© ESO, 2017

1. Introduction

A consistent description of the magnetic coupling between a magnetized coronal field and a dynamo-generated interior model is still lacking. Mean-field models usually employ a current-free description of the coronal field (Krause & Raedler 1980; Moffatt 1980), an assumption that is not compatible with an active corona.

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).

Moreover, as is well known, in order to obtain the photospheric field from Zeeman-Doppler imaging (ZDI), it is necessary to solve the strongly nonlinear inverse problem. This problem is very often beset by the presence of several local minima. The choice of a proper forward model is therefore essential to make sure we only consider solutions that are physically meaningful (Carroll et al. 2009; Stift & Leone 2017). We therefore hope that our analytical solution can also be useful in the ZDI reconstruction approach.

The structure of this article is the following: Sect. 1 contains the physical motivation of this work, Sect. 2 contains the basic equations of the model and their physical implications, and Sect. 3 is devoted to the conclusions.

2. Model

2.1. Basic equations

We assume that the field in the stellar interior admits a description in terms of the standard mean-field dynamo equation.

In the case of homogeneous, isotropic, and non-mirrorsymmetric turbulence, we can write the evolution equation for the mean magnetic field B(3)where α describes the turbulent flow, U is the mean flow, and η characterizes the turbulent (eddy) diffusivity. α and η are simple coefficients in our case, but they would be tensors if isotropy were not assumed. A mean field in a turbulent medium is defined as the expectation value of the total field in an ensemble of identical systems. In our case, the mean magnetic field can be seen as a time average of the total magnetic field over a timescale that is shorter than the long-term evolution of the field.

If U = 0, we have α = const., η = const., so an exact stationary solution can be obtained by employing the standard decomposition in toroidal and poloidal component (4)where Ψ = Ψ(r,ϑ,ϕ) and Φ = Φ(r,ϑ,ϕ) are scalar functions, B_T is the toroidal component, and B_P is the poloidal component (see Krause & Raedler 1980, for details). By inserting Eqs. (4)in (3), we easily find

which implies (6)It is convenient to employ the following decomposition: where x is the normalized stellar radius x = r/R, φ_nm(x) and ψ_nm(x) are the eigenfunctions of the radial elliptic problem defined by Eq. (5), and are the spherical harmonics functions.

By inserting Eqs. (7b)in (6), we obtain that the only regular interior solution is (8)where J_n(x) is the Bessel function of the first kind, and A_nm is a complex coefficient that must be determined assuming an exterior field configuration. For simplicity’s sake, we use A_nm = 1 whenever we need to normalize quantities such as the mean magnetic field B and the magnetic helicity A·B. By substituting Eqs. (8)in (5b), we find (Krause & Raedler 1980) (9)where B_nm is a constant, and C_α = α/η is the eigenvalue parameter of Eq. (5).

2.2. Adding a force-free corona

We now consider that the dynamo domain is surrounded by a corona. In the exterior, we therefore assume that the field satisfies the linear force-free condition, namely Eq. (2)with β = const. This is the state of minimum dissipation for a given amount of magnetic energy (Chandrasekhar & Woltjer 1958). As explained in Chandrasekhar & Kendall (1957), the general solution of Eq. (2)can be obtained by inserting Eqs. (4)in (2), thus obtaining the Helmholtz equation for the Φ function (10)and Ψ = βΦ in this case. The general solution of Eq. (10)reads (11)where C_β = βR, C_nm and D_nm are coefficients depending on n and m, and j_n(x) and y_n(x) are the spherical Bessel functions: (12)where J_n(x), Y_n(x) are the Bessel function of the first and second kind, respectively. We note that as r ≥ R, differently from what was discussed in Nakagawa (1973) and Priest (1982), the general solution of the linear force-free equations must include both the Bessel function of the first and second kind.

We impose the continuity of all the field components across the boundary, so that (13)where the notation [[ F] ] indicates the difference between the values assumed by the quantity F on the two different sides of the boundary. This condition implies that (14)where J is the current, and n is a unit vector perpendicular to the boundary. The continuity of the tangential component of the electric field E reads (15)In our case, Eq. (15)reads (16)which can be satisfied as long as η is not assumed to be infinite in the r>R domain (see P. H. Roberts in Proctor & Gilbert 1994).

After some algebra, it is finally possible to show that the continuity of the field components determines B_nm, C_nm and D_nm as a function of A_nm as follows: We exclude the possibility that C_α = C_β, as in this case the interior field would also be force free. It is worth noting that the surface toroidal field at the surface is in general non-zero, which was instead a necessary condition in the current-free case.

While in the current-free as well as in the vertical (i.e., purely radial) field case, the “quantization” condition for C_α is obtained by imposing the vanishing of the toroidal field on the stellar surface (Krause & Raedler 1980), in this case, the discrete turbulent spectrum is obtained by imposing the outer boundary condition at r = R_out.

