VI.
GENERATION OF SOLAR M A G N E T I C FIELDS
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
O R D E R A N D CHAOS IN T H E SOLAR
CYCLE
A.A. RUZMAIKIN
Institute of Terrestrial
Magnetism,
Ionosphere and Radio Wave
Propagation
Academy of Sciences
USSR
142092, Troitsk, Moscow Region, USSR
A B S T R A C T . Solar activity varying with an 11-year cycle is chaotic at large time scales. The
evidence comes from an analysis of observations of the sunspot number and of radioactive carbon.
Thereby an estimate of the dimension of the solar at tractor can be obtained.
The origin of the sunspots can be associated with the interactions of the regular, large-scale,
chaotic, and intermittent magnetic fields.
1. I n t r o d u c t i o n
Before t h e n i n e t e e n t h c e n t u r y solar activity was observed b y people as r a n d o m events, m a i n l y
t h r o u g h t h e a p p e a r a n c e of s u n s p o t s . H. Schwabe was t h e first w h o discovered t h e 11-year
cyclicity in t h e middle of t h e 19th century. Since t h a t t i m e solar scientists h a v e b e e n looking
for t h e o r d e r in t h e solar activity.
Until t h e present t i m e t h e s t a n d a r d p o i n t of view h a s been t h e following: T h e r e is order
in t h e large scales of space a n d t i m e , e.g. larger t h a n a t e n t h of t h e solar r a d i u s a n d one year.
For e x a m p l e an a n n u a l average of t h e M a u n d e r butterfly d i a g r a m for t h e s u n s p o t l a t i t u d e
d i s t r i b u t i o n represents order. T h e n t h e r e is disorder in t h e small scales, e.g. a single spot
a p p e a r s a n d d i s a p p e a r s randomly.
M o d e r n knowledge h a s cast new light on this issue. It a p p e a r s t h a t t h e activity becomes
chaotic a t large t i m e scales, say, h u n d r e d s of years. For e x a m p l e , t h e r e were n o s u n s p o t s
d u r i n g t h e epochs of t h e G r a n d M i n i m a , which occur r a n d o m l y in t i m e . T h e concept of
global stochasticity of solar activity was i n t r o d u c e d eight years ago ( R u z m a i k i n , 1981; see
also Zeldovich a n d R u z m a i k i n , 1983). T h e origin of chaotic b e h a v i o u r c a n b e associated with
a s t r a n g e a t t r a c t o r in t h e p h a s e space of t h e solar d y n a m i c a l s y s t e m .
At t h e same t i m e t h e r e exist some quasi-ordered s t r u c t u r e s a t small scales, for e x a m p l e
t h e network of g r a n u l a t i o n a n d mesogranulation.
In t h a t sense t h e active Sun is a kingdom of physics. Its p i c t u r e resembles t h e A r a b i a n
m a k a m a , a genre of l i t e r a t u r e t h a t mixes prose a n d verse, erudition a n d knavish trickery,
sermon a n d joke. Here I will t r y t o d e m o n s t r a t e this p i c t u r e by c r u d e strokes with t h e
M H D b r u s h . It is m a g n e t o h y d r o d y n a m i c because convection of t h e c o n d u c t i n g fluid a n d t h e
m a g n e t i c field is responsible for all active p h e n o m e n a on t h e Sun.
343
/. O. Stenflo (ed.), Solar Photosphere: Structure, Convection, and Magnetic Fields,
©1990 by the IAU.
343-353.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
344
2. S o l a r A t t r a c t o r
Let us first consider t h e processes averaged over a t i m e scale exceeding a year a n d a spatial
scale exceeding t h e basic scale of solar convection, t h e d i a m e t e r of a s u p e r g r a n u l e . T h e
m o t i o n s a n d m a g n e t i c fields evolving in t h e convection zone represent a n averaged solar
d y n a m i c a l s y s t e m . T h e p h a s e space of t h e d y n a m i c a l s y s t e m can be described, e.g. by t h e
c o m p o n e n t s of t h e m e a n (large-scale) m a g n e t i c field a n d t h e m e a n velocity field. A p a t h in
t h i s p h a s e space corresponds t o some solution of t h e averaged M H D e q u a t i o n s . F o r t u n a t e l y
it is not necessary t o investigate all p a t h s (solutions), because n o r m a l l y t h e r e a r e some
manifolds in phase space t o which all or almost all p a t h s a t t r a c t . It is sufficient t o find t h e
a t t r a c t o r s in order t o know t h e behaviour of t h e d y n a m i c a l s y s t e m . T h i s p r o b l e m is also
very difficult. However, some i m p o r t a n t characteristics of t h e a t t r a c t o r can easily be found
directly from t h e observational d a t a . T h e dimension of t h e a t t r a c t o r can for i n s t a n c e be
determined.
