LIST OF PARTICIPANTS, TITLES OF THE TALKS AND THE ABSTRACTS (where available)
Armbruster, D. Structurally stable heteroclinic cycles
and the dynamo dynamics
Alemany, D.
Ashwin, P.
Benevolenskaya, E. Two-component dynamical model
of the solar cycle
Bigazzi, A. Dynamo action in 3D homogeneous turbulence
Brachet, M. Are the Taylor-Green vortex and the von
Karman swirling flow good candidates for dynamo action?
Brandenburg, A. Simulations of large-scale dynamo
action
Brasch, W.
Brestensky, J. Rotating magnetoconvection in dependence
on various diffusive processes and boundary conditions
Brooke, J.M. Symmetries of the solar dynamo: comparing
theory with observations
Brummel, N.Linear
and nonlinear action in simply driven flows.
Burguete, J. Homogeneous dynamo: numerical analysis
of experimental von Karman type flows
Busse, F. Convection driven dynamos in spherical
shells
Cattaneo, F. Convectively driven dynamos
Chossat, P.
Covas, E. Nonlinear dynamics underlying axisymmetric
mean field dynamo models.
Cupal, I. 3D geodynamo in mean field approximation
Daviaud, F. VKS, an experimental fluid dynamo
Dormy, E. Interaction between numerical simulations
and asymptotic studies in the Earth's liquid core
Egbers, C. Thermal flow in a rotating spherical gap
with dielectrophoretic central force field
Emonet,T. Recent results on convectively driven dynamos
Fauve, S.
Ferriere, K. The galactic dynamo
Forest, C. Physics goals and initial results from
the Madison dynamo experiment
Frick, P. On nonstationary dynamo experiment in a
breaked torus
Fuchs, H. On self-killing and self-exciting dynamos
Gabov, A.
Gailitis, A., Riga dynamo experiment
Hejda, P. About the numerical method solving 3D geodynamo
in mean field approximation
Hollerbach, R. Magnetically induced shear layers
and jets
Hughes, D.W. Large and small scale dynamo action
Ivers, D. Dynamo problems in spherical and nearly
spherical geometries
Jault, D. Inviscid geodynamo models
Jones, C. Onset of convection and magnetoconvection
in rapidly rotating spherical geometry
Junk, M.
Kaiser, R. Mathematical problems related to antidynamo
theorems
Kleeorin, N. Magnetic helicity with zero and nonzero
mean magnetic field
Knobloch, E.
Kosovichev, A. Helioseismic tests of dynamo models
Kurts,J.Solar
activity cycle in phase synchronized with solar inertial motion
Kuzanyan, K. Asymptotic WKBJ-modelling of solar dynamo
waves
Lathrop, D.P. Towards a self-generating magnetic
dynamo: the role of turbulence
Laure, P. Generation of magnetic field in the Couette-Taylor
system
Lauterbach, R. Heteroclinic
cycles in systems with spherical symmetries
Leorat, J.
Matthews, P. Dynamo action in rotating convection
Melbourne, I.Magnetic
dynamos in nonlinear RMHD
Mueller, U. An experimental demonstration of a homogeneous
two scale dynamo
Nore, C. Dynamo
action in the Taylor-Green vortex: thresholds and saturation
Odier, P. Advection of a magnetic field by a turbulent
swirling flow
Oprea, I.Transition from kinematic to dynamic regime
in an intermediate convective dynamo
Otmianowska-Mazur, K. 3D MHD simulations of magnetic
dynamos in barred galaxies
Petrelis, F. Simple models for studying the dynamics
of fluid dynamos
Pinton, J-F.
Ponty, Y. Dynamo action in flow driven by shear and
convection
Proctor, M. Destabilization of transverse modes by
noise
Raedler, K-H.
Rieutord, M.$\alpha$-effect produced by inertial
modes in a spherical shell
Roberts, P. On the dynamics of the Earth dynamo
Rogachevskii, I. Nonlinear mean field dynamo: electromotive
force for anisotropic turbulence and intermediate nonlinearity
Rosaev, A.E. The variation of Earth's magnetic field
by tidal force
Ruediger, G. The simplest consistent dynamo: a non-rigidly
rotating sphere
Rucklidge, A. Heteroclinic model of geodynamo reversals
Ruzmaikin, A. The solar dynamo: old problems and
new challenges
Seehafer N. Dynamos in rotating and nonrotating convection
Silbert, M.
Simkanin, J. Hydromagnetic instabilities in non-uniformly
stratified layer in dependence on boundary conditions and Roberts number
Starchenko
Stepanov
Soward, A. Nonlinear $\alpha^2\omega$ waves in stellar
shells
Tavakol, R.K. Intermittency and fragility in axisymmetric
dynamo models
Tilgner, A. Precession driven flows and dynamos -
progress on the Karlsruhe experiment
Tobias, S.
Weiss, N. O.
Von Rekovski, B. Dynamos in accretion disks with
vertical winds.
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The problem of the onset of thermal convection in a rapidly rotating
sphere is a classical problem, but the asymptotic solution in the limit
of large Taylor number is not completely straightforward. Since the onset
of convection occurs close to a critical cylinder aligned with the rotation
axis, and of radius approximately one half the radius of the sphere, a
{\bf local} theory seems appropriate, and this was developed by Roberts
(1968) and Busse (1970). However, to compute the critical Rayleigh number
for onset, a global theory is required, which eliminates the possibility
of {\it phase-mixing}. The Roberts-Busse local theory still provides the
required dispersion relation, but the global criteria requires that phase-mixing
must vanish, so that a point in the complex $s$-plane must be isolated
at which $\partial \omega / \partial s=0$, where $\omega$ is the frequency
derived from the local dispersion relation and $s$ is distance from the
rotation axis. The method by which this double-turning point is found,
and hence how the critical Rayleigh number is found, will be discussed.
The asymptotic theory is compared with full numerical solutions of the
partial differential equations at Taylor numbers up to $10^{12}$;excellent
agreement is found. The technique can be extended to the case of magnetoconvection,
for simple magnetic fields, provided the Elsasser number is of order $O(E^{1/3})$,
where $E$ is the Ekman number. The magnetic case does, however, introduce
some new features arising from the boundary conditions. Further problems
can arise in the spherical gap problem (convection between concentric rotating
spheres) when the critical cylinder at which convection onsets is too close
to the inner sphere.
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The global structure of a self-excited magnetic field arising from the
magnetic shear instability has been simulated in spherical geometry by
a 3D fully nonlinear approach. In order to model the structure of an non-rigidly
rotating star we prescribe a rotation profile of the Brandt-type which
is steep in the outer regions but yields rigid rotation at the inner core.
