Geophys. J. Int. (2000) 140, F1^F4

Downloaded from https://academic.oup.com/gji/article/140/1/F1/2029834 by guest on 15 January 2024

FAST-TRACK PAPER
Is the geodynamo process intrinsically unstable?
K. Zhang1 and David Gubbins2
1 School of Mathematical Sciences, University of Exeter, EX4 4QJ, UK 2 School of Earth Sciences, Leeds University, Leeds, LS2 9JT, UK. E-mail: d.gubbins@earth.leeds.ac.uk
Accepted 1999 September 8. Received 1999 July 14; in original form 1999 March 2
SUMMARY Recent palaeomagnetic studies suggest that excursions of the geomagnetic ¢eld, during which the intensity drops suddenly by a factor of 5^10 and the local direction changes dramatically, are more common than previously expected. The `normal' state of the geomagnetic ¢eld, dominated by an axial dipole, seems to be interrupted every 30^100 kyr; it may not therefore be as stable as we thought. We have investigated a possible mechanism for the instability of the geodynamo by calculating the critical Rayleigh number (Rc) for the onset of convection in a rotating spherical shell permeated by an imposed magnetic ¢eld with both toroidal and poloidal components. We have found Rc to be a very sensitive function of the poloidal ¢eld at the very small Ekman number pertaining to the core. The magnetic Reynolds number, and therefore the dynamo action, is equally sensitive to the applied ¢eld because of its dependence on the di¡erence between the Rayleigh number and its critical value. This explains why numerical dynamo simulations at small Ekman number fail when similar magnetoconvection calculations succeed: the £uctuating magnetic ¢eld of the dynamo leads to rapid swings in convection strength that cannot be resolved numerically. The geodynamo may be unstable for the same reason, with the strength of convection varying wildly in response to the inevitable small changes in magnetic ¢eld. Frequent geomagnetic excursions may therefore be a manifestation of the instability arising from the core's very weak viscosity and the controlling e¡ects of the Earth's rotation.
Key words: geodynamo, magnetoconvection.

1 INTRODUCTION
Recent studies suggest that the Earth's magnetic ¢eld has fallen dramatically in magnitude and changed direction repeatedly since the last reversal 700 kyr ago (Langereis et al. 1997; Lund et al. 1998). These important results paint a rather di¡erent picture of the long-term behaviour of the ¢eld from the conventional one of a steady dipole reversing at random intervals: instead, the ¢eld appears to spend up to 20 per cent of its time in a weak, non-dipole state (Lund et al. 1998). One of us (Gubbins 1999) has suggested that this is evidence of a rapid natural timescale (500 yr) in the outer core, and that the magnetic ¢eld is usually prevented from reversing completely by the longer di¡usion time of the inner core (2^5 kyr). This raises a number of important but di¤cult questions for geodynamo theory. How can the geomagnetic ¢eld change so rapidly and dramatically? Can slight variations of the geomagnetic ¢eld a¡ect the dynamics of core convection signi¢cantly? If so, is the geodynamo process intrinsically unstable?
Of course, an ideal way to answer the above questions is to simulate the geodynamo directly (Glatzmaier & Roberts
ß 2000 RAS

1995; Glatzmaier & Roberts 1996; Kuang & Bloxham 1997; Jones et al. 1995). However, it is impossible to simulate the strong e¡ects of rotation in the Earth because it produces very small-scale solutions that vary rapidly with time (Zhang & Jones 1997). In this paper we argue, on the basis of results from magnetoconvection studies, that large swings in the geomagnetic ¢eld result from extreme sensitivity of core convection to changes in the poloidal geomagnetic ¢eld. Furthermore, this behaviour cannot be simulated by the present generation of geodynamo models, and may be the root cause of apparent numerical instabilities reported by some authors (e.g. Walker et al. 1998).
Core magnetohydrodynamics (MHD) is subtle because of the competing e¡ects of rotation and geomagnetic ¢elds. There are six major forces: Coriolis, Fc, buoyancy, Fb, magnetic (Lorentz), Fl, inertial, Fi, viscous, Fv, and pressure, Fp. These must be in balance at any instant of time:

FczFbzFlzFizFvzFp~0 X

(1)

In dimensionless form, with length measured by the core radius and time by the magnetic di¡usion time, the ratio of

F1

F2 K. Zhang and D. Gubbins
Coriolis to viscous forces is given by the Ekman number, E, which is extremely small [E~O(10{15)]; the buoyancy force is measured by the Rayleigh number, RE ; the Lorentz force by the Elsasser number, "; and the inertial forces by the Rossby number, Ro, which is also extremely small [Ro~O(10{7)]. Note that our Rayleigh number, RE, is the same as that used by Roberts (1968) and others in the problem of convection and di¡erent by a factor of E from the so-called modi¢ed Rayleigh number, Rmod~ERE . The pressure force, Fp, is passive in the Boussinesq approximation.
The small Ekman number E causes an intriguingly subtle balance among the six forces in eq. (1) that must be satis¢ed by a dynamic dynamo all the time. In non-magnetic convection a force balance is struck between buoyancy, pressure and viscous forces. This leads to very small length scales O(E1a3) and very large critical Rayleigh numbers, the smallest value of RE for which convection occurs, of order O(E{4a3) (Roberts 1968; Busse 1970). This makes numerical simulations di¤cult but possible at low Ekman number (E¦10{5, Sun et al. 1993). In magnetoconvection with an externally imposed magnetic ¢eld the primary force balance is magnetostrophicöbetween Coriolis, buoyancy, pressure and Lorentz forcesöprovided the magnetic ¢eld is strong enough [Elsasser number "~O(1)]. The viscous force is not required in the leading force balance; the solution can be large scale and therefore presents few numerical di¤culties. Why then is the dynamo calculation so much more di¤cult at low E, when the only di¡erence is that the ¢eld is self-generated rather than being imposed?

2 THE MAGNETOCONVECTION MODEL
We consider a spherical shell of electrically conducting Boussinesq £uid with constant thermal di¡usivity i, magnetic di¡usivity j, thermal expansion coe¤cient a and kinematic viscosity l in which convection is maintained by a uniform distribution of heat sources. The £uid is con¢ned in a spherical shell of inner radius ri and outer radius ro, with ri/ro~0X4. The whole system rotates with a constant angular velocity ). We assume that the inner and outer bounding spherical surfaces are stress-free and impenetrable, since it is well known that the choice of velocity boundary condition does not a¡ect the leading-order convection solution. Perfectly magnetic insulating boundaries are assumed at both the inner and outer bounding surfaces of the shell.
In the problem of magnetoconvection, we impose a largescale magnetic ¢eld upon the spherical shell of electrically conducting £uid. Our imposed axisymmetric magnetic ¢eld contains both toroidal and poloidal parts with dipole symmetry:

B~B0(BPzBT) ,

(2)

scaled so that jBPjmax~1 and jBTjmax~1. We use the same functional form of BT and BP as in a previous study of magnetic ¢eld instability (Zhang & Fearn 1994; Zhang & Fearn 1995).
Let us look at the form of the poloidal magnetic ¢eld BP as an example. Any mean poloidal ¢eld can be represented as a
linear combination of functions Hln(h, r) that are solutions of

(b2lnz+2)Hln(h, r)~0 ,

(3)

where bln are to be determined and spherical polar coordinates (r, h, 0) are used. The boundary conditions are to match the

potential ¢elds in the exterior of the shell that satisfy

+2Hln(h, r)~0

(4)

for r b ro or r ` ri, which yields

Hln(h, r)~Pl(cos h)[ jl(rbln)nl{1(robln){jl{1(robln)nl(rbln)] ,

(5)

where Pl(cos h) is the Legendre function, and jl(rbln) and nl(rbln) are the spherical Bessel functions of the ¢rst and second kinds. The parameter bln is then determined by the equation

jl (ribln)nl{1(robln){jl{1(robln)nl (ribln)~0 ,

(6)

where 0 ` bl1 ` bl2 ` bl3 F F F . Here n in bln re£ects the complexity of the poloidal ¢eld in the radial direction. We choose

the poloidal ¢eld with the largest scale with l~1 and n~1,





BP~

{r+2H11

z

1 r

L Lr

r2

LH11 Lr

rª

z

1 r

L Lh

L(rH11) Lr

hê

.

(7)

Our toroidal ¢eld BT is chosen in a very similar way. The magnetoconvection problem is characterized by ¢ve
independent dimensionless parameters: the Rayleigh number, RE, the Ekman number, E, the Prandtl number, Pr, the Roberts number, q, and the Elsasser number, ", all with the usual de¢nitions. Pr and q represent material properties of an electrically conducting £uid.
The primary objective of our calculation is to show that the MHD convection system is very sensitive to small variations of the poloidal ¢eld at small E, which can lead to rapid swings in convection strength that cannot be resolved numerically and which may lead to instabilities of a geodynamo model. To achieve this objective, we have neglected the inertial term Lu/Ltzu ? +u in the equation of motion by taking the large Pr limit (see also Glatzmaier & Roberts 1995). This can be justi¢ed on the basis that convection relevant to dynamo action is on a much longer timescale than the period of rotation. We take the Roberts number q~1 and ¢x the Elsasser number at "~10. This value of " ensures we are in the strong-¢eld regime, in which length scales are large and the main force balance is magnetostrophic. It is also typical of the values obtained from large-scale geodynamo calculations, which have "~O(10) based on the average ¢eld (Sarson et al. 1998).
We ¢x E, Pr, q and " and solve the equations of motion, heat and induction simultaneously for many di¡erent values of the Rayleigh number RE, to determine the smallest RE (which is referred to as Rc) at which convection can take place. Repeating the calculations for di¡erent values of , the strength of the poloidal ¢eld, gives the variation of Rc with .
Our simulation cannot reach the value of the Ekman number for the core, which is about 10{15. However, we can reach the asymptotic region for small E. Table 1 gives results for ~0. It shows the solution approaching a limit with Rc&12E{1 and drift rate C&8X5 as E?0. The scaling of Rc arises from the necessity of buoyancy to remain in the force balance.
We have simulated 30 solutions at small E by increasing  gradually from zero. The results are shown in Fig. 1 for the most unstable linear mode, which is m~1, except very close to ~0, when Rc for m~2 becomes comparable. Increasing  slightly from zero to 0.07 reduces the critical Rayleigh number Rc from Rc&12E{1 to Rc&1. The range of Rc is larger for smaller E: at E~10{15 it ranges from 1 to 1016, a huge e¡ect

ß 2000 RAS, GJI 140, F1^F4

Downloaded from https://academic.oup.com/gji/article/140/1/F1/2029834 by guest on 15 January 2024

Table 1. Results of linear magnetoconvection calculation for ~0, m~1, showing approach to an asymptotic limit as E?0. C is the dimensionless oscillation frequency of the solution (drift rate).

E

Rc

ERc C

1X0¾10{4 9X68¾104 9X69 7X02 5X0¾10{5 2X12¾105 10X1 7X61 1X0¾10{5 1X18¾106 11X9 8X12 5X0¾10{6 2X42¾106 12X1 8X25 1X0¾10{6 12X3¾106 12X3 8X32

for such a small change in the ¢eld. The corresponding drift rate changes from positive (eastwards) to negative (westwards). We therefore expect the amplitude and pattern of convection, and hence the size of Rm, also to change dramatically in response to small variations in the poloidal ¢eld.
3 IMPLICATIONS FOR NUMERICAL SIMULATIONS
Now consider the more complicated scenario when the magnetic ¢eld is self-generated by dynamo action. Di¡erential rotation can produce a large BT with dipole symmetry, and a poloidal ¢eld will arise from the action of radial motion on the toroidal ¢eld, or from small-scale £ows, with timescales of centuries.
Negative values of Rc in Fig. 1 correspond to magnetic instabilities, which draw their energy from the imposed magnetic ¢eld rather than the applied heat sources. Such a solution could not be maintained inde¢nitely if the ¢eld were generated from dynamo action, although it could occur temporarily as part of a time-dependent solution. The part of the curve with Rc§0 is therefore the most relevant for dynamo calculations.

Figure 1. The scaled critical Rayleigh number, E{1Rc, and the corresponding drift rate, C, are plotted against , the poloidal ¢eld strength. E~10{4 here, but Table 1 shows this is in the asymptotically small Ekman number regime.
ß 2000 RAS, GJI 140, F1^F4

Is the geodynamo process intrinsically unstable? F3
When the Ekman number is small, Rc can be wildly and sensitively dependent on the strength and form of the magnetic ¢eld. It can change from Rc~O(1020) when the magnetic ¢eld is weak (Busse 1970; Zhang 1992) to Rc&1 in the magnetostrophic regime. We show here that Rc is extremely sensitive to small variations in the magnetic ¢eld, particularly the poloidal ¢eld. It follows that Rm can swing rapidly through a wide range of values because of its dependence on (RE{Rc). It is di¤cult to anticipate the existence of a quasi-steady geodynamo if the whole system is so sensitive to small variations of the ¢eld and Rm varies so wildly. The results suggest that the dynamic geodynamo is intrinsically unstable and is characterized by a strong time dependence.
These results also provide a clue as to why numerical integrations of an Earth-like dynamo model characterized by small Ekman number (rather than hyperdi¡usivity with large e¡ective Ekman number at small wavelength) prove to be formidably di¤cult (Walker et al. 1998), while no such di¤culties arise in the corresponding non-magnetic problem (Sun et al. 1993). Although non-magnetic convection may be highly chaotic, the driving force that determines the average amplitude of convection, measured by (RE{Rc), is ¢xed and time-independent. In the geodynamo problem, because the Lorentz force enters eq. (1) together with a small Ekman number, the dynamic balance becomes highly variable even though RE is ¢xed. This inevitably leads to rapid variations and collapses of the magnetic ¢eld, and the many numerical problems that arise in computer simulations.
This discussion is based on linear simulations of the magnetoconvection, but the real problem is non-linear. However, we believe the dynamic behaviour discussed in this paper would be manifested in the non-linear problem because all the key elements of the dynamic force balance (1) in the £uid core at small Ekman number have been captured. We imagine a non-linear solution in which the applied ¢eld varies with time. The linear calculations reported here will not re£ect this time dependence, which changes the nature of the stability analysis. However, there is no reason to expect the strong dependence of Rc on Bp to change. We shall investigate the e¡ects of a time-dependent ¢eld in a future study.
We also imagine the ¢eld to be self-generated through a dynamo mechanism rather than imposed, which is much more di¤cult to investigate or quantify. Dynamo action occurs through non-linear interaction of the convection with the magnetic ¢eld via the term +¾(v¾B) in the induction equation, which may stabilize or destabilizethe system. A further study of the dependence of the generated ¢eld on the £ow is underway using kinematic dynamo theory.
4 IMPLICATIONS FOR THE EARTH'S MAGNETIC FIELD
This study was inspired by recent palaeomagnetic results, which suggest the geomagnetic ¢eld is rather unstable and undergoes collapses in strength and large changes in direction after a few tens of thousands of years. The interval of weak, non-dipolar ¢eld lasts only 2^5 kyr before the ¢eld grows once more to its typical modern strength and dipole-dominated character.
Current geodynamo simulations do not show such dramatic behaviour. We attribute the extra stability to the larger e¡ective viscosity in the numerical calculations, necessitated by the

Downloaded from https://academic.oup.com/gji/article/140/1/F1/2029834 by guest on 15 January 2024

Downloaded from https://academic.oup.com/gji/article/140/1/F1/2029834 by guest on 15 January 2024

F4 K. Zhang and D. Gubbins
limited temporal (and spatial) resolution o¡ered by even the largest computers available. High viscosity applied to smallwavelength convection will prevent it from reaching the high Rc regime in Fig. 1, thus limiting the swings in convective and dynamo power.
We therefore envisage a true geodynamo operating mainly in magnetostrophic balance with occasional collapses into a highRc regime. What happens then is a matter for speculation at the moment because the £ow would be small scale, rapidly timevarying and beyond present numerical resolution. Observations show clearly that geodynamo action continues and the largescale, magnetostrophic state is quickly re-established. The inner core may play a stabilizing role by giving the poloidal ¢eld a longer timescale based on electrical di¡usion rather than £uid advection (Hollerbach & Jones 1993; Gubbins 1999).
Our intention has been to isolate the e¡ect of a poloidal magnetic ¢eld on magnetoconvection in a rapidly rotating spherical system and to show it can have a dramatic e¡ect on the convection. We have therefore excluded other possible e¡ects such as those of di¡erential rotation, which may be important in understanding the exchange of angular momentum between core and mantle (Jault et al. 1988), and non-linear stability in the magnetostrophic approximation (e.g. McLean & Fearn 1999). The e¡ects of di¡erential rotation are expected to be of secondary importance in eq. (1) simply because the inertial term u ? +u can be neglected to leading order on the long timescale that is relevant to dynamo action.
The di¤cult theoretical question now posed is not why the geodynamo is so unstable, but why the large-scale magnetostrophic state is as stable as it is, persisting for tens of thousands of years or about one magnetic di¡usion time.
ACKNOWLEDGMENTS
This work was supported by PPARC grant GR/L22973 (KZ) and by NERC grant GR3/9741 (DG).
REFERENCES
Busse, F.H., 1970. Thermal instabilities in rotating systems, J. Fluid Mech., 44, 444^460.
Glatzmaier, G.A. & Roberts, P.H., 1995. A three-dimensional convective dynamo solution with rotating and ¢nitely conducting inner core and mantle, Phys. Earth planet. Inter., 91, 63^75.