2.1.
DIMENSION
T h e simplest a t t r a c t o r s have a n integer dimension. T h e a t t r a c t o r of dimension d = 0 corres p o n d s t o a p o i n t . If such an a t t r a c t o r were present a t t h e origin of t h e d y n a m i c a l system
u n d e r consideration, t h e Sun would not have any large-scale m a g n e t i c field a n d m o t i o n s . A
closed curve as a n a t t r a c t o r of dimension d = 1 is t h e limiting Poincaré cycle. It is exactly
t h e 11-year cycle if it can be stable in t i m e . T h e a t t r a c t o r of dimension d = 2, a so-called
2-torus, m a y describe t h e 11-year cycle as m o d u l a t e d by a secular variation. T h e trajectories
of t h e p a t h s a r e regular for a t t r a c t o r s of integer dimensions.
More typical a r e however a t t r a c t o r s of fractal dimensions. Crudely one can imagine an
a t t r a c t o r of fractal dimension, e.g. d = 3 - c, where e < 1, as a ball w i t h an infinite n u m b e r
of small holes in i t . A p a t h travelling across this manifold will certainly seem very irregular
or chaotic. A well k n o w n e x a m p l e is t h e Lorenz a t t r a c t o r (Lorenz, 1963), which h a s d = 2.09
( s o m e t h i n g m o r e t h a n a surface). T h e dimension of t h e a t t r a c t o r is a m e a s u r e of t h e regular
or chaotic b e h a v i o u r of t h e d y n a m i c a l s y s t e m .
Is it possible t o find t h e dimension from observational d a t a ? At a first glance it a p p e a r s
impossible w h e n one as usual only knows t h e t i m e behaviour of a single q u a n t i t y ( a projection
in p h a s e s p a c e ) . O n e may, however, h o p e t h a t a sequence of values sufficiently prolonged
in t i m e will reveal t h e basic properties of t h e a t t r a c t o r . Grassberger a n d P r o c a c c i a (1983)
suggested a m e t h o d of finding t h e a t t r a c t o r dimension b y using a t i m e series of t h e observed
q u a n t i t y x(t),x(t
+ r ) , . . . , x[t -f (TV — l ) r ] , w i t h a given t i m e i n c r e m e n t r . T h e a t t r a c t o r
sought for m u s t b e e m b e d d e d in a p h a s e space of a large dimension m , defined by t h e vectors
= [*(*). *(* + r ) , . . . , * ( * + ( m - l ) r ) ] ,
i = 1 , 2 , . . . , Ν - 1.
T h e dimension d of t h e a t t r a c t o r is defined as t h e exponent of t h e correlation integral w(r)
r for small distances r between t h e points
~
d
Ν
w h e r e Ν is t h e t o t a l n u m b e r of e x p e r i m e n t a l points used, a n d θ is t h e Heaviside function.
T h u s , t o find t h e dimension one needs t o isolate for small r a linear p a r t of t h e curve (log w
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
345
vs. l o g r ) , a n d t o d e t e r m i n e i t s inclination.
Applying t h i s m e t h o d t o t h e series of relative s u n s p o t n u m b e r s , a dimension close t o 2
has been o b t a i n e d ( K u r t h s , 1987; Makarenko a n d Ajmanova, 1989). T h e series of s u n s p o t
numbers is, however, t o o short t o include such features as t h e G r a n d Minima. I t is b e t t e r
to use m o r e e x t e n d e d sequences, such as t h e c a r b o n isotope C a b u n d a n c e in t h e a n n u a l
tree rings available over 9000 years (Suess, 1965, 1978; D a m o n a n d Sonett, 1989; Dergachev,
private c o m m u n i c a t i o n ) . T h e r a t e of C formation in t h e E a r t h ' s a t m o s p h e r e d u e t o t h e
b o m b a r d m e n t b y cosmic r a y particles is well anti-correlated with t h e n u m b e r of s u n s p o t s ,
as a result of t h e solar m a g n e t i c field acting a s a filter for t h e cosmic r a y s . T h u s C is a n
index of solar activity.
An e s t i m a t e of t h e a t t r a c t o r dimension from Suess's d a t a over 6000 years gives d « 3.3
(Gizzatulina et a l . , 1988), i.e., t h e dimension is close t o 3 . Because t h e 3-torus according
to t h e theory of stochastic dynamical systems is u n s t a b l e a n d will b e transformed i n t o a
strange a t t r a c t o r , t h e above result m e a n s t h a t t h e m e a n solar a t t r a c t o r is fractal, i.e., t h e
dynamical system is chaotic. T h e e s t i m a t e of t h e dimension c a n b e improved b y using m o r e
extended a n d a c c u r a t e d a t a . T h e first estimates of t h e a t t r a c t o r dimension indicate a low
dimension. T h i s also suggests t h a t t h e m e a n solar d y n a m i c a l system is very simple. It c a n
be described in t e r m s of only few effective m o d e s . Let u s consider t h e p r o b a b l e origin of
these m o d e s ( R u z m a i k i n , 1985).
