Stellar Dynamos in the Transition Regime: Multiple Dynamo Modes and Antisolar Differential Rotation

The mean magnetic field is prevailingly symmetric about the equator (quadrupolar) and shows cyclic behavior with poleward-migrating $\overline{B}$_ϕ, and polarity reversals at mid to high latitudes (Figure 1(a)). Detailed inspection of the solution reveals the presence of a cyclic and a stationary constituent, the latter being 2–2.5 times stronger (in rms values) than the former. We interpret these as two different, coexisting dynamo modes, $\langle \overline{{\boldsymbol{B}}}{\rangle }_{t}$ and ${\overline{{\boldsymbol{B}}}}^{\mathrm{cyc}}=\overline{{\boldsymbol{B}}}-\langle \overline{{\boldsymbol{B}}}{\rangle }_{t}$, respectively; see Figures 1(b)–(d) for the toroidal component of $\overline{{\boldsymbol{B}}}$^cyc at two depths, along with its dependence on radius and time at latitude +50° where it is strongest in rms value. Its topology is similar throughout the convection zone, and the poleward migration is present at all depths.

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We start our analysis by investigating meridional circulation and DR as shown in Figures 2(a)–(c). The former has a dominant, large, counterclockwise cell, producing a relatively strong (20 m s⁻¹) poleward flow near the surface at almost all latitudes. There is a slow equatorward return flow widely distributed in the bulk of the convection zone at mid to high latitudes. In the slow rotation regime, antisolar DR is often accompanied by a single cell counterclockwise meridional circulation. In contrast, in the regime of fast rotation, solar-like DR drives multicellular meridional circulation aligned with the rotation axis. The cell pattern in this run represents a transitional state between these two regimes (e.g., Käpylä et al. 2014; Featherstone & Miesch 2015; Karak et al. 2015).

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The DR profile shows a decelerated equator and accelerated polar regions at the surface; hence, it is broadly speaking antisolar, despite some regions of weakly solar-like DR at the bottom of the CZ. The pole-equator difference at the surface is comparable to runs with similar rotational influence (e.g., Karak et al. 2015; Warnecke 2018). However, the energy in the DR compared to the total kinetic energy, neglecting the rigid rotation, is smaller than in runs with slightly slower and faster rotation (Viviani et al. 2018). This is most likely because our run is very close to the actual AS–S transition. In the DR profile, we find two distinct features: at midlatitudes, there is a local minimum of Ω, which has also been found in simulations with about three times faster rotation. In these, the resulting shear drives a dynamo wave obeying the Parker–Yoshimura rule (e.g., Warnecke et al. 2014). Furthermore, we find strong negative shear in a layer near the surface at low latitudes.

Next, we look at the turbulent transport coefficients, for which we have used a slightly different definition than in previous work (Schrinner et al. 2007; Warnecke et al. 2018), see Appendix A for motivation and details. We begin by discussing ${\boldsymbol{\alpha }}$ and ${\boldsymbol{\gamma }}$, and compare them with their counterparts from a more rapidly rotating dynamo run with solar-like DR of Warnecke et al. (2018) in terms of the ratio of their extremal values. Regarding ${\boldsymbol{\alpha }}$ (see Figure 3), both α_rr and α_ϕϕ are 20% smaller than in Warnecke et al. (2018), while the meridional profiles are similar. Furthermore, α_θθ is nearly 30% larger and shows an opposite sign near the surface close to the equator. The corresponding ratios for the off-diagonal components α_rθ, α_θϕ, and α_rϕ are 1.9, 1.1, and 0.7, respectively. Moreover, α_rθ and α_rϕ show opposite signs at the equator near the surface. We associate these differences from Warnecke et al. (2018) with the milder rotational influence on convection, characterized by the Coriolis number, being roughly three times smaller in our run. The usage of the new definition of the turbulent transport coefficients could also have caused some of these differences, but this influence was checked to be very small by re-computing the coefficients for Warnecke et al. (2018) using the new convention. A detailed comparison is shown in Table 1 of Appendix A.

