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A test of the frozen-° ux approximation using a new geodynamo model
By Paul H. R obert s1 a n d G a r y A. G lat z ma ier2
1Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095, USA
2Earth Sciences Department, University of California, Santa Cruz, CA 95064, USA
The physics underlying the frozen-®ux approximation is reviewed. Recent highresolution geodynamo simulations, described here for the ­ rst time, a¬ord opportunities of testing the approximation, and the results from a very simple test are reported. This consists in evaluating the unsigned ®ux of magnetic ­ eld from the core at three successive epochs separated by about 150 years, and in showing that this changes by only ca. 3% in the intervals between the epochs. Because of the smallness of this change, which is created by the electromagnetic di¬usion excluded in the frozen-®ux approximation but present in the simulations, we argue that the approximation is useful in analogous analyses of the geomagnetic ­ eld over similar time-scales.
Keywords: geodynamo; computer simulations; frozen ° ux

1. Background to the approximation

The magnetic ­ eld, B , in a body of conducting ®uid such as the Earth’s core is governed by the induction equation

@B = r [v B @t

r B ];

(1.1)

and by

r B = 0:

(1.2)

Here v is the velocity of the ®uid and is its magnetic di¬usivity. Equation (1.2) is an initial condition; if it holds at time t = 0, it holds for all t according to (1.1).
Let us introduce a characteristic length-scale, L , for B and a characteristic ®ow speed, V . Dimensionless analysis of (1.1) shows that B evolves on two time-scales, the advection time-scale, v, and the time-scale, :

v = L =V ;

= L 2= :

(1.3)

The ratio of these time-scales is the magnetic Reynolds number,

R= = v=V L = :

(1.4)

In order of magnitude this is the ratio of the second term in (1.1) to the third and, if

R 1;

(1.5)

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we are encouraged to omit the third term, giving the induction equation for a perfectly conducting ®uid ( = 0):

@B = r @t

(v

B ):

(1.6)

Alfv´en’s frozen-®ux theorem follows from (1.6): `in a perfectly conducting ®uid, mag-

netic ®ux tubes are material volumes’, i.e. they move with the ®uid as though frozen

to it.

In applying these results to the geomagnetic ­ eld, we shall suppose that B at a

given point x in the core varies rapidly, on the time-scale v. By (1.1), the relative motion between material volumes and ®ux tubes is of order =L , which is small

compared with V by (1.5). The variation of B at x may therefore be visualized

through Alfv´en’s theorem as the rearrangement of pre-existing magnetic ®ux by the

®uid velocity v. But, if we move with a small element of ®uid volume V as it is

advected by the ®ow, we would ­ nd according to (1.1) that the ®ux threading V

changes slowly, on the time-scale

v. If, then, we follow V over a `time-scale

of observation’, O , that is of order or greater, we would ­ nd that the change in

the threading ®ux would be so substantial as to make the frozen-®ux approximation

(1.6) valueless. In short, inequality (1.5) is an insu¯ cient justi­ cation for replacing

(1.1) by (1.6). Necessary conditions for frozen ®ux to be applicable are

v

;O

:

(1.7)

When both these hold, Alfv´en’s theorem is useful both analytically and as a means of visualizing induction processes in the core.

2. Applicability of the approximation to the Earth’s core

The period of time over which data of su¯ cient quality exist to test (1.6), and to

make use of (1.6) if they survive the test, is not even as great as 100 years, though we

shall take O = 100 yr. The traditional way of estimating V for the core is through the westward drift of the ­ eld. This suggests that V = 5 10 4 m s 1 for features

of scale L 103 km, so that v 65 yr, which is of the same order as the eddy turnover time in the Glatzmaier{Roberts geodynamo simulations (see Glatzmaier &

Roberts 1996b, 1997). Taking = 2 m2 s 1 (see, for example, Braginsky & Roberts

1995, Appendix E), we ­ nd that R 250 and

1:6 104 yr. It appears that both

inequalities (1.7) are obeyed.

