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Rotating parametric instability in earlyEarthKeith Aldridge a , Ross Baker a & David Mcmillan aa Graduate Program in Physics and Astronomy , York University ,Toronto, CanadaPublished online: 18 Nov 2010.

To cite this article: Keith Aldridge , Ross Baker & David Mcmillan (2007) Rotating parametricinstability in early Earth, Geophysical & Astrophysical Fluid Dynamics, 101:5-6, 507-520, DOI:10.1080/03091920701562046

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Geophysical and Astrophysical Fluid Dynamics,Vol. 101, Nos. 5–6, October–December 2007, 507–520

Rotating parametric instability in early Earth

KEITH ALDRIDGE*, ROSS BAKER and DAVID MCMILLAN

Graduate Program in Physics and Astronomy, York University, Toronto, Canada

(Received 4 December 2006; in final form 31 May 2007)

The existence of Earth’s magnetic field both before and during the growth of the solid innercore is consistent with a geodynamo driven by a rotating parametric instability (RPI) in thefluid outer core. Here we confirm by laboratory experiment the existence of an RPI in both arotating fluid sphere and spherical shell through excitation by a travelling perturbation. An RPIis generated in each case by elliptically deforming the flexible, spherical boundary containingthe fluid. This externally imposed strain, corresponding to Earth’s Luni-solar strain of the semi-diurnal tide, couples low order rotational modes of the fluid to produce an RPI. The RPIcontinues to grow and decay as long as the boundary is perturbed. Our current laboratory workconfirms that the RPI that appears to exist with the present inner core could have existed whenthe core was much smaller, or even when there may have been no inner core present. Thus anRPI could have driven the geodynamo throughout Earth’s history.

Keywords: Geodynamo; Rotational parametric instability; Inner core paradox; Relativepaleointensity

1. Introduction

Evidence from paleomagnetic observations confirms that the geomagnetic field has

existed for at least 3.8Ga (McElhinny and Senanayake 1980). Accordingly, the

geodynamo that is required to maintain Earth’s magnetic field against dissipation

would need to have been running for that entire period. Compositional convection

requires that deposition of heavy components of the liquid core at the inner core-outer

core boundary must continue in order to have lighter buoyant components rise to

maintain the geodynamo against dissipation. Recent estimates of the age of the inner

core at about 2.5Ga (Labrosse et al. 2001) however, leave some 1.3Ga without a

mechanism of compositional convection to maintain the geodynamo, producing what

can be called the ‘inner-core paradox’.Numerical geodynamos were traditionally maintained by thermal convection since the

first successful model of Glatzmaier and Roberts (1995). Compositional convection was

invoked when it was appreciated that thermal convection processes were insufficient to

maintain the geodynamo (Glatzmaier and Roberts 1996). But the inner-core paradox

*Corresponding author. Email: keith@yorku.ca

Geophysical and Astrophysical Fluid Dynamics

ISSN 0309-1929 print/ISSN 1029-0419 online 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/03091920701562046

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requires that thermal convection be reconsidered if no inner core were present, albeit notan unlikely circumstance when thermal convection may have been much more vigorous.

Is there, however, an alternative mechanism that could maintain the geodynamothroughout Earth history that is insensitive to the presence of an inner core?A geodynamo driven by Luni-solar precession, once considered by Malkus (1968), wassubsequently ruled out essentially on energetic grounds by Loper (1975) and Rochesteret al. (1975). Not included in either of these models, however, were nonlinear terms thatcan produce a parametric instability. Indeed all that is needed to turn a laminar flow ofEarth’s conducting fluid core into fully developed turbulence is a small, but periodicstraining of the fluid streamlines. The first numerical geodynamo driven by precession,recently developed by Tilgner (2006), shows that mechanically driven flows can producedynamo action, although dipolar fields that reverse like Earth’s magnetic field have yetto be found.

If we suppose that a parametric instability were excited in Earth’s core, what evidencemight we find for this in Earth’s magnetic field? While there is little chance of detectingsuch an instability on short timescales, the growth and decay of an instability should bepreserved in the paleomagnetic record. Aldridge and Baker (2003) searched records ofrelative paleointensity from a site in the North Atlantic with a high sedimentation rateand took the temporal behaviour of the magnetic intensity as a proxy for the fluidvelocity (Moffatt 1978). Thus the growth and decay of the field was interpretedas a measure of the corresponding growth and decay of a parametric instability inEarth’s core. Subsequently these authors analyzed several sets of both single site recordsand composite stacks of relative paleointensity, finding that the timescales of growthand decay were consistent with what would be expected from straining of streamlines ofthe fluid core by the gradient of the Luni-solar gravitational field.

