2.2 The dynamo problem

The first term on right hand side of Equation (1

) represents the inductive action of the flow field, and thus can act as a source term for B; the second term, on the other hand, describes the resistive dissipation of the current systems supporting the magnetic field, and is thus always a global sink for B. The relative importances of these two terms is measured by the magnetic Reynolds number $Rm = uL/j$ , obtained by dimensional analysis of Equation (1

). Here $j$ , $u$ , and $L$ are “typical” numerical values for the magnetic diffusivity, flow speed, and length scale over which B varies significantly. The latter, in particular, is not easy to estimate a priori, as even laminar MHD flows have a nasty habit of generating their own magnetic length scales (usually $oc Rm -1/2$ at high $Rm$ ). Nonetheless, on length scales comparable to the large-scale solar magnetic field, $Rm$ is immense, and so is the usual viscous Reynolds number. This implies that energy dissipation will occur on length scales very much smaller than the solar radius.

The dynamo problem consists in finding/producing a (dynamically consistent) flow field u that has inductive properties capable of sustaining B against Ohmic dissipation. Ultimately, the amplification of B occurs by stretching of the pre-existing magnetic field. This is readily seen upon rewriting the inductive term in Equation (1 ) as

$\~/ × (u × B) = (B . \~/ )u - (u . \~/ )B - B(\ ~/ .u). (3)$

In itself, the first term on the right hand side of this expression can obviously lead to exponential amplification of the magnetic field, at a rate proportional to the local velocity gradient.

In the solar cycle context, the dynamo problem is reformulated towards identifying the circumstances under which the flow fields observed and/or inferred in the Sun can sustain the cyclic regeneration of the magnetic field associated with the observed solar cycle. This involves more than merely sustaining the field. A model of the solar dynamo should also reproduce

cyclic polarity reversals with a $~ 10 yr$ half-period,
equatorward migration of the sunspot-generating deep toroidal field and its inferred strength,
poleward migration of the diffuse surface field,
observed phase lag between poloidal and toroidal components,
polar field strength,
observed antisymmetric parity,
predominantly negative (positive) magnetic helicity in the Northern (Southern) solar hemisphere.

At the next level of “sophistication”, a solar dynamo model should also be able to exhibit amplitude fluctuations, and reproduce (at least qualitatively) the many empirical correlations found in the sunspot record. These include an anticorrelation between cycle duration and amplitude (Waldmeier Rule), alternance of higher-than-average and lower-than-average cycle amplitude (Gnevyshev-Ohl Rule), good phase locking, and occasional epochs of suppressed amplitude over many cycles (the so-called Grand Minima, of which the Maunder Minimum has become the archetype; more on this in Section 5 below). One should finally add to the list torsional oscillations in the convective envelope, with proper amplitude and phasing with respect to the magnetic cycle. This is a very tall order by any standard.

Because of the great disparity of time- and length scales involved, and the fact that the outer 30% in radius of the Sun are the seat of vigorous, thermally-driven turbulent convective fluid motions, the solar dynamo problem is rarely tackled as a direct numerical simulation of the full set of MHD equations (but do see Section 4.9 below). Most solar dynamo modelling work has thus relied on simplification - usually drastic - of the MHD equations, as well as assumptions on the structure of the Sun’s magnetic field and internal flows.