The kinematic dynamo action of spiralling convective £ows

SUMMARY We consider the kinematic production of magnetic ¢elds in a sphere by velocity ¢elds dominated by di¡erential rotation and spiralling convective cells. The high magnetic Reynolds number limit of Braginsky (1964) is considered and formulae are derived allowing an a -e¡ect parametrization of such £ows to be easily calculated. This permits an axisymmetric system to be investigated in parallel with the direct 3-D numerical computations. Good agreement between the asymptotic and 3-D calculations is found. The ‘spiralling’ property typical of convective motion in rotating spheres is important in terms of dynamo action; the di¡erential rotation coexisting with this feature is also vital. Indeed, it is the presence of both features which allows the analysis of Braginsky to be employed. With £ows approximating the columnar form anticipated for rapidly rotating convection, dynamo action is relatively easily achieved for all azimuthal wavenumbers; modes of di¡ering wavenumbers interact almost by a simple superposition. With £ows of more complex latitudinal form, the mutual interactions between modes become more complicated. For columnar-type £ows, dipole magnetic ¢elds are favoured when the sense of outward spiralling is prograde and the zonal £ow is eastwards, as is physically preferred.


INTRODUCTION
The production of the Earth's magnetic ¢eld was ¢rst linked to the homogeneous dynamo problem early in this century. Initial interest focused on the kinematic part of the problem, in which a velocity ¢eld is presupposed and its capacity to excite magnetic ¢elds is investigated. After initial di¤culties were overcome and the possibility of dynamo action by some simple, idealized £ows was established, the kinematic problem has become somewhat neglected; attention has focused on the more complex magnetohydrodynamic dynamo problem involving the non-linear interaction between ¢eld and £ow. The simpler kinematic problem remains imperfectly understood, however.
Several £ows of various types have been identi¢ed in the literature as being capable of self-excited dynamo action, but there is only a partial understanding of why these £ows, and not others, have this capacity. Explanations often cite the £ows' helicity, or behaviour observed in simpler problemsö £ux expulsion by closed-streamline circulation, or the concentration of £ux by high R m £ow, for exampleö but the relevance of these features is seldom truly established. The £ows that have been investigated kinematically have almost always been of rather simple type, mainly for numerical reasons. It is therefore worrying that highly complicated models of the full dynamo problem are being investigated, while the kinematic dynamo properties of £ows of even moderate complexity have hardly been studied in isolation.
Work on the dynamic problem has meanwhile progressed largely through the`intermediate' approach, where the dynamo action of the velocity is conveniently parametrized in terms of a so-called a-e¡ect. Whilst the a-e¡ect models may constitute a valid approximation to the full dynamo process, and the intermediate studies are still necessary and useful at the present stage of knowledge, it also remains somewhat unsatisfactory that the a-e¡ect is so widely, and arbitrarily, invoked.
In this paper we consider the kinematic problem for rather general £ows, of a type that might be excited by convection in rapidly rotating systems. We make use of the theory of magnetic-¢eld generation developed by Braginsky (1964) and applied by him to the Earth's core. This theory allows axisymmetric calculations appropriate to the high magnetic Reynolds number regime to be conducted in parallel with the 3-D numerical computations. The axisymmetric calculations are carried out using rigorously derived a-e¡ects, to some extent justifying their more arbitrary use elsewhere, and, it is hoped, leading to a better understanding of the a-e¡ect as a parametrization of the inductive action of quite general 3-D £ows.

MATHEMATICAL FORMALISM
The kinematic dynamo problem has been described by many authors, so we shall discuss it here only brie£y. Roberts (1994) gives a thorough review; Sarson & Gubbins (1996;henceforth SG) describe the problem more brie£y, with speci¢c details of the numerical method used here. This method has been highly satisfactorily benchmarked by Holme (1997).
The kinematic problem is described by the induction equation for the time-evolution of a magnetic ¢eld in an electrically conducting £uid, where the magnetic ¢eld B is solenoidal, and where the velocity uöhere assumed stationary and incompressible, and so also solenoidalöis prescribed. The magnetic Reynolds number arising from the non-dimensionalization, R m~k pUL, is de¢ned in terms (in the order given) of the magnetic permeability and electrical conductivity of the £uid, and of its velocity and length scales. We solve (1) in a unit sphere, r¦1, with an insulating exterior; see SG for details. Eq.
(1) is linear in B, and so can be solved for eigensolutions B ê exp (jt), with the complex growth-rate j as an eigenvalue. We are interested in self-sustained magnetic ¢elds, and so look for solutions with Re j~0. The critical magnetic Reynolds number, R c m , at which such a solution is obtainedöif it existsöis a measure of the ability of the £ow to sustain dynamo action. Solutions are also characterized by their frequency at criticality, u~Im j.