A series with different magnetic Reynolds numbers has been performed with
an increasing number of modes in spectral space. Starting from arbitrary
small perturbations the magnetic and kinetic energies grow by several orders
of magnitude as soon as a certain azimuthal resolution of at least $m=15$
has been used at a Reynolds number of order C$_{\Omega}=10^{5}$. Several
phases of the transition to turbulence are realized and interpretations
are given for the respective effects occurring at each stage. The resulting
magnetic field is highly nonaxisymmetric. The flow is almost axisymmetric
but shows a Kolmogorov-like behavior for small scales. The outer surface
of the shell is penetrated by magnetic field lines in spotlike regions
which are located mainly in the equatorial plane. The problem of angular
momentum transport is discussed in terms of the Shakura-Sunyaev viscosity-alpha,
which adopts values in the range $10^{-(3 ...5)}$. If the parameters are
used of a main-sequence star of spectral type A, then the resulting magnetic
fields are of order $10^{3...5}$ Gauss, very close to the values known
from the observations of Ap-stars. First results are reported for the realistic
case that a density stratification stabilizes the magnetorotational instability.
...
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I will present a model of supernova-driven turbulence in the interstellar
medium of our Galaxy, describe the threefold impact of this turbulence
on the large-scale Galactic magnetic field (alpha-effect, vertical advection,
and magnetic diffusion), and present recent numerical solutions of the
dynamo equation, which support the idea that the large-scale magnetic field
can be amplified through a combination of large-scale differential rotation
and supernova-driven turbulence.
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We have considered the issue of large- and small-scale dynamo action,
by conducting numerical simulations of dynamo action in a tall box. The
initial magnetic field strength is chosen to be sufficiently weak that
the magnetic field first evolves kinematically, before saturating in the
nonlinear regime. The magnetic field configuration in the dynamical regime
is shown to depend crucially on the choice of initial field. In certain
cases the growth and saturation of the large-scale magnetic field can be
attributed to an $\alpha$-effect arising from the interactions of a small-scale
flow and a small-scale field. To examine the nature of the $\alpha$-effect
in more detail and, in particular, its dependence on the mean field and
on the magnetic Reynolds number, we performed a further series of simulations
of dynamo action in a smaller box with an imposed uniform magnetic field.
The most significant finding is that the $\alpha$-effect has a strong dependence
on the magnetic Reynolds number; consequently the $\alpha$-effect is influenced
by a very weak mean magnetic field.
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Nonlinear $\alpha^2\Omega$-dynamo waves are considered in a thin turbulent,
differentially rotating convective stellar shell. Nonlinearity arises from
$\alpha$-quenching, while an asymptotic solution is based on the small
aspect ratio of the shell. Wave modulation is linked to a latitudinal-dependent
local $\alpha$-effect and zonal shear flow magnetic Reynolds numbers $R_\alpha
f(\theta)$ and $R_\Omega g(\theta)$ respectively; here $\theta$ is the
latitude. The essential picture developed is that of a modulated dynamo
wave whose amplitude varies spatially with $\theta$. The linear solution
is controlled by the properties of the double turning point $\theta_c$
of the ordinary differential equation for the mode amplitude. Significantly,
though $\theta_c$ is real and is located at the local dynamo number maximum
in the $\alpha\Omega$-dynamo limit $R_\alpha\to 0$, it migrates into the
complex $\theta$-plane once $R_\alpha\not=0$. Linear and weakly nonlinear
solutions are found over a limited range of $R_\alpha$ and their qualitative
properties are found to be largely similar to those for the $\alpha\Omega$-dynamo
limit. One significant astrophysical difference is the fact that the frequency
generally decreases with increasing $R_\alpha$. Thus $\alpha^2\Omega$-stellar
dynamos may occur with $\alpha\Omega$-dynamo wave characteristics but exhibit
significantly longer cycle times increased by a factor roughly two or more.
{}Finite amplitude dynamo waves, like those when $R_\alpha \to 0$, are
modulated by an envelope which evaporates smoothly at some low latitude
but is terminated abruptly by a front at a high latitude $\theta_F$. For
given non-zero $R_\alpha$ these frontal solutions are subcritical (a property
linked to the complex-value taken by $\theta_c$). This issue and the validity
of the frontal solutions is explored for small $R_\alpha$, when the solutions
may be regarded as modifications of pure $\alpha\Omega$-dynamo waves. {}For
sufficiently large $R_\alpha$ new low frequency modes are introduced that
are more closely identified with steady $\alpha^2$-dynamos localised near
the pole $\theta=\pi/2$. Up to four distinct finite amplitude states can
exist; these can be loosely characterised as $\alpha\Omega$-high frequency,
$\alpha^2\Omega$-medium frequency, $\alpha^2$-low frequency and $\alpha^2$-steady
modes. In view of the possible mode competition, we comment on the likely
realised physical state.
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The screw dynamo or so-called Ponomorenko
dynamo is able to generate a non-axisymmetric magnetic field in an axisymmetric
helical flow and is one of the simplest known dynamo models. Basically
a laminar helical flow like Couette-Poiseuille can produce the magnetic
field self-excitation. A generation of magnetic field is possible in other
way which is based on small-scale turbulence. A turbulent helicity (alpha-effect)
can amplify a magnetic field in absence of mean field flow. Both generation
mechanisms is supposed to act in several astrophysical object and the proposed
Perm dynamo experiment. We study interaction of generation mechanisms helical
large-scale laminar flow and small-scale turbulent helicity. The crucial
parameters of dynamo-process in a different regimes have a strong interest
especially in the frame of the Perm dynamo experiment. The simultaneous
action of these mechanisms has been studied in the context of this experiment,
taking properly into account inhomogeneities and anisotropies of the turbulence.
Depending on the relevant parameters they may indeed support or counteract
each other.
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A new experimental project on MHD-dynamo
problem is developing in the Institute of Continuous Media Mechanics of
Ural Branch of Russian Academy of Sciences, Perm, Russia. The experiment
is intended for study the magnetic field evolution in nonstationary turbulent
flow of liquid sodium. In contrast to existing experimental projects, which
have to realize the stationary flow of liquid sodium and require extremal
power, in this project the energy will be accumulated during relatively
long acceleration of the toroidal vessel (diameter about 1 m) with liquid
sodium and the large power screw flow will be obtained only during the
abrupt brake. Internal divertors will enable the required profile of velocity.
The expected flow can act as a dynamo provided its magnetic Reynolds $R_m$
number is expected to exceed several tens. In this talk we present the
results of experimental study of screw water flows in braked toroidal chanels,
the results of numerical simulations of the corresponding dynamo problem
which enables to optimize the parameters of the MHD apparatus and the general
design of the dynamo apparatus.
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The geodynamo process of magnetic
field generation occurs in the liquid spherical core of the Earth and is
described by 3D magnetohydrodynamic equations. Due to the complexity of
these equations, the approach to the problem is usually numerical. >From
the pioneering calculations of the kinematic dynamo in fifties to the most
advanced 3D hydronagnetic simulations in nineties, the numerical process
was based on the decomposition of the magnetic and velocity fields into
toroidal and poloidal parts. The functions were usually further expanded
into spherical harmonics, some of the 2D dynamo computations were carried
out by finite difference method. The latter approach was mostly applied
to the axisymmetric hydrodynamic models in magnetostrophic approximation.
