A&A 563, A100 (2014) DOI: 10.1051/0004-6361/201321960 c ESO 2014

Astronomy A&strophysics

Detailed mathematical and numerical analysis of a dynamo model
Aditi Sood and Eun-jin Kim
School of Mathematics and Statistics, University of Sheﬃeld, S3 7RH, UK e-mail: [smp11as;e.kim]@sheffield.ac.uk
Received 25 May 2013 / Accepted 31 December 2013
ABSTRACT
We investigate the role of nonlinear feedback by α-quenching, ﬂux losses, and feedback by diﬀerential rotations in dynamo. Speciﬁcally, by studying the nonlinear dynamo model analytically and numerically, we unfold how frequency p of magnetic ﬁeld, magnetic ﬁeld strength |B|, and phase φ are inﬂuenced by diﬀerent types of nonlinear feedback in the limit of a very weak mean and/or ﬂuctuating diﬀerential rotation. We ﬁnd that p and φ are controlled by both ﬂux losses with no inﬂuence by α-quenching when there is no back reaction because of ﬂuctuating diﬀerential rotation. We ﬁnd a similar eﬀect of poloidal ﬂux loss and toroidal ﬂux loss on p and |B| in the absence of a back reaction of shear. Their eﬀect becomes totally diﬀerent with the inclusion of this back reaction. Detailed investigations suggest that toroidal ﬂux loss tends to have more inﬂuence than poloidal ﬂux loss (with or without α-quenching) in the presence of ﬂuctuating shear. Furthermore, the eﬀect of α-quenching is boosted when combined with toroidal ﬂux loss, indicating that the dynamic balance of dynamo is optimized in the presence of both α-quenching and ﬂux loss. These results highlight the importance of nonlinear transport coeﬃcients and diﬀerential rotation in the regulation of a dynamo.
Key words. dynamo – stars: magnetic ﬁeld – stars: evolution – stars: solar-type – stars: rotation

1. Introduction
Over the decades it has been revealed through high-resolution observations that the magnetic ﬁeld on the solar surface is very structured, consisting of small-scale, rapidly changing magnetic ﬁelds, which seem to emerge randomly and are highly variable. At the same time, solar magnetic ﬁeld shows a remarkable degree of organization on a global scale, displaying the butterﬂy diagrams and polarity reversal with quite a regular cycle of 22 years. These observations can be understood in terms of the ampliﬁcation of magnetic ﬁeld by inductive motion of a conducting ﬂuid in the framework of magnetohydrodynamics (MHD) (e.g., Krause & Rädler 1980) by using a nonlinear dynamo theory. Speciﬁcally, the coexistence of regular and complex behaviour of solar magnetic ﬁeld can be due to the dynamic balance among various nonlinear interactions. It is thus important to understand the role of nonlinear interaction in the regulation of dynamo.
To understand the nonlinear dynamo process, dynamical systems are the simplest tools because they have the capability of exhibiting complex dynamics as (control) parameters of the system change. A notable study of dynamo was previously done by Weiss et al. (1984) by using dynamical model, who illustrated in detail the transition from regularity to chaos as the control parameter (the so-called dynamo number) varies. In particular, Jones et al. (1985) were able to explain the existence of aperiodic magnetic cycle in a dynamical system given by complex Lorenz system, even without an explicit stochasticity (e.g., in parameters, external forcing, etc.).
Based upon the traditional α − Ω dynamo (Parker 1955; Moﬀat 1978; Parker 1979), the dynamo number in this work, which controls the eﬃciency of dynamo action, is proportional to the product of α and Ω eﬀects, which generate a toroidal ﬁeld from a poloidal ﬁeld by cyclonic convection (α eﬀect) and a poloidal ﬁeld from a toroidal ﬁeld by shearing (Ω eﬀect).

There is also a dynamical model (Charbonneau 2005) based on ﬂux transport dynamo (another popular dynamo model), which involves the global transport of the magnetic ﬁelds via meridional circulations, as well as buoyant rise in magnetic ﬂux (Dikpati & Charbonneau 1999; Rempel 2006). Weiss et al. (1984) explained the existence of aperiodic magnetic cycles using a simple parameterized model. Since then, various studies have revealed that with a certain set of conditions, low-order models are able to capture the basic behaviour of solar and stellar magnetic ﬁelds (Tobias et al. 1995; Weiss et al. 2001; Mininni et al. 2001; Pontieri et al. 2003; Wilmot-Smith et al. 2005; Passos & Lopes 2008; Lopes & Passos 2009, 2011). A continuous eﬀort has been devoted to further developing dynamo models to explain interesting properties of solar magnetic activity, as well as solar diﬀerential rotational proﬁle, uncovered by high-resolution observational data (Yoshimura 1975, 1978; Jepps 1975; Ivanova & Ruzimikin 1977).
To understand the eﬀect of various nonlinear interactions that could be important in the evolution of stellar rotation and magnetic ﬁelds over the course of spin-down, we revisit a simple parameterized dynamo model that can serve as a full dynamo model. In fact, the observations of the evolution of magnetic ﬁelds and rotation of other stars of diﬀerent ages with diﬀerent rotation rates have been providing us with valuable information about the relation among rotation, diﬀerential rotation, and magnetic activity, which can be utilized to test a dynamo theory against observations for improvement, which can work not only for the Sun but also for other solar-type stars with diﬀerent rotation rates. The exploration of the relation between magnetic activity and rotation for a broad range of rotation rates is, however, a very challenging problem and cannot be done practically by full MHD simulations owing to the required computational demand (e.g., the resolution of a broad range of length and time scales, etc.). An illustrative theoretical model for a minimal number of key quantities is thus valuable for gaining

Article published by EDP Sciences

A100, page 1 of 12

A&A 563, A100 (2014)

insight into this problem. This is particulary the case in view of the limitations inherent in any model (e.g., in parameterizations of transport coeﬃcients in MHD simulations). We note that a simple parameterized model has attracted a lot of attention for fusion plasmas as a useful model for understanding regulation and bifurcation (L-H transition), which is crucial for improving plasmas (Zhu et al. 2013; Malkov et al. 2009; Kim & Diamond 2003).
The main purpose of this paper is to present detailed analysis of nonlinear dynamo models to elucidate the eﬀect/role of various nonlinear interactions that have been proposed in previous works and thus to identify self-regulatory behaviour in a viable solar/stellar dynamo model that can explain observations. To this end, we ﬁrst examine the nonlinear eﬀect through nonlinear transport coeﬃcients (α-quenching and ﬂux losses) and feedback of diﬀerential rotation on frequency p and magnetic ﬁeld strength |B|. Our focus would be the feedback mechanism of the Lorentz force on α-eﬀect, diﬀerential rotation and the enhanced magnetic dissipation through ﬂux losses. Considering observations, we then demonstrate that regulation by the “near” balance between the (nonlinear) generation and destruction of the magnetic ﬁelds is a desirable property of a successful model. In particular, this dynamic balance leads to the almost linear increase in frequency with rotation rate, ﬂattening of magnetic energy for high rotation, and quenching in total shear consistent with observation.
The remainder of this paper is organized as follows. In Sect. 2, we introduce our model. The detailed analysis of this model is provided step by step in Sects. 3 to 5: Sect. 3 discusses nonlinear eﬀects through transport coeﬃcients and Sect. 4 (Sect. 5) focuses on nonlinear eﬀects through mean (ﬂuctuating) diﬀerential rotation. We explain the implications of our results, together with key observational data in Sect. 6. Section 7 contains the conclusions.