Previous kinematic calculations (e.g. Bullard & Gellman 1954;Pekeris, Accad & Shkoller 1973;Kumar & Roberts 1975;Dudley & James 1989) have employed simple, large-scale velocities. We wish to study how di¡erent components of velocity interact in the production of a magnetic ¢eld, and assume velocities of the general form where we have introduced toroidal and poloidal vector harmonics following the Bullard & Gellman (1954) convention, in terms of the associated Legendre functions P m l (cos h). We drop the superscript`c' for the axisymmetric (m~0) components, the distinction between cosine and sine no longer being required.
The £ow (2) contains axisymmetric components of di¡erential rotation (t 0 1 ) and meridional circulation (s 0 2 ), as employed by Kumar & Roberts (1975;KR hereafter). These were retained following the conclusions of P. H.  on their importance for dynamo action in a-e¡ect models, and the con¢rmation of this property in 3-D calculations by SG. The s m l components, of both sine and cosine 0 dependence, are investigated for various combinations of l and m. The parameters simply control the relative magnitudes of the appropriate velocity components. In what follows we ¢x 0~1 , since the amplitude of the £ow is already de¢ned by R m .
Whilst it might seem limiting to restrict the £ow under consideration to non-axisymmetric poloidal harmonics, these harmonics should be the principal ones excited by buoyant convection in a system with radial gravity, so are of primary interest. Although toroidal harmonics can also potentially interact with poloidal harmonics to cause dynamo action in the high-R m regime, this possibility is not investigated in the current work.
The parameters p control the radial complexity (`number of cells') of the s m l components, allowing further freedom in the £ow prescribed. The cosine and sine variations in r introduce a sense of`spiralling' to the convective £ow, as observed in experiments (Carrigan & Busse 1983) and numerical simulations (e.g. Zhang 1992). The spiralling nature of the £ow is required for dynamo generation in the asymptotic limit of Braginsky (1964); this property is discussed in some detail in Section 5. The lowest power of r 4 appearing in s ms l (r) is, strictly, only appropriate for l¦3; to ensure regularity at the origin these functions should vary as O(r lz1 ). The forms (7)^(8) were used for all l, however, for numerical convenience. No di¤culties at the origin were encountered in practice; the weak singularity at an isolated point in the neighbourhood of which the amplitudes of all ¢elds are very low obviously has little e¡ect on the convergence of the numerical scheme.
In attempting to model the geodynamo, which generates a predominantly axisymmetric magnetic ¢eld and which is thought to be at relatively high R m , we may make use of the asymptotic analysis of Braginsky (1964). Braginsky considered the dynamo action of a system ordered as where u , B denote the axisymmetric components, and u', B' the non-axisymmetric. The axisymmetric parts are dominated by the zonal (0) components, with weaker meridional (poloidal) parts: In terms of our velocity (2), these expansions are appropriate only for 1 %1, m l %1. To leading order in R {1a2 m , the toroidal and poloidal parts of the azimuthally averaged induction equation can then be combined to give in terms of where a~1 Angle brackets denote azimuthal averaging, L 1 /L0 di¡erentiation with respect to 0 treating unit vectors as constants, and a hat indicates the inverse operation of inde¢nite integration with respect to 0. The subscript`p' denotes the`poloidal' part of a quantity [more strictly the meridional part, which can be identi¢ed with the poloidal component de¢ned by (4) only in the axisymmetric case]. The subscript`e' refers to the`e¡ective' variables, introduced as de¢ned above so that eq. (11) reduces to the simple form given, identical to that for a true axisymmetric magnetic ¢eld, but with the generation term involving a now present. For R m &1, expression (9) requires solutions to be dominantly axisymmetric; we refer to this state as the`nearly axisymmetric', or Braginsky, regime. The purely axisymmetric nature of the asymptotic limit facilitates detailed numerical surveys.
For a velocity of the form (2), with u'~ ms l s ms l z mc l s mc l (16) (considering only a single harmonic initially), the asymptotic prescriptions (13), (14) give in terms of f 3 (r)~(p ms' l p mc l {p mc' l p ms l )/r 2 , where p mcas l~s mcas l /t 0 1 and the primes denote di¡erentiation with respect to r. Here " 1~1 R m is a ¢nite quantity arising in the R m ??, 1 ?0 limit, following from expansion (10); likewise " ms l " mc l~ ms l mc l R m is ¢nite, from (9). Expressions (18) and (19) are given for Schmidt-normalized spherical harmonics, which we use throughout.
The above prescriptions can be used for any choice of l and m of interest. u ep can be expanded as a series of terms u ep~ 2l l'~2,even with the constants S l' determined from expression (18). Similarly we can rewrite (19) in the form Values of the coe¤cients S l' , A (1) l' , A (2) l' , for certain values of l and m, are given in Table 1. In the asymptotic limit the quantity " ms l " mc l appearing in the a-term can be taken as the critical parameter for dynamo excitation. Asymptotic calculations carried out for speci¢ed values of " 1 and " ms l " mc l approximate the 3-D state with R m~" ms l " mc l /( ms l mc l ) , for any chosen value of ms l mc l . The magnitude of ms l mc l determines the strength of the non-axisymmetric component, and hence thè distance from the asymptote'. The correspondence between the asymptotic and 3-D cases is described in more detail in SG and in KR, the latter work detailing a suite of calculations validating the procedure. Because of the non-linear nature of transformation (25), care must be taken in extrapolating conclusions from the Braginsky limit to the 3-D case. " 1 does not scale directly to 1 but depends upon R m also, and thus a ¢xed " 1 does not correspond to a ¢xed 1 , except in the case 1~0 . From the expansions (23) and (24) it is evident that u ep is symmetric and a anti-symmetric with respect to the equatorial plane. This property permits magnetic ¢elds to be either equatorially symmetric (quadrupolar) or anti-symmetric (dipolar), consistent with the symmetries permitted by the full 3-D £ow.
When two modes of di¡ering m are present, their individual contributions to the asymptote may simply be added; the prescriptions (12)^(14) do not provide any intermode interaction in this case. To conduct asymptotic calculations, we need then only ¢x the relative strengths of the two modes, m1s l1 m1c l1 /( m2s l2 m2c l2 ).   When £ow of a given azimuthal mode m has a more complex latitudinal structure than the single l harmonic considered above, the interaction between the various harmonics l 1 =l 2 must be taken into account. For we obtain the following terms, in addition to the terms arising from (18) and (19) for each l i individually: The terms are analogous to the earlier terms in (18)^(22); F i 1 (h) and f i 1 (r) are obtained from F i (h) and f i (r) by interchanging l 1 and l 2 . We can expand these as with S l' (r) and A l' (r) de¢ned as before. For (l 1 zl 2 ) odd, u ep is no longer symmetric with respect to the equatorial plane and a is no longer anti-symmetric, thus eliminating the possibility of purely dipolar or quadrupolar magnetic ¢elds. The relevant constants in the expansions of u ep and a are given in Table 2 for the interaction of an s 2 2 velocity with other m~2 harmonics with l¦6. The additional terms now also appear in the de¢nition of a. These latter terms can be treated as before; they vanish, however, if " ms l1 " mc l2~" mc l1 " ms l2 . As this is a reasonable simpli¢cation with which to begin studies, we restrict ourselves to this case in the calculations reported here.
Toroidal harmonics can also interact with poloidal harmonics to cause dynamo action in the high-R m regime. This possibility is not investigated in the current work.

FLOWS OF VARYING AZIMUTHAL MODE
Of considerable interest to the picture of`columnar' convection anticipated from the hydrodynamics of rapidly rotating systems (e.g. Busse 1970) is the selection and interaction of modes of varying azimuthal wavenumber m. The choice of m is a matter of even greater interest in magnetohydrodynamics, given the con£icting tendencies involved, with rotation favouring high m but magnetic constraints preferring smaller values. Magnetoconvection studies (e.g. Busse 1976Busse , 1983Fearn 1979a,b;Longbottom, Jones & Hollerbach 1995;Zhang 1995) suggest that for Elsasser numbers "~pB 2 /(2)o)öwhere B is a measure of the magnetic ¢eld strength, ) is the rotation rate, and o the £uid densityöin the range 1¦"¦10, as estimated for the geodynamo, m`O(10) is to be expected.
Within our formalism, we can approximate a columnar form for the £ow of wavenumber m by restricting the latitudinal structure to be sectorial (i.e. of degree l~m). In this section we restrict ourselves to such £ows and superpositions thereof. We will consider £ows of more complex form, with latitudinal structures l b m, in Section 4.
Considering only even m and starting our study, for numerical convenience, with the lowest m that allow multiple-mode interaction, we consider the special case of the velocity ¢eld (2) given by where we have adopted a simpli¢ed indexing of the parameters, consistent with the studies of KR and SG. The radial functions given by (5)^(8) are employed, but with an additional parameter, u, introduced to allow a phase-shift in 0 between the m~2 and m~4 modes: s 2s 2 (r)~r 4 (1{r 2 ) 2 cos p 1 r , s 2c 2 (r)~r 4 (1{r 2 ) 2 sin p 1 r , s 4s 4 (r)~r 4 (1{r 2 ) 2 (cos p 2 r cos u{ 5 / 4 sin p 2 r sin u) , s 4c 4 (r)~r 4 (1{r 2 ) 2 ( 4 / 5 cos p 2 r sin uz sin p 2 r cos u) .
When u~0, the functions given in Section 2 are regained. Table 2. Signed squares of the coe¤cients in the expansions of u ep and a, for the interaction of an s 2 2 £ow with other s m l components with m~2 and l¦6.  The forms of a and u ep produced by this £ow in the Braginsky limit are shown in Fig. 1; as well as showing the individual contributions from the m~2 and m~4 components, this ¢gure also shows the contributions from an m~6 mode, to illustrate the varying form of these quantities with increasing wavenumber.
Critical curves for the onset of dynamo action can easily be calculated in the asymptotic limit, and extrapolated to give approximations of the true 3-D behaviour, as described in Section 2. Since the 3-D calculations are more time-consuming to perform, the basic behaviour of the velocities is, wherever possible, investigated through such extrapolated results. 3-D calculations are described here only to illustrate individual solutions, or to investigate features (such as the phase-shift, u, introduced above) which cannot be investigated in the asymptote. More extensive 3-D calculations have been carried out, however, con¢rming the extrapolated results, and the validity of the asymptotic approximation in general. Failure to obtain a convincing 3-D solution in conjunction with the asymptotic result occurred only in very high R m cases, where the 3-D calculation was clearly breaking down due to inadequate numerical resolution. Fig. 2 shows critical curves for the m~4 mode in isolation, detailing solutions of both dipole and quadrupole symmetry. There are two notable features in the behaviour exhibited. First, the preference is for dipole solutions to be excited for R m b 0, quadrupole for R m`0 . (Negative R m simply constitutes a reversal in the sense of the velocity.) For these simple £ows this can be related to a change in sign of the product ms l mc l (determining the sense of spiralling of the convective cells) or of 0 (the sense of zonal £ow), with a simultaneous reversal of 1 (the meridional circulation). (See SG for a discussion of the lack of independence of the signs of the various i parameters and of R m .) Second, meridional circulation (controlled by 1 ) strongly in£uences the ease of excitation and the time dependence of the preferred solution. Both of these features are consistent with previous studies of a-e¡ect dynamos (e.g. P. H. . Fig. 2 also compares closely with results for the m~2 £ow studied in detail by SG. This is reasonable given the similar asymptotic forms arising for sectorial modes of di¡ering m (see Fig. 1). The axisymmetric magnetic ¢elds produced by the two modes are also similar. The most notable di¡erence between the two wavenumbers is an approximate`o¡set' in the 1 axis; 1~0 favours an oscillatory solution for m~2, a stationary solution for m~4, for example. This can be explained by the relative strength of the e¡ective meridional circulation, u ep , in the two cases (compare the coe¤cients in Table 1); this quantity acts analogously to the true meridional circulation. The importance of this feature, ¢rst highlighted by SG, is therefore con¢rmed here. Fig. 3 shows the critical curves with the m~2 and m~4 modes simultaneously present. The basic pattern is the same, with the detailed behaviour lying somewhere between the two individual cases. The lower values of R c m now obtained show that both modes are contributing towards dynamo action. That they should combine so e¡ectivelyöalmost as a simple addition of the e¡ectsöis not  surprising, however, given the similar way in which they act to generate magnetic ¢elds, re£ected in their similar asymptotic forms and their similar behaviour in isolation.
In the case considered above, the senses of spiralling of the two individual modes, controlled by the signs of 2 3 and of 4 5 , are the same. When the two modes spiral in the opposite sense, so that the product 2 3 4 5 is negative, dynamo action is impeded, as the higher values of R c m shown in Fig. 4 evince. This is not inconsistent with the e¡ects of the two modes being broadly superimposed, however. In this case the two individual a distributions in the asymptote are oppositely signed, and would in isolation excite magnetic ¢elds of di¡ering symmetry (dipole versus quadrupole). They might thus be expected to combine negatively in terms of dynamo action, and to excite ¢elds rather di¡erent from those obtained above. This is indeed observed.
The radial dependence of the two modes has hitherto been ¢xed in the form (46)^(49), with p 1~p2~3 n. As a preliminary investigation of the importance of this factor, various alternative p i were considered. Their e¡ect was found to be rather slight, however, with qualitatively identical behaviour being obtained in all cases, the variation being consistent with that obtained by SG upon such radial variations in the m~2 velocity alone. This was not unexpected, since the`cellular' velocity structure was chosen to allow eqs (13) and (14) to assume smooth large-scale forms for all values of p i . A more thorough investigation of the importance of the radial structure should consider modes without the imposed cos p i r/sin p i r form. Such a study has not yet been undertaken, although this point is further addressed in Section 5.
The most easily excited critical curves detailed above, obtained for the asymptotic system, have been con¢rmed by direct 3-D calculations. Signi¢cant discrepancies between the asymptotic and 3-D cases were found only at the higher values of R c m , where solutions of more complex ¢eld morphology required a ¢ner resolution than was available for the 3-D computations. The clear manner in which the 3-D numerical resolution deteriorated as these solutions were approached (through a continuous variation in parameters from well-established solutions), and the good agreement obtained in all other cases, give us no reason to doubt the asymptotic results.
In all cases the behaviour of the system could be understood through recourse to the asymptotic system, with dynamo action depending upon the distributions of a and u ep arising there in a relatively straightforward way. The presence of several modes of varying m is not in itself detrimental to dynamo action. The various modes can in fact be combined almost as a simple superposition. If the sense of spiralling of the two convective modes is the sameöthat is if 2 3 4 5 is positive, for the speci¢c velocity (45)öthen dynamo action can be obtained at lower R c m than for either mode alone. If the opposite is the case, however, the two modes can interact negatively, resulting in a higher R c m and a more complex magnetic ¢eld. Results consistent with all of the above have also been obtained with the addition of an m~6 £ow.
A 3-D dipole solution obtained with both m~2 and m~4 components of £ow present is shown in Fig. 5. The axisymmetric ¢eld (Figs 5a and b) is comparable with that obtained for the KR £ow (see KR or SG); the radial ¢eld (Fig. 5c) shows a somewhat more complicated non-axisymmetric structure, but the ¢eld remains dominantly large scale. The numerical convergence of this solution, and subsequent 3-D solutions shown, is addressed in Appendix A. The compatibility of modes of varying m in exciting magnetic ¢elds is not signi¢cantly a¡ected by an azimuthal phase-di¡erence u between the modes, as can be seen from Fig. 6. The di¡erence in R c m obtained for varying u is only of the order of 5 per cent. This value achieves a maximum when the strengths of the two modes are comparable, as is the case for the ¢gure shown; in other cases one mode acts as a relatively minor perturbation to the second, and the e¡ect is diminished. Dynamo action is slightly favoured when the phase di¡erence is such that equatorially outward £ow is concentrated in two large cells and two small cells, rather than in four equally sized cells.

MORE COMPLEX CELLULAR FLOWS
A convective £ow may be dominated by a given azimuthal wavenumber m, yet possess signi¢cant latitudinal structure within this mode. Whilst the preceding section restricted itself to non-axisymmetric £ows of purely sectorial (l~m) form, this section brie£y considers the e¡ect on dynamo action of the inclusion of components of £ow with l b m, such modes inevitably being present in any realistic convective velocity.
A simple special case of the velocity ¢eld (2) is again considered, u~ 0 t 0 1 z 1 s 0 2 z 2 s 2s 2 z 3 s 2c 2 z 6 s 2s l z 7 s 2c l , for various choices of l b 2. The radial functions (5)^(8) are retained. The prescriptions (27) and (28) allow us to incorporate the interaction between components of di¡ering degree into the axisymmetric parametrization, and the asymptotic results again agree satisfactorily with direct 3-D computations. The addition of an l~4 structure to the m~2 £ow is ¢rst considered. This composite £ow retains the equatorial symmetry of the original, avoiding additional computational requirements. The distributions of a and u ep arising in the Braginsky limit are shown in Fig. 7. Critical curves obtained for the s 2 4 mode in isolation are shown in Fig. 8; results for the combined £ow, with s 2 4 and s 2 2 modes both present, are given in Fig. 9.
The patterns of behaviour obtained with varying 1 di¡er signi¢cantly from those obtained for the purely sectorial m~2 £ow. This was to be anticipated, however, given the more complex latitudinal structures now present, evident in the asymptotic forms as well as in the original 3-D £ow. The magnetic ¢eld morphology produced is also somewhat more complex, as can be seen in Fig. 10, obtained from a 3-D calculation with both components of £ow present. In comparison with Fig. 5, the toroidal £ux now exhibits a Figure 7. The asymptotic forms of a (top row) and of the streamfunction of e¡ective poloidal £ow, u ep (bottom row), associated with the interaction of the velocities s 2 2 and s 2 4 . The left column shows the terms arising from s 2 4 in isolation (those for 2 2 are shown in Fig. 1), the centre column those arising purely from the interaction; the right column shows the net form, assuming 2 3~6 7 . sign-change in each hemisphere. The simultaneous presence of both modes need not inhibit dynamo action, however; the global minimum of R c m is slightly smaller than that obtained with either mode in isolation. Similar calculations carried out with the addition of an l~6 component of £ow con¢rm this behaviour. Calculations with such higher modes in isolation show that the magnetic ¢eld structure can change signi¢cantly as the di¡erence l{m is increased, however. When several such modes are combined we might then anticipate an e¡ect similar to that obtained in Section 3 for negative values of 2 3 4 5 ; although each component can in isolation excite magnetic ¢elds, their simultaneous presence can inhibit dynamo action.
