Draft version February 6, 2014Preprint typeset using LATEX style emulateapj v. 11/10/09

A DYNAMO MODEL OF MAGNETIC ACTIVITY IN SOLAR-LIKE STARS WITH DIFFERENT ROTATIONALVELOCITIES

Bidya Binay Karak1,2, Leonid L. Kitchatinov3,4 and Arnab Rai Choudhuri1

1Department of Physics, Indian Institute of Science, Bangalore 560012, India2Nordita KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

3Institute for Solar-Terrestrial Physics, Lermontov Str. 126A, Irkutsk 664033, Russia4Pulkovo Astronomical Observatory, St. Petersburg 176140, Russia

Draft version February 6, 2014

ABSTRACT

We attempt to provide a quantitative theoretical explanation for the observations that Ca II H/Kemission and X-ray emission from solar-like stars increase with decreasing Rossby number (i.e., withfaster rotation). Assuming that these emissions are caused by magnetic cycles similar to the sunspotcycle, we construct flux transport dynamo models of 1M⊙ stars rotating with different rotation periods.We first compute the differential rotation and the meridional circulation inside these stars from amean-field hydrodynamics model. Then these are substituted in our dynamo code to produce periodicsolutions. We find that the dimensionless amplitude fm of the toroidal flux through the star increaseswith decreasing Rossby number. The observational data can be matched if we assume the emissions togo as the power 3–4 of fm. Assuming that the Babcock–Leighton mechanism saturates with increasingrotation, we can provide an explanation for the observed saturation of emission at low Rossby numbers.The main failure of our model is that it predicts an increase of magnetic cycle period with decreasingrotation period, which is the opposite of what is found observationally. Much of our calculations arebased on the assumption that the magnetic buoyancy makes the magnetic flux tubes to rise radiallyfrom the bottom of the convection zone. On taking account of the fact that the Coriolis force divertsthe magnetic flux tubes to rise parallel to the rotation axis in rapidly rotating stars, the results donot change qualitatively.

Subject headings: dynamo – Sun: activity – Sun: magnetic fields – stars: activity

1. INTRODUCTION

All late-type stars have convection zones in their outerlayers and are expected to have magnetic activity dueto the dynamo action taking place there. Since the dy-namo action crucially depends on rotation (Parker 1955),the more rapidly rotating stars are likely to have moremagnetic activity. Amongst the important observationalsignatures of such stellar magnetic activity are the en-hanced Ca II H and K emission (Noyes et al. 1984a;Saar & Brandenburg 1999) as well as the X-ray emission(Pallavicini et al. 1981; Pizzolato et al. 2003; Wright etal. 2011), there being a good correlation between thesetwo kinds of emission (Schrijver et al. 1992). Noyes etal. (1984a) realized that the Rossby number (the ratioof the rotation period to the convective turnover time) isa particularly convenient parameter to classify the late-type stars. In spite of some scatter, Fig. 8 of Noyes etal. (1984a) shows that the Ca II H and K emission has afunctional dependence on the Rossby number—first in-creasing rapidly with faster rotation and then increasingmore slowly for stars rotating very fast. Wright et al.(2011) present a similar pattern for the X-ray emissionfrom the late-type stars, as seen in the right panel of theirFig. 2. They conclude that there is a power-law rela-tion between the X-ray emission and the Rossby numberfor slowly rotating stars (the index being about −2.70),whereas the X-ray emission saturates for rapidly rotatingstars.Apart from the suggestion that these observational

data can be qualitatively explained by assuming that the

dynamo action becomes stronger with rotation and thenprobably saturates at sufficiently high rotation, detailedcalculations so far have not been done. During the lastfew years, models of the solar dynamo have become in-creasingly realistic (see, for example, Charbonneau 2010;Choudhuri 2011; and references therein). The aim of thepresent paper is to extend our knowledge of the solardynamo to solar-like stars rotating at different rates andexplore whether the patterns of Ca II H/K and X-rayemission can be explained quantitatively.The flux transport dynamo model has emerged in re-

cent years as the most popular theoretical model ofthe Sun’s magnetic activity cycle. In this model, thetoroidal field is produced by the stretching of the poloidalfield by differential rotation at the bottom of the con-vection zone, where helioseismology has discovered aconcentrated layer of differential rotation known as thetachocline. The toroidal field produced at the bottomof the convection zone rises to the solar surface due tomagnetic buoyancy. The decay of tilted sunspot pairs atthe solar surface gives rise to the poloidal field by what iscalled the Babcock–Leighton mechanism (Babcock 1961;Leighton 1964) which has received strong observationalsupport recently (Dasi-Espuig et al. 2010; Kitchatinov &Olemskoy 2011a). The meridional circulation of the Sunplays a very important role in such dynamo models. It isobserved to be poleward at the solar surface and carriesthe poloidal field poleward with it. To avoid piling upof matter near the sun’s poles, the meridional circula-tion has to have an equatorward return flow through the

2

deeper layers of the convection zone. It is necessary fortheoretical models to have such an equatorward merid-ional circulation at the bottom of the convection zoneto cause the equatorward advection of the toroidal fieldgenerated there, in order to explain the appearance ofsunspots at lower latitudes with the progress of the solarcycle (Choudhuri et al. 1995).It is expected that solar-type stars rotating at different

rates will have similar flux transport dynamos operatingwithin their convection zones. As indicated in the lastparagraph, we need a detailed knowledge of the differ-ential rotation and the meridional circulation to modelsuch a flux transport dynamo. For the Sun, helioseis-mology has provided a detailed map of the differentialrotation (Schou et al. 1998), which is used in solar dy-namo models. Although helioseismology provides someinformation about the nature of the meridional circu-lation in upper layers of the solar convection zone, wehave no direct observational data about the return flowof meridional circulation through the deeper layers of theconvection zone (but see Schad et al. 2013 for some he-lioseismic evidence of this). Since the stable stratifica-tion in the radiation zone does not permit a considerablemeridional circulation, we expect the meridional circula-tion to remain confined within the convection zone. Byrequiring ∇.(ρv) = 0, one is able to come up with a rea-sonable distribution of the meridional circulation which,when used in solar dynamo models, gives results con-sistent with observations. Although some informationabout the differential rotation at the surface of a fewstars is now available (see, for example, Berdyugina 2005;Strassmeier 2009), we need detailed information aboutdifferential rotation throughout the convection zone of astar in order to construct realistic dynamo models. Suchinformation about differential rotation or meridional cir-culation is not available from observational data for anystars. So, in order to construct detailed stellar dynamomodels, we need to calculate the differential rotation andthe meridional circulation from theoretical analysis.Before the flux transport dynamo model of the solar

