Mon. Not. R. Astron. Soc. 306, 300±306 (1999)

q 1996 RAS

Non-axisymmetric solar magnetic fields

David MossMathematics Department, University of Manchester, Manchester M13 9PL

Accepted 1999 January 28. Received 1999 January 25; in original form 1998 December 4

A B S T R A C T

A simple non-linear, non-axisymmetric mean field dynamo model is applied to a

differentially rotating spherical shell. Two approximations are used for the angular velocity,

to represent what is now believed to be the solar rotation law. In each case, stable solutions

are found which possess a small non-axisymmetric field component. Although the model has

a number of obvious shortcomings, it may be relevant to the problem of the solar active

longitudes.

Key words: magnetic fields ± MHD ± Sun: activity ± Sun: magnetic fields.

1 I N T R O D U C T I O N

Dynamo models for the solar magnetic field are commonplace in

the literature, dating back to the pioneering papers of Parker

(1955), Steenbeck, Krause & RaÈdler (1966) and Leighton (1969).

Many of these models reproduced the generic features of the solar

field ± cyclic variation and predominantly equatorward migration

of the field pattern during a cycle ± but only by employing (as

recognized by the authors) unrealistic rotation laws and rather

arbitrary assumptions about the alpha-coefficient of mean field

dynamo theory. Two features of the solar magnetic field have gone

largely unaddressed in the context of dynamo models: the

presence of a poleward branch of the butterfly diagram at high

latitudes (but see some recent papers: e.g. RuÈdiger & Brandenburg

1995 and Tobias 1996), and the presence of active longitudes, an

intrinsically non-axisymmetric feature (e.g. Bai 1987, Jetsu et al.

1997, and references therein).

With the assumption of uniform rotational shear, or of angular

velocity constant on cylinders (both commonly made in earlier

work), non-axisymmetric magnetic fields are unstable in non-

linear mean field dynamo models. However, work on more

general mean field dynamo models suggests that with certain

arbitrarily chosen rotation laws it may be possible to have at least

a small non-axisymmetric field component co-existing with a

dominant axisymmetric component (e.g. RaÈdler et al. 1990; Moss,

Tuominen & Brandenburg 1991). Recent helioseismological

determinations of the angular velocity in the solar convection

zone are very different from either a constant shear law or angular

velocity constant on cylinders, suggesting instead that a better

approximation is that there is little radial variation in angular

velocity, except near the bottom of the convection zone (e.g.

Christensen-Dalsgaard & Schou 1988; Tomczyk, Schou &

Thompson 1995; Kosevichev et al. 1997). Thus the surface

latitudinal dependence of angular velocity persists through the

bulk of the convection zone. This suggests the possibility that the

winding up, and subsequent enhanced dissipation, by differential

rotation of non-axisymmetric field may be less marked than

hitherto believed, at least in some parts of the dynamo region.

Here we investigate a very simple non-linear (alpha-quenched)

solar dynamo model, which allows for the presence of a non-

axisymmetric field component. We use an analytically simple

rotation law that captures the main features of the more recent

observations, and also a more sophisticated interpolation on the

results of Kosovichev et al. (1997). We find that, for plausible

dynamo parameters, a small non-axisymmetric field can co-exist

with a dominant axisymmetric part. Many possible refinements

are omitted from our model ± the purpose of the paper is merely

to draw attention to that fact that simple dynamo models may in

principle be consistent with the presence of a relatively weak non-

axisymmetric component of magnetic field. This could be

associated with the phenomenon of active solar longitudes.

2 T H E R OTAT I O N L AW

Results from helioseismology (e.g. Christensen-Dalsgaard &

Schou 1988; Tomczyk, Schou & Thompson 1995; Kosovichev

et al. 1997) indicate a solar rotation law in which the angular

velocity is almost independent of radius, maintaining the surface

latitudinal dependence Vs(u), to near the bottom of the convection

zone (which occupies the region r0R # r # R, where r0 < 0:7);

r; u and f denote spherical polar coordinates. The angular

velocity then undergoes a sharp transition to a radially dependent

law. Undoubtedly there are details on smaller scales, but the

general behaviour can be captured by an interpolation formula of

the following form (cf. Charbonneau & MacGregor 1997):

