Dynamo generated toroidal magnetic fields in rapidly rotating stars
D. Moss^{1}  D. Sokoloff^{2}
1  School of Mathematics, University of Manchester,
Oxford Road, Manchester, M13 9PL, UK
2 
Department of Physics, Moscow University, 119992 Moscow, Russia
Received 26 November 2008 / Accepted 16 February 2009
Abstract
Aims. Results presented in the recent observational paper by Petit et al. (2008, MNRAS, 388, 80) suggest that, for solarlike stars, the largescale surface toroidal fields become strong for rotational periods less than about 12 d. We discuss this observation in the context of stellar dynamo theory, as a manifestation of a bifurcation in dynamo regimes occurring around this rotation period.
Methods. Working in the context of mean field theory, we first consider two options for such a bifurcation for a given stellar rotation law: (a) a bifurcation resulting in a sudden increase of the nearsurface toroidal field, with all other details of the model being kept unaltered; (b) a new type of surface boundary condition at the stellar surface, forced by internal reorganization of magnetic field with increasing rotation rate, which causes a moreorless abrupt increase in largescale surface toroidal field.
Results. Neither of these options seem to provide anything like the reported behaviour of surface toroidal field for plausible choices of parameters, although we cannot conclusively eliminate a possible role for the latter mechanism. We conclude that any such bifurcation most plausibly is associated with some reorganization in the stellar hydrodynamics as the stellar rotation rate increases. The simplest suggestion, i.e. a transition from a solarlike rotation law (with spokelike angular velocity contours through the bulk of the convection zone) to a law with quasicylindrical isorotation contours as thought to be appropriate to rapidly rotating stars, seems to reproduce the observed phenomenology reasonably well.
Conclusions. While we appreciate the manifold uncertainties in the available theory and observations, we thus suggest that the bifurcation deduced by Petit et al. (2008) is an observational manifestation of the transition between solarlike and quasicylindrical rotation laws, occuring near a rotational period of 12 d.
Key words: Sun: activity  Sun: magnetic fields  stars: magnetic fields  magnetic fields
1 Introduction
Historically the solar activity cycle was identified from sunspot data. Contemporary interpretation of the cycle identifies it as a magnetic phenomenon which involves toroidal as well as poloidal magnetic field components, where sunspots are considered as tracers for the toroidal magnetic field. Correspondingly, confrontation between theoretical findings and solar phenomenology based on sunspot data can only illuminate the situation from a very limited viewpoint. Specific dynamo models for the solar cycle suggest various relations between toroidal and poloidal dynamo generated solar magnetic fields (for example, phase shift between the components) which seem on first sight to be very attractive for confrontation between theory and observations. In practice however the available tracers for toroidal and poloidal solar magnetic fields are so physically different that their confrontation is quite nontrivial, and comparison between theory and observations is usually based primarily on the sunspot data. As far as it is known, the description of the solar cycle determined from data concerning tracers of poloidal magnetic field appears to be substantially different than the conventional description based on sunspot data (e.g. Obridko et al. 2006): specifically, the activity wave identified by the surface mean magnetic field appears as a system of standing waves rather than the traveling wave identified from the sunspot data.
Because the physical nature of stellar activity cycles appears predominantly similar to that of the solar cycle, it is natural to consider the relation between toroidal and poloidal magnetic fields in a wider stellar perspective. The observational identifications of toroidal and poloidal stellar magnetic components are then, however, even more indirect than those for the Sun. Suitable observational results concerning the relation between toroidal and poloidal magnetic components are obviously desirable to clarify the general situation; the recent paper by Petit et al. (2008) seems to provide some such information. In a study of solarlike dwarfs, Petit et al. (2008) estimate that largescale surface toroidal fields become strong for rotational periods less than about d, i.e. for . They consider the phenomenon as a bifurcation between regimes for slowly rotating stars, where the surface magnetic field is presumably predominantly poloidal, and rapidly rotating stars where the surface toroidal component becomes substantial.
Our aim here is to interpret this observational finding in the context of stellar dynamo theory. Such an interpretation is far from straightforward.
We can envisage, and discuss below, three broad possibilities; that the conventional surface boundary condition of the magnetic field needs modification (see, e.g., Moss 1977, in a rather different context), that there is a bifurcation in dynamo properties as the rotation rate (and thus dynamo number) increases, due solely to a reorganization of the dynamo (e.g. Brandenburg et al. 1989; Jennings & Weiss 1991), or that the increase in surface toroidal field is a consequence of a change of the underlying internal hydrodynamics. These possibilities are discussed in more detail in Sects. 4.14.3 respectively. There is also the possibility that the observed surface toroidal fields might be related to coronal fields with a systematic orientation, but even so their appearance at periods of about 12 d still needs explanation, and plausibly could still be associated with stronger interior toroidal fields, as discussed here. Our general conclusion, perhaps more important than the discussion of any particular dynamo model, is that the bifurcation suggested by Petit et al. has the potential to be very informative when confronting stellar dynamo theories with stellar activity observations.
2 Background
Observations of solartype stars suggest that
(1) 
where is the observed variation of rotation rate over the surface (not extending above midlatitudes) and . Thus
where , and is the equatorial rotation rate.
At the same time a number of observational studies are indicating the presence of largescale toroidal fields at the surfaces of latetype stars, and Petit et al. (2008) in a study of solarlike stars suggest the existence of a ``threshold'' for their appearance around d. Conventional dynamo models usually take as a boundary condition at the stellar surface, matching the interior field to a vacuum external field (but see Covas et al. 2005). There appear to be three immediate possible theoretical explanations of the appearance of strong surface toroidal field. It is possible that the zero boundary condition on the surface toroidal field is inappropriate, and the interior field should be fitted to a more general external forcefree field  this possibility attracted attention in the 1970 s in the context of CP star modelling (e.g. Milsom & Wright 1976; Moss 1977). An alternative is that there is a bifurcation in the field structure as the rotation period decreases, leading to a stronger subsurface toroidal field which becomes unstable and erupts through the surface. A third possibility is that the underlying stellar rotation law changes from a quasisolar form, appropriate to slower rotators, to a quasicylindrical form at higher rotation rates, also leading to a sudden increase in subsurface toroidal field strength. We study all the above possibilities in the framework of a conventional meanfield dynamo model in a stellar convective shell. Note that we thus only discuss the largescale magnetic field.
Figure 1: Equally spaced isorotation contours for a) the solarlike rotation law; b) the quasicylindrical rotation law. See text for details. 

