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5 Magnetic Flux Tubes and the Dynamo Problem

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5 Magnetic Flux Tubes and the Dynamo Problem 5 Magnetic flux tubes and the dynamo problem Manfred Schüssler and Antonio Ferriz-Mas The observed properties of the magnetic field in the solar photosphere and theoretical studies of magneto-convection in electrically well-conducting fluids suggest that the magnetic field in stellar convection zones is quite inhomogeneous: magnetic flux is concentrated into magnetic flux tubes embedded in significantly less magnetized plasma. Such a state of the magnetic field potentially has strong implications for stellar dynamo theory since the dynamics of an ensemble of flux tubes is rather different from that of a more uniform field and new phenomena like magnetic buoyancy appear. If the diameter of a magnetic flux tube is much smaller than any other relevant length scale, the MHD equations governing its evolution can be considerably simplified in terms of the thin- flux-tube approximation. Studies of thin flux tubes in comparison with observed properties of sunspot groups have led to far-reaching conclusions about the nature of the dynamo-generated magnetic field in the solar interior. The storage of magnetic flux for periods comparable to the amplification time of the dynamo requires the compensation of magnetic buoyancy by a stably stratified medium, a situation realized in a layer of overshooting convection at the bottom of the convection zone. Flux tubes stored in mechanical force equilibrium in this layer become unstable with respect to an undular instability once a critical field strength is exceeded, flux loops rise through the convection zone and erupt as bipolar magnetic regions at the surface. For parameter values relevant for the solar case, the critical field strength is of the order of 105 G. A field of similar strength is also required to prevent the rising unstable flux loops from being strongly deflected poleward by the action of the Coriolis force and also from ‘exploding’ in the middle of the convection zone. The latter process is caused by the superadiabatic stratification. The magnetic energy density of a field of 105 G is two orders of magnitude larger than the kinetic energy density of the convective motions in the lower solar convection zone. This raises serious doubts whether the conventional turbulent dynamo process based upon cyclonic convection can work on the basis of such a strong field. Moreover, it is unclear whether solar differential rotation is capable of generating a toroidal magnetic field of 105 G; it is conceivable that thermal processes like an entropy-driven outflow from exploded flux tubes leads to the large field strength required. The instability of magnetic flux tubes stored in the overshoot region suggests an alternative dynamo mechanism based upon growing helical waves propagating along the tubes. Since this process operates only for field strengths exceeding a critical value, such a dynamo can fall into a ‘grand minimum’ once the field strength is globally driven below this value, for instance by magnetic flux pumped at random from the convection zone into the dynamo region in the overshoot layer. The same process may act as a (re-)starter of the dynamo operation. Other non-conventional dynamo mechanisms based upon the dynamics of magnetic flux tubes are also conceivable. “chap05” — 2002/9/27 — 13:21 — page 123 — #1 124 M. Schüssler and A. Ferriz-Mas 5.1. Introduction The major part of the magnetic flux at the surface of the Sun is not distributed in a diffuse manner, but appears in discrete structures, namely magnetic flux tubes, of which sunspots are the most prominent manifestation (e.g. Zwaan, 1992). More than 90% of the mag- netic flux on the solar surface outside of sunspots is concentrated into intense flux tubes with a field strength of 1–2 kG and diameters between 500 and less than 100 km. These small magnetic elements are surrounded by plasma permeated by much weaker field. The filamentary state of the observed magnetic field is a consequence of the very large magnetic Reynolds number of the photospheric flows and the unstable (superadiabatic) stratification below the photospheric surface. The idea that the discrete appearance of the magnetic field extends further through the convection zone was already advanced at the beginning of the eighties; it is also supported by numerical simulations of magnetocon- vection (Galloway and Weiss, 1981; Parker, 1984; Nordlund et al., 1992; Proctor, 1992; Brandenburg et al., 1995). Simulations of the kinematics of magnetic fields in flows with chaotic streamlines also indicate the widespread formation of tube-like magnetic structures (e.g. Dorch, 2000). The concentration of magnetic flux into flux tubes has important consequences for its storage. Since a magnetic field gives rise to an effective magnetic pressure, a flux tube surrounded by almost field-free plasma easily becomes buoyant. Buoyancy can