Vs�u� � V0�1ÿ a cos2 uÿ b cos4 u�; �1�

V�r; u� � Vs�u�; r . rVR; �2�

V � V0�1ÿ f �r��Vs�uV� � f �r�Vs�u�; r0R # r # rVR; �3�where f(r) goes smoothly from 0 to 1 as r/R goes from r0 to rV. We

take

f �r� � x2�3ÿ 2x�; x � �r ÿ r0R�R�rV ÿ r0� ; �4�

Non-axisymmetric solar magnetic fields 301

although we have confirmed that a different choice of this function

makes little difference to the results. We put a � 0:126, b � 0:159

(e.g. Tassoul 1978). There are indications from helioseismology

that �rV ÿ r0�=�1ÿ r0� should be small (i.e. a narrow tachocline),

but for numerical reasons we adopt rV � 0:8. For most of the

calculations described below we take uV � p=3. Contours of

isorotation are shown in Fig. 1(a). We will refer to this as rotation

law I.

We also used a more sophisticated interpolation on the results of

Kosovichev et al. (1997) supplied by P. Heikkinen and M. Korpi

(private communication). Isorotation contours for this rotation law

(II) are shown in Fig. 1(b). With law II, the tachocline at the base

of the convection zone is rather less emphasized.

3 T H E M O D E L

We solve the standard alpha-quenched mean field dynamo

equation

­B

­t� 7� �aB� u� Bÿ h7� B� �5�

in a rotating spherical shell, r0R # r # R. We use the code

described by Moss, Tuominen & Brandenburg (1991), which does

not restrict the magnetic field B�r; u;f; t� to be axisymmetric. In

this code B � BP � BT, where

BP � 7� a�r; u; t�f̂� 7� 7�F�r; u;f; t�r̂; �6�

BT � b�r; u; t�f̂� 7�C�r; u;f; t�r̂ �7�are the poloidal and toroidal parts of the field. In this investigation

we make the simplifying assumptions that both a and h are

scalars, whilst bearing in mind that a more realistic model would

write them as tensor quantities. We take

a � a0 cos u=�1� B2�; �8�with a0 � constant. u � V� r comprises solely the rotational

velocity and h is the turbulent diffusivity, assumed to be uniform.

Boundary conditions for the solar dynamo problem are actually

quite uncertain. It is conventional to make the interior field fit

smoothly on to an external vacuum field at the surface r � R, and

we also use this condition. Note, however, that the field

immediately above the solar photosphere does not appear to be

a vacuum field, with the presence of coronal holes, local

emergence of toroidal flux, etc., and that we (in common with

almost all other investigators) have not studied the robustness of

our results with respect to changes in this boundary condition.

Likewise, there is considerable uncertainty attached to the

physical conditions near the base of the convection zone, r < r0R,

connected for example with issues such as the nature and extent of

the overshoot region. We adopt a simple `overshoot' condition,

corresponding to the field falling to zero at distance d beneath the

formal boundary so that, for example, d­a=­r � a at r � r0R,

with similar conditions imposed on the other variables (cf. Moss,

Mestel & Tayler 1990; Tworkowski et al. 1998). We put

d � 0:03R, but verify that the overall character of the results is

little altered by modest changes in d (including the case d � 0).

We find that an meridian plane resolution of 51� 101

meshpoints, uniformly distributed over the ranges r0 � 0:7R #r # R, 0 # u # p respectively, gives adequate resolution in

meridian planes. F and C are expanded as

F �XM

1

Fm�r; u� exp �imf�; �9�

C �XM

1

Cm�r; u� exp �imf�; �10�

and so the code allows a choice of the number of azimuthal modes,

m � 0; 1; :::;M, that are included. We find that taking M � 1 gives

a good approximation to the solutions found with larger values of

M, and so use this basic two-mode approximation.

