Modeling of differential rotation in rapidly rotating solartype stars
Abstract
We investigate differential rotation in rapidly rotating solartype stars by means of an axisymmetric mean field model that was previously applied to the sun. This allows us to calculate the latitudinal entropy gradient with a reasonable physical basis. Our conclusions are as follows: (1) Differential rotation approaches the TaylorProudman state when stellar rotation is faster than solar rotation. (2) Entropy gradient generated by the attached subadiabatic layer beneath the convection zone becomes relatively small with a large stellar angular velocity. (3) Turbulent viscosity and turbulent angular momentum transport determine the spatial difference of angular velocity . (4) The results of our mean field model can explain observations of stellar differential rotation.
1 Introduction
Our sun has an elevenyear magnetic activity cycle, which is thought to be sustained by the dynamo motion of internal ionized plasma, i.e., a transformation of kinetic energy to magnetic energy (Parker, 1955). Our understanding of the solar dynamo has significantly improved during the past fifty years, and some kinematic studies can now reproduce solar magnetic features such as equatorward migration of sunspots and poleward migration of the magnetic field (Choudhuri et al., 1995; Dikpati & Charbonneau, 1999; Küker et al., 2001; Charbonneau, 2005; Hotta & Yokoyama, 2010a, b). The most important mechanism of the solar dynamo is the effect, the bending of preexisting poloidal magnetic fields by differential rotation and the generation of toroidal magnetic fields. Thus, the distribution of the differential rotation in the convection zone is a significant factor for the solar dynamo. Using helioseismology, it has recently been shown that the solar internal differential rotation is in a nonTaylorProudman state (see review by Thompson et al., 2003), meaning the isorotation surfaces are not parallel to the axis.
Based on solar observations, it is known that Ca HK fluxes can be a signature of stellar chromospheric activity, and such chromospheric signatures are in correlation with magnetic activity. Wilson (1968, 1978) and Baliunas et al. (1995) discuss a class of stars that shows a periodic variation in Ca HK fluxes, which suggests that they have a magnetic cycle similar to our sun. It is natural to conjecture that such magnetic activity is maintained by dynamo action. Various studies have been conducted to investigate the relationship between stellar angular velocity and its latitudinal difference i.e., , where the suggested range of is (Donahue et al., 1996; Reiners & Schmitt, 2003; Barnes et al., 2005). This means that the angular velocity difference increases and the relative difference decreases with increases in the stellar rotation rate .
In this paper, we investigate differential rotation in rapidly rotating stars using a mean field framework. Our study is based on the work of Rempel (2005b), in which he suggests the importance of the role of the subadiabatic layer below the convection zone in order to maintain a nonTaylorProudman state in the Sun. The aim of this paper is to use a mean field model to analyze firstly the dependence of the morphology of differential rotation on stellar angular velocity, and secondly the physical process which determines the observable angular velocity difference . According to our knowledge, this is the first work which systematically discusses the application of Rempel’s (2005b) solar model to stars.
Other research adopts another approach to the use of mean field models for the analysis of differential rotation in rapidlyrotating stars (Kitchatinov & Rüdiger, 1995; Küker & Stix, 2001). In these studies, the nonTaylorProudman state is sustained by anisotropy of turbulent thermal conduction. This anisotropy is generated by the effects of stellar rotation on convective turbulence.
Threedimensional numerical studies on stellar differential rotation also exist (Brown et al., 2008; Miesch & Toomre, 2009). In these studies, they resolve stellar thermal driven convection and can calculate a selfconsistent turbulent angular momentum transport and anisotropy of turbulent thermal conductivity. The subadiabatic layer below the convection zone, however, is not included. The effects of anisotropy of turbulent thermal conductivity and the subadiabatic layer are discussed in this paper.
2 Model
Using numerical settings similar to those of Rempel’s (2005b), we solve the axisymmetric hydrodynamic equations in spherical geometry , where is the radius, and is the colatitude. The basic assumptions are as follows.

A mean field approximation is adopted. All processes on the convective scale are parameterized. Thus, the coefficients for turbulent viscosity, turbulent heat conductivity, and turbulent angular momentum transport are explicitly given in the equations.