Earth responds to the Luni-solar gradient in two primary ways. First, due to itsslightly oblate figure, Earth’s rotation axis precesses with an amplitude of 23.5 degreesat a period of approximately 25,800 yr. This precession produces a predominantshearing of fluid streamlines in Earth’s core, which will lead to a parametric instabilityif dissipation in the core is sufficiently small. Second, ellipsoidal distortion ofequipotentials is evident on Earth as the solid Earth tide. Straining of otherwisecircular streamlines in Earth’s core into ellipses also produces a travelling parametricinstability since the Lunar and Solar days are longer than the sidereal day. Thelaboratory experiments described in this paper are the counterparts of this travellingelliptical instability.

Strains, , that couple rotational core modes are proportional to the gradient ofthe Luni-solar gravitational field. These small strains, corresponding to millennialgeomagnetic time scales, have existed throughout Earth history. A kinematicgeodynamo has been shown possible if the flow has specific dynamical properties.Rotational modes, for example, which are coupled by elliptical strain, have theirvorticity coinciding with their velocity and can produce an -dynamo (Olson 1981).A dynamo driven by a travelling elliptical instability avoids the inner-core paradox aslong as the instability can be excited both with and without an inner core.

Properties of parametric instability of a rotating fluid, or rotating parametricinstability (RPI), were given recently in theoretical work by Kerswell (1993). Inertial orrotational modes of a contained, ideal fluid sphere have frequencies nmk¼!nmk/,where (0< nmk<2) and is the fluid’s rotation speed. The indices (n,m) givethe modal structure in the plane containing the rotation axis of the fluid, while

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k=0,1,2, . . . is the azimuthal wavenumber of the mode. Elliptical deformation ofthe fluid’s streamlines will couple two inertial modes with wavenumbers k1 k2¼ 2 andfrequencies nmk1 nmk2 ¼ 2, while shearing of streamlines will couple two inertialmodes with wavenumbers k1 k2¼ 1 and frequencies nmk1 nmk2 ¼ 1.

Work on parametric instabilities in a cylindrical geometry dates back to Gledzer et al.(1975) who studied elliptical instability in rotating elliptically deformed cylindricalcavities, and later by Vladimirov and Tarasov (1985). Later Malkus (1989) and morerecently Eloy et al. (2000) excited the elliptical instability in a rotating cylindrical cavityby deforming its external boundary through sets of rollers that were stationary in thelaboratory frame of reference. Work on parametic instability in an ellipsoidal cavitywas pioneered by Boubnov (1978). Subsequently elliptical instability was excited in afull sphere by Lacaze et al. (2004) using a stationary external perturbation and in 35%shells by Aldridge et al. (1997) and Seyed-Mahmoud et al. (2004) using a travellinginternal perturbation. Lacaze et al. (2005) have excited parametric instability with astationary, external perturbation in a 33% shell and found excellent agreement with amodel based on linear stability theory. In summary, the existence of parametricinstability has been confirmed under the following circumstances: excited by astationary perturbation in a full sphere and a 33% shell (Lacaze et al. 2004, 2005);and excited by a travelling perturbation in a 35% shell (Seyed-Mahmoud et al. 2004).The present work uses a travelling perturbation to confirm the existence of a parametricinstability in a 23% shell and a full sphere, which correspond to the geophysical caseof an early Earth. The previous result for travelling perturbation in a 35% shellcorresponds to the present Earth.

2. Experimental configuration

Excitation and measurement of an RPI in both a sphere and a spherical shell througha travelling perturbation determined the design of our experiment. Here, ellipticalstraining of the streamlines is produced by a retrograde-rotating set of rollers thatdeform the outer boundary, while the container itself rotates in a prograde sense.

When the perturbation counter-rotates at precisely the speed of the prograde rotationof the container, the perturbation is stationary with respect to the laboratory frame ofreference, and results pertaining to that scenario are discussed above. We areparticularly interested in the travelling perturbation since this case corresponds toplanetary applications of the Luni-solar tide.

A flexible-walled container, cast with Sylgard between two glass spheres (figure 1,left), is shown holding the working fluid and in operation (figure 1, right). The rubbertube connected to a vacuum pump removes air from the Sylgard compound prior tocuring. The completed apparatus, mounted on a table, is shown on the right and is filledwith the working fluid (water), which has a kinematic viscosity ¼ 1.004 cs(106m2 s1) at 20C. The green fluorescein dye, which makes the fluid appearslightly cloudy in the photograph, is used as a marker in the fluid. One of the rollers thatdeform the boundary to an elliptical shape is visible near the right of the shell and thedigital camera used to photograph the fluorescein dye is seen left of centre. The camerais attached to the perturbation frame and in the equatorial plane of the sphere andpointed at approximately 45 to the line joining the two rollers. The spherical cavity has

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an internal radius, R¼ 14.0 cm and a wall thickness of 3.0 cm. The deformation due theweight of the rubber shell and contained water is approximately 0.003 of the cavityradius or 1/10 of the perturbation amplitude. The polar regions of the container arefixed by its support frame.

Figure 1. Apparatus to excite parametric instability. Fabrication of elastic container (left) and completedsystem including perturbation (right) with inner core installed.

Ω

ω

Camera

Roller

Figure 2. Schematic showing travelling perturbation at frequency ! in oppposite sense to container rotatingat speed . Camera is fixed in perturbation system as shown by straight lines. The rollers deform theboundary elliptically up to ¼ 0.05.

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The container is set rotating in a prograde sense at rate rad s1, usually near2 rad s1. The perturbation system consists of two rollers that rotate in a retrogradesense at speed ! rad s1 and deform the outer boundary inward, producing a strain ", asillustrated in the schematic figure 2.

Both rotation systems are maintained by independent servo-controllers up to speedsof 7 rad s1. The deviations from the average speeds are 0.3% for the container and 3%for the perturbation motors. The counter-rotation starts after 3 spin-up times (3 60 s)of the main fluid body. Since the parametric instability develops as a wave whicheventually breaks, fluorescein dye was used as a marker for its development. Dye wasintroduced after 2 more spin-up times at the top of the container through a column ofwater so that it would reach a constant rate of descent before entering the cavity. Forthe sphere, the dye was released on axis, and with the inner core configuration, 2 cm offaxis. Otherwise linear filaments of dyed fluid then developed into a wave shape whichwas then photographed at a nominal rate of 30 frames per second by the cameramounted in the perturbation frame. Video images were sent by wireless connection to areceiver in the laboratory frame.

Previous experiments, reported by Seyed-Mahmoud et al. (2004) using the same styleof travelling perturbation, had found the parametric instability close to ¼! after anextensive search of this ratio in that vicinity. The present experiment followed that sameprocedure with results given below that are centred in the frequency ratio of these twospeeds, and where the instability was readily observed as a breaking wave.

3. Results

Onset of the RPI was confirmed for both a full sphere and a spherical shell of fluid byvisual observation of the fluorescein dye as it developed from a growing wave into aturbulent state. Digitally recorded images of this development were stored forsubsequent analysis to determine the rate of growth of the instability.

Shown in figure 3 is a sequence of six images taken over a 32 s interval for¼ 5.35 rad s1, !¼ 5.24 rad s1 and the strain ¼ 0.034, where the strain is defined asthe maximum radius minus the minimum radius produced by the deforming rollers,divided by the mean radius. The Ekman number E¼ (R2)1¼ 9.6 106.

Figure 3. Images of instability in a sphere; sequence runs left to right, top to bottom and spans 32 s.

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Emergence of the instability is apparent from the deflection of the wavelikedisturbance toward the horizontal direction until the wave finally breaks into a smallscale flow. A sample of this growth and collapse is shown in figure 3. After decay of thisflow the process is repeated and continues to grow and decay as long as theperturbation continues. The timing between successive growths appears to be anaperiodic sequence. An inner solid sphere with radius r¼ 3.2 cm was installed at thecentre of the flexible-walled sphere to produce a spherical shell of rotating fluid. Theabove experiment was repeated. Several sequences of growth and decay were found.

Shown in figure 4 is one sequence of six images taken over a 40 s intervalfor ¼ 5.35 rad s1, !¼ 5.24 rad s1 and the strain ¼ 0.034 and Ekman number,E¼ 9.6 106. The last panel of this figure shows the initial growth stage after theprevious growth and collapse. The instability was found at a slightly larger frequencyratio!1¼ 0.98 from that found by Seyed-Mahmoud et al. (2004) for the 35% shell.Linear stability theory predicts these frequencies to be unaffected by the shell ratio, butmeasured frequencies in our experiments using a travelling perturbation indicateotherwise.

Although not as vigorously developed, an RPI is clearly found in this spherical shellgeometry.

