2. The model
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 A and B. Nondimensionalizing in terms of the solar radius R and time , where is the maximum value of , and putting , , , and , results in a system of equations for and . The dynamo parameters are the two magnetic Reynolds numbers and , and the turbulent Prandtl number . is the solar surface equatorial angular velocity and . Thus and are the turbulent magnetic diffusivity and viscosity respectively, is fixed when is determined (see Sect. 3), but the value of is more uncertain. The density is assumed to be uniform.
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 B and vacuum boundary conditions for . The motivation for this is that the surface boundary condition is ill-defined, and there is some evidence that the more usual condition may be inadequate. This issue has recently been discussed at length by Kitchatinov et al. (2000), who derive `non-vacuum' boundary conditions on both B and .
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