The perturbations of the density and pressure associated with differential rotation are small, i.e., and . Here and denote the reference state density and pressure respectively, whereas and are the perturbations. We neglect the secondorder terms of these quantities. Note that the perturbation of angular velocity () and meridional flow (, ) are not small.

Since the reference state is assumed to be in an energy flux balance, the entropy equation includes only perturbations.
2.1 Equations
We do not use the anelastic approximation here. The equations in an inertial frame can be expressed as
(1)  
(2)  
(3)  
(4)  
(5) 
where is a constant value that represents the angular velocity of the rigidly rotating radiative zone. We set it as a parameter in Table 1. is the ratio of specific heats, with the value for an ideal gas being . is the coefficient of turbulent thermal conductivity. represents superadiabaticity, where (see §2.2). denotes gravitational acceleration. Following from this, the perturbation of pressure and pressure scale height are expressed as
(6)  
(7) 
is dimensionless entropy normalized by the specific heat capacity at constant volume . Turbulent viscous force follows from
(8) 
(9) 
(10) 
with the Reynolds stress tensor
(11) 
Here is the coefficient of turbulent viscosity and is the coefficient of the effect (Kitchatinov & Rüdiger, 1995), a nondiffusive angular momentum transport caused by turbulence. and are expected to have the same value, since both effects are caused by turbulence, i.e., thermal driven convection. We discuss this in more detail in §2.3. denotes the deformation tensor, which is given in spherical coordinates by
(12)  
(13)  
(14)  
(15)  
(16)  
(17) 
An expression for the effect () is given later. The amount of energy that is converted by the Reynolds stress from kinematic energy to internal energy is given by
(18) 
2.2 Background Stratification
We use an adiabatic hydrostatic stratification for the spherically symmetric reference state of , and . Gravitational acceleration is assumed to have dependence, since the radiative zone () has most of the solar mass. This is expressed as,
(19)  
(20)  
(21)  
(22) 
where , , , and denote the values at the base of the convection zone of density, pressure, temperature, pressure scale height and gravitational acceleration, respectively. In this study we use , with representing the solar radius (). We adopt solar values , , and , where is the Boltzmann constant, and is the mean particle mass. Fig. 1 shows the profiles of background density, pressure and temperature, and gravitational acceleration.
Although the real sun’s stratification is not adiabatic in the convection zone, our reference state is valid, since the absolute value of superadiabaticity is small. In order to include the deviation from adiabatic stratification, we assume superadiabaticity has the following profile:
(23) 
Here and denote the values of superadiabaticity in the overshoot region. and denote the position and the steepness of the transition toward the subadiabatically stratified overshoot region, respectively. Superadiabaticity in convection zone is define as
(24) 
where denotes the location of the upper boundary. We specify , , and in our simulations. is took as a free parameter. The entropy gradient can be expressed as
(25) 
The third term of eq. (5), , includes the effect of deviations from adiabatic stratification. The term indicates that an upflow (downflow) can make negative (positive) entropy perturbations in the subadiabatically stratified layers .
2.3 Diffusivity Profile
We assume the coefficients of turbulent viscosity and thermal conductivity to be constant within the convection zone, and these smoothly connect with the values of the overshoot region. We assume that the diffusivities only depend on the radial coordinate:
(26)  
(27)  
(28) 
with
(29)  
(30) 
where , and are the values of the turbulent diffusivities within the convection zone, and and are the values in the overshoot region. We specify , and , and we treat as a parameter. specifies the values of the turbulent diffusivities at , i.e., , and at . and are the widths of transition. We specify , and . As already mentioned, the coefficients for turbulent viscosity and the effect are different in our model from those of Rempel’s (2005b). There are two reasons for this. One is that we intend to investigate the influence of both effects on stellar differential rotation separately (see §4.2). The other reason is that the formation of a tachocline in a reasonable amount of time requires a finite value (though small) for the coefficient of turbulent viscosity even in the radiative zone, in which there is likely to be weak turbulence (Rempel, 2005b). Fig. 2 shows the profiles of , and .
