Powering Stellar Magnetism: Energy Transfers in Cyclic Dynamos of Sun-like Stars

We perform 3D MHD simulations of convective dynamo action coupled to a stable radiative interior where the anelastic MHD equations are solved for the motions of a conductive plasma in a rotating sphere (Jones et al. 2011). The anelastic approximation captures the effects of density stratification without having to resolve sound waves, which would severely limit the time step (Brown et al. 2012). In the MHD context, the anelastic approximation filters out fast magnetoacoustic waves but retains Alfvén waves.

The code ASH uses a pseudo-spectral method (Clune et al. 1999). The velocity ( v ), magnetic ( B ), and thermodynamic variables (entropy S, pressure P) are expanded in spherical harmonics Y_{ℓ m} (θ, ϕ) for their horizontal structure and in Chebyshev polynomials T_n (r) for their radial structure (Brun et al. 2004). The density (ρ), entropy, pressure, and temperature (T) are linearized about the spherically symmetric background values, denoted by the symbol (ˆ). The equations solved by ASH are (Brun et al. 2004)

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\cdot \hat{\rho }{\boldsymbol{v}}=0,{\boldsymbol{\nabla }}\cdot {\boldsymbol{B}}=0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\begin{array}{l}\hat{\rho }\left(\displaystyle \frac{\partial {\boldsymbol{v}}}{\partial t}+({\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}){\boldsymbol{v}}+2{{\boldsymbol{\Omega }}}_{* }\times {\boldsymbol{v}}\right)\\ \ \ \ =\,-{\boldsymbol{\nabla }}P+\rho {\boldsymbol{g}}+\displaystyle \frac{1}{4\pi }({\boldsymbol{\nabla }}\times {\boldsymbol{B}})\times {\boldsymbol{B}}+{\boldsymbol{\nabla }}\cdot {\boldsymbol{ \mathcal D }}\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\hat{\rho }\hat{T}\displaystyle \frac{\partial S}{\partial t}=-\hat{\rho }\hat{T}{\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}(\hat{S}+S)-{\boldsymbol{\nabla }}\cdot {\boldsymbol{q}}+{{\rm{\Phi }}}_{d}+\hat{\rho }\epsilon \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\boldsymbol{\nabla }}\times [{\boldsymbol{v}}\times {\boldsymbol{B}}-\eta {\boldsymbol{\nabla }}\times {\boldsymbol{B}}],\end{eqnarray} \tag{ 4 }$$

with the velocity field ${\boldsymbol{v}}={v}_{r}{\hat{{\boldsymbol{e}}}}_{r}+{v}_{\theta }{\hat{{\boldsymbol{e}}}}_{\theta }+{v}_{\varphi }{\hat{{\boldsymbol{e}}}}_{\varphi }$, the magnetic field ${\boldsymbol{B}}={B}_{r}{\hat{{\boldsymbol{e}}}}_{r}+{B}_{\theta }{\hat{{\boldsymbol{e}}}}_{\theta }+{B}_{\varphi }{\hat{{\boldsymbol{e}}}}_{\varphi }$, the angular velocity in the rotation frame ${{\boldsymbol{\Omega }}}_{* }={{\rm{\Omega }}}_{* }{\hat{{\boldsymbol{e}}}}_{z}$, ${\hat{{\boldsymbol{e}}}}_{z}$ the unit vector along the rotation axis, and g the magnitude of the gravitational acceleration. A volumetric heating term $\hat{\rho }\epsilon$ is also taken into account to approximate the generation of energy by nuclear reactions in the stellar core. The nuclear reactions are modeled very simply by assuming that $\epsilon ={\epsilon }_{0}{\hat{T}}^{{n}_{c}}$. By enforcing that the integrated luminosity of the star matches its known surface value, we can determine ₀ and n_c as listed in Table 7 of Brun et al. (2017). Note that only the low-mass star series of models (e.g., 0.5 and 0.7 M_⊙) require that heating source term, because their computational domain includes a portion of the nuclear energy generation core.

The diffusion tensor D and the dissipative term Φ_d are defined as

$$\begin{eqnarray*}&&{D}_{{ij}}=2\hat{\rho }\nu \left[{e}_{{ij}}-\displaystyle \frac{1}{3}{\boldsymbol{\nabla }}\cdot {\boldsymbol{v}}{\delta }_{{ij}}\right],\end{eqnarray*}$$

$$\begin{eqnarray*}&&{{\rm{\Phi }}}_{d}=2\hat{\rho }\nu \left[{e}_{{ij}}{e}_{{ij}}-\displaystyle \frac{1}{3}{\left({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}}\right)}^{2}\right]+\displaystyle \frac{4\pi \eta }{{c}^{2}}{{\boldsymbol{J}}}^{2},\end{eqnarray*}$$

with e_ij the stress tensor and J = c/4π ∇ × B the current density. The energy flux q is the sum of a radiation flux and of a turbulent entropy diffusion flux:

$$\begin{eqnarray*}&&{\boldsymbol{q}}={\kappa }_{r}\hat{\rho }{c}_{p}{\boldsymbol{\nabla }}(\hat{T}+T)+\kappa \hat{\rho }\hat{T}{\boldsymbol{\nabla }}S+{\kappa }_{0}\hat{\rho }\hat{T}\displaystyle \frac{\partial \hat{S}}{\partial r}{\hat{{\boldsymbol{e}}}}_{r},\end{eqnarray*}$$

with ν, κ, and η the effective eddy diffusivities of the momentum, heat, and magnetic field transport, κ_r the atomic radiation diffusion coefficient, κ₀ the effective thermal diffusivity acting only on the spherically symmetric (l = 0) entropy gradient, and c_p the specific heat at constant pressure.

Due to limitations in computing resources, current numerical simulations cannot capture all scales of solar convective motions and magnetic fields from global to atomic dissipation scales. The simulations described in this study resolve nonlinear interactions among a large range of scales but motions and magnetic fields still exist in solar-like stars on scales smaller than our grid resolution. Hence, our models should be considered as large-eddy simulations (LES) with parameterization to account for subgrid-scale (SGS) motions. The effective eddy diffusivities ν, κ, and η represent momentum, heat, and magnetic field transport by motions that are not resolved by the simulation. They are allowed to vary with radius but are independent of latitude, longitude, and time for a given simulation. In the simulations reported here, ν, κ, and η have the following profile:

$$\begin{eqnarray*}&&\nu (r)={\nu }_{\mathrm{bot}}+{\nu }_{\mathrm{top}}{f}_{\mathrm{step}}(r),\end{eqnarray*}$$

where

$$\begin{eqnarray*}\begin{array}{rcl}{f}_{\mathrm{step}}(r) & = & {\left(\hat{\rho }/{\hat{\rho }}_{\mathrm{top}}\right)}^{\alpha }[1-\beta ]f(r),\\ f(r) & = & 0.5(\tanh ((r-{r}_{t})/{\sigma }_{t})+1),\\ \beta & = & {\nu }_{\mathrm{bot}}/{\nu }_{\mathrm{top}}={10}^{-3},\end{array}\end{eqnarray*}$$

and with ν, ν_bot and ν_top in cm² s⁻¹ and r_t and σ_t in cm, as provided in Table 7 of Brun et al. (2017), and α is −0.5 for all cases. All models assumed a Prandtl number Pr = ν/κ of 0.25, so that κ can be directly obtained from the amplitude and profile of ν. The magnetic Prandtl number P_m = ν/η is equal to 1 or 2 depending on the case considered (see Table 3), so that η can as well be deduced from ν. These tapered profiles are chosen in order to take into account the much smaller subgrid-scale transport expected in the stably stratified radiative interior. A representative profile is shown in Figure 1. Their amplitudes are adapted for each rotation rate and stellar mass considered in order to achieve the best turbulent convective dynamo states while retaining a reasonable numerical resolution and computing effort (still, each model has used of the order of 8 to 10 million CPU hours spread over several years).

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The diffusivity κ₀ is set such as to have the unresolved eddy flux carrying the solar flux outward at the top of the domain (see Figure 2). It drops off exponentially with depth in order to avoid a large inward heat flux in the stable zone (see Miesch et al. 2000). Of course, there is some arbitrariness in choosing the exact shape and amplitude of our diffusivity profiles, and we do our best to limit their influence on the results reported here.

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The mass flux and magnetic vector fields are maintained divergenceless by a stream function formalism (Brun et al. 2004):

$$\begin{eqnarray}&&\hat{\rho }{\boldsymbol{v}}={\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times (W{\hat{{\boldsymbol{e}}}}_{r})+{\boldsymbol{\nabla }}\times (Z{\hat{{\boldsymbol{e}}}}_{r}),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\boldsymbol{B}}={\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times (C{\hat{{\boldsymbol{e}}}}_{r})+{\boldsymbol{\nabla }}\times (A{\hat{{\boldsymbol{e}}}}_{r}).\end{eqnarray} \tag{ 6 }$$

A perfect ideal gas equation is used for the mean state and the fluctuations are linearized:

$$\begin{eqnarray*}&&\hat{P}=(\gamma -1){c}_{p}\hat{\rho }\hat{T}/\gamma \end{eqnarray*}$$

$$\begin{eqnarray*}&&\rho /\hat{\rho }=P/\hat{P}-T/\hat{T}=P/\gamma \hat{P}-S/{c}_{p}\end{eqnarray*}$$

with γ = 5/3 the adiabatic exponent.

The anelastic MHD system of equations requires 12 boundary conditions (BCs). We use an impenetrable and stress-free BCs at the top and bottom of the domain, i.e.,

$$\begin{eqnarray*}&&{v}_{r}=\displaystyle \frac{\partial }{\partial r}\left(\displaystyle \frac{{v}_{\theta }}{r}\right)=\displaystyle \frac{\partial }{\partial r}\left(\displaystyle \frac{{v}_{\phi }}{r}\right)=0.\end{eqnarray*}$$

Magnetic BCs are perfectly conducting at the lower radial boundary and the magnetic field matches a potential field in the upper boundary: ${B}_{r}{| }_{{r}_{\mathrm{bot}}}=\tfrac{\partial }{\partial r}\left(\tfrac{{B}_{\theta }}{r}\right){| }_{{r}_{\mathrm{bot}}}=\tfrac{\partial }{\partial r}\left(\tfrac{{B}_{\varphi }}{r}\right){| }_{{r}_{\mathrm{bot}}}=0$ and ${B}_{r}{| }_{{r}_{\mathrm{top}}}={\rm{\nabla }}{\rm{\Psi }}\Rightarrow {\rm{\Delta }}{\rm{\Psi }}=0$, with r_top and r_bot, respectively, the radius of the top and bottom of the numerical domain and r_bcz that of the base of the convective layer (see Table 1).

Table 1. Global Properties on the Main Sequence of the Four Stars Used in Our ASH Dynamo Models

Mass	Radius	L*	Teff	Sp. T.	Mbcz	rbcz	${\hat{T}}_{\mathrm{bcz}}$	${\hat{\rho }}_{\mathrm{bcz}}$	${{\rm{\Delta }}}_{\mathrm{cz}}\hat{\rho }$	${{\rm{\Delta }}}_{f}\hat{\rho }$	rbot	rtop
(M⊙)	(R⊙)	(L⊙)	(K)		(M⊙, M*)	(R⊙, R*)	(K)	(g cm−3)	⋯	⋯	(R*)	(R*)
0.5	0.44	0.046	4030	K7	0.18, 0.36	0.25,0.56	4.3 × 106	14.0	42	193	0.13	0.95
0.7	0.64	0.15	4500	K4/K5	0.079, 0.11	0.42,0.66	3.0 × 106	2.1	50	605	0.32	0.97
0.9	0.85	0.55	5390	G8	0.042, 0.046	0.59,0.69	2.6 × 106	0.51	67	1013	0.38	0.97
1.1	1.23	1.79	6030	G0	0.011, 0.010	0.92,0.75	1.6 × 106	0.048	81	830	0.5	0.97

Note. All the listed values were computed with the CESAM stellar evolution code (Morel 1997). We adopt M_⊙ = 1.989 × 10³³ g, R_⊙ = 6.9599 × 10¹⁰ cm, and L_⊙ = 3.846 × 10³³ erg s⁻¹. The density ratios ${{\rm{\Delta }}}_{\mathrm{cz}}\hat{\rho }$ and ${{\rm{\Delta }}}_{f}\hat{\rho }$ are evaluated by forming the ratio between the value of the density, respectively, at the base of the convection and the top of the domain and at the bottom and the top of the domain.

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Finally, we maintain the entropy flux at the top and bottom. Keeping the values of $d\hat{S}/{dr}{| }_{{r}_{\mathrm{top}},{r}_{\mathrm{bot}}}$ fixed at all times in the simulations further implies that the fluctuating dS/dr is set to zero at both BCs.

The simulation is focused on the bulk convection zone, avoiding regions too close to the stellar surface. We include a stably stratified layer below the convective envelope, hence providing a realistic bottom boundary condition for the fields and flow that are allowed to be pumped down and to penetrate into the radiative interior. The code uses a realistic background stratification for the profiles of entropy ($\hat{S}$), density ($\hat{\rho }$), temperature ($\hat{T}$) derived from a one-dimensional solar-structure model CESAM (Morel 1997; Brun et al. 2002). Our starting point is the G- and K-star rotating convective 3D models published in Brun et al. (2017) (see also Matt et al. 2011; Brun et al. 2015 and Table 1).

