SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR
DYNAMO
PABLO D. MININNI, DANIEL O. GOMEZ∗ and GABRIEL B. MINDLIN
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,
Ciudad Universitaria, 1428 Buenos Aires, Argentina
(Received 10 September 2000; accepted 13 February 2001)
Abstract. The aim of this paper is to investigate the dynamical nature of the complexity observed in
the time evolution of the sunspot number. We report a detailed analysis of the sunspot number time
series, and use the daily records to build the phase space of the underlying dynamical system. The
observed features of the phase space prompted us to describe the global behavior of the solar cycle
in terms of a noise-driven relaxation oscillator. We find the equations whose solutions best fit the
observed series, which adequately describe the shape of the peaks and the oscillations of the system.
The system of equations obtained from this fitting procedure is shown to be equivalent to a truncation
of the dynamo equations. A linear transformation maps the phase space of these equations into the
phase space reconstructed from the observations. The irregularities of the solar cycle were modeled
through the introduction of a stochastic parameter in the equations to simulate the randomness arising
in the process of eruption of magnetic flow to the solar surface. The mean values and deviations
obtained for the periods, rise times and peak values, are in good agreement with the values obtained
from the sunspot time series.
1. Introduction
The solar cycle was discovered by Schwabe in 1843 from observations of the
sunspot number as a function of time. Schwabe (1843) determined for the solar
cycle a period of approximately 10 years, and also described an irregular behavior,
with fluctuations in the period and in the extent and intensity of the maxima.
In 1848 the Swiss astronomer Wolf introduced the criteria used to measure the
sunspot number, and established an international network of observatories which
register spatial and temporal information of sunspots on a daily basis. Since then,
the amount of information on the solar cycle grew steadily, together with the advent
of new and more reliable observational techniques. But even though these new
methods can provide better indexes of solar activity, the sunspot number so far is
the only one which allows us to study the long time dynamics of the solar cycle.
Sunspots correspond to large concentrations of magnetic field on the solar surface. The generation of this field is explained by the dynamo effect, which is
produced by the coupling of the magnetic field with the subphotospheric velocity
∗ Also at: Instituto de Astronomı́a y Fı́sica del Espacio, C.C. 67, Suc. 28, 1428 Buenos Aires,
Argentina
Solar Physics 201: 203–223, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
field. The Sun displays a differential rotation about its axis, with the equator rotating about 20% faster than the poles. According to the classical theory (Babcock,
1961; Leighton, 1969), this gradient stretches and writhes the magnetic lines in the
deep layers of the convective region, and as a consequence of kink instabilities the
field lines reach the surface to form the spots. The subsequent dynamics of sunspots
is governed by drift motions and diffusion, and by the gradual flux cancellation
with the pre-existing magnetic field. This alternation between buoyancy and flux
cancellation drives the global dynamics of the solar cycle (see, e.g., Priest, 1984;
Zeldovich and Ruzmaikin, 1990).
Recent observations from the Michelson Doppler Imager aboard the SOHO
spacecraft suggest that it is the radial velocity shear rather than the latitudinal
which is responsible for creating the strong toroidal magnetic field at the base of the
convection zone. There are also different explanations concerning the emergence
of field lines to the solar surface, which involve mechanisms such as currents generated by turbulent motions (Krause and Rädler, 1980), or buoyant instabilities of
magnetic flux tubes within the base of the convective region (see, e.g., Ferriz-Mas,
Schmitt, and Schüssler, 1994; Caligari, Moreno-Insertis, and Schüssler, 1995).
To explain the nature of the irregularities displayed by the solar cycle, two
different mechanisms have been invoked in the literature: chaos or stochasticity.
Although the presence of a chaotic attractor of reduced dimensionality is quite
appealing, there are still no firm indications of its existence. The current inability
to show clear evidence of a low dimensional chaotic attractor is essentially related
to the insufficient length of the sunspot database (Paluš and Novotná, 1999). On
the other hand, the spatial and temporal complexity of the convective cells in the
Sun can naturally be described considering the presence of stochastic coefficients
in the dynamo equations (for instance, Choudhuri, 1992). Within this framework,
the spatial and temporal irregularities observed in the time series can be described
as the end result of the stochastic process underlying the magnetic eruption to
generate the spots (Hoyng, 1993).
The aim of the present paper is to generate a simple model of the solar cycle. To
this end, we perform a detailed analysis of the daily and monthly sunspot number
time series. We used the daily records to build the phase space of the underlying
dynamical system. While early papers made embeddings of the data, many of them
used the sunspot number to reconstruct the phase space. It was recently shown
that the phase space cannot be reconstructed from intensity time series (see, e.g.,
Gilmore et al., 1997). Since the Wolf number is somehow related to the magnetic
field, we followed the method outlined in that paper to reconstruct a time series
proportional to the magnetic field.