2.4 The Effect
In this study we adopt the nondiffusive part of the Reynolds stress, called the effect. The effect transports angular momentum and generates differential rotation. The effect tensors are expressed as
(31)  
(32) 
where is the amplitude of the effect and is the inclination of the flux vector with respect to the rotational axis. We use for the amplitude of the effect the expressions
(33)  
(34) 
where . and are freeparameters. The value of needs to be equal to or larger than 2 to ensure regularity near the pole, so we set . The effect does not depend on , or , meaning it is a stationary effect. We emphasize that the effect depends on stellar angular velocity , since the effect is generated by turbulence and Coriolis force. The more rapidly the star rotates, the more angular momentum the effect can transport. The dependence of and on stellar angular velocity is discussed in §4.3.
2.5 Numerical Settings
Using the modified LaxWendroff scheme with TVD artificial viscosity (Davis, 1984), we solve Equations (1)(5) numerically for the northern hemisphere of the meridional plane in and . We use a uniform resolution of points in the radial direction and points in the latitudinal direction in all of our simulations. Each simulation run is conducted until it reaches a stationary state. All the variables , , , and are equal to zero in the initial condition. At the top boundary () we adopt stressfree boundary conditions for , and and set the derivative of to zero:
(35)  
(36)  
(37)  
(38) 
The boundary conditions for , and at the lower boundary () are the same as those at the top boundary. Differential rotation connects with the rigidly rotating core at the lower boundary, so we adopt there. At both radial boundaries, we set to make the right side of eq. (2) equal zero. At the pole and the equator ( and ) we use the symmetric boundary condition:
(39)  
(40)  
(41)  
(42)  
(43) 
Due to the low Mach number of the expected flows, a direct compressible simulation is problematic, so adopting the same technique as Rempel (2005b), we reduce the speed of sound by multiplying the right side of eq. (1) by . The equation of continuity is therefore replaced with
(44) 
The speed of sound then becomes times smaller than the original speed. We use in all our calculations. This technique can be used safely in our present study since we only discuss stationary states, so the factor becomes unimportant. The validity of this technique is carefully discussed by Rempel (2005b). We test our code by reproducing the results presented by Rempel (2005b) and check the numerical convergence by runs with different grid spacings. After checking and cleaning up at every time step, conservation of total mass, total angular momentum and total energy are maintained through the simulation runs.
3 Stellar Differential Rotation and the TaylorProudman Theorem
In this section, based on the work of Rempel (2005b), we explain how the subadiabatically stratified region can generate solarlike differential rotation. The component of the vorticity equation can be expressed as
(45) 
where , and the axis represents the rotational axis. The inertial term and the diffusion term are neglected. If the last term of eq. (45) is zero, meaning there is no variation in entropy in the latitudinal direction, then in a stationary state, which is the TaylorProudman state. Solarlike differential rotation is generated in four stages.

In the northern hemisphere, the effect transports angular momentum in the negative direction and generates a negative .

The negative generates a negative due to Coriolis force. This counterclockwise meridional flow corresponds to a negative (downflow) at high latitudes and a positive (upflow) at low latitudes.

As we mentioned in Section 2.2, downflow (upflow) generates positive (negative) entropy perturbations in the subadiabatically stratified layer beneath the convection zone (). Meridional flow can generate positive entropy perturbations at high latitudes and negative entropy perturbations at low latitudes. Therefore, becomes negative in the overshoot region.

The negative also keeps negative in a stationary state.
The profile of angular velocity in the convection zone is determined by a balance of angular momentum transport from meridional flow and a reduction in meridional flow from buoyancy force at the subadiabatic layer.
4 Results and Discussion
We run simulations for seventeen cases, with Table 1 showing the parameters for each case.
4.1 Stellar Differential Rotation
In this section, we discuss the cases with angular velocities up to 16 times the solar value (represented by ), placing an emphasis on the morphology of stellar differential rotation. Fig. 3 shows the results of our calculations which correspond to cases 15 in Table 1. It is found that the larger stellar angular velocity is, the more likely it is for differential rotation to be in the TaylorProudman state, in which the contour lines of the angular velocity are parallel to the rotational axis. To evaluate these results quantitatively, we define a parameter which denotes the morphology of differential rotation. We call it the NonTaylorProudman parameter (hereafter the NTP parameter), which is expressed as
(46) 
where is the angular velocity of the radiative zone. When the NTP parameter is zero, differential rotation is in the TaylorProudman state. Conversely, differential rotation is far from the TaylorProudman state with a large absolute value of the NTP parameter. The value of the NTP parameter with various stellar angular velocities is shown in Fig. 4. The NTP monotonically decreases with increases in stellar angular velocity. These results indicate that with large stellar angular velocity values, differential rotation approaches the TaylorProudman state. These results are counterintuitive, however, since we do not expect differential rotation to approach the TaylorProudman state with increasing stellar angular velocity values, since the effect, which is a driver of the deviation from the TaylorProudman state, is proportional to stellar angular velocity . These are the most significant findings of this paper, so hereafter in this section we discuss these unexpected results.
We next discuss the temperature difference between the equator and the pole at the base of the convection zone (). Since temperature is given as a function of entropy by
(47) 
and it is easier to measure than entropy, we use it here for discussing the thermal structure of the simulation results in the convection zone. Further, although it is mentioned in §3 that entropy gradient is crucial for breaking the TaylorProudman constraint, the temperature difference can be used as its proxy. Fig. 5 shows the relationship between stellar angular velocity and temperature difference at , where . Although the temperature difference monotonously increases with larger stellar angular velocity values, it is not enough to make the rotational profile largely deviate from the TaylorProudman state. This can be explained by using the thermal wind equation, which is a steady state solution of eq. (45):
(48) 
The inertial term and the diffusion term are neglected here. This equation indicates that, for a given value of the NTP, we need an entropy gradient proportional to . However, our simulation results show that , which means that as increases, the thermal driving force becomes insufficient to push differential rotation away from the TaylorProudman state. In other words, the latitudinal entropy gradient in rapidly rotating stars is so small that differential rotation stays close to the TaylorProudman state. In our model, meridional flow generates latitudinal entropy gradient at the base of the convection zone. It is conjectured that the insufficient thermal drive is due to a slow meridional flow.
We next investigate the dependence of meridional flow on stellar angular velocity. Fig. 6 shows the radial profile of latitudinal velocity at , using the results of cases 1, 2 and 9. In case 2, stellar angular velocity is twice that of case 1 (the solar value). In case 9, stellar angular velocity is equal to the solar value, and the amplitude of the effect is two times the value in case 1. Fig. 6 shows that meridional flow does not depend on stellar angular velocity, while it correlates with the effect. Considering eq. (34), the effect increases with larger values of stellar angular velocity, since the amplitude of the effect is proportional to . The reason why differential rotation in rapidly rotation stars is close to the TaylorProudman state is that meridional flow does not become fast with large stellar angular velocity values.
We interpret the result that the speed of meridional flow does not depend on stellar angular velocity in our model as follows. With large values of stellar angular velocity, more angular momentum is transported by the effect (Note that the effect is proportional to in equation (34)), so meridional flow obtains more energy from differential rotation. The energy gain does not result in an increase in speed because of the associated enhancement of the Coriolis force, which bends the meridional flow in the longitudinal direction. Another explanation is possible in terms of angular momentum transport. The angular momentum fluxes from both meridional flow and the Reynolds stress ( effect) must be balanced in a steady state. The former is proportional to and the latter is proportional to , where is the amplitude of meridional flow. Therefore, meridional flow does not depend on stellar angular velocity (Miesch, 2005). Our results (Fig. 6) indicate that with a larger stellar angular velocity (case 2), the above mechanism does not generate fast meridional flow. However, this does not occur when only the effect is large (case 9).
4.2 Angular Velocity Difference on the Surface
In this subsection we discuss angular velocity difference at the surface and the relationship between our results and previous observations. We conduct numerical simulations to investigate the physical process which determines (cases 1, 611). We define angular velocity difference as .
is determined by two opposing effects, a smoothing effect from turbulent viscosity and a steepening effect from the effect. In a stationary state these two effects cancel each other out. Latitudinal flux for turbulent viscosity and the effect can be written as and , respectively. Because these two have approximately the same value, can be estimated as
(49) 
where denotes the differential rotation region.
In order to confirm eq. (49), we conduct two sets of simulations, firstly varying the value of turbulent viscosity (), and secondly the amplitude of the effect (). Note that the setting for turbulent viscosity does not reflect a real situation, since the coefficients of turbulent viscosity and the effect should have a common value. Nonetheless, this is necessary for the purpose of our investigation. The simulation results are shown in Figures 7 and 8. We obtain and , which are consistent with eq. (49).
Fig. 9 shows the results of the dependence of on (Cases 15). Asterisks denote the difference at the surface between the equator and the pole, squares show the difference between the equator and the colatitude , and triangles are the difference between the equator and the colatitude . The difference at low latitudes (squares and triangles) monotonically increases with stellar angular velocity. However this is not the case for angular velocity difference between the equator and the pole (asterisk). As we discussed in §4.1, when stellar rotation velocity is large, the TaylorProudman state is achieved, meaning the gradient of angular velocity at the surface concentrates in lower latitudes. Due to this concentration, becomes smaller in Eq. (49) with larger values of . Thus, does not show an explicit dependence on . At low latitudes, is fixed and the angular velocity difference increases with stellar angular velocity. We obtain (between the equator and the colatitude : squares) and (between the equator and the colatitude : triangles). This indicates that decreases with stellar angular velocity. These results are consistent with previous stellar observations (Donahue et al., 1996; Reiners & Schmitt, 2003; Barnes et al., 2005).
4.3 Variation of effect and superadiabaticity
In this section, we discuss the dependence of meridional flow and differential rotation on free parameters. The parameter set is shown in Table 1 (cases 1217). At first we investigate the influence of the variation of the effect. The effect has two free parameters, i.e., amplitude and inclination angle (see §2.4). Amplitude is thought to become smaller with a larger stellar angular velocity, due to the saturation of the correlations such as and , where , and are the radial, latitudinal and longitudinal component turbulent velocities, respectively. Fig. 6 shows that meridional flow becomes slower with a smaller , keeping the value constant (Case 10). It is clear with the result of §4.1 that meridional flow becomes slow with a larger angular velocity when the variation of is included. Brown et al. (2008) reported this effect with their threedimensional hydrodynamic calculation. When meridional flow is slow, the entropy gradient generated by the subadiabatic layer is small, and differential rotation approaches the TaylorProudman state.
The inclination angle is thought to be small with large stellar angular velocity values, since the motion across the rotational axis is restricted (Kichatinov & Rüdiger, 1993). In case 12, differential rotation with a small inclination angle () is calculated. Other parameters are the same as case 1. The radial distribution of meridional flow is shown in Fig. 6. Meridional flow becomes faster with a smaller inclination angle. Because of the efficient angular momentum transport in the direction when the inclination angle is small, the second term on the right hand side of Eq. (45) is large. This generates a large , i.e. fast meridional flow.
In summary, we found that rapid stellar rotation causes two opposing effects on the speed of meridional flow. The speed is reduced by the suppression of , while it is enhanced by the angular momentum transport along the axial direction with a smaller . Although the results of the threedimensional calculation suggest that meridional flow becomes slower with a larger stellar angular velocity, our model cannot draw a conclusion about the speed of meridional flow in rapidly rotating stars.
Next we investigate the influence of superadiabaticity in the convection zone. In cases 1317, superadiabaticity in the convection zone . The differences of the NTP parameters with adiabatic and superadiabatic convection zones are shown in Fig. 10. The NTP parameter values with a superadiabatic convection zone are smaller than those with an adiabatic convection zone, since meridional flow in the superadiabatic convection zone makes the entropy gradient small. This result is suggested by Rempel (2005a). Note that the difference between the values of the NTP parameters with an adiabatic and those with a superadiabatic convection zone decreases as the stellar angular velocity increases, since the generation of entropy gradient by the subadiabtic layer becomes ineffective with a larger stellar angular velocity.
5 Summary
We have investigated differential rotation in rapidly rotating stars using a mean field model. This work is significant because it can be used as a base for further research on stellar activity cycles, which are most likely caused by the dynamo action of differential rotation in the stellar convection zone.
First, we investigated the morphology of differential rotation in rapidly rotating stars. Although more angular momentum is transported by convection with larger stellar angular velocity, the Coriolis force is stronger than in the solar case, so meridional flow does not be fast. In our model, meridional flow generates latitudinal entropy gradient in the subadiabatically stratified overshoot region. Since the meridional flow is not fast, the entropy gradient is insufficient to move differential rotation far from the TaylorProudman state in rapidly rotating stars. As a result, the differential rotation of stars with large stellar angular velocity is close to the TaylorProudman state.
The temperature difference between latitudes is probably controlled by two important factors, i.e., the subadiabatic layer below the convection zone and anisotropic heat transport caused by turbulence and rotation. We suggest that the former is important in slow rotators like the sun, and the latter in rapid rotators. The subadiabaticlayer effect is included in our model, while anisotropic heat transport is not. We found that the effect of the subadiabatic layer can generate a temperature difference in the solar case, which moderately increases with higher rotation speeds, and in case . The threedimensional simulations by Brown et al. (2008) include a selfconsistent calculation of anisotropy of turbulent thermal transport but not the subadiabatic layer at the bottom boundary. In their calculation is most likely smaller than in the solar case, since they cannot reproduce the solar differential rotation only with anisotropy of thermal transport. Also, in case , which is larger than the case with the subadiabatic layer. We speculate that anisotropic heat transport becomes more significant in rapidly rotating stars. There is also a possibility that our calculated entropy gradient at the base of the convection zone can be used as a boundary condition for a selfconsistent three dimensional simulation of stellar convection (Miesch et al., 2006). Note that differential rotation in rapidly rotating stars in Küker & Stix (2001) is not in the TaylorProudman state when anisotropy of turbulent thermal conductivity is included. A future study of the simultaneous effects of the attached subadiabatic layer beneath convection zone and anisotoropy of the turbulent thermal conductivity on stellar differential rotation would provide a better understanding of stellar differential rotation.
Next, we investigated angular velocity difference at the surface. The effect causes spatial difference in the rotation profile, while turbulent viscosity reduces the difference. Angular velocity difference is determined in eq. (49), which is then used to investigate differential rotation in rapidly rotating stars. Since stellar rotation is close to the TaylorProudman state, and the radiative core is rotating rigidly, differential rotation is concentrated at low latitudes with large stellar angular velocity. This concentration leads to a small in eq. (49). Therefore, only at low latitudes our model is consistent with stellar observations.
Our conclusions are as follows: (1) Differential rotation approaches the TaylorProudman state when stellar rotation is faster than solar rotation. (2) Entropy gradient generated by the attached subadiabatic layer beneath the convection zone becomes relatively small with a large stellar angular velocity. (3) Turbulent viscosity and turbulent angular momentum transport determine the spatial difference of angular velocity . (4) The results of our mean field model can explain observations of stellar differential rotation.
Our future work will focus on the stellar MHD dynamo. Several investigations have been conducted on the stellar dynamo using a kinematic dynamo framework (Dikpati et al., 2001; Charbonneau & Saar, 2001; Moss & Sokoloff, 2009; Jouve et al., 2010). Since, under such a framework, only the magnetic induction equation is solved using a given velocity field, solving a linear equation, such analysis does not give sufficient information on the strength of the dynamogenerated stellar magnetic field. To obtain the full amplitude of the stellar magnetic field, the feedback to the velocity field is required, i.e., an MHD framework. Adopting a similar approach to Rempel (2006), we can use the results of this paper to investigate the strength of the stellar magnetic field. Recent observations of the strength of the magnetic field generated by stellar differential rotation have been conducted using spectroscopy (e.g. Petit et al., 2008). A comparison of these observations and numerical calculations of the stellar dynamo could give new insight into the stellar magnetic field. Finally, our stellar MHD dynamo study would also contribute to the understanding of recent investigations into stellar magnetic cyclic activity periods (Noyes et al., 1984; Saar & Brandenburg, 1999).
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Case  

1  1  0  
2  1  0  
3  1  0  
4  1  0  
5  1  0  
6  1  0  
7  1  0  
8  1  0  
9  2  0  
10  0.5  0  
11  0.25  0  
12  1  0  
13  1  
14  1  
15  1  
16  1  
17  1 