J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Role of the Turbulent Magnetic Helicity in the
Non-Linear Behavior of Solar Dynamo
Irina KITIASHVILI1) and Alexander KOSOVICHEV2)
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA
1)

2)(Received: 5 September 2008 / Accepted: 29 November 2008)

Magnetic helicity plays important role in solar dynamo and magnetic ﬁeld variations. It is importantfor explaining the observed variations of the sunspot number variations during solar cycles of activity.
For comparison we consider the classical Parker’s dynamo model [1] and the Kleeorin-Ruzmaikin model[2]. We found that in the low-order approximation the Parker’s model (without magnetic helicity)
does not allows to reproduce the typical behavior of the sunspot number with a fast growth and slow
decay or obtain a chaotic solution, even in strong nonlinear regimes. The analysis of the KleeorinRuzmaikin model shows the existence of nonlinear periodic and chaotic solutions for conditions of
the solar convective zone. For this model we obtain proﬁles of the sunspot number variations, which
qualitatively reproduce the typical proﬁle of the solar cycles. We apply the Ensemble Kalman Filter
method and show the the sunspot data can be assimilated in the dynamo model. This opens perspectives
for estimating the physical state of the solar dynamo and for forcasting the solar activity cycles.
Keywords: Magnetohydrodynamics, turbulence, solar dynamo, solar activity, solar magnetic ﬁelds,
sunspots, data assimilation, Ensemble Kalman Filter method
Because of the conservation of the total helicity agrowth of the large-scale magnetic helicity due to the
dynamo action is compensated by the growth of thesmall-scale helicity of opposite sign [4]. Thus, thesmall- and large-scale magnetic ﬁelds grow together
and are mirror-asymmetrical and hence the conditionof magnetic helicity conservation is, perhaps, more severe for a restriction of the dynamo action than thecondition of the energy conservation, which leads toquenching of the kinetic helicity.
For modeling the solar cycle we consider nonlinear behavior of the Parker’s dynamo model (withoutmagnetic helicity) and the Kleeorin-Ruzmaikin model,
explicitly based on the idea of magnetic helicity conservation. For simplicity, we use a ”low-order model”
approach [5, 6], reducing the dynamo equations to asimple nonlinear dynamical system. The goal of thisresearch is to ﬁnd solutions, which can reproduce thebasic properties of the sunspot number variation. To
connect the dynamo model solutions with the sunspot

1. Introduction
Solar magnetic ﬁelds are generated by a dynamo
action in the turbulent plasma of the Sun’s interior.
One of the manifestations of solar magnetic activity
is the 11-year sunspot cycle, (Fig. 1), which is characterized by the fast growth and slowly decay of thesunspot number parameter. For an explanation of themagnetic ﬁeld generation Parker [1] proposed a simple
dynamo model, which describes the phenomenon as anaction of two factors: the diﬀerential rotation and cyclonic convective vortices. The mean-ﬁeld theory anddiscovery of the α-eﬀect give us a general descriptionof the process of magnetic ﬁeld generation [3].
The dynamo process is characterized by algebraic
and dynamic nonlinearities. The algebraic nonlinearity can be determined as inﬂuence of the magnetic ﬁeld
on ﬂuid motions and on the kinetic helicity. This results in quenching of the electromotive force and limits
the growth of the magnetic ﬁeld. The evolution of thesmall-scale magnetic helicity in the turbulent plasmacauses a dynamical nonlinearity in the dynamo process. The turbulent helicity conditionally can be divided into two parts: hydrodynamic and magnetic.
The kinetic helicity describes helical turbulent ﬂuid
motions; the magnetic helicity determines the order oftwisted magnetic ﬁeld lines. Due to the fact that the
kinetic helicity makes the magnetic ﬁeld small-scaled,
the back inﬂuence on the turbulent ﬂuid motions canrestrict the unlimited growth of the magnetic ﬁeld.
In the mean-ﬁeld approach the magnetic helicity
is separated into large- and small-scale components.

sunspot number

250
200
150
100
50
0

1775 1800 1825 1850 1875 1900 1925 1950 1975 2000

years

Fig. 1 Observed monthly sunspot number series for 1755
- 2007 yrs. from NGDC.

author’s e-mail: irinasun@stanford.edu

153©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

I. Kitiashvili and A. Kosovichev, Role of the Turbulent Magnetic Helicity in the Non-Linear Behavior of Solar Dynamo

number parameter we use two suggestions of Bracewell[7,8]: 1) the information about the periodical reversalsof the magnetic ﬁeld is included in the sunspot number
series by assigning alternating positive and negative
signs to the sunspot cycles, and 2) the sunspots number parameter is modeled in the form of a three-halfs
law: W ∼ B 3/2 , where W is the sunspot number, andB is the strength of the Sun’s toroidal magnetic ﬁeld.