1 4
1 4
1 4
2.2. 2 2 - Y E A R S O L A R C Y C L E A S A M A N I F E S T A T I O N O F D Y N A M O W A V E S
One of t h e possible effective modes of t h e m e a n solar d y n a m i c a l system is t h e 22-year magnetic cycle. T h e analysis of t h e a n n u a l m e a n values of t h e Wolf s u n s p o t n u m b e r s W for 19
of t h e 11-year cycles m a d e by G u d z e n k o a n d C h e r t o p r u d (1964) shows t h e existence of a
limiting cycle of n e a r elliptical form in p h a s e space (W, aW/at). T h i s analysis assumes t h e
existence of t h e derivative aW/at a n d hence excludes t h e possibility for t h e a t t r a c t o r t o b e
fractal (Makarenko a n d Ajmanova, 1989). However, as a s m o o t h a p p r o x i m a t i o n over a finite
time interval it looks reasonable a n d gives t h e possibility of e x t r a c t i n g o n e of t h e effective
modes of t h e d y n a m i c a l system.
T h i s t w o - p a r a m e t e r limiting cycle c a n b e explained by t h e m e a n field d y n a m o t h e o r y
(cf. K r a u s e a n d R ä d l e r , 1980). T h e m e a n m a g n e t i c field is excited a n d evolves i n t h e
solar convective zone d u e t o t h e differential r o t a t i o n ( V Q ) , m e a n helicity ( a ) , a n d t u r b u l e n t
diffusion (ντ)'OA
dt
αΒφ + ν V' Α,
:2
τ
r s i n 0 ( V O χ νΑ)
+
φ
ν ΨΒ ,
τ
φ
where A is t h e a z i m u t h c o m p o n e n t of t h e vector potential of t h e poloidal field
(B ,Be).
T h e solution h a s t h e form of waves p r o p a g a t i n g , w h e n t h e value of v>t/(&R) is small,
along t h e surfaces of constant angular velocity (Yoshimura, 1975). For i n s t a n c e , w h e n t h e
a n g u l a r velocity only depends o n t h e radial c o o r d i n a t e , t h e d y n a m o wave travels along t h e
solar surface:
r
Α = A(wt - k9),
Β
φ
= Β (ωί
φ
- k9 + ê).
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
346
T h e direction of p r o p a g a t i o n is d e t e r m i n e d by t h e sign of t h e dimensionless d y n a m o n u m b e r
D = Rç (sme)a ax(àQ/ar) /uj>.
T h e wave p r o p a g a t e s from t h e pole t o t h e e q u a t o r
w h e n D is negative, a n d t o t h e pole when D is positive. T h e frequency of t h e wave is
ω = ( & I . D I / 2 ) / , t h e p h a s e speed is ω/k = (|Z^|/2A;) ^ , a n d t h e g r o u p speed is a factor
of two smaller t h a n t h e p h a s e speed. T h e p h a s e shift between t h e poloidal a n d toroidal
c o m p o n e n t s of t h e m a g n e t i c field is δ, which is close t o π / 4 in t h e s t a n d a r d kinematic
m o d e l s of t h e solar d y n a m o (cf. P a r k e r , 1979; K r a u s e a n d R ä d l e r , 1980). In t h i s case t h e
r a d i a l c o m p o n e n t of t h e field is lagging b e h i n d Β ψ by 3π/4.
In t h e o t h e r idealized case when Ω is only d e p e n d e n t on 0, t h e d y n a m o wave p r o p a g a t e s
radially. It is of i m p o r t a n c e t o know t h e real distribution of t h e a n g u l a r velocity as well
as t h e m e a n helicity in t h e solar convection zone. According t o mixing l e n g t h models of
stellar convection zones ( S p r u i t , 1974), t h e t u r b u l e n t diffusivity can b e considered as almost
c o n s t a n t in t h e b u l k of t h e solar convection zone.
Makarov et al. (1987) (cf. also B r a n d e n b u r g a n d T u o m i n e n , 1988), have m a d e a n a t t e m p t
t o find t h e d y n a m o wave distribution on t h e solar surface by using t h e d e p t h dependence
of t h e a n g u l a r velocity o b t a i n e d from current helioseismology d a t a ( F i g . l ) a n d t h e m e a n
helicity e s t i m a t e d via t h e mixing-length theory. T h e y found t h a t t h e source for g e n e r a t i n g
t h e m e a n m a g n e t i c field, ar(dü/dr),
h a s t h r e e m a i n m a x i m a in t h e convection zone: n e a r
t h e solar surface a t 0.9Ä©, a t 0.8Ä©, a n d close t o t h e b o t t o m a t 0.7 Rq.