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Concerning the turbulent pumping (see Figure 3), γ_r has a similar magnitude, γ_θ is 40% weaker and γ_ϕ is 40% stronger than in Warnecke et al. (2018). Here, too, the new definition has no significant effect. Note also the different normalization we used for ${\boldsymbol{\gamma }}$. γ_r is upward everywhere except in the bulk of the convection zone at mid and high latitudes. γ_θ is equatorward (poleward) in the upper (lower) half of the convection zone. γ_ϕ is prograde near the surface and at midlatitudes near the bottom, and negative everywhere else. The magnitudes of all three components are around 0.3 ${u}_{\mathrm{rms}}^{{\prime} }$, where ${u}_{\mathrm{rms}}^{{\prime} }(r,\theta )=\langle {\boldsymbol{u}}{{\prime} }^{2}{\rangle }_{\phi t}^{1/2}$ is the local turbulent rms velocity in the meridional plane. The effective mean velocity resulting from ${\boldsymbol{\gamma }}$ is shown by its time average in Figures 2(e)–(g). The radial component, ${\overline{U}}_{r}^{\mathrm{eff}}$, is completely dominated by γ_r, leaving nearly no trace of the actual flow. γ_θ changes the sign of $\overline{U}$_θ only slightly below the surface and reduces its magnitude by around 30%. However, the meridional flow cells are completely destroyed, as shown by the flow lines in Figure 2(f). γ_ϕ is accelerating the equator and decelerating the polar region. The larger change in Ω^eff compared to Warnecke et al. (2018) is because γ_ϕ increases with decreasing Ω₀.

The reconstruction of the turbulent EMF $\overline{{\boldsymbol{ \mathcal E }}}$ based on Equation (4) shows reasonable agreement with $\overline{{\boldsymbol{u}}^{\prime} \times {\boldsymbol{b}}^{\prime} }$, see Appendix C. Therefore, we can confidently use the set of turbulent transport coefficients to describe the dynamo processes in this run.

As a first step in determining the possible dynamo mechanism, we compare the period of the magnetic field cycle with theoretical expectations. We compute the magnetic cycle period by Fourier transforming $\overline{B}$_ϕ at r = 0.98R and then averaging the spectra over latitude. As a result, we get P_cyc = (3.2 ± 0.3) yr, where the error is obtained from the width at half maximum.

The two main dynamo scenarios both make predictions for the dynamo cycle length P_cyc. The Parker–Yoshimura dynamo period is locally defined as (Parker 1955; Yoshimura 1975)

$$\begin{eqnarray}&&{P}_{\mathrm{PY}}=2\pi {\left|\displaystyle \frac{{\alpha }_{\phi \phi }{k}_{\theta }}{2}r\cos \theta {\partial }_{r}{\rm{\Omega }}\right|}^{-1/2},\end{eqnarray} \tag{ 7 }$$

where k_θ = 2π/(rΔθ) is the latitudinal wavenumber of the dynamo wave with Δθ = π/2 − θ₀. The justification of using only α_ϕϕ in Equation (7) is that the other contributions to the poloidal field generation are smaller.

The cycle period of an advection-dominated dynamo is related to the travel time of the meridional circulation from the equator to the pole, τ_MC, such that P_MC ≈ 2 τ_MC (Küker et al. 2001, 2019). Hence, in our notations, the expected cycle period can be written as

$$\begin{eqnarray}&&{P}_{\mathrm{MC}}=\displaystyle \frac{2r{\rm{\Delta }}\theta }{{\overline{U}}_{\mathrm{MC}}(r,\theta )}\end{eqnarray} \tag{ 8 }$$

where $\overline{U}$_MC is the temporal rms⁵ of the meridional flow at the location of the dynamo wave. Traditionally, advection-dominated dynamo models assume the meridional flow and the resulting migration to be significant near the bottom of the convection zone, which would correspond to setting r = 0.7 R, but in the present case it is not so straightforward to determine the location of the dynamo wave.