When modes of even and odd l are mixed, the symmetry of £ow which permitted separate dipole and quadrupole solutions is broken, and ¢elds asymmetric with respect to the equator are produced. The only such case we have investigated thoroughly is that given by (50) with l~3. The contributions to a and u ep arising for this £ow in the Braginsky limit are shown in Fig. 11. The results for the s 2 3 mode in isolation, and with both s 2 3 and s 2 2 modes present (only the latter case breaking the dipole/quadrupole dichotomy) are given in Figs 12 and 13 respectively.
The behaviour with varying 1 again di¡ers signi¢cantly from that obtained for the sectorial £ows considered in Section 3, stationary solutions now being preferentially excited at all parameter values considered. For the s 2 3 mode in isolation (Fig. 12), the importance of the sign of R m öand hence of the relative senses of spiralling and of zonal £owöto the preference for dipole or quadrupole modes can still be seen, however.
The s 2 3 velocity is no less capable of dynamo action than the sectorial £ow. Nor does the interaction of the two modes inhibit dynamo action, despite their opposing equatorial symmetries. Furthermore, the magnetic ¢elds excited by the composite £ow need not di¡er totally from those considered previously. Fig. 14 shows the asymmetric ¢eld obtained for such a velocity; the ¢eld remains large scale and well resolved, and in some respects comparable with that of Fig. 5.
The introduction of non-sectorial components of non-axisymmetric £ow need not inhibit dynamo action, therefore, at least not for the relatively small values of l{m considered here. The Braginsky limit system remains a good approximation for the dynamo action of these more complex 3-D £ows, and therefore remains a useful tool for exploring and comprehending their behaviour. The dynamo action of the non-sectorial £ows can be rather di¡erent from that considered previously. In all cases, however, the overall topology of £owöfor example the senses of spiralling and of zonal £owöremains an important determinant for the type of magnetic ¢eld excited.

DISCUSSION
In using the forms (7) and (8) for the non-axisymmetric components of £ow we have introduced a tilt to the radial motion. Although velocity ¢elds of this form di¡er considerably from the actual solutions for thermal convection in a rotating sphere, they exhibit some similarities for the cases of small l{m which we have emphasized in our numerical study. The tilt describes a variation of the azimuthal phase with radius, which mimics the variation of azimuthal phase with distance from the axis shown by convection columns. For ms l~ mc l~A , for example, the non-axisymmetric £ow associated with (7) and (8) can be described by s m l (r, 0)~Ar 4 (1{r 2 ) 2 sin (m0zpr) , making this phase relation explicit; the parameter p can then be directly linked to the strength (and sense) of the tiltedness. Although such tiltedness was not included in the original analyses of convection in rotating spheres by Roberts (1968) and Busse (1970), it was soon recognized as an important feature of convection in connection with the generation of mean zonal £ows (Gilman 1976;Busse & Hood 1982;Busse 1982). The spiralling of the columns was also demonstrated in the experiment of Carrigan & Busse (1983) and becomes dramatically enhanced with increasing Taylor numbers, as shown, for instance, in the computations of Zhang (1992).
In the present context the tilting of the non-axisymmetric cells of motion is a necessary ingredient of dynamo action. To make this point we generalize the radial function (8) to s mc l (r)~r 4 (1{r 2 ) 2 sin (przp 0 ) .
This expression reduces to the original form for p 0~0 , whilst in the case p 0~n /2 the tilt disappears and planes of symmetry of constant 0 are realized. Results for various values of p 0 are given in Table 3 for the s 4 4 velocity with 1~0 , 4~5~0 X04. Dynamo action could not be obtained for the case p 0~n /2. The table also shows the expected 1/ cos p 0 variation for the critical R c m in the asymptotic limit, since only the sin pr component of (52) interacts with the s ms l term in that limit. The good agreement with the predicted asymptotic variation shows that the`spiralling component' of the £ow is the essential one for magnetic ¢eld generation.