>From the numerical point of view, the magnetostrophic approximation is
characterized by the fact that the sharp changes of velocities in the boundary
layers are replaced by jumps and the velocity in the bulk of the core is
expressed in the form of integrals in {\bf z} direction. The solution thus
requires a transition between spherical and cylindrical coordinates. Alternative
methods that deal directly with three components of magnetic and velocity
fields were developed by Nakajima and Roberts ({\it Phys. Earth Planet.
Inter.}, {\bf 91}, 53-61) (1995)) and Hejda and Reshetnyak ({\it Studia
geoph. et geod.}, {\bf 43}, 319-325 (1999)). Whereas Hejda and Reshetnyak
kept the spherical coordinates, Nakajima and Roberts solved the problem
in cylindrical coordinates and avoided the transition between the coordinate
systems by a special mapping method. Formulation of the problem in above
mentioned curvilinear coordinates has following drawbacks:\\ - the differential
operators are singular at the axis of rotation (and in the center od c.s.
in spherical coordinates) - the grid is much more denser near the axis
of rotation (center of coordinate system); the computer memory and time
are thus wasted on the regions which are not of special interrest - extra
boundary conditions must be formulated on the artificial boundaries. In
the present contribution the application of finite element method will
be discussed. It will be shown that all the peculiarities connected with
the spherical geometry can be solved on the level of mesh generation. The
spherical mesh can be constructed with nearly uniformly distributed nodes.
Attention will be paid to the estimation of the mesh quality. As the equations
are expressed in cartesian coordinates, they contain no singular coefficients.
Moreover, the form of differential operators is much more simple then in
curvilinear coordinates. The method will be tested on some simple problems.
Eigenvalues of kinematic models will be found by inverse iterations and
compared with results obtained by different methods and authors.
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The solar magnetic cycle shows
a complicated multi-mode behaviour. At least, two components are found
in the distribution of the magnetic field on the solar photosphere. The
existence of these two components can be explained by a non-linear model
based on the Parker's dynamo theory with two sources. The properties of
this dynamical system are investigated numerically, and it shown that this
model can qualitatively reproduce the observed behaviour of the two dynamo
components.
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Helioseismology provides important
tests for dynamo theories by measuring the variations of the internal structure
and dynamics of the Sun with the activity cycle. Recent results from the
GONG network and MDI/SOHO space experiment have revealed correlated variations
of the zones of generation of the solar magnetic fields and zonal shear
flows in the convection zone. An attempt is made to detect the solar-cycle
variations in the tachocline region at the base of the convection zone,
which is believed to be the main cite of the solar dynamo. The helioseismic
measurements provide both input and test data for dynamo models.
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Numerical studies of a laminar
dynamo model have revealed two remarkable phenomena. A spherical body of
an electrically conducting incompressible fluid is considered which is
surrounded by free space. The fluid shows an inner motion due to a given
force and satisfies the no--slip condition at the boundary. A rotation
of the fluid body as a whole is admitted, too. The full interaction of
magnetic field and motion is taken into account. Starting from a fluid
motion capable of dynamo action and a weak magnetic field it was observed
that the growing magnetic field destroys the dynamo property of the motion
and then decays, and that the system ends up in a state with another motion
incapable of dynamo action and zero magnetic field. The finding of this
'self-killing' of dynamos implies a warning concerning simple parameterizations
of the back--reaction of the magnetic field on the motion as used, e.g.,
in mean--field dynamo theory. In another case with a motion unable to prevent
small magnetic fields from decay it proved to be possible that strong magnetic
fields deform it so that a dynamo starts to work which enables the system
to approach a steady state with a finite magnetic field. This state is
a genuine magneto--hydromagnetic one and has no kinematic counterpart.
A similar phenomenon has been discussed in the context of Balbus--Hawley
instability.
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It is shown that transverse perturbations
from structurally stable heteroclinic cycles can be destabilized by added
noise even when each individual fixed point of the cycle is stable to tranverse
modes. A necessary condition for this process is that the tranverse dynamics
is non-normal. The phenomenon is illustrated by a simple two-dimensional
switching model and by a simulation of a convectively driven dynamo.
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I will present an experimental
study of the magnetic field fluctuations generated in a turbulent flow
of liquid metal, in the presence of an externally applied field. We consider
the case of a weak `seed' field, so that the Lorentz forces do not modify
the flow. The velocity gradients induce magnetic field fluctuations at
all scales, the description of which pertains to the dynamics of a `passive
vector' in turbulence, in analogy to the passive scalar case. However this
passive vector dynamics involves stretching of magnetic field lines by
velocity gradients, analogous to stretching of vorticity lines, and is
thus at an intermediate level of complexity between passive scalar advection
and fully developed turbulence. In particular, stretching of magnetic field
lines by velocity gradients may overcome Joule dissipation and generate
a large scale magnetic field by amplification of weak initial disturbances--
this is the dynamo effect. We use the flow created in the gap between two
coaxial rotating disks, the von K\'arm\'an swirling flow. At small scales,
it is known to produce a very intense turbulence in a compact region of
space, allowing a study of statistical properties of the magnetic field,
such as scaling laws. At large scales, the mean flow acts upon the magnetic
field, modifying the field lines. In the case of counter-rotating disks,
this mean flow possesses many features, such as differential rotation or
poloidal and toroidal components, which are known to favor dynamo action.
In the co-rotating case, the flow contains a coherent vortex which acts
upon the magnetic field, such as to expulse it from its core. Experimental
results on these various aspects of the magnetic field dynamics will be
presented.
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The properties of both decaying
and forced homogeneous compressible MHD turbulence with helicity are investigated
via direct numerical simulations at resolutions up to $240^3$. Particular
stress is put on the detection and study of dynamo action and the relation
of the alpha-effect with properties of the flow such as the Helicity. The
inverse cascade of energy and its connection with the dynamo is also investigated
along with some topological properties of the resulting flow such as the
alignment properties of the fields.
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The solution of various forms of
the rotating Boussinesq/anelastic non-linear and linearised magnetohydrodynamic
(mass, momentum, induction and heat) equations in spherical and nearly-spherical
geometries is discussed. Linearisation of the equations may be about an
axisymmetric or three-dimensional (steady or possibly non-steady) basic
state. The prototype physical model consists of uniformly electrically-conducting
spherical solid-inner and spherical-shell fluid-outer cores, and an insulating
exterior. Various extensions have been considered, including no inner-core
or an insulating inner-core, a non-uniformly electrically-conducting mantle,
anistropic turbulent viscous or thermal diffusion, pressure buoyancy, non-spherically-symmetric
gravitation and departures from spherical boundaries. The equations are
discretised using scalar, vector and tensor spherical harmonic expansions
and toroidal-poloidal representations in angle, and second- or fourth-order
finite-differences on a variable grid or Chebychev collocation in radius.