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3 Rotation laws
If we take standard
dynamo theory,
then
and
.
We can construct a series of ``solar type''
dynamo models, with the convection zone occupying approximately the
outer 30% by radius of the star, and
increasing from
.
Taking a ``solarlike'' surface rotation law of the
form
where is polar angle and ``solar'' values are a=0.126, b=0.159, (cf. Tassoul 1978) we construct a plausible extrapolation between the surface r=R and the base of the overshoot region at r=r_{0}R of the form
Here is taken as , and corresponds to the polar angle at which the surface angular velocity is that of the base of the tachocline, and f(r) is a function that goes smoothly from unity at r=r_{0}R to zero at . We take r_{0}=0.64, and . Isorotation contours are shown in Fig. 1a. Thus .
This value of at is used to determine for all with both n=0.4 and n=0, using (2). Although this apparently gives values that are too large compared with some observations we note (i) the Sun is the only really welldetermined case; (ii) nonsolar values are estimated from a limited range of latitudes, not extending close to the poles, and so probably underestimate . The stars which really seem to have very small values are very rapid rotators, and so outside of the range of periods of immediate interest. In any case, if we have overestimated , then we have done so uniformly across the range of rotation periods, and any differential effects we observe should be valid.
For more rapid rotation we can expect the rotation law to be quasicylindrical,
and so adopt as an empirical law (cf. Covas et al. 2005)
where is the equatorial angular velocity and, rather arbitrarily, d=0.05, . Then . Isorotation contours are shown in Fig. 1b.
4 Results
4.1 Modification of surface boundary condition
We first summarize the general theory of toroidal fields in very
low density domains in the context of largescale stellar
magnetic fields. This follows quite closely
the discussion in Raadu (1971), Milsom & Wright (1976), Moss (1977).
When modelling stellar magnetic fields it is necessary to ensure that the
field in the interior joins smoothly onto a forcefree field in the
external, very low density, region. Splitting the magnetic field into
poloidal and toroidal parts,
,
the Lorentz
force can be written as
Restricting consideration to axisymmetric fields, the last term is identically zero, the first two are poloidal vectors and the third is toroidal. In this case .
The condition
can be satisfied by setting
and
;
this provides the
usual boundary condition on the interior field at r=R.
Configurations with
and
are
also possible. Then a necessary condition is that
,
which is the torquefree condition. If
is the ``streamfunction'' for
the poloidal field (so
,
where
is the toroidal
component of the vector potential,
),
so that
on fieldlines of poloidal field),
then
implies
where F is an arbitrary function (e.g. Lüst & Schlüter 1954). For example, Raadu (1971) showed that if , then k>2 for satisfactory behaviour at large distances. Thus the boundary condition at r=R could be replaced by (7)  of course the vacuum condition on the poloidal field would also need amendment to give with . Moss (1977) constructed models with steady fields relevant to magnetic CP stars with such boundary conditions, by arbitrarily specifying and solving the equation explicitly in r>R.
Figure 2: The dependence of on fractional radius r/R_{*}, at an arbitrary latitude. 