Among the important results of the present study, we find that the features of the
phase space allow us to describe the global behavior of the solar cycle in terms of
a rather simple relaxation oscillator in two dimensions. The absence of systematic
self-crossings in this phase space suggests that the complexity of the sunspot time
series does not arise as a consequence of chaos. On the other hand, the irregularities
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
205
Figure 1. Daily sunspot number from 1816 to the present (Zürich observatory).
of the solar cycle can be adequately modeled through the introduction of a stochastic parameter in the equations, to simulate the randomness associated to the process
of eruption of magnetic flow, caused by the rising and sinking of convective cells.
In Section 2 we describe the sunspot time series used for the present analysis,
and in Section 3 we enumerate the steps followed to build the associated phase
space. In view of the global features displayed by this phase space, we seek a
dynamical system which shares these features in Section 4, and perform a fitting
procedure to determine the parameters of the dynamical system in Section 5. As
mentioned earlier, the purpose of this work is to perform a mathematical analysis
of the data in order to come out with a simple model which imitates the sunspots
number time series. However, it is possible to derive a connection between our
equations and a spatial truncation of the dynamo equations, which are believed
to be the theoretical bases of the solar cycle. In Section 6 we show that our system of equations is equivalent to a drastic truncation of the dynamo equations.
In Section 7 we discuss how to model the irregularities observed in the sunspot
time series through the introduction of stochasticity in one of the parameters of our
relaxation oscillator. Finally, in Section 8 we summarize the main results obtained
in the present paper.
2. Daily and Monthly Time Series
Currently, there are more than thirty stations around the world carrying out the
daily count of sunspot number. The Greenwich and Zürich observatories gather all
this information, and keep track of the final index. These observatories publish the
daily sunspot number on a monthly basis (Withbroe, 1989).
Figure 1 shows the daily sunspot number from 1816 to the present, as recorded
by the observatory of Zürich. Figure 2 displays the monthly average of sunspot
numbers from 1750, performed by the same observatory.
The most reliable data are the daily measurements since 1849. Before that year,
the time series shows spurious fluctuations caused by weather and other factors,
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
Figure 2. Monthly average of sunspot numbers (Zürich observatory).
and also display seasonal variations. In addition, the changes introduced by Wolf
in the measurement index also added spurious structures to this time series (Foukal,
1990).
In the monthly time series shown in Figure 2, some of the minima do not reach
down to zero. This is a spurious effect introduced by the averaging procedure,
since the daily data show that the Sun goes through several days without spots in
all the minima covered by this dataset. Toward each minimum of activity, there
are typically more than fifty days without counts, and there can be more than ten
consecutive days with the surface of the Sun totally clean. This fact prompted us to
perform our analysis with the daily sunspot number.
In dynamo models, the physical magnitude of interest is the magnetic field. Our
aim is to build a dynamical system with attracting solutions able to reproduce the
observed data. Note that the sunspot number time series is somehow related to the
magnetic field.
According to dynamo theories, the number of sunspots is proportional to the
magnetic energy erupting to the photosphere. Many authors therefore choose this
number as a quadratic quantity in the magnetic field components (for instance,
Tobias, Weiss, and Kirk (1995) chose the toroidal component, while Platt, Spiegel,
and Tresser (1993) chose the poloidal one).
On the other hand, the solar magnetic field has been recorded for thirty years. It
is therefore worthwhile to establish a quantitative relationship between the sunspot
number and magnetic fields. In Figure 3 we show the sunspot number, and the
square of the mean magnetic field according to the Stanford observatory for the
last two cycles, both normalized by their respective maximum values. At short
time scales, the time series show differences due to the particular definitions of
these indexes. But for time scales of the order of the solar cycle, both curves are
remarkably similar and justify the choice of the sunspot number as a quantitative
indicator of the magnetic activity of the Sun.
Also, to show the dynamics of the magnetic field in an explicit fashion, it is customary to define the Bracewell number (Bracewell, 1953), as the sunspot number
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
207
Figure 3. Sunspot number (R) and the square of the mean magnetic field according to the Stanford
observatory (SMMF).
with a sign change at the beginning of each period. Therefore, this number displays
a period of 22 years with a sign change every 11 years.