2. Model equations

The nonlinear dynamo model under investigation is the dynamical system consisting of the seven coupled ordinary diﬀerential equations (ODE) (Sood & Kim 2013; Weiss et al. 1984),

(∂t + F2)A = 2DB, where

D=

D0 , F1

(1)

(∂t

+

F3)B

=

i(1

+

w0)A

−

1 iA∗w, 2

(2)

(∂t

+ ν0)w0

=

1 i(A∗B − 2

AB∗),

(3)

(∂t + ν)w = −iAB,

(4)

which are given in a dimensionless form. Here, A is the poloidal

magnetic ﬂux, B is the toroidal magnetic ﬁeld, w0 and w are mean and ﬂuctuating diﬀerential rotations with zero and twice

the frequency of A and B, respectively, which are generated by

a nonlinear back reaction of A and B. Poloidal magnetic ﬁeld A,

toroidal magnetic ﬁeld B, and ﬂuctuating diﬀerential rotation w

are complex, whereas mean diﬀerential rotation w0 is real. The complex conjugate of A is represented by A∗. We note that in

these dimensionless units, the total mean diﬀerential rotation is

given

by 1 + w◦

=

ΔΩ Ω

;

thus,

mean

diﬀerential

rotation

w0,

due

to the back reaction, appears with a negative sign to reduce the

total shear from 1 to 1 + w0 < 1. This is because the diﬀer-

ential rotation is inhibited by the tension in the magnetic ﬁeld

lines via Lorentz force (i.e., causing the quenching of Ω-eﬀect.).

Viscosity of mean and ﬂuctuating diﬀerential rotations are represented by ν and ν0, respectively, which are considered to be constant.
The parameter D in Eq. (1) involves the dynamo number D0 ∝ αΩ, which is the key (dimensionless) control parameter. And F1, F2, and F3 are “nonlinear” transport coeﬃcients, deﬁned as

F1 = 1 + κ1|B|2, F2 = 1 + λ1|B|2, F3 = 1 + λ2|B|2,

where κ1, λ1, and λ2 are constant parameters; F1 represents the α source term (i.e., helicity by the magnetic ﬁeld) due to the back
reaction by magnetic ﬁeld that was studied in previous works
including Pouquet et al. (1976) and Kleeorin et al. (1982). The
enhanced magnetic dissipation through the loss of toroidal and
poloidal magnetic ﬂuxes is represented by F2 and F3, respectively, which removes magnetic ﬂux from the region in which the dynamo operates. We note that this is totally diﬀerent from the assumption of reduction in magnetic dissipation used in some previous works. In this model, we assume that α is a parameter that increases with rotation as α ∝ Ω based on an αΩ dynamo, but we envision that α captures other physical mechanisms that increase linearly with Ω. Therefore, dynamo number D0 is scaled with rotation rate Ω as D0 ∝ Ω2 (Sood & Kim 2013).The proportionality constant between D0 and Ω2 is taken to be unity in the paper for simplicity. In this paper, our model focuses on the dynamic balance through F1 − F3. Other possible feedback, such as the Malkus-Proctor eﬀect (1975) or Λ-quenching (Kitchatinov et al. 1994), will be left for future work. It is impor-
tant to note that according to our non-dimensionlization, physi-
cal poloidal magnetic ﬁeld A is obtained by dividing our dimensionless poloidal magnetic ﬂux A in Eqs. (1)−(4) by rotation Ω.
Expecting the regulation in a nonlinear dynamo model,
which can weaken (suppress) the chaotic solution, we make an ansatz for the ﬁnite-amplitude solution to Eqs. (1)−(4) in the following system

A = aeipt, B = bei(pt + φ), w = ca2e2ipt,

(5)

where a > 0 and b > 0 are assumed to be positive real; c is a complex constant; p is real angular frequency; φ is the phase dif-
ference between toroidal b and poloidal a magnetic ﬁelds. Using ﬁnite amplitude solution of these forms in Eqs. (1)−(4) gives us the following ﬁve relations among a, b, φ, p, and w0

− p2

+

F2F3

=

−(2p2 + 2(4 p2

F2ν) a2 , + ν2)

(6)

(F2

+

F3)p

=

2D

−

pa2 ν0

+

(2F2 2(4

− p2

ν) +

p a2 ν2)

,

(7)

w0

=

− pa2 2ν0 D

,

(8)

a = 2Db ,

(9)

p2 + F22

tan φ = p ,

(10)

F2

where

D

=

D0 F1

,

F1

=

1 + κ1b2,

and

F2,3

=

1 + λ1,2b2.

In the

above

equations, is the parameter tracking the presence of ﬂuctuating

shear w as

=

1, w 0, 0, w = 0.

A100, page 2 of 12

A. Sood and E. Kim: Detailed mathematical and numerical analysis of a dynamo model

Therefore, when there is no ﬂuctuating shear ( = 0), Eqs. (6), (7) are simpliﬁed as

p = ± F2F3,

(11)

(F2

+

F3)

+

4D20b2 ν0F12F22(F2 +

F3)

F1

F2F3 = 2D0,

(12)

where Eqs. (11) and (9) were used in obtaining Eq. (12). It is
worth noting that neglecting nonlinear terms in the above equations (i.e., F1 = F2 = F3 = 1, w0 = 0) gives the linear dispersion relation for the onset of bifurcation as p = 1 and φ = π/4.
Interestingly, Eq. (11) shows that the frequency p of the mag-
netic ﬁeld is determined by the dissipation of both the magnetic
ﬁelds, given by the geometric mean of F2 and F3. Thus p can increase with b due to the enhanced magnetic dissipation given by F2 = 1 + λ1b2 or F3 = 1 + λ2b2. In contrast, p is not inﬂuenced by α-quenching (F1), demonstrating diﬀerent eﬀects on the frequency p of α-quenching (F1) and of ﬂux losses (F2,3). The ﬂux√losses F2 and F3 determine the phase φ between b and a as φ = F3/F2; that is, when the toroidal ﬂux loss F3 is stronger
(weaker) than poloidal ﬂux loss F2, it removes b more (less) efﬁciently than a, increasing (decreasing) the phase φ between b and a. w0 in Eq. (8) clearly shows that the sign of w0 is negative deﬁnite, always reducing the total mean diﬀerential rotation 1 + w0 < 1. Explicit examples of these behaviours are presented in the following Sects. 4, 5 through the detailed analysis of ﬁ-
nite amplitude solution by taking the diﬀerent limits of F1, F2, and F3 as well as w0, w → 0. This will elucidate the role of α-quenching, ﬂux losses, mean and ﬂuctuating diﬀerential rotation in self-regulatory behaviour of magnetic activity/cycle. Analytical predictions in Sects. 3−4 are then accompanied by
the conﬁrmation by numerical simulations. Speciﬁcally, for the
numerical parts of our paper, we solve Eqs. (1)−(4) by timestepping for all complex variable A, B, w, and w0 using ν = 0.5, ν0 = 35.0 for diﬀerent values of D0 between 1−400 (See Sood
& Kim 2013 for more details). We also note that for the numerical simulations, the values of the parameters κ1, λ1, and λ2 are assumed to have the same value, κ1 = λ1 = λ2 ≡ λ = 2.5.