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This sort of formalism appears to work reasonably well when investigating the time independent field structures of the CP stars, but encounters difficulties when considering oscillatory dynamo fields  there are difficult physical issues connected with the properties of a timevarying external toroidal field extending to large distances. Moreover, in the case being discussed, there is a direct link between toroidal and poloidal field components via the alpha term in the dynamo equation. For example, a dipolelike dynamo field (odd parity) has an even function with respect to the equator, whereas is odd (given that is antisymmetric). But then any plausible in (7) insists that be an even function, in contradiction to the assumption of an odd parity field structure. It is, of course, quite implausible that the very low density environs of a latetype star can alter significantly the gross structure of the internal magnetic field (``the tail does not wag the dog''). We conclude that the analysis is incomplete.
Putting all this on one side, the pragmatic question here is whether use of boundary condition such as (7) with the corresponding amendment to the poloidal field condition would change significantly the dependence of surface toroidal field on rotation rate. Very limited and approximate numerical experimentation (ignoring the modification to the poloidal boundary condition) did not support any strong effect, and a priori it is difficult to think of a strong physical argument that this would happen. Thus we did not pursue the matter further.
In summary, while it is clear that the surface boundary condition may be inappropriate, rigorous implementation of a more general boundary condition in the context of an oscillatory dynamo generated field requires resolution of difficult issues, which we have not pursued. Our intuition is that these issues may not be directly relevant to the problem addressed in this paper, but of course we cannot rule out such a possibility.
Figure 3: Dependence of measures of toroidal energy on rotation rate with fixed parity P=1 for n=0. Asterisks denote results with the solarlike rotation law (3), diamonds those with the quasicylindrical law (4). 

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4.2 Bifurcation due to more intensive dynamo action
We use the mean field dynamo model of Covas et al. (2005), implementing a naive alphaquenching nonlinearity rather than the feedback of the Lorentz force on the differential rotation as used there. We write , where is given in Fig. 2. The turbulent diffusivity is taken as a constant, , except in fractional radii , where it changes smoothly so that . We start with a ``solar'' model that is about 20% supercritical. Taking cm^{2} s^{1} gives the dynamo number . The marginal value of is then about 3.2, and so we take for this baseline model. Values of for larger values of are determined by scaling from these values. For each model we calculate the energies in the poloidal and toroidal components of the (axisymmetric) magnetic field in the entire dynamoactive region, the energies in the shell , and the energies in the immediately subsurface region, , respectively , , . Specifically calculations were performed for the solarlike rotation law for , and for the quasicylindrical rotation law with , for both n=0.4 and n=0.
Figure 4: Dependence of measures of toroidal energy on rotation rate with fixed parity P=1 for n=0.4. Notation is as in Fig. 3. 

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Figure 5: Dependence of measures of toroidal energy on rotation rate with free parity for n=0.4. Notation is as in Fig. 3. 