Because of its low kinematic viscosity, , the core is certainly in turbulent motion,

with kinetic and magnetic energies spread over many scales L . The use of the single

scale above is clearly somewhat simplistic. Considered as a function of eddy size,

the magnetic Reynolds number R should ultimately decrease with L so that (1.5)

is violated for all su¯ ciently small L . Let us estimate very approximately the scale

L c for which R(L c) = 1; roughly speaking, (1.6) is reasonable for L > L c and is unreasonable for L < L c. The energy spectrum of the core is unknown, but let us suppose (as suggested by the computer simulations to be described in x 3) that the

kinetic energy K` in wavenumber ` is approximately proportional to ` 2:4. (Here ` refers to the spherical harmonic order in the decomposition of the total energy,

poloidal and toroidal; see also (2.2).) This suggests that V / L 1:2 for eddies of scale

L , and that R / L 2:2 for these eddies. If R = 250 for L 103 km (see above), then

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R = 1 for L c 80 km. This is comparable with the smallest scale, ` = L, of B that

can be resolved on the core surface; L 12{13 (see below). For L c 80 km, we have

=L

2 c

=

100 yr. We conclude that both of the inequalities (1.7) are marginally

satis­ ed for ` = L and O 100 yr, and are increasingly well obeyed as ` decreases

from L. If this is true then, as suggested by Roberts & Scott (1965), (1.6) should be

a good approximation to (1.1) for the accessible length- and time-scales.

The Roberts{Scott idea encountered immediate di¯ culties. First, B and @B =@t

are known at the Earth’s surface, r = a, but (1.6) at best applies at S, the core

surface, r = c. In between lies the mantle, and it is necessary to extrapolate B

through this. Since the electromagnetic time constant of the mantle, M , is much

smaller than v (see, for example, Gubbins & Roberts 1987, x 3.3), RM

M = v 1.

This means that to a good approximation the mantle is an electrical insulator and,

if there are no other sources in the mantle, B is a potential ­ eld, B^ , not only on and

above the Earth’s surface but also within the mantle:

B^ = rV; for r > c;

(2.1)

where r2V = 0 and where the usual expansion in exterior spherical harmonics applies:

X 1 X`

V =a

[g`m(t) cos m + hm ` (t) sin m ](a=r)`+ 1P`m( ):

`= 1 m= 0

(2.2)

Here (r; ; ) are geocentric spherical coordinates, with = 0 as the north polar axis,

P`m are the Schmidt normalized Legendre functions and g`n and hm ` are the Gauss coe¯ cients.
In principle, we can evaluate B^ at the base of the mantle by carrying out the di¬er-

entiations (2.1) and setting r = c. In practice, di¯ culties arise. First, the assumption

that there are no other sources of B between r = a and r = c is incorrect; there is

signi­ cant permanent magnetism in the crust, which makes the sum (2.2) meaning-

less at r = c if taken beyond a cut-o¬ ` = L of order 12 to 13. Second, because of

inaccuracies and poor spatial coverage in the data, arise; these increase with `, and they produce errors

ienrrBo^rsthinatga`mreafnudrthhem ` r

inevitably enhanced,

by a factor proportional to (a=c)` that increases with `, during extrapolation to the

core surface. Nevertheless, when the series (2.2) is terminated at a value L 12, techniques have been developed through which B^ can be obtained on S with some

degree of con­ dence, as many studies (too numerous to reference here) attest.

A further di¯ culty concerns the no-slip conditions which, in the reference frame

rotating with the mantle (the frame we shall now employ until x 3), are

r v = 0; on S r v = 0; on S

(2.3) (2.4)

We have here used the radius vector r from the geocentre as the normal to S. If we applied (2.4) to (1.6), we would ­ nd that B^ is time-independent in the mantle frame. To resolve this absurdity, we have to consider the magnetohydrodynamics (MHD) of the core. The (molecular) viscosity, , of the core is thought to be smaller by a factor of about 106 than the (molecular) magnetic di¬usivity, (see, for example, Braginsky & Roberts 1995; De Wijs et al . 1998). Plausibly, therefore, there is a thin