3. Nonlinear effects through transport coefﬁcients

The α-eﬀect and ﬂux losses are the crucial transport coeﬃcients, capturing the overall eﬀects of small (unresolved) turbulence.
The feedback of growing magnetic ﬁelds alters the value of these transport coeﬃcients, making them dependent on magnetic ﬁelds. These nonlinear transport coeﬃcients provide a way
for a dynamo to self regulate. To understand the self regulation through these transport coeﬃcients, it is imperative to investigate the case in the limit of a very weak mean or ﬂuctuating diﬀerential rotation by assuming w0, w → 0. The relevant Eqs. (8)−(12) are then

p = ± F2F3,

(13)

(F2 + F3)F1 F2F3 = 2D0,

(14)

a = 2D0b ,

(15)

F1 p2 + F22

tan φ = p ·

(16)

F2

In the above system, we consider diﬀerent cases of F1, F2, and F3 and provide the scalings of p, b, and a with rotation.

In the remainder of the paper, the scaling of poloidal ﬁeld with Ω in physical units is obtained by dividing A by Ω because of
our nondimensionlization as noted in Sect. 2. Also note that here a = |A| and b = |B|.
1. F1 = F = 1 + λb2, F2 = F3 = 1. In the presence of α-quenching only with no ﬂux loss, Eqs. (13)−(16) are simpli-
ﬁed as

p = 1,

(17)

2(1 + λb2) = 2D0,

(18)

a = 2D0b , (1 + λb2)

(19)

tan φ = 1.

(20)

Equation (17) explicitly shows that while α-quenching (F1) provides nonlinear damping to saturate the growth of mag-

netic ﬁeld, it does not inﬂuence the frequency p = 1, which

is the value of the frequency at the onset of bifurcation to

a ﬁnite amplitude solution. Moreover, phase φ is not altered

by

α-quenching,

maintaining

a

linear

value

π 4

.

In

the

limit

of

D0 1, Eqs. (18)−(20) enable us to ﬁnd the scalings of a as

a ∼ Ω and and b as b ∼ Ω.

We now compare these analytical results by numerical simu-

lation of Eqs. (1)−(4). As noted in Sect. 2, we solve these equa-

tions by time stepping and obtain the frequency spectrum by

taking a FFT (fast Fourier transform) of the time series of dy-

namical variable B. Once we have the power spectrum we as-

sign colours according to the colour bar next to the ﬁgure. We

ﬁnd that analytical scalings are in perfect agreement with the re-

sults obtained from numerical simulations shown in Fig. 1 where

κ1 = λ = 2.5 is used. Speciﬁcally, Fig. 1a represents the intensity of frequency spectrum for diﬀerent values of Ω where

the high-to-low intensity is coded in yellow to black colours. A

strip of a bright yellow colour tracks the frequency of maximum

intensity, which hardly changes with Ω with the same scaling

(p = 1) as predicted above. We note that the frequency shown in

Fig. 1a is lower than the angular frequency p by a factor of 2π.

Figures 1b−1c show b and a as functions of Ω, where the scal-

ings similar to the analytic values can be obtained. These results

conﬁrm the validity of our ﬁnite amplitude solutions. We note

that a ∼ Ω implies that the scaling of poloidal magnetic ﬁeld in the physical unit is ∼Ω0.

2. F1 = 1, F2 = F3 = F = 1 + λb2. For simplicity, when poloidal ﬂux loss and toroidal ﬂux loss are enhanced to the same

degree while there is no α-quenching, Eqs. (13)−(16) become

p ∼ (1 + λb2), 2D0 = 2(1 + λb2)2, a ∼ 2D0/21/2(1 + λb2)3/2,

and tan φ = 1. Therefore, both ﬂux losses lead to the increase in

frequency p ∼ (1+λ1,2b2) with b, while their eﬀects are canceled

in

the

phase

shift

φ

=

π 4

.

The

increase

of

p

with

b

is

determined

consistently by the nonlinear dynamics, which sets the values of

b. This dynamical determination of p is reminiscent of the ﬂux

transport dynamo where p depends on meridional ﬂow, etc. In

the limit of D0 1, the scalings of p, b, and a are found to be

p

∼

Ω,

b

∼

Ω

1 2

,

and

a

∼

Ω

3 2

,

which

are

again

conﬁrmed

by

numerical simulations.

3. F1 = F2 = F3 = F = 1 + λb2. In this case α-quenching and ﬂux loss due to poloidal magnetic ﬁeld and toroidal mag-

netic ﬁeld are the same, equally contributing to the ﬂatten-

ing of magnetic ﬁelds for large rotation, Eqs. (13)−(16) give p ∼ F ∼ 1 + λb2, D0 = (1 + λb2)3, a = 2D0b/21/2(1 + λb2)3, and tan φ = 1. Clearly, poloidal ﬂux loss and toroidal ﬂux loss

both determine the maximum frequency p of the magnetic ﬁeld;

p increases with Ω, similar to the case where the eﬀect of both

A100, page 3 of 12

POWER

FREQUENCY OF MAGNETIC FIELD

A&A 563, A100 (2014) κ1=2.5,λ1=λ2=0.0
͉A͉

κ1=2.5,λ1=λ2=0.0
͉B͉

ROTATION RATE
(a) p as a function of Ω

Ω (b) |A| as a function of Ω

Ω (c) |B| as a function of Ω

Fig. 1. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A| and toroidal magnetic ﬁeld |B| as a function of rotation rate Ω for F1 = F, F2 = F3 = 1.0 for Case 1 in the fourth-order system.

κ1=0.0,λ1=λ2=2.5

κ1=0.0,λ1=λ2=2.5

͉B͉ ͉A͉

POWER

FREQUENCY OF MAGNETIC FIELD

ROTATION RATE
(a) p as a function of Ω

Ω
(b) |A| as a function of Ω

Ω
(c) |B| as a function of Ω

Fig. 2. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A| and toroidal magnetic ﬁeld |B| as a function of rotation rate Ω for F1 = 1.0, F2 = F3 = F for Case 2 in the fourth-order system.

κ1=λ1=λ2=2.5

κ1=λ1=λ2=2.5

͉A͉

͉B͉

POWER

FREQUENCY OF MAGNETIC FIELD

ROTATION RATE
(a) p as a function of Ω

Ω
(b) |A| as a function of Ω

Ω
(c) |B| as a function of Ω

Fig. 3. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A| and toroidal magnetic ﬁeld |B| as a function of rotation rate Ω for F1 = F2 = F3 = F for Case 3 in the fourth-order system.