)
m
max
1
2
ÄddV)
ι
1
2
ί
I
I
I
I
I
1
0.7
0.8
0.9
1.0
F i g u r e 1. T h e radial profile of t h e a n g u l a r velocity in t h e solar convection zone a p p e a r s t o
have regions with different sign of t h e g r a d i e n t .
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
347
latitude
Figure 2. Schematic representation of two m a g n e t i c waves on t h e solar surface ( t h e p o l a r a n d
equatorial d y n a m o waves, p r o d u c e d by t h e action of m e a n helicity a n d differential r o t a t i o n
of t h e t y p e given in F i g . l ) .
In t h e first region t h e value of a(dü/ar)
is positive, so t h a t t h e d y n a m o wave originating t h e r e
p r o p a g a t e s t o w a r d s t h e pole in each h e m i s p h e r e . T h e m a x i m u m of t h e wave a m p l i t u d e is
reached a p p r o x i m a t e l y a t a l a t i t u d e of 65°. T h i s wave p r o b a b l y manifests itself in t h e activity
of polar faculae m i g r a t i n g t o t h e pole. T h e second m a x i m u m , a t 0.8 R©, h a s a(dQ,/dr) < 0.
It is larger a n d creates a wave t h a t is t e n times larger in a m p l i t u d e , a n d which m i g r a t e s t o
t h e e q u a t o r . T h i s wave is n a t u r a l l y associated with t h e s t a n d a r d M a u n d e r butterfly d i a g r a m
of s u n s p o t activity (Fig. 2 ) . T h e t h i r d a n d deepest m a x i m u m of t h e source also creates a
wave p r o p a g a t i n g t o w a r d s t h e pole. T h i s wave is, however, r a t h e r weak on t h e solar surface
d u e t o t h e considerable d e p t h of t h e source. Makarov et al. (1987) have associated t h i s wave
with t h e a p p a r e n t e x t r a reversals of t h e poloidal m a g n e t i c field. Let us finally m e n t i o n an
i m p o r t a n t element of order, which was pointed o u t by Stenflo (1988). T h e observed o d d
r o t a t i o n a l l y s y m m e t r i c m o d e s of t h e solar m a g n e t i c field vary with t h e 22 y r period, while
t h i s period is absent for t h e even m o d e s . T h i s fact agrees with t h e s y m m e t r y a n d preferable
excitation of o d d m o d e s in solar d y n a m o theory.
2.3.
R O T A T I O N A L WAVES A N D M E R I D I O N A L F L O W S
O b s e r v a t i o n a l evidence have been found for zones of higher a n d lower r o t a t i o n a l velocities
( 3 - 6 m s " in a m p l i t u d e ) , p r o p a g a t i n g with a n 11-year period, s u p e r p o s e d on t h e general
differential r o t a t i o n ( L a B o n t e a n d Howard, 1982). T h e s t u d y of s u n s p o t m o t i o n s have shown
meridional flows with t h e s a m e periodicity ( T u o m i n e n et al., 1983).
1
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
348
T h e s t r o n g correlation between t h e m o t i o n s a n d t h e large-scale m a g n e t i c field suggests
t h a t t h e m o t i o n s are created due t o t h e action of d y n a m o waves. A simple p i c t u r e of a
r o t a t i o n a l wave p r o d u c e d by a m a g n e t i c d y n a m o wave h a s been d r a w n by Kleeorin and
R u z m a i k i n (1984 a) as follows:
T h e r o t a t i o n a l velocity variations produced by t h e m a g n e t i c field can be e s t i m a t e d from
t h e balance between t h e t u r b u l e n t a n d m a g n e t i c stresses,
d
f
ΐΐφ
Β Βφ
v
τ
T h e m a g n e t i c field can be considered as given, because t h e r o t a t i o n a l variations observed
are small. In p a r t i c u l a r , in t h e case t h a t Ω = Ω ( Γ ) , t h e field can b e w r i t t e n in t h e form
B
= 6 ( r , 0 ) c o s ( u r t - kO - π/2 - δ),
Β
=&0(r,0)cos(o;<-fc0),
r
φ
r
where t h e a m p l i t u d e distribution, frequency, a n d p h a s e are found by solving t h e d y n a m o
equations.
For t h e time-varying p a r t of t h e r o t a t i o n a l velocity one i m m e d i a t e l y o b t a i n s a wave with
t h e d o u b l e frequency,
η
φ
= u (r, Θ) cos(2u>t - 2k6 + π/2 - δ).