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boundary layer on S of Ekman-layer type in which viscous forces are signi­ cant. In the `mainstream’ beneath the layer, the ®ow is inviscid to leading order, and can obey (2.3) but not (2.4). The boundary-layer ®ow matches to the mainstream at a level indistinguishable from r = c that we shall call the `top of the core’; the boundary-layer ®ow also satis­ es conditions (2.3) and (2.4) on S. In this process it creates a small violation of (2.3) at the top of the core through a pumping process of the type associated with Ekman layers (see, for example, Greenspan 1968). We shall ignore this, and shall apply (2.3) at the top of the core, where the ®ow is therefore horizontal; we denote it by vH and term it the `sur­ cial velocity’. It is clear from (1.2) that, to a very good approximation, the normal component Br of B is continuous across the boundary layer and is therefore equal to B^ r on S.
The frozen-®ux approximation has been used on numerous occasions in e¬orts to determine vH. Because Br does not change across the boundary layer, it is attractive to work with the r-component of (1.6), which is

@B^ r @t

=

rH

(vrB^ H

B^ rvH):

Ignoring pumping by the boundary layer, we ­ nd from (2.5) that

(2.5)

@B^ r @t

+

rH

(B^ r vH) = 0;

which may also be written as

(2.6)

@B^ r @t

+ vH

rHB^ r =

B^ rrH vH:

(2.7)

The frozen-®ux approximation is disappointing in that it is impossible to derive

the sur­ cial velocity in full (or perhaps at all) from the ­ eld. There are serious

problems of non-existence and non-uniqueness that should be addressed. Concerning

non-existence, we introduce (following Backus 1968) the null ° ux curves, i.e. the curves on S on which B^ r is zero. An area of S that is enclosed by a null ®ux curve and in which B^ r has everywhere the same sign is called a null ° ux patch. There is one principal null ®ux curve, the magnetic equator, and several subsidiary null

®ux curves. It is easily shown from (2.6) that, since vH is ­ nite everywhere and in

particular on the null ®ux curves,

Z

d dt

B^ r dS = 0;
P

(2.8)

where the integral is taken over a null ®ux patch P and d=dt includes the time

evolution of P . This result is no more than an example of Alfv´en’s theorem applied

to the cross-section, P , of a ®ux tube emerging from the core. If we take all the null

®ux curves together, we see from (2.8) that

I

jB^ rj dS = constant for all t;

(2.9)

S

this integral of unsigned ° ux, jB^ rj, being taken over the entire core surface S. Of course, (2.9) does not imply that (2.8) holds for each individual null ®ux patch.

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There are two reasons why no straightforward analysis of the geomagnetic data will con­ rm (2.8), where by `straightforward analysis’ we mean one that ignores the frozen-®ux approximation. First, there are what we shall call errors of type 1: even if (2.8) were true, data analysis would not con­ rm it because of errors in, and the incompleteness of, available data and because of the di¯ culties inherent in extrapolating ­ elds accurately from the Earth’s surface to the core surface. Second (the error of type 2), because of electromagnetic di¬usion, (2.8) is not precisely true anyway. The basic tenet of Roberts & Scott (1965) was that error 1 is much more signi­ cant than error 2. If this is true, all claims that deviations from (2.8) have been discovered must be viewed with suspicion. Moreover, if error 2 is swamped by error 1, nothing is lost by eliminating error 2 totally through a non-straightforward data analysis in which (2.8) is imposed as a constraint.
The idea of adding constraints to geomagnetic analyses is, of course, nothing new. For example, analyses are often performed with the series (2.2) rather than the full representation that also includes the exterior harmonics; in other words, the analyses have been constrained a priori to exclude the exterior harmonics. The Roberts{ Scott proposal was merely to add a new form of constraint. Such analyses have been performed by Constable et al. (1993) and O’Brien et al . (1997). Not only is nothing lost by such analyses, but also something is gained, namely the certainty that a ­ nite vH exists. This vH is not unique. Roberts & Scott (1965) provided a simple example. If the sur­ cial velocity is purely toroidal, rH vH = 0, and (2.7) reduces to

@B^ r @t

+

vH

rHB^ r = 0:

(2.10)

If vH is one solution to (2.10) for the assigned B^ r and @B^ r=@t, then vH + r rHf is another, where f (B^ r) is any function that is constant on each constant-B^ r contour on S. A much more complete and satisfying discussion of non-existence and non-

uniqueness was later provided by Backus (1968). Several proposals have been made

to remove the non-uniqueness by supplementing the frozen-®ux approximation with

additional hypotheses, but we shall not review these.

