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Astron. Astrophys. 321, 151-158 (1997)

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2. The model

We consider two dynamo-active spheres, radii [FORMULA], [FORMULA] contained within a spherical computational volume of normalized radius unity. For many of the calculations we placed the origin of spherical polar coordinates ([FORMULA] at the centre of the computational volume. whose surface was thus tangent to both of the dynamo-active spheres. The separation of the centres, d, then satisfies [FORMULA], and the axis of the coordinate system is parallel to the orbital angular velocity vector. Fig. 1 shows a cross-section through a plane containing the rotation axis, and also a section through the equatorial plane. More realistically, the origin should be at the centre of mass of the system, which needs some assumption such as


where [FORMULA], [FORMULA] are the radius and mass of a component, in order to determine fully the geometry. (For equal radii, [FORMULA], the models are identical.) A relation such as this imposes a geometrical constraint on possible configurations; thus calculations with [FORMULA], and origin not at the centre of mass, should be regarded as perhaps less realistic. However, trials show that results do not differ generically between these slightly different configurations. In relation (1), we take [FORMULA] for non-contact, and [FORMULA] for contact, configurations, corresponding approximately to the standard relations for detached and contact lower main sequence systems. An alternative assumption to that described below is that [FORMULA], which removes the distinction between use of the two different origins of coordinates as discussed above. Test calculations with this assumption give similar behaviour to those with [FORMULA], for rather larger values of [FORMULA].

[FIGURE] Fig. 1. Schematic cross-section through line of centres and rotation axis. The outer circle represents the intersection of the cross-section with the surface of the computational volume. When O is at the centre of the volume, this circle is tangent to both the circles radius [FORMULA] and [FORMULA].

A conventional [FORMULA] -effect is taken in the form [FORMULA] inside the dynamo-active regions, where [FORMULA] is a constant, and [FORMULA] outside. Thus, within the computational sphere, we can write [FORMULA], where


The coefficients [FORMULA] can readily be determined analytically when [FORMULA], but in general were evaluated numerically. A standard [FORMULA] -quenching nonlinearity,


was used to limit the solutions at finite amplitude. The non-dimensional dynamo parameter is [FORMULA], where R is the radius of the computational sphere (Fig 1), and [FORMULA] is the diffusivity, assumed constant throughout the computational sphere. Rotation is assumed uniform, and so there is no corresponding dynamo number to quantify differential rotation - this is strictly an ` [FORMULA] dynamo'. On the surface of the computational volume, vacuum boundary conditions are applied. In general, [FORMULA] will contain contributions from all azimuthal Fourier modes, see Eq. (2). Correspondingly, the nonlinearity (3) means that the dynamo field will in general be of the form


ie all Fourier modes will be present in [FORMULA] also.

Thus the system has three important parameters: [FORMULA], [FORMULA], [FORMULA]. Comparison between solutions with different values of [FORMULA], [FORMULA] is complicated by the changing value of [FORMULA] for which the dynamo is first excited, [FORMULA] say. This can be thought of as being, in part at least, due to the relative volume where [FORMULA] decreasing as the separation increases (ie as [FORMULA] and [FORMULA] decrease). We did not attempt to derive systematically [FORMULA] values.

For the case where the binary radii are equal, certain symmetries in the solution can be predicted. In this case [FORMULA] contains only terms proportional to [FORMULA], [FORMULA]. Thus dynamo solutions fall into two azimuthal symmetry families, with the field written symbolically as




This follows directly from the form of nonlinearity (3). For given parameters either or both or neither of these families may be excited. We can refer to family (5) as of type E, and to (6) as of type O. If both are simultaneously excited (O+E), then there will be no direct modal interactions. However the nonlinearity (3), being local, will then give an indirect interaction between odd and even field components, the odd symmetry field being able to quench the dynamo growth of the even in a spatially nonuniform manner, and vice versa.

In the more general case of unequal radii, where all Fourier components of [FORMULA] are non-zero, the magnetic field will also contain all Fourier components (type W, say). However, if the radii are comparable, but not equal, the even-m contributions to [FORMULA] will still dominate. If the nearby solution for strictly equal radii is of odd or even type with respect to m, then it can be expected that the solution for unequal radii will be dominated by this solution, with only a relatively small component of the opposite symmetry type present. If [FORMULA], [FORMULA] are respectively the total magnetic energy and that contained in the axisymmetric part of the field, then [FORMULA], [FORMULA], is a measure of the global azimuthal symmetry. The energies, [FORMULA], in the m th mode can also be calculated.

The other useful global symmetry parameter is the parity. If [FORMULA], [FORMULA], denote fields with radial components that are respectively odd, even with respect to the plane [FORMULA], and [FORMULA], [FORMULA] are the corresponding parts of the total magnetic energy, then [FORMULA], with [FORMULA] (cf Moss et al 1991). Parity parameters [FORMULA] can also be defined for the individual azimuthal Fourier modes.

In the limit [FORMULA], the spheres are almost superimposed, and the overall dynamo-active volume approaches a sphere. The solutions should then approach the known solution for a sphere. We computed the marginal value of [FORMULA] when [FORMULA], [FORMULA], to be approximately 7.75, with [FORMULA]. This is consistent with the standard marginal value of [FORMULA] for the first bifurcation from the trivial solution to a dipolar axisymmetric field for the spherical [FORMULA] dynamo.

We note that the problem could have been tackled by taking the line of centres as the axis [FORMULA] of the polar coordinate system, with [FORMULA] within the dynamo-active regions. A trial calculation with [FORMULA] was found to agree reasonably well with results obtained with the geometric configuration used above, but this approach seemed less flexible, and so was not pursued.

The code used is an adaption of that described in Moss et al (1991). An explicit [FORMULA] grid in r and [FORMULA] is used, together with a modal expansion in [FORMULA], including modes [FORMULA]. Experience suggested that [FORMULA], [FORMULA], [FORMULA] was adequate for the parameters used. The dimensionless time, [FORMULA], is measured in units of [FORMULA].

We note that, despite some perhaps superficial similarities, this work does not describe a dynamo of Herzenberg type (Herzenberg, 1958). In our case the spin axes are assumed to be strictly aligned, and the components do not spin relative to one another. Some related issues connected with the Herzenberg dynamo are discussed in Brandenburg, Moss and Soward, in preparation.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998