Adv. Space Res. Vol. 6, No. 8, pp. 19-28. I986 0273-[17" <6 50.00 - .50 Printed in Great Bntain. All rights reserved. Copyhght ~ COSPAR

NUMERICAL SIMULATIONS OF SOLAR AND STELLAR DYNAMOS

D. J. Galloway

Department of Applied Mathematics, Australia

Universi~ of Sydney, N.S.W. 2006,

ABSTRACT

Numerical calculations in dynamo theory can be performed with several ends in mind. The most theoretical approach seeks to investigate simple models in order to clarify fundamental processes of dynamo theory (for example, the transition to the highly-conducting limit). The most practical approach attempts plausibly to parametrise the more troublesome physical processes (e.g. turbulence, via the alpha-effect), thereby generating a simpler, astrophys- ically versatile, set of model equations. By an appropriate choice of parameters, these models can reproduce the global aspects of the solar cycle; they have also been used for other late-type stars, but often they contain too many free parameters to have much predic- tive power. The most ambitious approach takes a set of fundamental equations and integrates them numerically in three dimensions. So far, only the Sun has been modelled in this way, and the results fail to reproduce many important aspects of the solar cycle. This may be due to inadequate treatment of the small scales (these must still be parametrised due to severe limitations on numerical resolution), or it may reflect the possibility that the right input physics has yet to be included.

i. INTRODUCTION

Spiegel /i/ has classified theoretical astrophysicists according to the political basis from which they attack a given problem. The subject-matter of this paper provides a classic illustration of his analysis; depending (amongst other things) on whether an author hails from a mathematics institute or an observatory, so his dynamos may embody the politics of Attila the Hun or Trotsky. On the right wing are those who retreat from the enormous diffi- culty of the full dynamo problem and solve instead much simpler problems designed to teach the theoretician to understand more about the fundamental processes. The advantage of this approach is that adequate mathematical and computational rigour are usually possible; the drawback is that only situations remote from real stars can be dealt with. Workers on the left wing try to simulate or reproduce actual astrophysical objects; if a difficult physic-

al process has to be included it is described by one or more free parameters. This approach has much more contact with observation, but often so many free parameters are introduced that the theory cannot predict very much, and when several assumptions are stacked on top of each other, credibility is strained.

Recently, the advent of supercomputers has fostered a third, 'Realpolitik' approach; in- stead of retreating from the difficulties of the full problem, its practitioners meet them head-on, and attempt to include as much as possible of the real physics in a computer code which solves the full equations whilst keeping parametrisation to a minimum. Disconcerting- ly, although this approach is undoubtedly the worthiest, it has so far failed to reproduce most of the important observational features of the only star for which models have been run so far, the Sun. Furthermore, it is expensive, and only open to those with access to the world's largest machines.

Here no attempt will be made to discuss numerical models with extensive parametrisations (see /2/ for a brief review) - the author did, after all, originate in the country that brought you Margaret Thatcher. Section 2 will describe calculations performed with the object of studying dynamo theory in its own right, and section 3 describes results from large-scale simulations designed to reproduce the Sun. Section 4 concludes by ex~T.ining why there is a discrepancy between these calculations and solar observations, and makes some re- marks about the prospects for improving this.

2. THEORETICAL COMLD U"FAT I ON S

These attempt to answer fundamental questions such as 'what are the most important prerequi- sites for a dynamo?', 'is rotation a necessary ingredient for an efficient dynamo?', 'what

19

20 D.J . Gallo~a~

kinds of kinetic and magnetic energy spectra result when a dynamo is present?', and so forth.

Most co--only, incompressible velocity fields are considered, meaning, for cases where con-

vection is the driving force, that ~he Boussinesq approximation is used.

Meneguzzi, Frisch and Pouquet* /3/ have performed computations of a dynamo due to homogene-

ous incompressible turbulence in three dimensions. The equations solved are

~B _c. = ?A(uAB) + ~?2B (induction eauation) ~t . . . .