The equations are solved using eigenvalue, critical-value or time-stepping
techniques. These codes have been extensively tested against many known
analytical and numerical solutions, including kinematic dynamo problems
and thermal-/magneto-convection problems. The main problems of interest
are topographic effects on the convection, pressure buoyancy, locking of
convection to a non-spherically-symmetric gravitation varying in colatitude
and/or azimuth and turbulent viscous and thermal diffusivities due to the
effects of the magnetic field and rotation.
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A spherical, liquid socium dynamo
experiment has been constructed at the Univerisity of Wisconsin. The experiment
is constructed with the goals of (1) observing self-excited magnetic eigenmodes
and subsequent saturation, of the variety proposed by Gubbins, and Dudley
and James; (2) experimental observation of turbulent alpha and beta effects
prior to self-excitation; and (3) studies of MHD turbulence and the possibility
of a self-excited, small-scale, turbulent dynamo of the type proposed by
Kraichnan and Iroshnikov. The experiment consists of a 1 m diameter sphere
filled with liquid sodium and two 100 Hp motors driving two impellers in
the sodium. Extensive experiments have been performed in a dimensionally
identical water experiment, where velocity fields have been measured with
Laser Doppler Velocimetry. MHD modelling indicates the measured velocity
fields lead to self-excited eigenmodes in the sodium experiment. The infrastructure
of the sodium laboratory is completed and experiments are to begin in spring,
2000. Initial results will be presented.
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Modelling of the geodynamo mechanism
is hindered by the small values of both the Ekman number and the magnetic
Prandtl number in the Earth's core. On the other hand, the magnetic field
is expected to be predominantly large scale, enabling large scale simulations.
I have thus endeavoured to study numerically the balance between the Coriolis
acceleration and the Lorentz force density, while all other accelerations
and viscous forces are ignored. A basis of divergence-free velocity field
densities obeying the no-penetration condition at the boundaries is defined
to calculate a ``magnetic wind'' from a known distribution of magnetic
forces. Projecting onto this basis allows to invert the Coriolis operator.
Then, the inferred velocity is inserted in the induction equation to update
the magnetic field. I will report on the difficulties of this approach
and on my most recent results. Finally, the calculated models are used
to show how the frozen-flux description becomes valid, at the core surface,
with increasing magnetic field strength. This question is of importance
because the frozen-flux hypothesis is widely used to infer core dynamics
from magnetic field measurements.
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We study simple models of fluid
dynamos in order to characterize the bifurcation that leads to the dynamo
effect (stationary or oscillatory, supercritical or subcritical) and to
understand the energy or power balances in the saturated regimes (ratio
of kinetic to magnetic energy, ratio of viscous to Joule dissipation).
We discuss the experimental realisation of these simple fluid dynamos.
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Dynamo action is demonstrated numerically
in the forced Taylor-Green vortex \cite{BMONMU83} made up of a confined
swirling flow composed of a shear layer between two counter--rotating eddies.
This flow corresponds to a standard experimental set--up in the study of
turbulence called the Von Karman swirling flow. The critical magnetic Reynolds
number above which the dynamo sets in depends crucially on the fundamental
symmetries of the TG vortex. These symmetries can be broken by introducing
a scale separation in the flow, or by letting develop a small non--symmetric
perturbation which can be either kinetic and magnetic, or only magnetic
\cite{NBPP97}. The nature of the boundary conditions for the magnetic field
(either conducting or insulating) is essential in selecting the fastest
growing mode. We also present cases where a long term magnetic field produced
by dynamo action saturates and discuss the saturation energetics. Implications
of our results to a planned laboratory experiment using sodium liquid in
the Von Karman swirling flow (Daviaud, Fauve, Pinton, private communication)
will be discussed.
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The Taylor-Green (TG) vortex is
a standard turbulent flow used in numerical computations. It is related
to the experimental Von Karman swirling flow. The relation between the
experimental flow and the TG vortex is a similarity in overall geometry:
a shear layer between two counter-rotating eddies. The TG vortex, however,
is periodic with free-slip boundaries while the experimental flow is contained
inside a tank between two counter-rotating disks. Experiments in Sodium
are planned (Fauve, private communication), for a Von Karman swirling flow
in which the magnetic Reynolds number may be close to the critical value
$R^m_c$ above which a dynamo sets in. We will show that a forced Taylor-Green
vortex is consistent with a long term magnetic field produced by dynamo
action and find the critical magnetic Reynolds numbe. A number of symmetries
of the TG vortex are found to be {\it spontaneously broken}, in the sense
that a small non-symmetric component of the initial data will grow and
eventually completely break the symmetry of the solution. The relation
between the TG and the Von Karman Dynamos and the effect of boundaries
will also be discussed.
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A numerical investigation is presented
of kinematic dynamo action in a dynamically driven fluid flow. The fluid
lies in a rotating plane layer and all motions are taken to be two-dimensional,
that is, depending on two spatial coordinates, together with time. The
base of the layer is given a constant velocity (the top remaining at rest)
and this produces a shear flow in the form of a spiral Ekman layer localised
close to the lower boundary. This basic state is destabilised by a convective
instability through heating the base of the plane layer, or by a purely
hydrodynamic instability of the Ekman layer flow. Kinematic dynamo action
is studied in the flows that result in various regimes. When the Ekman
layer flow is destabilised by the hydrodynamic Ekman layer instability,
the fluid flow that results is steady in an appropriately chosen moving
frame, and takes the form of a row of cats' eyes. Kinematic magnetic field
growth is observed, with some modes growing by the Ponomarenko dynamo mechanism,
while other modes appear to be associated with stagnation points and heteroclinic
separatrices. When the Ekman layer flow is destabilised thermally, far
away from the onset the convective instability creates a flow that is intrinsically
time-dependent, that is, is unsteady in any moving frame. The magnetic
field is concentrated in magnetic sheets situated around the convective
cells and evidence for fast dynamo action is obtained. The presence of
an Ekman layer close to the bottom boundary of the layer breaks an up--down
symmetry and localises the magnetic field near the lower boundary.