In summary, to obtain a time series proportional to the spatially averaged magnetic field, we consider the sunspot number as quadratic in the magnetic field and
therefore take the square root of the sunspot number series. Furthermore, following
Bracewell (1953), we change sign at each minimum. Note that the time series thus
obtained is not the Bracewell number, but is such that its square corresponds to
the sunspot time series. The location of solar activity minima was determined
by filtering the time series with a low-pass filter to eliminate the high-frequency
components, and comparing the values thus obtained with those tabulated by the
Zürich observatory. Figure 4 shows the time series obtained after applying this
method.
The sunspot number has a considerable level of high-frequency fluctuations.
We need to smooth out the series and calculate its derivatives in order to build
the underlying phase space. Since the high-frequency components of the magnetic
activity of the Sun are not well displayed by the sunspot number, we can get a
smoothed time series using a frequency filter. In Figure 4 we show a filtered time
series in which we eliminated all the frequencies whose amplitudes were smaller
than 2% of the amplitude of the fundamental mode, and the spurious daily oscillation was removed using a low-pass filter. Hereafter, this time series will be referred
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
Figure 4. Time series proportional to the magnetic field obtained from the daily sunspot number
(BR ), and smoothed version of the original time series, B(t).
Figure 5. Square of time series (B 2 ) and the daily sunspot number (R).
to as B(t). As mentioned above, the square of this time series, B(t)2 , is a smoothed
version of the original sunspot series (see Figure 5).
3. The Phase Space
The daily time series of B(t) has more than 60 000 data. It is a typical task in natural
sciences to face problems starting from the knowledge of a scalar time series data.
If the law behind it is deterministic, there will be a finite number of points such that
its knowledge allows us to predict the immediate future. The procedure of creating
a multivariate environment from the scalar data is known as embedding, and as it
was stated before, the first requirement of this procedure is to guarantee that each
point in the constructed phase space has a unique future. The second requirement
for generating a phase space is the preservation of differential information: points
close in the reconstructed phase space will be mapped closely. With these two
requirements in mind, we know that if systematic crossings exist in a conjectured
phase space (i.e., groups of points in a box are mapped into different groups of
points as time evolves), the intersections in phase space can be eliminated increasing the dimension of the embedding. On the contrary, if the self-intersections of
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
209
the reconstructed flow are not systematic, no finite increase of the dimensionality
will allow us to generate a deterministic flow associated with the data (Mindlin and
Solari, 1995; Mancho, Duarte, and Mindlin, 1996).
To obtain a simple dynamical system of the solar cycle we reconstructed the
phase space using an embedding (Mindlin, Merener, and Boyd, 1998; Gilmore,
1998, for a review). The temporal derivative of B(t) was calculated using a finite
difference formula
Ḃ =
B(ti+h ) − B(ti−h )
+ O(h2 ) .
ti+h − ti−h
(1)
Even though the dimension of the embedding is not known, and would justify an
analysis of the degrees of freedom of the data, the number of periods is insufficient
for such analysis. On the other hand the simplicity of the series suggests that a
phase space of two dimensions is enough to describe the system. Figure 6 shows
the phase space using a step h of 200 days.
At first glance, the presence of self-crossings of the trajectory in phase space
seems to justify an embedding in a phase space with more dimensions. However,
the crossings observed in Figure 6 are not systematic and can be regarded as caused
by the addition of noise to a simple limit cycle rather than projection effects from
a larger phase space. In order to filter out noise and identify the attractor of the
underlying dynamical system, we performed the following analysis.
Differentiation increases the ratio between noise and the original signal. As a
result, when the step h is reduced, noise increases noticeably, but for large enough
step sizes, the system shows a quite rectangular limit cycle with slow dynamics
along the horizontal sides and a much faster evolution along the vertical sides
(Figure 6). For each region, we calculated the slope of the straight line that best
adjusts the data using a least-squares fit, for step sizes decreasing from 1600 to 200
days. In Figure 7 we show the convergence of the slope for small steps in one of
the phase space regions (region B3 , see Figure 6).
The slopes converge rather gently in the four regions. Therefore, the approximation of the cyclic attractor displayed in Figure 6 by a parallelogram seems justified.
Since for step sizes of 200 days the slopes converge to well-defined values (see
Figure 7), we used this time delay to perform our analysis.
The rectangular shape of the cycle, together with the slow dynamics in its lateral sides and the much faster evolution in its horizontal sides, suggests that this
dynamical system can be imitated by a relaxation oscillator.
4. A Paradigmatic Relaxation Oscillator
We look for a dynamical system with an attracting solution sharing the features
found in the time series data. The two key elements that we want to reproduce
are: (a) the difference of time scales in different portions of the trajectory, and
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Figure 6. Phase space using a step of 200 days. The straight lines, labeled as B1 , B2 , B3 , B4 ,
correspond to the best linear fit in each sector of the limit cycle.