The MHD models are initialized from their equivalent progenitor hydrodynamical models in which a small magnetic field perturbation is introduced in the convective envelope (many orders smaller than the final magnetic field observed in the simulation). In that hydrodynamic study we published 15 simulations covering four mass bins and four rotation rates. We have models for stellar masses 0.5, 0.7, 0.9, and 1.1 M_⊙ and rotation rates ranging from 1/8 to 5 times the solar rotation rates. In keeping with the naming nomenclature of Brun et al. (2017), we name our model such as to indicate the mass of the star and its rotation rate. The models are named MAxrm, where "A" is the mass of the star and "r" the rotation rate of the star (in solar rotation rate). The index "x" indicates slow/antisolar (x = s) and prograde (x = R) DR models (except model M11R1m, which is also antisolar) and "m" stands for magnetism, to distinguish between the hydrodynamic progenitor published in 2017 and their MHD dynamo counterparts considered in this study.

The models have a numerical resolution of (N_r , N_θ , N_ϕ ) 769 × 256 × 512 except for few cases in the M09m and M11m series rotating at Ω_* = 3 or 5Ω_⊙, where N_θ is 512 and N_ϕ is 1024.

In Figure 2 we show in (a) an example of the radial dependence of the entropy gradient, and in (b) the temporal and azimuthally averaged radial energy flux balance as luminosities for the model with 1.1 solar mass and 3 times the rotation rate of the Sun. We note the sharp increase of the stratification at the base of the convective envelope that is coherent with the stiff radiative interior found in main-sequence solar-like stars. Such a realistic interface, as opposed to an impenetrable wall, allows the convective motions to overshoot beyond the radius where the entropy gradient changes sign. By doing so they generate internal gravity waves and pump magnetic field. Because we are in this study mostly interested in the magnetic state of our simulation, we refer the reader to the following multidimensional studies of internal gravity wave generation in solar-like stars (Rogers & Glatzmaier 2006; Brun et al. 2011; Alvan et al. 2014, 2015). Turning to the radial flux balance, we note that the enthalpy flux (dashed–triple-dotted line) dominates energy transport in most of the convective envelope. The diffusive fluxes (radiative in the bottom half of the computational domain (long dashed) and unresolved near the top (dotted line)) carry the stellar luminosity at each end of the domain. We note an inward kinetic energy flux (dashed–dotted line) reaching about 10% of the star's luminosity, as is common to find in stratified convection simulations. The Poynting and viscous fluxes account for less than 1% of the radial energy balance. Finally, we note the negative enthalpy flux near the base of the convective envelope, which is compensated by a local increase of the radiative flux, such as to reach a satisfactory radial energy balance and thermal equilibrium.

As indicated above, we initiate each of the 15 dynamo simulations from mature, relaxed hydrodynamics convective states and introduce a seed magnetic field in the convective envelope only. These hydrodynamical progenitors have been run long enough to reach a statistically stationary state in the convection zone and a well established rotation profile. They possess a genuinely established tachocline, defined as the transition between DR in the outer convective envelope to solid-body rotation in their stable radiative interior, leading to regions with strong shear (Spiegel & Zahn 1992). The tachocline plays an important role in the dynamo process of magnetic field generation in solar-like stars as reported in simulations performed by several authors (Glatzmaier 1985b; Browning et al. 2006; Racine et al. 2011; Masada et al. 2013; Lawson et al. 2015; Guerrero et al. 2016).

The main parameters of the models are listed in Tables 2 and 3. The density scale heights between the top and the base of the convection zone and between the top and the bottom of the model are defined as ${N}_{{\hat{\rho }}_{\mathrm{bcz}}}=\mathrm{ln}({\hat{\rho }}_{\mathrm{out}}/{\hat{\rho }}_{\mathrm{bcz}})$ and ${N}_{\hat{\rho }{tot}}=\mathrm{ln}({\hat{\rho }}_{\mathrm{out}}/{\hat{\rho }}_{\mathrm{in}})$. For the M05 model, ${N}_{{\hat{\rho }}_{\mathrm{bcz}}}=3.25$ and ${N}_{{\hat{\rho }}_{\mathrm{tot}}}=4.70$; M07 model, ${N}_{{\hat{\rho }}_{\mathrm{bcz}}}=3.48$ and ${N}_{{\hat{\rho }}_{\mathrm{tot}}}=5.78$; M09 model, ${N}_{{\hat{\rho }}_{\mathrm{bcz}}}=3.31$ and ${N}_{{\hat{\rho }}_{\mathrm{tot}}}=5.99$; and M11 model, ${N}_{{\hat{\rho }}_{\mathrm{bcz}}}=3.28$ and ${N}_{{\hat{\rho }}_{\mathrm{tot}}}=5.60$. The convective flows at the middle of the convective envelope vary from 5 m s⁻¹ up to about 300 m s⁻¹ in our sample, and the convective turnover time from 7 (case M11R1m) to 222 (case M05R3m) days. The surface DR between the equator and latitude 60° varies from −102 to +278 nHz in our sample, and we will study its maintenance in detail in Section 4.1. In the middle of the convection zone, the rms magnetic field typically varies from 1 to 70 G in our models. It is found to be maximum close to the bottom of the convective envelope, where the large-scale shear efficiently powers the dynamo, and the magnetic field can be stored in the tachocline close to the convective-radiative boundary.

Table 2. Models Dimensional Characteristics

	Ω*	${\tilde{V}}_{r}$	${\tilde{V}}_{\theta }$	${\tilde{V}}_{\phi }$	ΔΩ	${\tilde{B}}_{r}$	${\tilde{B}}_{\theta }$	${\tilde{B}}_{\phi }$	τc	τν	τκ	τη
	(Ω⊙)	(m s−1)	(m s−1)	(m s−1)	(nHz)	(G)	(G)	(G)	(days)	(yr)	(yr)	(yr)
					MHD (HD)
M05Sm	1/8	13.52	12.19	29.35	−23 (−24)	13.54	15.01	24.87	102.23	15.78	3.94	15.78
M05R1m	1	7.27	7.39	29.72	112 (129)	15.92	15.66	40.92	190.03	37.33	9.33	74.65
M05R3m	3	6.21	6.80	56.85	200 (85)	10.32	9.34	25.91	222.49	64.65	16.16	64.65
M05R5m	5	6.95	4.69	6.59	9 (146)	39.36	49.61	70.71	198.79	64.65	16.16	64.65
M07Sm	1/4	25.44	18.14	30.35	−53 (−32)	11.85	11.05	17.72	62.82	4.50	1.13	4.50
M07R1m	1	16.34	14.48	44.72	111 (120)	5.28	5.00	8.12	97.82	8.22	2.06	16.45
M07R3m	3	14.74	11.21	38.41	68 (187)	29.62	33.21	67.96	108.46	14.24	3.56	28.49
M07R5m	5	13.42	11.55	14.34	−2 (223)	35.85	42.33	54.71	119.11	18.39	4.60	18.39
M09Sm	1/2	53.51	36.80	48.98	−36 (−25)	1.70	1.66	1.68	35.83	2.72	0.68	2.72
M09R1m	1	38.74	35.32	68.55	102 (108)	2.32	2.44	3.17	49.50	3.86	0.97	7.72
M09R3m	3	30.61	32.42	148.70	265 (288)	1.07	1.07	1.93	62.64	6.67	1.67	6.67
M09R5m	5	27.94	19.74	56.43	76 (338)	20.33	19.82	47.02	68.62	7.18	1.80	7.18
M11R1m	1	130.77	93.56	140.61	−102 (−131)	12.69	11.81	13.06	16.67	1.46	0.37	2.93
M11R3m	3	90.23	81.57	272.53	278 (291)	4.49	4.66	6.83	24.17	2.54	0.63	2.54
M11R5m	5	88.50	61.74	88.73	109 (435)	18.63	18.48	32.93	24.63	2.62	0.65	3.27

Note. Characteristic velocities, differential rotation, magnetic fields, and timescales are listed, using averages over a small interval of 0.01R_⋆ at the middle of the convective envelopes (unless stated otherwise). The differential rotation is taken between latitude 60° and the equator at the surface of the models (see Section 4.1). Likewise, the total magnetic flux is computed at the surface of the models and averaged over at least one magnetic cycle for the cyclic cases (see Section 5.4). The rms velocity and magnetic field are $\tilde{v}=\left({\tilde{V}}_{r}^{2}+{\tilde{V}}_{\theta }^{2}+{\tilde{V}}_{\phi }^{2}\right)$ and $\tilde{B}=\left({\tilde{B}}_{r}^{2}+{\tilde{B}}_{\theta }^{2}+{\tilde{B}}_{\phi }^{2}\right)$. Here, ${\tau }_{c}=D/\tilde{{v}_{r}}$ is the overturning convection time, and the dissipation timescales are defined as τ_x = D²/x with x ∈ [ν, κ, η], where D = r_top − r_bcz the thickness of the convective layer that differs for each mass bin.

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Our sample of simulations were designed to operate in a relatively homogeneous turbulent Reynolds number regime, as seen in the first column of Table 3. The supercriticality degree can be characterized by the Rayleigh number achieved in our models, compared to a critical Rayleigh number for the onset of convection. Such a modified Rayleigh number was proposed by Takehiro et al. (2020) and is listed in column 6 of Table 3. All our models exhibit a Rayleigh number at least five times larger than the critical Rayleigh number. We have run these models as long as we could while maintaining a reasonable numerical cost to achieve the large parameter study presented here and while computing the models showing a magnetic cycle over several decades. In this study we use the Rossby number as a measure of the influence of rotation on the flows maintaining the DR as well as power dynamos. Several Rossby number definitions have been proposed in the community, and we have computed the fluid Rossby number Ro_f, the convective Rossby number Ro_c, and the stellar Rossby number Ro_s in Table 3. We refer the reader to Brun et al. (2017) and their Appendix for a more in-depth discussion of these various definitions of the Rossby number. We will focus here on the fluid Rossby number Ro_f and note that the other two are related to Ro_f through a nearly linear relationship. The fluid Rossby number decreases with rotation rate and increases with mass and varies from 0.07 to 2.35 in our sample of models. This range nicely covers the transition from solar-like to antisolar DR regimes (the transition is nearly at Ro_f ≃ 1), and our smallest Rossby number (namely model M05R5m) is close to the expected fast rotators' saturation regime.

We now briefly present one representative cyclic dynamo solution before entering into a more detailed analysis of our dynamo simulations ensemble in Sections 4–6.

We analyze the DR of the simulations that results from the angular momentum redistribution occurring mostly in the convection zone. The panels of Figure 5 show a meridional cut of the axisymmetric DR averaged over 10 overturning convective times, defined as ${\tau }_{c}={\int }_{{r}_{\mathrm{bcz}}}^{{r}_{\mathrm{top}}}{dr}/\tilde{{v}_{r}}$ (see Table 2).

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We observe that for the simulations M05Sm, M07Sm, M09Sm, and M11R1m, there is an antisolar DR, with the poles rotating faster than the equator, like their hydrodynamical counterparts (see Figure 6 and also Brun et al. 2017). The cases rotating at an intermediate rotation rate show a solar-like DR. Finally, the cases rotating the fastest (R5 series) show almost no DR (in particular for cases M05Sm and M07Sm). This constitutes a big difference with their hydrodynamical counterpart cases. The magnetic field here had a major impact, with almost solid-body rotation imposed throughout the convective envelope. There is little asymmetry in the profiles between the northern and southern hemispheres, as expected when the average is performed over an interval long enough with respect to the convective overturning time (except for M05Sm for which the rotational constraint is the weakest and the longitudinal average less meaningful). Figure 6 also displays radial cuts of the rotation for the MHD cases (blue lines) and hydrodynamic progenitor cases (gray lines). In cases rotating 1, 3, and 5 times the solar rotation rate (bottom three rows), the velocity range in latitude (different styles of line) is generally reduced in the presence of a magnetic field. This effect is observed to be stronger as the rotation rate increases. Conversely, the effect of the magnetic field is mild for the slowly rotating cases (upper row), except on the slowly rotating case M11R1m, which still shows some degree of magnetic feedback on its DR. As one may expect, the radial gradient of the DR near the tachocline is generally weaker in all MHD simulations compared with hydrodynamic progenitors. This points to a magnetic feedback of the dynamo field on the DR itself, a feedback that is observed to strengthen as the rotation rate increases.

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We have calculated the surface latitudinal DR ΔΩ for each model, defined as the difference between the equator and 60° latitude. A positive value thus denotes a solar-like DR and a negative value an antisolar DR. We report these values for the magnetic cases as well as the hydrodynamic progenitors in Table 2 (fourth column).

The DR of our sample spans a range between −102 and +278 nHz, with some fast-rotating models presenting an extremely weak DR, like M05R5m with ΔΩ = 9 nHz. We find that the absolute DR generally weakens in MHD models compared to their hydrodynamic progenitors, as expected from the radial profiles shown in Figure 6. This is particularly striking for fast rotators such as M09R5m, which goes from 338 nHz in hydro to 76 nHz in MHD.