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The magnetic helicity is a fundamental
quantity in magnetohydrodynamics because it is conserved in the limit of
infinite electrical conductivity. The magnetic helicity determines the
magnetic part of the $ \alpha $-effect. The latter is of great importance
in view of nonlinear magnetic dynamo. The evolution of the magnetic helicity
tensor for a nonzero mean magnetic field in an anisotropic turbulence is
studied. It is shown that the isotropic and anisotropic parts of the magnetic
helicity tensor have different characteristic times of evolution. The time
of variation of the isotropic part of the magnetic helicity tensor is much
larger than the correlation time of the turbulent velocity field. The anisotropic
part of the magnetic helicity tensor changes for the correlation time of
the turbulent velocity field. The mean turbulent flux of the magnetic helicity
is calculated as well. It is shown that even a small anisotropy of turbulence
strongly modifies the flux of the magnetic helicity. It is demonstrated
that the tensor of the magnetic part of the $ \alpha $-effect for weakly
inhomogeneous turbulence is determined only by the isotropic part of the
magnetic helicity tensor. The magnetic helicity in the case of zero mean
magnetic field is also discussed. It is demonstrated that the two-point
correlation function of the magnetic helicity in the case of zero mean
magnetic field has anomalous scalings for both, compressible and incompressible
turbulent helical fluid flow. The magnetic helicity in the limit of very
high electrical conductivity is conserved. This implies that the two-point
correlation function of the conserved property does not necessarily have
normal scaling. The reason for the anomalous scalings of the magnetic helicity
correlation function is that the magnetic field in the equation for the
two-point correlation function of the magnetic helicity plays a role of
a pumping with anomalous scalings. It is shown also that when magnetic
fluctuations with zero mean magnetic field are generated the magnetic helicity
is very small even if the hydrodynamic helicity is large.
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A nonlinear electromotive force
for an anisotropic turbulence in the case of intermediate nonlinearity
is derived. The intermediate nonlinearity implies that the mean magnetic
field is not enough strong in order to affect the correlation time of turbulent
velocity field. The nonlinear mean-field dependencies of the hydrodynamic
and magnetic parts of the $ \alpha $ effect, turbulent diffusion, turbulent
diamagnetic and paramagnetic velocities for an anisotropic turbulence are
found. It is shown that the nonlinear turbulent diamagnetic and paramagnetic
velocities are determined by both, an inhomogeneity of the turbulence and
an inhomogeneity of the mean magnetic field $ {\bf B} .$ The latter implies
that there are additional terms in the turbulent diamagnetic and paramagnetic
velocities $ \propto {\nabla} B^{2} $ and $ \propto ({\bf B} \cdot {\nabla})
{\bf B} .$ These effects are caused by a tangling of a nonuniform mean
magnetic field by hydrodynamic fluctuations. This increases inhomogeneity
of the mean magnetic field. It is also shown that in an isotropic turbulence
the mean magnetic field causes an anisotropy of the nonlinear turbulent
diffusion. Two types of nonlinearities in magnetic dynamo determined by
algebraic and differential equations are discussed. Nonlinear systems of
equations for axisymmetric $ \alpha \Omega $ dynamos in both, spherical
and cylindrical coordinates are derived. Nonlinear galactic dynamo in the
thin-disc approach is discussed.
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Systematic observations of tracers
of the solar magnetic activity such as sunspots, sunspot groups, active
regions, polar faculae etc. indicate that this activity has a form of two
travelling waves: low-latitudinal equatorward and high-latitudinal poleward
waves in each given hemisphere. The simplest mechanism of this field generation
can be given in the framework of mean field dynamo theory. The corresponding
dimensionless number (field regeneration rate) is large, which can be used
in asymptotic analysis of the governing equations. The use of WKBJ enables
us to resolve analytically short waves in thin layers. Even a kinematic
approach yields general structure of the solution and fixes the frequency
of the cycle. Provided knowledge on the generation sources ($\alpha$ and
$\Omega$-effects) we are able to outline the shape of the dynamo wave.
Nonlinear development of such solution retains some properties of the linear
wave but includes finite wave amplitudes and refines the structure of the
wave front. Asymptotic solution of the weakly-nonlinear problem yields
the transition (bifurcation) to the finite amplitude state and enables
some relation between the period and amplitude of the dynamo wave. Recent
advances in helioseismology provide reliable information on the internal
rotation of the Sun. These data allow us to estimate gradients of angular
velocity , i.e., the strength of the $\Omega$-effect. The 2-D asymptotic
dynamo model reveals spatial locations of the maxima dynamo waves in the
solar convective zone. Further studies are focused on the interaction between
the two waves in one given hemisphere as well as the waves in both the
hemispheres with each other. This yields splitting the frequency of the
cycle for dipole and quadrupole modes, and thus causes global modulation
of the solar cycle. Further nonlinear development of these studies should
involve not only algebraic but dynamic nonlinear back-reaction of the magnetic
field to rotation and convection (e.g., Malkus-Proctor effect). These produces
complicated multiperiodic oscillatory behaviour of mean fields. These theoretical
considerations are subject to comparison with available observations.
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We study the dynamo activity of
Boussinesq convection in a horizontal plane electrically conducting fluid
layer. First the case of nonrotating convection is considered. For the
primary convection rolls no dynamo action is found. However, the skewed
varicose instability leads from the rolls to a pattern of asymmetric squares
which is capable of kinematic dynamo action. For these squares, with rising
or with descending motion in the center (and descending or rising motion
near the boundary) the up-down reflection symmetry is spontaneously broken.
They possess a net kinetic helicity and are generated by two unstable checkerboard
(symmetric square) patterns and their nonlinear interaction. The checkerboard
patterns possess no net helicity and show no dynamo action. The asymmetric
squares are only kinematic, not nonlinear dynamos. Nonlinearly these kinematic
dynamos kill themselves by forcing the solution into the basin of attraction
of purely hydrodynamic roll patterns (with wave numbers smaller than that
of the original rolls). However, the asymmetric squares become nonlinear
dynamos if rotation is added to the system.
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The understanding of thermal convection
in spherical gaps under a central force field is important for large scale
geophysical motions. Neglecting the magnetic field, the dielectrophoretic
force can be used to produce a central force field under microgravity conditions.
In a space experiment, currently under construction, thermal convection
in a rotating spherical gap with heated inner sphere and cooled outer sphere
will be visualized by a Wollaston interferometer. High voltage is used
to produce a dielectrophoretic central force field in the gap. The parameters
are chosen in analogy to the convection in the earth's inner core. The
experiment and its restrictions will be presented as well as numerical
predictions for the expected flows. The axial-symmetric flow is calculated
on a staggered grid with a finite volume method. The conjugate-gradient
method with a pre-conditioner accelerates the approximation. In azimuthal
direction a spectral analysis allows a three-dimensional simulation for
spherical shells with large radius ratios.
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It is well known that dimensionless
numbers relevant to geodynamo modeling are out of reach of present numerical
computations. Difficulties are raised by the extremely small values of
both the Ekman number ($10^{-15}$) and the magnetic Prandtl number ($10^{-6}$).
Asymptotic techniques however can lead to a description of the flow and
of magnetic induction, inside boundary and shear layers. These studies
give a nice way to validate numerical simulations. We concentrate here
on simplified problems and try to achieve a satisfactory agreement with
asymptotic studies. The first problem we consider is an apparently simple
axisymmetrical problem, where all motions are generated by differentially
rotating boundaries in the presence of an imposed magnetic field. In the
non-magnetic case, we show for the first time how the solution tends, with
decreasing Ekman numbers, to the asymptotic limit of Proudman(1956). We
then show the first good quantitative agreement with Stewartson's (1966)
description of the shear layer near the tangent cylinder. We then consider
the MHD flow in the same geometry with an imposed dipolar magnetic field.