Figure 7. Convergence of the slope in one of the regions of phase space (region B3 ).
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
211
(b) the rectangular shape of the average cycle. These two features together, can
be modeled by a dynamical system displaying relaxation oscillations. While other
nonlinear relaxation oscillators have been proposed to explain the nature of the
solar cycle (see, e.g., Weiss, Cattaneo, and Jones, 1984; Spiegel and Wolf, 1987;
Schmalz and Stix, 1991, and references therein), these models usually have three
or more degrees of freedom and irregularities arise as the result of deterministic
chaos. The features observed in our reconstructed phase space, however, suggest
that a non-chaotic system driven by stochasticity is also a valid scenario to describe
the complexity displayed by the dataset.
Let us analyze the paradigmatic (and probably simplest) relaxation oscillator
described in the literature: the Van der Pol oscillator. This dynamical system describes an autonomous oscillator where the friction coefficient depends on the
amplitude of the oscillations,
ẋ = −y − µx ξ x 2 − 1 ,
(2)
ẏ = 2 x .
For positive values of µ, the origin is a repellor. The trajectories in phase space
depart spiraling off the origin until they reach a closed orbit, the so-called limit
cycle. In a similar fashion, a trajectory starting outside the limit cycle, spirals with
decreasing radius toward the attracting cycle.
The shape of the limit cycle depends strongly on the parameters. For certain
values of these parameters, the temporal evolution of the system shows two time
scales, like the ones observed in the solar cycle time series. By performing a best
fit of the parameters, the shape of the limit cycle obtained from the observed series
(see Figure 6) can be correctly described by a Van der Pol oscillator.
Since this set of equations is written in terms of two variables and only one of
them is proportional to the observed time series, we used the so-called standard
form to reconstruct the vector field.
5. The Standard Form
For a given time series v(t), the underlying dynamical system can be formally
written in the form
u̇ = v ,
v̇ = w ,
..
.
(3)
ż = F (u, v, w, . . . , z) .
This is the so-called standard form of the dynamical system, where F is the
standard function (Gouesbet, 1991; Mindlin, Merener, and Boyd, 1998). A nonlinear change of coordinates allows us to obtain the standard form of any dynamical
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
system. We can therefore build the original phase space by appropriately setting the
free parameters in the standard function F . Changing coordinates in Equations (2),
u = −y − µ(ξ x 3 − x) ,
(4)
we obtain
ẋ = u ,
u̇ = − 2 x − µ u (3ξ x 2 − 1) ,
(5)
where the standard function is F (x, u) = − 2 x − µu(3ξ x 2 − 1). Assuming that
x corresponds to the observed time series B(t), we can compute u and u̇ following
a finite difference scheme,
u=
x(ti+h ) − x(ti−h )
,
ti+h − ti−h
u̇ = 4
(6)
x(ti+h ) − 2x(ti ) + x(ti−h )
.
(ti+h − ti−h )2
(7)
To set the free parameters that best fit the observational data, we randomly took
30 000 data from the time series B(t) and minimized the mean square error between
F and u̇ using a simplex method. As a result, we obtained an adjustment with
χ 2 = 0.4 per point and
= 0.2993 ,
µ = 0.2044 ,
ξ = 0.0102 .
(8)
To give a quantitative measure of the goodness-of-fit we used the probability Q
(Press et al., 1986), from which we obtain that a probability of 99% corresponds
to χ 2 = 0.98 per point. Therefore, the quality of our fit (χ 2 = 0.4) is remarkably
good.
In Figures 8 and 9 we overlay the observed time series B(t) and the trajectory
of the Van der Pol oscillator with the set of parameters listed in Equations (8).
We find that the Van der Pol oscillator is indeed a reasonable dynamical system
to describe the main features of the evolution of the variable B(t). For instance, this
set of equations models quite adequately the shape of the peaks and the relaxation
oscillations of the system, as well as the fast increase and slow decrease in the
sunspot number.
Taking the square of the time series we obtain the sunspot number. The equations fit the shape of the peaks, reaching each maximum in approximately 4 years
and decaying in 6 years, which agrees fairly well with the average times observed
in the observed sunspot number series.
In Figure 10 we show the frequency spectrum for the time series B(t) and for
the time series integrated from the Van der Pol equations. The theoretical spectrum
correctly fits the shape of the peak of 22 years, and both spectra remain similar
down to frequencies corresponding to periods of about one year.