2. Formulation of the Dynamo Models
It is well-known that the induction equation in themean-ﬁeld approximation [9] for the case for isotropic
turbulence is written∂ �B�= ∇×(�v� × �B� + α �B� − η∇ × �B�) , (1)∂twhere �B� represents the averaged over longitude magnetic ﬁeld, �v� represents mean global-scale motions in
the Sun (such as the diﬀerential rotation), parameterα is helicity, η describes the sum of the turbulent andmolecular magnetic diﬀusivity η = ηt + ηm (usuallyηm � ηt ).
For describing the average magnetic ﬁeld, following [1], we choose a local coordinate system, xyz,
where axis z will represents the radial coordinate, axis
y is the azimuthal coordinate and axis x coincides withcolatitude (Fig. 2). Hence, the vector of the meanﬁeld, �B�, can be represented as
�B� = B(x, y, z)ey + ∇ × [A(x, y, z)ey ] ,(2)where B(x, y) is the toroidal component of magnetic
ﬁeld, A(x, y) is the vector-potential of the poloidal
ﬁeld. Assuming that �v� = υy (x)ey (rotational component) we can write the dynamical system describing
Parker’s model of the α-dynamo [1] in the standard
form:
∂A
∂t
∂B
∂t

= αB + η∇2 A=

G∂A+ η∇2 B,∂x

Assuming that the coeﬃcients are constants andseeking a solution of the model in the form (A, B) ∼(A0 , B0 ) exp[i(kx − ωt)], we ﬁnd the well-known result [1] that a pure periodic solution exists if D =αG/η 2 k 3 = 2, where D is the so-called ”dynamo number”. The solutions grow in time for |D| > 2, anddecay for |D| < 2.
According to the periodic solution for Parker’s
model, proﬁles of variation for toroidal and poloidal
components of magnetic ﬁeld are sinusoidal. Therefore, model solar cycle has symmetric proﬁle and is different from the mean proﬁle of the solar cycle. Then,for deformation of a proﬁle and obtaining chaotic solutions we increase the dynamo number, and following a standard procedure, we include a nonlinearity(α-quenching) as α/(1 + ξB 2 ) [10, 11], where ξ is aquenching parameter, which limits the growth of themagnetic ﬁeld amplitude. However, our numericalcalculation showed that in the nonlinear regime thevariations of the toroidal ﬁeld and the sunspot number are also periodic and similar to the classical caseof the linear harmonic solution for |D| = 2. Thus,in the one-mode approximation the classical Parker’s
dynamo model even in nonlinear cases gives only periodic oscillatory solutions, and, therefore, cannot explain the observed variations of the sunspot numberin the solar cycles.
For creating chaotic variations of the magnetic
ﬁeld it is necessary to add to the Parker’s model athird equation describing variations of the magnetic
helicity and its interaction with the large-scale magnetic ﬁeld [2, 12]:
�
�
∂αm
Q
αm
α
2
=
,(5)
�B� (∇ × �B�) − �B� −
∂t
2πρ
η
T
where Q ∼ 0.1, T is characteristic time of the magnetic diﬀusion. Equation (5) is written for the caseof an uniform turbulent diﬀusion, and when the magnetic Reynolds number is large, η ≈ ηt .
For analysis of the Kleeorin-Ruzmaikin model wetransform equations (3)-(5) into a nonlinear dynamical system in nondimensional variables. Following
the approach of Weiss [6], we average the system of
equations (3)-(5) in a vertical layer to eliminate zdependence of A and B and consider a single Fouriermode propagating in the x-direction assuming A =A(t)eikx , B = B(t)eikx .
Note that the formulation and the interpretationof solutions of the simpliﬁed system are not straightforward because it does not adequately describes nonlinear coupling of the spatial harmonics. So, dominantmodes of the toroidal ﬁeld in the case of the solar dynamo are described by the harmonics [5], which are
antisymmetric with respect to the equator (in accordance with the Hale law), sin(kx), where x is colatitude, and the wavenumber, k, is even: k = 2, 4, ...

