The Multivariate Extension of the Lomb-Scargle Method

The common methods of spectral analysis for multivariate (n-dimensional) time series, like discrete Frourier transform (FT) or Wavelet transform, are based on Fourier series to decompose discrete data into a set of trigonometric model components, e. g. amplitude and phase. Applied to discrete data with a finite range several limitations of (time discrete) FT can be observed which are caused by the orthogonality mismatch of the trigonometric basis functions on a finite interval. However, in the general situation of non-equidistant or fragmented sampling FT based methods will cause significant errors in the parameter estimation. Therefore, the classical Lomb-Scargle method (LSM), which is not based on Fourier series, was developed as a statistical tool for one dimensional data to circumvent the inconsistent and erroneous parameter estimation of FT. The present work deduces LSM for n-dimensional data sets by a redefinition of the shifting parameter \tau, to maintain orthogonality of the trigonometric basis. An analytical derivation shows, that n-D LSM extents the traditional 1D case preserving all the statistical benefits, such as the improved noise rejection. Here, we derive the parameter confidence intervals for LSM and compare it with FT. Applications with ideal test data and experimental data will illustrate and support the proposed method.Comment: to be publishe

Four-frequency solution in a magnetohydrodynamic Couette flow as a consequence of azimuthal symmetry breaking

The occurrence of magnetohydrodynamic (MHD) quasiperiodic flows with four fundamental frequencies in differentially rotating spherical geometry is understood in terms of a sequence of bifurcations breaking the azimuthal symmetry of the flow as the applied magnetic field strength is varied. These flows originate from unstable periodic and quasiperiodic states with broken equatorial symmetry but having four-fold azimuthal symmetry. A posterior bifurcation gives rise to two-fold symmetric quasiperiodic states, with three fundamental frequencies, and a further bifurcation to a four-frequency quasiperiodic state which has lost all the spatial symmetries. This bifurcation scenario may be favoured when differential rotation is increased and periodic flows with $m$-fold azimuthal symmetry, $m$ being product of several prime numbers, emerge at sufficiently large magnetic field.Comment: 8 pages, 7 figures, published in Phys. Rev. Le

Long term time dependent frequency analysis of chaotic waves in the weakly magnetized spherical Couette system

The long therm behavior of chaotic flows is investigated by means of time dependent frequency analysis. The system under test consists of an electrically conducting fluid, confined between two differentially rotating spheres. The spherical setup is exposed to an axial magnetic field. The classical Fourier Transform method provides a first estimation of the time dependence of the frequencies associated to the flow, as well as its volume-averaged properties. It is however unable to detect strange attractors close to regular solutions in the Feigenbaum as well as Newhouse-Ruelle-Takens bifurcation scenarios. It is shown that Laskar's frequency algorithm is sufficiently accurate to identify these strange attractors and thus is an efficient tool for classification of chaotic flows in high dimensional dynamical systems. Our analysis of several chaotic solutions, obtained at different magnetic field strengths, reveals a strong robustness of the main frequency of the flow. This frequency is associated to an azimuthal drift and it is very close to the frequency of the underlying unstable rotating wave. In contrast, the main frequency of volume-averaged properties can vary almost one order of magnitude as the magnetic forcing is decreased. We conclude that, at the moderate differential rotation considered, unstable rotating waves provide a good description of the variation of the main time scale of any flow with respective variations in the magnetic field.Comment: 12 pages, 9 figures and 2 tables. Accepted for Physica D: Nonlinear Phenomen

Intermittent chaotic flows in the weakly magnetised spherical Couette system

Experiments on the magnetised spherical Couette system are presently being carried out at Helmholtz-Zentrum Dresden-Rossendorf (HZDR). A liquid metal (GaInSn) is confined within two differentially rotating spheres and exposed to a magnetic field parallel to the axis of rotation. Intermittent chaotic flows, corresponding to the radial jet instability, are described. The relation of these chaotic flows with unstable regular (periodic and quasiperiodic) solutions obtained at the same range of parameters is investigated.Peer ReviewedPostprint (published version

Report on the Survey 2012 amongst doctoral candidates within the Helmholtz Association: Created, carried out and evaluated by the Helmholtz Juniors, the PhD representatives of the Helmholtz Association

The Helmholtz Juniors are the PhD students‘ network of the German Helmholtz-Association (HGF). Their main mission is to intensify collaboration between the PhD students of the different Helmholtz research centers and improvement of the PhD education. In order to represent the interest of the PhD students at the Helmholtz Association, we need to have precise and up-to-date knowledge about the working conditions, problems and wishes of PhDs. This survey is a crucial basis. In the report, firstly we provide information about the background of the participants. Secondly we address four main topics of interest, namely PhD project planning, the income situation of PhD students, conditions for starting a family during the time as PhD student and the situation of students of foreign nationalities within the HGF. And thirdly we report results regarding the Helmholtz graduate schools

Chaotic wave dynamics in weakly magnetised spherical Couette flows

Direct numerical simulations of a liquid metal filling the gap between two concentric spheres are presented. The flow is governed by the interplay between the rotation of the inner sphere (measured by the Reynolds number Re) and a weak externally applied axial magnetic field (measured by the Hartmann number Ha). By varying the latter a rich variety of flow features, both in terms of spatial symmetry and temporal dependence, is obtained. Flows with two or three independent frequencies describing their time evolution are found as a result of Hopf bifurcations. They are stable on a sufficiently large interval of Hartmann numbers where regions of multistability of two, three and even four types of these different flows are detected. The temporal character of the solutions is analysed by means of an accurate frequency analysis and Poincar\'e sections. An unstable branch of flows undergoing a period doubling cascade and frequency locking of three-frequency solutions is described as well.Comment: 32 pages, 12 figures and 3 table

Experimental investigation of the return flow instability in magnetic spherical Couette flow

We conduct magnetic spherical Couette (MSC) flow experiments in the return flow instability regime with GaInSn as the working fluid, and the ratio of the inner to the outer sphere radii $r_{\rm i}/r_{\rm o} = 0.5$, the Reynolds number ${\rm Re} = 1000$, and the Hartmann number ${\rm Ha} \in [27.5,40]$. Rotating waves with different azimuthal wavenumbers $m \in \{2, 3, 4\}$ manifest in certain ranges of ${\rm Ha}$ in the experiments, depending on whether the values of ${\rm Ha}$ were fixed or varied from different initial values. These observations demonstrate the multistability of rotating waves, which we attribute to the dynamical system representing the state of the MSC flow tending to move along the same solution branch of the bifurcation diagram when ${\rm Ha}$ is varied. In experiments with both fixed and varying ${\rm Ha}$, the rotation frequencies of the rotating waves are consistent with the results of nonlinear stability analysis. A brief numerical investigation shows that differences in the azimuthal wavenumbers of the rotating waves that develop in the flow also depend on the azimuthal modes that are initially excited