of “looking at orbitals ” in different ways is still an up-to-date problem and appears as a necessary complement for a better understanding of a chemical structure.

Molecular Hartree-Fock equations for iteration-variation calculations in momentum space

SummaryMolecular Hartree-Fock equations for iteration-variation calculations in momentum space based on trial functions expressed as linear combinations of spherical Gaussian functions are obtained. They are applied to the hydrogen molecule to test their validity. The significant improvements brought by a first iteration are accessed through an asymptotic analysis.

Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm

Abstract The operator corresponding to the Hartree-Fock (HF) equation for cations with only one electron is decomposed onto an orthonormal wavelet basis. The Beylkin, Coifman, Rokhlin (BCR) algorithm is used to represent matrices in a sparse form called the Non-Standard (NS) form. For the sake of simplicity, only one-dimensional schemes of the operator are treated in this paper. The validity of the representation is tested by applying the operator to various functions, also expressed in a NS form, and the results are compared with the exact solution, the Slater function; the energy value is chosen as a comparative criterion.

Spectra and filtering: a clarification

Filtering methods have been introduced in the early nineties, in 1988 by Farge et al. for a wavelet filtering, in 1987 by Benzi et al. and in 1994 by Borue for a direct cut-off filtering. The aim of these methods is to filter the velocity and/or the vorticity fields of two-dimensional turbulence experiments in order to enhance the various components of the fluid. Using this king of methods allows us to separate the vortices from the background essentially composed by vorticity filaments. We have also shown the ability of the wavelet packets in performing this filtering. However, we had underestimated, like Borue in 1994, the influence of the filtering process in the computation of the energy and/or enstrophy spectra. We will show in the present paper how the introduction of discontinuities due to the filtering process can subsequently modify the spectra.

Electronic structure of H2 and HeH+ computed directly in momentum space

The possibilities of one‐step iterative computations in momentum space previously developed for neutral and charged atoms are now assessed to two‐electron diatomic molecular case, the helium hydride positive ion HeH+. The improvements induced in its ground state doubly occupied orbital initially approximated by different primitive Gaussian functions of increasing quality are analyzed through the calculation of test quantities such as the momentum density, 〈p−1〉, 〈p2/2〉, and 〈p4〉. Moreover the structure of the equation leading to the first iterate for HeH+ suggests that the procedure could be generalized to polyatomic systems.The possibilities of one‐step iterative computations in momentum space previously developed for neutral and charged atoms are now assessed to two‐electron diatomic molecular case, the helium hydride positive ion HeH+. The improvements induced in its ground state doubly occupied orbital initially approximated by different primitive Gaussian functions of increasing quality are analyzed through the calculation of test quantities such as the momentum density, 〈p−1〉, 〈p2/2〉, and 〈p4〉. Moreover the structure of the equation leading to the first iterate for HeH+ suggests that the procedure could be generalized to polyatomic systems.

Wavelet-based analysis of enstrophy transfers in two-dimensional turbulence

Two-dimensional turbulence admits two different ranges of scales: a direct enstrophy cascade from the injection scale to the small scales and an inverse energy cascade at large scales. It has already been shown in previous papers that vortical structures are responsible for the transfers of energy upscale while filamentary structures are responsible for the forward transfer of the enstrophy. Here we propose an original mathematical tool, the interaction function, for studying the space localization of the enstrophy fluxes. It is defined using an orthogonal two-dimensional wavelet decomposition.