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
213
Figure 8. Trajectories of the observed time series B(t) and of the Van der Pol oscillator x(t) in phase
space.
Figure 9. Observed times series and the Van der Pol oscillator.
Figure 10. Frequency spectrum for the time series B and for the time series integrated from the Van
der Pol equations.
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
Figure 11. Statistics of the difference between the two series (B − x), and the Gaussian curve that
best fits the histogram (G).
The spectrum of the observational series also shows a broad peak centered at
a period of 30 days. This peak and the increase in the power of the spectrum,
are caused by a systematic error in the measurements due to the rotation of the
Sun, and are not related to the dynamics of the magnetic fields (see, for instance,
Salakhutdinova, 1998, for a detailed discussion of this effect).
The difference between the observed time series B(t) and the one generated by
the Van der Pol equations displays a Gaussian distribution. In Figure 11 we show
the statistics of the difference between the two series and the Gaussian curve that
best fits this histogram. This result suggests that the observed time series could
be approximated by a deterministic process of low dimensionality plus a Gaussian
stochastic process.
6. The Dynamo Equations
Babcock (1961) put forward a synthesis of twenty years of measurements and built
a simple model of the Sun’s magnetic field structure which describes the 11-year
activity cycle and provides the conceptual framework for sunspot cycle analysis.
According to Babcock’s model (Babcock, 1961; Foukal, 1990) the differential rotation stretches the magnetic lines in the deep layers of the Sun, thus transforming
kinetic energy into magnetic energy. Once the intensity of the essentially toroidal
magnetic field surpasses a critical level, the field erupts to the surface to form active
regions, due to the combined action of magnetic buoyancy and kink instabilities. In
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
215
the last stage of the cycle, the sunspots, which are the footpoints of bipolar active
regions generated by this eruption process, migrate in latitude until they cancel
with the pre-existing global magnetic field.
In dynamo theories, the process of magnetic eruption to generate the spots is
the so called α-effect, and it was first postulated by Parker (1955). The action
of the Coriolis force on the buoyant convective cells introduces a net helicity on
the flow, which is responsible for the tilt of the rising field lines. This process
therefore converts toroidal magnetic field into poloidal magnetic field. Several
scenarios have been proposed (Leighton, 1969; Krause and Rädler, 1980; Caligari, Moreno-Insertis, and Schüssler, 1995) to explain the nature of the α-effect,
involving turbulent motions at the base of the convective zone and buoyancy forces,
demonstrating that solar-like magnetic field solutions can be obtained by judicious
choices of its mathematical expression in the dynamo equations.
The complete resolution of the solar dynamo involves solving the induction
equation, coupled with the Navier–Stokes equation for the velocity field. The equations obtained from the phase space reconstruction must be compatible with a
spatial truncation of these MHD equations. There is no consensus in the literature
with regard to the expression for the saturation term. Many nonlinear mechanisms
have been proposed, mainly in the form of α-quenching and magnetic buoyancy.
As an example we choose a particular form of quenching and buoyancy and show
that there exists a linear transformation which maps the variables of the Van der
Pol equations into the components of the magnetic fields. Note that other linear
transformations, or even nonlinear transformations which preserve the topology of
the phase space, can be constructed to map the Van der Pol equations into different types of nonlinearities in the dynamo equations. Therefore, the phase space
reconstructed from observations cannot be used to identify the correct saturation
effect.
Although the connection between our semi-empirical nonlinear oscillator and a
spatial truncation of the dynamo equations is in itself an important result, we want
to emphasize that the main goal of the present analysis was to build a standard
vector field equation which fits the observed time series. We believe that this mathematical model is therefore important, since theoretical models can be tested against
the standard form. Within this framework, the change of variables suggested below
provides the connection with one of the possible theoretical models.
In spherical coordinates, we write the induction equation (see, e.g., Priest, 1984)
as
Bφ
∂Bφ
+ rv p · ∇
= r(∇ × Ap ) · ∇ω + η(∇ 2 − r −2 )Bφ − γ Bφ 3 ,
∂t
r
(9)
1
∂Ap
2
−2
+ vp · ∇(rAp ) = αBφ + η(∇ − r )Ap ,
∂t
r
where Bφ is the toroidal component and Bp = ∇ × (Ap φ̂) the poloidal component of the magnetic field. The coefficient η is the magnetic diffusivity and
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
r = R
sin θ, where R
is the solar radius. The poloidal component of the mean
velocity field is v p , and the angular velocity ω describes the differential rotation
of the sun. The process responsible for the conversion of toroidal into poloidal
magnetic field is described by the coupling constant α in the induction equation
(see, e.g., Priest, 1984). The constant γ describes the removal of flux by magnetic
buoyancy ((Leighton, 1969; Yoshimura, 1975; Jones, 1982).
