List of Footnotes

1 Equation (2View Equation) is written here in a frame of reference rotating with angular velocity Ω, so that a Coriolis force term appears explicitly, while the centrifugal force has been subsumed into the gravitational term.
2 Note, however, that an axisymmetric flow can sustain a non-axisymmetric magnetic field against resistive decay.
3 Helioseismology has also revealed the existence of a significant radial shear in the outermost layers of the solar convective envelope. Even if the storage problem could be somehow bypassed, it does not appear possible to construct a viable solar dynamo model relying exclusively on this angular velocity gradient (see, e.g., Dikpati et al., 2002; Brandenburg, 2005, for illustrative calculations).
4 Models retaining both α-terms are dubbed 2 α Ω dynamos, and may be relevant to the solar case even in the C α ≪ CΩ regime, if the latter operates in a very thin layer, e.g. the tachocline (see, e.g., DeLuca and Gilman, 1988Jump To The Next Citation Point; Gilman et al., 1989; Choudhuri, 1990); this is because the α-effect gets curled in Equation (25View Equation) for the mean toroidal field. Models relying only on the α-terms are said to be 2 α dynamos. Such models are relevant to dynamo action in planetary cores and convective stars with vanishing differential rotation (if such a thing exists).
5 These are not “waves” in usual sense of the word, although they are described by modal solutions of the form exp(ik ⋅x− ωt).
6 Although some turbulence model predict such higher-order latitudinal dependencies, the functional forms adopted here are largely ad hoc, and are made for strictly illustrative purposes.
7 Mea culpa on this one...
8 For this particular choice of α, η, and Ω profiles, solutions with negative C α are non-oscillatory in most of the [Cα,CΩ,Δ η] parameter space. This is in agreement with the results of Markiel and Thomas (1999Jump To The Next Citation Point).
9 We largely exclude from the foregoing discussion mathematical toy-models that aim exclusively at reproducing the shape of the sunspot number time series. For recent entry points in this literature, see, e.g., Mininni et al. (2002).
10 Dynamo saturation can also occur by magnetically-mediated changes in the “topological” properties of a turbulent flow, without significant decrease in the turbulent flow amplitudes; see Cattaneo et al. (1996) for a nice, simple example.
11 This effect has been found to be the dominant dynamo quenching mechanism in some numerical simulations of dynamo action in a rotating, thermally-driven turbulent spherical shell (see, e.g., Gilman, 1983), as well as in models confined to thin shells (DeLuca and Gilman, 1988).