cycle was proposed in 1990s (Wang et al. 1991; Choud-huri et al. 1995; Durney 1995), the importance of merid-ional circulation in the dynamo process was not generallyrecognized and there were some early efforts of construct-ing αΩ models of stellar dynamos without including themeridional circulation (Belvedere et al. 1980; Branden-burg et al. 1994). Jouve et al. (2010) made the first com-prehensive attempt of constructing flux transport mod-els of stellar dynamos by using differential rotation andmeridional circulation on the basis of 3-D hydrodynamicsimulations. Isik et al. (2011) assume an interface αΩdynamo to generate magnetic fields in stars and use themeridional circulation at the stellar surface only to ad-vect the magnetic flux that has emerged there.Using a mean field model of turbulence in the con-

vection zone, Kitchatinov & Rudiger (1995) developed amodel of differential rotation in solar-type stars. Thismodel was improved and extended by Kitchatinov &Olemskoy (2011b; hereafter KO11b) to calculate differen-tial rotation profiles of main-sequence dwarfs having dif-ferent masses and different rotation periods. This modelautomatically gives rise to a meridional circulation whichis essential for angular momentum balance. For a solarmass star with solar rotation period, this model gives

a differential rotation profile (see Fig. 1 in KO11b) re-markably close to what is found in helioseismology. Theaccompanying meridional circulation consists of a sin-gle cell in the convection zone with poleward flow nearthe surface and equatorward flow near the bottom of theconvection zone. In the present work, we use this modelto compute the differential rotation and the meridionalcirculation of solar-like stars having different rotation pe-riods. Then we give these as inputs in a dynamo modelbased on the code Surya developed in Indian Instituteof Science (Nandy & Choudhuri 2002; Chatterjee et al.2004; Karak 2010; Karak & Choudhuri 2011).In order to avoid too many complications in this initial

exploratory paper, we restrict ourselves only to stars ofmass 1M⊙. Since stellar rotation slows down with age(Skumanich 1972), the sequence of solar mass stars withdifferent rotation periods can also be viewed a sequenceof stars having different ages. The differential rotationand the meridional circulation of such stars having differ-ent rotation periods are first obtained from the model ofKO11b. As the star is made to rotate faster, the merid-ional circulation is found to be confined to the edges ofthe convection zone (the poleward flow in a narrow layernear the surface and the equatorward flow in a narrowlayer near the bottom). However, we found that evensuch a meridional circulation is able to sustain a fluxtransport dynamo and we have been able to constructmodels of the dynamo operating in 1M⊙ stars havingdifferent rotation periods by putting the appropriate dif-ferential rotation and the appropriate meridional circula-tion in the dynamo code. The next challenging questionis to connect the results of the dynamo calculation withthe observed emission in Ca II H/K and X-ray.In order to limit the growth of the magnetic field gen-

erated by the dynamo, it is necessary to include the non-linear feedback of the growing magnetic field on the dy-namo. The simplest way of doing this is to include aquenching in the α parameter describing the generationof the poloidal field. If the quenching is of such naturethat the dynamo action gets quenched when the toroidalfield is stronger than B0, then the maximum value of thetoroidal field hovers around B0 and the total toroidal fluxin the convection zone at an instant would be fB0R

2s,

where f is usually much smaller than 1. As the toroidalfield changes sign, f is expected to vary in a periodicfashion going through positive and negative values. Letfm be the amplitude of f , implying that the maximumtoroidal flux in the convection zone is fmB0R

2s. In our

dynamo simulations of 1M⊙ stars with different rotationperiods, we find that fm increases with decreasing rota-tion periods. In other words, stars rotating faster pro-duce more magnetic flux. The emissions in Ca II H/Kor X-ray, which depend on the overall magnetic activi-ties of the stars, are expected to increase with increasingfm. Since the emissions (especially the X-ray emission)often arise from magnetic reconnection involving the in-teraction of one magnetic flux system with another, onemay naively expect that the emissions may go as f2

m.One of the aims of the present study is to check whetherthe observed dependence of the emissions on the Rossbynumber can be explained on the basis of such assump-tions.One assumption used in most of the dynamo models

developed so far is that magnetic buoyancy makes the

3

toroidal field at the bottom of the convection zone to riseradially through the convection zone to the solar surface.However, in the case of stars rotating sufficiently fast,the Coriolis force makes flux tubes rise parallel to therotation axis rather than radially (Choudhuri & Gilman1987; Choudhuri 1989; D’Silva & Choudhuri 1993; Fanet al. 1993). This happens when the rotation period isshorter than the dynamical time scale of flux tube rise. Ifthe dynamical time scale of flux tube rise is comparableto the turnover time of convection (which is the case forthe Sun), then we would expect the Coriolis force to bedominant and make flux tubes rise parallel to the rota-tion axis when the Rossby number is less than 1. It isintriguing that the nature of dependence of the emissionon Rossby number changes around such a value, as seenin Fig. 8 of Noyes et al. (1984a) and Fig. 2 of Wright etal. (2011). The usual explanation given for such satura-tion is that the dynamo saturates when the rotation isfast. One important question is whether a change in thenature of the dynamo due to the change in the nature ofmagnetic buoyancy could also be behind this. Dopplerimaging shows that some fast rotating stars have polarspots (Vogt & Penrod 1983; Strassmeier et al. 1991),which are believed to be caused by the Coriolis force di-verting the rising flux tubes to high latitudes (Schussler& Solanki 1992). But, to the best of our knowledge, noprevious calculation has been done to explore how thenature of the dynamo changes when the magnetic fluxrises parallel to the rotation axis rather than radially.We present some such calculations and show that the dy-namo becomes less efficient and generates less magneticflux when the toroidal field is assumed to rise parallel tothe rotation axis.One important question connected with the theory of

stellar dynamos is to obtain periods of magnetic cycles.Thanks to the program of the Mount Wilson Observatoryfor monitoring Ca H/K emission from several solar-likestars for many years, the activity cycles of many solar-like stars have been discovered (Wilson 1978; Baliunaset al. 1995). There is evidence that stars with longer ro-tation periods have longer activity cycle periods. Thiswas first reported by Noyes et al. (1984b), who pointedout that this can be easily explained for a linear αΩ dy-namo on the basis of some scaling arguments (see alsoBrandenburg et al. 1998). However, in flux transport dy-namos, the cycle period is determined essentially by thetime scale of the meridional circulation. Ironically, fluxtransport stellar dynamos have difficulty in explainingthe observed increase of cycle period with the increase ofrotation period (or with the decrease of rotation rate).This was pointed out by Jouve et al. (2010, see the sec-ond panel of their Figure 4), who discussed various waysof getting around this difficulty. We also encounter thisdifficulty in our calculations. However, in the presentpaper, we do not discuss possible mechanisms of solvingthis difficulty.The mathematical formulation of the flux transport

dynamo model is described in the next Section. In §3we present the results obtained by assuming that theflux tubes rise radially through the convection zone dueto magnetic buoyancy. Then we discuss in §4 how ourresults get modified when we allow the Coriolis force tomake the flux tubes rise parallel to the rotation axis. Ourconclusions are summarized in §5.