As the poloidal magnetic field increases, the corresponding Lorentz force will
tend to inhibit convection. Since in mean field theories the value of α depends on
the helicity of the velocity field (Krause and Rädler, 1980), increasing the magnetic field causes α to decrease. We can replace the Navier–Stokes equation by a
phenomenological equation to describe the evolution of α. As an example, we can
introduce the back-reaction of the magnetic field by splitting α in Equations (9)
into (see, for instance, Zeldovich and Ruzmaikin, 1990)
τα
Bφ 2 ,
(10)
α = α0 −
4πρL0 2
where L0 is the depth of a thin layer at the bottom of the convection zone and τα a
characteristic dissipation time. Mean field theory is often used within the context of
the solar dynamo. However, several limitations to this approach have been pointed
out (Dikpati and Charbonneau, 1999, for a review): (1) it has problems to produce
strong toroidal fields, (2) the first-order smooth approximation (FOSA) involved
in its derivation can hardly be fulfilled in the Sun, (3) although α is known to
be anisotropic, it is usually assumed to be a scalar quantity. Recent numerical
experiments have suggested that a class of mean field dynamos known as interface
dynamos can avoid some of this difficulties (Charbonneau and MacGregor, 1996).
Also, alternative approaches to avoid FOSA have been proposed (Blackman and
Field, 1999).
It is convenient to write down the equations in dimensionless units. To this
end, we define dimensionless variables with the aid of a typical velocity α0 and a
longitude L0 ,
B=
Bφ
,
α0
(11)
A=
Ap
,
L 0 α0
(12)
where
√ T0 is a characteristic time given by L0 /α0 , and Bφ and Ap were divided
by 4πρ to express them in velocity units. Following Tobias, Weiss, and Kirk
(1995) we intend to associate the dimensionless toroidal magnetic field B with the
time series B(t) derived from the sunspot number time series and introduced in
Section 2.
The magnetic Reynolds number is written as R = α0 L0 /η, and the angular
rotation of the Sun can be written as ' = ωT0 . The characteristic relaxation time
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
217
for α in dimensionless units is τ = τα /T0 and the poloidal component of the mean
velocity field is v = vp /α0 .
To obtain a set of ordinary differential equations, we simply replaced the spatial
derivatives by 1/L0 in Equations (9), and using v 1/R we finally obtain
Ȧ = vA + 1 − τ B 2 B ,
(13)
'
Ḃ = vB + A − γ0 B 3 ,
(
where ( = L0 /R
< 0.1. The nonlinear term proportional to τ is responsible
for the saturation in the conversion of toroidal into poloidal magnetic field (αquenching). The term proportional to γ0 (where γ0 = γ α0 ) describes the saturation
effect of magnetic buoyancy.
In the Appendix, we show that there exists a linear transformation which maps
this system into the Van der Pol equations obtained from the time series (see Sections 4 and 5). This transformation provides a link between the three parameters of
the dynamical system (see Equations (8)) with the physical magnitudes involved
in these simplified evolution equations (Equations (13)). More specifically, in the
Appendix we show that the following relations must be satisfied: vp ≈ 0.1 α0 ,
ωR
≈ 0.8 vp and τα /γ ≈ L0 /α0 2 . Note that these relationships are just conditions to ensure the equivalence between the equations. Using typical values for the
differential rotation, the mean magnetic field and the mass density at the convection
zone, these relations yield poloidal and toroidal velocities of a few hundred meters
per second, γ of the order of ten kilometers per second, and α0 of the order of
one kilometer per second. For instance, according to mean field theory, the value
of α0 can be roughly compared with the convective velocity of the photosphere,
of 1 km s−1 . The rising rate of a buoyant flux tube can be roughly compared with
the Alfvén velocity (Parker, 1979), which gives a velocity of the same order in a
flux tube at the solar surface. However, note that a poloidal velocity of one hundred
meters per second is five times larger than the observed meridional flow at the solar
surface (about 20 m s−1 ).
7. Irregularities of the Solar Cycle
Numerous attempts to infer the nature of the irregularities displayed by the solar
cycle have been reported in the literature. As mentioned in the Introduction, these
attempts relied on either of two rather different mechanisms: chaos or stochasticity.
