4.1 Model ingredients

All kinematic 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).

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 Ω(r,𝜃) and magnetic diffusivity η (r ):

[ ( )] Ω-(r,𝜃) ΩS(𝜃)-−-ΩC-- r −-rc ΩE = ΩC + 2 1 + erf w , (15 )
with
2 4 ΩS (𝜃) = 1 − a2 cos 𝜃 − a4 cos 𝜃, (16 )
and
[ ( )] η(r) 1-−-Δ-η r −-rc ηT = Δ η + 2 1 + erf w . (17 )
With appropriately chosen parameter values, Equation (15View Equation) 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-latitudes3. This rotational transition takes place across a spherical shear layer of half-thickness w coinciding with the core-envelope interface at rc∕R⊙ = 0.7 (see Figure 5View Image, with parameter values listed in caption). As per Equation (17View Equation), a similar transition takes place with the net diffusivity, falling from some large, “turbulent” value ηT in the envelope to a much smaller diffusivity ηc in the convection-free radiative core, the diffusivity contrast being given by Δ η = ηc∕ηT. Given helioseismic constraints, these represent minimal yet reasonably realistic choices.

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 in panel B of Figure 5View Image, for a tachocline of half-thickness w ∕R ⊙ = 0.05, the mid-latitude latitudinal shear at r∕R ⊙ = 0.7 is comparable in magnitude to the equatorial radial shear; its potential contribution to dynamo action should not be casually dismissed.

View Image

Figure 5: Isocontours of angular velocity generated by Equation (15View Equation), with parameter values w ∕R = 0.05, ΩC = 0.8752, a2 = 0.1264, a4 = 0.1591 (Panel A). The radial shear changes sign at colatitude 𝜃 = 55 ∘. Panel B shows the corresponding angular velocity gradients, together with the total magnetic diffusivity profile defined by Equation (17View Equation) (dash-dotted line). The core-envelope interface is located at r∕R ⊙ = 0.7 (dotted lines).

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