The first effects of the Lorentz force is to smooth the change in angular
velocity at the tangent cylinder. As the Elsasser number is further increased,
the Proudman-Taylor constraint is violated, Ekman layers are changed into
Hartmann layers, shear at the inner sphere boundary vanishes, and the flow
tends to a bulk rotation together with the inner sphere. Unexpectedly,
for increasing strength of the field, we observed a super rotation (the
fluid's angular velocity reaches a maximum inside the fluid volume) localized
inside an equatorial torus limited by field lines of the imposed magnetic
field. At a given field strength, the amplitude of this phenomenon depends
on the Ekman number and tends to vanish in the magnetostrophic limit. This
demonstrates the decisive role played by the Ekman number on the solutions
(even with important magnetic effects). We then concentrate on the asymptotic
description of this feature for large Hartmann numbers. An equation has
been derived by Soward (University of Exeter), for this MHD shear layer.
We present the numerical solution of this equation and a comparison with
finite Hartmann number solutions. Finally, we study the properties of the
onset of convection in a rotating spherical shell. We study no-slip and
free slip boundary conditions and consider different heatings~: uniform
heat sources (widely used in previous studies, and in asymptotic studies)
and imposed differential temperature gradient. We present a comparison
with the asymptotic studies of (Jones et al. 2000 and Roberts 1968/Busse
1970) and show how the pattern of motions at the onset of convection strongly
depends on the heating considered. We suggest consequences for numerical
simulations of the geodynamo at moderate Ekman numbers.
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The stability study of the hydromagnetic
systems is a significant part of the geodynamo theory. The linear magnetoconvection
and its dependence on boundary conditions grants a lot of useful results
till now. Marginal instabilities study follows in the model of rapidly
rotating non-uniformly stratified fluid layer permeated by an azimuthal
magnetic field growing linearly with distance from the vertical rotation
axis. More cases of non-uniform stratification are investigated and compared
with uniform unstably stratified layer case. The boundaries are perfectly
thermally conducting and rigid (no-slip). The dependence on electromagnetic
boundary conditions is investigated, i.e. both boundaries are perfect conductors
or finitely conducting as well as in the Earth's core. The significant
parameter of the presented analysis is Roberts number $q\,\equiv\,\kappa\,/\,\eta\,=\,\tau_{\eta}\,/\,\tau_{\kappa}$
(where $\tau_{\eta}\,,\,\tau_{\kappa}$ are times of ohmic and thermal diffusion),
which expresses the coupling of two different processes - thermal and magnetic
diffusion. There are two qualitatively different sets of results (from
the physical point of view): (1) $q\,\ll\,1,$ (i.e. $\tau_{\eta}\,\ll\,\tau_{\kappa}$).
The developed instabilities are extremely slow (of frequencies $\approx
\tau_{\kappa}^{-1}$) and are thermally or magnetically driven. (2) $q\,\ge\,O(1),$
(i.e. $\tau_{\eta}\,\ge\,\tau_{\kappa}$). The developed instabilities of
the MAC waves type (Soward 1979) are characteristic by period typical for
geomagnetic secular variations and they are determined by dynamical balance
of Magnetic, Coriolis and Archimedean forces. The hydromagnetic system
is able to change into the state of double scale magnetoconvection, i.e.
there are more discrete values of Elsasser number, $\Lambda$, for which
two instabilities of different critical radial wave numbers with corresponding
frequencies coexist at the same Rayleigh number. The developed instabilities
and the mean electromotive force, which is produced by these ones, are
in both cases dependent on electromagnetic boundary conditions.
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Marginal instabilities in horizontal
uniformly stratified rapidly rotating fluid planar layer permeated by the
azimuthal magnetic field, linearly growing with distance from the vertical
rotation axis, are investigated. Arising linear magnetoconvection is studied
in viscous and inviscid fluid in dependence on Elsasser number, ${\cal
O}(10^{-2})\leq\Lambda \leq {\cal O}(10^3)$, for two sets of Roberts number,
$q={\cal O}(1)$ and $q\ll 1$. Both boundaries are perfectly thermally conducting
with two types of electromagnetic and/or mechanical boundary conditions.
Their electrical conductivities are either infinite or finite Earthlike
ones. Each boundary is only either stress free or no-slippery for viscous
fluid of various Ekman number, ${\cal O}(10^{-7})\leq E\leq {\cal O}(10^{-3})$.
Numerical results were confirmed by asymptotic analysis which indicated
physical interpretation, too. Almost all modes of instabilities correspond
to Soward (1979) model of the simplest boundary conditions (of stress free
perfect electric conductors), and propagate azimutally eastward or westward.
Westward modes are always preferred with some exceptions of magnetically
driven instabilities for larger $\Lambda={\cal O}(10^3)$. Case $q\ge{\cal
O}(1)$, in distinction with case $q\ll 1$, gives the instabilities of MAC-waves
kind with frequencies typical for geomagnetic secular variations much greater
than frequencies ${\cal O}(\tau_{\kappa}^{-1})$ corresponding to thermal
diffusion time for case $q\ll 1$. Due to viscosity various competitive
modes arise. Their preference is strongly dependent on $q$ and $E$. Critical
frequencies of MAC waves are most sensitive on Roberts number for $q\approx
1$ and lose this sensitivity for increasing $q$ at $q\approx 2.$ Greater
frequencies of MAC waves induce their sensitivity to viscosity for greater
$\Lambda > {\cal O}(1),$ too. This sensitivity for $\Lambda = {\cal O}(1),$
is apparent only for greater azimuthal wave numbers $m\geq{\cal O}(10)$.
However, the preferred modes are not influenced by viscosity and they are
sensitive only to electromagnetic boundary conditions. Many qualitative
differencies in the dependences on $q$ and $E$ do exist for infinitely
and finitely conducting boundaries. E.g. MC-waves were found only for finitely
electrically conductive boundaries, and for sufficiently strong field only
with $m=1$. They almost do not depend on Roberts number $q$, but the direction
of their propagation strongly depends on Ekman number. The eastward MC-waves
do occur at very small Ekman number, $0\leq E \leq {\cal O}(10^{-7})$,
as well as at small $E \geq {\cal O}(10^{-3})$, while the westward MC-waves
do occur only at small $E \geq {\cal O}(10^{-3})$.
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Eurico Covas has an extensive abstract
that covers various aspects of the joint work we have been doing over the
last 3 years. We will talk on different aspects of these works. One of
us will focus on the mathematical results and the other on the confirmation
of these results in various families of axisymmetric mean field dynamo
models.