In linear theory, the eigenmodes of equation (5) have fields that

possess either symmetry or antisymmetry with respect to the

equatorial plane. These symmetries will be denoted by S and A

respectively, and we shall also refer to even (S) and odd (A) type

solutions. We can classify the gross properties of our non-linear

solutions in terms of the quantities Ei;Pi; i � 0; 1; :::;M: Ei is the

energy in the azimuthal mode m � i, and Pi is the parity parameter

for this mode,

Pi � ESi 2 EA

i

ESi � EA

i

; �11�

where ESi ;E

Ai are the energies in the part of the field with

symmetry/antisymmetry with respect to the rotational equator (see

e.g. Moss, Tuominen & Brandenburg 1991). Thus, for example,

P0 � �1=ÿ 1 corresponds to quadrupolar-/dipolar-like parity of

the axisymmetric part of the field. These cases are also referred to

as being of (pure) even/odd parity.

Following non-dimensionalization in terms of the length R and

the time R2/h , the controlling parameters are

Ca � a0R

h; CV � V0R2

h; �12�

q 1999 RAS, MNRAS 306, 300±306

Figure 1. Isorotation contours for (a) rotation law I, and (b) rotation law II.

302 D. Moss

in addition to the parameters rV and uV of the rotation law (1)±

(3). We take V0 to be the equatorial rotation rate.

The values of a0 and h are ill-known, but, with plausible values

a0 , 103 cm s21, h , 5� 1011 ±5� 1012 cm2 s21 and R < 7�1010 cm; then Ca , 10±100, CV , 2� 103 ±2� 104. As remarked,

these estimates are quite uncertain, but provide a starting point for

our investigations.

4 R E S U LT S

4.1 Rotation law I

4.1.1 Pure parity solutions

We first study pure parity solutions, with pre-imposed odd parity

for the axisymmetric part of the field (A0: dipole-like) and even

parity for the m � 1 part (S1: perpendicular dipole-like). We

implement this choice by prescribing initial conditions with the

m � 0 and 1 parts of the initial fields having these parities, but

otherwise of arbitrary form. We illustrate our solutions by taking

Ca � 25; CV � 104; and uV � p=3; rV � 0:8.

q 1999 RAS, MNRAS 306, 300±306

Figure 3. Butterfly diagram for the axisymmetric part of the toroidal field just below the surface r � R, for the pure parity solution, P0 � ÿ1, using rotation

law I and Ca � 25, CV � 104.

Figure 2. Variation with time of the energies E0 (upper) and E1 (lower) in

the m � 0 and 1 parts of the field for the pure parity calculation (P0 � ÿ1,

P1 � �1), with rotation law I and Ca � 25;CV � 104. Note that E1 has

been multiplied by a factor of 100 in this figure.

Non-axisymmetric solar magnetic fields 303

In Fig. 2 we show the variation with time of the energy in the

m � 0 and 1 parts of the field. Both E0 and E1 vary regularly with

time. The butterfly diagram of the toroidal field immediately

below the surface of the model is shown in Fig. 3. The bulk of the

field displays equatorward migration, with the merest hint of a

poleward branch.

At the surface, the non-axisymmetric part of the field is

concentrated at high latitudes. The non-axisymmetric field pattern

migrates longitudinally with a period slightly longer than the

rotation period 2p/V0.

The general nature of the results is robust with respect to

modest changes in the controlling parameters Ca , CV and rV. If

CV * 2� 104 then the non-axisymmetric part of the solution

disappears, and if CV & 5� 103 there is no migration of the field

pattern during the cycle (butterfly diagram). If uV is reduced

significantly from the value p/3, the equatorward migration again

disappears for those values of Ca and CV for which a stable non-

axisymmetric field is present. Modest changes in the overshoot

boundary condition parameter d, including reducing it to zero

(so that the boundary conditions at r � r0R become

a � b � F � C � 0), do not cause any significant changes to

the solutions.

The ratio ,E1./,E0. decreases with increasing CV at given

Ca , and with decreasing Ca at fixed CV (angular brackets denote

time averages). It is hard to compare this result with the empirical

evidence. The more rapidly rotating solar-type stars appear to be

more active, and to have more marked non-axisymmetric surface

structures, which for given values of the diffusivity, alpha-effect

and rotation law would be contrary to the trends found here.

However, it is quite plausible, for example, that differential

rotation is reduced at higher angular velocities (e.g. RuÈdiger

1989), which would favour the generation of non-axisymmetric

magnetic fields. There is some support for this idea from

Henry et al. (1995), who suggest a trend towards solid body

rotation as angular velocities increase amongst chromospherically

active stars.