ﬂux losses is considered; a and b both increase with rotation rate

(cf. Fig. 3a to 3c). Interestingly, the phase between a and b re-

mains unaltered due to the equal amount of F2 and F3, with its

value φ

=

π 4

.

In

the

limit

of

D0

1,

we

ﬁnd

that

p

∼

Ω

2 3

,

a

∼

Ω,

and

b

∼

Ω

1 3

.

Numerically

found

scalings

are

almost

similar for p, b, and a (cf. Table 1). Thus, equal amount of

α-quenching, enhanced poloidal ﬂux loss and toroidal ﬂux loss

lead to stronger reduction in magnetic ﬁeld for a higher rotation

rate, which can provide strong dynamic balance in a nonlinear

dynamo.

4. F1 = F2 = F = 1 + λb2, F3 = 1. In the presence of only α-

quenching and enhanced poloidal magnetic ﬂux loss, the various scalings are simpliﬁed as p ∼ (1 + λb2)1/2, 2D0 = (2 + λb2)2

(1 + λb2)1/2, a = 2D0b/(2 + λb2)1/2(1 + λb2)2, tan φ ∼ 1/(1 + λb2)1/2. Thus, φ decreases for large b (or large D0), reducing

phase shift between b and a to zero. Furthermore, in the limit of

D0

1,

these

give

us

p

∼

Ω2 5

,

b

∼

Ω2 5

,

and

a

∼

Ω4 5

.

5. F1 = F3 = F = 1 + λb2, F2 = 1. This is the case where α-quenching and ﬂux loss due to toroidal magnetic ﬁeld are in action; Eqs. (13) to (16) give p ∼ (1 + λb2)1/2, 2D0 = (1 + λb2)5/2, a = 2D0b/(2 + λb2)1/2(1 + λb2)2, and tan φ = 1. Clearly,

frequency p of magnetic ﬁeld is inﬂuenced by toroidal magnetic

ﬁeld b. For D0 1, the scalings of p, b, and a are found to be

p

∼

Ω

2 5

,

b

∼

Ω

2 5

,

and

a

∼

Ω

6 5

.

Compared

to

the

Case

4

above,

we see that change from toroidal to poloidal ﬂux loss only aﬀects

the scaling of a with Ω.

A100, page 4 of 12

A. Sood and E. Kim: Detailed mathematical and numerical analysis of a dynamo model

Table 1. Scaling exponents for p, |B|, and |A| for Ω ≥ 5 in the cases of the fourth-order system where p = Ωξ, |B| = Ωβ, and |A| = Ωγ.

Transport coeﬃcients ξ

β

γ

F1 = F, F2 = F3 = 1 F1 = 1, F2 = F3 = F F1 = F2 = F3 = F F1 = F2 = F, F3 = 1 F1 = F3 = F, F2 = 1 F1 = F2 = 1, F3 = F F1 = F3 = 1, F2 = F

0 1.0 1.0 1.0 0.50 1.55 0.66 0.41 1.08 0.41 0.47 0.84 0.41 0.47 1.55 0.67 0.69 2.03 0.67 0.69 1.35

6. F3 = F = 1 + λb2, F1 = F2 = 1. Here ﬂux loss due to toroidal magnetic ﬁeld tames the amplitude of b for large Ω, turning Eqs. (13)−(16) into the form p = (1 + λb2)1/2, 2D0 = (2 + λb2)(1 + λb2)1/2, a = 2D0b/(1 + λb2)1/2, and tan φ ∼ (1 + λb2)1/2. Therefore, toroidal ﬂux loss increases the

frequency according to the strength of b, which is determined

dynamically. Also, the phase φ between a and b increases, ap-

proaching

π 2

as

D0

1, conﬁrming that the preferential removal

of toroidal ﬂux over poloidal ﬂux increases the phase. In the limit

of D0

1,

we

ﬁnd

power-law

scalings

p

∼

Ω

2 3

,

b

∼

Ω

2 3

,

and

a ∼ Ω2.

7. F2 = F = 1 + λb2, F1 = F3 = 1. Flux loss due to poloidal ﬁeld determines the frequency, simplifying Eqs. (13)−(16) as p = (1 + λb2)1/2, 2D0 = (2 + λb2)(1 + λb2)1/2, a = 2D0b/(1 + λb2)(2 + λb2)1/2, tan φ = (1 + λb2)−1/2. In the limit of D0 1, the scalings are found as p ∼ Ω2/3, b ∼ Ω2/3 and a ∼ Ω4/3, tan φ ∼ Ω−2/3. Therefore, compared to Case 6 above, the diﬀer-

ence is seen in the scalings of a and φ; that is, reduction in the

scaling of a and φ with Ω is observed, contrary to the case where

only saturation mechanism is ﬂux loss due to toroidal magnetic

ﬁeld.

To summarize, the eﬀects of α-quenching F1 and ﬂux losses F2 and F3 are shown to be diﬀerent; p is dynamically deter-
mined by b through ﬂux loss F2 and F3, but with no inﬂuence of α-quenching F1; that is, p = 1 with α-quenching, while p increases with Ω due to ﬂux loss. The eﬀects of F2 and F3 are almost similar, leading to the same scalings of p and b with Ω.

The phase φ is determined by the ratio of F2 and F3. In all cases, b is always found to increase with Ω as b ∝ Ωβ where β takes

the value between 1/3 and 1. The largest β = 1 is found in the

case of α-quenching only, while smallest β = 0.4 in the pres-

ence of nonlinear terms of same magnitude. Thus, in this fourth-

order system, b keeps growing with Ω, without ﬂattening for

large Ω. Numerical values of scaling exponents are summarized

in Table 1.

4. Nonlinear effect through mean differential
rotation
We now incorporate the nonlinear eﬀect due to the modiﬁcation of diﬀerential rotation by magnetic ﬁelds and examine the role of mean diﬀerential rotation in self-regulatory behaviour. Here, we exclude the ﬂuctuating diﬀerential rotation in order to identify the eﬀect of mean diﬀerential rotation. The eﬀect of ﬂuctuating diﬀerential rotation is presented in Sect. 5. The main equations are Eqs. (6)−(10) for w = 0, and ν0 = 35 is chosen for numerical simulations.
1. F1 = F2 = F3 = 1. Even in the absence of α-quenching or any ﬂux loss, the back reaction of mean diﬀerential rotation w0

provides the saturation mechanism to ensure a ﬁnite-amplitude solution, reducing Eqs. (6)−(10) to

p = ±1,

(21)

b = D−0 1/2ν10/2(1 − D−0 1)1/2,

(22)

a = D10/2ν10/2 (2(1 − D−0 1)),

(23)

w0

=

−1

+

1 D0

,

(24)

tan φ = 1.