We investigate in Figure 7 the DR trends with respect to the rotation rate (left panel) and rotation period (right panel). The DR of the hydrodynamic progenitors and of the MHD cases are respectively shown in small semitransparent and large opaque symbols. The shape of the symbol labels the rotation of the model, and the color the mass of the modeled star, as indicated in the legend. In the right panel, we compare the model DR to the DR in the Kepler sample obtained by Reinhold & Gizon (2015) (shown as black dots). The dotted lines correspond to their estimated observational detection limits. We first note that the absolute value of our DRs agrees well with the observed values. In addition, the DR range in our sample increases as the rotation period decreases, like what is observed in the Kepler satellite sample. Several of our models nevertheless lie outside the observed values: the three antisolar DRs on the right (triangles) and two of our fast-rotating models. Several reasons can explain this discrepancy. Slowly rotating stars could produce very few starspots or even no starspots at all (see, for instance, van Saders et al. 2019), making their DR impossible to detect with photometry. Another possibility is that they lie outside the presently detectable limit with the Kepler data, due to their long rotational period (up to about 200 days for our most slowly rotating model). Finally, the two fast-rotating models (M05R5m and M07R5m) show very weak DRs due to magnetic feedback, which are outside the detection limits of Kepler (dotted black lines).

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The left panel of Figure 7 shows the DR trend with the rotation rate. Using only the hydrodynamic progenitors, we previously showed that the DR scales as Ω^0.66 (Brun et al. 2017). Blindly trying to fit such a power law to the MHD sample, we find that the exponent reduces to Ω^0.46. This weaker dependency is expected due to the magnetic feedback on the DR through the Lorentz force. It also agrees better with the observational trends, which are still quite uncertain and were found to vary from 0.2 (Balona & Abedigamba 2016 for G stars), 0.3 (Reinhold et al. 2013 for cool stars), to even 0.7 (Donahue et al. 1996 for F-K stars). Looking more closely at our sample on the left panel of Figure 7, it clearly appears that a power-law fit is a poor representation of the DR in our sample. Rather, we see that ΔΩ increases with Ω for slow rotators, while it dramatically drops for fast rotators due to the magnetic feedback. Following Saar (2011), we recast in Figure 8 the DR trend in terms of relative DR ΔΩ/Ω with respect to the fluid Rossby number Ro_f (for the different definitions of Rossby number used in this work, see the caption of Table 3 or the Appendix A of Brun et al. 2017). We find a trend that is very similar to the observational trend reported by Saar (2011) (shown by the dashed line in Figure 8): ΔΩ/Ω is roughly constant for inverse Rossby numbers lower than a certain threshold (here ${{Ro}}_{f}^{-1}\lesssim 5$), and it drops for fast rotators as ${\rm{\Delta }}{\rm{\Omega }}/{\rm{\Omega }}\propto {{Ro}}_{f}^{p}$. Saar (2011) proposed that p = 2, but here our sample agrees with a somewhat large range p ∈ [2, 6]. Additional models with even higher turbulence level are required to confirm the exact amplitude of the drop in DR contrast found in the fast-rotating cases. Finally, our sample also shows some hint of an increase of ΔΩ/Ω at large Rossby numbers, which is outside the observable constraints for now. It would be interesting to search observationally for candidate solar-like stars possibly possessing such antisolar rotation states.

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In Brun et al. (2017), we have proposed that the DR could follow two power laws with respect to the Rossby number and the stellar mass. Here, we find that the DR is weakened at high Rossby number, and therefore we do not recover a simple power-law trend, as we saw in Figure 7. We can nevertheless attempt to fit such a combined power law on a subsample of our models, excluding the fast-rotating case but retaining the slow rotators. We obtain in this way

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Omega }}=107{{Ro}}_{{\rm{f}}}^{-0.73\pm 0.13}{\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}}^{1.93\pm 0.42}\,\mathrm{nHz}\,\ (\mathrm{HD}),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Omega }}=84{{Ro}}_{{\rm{f}}}^{-0.40\pm 0.20}{\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}}^{0.78\pm 0.62}\,\mathrm{nHz}\,(\mathrm{MHD}).\end{eqnarray} \tag{ 8 }$$

In the MHD case, we find again that the DR is less sensitive to both the Rossby number and the stellar mass. The power-law fit is nevertheless questionable here, as the range covered by our Rossby numbers and masses is quite small. We have nevertheless included the results of the fit here to compare with the purely hydrodynamic case (Brun et al. 2017). We can conclude here that the clear trend in stellar mass and effective temperature found in the hydrodynamic study (Brun et al. 2017) is less significant when magnetism is taken into account, but overall we see a better agreement with observations of the dynamo models compared to their hydrodynamical progenitors.

The MHD simulations therefore show that the magnetic field changes the angular momentum redistribution, especially for fast-rotating stars. In the next section, we perform a detailed analysis of this balance for four representative models.

We can better understand how the DR profiles are achieved in the dynamo models by identifying the main physical processes responsible for redistributing angular momentum within rotating convective shells. Our choice of stress-free and potential-field BCs at the top and stress-free and perfect conductor BCs at the bottom of the computational domain have the advantage that no net external torque is applied, and thus angular momentum is conserved. We can assess the transport of angular momentum by considering the mean radial (${{ \mathcal F }}_{r}$) and latitudinal (${{ \mathcal F }}_{\theta }$) angular momentum fluxes, applying the procedure used in Brun et al. (2004). Starting from the ϕ component of the momentum equation expressed in conservative form and averaged in time and longitude:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial ({r}^{2}{{ \mathcal F }}_{r})}{\partial r}+\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial (\sin \theta {{ \mathcal F }}_{\theta })}{\partial \theta }=0,\end{eqnarray} \tag{ 9 }$$

involving the mean radial angular momentum flux

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal F }}_{r}=\hat{\rho }r\sin \theta \left[-\nu r\displaystyle \frac{\partial }{\partial r}\left(\displaystyle \frac{\hat{v}}{r}\right)+\widehat{{v}_{r}^{{\prime} }{v}_{\phi }^{{\prime} }}\right.\\ \ \ \left.+\,{\hat{v}}_{r}({\hat{v}}_{\phi }+{\rm{\Omega }}r\sin \theta )-\displaystyle \frac{1}{4\pi \hat{\rho }}\widehat{{B}_{r}^{{\prime} }{B}_{\phi }^{{\prime} }}-\displaystyle \frac{1}{4\pi \hat{\rho }}{\hat{B}}_{r}{\hat{B}}_{\phi }\right],\end{array}\end{eqnarray} \tag{ 10 }$$

and the mean latitudinal angular momentum flux

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal F }}_{\theta }=\hat{\rho }r\sin \theta \left[-\nu \displaystyle \frac{\sin \theta }{r}\displaystyle \frac{\partial }{\partial \theta }\left(\displaystyle \frac{{\hat{v}}_{\phi }}{\sin \theta }\right)\right.\\ \ \ \left.+\,\widehat{{v}_{\theta }^{{\prime} }{v}_{\phi }^{{\prime} }}+{\hat{v}}_{\theta }({\hat{v}}_{\phi }+{\rm{\Omega }}r\sin \theta )-\displaystyle \frac{1}{4\pi \hat{\rho }}\widehat{{B}_{\theta }^{{\prime} }{B}_{\phi }^{{\prime} }}-\displaystyle \frac{1}{4\pi \hat{\rho }}{\hat{B}}_{\theta }{\hat{B}}_{\phi }\right].\end{array}\end{eqnarray} \tag{ 11 }$$

In these equations, the terms on the right-hand side represent, for both fluxes, contributions respectively from viscous diffusion (which we denote as ${{ \mathcal F }}_{r}^{\mathrm{VD}}$ and ${{ \mathcal F }}_{\theta }^{\mathrm{VD}}$), Reynolds stresses (${{ \mathcal F }}_{r}^{\mathrm{RS}}$ and ${{ \mathcal F }}_{\theta }^{\mathrm{RS}}$), meridional circulation (${{ \mathcal F }}_{r}^{\mathrm{MC}}$ and ${{ \mathcal F }}_{\theta }^{\mathrm{MC}}$), Maxwell stresses (${{ \mathcal F }}_{r}^{\mathrm{MS}}$ and ${{ \mathcal F }}_{\theta }^{\mathrm{MS}}$), and large-scale magnetic torques (${{ \mathcal F }}_{r}^{\mathrm{MT}}$ and ${{ \mathcal F }}_{\theta }^{\mathrm{MT}}$). The Reynolds stresses are linked to correlations of the fluctuating velocity components coming from organized tilts within the convective structures, especially in the downflow plumes. Likewise, the Maxwell stresses are associated with correlations of the fluctuating magnetic field components due to the twist and tilt of the dynamo-generated magnetic structures.

In Figure 9 we show the components of ${{ \mathcal F }}_{r}$ and ${{ \mathcal F }}_{\theta }$ for the M07 case series, having integrated over colatitude and radius as follows:

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{r}(r) & = & {\displaystyle \int }_{0}^{\pi }{{ \mathcal F }}_{r}(r,\theta )\,{r}^{2}\sin \theta \,d\theta ,\\ {{\rm{\Phi }}}_{\theta }(\theta ) & = & {\displaystyle \int }_{{r}_{\mathrm{bot}}}^{{r}_{\mathrm{top}}}{{ \mathcal F }}_{\theta }(r,\theta )\,r\sin \theta \,{dr}.\end{array}\end{eqnarray} \tag{ 12 }$$

Thus, Φ_r represents the net angular momentum flux through horizontal shells at different radii and Φ_θ represents the net flux through cones at different latitudes. This representation is helpful in assessing the direction and amplitude of angular momentum transport within the computational domain by each component of ${{ \mathcal F }}_{r}$ and ${{ \mathcal F }}_{\theta }$.

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For each of the four cases, we display Φ_r on the left panel and Φ_θ on the right panel, both normalized by ${R}_{* }^{2}$. Turning to the radial angular momentum transfer, we first note a very good overall radial balance. We find that the Reynolds stresses (green dashed–dotted curves) transport angular momentum outward in all the low Rossby number models. By contrast, M07S, the slowly rotating case, has the Reynolds stresses transporting angular momentum inward. The viscous diffusion and Maxwell stresses oppose this transport, tending to rigidify the rotation state in the radial direction. The meridional circulation has one large cell per hemisphere for the M07Sm case (see Section 4.3). It opposes the Reynolds stresses, but as the rotation rate increases and the Maxwell stresses gain in amplitude, it changes in profiles and direction to yield a radial balance of angular momentum, from the angular momentum equation. Note that the mean large-scale magnetic torques (black dotted line) have little influence on the overall radial angular momentum balance.

Considering now Φ_θ , we can assess the balance of latitudinal angular momentum transport. We first notice that the Reynolds stresses (green curves) are systematically equatorward in both hemisphere (positive in the northern hemisphere and negative in the southern one). Because most cases exhibit a very good latitudinal balance, as demonstrated by the solid black curve, these Reynolds stresses must be nicely counterbalanced. A quick survey of the right panels for all four models indicates that many contributors act depending on the Rossby number of the simulations. For the slowly rotating case (upper-left corner), we see that it is mostly the meridional circulation (cyan dashed curve) that does most of the work (we defer the reader to Section 4.3 for a discussion of the meridional circulation patterns in the various dynamo cases). By contrast, magnetic terms do not play much role in the case of M07Sm. For M07R1m (right-top corner) it is now the viscous diffusion that plays that role of opposing the Reynolds stresses. For that case, the meridional circulation is not doing much, but we do see a 20% contribution of the large-scale magnetic torques, the Maxwell stresses being still weak. As the Rossby number decreases and the dynamo action becomes more intense, we see that the magnetic terms start influencing the latitudinal angular momentum transport more and more, tending to oppose the Reynolds stresses. It is particularly noticeable for the Maxwell stresses. They are the dominant player for the M07R3m case (bottom-left corner), helped by the large-scale magnetic torques. In that case the meridional circulation is somewhat helping the Reynolds stresses, notably at low latitudes near the equator. For the M07R5m case, the story becomes less clear; except for the Reynolds stresses, all terms fluctuate and sometimes oppose or reinforce the turbulent stresses. Maxwell stresses still play an important role as does the meridional circulation. In that model, the DR has been so significantly quenched by dynamo action that it is not surprising the trends are less clear and systematic. In summary, in most cases the transport of angular momentum by Reynolds stresses are opposed by a combination of meridional circulation, viscous stresses, and Maxwell stresses.

The meridional flow patterns are also affected by the presence of magnetism in our set of models, especially for the fast-rotating cases. We immediately note that the meridional circulation is indirectly modified by magnetism (as will be made clear in Section 6.2). Indeed, magnetic stresses play a negligible role in setting the meridional flows in our models, and the differences we observe compared to the hydrodynamical counterparts originate from changes in the DR (see, e.g., Passos & Charbonneau 2014).

We illustrate the meridional flow pattern achieved in the M07m set of simulations in Figure 10. The slow-rotating case (first panel) is very similar to its hydrodynamic progenitor, with a well-defined circulation cell in each hemisphere. Both cells circulate from the equator to the pole at the surface and from the pole to the equator at the base of the convective envelope. The second model rotating at the solar rate (second panel) is also similar to its hydrodynamical progenitor and shows a more complex circulation profile. These are consistent with previous numerical experiments by, e.g., Karak et al. (2015). It consists of stacked cells elongated along the rotation axis outside the tangent cylinder and two counterrotating cells in each hemisphere at high latitude. Finally, the fast-rotating models (third and fourth panels) exhibit a peculiar meridional circulation pattern concentrated at the equator, with two stacked transequatorial cells (see, e.g., Simitev & Busse 2009). These profiles can be understood as follows. In these models, the DR is strongly quenched by magnetic feedback as seen in the previous section. In particular, the radial shear of DR vanishes at the equator as seen in Figure 6. As a result, gyroscoping pumping (Miesch et al. 2006; McIntyre 2007; Featherstone & Miesch 2015) dramatically weakens along the equator, and the resulting meridional circulation is both very weak (this can be seen in the drop of meridional flow kinetic energy in Table 5) and mainly driven by the remaining latitudinal shear. This leads to two meridional cells crossing the equator, as seen in the last panels of Figure 10. Having presented the large-scale flows achieved in the simulations, we now turn to discussing their magnetic properties.