The importance of the parameter p for dynamo action was explained originally (Kumar & Roberts 1975) in terms of the`number of cells in radius', in analogy with Cartesian periodic dynamos (e.g. G. O. . Here we suggest that it is perhaps better Figure 9. Critical stability curves R c m as a function of 1 , from extrapolations of the asymptotic system with l~4, p 1~p2~3 n, 2~3~6~7~0 X04, from calculations with 40 grid points, truncated at degree 14. Details of the plot as for Fig. 2. interpreted in terms of the tiltedness of the £ow. It is worth noting though that whilst the absence of tilt prohibits dynamo action in the Braginsky limit for the simple l~m case investigated above, it need not necessarily do so when modes with l b m are present. The terms given by (42)^(44) then allow for non-zero a, if " ms l1 " mc l2 =" mc l1 " ms l2 . Flows that resemble convection in rotating spheres should not be stationary, as we have prescribed, but rather drift with respect to the rotating reference frame. A uniformly drifting £ow can be modelled simply by the addition of an overall solid body rotation; by augmenting t 0 1 (r) by jr 2 in (5), we e¡ectively solve for the ¢eld and £ow in a system co-rotating at angular velocity {j. The addition of such a drift was ¢rst considered by KR, who found the e¡ect to be slight. This was also the case for the other £ows investigated here. Although this technique restricts us to a single drift rate applying uniformly to all components of £ow, the result of Section 3, that an azimuthal phase di¡erence between modes does not greatly a¡ect dynamo action, suggests that the same conclusion should also apply for more complexly vacillating £ows.
It is worth pointing out that the critical values of R m associated with these £ows need not be particularly high, despite their connection with the high-R m theory of Braginsky. Solutions with R c m *O(100) can be attained by using stronger non-axisymmetric £ows than those detailed to date. Table 4 shows the variation of R c m with 4~5 for the l~m~4 £ow. Here we have also calculated a normalized critical magnetic Reynolds number, R c m 1 öusing the root mean square velocity, u rms , as the velocity scale Uöas a more suitable measure of the true ability of these £ows to excite dynamo action. [The amplitude of our velocities was previously ¢xed by 0~1 in (2).] Thus these solutions need not be considered inapplicable to the geodynamo, where an R m of O(100) is to be anticipated (e.g. Gubbins & Roberts 1987).
As the strength of the non-axisymmetric £ow is increased in Table 4, we move further from the Braginsky asymptote, of course. This can be seen by comparing R c m with values extrapolated from the asymptotic calculation via (25), R c m (extX). (The relevant critical Table 3. R c m with varying p 0 , for the s 4 4 convective £ow with p~3n, 1~2~3~0 , 4~5~0 X04, with 150 radial grid points and all relevant harmonics out to degree and order 20. The bottom line shows the variation expected from the asymptotic theory. value of " 4 " 5 in the asymptote is 4.336.) Even at 4~5~0 X04, the extrapolated value di¡ers by 16 per cent from the true value; this deviation increases with the strength of the non-axisymmetric £ow. For the higher values of 4~5 considered, the validity of the asymptotic approximation is clearly breaking down, with R c m increasing with these parameters, at odds with the asymptotic theory. This is hardly surprising, as the strength of the non-axisymmetric £ow, re£ected in the variation of u rms , is clearly now violating the  Figure 11. The asymptotic forms of a (top row), and of the streamfunction of e¡ective poloidal £ow, u ep (bottom row), associated with the interaction of the velocities s 2 2 and s 2 3 . The left column shows the terms arising from s 2 3 in isolation (those for s 2 2 are shown in Fig. 1), the centre column those arising purely from the interaction; the right column shows the net form, for 2 3~6 7 .
Braginsky regime scaling (9). Despite the relatively large discrepancies in R c m , however, the axisymmetric magnetic ¢eld morphology remains similar to its asymptotic form up to rather high values of 4~5 ; the asymptotic theory thus remains useful for these £ows, even if it is no longer quantitatively accurate.
Since the asymptotic system cannot entirely explain the variation in dynamo action for these velocities with signi¢cant nonaxisymmetric parts, it is of interest to consider alternative measures which might be of use. Previous workers have tried to relate the ability of £ows to act as dynamos to the helicity, h~u . v, where v~+|u is the vorticity (e.g. Nakajima & Kono 1991;Love & Gubbins 1996); this quantity is important for magnetic ¢eld generation in the two-scale theory of Steenbeck, Krause & RÌdler (1966). The helicity has most commonly been considered in a volume-integrated form. Here we consider a normalized form of this quantity, H~ u . v dV /(u rms u rms ) .