(3)
(4)

where G = ∂ �υy � /∂z is the rotational shear.

Fig. 2 Local Cartesian reference coordinate system.

154

I. Kitiashvili and A. Kosovichev, Role of the Turbulent Magnetic Helicity in the Non-Linear Behavior of Solar Dynamo

The ﬁrst k = 2 harmonic has the largest growth rate.
We retain only this mode in our dynamical model.
Then equations (3)-(5) in nondimensional variables
became [13]
dA
dt
dB
dt
dαm
dt

pear in the variations of the vector potential, A, ofthe poloidal ﬁeld. The evolution of the magnetic helicity represents a relatively smooth growth followed
by a sharp decay. The helicity has maxima when
the toroidal ﬁeld is zero. In these calculations thevalue of parameter ν, which describes damping rateof magnetic helicity and depends on the turbulence
spectrum and the dissipation though helicity ﬂuxes,
is of the order of unity. Finally, the variations of thesunspot number, W , with the amplitude increase arecharacterized by higher peaks and shorter rising times(Fig. 3d). Note that in the sunspot number proﬁlewe can recognize the well-known general properties ofthe sunspot number proﬁle with the rapid growth at
the beginning of a cycle and a slow decrease after the
maximum.
With the increase of |αh D0 | (|αh D0 | > 2) the
proﬁle of magnetic ﬁeld variations continue to deformand can become unstable with very steep variationsof the magnetic ﬁeld. The solution can be stableagain if we enhance the back reaction by increasing thequenching parameter. We use the following quenching formula for the kinetic part of helicity, αh , [12]α = αh /(1 + ξB 2 ) + αm . Thus we always have a
possibility for selecting ξ to obtain periodic nonlinear
solutions.
The transition of the periodic to chaotic solutions
occurs when the dynamo number, |αh D0 |, increases.
In the transition regime the cycle amplitude becomes
modulated: it slowly increases with time, and thensuddenly and very sharply declines, and then start
growing again [13].

= DB − A,= iA − B,
=

(6)

��
�
�
−ναm + AB − D B 2 − λA2 ,where D = D0 α and α = αh + αm are the nondimensional dynamo number and total helicity that is
the sum of kinetic and magnetic helicities, D0 =2
α0 Gr3 /η 2 , α0 = 2QkυA/G, r is a layer radius, υA is
the Alfvén speed, ν = T0 /T , T0 is characteristic timeof the turbulent diﬀusion [2] and λ = (k 2 η/G)2 =Rm−2 , k is a characteristic wavelength, Rm is themagnetic Reynolds number.

3. Periodic and chaotic solutions
In order to estimate the range of parameters of theKleeorin-Ruzmaikin model (3)-(5), and for modelingthe solar cycle we used the standard model of the interior structure of the Sun for the top, bottom and middle areas of the convective zone. The key parameter ofthe model is the dynamo number D = D0 αh , because
its magnitude determines behavior of the magnetic
ﬁeld, which depends on the rotational velocity andmagnetic ﬁeld strength. Parameter λ determines theinﬂuence of vector-potential A on variations of magnetic helicity αm . From our estimates it follows thatfor the solar conditions λ ≤ 10−4 . Consequently, we
can neglect the term with λ. The nondimensional parameter, ν, describes the ratio of two characteristic
times of turbulent (T0 ) and magnetic (T ) diﬀusivities. In the absence of helicity ﬂuxes the value of thedamping parameter ν is small. However, in reality thehelicity ﬂuxes increase the dissipation rate. This canbe modeled as a damping term, which increases theeﬀective value of ν [14]. Because of this the value ofν is to some extent uncertain.
Thus, for the Kleeorin-Ruzmaikin model, given
by equations (3)-(5), the linear instability conditionis also |D| ≡ |αh D0 | > 2. However, in this case the
proﬁle of the periodic solutions is not sinusoidal, anddepends on the initial conditions, A0 and B0 . For
higher initial values the amplitude of the nonlinear
oscillations in the stationary state is higher. However,the shapes of the oscillation proﬁles are similar.
Figure 3 illustrates solutions for the model of
Kleeorin-Ruzmaikin, and the corresponding variationsof the sunspot number for diﬀerent initial conditions.
As mentioned, changes of initial values for magnetic
ﬁeld components A0 and B0 leads to very similar proﬁles. In high amplitude cases, dual peaks may ap-