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We find three dynamo states in our sample of 15 MHD models: long (decadal) magnetic cycles, short (yearly) magnetic cycles, and stable magnetic wreaths concentrated close to the bottom of the convection zones. These three states are illustrated in Figure 11 with models M07R5m, M09R3m, and M09R1m.

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Let us first focus on the decadal cycles, as the one found for M09R3m (see middle left panels in Figure 11). In this model, we find that the global magnetic field of the star reverses with a period of 10 yr (see first column in Table 4). The averaged azimuthal field at the bottom of the convection zone presents a solar-like butterfly diagram, with both polar and equatorial branches. The magnetic field is generally consistent with dipolar symmetry, with the azimuthal field of inverse polarities in each hemisphere. We also see some departures from hemispheric symmetry (for instance around t = 42 yr). The azimuthal field is found to be concentrated at the base of the convective envelope and in the tachocline, where the radial shear of Ω is maximized, as shown in the time–radius and meridional diagrams. It develops over a relatively large latitudinal extent, as shown by the active latitudinal band reported in the fourth column of Table 4. We find this band to be centered at higher latitudes the slower the model rotates for low and intermediate Rossby numbers. Conversely, this activity band moves to high latitudes for models with antisolar DR. The averaged radial magnetic field at the surface is also found to reverse with the same timescale. At the surface, the migration branches are nevertheless not as clear as deep within the convection zone in this case. Strugarek et al. (2017, 2018) also found similar deeply seated cycles using the EULAG code, as well as Augustson et al. (2015) using the ASH code. The cyclic behavior in their results originates from the nonlinear magnetic feedback of the large-scale Lorentz force onto the DR. This weakens the source of the mean toroidal field that decreases and reverses, while the associated poloidal field closely follows due to the sign inversion of the electromotive force. We find the same mechanism in this new sample of simulation with the ASH code. Indeed, the DRKE cyclic variation observed in Figure 3 compensates the magnetic energy cyclic variation, pointing toward a magnetic cycle determined by the Lorentz force feedback. This fascinating dynamo regime sustaining a long decadal magnetic cycle, because of the existence of a subtle nonlinear feedback loop between the large-scale shear and the toroidal magnetic field, is therefore confirmed by the present study using a different numerical code than Strugarek et al. (2017). We stress that its existence can be unveiled here only because we consider fully nonlinear convective dynamos, with a self-consistent DR maintenance and magnetic field generation.

Table 4. Magnetic Properties of the Modeled Dynamos

	Long Cycle	Short Cycle	δt Ω	Active lat.	Φtot	Brms	Br,dip	BL,surf	fdip	fquad
	(y/[yr]/n)	(y/[yr]/n)	[nHz]	[°–°]	[1024 Mx]${}_{\min }^{\max }$	[G]${}_{\min }^{\max }$	[G]${}_{\min }^{\max }$	[G]${}_{\min }^{\max }$
M05Sm	n	n	2.4	[43–45]	${3.0}_{2.6}^{3.3}$	${793}_{683}^{884}$	${191}_{156}^{217}$	${357}_{296}^{407}$	0.21	0.22
M05R1m	13.6 ± 5.7	1.2 ± 0.6	2.1	[31–31]	${1.9}_{1.3}^{2.5}$	${575}_{343}^{805}$	${58}_{17}^{99}$	${158}_{88}^{234}$	0.05	0.25
M05R3m	21.4 ± 9.4	0.5 ± 0.2	2.1	[44–45]	${2.1}_{1.0}^{3.4}$	${552}_{297}^{830}$	${38}_{5}^{93}$	${146}_{69}^{215}$	0.28	0.37
M05R5m	n	1.8 ± 1.0	3.0	[49–55]	${8.6}_{5.7}^{9.1}$	${1698}_{1131}^{1851}$	${196}_{51}^{236}$	${819}_{426}^{929}$	0.19	0.35
M07Sm	n	n	5.4	[51–53]	${4.8}_{4.5}^{5.2}$	${575}_{524}^{648}$	${136}_{130}^{142}$	${306}_{279}^{340}$	0.29	0.38
M07R1m	6.2 ± 1.1	1.4 ± 1.3	3.7	[26–35]	${8.6}_{5.8}^{11.7}$	${949}_{616}^{1325}$	${200}_{100}^{266}$	${439}_{261}^{601}$	0.38	0.48
M07R3m	y	2.5 ± 0.8	4.1	[21–22]	${8.7}_{3.2}^{13.0}$	${972}_{127}^{1951}$	${157}_{40}^{251}$	${340}_{198}^{573}$	0.12	0.40
M07R5m	n	1.0 ± 0.7	1.2	[55–58]	${18.4}_{15.3}^{20.2}$	${1597}_{1320}^{1879}$	${181}_{52}^{351}$	${925}_{699}^{1072}$	0.35	0.48
M09Sm	n	n	13.1	[72–73]	${1.8}_{1.7}^{1.8}$	${109}_{97}^{123}$	${53}_{49}^{55}$	${60}_{55}^{66}$	0.60	0.17
M09R1m	n	n	9.0	[20–21]	${1.1}_{0.9}^{1.5}$	${68}_{58}^{100}$	${11}_{9}^{16}$	${23}_{18}^{33}$	0.23	0.29
M09R3m	9.9 ± 1.8	0.9 ± 0.6	9.0	[24–26]	${2.2}_{0.3}^{4.0}$	${133}_{15}^{261}$	${10}_{0.9}^{23}$	${47}_{6}^{91}$	0.16	0.27
M09R5m	n	1.3 ± 0.7	9.5	[30–35]	${13.4}_{10.7}^{18.7}$	${657}_{485}^{970}$	${274}_{221}^{400}$	${392}_{284}^{608}$	0.35	0.44
M11R1m	n	n	12.4	[46–47]	${14.5}_{13.2}^{15.8}$	${589}_{544}^{650}$	${10}_{0.8}^{23}$	${184}_{161}^{197}$	0.06	0.31
M11R3m	4.9 ± 0.9	n	11.5	[20–21]	${2.7}_{0.7}^{7.1}$	${86}_{20}^{226}$	${11}_{0.7}^{31}$	${55}_{13}^{160}$	0.22	0.29
M11R5m	.	.	39.0	[52–77]	${53.1}_{41.6}^{57.7}$	${1208}_{980}^{1329}$	${713}_{528}^{875}$	${809}_{596}^{986}$	0.51	0.11

Note. The first column indicates the presence or absence of a long (decadal), deeply seated magnetic cycle. When the time series were long enough to identify a cycle period unambiguously, its value is given with error bars. Otherwise, the existence of such a cycle is indicated by a yes ("y") and its absence by a no ("n"). The second column shows the same for the short magnetic cycle that we identify in the upper convection zone near the equator. We do not indicate this information for model M11R5m, which was not run long enough to determine the existence or absence of magnetic cycles. The third column indicates the amplitude of the torsional oscillations at the surface in nHz (see Section 5.3). The fourth column shows the active latitudinal band at the bottom of the convection zone, based on the azimuthally averaged and temporally varying azimuthal field straddling the base of the convection zone. The fourth column shows the total magnetic flux at the surface, in units of 10²⁴ Mx, with the minimum and maximum as the subscript and superscript (see Section 5.4). The three next columns show the root-mean-squared surface field in Gauss, the surface dipole in Gauss, and the surface large-scale radial field B_L,surf (taken for l < 5) in Gauss with the same layout (see Section 7). Finally, the last two columns show the fractions of the large-scale dipole (f_dip) and quadrupole (f_quad), as defined in Section 5.5.

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Still, we have attempted to interpret our simulations through mean-field dynamo theory by inverting the α tensor and its antisymmetric part γ by means of the singular-value decomposition (SVD) technique (see Augustson et al. 2015; Simard et al. 2016). The details of this procedure are given in Appendix B. One can then use the derived $\bar{\alpha }$ profile to compute the Parker–Yoshimura rule (Parker 1955; Yoshimura 1975) and assess the consistency of a mean-field approach with our 3D turbulent model. We therefore compute

$$\begin{eqnarray}&&{S}_{\theta }=-\lambda \alpha {\partial }_{r}\left({\rm{\Omega }}/{{\rm{\Omega }}}_{0}\right),\end{eqnarray} \tag{ 13 }$$

where $\lambda =r\sin \theta$. The time–latitude variations of S_θ are shown at the base of the convection zone of M09R3m in the left panel of Figure 12, with red/white denoting a southward migration rule and blue/black a northward migration rule. We overlay contours of B_ϕ as black contours (solid/dashed denoting positive/negative contours) in the left panel. We see that the derived Parker–Yoshimura dynamo wave rule does not agree with the observed latitudinal propagation, which strengthens our interpretation in terms of a cycle dominated by the Lorentz force feedback on the DR itself.

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We also find another type of cyclical behavior in our sample of models: short cycles, which seem to preferentially be sited close the equator and in the upper part of the convection zone. Such types of cycles have already been reported in previous publications with numerical models (Beaudoin et al. 2016; Käpylä et al. 2016; Strugarek et al. 2018) and could be reminiscent of the possible quasi-biennial oscillations observed in the Sun (Broomhall et al. 2012; Simoniello et al. 2013). They oscillate on a yearly timescale as shown in the second column of Table 4. Short cycles are interestingly found in almost all of our models, except the slowly rotating cases. Two short cycles are illustrated in Figure 11 for cases M07R5m and M09R3m. In the former fast-rotating case, no long deeply seated cycle is observed, and the short cycle clearly appears in both the latitude–time and radius–time diagrams. In the case M09R3m, both types of cycle are found at the same time, and the short cycle appears clearly once the signal of the long cycle is removed (see zoomed panel). The short cycles are found to always show a poleward propagation branch and to be concentrated close to the equator. We have performed the same SVD analysis and show the Parker–Yoshimura rule S_θ (Equation (13)) for model M07R5m, which is shown in the right panel of Figure 12. In this case, the analysis is carried out in the upper part of the convective envelope, and contours of B_r are overlaid above the propagation rule. The Parker–Yoshimura rule is found here to be in good qualitative agreement with the poleward branch, suggesting that an α − Ω or an α² − Ω dynamo could be at the source of this type of cycle. The short cycles furthermore embed much less magnetic energy than the deeply seated ones, and we do not find any clear DRKE beating associated with them. As a result, we find that the two types of cyclical behaviors likely originate from two different dynamo processes: the deep-seated cycle from the large-scale feedback loop between the magnetic field and the DR through Maxwell torques, and the short cycles from the standard α − Ω or α² − Ω dynamo loop. Finally, short cycles were also reported in the study of Strugarek et al. (2018), with the same type of localization within the convective envelope. In this previous study, the short cycles were only found at small Rossby number, i.e., for the fast- rotating cases. Here we find short magnetic cycles much more ubiquitously in our sample models, as they only disappear at large Rossby numbers. It is possible that the coarse resolution used in Strugarek et al. (2018) with the EULAG code prevented models at intermediate Rossby number to develop such magnetic cycles. Additional modeling effort pushing the turbulence level of the simulations is required to properly assess this point, which is left for future work.

Finally, some models in our sample do not present any cyclical behavior. Instead, they sustain a steady dynamo with stable magnetic wreaths within their convective envelope and tachocline. This is the case, for instance, with model M09R1m shown in the lower panels of Figure 11. We obtain such solutions only in the high Rossby number regime, close to and above the transition toward an antisolar DR.

To summarize, we find that the different types of cyclical behaviors exist in specific Rossby number ranges in our sample. We illustrate this in Figure 13 where we follow Gilman (1983) and show DRKE/KE as a function of Ro_f in our set of models. Short cycles are found for Ro_f ≲ 0.42, deeply seated solar-like cycle for 0.15 ≲ Ro_f ≲ 0.65, and steady magnetic fields for Ro_f ≳ 1.0. The exact boundaries between these cyclical behaviors regimes are not precisely defined and may depend on a number of factors. First, let us note that the same trend was found in Strugarek et al. (2018) with the EULAG code, as shown by the colored stars also plotted in Figure 13. This is very important because it again demonstrates that the results discussed in this study are not code or setup dependent, but the results of genuine nonlinear convective dynamo action in a rotating spherical shell. It confirms that the Rossby number is one of the key parameters to characterize the various dynamo states found in the literature and that cyclic convective dynamo solutions clearly exist in a parameter regime that our study helps to refine. The transitions between the different types of cycles were found at slightly different Rossby numbers, possibly due to different Reynolds, Prandtl, and Rayleigh numbers regimes achieved in the two ensemble of simulations. Indeed, Nelson et al. (2013) showed that fast-rotating models exhibiting stable wreaths of magnetism (Brown et al. 2010) could produce reversals when the Reynolds and Rayleigh numbers are increased. Because the Rossby number of the more turbulent models nevertheless changes significantly as well, it is therefore unclear whether this can be attributed to a fundamental change in the dynamo action or if it is the consequence of a change in Rossby number. Fundamental exploration aimed at predicting the Rossby number of turbulent numerical experiments such as Anders et al. (2019) is very promising in that respect and needs now to be extended to the full MHD regime. For the time being, we can conclude here that qualitatively the different regimes highlighted by our simulations are robust, yet simulations at much higher turbulent levels are required to assess the exact regime boundaries. Please note that case M11R5m is sometimes omitted in ensemble analysis in Sections 5 and 6 because it is not as well numerically converged as all the other cases and can sometimes be an outlier in some analysis. This does not impact our conclusions in any of the results reported in the paper.