The developments just described are all based on the idea that the frozen-®ux

approximation is an adequate way of describing the evolution of B in the mainstream

beneath the boundary layer, i.e. that not only but also can be assumed to be

zero at leading order. Since the geodynamo problem concerns countering magnetic

di¬usion by motional induction, how can it be right to use (1.6) instead of (1.1) in

the mainstream? Recently Love (1999) has given two ingenious kinematic examples

to show how (1.6) leads to vH quite di¬erent from the actual sur­ cial velocity of the ®ow maintaining the dynamo. The ­ rst of these is steady, and therefore does not

satisfy the requirement of x 1 that B evolves on the v time-scale. The same seems to be true of Love’s (1999) second model, though both v and B are time dependent

in that case. Since we have no ambitions in this paper to infer vH, we shall not enter into these controversial matters here.

Other doubts about the value of the frozen-®ux approximation hinge on whether

errors of type 2 are smaller than those of type 1. This is a delicate matter for, while

error 2 must be greater for a small-scale patch than a large-scale patch because its

electromagnetic time constant is smaller, error 1 is also greater because the patch

is described by higher harmonics ` of the ­ eld, the errors in which are much larger,

especially after extrapolation to S. In claiming that they have strong evidence that

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new ®ux is emerging from the core into the mantle from a secular variation centre under and to the west of Africa, Bloxham & Gubbins (1986) are implying that error 2 can be detected despite error 1. If there is a strong subsurface toroidal ­ eld in the core, it is easy to imagine that a poloidal upwelling, in which rH vH 6= 0, will push toroidal ­ eld through S and into the mantle. Several models of such a process have been constructed (Coulomb 1954, 1955; Allan & Bullard 1958, 1966; Hide & Roberts 1961; Nagata & Rikitake 1961; Rikitake 1967; Bloxham 1986; Drew 1993). There is no doubt that such processes can and must occur in the Earth, but it is less clear how far they discredit the frozen-®ux approximation. At the time the early papers just referenced were written, it appeared that they posed a serious threat to the frozen®ux approximation, but in recent simulations of the geodynamo the toroidal ­ elds have been comparable with the poloidal ­ elds, not much larger as had previously been expected (see, for example, Glatzmaier & Roberts 1996b, 1997). These geodynamo simulations also suggest that toroidal ­ elds are relatively weak near the core{mantle boundary compared with those near the inner-core boundary. The creation of new poloidal ®ux in the mantle by upwellings in the core may therefore not be as rapid as had been thought.
A further concern centres on the e¬ects of small-scale turbulence on the large-scale ­ eld and ®ow (belonging to ` L). The EMF v B created by small-scale v and B has a part that alters the large-scale ­ eld. The way it does so de­ nes the subject of mean ¯eld electrodynamics (see, for example, Krause & Radler 1980). The large-scale EMF from the turbulence is parametrized as B r B (for example), where B is now the large-scale ­ eld; B de­ nes the so-called -e¬ect and represents a turbulent enhancement of the molecular di¬usivity . Braginsky & Meytlis (1990) argue cogently that these terms are insigni­ cant in the MHD of the Earth’s core.
The situation is di¬erent for the turbulent transport of momentum. As we have seen, the molecular viscosity of the core is about 106 times smaller than , and turbulence is required, and exists, to transport momentum. The Reynolds analogy provides an approximate way of parametrizing this process, through the introduction of a `turbulent viscosity’ t of order . Turbulent transport alters the structure of boundary layers, an e¬ect that has been observed, for example, in ocean physics (see Hunkins 1966). Because t , the boundary layer on S may more closely resemble an Ekman{Hartmann layer (Gilman & Benton 1968) than the Ekman layer invoked earlier. If so, r B will change signi­ cantly across the layer, so that r B at the top of the core will di¬er from r B^ on S. The tangential components of (1.6) will then not be immediately useful in inferring vH. The boundary layer is thin, however, so that the validity of (2.6) is not a¬ected.
Questions similar to these arise in numerical simulations, where the e¬ect of the unresolved scales of v and B on the resolved scales may require parametrization. This matter will arise below, where we shall suppose (as above) that the molecular EMF need not be modi­ ed. We shall ­ nd it necessary, however, to increase even beyond t .