The chaotic approach consists of modeling the dynamics of the system by a
set of equations of reduced dimensionality, including nonlinearities able to display
chaotic behavior for reasonable values of the relevant parameters (Zeldovich and
Ruzmaikin, 1990; Knobloch and Landsberg, 1995; Knobloch, Tobias, and Weiss,
1998, and references therein). Although the idea of describing the solar cycle
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P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
through a chaotic attractor of reduced dimensionality seems appealing, its very existence is still being debated. For instance, Weiss, Cattaneo, and Jones (1984) have
observed chaos for a highly truncated version of the dynamo equations, but such
chaotic behavior disappeared as the phase space of the system was expanded to
include more variables. Also, searches for signatures of a low-dimensional chaotic
attractor in the sunspot time series are not conclusive due to insufficient number
of data (Carbonell, Oliver, and Ballester, 1994). Standard algorithms to compute
Lyapunov exponents and correlation dimension of time series have been found
unreliable when applied to datasets which are not sufficiently long (Paluš and
Novotná, 1999).
As an alternative approach to explain the irregularities observed in the solar
cycle, the scenario of a stochastically driven solar dynamo has been studied in a
number of recent papers (for instance, Choudhuri, 1992; Moss et al., 1992; Hoyng,
1993; Ossendrijver and Hoyng, 1996). If the averaging procedure involved in mean
field theory is properly considered, the presence of stochastic coefficients in the dynamo equations cannot be neglected. In the particular case of the solar dynamo, the
rising and sinking fluid motions in the convection zone play a crucial role in driving
the system through the so-called α-effect. The spatial and temporal complexity of
these convective cells can naturally be described by adding a stochastic part to the
velocity field, which in turn will drive stochastic magnetic field components.
Within this framework, the spatial and temporal irregularities observed in the
magnetic eruption to generate the spots, are adequately described as the end result
of this stochastic process. When the toroidal magnetic field increases and reaches a
critical value, there is a high probability for the spot to be generated. However,
the instability also has a non-zero probability of occurrence for smaller values
of the field as well. The action of the Coriolis force on the buoyant convective
cells introduces a net helicity on the flow, which is responsible for the tilt of the
rising field lines, which in turn causes the parameter α to become a stochastic
variable itself (for instance, Choudhuri, 1992). Leighton (1969) also introduces
stochasticity in his model by including a random factor in the characteristic time
τα responsible of the α-effect, and Barnes, Sargent, and Tryon (1980) use random
noise to simulate sunspot cycles
Since ξ in Equations (5) and τ in the truncation of the dynamo equations are
linearly related (see Appendix), within the framework of our much simpler theoretical model the conversion of toroidal field into poloidal field is modeled through
the nonlinear term proportional to ξ . Following Leighton (1969) and Choudhuri
(1992), this induces a separation of ξ into a mean value and a stochastic component,
ξ = ξ0 + r ξs ,
(14)
where ξs is assumed to be a zero mean Gaussian stochastic process, describing
white noise with dispersion equal to unity. Therefore, the dimensionless parameter r is the r.m.s. value of the stochastic part of ξ . The stochastic Van der Pol
Equations (5) can now be written in the following form:
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
219
Figure 12. Observed times series and the stochastic Van der Pol oscillator.
Figure 13. Square of the yearly averaged stochastic Van der Pol oscillator.
ẋ = u ,
u̇ = − 2 x − µ u 3(ξ0 + r ξs )x 2 − 1 .
(15)
The value of ξs was randomly changed with a correlation time that was chosen
between 1 and 60 days (which samples typical correlation times from granulation
to giant cells), and for each value we choose the value of r which fits the observed
deviation in the mean amplitude. However, changing the correlation time did not
change our results appreciably (Mininni, Gómez, and Mindlin, 2000). As an example, in Figures 12 and 13 we show the result of integrating stochastic Van der Pol
equations for a period of 150 years, with a correlation time of 1 day and r = 0.07.
Note that as the correlation time is increased, the value of r required to generate
the same variability is reduced. For a correlation time of 45 days, r ≈ 0.01. Since
ξ0 ≈ 0.01 (see Equations (8)), we obtain that the signal-to-noise ratio for ξ is
of order unity, in good agreement with observations (Kuzanyan, Bao, and Zhang,
1999) and theoretical estimates (Choudhuri, 1992).
Table I shows the mean values and deviations for the observational data, and for
the theoretical dynamical system driven by noise. Note that the theoretical dispersions of all the parameters listed in Table I (i.e., period, rise time and maximum)
have been adjusted with only one parameter (r).
220
P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
TABLE I
Mean values and deviations for the observational data (daily
sunspot number), and for the theoretical dynamical system
driven by noise.
