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MAGNETOHYDRODYNAMICS: Solar and stellar magnetic fields

Research in Magnetohydrodynamics:

• The thin-flux tube approximation

• Shocks in magnetic flux tubes

• Stability analysis of toroidal flux tubes in stellar convection zones

• Basic ideas for a stellar dynamo based on magnetic flux tubes

• Publications

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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.

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).

The stability analysis of toroidal flux tubes is the key for treating three different (but related) problems:

• (1) 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 observations.

• (3) 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 force.

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.

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 unstable.

• 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 active regions.

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 proper motion).

• The long intervals (approx. 50-100 years) with absence of activity in the form of sunspots (grand minima or Maunder minima).

• 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 low activity.

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