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MAGNETOHYDRODYNAMICS: Solar and stellar magnetic fields
Research in Magnetohydrodynamics:
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The thin-flux tube approximation
A large number of theoretical studies on the structure and
dynamics of magnetic flux tubes make use of the ``thin flux tube
approximation'', based on an expansion approach about the axis of
the tube, which permits the reduction of the full MHD problem to a
mathematically more tractable set of equations.
For a vertical, axisymmetric magnetic flux tube the physical
quantities are expanded in the radial coordinate about the axis of
symmetry and the power series are inserted into the MHD equations
written in cylindrical coordinates. The assumption of axial
symmetry significantly reduces the number of unknowns and
equations. The closure of the system is provided by appropriate
boundary conditions. By retaining only the zeroth- and
first-order terms, the `classical' thin flux tube approximation
(Defouw, 1976; Roberts & Webb, 1978) is recovered, but in order
to include twisted magnetic fields and azimuthal flows, the
expansion had to be extended to higher orders. As applications,
the axisymmetric wave modes of a magnetic cylinder were studied.
The formalism was also applied to derive the magnetostatic
equations governing the equilibrium structure of an axisymmetric,
vertical flux tube embedded in a stratified atmosphere.
Shocks in magnetic flux tubes
The properties of MHD shock waves in the framework of the thin
flux tube approximation were investigated. It was shown that the
properties of shock waves confined to flux tubes exhibit many
analogies with those of slow MHD shocks in extended media. The
analogies and differences with purely hydrodynamic shocks were
also examined. In particular, it was shown that the sub- or
supercritical character of the flow velocity with respect to the
sound, Alfvén and cusp speeds is derivable from thermodynamic
considerations only, as for HD shocks, in contrast to general MHD
shocks, for which the evolutionary conditions have to be applied.
The theory of shock waves in thin flux tubes is not only of
interest in connection with concentrated magnetic structures in
stellar atmospheres. Its understanding is conceptually important
from both physical and mathematical point of view since the flux
tube provides one of the simplest forms of equations governing the
dynamics of a magnetized plasma confined by an external pressure
and subject to a permanent constraint (internal gas pressure
variations are related to internal magnetic field variations).
Stability analysis of toroidal flux tubes in stellar
The stability analysis of toroidal flux tubes is the key for
treating three different (but related) problems:
It permits to calculate the maximum field stength that can be
stored in mechanical equilibrium in a stellar interior. As a
consequence of our stability analysis, we now support the idea
that the fields giving rise to solar active regions originally
have a strength of the order of 105 G.
- (2) It yields the initial condition for the numerical
simulations of the rise of unstable flux tubes from the bottom of
the convection zone to the surface. The latitudes of emergence
resulting from the simulations show good agreement with the
It permits to compute an -effect due to the instabilities
undergone by a buoyant flux tube once its field strength has
exceeded a critical value. The alpha-effect of dynamo theory is
the inductive effect that regenerates the toroidal field from the
poloidal field component; traditionally, this effect has been
associated with turbulence in a rotating medium. We show that it
is possible to have an alpha-effect entirely produced by the
combination of an instability (due buoyancy) with the Coriolis
Through point (3) above we connect the stability analysis of the
equilibrium of toroidal magnetic fields in a rotating medium with
dynamo theory. In the meantime, a consistent model for the
storage of magnetic flux and the formation of active regions that
is consistent with their basic properties has emerged.
Basic ideas for a stellar dynamo based on magnetic flux tubes
In collaboration with Dieter Schmitt (Göttingen) and with
Manfred Schüssler (Freiburg) I am currently working on the
development of a new picture of solar/stellar dynamos based on the
assumption that the magnetic fields in stellar convective zones
are highly intermittent.
The main features of our dynamo picture are:
The flux tubes are stored in mechanical equilibrium in a layer of
overshooting convection between the bottom of the convection zone
and the radiative region.
Differential rotation (omega-effect), and possibly other
mechanisms such as the explosion of `weak' flux tubes in the
convection zone (Moreno-Insertis et al. 1995, ApJ 452, 894)
intensify the magnetic field until the tubes become buoyantly
The non-axisimmetric modes of unstables flux tubes may give rise
to an alpha-effect through the generation of a mean electric field
due to the perturbations of velocity and magnetic field.
Above a threshold value of the magnetic field, the flux tubes
become so unstable (growth-times of the order of a few weeks or
even less) that they leave the overshoot region and rise through
convection zone towards the stellar surface to emerge as bipolar
Up to date we have been able to expain the following observational
facts with our model:
The restriction of sunspots to two belts of
latitude (5 deg to 40 deg) above and below the equator.
The tilt angle of the active regions main axis
(by typically 10 deg) with respect to the equator.}
The asymmetry between preceding and following
part of the active region (as regards morphology, stability and
The long intervals (approx. 50-100 years) with absence of
activity in the form of sunspots (grand minima or
In a letter in A&A we have shown that the alternation
between intervals of cyclic activity and grand minima like the
Maunder minimum can be explained by an on-off intermittency
arising in a dynamo driven by the instability of magnetic flux
tubes in the overshoot layer at the bottom of the convection zone.
The combination of a threshold in field strength for dynamo action
and random fluctuations due to magnetic fields from a turbulent
convection zone leads to activity cycles with strong amplitude
variations and the occasional appearance of grand minima of very
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25 April 2005