We start by using the measured radial DR in Equation (7), and meridional flow in Equation (8), and obtain for the averages over the convection zone $\langle {P}_{\mathrm{PY}}{\rangle }_{r\theta }=2.2\,\mathrm{yr}$ and $\langle {P}_{\mathrm{MC}}{\rangle }_{r\theta }=8.2\,\mathrm{yr}$. Using the meridional circulation in the lower quarter of the convection zone only, we obtain, instead, $\langle {P}_{\mathrm{MC}}{\rangle }_{\delta r\theta }=63.8\,\mathrm{yr},$ where δr goes from 0.7R to R/4. Considering the relevant role of the turbulent pumping, especially in $\overline{U}$_r, we also calculated the periods using the effective velocity, that is adding the contributions of turbulent pumping to the measured large-scale flows, obtaining $\langle {P}_{\mathrm{PY}}^{\mathrm{eff}}{\rangle }_{r\theta }\,=2.0\,\mathrm{yr}$, $\langle {P}_{\mathrm{MC}}^{\mathrm{eff}}{\rangle }_{r\theta }=5.6\,\mathrm{yr}$, and $\langle {P}_{\mathrm{MC}}^{\mathrm{eff}}{\rangle }_{\delta r\theta }=22.0\,\mathrm{yr}.$ The Parker–Yoshimura periods are less affected than those from meridional circulation, as the ${\boldsymbol{\gamma }}$ contribution is more significant for the meridional circulation than for the DR. In conclusion, the Parker–Yoshimura periods are consistent with the measured magnetic cycle, while advection by meridional flow cannot explain it.

If the mean magnetic field was advected by the meridional flow or its effective counterpart, one would not be able to explain poleward migration virtually everywhere within the convection zone. This becomes evident from Figures 2(b) and (f), where equatorward flows are present. Whether the meridional circulation is able to overcome diffusion, can be assessed by help of the corresponding dynamo number (or turbulent magnetic Reynolds number)

$$\begin{eqnarray}&&{C}_{U}={\rm{\Delta }}r\,\langle {\overline{U}}_{\mathrm{MC}}{\rangle }_{r\theta }/\langle \mathrm{Tr}\left\{{\boldsymbol{\beta }}\right\}{\rangle }_{r\theta },\end{eqnarray} \tag{ 9 }$$

where $\mathrm{Tr}$ {·} indicates the trace. The time-averaged value for the measured mean and the effective mean flow are 0.2 and 0.6, respectively. Values below unity imply that the (effective) flow cannot overcome diffusion, not even with ${\boldsymbol{\gamma }}$ included; therefore, the advection-dominated dynamo scenario is not applicable here. However, the obtained values indicate that the meridional circulation may not be completely negligible in the magnetic evolution.

The prediction for the Parker–Yoshimura wave propagation direction given by (Yoshimura 1975)

$$\begin{eqnarray}&&{\boldsymbol{\xi }}(r,\theta )=-{\alpha }_{\phi \phi }{\hat{{\boldsymbol{e}}}}_{\phi }\times {\boldsymbol{\nabla }}{\rm{\Omega }},\end{eqnarray} \tag{ 10 }$$

is depicted in Figures 2(d) and (h) for the shear from Ω and Ω^eff, respectively. Near the bottom of the convection zone, where also the cyclic field is strongest, ${\boldsymbol{\xi }}$ is poleward at almost all latitudes, which would agree with the actual field propagation. In the bulk of the convection zone, however, the predicted direction is equatorward, failing to explain the actual migration. Hence, neither the Parker–Yoshimura dynamo wave nor the advection-dominated dynamo alone can be responsible for the oscillating poleward-migrating magnetic field throughout the convection zone.

