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Results. The dynamo number is a decreasing function of β. Moreover, if the current is parallel to the field (β > 0), the dynamo number is smaller than in the force-free case. In contrast, for (β
However, direct numerical simulations are not able to resolve the surface layers where the pressure scale height is very small and the Lorentz force cannot be neglected. Nonetheless, recent models have been used to estimate the effect of a nearly force-free corona on controlling the emergence of flux from lower layers (Warnecke et al. 2011), on the injection of magnetic twist in the heliosphere (Warnecke et al. 2012), and on a global dynamo (e.g., Warnecke et al. 2016), concluding that the presence of a corona cannot be neglected.
Common models of coronal field based on a potential field (Nash et al. 1988), (1)or on a force-free field, (2)where b is the total magnetic field and β = β(x) is a scalar function, are often used to describe chromospheric observations (Wiegelmann & Sakurai 2012). On the other hand, the coupling with the interior is always neglected. In the framework of mean-field dynamo theory, a new proposal has been presented in Bonanno (2016), where a consistent coupling with an α2Ω dynamo model for the interior was achieved under the assumption that the coronal field is harmonic. On the other hand, there are no known analytical solutions in spherical symmetry of a dynamo-generated interior field coupled with a force-free exterior. The aim of this paper is to present such a solution for a linear force-free field. Although our solution has been obtained for a very idealized case (non-helical homogeneous turbulence), some of its features are in agreement with the findings of Bonanno (2016).
The analytical linear force-free solution presented in this paper has been obtained by coupling a corona with a dynamo-generated field in the interior. Although it is a highly idealized situation, it shows several interesting features. The most important property of the solution is the endowment of a new dependence of the dynamo number on the strength and the topology of the force-free field as parametrized by the parameter β. Positive β produces smaller dynamo numbers, while negative β renders the dynamo more difficult to excite. This is in agreement with the harmonic atmosphere model in Bonanno (2016), as Beltrami fields are also harmonic, while the reverse is not true in general. The toroidal field is non-zero at the surface, and therefore it could be important to implement this solution in the ZDI regularization procedure. Because the coupling with the interior has significantly reduced the number of free parameters that are required to specify the field for each harmonics, the space of possible solutions could be significantly reduced. We hope to discuss possible physical application of our solution in a forthcoming paper, where we will extend our approach to the non-stationary dynamo case. 2ff7e9595c
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