134 research outputs found
An optimal scale separation for a dynamo experiment
Scale separation between the flow and the magnetic field is a common feature
of natural dynamos. It has also been used in the Karlsruhe sodium experiment in
which the scale of the magnetic field is roughly 7 times larger than the scale
of the flow [R. Stieglitz and U. M\"uller, Phys. Fluids 13, 561 (2001)].
Recently, Fauve & P\'etr\'elis ["Peyresq lectures on nonlinear phenomena", ed.
J. Sepulchre, World Scientific, 1 (2003)] have shown that the power needed to
reach the dynamo threshold in a dynamo experiment increases with the scale
separation in the limit of large scale separation. With a more elaborate method
based on subharmonic solutions [F. Plunian and K.-H. R\"adler, Geophys.
Astrophys. Fluid Dynamics 96, 115 (2002)], we show, for the Roberts flow, the
existence of an optimal scale separation for which this power is minimum.
Previous results obtained by Tilgner [Phys. Lett. A 226, 75 (1997)] with a
completely different numerical method are also reconsidered here. Again, we
find an optimal scale separation in terms of minimum power for dynamo action.
In addition we find that this scale separation compares very well with the one
derived from the subharmonic solutions method.Comment: 6 pages, 2 figure
Cascades and dissipation ratio in rotating MHD turbulence at low magnetic Prandtl number
A phenomenology of isotropic magnetohydrodynamic turbulence subject to both
rotation and applied magnetic field is presented. It is assumed that the triple
correlations decay-time is the shortest between the eddy turn-over time and the
ones associated to the rotating frequency and Alfv\'en wave period. For
it leads to four kinds of piecewise spectra, depending on the four parameters,
injection rate of energy, magnetic diffusivity, rotation rate and applied
field. With a shell model of MHD turbulence (including rotation and applied
magnetic field), spectra for are presented, together with the ratio
between magnetic and viscous dissipation.Comment: 5 figures, 1 table, appear in PR
Shell Models of Magnetohydrodynamic Turbulence
Shell models of hydrodynamic turbulence originated in the seventies. Their
main aim was to describe the statistics of homogeneous and isotropic turbulence
in spectral space, using a simple set of ordinary differential equations. In
the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged
based on the same principles as their hydrodynamic counter-part but also
incorporating interactions between magnetic and velocity fields. In recent
years, significant improvements have been made such as the inclusion of
non-local interactions and appropriate definitions for helicities. Though shell
models cannot account for the spatial complexity of MHD turbulence, their
dynamics are not over simplified and do reflect those of real MHD turbulence
including intermittency or chaotic reversals of large-scale modes. Furthermore,
these models use realistic values for dimensionless parameters (high kinetic
and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing
extended inertial range and accurate dissipation rate. Using modern computers
it is difficult to attain an inertial range of three decades with direct
numerical simulations, whereas eight are possible using shell models. In this
review we set up a general mathematical framework allowing the description of
any MHD shell model. The variety of the latter, with their advantages and
weaknesses, is introduced. Finally we consider a number of applications,
dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the
Hall effect.Comment: published in Physics Report
Intermittency in the homopolar disk-dynamo
We study a modified Bullard dynamo and show that this system is equivalent to
a nonlinear oscillator subject to a multiplicative noise. The stability
analysis of this oscillator is performed. Two bifurcations are identified,
first towards an `` intermittent\rq\rq state where the absorbing (non-dynamo)
state is no more stable but the most probable value of the amplitude of the
oscillator is still zero and secondly towards a `` turbulent\rq\rq (dynamo)
state where it is possible to define unambiguously a (non-zero) most probable
value around which the amplitude of the oscillator fluctuates. The bifurcation
diagram of this system exhibits three regions which are analytically
characterized
An optimal scale separation for a dynamo experiment.
Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly 7 times larger than the scale of the flow [1]. Recently, Fauve & P Ìetr Ìelis [2] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [3], we show, for the Roberts flow [4], the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [5] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method [3]
Rossby waves and -effect
Rossby waves drifting in the azimuthal direction are a common feature at the
onset of thermal convective instability in a rapidly rotating spherical shell.
They can also result from the destabilization of a Stewartson shear layer
produced by differential rotation as expected in the liquid sodium experiment
(DTS) working in Grenoble, France. A usual way to explain why Rossby waves can
participate to the dynamo process goes back to Busse (1975). In his picture,
the flow geometry is a cylindrical array of parallel rolls aligned with the
rotation axis. The axial flow component (the component parallel to the rotation
axis) is (i) maximum in the middle of each roll and changes its sign from one
roll to the next. It is produced by the Ekman pumping at the fluid containing
shell boundary. The corresponding dynamo mechanism can be explained in terms of
an -tensor with non-zero coefficients on the diagonal. In rapidly
rotating objects like the Earth's core (or in a fast rotating experiment),
Rossby waves occur in the limit of small Ekman number (). In
that case, the main source of the axial flow component is not the Ekman pumping
but rather the ``geometrical slope effect'' due to the spherical shape of the
fluid containing shell. This implies that the axial flow component is (ii)
maximum at the borders of the rolls and not at the centers. If assumed to be
stationary, such rolls would lead to zero coefficients on the diagonal of the
-tensor, making the dynamo probably less efficient if possible at all.
Actually, the rolls are drifting as a wave, and we show that this drift implies
non--zero coefficients on the diagonal of the -tensor. These new
coefficients are in essence very different from the ones obtained in case (i)
and cannot be interpreted in terms of the heuristic picture of Busse (1975)
Parametric instability of the helical dynamo
We study the dynamo threshold of a helical flow made of a mean (stationary)
plus a fluctuating part. Two flow geometries are studied, either (i) solid body
or (ii) smooth. Two well-known resonant dynamo conditions, elaborated for
stationary helical flows in the limit of large magnetic Reynolds numbers, are
tested against lower magnetic Reynolds numbers and for fluctuating flows (zero
mean). For a flow made of a mean plus a fluctuating part the dynamo threshold
depends on the frequency and the strength of the fluctuation. The resonant
dynamo conditions applied on the fluctuating (resp. mean) part seems to be a
good diagnostic to predict the existence of a dynamo threshold when the
fluctuation level is high (resp. low).Comment: 37 pages, 8 figure
Oscillating Ponomarenko dynamo in the highly conducting limit
This paper considers dynamo action in smooth helical flows in cylindrical
geometry, otherwise known as Ponomarenko dynamos, with periodic time
dependence. An asymptotic framework is developed that gives growth rates and
frequencies in the highly conducting limit of large magnetic Reynolds number,
when modes tend to be localized on resonant stream surfaces. This theory is
validated by means of numerical simulations.Comment: 12 pages, 4 figure
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