To understand the failure of the simple dynamo scenarios in explaining cycles and migration of the field, we finally turn to computing the terms contributing to the magnetic field generation in detail. We present the contributions of the Ω and α effects, that is, of $\overline{{\boldsymbol{B}}}$ · ${\boldsymbol{\nabla }}$Ω and ${\boldsymbol{\nabla }}$ × (${\boldsymbol{\alpha }}$ · $\overline{{\boldsymbol{B}}}$), in terms of their temporal rms values in Figure 4 employing the total magnetic field (upper row), and show the corresponding temporal rms magnetic fields in the lower row. The two leftmost (rightmost) columns show the generators for the poloidal (toroidal) magnetic field. From the magnitudes of the toroidal generators, it is evident that the α effect is equally important, or even dominant over the Ω effect. Hence, the generation of the toroidal field by the α effect is more efficient than by the Ω effect, suggesting an α²Ω or even an α² dynamo mechanism for the observed dynamo.

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The Ω effect generates toroidal field efficiently at low latitudes near the surface and at midlatitudes in the bulk of the convection zone, coinciding with the surroundings of the local minimum of Ω. The α effect is strongest near the surface, but shows also toroidal field generation around the local minimum of Ω. The patches of strong rms toroidal field, however, overlap only partially with its generators, and its profile is clearly offset deeper into the convection zone. One reason might be the radial-field boundary condition, which suppresses any toroidal field near the surface. The α effect generates poloidal field mostly at high latitudes at all depths of the convection zone, although there are also regions of strong field generation close to the surface near the equator. The high-latitude field generator profiles match qualitatively better to the rms poloidal field distribution than to that of the toroidal field, but still the match is very incomplete.

The mismatch between the generators and the actual field distribution indicates that our conclusion of the generating mechanism being a simple α²Ω or α² dynamo is not a very solid one, and that other dynamo effects might be at play. For example, we find that the δ (Rädler) effect may also redound to the driving of the dynamo. Its contribution, shown in Figure 6 in Appendix A, is significant at midlatitudes near the surface for the poloidal field (panels a–b) and at the surface at all latitudes for the toroidal field (panel c). Particularly in the latter case, the contribution of ${\boldsymbol{\delta }}$ is strong in the same regions as the α and Ω effects and with roughly the same magnitude. However, this effect, in its simplest form in a shear flow, is known to lead to stationary solutions (Brandenburg & Subramanian 2005). Hence, its role for the oscillatory dynamo mode is likely to be negligible. How the δ effect contributes to the magnetic field generation needs to be analyzed in detail using mean-field simulations.

The study of Warnecke (2018) covers parameter regimes very close to the one explored here, but all of these solutions appear to exhibit only stationary or temporally irregular modes. This draws attention to the role of the wedge assumption used in that study. There, the computational domain covers only π/2 in azimuth, instead of the full 2π interval here, being virtually the only difference between these two studies. Our interpretation is that there are various dynamo modes excited with very similar critical dynamo numbers. In terms of dynamo theory, the coexistence of a steady and an oscillating field constituent can be understood as follows: sufficiently overcritical flows enable the growth of more than one dynamo mode. Under the assumption of steady mean flows and statistically stationary turbulence, the time dependence of these eigenmodes is exponential with an, in general, complex increment. It is well conceivable that a nonoscillating and an oscillating mode are both excited and even continue to coexist in their nonlinear stage, although their kinematic growth rates were different.

As mentioned in Schrinner et al. (2007) and Warnecke et al. (2018), there is some arbitrariness in deriving the transport coefficients (see Equation (4)) from the (noncovariant) tensors $\tilde{{\boldsymbol{a}}}$ and $\tilde{{\boldsymbol{b}}}$ defined by

$$\begin{eqnarray}&&{\overline{{ \mathcal E }}}_{\kappa }={\tilde{a}}_{\kappa \lambda }{\overline{B}}_{\lambda }+{\tilde{b}}_{\kappa \lambda r}{\partial }_{r}{\overline{B}}_{\lambda }+{\tilde{b}}_{\kappa \lambda \theta }{\partial }_{\theta }{\overline{B}}_{\lambda },\ \kappa ,\lambda =r,\theta ,\phi ,\end{eqnarray} \tag{ 11 }$$

