Timescales of geomagnetic secular acceleration in satellite field models and geodynamo models

Geophysical Journal InternationalGeophys. J. Int. (2012) 190, 243–254 doi: 10.1111/j.1365-246X.2012.05508.x

GJI

Geo

mag

netism

,ro

ckm

agne

tism

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Timescales of geomagnetic secular acceleration in satellite fieldmodels and geodynamo models

U. R. Christensen,1 I. Wardinski2 and V. Lesur2

1Max-Planck-Institut fur Sonnensystemforschung, Katlenburg-Lindau, Germany. E-mail: [email protected] Centre Potsdam, GFZ German Research Centre for Geosciences, Section 2.3: Earth’s Magnetic Field, Germany

Accepted 2012 April 13. Received 2012 April 12; in original form 2011 December 22

S U M M A R YMagnetic satellite data from the last decade allow to model geomagnetic secular acceleration,the second time derivative of the field, in a highly precise manner. Robust estimates of thesecular acceleration (SA) are obtained by using order six B-Splines as representation of thefield variability, which in turn allows us to estimate the characteristic SA timescale, τ SA. Weconfirm a recent finding that τ SA is of order 10 years and fairly independent of the sphericalharmonic degree n. This contrasts with the characteristic timescale of geomagnetic secularvariation τ SV, which is a decreasing function of n and is >∼100 yr for n ≤ 5. Conceivably theSA timescale might be related to short-term processes in the core, distinct from convectiveoverturn whose timescale is reflected by τ SV. Previously it had been shown that dynamosimulations reproduce the shape of the secular variation (SV) spectrum and, provided theirmagnetic Reynolds number Rm has an Earth-like value of order 1000, also the absolute valuesof τ SV. The question arises if dynamo simulations can capture the observed timescales ofgeomagnetic SA. We determined τ SA(n) for a set of dynamo models, covering a range of valuesof the relevant control parameters. The selection of models was based on the morphologicalsimilarity of their magnetic fields to the geomagnetic field and not on criteria related to the timedependence of the field, or on any aspect of the spectra associated with their field variation.We find that τ SA depends only weakly on n up to degree 10, but for larger n it asymptoticallyapproaches the 1/n-dependence that is also found for τ SV(n). The acceleration timescale atlow n varies with magnetic Reynolds number more strongly than τ SV and may also dependon magnetic field strength. For an Earth-like Rm ≈ 1000, τ SA is of order 10 yr for n � 2–10,as found in the field models from satellite data. A simple scaling analysis based on the frozenflux assumption for magnetic variations suggests two contributions to the SA, an advectivepart that scales with velocity U and has a length scale dependence corresponding to n−1,and a part that depends on the acceleration of the flow U without explicit dependence on thelength scale. Their combination can explain the spectral shape of τ SA(n) in numerical models,with the latter term dominating at n < 10. The characteristic timescale of acceleration ofthe near surface flow U/U correlates with τ SA in the different numerical models and is ofthe same order as τ SA. This suggests that the observed 10 yr timescale of geomagnetic SAreflects the characteristic time of core flow acceleration. To explain the geomagnetic SV andSA timescales, we find that the rms velocity near the core surface must be 18 km yr−1 and therms flow acceleration approximately 2 km yr−2, although a statistical analysis of the inductionequation suggests that most of the latter may occur at flow scales corresponding to harmonicdegrees n > 12. The ability of dynamo models to match simultaneously SV and SA timescalessuggests that dynamic processes in the core at the decadal timescale are not fundamentallydifferent from those at the centennial timescale.

Key words: Dynamo: theories and simulations; Rapid time variations; Satellite magnetics.

1 I N T RO D U C T I O N

The internal magnetic field of the Earth varies on timescales thatrange from 100 Myr to less than a year, where the extremes are

represented by the variable length of magnetic chrons at one endand rapid changes that are associated, for example, with geomag-netic jerks, at the other end. The variation of the magnetic field ondecadal to centennial timescales is of particular interest to infer the

C© 2012 The Authors 243Geophysical Journal International C© 2012 RAS

244 U. R. Christensen, I. Wardinski and V. Lesur

fluid flow near the top of Earth’s core (e.g. Holme 2007). Secu-lar variations (SVs) with characteristic timescales of one or a fewcenturies probably reflect the convective overturn time of the core.Field variations at the shortest observable timescales of the order ofone decade or less could possibly be related to wave-like motions(e.g. Gillet et al. 2010) that are only indirectly related to the dynamoprocess.

The successful operation of the satellites Ørsted and CHAMP formore than a decade enables the analysis of short-term variations ofthe dynamo field with an accuracy that has not been reached before.From these data several internal field models have been derived,such as recently the CHAOS-3 model (Olsen et al. 2010) and theGRIMM2 model (Lesur et al. 2010a). In these models the timedependence of the Gauss coefficients of the internal geomagneticfield is parametrized by high-order spline functions. The high spatialand temporal coverage of the satellite data and the smoothness ofthe temporal representation in the field models allows us to not onlydetermine the first time derivative (SV) of the field components, butalso their second time derivative (secular acceleration, SA).