4.1.2 Stability of solutions

We also solve equation (5) starting with a completely arbitrary

mixture of m � 0 and 1 field components, i.e. without prescribing

the initial parities. Starting from a variety of initial conditions, the

only stable solution found is a mixed parity limit cycle, with both

,P0. and ,P1. near to �1. The behaviour of such a solution,

with Ca � 25 and CV � 104, is illustrated in Fig. 4. Thus, unlike

the observed solar field, the stable axisymmetric component has

near-quadrupolar parity. It is known that, in comparatively thin

shells, the standard axisymmetric mean field dynamo with a

constant shear rotation law has very similar excitation conditions

for both dipolar and quadrupolar solutions (e.g. Roberts 1972),

and non-linear mixed parity solutions are also found (Brandenburg

et al. 1990; Tworkowski et al. 1998). Thus this particular feature

may be of little significance in this sort of model. In other respects

this oscillating parity solution is quite similar to the P0 � ÿ1,

P1 � �1 solution with fixed parity discussed in Section 4.1.1.

4.2 Rotation law II

We have repeated our investigation using rotation law II. The

properties of the solutions found are quite similar in some ways to

those with rotation law I. There is again a parameter range in

which a non-axisymmetric field component is stable. Such

solutions are found at similar values of Ca to those of Section

4.1, but at rather smaller values of CV. Specifically, the non-

axisymmetric part of the solution is unstable if CV * 2:5� 103.

However, with law II, the stable solutions have an axisymmetric

field of strictly A0 type (P0 � ÿ1), and non-axisymmetric fields

are of strictly even parity, P1 � ÿ1 (S1). The general properties of

the non-axisymmetric part of the solution are similar to those

described in Section 4.1 ± in particular, the surface non-

axisymmetric field is concentrated at high latitudes, although

not quite as strongly (see Fig. 5). The longitudinal migration

pattern of the non-axisymmetric dynamo wave is again a little

longer than the rotation period. [This can be compared with the

estimates given by Bai (1987) and Jetsu et al. (1997), which

suggest periods for the non-axisymmetric features that are either

close to, or in one case about 15 per cent less than, the equatorial

rotation period.] Note that the butterfly diagram for the

axisymmetric component does not show equatorial migration at

lower latitudes with this rotation law: see Fig. 6.

5 C O N C L U S I O N S

We have demonstrated that the solar dynamo problem possesses

q 1999 RAS, MNRAS 306, 300±306

Figure 4. Mixed parity solution: Ca � 25, Cv � 104, rotation law I.

Bottom panel: variation with time of the energies E0 (upper) and E1

(lower) in the m � 0 and 1 parts of the field. Note that the value of E1 has

been multiplied by 100. Top panel: variation with time of the parities P0

(lower) and P1 (upper).

304 D. Moss

q 1999 RAS, MNRAS 306, 300±306

Figure 5. Contours of equal strength of the radial component of the non-axisymmetric part of the surface field: Ca � 30, CV � 2000, rotation law II. Poles

are at top and bottom; longitude runs horizontally.

Figure 6. Butterfly diagram for the axisymmetric part of the toroidal field just below the surface r � R, for Ca � 30 and CV � 2000, using rotation law II.

Non-axisymmetric solar magnetic fields 305

stable solutions in which a small non-axisymmetric component of

magnetic field, of approximately `perpendicular dipole' form, can

co-exist with the dominant axisymmetric component.

Apart from the adoption of rotation laws that possess the main

features of what is now believed to be the solar rotation law, the

model makes no claims to sophistication. In particular, the mean

field transport coefficients a and h are scalars, and a0 (equation 8)

and h are constants.

Moreover, the non-linearity is a crude form of alpha-quenching,

neglecting for example any effects of buoyancy, the torque exerted

by the Lorentz force, and any meridional circulation. Also the

possibility of h -quenching is ignored. In principle, a representa-

tion of most of these mechanisms could be included in a more

sophisticated model,

These omissions and deficiencies may account, in part at least,

for the obviously unsatisfactory features of the models. In

particular, the cycle period of the axisymmetric field is

unsatisfactorily short with the parameters used (about 10 per

cent of the solar cycle period when CV � 104). Further, the

butterfly diagram even for rotation law I does not reproduce in

detail that of the observed field, and that for law II is quite unsolar.