which form the immediate outcome of the test-field method. Here, we specify a choice, different from the one employed earlier (see Schrinner et al. 2007; Warnecke et al. 2018; Warnecke 2018), and characterized by a maximum of vanishing components in ${\boldsymbol{\kappa }}$. As a consequence, the role of ${\boldsymbol{\kappa }}$ in the turbulent EMF $\overline{{\boldsymbol{ \mathcal E }}}$ is reduced, while mainly that of ${\boldsymbol{\beta }}$ is enhanced. This is motivated by the difficulty to interpret ${\boldsymbol{\kappa }}$ physically, whereas ${\boldsymbol{\beta }}$ clearly stands for turbulent dissipation. As a meaningful side effect, the diagonal elements of the latter become equal for isotropic turbulence. Furthermore, localized appearances of negative definite ${\boldsymbol{\beta }}$, which are destructive to mean-field modeling, become more visible as less of the diffusive contributions (ideally none) are "hidden" in ${\boldsymbol{\kappa }}$. Thus, removing the negative definiteness in the redefined ${\boldsymbol{\beta }}$ has better prospects to render the mean-field model feasible.

In Equation (11), the components ${\tilde{b}}_{\kappa \lambda \phi }$ do not appear as all ϕ derivatives vanish. They show up in the definitions of ${\boldsymbol{\alpha }}$, ${\boldsymbol{\beta }}$, etc. though, but setting them arbitrarily cannot change $\overline{{\boldsymbol{ \mathcal E }}}$. Here, we choose ${\tilde{b}}_{\kappa \lambda \phi }=-{\tilde{b}}_{\kappa \phi \lambda }$, in contrast to Schrinner et al. (2007) who set ${\tilde{b}}_{\kappa \lambda \phi }=0$. Then we arrive at the following expressions for the transport coefficients, where underlines indicate new or altered terms in comparison to Schrinner et al. (2007):