2. FLUX TRANSPORT DYNAMO MODEL

We assume the magnetic field to be axisymmetric andwrite it in the following form:

B = ∇× [A(r, θ)eφ] +B(r, θ)eφ (1)

where Bp = ∇× [A(r, θ)eφ] is the poloidal component ofthe magnetic field and B(r, θ) is the toroidal component.Then, in the flux transport dynamo model, we solve thefollowing equations to study the evolution of the mag-netic fields:

∂A

∂t+

1

s(vp.∇)(sA) = η

(

∇2 −1

s2

)

A+ S(r, θ;B), (2)

∂B

∂t+

1

r

[

∂

∂r(rvrB) +

∂

∂θ(vθB)

]

= η

(

∇2 −1

s2

)

B

+s(Bp.∇)Ω +1

r

dη

dr

∂(rB)

∂r, (3)

where s = r sin θ.Here vp = vr r + vθ θ is the meridional circulation ve-

locity, whereas Ω is the angular velocity, both vp andΩ being functions of r and θ. The coefficient η is theturbulent magnetic diffusivity. The Babcock–Leightonmechanism for generating the poloidal field is encapturedthrough the source term S(r, θ;B) in (2). We shall dis-cuss below how we specify η, S(r, θ;B), Ω and vp. Oncethese parameters are given, we have to solve (2) and (3)within a hemisphere of the convection zone of the star.Since the bottom of the convection zone for solar-likestars is at 0.7Rs, we carry on the numerical integrationof (2) and (3) within the range 0.55Rs < r < Rs and0 < θ < π/2, the bottom of the integration region beinga little bit below the bottom of the convection zone. Wenow come to the boundary conditions which we use. Theboundary conditions at the pole are A = 0, B = 0, andat the equator are ∂A/∂r = 0, B = 0 which force a dipo-lar symmetry, whereas at the lower boundary we takeA = 0, B = 0. Above the upper boundary, we assume themagnetic field to be a potential field. The upper bound-ary condition we use is B = 0 and A smoothly matchesthis potential field across the boundary (see Dikpati &Choudhuri 1995 and Chatterjee et al. 2004 for details).We use the code Surya developed in Indian Institute ofScience to solve (2) and (3) with these boundary condi-tions (Nandy & Choudhuri 2002; Chatterjee et al. 2004;Karak 2010; Karak & Choudhuri 2011).We now come to the specification of the various param-

eters. We use the angular velocity profile Ω(r, θ) and themeridional circulation vp computed from the mean-fieldhydrodynamics model of KO11b. The numerical modelsolves the system of three joint equations for angular ve-locity, meridional flow and entropy in convection zoneof a solar-type star. The only tunable parameter of themodel is the measure of anisotropy, Cχ, of turbulent heattransport,

Cχ =(

χ‖ − χ⊥

)

/χ⊥, (4)

where χ‖ and χ⊥ are the eddy thermal diffusivities alongthe rotation axis and normal to the axis, respectively.The anisotropy is induced by rotation. The value ofCχ = 1.5 gives close agreement with helioseismology.The Cχ-parameter was, therefore, fixed to this value

4

and not varied in stellar applications. The eddy viscosi-ties and diffusivities are not prescribed but expressed interms of the entropy gradient, and the entropy is one ofthe dependent variables of the model. In order to com-pute the differential rotation, the model needs the struc-ture of a star to be specified. We use the EZ code of stel-lar evolution by Paxton (2004) to specify the structure of1M⊙ star as function of age. Then, the gyrochronologyrelation by Barnes (2007) was used to identify rotationrate for a star of given age. This provides a sequence ofmodels for differential rotation and meridional flow in theSun of different ages. More details about this modelingof stellar differential rotation can be found in KO11b andKitchatinov & Olemskoy (2012a). All the computationsgive solar-type rotation with the equator rotating fasterthan poles. Only with unrealistically slow rotation (with,say, Prot = 60 days) and artificially reduced anisotropyof heat transport can the anti-solar rotation be found(Kitchatinov & Olemskoy 2012a). A similar conclusionhas been reached by Gastine et al. (2014).We point out that the model of KO11 does not include

overshooting and provides the differential rotation onlyabove 0.72Rs. We assume the core of the star below0.7Rs to rotate with the constant angular velocity Ωcore

corresponding to the rotation period used for the partic-ular case. To produce a smooth fit between 0.7Rs and0.72Rs, we use the following procedure. Let Ωmodel(r, θ)be equal to Ω given by the model of KO11b above 0.72Rs,whereas below 0.72Rs we assume Ωmodel(r, θ) to be equalto the value of Ω at (r = 0.72Rs, θ) from the model. Wenow take the angular velocity to be given by the followingexpression

Ω(r, θ) = Ωmodel(r, θ) +1

2[Ωcore − Ωmodel(r = 0.72Rs, θ)][

1− erf

(

r − 0.71Rs

0.01Rs

)]

(5)

This expression of angular velocity implies a strong dif-ferential rotation between 0.7Rs and 0.72Rs, which is ourtachocline.In this paper, we carry on calculations for 1M⊙ stars

having rotation periods of 1, 2, 3, 4, 5, 7, 10, 15, 20,25.38 (solar value) and 30 days. Figure 1 shows the an-gular velocity distributions of stars with rotation periodsof 1, 5, 15 and 30 days. It is clear that the angular ve-locity tends to be constant over cylinders when the ro-tation is fast, whereas there is a tendency of it beingconstant over cones for rotation periods comparable tothe solar rotation period and longer. For all the caseswe computed, the meridional circulation always consistsof a single cell with poleward flow near the surface andequatorward flow at the bottom of the convection zone.For faster rotations (i.e. for shorter rotation periods), themeridional circulation tends to be confined to the periph-eries of the convection zone. In Figure 2, we show vθ asfunction of r at the latitude 45 for rotation periods of 1,5, 15 and 30 days. We point out again that the model ofKO11b does not include overshooting in the tachocline.As a result, the meridional circulation abruptly falls tozero at the bottom of the convection zone.The only remaining parameters to be specified are the

turbulent magnetic diffusivity η and the source termS(r, θ;B). We specify them in such a way that the code

(a)30d

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

300

350

400

(b)15d

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

650

700

750

(c)5d

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

2220

2260

2300

2340

(d)1d

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.152

1.154

1.156

1.158

x 104

Fig. 1.— The distribution of angular velocity in the poloidalplanes of stars with rotation periods 30, 15, 5 and 1 days. Therotational frequencies in nHz are indicated by the different colors.Note that the lower boundary at about 0.72Rs of the KO11b modelis smoothed using Eq. (5) to form a tachocline-like shear layer.

0.7 0.75 0.8 0.85 0.9 0.95 1−28

−24

−20

−16

−12

−8

−4

0

4

8

12

Radius (r/Rs)

v θ (m

/s)

Fig. 2.— The component vθ (in m s−1) of meridional circulationat 450 latitude of four different stars. Solid (red), dashed (black),dash-dotted (blue) and dot-pointed (magenta) lines correspond tostars with rotation periods 30, 15, 5 and 1 days respectively.

Surya gives stable periodic solutions for solar-like starsover the wide range of rotational velocities we are con-sidering. So we have specified them in a way somewhatdifferent from what we had done in some of our earliercalculations (Nandy & Choudhuri 2002; Chatterjee etal. 2004; Karak & Choudhuri 2011, 2012, 2013). Withour earlier specifications of η and S(r, θ;B), we had beenable to reproduce various aspects of the solar cycle ex-tremely well. However, we now find that these earlierspecifications do not give stable periodic solutions whenwe use the angular velocity profile and the meridional cir-culation appropriate for very short rotation periods. Sowe use somewhat different specifications of turbulent dif-fusivity and magnetic buoyancy, which are very similarto what is done in other flux transport dynamo models(e.g., Munoz-Jaramillo et al. 2009; Hotta & Yokoyama2010). When we use these specifications for the case ofsolar rotation period, we find the activity cycle periodto be around 6.5 yr instead of 11 yr (see Choudhuri etal. 2005 for a discussion) and the butterfly diagram alsodoes not look very solar-like. However, here we are in-

5

terested more in finding out how the behavior of the dy-namo changes with different rotational velocities, ratherthan matching the observational data for only one valueof rotational velocity (the solar value) for which we havedetailed observational data. So we have used a modelin which we can hold the other things invariant whilechanging the angular velocity profile and the meridionalcirculation for different rotation periods. We hope thatthe results obtained with this model at least qualitativelycaptures the changing behavior of the dynamo with dif-ferent rotation periods.We take the turbulent magnetic diffusivity to be a func-

tion of r alone having the following form:

η(r) = ηRZ +ηSCZ

2

[

1 + erf

(

2r − rBCZ

dt

)]

+ηsurf2

[

1 + erf

(

r − rsurfd2

)]

(6)

with rBCZ = 0.7Rs, dt = 0.03Rs, d2 = 0.05Rs, rsurf =0.95Rs, ηRZ = 5×108 cm2 s−1, ηSCZ = 5×1010 cm2 s−1,and ηsurf = 2× 1012 cm2 s−1. The radial dependence ofthe turbulent diffusivity is shown in Figure 3. Note thatthe diffusivity is weaker in the lower half of the convectionzone compared to its value in the upper layers and itfalls to a very low value below the convection zone. Itis very similar to the diffusivity profile used by Hotta &Yokoyama (2010).

0.6 0.7 0.8 0.9 1

109

1010

1011

1012

Radius (r/Rs)

η (c

m2 /s

)

Fig. 3.— Variation of turbulent diffusivity η with stellar radiusused in our dynamo model.

The source term S(r, θ;B) captures the Babcock–Leighton mechanism of generation of the poloidal fieldnear the stellar surface from the decay of tilted bipolarstarspots. We use the following form for this term:

S(r, θ;B) =α(r, θ)

1 + (B(rt, θ)/B0)2B(rt, θ), (7)

where B(rt, θ) is the value of the toroidal field at latitudeθ averaged over the tachocline from r = 0.685Rs to r =0.715Rs. We take

α(r, θ) =α0

4

[

1 + erf

(

r − r4d4

)][

1− erf

(

r − r5d5

)]

× sin θ cos θ (8)

with r4 = 0.95Rs, r5 = Rs, d4 = 0.05Rs, d5 = 0.01Rs.These parameters ensure that α(r, θ) is non-zero onlyin a thin layer near the surface, making the Babcock–Leighton mechanism operative only near the stellar sur-face. Following many previous authors, we carry on anon-local treatment of magnetic buoyancy by making theBabcock–Leighton mechanism operate on the magneticfield B(rt, θ) in the tachocline. We have also included α-quenching. While the α-quenching is easier to interpretfor the traditional α-effect based on helical turbulence(Parker 1955; Steenbeck, Krause & Radler 1966), we ex-pect such quenching to be present even in the Babcock–Leighton mechanism, since a stronger magnetic field re-duces the relative importance of the Coriolis force com-pared to the magnetic buoyancy, thereby reducing thetilt of the emerging starspot pair (D’Silva & Choudhuri1993). The factor B0 appearing in the quenching is theonly nonlinearity in our model. Since the dynamo ac-tion is suppressed when the toroidal field exceeds thisvalue, we expect the maximum value of the toroidal fieldin the tachocline not to exceed B0 substantially. Thefactor α0 in (8) determines the strength of the Babcock–Leighton process. For shorter rotation periods, the Cori-olis force is stronger, making the Babcock–Leighton pro-cess also stronger by making the tilts of bipolar starspotslarger. Taking the Babcock–Leighton process to be in-versely proportional to the rotation period T , we write

α0 = α0,sTs

T, (9)

where Ts is the solar rotation period and α0,s is the valueof α0 for the solar case, which we take α0,s = 1.6 cm s−1.We shall see in §3 that the magnetic activity keeps on ris-ing for shorter rotation periods when we use (9), insteadof being saturated as seen in observational data. Onepossible reason behind the saturation seen in the obser-vational data for short rotation periods is that the dy-namo action gets saturated for very fast rotations. Thiscan be phenomenologically included by replacing (9) by

α0 =α0,s

β

[

1− exp

(

−βTs

T

)]

. (10)

When we use such an expression, the Babcock–Leightonmechanism saturates when T ≪ βTs and we get back (9)when T ≫ βTs.Finally, it should be noted that magnetic buoyancy is

included in (5) on the assumption that flux tubes riseradially. As we pointed out, the strong Coriolis forcemay make flux tubes parallel to the rotation axis whenthe rotation period is sufficiently short. We shall considerthis possibility in §4, where we shall have to modify (7).

3. RESULTS FOR RADIAL RISE OF MAGNETIC FLUX

We now present the results obtained by using thesource term of the form (7), which implies that magneticflux rises radially due to magnetic buoyancy. We first dis-cuss results in which the Babcock–Leighton mechanismis assumed to vary with rotation period as (9) withouta saturation for rapid rotations as included in (10). To-wards the end of this Section, we shall point out how ourresults get modified on including the saturation.We first compute the differential rotation and the

meridional circulation for 1M⊙ stars with rotation pe-

6

riods of 1, 2, 3, 4, 5, 7, 10, 15, 20, 25.38 (solar value)and 30 days. Then we run our dynamo code for all thesecases. We find that our code relaxes to give periodic solu-tions corresponding to activity cycles in all these cases.The butterfly diagrams obtained for the rotation peri-ods of 1, 5, 15 and 30 days are shown in Figure 4. Thecontours indicate the values of the toroidal field at thebottom of the convection zone, whereas the colors indi-cate the values of the radial field at the stellar surface.Since starspots are expected to form when the toroidalfield at the bottom of the convection zone is sufficientlystrong, the contours can be taken to indicate the butter-fly diagrams of starspots appearing on the surface of thestar. While these butterfly diagrams tend to be confinedto lower latitudes when the rotation is fast, they extendto higher latitudes for slow rotators like the Sun. Whilethis does not fit the solar observations in detail, we findthat many of the broad features of solar observations arereproduced. At lower latitudes in all the cases shownin Figure 4, we find an equatorward propagation. Thetoroidal field produced at the bottom of the convectionzone is advected by the equatorward meridional circula-tion there and the poloidal field produced from it by theBabcock–Leighton mechanism also shows a tendency ofequatorward shift. On the other hand, we find a pole-ward propagation in the higher latitudes. The poloidalfield produced near the surface by the Babcock–Leightonmechanism is advected by the meridional circulation inthe poleward direction at higher latitudes. Since diffu-sion is important in our dynamo model and the poloidalfield diffuses to the bottom of the convection zone wherethe toroidal field is produced from it (Jiang et al. 2007),we see that the toroidal field also shows a tendency to-wards poleward shift at higher latitudes. This overalltendency of equatorward propagation in low latitudesand poleward propagation in high latitudes is consistentwith solar observational data. It may be noted that evenin detailed solar dynamo calculations it is nontrivial toconfine the butterfly diagram to lower latitudes. Nandy& Choudhuri (2002) pointed out that a meridional cir-culation penetrating below the bottom of the convectionzone helps in confining sunspot eruptions to low lati-tudes. Such a penetrating meridional circulation is usedin the majority of solar dynamo papers from our group,but that is not the case in this paper. Some recent au-thors (Munoz-Jaramillo et al. 2009; Hotta & Yokoyama2010) used an expression for the α-coefficient which isstrongly suppressed in high latitudes, which is not donehere.It may be noted that the activity cycle periods for

slow rotators like the Sun are somewhat shorter thanthe sunspot cycle periods. The reason behind this wouldbe obvious on comparing Fig. 2 of KO11b with Fig. 2 ofChatterjee et al. (2004). It is clear that the equatorwardreturn flow of the meridional circulation in the modelof KO11b is more concentrated compared to the merid-ional circulation we used in our solar dynamo models(Chatterjee et al. 2004; Karak 2010; Karak & Choud-huri 2011). As a result, the equatorward flow at thebottom of the convection zone in the model of KO11bwhich we use here is stronger than what it was in ourearlier dynamo calculations. Since the cycle period in aflux transport dynamo decreases with increasing merid-ional circulation (Dikpati & Charbonneau 1999), it is not

surprising that the meridional circulation based on themodel of KO11b makes the activity cycle period some-what shorter than what it is for the Sun. Kitchatinov &Olemskoy (2012b) found that the dynamo model basedon their meridional circulation gives a cycle period closerto the sunspot period on including diamagnetic pump-ing, which is not included in the present calculations. Insummary, the model developed in this paper, while ap-plied to the solar case, may not fit all the observationaldata in quantitative detail, but the various qualitativefeatures are broadly reproduced. The main advantage ofthe dynamo model presented in this paper is that it givesperiodic activity cycles over a wide range of rotation pe-riods of solar-like stars. We believe that this model givesa good idea of the general trend in the behavior of thedynamo when the rotation period is changed.For all the rotation periods for which we have car-

ried out dynamo calculations, we study how the totaltoroidal flux through the convection zone changes withtime. Writing the total toroidal flux as fB0R

2s, we take f

as a measure of the total toroidal flux. Figure 5 shows thevariations of f with time for the rotation periods of 1, 5,15 and 30 days. We find that the flux varies periodicallygoing through positive and negative values, as expectedfrom the fact that the toroidal field changes its directionfrom one half-cycle to the next half-cycle. It is clear fromFig. 5 that the amplitude fm of the toroidal flux increaseswith decreasing rotation period (i.e. for faster rotators).To have an idea of the nature of the toroidal field gen-erated by the dynamo, Figure 6 gives the distributionsof the toroidal field at the instants when its value in thetachocline is maximum.We expect more emission in Ca II H/K or in X-ray

when there is more magnetic flux. Since the productionof the emission usually involves magnetic reconnectionof one flux system with another, we may naively expectthe emission to go as the square of the magnetic flux, i.e.as f2

m. Now we explore how f2m changes with the rota-

tion period and whether this is consistent with the ob-servational data of Ca II H/K and X-ray emissions fromsolar-like stars. Noyes et al. (1984a) discovered that allthe data points for Ca II H/K emission lie in a narrowrange if one plots the emission against the Rossby num-ber rather than the rotation period. In the present study,we have carried out calculations for stars with the samemass 1M⊙, for which the convective turnover time willnot vary much with the rotation period. Although in thepresent case it would be completely satisfactory to studythe variations of f2

m with the rotation period, we dividethe rotation period by the convective turnover time toobtain the Rossby number Ro. The convective turnovertime was estimated from the local mixing-length rela-tions (cf. Eq.(8) of Kitchatinov & Olemskoy (2012a)).Turnover time value for the middle of convection zone(r = 0.86Rs) was then used to define the Rossby number.We study the variations of f2

m with the Rossby numberRo so that our results can be directly compared withthe observational data. The straight line (dot-dashed) inFigure 7 shows how f2

m varies with the Rossby numberRo. Since a straight line in this log-log plot is a good fit,we conclude that there is a power-law relation betweenf2m and the Rossby number Ro:

f2m ∝ Ro−δ. (11)

7

Time (years)

Latit

ude

0 10 20 30

20

40

60

80

−0.02

−0.01

0.00

0.01

0.02

Time (years)

Latit

ude

0 10 20 30

20

40

60

80

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

Time (years)

Latit

ude

(a)30 d

(b)15 d

(c)5 d

(d)1 d

0 10 20 30

20

40

60

80

−0.08

−0.04

0.00

0.04

0.08

Time (years)

Latit

ude

0 10 20 30

20

40

60

80

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

Fig. 4.— Butterfly diagrams of stars with rotation periods 30, 15, 5 and 1 days. Blue (solid) contours correspond to the positive valuesof the toroidal field at the bottom of the convection zone, whereas red (dashed) contours correspond to the negative values of the toroidalfield. The greyscale on the background shows the radial field (in the unit of B0) on the stellar surface; white denotes the positive valueand black denotes the negative.

8

0 5 10 15 20 25 30 35−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (years)

Flu

x (f

)

Fig. 5.— Total flux f of the toroidal field calculated over thewhole convection zone of stars. The solid (red), dashed (black),dash-dotted (blue), and dot-pointed (magenta) lines correspond tostars with rotation periods 30, 15, 5 and 1 days respectively.

15d

1d

0 0.2 0.4 0.6 0.8 1

30d

0.2

0.4

0.6

0.8

1

−5

−4

−3

−2

−1

0

1

5d

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Fig. 6.— A snapshot of the toroidal field (background colors)and the poloidal field (contours) when the toroidal field in thetachocline reaches the maximum value. Magnetic fields are in theunit of B0. The four panels correspond to stars with the fourdifferent rotation periods 30, 15, 5 and 1 days. Note that samescales for the toroidal field (color) and the poloidal field (contours)are used in all the panels.

Our simulations give the value δ ≈ 1.3, which is the slopeof the straight line in Figure 7.The Ca II H/K data presented in Fig. 8 of Noyes et al.

(1984a) or the X-ray data presented in Fig. 2 of Wrightet al. (2011) can be fitted with a power law for stars withlong rotation periods. We can take the luminosity as

L ∝ Ro−γ (12)

for long rotation periods. While the canonical value of γis taken to be around 2, Wright et al. (2011) propose ahigher value of 2.70. From (11) and (12), we conclude

L ∝ f2γ/δm (13)

If γ is taken to be 2, then we find L to go as a power ofabout 3 of fm. On the other hand, if γ is 2.70, then Lwould go as a higher power of fm. It appears from ouranalysis that L goes as a higher power of fm than thepower 2 expected from very naive considerations. How-ever, we are happy that, in spite of many uncertainties

−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−7

−6

−5

−4

−3

−2

log(Rossby no)

log(

f m2)

20

15

710

30Sun

54

321

Fig. 7.— The theoretically computed quantity f2m as a function of

the Rossby number, fm being the dimensionless amplitude of thetoroidal flux over the whole convection zone. The values shownby red color (rectangle) correspond to the case when α0 is variedaccording to Eq. (9), whereas the values shown by blue color (star)correspond to the case when α0 is varied according to Eq. (10).

in our model, we get the general trend. There are waysof bringing the theoretical model closer to the observa-tional data. We shall make some comments on this inthe Conclusion. In this first exploratory study, we justpresent results which follow from the most obvious con-siderations.As we already mentioned, the observational data show

a trend of saturation for stars rotating very fast for whichthe Rossby number is less than 0.1 (Wright et al. 2011).In our theoretical model, we do not get any such satura-tion if the strength of the Babcock–Leighton mechanismis taken to be given by (9). We now carry on some calcu-lations using (10) instead of (9). We take βTs = 3.6257days. Then we would expect a saturation for stars hav-ing rotation periods shorter than 10 days. We indeed findthat, on using (10) instead of (9), stars with short rota-tion periods produce less toroidal flux. Figure 8 showsa butterfly diagram for the rotation period 5 days. Thishas to be compared with one of the butterfly diagramsin Figure 4. The dashed line in Figure 7 shows how f2

mvaries with the Rossby number on using (10) instead of(9). It is clearly seen in Figure 7 that on including asaturation in the Babcock–Leighton mechanism there isa tendency of f2

m growing more slowly and reaching sat-uration, implying that the emission also would be sat-urated for fast rotators in accordance with the observa-tional data. We thus conclude that our theoretical modelis in qualitative agreement with the broad features of theobservational data.At last, we come to the question how the activity cycle

period varies with the rotation period. Figure 9 showshow the activity cycle period changes with the rotationperiod, the blue open circles giving the results obtainedby using (9) and the red solid circles giving results ob-tained by using (10). We find that the cycle period in-creases with decreasing rotation period. It is not difficultto give a physical explanation of this. As we considerstars rotating faster, the meridional circulation tends tobecome more confined near the edges of the convectionzone, as seen in Figure 2. Since such a meridional cir-culation is less effective in advecting the magnetic fields,

9

Time (years)

Latit

ude

0 10 20 30

20

40

60

80

−0.06

−0.04

−0.02

0.0

0.02

0.04

0.065 d

Fig. 8.— Butterfly diagram of a star with rotation period of 5days. In this case the strength of the Babcock-Leighton α is takenaccording to Eq. (10).

we find longer cycle periods for shorter rotation peri-ods. This theoretical result goes against the observa-tional trend that stars with longer rotation periods tendto have longer activity cycles (Noyes et al. 1984b). Wepoint out that Jouve et al. (2010) also found an increasein cycle period with decreasing rotation period, exactlylike what we have found, contrary to observations. How-ever, Do Cao & Brun (2011) have found a solution of thisby adding arbitrarily large latitudinal turbulent pump-ing in rapidly rotating stars. We believe that some im-portant physics is still missing from our stellar dynamomodels. In Conclusion we shall discuss some possibilitiesof closing the gap between observations and theoreticalresults.

0 5 10 15 20 25 30

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

Rotation period (d)

Cyc

le p

erio

d (y

r)

Fig. 9.— Variations of the stellar activity cycle period (in years)with rotation period (in days). The blue open circles show the casewhen α0 is varied according to Eq. (9,) whereas red filled circlesshow the case when it is varied according to Eq. (10)

4. RESULTS FOR THE RISE OF MAGNETIC FLUXPARALLEL TO THE ROTATION AXIS

The rise of magnetic flux tubes through the solar con-vection zone has been studied extensively on the basisof the thin flux tube equation (Spruit 1981; Choudhuri

1990). When the rotation period is less than the dy-namical time scale of rise due to magnetic buoyancy, theCoriolis force diverts the flux tubes to rise parallel tothe rotation axis (Choudhuri & Gilman 1987; Choud-huri 1989; D’Silva & Choudhuri 1993; Fan et al. 1993).Some mechanisms have been suggested for suppressingthis effect of the Coriolis force, such as the strong in-teraction with the surrounding turbulence in the con-vection zone (Choudhuri & D’Silva 1990) or the initia-tion of the Kelvin–Helmholtz instability inside the fluxtube (D’Silva & Choudhuri 1991). However, these mech-anisms are effective only if the cross-section of the fluxtube is rather small. Some of the starspots are muchlarger than sunspots, so these mechanisms are unlikelyto be very important. Also rapidly rotating stars tendto have polar spots—presumably diverted there by theaction of the Coriolis force (Schussler & Solanki 1992).Both the Ca II H/K emission and X-ray emission from

solar-like stars tend to saturate when the Rossby numberis less than about 0.1, as seen in Fig. 8 of Noyes et al.(1984a) or Fig. 2 of Wright et al. (2011). This is alsoapproximately the Rossby number below which the rota-tion period is shorter than the dynamical time scale. Onequestion which we explore here is whether the observedsaturation could be caused by the effect of the Coriolisforce. In all the calculations of the previous Section, wetook the source function to be given by (7), which im-plied that the flux tubes rose radially. Now, when therotation period is less than 15 days, we replace (7) by

S(r, θ;B) =α(r, θ)

1 + (B(rt, θb)/B0)2B(rt, θb), (14)

where B(rt, θb) is the value of the toroidal field averagedover the tachocline from r = 0.685Rs to r = 0.715Rs notat the latitude θ where the source function is calculated,but at the latitude θb from which a rise parallel to therotation axis would bring to flux tube at the latitude θwhen it reaches the stellar surface. We obviously have

rb sin θb = Rs sin θ, (15)

where rb is the value of r at the bottom of the convectionzone. When we calculate the source function for starswith rotation periods shorter than 15 days, we now use(14) with θb given by (15). Since θb cannot be largerthan π/2, it is clear from (15) that the source functionvanishes when θ is larger than sin−1(rb/Rs). In otherwords, the source function S(r, θ;B) is non-zero only inhigh latitudes and, in accordance with (2), the poloidalfield generation takes place only in high latitudes.In §3 we presented dynamo calculations for 1M⊙ stars

with rotation periods of 1, 2, 3, 4, 5, 7, 10, 15, 20, 25.38(solar value) and 30 days. In the cases of rotation periodsof 15, 20, 25.38 (solar value) and 30 days (slow rotators),the effect of the Coriolis force is not expected to be verystrong. So the magnetic flux would rise radially and theresults of §3 would not change even if we take the Coriolisforce into account. Only for stars with rotation periods of1, 2, 3, 4, 5, 7 and 10 days (fast rotators), we expect themagnetic flux to rise parallel to the rotation axis whenthe effect of the Coriolis force is included. So we nowcarry on calculations only for stars with these rotationperiods by using the source function given by (14) and(15) rather than (7).

10

Figure 10 presents the butterfly diagram for the rota-tion period of 5 days. On comparing with the butterflydiagram for this rotation period based on the assumptionof radial rise, as presented in Figure 4, we find that themagnetic fields are now more confined in the higher lat-itudes. This is certainly expected, given that the sourcefunction now vanishes at low latitudes. We also calculatethe dimensionless toroidal flux amplitude fm for all thefast rotators by treating the source function according to(14) and (15). Figure 11 shows a plot of f2

m as a func-tion of the Rossby number, in which the values of fm forslow rotators (rotation periods of 15, 20, 25.38 and 30days) are the same as used in Figure 7, but for fast rota-tors (rotation periods of 1, 2, 3, 4, 5, 7 and 10 days) fmis calculated by taking the source function to be givenby (14) and (15). For comparison we have overplottedthe earlier results of the radial rise on this plot. A lookat Figure 11 shows that the line joining the fast rota-tors gets slightly shifted below when the rise parallel tothe rotation axis due to the Coriolis force is taken intoaccount. This means that the amount of toroidal fluxproduced in the fast rotators is somewhat less when themagnetic flux is assumed to rise parallel to the rotationaxis. This is in agreement with what we expect on theground that the magnetic fields now occupy a smallerregion (only the high latitudes) of the stellar convectionzone.Although the line for fast rotators is slightly displaced

with respect to the line for slow rotators in Figure 11,we see the same trend of f2

m increasing with the Rossbynumber even for fast rotators that we see for slow rota-tors. One of our aims was to check whether the effectof the Coriolis force can explain the saturation of Ca IIH/K and X-ray emission for low Rossby number. FromFigure 11 we conclude that, although the rise parallel tothe rotation axis due to the Coriolis force causes a de-crease in the flux, the general trend of f2

m increasing withdecreasing Rossby number is not halted by the Coriolisforce. We presumably need something like the saturationof the Babcock–Leighton mechanism for fast rotation asgiven by (10) in order to explain the observed saturation.

Time (years)

Latit

ude

82 87 92 97 102 107 112 117

10

20

30

40

50

60

70

80

−0.06

−0.04

−0.02

0.0

0.02

0.04

0.06

Fig. 10.— Butterfly diagram of a star with rotation period of 5days. In this case the source function is given by Eqs. (14) and(15) corresponding to magnetic flux rising parallel to the rotationaxis.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

log(Rossby no)

log(

f m2)

Sun30

20

15

10

7

54

3

2

1

Fig. 11.— The dashed line (magenta points) is the same as inFigure 7, whereas the solid line (blue points) is for the case wherethe source function is given by Eqs. (14) and (15) for rapidly ro-tating stars with rotation periods less than and equal to 10 days.