B
1.5
1.0
0.5
-0.5
-1.0
-1.5

Α

(a)
1

t

t
185

190

185

195

190

195

-1
-2

αm

(c)

W

(d)

2.0

1
t
-1
-2
-3
-4

(b)

2

185

190

195

1.5
1.0
0.5
t
188 190 192 194 196 198

Fig. 3 Variations of the magnetic ﬁeld for the middle
convective zone αh D0 = −2: ν = 1.28, αh =2.439, D0 = −0.82 for diﬀerent initial conditions:
B0 = 4i, A0 = −0.01i (dotted curve), B0 = 4i,
A0 = −i (dashed curve) and B0 = 1 + 4i, A0 = −i(black curve): a) toroidal component, B; b) vectorpotential, A, of the poloidal magnetic ﬁeld; c) magnetic helicity variations; d) evolution of the model
sunspot number.

155

I. Kitiashvili and A. Kosovichev, Role of the Turbulent Magnetic Helicity in the Non-Linear Behavior of Solar Dynamo

B

(a)

2

(b)

270

280

290

300

-4
(c)

m

t
260

270

280

290

300

4

t
-10
-20

(a)

5

260

270

280

290

300

Amplitude

260
-2

-50

A
30
20
t 10

(d)

W
10

3
2

8
6

-100

4

-150

2

1
0.4

t
260

270

280

290

0.6

0.8

1.0

1.2

16
14

300

Amplitude

Fig. 4 Example of the chaotic solution for: a) toroidal ﬁeldB, b) vector-potential A, c) magnetic helicity αm ,d) model sunspot number W and e) the phase portrait of the magnetic components.

1.4

1.6
(b)

12
10
8
6
4

In the case of signiﬁcant deviations from the condition of linear stability the solutions become chaotic
for all variables of the dynamical system. Figure 4
shows an example of chaotic variations for the middle convective zone parameters: ν = 1.28, λ =1.23 × 10−6 , D0 = −0.82, αh = 3.2, ξ = 3.9 × 10−3
for the magnetic ﬁeld components, the magnetic helicity and the sunspot number parameter [13]. In the
chaotic solutions, the peaks of the toroidal magnetic
ﬁeld, B (Fig. 4a) strongly correlate with the peaksof the vector-potential, A, and the magnetic helicity,
αm , (Fig. 4b, c). The growth of the toroidal ﬁeld also
leads to strengthening of the poloidal ﬁeld and strong
ﬂuctuations of the magnetic helicity.
Now we can see from Figs. 3d and 4d that the
proﬁles of the model sunspot number variations qualitatively describe the mean proﬁle of the solar cycles. The next important characteristic of the solar
cycles is the relationship between the amplitude andthe growth time. Fig. 5 shows this relationship forsome periodical solutions (panel a), four chaotic solutions (panel b) and properties for the real 23 solar
cycles (panel c). In the case of the relationship for theperiodic solutions (panel a) the circles show a sequence
for a ﬁxed value of the kinetic helicity, αh = 2.44
and the dynamo number varying from -7 to -0.82, thecrosses show the case of ﬁxed D0 = −0.82 and varying αh , from 2.44 to 3. The size of the crosses andcircles is proportional to the corresponding values of|D0 | and αh ). The four chaotic solutions shown inpanel b) are obtained for D0 = −0.82 and diﬀerent
values of the kinetic helicity: αh = 2.8 (black circles),αh = 3 (empty circles)and αh = 3.2 (stars). The
time scales are non dimensional. The bottom panel(Fig. 5c) shows the observed amplitude-growth time
properties of the solar cycles of 1755 - 2007. Thus,
all three panels demonstrate that the growth time is

2
0

0.4

0.6

0.8

220
200

1.0

1.2

1.4
(c)

Amplitude

180
160
140
120
100
80
60
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Growth time

Fig. 5 Relationships between the amplitude of the model
sunspot number and the growth time for a) periodic
solutions; b) chaotic solutions (for D0 = −0.82 anddiﬀerent values of the kinetic helicity); and c) for
real solar cycles.

shorter for stronger cycles.