Power spectra as function of spherical harmonic degree have beenintroduced by Mauersberger (1956) and Lowes (1974) to charac-terize the magnetic field. At the core–mantle boundary (CMB), thespectrum of the non-dipole field is slightly reddish up to the highestharmonic degree for which the core field can be confidently deter-mined (around 13). Similarly, power spectra can be calculated for SVand SA. In these cases the spectra at the CMB are blue (Alldredge1984), that is, the power increases with harmonic degree. At a givenspherical harmonic degree n the square root of the ratio between thepower in the field to that in the SV defines a characteristic timescaleof SV τ SV(n), which can be understood as the correlation time ofthe field (Hulot & Le Mouel 1994). At low degrees τ SV is typicallyin the order of one to two centuries and has a trend that is found todecrease with increasing degree n. The precise dependence on n isdebated. Holme & Olsen (2006) and Holme et al. (2011) find thatτ SV(n) ∝ n−3/2 fits the timescales in the satellite models. Exclud-ing the dipole component, Christensen & Tilgner (2004) suggesteda weaker dependence τ SV(n) ∝ 1/n. They found this dependenceon n in numerical dynamo models and determined that the SVtimescale, normalized with the magnetic diffusion time, varies withthe inverse of the magnetic Reynolds number Rm = UD/η, whereU is the rms flow velocity, D the thickness of the convecting shelland η the magnetic diffusivity. For a plausible value of Rm ≈ 1000the dynamo models reproduce the observed SV timescales, whichsuggests that they are basically controlled by the advection timeD/U ≈ 100–200 yr. Lhuillier et al. (2011) reaffirmed the 1/n de-pendence for the non-dipole field in dynamo models and found thatit is compatible with the geomagnetic gufm1 model for the period1840–1990 (Jackson et al. 2000) and with recent satellite models.

The ratio between the power in the SV and in the SA can be usedto define the characteristic timescale of SA, τ SA(n). Lesur et al.(2008) originally found for the GRIMM satellite model that τ SA

decreases with increasing n. Using the most recent model of theCHAOS series, Holme et al. (2011) found that τ SA is independentof the spherical harmonic degree and is of order 10 years. The sameresult is obtained for the GRIMM2 model (Lesur et al. 2010a). Inthis work we also confirm this for the latest model of the GRIMMseries (GRIMM3; Lesur, in preparation) with a more fine-grainedtemporal representation compared to GRIMM2.

The inferred SA timescale of 10 years is significantly shorterthan the typical SV timescale of 100 years or more at the samespherical harmonic degree. Holme et al. (2011) speculate that τ SV

and τ SA may measure different processes that govern the long-term

evolution of the core flow and its decadal variations, respectively,and pose the question if dynamo models would be capable to alsomatch the SA timescales. If the short-term dynamics of the Earth’score is dominated by magnetohydrodynamic waves and torsionaloscillations (Gillet et al. 2010), dynamo models may in fact beincapable of capturing the timescale of SA. In this case τ SA shouldbe related to the timescale D/UA, where UA = B/

√ρμ is the

Alfven velocity with B the characteristic field strength in the core,ρ its density and μ magnetic permeability. For a characteristic fieldstrength of a few mT in the core (Gillet et al. 2010) the Alfvenvelocity is at least an order of magnitude faster than the flow velocityand D/UA is <∼10 yr. However, in present dynamo models the flowand Alfven velocities are of the same order (Wicht & Christensen2010) and the advective timescale and that of magnetic waves aresimilar.

The purpose of this paper is first to reanalyse the most recent satel-lite internal field models of the Earth concerning the timescales ofSA and their dependence on spherical harmonic degree (Section 2).Then, in Sections 3 and 4 we study SA timescales in a range ofnumerical geodynamo models and their dependence on the modelparameters. Since we find that some models are able to match theobserved acceleration time, we develop in Section 5 a simple theorythat explains the acceleration timescale and its dependence on har-monic degree in terms of the fluid flow and its acceleration. We alsopresent a statistical analysis of the induction equation to determinethe relative contributions of flow at large and small length scales tothe SV and SA at low spherical harmonic degree.

2 S E C U L A R A C C E L E R AT I O N I NG E O M A G N E T I C M O D E L S

Here we analyse a set of recent geomagnetic models for the timespan 1957–2010 in terms of their magnetic SA. The set of fieldmodels consists of C3FM2 (Wardinski & Lesur 2012), a model cov-ering the period 1957–2008 and based on geomagnetic observatorydata, CHAOS-4a (Holme et al. 2011), which is covering the period1997–2010 and mainly made of satellite data but also includes somegeomagnetic observatory data, and GRIMM3 (Lesur et al., in prepa-ration), covering the period 2001–2010. We analyse two version ofGRIMM3, one version, GRIMM3-V1, is built using geomagneticobservatory and satellite data, and GRIMM3-V2 is entirely basedon geomagnetic satellite data.

Commonly, the internal magnetic field is expressed as negativegradient of the scalar potential �, which is described via an expan-sion in spherical harmonic functions in terms of the Gauss coeffi-cients gm

n , hmn :

� = a∞∑

n=1

(a

r

)n+1 n∑m=0

Pmn (cosθ )

[gm

n cos(mφ) + hmn sin(mφ)

], (1)

where r is radius from the centre of the Earth, a the mean Earthradius, θ colatitude, φ longitude and Pm

n are Schmidt-normalizedassociated Legendre functions.

In current geomagnetic field models the Gauss coefficients andtheir time derivatives are given as continuous functions in time. Thefirst and second time derivatives, gm

n , hmn and gm

n , hmn , describe the

SV and the SA, respectively. The models are truncated at differ-ent maximum spherical harmonic degree N ; C3FM2 at N = 14,GRIMM3 at N = 18 and CHAOS-4a at N = 20. However, this setof models is characterized by the same temporal representation ofthe Gauss coefficients. These are expanded in time using order six

C© 2012 The Authors, GJI, 190, 243–254

Geophysical Journal International C© 2012 RAS

Geomagnetic secular acceleration 245

B-splines Mk(t):

gmn (t) =

K∑k=1

gm,kn Mk(t) , hm

n (t) =K∑

k=1

hm,kn Mk(t). (2)