~u Vp 1 ~+ u.?u = - --+ (?AB)AB + UTaU + f 3t - p UO ~ - -

(moment,~n equation, incl%ding effect of Lorentz force)

V.u = 0 (incompressibility)

V.B = 0

Here the symbols have their conventional meanings. There are two diffusivities - ~ (magnet-

ic), and ~ (viscous). The motion is sustained by a force f derived from the computer's

random number generator, sometimes subjected to certain constraints. The geometry is that

of a 2~-periodic cubical pattern and is thus space-filling. The equations are solved by a

pseudo-spectral scheme - nearly all dynamo calculations have used either spectral or mixed

spectral/finite difference methods, on the basis that if it is at all possible to use a

spectral scheme, conventional wisdom says it should be rather more accurate than finite

differencing.

Before solving the above equations numerically, they are scaled so that the effects of vari-

ous terms are represented by non-dimensional quantities. Important for us will be the ordi-

nary (kinetic) Reynolds number

UL Re

and the magnetic Reynolds number

UL R E m

U being a typical velocity scale, L a typical length. It is here that the most fundamental

limitation of the computational approach becomes evident. This is connected with the idea

of scale separation. When Re or R m are very large, the nature of the flow is i] to become

turbulent and ii] to establish spontaneously layers of very small length scale where the

variables change very quickly. Sometimes the positions of these 'boundary layers' are pre-

dictable, sometimes they occur at random within the body of the flow. Theoretical models of

these structures are known, and their thicknesses are often of the order of Re ½ or Rm ~ times

L. To compute the flow properly, without recourse to some auxiliary modelling of these small

scales, they must be numerically resolved

In a star, values of Re (or R m) based on the radius are not uncommonly i0 I° - 102o . Thus of

the order of 105 - i0 I° grid points would be necessary in any one dimension to even begin to

see these features. Of course, this is impossible, and in practice people compute up to the

highest Re they can manage, and then hope they have reached some kind of asymptotic regime

which can be extrapolated to stellar values. The computable values currently lie around 500,

so the extrapolation involves a considerable act of faith. Some kind of modelling of the

small scales seems inevitably to be necessary.

Meneguzzi et al.'s calculations used resolution up to (64) 3 , at which stage the storage

required exceeded the main memory of a 1 ~Mword Cray-l, necessitating going out of core,

with a consequent severe reduction in speed and an expenditure of several tens of Cray hours.

The diffusivity ratio u/~ ('magnetic Prandtl number') was set to i; thus R m and Re were

equal and reached about i00. Results are reproduced in figure i, t~ken from their paper.

Note that dynamo action apparently occurs whether or not the driving f is helical - a crucial matter, since the parametrised 'e~' dynamo theories normally demand helicity as an essential

ingredient. Actually, it is debatable whether the computations have been run long enough to

settle down, since the magnetic diffusion time is of order R~ times the eddy turnover time.

However, the authors report that lower resolution computations continued for longer physical

times show the same effect. Even with (64) 3 resolution it is hard to achieve adequate scale

separation between phenomena occurring on the global scale and those at the diffusive cut-

off. This is exacerbated by the spatial intermittency of the magnetic field (fig. ib).

* It should be noted that the last two authors are mavericks in that although featured in

this section they in fact belong to an observatory.

Numerical Simulations of Dynamos 21

EV

,L

EM

(a) (b)

Fig. 1. Turbulent dynamo with non-helical driving: Reynolds numbers are

R V = R M ~ i00. (a) Temporal variation of kinetic (E V) and magnetic (E M) energy.

The time unit is the eddy turnover time. (b) Spatial intermittency of B - field

at t = 23. The shaded regions have l~I within less than 5% of its maximum value

(from /3/).

Recently Meneguzzi /4/ has extended these calculations to the case where the driving is pro-

duced by a thermal buoyancy force due to an imposed temperature gradient in a gravitational

field. The possibility of a uniform added rotation was also included. It was found that

i] for sufficiently small magnetic diffusivity (sufficiently high R m) convection drives a

dynamo; ii] the magnetic field is again highly intermittent; and iii] even without rota-

tion, there is still a dynamo. This last is surprising because rotation is usually thought

to be a necessary ingredient of a stellar dynamo, and one again wonders if it would persist

if the calculations could be continued longer.