(25)

Equation (21) immediately shows that the frequency of magnetic

ﬁeld is independent of rotation rate, with no inﬂuence of w0. In

the limit of a high rotation rate, i.e. D0 1, Eqs. (22)−(25)

give us analytical scalings b ∼ Ω−1, a ∼ Ω and total shear

1 + w0 ∼ Ω−2 1 for Ω 1. The phase angle between

poloidal magnetic

ﬁeld a and b keeps

its

linear

value

of φ

=

π 4

(cf. Eq. (25)). We note that the dependence of the total shear on

rotation rate Ω−2 shows signiﬁcant shear quenching. These re-

sults are conﬁrmed by numerical simulations. Figure 4a shows

one single dominant frequency, p ∼ 1, that hardly changes with

rotation. A red broad band of weak intensity around the local-

ized frequency of maximum intensity increases its width initially

and then decreases for higher rotation. This behaviour arises be

cause shear quenching due to Lorentz force is too strong, leading

to signiﬁcant quenching of magnetic ﬁeld strength b for Ω 1.

This severe shear quenching is shown in Fig. 4d, where total

shear decreases rapidly and approaches zero for a higher rotation

rate (Ω ≥ 12). Numerically, scalings of |B|, |A| and total shear 1 + w0 for high rotation are found to be a ∼ Ω1.02, b ∼ Ω−0.98, 1 + w0 ∼ Ω−2.0 ∼ D−0 1, which are similar to analytically predicted values, noted above.
2. F1 = F = 1 + λb2, F2 = F3 = 1. In the presence of α-quenching alone with no ﬂux loss, Eqs. (6) to (10) become

p = ±1,

(26)

F2 +√Dν00b2 = D0F,

(27)

a = 2D0b ,

(28)

F

w0

=

−

D0b2 F ν0

,

(29)

tan φ = 1.

(30)

Again, the frequency p is independent of rotation rate.
Analytical scalings are obtained from Eqs. (26)−(30) for the
limit of the high rotation rate. In the limit of the high rotation rate, D0 1, we obtain b ∼ Ω−1, a ∼ Ω, and (1 + w0) ∼ Ω−2. The total shear 1+w0 ∼ Ω−2 1 for large Ω implies that Lorentz force generates strong mean diﬀerential rotation w0 < 0, causing
the signiﬁcant reduction in the magnetic ﬁeld. 3. F2 = F3 = F = 1 + λb2, F1 = 1. Equations (6) to (10)
give us

p = ±F,

(31)

F4 +√Dν00b2 = D0F2,

(32)

a=

2D0b F 3/2

,

(33)

w0

=

−

D0b2 F 2 ν0

,

(34)

tan φ = 1.

(35)

A100, page 5 of 12

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A&A 563, A100 (2014)
0

κ1=λ1=λ2=0.0

͉A͉

ROTATION RATE
(a) p as a function of rotation rate. κ1=λ1=λ2=0.0
͉B͉

Ω
(b) |A as a function of Ω
κ1=λ1=λ2=0.0

1+ω0

Ω
(c) |B| a function of Ω

Ω
(d) Total shear as a function of rotation rate.

Fig. 4. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A|, toroidal magnetic ﬁeld |B| and total shear as a function of rotation rate Ω for F1 = F2 = F3 = 1 for Case 1 in the ﬁfth-order system.

Thus, p changes as b changes with Ω (cf. Eq. (31)) due to non-

linear ﬂux losses F2 and F3. In the limit of the high rotation rate, D0 1, we ﬁnd b ∼ Ω−1 1, 1 + λb2 ∼ 1, a ∼ Ω, and 1 + w0 ∼ Ω−2. The decline in toroidal magnetic ﬁeld with Ω is

again due to the severe shear quenching present in the system.

On the other hand, the phase shift between a and b remains as

π 4

owing

to

equal

amount

of

the

toroidal

and

poloidal

magnetic

ﬂuxes.

4. F1 = F2 = F3 = F = 1 + λb2. Equations (6)−(10) are

simpliﬁed as

p = ±F,

(36)

F6 +√Dν00b2 = D0F3,

(37)

a=

2D0b F 3/2

,

(38)

w0

=

−

D0b2 F 2 ν0

,

(39)

tan φ = 1.

(40)

In this case we numerically ﬁnd that the presence of F1, F2, and F3 leads to interesting dynamics for large Ω. For instance, the system bifurcates at Ω = 19.7 with a sudden change in p, |B|, |A|, and total shear (cf. Figs. 5a−5d). This result suggests a
possibly dramatic change in the dynamo owing to a strong mean
shear. Its physical relevance in the evolution of solar spin-down
will be addressed in future work.

5. F1 = F2 = F = 1 + λb2, F3 = 1. The presence of α-quenching and nonlinear poloidal ﬂux loss reduces

Eqs. (6)−(10) to

√

p = ± F,

(41)

b2 = 2−1D−0 1F6(1 + F−1)(F1/2 − 1 − F−1),

(42)

a = 2√D0b ,

(43)

F2 F + 1

w0

=

− pa2 F 2ν0 D0

,

(44)

tan φ = F−1/2.

(45)

We note that p is inﬂuenced by b as p ∼ b (cf. Eq. (41)). Again,
analytical scalings are obtained within the limit of a very high rotation rate, i.e., D0 1 as p ∼ Ω2/5, b ∼ Ω2/5, a ∼ Ω4/5, and (1 + w0) ∼ Ω−2, whereas the phase shift between a and b decreases as rotation rate increases. The analytical scalings for p, |B|, and |A| are the same as for those obtained in the fourth-order system in the case of α-quenching and ﬂux loss. Numerically
found scalings are very similar to these analytical results. Somewhat similar results are found for diﬀerent cases, as
brieﬂy summarized as follows. For F1 = F3 = F, F2 = 1, and F3 = F, F1 = F2 = 1, we ﬁnd that frequency p and phase shift are inﬂuenced by b. For F2 = F, F1 = F3 = 1, p and phase shift are the same as in Case 5. All the numerical results for high rotation, Ω ≥ 5 are listed in Table 2.

A100, page 6 of 12

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A. Sood and E. Kim: Detailed mathematical and numerical analysis of a dynamo model
κ1=λ1=λ2=2.5
͉A͉

ROTATION RATE
(a) p as a function of rotation rate. κ1=λ1=λ2=2.5
͉B͉

Ω
(b) |A| as a function of Ω
κ1=λ1=λ2=2.5

1+ω0

Ω
(c) |B| a function of Ω

Ω
(d) Total shear as a function of rotation rate.

Fig. 5. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A|, toroidal magnetic ﬁeld |B| and total shear as a function of rotation rate Ω for F1 = F2 = F3 = F for Case 4 in the ﬁfth-order system.

Table 2. Scaling exponents for p, |B|, |A|, and 1 + w0 for Ω ≥ 5 in the cases of the ﬁfth-order system where p = Ωξ, |B| = Ωβ, |A| = Ωγ, and 1 + w0 = Ωδ.