The knot spacing of the temporal basis functions of satellite basedmodels, GRIMM3 and CHAOS-4a, is 6 months, whereas it is 1.2 yrfor C3FM2. Such short interval between knots allows us to capturerapid SV and SA. However, for a reasonable estimation of the SAit is required that excessive temporal complexity is penalized inthe inversion. This can be done by requiring that for a given misfitto the data a norm involving the third time derivative of the radialcomponent Br of the field over the core surface is minimized (Lesuret al. 2010a):∫ t2

t1

∫S(c)

(∂3 Br

∂t3

)2

d dt, (3)

where S(c) is a spherical surface of radius c = 3485 km, the es-timated radius of the core, and t1, t2 are the model start and endepoch, respectively. This is the case for the CHAOS and GRIMMmodels, where also ∂2Br/∂t2 is minimized at the endpoints of theirtime validity periods. A different approach is followed in the makingof C3FM2 (Wardinski & Lesur 2012). This model is built by coesti-mating the core surface flow and the magnetic field at the CMB fromgeomagnetic observations under the condition that changes of themagnetic field are entirely due to the advection of the magnetic field(frozen flux condition, Roberts & Scott 1965; Lesur et al. 2010b).C3FM2 is regularized by minimizing the temporal variability of thecore surface flow.

The spectral power in the field at degree n at radius r is given by

W (n) = (n + 1)(a

r

)2n+4 n∑m=0

[(gm

n

)2 + (hm

n

)2]

, (4)

and is usually either evaluated at the Earth’s surface (r = a) or at theCMB at r = c. The sum of all W (n) is the mean squared magneticfield strength at the radius under consideration. In an equivalentway, the power spectra for the SV

W ′(n) = (n + 1)(a

r

)2n+4 n∑m=0

[(gm

n

)2 + (hm

n

)2]

(5)

and the SA

W ′′(n) = (n + 1)(a

r

)2n+4 n∑m=0

[(gm

n

)2 + (hm

n

)2]

(6)

can be defined.The SV timescales are then obtained as

τSV(n) =√

W (n)/W ′(n) (7)

and the SA timescales as

τSA(n) =√

W ′(n)/W ′′(n) . (8)

The timescales are independent of the radius to which the powerspectra refer. They can be calculated for a given epoch, or morerobustly, by using time averages of the power spectra when dataof sufficient quality are available over an extended period of time(Voorhies 2004; Lhuillier et al. 2011).

Taken by itself, the temporal regularization acts on all sphericalharmonic degrees in the same way, as the spectral power at degreen of the third time derivative contributes to the regularization con-dition (eq. 3) equally for all n. However, effectively the temporalregularization damps the internal field SA more strongly at high

degrees than it does at low degrees. This is suggested by the dropof power for degrees n > 10 in the SA spectra at the CMB ofGRIMM-2 (Lesur et al. 2010a) and CHAOS-4 (Holme et al. 2011),which is in contrast to the increase of power in the SV at thesedegrees. The reason for the stronger damping of fast changes ofthe internal field at high degrees is that the misfit to the data isevaluated at the points of observation, that is, the satellite altitudeor the Earth’s surface, whereas the third time derivative of the fieldis minimized at the CMB. Because of the geometric decrease ofthe field with radius according to r−(n +2), a certain change in theacceleration on the CMB (which is penalized by the regularizationcondition) can contribute less to reduce the data misfit at high nthan it can at low n. Therefore, for a field component with high n,the inversion tends to set its acceleration close to the average value,with little change during the modelled time interval. The accelera-tion is underestimated because its temporal average is smaller thanthe rms value. This effect is obvious in the IGRF-05 candidates (e.g.Lesur et al. 2005; Olsen et al. 2005) where the field models were de-rived using a Taylor expansion to describe their temporal behaviour,setting by construction the third time derivatives to zero. In thesecandidates the estimated acceleration is significantly weaker than inGRIMM3 or CHAOS-4.

Potentially, a too strong regularization could affect the SA also atlong wavelength. There is no indication that the large-scale accel-eration is seriously underestimated in the field models used in thisstudy. They all fit remarkably well the SV observed in observato-ries. Very fine details of geomagnetic jerks are perhaps not capturedby the models, because these are related to uncertain small spatialscales of SA. However, during jerks the SA is typically discontinu-ous but not necessarily large. We find no evidence that the observed(long wavelength) magnetic field presents much stronger amplitudesof the acceleration than those seen in the models. Fig. 1 shows theobserved and modelled SV at an observatory site (Niemegk, Ger-many). All known geomagnetic jerks are reasonably well recov-ered by the field models. Because of the fact that in the modellingschemes of GRIMM3 and C3FM2 external field contributions areremoved from the observations the models deviate from the data bya few nT/yr.

While analysing C3FM2, we found that the variance in the SAspectra can be significantly reduced by only considering the spectrafrom 1980 to 2008. The high variance of the spectra might be causedby the sparser spatial coverage of geomagnetic observations towardsearlier epochs of model C3FM2. The data distribution during theperiod 1980–2008 stayed nearly constant and therefore may notcause variability. Fig. 2 shows the SV and SA timescales averagedover the period 1980–2008. The SV timescale τ SV is fitted by alinear function that varies as 1/n:

τSV(n) = τ oSV/n, (9)

and the SA timescale τ SA is assumed to be represented by a constant

τSA(n) = τ oSA . (10)

In the fitting procedure the contribution of the first sphericalharmonic degree has been excluded, mainly because it differsstrongly for different geomagnetic field models, such as GRIMM3and CHAOS-4a. Very likely, this is caused by different modellingphilosophies for external field variations and their induced coun-terparts in the lithosphere and mantle. Because of the nature ofthese signals, they are most prominent in the dipole term. As aconsequence such differences affect the estimates of the first degreegeomagnetic field time derivatives, that is, estimates of SV and SA.Furthermore, we do not consider the SA for spherical harmonic

C© 2012 The Authors, GJI, 190, 243–254



Figure 1. Comparison between the observed (grey dots) and the mod-elled secular variation of C3FM2 (solid line), GRIMM3 (dotted line) andCHAOS-4a (dash-dotted line) at Niemegk. The vertical lines, marked withthe characters A–H, indicate occurrence times of geomagnetic jerks. Notethat the scatter around the solid lines are mainly caused by residual externalfield variation.

Figure 2. Secular variation (SV) timescales (squares) and secular accel-eration (SA) timescales (circles) versus spherical harmonic degree for thegeomagnetic field model C3FM2. The error bars indicate the temporal vari-ability of the SV and SA timescales during the period 1980–2008. Bestfitting lines for an assumed 1/n dependence of the SV (solid line) and con-stant SA (dash-dotted line) excluding contributions of the first sphericalharmonic degree.

Figure 3. Secular variation (SV) timescale (squares) and secular acceler-ation (SA) timescales (circles) versus spherical harmonic degree for theGRIMM3 V1 geomagnetic field model. The error-bars indicate the tempo-ral variability of the SV and SA timescales during the period 2001–2010.Best fitting lines for an assumed 1/n dependence of the SV (solid line) andSA (dash-dotted line) excluding contributions of the first spherical harmonicdegree.

degree n > 11 in our analysis. We consider estimates of the SAbeyond this degree as becoming unreliable and mostly dependingon modelers’ a priori beliefs.

The error-bars in Fig. 2 indicate the temporal variability of theSV and SA timescales. Most notable is the variability of the SAtimescales of C3FM2 at degrees two and three. It becomes lessfor higher spherical harmonic degrees. In contrast, the variabilityof the SA timescale derived from GRIMM3 (Fig. 3) appears tobe less pronounced at low degrees and increases towards higherdegrees. Although this is not fully understood, it could be thatapplying different regularization constraints, that is, minimizing thetemporal core surface flow variability in C3FM2 versus minimizingthe third time derivative of Br in CHAOS-4 and GRIMM3, may playa role. The temporal variability of the SV timescales of the two fieldmodels falls into the same range.

We also note in Fig. 3 a slight increase of the accelerationtimescale at n ≥ 8. This effect is more likely due to the regularizationapplied rather than to the degree dependence of the timescale. Asargued earlier, at high degrees the regularization damps the SA morestrongly and the associated timescales are slightly overestimated.

Table 1 lists the fitted timescales τ oSV and τ o

SA (eqs 9 and 10)of the geomagnetic field models. The values agree within theiruncertainties among the models, only τ o

SA derived from C3FM2 de-viates significantly from the values obtained from satellite-basedfield models. This may have several causes. One is the differentkind of regularization applied. Minimizing changes in the core flowas in C3FM2 may damp SA more strongly than minimizing the thirdtime derivative of the field. This is supported by the close relationbetween core flow acceleration and magnetic acceleration that wefind in the dynamo models (see later). Furthermore, the C3FM2model is entirely based on observatory data and therefore the spa-tial resolution is less than for the satellite epoch models. In otherwords, the model SV and SA beyond degree 6 is controlled rather

Table 1. Estimates of the secular variation and secular accel-eration timescales at low n derived from geomagnetic fieldobservations.

Model Model period τ oSV (yr) τ o

SA (yr)

C3FM2 1980–2008 431.5 ± 59.0 17.1 ± 1.1CHAOS 4a 1997–2010 416.9 ± 52.7 11.6 ± 0.9GRIMM3 V1 2001–2010 471.4 ± 59.1 12.4 ± 0.9GRIMM3 V2 2001–2010 434.4 ± 59.6 12.7 ± 0.9

C© 2012 The Authors, GJI, 190, 243–254



strongly by the a priori constraints of temporal and spatial smooth-ness. Due to the wider spline knot spacing and therefore strongertemporal averaging, the SA might be slightly underestimated (andhence the acceleration time overestimated), as discussed earlier. Be-sides, as the SV and SA beyond degree 6 are strongly affected bythe model regularization, it is likely that the related uncertainty isunderestimated. Placing a larger uncertainty on τ o

SA derived fromC3FM2 may bring this value in agreement to the values derivedfrom satellite epoch field models. However, the cause of the highervariability of low degree terms of C3FM2 cannot be related to themodel regularization.

3 DY NA M O M O D E L S

We analyse 27 different dynamo models in rotating spherical shellswith outer radius ro and inner radius ri = 0.35ro. Convection isdriven by a uniform fixed buoyancy flux at the inner boundary. Theflux on the outer boundary is zero and a homogeneous volumetricsink balances the source (Kutzner & Christensen 2000). This de-scribes a situation where the heat flow at the CMB is exactly theflux that can be transported conductively along the adiabatic gradi-ent, while deeper in the core a compositional flux and superadiabaticheat flux associated with the release of light elements and latent heatat the top of the growing inner core drive convection. We use no-slipboundary conditions at the inner and outer boundaries and assumean insulating inner core and mantle. We solve the non-dimensionalequations as given in Kutzner & Christensen (2002), using the mag-netic diffusion time Tmag = D2/η as the basic timescale, where D =ro − ri is the shell thickness and η is magnetic diffusivity. We as-sume for the Earth’s core η = 1.33 m2 s−1, which results in Tmag =121 500 yr, the value that we use to rescale non-dimensional time tophysical time. Boundary conditions correspond to model type 9 inKutzner & Christensen (2002).

We vary three of the four basic control parameters, the Ekmannumber E, magnetic Prandtl number Pm and Rayleigh number Ra,which are defined as

E = ν

D2, (11)

Pm = ν

η, (12)

Ra = gFi D2

κ2ρν, (13)

where ν is kinematic viscosity, the angular frequency of rotation,g gravity at ro (assumed to vary linearly with radius), Fi the totalanomalous mass flux at ri, κ the thermal or compositional diffusivityand ρ the density. We keep the Prandtl number Pr = ν/κ fixed toone.

Most of the models have been reported before (Christensen et al.2009, 2010), but for the purpose of this study the runs have beencontinued for at least 10 advection times D/U , equivalent to atleast 1200 years of real time, to obtain high-cadence records ofthe magnetic field variation. Tests with longer intervals in a fewcases indicated that this is sufficient for estimating the timescalesof magnetic field change to within a few per cent. The total runtime of all models was larger than 50 advection times, which isnormally sufficient for the dynamo to reach an equilibrated state.We selected cases that are rated to be at least marginally ‘Earth-like’ in terms of their magnetic field morphology according to thecriteria by Christensen et al. (2010). An important criterion is, for

example, that the ratio of non-dipole to dipole contributions tothe field at the top of the dynamo is similar to that in the historicalgeomagnetic field. To cover a broad range of values for the magneticReynolds number Rm at low Ekman number, we included one case(at E = 10−5 and Pm = 2) that fails the morphological criteriaby having a field that is too dipolar and equatorially symmetric.The field variation timescales are in agreement with the scalinglaws discussed later also for this case. Our set of models covers adecent range of values for relevant parameters, that is, one orderof magnitude for Rm, more than one for Pm and two orders ofmagnitude for E. The temporal behaviour of the model field or anyaspect of the related spectra have not been selection criteria.

During the model runs we evaluated for each harmonic modeof the poloidal magnetic field at the outer boundary the first andsecond time derivatives by finite differencing, using time steps oftypically 0.5 yr. This may miss very rapid variations, but tests withshorter time steps did not show significant differences. Also, inter-nal geomagnetic variations with timescales much less than a yearare probably unobservable because they are damped in the weaklyconducting mantle. At each of these time steps the degree power inthe field, in the SV and in the SA according to eqs (4)–(6) was cal-culated. Time averages of the respective power spectra, taken overthe full time interval for which high-cadence records have been ob-tained, were then used to calculate the SV times (eq. 7) and the SAtimes (eq. 8).

We also monitored the power (kinetic energy) V (n) in the flowfield u(re, θ , φ) and the power V ′(n) in the flow accelerationu(re, θ, φ) near the outer boundary as a function of spherical har-monic degree n. They are evaluated ‘at the top of the free stream’below the viscous Ekman layer near the outer boundary at re, whichwas taken as the radius of the closest numerical gridpoint that hasa distance of at least 1.5D

√E from the outer boundary. We de-

termined for each model the flow acceleration timescale (Voorhies1995)

τflow =√

V/V ′ , (14)

where for simplicity only the time averages of the total power(summed over all degrees) have been used.

4 M O D E L R E S U LT S

In Table 2 we list the model parameters and some model results,such as the magnetic Reynolds number measuring the mean flowvelocity and the Elsasser number as measure for the magnetic fieldstrength

� = B2

ρμη, (15)

where B is the rms magnetic field strength inside the dynamo. The‘master timescales’ of SV, τ o

SV, (eq. 9) and of SA τ oSA, τ∞

SA, are givenin magnetic diffusion time units. The precise definition of the lattertwo is given in the next section (eq. 17). Below we analyse one ofour models, which fits various properties of the geomagnetic fieldincluding the SV and SA timescales well, in more detail and call itthe reference model. It is highlighted in bold face in Table 2.

4.1 Spatial spectrum of secular acceleration times

Fig. 4 shows as function of spherical harmonic degree the timescalesof SV and SA for the reference model. It has a magnetic Reynoldsnumber of order 1000 and shows a semblance with the morpho-logical properties of the geomagnetic field that is rated very high

C© 2012 The Authors, GJI, 190, 243–254



Table 2. Model parameters and results.

E Pm Ra Rm � τ oSV τ o

SA τ∞SA τflow

10−3 12 1.2 × 106 195 27.6 0.0149 0.00217 0.0096 0.0012710−3 12 2.0 × 106 283 26.9 0.0133 0.00127 0.0083 0.00099

3 × 10−4 1.5 6.0 × 107 184 1.93 0.0178 0.00107 0.0083 0.000953 × 10−4 2 4.5 × 107 203 2.89 0.0162 0.00090 0.0077 0.000953 × 10−4 3 1.5 × 107 149 4.98 0.0226 0.00148 0.0117 0.001413 × 10−4 3 7.0 × 107 376 7.62 0.0104 0.00041 0.0048 0.00056

10−4 1 1.0 × 109 311 4.00 0.0096 0.00047 0.0039 0.0003910−4 2 1.15 × 108 192 5.48 0.0194 0.00090 0.0068 0.0008910−4 2 5.0 × 108 440 6.97 0.0079 0.000256 0.00278 0.0003210−4 2 1.0 × 109 594 16.5 0.0056 0.00032 0.00232 0.00022410−4 5 5.0 × 109 1021 25.2 0.0039 0.000139 0.00143 0.00012410−4 10 3.5 × 107 436 39.1 0.0072 0.00072 0.0037 0.0003810−4 10 8.0 × 107 680 54.9 0.0051 0.00042 0.0024 0.00022

3 × 10−5 0.5 5.0 × 109 239 1.72 0.0142 0.00047 0.0032 0.000363 × 10−5 1 5.0 × 109 449 6.10 0.0086 0.000257 0.00245 0.0002813 × 10−5 2.5 3.0 × 109 816 15.2 0.0044 0.000160 0.00133 0.0001303 × 10−5 2.5 5.0 × 109 1030 14.5 0.0040 0.000094 0.00104 0.0000773 × 10−5 3 7.0 × 108 416 17.1 0.0091 0.00049 0.00289 0.0002673 × 10−5 4 3.5 × 108 378 19.9 0.0091 0.00061 0.0035 0.000323 × 10−5 5 5.0 × 109 1980 38.1 0.0023 0.000051 0.00055 0.000040

10−5 0.5 2.0 × 1010 310 3.06 0.0136 0.00053 0.00240 0.00026810−5 0.8 1.0 × 1011 973 13.2 0.0038 0.000104 0.00096 0.00008010−5 1 5.0 × 109 317 3.46 0.0126 0.00040 0.0030 0.0003310−5 1 1.0 × 1010 441 4.75 0.0098 0.000257 0.00198 0.00020910−5 2 8.0 × 108 231 3.06 0.0139 0.00075 0.0043 0.0005610−5 3 6.0 × 109 749 28.4 0.0049 0.000232 0.00160 0.00011110−5 4 1.0 × 109 374 20.4 0.0083 0.00059 0.0035 0.000285

Figure 4. Mean values of secular variation (squares) and secular accelera-tion (circles) timescales versus spherical harmonic degree for the dynamomodel with E = 3 × 10−5, Pm = 2.5, Ra = 5 × 109, in magnetic diffusiontime units on the right axis and in years on the left axis. Error bars show therange between the tenth to the ninetieth percentile of the set of timescalesdetermined separately for 10 yr data segments. Best fitting lines for an as-sumed 1/n dependence (SV) and a fit given by eq. (17) (SA) exclude thedipole.

according to the criteria of Christensen et al. (2010). A snapshot ofits field structure is found in Buffett & Christensen (2007). Squaresand circles in Fig. 4 indicate the timescales based on the powerspectra averaged over approximately 5000 yr model run time. In ad-

dition, τ SV(n) and τ SA(n) have been calculated separately for each10 yr segment of the model run. Error bars indicate the range fromthe tenth to the ninetieth percentile of the respective values, that is,10 per cent of the values based on a 10 yr interval fall below theerror bar and 10 per cent fall above the error bar.

As was previously found for numerical dynamo models, the SVtime varies approximately as 1/n when the dipole (n = 1) is excluded:

τSV(n) = τ oSV/n . (16)

The master timescale τ oSV that is obtained from a fit in the range

2 ≤ n ≤ 48 is 481 yr, which is slightly larger than the geomagneticvalue for the satellite epoch (Lhuillier et al. 2011, and this work)and almost exactly agrees with the value by Lhuillier et al. (2011)for the gufm1 model in the time period 1840–1990.

The SA times obviously do not follow a simple power law de-pendence on n. At high spherical harmonic degree τ SA(n) seemsto asymptotically approach a 1/n dependence, whereas towards lowdegree the distribution flattens, varying only by 35 per cent betweenn = 1 and n = 10. This has been parametrized in the followingform:

τSA(n) =[

1

(τ oSA)2

+(

n

τ∞SA

)2]−1/2

, (17)

where τ oSA is the acceleration timescale at low n and τ∞

SA is the mastertimescale describing the approximate dependence at high n, whereτSA(n) ≈ τ∞

SA/n. The two parameters have been determined for eachmodel by a best fit in the range n = 2–48. Although the dipole wasagain excluded from the fit, in contrast to the SV timescale it is inline with the acceleration timescales of other low-degree harmonicsfor many models, such as in the reference case shown in Fig. 4.In the reference model, τ o

SA is 11.4 yr, that is, very similar to theacceleration timescale obtained from satellite field models at lowharmonic degree (Holme et al. 2011, and this work).

C© 2012 The Authors, GJI, 190, 243–254



Figure 5. Secular variation (SV) (upper data) and low-degree secular accel-eration (SA) timescale (lower data) versus magnetic Reynolds number forall dynamo models, in magnetic diffusion time units on the right axis andin years on the left axis. The Ekman number is keyed by symbol shape andmagnetic Prandtl number by shading, with white for Pm < 1, light grey forPm = 1, darker grey to black for Pm > 1. Lines are for an assumed Rm−1

dependence in case of SV and for the best fitting exponent Rm−1.29 in thecase of SA.

4.2 Parameter dependence of acceleration timescale

In Fig. 5 we plot the master timescale of SV τ oSV and the low-degree

SA timescale τ oSA for all models versus the magnetic Reynolds

number. As was found before, an inverse dependence on Rm fits themodel results for τ o

SV very well:

τ oSV = 3.61 Tmag Rm−1 . (18)

Models with a magnetic Reynolds number of order 1000 match theobserved geomagnetic SV time.

The low-degree SA timescale shows a slightly stronger depen-dence on Rm, with a best fit given by

τ oSA = 0.98 Tmag Rm−1.29 , (19)

however, the quality of the fit is inferior compared to the fit of theSV timescale. In particular, models with a high magnetic Prandtlnumber (dark symbols) tend to fall above the fitting line, whereasmodels with low magnetic Prandtl number (white or light grey) aremore on the low side. For a magnetic Reynolds number of 1000,eq. (19) gives an acceleration timescale of 15.7 yr, not much higherthan the acceleration time found in the geomagnetic satellite models.

In the models the viscous boundary layer is much thicker thanit is in the Earth. Magnetic field variations propagate partly by dif-fusion across the boundary layer and very high frequencies, whoseskin depth is less than or similar to the boundary layer thickness, areattenuated. Olson et al. (2012) studied temporal spectra of dipolefluctuations in dynamo models and found that at very high frequen-cies they are somewhat more energetic below the viscous boundarylayer than they are on the outer boundary. In non-dimensional termsthe nominal boundary layer thickness is E1/2 and the skin depth is(2τ )1/2 when we use the timescale τ for inverse angular frequency.

Significant attenuation can be excluded when E � 2τ . For a fewmodel cases we find that this condition is only marginally satisfiedwhen we use τ o

SA for τ . Their acceleration timescales might thusbe overestimated because of damping of the highest frequencies inthe Ekman layer. However, we do not find systematically higher SAtimescales in the four cases with E > 0.5τ . Excluding these casesfrom the fit (eq. 19) does not significantly alter the result. We there-fore conclude that the too thick Ekman layer in the models has nostrong affect on the low-degree SA timescale, although the shorterhigh-degree timescales may be somewhat affected.

The significant scatter for τ oSA in Fig. 5 suggests a dependence on

some other parameter besides Rm. As expected, a three-parameterfit of the form

τ oSA = const Rmα Pmβ Eγ (20)

reduces the scatter. The best fitting exponent for the Ekman number,γ = −0.03 is not significantly different from zero, whereas themagnetic Prandtl number has some influence (β = 0.38). However,an even better reduction of the misfit is obtained by expressingτ o

SA in terms of powers of Rm and of the magnetic field strength,relative to the Coriolis force, as measured by the Elsasser number�. Because a better fit is obtained with fewer parameters, this ispreferable compared to the fit by eq. (20). A parametrization of τ o

SA

as a function of Rm and � is also more convenient, because thesetwo parameters match the Earth values in (some) dynamo models,whereas no model matches those of E or Pm. A nearly optimal fitis given by

τ oSA = 3.62 Tmag Rm−5/3�5/12, (21)

and is shown in Fig. 6. A strong magnetic field can conceivablyincrease the SA timescale through its damping effect on magneto-hydrodynamic turbulence, although it is not obvious how this couldwork in detail and so far the exponents in eq. (21) are purely em-pirical. For a magnetic field strength inside the core of 2.5–4 mT(Buffett 2010; Gillet et al. 2010) and standard values for other pa-rameters, the Elsasser number of the Earth’s core is in the range5–12. Using this together with Rm = 1000 in eq. (21) leads to alow-degree SA time of 8.6–12.8 yr, in good agreement with thevalues derived from observations.

Figure 6. Low-degree secular acceleration timescale versus combinationof magnetic Reynolds number and Elsasser number; see Fig. 5 for furtherexplanation. The fitting line has a slope of −5/3.

C© 2012 The Authors, GJI, 190, 243–254



Figure 7. High-degree secular acceleration timescale versus magneticReynolds number; see Fig. 5 for further explanation. The fitting line hasa slope of −10/9.

Fig. 7 shows the variation of the high-degree ‘master’ timescaleof SA with magnetic Reynolds number. The best power-law fit isobtained for an exponent slightly less than −1

τ∞SA = 2.49 Tmag Rm−10/9 . (22)

A two-parameter fit involving the Elsasser number does not reducethe scatter to the same degree as it does in case of τ o

SA.

5 A F RO Z E N F LU X T H E O RY F O RS E C U L A R A C C E L E R AT I O N

5.1 Scaling analysis

Here we present a simple scaling theory for the magnetic SA timebased on the concept of frozen magnetic flux. It is assumed thaton short timescales the contribution of magnetic diffusion to thechange of the magnetic field can be neglected. The time derivativeof the radial magnetic field at the outer boundary of the dynamo isthen given by the equation (Roberts & Scott 1965; Holme 2007)

Br = −∇h · (uh Br) , (23)

where uh is the horizontal flow velocity and ∇h · . . . is the horizontaldivergence operator. We denote by U , B and B the characteristicamplitude of velocity, magnetic field and SV, respectively, and relatethe length scale � that is associated with the horizontal divergence tothe spherical harmonic degree, �∝ D/n. Inserting these character-istic values into eq. (23) and dividing by B we obtain the followingscaling equation

B

B∝ nU

D∝ τ−1

SV . (24)

Multiplying with the magnetic diffusion time Tmag = D2/η gives thescaling relation for the inverse non-dimensional SV time

Tmag

τSV∝ n Rm , (25)

which has the same dependence on harmonic degree and magneticReynolds number as found empirically for the dynamo models.

Figure 8. Low-degree secular acceleration timescale versus accelerationtimescale of the flow at the top of the free stream. Times in magneticdiffusion units on the right and top, in years on left and bottom scale.

Differentiating eq. (23) with respect to time we obtain the frozenflux equation for the SA of the field

Br = −∇h · (uh Br ) − ∇h · (uh Br) . (26)

Denoting the characteristic values of secular magnetic accelerationand flow acceleration by B and U , respectively, we obtain

B

B∝ nU

D+ nU

D

B

B. (27)

Multiplying with Tmag and making use of eq. (24) to eliminate B/Bwe obtain for the inverse non-dimensional SA timescale

Tmag

τSA∝ n Rm + Tmag

U

U. (28)

The first term describes the acceleration of the field by advectionof changing magnetic flux. It has the same dependence on harmonicdegree and magnetic Reynolds number as the SV timescale andshould dominate at large n. This agrees with the 1/n-dependencefound for the high-degree acceleration timescale τ∞

SA. The predictedinverse dependence on magnetic Reynolds number at high n is lessclear in the dynamo models (Fig. 7) than it is in the case of τ SV, butis at least approximately satisfied.