A second problem rather simpler than the above is to investigate whether chaotic motions are

in any way helpful for dynamo action. This has been raised particularly by Arnold,

Zeldovich, and co-workers /5,6/, and the related question of whether there is a 'fast'

dynamo (see the definition in Soward's contribution) has been studied both there and in

Moffatt and Proctor /7/. Chaos is here to be understood in the sense that neighbouring

fluid particles diverge exponentially with time, so that, were the field to be frozen in,

field lines would be stretched to an extreme degree. Extreme gradients are also generated;

these promote diffusion even when the diffusivity is low, and in the limit R m + ~, it is not

clear whether or not the stretching wins out over this enhanced diffusion.

A simple flow which apparently has chaotic flow lines /8,9,10/ is the helical Beltrami

(TAu = U) flow

U = (A sin z + C cos y, B sin x + A cos z, C sin y + B cos x)

The kinematic dynamo action of this flow has been investigated numerically by Arnold and

Korkina /6/ and Galloway and Frisch /ii/, for cases where the field generated has the same

scale as the motion (scale separated cases have been considered analytically by G.O. Roberts

/12/, Childress /13/, and Childress and Soward /14/.

The problem is again 2~-periodic, and the induction equation

aB 1 = ?^(u^B) + -- ?2B

at - - R m -

(after scaling) is solved with R m as high as possible. The velocity field is not allowed to

change with time, so attention is restricted to the initial phase of field growth, where the

Lorentz force is negligible (the 'kinematic' problem).

Arnold and Korkina /6/ showed that for the case A = B = C = l, dynamo action appears for R m - 8, but disappears for R m { 17.5. They used a spectral method, but assumed a time

dependence e ot and then solved for the eigenvalue ~ using an inverse iteration method.

Galloway and Frisch /ii/, using a much larger computer, and a time-stepping spectral method,

showed that dynamo action reappears in a different form at R m ~ 27.5, and persists up to

El D. ,I. Oallm~a~

R m ~ 450, beyond which it is not possible to compute reliably, due to reso!uti3n limitations. Other values of A, B and C were also used. Results for the field structure are shown in figure 2; the magnetic energy, is concentrated into well-defined cigar-shaped features.

Fig. 2. Magnetic field generated by a chaotic flow. (A = B = C = l; R m = 450.) Stereoscopic plot of surfaces where B 2 exceeds 25% of its maximum value (from /ii/).

Some conclusions are-

i] the field is highly intermittent, with concentrations into flux ropes or sheets of

thickness ~½ × global scale

ii] with this (very simple) 3-D problem, an acceptable resolution of scale-separation

begins to be achieved, though one would wish for more

iii] dynamo action is the rule rather than the exception, though there are some curious

windows in R m where it vanishes

iv] structures where the field is strong appear to be embedded in or at the edges of regions where the flow is chaotic (and they are often associated with stagnation

points)

v] integrable (non-chaotic) flows with ABC = 0 gives slow dynamos (see Soward, this volume). It is possible but not proven that certain flows with ABC ~ 0 give fast dynamos. Since it now looks as though even non-chaotic flows can give fast dynamos,

the importance of this question may have been overrated.

Finally in this theoretical section, reference should be made to calculations on magneto- convection which, though they do not in themselves produce dynamos, may contain results essential for an understanding of the detailed working of the dynamo process. Computations have shown that convective eddies tend to expel magnetic flux and concentrate it in regions where the flow is strongly convergent; if the field thereby becomes locally so strong that the Lorentz force is appreciable, the motion becomes suppressed in the neighbourhood of the concentrations. These undoubtedly important effects are reviewed in /15,16,17/. In addi- tion, topological asymmetries in the convection pattern (on the Sun, the updraughts appear isolated whereas the downdraughts form a continuous neb~ork) may force magnetic flux to the base of the convection zone, thereby causing the dynamo to operate principally in this region /18,19/. This process is known as topological pumping. Computations have been per- formed to study it /18,20,21/, but are currently incomplete because the competing effects of magnetic buoyancy /19,22/ have never been considered simultaneously.