Transport coeﬃcients
F1 = F2 = F3 = 1 F1 = F, F2 = F3 = 1 F1 = 1, F2 = F3 = F F1 = F2 = F3 = F F1 = F2 = F, F3 = 1 F1 = F3 = F, F2 = 1 F1 = F2 = 1, F3 = F F1 = F3 = 1, F2 = F

ξ
0 0 −0.99 varies 0.39 −0.43 −0.30 −0.57

β
−0.98 −1.39 −1.56 varies 0.45 −1.45 −1.02 −1.62

γ
1.02 1.42 1.30 varies 0.93 1.66 1.20 1.30

δ
−2.0 −3.46 −4.30 varies −0.16 −4.40 −2.76 −3.21

found to be decreased with Ω after an initial increase in all cases except F1 = F2 = F, F3 = 1, which is very similar to the numerical results found for F1 = F2 = F, F3 = 1 in Sect. (3). But in certain cases where the combined action of F1 = F2 = F3 = F is considered, A/Ω grows faster than |B| for higher Ω.
In summary, the mean diﬀerential rotation generated by
the Lorentz force severely inhibits the dynamo with signiﬁcant quenching of magnetic ﬁelds for large Ω. That is, without ﬂuctuating diﬀerential rotation, quenching in mean diﬀerential rota-
tion tends to become severe, possibly shutting down the dynamo for large Ω. This thus suggests that self regulation of diﬀerential rotation through ﬂuctuating diﬀerential rotation is needed to
prevent it. This will be shown in the next section.

We now compare these results with those obtained in the
fourth-order system. The similarity is found in the frequency
behaviour of p that is controlled by the ﬂux loss through F2 and F3, without being inﬂuenced by α-quenching and also in phase shift, which is determined by ratio of F2 and F3. Also, it is noticed that p does not change its value with rotation rate for F1 = F2 = F3 = 1 and F1 = F, F2 = F3 = 1, bearing a resemblance to Case 1 in Sect. 3. The similar role of F2 and F3 in the fourth-order system is, however, broken owing to mean diﬀerential rotation. The agreement between analytical
scalings and numerical scalings becomes less accurate than the
fourth-order system, with only two cases of a close match (e.g., F1 = F2 = F3 = 1 and F1 = F2 = F, F1 = 1). Both b and a are

5. Nonlinear effect through ﬂuctuating differential
rotation
The goal of this section is to understand the eﬀect of ﬂuctuating diﬀerential rotation by assuming w0 = 0 in Eqs. (6)−(10). Since, analytical solutions are not particularly illuminating, Eqs. (6)−(10) are solved numerically to examine how the scalings of a, b, and p change with Ω in diﬀerent cases. We choose ν = 0.5 for these simulations to highlight the eﬀect of nonlinear terms in taming the chaotic property of the system.
I. F1 = F2 = F3 = 1: this is the case where dynamo saturates solely due to the back reaction of ﬂuctuating diﬀerential rotation with no nonlinear feedback through transport coeﬃcients.

A100, page 7 of 12

POWER

FREQUENCY OF MAGNETIC FIELD

A&A 563, A100 (2014)

κ1=λ1=2.5

͉A͉

ROTATION RATE
(a) p as a function of rotation rate. κ1=λ1=2.5
͉B͉

Ω
(b) |A| as a function of Ω
κ1=λ1=2.5

1+ω0

Ω
(c) |B| a function of Ω

Ω
(d) Total shear as a function of rotation rate.

Fig. 6. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A|, toroidal magnetic ﬁeld |B| and total shear as a function of rotation rate Ω for F1 = F2 = F, F3 = 1 for Case 5 in the ﬁfth-order system.

Figure 7a shows the frequency spectra for diﬀerent values of Ω. We can see a broad band of frequency p of a magnetic ﬁeld of high intensity, which increases very rapidly with Ω. This broad band is a manifestation of the presence of many diﬀerent frequencies in the system as a result of chaotic dynamics. A lo-
calized band of lower frequencies is also noticeable, which increases with Ω. The b and a plotted in Figs. 7b and 7c both show their rapid increase with Ω with small ﬂuctuations appearing for large Ω. Strong chaos in the system is caused by ﬂuctuating differential rotation (cf. Figs. 7a to 7c).
II. The presence of α-quenching or either ﬂux loss tames the chaos, leading to a ﬁnite amplitude solution with a localized frequency. For diﬀerent values of F1, F2, and F3, the scaling results of dynamical variables for large Ω are summarized in Table 3. The scalings p, |B|, and |A| are highly variable with Ω, especially for small Ω. We thus study these as a function of Ω by computing local scalings and plot them in Figs. 8a−8c. For diﬀerent combinations of F1, F2, and F3, we plot results using diﬀerent colours/linestyles in Figs. 8a−8c. We denote the local scaling exponents of p, |B|, and |A| as ξ1, β1, and γ1, respectively. Clearly, the change in local scalings of p, |B|, and |A| become less variable for high-rotation Ω ≥ 5 (cf. Figs. 8a−8c). It is also interesting to see that the scalings of p are more robust because they vary less than those in |B| and |A|.
We ﬁrst examine the eﬀect of the inclusion of ﬂux losses in the presence of nonlinear α-eﬀect. In the case of α-quenching only (i.e., for F1 = F, F2 = F3 = 1), |B| grows rapidly with Ω,

Table 3. Scaling exponents for p, |B|, and |A| for Ω ≥ 5 in the cases of the sixth-order system where p = Ωξ, |B| = Ωβ, and |A| = Ωγ.

Transport coeﬃcients ξ

β

γ

F1 = F, F2 = F3 = 1 F1 = F2 = F, F3 = 1 F1 = F2 = F3 = F F1 = F3 = F, F2 = 1 F1 = 1, F2 = F3 = F F1 = F2 = 1, F3 = F F1 = F3 = 1, F2 = F

0.63 0.73 0.71 0.57 0.50 0.78 0.78 0.40 1.01 0.81 0.36 0.95 1.20 0.52 1.39 1.20 0.44 1.24
−−−

whereas p and |A| increase very slowly with Ω (see Figs. 8a−8c).
When nonlinear toroidal ﬂux loss is added to this case (i.e., for F1 = F3 = F, F2 = 1), β1 (local scaling of magnetic ﬁeld strength |B|) is decreased by approximately half, while ξ1 (local scaling of frequency p of magnetic ﬁeld) and γ1 (local scaling of poloidal magnetic ﬁeld strength |A|) become larger. The
inclusion of nonlinear poloidal ﬂux loss to this case (i.e., for F1 = F2 = F3 = F) increases the β1 ∼ β by 10% with a powerlaw scaling of β = 0.40 (cf. Table 3) in comparison with the value of β1 ∼ β = 0.36 in the case F1 = F3 = F, F2 = 1. We note that the reduction in the value of ξ1 ∼ ξ is 4% and γ1 ∼ γ is 6%. Thus, results suggest that including nonlinear poloidal ﬂux
loss in the presence of nonlinear α and toroidal ﬂux loss reduces the scaling of p with Ω, while increases the scaling of |B|. This
demonstrates that poloidal ﬂux loss does not contribute much to

A100, page 8 of 12

FREQUENCY OF MAGNETIC FIELD POWER

A. Sood and E. Kim: Detailed mathematical and numerical analysis of a dynamo model

κ1=λ1=λ2=0.0

κ1=λ1=λ2=0.0

͉A͉

͉B͉

ROTATION RATE
(a) p as a function of Ω

Ω
(b) |A| as a function of Ω

Ω
(c) |B| as a function of Ω

Fig. 7. Frequency of maximum intensity p, poloidal magnetic ﬁeld |A| and toroidal magnetic ﬁeld |B| as a function of rotation rate Ω for F1 = F2 = F3 = 1 for Case 1 in the sixth-order system.