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When we examine results for a fixed rotation law (i.e. only diamonds or only asterisks in the figures), we see no evidence for a sudden bifurcation, e.g. to a branch with markedly higher energies, as the rotation speed increases. The only exception is in Fig. 3 (rotation law (3)) at . However the magnetic fields of the models with are steady, and so irrelevant to our investigation.
Table 1: Mean parity versus rotation rate for models with solarlike rotationlaw and . All models are singly periodic.
4.3 Transition from one rotation law to the other
If we now consider the results for the two rotation laws together, the graphs of for both n=0 and n=0.4 show a displacement by a factor of about 4 at , and there is a similar, smaller, discontinuity in the graphs for , see Figs. 3 and 4. The other measures of magnetic energy are almost continuous when switching from one rotation law to the other.
Thus Figs. 35 give a hint that the bifurcation under discussion can be associated with a transition from a solarlike rotation law with approximately radial contours through most of the convection zone to a quasicylindrical law. Of course in ``reality'' a gradual transition from a rotation law appropriate to slow rotators to a rapidrotator law could be expected. However, unless the transition is over a wide range of rotation speeds, the above treatment should be adequate to illustrate the behaviour of field energies as increases. In principle, it would be trivial to compose a synthetic rotation law that was solarlike for say and quasicylindrical for , where d, with a smooth transition between. Then our results presented in Sect. 4.2 would be valid outside of this transition range, and would suggest a significant jump in toroidal field, especially in the measures and , as the rotation law changes.
5 Discussion
Our results presented in Figs. 35 show that there is no evidence for marked bifurcations in magnetic energy for given hydrodynamic flows that do not change with rotation rate  i.e. the relations defined by the asterisks and diamonds separately do not show any sudden changes of slope (except for the two most rapidly rotating models of Fig. 3 with the solarlike rotation law, where there has been a bifurcation from oscillatory to (physically irrelevant) steady solutions).
On the other hand there is a distinct jump in the largescale toroidal energy between the relations for solarlike and quasicylindrical rotational laws. If we believe that these rotation laws are moreorless appropriate physically, then we have a possibility of identifying the origin of the observed appearance of largescale surface toroidal fields around d, namely that it is a result of a change in the internal azimuthal flows, as the TaylorProudman constraint becomes important at shorter rotational periods. We realise that our conclusions may be rather model dependent, but would stress in conclusion our belief that observations such as those of Petit et al. (2008) have considerable potential for illuminating and discriminating between various mechanisms for magnetic field generation.
Acknowledgements
D.S. is grateful to financial support from RFBR under grant 070200127a. We thank the referee, A. Brandenburg, for his constructive comments.
References
 Brandenburg, A., Krause, F., Meinel, R., Moss, D., & Tuominen, I. 1989, A&A, 213, 411 [NASA ADS] (In the text)
 Covas, E., Moss, D., & Tavakol, R. 2005, A&A, 429, 657 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
 Jennings, R. L., & Weiss, N. O. 1991, MNRAS, 252, 249 [NASA ADS] (In the text)
 Lüst, R., & Schlüter, A. 1954, Z. Astrophys., 34, 263 [NASA ADS] (In the text)
 Milsom, F., & Wright, G. A. 1976, MNRAS, 174, 307 [NASA ADS] (In the text)
 Moss, D. 1977, MNRAS, 178, 51 [NASA ADS] (In the text)
 Obridko, V. N., Sokoloff, D. D., Kuzanyan, K. M., Shelting, B. D., & Zakharov, V. G. 2006, MNRAS, 365, 827 [NASA ADS] [CrossRef] (In the text)
 Petit, P., Dintrans, B., Solanki, S. K., et al. 2008, MNRAS, 388, 80 [NASA ADS] [CrossRef] (In the text)
 Tassoul, J.L. 1978, Theory of Rotating Stars (Princeton, N.J.: Princeton Univ. Press) (In the text)
 Raadu, M. 1971, Ap&SS, 14, 464 [NASA ADS] [CrossRef] (In the text)
All Tables
Table 1: Mean parity versus rotation rate for models with solarlike rotationlaw and . All models are singly periodic.
All Figures
Figure 1: Equally spaced isorotation contours for a) the solarlike rotation law; b) the quasicylindrical rotation law. See text for details. 

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In the text 
Figure 2: The dependence of on fractional radius r/R_{*}, at an arbitrary latitude. 

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In the text 
Figure 3: Dependence of measures of toroidal energy on rotation rate with fixed parity P=1 for n=0. Asterisks denote results with the solarlike rotation law (3), diamonds those with the quasicylindrical law (4). 

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In the text 
Figure 4: Dependence of measures of toroidal energy on rotation rate with fixed parity P=1 for n=0.4. Notation is as in Fig. 3. 

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In the text 
Figure 5: Dependence of measures of toroidal energy on rotation rate with free parity for n=0.4. Notation is as in Fig. 3. 

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In the text 
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