[Plausibly, the butterfly diagram might be sensitive to a change in

the simple cos u dependence of the alpha-effect (equation 8)

(cf. RuÈdiger & Brandenburg 1995).] Also the concentration of the

non-axisymmetric surface field to high latitudes in both cases

seems unrealistic in the solar context, although high-latitude

`polar spots' are commonly deduced to occur on other late-type

stars (e.g. Rice, Strassmeier & Linsky 1996). (Thus the non-

axisymmetric part of the field, although described as `perpendi-

cular dipole-like', is dominated by higher multipoles with m � 1

and odd parity with respect to the equator.)

Of course, the `butterfly diagram problem' is of long standing,

and has only been solved in the past by making arbitrary and

unrealistic assumptions about the rotation law. The problem of

too-short cycle periods is also shared with many earlier mean field

models in which the transport coefficients do not vary with radius,

but may be alleviated by allowing h to be reduced near the bottom

of the convection zone where the turbulence becomes weaker (e.g.

MacGregor & Charbonneau 1997; Charbonneau & MacGregor

1997). Another possibility is that enhanced h -quenching where

the toroidal field is strong may again provide a reduced effective

diffusivity near the bottom of the convection zone (e.g. Tobias

1996). The introduction of anisotropies in a and h may also play

a role in resolving this problem. The effect of such modifications

on the properties of the non-axisymmetric part of the solution,

such as the surface field structure, is unclear. It is known that an

anisotropy in a can favour non-axisymmetric field generation to

some extent (e.g. RuÈdiger & Elstner 1994).

We note that our dynamos operate in a substantially super-

critical regime. Our viewpoint is that it is valid to investigate

dynamo parameters that are physically reasonable, provided that

the structure of the resulting magnetic fields is broadly plausible.

For example, the axisymmetric fields in the models discussed here

are of global scale and of quite unremarkable structure. Sunspot

frequency is known to have been strongly reduced in the late 17th

and early 18th centuries (Maunder 1890), and the occurrence of

similar `Maunder minima' has been inferred in other solar-type

stars (e.g. Baliunas & Soon 1995). It may be that the presence of

intermittency in the long-term solar record is an indication of

significant supercriticality of the solar dynamo.

We must also point out that we have ignored the controversy

about whether alpha-quenching in traditional form would limit the

field strength at far below equipartition values (e.g. Vainshtein &

Cattaneo 1992). One possible resolution, which is not incom-

patible with the spirit of these calculations, is that the alpha-effect

may only be effective in the middle or upper part of the convective

shell, where the field strength is expected to be relatively small in

models with a narrow tachocline.

In summary, we have demonstrated that weak large-scale non-

axisymmetric field structures may be consistent with non-linear

mean field models of the solar dynamo. In this there are some

similarities with non-linear dynamo solutions in thick shells or

spheres, where the non-linearity is either an alpha-quenching

(Moss, Tuominen & Brandenburg 1991) or via a parametrized

form of the Navier±Stokes equation (Moss et al. 1995). In each of

these cases, the rotation law is also far from a uniformly

distributed radial shear law. The non-axisymmetric parts of the

surface magnetic fields in these models also have the property that

they are concentrated at high latitudes (although not to the extent

found in the models of this paper). It has been suggested that non-

axisymmetric dynamo solutions such as these may be relevant to

the presence of marked non-axisymmetric structures on the

surfaces of late-type rapidly rotating stars (e.g. Jetsu 1993; Jetsu,

Pelt & Tuominen 1993, and references therein). It is an interesting

speculation that the solar active longitudes may be a manifestation

of the same phenomenon. Although the limited results presented

here certainly do not provide a satisfactory model to explain the

solar active longitudes, it appears that a more refined and

sophisticated model of this nature might be relevant.

AC K N OW L E D G M E N T S

The author is grateful to P. Heikkinen and M. Korpi for providing

the interpolation on the SOHO rotation data. The careful report of

an anonymous referee improved considerably the presentation of

the results. Support from the EC Human Capital and Mobility

(Networks) grant No. ERBCHRXCT940483 is acknowledged.

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This paper has been typeset from a TEX=LATEX file prepared by the

author.

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