the regulation of the growth of toroidal magnetic ﬁeld strength with Ω.
Second, we focus on the eﬀect of ﬂux losses when F1 = 1. The equal amount of F2 and F3 increases the scaling of p, |B|, and |A| to a value ξ1 ∼ ξ = 1.20, β1 ∼ β = 0.52, and γ1 ∼ γ = 1.39, respectively. In comparison, toroidal ﬂux loss (i.e., for F3 = F, F2 = F1 = 1) does not change the scaling exponent of p, but it is noticed to reduce the scaling of |B| and |A| with scaling exponents β1 ∼ β = 0.44 and γ1 ∼ γ = 1.24 in comparison with both ﬂux losses case. Figure 8a shows local scaling of p as a function of Ω for F2 = F3 = F and F3 = F. For F1 = F3 = 1, F2 = F, p, |b| and |a| show ﬂuctuation for a higher rotation rate, as can be seen from Figs. 8a−8c, where local scalings of p, |B|, and |A| are highly variable as Ω changes, indicating that poloidal ﬂux loss is not eﬃcient in reducing the eﬀect of ﬂuctuating diﬀerential rotation. Therefore, these results suggest that nonlinear toroidal magnetic ﬂux loss is more important than poloidal ﬂux loss in taming the dynamo.
In summary, the study of all cases of the sixth-order system demonstrates that the presence of ﬂuctuating diﬀerential rotation leads to magnetic ﬁeld |B|, which keeps increasing with Ω for high rotation with the scaling in the range [0.36, 0.73] (cf. Table 3). However, this increase in magnetic ﬁeld is less rapid compared to the case of the fourth-order system with no ﬂuctuating and no mean diﬀerential rotation. For instance, in the fourth-order system, the scaling of magnetic ﬁeld |B| with Ω are [0.41, 1.0] (cf. Table 1). Toroidal ﬂux loss is more important in reducing β1 as a nonlinear term than poloidal ﬂux loss. These results show that in the sixth-order system various saturation mechanisms, especially, α-quenching and toroidal ﬂux loss, are able to slow down the growth of magnetic ﬁeld with Ω to some degree, but not strongly enough to ﬂatten the magnetic ﬁeld for large Ω. Compared to the results in the fourth-order system, the opposite eﬀects of mean diﬀerential rotation and ﬂuctuating differential rotation shown in Sects. 4 and 5, respectively, suggest that the combined eﬀect of both mean and ﬂuctuating diﬀerential rotation is beneﬁcial for the self-regulatory behaviour of dynamo. These opposite eﬀects of mean and ﬂuctuating diﬀerential rotation also make the eﬀects of toroidal and poloidal ﬂux loss equally important in the seventh-order system, as discussed in Sect. 6.
6. Implications and key observations
Ever more improved observational data of the evolution of magnetic ﬁelds and rotations of various stars of diﬀerent age and

spectral types with diﬀerent rotation rates have been revealing valuable information about the relation among rotation, diﬀerential rotation, and magnetic activity. First, the cycle period Pcyc of magnetic ﬁelds of the solar type stars is shown to depend upon the stellar rotation period Prot as Pcyc ∝ Pβrot, with scaling exponent β = 1.25 ± 0.5 (Noyes et al. 1984); e.g., β = 0.8 for the active branch and β = 1.15 for the inactive branch (Saar & Brandenburg 1999; Charbonneau & Saar 2001; Saar 2002). Second, the magnetic activity (a measure of the strength of magnetic ﬁelds) has been observed to increase with stellar rotation but tends to saturate for high rotation (with Prot < 3 day). Third, the diﬀerential rotation ΔΩ is observed to increase with the stellar rotation rate Ω, while the relative diﬀerence ΔΩ/Ω decreases with Ω. The precise dependence of diﬀerential rotation on rotation is quite uncertain since ΔΩ ∝ Ωn where 0 < n < 1 (Fröhlich et al. 2009; Hotta & Yokoyama 2011; Donahue et al. 1996; Reiners & Schmitt 2003; Barnes et al. 2005). Furthermore, the time variation of diﬀerential rotation has been reported in recent work (e.g., Fröhlich et al. 2012; Hotta & Yokoyama 2011).
Therefore, a main challenge in dynamo theory is to explain these observations. For instance, the almost linear relation between the frequency of magnetic ﬁelds and stellar rotation could be explained within the framework of linear dynamo theory, for instance, by the linear dispersion relation in α − Ω dynamo. However, observed magnetic ﬁelds are manifestations of nonlinear saturation, making the understanding of how stellar magnetic ﬁelds apparently manage to almost maintain a linear dependence of their frequency on rotation rate as one of the fundamental questions in any dynamo theory.
Our results from Sects. 3−5 above show that none of the cases considered are consistent with all these observations. Towards understanding these observations, Sood & Kim (2013) have numerically studied the seventh-order system, given by Eqs. (1)−(4), and performed a thorough parameter study on the choice of nonlinear quenching eﬀects of F1, F2, and F3 by using ν = 1 and ν0 = 35. From their investigation, they proposed a minimal model that can reproduce the observed relation between magnetic frequency and rotation rate. In particular, they highlight the importance of the regulatory behaviour of dynamo through a nonlinear balance between the generation and destruction of the magnetic ﬁelds. They also demonstrate the importance of both mean and ﬂuctuating diﬀerential rotations in achieving dynamical balance among magnetic ﬁelds. We now show how, through the balance among diﬀerent nonlinear eﬀects of transport coeﬃcients and mean and ﬂuctuating diﬀerential rotations, this regulatory behaviour arises in the seventh-order

A100, page 9 of 12

A&A 563, A100 (2014)

5

F1 = F2 = F3 = F

F1 = F, F2 = F3 = 1

4

F1 = F2 = F, F3 = 1

F3 = F, F1 = F2 = 1

F1 = F3 = F, F2 = 1

F2 = F, F1 = F3 = 1

3

F2 = F3 = F, F1 = 1

3.0

F2 = F, F1 = F3 = 1

F1 = F, F2 = F3 = 1

2.5

F1 = F2 = F, F3 = 1 F1 = F3 = F, F2 = 1

F3 = F, F1 = F2 = 1 F2 = F, F1 = F3 = 1

F2 = F3 = F, F1 = 1

2.0

ξ1 β1
γ1

2

1.5

1

1.0

0

0.5

−1

0

5

10

15

20

Ω

(a) ξ1 as a function of Ω

0.0

0

5

10

15

20

Ω

(b) γ1 as a function of Ω

3.0

F2 = F, F1 = F3 = 1

F1 = F, F2 = F3 = 1

2.5

F1 = F2 = F, F3 = 1 F1 = F3 = F, F2 = 1

F3 = F, F1 = F2 = 1 F2 = F, F1 = F3 = 1

F2 = F3 = F, F1 = 1

2.0

1.5

1.0

0.5

0.0

0

5

10

15

20

Ω

(c) β1 as a function of Ω

Fig. 8. Local scalings ξ1 for p, γ1 for |A|, and β1 for |B| are plotted as functions of rotation rate for diﬀerent combinations of F1, F2, and F3, where for diﬀerent combinations, diﬀerent colours/linestyles are assigned as: for F1 = F2 = F3 = F black/solid line, F1 = F2 = F, F3 = 1 green/dot, F1 = F3 = F, F2 = 1 gold/dash, F1 = 1, F2 = F, F3 = F pink/dash dot, F1 = F, F2 = F3 = 1 turquoise/dash dot dot, F1 = F2 = 1, F3 = F red/long dash, F1 = F3 = 1, F2 = F turquoise/“+”.

system when F1 ∼ F2 ∼ F3. To demonstrate how the results in Sects. 4−6 are altered in this seventh-order system, we use the same parameter values ν = 0.5 and ν0 = 35 used in Sects. 4−6.

