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Nonlin. Processes Geophys., 13, 413423, 2006
www.nonlin-processes-geophys.net/13/413/2006/
� Author(s) 2006. This work is licensed
under a Creative Commons License.
Nonlinear Processes
in Geophysics
GEOFLOW: simulation of convection in a spherical shell under
central force eld
P. Beltrame
1,*
, V. Travnikov
1
, M. Gellert
1
, and C. Egbers
1
1
Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology, Siemens-Halske-Ring 14,
03046 Cottbus, Germany
*
now at: Max-Planck-Institut, Dresden, Germany
Received: 4 August 2005 Revised: 26 April 2006 Accepted: 26 April 2006 Published: 15 August 2006
Part of Special Issue Turbulent transport in geosciences
Abstract.
Time-dependent dynamical simulations related to
convective motion in a spherical gap under a central force
eld due to the dielectrophoretic effect are discussed. This
work is part of the preparation of the GEOFLOW-experiment
which is planned to run in a microgravity environment. The
goal of this experiment is the simulation of large-scale con-
vective motion in a geophysical or astrophysical framework.
This problem is new because of, on the one hand, the nature
of the force eld (dielectrophoretic effect) and, on another
hand, the high degree of symmetries of the system, e.g. the
top-bottom reection. Thus, the validation of this simulation
with well-known results is not possible. The questions con-
cerning the inuence of the dielectrophoretic force and the
possibility to reproduce the theoretically expected motions
in the astrophysical framework, are open. In the rst part, we
study the system in terrestrial conditions: the unidirectional
Earths force is superimposed on the central dielectrophoretic
force eld to compare with the laboratory experiments dur-
ing the development of the equipment. In the second part,
the GEOFLOW-experiment simulations in weightless condi-
tions are compared with theoretical studies in the astrophys-
ical frameworks, in the rst instance a uid under a self-
gravitating force eld. We present complex time-dependent
dynamics, where the dielectrophoretic force eld causes sig-
nicant differences in the ow compared to the case that does
not involve this force eld.
1
Introduction
The present paper shows results of investigations of the inu-
ence of a radial force eld, produced by the dielectrophoretic
effect (Pohl , 1978) in spherical Rayleigh-B�enard convection
using a three-dimensional code and bifurcation analysis. It is
Correspondence to: P. Beltrame
(beltrame@mpipks-dresden.mpg.de)
the preparatory work for an experimental set-up: the convec-
tive motion in a spherical gap under the inuence of an ar-
ticial central force eld. This experiment is planned to run
on the ISS (International Space Station) under microgravity
conditions. Experimental details can be found in Egbers et al.
(2003). The experimental cell is formed by an outer glass
sphere, which can be cooled, and an inner sphere, which can
uniformly heated within (Fig. 1). The temperature difference
is maintained constant with T
1
>T
2
. The central force eld is
generated by applying a high voltage ( 10 kV) between in-
ner and outer sphere. Using a dielectric uid (silicon oil), the
resulting central dielectrophoretic force eld is proportional
to 1/r
5
. Currently, three different viscosities of uid silicone
oils and three different inner radii (R
1
) are available for the
experiment (Table 1), resulting in three values of the Prandtl
number, P r, and also three aspect ratios, , respectively. The
central Rayleigh number Ra
c
can vary over a large range
by varying the voltage (Table 1).
The possible nondimensional parameters have similar val-
ues to the Earths mantle ones, in particular the aspect ra-
tio, the Rayleigh number and, in both cases (GEOFLOW and
Earths mantle), P r
1. The aspect ratio of the Earths outer
core (
e
=
0.34) is close to the GEOFLOWs ratio too. But the
Prandtl number (0.1<P r<10) is smaller than for the experi-
ment and the very large Rayleigh number (Ra
e
>
10
26
) cannot
be achieved in the experiment. However, the rapid decay of
the dielectrophoretic force eld (1/r
5
variations) can better
represent the gravity eld of the outer core (1/r
2
variations)
than the linear variation of the Earths mantle gravity eld.
Although the GEOFLOW-experiment allows the system
to rotate, we consider here the non-rotating case in contrast
to the Earths case where the Taylor number plays a relevant
role. This limiting case is motivated, on one hand, by the rich
dynamics expected and, on another hand, by the possibility
to interpret the results in theoretical way using group theory
for the spherical symmetry: the O(3) group.
Published by Copernicus GmbH on behalf of the European Geosciences Union and the American Geophysical Union. 414
P. Beltrame et al.: GEOFLOW: simulation
Fig. 1.
Set-up of the GEOFLOW-experiment.