3. Application of the approximation to numerical simulations
The uncertainties introduced by the data errors and their incompleteness, combined with the di¯ culties of extrapolation to the core surface, make it hard to decide how seriously the frozen-®ux approximation is violated in the Earth’s core during the
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time O 100 yr over which it has been closely observed. Recently, however, there have been several numerical simulations of the geodynamo that have produced ­ elds that have resembled the geomagnetic ­ eld both in structure and evolution (see, for example, Glatzmaier & Roberts 1996b, 1997). Since simulations directly provide B on the core surface, they completely eliminate errors of type 1, and they do so in an Earth-like context. The idea of using such simulations to test the frozen-®ux approximation is attractive and has been attempted by Glatzmaier & Roberts (1996a), with encouraging results. But these tests can be criticized because the simulations on which they are based required hyperdi¬usion in order to secure numerical convergence, and hyperdi¬usion of B violates frozen ®ux. Recently, however, we have produced a very high-resolution geodynamo simulation without hyperdi¬usion, which is quite suitable for testing frozen ®ux. Results from this model are presented here for the ­ rst time. We know a priori that type-2 errors must be present, so that (2.8) cannot be precisely true. The test of the frozen-®ux hypothesis must therefore be modi­ ed. It becomes `how well is (2.8) obeyed’ ? We shall ask only `how well is the unsigned ®ux (2.9) from the core conserved’ ? Since (2.9) does not imply (2.8), this is a more limited objective.
We solve the MHD convection equations in the form proposed by Braginsky & Roberts (1995) and in a frame of reference in which the total angular momentum of Earth is zero. This means that the mantle and therefore the core{mantle boundary are in slow solid-body rotation, so although (2.4) is satis­ ed in the frame of the mantle, it is not in our frame of reference. This rotation, which is quite small (of order 3 10 4 deg yr 1), has not been removed in the ­ gures presented here. All di¬usivities, with the exception of , are assumed to be 7 m2 s 1. Because we have used = 7 m2 s 1 rather than the geophysically more realistic 2 m2 s 1, we shall, in what follows, reinterpret time-scales by multiplying them by the ratio, 3.5, of these two values of , so that all velocities will be reduced by the same factor. For example, the scaled rotation of the core boundary is of order 10 4 deg yr 1. Nevertheless, we recognize that such a rescaling of the MHD equations strictly requires the rotation of Earth to be reduced by the same factor, and this has not been done. Typical core velocities V are of order 3 10 4 m s 1, and the overturning time of large-scale core eddies is ca. 200 yr. These values are consistent with those used in x 2 to motivate the frozen-®ux approximation. As in our earlier simulations, it is necessary to use a viscosity even larger than ; after scaling it is 1500 m2 s 1.
Glatzmaier & Roberts (1996b) used the spectral transform technique in their numerical work. In their 1996 simulation, all harmonics were included in the expansion of v and B up to ` = m = 21. In the present highly resolved model, a trapezoidal truncation is employed: 120 Fourier modes (0 m 119) in longitude are included and, for each m, all values of ` are present from m to 120 + m. The largest value of ` appearing anywhere is therefore ` = 239, which occurs for m = 119. The Chebychev expansion in r was up to 48 in the ®uid core and up to 32 in the solid core. The integration was initiated from a case of low resolution.
The greater spatial resolution of the new model (60 times as many spherical harmonics) required a small numerical time-step (7 days) because of the demands of the magnetic Courant condition. The computational expense is very great, and so far the model has been integrated for only about 1400 years of simulated time. This was far enough for all the transients connected with the initialization of the computation to have disappeared. This is con­ rmed by ­ gure 1, which shows the magnetic and