In particular, following Bonanno (2016), we assume that at r = R_out the radial component of the field is dominant, so that (18)This is consistent with the presence of a stellar wind in the solution found by Parker (1958).

It is easy to show that Eq. (18)implies the following equation to hold:

(19)

Clearly, the eigenvalues for a given n but different m coincide, meaning that there is degeneration with respect to m as Eq. (19)does not depend on m. It is interesting to study the “quantized” spectrum for various values of β and R_out. While in the limit β → 0 it is possible to show that Eq. (19)reproduces the well-known textbook solution for the current-free case (see Krause & Raedler 1980), in general, the β-dependence modifies the standard current-free solution.

In Fig. 1 the zeroes of Eq. (19)are displayed for β = 0.1 and R_out = 2R and various values of n, while in Figs. 2 and 3 the dependence of C_α on C_β is displayed for n = 1 (red) and n = 2 (black), while the blue line represents the n = 1 solution with R_out = 3R.

It is clear that C_α is a decreasing function of C_β, and this pattern is substantially unchanged for any value of n. Moreover, a positive C_β produces values of C_α that are lower than in the current-free case, while negative values of C_β lead to greater values of C_α. In other words, if the current in the corona is parallel to the field, the dynamo is easily excited, while if the current is antiparallel, the dynamo condition is more difficult to attain.

Explicit values for n = 1 are displayed in Table 1.

Fig. 1 Various zeroes of Eq. (19)are displayed for Rout = 2R, Cβ = 0.1 and n = 1 (black), n = 2 (red) and n = 3 (blue). The normalization is given by Anm = 1.

Fig. 2 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of Cβ for Rout = 2R. In blue we depict the n = 1 eigenvalue for Rout = 3R.

Fig. 3 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of Cβ for Rout = 2R for negative values of Cα. In blue we depict the n = 1 eigenvalue for Rout = 3R.

Fig. 4 Radial dependence of the magnetic helicity A·B for the n = 1 mode (solid line) for Cβ = 0.8 (black) and Cβ = 0.1 (red), and for the n = 2 mode (dashed lines) at θ = π/ 4. The magnetic helicity is normalized using Anm = 1.

The solution presented in this article in principle allows explicitly computing the magnetic helicity flux across the stellar boundary, which is non-zero for a non-vanishing C_β. The volume-integrated magnetic helicity A·B is maximum at the inner boundary, and it decays to zero at the outer boundary, as is shown in Figs. 4 and 5. This is not a coincidence: it is possible to show that our outer boundary condition (18)amounts to the condition of no helicity flux across the outer boundary.

2.3. Magnetic field components

We can provide an explicit expression for the magnetic field in the outer part of the domain.

Reality condition requires in Eq. (17). Therefore Eq. (11)reads

(20)

where for β> 0, and are arbitrary real coefficients, and

Moreover, are the normalized associated Legendre functions of degree n, order m, and argument μ = cosθ. In particular, C_α and C_β are not arbitrary, but are linked via the quantization condition given by Eq. (19).

Fig. 5 Global radial dependence of the angle averaged magnetic helicity ∫A·B dθdφ for the n = 1 mode (black) mode and for the n = 2 mode (red) for Cβ = 0.5. Note the presence of a small tail of non-zero helicity extending up to r = 1.5R. The magnetic helicity is normalized using Anm = 1.

We thus have

(23)

3. Conclusion

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.

Acknowledgments

We thank Axel Brandenburg for comments and useful discussions. F.D.S. acknowledges the Swedish Research Council International Postdoc fellowship for support, Giuseppe Jacopo Guidi for discussions, and the INAF Astrophysical observatory of Catania for the warm hospitality that allowed this work to be carried out.

References

All Tables

All Figures

	Fig. 1 Various zeroes of Eq. (19)are displayed for Rout = 2R, Cβ = 0.1 and n = 1 (black), n = 2 (red) and n = 3 (blue). The normalization is given by Anm = 1.
In the text

	Fig. 2 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of Cβ for Rout = 2R. In blue we depict the n = 1 eigenvalue for Rout = 3R.
In the text

	Fig. 3 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of Cβ for Rout = 2R for negative values of Cα. In blue we depict the n = 1 eigenvalue for Rout = 3R.
In the text

	Fig. 4 Radial dependence of the magnetic helicity A·B for the n = 1 mode (solid line) for Cβ = 0.8 (black) and Cβ = 0.1 (red), and for the n = 2 mode (dashed lines) at θ = π/ 4. The magnetic helicity is normalized using Anm = 1.
In the text

	Fig. 5 Global radial dependence of the angle averaged magnetic helicity ∫A·B dθdφ for the n = 1 mode (black) mode and for the n = 2 mode (red) for Cβ = 0.5. Note the presence of a small tail of non-zero helicity extending up to r = 1.5R. The magnetic helicity is normalized using Anm = 1.
In the text