4.

Assimilation of Sunspot Data into
the Dynamo Model

Despite the fact that the Kleeorin-Ruzmaikin dynamo model reproduces the basic properties of a solar cycle, it has signiﬁcant deviations from the actual
sunspot data. This problem is related to inaccuracies of the model, which contains various assumptions
and approximations, unknown initial conditions, anderrors of the observational data. Data assimilation
methods allow us to use the observed information andobtain the best estimate of the model physical state.
In the case of non-linear systems the Ensemble

156

sunspot number

I. Kitiashvili and A. Kosovichev, Role of the Turbulent Magnetic Helicity in the Non-Linear Behavior of Solar Dynamo

220
200
180
160
140
120
100
80
60
40
20
0

19
3

18

21 22

4
8
1

2

11
17

9
10
13

5

1760

1780

1800

6

1820

1840

1860

16

12

14

1880

1900

7

23
20

15

1920

1940

1960

1980

2000

years
Fig. 6 Results of assimilation for the known observational data of the annual sunspot number (circles). The grey curve
shown the reference solution (without assimilation analysis), and the black curve shows the EnKF estimate of thesunspot number variations, obtained from the data and Kleeorin-Ruzmaikin dynamo model. Numbers from 1 to 23
are the conventional numbering of the sunspot cycles.

For comparison, we consider the classical Parker’s dynamo model [1] and the Kleeorin-Ruzmaikin model [2].
We ﬁnd that the Parker’s model (which does not take
into account the magnetic helicity) does not allow reproducing the typical behavior of the sunspot number with a fast growth and slow decay or obtaining achaotic solution, even in strong nonlinear regimes.
The analysis of the Kleeorin-Ruzmaikin model,
which describes the evolution of the magnetic helicity based on the balance between the large-scale andturbulent magnetic helicities, shows the existence of
nonlinear periodic and chaotic solutions for conditionsof the solar convective zone. For this model we obtained the proﬁles of the sunspot number variations,which qualitatively reproduce the typical proﬁle of thesolar cycles. Also, the Kleeorin-Ruzmaikin model has
been used for assimilating the sunspot data by applying the Ensemble Kalman Filter method. The initial
results show that this assimilation method works forthis model reasonably well, and can be used for predicting solar sunspot cycles.
This work was supported by the Center for Turbulence Research (Stanford) and the International Space
Science Institute (Bern).

Kalman Filter (EnKF) is shown to be an eﬀective
method for data assimilation [15]. This method isbased on the standard Kalman Filter and includes astatistical analysis of ensembles of possible solutions.
The main diﬀerence of the EnKF from the standard
Kalman Filter is in using for the analysis an ensembleof possible states of a system, which can be generated
by the Monte Carlo simulations.
Figure 6 shows the result of assimilation of theannual sunspot number data (circles) into KleeorinRuzmaikin model by the EnKF method. For comparison the ﬁgure shows an exact solution of the model(grey curve) without assimilation and the best estimate of the sunspot number variations (black curve)
obtained by the EnKF method from the dynamo
model calculations. It reproduces well the observational data (Fig. 1). The errors are estimated bymodeling the model uncertainties by a random forcing
function and assuming a 14% error in the data. This
result provides a basis for developing physics-based
methods of predicting the solar activity cycles.

5. Conclusion
In the mean-ﬁeld theory the dynamo process is
described in terms of plasma helicity, which has twocomponents: kinetic, caused by helical motions, andmagnetic. The dynamo theories often include only
the kinetic part. However, it has been suggested thatthe magnetic helicity may also play a signiﬁcant role.
Using a low-order dynamical system approach we examine the inﬂuence of the kinetic and magnetic helicities on the non-linear ﬂuctuations of the dynamogenerated magnetic ﬁeld in the conditions of the solar
plasma, and compare these with the sunspot number variations observed during the solar 11-year cycles.

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