## 2. The modelWe shall assume that the gross features of the large scale solar magnetic field can be described by a mean field dynamo model, with the standard equation Here , the term proportional to represents the effects of turbulent diamagnetism, and the velocity field is taken to be of the form , where , is a prescribed underlying rotation law and the component satisfies where is the operator
and
is the induction constant. The
source of the sole nonlinearity in the dynamo equation is the feedback
of the azimuthal component of the Lorentz force (Eq. (2)), which
modifies only slightly the underlying imposed rotation law, but thus
limits the magnetic fields at finite amplitude. The assumption of
axisymmetry allows the field to be
split simply into toroidal and poloidal parts,
, and Eq. (1) then yields two scalar
equations for When attempting to model astrophysical systems, boundary conditions
are often rather ill-determined. We try to make physically motivated
choices. For our inner boundary conditions we chose
, ensuring angular momentum
conservation, and an overshoot-type condition on
(cf. Moss & Brooke 2000). At the
outer boundary, we used an open boundary condition
on Equations (1) and (2) were solved using the code described in Moss & Brooke (2000) (see also Covas et al. 2000) together with the above boundary conditions, over the range , . We set ; with the solar convection zone proper being thought to occupy the region , the region can be thought of as an overshoot region/tachocline. In the following simulations we used a mesh resolution of points, uniformly distributed in radius and latitude respectively. In this investigation, we took to be given in by an interpolation on the MDI data obtained from 1996 to 1999 (Howe et al. 2000a). For we took , where (cf. Rüdiger & Brandenburg 1995) and for with cubic interpolation to zero at and , with the convention that and . Also, in order to take into account the likely decrease in the turbulent diffusion coefficient in the overshoot region, we allowed a simple linear decrease from at to in . © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |