zotero/storage/YZDESLVH/.zotero-ft-cache

Geophys. J. Int. (2008) 173, 79–91

doi: 10.1111/j.1365-246X.2007.03707.x

GJI Geomagnetism, rock magnetism and palaeomagnetism

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Implication of kinematic dynamo studies for the geodynamo
David Gubbins
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK. E-mail: gubbins@earth.ac.uk
Accepted 2007 December 3. Received 2007 December 1; in original form 2006 October 3
SUMMARY In the kinematic dynamo problem Maxwell’s equations are solved for the magnetic ﬁeld given a prescribed ﬂuid velocity. Although no dynamic equations are involved, it does provide an accurate link between the magnetic ﬁeld and ﬂuid velocity and can therefore be used to infer something about the ﬂow underlying the observed geomagnetic ﬁeld. In this sense it complements the commonly used frozen-ﬂux theory for inverting secular variation for core ﬂow, in which electrical diffusion is neglected, and can be used to show up some of the strengths and weaknesses of the frozen-ﬂux approximation. It might be thought that kinematic models have been superceded by dynamic models that include the momentum and heat equations, but this is not the case. Even the biggest numerical simulation cannot approach the correct parameters for the Earth’s core, and the classes of ﬂows that result are, in fact, quite restricted as well as being too complex for simple physical interpretation. A variety of simple ﬂows have been studied for dynamo action; of particular interest here are a broad class of ﬂows, based loosely on extensions of the early simple choice of Bullard & Gellman and to some extent representative of what might be generated by convection in the Earth’s rapidly rotating core: this paper reviews the implications of the solutions for geomagnetism. The non-axisymmetric ﬂows mimic convection rolls in a rotating sphere, the axisymmetric poloidal ﬂows describe meridional circulation, a likely secondary ﬂow, and the axisymmetric toroidal ﬂow is a simple differential rotation. Helicity, which seems to be important for dynamo action, is related to spiralling of the rolls. Dipole, rather than quadrupole, ﬁelds are preferred when spiralling is eastward and differential rotation westward at the surface. Magnetic ﬂux tends to be concentrated at stagnation points of the ﬂow, and dynamo action fails when this concentration becomes so intense that steep gradients develop so as to enhance energy loss by diffusion. On the surface these stagnation points are centres of downwelling that concentrate vertical ﬂux. Flux concentration plays an important role in determining the magnetic ﬁeld’s symmetry: for example, concentration of vertical ﬂux in high latitudes favours dipolar ﬁelds while concentration near the equator, where dipole symmetry requires a change of sign, favours quadrupole symmetry. Steady ﬂuid ﬂow can only produce steady or oscillatory solutions to the kinematic problem at onset of instability. Steady solutions are preferred in three dimensions, in contrast to predictions of axisymmetric mean-ﬁeld dynamo theory, where oscillatory solutions are the most common. Oscillatory solutions are preferred when poloidal and toroidal ﬁeld coincide, which is unlikely to happen in the Earth because poloidal ﬁeld is concentrated at high latitudes around the tangent cylinder while the toroidal ﬁeld is probably strongest in mid-latitudes. Geomagnetic reversals are not oscillatory in nature and, therefore, require time-dependent ﬂow, but kinematic examples show that only a tiny change in ﬂow is needed to produce a realistic geomagnetic reversal. Linear modes of the induction equation of all symmetries are beginning to guide work on the dynamics of the geodynamo.
Key words: Dynamo: theories and simulations; Geomagnetic excursions; Reversals: process, time scale, magnetostratigraphy.

1 INTRODUCTION The geomagnetic ﬁeld is generated by a dynamo process operating in the liquid iron core. We advance our understanding of this process
C 2008 The Author Journal compilation C 2008 RAS

by studying simple models of convection of electrically conducting ﬂuid in a rapidly rotating sphere surrounded by an electrical insulator. Although the basic features of the geomagnetic ﬁeld seem simple—a steady dipole aligned with the spin axis—the dynamo
79

80 D. Gubbins
theory is not simple and mostly involves numerical modelling. Complexity is necessitated because a homogeneous sphere of conducting ﬂuid must regenerate magnetic ﬁeld by the same mechanism as an engineering dynamo, which is multiply connected with a complicated geometry. Several results, such as Cowling’s theorem which states that no dynamo can generate a magnetic ﬁeld that is symmetric about an axis, can be lumped roughly into the statement that ‘nothing simple works’. This is unsatisfactory: theories should be elegant but we persist with the dynamo theory because ‘nothing else works’.
Early studies focused on the basic question of whether any homogeneous sphere of ﬂuid can generate a magnetic ﬁeld by addressing the kinematic problem in which the ﬂuid velocity is speciﬁed and Maxwell’s equations are solved for the magnetic ﬁeld (Elsasser 1946; Bullard & Gellman 1954; Backus 1958; Herzenberg 1958). Dynamo action is declared if the magnetic energy persists indeﬁnitely. In practice, this usually involves solving for the fastest growing mode of the linear induction equation in the form B exp pt and searching for (p) ≥ 0. Recently, Livermore & Jackson (2004a) have introduced a different approach based on the transient growth of magnetic energy, M. If M decays with time initially for all candidate magnetic ﬁelds the system is stable; an initial growth of M may indicate instability or a transient that ultimately decays. Thus (p) < 0 is a necessary condition for stability while dM/dt > 0 is a necessary condition for instability. The distinction becomes important when considering time-dependent ﬂows or turbulent ﬂows in which only the statistical properties of the ﬂow are known. The eigenvalue method for determining linear stability breaks down in these cases (except in the case of time-periodic ﬂows, see Section 3.3.2) but the energy method still gives information about possible dynamo action. Transients that grow initially but then decay under induction by one steady ﬂow may continue to grow when the ﬂow changes at a later time, giving dynamo action when linear stability analysis of individual snapshots of the ﬂow ﬁnd only decaying modes at every stage. Backus’ early dynamo worked in this way (Backus 1958), but the idea of using time-dependent ﬂows was not revisited until recently.
Progress on the dynamics was made through simpliﬁed studies of rotating convection, ﬁrst without and then with an imposed magnetic ﬁeld, avoiding the need for the system to generate its own magnetic ﬁeld (e.g. Roberts 1968; Busse 1970; Zhang 1999). Eventually computer power grew to the point when it became possible to explore fully self-consistent dynamo models by solving the coupled equations of momentum, heat, and induction (Zhang & Busse 1990; Glatzmaier & Roberts 1995; Olson et al. 1999; Jones 2000). These models involve disarmingly few input parameters: measures of the heating (Rayleigh number), rotation (Ekman number), diffusion (Prandtl numbers) and the boundary conditions, but Earth-like values remain beyond foreseeable computer power. Even simpliﬁed choices of parameters yield magnetic ﬁelds that are quite Earthlike; unfortunately they are also chaotic with considerable spatial complexity and are, therefore, hard to understand.
In this paper, I review results for the kinematic dynamo problem and their implications for the geodynamo. This return to kinematics needs some justiﬁcation in view of the current success of dynamic models. There are a number of advantages.
(i) It complements work on core ﬂows using the frozen ﬂux hypothesis. The induction equation connects the magnetic ﬁeld to the ﬂuid velocity in the core. This is central to using geomagnetic observations to infer core ﬂow and understand the dynamics. The usual approach to ﬁnding core ﬂow is to neglect diffusion and use Alfve´n’s

theorem of frozen-in magnetic ﬁeld to determine core motion from secular variation (Roberts & Scott 1965). This approximation, almost by deﬁnition, eliminates the slow, steady ﬂows responsible for generating the magnetic ﬁeld because the dynamo is a balance of diffusion and advection (Love 1999). The frozen-ﬂux approximation is valid only if the ﬂow varies on a timescale that is longer than that of the magnetic ﬁeld but shorter than the diffusion time (Braginsky & Le Moue¨l 1993; Gubbins & Kelly 1996; Gubbins 1996). Diffusion must be included if we are to determine the component of core ﬂow that remains steady for a magnetic diffusion time (25 kyr) or longer, and probably for some historical secular variation events as well.
It is now possible to examine the accuracy of the frozen ﬂux approximation using synthetic SV data obtained from dynamic models; Rau et al. (2000) have done this and found the method reproduced the surface ﬂow adequately; Aubert et al. (2007) have also used output from dynamic dynamo calculations in a core motion study. This does not mean, however, that the method will work for the Earth. Current geodynamo models are a long way from reality because they employ parameter values that are very different from those of the Earth. While they reproduce some aspects of geomagnetic behaviour surprisingly well, they do not reproduce all behaviour faithfully. Most importantly in the present context they rarely, if ever, produce the kind of ﬂux expulsion now underway in the Southern Hemisphere, which is the very feature showing present departure from frozen ﬂux in the Earth. This restriction on the parameter values that can be explored numerically produces a rather restricted class of ﬂows that may be unrepresentative of the Earth. The key to validity of frozen ﬂux lies in having a rather precise set of timescales, with the ﬁeld changing rather rapidly on the diffusion timescale. Dynamic models are necessarily run at high Ekman number and high Rayleigh number, putting them into a chaotic regime that would favour frozen ﬂux. Since only the induction equation is involved in determining the evolution of the ﬁeld and its inversion for core ﬂow by frozen ﬂux, all the dynamic equation does is to restrict the possible ﬂow. It is better, therefore, to examine the validity of frozen-ﬂux theory using a kinematic approach with plausible ﬂows.
(ii) Long-time integrations are unnecessary. Solutions of the induction equation with steady ﬂow vary exponentially with time and a positive real part to the exponent guarantees dynamo action. This is a distinct computational advantage over non-linear dynamo simulations that can only demonstrate dynamo action by time integration. The kinematic approach loses much of its attraction when the ﬂow is time dependent because then time stepping is also needed. An exception is the study of time-periodic ﬂows, when the magnetic ﬁeld varies with time as an exponential multiplied by a periodic function with the same period as the velocity (Willis & Gubbins 2004).
(iii) The induction equation must be satisﬁed by any dynamo. Solutions are relevant if the ﬂuid ﬂow also satisﬁes the dynamical equations. Finding ﬂows that produce Earth-like magnetic ﬁelds and then determining the dynamics that might produce them is a valid, observation-led approach. A good example of this is the concentration of magnetic ﬂux by downwelling. This kinematic result (Hutcheson & Gubbins 1994) was used to propose downwelling induced by cold regions of the mantle, a dynamic result ﬁrst conﬁrmed in non-magnetic convection by Zhang & Gubbins (1996), later in dynamical dynamo runs (Glatzmaier et al. 1999; Bloxham 2000; Christensen & Olson 2003), and ﬁnally in a solution with narrow downwelling plumes locked to the boundary (Gubbins et al. 2007). Another example is the observed switch between steady and oscillatory solutions when the meridional circulation is changed in
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Implication of kinematic dynamo studies for the geodynamo 81

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a kinematic solution (Gubbins & Gibbons 2002; Willis & Gubbins 2004), which led to new interpretation of a reversal in a dynamical dynamo (Sarson & Jones 1999; Sarson 2000). In a more quantitative approach Sarson (2003) has modiﬁed a class of ﬂows studied extensively for kinematic dynamo action so that it satisﬁes the thermal wind equation, a ﬁrst step towards determining a dynamical dynamo from kinematic solutions that produce magnetic ﬁelds of a desired type.
(iv) The kinematic approach amounts to exploring solutions of the induction equation, or ﬁnding the modes of the induction operator. The modes could be useful in exploring the full dynamical solution because any candidate magnetic ﬁeld can be expanded in the set of modes. It remains to be seen how successful this approach will be given the numerical effort required to ﬁnd the modes in the ﬁrst place. For example, the Backus dynamo (Backus 1958) is a simple example of a kinematic dynamo with time dependent ﬂow: the same steady ﬂow is switched on and off periodically. The general solution for this time-dependent ﬂow can be represented in terms of modes of the induction equation when the ﬂow is turned on and by the decay modes when the ﬂow is turned off. Each mode decays or grows exponentially with time, so the time dependence can be followed by applying the growth or decay factors and transforming between the two bases each time the ﬂow is switched. The numerical scheme is only efﬁcient when most of the modes decay very quickly and it is only necessary to use a small number of modes. In another example, Melbourne et al. (2001) presented a simple dynamical system exhibiting a heteroclinic cycle as a model for geomagnetic reversal behaviour. The system possesses three saddle points and cycles between them, remaining near a saddle for a long time before switching to another saddle. The saddles represent solutions of the induction equation with different symmetry (for example, steady and oscillatory modes) and the transition between them a breakdown from one symmetry to another. The study requires prior knowledge of modes of the induction equation for suitable ﬂows.
Kinematic theory is limited in two major respects: we do not know the ﬂow in the core, and an impossibly large class of ﬂows would need to be explored in order to establish a deﬁnitive connection between the ﬂuid ﬂow and the observed ﬁeld. Nevertheless, generic results can be drawn from a limited study and these may be tested by appropriate dynamical studies.
Recent results from a number of kinematic dynamo studies are based on the ﬂow chosen by Kumar & Roberts (1975). This ﬂow has the advantage of generating a magnetic ﬁeld with a fourfold symmetry that resembles that of the Earth. The resemblance is fortuitous because the original authors were simply building ﬂows they thought would act as dynamos, guided by the previous attempts of Bullard & Gellman (1954) and Lilley (1970) and the Braginsky theory (Braginsky 1964). The model has convective downwelling on the great circle φ = ±90◦ that tends to concentrate surface ﬂux; in the core the surface ﬂux is concentrated around the Paciﬁc. This similarity adds real geophysical relevance to a whole class of ﬂows based on the original Bullard–Gellman ﬂow. The ﬂows have been extended in the recent studies of Nakajima & Kono (1991, 1993), Hutcheson & Gubbins (1994), Sarson & Gubbins (1996), Sarson & Busse (1998), Gubbins et al. (2000a,b), Gubbins & Gibbons (2002) and Sarson (2003).
The theory and method of solution is described in Section 2. Section 3 reviews the generic results relevant for the Earth’s magnetic ﬁeld. The results are interpreted in terms of dynamo action, ﬁeld morphology and time dependence in Section 4.

2 T H E O RY

In the kinematic dynamo problem the ﬂuid velocity v is speciﬁed and the magnetic ﬁeld is governed by the induction equation:

∂B ∂t

=

Rm ∇

× (v ×

B) + ∇2 B,

(1)

where Rm = μσ Vc is the dimensionless magnetic Reynolds number, μ the magnetic permeability, σ the electrical conductivity, and c the linear dimension. Eq. (1) is solved in a sphere of radius c with
insulating boundary conditions, so that B must match a potential external ﬁeld at r = c.
When v is independent of time solutions of (1) take the form B = B0 exp ( p + iω)t. The growth rate p and oscillation frequency ω may be found efﬁciently as an eigenvalue problem. Dynamo action is said to occur if the growth rate p(Rm) ≥ 0. The critical magnetic Reynolds number Rcm yields zero net growth in B: p(Rcm) = 0. When v is time dependent (1) must be solved by time stepping unless v is periodic in time with period T, when solutions take the form Bq exp[qt] where Bq is periodic with period T and q is a Floquet parameter. The eigenvalue problem may then be solved efﬁciently
by a few time integrations of one period (Willis & Gubbins 2004).
Many different geometries and different ﬂows have been con-
sidered; spherical geometry is the most relevant to the geodynamo
and the ﬂows are usually expressed in terms of vector spherical
harmonics

v=

mt l

t

m l

+

m l

s

slm

,

(2)

lmst

where tml , sml are toroidal and poloidal vector spherical harmonics:

t

m{c/s} l

=

∇

×

tlm (r )Plm (cos θ ){cos / sin}(mφ)er

,

(3)

slm{c/s} = ∇ × ∇ × slm (r )Plm (cos θ ){cos / sin}(mφ)er ,

(4)