be a very efficient mechanism for removing magnetic flux and bringing it to the solar surface (Parker, 1955a, 1975). Simplified estimates as well as detailed numerical simulations show that flux tubes with a field strength of the order of equipartition with the energy density of convective motions (or larger) rise to the solar surface within one month or less (see, e.g. Moreno- Insertis, 1992, 1994, and references therein); this time scale is much shorter than the 11- year characteristic time for field generation and amplification by the dynamo process. Such rapid buoyant flux loss is suppressed if the toroidal magnetic flux is generated and stored within a subadiabatically stratified (i.e. convectively stable) layer of overshooting convection below the convection zone proper (see, e.g. Spiegel and Weiss, 1980; Galloway and Weiss, 1981; van Ballegooijen, 1982a,b, 1983; Schüssler, 1983; Schmitt et al., 1984; Tobias et al., 1998). The stable stratification impedes radial motion and allows flux tubes to find an equilibrium position with vanishing buoyancy (Moreno-Insertis et al., 1992; Ferriz-Mas, 1996). In the past decade, studies of the equilibrium, stability, and dynamics of magnetic flux tubes have led to the conclusion that the field strength of the dynamo-generated magnetic field stored at the bottom of the solar convection zone is of the order of 105 G, an order of magnitude larger than the (equipartition) field strength following from setting the magnetic energy density equal to the kinetic energy density of convection or differential rotation (e.g. Moreno-Insertis, 1992; Schüssler, 1996; Fisher et al., 2000). Apart from the obvious question concerning the origin of such a strong field, this results also raises doubts with regard to the traditional view of the ‘turbulent’ dynamo process. Such strong fields cannot be considered in the framework of a kinematical theory and the turbulence is probably strongly affected in the presence of a super-equipartition field. In this contribution we give an overview of our present understanding of the magnetic field dynamics in the solar convection zone and its implications for the dynamo theory of the large-scale solar magnetic field. We begin with a short description of the commonly used thin flux tube approximation (Section 5.2) and then discuss the flux tube dynamics in the convection zone in Section 5.3. This concerns the storage of magnetic flux (Section 5.3.1), “chap05” — 2002/9/27 — 13:21 — page 124 — #2 Magnetic flux tubes and the dynamo problem 125 the instability and rise of magnetic flux tubes (Section 5.3.2), as well as the origin of the strong super-equipartition toroidal field (Section 5.3.3). Section 5.4 is devoted to discussing a dynamo model based on magnetic flux tubes. We consider the α-effect resulting from the undular instability of a toroidal flux tube in a rotating star (Section 5.4.1) and describe the results of a simple nonlinear dynamo model based upon these ideas. A possible consequence of such a model is the appearance of extended periods with no strong magnetic field; such periods could be related to the ‘grand minima’ of solar activity such as the Maunder minimum in the seventeenth century. In Section 5.5 we summarize the main results. 5.2. Thin flux tubes The study of the structure and dynamics of magnetic fields in stars is a complex mathematical problem due to the nonlinear nature of the magnetohydrodynamic (MHD) equations, to the large hydrodynamic and magnetic Reynolds numbers leading to strong structuring of the magnetic field, and to the stratification of the plasma. On the other hand, a filamentary nature of magnetic fields in the convection zone allows for a considerable simplification of the MHD equations if the diameter of a flux tube is small compared to all other relevant length scales (scale heights, radius of curvature, wavelengths of MHD waves, etc.). The flux tube is idealized as a bundle of magnetic field lines, which is separated from its non-magnetic environment by a tangential discontinuity. It is assumed that instantaneous, lateral pressure balance is maintained between the flux tube and the exterior. Difussion of the magnetic field due to Ohmic resistance is neglected. The formal thin flux-tube approximation has been developed with different degrees of generality by Defouw (1976); Roberts and Webb (1978); Spruit (1981) and Ferriz-Mas and Schüssler (1989, 1993). This approximation permits the reduction of the full set of MHD equations to a mathematically more easily tractable form while retaining the full effects of the compressibility of the plasma, the magnetic Lorentz force and gravity. In physical terms, the approximation amounts to describing the dynamics of a magnetic flux tube as the motion of a quasi-one-dimensional continuum in a three- dimensional environment. A magnetic tube with a flux of 3 × 1021 Mx (corresponding to a medium-size sunspot) and a field strength of 105 G would have a diameter of approximately 2000 km.
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