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Dynamo action, which converts kinetic
energy into magnetic energy, is the manifestation of the coupling between
kinetic and magnetic excitations in a conducting fluid. The occurence of
the dynamo action is not questionable,but the nonlinear regimes are very
poorly known. As for hydrodynamic turbulence, the experimental approach
could represent an efficient tool to study the nonlinear effects in MHD
flows. Some numerical examples have shown that dynamo action is present
in flows at magnetic Reynolds numbers $Rm = \lambda V_{max} L_{max} = 100$,
say, where $1/\lambda$ is the magnetic diffusivity and $V_{max},L_{max}$
are respectively the maximal speed of the flow and a characteristical length
of the conducting volume. Using the best available fluid conductor ($1/\lambda=
10 m^2 /s$), liquid sodium at 150$^\circ$C, the condition $Rm = 100$ implies
that $V_{max} L_{max} = 10 m^2/s$, which represents the main technical
challenge to be achieved by any experimental fluid dynamo. We will here
concentrate on the feasability study of a peculiar experimental fluid dynamo.
Note also that using liquid sodium as a conducting fluid, the kinetic Reynolds
number of the flow is about $10^5 Rm$, which shows that it is in a regime
of fully developped turbulence. Our approach is to try various forcing
mechanisms and geometries in water models to measure the velocity fields
which are then introduced in the numerical computation of a kinematic dynamo
problem. The experimenatal flows are von Karman type, produced between
two counter-rotating disks in a cylindrical container. Using laser doppler
and pulsed doppler ultrasonic velocimetries we can retrieve the temporal
mean velocity and the turbulence rates at each spatial point. The kinematic
approach gives relatively fast answers. Looking at the magnetic energy
evolution $E_{B}(t)= e^{\sigma t}$, the dynamo effect appears when this
energy grows without external excitation in the form of a magnetic field:
$\sigma > 0$ means dynamo effect, and $\sigma < 0$ no dynamo effect.
The influence of various parameters, such as the poloidal to toroidal component
ratio of the velocity field $V_{pol} / V_{tor}$, an external magnetic field,
the influence of time dependent flows or an external conducting layer are
being tested.
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In the first run of Riga Dynamo
experiment an intense flow of liquid sodium produced by an outside driven
propeller have generated a slowly growing magnetic field eigenmode. For
a slightly decreased flowrate the observed field is slowly decaying. The
measured results correspond satisfactory with theoretical predictions for
the growth rates and frequencies. In the report will be presented computational
base, optimisation, the detailed design of the experiment, current results
and next experimental steps.
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Sunspot records dating back to
the $17^{\rm th}$ century show an aperiodic cyclic activity. These records
show also that this cyclic behaviour was interrupted by epochs of reduced
activity, such as the M\"aunder minimum, with intermediate time scales
of $\sim 10^2$ years. Furthermore, there is indirect evidence from the
study of proxy indicators, such as $^{14}C$ and $^{10}Be$, which demonstrate
that such epochs of reduced activity persisted back in time, at least for
the last 10\,000 years. In addition, observations of magnetic activity
in solar--type stars over the last three decades provide evidence that
similar variability can also be present in such stars. This latter class
of observations seem to also indicate that similar solar--type stars can
exhibit a range of qualitative distinct magnetic behaviours. Such variabilities
in the Sun and solar--type stars are thought to be due to dynamos operating
in or near the base of stellar convective interiors. Motivated by the above
observational evidence and bearing in mind that the underlying regimes
in such stars are bound to be non--linear, an attempt is made in this Thesis
to study possible mechanisms for such variability by making a study of
the generic non--linear dynamics underlying axisymmetric dynamo models.
Emphasis on generic features --- namely the presence of invariant subspaces
as well as two other technical assumptions generically satisfied by such
dynamo models --- is essential, given the unavoidably approximate nature
of such models. This study reveals a number of novel results, including
a new form of intermittency, referred to as {\em in--out intermittency}
--- together with its precise signatures and scalings --- as well as several
forms of {\em final state sensitivity}, including the presence of regions
of parameter space possessing sensitivity to initial conditions and parameter
values. By employing extensive simulations of axisymmetric ODE and PDE
mean--field dynamo models, it is shown that these predictions do indeed
occur. In particular, it is shown that in--out intermittency does indeed
occur in such models, hence substantiating an old proposal --- the {\em
intermittency conjecture} --- according to which dynamical intermittency
could account for M\"aunder--type variability. In addition, it is shown
that more than one type of intermittency can occur in such settings which
demonstrates that a more appropriate proposal would be that of {\em multiple
intermittency}, each with their own precise signatures and scalings. Furthermore,
it is shown that such models can possess multiple coexisting solutions
(attractors), as well as what is referred to as {\em fragility}, i.e.\
the property that small changes in the initial conditions as well as in
the parameters and the details of the models can produce qualitative changes.
If present in real stars, this could, in principle, account for the observational
evidence for similar solar--type stars having qualitatively different modes
of magnetic behaviour. We call this the {\em fragility conjecture}. Given
the genericity of such non--linear behaviours, it is likely that these
phenomena will persist in more realistic models. Finally given the related
precise signatures and scalings, it is in principle possible to make a
comparison with the observational and proxy data.
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We consider numerically the problem
of spherical Couette flow in an electrically conducting fluid, and impose
a strong magnetic field aligned with the axis of rotation. We show that
the resulting flow depends dramatically on whether the boundaries are taken
to be electrically insulating or conducting. In both cases the so-called
tangent cylinder, the cylinder circumscribing the inner sphere and aligned
with the axis of rotation, plays a crucial role, but the details of what
happens on it are very different in the two cases. For insulating boundaries,
the flow consists of a shear layer right on the tangent cylinder, with
the fluid at rest outside, and in essentially solid-body rotation at a
rate $\Omega/2$ inside. In sharp contrast, for conducting boundaries, the
flow consists of a powerful counter-rotating jet just outside the tangent
cylinder. The thickness of both the shear layer and the jet scale as $M^{-1/2}$,
where the Hartmann number $M=B_0 L(\sigma /\nu\rho)^{1/2}$ measures the
strength of the imposed field. However, whereas the jump across the shear
layer remains constant at $\Omega/2$, the magnitude of the jet {\it increases},
roughly as $M^{0.6}$, and thus exceeds $\Omega$ for sufficiently large
$M$. Having obtained these --- so far purely axisymmetric --- solutions,
we next compute the onset of non-axisymmetric instabilities, and find that
the critical Reynolds number scales as $M^{0.66}$ for the shear layer and
as $M^{0.16}$ for the jet. The (fully three-dimensional) nonlinear equilibration
of these instabilities will also be considered. Finally, time permitting,
we will consider what happens if dipole or quadrupole or other non-uniform
fields are imposed instead.
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Turbulent flow of liquid sodium
is driven toward the transition to self-generating magnetic fields. The
approach toward the transition is monitored with decay measurements of
pulsed magnetic fields. These measurements show significant fluctuations
due to the underlying turbulent fluid flow field. This talk presents experimental
characterizations of the fluctuations in the decay rates and induced magnetic
fields. These fluctuations have a significant implications on the transition
to self-generation which should occur at larger magnetic Reynolds number.