3. GLOBAL SIf.IULATIONS OF THE SOLAR DYNAMO

The aim of this approach is to solve~ with as little approximation as possible, the full

equations describing the Sun's convection zone, so that convective motions are produced with in a spherical shell. These interact with a rotation field to produce the solar differen- tial rotation, and the resulting total velocity then generates a magnetic field which, in turn, is allowed to react back on the motion via the Lorentz force. The hope is that this highly self-consistent approach will reproduce the main features of the solar cycle, and the

Numerical Simulations of D> namos 23

\ /,- >,:

c>

.. ,','~ L ._~.~ ,,,, .~.~ :''. ._"rT~. \ : q, ~,:

• -Y :- ""C, " ~i

Fig. 3. Poincare sections of particle trajectories crossing the planes z = r~/4 (r = 0,1,..-,7). One particle only is tracked; its successive intersections

suffice to delineate a connected region where the flow is chaotic. Superimposed are contours of intensity of B 2 in the same plane, normalised to the maximum over the whole cube. Note the clear relation between the strong B features and the chaos (A = B = C = i; the field is for R m = 35) (from /ii/).

same codes can then be used to investigate dynamo activity on other stars.. Two classes of

model have been developed so far: the Boussinesq (almost-incompressible) calculations of Gilman and Miller /23/ and Gilman /24/, and the compressible (anelastic) calculations of Glatzmaier /25,26,27/.

The former models take V.u = 0, V-B = 0, and in addition solve the induction equation, the Navier-Stokes equation (including the Lorentz force jAB), and an advection-diffusion equa- tion describing heat transfer. The calculations are conducted in a thick spherical shell intended to represent the Sun's convection zone, and a variety of boundary conditions have been used, based on a mixture of physical relevance and computational expediency (see the original papers for details, but note that the policy of varying the boundary conditions, to see what their effect is, is praiseworthy).

The numerical methods are spectral in the azimuthal direction (~), and finite-difference in meridional planes (r,@). The latter causes some numerical problems near the poles, which have to be either omitted from the calculation or subject to some kind of filtering. The rotation axis runs from the south to north poles. The calculations are fully three-

24 D.J . Gallov~a~

dimensional and time-dependent; for this reason ~_hey demand im/nense computational resources

When the equations are scaled, the solutions are found to depend on five non-dimensional parameters: in the standard notation of Chandrasekhar /28/ these are a Rayleigh number

g~iTd3/<v measuring ~he strength of ~ne the_~mal forcing; a Taylor number ~2d~/vz giving the strength of rotation; two Prandtl n'~.nbers p = v/< and ~/<, referred to as Q by Gilman, which are diffusivity ratios, and finally a cell size parameter describing the depth of the shell.

The procedure is: adjust Cne non-magnetic parameters until the model represents solar obser- vations (correct differential rotation, lack of observable pole-equator temperature differ- ence) with the jAB force switched off. (Note that this generates convective 'giant cell' velocities significantly larger than Cne observed upper limits.) Then choose a value of ~/< and add a seed magnetic field to see if it grows with time, and if it ends up resembling solar bebaviour.

For numerical reasons (the inability to separate scales) the true solar diffusivities cannot

be used. Instead, much higher values ("eddy coefficients") are adopted, the justification being that for a global simulation small-scale, turbulent motions should be modelled like this anyway. Thus the kinetic or magnetic Reynolds numbers in these calculations end up being of order i00, compared with solar values many orders of magnitude higher.

Typical nudnerical resolution is 16 r-points, 60 @-points, and 24 ¢-modes, though any one of these can be increased at Cqe expense of the others. Calculations then take several tens of Cray-hours to simulate times comparable with the solar cycle.

Results (figs. 4 and 5) may be sun~arised as follows:-

i] the Sun's differential rotation can be reproduced, though not without giant cell velo- cities substantially higher Cnan anything observed

ii] when N/< is chosen small enough, dynamo action apparently occurs

iii] the resulting dynamos are much less systematically regular than the Sun's - it is hard to discern anything with the rigour of Hale's polarity laws, though the toroidal fields do s/now a preference for a parity antisymmetric about the equator

iv] the B-fields show, if anything, propagation away from the equator at the surface (rather than towards, as obse~ved in the sunspot cycle)

v] oscillations in magnetic ener~] occur, but their periods are much faster than the solar cyc{e (i year rather than 22), and seem to reflect the turnover time of the large eddies.