The behaviour of frequency of magnetic ﬁeld and magnetic

ﬁeld strength with rotation rate in this seventh-order system is shown in Figs. 9a−9b. Figure 9a shows an almost linear increase

in the frequency p of maximum intensity (depicted in yellow)

with rotation rate. A red band of the localized frequency of max-

imum intensity around p is found to gradually increase its width with rotation rate. The power-law dependence of p on Ω is found to be p ∼ Ω0.82 for a higher rotation rate and agrees with observations. The ﬂattening of magnetic ﬁeld strength |B| for a higher

rotation rate can be seen in Fig. 9b, which is also consistent with

observations.

To investigate the balance, in Figs. 10a−10b, we plot the

magnitude of various nonlinear terms appearing in Eqs. (1), (2)

for D0 = 2, which are |2DB|, |F2A|, |i(1 + w0|)A, |F3B|, and

|

1 2

iA∗w|.

Figure

10a

shows

that

there

is

a

balance

between

the

A100, page 10 of 12

α-quenching term |2DB| and poloidal ﬂux loss term |F2A|, de-

picted in black and red, respectively. This thus indicates that

the dissipation and the generation of magnetic ﬁeld are al-

most balanced in Eq. (1). Figure 10b shows that the genera-

tion of poloidal ﬁeld due to shear |w0A| (illustrated in black)

is in balance with toroidal ﬂux loss |F3B| (represented in red).

Fluctuating

diﬀerential

rotation

|

1 2

iA∗

w|

depicted

in

blue

is

less

dominant in this case. Thus, the two main nonlinear terms in

Eq. (2) are of similar magnitude.We compare the nonlinear terms

in Eqs. (1) and (2) for all other cases in seventh-order systems

and conﬁrm that the dynamical balance is best achieved for this

seventh-order system with F1 = F2 = F3.

7. Conclusions
We studied nonlinear dynamical model analytically and numerically to elucidate the eﬀects of nonlinear terms in the regulation of a dynamo. In the fourth-order system, ﬂux losses

A. Sood and E. Kim: Detailed mathematical and numerical analysis of a dynamo model

(a) p as a function of rotation rate Ω.

(b) Magnetic ﬁeld strength |B| as a function of rotation rate Ω.

Fig. 9. Frequency p of magnetic ﬁeld and toroidal magnetic ﬁeld |B| are plotted as a function of rotation rate Ω for F1 = F2 = F3 = F in case of the seventh-order system for ν = 0.5 and ν0 = 35.0.

(a) Dynamical balance between nonlinear terms on R.H.S. of Eq. (1).

(b) Dynamical balance between nonlinear terms on R.H.S. of Eq. (2).

Fig. 10. Time series of diﬀerent nonlinear terms for F1 = F2 = F3 = F in case of the seventh-order system for ν = 0.5 and ν0 = 35.0.

determine the frequency p of magnetic ﬁeld, without any inﬂuence of α-quenching; that is, α-quenching does not aﬀect the frequency p. The eﬀect of ﬂux losses due to poloidal magnetic ﬁeld and toroidal magnetic ﬁeld is the same on p and |B| irrespective of the presence or absence of α-quenching. Phase φ
is determined by the ratio of F2 and F3. We observed the increase in p and |B| with Ω in all cases, with power-law scalings ξ ∼ [0, 1.0] and β ∼ [0.4, 1.0], respectively. We ﬁnd good agree-
ment between analytical and numerical results in this system.
The presence of shear due to mean diﬀerential rotation, that
is, in the ﬁfth-order system, frequency p keeps a constant value p = 1 for diﬀerent rotations when the α-eﬀect is quenched or when there are no transport coeﬃcients present in the system.
That is, again the frequency p is determined by both ﬂux losses
and phase is determined by the ratio of F2 and F3, which is similar to the fourth-order system. But the eﬀects of F2 and F3, with or without α-quenching, are no longer the same on p and |B| in contrast to the fourth-order system. We ﬁnd growth of p and |B| in the presence of an equal combination of α-eﬀect and poloidal ﬂux loss (i.e., F1 = F2 = F, F3 = 1). The quenching of magnetic ﬁeld |B| and p is more signiﬁcant when the nonlinear α-eﬀect and toroidal ﬂux loss are taken together (i.e., F1 = F3 = F, F2 = 1). The decreasing behaviour of p and |B| in almost all cases is due
to severe shear quenching, which is responsible for the shutting
down of dynamo action.

In the sixth-order system, we present local scalings with rotation rate to demonstrate the eﬀect of nonlinear terms. Here, unlike the fourth- and the ﬁfth-order systems, α-quenching (i.e., F1) inﬂuences p, while p and |B| behave diﬀerently when the effect of either ﬂux loss is considered in the presence or absence of α-quenching. Interestingly, in this case, p and |B| always increase with Ω despite the feedback from ﬂuctuating diﬀerential rotation. Toroidal ﬂux loss is shown to be more important than
poloidal ﬂux loss. The results observed for the ﬁfth-order and the
sixth-order systems thus identify the pivotal role of the combined eﬀect of both mean and ﬂuctuating diﬀerential rotation in the self regulation of a dynamo to work near marginal stability. We also
note that the self regulation between generation and dissipation
of magnetic ﬁelds is a prerequisite for working of the dynamo. In all cases, the scaling of p is less sensitive to the change in Ω or to the diﬀerent combinations of nonlinear transport coeﬃcients.
Finally, we presented how this self-regulatory behaviour
comes about in the seventh-order system in the presence of an equal combination of α-quenching, poloidal ﬂux loss, and toroidal ﬂux loss (Sood & Kim 2013), where α-quenching, toroidal ﬂux loss and poloidal ﬂux loss are equally important. While our work is based on the α − Ω dynamo model, similar regulation of the generation and the dissipation of magnetic ﬁelds not only between nonlinear transport coeﬃcients but also between mean and ﬂuctuating diﬀerential rotation is likely to

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A&A 563, A100 (2014)

persist in other models. The extension of this work to other models will be addressed in a future paper.
It will also be of great interest to utilize our model to understand the spin down of solar type stars (Leprovost & Kim 2010; Kepens et al. 1995) by using a magnetic ﬁeld that is consistently obtained from a dynamo model (work in progress).
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