Analytic meanfield α^{2}dynamo with a forcefree corona
^{1} INAFOsservatorio Astrofisico di Catania, 95125 Catania, Italy
email: alfio.bonanno@oact.inaf.it
^{2} Department of Geology & Geophysics, Yale University, New Haven, CT 06511, USA
^{3} Nordita, Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden
email: fabio@nordita.org
Received: 7 June 2017
Accepted: 24 July 2017
Context. Stellar dynamos are affected by boundary conditions imposed by stellar coronae. Under some approximations, it is possible to find analytical solutions. Interior dynamo models often consider a currentfree corona without taking into account the constraints imposed by the presence of currents in the corona.
Aims. We aim to analytically evaluate the effect of coronal currents and of an outer boundary condition on the efficiency of an α^{2} dynamo. We intend to estimate the change in geometry and dynamo excitation numbers with respect to the currentfree case.
Methods. We analytically solved the turbulent dynamo induction equation for a homogeneous, nonmirror symmetric turbulence in a spherical domain surrounded by a linear forcefree corona with the mean magnetic field B satisfying ∇ × B = βB.
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 forcefree case. In contrast, for (β < 0), the dynamo number is greater than in the forcefree case.
Conclusions. Currents in the corona need to be taken into account because they affect the condition for excitation of a dynamo.
Key words: dynamo / stars: coronae / stars: magnetic field
© ESO, 2017
1. Introduction
A consistent description of the magnetic coupling between a magnetized coronal field and a dynamogenerated interior model is still lacking. Meanfield models usually employ a currentfree 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 forcefree 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 forcefree 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 meanfield dynamo theory, a new proposal has been presented in Bonanno (2016), where a consistent coupling with an α^{2}Ω 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 dynamogenerated interior field coupled with a forcefree exterior. The aim of this paper is to present such a solution for a linear forcefree field. Although our solution has been obtained for a very idealized case (nonhelical 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 ZeemanDoppler 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 meanfield dynamo equation.
In the case of homogeneous, isotropic, and nonmirrorsymmetric 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 longterm 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 forcefree 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 forcefree 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 forcefree 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 nonzero, which was instead a necessary condition in the currentfree case.
While in the currentfree 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:
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 wellknown textbook solution for the currentfree case (see Krause & Raedler 1980), in general, the βdependence modifies the standard currentfree 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 currentfree 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 R_{out} = 2R, C_{β} = 0.1 and n = 1 (black), n = 2 (red) and n = 3 (blue). The normalization is given by A_{nm} = 1. 

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Fig. 2 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of C_{β} for R_{out} = 2R. In blue we depict the n = 1 eigenvalue for R_{out} = 3R. 

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Fig. 3 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of C_{β} for R_{out} = 2R for negative values of C_{α}. In blue we depict the n = 1 eigenvalue for R_{out} = 3R. 

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

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The solution presented in this article in principle allows explicitly computing the magnetic helicity flux across the stellar boundary, which is nonzero for a nonvanishing C_{β}. The volumeintegrated 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
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).
First four positive and negative eigenvalues for n = 1 for C_{α} for various values of C_{β}.
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 C_{β} = 0.5. Note the presence of a small tail of nonzero helicity extending up to r = 1.5R. The magnetic helicity is normalized using A_{nm} = 1. 

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We thus have
3. Conclusion
The analytical linear forcefree solution presented in this paper has been obtained by coupling a corona with a dynamogenerated 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 forcefree 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 nonzero 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 nonstationary 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.
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All Tables
First four positive and negative eigenvalues for n = 1 for C_{α} for various values of C_{β}.
All Figures
Fig. 1 Various zeroes of Eq. (19)are displayed for R_{out} = 2R, C_{β} = 0.1 and n = 1 (black), n = 2 (red) and n = 3 (blue). The normalization is given by A_{nm} = 1. 

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In the text 
Fig. 2 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of C_{β} for R_{out} = 2R. In blue we depict the n = 1 eigenvalue for R_{out} = 3R. 

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In the text 
Fig. 3 Eigenvalues for the n = 2 (black) and for the n = 1 mode (red) as a function of C_{β} for R_{out} = 2R for negative values of C_{α}. In blue we depict the n = 1 eigenvalue for R_{out} = 3R. 

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

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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 C_{β} = 0.5. Note the presence of a small tail of nonzero helicity extending up to r = 1.5R. The magnetic helicity is normalized using A_{nm} = 1. 

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In the text 