To obtain a perfect spherical symmetry, we have also
neglected the thin axis supporting the inner sphere in the
GEOFLOW-experiment as shown in Fig. 1. The thin axis
will be taken into account in future works as a perturbation
of the perfect case. Besides theoretical aspects, without this
symmetry we could not use a pseudo-spectral method for the
numerical computation, and the CPU time would then in-
crease dramatically.
First results of numerical investigations, corresponding to
convection in the rotating or non-rotating spherical gap under
weightlessness conditions are published in Travnikov et al.
(2003) and Travnikov (2004). These papers deal with the
calculation of the basic ow, stability analysis and point out
that the GEOFLOW-experiment can reproduce the differ-
ent steady-states and rotating waves, which arise for a self-
gravitating case (1/r
2
force eld). Nevertheless, it is very
difcult to validate these results with terrestrial experiments
or well-known results, because the Earths gravity eld has
a non-negligible inuence, in particular for the non-rotating
case where a lot of symmetry is broken. That is why we aim
at simulating the system under two forces: the axial gravity
force and the central dielectrophoretic force. Furthermore,
we take the opportunity to point out the effect of the di-
electrophoretic force compared with experimental/numerical
work without this eld (Futterer et al., 2004).
The second part of this paper considers the weightlessness
case and it focuses on the comparison between the motion
due to the dielectrophoretic eld (1/r
5
radial dependence)
and the central gravity eld (1/r
2
radial dependence), which
corresponds, for example, to the Earths outer core. The 1/r
2
case has shown very rich dynamics, in particular the occur-
rence of motion reversals (Friedrich and Haken, 1986). This
motion has astrophysical relevance because it can help to
understand quasi-periodic phenomena such the Earths mag-
netic eld reversal. Because Chossat and Guyard (1996) have
pointed out that these reversal motions (or heteroclinic cy-
cles) are due to the spherical symmetry, we expect such dy-
namics for the 1/r
5
eld force too. The requirements on the
GEOFLOW parameters which lead to possible heteroclinic
cycles, are determined in Beltrame et al. (2003a) and Bel-
trame and Egbers (2004). The expected dynamics can be the
same type as for the 1/r
2
case (Beltrame and Egbers, 2005)
or can be new types of dynamics (Beltrame, 2006a
1
). In any
cases, the dynamics are poorly known for both astrophysi-
cal (1/r
2
) and dielectrophoretic (1/r
5
) elds. Beyond these
theoretical results, we will check the range of parameters for
which these dynamics can be observed in GEOFLOW frame-
work.
Part I
Terrestrial conditions
Four main non-dimensional numbers are necessary to de-
scribe the phenomenon: the radius ratio =
R
1
R
2
, the Prandtl
number P r=


, the Rayleigh number Ra
g
=
g T R
3
2 mea-
suring the gravity force and the central Rayleigh number
Ra
c
=
2
0 r
0 V
2
T
measuring the dielectrophoretic force (
0
is the vacuum dielectric constant.) The notations are as fol-
lows: R
1
and R
2
are the radii of both spheres, is the coef-
cient of volume expansion, the viscosity, the thermal con-
ductivity and
0
the density. Furthermore,
r
is the dielectric
constant, V the effective voltage and the dielectric variabil-
ity. This last constant is related to the dielectric constant lin-
ear dependence on the temperature: =
0 r
(
1 (T
1 T
2
))
.
The investigation is performed for =0.5, P r=42.81 (sil-
icone oil M1). The temperature difference varies between
T =
2 and
T =
8 K. The ow structure then depends on
the voltage V for xed
T
(Ra
g
).
The goal of this part is to perform a numerical investiga-
tion of the inuence of a fast oscillating electric eld on the
convective ow in the spherical gap in a terrestrial laboratory.
2
Mathematical background
2.1
Basic equations
We consider an incompressible, Newtonian uid under the
Boussinesq approximation.
The force acting on the vol-
ume element of the dielectric medium, consists of three
parts: Coulomb force F
c
= f r
E
(
f r
free charge density),
dielectrophoretic force F
d
=
1
2
E
2
and the gradient part
1
Beltrame, P.: Intermittency between the Modes 3 and 4 near
the onset of convection in a spherical shell under dielectrophoretic
force, J. Adv. Space Res., in review, 2006a.
Nonlin. Processes Geophys., 13, 413423, 2006
www.nonlin-processes-geophys.net/13/413/2006/ P. Beltrame et al.: GEOFLOW: simulation
415
Table 1.
Mean physical and nondimensional parameters of GEOFLOW-experiment.
Experiment parameters
Inner radius
R
1
8.113.5
mm
Outer radius
R
2
27
mm
Temperature Difference
T
26
K
Vol