15 research outputs found
A comparison of high-order time integrators for thermal convection in rotating spherical shells
A numerical study of several time integration methods for solving the threedimensional
Boussinesq thermal convection equations in rotating spherical shells
is presented. Implicit and semi-implicit time integration techniques based on
backward differentiation and extrapolation formulae are considered. The use of
Krylov techniques allows the implicit treatment of the Coriolis term with low
storage requirements. The codes are validated with a known benchmark, and
their efficiency is studied. The results show that the use of high order methods,
especially those with time step and order control, increase the efficiency of the
time integration, and allows to obtain more accurate solutions.Postprint (author’s final draft
Two computational approaches for the simulation of fluid problems in rotating spherical shells
Many geophysical and astrophysical phenomena such as magnetic fields generation, or the differential rotation observed in the atmospheres of the major planets are studied by means of numerical simulations of the Navier-Stokes equations in rotating spherical shells. Two different computational codes, spatially discretized using spherical harmonics in the angular variables, are presented. The first code, PARODY, solves the magneto-hydrodynamic anelastic convective equations with finite a difference discretization in the radial direction. This allows the parallelization on distributed memory computers to run massive numerical simulations of second order in time. It is mainly designed to perform direct numerical simulations. The second code, SPHO, solves the fully spectral Boussinesq convective equations, and its variationals, parallelized on shared memory architectures and it uses optimized linear algebra libraries. High-order time integration methods are implemented to allow the use of dynamical systems tools for the study of complex dynamics.Postprint (published version
Two computational approaches for the simulation of fluid problems in rotating spherical shells
Many geophysical and astrophysical phenomena such as magnetic fields generation, or the differential rotation observed in the atmospheres of the major planets are studied by means of numerical simulations of the Navier-Stokes equations in rotating spherical shells. Two different computational codes, spatially discretized using spherical harmonics in the angular variables, are presented. The first code, PARODY, solves the magneto-hydrodynamic anelastic convective equations with finite a difference discretization in the radial direction. This allows the parallelization on distributed memory computers to run massive numerical simulations of second order in time. It is mainly designed to perform direct numerical simulations. The second code, SPHO, solves the fully spectral Boussinesq convective equations, and its variationals, parallelized on shared memory architectures and it uses optimized linear algebra libraries. High-order time integration methods are implemented to allow the use of dynamical systems tools for the study of complex dynamics.Postprint (published version
Continuation and stability of rotating waves in the magnetized spherical Couette system: Secondary transitions and multistability
Rotating waves (RW) bifurcating from the axisymmetric basic magnetized
spherical Couette (MSC) flow are computed by means of Newton-Krylov
continuation techniques for periodic orbits. In addition, their stability is
analysed in the framework of Floquet theory. The inner sphere rotates whilst
the outer is kept at rest and the fluid is subjected to an axial magnetic
field. For a moderate Reynolds number (measuring inner
rotation) the effect of increasing the magnetic field strength (measured by the
Hartmann number ) is addressed in the range
corresponding to the working conditions of the HEDGEHOG experiment at
Helmholtz-Zentrum Dresden-Rossendorf. The study reveals several regions of
multistability of waves with azimuthal wave number , and several
transitions to quasiperiodic flows, i.e modulated rotating waves (MRW). These
nonlinear flows can be classified as the three different instabilities of the
radial jet, the return flow and the shear-layer, as found in previous studies.
These two flows are continuously linked, and part of the same branch, as the
magnetic forcing is increased. Midway between the two instabilities, at a
certain critical , the nonaxisymmetric component of the flow is
maximum.Comment: Published in the Proceedings of the Royal Society A journal. Contains
3 tables and 12 figure
Modulated rotating waves in the magnetized spherical Couette system
We present a study devoted to a detailed description of modulated rotating
waves (MRW) in the magnetized spherical Couette system. The set-up consists of
a liquid metal confined between two differentially rotating spheres and
subjected to an axially applied magnetic field. When the magnetic field
strength is varied, several branches of MRW are obtained by means of three
dimensional direct numerical simulations (DNS). The MRW originate from parent
branches of rotating waves (RW) and are classified according to Rand's (Arch.
Ration. Mech. Anal 79:1-37, 182) and Coughling & Marcus (J. Fluid Mech.
234:1-18,1992) theoretical description. We have found relatively large
intervals of multistability of MRW at low magnetic field, corresponding to the
radial jet instability known from previous studies. However, at larger magnetic
field, corresponding to the return flow regime, the stability intervals of MRW
are very narrow and thus they are unlikely to be found without detailed
knowledge of their bifurcation point. A careful analysis of the spatio-temporal
symmetries of the most energetic modes involved in the different classes of MRW
will allow in the future a comparison with the HEDGEHOG experiment, a
magnetized spherical Couette device hosted at the Helmholtz-Zentrum
Dresden-Rossendorf.Comment: Contains 3 tables and 8 figures. Published in the Journal of
Nonlinear Scienc
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
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
Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature
This article proposes spectral numerical methods to solve the time evolution
of convection problems with viscosity strongly depending on temperature at
infinite Prandtl number. Although we verify the proposed techniques just for
viscosities that depend exponentially on temperature, the methods are
extensible to other dependence laws. The set-up is a 2D domain with periodic
boundary conditions along the horizontal coordinate. This introduces a symmetry
in the problem, the O(2) symmetry, which is particularly well described by
spectral methods and motivates the use of these methods in this context. We
examine the scope of our techniques by exploring transitions from stationary
regimes towards time dependent regimes. At a given aspect ratio stable
stationary solutions become unstable through a Hopf bifurcation, after which
the time-dependent regime is solved by the spectral techniques proposed in this
article.Comment: 17 pages, 7 figure
The onset of low Prandtl number thermal convection in thin spherical shells
This study considers the onset of stress-free Boussinesq thermal convection
in rotating spherical shells with aspect ratio ( and
being the inner and outer radius), Prandtl numbers , and Taylor numbers . We are
particularly interested in the form of the convective cell pattern that
develops, and in its time scales, since this may have observational
consequences. For a fixed and by decreasing from
0.1 to a transition between spiralling columnar (SC) and
equatorially-attached (EA) modes, and a transition between EA and equatorially
antisymmetric or symmetric polar (AP/SP) weakly multicellular modes are found.
The latter modes are preferred at very low . Surprisingly, for the unicellular polar modes become also preferred at
moderate because two new transition curves between EA and
AP/SP and between AP/SP and SC modes are born at a triple-point bifurcation.
The dependence on and of the transitions is studied to
estimate the type of modes, and their critical parameters, preferred at
different stellar regimes.Comment: Accepted for publication in Physical Review Fluids. Contains 17
pages, 8 figures and 3 tables. Added brief erratum correcting values used for
estimates of neutron star ocean viscosit