86 research outputs found

### Predictive Scaling Laws for Spherical Rotating Dynamos

State of the art numerical models of the Geodynamo are still performed in a
parameter regime extremely remote from the values relevant to the physics of
the Earth's core. In order to establish a connection between dynamo modeling
and the geophysical motivation, {it is necessary to use} scaling laws. Such
scaling laws establish the dependence of essential quantities (such as the
magnetic field strength) on measured or controlled quantities. They allow for a
direct confrontation of advanced models with geophysical {constraints}.
(...)
We show that previous empirical scaling laws for the magnetic field strength
essentially reflect the statistical balance between energy production and
dissipation for saturated dynamos. Such power based scaling laws are thus
necessarily valid for any dynamo in statistical equilibrium and applicable to
any numerical model, irrespectively of the dynamo mechanism.
We show that direct numerical fits can provide contradictory results owing to
biases in the parameters space covered in the numerics and to the role of a
priori hypothesis on the fraction of ohmic dissipation.
We introduce predictive scaling laws, i.e. relations involving input
parameters of the governing equations only. We guide our reasoning on physical
considerations. We show that our predictive scaling laws can properly describe
the numerical database and reflect the dominant forces balance at work in these
numerical simulations. We highlight the dependence of the magnetic field
strength on the rotation rate. Finally, our results stress that available
numerical models operate in a viscous dynamical regime, which is not relevant
to the Earth's core

### The dynamo bifurcation in rotating spherical shells

We investigate the nature of the dynamo bifurcation in a configuration
applicable to the Earth's liquid outer core, i.e. in a rotating spherical shell
with thermally driven motions. We show that the nature of the bifurcation,
which can be either supercritical or subcritical or even take the form of isola
(or detached lobes) strongly depends on the parameters. This dependence is
described in a range of parameters numerically accessible (which unfortunately
remains remote from geophysical application), and we show how the magnetic
Prandtl number and the Ekman number control these transitions.Comment: 16 pages, 14 figure

### Toward an asymptotic behaviour of the ABC dynamo

The ABC flow was originally introduced by Arnol'd to investigate Lagrangian
chaos. It soon became the prototype example to illustrate magnetic-field
amplification via fast dynamo action, i.e. dynamo action exhibiting
magnetic-field amplification on a typical timescale independent of the
electrical resistivity of the medium. Even though this flow is the most
classical example for this important class of dynamos (with application to
large-scale astrophysical objects), it was recently pointed out (Bouya Isma\"el
and Dormy Emmanuel, Phys. Fluids, 25 (2013) 037103) that the fast dynamo nature
of this flow was unclear, as the growth rate still depended on the magnetic
Reynolds number at the largest values available so far $(\text{Rm} = 25000)$ .
Using state-of-the-art high-performance computing, we present high-resolution
simulations (up to 40963) and extend the value of $\text{Rm}$ up to $5\cdot10^5$ . Interestingly, even at these huge values, the growth rate of the
leading eigenmode still depends on the controlling parameter and an asymptotic
regime is not reached yet. We show that the maximum growth rate is a decreasing
function of $\text{Rm}$ for the largest values of $\text{Rm}$ we could achieve
(as anticipated in the above-mentioned paper). Slowly damped oscillations might
indicate either a new mode crossing or that the system is approaching the limit
of an essential spectrum

### Transition between viscous dipolar and inertial multipolar dynamos

We investigate the transition from steady dipolar to reversing multipolar
dynamos. The Earth has been argued to lie close to this transition, which could
offer a scenario for geomagnetic reversals. We show that the transition between
dipolar and multipolar dynamos is characterized by a three terms balance (as
opposed to the usually assumed two terms balance), which involves the
non-gradient parts of inertial, viscous and Coriolis forces. We introduce from
this equilibrium the sole parameter ${{\rm Ro}}\,{{\rm E}}^{-1/3} \equiv {{\rm
Re}}\,{{\rm E}}^{2/3}$, which accurately describes the transition for a wide
database of 132 fully three dimensional direct numerical simulations of
spherical rotating dynamos (courtesy of U. Christensen). This resolves earlier
contradictions in the literature on the relevant two,terms balance at the
transition. Considering only a two terms balance between the non-gradient part
of the Coriolis force and of inertial forces, provides the classical ${{\rm
Ro}}/{\ell_u}$ (Christensen and Aubert, 2006). This transition can be
equivalently described by ${{\rm Re}} \, {\ell^{2}_u}$, which corresponds to
the two terms balance between the non-gradient part of inertial forces and
viscous forces (Soderlund {\it et al.}, 2012).Comment: 14 pages, 4 figure