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We have calculated the period of the short and long cycles and reported their values in the second and third columns of Table 4. We use the approach initially followed by Käpylä et al. (2016) and Strugarek et al. (2018) and rely on an empirical-mode decomposition method (Luukko et al. 2015) to identify quasi-periodic signals. Five of our models clearly exhibit a deeply seated long cycle that can be identified by eye. We were nevertheless able to calculate accurately the associated period for four of them. The cycle period of the fourth model would require integration times at least twice as long to be identified. This would require an even more massive numerical effort and will be explored in future work. Still we can deduce with some confidence what characterizes this long-cycle nonlinear dynamo case. Conversely, the short cycles take place higher up in the convective envelope and their short periods allow us to determine the cycle periods for all the models exhibiting them. The error bars on the cycle periods are directly estimated with the empirical-mode decomposition method, as explained in Strugarek et al. (2018).

The left panel of Figure 14 shows the cycle periods (in years) as a function of the rotation period (in days) of our models. We report both short and long cycles here, respectively, in blue circles and red circles. We have also added the cycles found with the EULAG code and reported in Strugarek et al. (2017, 2018) as red and blue stars. Finally, we have overlaid the detected cycles of distant stars reported by Böhm-Vitense (2007) as gray squares, as well as the Sun right in the middle of the figure. Our three identified long cycles are achieved by models with different masses, which make their direct comparison subject to caution in a (P_cyc, P_rot) diagram. Overall, we do not recover the dichotomy between active and inactive branches as initially proposed by Saar & Brandenburg (1999) and Böhm-Vitense (2007). Rather, our sample of models combining the ASH and EULAG simulations spans the whole diagram, including the hypothetical gap where the Sun stands.

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Using the EULAG sample of simulations only, we have previously shown that the cycle period is controlled by the effective Rossby number achieved by the simulated convection zone (Strugarek et al. 2017). This is shown for the long and short cycles in the middle and right panels of Figure 14. Here we find that our new ASH simulations are compatible with the trends obtained with the EULAG sample, which strengthens the similarities between the modeled dynamos in our two studies. This is moreover remarkable as the ASH simulations include a tachocline and a deeper radiative layer, whereas the EULAG sample considered only an isolated convective shell.

The fact that the cycle period seems to decrease with the Rossby number has also been reported by other research groups using yet another code (see, e.g., Warnecke 2018). So far, only one study relying on 3D turbulent simulations (Guerrero et al. 2019) has shown some evidence for the cycle period increasing with rotation period. We believe this is due to how their DR scales with rotation rate. Indeed, their simulations exhibit a DR that strengthens as the rotation rate decreases (i.e., the rotation period increases). This is at odds with all the aforementioned studies (including the present work), where we find it to increase with the rotation rate up to a point where magnetic feedback strongly back-reacts to suppress it. We suspect that the thermal treatment of the radiative-convective interface may produce this effect in the work of Guerrero et al. (2019), albeit additional analyses are required to confirm this interpretation. Finally, it is worth noting that more complex dynamo states have also been reported in a similar Rossby number regime with the PENCIL code by Viviani et al. (2019). This again warrants caution in the interpretation of simulation results at moderate Reynolds number and highlights the need to achieve more turbulent regimes in future work to confirm our trends.

We observe clear and strong torsional oscillations δ_t Ω (Basu & Antia 2019) in all our models that exhibit a long, deeply seated cycle. Torsional oscillations take the form of a modulation of the azimuthally averaged rotation rate Ω(r, θ, t) in depth, latitude, and time. We illustrate the torsional oscillations at the base of the convection zone of model M09R3m in Figure 15. The torsional positive/negative oscillations are shown in red/blue in nHz as a function of time and latitude. We have overlaid isocontours of B_ϕ in black (plain lines correspond to 4000 G, dashed lines to −4000 G). The torsional oscillations are observed to be in phase with long magnetic cycle. At cycle minimum (in between the black contours), the poles are rotating slower (blue) and the equator faster (red), meaning that the latitudinal DR is strengthened as the magnetic field weakens and the associated magnetic torque stops inhibiting it. During cycle maximum, the opposite situation occurs, and the DR is found to decrease substantially. We observe torsional oscillations very similar to what were found with EULAG simulations by Strugarek et al. (2017, 2018) and previous ASH simulations by Nelson et al. (2013) and Augustson et al. (2015). In all these simulations, the torsional oscillations are found to play a major role in producing the deeply seated cycle. This is reassuring because such nonlinear interplay between the flow and field seems independent of setup details such as BCs or numerical schemes. Moreover, torsional oscillations in our models are very energetic: they reach more than 20 nHz at the base of the convective envelope in model M09R3m, and their energy corresponds to the energy variations in the total magnetic energy (ME) seen in Figure 3. As a result, we find they play an active role in allowing deeply seated cycles by reversing locally ∂Ω/∂θ and hence generating a toroidal field of opposite sign.

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We have also searched for torsional oscillations at the locations of short magnetic cycles, i.e., at the surface and close to the equator of fast-rotating models. We find a temporal modulation of the local rotation rate at the surface in all our models. We have nevertheless not found any evidence for a correlation between these temporal variations and the short cycles themselves. This confirms that a different dynamo process sustains the short cycles, which is likely related to a more standard α − Ω mechanism as we have seen in Section 5.1.

Finally, we have characterized the surface torsional oscillations in all our models and reported in Table 4 the average values of δ_t Ω within the activity band identified in Table 4. The surface torsional oscillations range from about 1 to 39 nHz in our sample of simulations, which corresponds to 0.4%–6% of the model rotation rates. Torsional oscillations associated with short cycles are found to be very weak, and the ones associated with the long cycle to be prominent deep inside the convective envelope. As a result, we do not observe any strong correlation between the amplitude of the surface torsional oscillations and the Rossby number of our models: a linear regression gives ${\delta }_{t}{\rm{\Omega }}/{{\rm{\Omega }}}_{\star }\propto {{Ro}}_{{\rm{f}}}^{1.1\pm 0.15}\simeq {{Ro}}_{{\rm{f}}}$.

To further assess the magnetic properties of a dynamo solution, we display in Figure 16 for three representative cases (M05R1m, M09R3m, M11R1m) various measures of the magnetic flux available at the top boundary layer, that is, Φ_{+ and} Φ₋, the magnetic fluxes for ${B}_{r}{| }_{r={r}_{\mathrm{top}}}\gt 0$ and ${B}_{r}{| }_{r={r}_{\mathrm{top}}}\lt 0$, respectively, the total flux Φ_tot = ∣Φ₊∣ + ∣Φ₋∣, the net flux Φ_net = Φ₊ + Φ_−, and the southern Φ_S and northern Φ_N hemispheric fluxes (i.e., integrated only over the northern and southern hemispheres, respectively). First, we see the very good conservation of the divergenceless nature of the magnetic field, with Φ_net being systematically null (so implying that Φ₋ = −Φ₊, as clearly evident). This is the direct consequence of solving the induction equation via a poloidal−toroidal field decomposition (see Equation (6)). Likewise, the two hemispherical measures of Φ have opposite signs, but a much smaller amplitude than Φ₊ and Φ₋ by about a factor of 10. This is likely due to a highly structured magnetic field because for an axial dipole they are expected to be equal. When adding up the absolute value of Φ₊ and Φ₋, we can assess the total amount of magnetic flux generated by the dynamo. We find fluxes from 10²⁴ to 10²⁵ Mx, which are in good agreement with values observed in the Sun (see for instance Figure 3 of Schrijver & Harvey 1994). We also note that in the M05R1m and M09R3m cases, both of which possess a clear and long magnetic cycle, the temporal modulation of the magnetic fluxes is obvious. In the M05R1m case, the modulation is about a factor of 2 from the minimum to maximum of activity. In case M09R3m, it reaches almost a factor of 8 (compared to 5 for the Sun). Here again the larger mass (luminosity) of M09R3m and its higher rotation rate leads to a larger temporal modulation of the magnetic energy and hence the magnetic flux. Finally, for the steady dynamo case M11R1m, possessing an antisolar DR, a very small magnetic flux variability is observed. However, it is the model with the highest value of the total magnetic flux, reaching about 10 times what is observed in the present Sun.

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We furthermore see a tendency for Φ_tot to increase with both stellar mass and rotation rate, in good agreement with the level of magnetic energy found in the simulations. However, more robust tendencies appear on the rotation when one considers only the model with Ro_f < 1. They are interestingly compatible with a simple linear dependency, with ${{\rm{\Phi }}}_{\mathrm{tot}}\simeq 2.3\,{{\rm{\Omega }}}_{\star }^{0.84\pm 0.42}$ for the rotation rate. When considering how the total magnetic flux scales with rotation rate ${{\rm{\Phi }}}_{\mathrm{tot}}\propto {{\rm{\Omega }}}_{* }^{n}$, different values from n = 1.2 (Saar 2001) to n = 2.8 (Schrijver et al. 2003) have been proposed (Rempel 2008). In our study we find a tentative scaling with the fluid Rossby number as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{tot}}\simeq 1.19\,{{Ro}}_{{\rm{f}}}^{-0.88\pm 0.31}{10}^{24}\mathrm{Mx},\end{eqnarray} \tag{ 14 }$$

as shown in Figure 17, where the time-averaged total flux of each model is considered (see also Table 2). Our models depart significantly from this trend when their Rossby number exceeds one, indicating a possible change for very slowly rotating stars. In this regime, our sample of models suggests that the total magnetic flux increases with Rossby number, as shown by the dashed–dotted line. Additional models at large Rossby numbers are required to fully characterize this regime properly, which we leave for future work. To summarize, we find that the total magnetic flux follows a trend compatible with the one from Saar (2001) for intermediate and small Rossby numbers and that this trend reverses for slow rotators (Ro_f > 1).

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We now turn to considering how the change of DR state as a function of the Rossby number may influence the relative amplitude of the dynamo modes. We have seen in the previous sections that as we vary the Rossby number, the type of dynamo solution changes, going from steady for large Rossby numbers to long-period cyclic solutions for intermediate values of the Rossby number, to fast cyclic solutions for low Rossby numbers.

Such a variation of the temporal behavior of the dynamo solutions may or may not be associated with a change in dominant field geometry. This is of particular importance because it has been recently claimed by van Saders et al. (2016, 2019), Metcalfe & van Saders (2017), and Hall et al. (2021) that the Sun and solar-like stars older than the Sun may be undergoing a magnetic activity transition around a Rossby number of 1 (see Lorenzo-Oliveira et al. 2018 for an alternative view). In particular, they argue that the wind-braking efficiency may be collapsing around that rotational state transition. This would result in stars rotating more rapidly than what the Skumanich law or gyrochronology would have predicted (Skumanich 1972; Barnes 2003). If, for instance, a collapse of the large-scale dynamo modes (mainly dipole and quadrupole) would occur after transiting to antisolar DR, this would provide a very simple explanation, as it is well known that the most efficient wind braking for Sun-like stars is found for the simplest magnetic field geometry (Kawaler 1988; Réville et al. 2015; Finley & Matt 2018). In order to assess if such a change of magnetic geometry occurs at or near the R_of ∼ 1 limit, we will use a measure called f_dip, which was introduced by Christensen & Aubert (2006), and that permits the assessment of the energy content of the dipolar field with respect to the first 12 magnetic modes. We also introduce f_quad, using the same principle, as a quadrupolar field configuration is still quite efficient at spinning down a star via its associated wind braking. Both are defined as

$$\begin{eqnarray}&&{f}_{\mathrm{dip}}=\displaystyle \frac{{\sum }_{m}{\left({a}_{1,m}\right)}^{2}}{{\sum }_{l\lt 12,m}{\left({a}_{l,m}\right)}^{2}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{quad}}=\displaystyle \frac{{\sum }_{m}{\left({a}_{2,m}\right)}^{2}}{{\sum }_{l\lt 12,m}{\left({a}_{l,m}\right)}^{2}}\end{eqnarray} \tag{ 16 }$$

where a_l,m are the spherical harmonics coefficient of the radial magnetic field at the upper boundary (surface) of our models.

In Figure 18 we show f_dip (left panel) and f_quad (right panel) from 32 dynamo cases: the 15 cases analyzed in details in this paper, to which we add the 17 published in Strugarek et al. (2018), using the EULAG-MHD code (Smolarkiewicz & Charbonneau 2013). This allows us to extend our database and to compare nonlinear dynamo solutions obtained with two different MHD codes using very different numerical techniques, hence giving us confidence that the trend found in our simulations is not due to a given code. We observe a relatively good agreement between the ASH and EULAG databases for f_dip and surprisingly find that the EULAG set of simulations produces systematically a weaker f_quad compare to the ASH database. In both series we find a weak trend for a decrease of f_dip and f_quad with the Rossby number. Nevertheless, we do not find any hint of a collapse of f_dip or f_quad when the Rossby number exceeds 1 and the DR realized in the simulations becomes antisolar. The weak decreasing trend is not significant enough to explain the stalling of stellar wind braking advocated by van Saders et al. (2016) and Metcalfe et al. (2016). Hence, it seems unlikely that field geometry is the source of the wind-braking regime change for old solar-type stars. This is in agreement with the observational study of Vidotto et al. (2016), who have analyzed spectropolarimetric inversion for a suite of Sun-like stars, and they too did not find a collapse of the dipole strength as they crossed the Ro_f = 1 limit. So if such a stalling of stellar spin-down occurs, it must come from another mechanism (see Section 7).

