Simple solar dynamo models with variable ex and w effects

1997MNRAS.288..551R

Mon. Not. R. Astron. Soc. 288, 551-564 (1997)

Simple solar dynamo models with variable ex and w effects Colin B. Roald 1* and John H. Thomas 1,2* 1Department of Physics and Astronomy and C. E. K. Mees Observatory, University of Rochester, Rochester, NY 14627, USA 2Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627, USA

Accepted 1997 January 14. Received 1996 December 30; in original form 1996 July 8

ABSTRACT We examine a pair of radially averaged, pseudo-Cartesian dynamo models, one with a dynamically variable a effect, and the other with a dynamically variable w effect. Our models are kept deliberately minimal in order to permit extensive numerical analysis, including in particular direct solution of periodic orbits and continuation of unstable branches. Despite some differences in the formulations used, the bifurcation structure of the variable-w model is found to resemble closely that computed by Jennings & Weiss, whereas that of the variable-a model matches the structure computed by Schmalz & Stix somewhat less well. Our two systems, however, resemble each other not at all, despite having the same linearized form. Quenching of the a and w effects is confirmed to have dramatically different effects, with w-quenching perhaps producing better resemblance to the Sun. Also, we find the variable-a model to contain periodic solutions for positive dynamo number that do not show poleward propagation, contrary to conventional wisdom.

Key words: chaos - MHD - Sun: activity - Sun: magnetic fields - Sun: rotation - stars: magnetic fields.

1

INTRODUCTION

Ever since the pioneering work of Parker (1955), most efforts to understand the solar magnetic cycle have been based on the socalled aw dynamo described by the mean-field induction equation. The mainstream of dynamo investigation has since been followed by authors attempting to take advalltage of the rapid increase in available computing power in recent decades to handle ever more complex dynamo models. Typical recent improvements include fully spherical treatments (e.g., Belvedere, Pidatella & Proctor 1990), treatments in two spatial dimensions, resolving the radial coordinate (e.g., Dikpati & Choudhuri 1994), and even supercomputer simulations trying to resolve the convective flow (Brummell, Cattaneo & Toomre 1995 present a review). The need for this development is plain: even the most sophisticated models are still barely a start at describing a system as vast and active as the Sun. Nevertheless, an important and interesting body of work on the non-linear aw dynamo explores deliberately simplified models which still contain most of the essential physics. Keeping the equations abbreviated allows the majority of processor time to be devoted to far more extensive numerical analysis and exploration of the systems than is possible with the 'state-of-the-art' models. The goal is to use comprehensive knowledge of the simpler systems to guide the investigation of more complex ones. *E-mail: [email protected] (CBR); rochester.edu (JHT)

[email protected].

Several more recent efforts along these lines have attempted to model the non-linear feedback of the magnetic field on the differential rotation (the w effect) and the helical convective turbulence (the a effect) in some plausible way, short of solving the full dynamical equations of motion. Rapid developments in the theory of non-linear systems and chaos have spawned efforts to understand the irregular modulation of the solar cycle in a deterministic way. (See the review by Weiss 1994.) Our choice of model follows those of Weiss, Cattaneo & Jones (1984) and Schmalz & Stix (1991) in describing the non-linear effects limiting the growth of the dynamo by independent partial differential equations, rather than prescribed functional expressions for the magnetic 'quenching' of the a and w effects. The resulting set of non-linear partial differential equations may then be approximated by a finite set of coupled non-linear ordinary differential equations by means of a truncated Fourier-Galerkin expansion, i.e., an expansion in basis functions each of which individually satisfies the boundary conditions (but not the differential equations). Weiss et al. specify a dynamical equation for w, whereas Schmalz & Stix specify a dynamical equation for a. Here we specify similar - although not identical - dynamical equations for both w and a and study separately the effects of variable w and variable a in order to compare their contributions to the non-linear behaviour of the dynamo. In treating this class of models, one line of this work has gone directly to the lowest possible truncations of the partial differential systems (Weiss et al. 1984; Jones, Weiss & Cattaneo 1985; Weiss 1993; Tobias, Weiss & Kirk 1995). Here, we take a somewhat

© 1997 RAS

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1997MNRAS.288..551R

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552

different approach, similar to that of Schmalz & Stix (1991), in seeking to carry the truncation to high enough order to achieve convergence of the non-linear behaviour of the discrete system, which then, presumably, represents the true behaviour of the continuous system. In doing so, we find very significant changes in the behaviour of the system with increasing truncation order at low orders before convergence is achieved.

2

and we now assume Lur « (RILA8 )Lu8 (which is plausible given that we have already assumed RILA8 » 1). This is equivalent to neglecting the latitudinal velocity shear, and leaves us with aA' (VxA*f/!)'Vu' =R"aif' A

(10)

where",* = au*/ar is the radial shear. The dynamo equations (4 and 5) simplify to aA* at'

MODELS

",'

= a'B*

1JT a 2A* a02 '

(11)

+ R2

2.1 Derivation We begin with the MHD mean-field induction equation (e.g., Moffatt 1978 equation 9.1):

aBO

-a = v x (u' xB') + v x (a'B') + 1JT v t'

2B',

(1)

where B' is the magnetic field, U' is the fluid velocity, 1JT is the turbulent magnetic diffusivity, and all asterisks indicate dimensional quantities. Assuming an axisymmetric dynamo, all variables are independent of the spherical coordinate f/! (a/af/! = 0), and we can write (2) B' = B'~ + V x (A·~). By axisymmetry and because the field must be single-valued at the poles, we have boundary conditions

A'

= B' = 0 at 0 = 0, 'IT.

(V2

1), r2

aA • at*,

=a

aBo at'

=(VXA'~).VU'+1JT(V2 _ _~-)B" 2

+ 1JT

-

V2 -

sin2 0 A,

r2 sm 0

I) la

a*

= aa[cos 0 + a(O, t')],

(13)

",*

= "'0 [sin 0 + ",(0, t·)].

(14)

Here the factor cosO in (13) is chosen so that the undisturbed a effect varies with latitude in proportion to the Coriolis force and changes sign across the equator. Following Jennings (1991), the factor sinO in (14) is chosen to represent the spherical geometry, which should cause the '" effect to vanish at the poles. For boundary conditions, we take

2

(5)

Similarly, ,aA' AA' Vx(A'f/!) = - - - 0 - , r r A

= '" = 0 at 0 = 0, 'IT.

(15)

For the dynamical equation for a, we assume that a diffuses by turbulent viscosity and is quenched (or generated) by the Lorentz force. The Lorentz force term is chosen so that the non-linear terms in the system conserve energy - i.e., represent only exchange of energy between the magnetic fields and the fluid flow (represented by a' and ",'). Our equation for a is then (16)

(6)

r2 sin2 0 = R2 a02 .

a

(4)

Now we assume the dynamo operates in a thin shell of radius R to make the quasi-Cartesian approximation that A' (B') varies over length-scale LA (L B) much shorter than R, and so we neglect 1I? sin2 0 compared with V2. We thus implicitly discard the regions immediately surrounding the poles. Next, we assume without justification that A' (B') varies with 0 on length-scale LA8 (LB8 ) much faster than with r on length-scale LAr (LBr ). This leads us to

(

Following Schmalz & Stix (1991), we now break a' into steady and time-variable parts, demanding that the variable part contribute no net energy to the system. We treat ",' similarly.

(3)

We also assume a purely azimuthal mean flow, u· = u·~. (In doing so, we exclude effects arising from meridional circulation, as in e.g. Choudhuri, Schussler & Dikpati 1995.) With these substitutions into (1), the terms can be grouped as f/!-components and terms that are curls of f/!-components (and which hence contain only r- and O-components). After setting the two groups separately equal and uncurling the latter, we have the spherical dynamo equations, •B'

(12)

where PT is the turbulent viscosity and p is the density, both taken to be constant (clearly, another drastic approximation). Similarly, we write a dynamical equation for", in the form

"'0

a",

')B' + Ji.2 a

1 (aA at* = - 4'ITpR3 aif

"'OPT

2

",

a02 .

(17)

Weiss et al. (1984) considered a similar equation for", in a singlemode truncation. We should note that the non-linear terms in equation (16) differ from the form used by Schmalz & Stix (1991). On dimensional grounds we have taken the energy density associated with the poloidal field to be proportional to (lIR aAlaO)2 as the square of a magnetic field, whereas they have used merely A2. The complete energy density in our model is

(7)

ao

(18)

and so A

(V xA·f/!)·Vu'

1 aA' au'

A' au'

= -;:aifTr - 7aij'

(8)

These terms are of approximate magnitude

A' u·

A' •

(VxA'f/!)'Vu' ~----~ LA8 Lur R Lun ' A

The full energy conservation argument may be found in Appendix A. We now rescale all our asterisked variables to reduce the system to dimensionless form, using the following transformations:

(9)

(19)

© 1997 RAS, MNRAS 288, 551-564

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System

1997MNRAS.288..551R

Simple solar dynamo models where

Bo =

1JTy'41TP R .

(20)

553

In the latter, we ignore magnetic feedback on the lX effect and treat only A, Band w as dynamical variables. To do so, we set the variable part of the lX effect to zero in equation (13). In this case, we want to rescale by redefining

Our system reduces to the non-dimensional form

Bo = RwoV4'ITp.

aA aZA at = D(cos 0+ a)B + ao 2 ' aB. aA a2 B at= (smO+w) ao + aoZ ' aa 1 (aZA) aZa at = 1) aoZ B + v aoZ ' aw

1

at=-~

(21)

This moves the magnetic Reynolds number into the equation for lX, and again we can discard it. Equations (21)-(24) now reduce to

(22)

aA

(23)

(aA) ao B+v aaozw' 2

(24)

Here, v = VTI1JT is the turbulent magnetic Prandtl number, 1) = aowoR311J is the dynamo number and 'Rw = R ZWOI1JT is the magnetic Reynolds number of the differential rotation. The sign of D is determined by the sign taken for the differential rotation, wo0 We will consider the three quantities v, 1) and 'Rm to be free parameters. In this paper, we recover dimensional values for our figures by assuming the values listed in Table 1 for our scale variables. Since the actual values of these variables for the Sun are uncertain in some cases by orders of magnitude, specific numbers we report in gauss and years should be interpreted with caution. Nevertheless, we report them with units anyway, feeling that doing so may be helpful, and is unlikely to hurt. Note that given these parameters, the scale for the a effect is determined by the choice of dynamo number, ao = 1)lwoR3. Since 1) and Wo always have the same sign, ao is always positive.

i

1Ji

2.2

at =

DB cos 0 +

aZA

aoZ '

(29)

aB aA aZB at = (sinO + w) ao + aoz '

(30)

aw (aA) at=ao B+v aZw aoz '

(31)

We will call this set of equations the variable-w system. The variable-a and variable-w systems each contain just two free parameters, v and 1).

2.3

Spectral expansion

In order to make use of the methods of non-linear dynamics, we approximate the partial differential equations (21)-(24) by a set of ordinary differential equations via a Fourier-Galerkin expansion; that is, we expand each of the dependent variables in a sine series whose individual terms satisfy the boundary conditions (3) and (IS). Thus, N

A(O,f) '= LAn(f) sin nO,

(32)

n=l

and similarly for B, a and W. Substituting these series into equations (2S)-(27) and exploiting the orthogonality of the funcon [0, 'IT1 then leaves us with the following system of tions sin coupled, non-linear ordinary differential equations approximating the variable-a system:

nO

Variable-a and variable-w versions

We will be primarily considering two subsets of this system of equations. In the first, we will ignore the magnetic feedback on the differential rotation and treat only A, B and a as dynamical variables. That is, we set the variable part of the differential rotation, w, to zero in equation (14). The system (21)-(24) then reduces immediately to

aA aZA at = 1)(cos 0 + a)B + aoZ' aB . aA a2 B at = smO ao + aoZ '

(28)

(33)

(34) (3S)

(2S) (26)

All indices range from 1 to N, and coefficients out of this range (as produced in some of the sums) are set to zero. The coupling function F is

(27) F nml

We will call this set of equations the variable-a system; it differs from equations (10)-(12) of Schmalz & Stix (1991) by the factor of sin 0 in (26) and by the presence of the second derivative aZ z operating on A in (27).

lao

= { 1T(n+m+l)(n+m 8nzl(~

m+l)(n-m-l)

o

if n + m + I = odd . . otherwIse (36)

Similarly the variable-w system expands to

.

D

Z

An = 2 (Bn- 1 + Bn +1) - nAn' Table 1. Values of scale variables. R Wo 'IT P

5 X 1010 em :!:2x 10-6 S-1 1013 cm2 s-1 0.1 gcm- 3

En

= ~ [en -

I)A n _ 1

-

+ 2"1~ ~ mAm(wn+m -

(37)

(n + I)A n+ 1] W m-

2

n + wn- m) - n Bn,

(38)

m

Wn

=-~

L mAm(Bn+m - Bm_n + Bn- m) -

vn ZWn-

m

© 1997 RAS, MNRAS 288, 551-564

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(39)

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C. B. Roald and 1. H. Thomas

The presence of the double sum makes the variable-a system considerably more computationally expensive to integrate.

Table 2. Symmetries possessed by solution types. Class

2.4 Symmetries

Trivial Steady

We say that a system of equations possesses a symmetry if there exists an operation that, acting on any solution of the system, will produce another solution to the system. We say that a solution is symmetric if performing a symmetry operation maps it into itself. Equations (21)-(24) possess two important symmetries on reflections about the equator (Jennings & Weiss 1991). The first, the quadrupole symmetry, describes solutions with toroidal field components of even parity about the equator: q : (0, t) -+ ('IT - 0, t), (A,B, a,w) -+ (-A,B, -a, w).

Symmetries

Multiplicity

0

D2h

qs ds

{e,q,s, tq} {e,d,s, t d } Ie,s} {e,q, t,., t d } {e,q} {e,d, t,., t q } {e,d} Ie, t q } Ie, td} {e,t,.} {e}

1 2 2 4 1 2

Label

ms Periodic

qJ q2 dJ d2 rnq

md mr

(40)

a

2 2 2 2 4

This is paired with the dipole symmetry, which describes solutions with toroidal field of odd parity about the equator: d: (0, t) -+ ('IT - 0, t), (A, B, a, w) -+ (A, -B, -a, w).

(41)

Together these symmetries generate the four-element dihedral group D2 , which also includes the reversal symmetry 7 = dq, 7: (O,t) -+ (0, t), (A,B, a, w) -+ (-A, -B,a, w),

and the identity e

(42)

= rr,

e: (O,t)-+ (0, t), (A,B,a,w)-+(A,B,a,w).

(43)

In the spectral expansion, these operations become

q : An(t) -+ ( - t An(t), Bn(t) --+ ( - t+ 1Bit), an(t) --+ ( - tan(t), wn(t) -+ ( - t+1wn(t),

(44)

d : An(t) -+ ( - t+ 1An(t), Bn(t) --+ ( - t Bn(t), an(t) --+ ( - tan(t), wn(t) -+ ( - t+1wn(t), 7 :

(45)

An(t) -+ -An(t), Bn(t) -+ -Bn(t), an(t) -+ an(t), Wn(t) -+ Wn(t),

(46)

e : An(t) --+ An(t), Bn(t) -+ Bn(t), an(t) -+ Cln(t), Wn(t) -+ Wn(t)·

(47)

Note that solutions posses~ing q can have non-zero Aeven, Bodd ' while those with d, non-zeru A odd , Beven. To classify the symmetries of periodic solutions, we must also introduce the phase shift symmetry,

s: (O,t)-+(O,t+~P), (A,B,a,w)-+(A,B,a,w),

(48)

where P is the period. When this symmetry operation is applied to an equilibrium solution, we may take P to have any arbitrary value. (This prescription can be made rigourous by introducing a Lie group; see Moore, Weiss & Wilkins 1991; Proctor & Weiss 1993.) The s symmetry is then possessed only by equilibria. When combined with the time-independent symmetries q and d, it generates the eight-element orthorhombic group D2h = D2 ®~. The additional elements are tq

= sq : (0, t) --+ ('IT -

0, t + ~P),

(A,B,a,w)-+ (-A,B, -a,w), td

= sd:

(O,t)-+ ('IT - O,t +~P), (A,B, a, w)

t,

= S7:

(49)

--+

(A, -B, -a,w),

(50)

(O,t)--+ (O,t+~P), (A,B, Cl, w) --+ (-A, -B, a,w).

(51)

The spectral expansions follow immediately from (44)-(47). A distinct type of symmetric solution exists for each subgroup of D 2h . The subgroups representing equilibrium (or stationary) and periodic solutions are listed in Table 2. Solutions with broken symmetries q or d generally exist in complementary pairs, mapped into each other by the broken symmetry. (The exceptions are the ql and dl solutions, in which cases the broken symmetry is equivalent to a phase shift.) The number of distinct complementary solutions existing for a given symmetry type is listed as 'Multiplicity' in the table. See Fig. 1 for the appearance of these symmetries in phase-space. In principle, rs = {e, 7, s, t,} is also a valid symmetry subgroup. However, we neglect it since the only solution that can possess 7 is the trivial solution, A = B = a = w = 0, which has the full D2h symmetry. This analysis of symmetries generally follows Jennings & Weiss (1991), although we note that they did not distinguish between the ql and q2 varieties of the quadrupole periodic symmetry, and likewise between the dl and d2 varieties of the dipole periodic symmetry. Physically, the difference is that ql and dl describe fields that completely reverse each half-cycle, while q2 and d2 describe oscillations superimposed on some non-zero mean field.

3

NUMERICAL METHODS

Our bifurcation diagrams were computed by use of Doedel, Wang & Fairgrieve's (1995) numerical package AUT094, which implements a pseudo-arclength continuation scheme for equilibrium and periodic solutions (Doedel, Keller & Kemevez 1991a,b). In this implementation, equilibrium solutions are found by application of the Newton-Raphson root-finding algorithm to the right-hand sides of the dynamical systems. Periodic solutions are computed using the method of orthogonal collocation, in which the trajectory is approximated piecewise by polynomials. Solving for the orbit becomes a boundary value problem on the interval t E [0, P], with the period P as an additional variable. This additional degree of freedom is balanced by a condition on the otherwise arbitrary phase of the orbit. The pseudo-arclength method computes a solution branch by using the direction of the branch in phaseparameter space to obtain the initial guess at its value at the next step along the branch. At each point where the algorithm evaluates the system, it also computes a set of bifurcation functions, which change sign when a bifurcation occurs. When this detects a bifurcation, the precise location of the bifurcation is interpolated by means of the false position method on the bifurcation function. © 1997 RAS, MNRAS 288, 551-564

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1997MNRAS.288..551R

Simple solar dynamo models

555

q2

...... c (]) c

o a. E o .------o ,, ~~' (])

o

a.

,,

d1

~~

............

,------'

:::J :10-

"'C

CO

:::J C'"

-A1 (dipole component) Figure 1. Sample of each type of symmetric solution, plotted in A \ -B \ phase-space. From equations (44) and (45) we see that quadrupole solutions have nonzero B \ ; dipole, A \. Symmetry operation q is here represented by reflection over the B\ axis; d, A \ axis. Complementary orbits related by broken symmetries are dashed.

(AUTO is actually written to use the secant method. We found this to be unreliable for this problem and modified the appropriate routine.) AUTO also includes branch-switching algorithms used to obtain initial points on bifurcating branches. Unfortunately, we are aware of no practical method to trace quasi-periodic orbits created from secondary Hopf bifurcations. If they happen to be stable, some information can be gained by straightforward integration of the ODEs by, for example, the Runge-Kutta method. The difficulty here is that transients often decay very slowly, particularly near bifurcations (where we are most interested in the result), making it a laborious and uncertain process to classify solutions. And for unstable quasi-periodic branches, nothing can be done. Period-doubling bifurcations are also difficult to follow, simply for pragmatic reasons. After each doubling, the number of mesh points we need to compute along the orbit also doubles. This quickly becomes computationally prohibitive, both in time and in memory demands. Generally we were able to follow branches produced by only the first, or at most the second, period-doubling bifurcation. As a measure of chaotic activity, we have computed Lyapunov exponents by the ODE algorithm (Wolf et al. 1985). Given a starting point on a trajectory, we define an orthonormal 'frame' of vectors about it, and allow them to evolve under the action of the dynamical system. About once per orbit, we re-orthogonalize the vectors by means of the Gram-Schmidt renormalization algorithm, in order to keep the vectors from diverging in magnitude and collapsing in on each other along the direction of most rapid growth. The cumulative growth is tracked for many orbits; the limiting values define the Lyapunov exponents for the orbit.

4

RESULTS

Our results are summarized in the bifurcation diagrams presented in Figs 2 and 3, with the locations and properties of the bifurcation points recorded in Tables 3 and 4. We have examined both positive and negative dynamo numbers, even though it is commonly assumed that the Sun operates with negative 'D. Partly we do this for completeness, but also because, in light of the results of Choudhuri et al. (1995), it seems that meridional circulation can produce equatorward propagation of sunspots even from a poleward propagating (positive 'D) dynamo wave.

4.1

Convergence

While we are ultimately interested only in the solutions to the partial differential systems (25)-(27) and (29)-(31), we actually can only solve the ODE approximations to these systems, equations (33)-(35) and (37)-(39). Our goal, therefore, is to ensure that performing the spectral expansion has as little effect on the dynamics of the system as possible, which we attempt to achieve by raising the truncation order N until further increases have negligible effect on the behaviour of the system. In practice, we have contented ourselves with computing behaviour at as large a truncation order as we could easily handle, and then checking as many points as feasible at a value of N larger again by a half. However, since the variable-a system is more computationally expensive to deal with, we were not able to treat it using as many modes as we could use for the variable-w system (N = 10,16 against N = 16,24), and we were forced to confine ourselves to a more restricted domain. We believe that we have

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C. B. Roald and J. H. Thomas

4 stable equilibrium - - unstable equilibrium stable periodic - - unstable periodic X pitchfork OHo f

B

~

3

;~

12

8

1 4

o

-6000

-4000

-2000

00

2000

4000

o

6000

Dynamo Number, D Figure 2. Bifurcation diagram for variable-w system, -6000 < 1J < 6000, N = 16, v = 0.5. Each line in this diagram represents a physically distinct solution to the equations; different solutions are identified by the value of their poloidal magnetic field, averaged over latitude and time. Each solution branch is labelled with its symmetry. Bifurcation points, where new branches are created, are marked. Only equilibrium and periodic solutions are shown; quasiperiodic orbits were not computed. Arrows mark the existence of these branches. Crowded regions of the diagram are shown expanded in the upper right.

achieved reasonably good convergence in our computational domains, excepting possibly the period-doubling branches in the variable-a system; see Tables 3 and 4. We can make a few comments about the effects of overtruncation. Obviously, truncation quantitatively affects the location of bifurcation points in parameter space. This sort of effect does not particularly worry us, as we have no illusions of using models this simple to make numerical predictions anyway. More seriously, overtruncation can qualitatively change the structure of the bifurcation diagram. That is, branches can cease to exist, or connect in different sequences, or entirely new branches can appear. Chaos occurs at much lower dynamo numbers. Indeed, Schmalz & Stix (1991) point out that at the lowest truncation order (N = 2) their system is equivalent to the Lorenz system (e.g., Sparrow 1982); this is true of both of our systems as well. However, at higher truncations our systems do not resemble the Lorenz system at all, and neither do they much resemble each other. That is to say, the overtruncated behaviour is no predictor of the actual behaviour of the partial differential system. Physically, the reason for the qualitative change in behaviour of overtruncated systems appears to lie in the fact that the effect of diffusion is concentrated in the higher order modes (Marcus 1981). Discarding those modes thus limits the ability of the system to dissipate the energy pumped in by the dynamo cycle. In this light, it

is not surprising that overtruncated systems go quickly chaotic with increasing dynamo efficiency, V. Some branches are more affected by truncation than others; convergence is not even across the whole bifurcation structure. We believe this is simply the result of varying degrees of spatial complexity: more modes should certainly be necessary to reasonably approximate solutions with finer structure. We also note that in general, convergence in N happens more quickly for smaller IVI. This is natural; it should be expected that the more energetic, larger IVI systems will be able to build up more fine structure before reaching a balance with diffusion.

4.2 Variable-w system Fig. 2 is the bifurcation diagram for the variable-w system, which shows the number and type of solutions existing as a function of the dynamo number, -6000 < V < 6000, for fixed Prandtl number p = 0.5. This value of p was set to match that used by Schmalz & Stix (1991), to make direct comparison possible. We choose to plot the mean value of the poloidal field over the orbit as the ordinate identifying each branch, simply because we find this variable to be in this case the most effective at distinguishing the branches from each other. © 1997 RAS, MNRAS 288, 551-564

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1997MNRAS.288..551R

Simple solar dynamo models 1.2

(a)

557

D

stable equilibrium - - unstable equilibrium stable periodic - - unstable periodic Xpitchfork OHopf eriod-doublin

0.8 ..-.


\) ....

\ .... \ ",

Contours (kG)

5

(c)

,

6' ..

10 15 Time (years)

20

w· Contours (x 10-8 s-')

90r-~~~-,~~~~T-~~~~~~~,

,

\

_4·

·········-4···························

'0:.

( . 'C!.

\, If

....., \

...

45

"

I

:'--': ~ ~ . ' J:,!.'

\

.' :;i .. : \

i "

o

20

,

,

45

10 15 Time (years)

5

90

\

0'..

1;1

o

,.


:: I

I ., J

:;::;

'/

L-~~'-'-~L-~"-~~'--'-'~~~~~~~'-'~-L-~~

- 90


0

:.i:)··) :.... ::.;:~~:. : :.....:.:.;. ;.) .~ :...:.:;:f()· : L.. :.~~. ......

,t'

J

I

CJ ::J

/
.

CJ)

+-'

5

:-',

45

OJ

~


+-'

25

15 20 Time (years)

w· Contours (x 10-8 s-')

(b)

(b)

10

5

•

CJ ::J

... ...:.:;:'

.:'

+-'

a -45 -'

-45

!'"

····-4···· -90L-~~~~~~~~~~~~~~L-~~~

5

10

15

Time (years) Figure 4. Contour plots of the variation in time of toroidal magnetic field strength for the stable periodic branches, variable-w system, at v = 0.5, N = 24: (a) mixed quadrupole (mq) branch, 1] = -1011; (b) pure dipole (dJ) branch, 1] = -3027; (c) pure quadrupole (qJ) branch, 1] = -4016.

o

5

10 Time (years)

15

Figure 5. Contour plots of the variation in time of differential rotation w * for the stable periodic branches, variable-w system, at v = 0.5, N = 24: (a) mixed quadrupole (mq) branch, 1] = -1011; (b) pure dipole (dI) branch, 1] = -3027; (c) pure quadrupole (qI) branch, 1] = -4016.

© 1997 RAS, MNRAS 288, 551-564

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System

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C. B. Roald and 1. H. Thomas 8 tO