The prime responsibility for the discrepancy between these models and the Sun was felt to lie with the restriction to the Boussinesq case. Thus Glatzmaier /25,26/ treated the an- elastic case where the fluid is fully stratified, with several scale heights across the shell, but sound waves are filtered out, permitting calculations which would otherwise have

a prohibitively small timestep. The full equations are given in /25/; several different num.erical schemes were tried, and eventually a spherical harmonic expansion was chosen, with the vertical (r) dependence expanded in Chebyshev polynomials. This has the disadvantage that there is no fast Legendre transform, but it avoids the problems at the poles referred to earlier. The typical resolution was similar to Gilman's, but the method, being fully spectral, should be somewhat more accurate. Glatzmaier /26/ also uses a subgrid-scale eddy diffusivity though the difference this m~<es is not much discussed.

The results (fig. 6) are unfortunately more or less as for the earlier calculations. In

particular, giant cells still occur with velocities that are too large, propagation of sur- face fields is towards the poles, and typical periods are too short.

Several authors have suggested the dynamo may operate principally at the base of the solar convection zone (e.g. /19,29/) - perhaps in the penetration region, perhaps confined there by topological pumping. Glatzmaier's latest calculations attempt to model this with a shell extending somewhat down into a subadiabatic region. In this case, the numerical difficult- ies are more extreme, and for this reason the magnetic field is only allowed to interact with convection in the lower part of the shell. In the outer part, the convective and thermodynamic aspects are solved fully, but the magnetic field is assumed to be a potential field (with no Lorentz force), and is__matched to the self-consistent field calculated lower down. The results (fig. 71 are somewhat encouraging, in that

i] the toroidal field components propagate slowly towards the equator at the base of the convection zone

Numerical Simulations of Dynamos 25

"~ ~.z- ,o ' . r.,o'. ~.~. o., r

L o . . . . . . . . . . . . . . . . . l

Tolal M~Ir, eli¢ Field

~ L

i To, oido I Field

zo t0" ~

I0(~(~0 I041~I0 10060 11440 I19~:1~ 1 2 ~ 126~I0 13~SO 13a40 14]Z0 14800 15ZeO 15760

TIME STEPS

Fig. 4. Energy traces for a dynamo computed by Gilman, for Q = 1.7. Note dimen- sional time scale for 1 year on the middle right of figure (from /24/).

,.... ;.:.~,....~..

. ,< • . •

8 1 6 0 8 3 2 0 8 4 8 0 8 6 4 0 8 8 0 0 8 9 6 0

R = 2 x l O 6, T = I O 7, P = I , Q = 1 . 5

Fig. 5. Meridian cross sections of toroidal field contours for Q = 1.5 and anti- symmetric initial toroidal field, showing evolution through half a magnetic cycle. Solid contours indicate field into the figure, dashed contours field out of figure. Note that new cycles begin near the equator and migrate towards the poles roughly parallel to the axis of rotation (from /24/).

ii] the poloidal field propagates initially towards the equator, but then stops. Glatzmaier points out that this would ultimately arrest the propagation of the toroidal field

iii] the calculations are numerically too demanding to be followed for complete cycles, but the initial tendencies are not incons~tent with a longer (22-year) cycle.

There is thus some hope that if similar calculations can be performed on the next generation of supercomputers, something resembling the Sun's behaviour will result. Ultimately, though, it ought to be the model itself which predicts just where the dynamo operates.

26 D.J . Gallowa~

it0 ~-: :. C

-90 - . ~ "

RADIAL VELOCITY

RADIAL MAGNETI C F IELD

90" ....'. ..... : ....... ' ....... '. ....... ' ....... i

LONGITUDE 760

Fig. 6. Contours of the radial components of the velocity (upper plot) and the magnetic field (lower plot) in a spherical surface just below the top boundary, for a dynamo computed by Glatzmaier. Solid (broken) contours represent outward (inward) directed fields. The zero contour is solid (from /25/).