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In summary, we have shown in Section 5 that the dynamo solutions presented in this study possess very interesting magnetic properties that agree very well with observations and other theoretical studies. In particular, we have confirmed the key role of the Rossby number (and magnetic Reynolds number) in determining the type of dynamo realized. Now we wish to characterize better their energy content and how energies flows back and forth from kinetic to magnetic reservoirs.

We now turn to discussing the global energy content in the convective envelope of the 15 dynamo cases presented in this study. In Table 5 we list the kinetic (KE) and magnetic (ME) energy densities and their axisymmetric and nonaxisymmetric components (see their definition in Appendix A and Brun et al. 2004). We first notice that as we increase the stellar mass, the KE is found to slightly decrease. This is due to the lower averaged mean density due to the shallower convective envelope in more massive stars. The averaged density over the simulated convective envelopes varies from 4 to 0.05 g cm⁻³ when going from models M05m to M11m, so a drop by a factor of 80. This is in part compensated by the higher luminosity (convective velocity) of the more massive stars, leading to values of KE in the range of 10⁶–10⁷ erg cm⁻³. Note that the total KE (i.e., the energy density multiplied by the volume) increases with stellar mass due to the much larger volume occupied by larger-mass stars. If we now decompose KE into its axisymmetric poloidal (MCKE) and toroidal (DRKE) and nonaxisymmetric (CKE) components, we can further understand how the energy is being distributed in the various models. First, as is often the case, MCKE is found to play a minor role in all models independently of their mass or rotation rates. In most cases, MCKE is of the order of 10⁴ erg cm⁻³ so about 1% or less of KE. This results in DRKE and CKE being the dominant components. Analyzing these two components, a clear trend is observed in common to all masses. As the rotation rate is increased, going from a Rossby number greater than 1 to a value less than about 0.1, we note that DRKE first increases to constitute up to 96% of KE. This means that most of the kinetic energy is in the DR with both strong latitudinal and radial shear across the convective envelope and at its base (we refer the reader to Section 4.1 where the angular velocity profiles of each model is discussed in details). Such a behavior is similar to what was observed in the purely hydrodynamic progenitors published in Brun et al. (2017). Hence, up to a certain rotational influence, the presence of dynamo-generated magnetic fields in the simulations does not modify significantly the trends observed before in the hydrodynamic cases. As a direct consequence, CKE is found to contribute less and less to the overall dynamics. CKE is found to be dominant for the slowly rotating cases, their convective motions having little azimuthal mean. As the Rossby number is decreased and the rotational influence on convective motions made stronger, we see that CKE drops to less than a few percent of the total KE. However, this is not the case when the rotational influence increases even further. For all the fastest cases with the smallest Rossby numbers, we notice a sudden drop of DRKE both in percentage and absolute value, while CKE contributes relatively more to KE (but KE also undergoes a decrease of its amplitude). This is due to the strong feedback of the Lorentz force on the DR, a phenomenon often called Ω quenching (Glatzmaier 1985a; Brun 2004; Brun et al. 2005; Karak et al. 2015) and seen only in global spherical rotating models by similitude to α quenching (Blackman & Field 2001; Brun et al. 2004; Subramanian & Brandenburg 2004) found in most local dynamo simulations (at the origin of the interface dynamo paradigm; Parker 1993; Mason et al. 2008) and characterized in our simulations by the absolute concomitant drop of CKE. This significant drop of DRKE or "Ω quenching," accompanied by a smaller decrease of CKE or "α quenching," leads to a strong decrease of KE. This confirms that dynamo simulations do not have the same rotational dependence as the purely hydrodynamic cases. Because most solar-like stars are likely to have magnetic fields, such a finding indicates that scaling laws derived in this work will likely be more accurate when compared to observations. Because the influence of magnetic field becomes more and more dominant as we lower Ro_f, it is also instructive to analyze how the magnetic energy content evolves as well.

In Table 5, we also provide the value of the ME densities (total magnetic energy (ME), axisymmetric poloidal (PME) and toroidal (TME) components, and nonaxisymmetric components (FME)). Here there are some surprises given what we just discussed for their KE counterparts MCKE, DRKE and FKE. First, the axisymmetric poloidal component PME contributes more to total ME than MCKE contributes to KE. It often represents a few percent of ME, and in one case, M11R5m, it is even found to be dominant. Interestingly, PME is found to reach its lowest values for intermediate rotators close to the Ro_f = 1 regime. In Ro_f > 1, we find that PME rises again, confirming the trend we observed on the total magnetic flux in Section 5.4. TME somewhat follows DRKE: it first increases with rotation rate, with more and more energy being pumped by the large-scale shear into toroidal ME via the dynamo Ω effect and also via complex convective motions. TME can reach values between 80% and 85% of the total ME. However, the Lorentz force feedback is so strong past a certain point that the large shear is quenched (the feedback destroying its generating source). In most of these highly rotationally constrained cases, the magnetic energy is found in the nonaxisymmetric magnetic field. These trends are also illustrated in Figure 19. It is worth noting that the three magnetic energies show an overall similar trend: the total energy density ME decreases with increasing Rossby number until Ro_f ≃ 1. The four models at Ro_f > 1 then exhibit a large scatter, and only PME shows an unambiguous increase with Rossby number in this regime. We also see a hint of a saturation and possibly a slight decrease of TME at low Rossby numbers. Additional simulations at even lower Rossby numbers are required to confirm this trend, which is to be expected based on the observed saturation of magnetic activity for fast rotators (see, e.g., Wright et al. 2011; Reiners et al. 2014). In all panels, we have indicated the inverse Rossby number trend as a gray dashed line. We remark that the three magnetic energy densities are all compatible with the ${{Ro}}_{{\rm{f}}}^{-1}$ trend at intermediate Rossby number, as expected from standard dynamo scaling laws in this regime (see Augustson et al. 2019). This translates into a bulk magnetic field ${B}_{\mathrm{bulk}}\propto {\mathrm{ME}}^{1/2}\propto {{Ro}}_{{\rm{f}}}^{-0.5}$. We note that this scaling does not necessarily translate into the same scaling for the surface large-scale magnetic field, as will be made clear in Section 7.

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The relative energies (shown as percentages in Table 5) also present interesting trends. We note first that in the slowly rotating cases, ME is only a few percent of KE. As we lower Ro_f, this value increases to reach equipartition by a subtle combination of both ME increasing while KE first increases and then decreases, as we have just seen. These variations inevitably lead to the fact that for the fastest rotating cases ME is even larger than KE and the simulations are in a so-called global super-equipartition state. This is very interesting, because it means that the kinetic energy in the convective envelope is not the maximal value that the ME can reach. This is due to a change in the force balance in the Navier–Stokes equation between the Lorentz, inertia, buoyancy, and Coriolis forces. As the rotation rate is increased and the Coriolis force becomes stronger and stronger, the balance at first shifts from being between mostly inertia and Lorentz force to a magnetostrophic state that implies a balance between Lorentz and Coriolis forces. We refer the reader to these following studies for more detailed discussions of dynamo scaling laws (Christensen 2010; Davidson 2014; Oruba & Dormy 2014; Brun et al. 2015; Augustson et al. 2019).

Overall, we see that the dynamo states reached in our 15 cases do not show a strong difference as a function of mass, at least in the range studied here. However, both in terms of amplitude of the magnetic field and in the time variability of the magnetic field (cyclic, unsteady or steady solutions), we confirm that rotation plays a key role in determining the type of dynamo found in our simulations. We also note that the mean axisymmetric magnetic fields are not negligible in most of the models, often reaching values of 5% of the total energy content for the poloidal field and a large fraction for the toroidal magnetic field. For the latter, this has important consequences for the energy made available for the various magnetic phenomena occurring at the surface of solar-like stars (see Section 7).

Note that we did not look for hysteresis around the Ro_f = 0.1 limit by running various cases with different values of the seed magnetic field, as was done in some geophysical dynamo studies (Schrinner et al. 2014). We consider stars to acquire their magnetic field through a complex formation process, in which the seed magnetic field is likely very weak (interstellar medium magnetic field amplitude are on averaged about 10–100 μG) and that starting the dynamo process with a weak seed field is the most likely scenario (Emeriau-Viard & Brun 2017). However, some studies have shown that weak and strong dynamo branches may exist under certain initial conditions (weak or strong seed magnetic field; Charbonneau 2004) or parameters such as the magnetic Prandtl number (Simitev & Busse 2009; Petitdemange 2018). Such weak or strong dynamo branches may explain some observed magnetic and rotational states seen in M dwarfs (Morin et al. 2011). Because this would depend on the local astrophysical context, we have decided to focus on the most common case of a weak seed magnetic field and refer the reader to these other complementary studies.

Having discussed how the KE and ME are distributed in our various models, we wish to go further in understanding exactly how these subtle balances come about. For this purpose we have computed the details of the energy transfers in our models, focusing on the mean axisymmetric components MCKE, DRKE, PME, and TME because large-scale fields and flows are of key astrophysical interest.

In this section we discuss the various energy transfers occurring in a rotating magnetized convective envelope. We refer the reader to Appendix A for the detailed derivation of the energy transfer equations, in which we have followed Starr & Gilman (1966) and Rempel (2006), generalizing their derivation to global 3D spherical geometry. We focus here on the energy budget for the mean (axisymmetric) fields in the convective envelope of our models. We decompose energies into toroidal (along the azimuth) and poloidal (in the meridional plane) components. The budgets can be summarized as

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{DRKE} & = & \mathop{\underbrace{{Q}_{\mathrm{RS}}^{\mathrm{DR}}}}\limits_{{\rm{Reynolds}}\,{\rm{stress}}}\mathop{\overbrace{-{Q}_{{\rm{\Omega }}}}}\limits^{{\rm{Omega}}\,{\rm{effect}}}\mathop{\underbrace{-{Q}_{C}}}\limits_{{\rm{Coriolis}}\,{\rm{force}}}\\ & & \times \,\mathop{\overbrace{-{Q}_{\mathrm{MS}}^{\mathrm{DR}}}}\limits^{{\rm{Maxwell}}\,{\rm{stress}}}\mathop{\underbrace{-{Q}_{\nu }^{\mathrm{DR}}}}\limits_{{\rm{Viscosity}}}\mathop{\overbrace{+{C}^{\mathrm{DR}}}}\limits^{{\rm{Curvature}}}\mathop{\underbrace{-{S}^{\mathrm{DR}}}}\limits_{{\rm{Boundaries}}},\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{MCKE} & = & \mathop{\underbrace{{Q}_{\mathrm{RS}}^{\mathrm{MC}}}}\limits_{{\rm{Reynolds}}\,{\rm{stress}}}\mathop{\overbrace{+{Q}_{\mathrm{TM}}^{\mathrm{MC}}}}\limits^{{\rm{Mixed}}\,{\rm{advection}}}\mathop{\underbrace{+{Q}_{\mathrm{MS}}^{\mathrm{MC}}}}\limits_{{\rm{Maxwell}}\,{\rm{stress}}}\\ & & \times \,\mathop{\overbrace{+{Q}_{{\rm{\nabla }}P}}}\limits^{{\rm{Pressure}}\,{\rm{work}}}\mathop{\underbrace{+{Q}_{C}}}\limits_{{\rm{Coriolis}}\,{\rm{force}}}\mathop{\overbrace{-{Q}_{\mathrm{PM}}^{\mathrm{MC}}}}\limits^{{\rm{Mixed}}\,{\rm{stresses}}}\\ & & \mathop{\underbrace{-{Q}_{b}}}\limits_{{\rm{Buoyancy}}}\mathop{\overbrace{-{Q}_{\nu }^{\mathrm{MC}}}}\limits^{{\rm{Viscosity}}}\mathop{\underbrace{+{C}^{\mathrm{MC}}}}\limits_{{\rm{Curvature}}}\mathop{\overbrace{-{S}^{\mathrm{MC}}}}\limits^{{\rm{Boundaries}}},\end{array}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{TME} & = & \mathop{\underbrace{{Q}_{{\rm{\Omega }}}}}\limits_{{\rm{Omega}}\,{\rm{effect}}}\mathop{\overbrace{+{Q}_{\mathrm{emf}}^{\mathrm{TM}}}}\limits^{{\rm{Elec.mot.force}}}\\ & & \times \,\mathop{\underbrace{-{Q}_{\mathrm{TM}}^{\mathrm{MC}}}}\limits_{{\rm{Mixed}}\,{\rm{advection}}}\mathop{\overbrace{-{Q}_{\eta }^{\mathrm{TM}}}}\limits^{{\rm{Ohmic}}\,{\rm{diffusion}}}\mathop{\underbrace{+{C}^{\mathrm{TM}}}}\limits_{{\rm{curvature}}}\mathop{\overbrace{-{S}^{\mathrm{TM}}}}\limits^{{\rm{Boundaries}}},\end{array}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{PME} & = & \mathop{\underbrace{{Q}_{\mathrm{PM}}^{\mathrm{MC}}}}\limits_{{\rm{Mixed}}\,{\rm{stresses}}}\mathop{\overbrace{+{Q}_{\mathrm{emf}}^{\mathrm{PM}}}}\limits^{{\rm{Elec.mot.force}}}\\ & & \times \,\mathop{\underbrace{-{Q}_{\eta }^{\mathrm{PM}}}}\limits_{{\rm{Ohmic}}\,{\rm{diffusion}}}\mathop{\overbrace{+{C}^{\mathrm{PM}}}}\limits^{{\rm{Curvature}}}\mathop{\underbrace{-{S}^{\mathrm{PM}}}}\limits_{{\rm{Boundaries}}},\end{array}\end{eqnarray} \tag{ 20 }$$

where all the different terms are detailed in Appendix A. We have computed individually each of the terms and show them normalized to the stellar luminosity in Figure 20, as a function of the fluid Rossby number of the models. For each model, we have averaged the balances (17)–(20) over typically 100 convective turnover time τ_c such that the sum of the terms is close to zero. Cyclic cases show large variations of the energy balance (we return to this point hereafter); in these cases, we averaged on a shorter time span chosen at cycle maximum. In addition, we have tabulated the transfers for three representative cases in Table 6 in units of both %L_⋆ and %L_⊙.

