6.2 What limits the amplitude of the solar magnetic field?

The amplitude of the dynamo-generated magnetic field is almost certainly restricted by the backreaction of Lorentz forces on the driving fluid motions. However, as outlined in Section 5.3.1, this backreaction can occur in many ways.

Helioseismology has revealed only small variations of the differential rotation profile in the course of the solar cycle. The observed variations amount primarily to an extension in depth of the pattern of low-amplitude torsional oscillations long known from surface Doppler measurements (but see also Basu and Antia, 2001 ; Toomre et al., 2003 ). Taken at face value, these results suggest that quenching of differential rotation is not the primary amplitude-limiting mechanism, unless the dynamo is operating very close to criticality. Once again the hope is that in the not-too-distant future, helioseismology will have mapped accurately enough cycle-induced variations of differential rotation in the convective envelope and tachocline, to settle this issue. It is also worth noting that mechanisms for generating torsional oscillations that are not directly relying on the Lorentz force are also possible (see Spruit, 2003 ).

Algebraic quenching of the $a$ -effect (or $a$ -effect-like source terms) is the mechanism most often incorporated in dynamo models. However, this state of affairs usually has much more to do with computational convenience than commitment to a specific physical quenching mechanism. There is little doubt that the $a$ -effect will be affected once the mean magnetic field reaches equipartition; the critical question is whether it becomes quenched long before that, for example by the small-scale component of the magnetic field. The issue hinges on helicity conservation and subtleties of flow-field interaction in MHD turbulence, and remains open at this writing. For recent entry points into this very active area of current research, see Cattaneo and Hughes (1996 ); Blackman and Field (2000 ); Brandenburg and Dobler (2001 ).

Magnetic fields are buoyantly unstable in the convective envelope, and so will rise to the surface on time scales much shorter than the cycle period (see, e.g., Parker, 1975 ; Schüssler, 1977 ; Moreno-Insertis, 1983 , 1986 ). This is in fact the primary reason why most contemporary dynamo models of the solar cycle rely on the velocity shear in the tachocline to achieve toroidal field amplification. If the dynamo were to reside entirely in the convective envelope, then this would be an important, perhaps even dominant, amplitude limiting mechanism (see Schmitt and Schüssler, 1989 ; Moss et al., 1990 ). If, on the other hand, toroidal field amplification takes place primarily at or beneath the core-envelope interface, then it is less clear whether or not this mechanism plays a dominant role, but this remains to be confirmed by detailed calculations.

A related question is whether the destabilization, rise and surface emergence of sunspot-forming toroidal flux ropes amounts to magnetic flux loss; the answer depends on the fate of the emerged portion of the flux ropes, namely if and how it disconnects from the deep-seated magnetic flux system. Because some stretching can also take place in the course of the buoyant rise across the convective envelope, it may even be that rising flux ropes amplify the deep-seated magnetic field, as nicely demonstrated by the numerical calculations of Rempel and Schüssler (2001 ). Estimates of the total magnetic flux emerging at the surface of the Sun in the course of a solar cycle also amount to a small fraction of the amount of magnetic flux produced in the tachocline, for most current solar dynamo models. This suggests that flux loss due to buoyant rise of toroidal flux ropes is not the dominant amplitude-limiting mechanism.