TOROID£L MAGNETIC FIELD

Fig. 7. A sequence of toroidal magnetic field profiles (solid (broken) contours represent fields in direction of increasing (decreasing) longitude) spanning four years of simulated time, for Glatzmaier's dynamo at the base of the convection zone (from /27/).

4. PROBLE~ ~;D ~ROSPECm_S--

Why, on the whole, do the global simulations fail to reproduce the Sun? Several reasons have been suggested, principally by the authors themselves.

First, the velocity fields produced by the non-magnetic parts of the calculation may not be

N u m e r i c a l S i m u l a ~ i o n ~ o . ~ D; n , m ~ o , 27

the right ones. In reproducing the observed differential rotation, global convective 'giant cell' velocities are produced which are apparently considerably above ~e observed upper

limits; nor is there ~ny obse~zational evidence for the 'cartridGe belt' pattern (fig. 6)

which appears because ~ne stron G rotation (low Rossby n~ber) organises the flow in Taylor columns. This may be due to the excessive viscosities that the models must assume in order to function at all (V~n Ballegooijen /30/ has argued on the basis of a giant cell model that the viscosities should be at least two orders of magnitude smaller). Note that this ~iffi- culty is indirectly a question of scale separation: the model must have a high viscosity in order to avoid the for~ation of thin structures which cannot be resolved.

The second possible shortcoming is related. The process of flux expulsion leads to the for- mation of flux ropes, with a consequent suppression of motion in their neighbourhood as the

field strength approaches the equioartition value B~ = (ueDUconv~ctive)½. This process has been considered analytically (see e.g. /15/') and it is known that the structures have charac-

Rm ½. If they are not resolved properly, the ability of the fluid to teristic thickness

twist the flux may be overestimated, because the inability of the motion to penetrate the regions of strong field is not included adequately. When this happens, the effective 'twist- ing parameter' ~ (in the language of e~ dynamo theory) may be too large /24/.

The reason for the poleward propagation of fields is apparently that e(~/3r) has the wrong sign (here ~ is the rotation rate) - the computations find ~/3r > 0 over most of the zone, whereas ~w dynamos function correctly because they assume 3~/~r < 0. According to Glatzmaier

/27/, the observations support ~he computational result; furthermore he states that the effective e changes sign near the base of the convection zone, thus causing ~(5~/~r) to have the right sign for equatorward propagation there.

There is a language problem in that the results of global simulations are often expressed in spectral or statistical language, whereas other physicists talk of processes (e.g. magnetic buoyancy, topological pumping, etc.). In principle, these effects should all be present in Glatzmaier's simulations, but it is hard to identify them explicitly. For this reason we are still ignorant of the basic non-linear mechanism which limits the growth of the solar cycle - is it the 'cut-off e' effect, the global effects of the Lorentz force on the differ- ential rotation profile, instability due to magnetic buoyancy, or something else?

We may be lucky: the extra resolution if Glatzmaier runs his code on a Cray-2 may just swing the balance. But resolution cannot be increased appreciably in the foreseeable future.

To double the resolution in each direction involves 8 times as much storage, and furthermore the timestep must also be reduced by 2. So the machine requirement is 16 times as much. Even such an increase is probably still too small to separate scales effectively. When we talk about increasing the resolution tenfold, we need machines four orders of magnitude more powerful than those (some of us) have access to today.

Ultimately, it may be crucial to have better models for the small- and intermediate-scale

processes as an input to the global simulations. Such work can be done both theoretically (as in Soward's paper), and computationally, and provides scope for those without access to the largest machines. Another asset would be the development of more supple global simula- tion codes, which could compare a variety of models derived from different physical hypo- theses without consuming hundreds of Cray CPU hours. A comparison with observations of other stars would then be possible, and extremely useful for deciding between theories. Meanwhile, the task for anyone embarking on a large-scale simulation is to think of a way to avoid merely reproducing the results of Gilman and Glatzmaier.

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28 D . J . G~dlo~a~

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