Glatzmaier, G.A. & Roberts, P.H., 1996. An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection, Physcica D, 97, 81^94.
Gubbins, D., 1999. The distinction between geomagnetic excursions and reversals, Geophys. J. Int., 137, F1^F3.
Hollerbach, R. & Jones, C.A., 1993. In£uence of the Earth's inner core on geomagnetic £uctuations and reversals, Nature, 365, 541^543.
Jault, D., Gire, C. & Le Moue« l, J.-L., 1988. Westward drift, core motions and exchanges of angular momentum between core and mantle, Nature, 333, 353^356.
Jones, C.A., Longbottom, A.W. & Hollerbach, R., 1995. A selfconsistent convection driven geodynamo model, using a mean ¢eld approximation, Phys. Earth planet. Inter., 92, 119^141.
Kuang, W. & Bloxham, J., 1997. An Earth-like numerical dynamo model, Nature, 389, 371^374.
Langereis, C.G., Dekkers, M.J., de Lange, G.J., Paterne, M. & van Santvoort, P.J.M., 1997. Magnetostratigraphy and astronomical calibration of the last 1.1 Myr from an eastern Mediterranean piston core and dating of short events in the Brunhes, Geophys. J. Int., 129, 75^94.
Lund, S.P., Acton, G., Clement, B., Hastedt, M., Okada, M. & Williams, R., 1998. Geomagnetic ¢eld excursions occurred often during the last million years, EOS, Trans. Am. geophys. Un., 79, 178^179.
McLean, D.R. & Fearn, D.R., 1999. The geostrophic nonlinearity and its e¡ect on magnetic instability, Geophys. Astrophys. Fluid Dyn., 90, 79^112.
Roberts, P.H., 1968. On the thermal instability of a self-gravitating £uid sphere containing heat sources, Phil. Trans. R. Soc. Lond. A, 263, 93^117.
Sarson, G.R., Jones, C.A. & Longbottom, A.W., 1998. Convection driven geodynamo models of varying Ekman number, Geophys. Astrophys. Fluid Dyn., 88, 225^259.
Sun, Z.P., Schubert, G. & Glatzmaier, G.A., 1993. Banded surface £ow maintained by convection in a model of the rapidly rotating giant planets, Science, 260, 661^664.
Walker, M.R., Barenghi, C.F. & Jones, C.A., 1998. A note on dynamo action at asymptotically small Ekman number, Geophys. Astrophys. Fluid Dyn., 88, 261^275.
Zhang, K., 1992. Spiralling columnar convection in rapidly rotating spherical £uid shells, J. Fluid Mech., 236, 535^556.
Zhang, K. & Fearn, D., 1994. Hydromagnetic waves in a rotating spherical shell generated by toroidal decay modes, Geophys. Astrophys. Fluid Dyn., 77, 123^147.
Zhang, K. & Fearn, D., 1995. Hydromagnetic waves in a rotating spherical shell generated by poloidal decay modes, Geophys. Astrophys. Fluid Dyn., 81, 193^209.
Zhang, K. & Jones, C.A., 1997. The e¡ect of hyperviscosity on geodynamo models, Geophys. Res. Lett., 24, 2869^2872.

ß 2000 RAS, GJI 140, F1^F4