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magnetic kinetic

log energy

0

50

100

150

200

239

spherical harmonic degree

Figure 1. Power spectra of the magnetic energy (upper curve) and kinetic energy (lower curve) integrated throughout the ° uid core at epoch 2, as a function of ` for 0 < ` < 240. The spectra are given logarithmically, in arbitrary units which cover 10 decades; the kinetic energy is consistently about a thousand times less than the magnetic energy. The representation of the ¯eld and ° ow used to obtain these results employed a trapezoidal truncation; 120 Fourier modes (0 m 119) in longitude are included and, for each m, all values of ` are present from m to 120 + m. The largest value of ` appearing anywhere is therefore ` = 239, which occurs for m = 119.

kinetic energies (relative to the rotating frame of reference) integrated throughout the ®uid core as a function of the spherical harmonic degree `. It was not, however, long enough for the system to have evolved far from its initial state, which was quite soon after a magnetic dipole reversal had taken place. This may be seen both from the comparatively small energy in the ` = 1 ­ eld (­ gure 1), and later from the untypically large deviation of the geomagnetic equator from the geographical equator in one large band of longitude. The fact that the ­ eld is transitional in no way detracts from its usefulness in testing the frozen-®ux approximation, quite the reverse in fact. Since the unsigned ®ux changes more rapidly during transition than between reversals, these ­ elds provide a particularly stringent test of the approximation. Figure 1 also demonstrates that the solution has converged satisfactorily, without hyperdiffusion. The magnetic energy is about three orders of magnitude greater than the kinetic energy, and has a broad peak between harmonic degrees ` = 5{15.
We focus on three snapshots of the solution, each separated from its neighbour by approximately 150 years of simulated time ( O = 150 yr). We shall refer to these

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(a)

(d)

1117

(b)

(e)

(c)

(f)

Figure 2. Contours of equal B^r on the core surface (a){(c) and on the Earth’ s surface (d){(f ). (a); (d) epoch 1; (b); (e) epoch 2; (c); (f ) epoch 3. The truncation level L is 12. The bold full curves are magnetic equators, B^r = 0. On the continuous curves B^r > 0; on the dashed curves B^r < 0. The contour interval is 7 G for the core surface and 0.1 G for the Earth’ s surface.
as `epoch 1’, `epoch 2’ and `epoch 3’. (Because of the infrequency of data dumps, separations at the shorter O 100 yr intervals were not available.) Figure 2 shows, in equal-area projections, contours of constant B^ r on the Earth’s surface and on the core surface for each epoch. Figure 2 is drawn for L = 12, and ­ gure 3 presents the same plots for L = 72. It is striking how much of the small-scale structure, associated with large `, is lost at the Earth’s surface through the geometric attenuation factor (c=a)`+ 2 (see (2.2)). Using epoch 3 as an illustration, ­ gure 4 shows how structure develops on the core surface as L is increased. The changes from one panel to the next tend to diminish as L increases; the ­ elds at L = 60 and L = 72 are very similar. This is also consistent with the results shown in table 1, where the maximum and minimum B^ r are given for the same six values of L as ­ gure 4.
These ­ gures underscore the limitations imposed by geometry in extrapolating B^ r downwards; it is quite di¯ cult to distinguish the plots of B^ r at the core surface for the L = 12 truncation from those for the L = 72 truncation at the same epoch, even though the ­ elds at the core surface are very di¬erent. (Indeed, the same may be said at the L = 12 and L = 24 levels.) Certainly, the same must be true for the Earth. Estimates of B^ r at the Earth’s core{mantle boundary using L = 12 are necessarily of large scale, but these highly ­ ltered results do not imply that the actual core{mantle boundary ­ eld is dominantly of large scale. Likewise, the sur­ cial ®ow, vH, based on these highly ­ ltered core{mantle boundary-­ eld structures will also tend to be of
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(a)

(d )

(b)

(e)

(c)

(f)

Figure 3. Contours of equal B^r on the core surface (a){(c) and on the Earth’ s surface (d){(f ).
(a); (d) epoch 1; (b); (e) epoch 2; (c); (f ) epoch 3. The truncation level L is 72. The bold full curves are magnetic equators, B^r = 0. On the continuous curves B^r > 0; on the dashed curves B^r < 0. The contour interval is 7 G for the core surface and 0.1 G for the Earth’ s surface.

large scale but, if B^ r at the core{mantle boundary were accurately known to ` = 239 as in the simulation, the sur­ cial ®ow would be dominated by small scales.

According to the frozen ®ux approximation, ­ eld topology cannot change on the

core surface. However, when we compare panels ­ gure 2a{c at the L = 12 truncation

and ­ gure 3a{c at L = 72 we see this is only true for the large null ®ux patches.

This is only to be expected since L = 72 corresponds to L much smaller than 80 km,

and for L = 72 is much less than 100 years. Another interesting feature of the

high-L plots is the presence of small regions of very high ®ux, `core spots’, somewhat

analogous to the sunspots observed on the solar surface.

Turning next to the question of the conservation (2.9) of unsigned ®ux, we divided the core surface S into 180 180 `rectangles’ of equal surface area, and summed B^ r and jB^ rj from each, to provide estimates of
I

F = B^ r dS; IS

(3.1)

U = jB^ rj dS:
S

(3.2)

The ®ux integral, F , should be zero by (1.2), and its numerical value gives some
feeling for the accuracy to which the unsigned ®ux, U , has been obtained. We found that jF j 10 4U . A further test was carried out in which S was divided into 360 360 rectangles, and the results did not di¬er to the accuracy shown in table 2,

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(b)

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(c)

(d )

(e)

(f)

Figure 4. Contours of equal B^r on the core surface for epoch 3 at di® erent truncation levels: (a) L = 12; (b) L = 24; (c) L = 36; (d) L = 48; (e) L = 60; (f ) L = 72. The bold full curve is the magnetic equator, B^r = 0. On the continuous curves B^r > 0; on the dashed curves B^r < 0. The contour interval is 7 G.
Table 1. Maxima and minima of B^r on the core surface as functions of the truncation level, L, for epoch 3 (Unit of B^r = 1 G.)

L

12 24 36 48 60 72

B^r;m ax 26 42 46 49 49 49

B^r; m in 26 44 64 70 76 76

where U is given as a function of the cut-o¬ L for the three epochs. Beyond L = 36, the unsigned ®ux does not change to the accuracy shown. The averages of jB^ rj over the core surface for the three epochs are, respectively, 6.4, 6.2 and 5.9 G.
It may be seen from table 2 that, at the geomagnetic (L = 12) truncation, the change in U over 150 years is only ca. 3%, or ca. 2% per century, corresponding to ­ eld changes on the core surface of order 0.1 G. In the corresponding geophysical context, the question to be answered is whether this type-2 error (to use our earlier terminology) is large or small compared with the type-1 errors arising from inaccuracies in, and the incompleteness of, the available data, and the di¯ culties of extrapolation to the core surface. The error of extrapolation may be roughly assessed from the simulation by asking how much of the U obtained at the L = 36 truncation has been omitted in the L = 12 truncation. Comparing the second and fourth columns of
Phil. Trans. R. Soc. Lond. A (2000)

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P. H. Roberts and G. A. Glatzmaier

Table 2. Evolution of the unsigned ° ux, U , from the Earth’s core for three truncation levels L (Unit of ° ux is 101 8 G cm2 .)

epoch L = 12 L = 24 L = 36

1

9.3

9.6

9.7

2

9.0

9.4

9.5

3

8.7

9.0

9.0

table 2 it appears that this is at least as great as the variations within column 1 (the type-2 error). Since this is only one part of the type-1 error, we conclude that the frozen-®ux approximation provides a way of modelling the geomagnetic ­ eld that is acceptable in view of the greater type-1 errors inherent in modelling the geomagnetic ­ eld on the core surface. This also supports the approach of Constable et al . (1993) and O’Brien et al . (1997) to geomagnetic ­ eld analysis.
This work was supported by the National Science Foundation under grants EAR97-25627 and EAR99-02969 and by the SDSC and NCSA Supercomputing Centers. It was also supported by the Institute of Geophysics and Planetary Physics, the University of California Partnership Initiatives Program and the Advanced Computing Laboratory, all at the Los Alamos National Laboratory. We are grateful to Maureen Roberts for preparing the ¯gures, and the referees for their criticisms.

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