(53) Figure 12. Critical stability curves R c m as a function of 1 , from extrapolations of the asymptotic system with l~3, p 2~3 n, 2~3~0 , 6~7~0 X04, from calculations with 40 grid points, truncated at degree 14. Details of the plot as for Fig. 2. Figure 13. Critical stability curves R c m as a function of 1 , from extrapolations of the asymptotic system with l~3, p 1~p2~3 n, 2~3~6~7~0 X04, from calculations with 40 grid points, truncated at degree 14. Details of the plot as for Fig. 2. The integral is carried out over a single hemisphere (the northern) since the localized helicity is equatorially anti-symmetric. Table 4 shows this quantity with varying 4~5 , and also shows the anticipated variation of R c m with the latter parameters, assuming a linear relationship between H and R c m and taking the value at 4~5~0 X04 as a reference. Although the relationship appears reasonable for the smallest values of 4~5 , the agreement across the whole range is poor. Indeed, no convincing relationship between R c m and H could be established for all values of 4~5 considered, given that the helicity remains monotonic in these parameters, whereas R c m does not. This is perhaps not surprising; 3-D dynamo action is a complicated process, and no single quantity, no matter how useful in certain cases, should be expected to describe it in general.

CONCLUSIONS
For £ows of the type considered, dominated by strong zonal £ows and with spiralling convective cells, the high R m asymptotic limit of Braginsky (1964) provides a good approximation for the production of a magnetic ¢eld by a large variety of otherwise diverse 3-D velocities. This allows axisymmetric calculations to be made in parallel with the 3-D calculations, greatly reducing the computational workload involved in surveying such £ows for dynamo action. Only for £ows deviating strongly from the Braginsky regime scalings does the asymptotic system fail to model the 3-D behaviour qualitatively.
For £ows of relatively simple latitudinal structure, resembling the columnar form preferred by convection in rapidly rotating systems, dynamo action is relatively easily attained for all azimuthal wavenumbers m. The form of magnetic ¢eld excited does not vary greatly with wavenumber. The interaction between modes of di¡erent wavenumber is correspondingly straightforward, and can almost be viewed as a simple superposition. Whilst our calculations show that modes spiralling in opposite senses can interact to the detriment of dynamo action, it is reasonable to expect that the sense of spiralling of a real convective £ow be determined by physical factors, such as the Coriolis force and the boundary curvature, which act similarly for all wavenumbers. For £ows of more complicated latitudinal form, dynamo action is still relatively easily achieved; the resultant magnetic ¢elds can be quite di¡erent in structure, however, and the interaction between di¡ering modes can therefore be quite complex.
Both the strong zonal £ows and the spiralling structure of the convective cells are important for the dynamo action of all these velocities. These features correspond closely to those demonstrated by Zhang (1992) to dominate in a numerical study of convection in the rapidly rotating, moderately low Prandtl number regime appropriate to the Earth. The type of magnetic ¢eld preferentially excitedödipolar or quadrupolaröcan be linked to the senses of the spiralling and of the zonal £ow. In this respect it is also encouraging that the prograde spiralling and prograde (eastwards) zonal £ows found in most numerical simulations of spherical convection are of the sense required for the preferential excitation of dipolar magnetic ¢elds.
The importance of meridional circulation for the dynamo action of these £owsöanticipated from the au models of P. H. , and previously noted in 3-D by SGöis once more apparent. Mechanisms which can induce such circulation are therefore of some interest for the geodynamo; Ekman suction is one obvious candidate (e.g. Fearn 1994). The important role played by the e¡ective meridional circulation arising from the non-axisymmetric convection, considered in detail by SG, makes this an issue of secondary importance, however.
In cases where dynamo action is obtained only at very high values of R c m öwhere the resultant magnetic ¢eld is typically of rather complex morphologyönumerical di¤culties are often encountered in the 3-D calculations. Such cases are particularly common for composite £ows, when the various components would in isolation produce magnetic ¢elds of rather di¡erent form. That such numerical problems are encountered even for the simple idealized £ows employed here highlights the need for a more complete understanding of these kinematic processes, before more complicated 3-D dynamical calculations can be con¢dently undertaken. The simpler asymptotic system remains well-behaved in such cases, however, and might usefully be employed as a diagnostic tool, allowing di¤culties of this nature to be identi¢ed. Velocity ¢elds which cause numerical di¤culties are often characterized by spatially complex distributions of a and u ep in the asymptotic limit, for example.
Further work in elucidating the inductive action of a wider range of velocities is required, however. General non-axisymmetric toroidal £ows should be incorporated into the analysis. More complex forms of di¡erential rotation should also be considered, introducing the further possibility of dynamo action in layers of concentrated regeneration, as elucidated by Braginsky (1964). The simple velocity ¢elds arbitrarily employed might then be replaced by speci¢c £ows arising from convective models. Although this has not yet proven practical, the kinematic conclusions obtained with our idealized velocities remain useful in understanding the general mechanisms occurring in magnetic ¢eld excitation, and in diagnosing some of the di¤culties arising in this process.

ACKNOWLEDGMENTS
This work was begun when GRS was at the UniversitÌt Bayreuth, supported by a grant from the EC`Non-linear phenomena and complex systems' network. He is currently supported by UK PPARC grant GR/K06495.