Van der Pol
Solar cycle
Period
(years)
Rise time
(years)
Maximum value
(sunspots)
10.7 ± 0.4
10.7 ± 0.8
4.3 ± 0.4
4.2 ± 0.4
152 ± 38
113 ± 40
8. Discussion
In this paper we present a simple model of the solar cycle, performing a detailed
analysis of the daily and monthly sunspot number time series. We used the daily
sunspot number to build the phase space of the underlying dynamical system. We
take the square root of the sunspot number series and change sign at each minimum
to generate a time series approximately proportional to the global magnetic field.
We smooth out the series using a frequency filter, and calculated its derivatives in
order to build the underlying phase space.
The lack of systematic self-crossings in the reconstructed phase space allows
us to propose a two-dimensional phase space to describe the dynamics of the solar
cycle. The features of the phase space thus obtained support a description of the
global behavior of the solar cycle in terms of a Van der Pol oscillator. We adjusted
the free parameters in the equations using the so-called standard form. We find
that the Van der Pol oscillator that best fits the observed series is able to describe
the main features of the observations quite adequately, with very few degrees of
freedom and thus few parameters. For instance, the equations correctly fit the shape
of the peaks and the spectrum of the observational series. Furthermore, this set
of equations was shown to be equivalent to a truncated version of the dynamo
equations.
The irregularities of the solar cycle were reasonably modeled through the introduction of a stochastic parameter in the equations, which reflects the randomness
of the rising and sinking convective cells. The mean values and deviations obtained
from the theoretical model for the rising times, periods and peak values, are in good
agreement with the corresponding values obtained from the observations.
SIMPLE MODEL OF A STOCHASTICALLY EXCITED SOLAR DYNAMO
221
Appendix
In this Appendix, we show the linear equivalence of Equations (13) and the Van der
Pol equations obtained from the time series. Note that other transformations can be
constructed to include different type of nonlinearities into the dynamo equations.
To accomplish this task, we build a linear transformation to transform the terms
in Equations (13) into the terms of the standard form of the Van der Pol equations (2). If x is proportional to the toroidal component of the magnetic field,
A
a b
x
=
.
(16)
B
c 0
y
Applying this transformation in Equations (13) we obtain the following equations:
'b
'a
x+
y − γ0 c 2 x 3 ,
ẋ = v +
(c
(c
(17)
3
c 'a 2
'a
2
3
y+
−
x + γ0 ac − τ c x .
ẏ = v −
(c
b
(bc
To preserve only the terms displaying the same nonlinearity as Equations (2),
we must satisfy the following relations:
µ = 2v ,
(18)
v2 −
'
= 2 ,
(
(19)
v=
'τ
,
(γ0
(20)
√
and also a = τ c/γ0 , b = −(c/ ', and c = (µξ /γ0 ). Replacing in Equations (17)
we finally obtain the Van der Pol equations.
From Equations (18)–(20) is straightforward to obtain an estimate of the physical parameters involved in Equations (13). Using the values for µ, (, and obtained from the observed time series, and replacing the dimensionless variables
with the physical magnitudes involved in the dynamo equation, we finally obtain
vp ≈ 0.1 α0 ,
(21)
ωR
≈ 0.8 vp ,
(22)
L0
τα
≈ 2 .
γ
α0
(23)
222
P. D. MININNI, D. O. GOMEZ AND G. B. MINDLIN
From sunspot data, the angular velocity of the differential rotation can be estimated by 0.55 sin2 φ µrad s−1 (Javaraiah and Komm, 1993) which gives poloidal
and toroidal velocities of a few hundred meters per second, and α0 of the order
of one kilometer per second. For instance, according to mean field theory, this
value can be roughly compared with the convective velocity in the photosphere of
1 km s−1 . Note that a poloidal velocity of one hundred meters per second is five
times larger than the observed meridional flow at the solar surface (about 20 m s−1 ).
This can be due to the evaluation of the differential rotation at the solar surface,
while observations suggests that in the base of the convective zone the latitudinal
shear is much smoother. Considering L0 ≈ 0.1 R
and a characteristic dissipation
time τα between 1 and 30 days, Equation (23) yields an estimate for γ ranging
between 0.3 and 10 km s−1 . As a crude estimate, the rising rate of a buoyant flux
tube should be compared with the Alfvén velocity (Parker, 1979), which gives
a velocity of the same order in a flux tube at the solar surface. Note that these
conditions arise just to ensure the equivalence between the Van der Pol and the
truncated dynamo equations.
Acknowledgements
The authors wish to thank an anonymous referee for his/her fruitful comments
and suggestions. We acknowledge financial support from the University of Buenos
Aires (grant TX065/98).
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