### Mechanisms of Planetary and Stellar Dynamos

We review some of the recent progress on modeling planetary and stellar
dynamos. Particular attention is given to the dynamo mechanisms and the
resulting properties of the field. We present direct numerical simulations
using a simple Boussinesq model. These simulations are interpreted using the
classical mean-field formalism. We investigate the transition from steady
dipolar to multipolar dynamo waves solutions varying different control
parameters, and discuss the relevance to stellar magnetic fields. We show that
owing to the role of the strong zonal flow, this transition is hysteretic. In
the presence of stress-free boundary conditions, the bistability extends over a
wide range of parameters.Comment: Proceedings of IAUS 294 "Solar and Astrophysical Dynamos and Magnetic
Activity" Editors A.G. Kosovichev, E.M. de Gouveia Dal Pino, & Y.Yan,
Cambridge University Press, to appear (2013

### Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

A comprehensive study of the classical linear spin-down of a constant density
viscous fluid (kinematic viscosity \nu) rotating rapidly (angular velocity
\Omega) inside an axisymmetric cylindrical container (radius L, height H) with
rigid boundaries, that follows the instantaneous small change in the boundary
angular velocity at small Ekman number $E=\nu/H^2\Omega \ll 1$, was provided by
Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering
inertial waves together with the dominant spin-down of the quasi-geostrophic
(QG) interior flow on the $O(E^{-1/2}\Omega^{-1})$ time-scale. On the longer
lateral viscous diffusion time-scale $O(L^2/\nu)$, the QG-flow responds to the
$E^{1/3}$-side-wall shear-layers. In our variant the side-wall and top
boundaries are stress-free; a setup motivated by the study of isolated
atmospheric structures, such as tropical cyclones, or tornadoes. Relative to
the unbounded plane layer case, spin-down is reduced (enhanced) by the presence
of a slippery (rigid) side-wall. This is evinced by the QG-angular velocity,
\omega*, evolution on the O(L^2/\nu) time-scale: Spatially, \omega* increases
(decreases) outwards from the axis for a slippery (rigid) side-wall;
temporally, the long-time ($\gg L^2/\nu)$ behaviour is dominated by an
eigensolution with a decay rate slightly slower (faster) than that for an
unbounded layer. In our slippery side-wall case, the $E^{1/2} \times E^{1/2}$
corner region that forms at the side-wall intersection with the rigid base is
responsible for a $\ln E$ singularity within the $E^{1/3}$-layer causing our
asymptotics to apply only at values of E far smaller than can be reached by our
Direct Numerical Simulation (DNS) of the entire spin-down process. Instead, we
solve the $E^{1/3}$-boundary-layer equations for given E numerically. Our
hybrid asymptotic-numerical approach yields results in excellent agreement with
our DNS.Comment: 33 pages, 10 figure

### Dipole Collapse and Dynamo Waves in Global Direct Numerical Simulations

Magnetic fields of low-mass stars and planets are thought to originate from
self-excited dynamo action in their convective interiors. Observations reveal a
variety of field topologies ranging from large-scale, axial dipole to more
structured magnetic fields. In this article, we investigate more than 70
three-dimensional, self-consistent dynamo models obtained by direct numerical
simulations. The control parameters, the aspect ratio and the mechanical
boundary conditions have been varied to build up this sample of models. Both,
strongly dipolar and multipolar models have been obtained. We show that these
dynamo regimes can in general be distinguished by the ratio of a typical
convective length scale to the Rossby radius. Models with a predominantly
dipolar magnetic field were obtained, if the convective length scale is at
least an order of magnitude larger than the Rossby radius. Moreover, we
highlight the role of the strong shear associated with the geostrophic zonal
flow for models with stress-free boundary conditions. In this case, the above
transition disappears and is replaced by a region of bistability for which
dipolar and multipolar dynamos co-exist. We interpret our results in terms of
dynamo eigenmodes using the so-called test-field method. We can thus show that
models in the dipolar regime are characterized by an isolated 'single mode'.
Competing overtones become significant as the boundary to multipolar dynamos is
approached. We discuss how these findings relate to previous models and to
observations.Comment: 35 pages, 16 figure

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