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Table 6. Dominant Energy Transfer Terms for Three Representative Cases (M07R5m—Low Rossby Number, M09R3m—Moderate Rossby Number, and M09R1m—High Rossby Number)

	M07R5m		M09R3m		M09R1m
	[%L⋆]	[%L⊙]	[%L⋆]	[%L⊙]	[%L⋆]	[%L⊙]
− QC	0.38	0.06	2.96	1.63	1.35	0.75
${Q}_{\mathrm{RS}}^{\mathrm{DR}}$	4.55	0.61	12.06	6.63	6.93	3.86
$-{Q}_{\nu }^{\mathrm{DR}}$	−0.53	−0.08	−13.97	−7.69	−7.43	−4.08
− QΩ	−0.43	−0.06	−0.84	−0.46	−0.004	−0.002
$-{Q}_{\mathrm{MS}}^{\mathrm{DR}}$	−3.84	−0.58	−0.70	−0.39	−0.05	−0.03
CDR	0.20	0.03	−0.26	−0.14	−0.50	−0.28
QC	−0.38	−0.06	−2.96	−1.63	−1.35	−0.75
${Q}_{\mathrm{RS}}^{\mathrm{MC}}$	0.15	0.02	0.29	0.16	0.15	0.08
$-{Q}_{\nu }^{\mathrm{MC}}$	−0.08	−0.01	−0.83	−0.46	−1.17	−0.64
Q∇P	1.36	0.20	2.65	1.46	2.17	1.20
− Qb	−0.87	−0.13	0.50	0.28	0.29	0.16
${Q}_{\mathrm{MS}}^{\mathrm{MC}}$	−0.17	−0.03	−0.03	−0.02	−0.002	−0.001
CMC	−0.01	−0.001	0.19	0.10	0.01	0.01
$-{Q}_{\eta }^{\mathrm{TM}}$	−0.44	−0.07	−0.54	−0.29	−0.004	−0.002
${Q}_{\mathrm{emf}}^{\mathrm{TM}}$	0.11	0.02	0.06	0.04	0.008	0.005
QΩ	0.43	0.06	0.84	0.46	0.004	0.002
$-{Q}_{\eta }^{\mathrm{PM}}$	−0.40	−0.06	−0.20	−0.11	−0.003	−0.002
${Q}_{\mathrm{emf}}^{\mathrm{PM}}$	0.42	0.06	0.27	0.15	0.005	0.003

Note. The strongest transfers for each case and each energy are identified in bold font. The four blocks of rows correspond in order to (i) the differential rotation kinetic energy balance (Equation (17)), (ii) the meridional circulation kinetic energy balance (Equation (18)), and (iii) the toroidal magnetic energy balance (Equation (19)) and the poloidal magnetic energy balance (Equation (20)). Some transfer terms are tiny and have thus been omitted from the table.

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The DR (upper-left panel of Figure 20) is always sustained primarily by Reynolds stresses in the models (as discussed in Section 4.2), with a dominant contribution of the radial component $\overline{{v}_{r}^{\prime} {v}_{\phi }^{\prime} }$ over the latitudinal component $\overline{{v}_{\theta }^{\prime} {v}_{\phi }^{\prime} }$. The cases exhibiting antisolar DR (Rossby number larger than 1) present a reversal of the latter term, showing that the latitudinal component of the Reynolds stress is detrimental to the DR KE in these cases. The magnetic contributions Q_Ω (blue) and ${Q}_{\mathrm{MS}}^{\mathrm{DR}}$ (red) start playing a significant role for fast-rotating cases (low Rossby numbers; see model M07R5m in Table 6), sometimes even dominating completely viscous dissipation (${Q}_{\nu }^{\mathrm{DR}}$, purple). In all cases the magnetic contributions tend to oppose the DR, as seen in Section 4.1. The power associated with the maintenance of DR can reach about 30% of the stellar luminosity and drops at minimum to about 4% in our sample of models. We remark that simulations with fluid Rossby numbers around Ro_f ∼ 0.2 achieve the most powerful maintenance of DR that can reach values up to 17% of the solar luminosity. At larger Rossby numbers, the star does not rotate fast enough and the DR is weakly maintained. At lower Rossby numbers, the magnetic feedback from the dynamo field is so efficient that the power associated with the maintenance of DR decreases significantly.

The meridional circulation energy balance (upper right panel of Figure 20) is dominated by a balance between the work of pressure (Q_∇P , cyan), buoyancy (Q_b , blue-green) and Coriolis (Q_c , green) forces (see also Table 6 where the dominant transfer terms are highlighted in bold font). The latter almost always remains negative, indicating an energy transfer from the meridional flow to the DR when models are in a steady state. Viscous dissipation (purple) plays a much lesser role for MCKE compared to DRKE, and magnetic contributions can be considered as negligible, except maybe for small Rossby number cases possessing transequatorial meridional cells (see Figure 10). We find that the relative contribution of buoyancy and pressure gradients varies from model to model and also varies in time for each model. We believe that is due to the anelastic approximation used in this study, and expect that a Lantz–Braginsky formulation (Brown et al. 2012) would lead to more systematic relative contributions of these two important terms for MCKE. Finally, we note that the power associated with the meridional circulation maintenance increases with Rossby number and does not go above 15% of the stellar luminosity in our sample.

Let us now turn to the power sustaining magnetism in our models. The toroidal (TME) and poloidal (PME) magnetic energy budgets are shown in the left and right lower panels of Figure 20. We immediately note that the power sustaining magnetism corresponds at maximum to 3% of the stellar luminosity in our sample for TME. This corresponds to an absolute maximum of 6% of the solar luminosity. A very large amount of power is therefore indeed channeled to sustain the large toroidal ME reservoir that the dynamo builds up in the simulations. Hence, it is expected that a significant proportion of this large magnetic energy reservoir will be accessible to trigger various surface magnetic activity events (Shibata et al. 2013). The power associated with PME is a bit weaker but still reaches up to 0.4% of the stellar luminosity. We find again that the most powerful transfers occur around R_of ∼ 0.2. The power involved saturates for lower Rossby numbers, which is reminiscent of the saturation of magnetic activity observed in the X-ray luminosity of fast-rotating stars (e.g., Wright et al. 2011). It slowly drops for large Rossby numbers, but the power maintains a value of at least 0.01% of the stellar luminosity even in our most slowly rotating models. These figures are in good qualitative agreement with the value of 0.1% found for the Sun by Rempel (2006) using 2.5D mean-field dynamo models. Let us stress again that with values ranging in our sample between 0.01% and 3% of the star's luminosity, this is a massive reservoir of ME extracted by dynamo action.

The PME balance is relatively straightforward: it is sustained primarily by the turbulent electromotive force originating from the convective motions (${Q}_{\mathrm{EMF}}^{\mathrm{PM}}$, yellow) and opposed by ohmic dissipation (purple). Mixed stresses involving the mean meridional flow (${Q}_{\mathrm{PM}}^{\mathrm{MC}}$, salmon) are not observed to play any major role here. The TME balance is slightly more complex. In most of our models, it is primarily sustained by the Omega effect (Q_Ω, blue), and saturated by ohmic dissipation (purple). Interestingly, we find that the role of the turbulent electromotive force can change from one model to the other (see Table 4), and it can even change sign with time in our cyclic solutions.

This is highlighted in Figure 21 where we observe how the various transfer terms for TME vary during one long cycle for model M09R3m in the left panel (TME is overplotted in black) and one short cycle for model M07R5m in the right panel. First, we observe that the amplitude of the transfers vary by an order of magnitude along the long cycle (left panel), being maximum when the magnetic energy is maximum as one may expect. We also see that electromotive force (yellow) plays a dominant role when TME increases right after cycle minimum and then switches sign and draws energy from TME when TME decreases. This striking behavior is at odds with the classical picture of constant-in-time parameterization of mean-field coefficients. It furthermore supports our interpretation that the dynamo processes behind the decadal magnetic cycles observed in some models involve a complex interplay between sources and sinks of magnetic energy that vary at different stages of the cycle. This is important because it reinforces the conclusions drawn in Section 5 about the special nature of the long-cycle period dynamo simulations presented in this study. We also see that the short cycle (right panel) behaves differently than the long cycle on the left. In model M07R5m, the electromotive force sometimes equates or even dominates the Ω effect while still being balanced by ohmic dissipation. In this case, the amplitude of the transfer terms vary much less with time, and we recover a behavior expected for α² − Ω dynamos. These simulations could therefore be categorized either as α − Ω or α² − Ω dynamos depending on the phases of evolution. We observe that the SVD analysis discussed in Section 5.1 and Appendix B shows coherent results when we take into account these temporal variations of the production terms, as shown in Figure 21. Given the highly time-dependent nature of these nonlinear convective dynamo simulations, the analysis presented in this section about their dynamical properties is more robust than the SVD decomposition we performed in Appendix B as a companion analysis, because it does not assume any scale-separation approximation.

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We follow the approach of Starr & Gilman (1966) and write the energy budgets in the following way (see the schematic in Figure 23):

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{DRKE} & = & {Q}_{\mathrm{RS}}^{\mathrm{DR}}-{Q}_{{\rm{\Omega }}}-{Q}_{C}-{Q}_{\mathrm{MS}}^{\mathrm{DR}}\\ & & -\,{Q}_{\nu }^{\mathrm{DR}}+{C}^{\mathrm{DR}}-{S}^{\mathrm{DR}},\end{array}\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\mathrm{MCKE} & = & {Q}_{\mathrm{RS}}^{\mathrm{MC}}+{Q}_{\mathrm{TM}}^{\mathrm{MC}}+{Q}_{\mathrm{MS}}^{\mathrm{MC}}+{Q}_{{\rm{\nabla }}P}\\ & & +\,{Q}_{C}-{Q}_{\mathrm{PM}}^{\mathrm{MC}}-{Q}_{b}-{Q}_{\nu }^{\mathrm{MC}}+{C}^{\mathrm{MC}}-{S}^{\mathrm{MC}},\end{array}\end{eqnarray} \tag{ A4 }$$

$$\begin{eqnarray}&&{\partial }_{t}\mathrm{TME}={Q}_{{\rm{\Omega }}}+{Q}_{\mathrm{emf}}^{\mathrm{TM}}-{Q}_{\mathrm{TM}}^{\mathrm{MC}}-{Q}_{\eta }^{\mathrm{TM}}+{C}^{\mathrm{TM}}-{S}^{\mathrm{TM}},\end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray}&&{\partial }_{t}\mathrm{PME}={Q}_{\mathrm{PM}}^{\mathrm{MC}}+{Q}_{\mathrm{emf}}^{\mathrm{PM}}-{Q}_{\eta }^{\mathrm{PM}}+{C}^{\mathrm{PM}}-{S}^{\mathrm{PM}}.\end{eqnarray} \tag{ A6 }$$

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In all that follows, quantities are separated into mean and fluctuating components through

$$\begin{eqnarray}&&A=\bar{A}+A^{\prime} ,\end{eqnarray} \tag{ A7 }$$

and the corresponding terms in the original derivation of Starr & Gilman (1966) are given in bold "SG66: [XX]" labels at the end of each equation, where XX is the term or equation number in Starr & Gilman (1966). Note that we have extra curvature terms C^x due to our choice of spherical coordinates.