3.1. Quantitative measure of instability growth rate

Growth of this wave, which is stationary in the roller frame of reference, was foundthrough image analysis of a sequence of frames captured during the wave growth.The code ilinefit is a Matlab code developed to identify and quantify the growth of thiswave and hence the RPI via the analysis of video imagery obtained during runs ofthe experiment.

While a particular run is in progress, fluorescein dye is released into the fluid asdescribed above and illuminated. The RPI reveals itself when originally linear, verticaldye traces show a sinusoidal wave growing in amplitude. The wavelength of thedeveloping instability is about 40% of the cavity radius R. It is the slope of the sinusoidthat we use as a proxy for RPI amplitude: a straight line has zero slope showing no

Figure 4. Images of instability in a shell; sequence runs left to right, top to bottom and spans 40 s.

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RPI development, while the wave, near breaking, bends over horizontally so has a very

large slope.The digital video camera attached to the experimental apparatus films each

experiment and the stored video file is exported as a colour image sequence. It is this

image sequence that provides input data for ilinefit, via a user supplied list of selected or

consecutive images.The code ilinefit may be used in a number of ways, including: scrolling through a list

to view an image sequence frame by frame; sub-sampling the image sequence and

producing a new list of images for analysis; or analyzing each image in the list for RPI

growth. In analysis mode, the user is prompted to set a ‘crop region’, in which the

observations will be made. Analysis in the crop-region is necessary because the features

we wish to examine usually occupy a relatively small portion of the original image and

cropping eliminates the interference of other bright features.The dye-streaks show up as bright areas in the grayscale image and its cropped

counterpart. We are thus interested in approximating the slope of a bright curvilinear

feature in the crop-region. The grayscale brightness in the crop-region is normalized to

the interval [0, 1] and all pixels having a brightness that exceeds a preset threshold are

determined. The resulting cloud of super-critical points then approximates the shape

and orientation of the bright feature and a linear polynomial fit to this collection of

points produces an approximation to the slope. If the cloud of points does not

satisfactorily represent the shape of the feature, a different threshold value may be

selected. In practice though, we compute slopes for several preset threshold values

simultaneously. At this point in the analysis of the current image, a second window

shows the normalized crop-region of the image, the collection of super-critical points

and fitted lines for each threshold value as seen in figure 5.Meanwhile a rectangle outlining the crop-region and all fitted lines (colour-coded to

threshold values) are superimposed on the full, original image seen here in figure 6.

The video time of the image, all slopes and intercepts, and the coordinates of the

crop-region are saved to disk.A second list of options is then available: pick a fit from the results for the

different thresholds; repeat analysis of the current image; or discard the current image.

The first option is used to find the ‘optimal fit’, where we select the best time series of

super-critical points or best slope representation from the different threshold values.

After picking, ilinefit proceeds to the next image in the list and produces a new set of fits

using the same crop-region.If the user is not satisfied with the collection of super-critical points and slopes for

the current image, they can chose to repeat the analysis, in which a new crop-region

must be selected. This feature is useful when the wave being analyzed moves out of the

crop-region after several cycles.The ilinefit algorithm was applied to video records similar to those of figure 3.

As the travelling perturbation remained on throughout the experiment, the instability

develops, decays and grows again, as expected. Plotted in figure 7 is the amplitude

of the wave expressed as the natural log of the inverse wave slope against time.

Two growths are shown with decay taking place during the interval between the

two growths.Shown in figure 8 are the measured amplitudes of the growing instability in the 23%

shell for the same experimental parameters as those for figure 4. A second interpretation

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Image threshold: 0.85; slope = 2.4001

10 20 30

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Image threshold: 0.825; slope = 2.3024

10 20 30

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Image threshold: 0.8; slope = 2.2011

10 20 30

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Image threshold: 0.775; slope = 2.1467

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Image threshold: 0.75; slope = 2.0232

10 20 3010 20 30

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Slope stats: µ = 2.2147; σ = 0.14454

Figure 5. Sample fits to slope of wave for various threshold values.

100 200 300 400 500 600 700 800

50

100

150

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250

300

350

400

450

Figure 6. Fitted lines to wave steepness in sample image.

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20 30 40 50 60 70 80 90 100 110−2

−1.5

−1

−0.5

0

0.5

1Full sphere

Time (s)

Am

plitu

de

τ = 18.60 s

τ = 10.50 s

Figure 7. Sample growths of instability after the perturbation remained turned on for several cycles ofgrowth and decay. Only growths are recorded and values are measured e-folding times of the instability’sgrowth.

20 30 40 50 60 70 80 90 100 110−2

−1.5

−1

−0.5

0

0.5

1Spherical Shell

Time (s)

Am

plitu

de

τ = 7.36 s

Figure 8. Growth of instability in 23% shell.

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of the data of figure 8 is shown in figure 9 which illustrates what can be termed a two-stage growth.

4. Oceanic records of relative paleointensity

If the instability found in our laboratory experiments is excited in Earth’s fluid corethrough straining of the core by the gradient of the Luni-solar gravity field, evidenceof this fact should be found in Earth’s paleomagnetic field. We assume here that thespectral properties of the observed magnetic field will be a proxy for that of the velocity(Moffatt 1978). Reported by Baker et al. (2006) are six sequences of paleomagneticintensity that those authors have searched for evidence of RPI in Earth’s core. Threesequences (WCB, ODP983, ODP1062) were from single sites while three were fromcomposite stacks (NAPIS-75, GLOPIS-75, SINT2000) comprising several records. Alldata studied was from the interval 0–80 ka. Significant errors in timing as well asamplitude will certainly result and these have been considered in Baker et al. (2006).Also studied in that paper are constraints on the use of stacked records,in comparison with single site composites. Details of the sources of this data aregiven in Baker et al. (2006).

4.1. Growth and decay of relative paleointensity

Exponential growths and decays of intensity would be expected if these changes inamplitude were due to an RPI in Earth’s core. Baker et al. (2006) identified severalintervals of growth and decay and found best fitting exponentials for each of thesegments they identified. After examining the three composite stacks by computingcoherence spectra between pairs of records constituting the stacks, Baker et al. (2006)

20 30 40 50 60 70 80 90 100 110−2

−1.5

−1

−0.5

0

0.5

1Spherical Shell, 2 Stage Growth

Time (s)

Am

plitu

de

τ = 3.91 s

τ = 20.81 s

Figure 9. Example of two-stage growth of instability in 22% shell.

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found that no reliable information could be obtained from the SINT2000 stack for timeintervals shorter than about 20 kyr. Thus smoothing through the stacking of some34 individual records has removed any possible evidence of the signature expected forparametric instability.

4.2. Modeling the data

From linear stability theory (Kerswell 1993) the observed growth rate is ( E1/2)while the observed decay is E1/2 where is the perturbation amplitude, E is theEkman number and is the fluid’s rotation speed. , are typically of order unity.While bulk dissipation will certainly occur at rate E, it is the boundary layerdissipation at rate E1/2, corresponding to the decay of the modes driving thedynamo, that are relevant here. Accordingly, an ideal growth rate, can be estimatedby summing sequential growths and decays.

4.3. Predictions from linear stability theory (LST)

We can calculate an ideal growth rate expected for Earth’s core for each of two sourcesof straining in the core. The semi-diurnal tide is expected to give a predominantlyelliptical strain, while precession of the mantle will produce mainly a shearing in thecore (Kerswell 1993). Table 1 lists the ideal growth rates expected from these twosources and expresses the reciprocal of these rates or e-folding times in thousands ofyears. Clearly the very small tidal strains and very slow precessional rates produced bygradients of Luni-solar gravity translate into long time scales that are typical of the oneswe see in paleomagnetic data. In the next section, the rates found from analysingrelative paleomagnetic intensities will be converted to ideal e-folding times forcomparison with those predicted from linear stability theory (LST).

4.4. Observations and predictions from LST

Reported ideal growth rates for each of 5 paleomagetic intensity sequences yieldedmean and standard deviations in kyr are: WCB (2.5 2.5), ODP983 (3.5 1.9),OPD1062 (2.1 1.2), NAPIS-75 (3.4 0.6), GLOPIS-75 (3.8 0.9) (Baker et al. 2006).

From the growths given above, the mean ideal e-folding time, weighted with inversevariances from these observations, is 3.3 0.44 kyr. Estimates of Ekman numbersfrom the decay portions of the records, using ¼ 2.62 are WCB (5.5 1015),ODP983 (1.0 1015), OPD1062 (3.3 1015), NAPIS-75 (1.1 1015), GLOPIS-75(0.7 1015) (Baker et al. 2006). which are consistent with other low estimates of core

Table 1. Predictions of ideal growth rates and e-folding times for tidal (elliptical) and shear(precession) strains in Earth.

Type of Deformation Ideal growth Idealperturbation amplitude rate e-folding time

Elliptical 8.5 108 3.1 1012 s1 10 kyrShear 4.0 108 1.7 1012 s1 18 kyr

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Ekman numbers. Coresponding estimates of kinematic viscosity for the core using thecore radius as the length scale give WCB (4.9 106), ODP983 (0.90 106), OPD1062(2.9 106), NAPIS-75 (0.98 106), GLOPIS-75 (0.64 106)m2 s1, similar to thoseof liquid metals. Given the very large range of estimates for core viscosity(Lumb and Aldridge 1991), this is not a profound result. It is however, not inconsistentwith the presence of an RPI in Earth’s core produced by gradients in Luni-Solar gravity.

4.5. Evidence for elliptical instability in Earth’s core

Laboratory experiments reported here for elliptical strains ¼ 0.034 should haveproduced an ideal e-folding time of about 11 s while observation gave 10.3 s as shown incolumn 1 of table 2. The estimate of the ideal e-folding time from these sameobservations, found by combining adjacent growth and decay compares favourablywith the ideal e-folding times predicted by LST. Note that this method uses both agrowth and a decay cycle to estimate an ideal e-folding time. Earth’s response comparedto LST is found from our observations, which for example yield 3.4 kyr from thepaleointensity data compared to 10 kyr for LST. For Earth estimates from LST we used¼ 0.5 since Earth’s slightly oblate outer core is essentially spherical in the presentapproximation.

5. Discussion and conclusion

Parametric instability has been excited in a fluid sphere contained by a flexible outerwall and in a spherical shell using a travelling perturbation. These experiments confirmthe existence of this instability in the sphere and spherical shell geometry (Lacaze et al.2004, 2005) already observed for a stationary perturbation. While the experimentsreported here are consistent with expectation from linear stability theory, the existenceof a second growth stage reported here for the 22% shell is indicative of a finiteamplitude stage of growth.

A 23% shell corresponds to an inner core of age approximately 1.6Ga. Both unstableflows were initiated by externally straining the flexible outer boundary containing thefluid. In earlier experiments RPI was excited in a spherical shell by elliptically strainingthe shell’s inner boundary (Seyed-Mahmoud et al. 2004). Thus RPI in a fluid shell hasbeen observed through excitation of both inner and outer boundaries. Forced strainingthroughout the shell as in Earth via tidal straining, however, can only be approximated inour laboratory experiments through boundary deformations. It is also noteworthy that,unlike in our experiments, tidal deformation of the lower mantle would not play a role inthe straining of core fluid. Finally, a caution must be made concerning the nature of

Table 2. Ideal e-folding times observed and predicted for laboratoryexperiments and Earth’s core.

Laboratory Earth’s core(¼ 0.034) (¼ 8.5 108)

LST 11.0 s 10 kyrFrom observations 10.3 1.8 s 3.6 0.6 kyr

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inertial modes at very low Ekman numbers expected in Earth’s fluid core. Indeed, thePoincare problem that describes the fluid pressure is hyperbolic in space but has specifiedboundary conditions, making it ill-posed in the Hadamard sense so that analyticalsolutions are not guaranteed. While there is no problem finding such solutions for a fullsphere (Greenspan 1969), such is not the case for a spherical shell (Rieutord et al. 2001).At lowEkman numbers complex behaviour results in spherical shells with the appearanceof limit cycles and an increasingly important role of characteristic surfaces in the fluid.

With the proviso that the above cautions allow, we conclude that a geodynamodriven by RPI would be independent of the presence of an inner core (Labrosse et al.2001). This conclusion has been drawn from the laboratory experiments reported hereusing a travelling perturbation and from other recent experiments that have shown theexistence of parametric instability in both a sphere and spherical shell geometriesexcited by a stationary perturbation (Lacaze et al. 2004, 2005). Analysis of records ofrelative paleointensity from oceanic cores over the past 80 kyr lend support to thepossibility of a geodynamo driven by parametric instability. Implicit in our conclusionis that the signatures found to be consistent with an elliptical instability in Earth’s corewould also exist at earlier epochs of the Luni-solar gravity field. When sufficientpaleointensity data from those epochs are available it will become possible to use thatdata to build a consistent model for Earth and Moon orbital elements with coreproperties that determine growth and decay of a rotating parametric instability.

Acknowledgements

We are grateful to Jerome Noir and an anonymous referee for several importantcomments which we believe helped us greatly improve our manuscript. Funding for thiswork came from a Discovery grant awarded to KDA by the Natural Sciences andEngineering Research Council of Canada.

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