Specifically, we predict that the transition will show intermittency.
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It is well known that the symmetry
group of a system has important consequences for its dynamical behaviour.
The symmetry group of the axisymmetric mean-field dynamo equations and
its effects on the bifurcation sequence as the dynamo number increases
has been the subject of recent study. There is strong evidence that there
was a major departure from equatorial symmetry in the distribution of sunspots
at the end of the Maunder minimum. However, the equatorial symmetry of
the sunspot cycle over the last 200 years has been the subject of some
debate.It is clear from the records that there are large departures from
equatorial symmetry over timescales of a few years, and evidence that whole
cycles have a dominant hemisphere which can change between cycles. It is
not agreed, however, to what extent this is a statistical effect of a noisy
component of the sunspot spatiotemporal distribution, or whether there
are cyclic changes in the symmetry of the solar field over a time scale
of several solar cycles, as predicted by several recent dynamo models.
This paper addresses this question by combining insights from the dynamics
of symmetric systems, with robust tools for identifying multi-periodicity
in complex and noisy time series. We use a time series which supplies the
latitude and longitude of daily sunspot observations from 1853-1996. This
series extends the Greenwich Photoheliographic Records backwards from 1876
by using the observations of Carrington and Sp\"{o}rer and forwards from
1976 by using the SOON records. An important result of this analysis has
been the identification of a long-period oscillation of the solar magnetic
equator (defined as the average of the sunspot numbers weighted by latitude)
on a period of 90 years, which is the period most closely associated with
the Gleissberg cycle. These methods also show that the cycle length also
varies on this timescale in accordance with other studies based on the
total sunspot counts (without reference to the spatial distribution of
the spots). We discuss this finding in the light of two possible models
of changes in the symmetry of the solar field, the parity modulation associated
with the Type I modulation of Knobloch and Landsberg (1996) and the newly
identified form of intermittency described as ``spiralling'' or ``in-out''
intermittency. There will also be discussion of the problems of identifying
the spatial behaviour of the sun's magnetic field from such a limited time
series. Regular observations of sunspot position are not available before
1853, apart from the French observations during the Maunder minimum (1660-1719).
Proxy records, such as $^{10}Be$ and $^{14}C$ records, can be extended
back for much longer but do not contain information as to the equatorial
symmetry of the solar field.
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It is well known, that all greatest
planets of Solar system have a magnetic field. The Mars have weak magnetic
field too, the Venus, most similar to Earth planet - in contrary have not
it. At recent time magnetic fields of Galilean satellites of Jupiter are
discovered. The presence of magnetic properties on celestial bodies so
different nature is non-direct evidence of external nature of them. The
most possible way of explanation of geomagnetic field inversion may be
following. The nature of magnetic field related with electric current in
Earth's core. On the other hand, the west drift of non-dipole component
of this field with velocity 0.2 rad/year is well known. The most simply
explanation of this fact is in differential rotation of core relatively
daily Earth's surface. Really, such phenomenon - more faster rotation of
inner core was discovered recently. On the other side, the increasing of
day and, as followed, decreasing of Earth's surface rotation rate is well
known. So, in accordance with orbital moment's conservation law, the removing
Moon from Earth and Moon's velocity decreasing take place. However, due
to Earth's orbit changes by planets perturbations (eccentricity decreasing),
the Moons acceleration is observed. The eccentricity show oscillation variations
- there are its increasing epoch sometime in past. According of them, the
epoch of more strong and more slow decreasing Earth's rotation rate change
one another. The core have more inertia relatively outer Earth's layers,
it may be described mathematically through 'angle of late'. At present
time core keep in memory epoch of more slow, then now epoch of decreasing
Earth's rotation rate. It explain inner core leading over daily surface
and west drift on secular variations of magnetic fields. The reversed situation
is according of inverse epoch, when core rotated more slow then daily surface.
It possible, when Earth's orbit eccentricity increased. For complete model
the description of way of charges separation is required. Maybe, high temperature
or phase changes is able to provide necessary polarization.
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The screw dynamo or so-called Ponomorenko dynamo is able to generate
a non-axisymmetric magnetic field in an axisymmetric helical flow and is
one of the simplest known dynamo models. Basically a laminar helical flow
like Couette-Poiseuille can produce the magnetic field self-excitation.
A generation of magnetic field is possible in other way which is based
on small-scale turbulence. A turbulent helicity (alpha-effect) can amplify
a magnetic field in absence of mean field flow. Both generation mechanisms
is supposed to act in several astrophysical object and the proposed Perm
dynamo experiment. We study interaction of generation mechanisms helical
large-scale laminar flow and small-scale turbulent helicity. The crucial
parameters of dynamo-process in a different regimes have a strong interest
especially in the frame of the Perm dynamo experiment. The simultaneous
action of these mechanisms has been studied in the context of this experiment,
taking properly into account inhomogeneity and anisotropy of the turbulence.
Depending on the relevant parameters they may indeed support or counteract
each other.
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We will give a short overview over the properties
of structurally stable heteroclinic cycles in the context of the dynamics
of spherical convection problems. A new low dimensional model linking the
heteroclinic structure of the convection problem to the hetereoclinic orbits
of the dynamo problem is presented. In addition an intermediate PDE-ODE
model of the interaction of the induction equation with the center manifold
of the convection problem is introduced. Preliminary simulations for both
models are discussed and the existence of heteroclinic cycles with nonzero
magnetic field is shown. The transition from the kinematic to the dynamic
regime is also analysed.top
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The problem of convection driven by thermal buoyancy in rotating, self-gravitating
rotating spherical fluid shells and the problem of the generation of magnetic
fields by such flows has been investigated through numerical simulations.
A formulation with a minimum set of parameters representing the most important
physical conditions of planetary dynamos has been chosen in order to obtain
a general view of the dependence of the structures of velocity and magnetic
fields on the parameters of the problem. While most of the dynamos exhibit
a spatio-temporally chaotic nature, stationary and time periodic dynamo
solutions with a characteristic azimuthal wavenumber can be obtained if
the magnetic Prandtl number is of the order 10 or larger. Convection without
magnetic field exhibits nearly periodic relaxation oscillations in which
convection columns generate a differential rotation which in turn shears
off the convecting eddies. After the viscous decay of the differential
rotation the process repeats itself. In the presence of a magnetic field
the oscillations are shortened through the damping of the differential
rotation by magnetic stresses. Magnetic fields exhibit dipolar, quadrupolar
or hemispherical structures as long as convection in the polar regions
of the spherical fluid shells is inhibited owing to a sufficiently high
rate of rotation. More complex magnetic fields with more random properties
are obtained when the Rayleigh number $R$ is increased. Owing to flux expulsion
from the convection a filamentary structure becomes predominant. The strength
of the magnetic field tends to saturate at a value of the Elsasser number
of the order unity, if the latter number is appropriately defined through
the average work done by Lorentz and Coriolis forces. top