The various terms of Equation (A3) are

$$\begin{eqnarray}&&{Q}_{\mathrm{RS}}^{\mathrm{DR}}=\displaystyle \int \hat{\rho }\left[\overline{v{{\prime} }_{r}v{{\prime} }_{\phi }}{\partial }_{r}\overline{{v}_{\phi }}+\overline{v{{\prime} }_{\theta }v{{\prime} }_{\phi }}\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{v}_{\phi }}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[6c+6d]}}\end{eqnarray} \tag{ A8 }$$

$$\begin{eqnarray}&&{Q}_{{\rm{\Omega }}}=\int \displaystyle \frac{1}{4\pi }\overline{{B}_{\phi }}\left[\overline{{B}_{r}}{\partial }_{r}\overline{{v}_{\phi }}+\displaystyle \frac{\overline{{B}_{\theta }}}{r}{\partial }_{\theta }\overline{{v}_{\phi }}\right]{dV},\,{\rm{}}\,{\bf{SG66:}}\,{\bf{[5a+5b]}}\end{eqnarray} \tag{ A9 }$$

$$\begin{eqnarray}&&{Q}_{C}=\int 2{\rm{\Omega }}\hat{\rho }\overline{{v}_{\phi }}\left[\cos \theta \overline{{v}_{\theta }}+\sin \theta \overline{{v}_{r}}\right]{dV},\,{\rm{}}\,{\bf{SG66:}}\,{\bf{[6a+6b]}}\end{eqnarray} \tag{ A10 }$$

$$\begin{eqnarray}&&\displaystyle \begin{array}{c}{Q}_{\mathrm{MS}}^{\mathrm{DR}}=\int \frac{1}{4\pi }\overline{B{{\prime} }_{r}B{{\prime} }_{\phi }}{\partial }_{r}\bar{{v}_{\phi }}+\frac{1}{4\pi }\frac{\overline{B{{\prime} }_{\theta }B{{\prime} }_{\phi }}}{r}{\partial }_{\theta }\bar{{v}_{\phi }}{dV},\\ \qquad \,\,{\bf{SG66:}}\,{\bf{[7a+7b]}}\end{array}\end{eqnarray} \tag{ A11 }$$

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{\nu }^{\mathrm{DR}}=\displaystyle \int \nu \hat{\rho }\left\{{\left[r{\partial }_{r}\left(\displaystyle \frac{\overline{{v}_{\phi }}}{r}\right)\right]}^{2}+{\left[\displaystyle \frac{\sin \theta }{r}{\partial }_{\theta }\left(\displaystyle \frac{\overline{{v}_{\phi }}}{\sin \theta }\right)\right]}^{2}\right\}{dV},\\ {\rm{}}\,{\bf{SG66:}}\,{\bf{[Fz]}}\end{array}\end{eqnarray} \tag{ A12 }$$

$$\begin{eqnarray}&&\begin{array}{l}{C}^{\mathrm{DR}}=\displaystyle \int -\displaystyle \frac{\hat{\rho }\overline{{v}_{\phi }}}{r}\left[\overline{{v}_{r}{v}_{\phi }}+\cot \theta \overline{{v}_{\theta }{v}_{\phi }}\right]\\ \,+\displaystyle \frac{1}{4\pi }\displaystyle \frac{\overline{{v}_{\phi }}}{r}\left[\overline{{B}_{r}{B}_{\phi }}+\cot \theta \overline{{B}_{\theta }{B}_{\phi }}\right]{dV},\end{array}\end{eqnarray} \tag{ A13 }$$

$$\begin{eqnarray}&&{S}^{\mathrm{DR}}=-\displaystyle \frac{1}{4\pi }{\int }_{r={R}_{\mathrm{top}}}\overline{{B}_{r}{B}_{\phi }}\overline{{v}_{\phi }}{dS}.{\rm{}}\,{\bf{SG66:}}\,{\bf{[Eq.}}\,{\bf{6]}}.\end{eqnarray} \tag{ A14 }$$

Q_C was defined previously in Equation (A10). The remaining terms in Equation (A4) are

$$\begin{eqnarray}&&\begin{array}{c}{Q}_{\mathrm{RS}}^{\mathrm{MC}}=\int \hat{\rho }\overline{v{{\prime} }_{r}v{{\prime} }_{\theta }}\left[{\partial }_{r}\bar{{v}_{\theta }}+\frac{1}{r}{\partial }_{\theta }\bar{{v}_{r}}\right]\\ +\,\hat{\rho }\bar{v{{\prime} }_{r}^{2}}{\partial }_{r}\bar{{v}_{r}}+\hat{\rho }\bar{v{{\prime} }_{\theta }^{2}}\frac{1}{r}{\partial }_{\theta }\bar{{v}_{\theta \,}}{dV},\,{\bf{SG66:}}\,{\bf{[3a+3b+3c]}}\end{array}\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{\mathrm{TM}}^{\mathrm{MC}}=\displaystyle \int \displaystyle \frac{1}{8\pi }{\overline{{B}_{\phi }}}^{2}\left[\displaystyle \frac{1}{{r}^{2}}{\partial }_{r}\left({r}^{2}\overline{{v}_{r}}\right)+\displaystyle \frac{1}{r\sin \theta }{\partial }_{\theta }\left(\sin \theta \overline{{v}_{\theta }}\right)\right]{dV},\\ {\bf{SG66:}}\,{\bf{[4a]}}\end{array}\end{eqnarray} \tag{ A16 }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{MS}}^{\mathrm{MC}} & = & \displaystyle \int \displaystyle \frac{1}{8\pi }\left[-\overline{B{{\prime} }_{r}^{2}}+\overline{B{{\prime} }_{\theta }^{2}}+\overline{B{{\prime} }_{\phi }^{2}}\right]{\partial }_{r}\overline{{v}_{r}}\\ & & +\,\displaystyle \frac{1}{8\pi }\left[\overline{B{{\prime} }_{r}^{2}}-\overline{B{{\prime} }_{\theta }^{2}}+\overline{B{{\prime} }_{\phi }^{2}}\right]\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{v}_{\theta }}\\ & & -\,\displaystyle \frac{1}{4\pi }\overline{B{{\prime} }_{r}{B}_{\theta }^{{\prime} }}\left[{\partial }_{r}\overline{{v}_{\theta }}+\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{v}_{r}}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[8a]}}\end{array}\end{eqnarray} \tag{ A17 }$$

$$\begin{eqnarray}&&{Q}_{{\rm{\nabla }}P}=\int -\left[\overline{{v}_{r}}{\partial }_{r}\overline{p}+\overline{{v}_{\theta }}\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{p}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[1a]}}\end{eqnarray} \tag{ A18 }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{PM}}^{\mathrm{MC}} & = & \displaystyle \int \displaystyle \frac{1}{4\pi }\left[\overline{{B}_{r}}\overline{{v}_{\theta }}-\overline{{v}_{r}}\overline{{B}_{\theta }}\right]\\ & \cdot & \left[-{\partial }_{r}\overline{{B}_{\theta }}+\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{B}_{r}}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[9a]}}\end{array}\end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray}&&{Q}_{b}=\int \bar{\rho }g\overline{{v}_{r}}{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[2a]}}\end{eqnarray} \tag{ A20 }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\nu }^{\mathrm{MC}} & = & \displaystyle \int 2\,\nu \hat{\rho }\left\{{\left[{\partial }_{r}\overline{{v}_{r}}-\displaystyle \frac{1}{3}\overline{({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}})}\right]}^{2}\right.\\ & & +\,{\left[\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{v}_{\theta }}+\displaystyle \frac{\overline{{v}_{r}}}{r}-\displaystyle \frac{1}{3}\overline{({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}})}\right]}^{2}\\ & & +\,{\left[\displaystyle \frac{\overline{{v}_{r}}}{r}+\displaystyle \frac{\overline{{v}_{\theta }}\cos \theta }{r\sin \theta }-\displaystyle \frac{1}{3}\overline{({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}})}\right]}^{2}\\ & & +\,\left.\displaystyle \frac{1}{2}{\left[\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{v}_{r}}+r{\partial }_{r}\left(\displaystyle \frac{\overline{{v}_{\theta }}}{r}\right)\right]}^{2}\right\}{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[Fm]}}\end{array}\end{eqnarray} \tag{ A21 }$$

$$\begin{eqnarray}\begin{array}{rcl}{C}^{\mathrm{MC}} & = & \displaystyle \int \displaystyle \frac{\hat{\rho }}{r}\left[\overline{{v}_{r}}\left(\overline{{v}_{\theta }^{2}}+\overline{{v}_{\phi }^{2}}\right)-\overline{{v}_{\theta }}\left(\overline{{v}_{r}{v}_{\theta }}-\cot \theta \overline{{v}_{\phi }^{2}}\right)\right]\\ & & -\,\displaystyle \frac{\overline{{v}_{r}}}{4\pi r}\left(\overline{{B}_{\theta }^{2}}+\overline{{B}_{\phi }^{2}}\right)+\displaystyle \frac{\overline{{v}_{\theta }}}{4\pi r}\left(\overline{{B}_{r}{B}_{\theta }}-\cot \theta \overline{{B}_{\phi }^{2}}\right)\\ & & +\,\displaystyle \frac{1}{8\pi r}\overline{B{{\prime} }^{2}}\left(2\overline{{v}_{r}}+\cot \theta \overline{{v}_{\theta }}\right){dV},\end{array}\end{eqnarray} \tag{ A22 }$$

$$\begin{eqnarray}&&{S}^{\mathrm{MC}}=-\displaystyle \frac{1}{4\pi }{\int }_{r={R}_{\mathrm{top}}}\overline{B{{\prime} }_{r}{B}_{\theta }^{{\prime} }}\overline{{v}_{\theta }}{dS}.{\rm{}}\,{\bf{SG66:}}\,{\bf{[Eq.}}\,{\bf{8]}}.\end{eqnarray} \tag{ A23 }$$

Q_Ω and ${Q}_{\mathrm{TM}}^{\mathrm{MC}}$ were defined previously in Equations (A9) and (A16). The remaining terms in Equation (A5) are

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{Q}_{\mathrm{emf}}^{\mathrm{TM}} & = & \displaystyle \int \displaystyle \frac{1}{4\pi }\left[\overline{B{{\prime} }_{\phi }v{{\prime} }_{r}}-\overline{B{{\prime} }_{r}v{{\prime} }_{\phi }}\right]{\partial }_{r}\overline{{B}_{\phi }}\\ & & +\,\displaystyle \frac{1}{4\pi }\left[\overline{B{{\prime} }_{\phi }v{{\prime} }_{\theta }}-\overline{B{{\prime} }_{\theta }v{{\prime} }_{\phi }}\right]\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{B}_{\phi }}{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[4b}}\,{\boldsymbol{+}}\,{\bf{4c]}}\end{array}\end{eqnarray} \tag{ A24 }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\eta }^{\mathrm{TM}} & = & \displaystyle \int -\displaystyle \frac{\overline{{B}_{\phi }}}{4\pi r}\left[{\partial }_{r}\left\{\eta {\partial }_{r}\left(r\overline{{B}_{\phi }}\right)\right\}\right.\\ & & +\,\left.\displaystyle \frac{\eta }{r}{\partial }_{\theta }\left\{\displaystyle \frac{1}{\sin \theta }{\partial }_{\theta }\left(\sin \theta \overline{{B}_{\phi }}\right)\right\}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[Jz]}}\end{array}\end{eqnarray} \tag{ A25 }$$

$$\begin{eqnarray}\begin{array}{rcl}{C}^{\mathrm{TM}} & = & \displaystyle \int -\displaystyle \frac{\overline{{B}_{\phi }}}{4\pi r}\left[\overline{{B}_{r}{v}_{\phi }}+\cot \theta \overline{{B}_{\theta }{v}_{\phi }}\right.\\ & & \left.-\,(\overline{{B}_{\phi }{v}_{r}}+\cot \theta \overline{{B}_{\phi }{v}_{\theta }})\right]{dV},\end{array}\end{eqnarray} \tag{ A26 }$$

$$\begin{eqnarray}&&{S}^{\mathrm{TM}}=-\displaystyle \frac{1}{4\pi }{\int }_{r={R}_{\mathrm{top}}}\overline{B{{\prime} }_{r}v{{\prime} }_{\phi }}\overline{{B}_{\phi }}{dS}.{\rm{}}\,{\bf{SG66:}}\,{\bf{[Eq.}}\,{\bf{7]}}.\end{eqnarray} \tag{ A27 }$$

${Q}_{\mathrm{MC}}^{\mathrm{PM}}$ have already been defined in Equation (A19). The remaining terms in Equation (A6) are

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{emf}}^{\mathrm{PM}} & = & \displaystyle \int \displaystyle \frac{1}{4\pi }\left[\overline{B{{\prime} }_{r}v{{\prime} }_{\theta }}-\overline{B{{\prime} }_{\theta }v{{\prime} }_{r}}\right]\\ & \cdot & \left[-{\partial }_{r}\overline{{B}_{\theta }}+\displaystyle \frac{1}{r}{\partial }_{\theta }\overline{{B}_{r}}\right]{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[9b]}}\end{array}\end{eqnarray} \tag{ A28 }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\eta }^{\mathrm{PM}} & = & \displaystyle \int \displaystyle \frac{\overline{{B}_{r}}}{4\pi {r}^{2}\sin \theta }\eta {\partial }_{\theta }\left\{\sin \theta \left[{\partial }_{r}\left(r\overline{{B}_{\theta }}\right)-{\partial }_{\theta }\overline{{B}_{r}}\right]\right\}\\ & & -\,\displaystyle \frac{\overline{{B}_{\theta }}}{4\pi r}{\partial }_{r}\left\{\eta \left[{\partial }_{r}\left(r\overline{{B}_{\theta }}\right)-{\partial }_{\theta }\overline{{B}_{r}}\right]\right\}{dV},{\rm{}}\,{\bf{SG66:}}\,{\bf{[Jm]}}\end{array}\end{eqnarray} \tag{ A29 }$$

$$\begin{eqnarray}&&{C}^{\mathrm{PM}}=\int \displaystyle \frac{\overline{{B}_{\theta }}}{4\pi r}\left[\overline{{B}_{\theta }{v}_{r}}-\overline{{B}_{r}{v}_{\theta }}\right]{dV},\end{eqnarray} \tag{ A30 }$$

$$\begin{eqnarray}&&{S}^{\mathrm{PM}}=-\displaystyle \frac{1}{4\pi }{\int }_{r={R}_{\mathrm{top}}}\overline{{B}_{r}{v}_{\theta }}\overline{{B}_{\theta }}{dS}.{\rm{}}\,{\bf{SG66:}}\,{\bf{[Eq.}}\,{\bf{9]}}.\end{eqnarray} \tag{ A31 }$$