(r, θ , φ) are spherical coordinates, Plm are Schmidt-normalized associated Legendre functions, and er is the unit radial vector. Flows consistent with convection inﬂuenced by rotation are expected to be dominated by symmetry about the equator, which requires the toroidal harmonics to have l − m odd and poloidal harmonics to have l − m even (Gubbins & Zhang 1993). This amounts to invariance under the transformation θ → π − θ. Other symmetries have been considered and may prove important in geomagnetic ﬁeld behaviour. Generated magnetic ﬁelds have the same or lower symmetry than the underlying ﬂow. For example, ﬂow with this ‘equatorial’ symmetry may generate magnetic ﬁelds that are either antisymmetric about reﬂection in the equatorial plane, the so-called dipolar family, or symmetric about reﬂection in the equatorial plane, the quadrupole family. The different symmetries are usually expressed in terms of the leading harmonic in the poloidal ﬁeld, l = 1, m = 0 or l = 2, m = 0. Solutions with different symmetry are linearly independent in the kinematic problem because the induction eq. (1) is linear; any linear combination is also a solution. The full dynamical dynamo problem is non-linear and solutions may not be combined, but they are separable and may exist in isolation. This adds relevance to the study of independent solutions of the kinematic problem.
The radial functions tlm(r) and slm(r) must be chosen to satisfy the boundary conditions and continuity conditions at the origin (although this has not always been followed); they are otherwise arbitrary. Most authors have chosen simple polynomial and trigonometric functions, sometimes modiﬁed by a taper function near the boundary. Some authors have used spherical Bessel functions, which amounts to expanding the ﬂow in the natural decay modes of the

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82 D. Gubbins

(a)

(b)

(c)

Figure 1. Fluid ﬂow deﬁned by eq. (5). (a) contours of vφ , controlled by parameter 0, in meridian section (b) streamlines of meridional circulation ( 1) in meridian section (c) streamlines of convection rolls ( 2, 3) in equatorial section.

sphere, because they optimize the helicity. The wide variety of radial function emphasizes the arbitrary nature of the kinematic problem but it is, in fact, no more or less arbitrary than the use of vector harmonics, which have no better case for admission other than that they satisfy the solenoidal and continuity conditions. We seek robust, generic results that should be independent of the particular choice of radial function, but this can never be guaranteed. Dudley & James (1989) review most of these ﬂows.
The most intensive study of a single class of ﬂows has been conducted for generalizations of the Kumar–Roberts dynamo (Kumar & Roberts 1975) itself a development of the failed Bullard–Gellman dynamo. The ﬂow is deﬁned as:

v=

0

t

0 1

+

1s02 +

2

s

2c 2

+

3

s

2s 2

,

(5)

with radial functions deﬁned by

t10(r ) = r 2(1 − r 2),

s20(r ) = r 6(1 − r 2)3,

s22c(r ) = r 4(1 − r 2)2 cos(nπr ),

s22s (r ) = r 4(1 − r 2)2 sin(nπr ).

(6)

The ﬁrst harmonic represents differential rotation about the coordi-

nate axis (which is implicitly assumed to be also the rotation axis),

the second meridional circulation, and the last two convective over-

turn. These three constituents of the ﬂow are shown in Fig. 1; in

some sense they are the simplest ﬂows that can generate the basic

features of the Earth’s magnetic ﬁeld and mimic convection in a

rotating sphere. Bullard & Gellman (1954) used only harmonics

t

0 1

and s22c; Lilley (1970) added s22s to remove the plane of symmetry

found to prevent dynamo action in the Braginsky limit (Braginsky

1964) (we would now say it adds helicity to the ﬂow); and Kumar &

Roberts (1975) added s02 because it was found to promote steady solutions in mean-ﬁeld αω dynamos (Roberts 1972b). Sarson & Busse

(1998) added higher wavenumbers to the convective part (sml ) in order to study the effect of more convection rolls and non-sectorial

(l = m) behaviour. Sarson (2003) further modiﬁed the ﬂow so that it

satisﬁed the thermal wind equation, bringing the ﬂow closer to what

might be driven by temperature differences. This involves removing

the meridional circulation ( 1 = 0) and adding toroidal harmonics t23{c/s} based on the radial functions

t32{s/c}

=

±4 5

d −3 dr r

s22{c/s } .

(7)

Initial studies focused on values of the parameters { i} close to the asymptotic regime studied by Braginsky (1964), the rather complicated limit Rm → ∞ with 1 Rm, 2R1m/2, and 3R1m/2 remaining ﬁnite (Nakajima & Kono 1991; Hutcheson & Gubbins 1994; Sarson
& Gubbins 1996). This allowed comparison with existing solutions

for the simpler, axisymmetric, Braginsky mean ﬁeld equation, identical with the αω equation of mean ﬁeld electrodynamics (Steenbeck et al. 1966; Roberts 1972b).
This choice of ﬂow has another advantage for studying the geodynamo: the symmetry of periodic repetition in φ (only m = 2 terms are present). The present geomagnetic ﬁeld has approximately this symmetry, and there is palaeomagnetic evidence to suggest that the same symmetry has persisted for a long time. Not all choices of the
i are independent because of this additional symmetry. The transformations φ → φ + π/2 and φ → −φ show the equivalence of the four combinations:

( 0 1 2 3), ( 0 1 − 2 − 3), (− 0 1 − 2 3),

(− 0 1 2 − 3).

(8)

(Dudley & James 1989). The four different solutions for B separate because of these symmetries, depending on whether they change sign under reﬂection in the equatorial plane or rotation through an angle π about the coordinate axis. The four symmetries are denoted by their leading poloidal vector spherical harmonic: axial dipole Da (led by l = 1, m = 0), axial quadrupole Qa(l = 2, m = 0), equatorial dipole De(l = 1, m = 1), and equatorial quadrupole Q e(l = 2, m = 1) [notation of Holme (1997)]. The same symmetries apply to the higher wavenumbers and thermal wind ﬂows studied by Sarson & Busse (1998) and Sarson (2003). Many studies have concentrated on the case 2 = 3 to optimize helicity in the convective overturn. Changing the sign of Rm reverses the ﬂow, so the second of these combines with Rm → −Rm to produce the combination (− 0 −
1 2 3). If we set 2 = 3, we can also restrict Rm to positive values and still explore the full range of dynamo solutions deﬁned by the two parameters 0 and 1.
Rm must be deﬁned consistently to compare dynamo action between different ﬂows, particularly when comparing the Braginsky regime with more balanced ﬂows.
The Braginsky regime is completely dominated by differential rotation, so one has to be careful to state whether Rm is based on the small but essential radial ﬂow or the much larger azimuthal ﬂow. In the ﬁrst comprehensive study of kinematic dynamo action of this class of ﬂows outside the Braginsky regime, Gubbins et al. (2000a) deﬁned Rm so that the total kinetic energy of the ﬂow is unity:

v2dV

=α

2 0

+

β

2 1

+

γ

2 2

+

δ

2 3

=

1,

(9)

where the scalars α, β, γ , δ are integrals of the radial functions (6). The fraction of energy in the meridional circulation is then

M

= sgn(

1)β

2 1

,

(10)

and that for differential rotation

D

= sgn(

0)α

2 0

.

(11)

The remaining energy lies in convection or radial overturn

C = 1 − |M| − |D|.

(12)

The entire class of ﬂows is deﬁned by the diamond |M| + |D| ≤ 1. D > 0 gives westward surface ﬂow; M > 0 gives surface ﬂow towards the equator and away from the poles.
The Braginsky limit becomes:

Rm → ∞, |D| → ±1, |M| → 0

(13)

in such a way that

1 − |D|2 = K |M|,

(14)

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Implication of kinematic dynamo studies for the geodynamo 83

For time-periodic ﬂows the solutions are found by integration in

time. Provided we integrate for long enough the fastest growing

Da Qa

mode will ultimately dominate (the slowest decaying mode domi-

De

nates if there are no growing modes). We can do better than this

by restarting the time integration after each period T. An arbitrary

initial solution is time-stepped for one period, then restarted with a

solution orthogonal to the ﬁrst. Repeated integration and orthogonal-

ization after each cycle soon leads to a set of modes that accurately

represents the fastest growing (or slowest decaying) solution. The

method is fast and requires very little memory, unlike IRAM, which

must store a large banded matrix. This matrix-free Krylov subspace

method is described further in Willis & Gubbins (2004).

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Figure 2. Dynamo action of two-parameter family of ﬂows. Blank squares do not support dynamo action. Greyscale gives solution symmetry with lowest Rcm. White lines are where oscillatory solutions have been found corresponding to oscillatory solutions in the Braginsky limit. After Gubbins & Gibbons (2002).
where K is a ﬁnite non-zero constant. Eq. (13) describes quadratics in (D, M) parameter space (Fig. 2) passing through each of the limit points D = ±1.
Other special cases of (5) have been the focus of attention, usually with somewhat different radial functions (Dudley & James 1989; Holme 2003). Axisymmetric ﬂows with m = 0, on the boundary of the diamond in Fig. 2, cannot generate axisymmetric magnetic ﬁelds because of Cowling’s theorem but they can generate nonaxisymmetric magnetic ﬁelds. Solutions have the form B ∝ exp imφ and each mode can be represented by a sum of harmonics with just one value of m. This makes for a much simpliﬁed set of equations and an easy numerical solution, which is why they have received so much attention. The dominant m = 1 ﬁeld is an equatorial dipole. If one wants to ﬁt this to a geodynamo one could rotate the whole dynamo through 90◦ to give a dominant axial dipole, but the ﬂow would not obey the symmetry expected in a rotating system.
Eq. (1) is solved numerically by expanding v, B (both are solenoidal, ∇ · v = ∇ · B = 0) in vector spherical harmonics and approximating the radial derivatives by some suitable ﬁnite difference or collocation scheme. The resulting algebraic eigenvalue problem is solved by standard methods, the Iteratively Restarted Arnoldi Method (IRAM) of (Lehoucq et al. 1998) being the most successful general method available. Dynamo action can never be established rigorously by numerical means. This statement goes beyond the usual cautionary remark for numerical rather than analytical solutions because in practice we increase Rm until p becomes positive or until the numerical scheme is incapable of representing the solution. If no dynamo exists within the accessible range of Rm we still cannot rule out dynamo action at higher Rm. The edges of each dynamo region in Fig. 2, where dynamo action fails, present the most severe numerical convergence problems. At these boundaries Rcm typically rises dramatically, B becomes spatially complex, and dynamo action is lost with little warning. Further details of the numerical methods are described in Gubbins et al. (2000a).
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3 RESULTS
3.1 Dynamo action
Dynamo practitioners have two opposing experiences, either that no obvious choice of ﬂow works as a dynamo (viz. Bullard & Gellman 1954; Lilley 1970), or that almost all sufﬁciently complicated ﬂows act as dynamos e.g. Roberts (1972a). The Kumar–Roberts class of ﬂows were, therefore, studied to determine what proportion act as dynamos. Over half of the ﬂows with 2 = 3 generate magnetic ﬁeld of one symmetry or another; many generate ﬁelds of 2 symmetries or even 3 with different Rcm; and a few generate two distinct symmetries with identical Rcm—for example the ones lying on the boundary lines between zones in Fig. 2.
The sides of the diamond in Fig. 2 have C = 0 and ﬂows that are axisymmetric. They cannot generate Da or Qa ﬁelds because of Cowling’s theorem, which states that no axisymmetric ﬁeld can be sustained by dynamo action, but De and Qe ﬁelds are possible. None have been found with the original Kumar–Roberts choice of radial functions but similar ﬂows were studied by Bullard & Gubbins (1977) and Dudley & James (1989) using slightly different radial functions. Both found De ﬁelds for some of their ﬂows. Bullard & Gubbins (1977) noted strongly sheared magnetic ﬁelds near the outer boundary with their ﬁrst choice of ﬂow, which did not produce dynamo action. Adding an outer layer of stagnant ﬂuid did produce dynamo action. The stagnant conducting layer reduced ﬁeld gradients near the outer boundary, thus reducing energy loss by diffusion. Dudley & James (1989) used very much simpler radial functions and obtained dynamo action without a stagnant layer. Holme (1997) modiﬁed the Kumar–Roberts ﬂow by changing the s02 radial function to r3(1 − r2)2, a close approximation to the Dudley & James (1989) trigonometric function, and obtained De solutions. Love & Gubbins (1996b) addressed the problem of choosing suitable radial functions by developing an optimization technique to ﬁnd ﬂows that produce smooth, and therefore well converged, magnetic ﬁelds. Holme (2003) used an optimization technique to explore the efﬁciency of radial functions in promoting dynamo action and found a spatial separation of poloidal and toroidal shear helped dynamo action; he concluded the Dudley & James ﬂow is close to optimal.
The line D = 0 has purely poloidal motion. It is well known that purely toroidal motion cannot sustain a magnetic ﬁeld (Bullard & Gellman 1954), and a similar antidynamo theorem has been conjectured for purely poloidal motions; this numerical demonstration shows the conjecture not to be true (Love & Gubbins 1996b). The solution at the centre has no differential rotation and generates a De ﬁeld (Holme 1997). No convincing dynamo exists at D = M = 0 with the original Kumar–Roberts ﬂows, although recent exploration has revealed a number of oscillatory De and Qe solutions near the centre of the diamond (Gubbins 2008).

84 D. Gubbins
The line M = 0 has no meridional circulation; the dynamo studied by Lilley (1970) is on this line close to D = 1. Lilley’s choice lies just outside the dynamo region, and a tiny change in his ﬂow produces a successful dynamo. Unfortunately, the growing solution he reported was a numerical artefact. Bullard & Gellman (1954) had M = 0 with
3 = 0 and also reported a growing solution that turned out to be a numerical artefact.

3.1.1 Helicity

Helicity is thought to promote dynamo action by converting toroidal to poloidal ﬁeld. Bullard & Gellman (1954) envisaged pushing and twisting of the toroidal ﬁeld by helical ﬂow to put energy into the dipole ﬁeld, but by omitting the s22s harmonic they lost the helicity. The α-effect of the Braginsky dynamo requires both s22c and s22s harmonics and is related to the axisymmetric average of the helicity. Pekeris et al. (1973), in an early study of a dynamo with harmonics s22c, t22c, chose Beltrami ﬂows with v parallel to ∇ × v in order to maximize the helicity. Beltrami ﬂows have radial functions based on spherical Bessel functions. Nakajima & Kono (1993) made a thorough study of the effect of helicity on kinematic dynamos, including the Pekeris dynamo, and concluded that the dynamo was more efﬁcient for larger mean helicity.
Most studies have concentrated on the case 2 = 3, which optimizes the helicity. Kumar & Roberts (1975) found numerical convergence was best in this case, and Hutcheson & Gubbins (1994) found that when 2 and 3 differ by too much the dynamo fails. The ﬂows near the Braginsky limit points D = ±1 act as αω dynamos, the link being formal in the appropriate limit. Flows near the centre line D = 0 might be expected to behave like α2 dynamos, and to some extent they do. Steenbeck & Krause (1969b) [translated in Roberts & Stix (1971)] originally proposed that the Earth had an α2 dynamo because its ﬁeld was steady, while the Sun had an αω dynamo because its ﬁeld was oscillatory.
Maximizing the mean helicity does not always lead to enhanced dynamo action. For the Kumar–Roberts dynamo the helicity of the ﬂow depends only on the convective part of the motion and is proportional to the product 2 3. There is an axisymmetric part and a non-axisymmetric part:

h = v · ∇ × v = 2 3[H0(D, M) + H1(D, M) cos 2φ].

(15)

Surprisingly, Love & Gubbins (1996a) found that the most efﬁcient dynamo (the one with lowest Rcm normalized by the kinetic energy) on the D = 0 line was not at the centre M = 0, where the helicity is
a maximum for the kinetic energy available, but at a point where the
non-axisymmetric part of the helicity [H 1 in (15)] is a maximum. This appears to hold rather generally, well away from the Braginsky
limit. Fig. 3 shows the non-axisymmetric part of the helicity compared to Rcm for the Da symmetry solutions. It seems that axisymmetric helicity, in which the ﬂow pushes and twists the toroidal ﬁeld in the same sense for all φ, does not lead to reconnection of the ﬁeld
to reinforce the poloidal ﬁeld (Love & Gubbins 1996a).

3.1.2 Effect of stagnation points and concentration of magnetic ﬂux on dynamo action
The boundaries between dynamo and non-dynamo regions in Fig. 2 are usually characterized by a rapidly rising critical magnetic Reynolds number and concentration of magnetic ﬂux into a few isolated spots. Flux concentration is independent of the sign of B and is often enhanced at stagnation points of the ﬂow, where magnetic ﬁeld may be carried in from many directions and, in some circumstances,

F

C D
E

B G

A

M

0

Rmc 1200

(a)

D

M

(b)

D

Figure 3. (a) Flows giving dynamo action with Da symmetry. Letters A–G
label distinct regions of dynamo action separated by areas where no dynamos have been found. Rcm as shown by the grey scale at bottom right. (b) Non-axisymmetric helicity of the ﬂow, H 1(D, M) in (15). Note the correspondence with Rcm, suggesting H 1 promotes dynamo action. After Gubbins et al. (2000a).

is limited only by diffusion. At high Rm the ﬂux is concentrated into narrow zones with diffusion enhanced accordingly. Since the only degree of freedom in this kinematic calculation is to change Rm, there comes a point at which increasing Rm enhances diffusion more rapidly than enhancing the dynamo action, the dynamo fails, and the growth rate becomes a decreasing function of Rm. Stagnation points are, therefore, central to determining the dynamo action. On the surface, stagnation points are identiﬁed as centres of upwelling and downwelling; ﬂux tends to be concentrated over downwellings.

3.1.3 Boundary effects A major problem with early searches for kinematic dynamo action lay with the choice of radial function. The work of G. O. Roberts
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on periodic dynamos in inﬁnite media showed that dynamo action would be possible if the ﬂow took on a cellular form with ﬁeld regeneration taking place by interaction between cells away from the insulating boundary (Roberts 1972a). This led to the choice of multiple cells in radius, as in Gubbins (1972) and the Kumar–Roberts dynamo [eqs (6) with n > 1]. Bullard & Gubbins (1977) identiﬁed a current sheet developing near the insulating boundary of many single-cell dynamos and attributed loss of dynamo action to enhanced ohmic diffusion near the boundary. They achieved dynamo action by adding a stagnant layer of ﬂuid around the region of motion; this allows electric currents to ﬂow out of the region where ﬁeld is generated, reduces the diffusion, and allows dynamo action. There is no hard-and-fast rule about the advantages of a stagnant layer, it depends on whether the ﬁeld regeneration leads to production of radial current density near an insulating boundary. Changing boundary conditions from insulating to perfectly conducting (where the current density is required to be normal rather than parallel to the boundary) usually improves dynamo action.
Dudley & James (1989) obtained dynamo action with axisymmetric ﬂows based on very simple radial functions. This model has received considerable attention because of its simplicity but numerical convergence in the radial direction is slow, presumably because of enhanced surface currents [although note Livermore & Jackson (2004b) attribute the difference to differences in shear at the origin]. This high sensitivity of dynamo action to changes in radial function is another example of the sudden change from dynamo generation to dynamo failure seen at the edges of dynamo regions in Fig. 2.
The problem of boundary-enhanced diffusion has surfaced more recently with the attempt to obtain dynamo action in the laboratory, where magnetic Reynolds numbers are necessarily restricted to low values and dynamo action must be made as efﬁcient as possible. Kaiser & Tilgner (1999) studied a Ponomarenko dynamo surrounded by stagnant ﬂuid. They found cases where dynamo action was impaired as well as enhanced, but concluded that laboratory models would beneﬁt from leaving some of the ﬂuid stationary. These conclusions were conﬁrmed in a separate study by Stefani et al. (2006b).
3.2 Spatial symmetry selection
The geophysically relevant solutions have Da symmetry and are steady. Other symmetries may be relevant to the Sun, which has an oscillatory magnetic ﬁeld reversing every 11 yr, or some of the other planets. Magnetic ﬁelds of Uranus and Neptune have been discussed in this context by Holme (1997), who found that Da ﬁelds are preferred only in the presence of strong differential rotation. Fig. 1 shows that Da ﬁelds are favoured by D > 0. There are two possible explanations of this, described in the next 2 subsections.
3.2.1 Sign of αω or dynamo number
Magnetic ﬁeld generation by αω dynamos, whether Braginsky or mean-ﬁeld, depends on the product of the two magnetic Reynolds numbers, Rα and Rω, known as the dynamo number (Parker 1979). The αω dynamo equations are axisymmetric and attention has focused almost exclusively on Da and Qa axisymmetric solutions. The Da solutions are favoured by positive dynamo numbers, the Qa solutions by negative dynamo numbers [see Parker (1979) for an extensive discussion]. Parity selection was also resolved by Proctor (1977a,b), who showed further that Da and Qa ﬁelds would be generated at similar dynamo numbers. The sign of the dynamo number

depends simply on the sign of the ﬂow because α is quadratic and ω linear in v. The symmetry relations (8) show that it is possible to change the sign of D ( 0) and α ( 2 3) and obtain a dynamo solution with the same Rcm. All these transformations preserve the dynamo number.
In the Kumar–Roberts dynamo these results are born out by 3-D calculations near the Braginsky limit points. The dynamo number is proportional to the product DM; Da solutions dominate for D > 0, while Qa solutions dominate for D < 0. This reﬂects the result from Parker’s simple αω theory, where parity selection is governed by the sign of the product DM. The 3-D results are not fully symmetric with respect to M but become more symmetric when the effective meridional circulation of the full Braginsky theory is used. Fig. 2 shows the crossover of parity at slightly negative values of D. Proctor (1977a) used the solutions of the adjoint αω dynamo problem with modiﬁed boundary conditions to determine symmetry selection and the critical dynamo number; in the 3-D problem the adjoint induction operator is formed simply by change of sign of v and change of boundary conditions (Gibson & Roberts 1966), suggesting that a change of sign of both D and M is needed to the change symmetry of the solution. This has been conﬁrmed using ﬂows with stagnant upper layers to reduce and effectively remove the effects of the boundary conditions, which are different for the adjoint problem (Sarson & Gubbins 1996). Fig. 2 conﬁrms the switch from Da to Qa with D → − D, M → − M near the limit points D = ±1.
3.2.2 Effects of stagnation points and concentration of magnetic ﬂux on symmetry selection
We have already seen that ﬂux concentration can impair dynamo action. It can also inﬂuence symmetry selection. Each symmetry places constraints on the magnetic ﬁeld, and a symmetry will not be favoured if the ﬂow is such as to generate strong components of ﬁeld in places forbidden by symmetry. Similarly, a symmetry will be favoured if it allows ﬁeld components in places where it is concentrated by the ﬂow. Flows that generate more than one symmetry generally do not have strong concentrations of ﬂux.
For example, ﬂows in the centre of the diamond in Fig. 2, those with small D and M, tend to generate strong radial ﬁeld Br near the surface at the downwelling limbs of the convection (Gubbins et al. 2000b). Da and Qe require a change of sign of Br across the equator, which can only be accommodated with steep ﬁeld gradients at the equator and high diffusion. Meridional circulation tends to shift surface radial ﬁeld towards the poles, which inhibits De symmetry and favours Da symmetry. Differential rotation promotes axisymmetric ﬁelds and, therefore, favours both Da and Qa. The sign of D determines whether surface ﬁeld is concentrated on the equator, favouring Qa solutions, or towards the poles, favouring Da solutions (Gubbins et al. 2000b). The result is the same as discussed in the previous section in connection with the sign of the dynamo number; it provides a more geometrical explanation of the effect. Flux concentration is easiest to see in the surface radial ﬁeld and is the most important for geophysical interpretation, but ﬂux concentration of other components within the core appears to have similar effects (Gubbins et al. 2000b).
3.2.3 Preference for axial symmetry?
It is possible the top of the Earth’s core is stably stratiﬁed, either because the temperature gradient is below the adiabatic gradient or because light elements have accumulated in the upper core. The

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86 D. Gubbins
kinematic results described above show that this may not be detrimental to dynamo action, quite the opposite, but it may have a strong inﬂuence on the geomagnetic ﬁeld at the core surface. A stably stratiﬁed region will not be stagnant because horizontal ﬂows are possible; they will be driven by lateral variations in density and by the magnetic force. Love (2000) has explored the effect of toroidal motions in a layer above a kinematic dynamo driven by the Kumar– Roberts ﬂow. His primary aim was to explain the magnetic ﬁeld of Saturn, which has a high degree of axial symmetry, but his results could also apply to the Earth. He found that simple differential rotation often enhances the axisymmetric part of the ﬁeld but not always. For the Earth, a layer of purely toroidal ﬂow at the top of the core could improve the match of dynamo model ﬁelds with the observed geomagnetic ﬁeld.
Livermore & Jackson (2006) applied the energy method to a variety of ﬂows that have been studied for kinematic dynamo action. They examine magnetic energy growth and decay for transients for candidate magnetic ﬁelds and plot the envelope function, the maximum energy achieved by any ﬁeld at a particular time. The important ﬁelds for dynamo action are the ones for which energy persists the longest, rather than those that grow the most in a short initial burst. They ﬁnd axisymmetric ﬁelds persist the longest, and suggest this is why axisymmetric ﬁelds are generally the dominant component of real dynamos. They also suggest that axisymmetric ﬁelds may be dominant in a growth phase of a dynamo, such as when the Earth’s ﬁeld is recovering from a reversal (Livermore & Jackson 2004b). They do not give any examples of working dynamos to back up this suggestion. Their results could explain the experiment of Pefﬂey et al. (2000), who used ﬂows similar to those of Dudley & James (1989) and found the dominant magnetic modes to be axisymmetric. This was not a dynamo experiment and all the modes decayed, but the dominance of axial symmetry appeared contrary to Cowling’s theorem. Tilgner (2002) showed the results were consistent with kinematic theory, but the prevalence of axial symmetry suggests the energy method is effective in predicting the behaviour of real experiments.
3.3 Time dependence
3.3.1 Dynamo waves
Steady ﬂows can only produce steady or oscillatory magnetic ﬁelds. Early studies of mean ﬁeld αω dynamos suggested that oscillatory solutions were preferred (Steenbeck & Krause 1969a), but later simple models showed that steady solutions were possible on separation of α and ω in space (Stix 1973; Deinzer et al. 1974; Gibbons 1998) or on addition of meridional circulation (Roberts 1972b). Similar results hold for 3-D kinematic dynamos: oscillatory solutions arise when regions of strong poloidal and toroidal components of ﬂow are in the same place, and oscillatory solutions only arise for a very narrow range of meridional circulation around the zero in effective meridional circulation in the Braginsky limit (Gubbins & Gibbons 2002): the zone of oscillatory solutions in Fig. 2 has thickness less than 10−2 in M. The absence of oscillatory 3-D solutions is remarkable and in stark contrast to the axisymmetric αω results. The explanation is simple: it is easier to separate poloidal and toroidal parts in three dimensions than 2.
The transition from steady to oscillatory solutions can involve either a change of mode or a pair of steady modes coalescing into a complex conjugate pair (eigenvalues of a real operator are either real or occur in complex conjugate pairs). Three possibilities

p A

E

D

F

Rm C

B

p A

(a) p

E B
Rm C

A D

D

B

Rm

C (b)

(c)
Figure 4. Schematic examples of growth rate curves. Solid lines indicate real growth rates, dotted lines complex growth rates (oscillatory solutions). (a) A pair of real eigenvalues A and B coalesce to a complex conjugate pair at C, which revert to a pair of real at D. The physically realisable solution is the one with positive growth rate at lowest Rm, which depends on whether p ≥ 0. The oscillatory mode is realized in the ﬁgure as drawn; any change in ﬂuid velocity that places C above, or D below, the line p = 0 will produce a steady realisable solution. (b) Two separate modes; AB is steady and CD oscillatory. As drawn AB is preferred but a small change in ﬂow can shift CD to make it unstable ﬁrst. (c) ABC is steady and preferred as shown. If a small change in ﬂow places B below p = 0 the oscillatory mode DF becomes the only dynamo mode.

are sketched in Fig. 4. When the same mode changes from real to complex the generated ﬁeld tends to be similar at certain parts of the cycle. Flux in the oscillatory solutions tends to migrate northsouth rather than azimuthally, with ﬂux reversal taking place by new patches forming at the equator (poles) and migrating towards the poles (equator). For some ﬂows both steady and oscillatory solutions are generated with the same critical magnetic Reynolds number.

3.3.2 Time-dependent ﬂow
An interesting result is that time-dependent ﬂows can be better dynamos than steady ﬂows, in the sense that the growth rate is greater than the mean growth rate of the constituent snapshots of the ﬂow (Gog et al. 1999; Normand 2003; Willis & Gubbins 2004). This could arise, for example, from movement of the stagnation points to reduce ﬂux concentration and its attendant enhanced diffusion (see Section 3.1.2). It is made possible by non-orthogonality of the modes of the induction equation: if the equation were self-adjoint then varying the ﬂow would not assist dynamo action. With non-orthogonal modes a transient may grow initially but then decay because it is a combination of decaying modes, but as the ﬂow changes so does the spectrum of modes. The ﬁeld is now a different combination of modes and a new transient may grow, leading to dynamo action. If the modes were orthogonal and all decaying the ﬁeld would inevitably continue to decay. Livermore & Jackson (2006) used the energy method to explore growth of transients and ﬁnd them to be
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quite robust to changes in ﬂow, suggesting they may hold clues to the form of the generated ﬁeld.
Steady ﬂow is also a great restriction on the possible time behaviour of the magnetic ﬁeld: it can never produce an Earth-like polarity reversal, for example. Periodic ﬂows are rather special but they allow us to explore some of the possible behaviour of time dependent ﬂow. In particular, it allows us to simulate reversals. If v is periodic in time with period T the solution for B takes the form Bq(t) exp [qt] with Bq (t + T ) =Bq (t). Fourier expansion in time of the periodic quantities is the usual analytical method of solution, but a much more efﬁcient numerical method is to time-step one period for a sequence of orthogonal starting conditions (Willis & Gubbins 2004).
Periodic, time-dependent ﬂows are speciﬁed by allowing the velocity parameters D, M to be periodic functions of time, deﬁning a closed curve in the parameter space of Fig. 2. The narrowness of the band of oscillatory solutions means that only very slight changes in ﬂuid velocity are required to change from steady to oscillatory solutions. If the orbit in (D, M)-space is designed to cross or just enter the oscillatory region it is possible to obtain a solution that is almost steady for most of the orbit and oscillates for only a short time. If the period of the orbit is chosen so that the ﬂow remains in the oscillatory region for about half a cycle the ﬁeld reverses. The solution is periodic with the ﬁeld remaining almost stationary for most of the cycle and the reversal taking place in a short time.
Stefani & Gerbeth (2005) have studied reversals in mean-ﬁeld αω dynamos with equilibration forced by alpha-quenching (Stefani et al. 2006a). They also ﬁnd reversal behaviour where ﬂows produce both steady and oscillatory kinematic solutions at the same magnetic Reynolds number—places where the growth rate curve for a pair of complex eigenvalues bifurcates into two curves for real eigenvalues as in Fig. 4(a). For high Rm they ﬁnd these bifurcation points tend to cluster around the zero growth-rate axis, suggesting high Rm-dynamos are prone to reversal behaviour. The mean-ﬁeld approximation is a limitation and, like any kinematic dynamo, it suffers from the arbitrary choices of α and ω. The relative scarcity of oscillatory solutions in 3-D kinematic dynamos (see Fig. 2) would suggest this argument may be restricted to mean-ﬁeld dynamos.

4 IMPLICATIONS FOR THE GEODYNAMO AND DYNAMIC MODELS
The kinematic dynamo can help understand the geodynamo in two ways: ﬁrst, it provides an interpretation of the observed surface magnetic ﬁeld in terms of the underlying core ﬂow, and secondly it provides physical insights into the dynamo mechanism that can guide further work on the forces driving the ﬂow.

4.1 Main ﬁeld

The clearest connection between ﬂow and surface ﬁeld in the dy-

namo solutions discussed here is concentration of vertical ﬂux by

ﬂuid downwelling. This is easy to see for steady ﬂows but harder

in dynamic solutions where the ﬂow is often chaotic and varying

too much in time for the effect to show up in anything other than a

time average. The effect has now been found in a quasi-steady dy-

namic model dominated by lateral variations in heat ﬂux or temper-

ature at the outer boundary (Gubbins et al. 2007; Willis et al. 2007;

Sreenivasan & Gubbins 2007). These solutions have heat ﬂux

boundary conditions determined by lower mantle seismic tomog-

raphy

or

a

single

Y

2 2

spherical

harmonic,

and

narrow

downwellings

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coincide with high radial ﬂux or cold spots on the boundary. In the tomographic case the ﬂux spots match the 4 main lobes of the present geomagnetic ﬁeld well, indicating a strong mantle inﬂuence on the geomagnetic ﬁeld. The opposite effect, absence of radial surface ﬂux above upwelling, is also observed. The absence of surface ﬂux within the tangent cylinder in the geomagnetic ﬁeld suggests upwelling there, which places a constraint on the vigour of convection in the whole core (Sreenivasan & Jones 2006). Downwelling near the equator, where with Da symmetry there is no surface radial ﬁeld to concentrate, tilts the toroidal ﬁeld below the surface to produce a typical cloverleaf pattern on the equator. The effect is weak or absent when differential rotation is large, presumably because azimuthal ﬂow dominates the downward ﬂow. Such patterns are often seen in dynamic models and there is a hint of this pattern on the core surface beneath Indonesia (Bloxham & Gubbins 1985), but nothing as strong as is usually seen in the models. This absence may be because of strong azimuthal ﬂow or weak vertical motion near the core mantle boundary resulting from local stratiﬁcation, as discussed in Section 3.2.3
The primary feature of the geomagnetic ﬁeld is the dominant axial dipole, which has persisted throughout Earth’s history. The kinematic models reported on here require signiﬁcant differential rotation to produce axial dipole symmetry, solutions with weak differential rotation and meridional circulation being dominated by De and Qe symmetry. Differential rotation tends to produce a large, axisymmetric toroidal ﬁeld that promotes axisymmetric poloidal ﬁelds. Da symmetry is preferred when the differential rotation leads to surface azimuthal drift in the same sense as the spiralling of the convection rolls. In the Earth the surface drift is predominantly westward, so we should expect a predominantly eastward spiralling, but this is not seen clearly in the shape of the main lobes of the geomagnetic ﬁeld. This may be because what we observe is a blurred combination of several individual rolls that are obscured by the limited resolution of core ﬁeld maps, it may be that convection is suppressed near the CMB and we observed the blurred effect of a more complex ﬁeld at depth, or it may simply be that spiralling is weak under core conditions.
Axial dipole symmetry is also favoured by ﬂux concentration at high latitudes because no sharp ﬁeld gradients are needed. In dynamic models ﬂux tends to be concentrated outside the tangent cylinder, near 70◦N and S, also favouring Da symmetry. However, the kinematic dynamos discussed here have no inner core yet still lead to ﬂux concentration at high latitudes. There is no reason, therefore, to expect any dramatic change in geomagnetic ﬁeld morphology in earlier times when the inner core may have been smaller. Dynamical studies tend to conﬁrm this [for example, Bloxham (2000)].
Spiralling of the rolls not only provides the helicity required by the Braginsky theory, it also provides a succession of rolls that overlap with radius [Fig. 1(c)], enabling the reconnection of small scale ﬁelds within each roll to reinforce the large scale ﬁeld, as happens in turbulent or periodic dynamos. Such intuition, based on the Braginsky theory, remains useful for large areas of the parameter space deﬁning the 3-D ﬂows, and is an important aid to understanding dynamo action in general.
The early picture of a geomagnetic ﬁeld with a dipole moment remaining stationary for several hundred thousand years between polarity reversals has been shown by ﬁne resolution paleointensity studies to be false [e.g. Guyodo & Valet (1999)]: the geomagnetic ﬁeld strength is continually ﬂuctuating by a large fraction of its maximum value and never remains constant for very long. This suggests the core ﬂow is also continually changing. Kinematic studies show that steady ﬂows make poor dynamos because they contain

88 D. Gubbins
stagnation points; it is also worth noting that some of the dynamos driven by time-dependent, oscillatory ﬂows studied by Willis & Gubbins (2004) had lower critical magnetic Reynolds numbers than any dynamo driven by a stationary, instantaneous ﬂow in the cycle [see also Gog et al. (1999)]. Most dynamic dynamo models have chaotic ﬂows that are highly time dependent and, therefore, avoid this problem: it seems that any sufﬁciently chaotic and vigorous ﬂow will generate a magnetic ﬁeld. The mixing properties of these time dependent ﬂows ensure that severe ﬂux concentrations do not arise.
It is worth asking how dynamic solutions avoid severe diffusion arising from ﬂux concentration—does the Lorentz force reduce the ﬂow speeds in these locations or does it change the pattern of ﬂow? In the limited number of studies undertaken it would seem that the ﬂow pattern is changed, moving or removing the stagnation points; the Lorentz force rarely becomes strong enough to oppose the ﬂow directly before diffusion associated with the ﬂux expulsion becomes too large. The dynamo instability appears to be stronger when the system has the extra degrees of freedom associated with a changing ﬂow, as in the dynamical case, as opposed to a ﬁxed ﬂow, as in the kinematic case. The study of dynamo action by steady ﬂows becomes highly relevant if the mantle inﬂuences the core ﬂow strongly. The boundary-locked regime of Gubbins et al. (2007) was hard to ﬁnd and only exists for a small range of the accessible parameters. This is partly because increasing the lateral buoyancy parameter, which controls the amplitude of lateral variations of heat ﬂux on the boundary, ultimately causes the dynamo to fail. Usually dynamo failure occurs when lateral heat ﬂux variations become comparable with the mean vertical heat ﬂux, and may be associated with a locked ﬂow that is nearly steady and has stagnation points.
4.2 Secular variation: frozen-ﬂux theory
Ever since the time of Halley our changing magnetic ﬁeld has been used to infer mass motion inside the Earth. The idea was formalized in Backus’ (1968) theory by invoking induction near the core surface and neglecting diffusion. The theory has been used many times in different guises to infer ﬂuid motion at the core surface. Estimates are required of both Br and its time derivative B˙ r [or a record of Br(t) over an interval of time] downward continued to the CMB. Dynamo action requires a balance between induction and diffusion, and it is obvious that frozen-ﬂux theory, by setting diffusion to zero, cannot return the ﬂow responsible for dynamo action directly: inferences made about the geodynamo from maps of core motions must be indirect at best. A circular argument underlies our basic estimate of the core’s magnetic Reynolds number because it relies on a velocity estimate based on observed secular variation, which in turn relies on neglect of diffusion!
Kinematic dynamos provide many examples where frozen-ﬂux fails. Most obvious are the steady solutions, where surface ﬂow inversions relying on frozen-in ﬁelds inevitably yield zero ﬂow. Love (1999) gives examples of kinematic dynamos with slowly varying magnetic ﬁelds for which frozen-ﬂux yields completely misleading ﬁelds. Kelly (1996) provides examples of time-varying magnetic ﬁelds generated by ﬂuid ﬂow in a spherical shell with magnetic ﬁeld obtained from a kinematic dynamo solution ﬁxed at the base. At time t = 0 the ﬂow in the shell jumps from zero to v, generating a time-varying magnetic ﬁeld at the upper surface that is inverted for the ﬂow using frozen-ﬂux theory. Initially the inversion gives a ﬂow that is approximately correct but it weakens with time and eventually reduces to zero as the transients die away and the magnetic ﬁeld becomes stationary. Stopping the ﬂow at this point generates

more transients but now frozen-ﬂux theory yields −v rather than the

correct v = 0! These examples show that the results from frozen-

ﬂux theory must be treated with caution and will depend critically

on the time history of the ﬂow.