5. CONCLUSION

Following the success of the flux transport dynamomodel in explaining various aspects of the solar cycle(Charbonneau 2010; Choudhuri 2011), we explore thetheoretical possibility that similar flux transport dy-namos operate in solar-like stars. We need profiles ofthe differential rotation and the meridional circulationto construct flux transport dynamo models. So we firstcompute these flows inside 1M⊙ stars rotating at differ-ent rates by using the mean-field hydrodynamics modelof KO11b. Then we use our dynamo code Surya to con-struct dynamo models of these stars. Our results arein qualitative agreement with many aspects of observa-tional data. For example, we find that the dimensionlessamplitude fm of the toroidal flux increases with increas-ing rotation. We naively expect the Ca H/K and X-rayemissions to go as f2

m. We find that we can match theobservational data if we assume the emissions to go assomewhat higher powers of fm. However, if fm were toincrease more rapidly with rotation than what is pre-dicted by our present model, then we see from (11) and(13) that it may be possible to make the emission L go asf2m. We point out that we have assumed the Babcock–Leighton mechanism strength to go as T−1 as seen in(9). If we assume this strength to increase faster withincreasing rotation, then we would expect to find thepower law index in (11) steeper. As seen from (13), thiswould lead to a weaker dependence of L on fm. Giventhe many uncertainties in the theoretical model, we donot attempt such fine tuning in this paper. We merelyshow that the simplest possible theoretical model quali-tatively gives the general trend seen in the observationaldata. Allowing magnetic flux to rise parallel to the ro-tation axis due to the Coriolis force when the rotation isfaster does not change the results qualitatively.One disagreement with observational data is that our

model predicts that the magnetic cycles have longer pe-riods when rotation periods are shorter. As we pointedout, the theoretical model of Jouve et al. (2010) alsohad this difficulty. We have discussed the reason behindthis. The model of KO11b predicts that the meridionalcirculation is more confined to the peripheries of the con-vection zone as the star rotates faster and is less effec-tive in advecting the magnetic fields. This less effective

11

meridional circulation makes the period of the flux trans-port dynamo longer. The decrease of cycle period withfaster rotation probably implies that the meridional cir-culation remains more effective in faster rotating starsthan what our present model suggests. We have takenthe anisotropy Cχ given by (4) to be equal to 1.5 in allour calculations. This value of Cχ gives a good agreementwith helioseismology for a 1M⊙ star with solar rotation.However, it is certainly possible that this anisotropy in-creases with faster rotation. A stronger anisotropy maymake the meridional circulation more effective in fasterrotating stars and thereby decrease the dynamo cycle pe-riod. We plan to explore these possibilities in future. Atpresent, even our understanding of the meridional circu-lation of the Sun is fairly incomplete. There have beenrecent claims that the meridional circulation of the Sunmay have a multi-cell structure (Zhao et al. 2013; Schadet al. 2013). Hazra et al. (2014) have shown that a fluxtransport dynamo can still work with a multi-cell merid-ional circulation as long as there is an equatorward flowat the bottom of the convection zone. All the calculationsin this paper, however, are based on simple single-cellmeridional circulation predicted by the model of KO11bon assuming Cχ = 1.5.By comparing observations with the results of our pre-

liminary theoretical studies, we find that magnetic ef-fects probably grow somewhat faster with rotation thanwhat is suggested by our present calculations. Only ifthe dimensionless amplitude fm of the toroidal flux in-creases faster with rotation, we would be able to makeemission go as f2

m. Probably flows inside faster rotat-ing stars also remain stronger than what is suggestedin the present model, to ensure that we have faster dy-namos with shorter cycle periods. We must rememberthat in the present dynamo model the generation of thepoloidal field from helical turbulent (the so-called α-effect) is not included, since we know from observationsthat the Babcock-Leighton process is the major sourceof the poloidal field in the Sun. However, if this is notso true for the rapidly rotating stars and if the α-effectstarts contributing to the poloidal field generation in ad-dition to the Babcock-Leighton process, then that canmake the stellar activity stronger and can also make thecycle periods shorter. Another thing to remember is thatwe do not vary the turbulent diffusivity in all the cal-culations. However, if the turbulence is weaker in theyoung rapidly rotating stars, then the weaker turbulentdiffusivity can make the dynamo stronger. Therefore,one of the aims of any future study should be to ex-plore various effects that may make magnetic activitygrow faster with rotation than what we have found. Theencouraging thing is that the present calculations showthe qualitative trend of magnetic activity increasing withrotation. In what ways the manifestations of strongermagnetic activity may differ from the manifestations ofsolar activity is another important question to be ad-dressed. Some fast-rotating solar-like stars are known to

have starspots much larger than sunspots (Strassmeier2009). Some such stars have superflares which are muchmore energetic than typical solar flares (Maehara et al.2012). One related question is whether solar flares sub-stantially more energetic than the flares recorded so farare possible in the Sun (Shibata et al. 2013).In this paper, we have restricted ourselves to studying

only the regular aspects of stellar activity cycles. The ob-servational data of stellar activity presented by Baliunaset al. (1995) show that many stellar cycles show strongirregularities. Constructing theoretical models of irregu-larities of the solar cycle has been a major research ac-tivity in the field of solar dynamo theory in recent years.Studying the irregularities of stellar cycles presumablywill be a fertile research field for the future. One in-triguing question is whether irregularities of stellar cy-cles show patterns similar to solar cycle irregularities.One important aspect of the solar cycle irregularities isthe Waldmeier effect (Waldmeier 1935) that stronger cy-cles tend to have shorter rise times. The data of Baliu-nas et al. (1995) do not cover a long enough time inter-val to conclusively ascertain whether stellar cycles alsoshow the Waldmeier effect. However, for a few stars, thetime variation plots of Ca H emission presented by Bal-iunas et al. (1995) cover several cycles. If we carefullylook at the Baliunas et al. (1995) plots of some stars—notably HD 103095 (in Fig. 1e), HD 149661 and HD26965 (in Fig. 1f), HD 4628, HD 201091 and HD 32147(in Fig. 1g)—we see tantalizing hints that stronger cyclestend to rise faster, suggesting that the Waldmeier effectis present in stellar dynamos as well. Karak & Choud-huri (2011) have shown how fluctuations in the merid-ional circulation can give rise to the Waldmeier effect ina flux transport dynamo. The tentative hint of the Wald-meier effect in stellar cycles certainly suggests that themeridional circulations inside solar-like stars also prob-ably have large fluctuations. Apart from fluctuations inthe meridional circulation, the other source of irregular-ities in the solar cycle is fluctuations in the Babcock–Leighton process (Choudhuri et al. 2007; Choudhuri &Karak 2009; Olemskoy et al. 2013). The combined fluc-tuations in the meridional circulation and the Babcock–Leighton mechanism can explain various aspects of grandminima in solar activity rather elegantly (Choudhuri &Karak 2012; Karak & Choudhuri 2013). The data of Bal-iunas et al. (1995) show grand minima phases of severalstars. Presumably the physics behind these stellar grandminima is the same as the physics behind solar grandminima. We hope that detailed studies of irregularitiesin stellar dynamos will be carried out in future.

LLK is thankful to the Russian Foundation for Ba-sic Research (projects 12-02-92691 Ind and 13-02-00277).This work was initiated during ARC’s visit to Irkutskfunded by a DST-RFBR Indo-Russian Exchange Pro-gram. A partial support was provided by the JC BoseFellowship awarded to ARC by Department of Scienceand Technology (DST), Government of India.

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