4.1 Model ingredients

All solar dynamo models have some basic “ingredients” in common, most importantly (i) a solar structural model, (ii) a differential rotation profile, and (iii) a magnetic diffusivity profile (possibly depth-dependent). Meridional circulation in the convective envelope, long considered unimportant from the dynamo point of view, has gained popularity in recent years, initially in the Babcock-Leighton context but now also in other classes of models.

Helioseismology has pinned down with great accuracy the internal solar structure, including the internal differential rotation, and the exact location of the core-envelope interface. Unless noted otherwise, all illustrative models discussed in this section were computed using the following analytic formulae for the angular velocity $_O_(r,h)$ and magnetic diffusivity $j(r)$ :

with

and

With appropriately chosen parameter values, Equation (17

) describes a solar-like differential rotation profile, namely a purely latitudinal differential rotation in the convective envelope, with equatorial acceleration and smoothly matching a core rotating rigidly at the angular speed of the surface mid-latitudes⁴ . This rotational transition takes place across a spherical shear layer of half-thickness $w$ coinciding with the core-envelope interface at $r /R = 0.7 c o.$ (see Figure 5

, with parameter values listed in caption). As per Equation (19

), a similar transition takes place with the net diffusivity, falling from some large, “turbulent” value $jT$ in the envelope to a much smaller diffusivity $jc$ in the convection-free radiative core, the diffusivity contrast being given by $Dj = jc/jT$ . Given helioseismic constraints, these represent minimalistic yet reasonably realistic choices.

Figure 5:

Isocontours of angular velocity generated by Equation (17

), with parameter values $w/R = 0.05$ , $_O_C = 0.8752$ , $a2 = 0.1264$ , $a4 = 0.1591$ (Panel A). The radial shear changes sign at colatitude $h = 55o$ . Panel B shows the corresponding angular velocity gradients, together with the total magnetic diffusivity profile defined by Equation (19

) (dash-dotted line). The core-envelope interface is located at $r/Ro . = 0.7$ (dotted lines).

It should be noted already that such a solar-like differential rotation profile is quite complex from the point of view of dynamo modelling, in that it is characterized by three partially overlapping shear regions: a strong positive radial shear in the equatorial regions of the tachocline, an even stronger negative radial shear in its the polar regions, and a significant latitudinal shear throughout the convective envelope and extending partway into the tachocline. As shown on Panel B of Figure 5

, for a tachocline of half-thickness $w/Ro. = 0.05$ , the mid-latitude latitudinal shear at $r/Ro . = 0.7$ is comparable in magnitude to the equatorial radial shear; its potential contribution to dynamo action should not be casually dismissed.

Ultimately, the magnetic diffusivities and differential rotation in the convective envelope owe their existence to the turbulence therein, more specifically to the associated Reynolds stresses. While it has been customary in solar dynamo modelling to simply assume plausible functional forms for these quantities (such as Equations (17 , 18 , 19 ) above), one recent trend has been to calculate these quantities in an internally consistent manner using an actual model for the turbulence itself (see, e.g., Kitchatinov and Rüdiger, 1993 ). While this approach introduces additional - and often important - uncertainties at the level of the turbulence model, it represents in principle a tractable avenue out of the kinematic regime.