0
T h e d i s t r i b u t i o n of t h e wave a m p l i t u d e , uo ~ brbφ, proves t o be m o r e homogeneous as
c o m p a r e d w i t h t h e distribution of t h e a m p l i t u d e of t h e m a g n e t i c field. Let t h e a z i m u t h
c o m p o n e n t of t h e m a g n e t i c field (manifested on t h e surface in t h e form of s u n s p o t s ) be
c o n c e n t r a t e d t o m i d - l a t i t u d e s , e.g. as αφ ~ sin 20, on t h e solar surface, a n d let t h e radial
c o m p o n e n t increase t o w a r d s t h e poles, e.g. as b ~ s i n 0 . T h e n uo ~ sin θ c o s θ is distributed
m o r e homogeneously over t h e l a t i t u d e s t h a n t h e field. T h e p h a s e of t h e m a x i m u m of t h e
r o t a t i o n a l wave is shifted as c o m p a r e d w i t h t h e p h a s e of t h e m a x i m u m m a g n e t i c flux (defined
by
T h e shift is 1/8 of t h e wavelength of t h e r o t a t i o n a l wave if δ
π/4, as t h e kinematic
d y n a m o t h e o r y predicts. T h i s consequence of o u r simple m o d e l is in good agreement with
t h e observations of Howard a n d L a B o n t e .
For m o r e details concerning t h e m a g n e t i c action on t h e r o t a t i o n a n d meridional m o t i o n ,
we refer t o t h e p a p e r s by Rüdiger et al. (1986) a n d Kleeorin a n d R u z m a i k i n (1989).
2
r
\ΒφI).
2.4.
=
CAUSES F O R T H E STOCHASTIC BEHAVIOUR O F SOLAR ACTIVITY
T h e second effective m o d e of t h e m e a n solar d y n a m i c a l s y s t e m can be identified with t h e
secular m o d u l a t i o n of t h e 22-year solar cycle. Nonlinear b e a t s between o d d (dipole type)
a n d even ( q u a d r u p o l e t y p e ) m a g n e t i c fields m a y provide a m e c h a n i s m for t h e m o d u l a t i o n
(Kleeorin a n d R u z m a i k i n , 1984 b ) . M a g n e t i c fields of b o t h symmetries in t h e form of waves
with closely spaced frequencies may be excited by d y n a m o action. T h e b e a t s strongly comp e t e with t h e synchronization t h a t t e n d s t o equalize t h e frequencies. P e r h a p s t h i s causes
t h e observed irregularity of t h e secular m o d u l a t i o n .
A n o t h e r reason for this irregular variation is a possible decay of t h e m a g n e t i c field t o zero.
Actually t h e existence of a n o n - m a g n e t i c sun does n o t contradict t h e equilibrium conditions
or o t h e r fundamentals. It m a y even be profitable from some p o i n t of view. However, the
non-magnetic state is unstable due t o t h e d y n a m o n u m b e r ( t h e a m p l i t u d e of t h e source for
t h e g e n e r a t i o n of m a g n e t i c fields in t h e solar convection zone) being sufficiently large.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
349
T h e m a g n e t i c field grows when t h e d y n a m o n u m b e r is large. However, t h e growing field
smoothes o u t t h e a n g u l a r velocity gradients a n d removes t h e p r e d o m i n a n c e of left-handed
vortices over t h e r i g h t - h a n d e d ones, i.e., it reduces t h e m e a n helicity. As a result t h e effective
d y n a m o n u m b e r decreases a n d t h e d y n a m i c a l system can fall down i n t o a n o n - m a g n e t i c s t a t e .
Some simple models d e m o n s t r a t i n g such a behaviour h a v e been c o n s t r u c t e d ( R u z m a i k i n ,
1981; Weiss et al., 1984; Malinetsky et al., 1986). In t h e first p a p e r b a s e d on a Lorenz
type m o d e l a n e s t i m a t e of t h e characteristic t i m e t o s t a y in t h e vicinity of t h e n o n - m a g n e t i c
s t a t e is given by r(D — D )" ^ ,
where τ is t h e decay t i m e of t h e m e a n helicity, a n d D
is
t h e value of t h e d y n a m o n u m b e r a t which excitation of d y n a m o waves s t a r t s . In t h e m o r e
developed model by Malinetsky et al. (1986) it is shown how g r o u p s of regular 22-years
oscillations a r e s e p a r a t e d by deep m i n i m a (Fig.3). T h e b e h a v i o u r of t h e s y s t e m in p h a s e
space is similar t o t h e d y n a m i c s of t h e 3-torus.
1
2
cr
cr
30!
20i0<
0-
20 t
-/0
-20
Figure 3 .
(1986).
3.
T i m e evolution of t h e m e a n m a g n e t i c field in t h e model of Malinetsky et al.
I n t e r m i t t e n c y o f solar m a g n e t i c fields
Together with t h e m e a n m a g n e t i c field a fluctuating, small-scale m a g n e t i c field is excited.
Observations provide evidence t h a t t h e field is c o n c e n t r a t e d in t h i n m a g n e t i c flux t u b e s .
T h e fluctuating field can also be associated with t h e origin of t h e s u n s p o t s . Let us consider
briefly t h e s e two p r o b l e m s .
3.1.
DISTRIBUTION OF T H E FLUCTUATING MAGNETIC FIELD
Let hi(r,t)
represent t h e deviation of t h e m a g n e t i c field from i t s m e a n value, a n d
w(r) =< / i , ( r i ) / i i ( r ) > ,
r = \τχ - r |
2
2
be a space correlation function of t h e field.
As was shown by Kleeorin et al. (1986) in t h e framework of a k i n e m a t i c d y n a m o m o d e l
w i t h a velocity field having a short correlation scale, a s s u m i n g a m a g n e t i c Reynolds n u m b e r
of t h e o r d e r of 1 0 typical for t h e solar convection zone, two m o d e s of t h e fluctuating m a g n e t i c
field can be excited (Fig.4).
T h e first m o d e corresponds t o a simple loop h a v i n g a d i a m e t e r c o m p a r a b l e t o t h e correlation scale of t h e t u r b u l e n t convection / ~ 1 0 k m ( t h e size of a s u p e r g r a n u l e ) , a n d a
8
4
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
350
F i g u r e 4. T h e m o d e s of t h e fluctuating m a g n e t i c field t h a t can be excited by d y n a m o action
in t h e solar convection zone.
thickness of IR^^
~ 1 k m . T h e second m o d e is associated with a m o r e complicated loop
h a v i n g a subset of smaller loops of d i a m e t e r IRZ%^ ~ 100 k m . T h i s scale m a y b e c o m p a r e d
with t h e dimensions of observed flux t u b e s (Muller, 1985).
It is f u r t h e r necessary t o t a k e i n t o account t h e cell s t r u c t u r e of t h e solar convection. It
causes advection a n d c o n c e n t r a t i o n of t h e m a g n e t i c field t o t h e cell b o u n d a r i e s (Galloway et
al., 1977).
T h e d i s t r i b u t i o n of t h e m a g n e t i c field in a t u r b u l e n t flow with a large m a g n e t i c Reynolds
n u m b e r should be very inhomogeneous from a f u n d a m e n t a l p o i n t of view (Molchanov et al.,
1985). T h e m a g n e t i c field in such a flow is c o n c e n t r a t e d i n t o i n t e r m i t t e n t b u n d l e s with large
regions in b e t w e e n with a relatively weak field. T h u s t h e a p p e a r a n c e of solar m a g n e t i c flux
t u b e s can be considered as a n a t u r a l manifestation of such a b e h a v i o u r of t h e m a g n e t i c field
d u e t o t h e action of r a n d o m m o t i o n s of t h e highly c o n d u c t i n g fluid.
2
3.2.
ON T H E ORIGIN O F S U N S P O T S
A s u n s p o t is a region of reduced t e m p e r a t u r e (by a t most 1500 K ) . It originates r a n d o m l y
in t h e vicinity of t h e m a x i m u m of t h e large-scale m a g n e t i c field t h a t evolves w i t h t h e solar
cycle. T h e t o t a l a r e a covered by s u n s p o t s over t h e 11-year cycle does n o t exceed 2 % of t h e
solar surface.
S u n s p o t s t h u s a p p e a r as a set of t e m p e r a t u r e s p o t s . However, t h e y a r e n o t like t e m p e r a t u r e s p o t s in a t u r b u l e n t flow, w h e r e excesses a n d deficits of t h e t e m p e r a t u r e a r e equally
abundant.
T h e origin of s u n s p o t s should be associated with t h e m a g n e t i c field. We recall t h a t solar
m a g n e t i c fields were discovered by G. Hale in s u n s p o t s . A p a i r of s u n s p o t s is n o r m a l l y
connected b y a loop of t h e a z i m u t h a l m a g n e t i c field emerging t h r o u g h t h e solar surface,
p r o b a b l y d u e t o b u o y a n c y ( P a r k e r , 1979). It is, however, n o t clear why s e p a r a t e loops a n d
n o t t h e whole l a t i t u d e belt of t h e a z i m u t h a l field e m e r g e . Let us n o t e t h a t t h e m a g n e t i c field
is r a t h e r irregular even inside s u n s p o t s , a l t h o u g h t h e s a m e field direction is p r e d o m i n a n t
inside t h e s p o t s ( B r a y a n d L o u g h e a d , 1969).
It is n a t u r a l t o associate t h e origin of s u n s p o t s w i t h t h e i n t e r m i t t e n c y of t h e solar m a g n e t i c
field, i.e., with r a n d o m , infrequent b u t intense concentrations of t h e fluctuating m a g n e t i c
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
351
field g e n e r a t e d b y r a n d o m m o t i o n s . Deviations from t h e m e a n m a g n e t i c field occur b o t h t o
increase a n d t o decrease t h a t field. However, only after a n increase t h e t o t a l m a g n e t i c field
( m e a n field plus fluctuation) can reach t h e critical value t h a t m a k e s it float d u e t o b u o y a n c y
(Fig.5).
Figure 5. T h e s u m of t h e m e a n < H > a n d t h e i n t e r m i t t e n t , fluctuating field h m a y reach
a critical value t o allow t h e field t o e m e r g e from t h e solar convection zone.
T h e d i s t r i b u t i o n of t h e i n t e r m i t t e n t m a g n e t i c field is close t o a log-normal d i s t r i b u t i o n
(Zeldovich et al., 1987):
P(ln#) =
-exp
-(lnff-q)
2σ
2
2
where a = < ΙηϋΓ > a n d σ = < ( I n i f — a ) > . T h e symbol < > m e a n s ' m e a n v a l u e ' . T h e
successive statistical m o m e n t s grow w i t h increasing o r d e r p:
2
2
To eliminate t h e u n k n o w n value of a it is b e t t e r t o use d i s t r i b u t i o n for t h e normalized
m a g n e t i c field H = Ej < H >,
n
d e t e r m i n e d by t h e single p a r a m e t e r σ.
T h e p r o b a b i l i t y for t h e normalized field t o exceed a critical value ζ is
Prob
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
352
E q u a t i n g t h i s probability with t h e relative area covered by s u n s p o t s ( 1 % of t h e t o t a l a r e a of
t h e S u n ) one can o b t a i n an e s t i m a t e for t h e p a r a m e t e r σ. In principle t h i s p a r a m e t e r should
be d e t e r m i n e d from theory.
References
B r a n d e n b u r g , A. a n d T u o m i n e n , I. (1988) 'Variation of m a g n e t i c field a n d flows d u r i n g t h e
solar cycle', C O S P A R p a p e r No. 12.4.6, E s p o o , F i n l a n d .
Bray, R . J . a n d L o u g h e a d , R . E . (1964), Sunspots, C h a p m a n a n d Hall L t d . , L o n d o n .
D a m o n , P. a n d S o n e t t , C P . (1989) ' T h e s p e c t r u m of r a d i o c a r b o n ' , in P r o c . I n t . Conf. ' S u n
in T i m e ' , A r i z o n a Univ. P r e s s , Tucson.
E d d y , J . A . (1976) ' T h e M a u n d e r M i n i m u m ' , Science 1 9 2 , 1189-1202.
Galloway, D . J . , P r o c t o r , M . R . E . , a n d Weiss, N . O . (1977) ' F o r m a t i o n of t h e intense m a g n e t i c
fields n e a r t h e surface of t h e S u n ' , Nature 2 6 6 , 686-692.
Gizzatulina, S.M., R u z m a i k i n , A.A., Rukavishnikov, V . D . , a n d T a v a s t s h e r n a , K . S . (1988)
' R a d i o c a r b o n evidence of global stochasticity of solar activity', P r e p r i n t 40 (794), IZMIR A N , Moscow ( s u b m i t t e d t o Solar P h y s . ) .
G r a s s b e r g e r , P. a n d Procaccia, I. (1983) 'Measuring t h e strangeness of s t r a n g e a t t r a c t o r s ' ,
Physica D 9 . 189-208.
G u d z e n k o , L.L. a n d C h e r t o p r u d , V . E . (1964) 'Some d y n a m i c a l properties of solar a c t i v i t y ' ,
Soviet Astron. 4 1 , 697-706.
Kleeorin, N.I. a n d R u z m a i k i n , A . A . (1984 a) ' O n t h e n a t u r e of 11-year torsional oscillations
of t h e S u n ' , Pisma Astron.Zh.
( U S S R ) 1 0 , 925-935.
Kleeorin, N . I . a n d R u z m a i k i n , A . A . (1984 b ) 'Mean-field d y n a m o with cubic non-linearity',
Astron. Nachr. 3 0 5 , 265-275.
Kleeorin, N . I . a n d R u z m a i k i n , A.A. (1989) 'Large-scale flows excited by m a g n e t i c fields in
t h e solar convective zone', P r e p r i n t , I Z M I R A N , Moscow.
Kleeorin, N.I., R u z m a i k i n , A.A., a n d Sokoloff, D . D . (1986) 'Correlation p r o p e r t i e s of selfexciting fluctuating magnetic fields', P r o c . V a r e n n a - A b a s t u m a n i Int.School ' P l a s m a Astrophysics', E S A SP-251-ISSN 0379-6566; (1988) Kinematics
and Physics of Celestial
Bodies 4 , 28-35.
K r a u s e , F . a n d Rädler, K.-II. (1980) Mean-Held magnetohydrodynamics
and dynamo
theory,
Akademie-Verlag, Berlin.
K u r t h s , J . (1987) ' E s t i m a t i n g p a r a m e t e r s of a t t r a c t o r s in s o m e astrophysical t i m e series', in
M . Forkas (ed.), P r o c . I n t . Conf. on Nonlinear Oscillations, B u d a p e s t , pp.664-667.
L a B o n t e , B . J . a n d H o w a r d , R . (1982) 'Torsional waves on t h e sun a n d t h e activity cycle',
Solar Phys. 7 5 , 161-178.
Lorenz, E . N . (1963) 'Deterministic nonperiodic flow', J. Atmosph.
Sei. 2 0 , 130-141.
M a k a r e n k o , N . G . a n d Ajmanova, G . K . (1989) 'K entropy a n d dimension of solar a t t r a c t o r ' ,
s u b m i t t e d t o P i s m a A s t r o n . Zh.
Makarov, V . l . , R u z m a i k i n , A . A . , a n d Starchenko, S.V. (1987) ' M a g n e t i c waves of solar
a c t i v i t y ' , Solar Phys. I l l , 267-277.
Malinetsky, G . G . , R u z m a i k i n , A.A., a n d Samarsky, A . A . (1986) Ά m o d e l of longperiodic
variations of solar a c t i v i t y ' , P r e p r i n t N o . 170, Keldysh Inst, of A p p l . M a t h . , Moscow.
Molchanov, S.A., R u z m a i k i n , A.A., a n d Sokoloff, D . D . (1985) ' K i n e m a t i c d y n a m o in r a n d o m
flow', Sov. Phys. Uspekhy 1 4 5 , 593-628.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326
353
Morfill, G. and Voges, W . (1989) 'The solar cycle: Deterministic or chaotic', in Proc. Int.
Conf. 'Sun in Time', Arizona Univ. Press, Tucson.
Muller, R. (1985) 'The fine structure of the quiet Sun', Solar Phys. 1 0 0 , 237-255.
Parker, E.N. (1979) Cosmic Magnetic Fields, Clarendon Press, Oxford.
Rüdiger, G., Tuominen, I., Krause, F., and Virtanen, H. (1986) 'Dynamo-generated flows in
the Sun: 1. Foundation and first results', Astron. Astrophys.
1 6 6 , 306-318.
Ruzmaikin, A . A . (1981) 'Solar cycle as strange attractor', Comments on Astrophys.
9, 85-93.
Ruzmaikin, A.A. (1985) 'The solar dynamo', Solar Phys. 100, 125-140.
Spruit, H.C. (1974) Ά model of the solar convection zone', Solar Phys. 3 4 , 277-290.
Stenflo, J.O. (1988) 'Global wave patterns in the Sun's magnetic field', Astrophys.
Space
Sei. 1 4 4 , 321-336.
Suess, H.E. (1965) 'Secular variations of cosmic-ray produced carbon 14 in the atmosphere
and their interpretations', J. Geophys. Res. 7 0 , 5937-5952.
Suess, H.E. (1978) Radiocarbon
2 0 , 1-18.
Tuominen, J., Tuominen, I., and Kyrolinen, J. (1983) 'Eleven-year cycle in solar rotation
and meridional motions as derived from the positions of sunspot groups', Mon.
Not.
Royal Astron. Soc. 2 0 5 , 691-704.
Weiss, N.O., Cattaneo, F., and Jones, C.A. (1984) 'Periodic and aperiodic dynamo waves',
Geophys. Astrophys.
Fluid. Dyn. 3 0 , 305-341.
Yoshimura, H. (1975) 'Solar cycle dynamo wave propagation', Astrophys.
J. 2 0 1 , 740-748.
Zeldovich, Ya.B. and Ruzmaikin, A.A. (1983) 'Dynamo problems in astrophysics',
Astrophys.
and Space Phys. Rev. 2, 333-383.
Zeldovich, Ya.B., Molchanov, S.A., Ruzmaikin, A.A., and Sokoloff, D . D . (1987) 'Intermittency in random medium', Sov. Phys. Uspekhi. 1 5 2 , 3-32.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 18 Jun 2017 at 15:22:11, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0074180900044326