The problem with Backus’ theory is that the limit η → 0 is sin-

gular: setting η = 0 is, therefore, likely to yield artiﬁcial results.

The problem is made clearer by considering the ﬁrst order correc-

tion to the frozen-ﬂux solution, which turns out to depend on the

ratio

(η/ω)

1 2

rather

than

η,

where

ω

is

the

frequency

of

time

vari-

ations of Br (Braginsky & Le Moue¨l 1993; Gubbins 1996). The approximation is clearly wrong for steady ﬁelds, where ω = 0. The

steady motion theorem of Voorhies & Backus (1985), which asserts

that ﬂows found by frozen-ﬂux theory are unique provided the ﬂow

remains steady, is ﬂawed for the same reason (Gubbins & Kelly

1996). The frozen-ﬂux approximation is only valid if the magnetic

ﬁeld changes sufﬁciently rapidly in time, yet it is only useful if the

ﬂow changes sufﬁciently slowly in time to allow an estimate to be

made.

In practice secular variation must be estimated by differencing

magnetic ﬁeld measurements made over an interval of time sufﬁ-

ciently long to reduce the errors incurred by the differencing—about

a decade for ground measurements but signiﬁcantly less for the data

from the new generation of satellites. This deﬁnes the shortest time

scale that can be studied. We may be lucky in having rapid geo-

magnetic changes, such as jerks, that appear to be consistent with

slower changes in ﬂuid ﬂow (Zatman & Bloxham 1997; Bloxham

et al. 2002). Success with frozen-ﬂux inversions of dynamical geo-

dynamo simulations could also be attributed to rapid variations in

ﬁeld associated with rather slow changes in ﬂow, perhaps as a re-

sult of advection of small scale magnetic ﬁeld. However, the main

evidence for diffusion at the surface of the Earth’s core comes from

changes in the reverse ﬂux patches in the southern hemisphere dur-

ing the last century or so, where upwelling appears to bring toroidal

ﬂux to the surface (Gubbins 2007). Diffusion may, and probably

does, act as strongly elsewhere where there is upwelling but we can-

not discriminate between diffusion and induction unless a patch of

ﬂux is surounded by a null-ﬂux curve where Br = 0.

4.3 Excursions and reversals
The present fall in dipole moment is related to reversal of ﬂux in the south Atlantic region and is thought by some to be the beginning of a geomagnetic excursion or even a reversal. The present ﬂux reversal is almost certainly caused by expulsion of toroidal ﬁeld from deeper within the core, which is rarely seen in dynamic models of the geodynamo. It appears similar to the development of a dynamo wave in the Braginsky regime, in which ﬂux reverses near the equator to produce a patch that migrates poleward (Gubbins & Sarson 1994; Gubbins & Gibbons 2002). The present development in the south Atlantic could be modelled as part of a dynamo wave cycle; a change in ﬂow is required to prevent the ﬁeld from oscillating regularly but the change need only be small in order to produce the desired effect. Consider a ﬂow with parameters on the boundary between oscillatory and steady solutions in Fig. 2, on the line emanating from the D = 1 Braginsky limit point. Such a ﬂow generates oscillatory and stationary magnetic ﬁelds at the same critical magnetic Reynolds number. Changing the ﬂow slightly favours the steady solution, changing it again will favour the oscillatory solution. By small, slow changes in time it is possible to construct a time-dependent ﬂow that produces reverse ﬂux patches as part of a dynamo wave (or overstability) but reverts to the steady ﬁeld before
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Implication of kinematic dynamo studies for the geodynamo 89

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progressing far through the wave cycle. Such events may be studied kinematically by constructing time-periodic ﬂows, where the repeat period of the ﬂow is much longer than the oscillation period of the dynamo wave. Leaving the ﬂow in the oscillatory state for longer would simulate an excursion, and leaving it for half a cycle produces a reversal (Willis & Gubbins 2004).
The symmetry of the dynamo equations under change of sign of ﬁeld has always meant reversed states were possible, the interesting question is how the geodynamo changes from one state to the other. From a dynamic point of view this may be achieved by symmetry breaking, where a solution of one symmetry becomes unstable to a solution of another symmetry. The critical points determining such evolution are those where solutions with more than one symmetry exist, such as the boundaries between preferred symmetries in Fig. 2. For example, ﬂows with M < 0 and small D lie close to the boundary between Da and Qa preferred solutions. A time-dependent solution with ﬂow in this region would produce a combination of the symmetries, with Da giving way to Qa, possibly producing hemispheric oscillations of the sort studied recently in dynamic calculations by Busse & Simitev (2006). A reversed Da may eventually be established, the transition ﬁeld indicating the nature of the reversal mechanism. The transition via an oscillatory solution is an alternative symmetry breaking (the symmetry in this case being invariance under translation in time); it seems a more likely candidate for geomagnetic reversals than Qa symmetry-breaking because it produces systematic VGP paths in the transition ﬁeld, as is sometimes observed (Love & Mazaud 1997). However, static ‘intermediate states’ occurring during transition, as have been proposed by Hoffman (2000) and others, might point to a different symmetry; and the same VGP path recorded at all sites would point to a the presence of a De mode. Stefani & Gerbeth (2005) model asymmetrical reversals, as proposed by Valet & Meynadier (1993), with a mean-ﬁeld dynamo, but the observations are disputed (Kok & Tauxe 1996; Mazaud 1996; Laj et al. 1996) and it is perhaps too early to read much into the comparison. The kinematic approach to reversals may, however, help interpret observed transition ﬁelds when data improve.
5 CONCLUSIONS
Kinematic studies provide a useful bridge between observation and dynamical theory by indicating the type of ﬂow in the core required to produce certain magnetic behaviour. They are limited because the ﬂuid ﬂow is always to some extent arbitrary, and only a small fraction of possible ﬂows can be studied. Nevertheless, some generic conclusions may be drawn about the type of ﬂow and dynamics responsible for generating a magnetic ﬁeld with some of the properties of the geomagnetic ﬁeld.
First, dynamo action requires helicity in the form deﬁned by Braginsky’s (Braginsky 1964). Furthermore, this helicity is related to the spiralling of convection rolls, which is known to occur in rotating convection but has not yet been observed in the geomagnetic ﬁeld. The Braginsky limit as originally applied involved implausibly large toroidal ﬁelds, but the asymptotics hold well away from the limit and can be usefully applied to more geophysically realistic kinematic dynamos. Magnetic ﬂux is concentrated at stagnation points in the ﬂow, suggesting that the centres of the four main lobes of the geomagnetic ﬁeld are centres of downwelling.
Secondly, the geomagnetic ﬁeld’s dipolar structure requires a speciﬁc combination of westward drift and eastward spiralling, corresponding to a positive dynamo number, as indicated by convection studies. Furthermore, stagnation points that concentrate ﬂux in high

latitudes also favour axial dipole ﬁelds over axial quadrupole or equatorial dipole ﬁelds because the axial dipole requires a change of sign of radial ﬁeld at the equator.
Finally, although geomagnetic-type reversals in which the ﬁeld retains the same polarity for long times between rapid reversals are not possible with steady ﬂow, they can be produced by ﬂow that changes very slightly, by less than 0.1 per cent in kinetic energy. The relatively infrequent nature of reversals and absence of oscillations is attributed to the spatial separation of toroidal and poloidal ﬁelds, the latter being strongest at high and low latitudes and the former, while not directly observed, being probably strongest in mid-latitudes.
ACKNOWLEDGMENTS
This work was partially supported by NERC consortium grant O/2001/00668. The author would like to thank the Shanghai Astronomical Observatory for their support and hospitality while ﬁnalizing this paper.
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