$$\begin{eqnarray}&&{\alpha }_{{rr}}={\tilde{a}}_{{rr}}-{\tilde{b}}_{r\theta \theta }/r\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\alpha }_{r\theta }={\alpha }_{\theta r}=\displaystyle \frac{1}{2}\left({\tilde{a}}_{r\theta }+{\tilde{a}}_{\theta r}+({\tilde{b}}_{{rr}\theta }-{\tilde{b}}_{\theta \theta \theta })/r\right)\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\alpha }_{r\phi }={\alpha }_{\phi r}=\displaystyle \frac{1}{2}\left({\tilde{a}}_{r\phi }+{\tilde{a}}_{\phi r}-(\underline{{\tilde{b}}_{r\phi r}+\cot \theta \,{\tilde{b}}_{r\phi \theta }}+{\tilde{b}}_{\phi \theta \theta })/r\right)\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\alpha }_{\theta \theta }={\tilde{a}}_{\theta \theta }+{\tilde{b}}_{\theta r\theta }/r\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\alpha }_{\theta \phi }={\alpha }_{\phi \theta }=\displaystyle \frac{1}{2}\left({\tilde{a}}_{\theta \phi }+{\tilde{a}}_{\phi \theta }-(\underline{{\tilde{b}}_{\theta \phi r}+\cot \theta \,{\tilde{b}}_{\theta \phi \theta }}-{\tilde{b}}_{\phi r\theta })/r\right)\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\alpha }_{\phi \phi }={\tilde{a}}_{\phi \phi }-(\underline{{\tilde{b}}_{\phi \phi r}+\cot \theta \,{\tilde{b}}_{\phi \phi \theta }})/r\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\gamma }_{r}=\displaystyle \frac{1}{2}\left({\tilde{a}}_{\phi \theta }-{\tilde{a}}_{\theta \phi }+(\underline{{\tilde{b}}_{\theta \phi r}+\cot \theta \,{\tilde{b}}_{\theta \phi \theta }}+{\tilde{b}}_{\phi r\theta })/r\right)\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\gamma }_{\theta }=\displaystyle \frac{1}{2}\left({\tilde{a}}_{r\phi }-{\tilde{a}}_{\phi r}+({\tilde{b}}_{\phi \theta \theta }-\underline{{\tilde{b}}_{r\phi r}-\cot \theta \,{\tilde{b}}_{r\phi \theta }})/r\right)\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\gamma }_{\phi }=\displaystyle \frac{1}{2}\left({\tilde{a}}_{\theta r}-{\tilde{a}}_{r\theta }-({\tilde{b}}_{{rr}\theta }+{\tilde{b}}_{\theta \theta \theta })/r\right)\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\beta }_{{rr}}=-\underline{1}\cdot {\tilde{b}}_{r\phi \theta }\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{\beta }_{r\theta }={\beta }_{\theta r}=\underline{\displaystyle \frac{1}{2}}({\tilde{b}}_{r\phi r}-{\tilde{b}}_{\theta \phi \theta })\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\beta }_{r\phi }={\beta }_{\phi r}=\displaystyle \frac{1}{4}(-\underline{2}{\tilde{b}}_{\phi \phi \theta }+{\tilde{b}}_{{rr}\theta }-{\tilde{b}}_{r\theta r})\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\beta }_{\theta \theta }=\underline{1}\cdot {\tilde{b}}_{\theta \phi r}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\beta }_{\theta \phi }={\beta }_{\phi \theta }=\displaystyle \frac{1}{4}(\underline{2}{\tilde{b}}_{\phi \phi r}+{\tilde{b}}_{\theta r\theta }-{\tilde{b}}_{\theta \theta r})\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\beta }_{\phi \phi }=\displaystyle \frac{1}{2}({\tilde{b}}_{\phi r\theta }-{\tilde{b}}_{\phi \theta r})\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\delta }_{r}=\displaystyle \frac{1}{4}({\tilde{b}}_{\theta \theta r}-{\tilde{b}}_{\theta r\theta }+\underline{2}{\tilde{b}}_{\phi \phi r})\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{\delta }_{\theta }=\displaystyle \frac{1}{4}({\tilde{b}}_{{rr}\theta }-{\tilde{b}}_{r\theta r}+\underline{2}{\tilde{b}}_{\phi \phi \theta })\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{\delta }_{\phi }=\underline{-\displaystyle \frac{1}{2}}({\tilde{b}}_{r\phi r}+{\tilde{b}}_{\theta \phi \theta })\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\kappa }_{{irr}}=-{\tilde{b}}_{{irr}}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{\kappa }_{{ir}\theta }={\kappa }_{i\theta r}=-\displaystyle \frac{1}{2}({\tilde{b}}_{{ir}\theta }+{\tilde{b}}_{i\theta r})\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{\kappa }_{{ir}\phi }={\kappa }_{i\phi r}=\underline{0}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{\kappa }_{i\theta \theta }=-{\tilde{b}}_{i\theta \theta }\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{\kappa }_{i\theta \phi }={\kappa }_{i\phi \theta }=\underline{0}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\kappa }_{i\phi \phi }=0.\end{eqnarray} \tag{ 35 }$$

The results from the new definition are shown in Figure 3 for ${\boldsymbol{\alpha }}$ and ${\boldsymbol{\gamma }}$ and in Figure 5 for the six independent components of ${\boldsymbol{\beta }}$, the vector ${\boldsymbol{\delta }}$ (first three columns), and for the nine independent nonzero components of ${\boldsymbol{\kappa }}$ (last three columns). ${\boldsymbol{\beta }}$, ${\boldsymbol{\delta }}$, and ${\boldsymbol{\kappa }}$ are normalized by η_t0 = ${u}_{\mathrm{rms}}^{{\prime} }$α_MLTH_p/3, where α_MLT = 5/3 is the mixing-length parameter and H_p = −1/∂_r ln p is the pressure scale height. The terms contributing to the magnetic field evolution from the δ (Rädler) effect, using the new definition, are shown in Figure 6.

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