19 research outputs found
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 -fold azimuthal symmetry, 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 magnetized 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 ¿¿) 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 analyzed by means of an accurate frequency analysis and Poincaré sections. An unstable branch of flows undergoing a period doubling cascade and frequency locking of three-frequency solutions is described as well.
One of the paradigms of magnetohydrodynamic flows in spherical geometry is the magnetized spherical Couette flow. An electrically conducting liquid is confined between two differentially rotating spheres and is subjected to a magnetic field. Despite its simplicity, this model gives rise to a rich variety of instabilities, and it is also important from an astrophysical point of view. The present study advances the knowledge of the dynamics of this problem by describing it in terms of dynamical systems theory, a rigorous mathematical way to understand time dependent behavior of natural systems.Peer ReviewedPostprint (author's final draft
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 , the Reynolds
number , and the Hartmann number .
Rotating waves with different azimuthal wavenumbers
manifest in certain ranges of in the experiments, depending on
whether the values of 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 is varied. In experiments with both fixed and varying , 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