Most interesting for a comparison with the observed SA timescaleis the second term on the right-hand side of eq. (28). It is (formally)independent of the length scale or spherical harmonic degree andshould dominate at low degrees. It predicts that the low-degreemagnetic acceleration timescale is proportional to the timescaleof flow acceleration (eq. 14). In Fig. 8 we tested this directly byplotting for the dynamo models the magnetic timescale against theflow timescale. Despite some scatter a linear relation is confirmed,with

τ oSA = 1.35 τflow . (29)

5.2 Source of low-degree secular variation andacceleration

The SV and SA result from the non-linear interaction of the flowfield and the magnetic field, or their time derivatives, as described by

C© 2012 The Authors, GJI, 190, 243–254



the terms on the right-hand side of eqs (23) and (26), respectively.Expanding all fields into their spherical harmonic components, theSV (acceleration) of a given component results from a large numberof combinations of velocity components and magnetic field compo-nents. SV (SA) at some low degree ns will thus result in part fromthe interaction of velocity and magnetic field components that areresolvable by geomagnetic observations, say, with degree n ≤ 12,and in part from the interaction of small-scale components with n >

12. The contribution of unresolved small scales to the SV at low andintermediate degrees has been identified as a major limitation forinferring the flow pattern at the top of the core (Hulot et al. 1992;Eymin & Hulot 2005). In the following, we attempt to estimate howlarge the relative contribution from the unresolved components maybe. To this end, we first want to answer the question of how much aflow component of a given degree nv, which interacts with a mag-netic field of a known spectrum, contributes in a statistical sense tothe SV at some low degree ns.

The individual contribution of a certain combination can be de-scribed in terms of the Adams (or Gaunt) and Elsasser integrals(James 1973), which must satisfy certain selection rules to havenon-zero values. For example, the interaction of a velocity compo-nent of degree nv with a field component of degree nb can contributeto the field change at ns only when |nv − nb| ≤ ns. Hence, for lowvalues of ns, only the interaction of flow components with magneticfield components that lie in the same narrow spectral band con-tribute. Instead of going through the tedious work of considering allpossible combinations of flow and field components and determin-ing their contribution to SV by evaluating the Gaunt and Elsasserintegrals, we use the dynamo code to assess the contribution of flowat degree nv to SV at ns in a statistical sense. We impose a toroidalor poloidal flow field of a single degree nv with unit power V (nv) =1, let it interact with a magnetic field that has a white spectrumwith unit power at all degrees [W (nb) = 1 for all nb], and recordthe power W ′(ns) of B at various degrees up to ns = 8. We set themagnetic diffusivity to a very small value so that its contributionto the field change is negligible. Actual dynamo magnetic powerspectra are not white, but usually they are smooth, and within thenarrow band nv ± ns in which a flow component of degree n inter-acts with B, the magnetic power is considered as nearly constant.Because we are interested in statistical properties, 200 different re-alizations of a random flow field of degree nv with random magneticfields are evaluated and the power in the field change is summedover harmonic order m and averaged over the different realizations.An implicit assumption for the validity of this approach is that theflow field and the magnetic field are not strongly correlated. This isprobably not strictly true, for example, Finlay & Amit (2011) find aslight preference for flow lines and contours of constant Br to eitheralign or be perpendicular to each other in different regions at thesurface of dynamo models.

Fig. 9 shows the resulting SV power P for toroidal flow of differentdegrees nv from one to 30. As long as nv > ns the power dependson 1/nv, that is, flow of higher degree makes a smaller contribution.For nv < ns the contributions are independent of the degree of theflow. The results are well fitted by

P(ns, nv) = 0.48n2

s (2ns + 3)/5

max(nv, ns). (30)

The power increases with ns because more combinations of flowand magnetic field components contribute at larger ns. The reasonsfor the precise analytical form of the dependence are not clear, but acubic dependence on ns for ns < nv was derived by Voorhies (2004)for a model assuming that the SV is caused by small-scale flow

Figure 9. Power of secular variation at degree ns resulting statistically fromthe interaction of a toroidal flow of degree nv with a magnetic field witha white power spectrum. Symbols are for different ns, increasing from onefor squares on the lowermost line to six for left-pointing triangles on theuppermost line. Lines show the fits given by eq. (30).

near the top of the core. In the case of poloidal flow, the generalpattern is similar but the dependence of P on nv is more complexthan in the toroidal case, in particular for values nv close to ns. Coreflow inversion (e.g. Voorhies 1986) suggests that by far most of thekinetic energy of the near-surface flow is in the toroidal part, andwe find the same in the numerical models. Therefore, we ignore thecomplexity associated with the poloidal flow here.

Fig. 10 shows the time-average power in B, B, B, U and U forthe reference dynamo model. At low degree both the SV power andthe SA power increase strongly with harmonic degree, roughly inparallel with each other. For SA the increase with harmonic degreeis in agreement with the form found for P(ns) in eq. (30) (brokenlines in Fig. 10), whereas for SV at degrees n > 2 it is somewhatless steep. For the recent geomagnetic field Holme et al. (2011)found W ′(n) ∝ n(n + 1). In the dynamo model this is approximatelysatisfied at n > 6 (dash-dotted line) but not at low degrees.

We now use the velocity spectrum and the magnetic field spec-trum in Fig. 10 together with the rule on the interaction of flow andmagnetic field given by eq. (30) to estimate the relative contribu-tions of large- and small-scale flow to SV at low degrees ns in themodel. We do this by calculating

nmax∑nv=nmin

P(ns, nv) V (nv) W (nv) (31)

for the ranges of nv from 1 to 12 and from 13 to 133 (the cut-off degree of the reference dynamo model). Here we assume thatW (nv) is representative for the power of the magnetic field in therelevant band of interaction nv ± ns. We find that the low-degreeSV derives to nearly equal amounts from the large-scale flow(nv < 13), which contributes between 50 and 54 per cent of thetotal at various degrees ns up to eight, and the small-scale flow(nv > 12), contributing 46–50 per cent.

The two terms on the right-hand side of the frozen flux equationfor the SA (eq. 26) are formally equivalent to the right-hand side of

C© 2012 The Authors, GJI, 190, 243–254



Figure 10. Time-average power spectra of the magnetic field on the outerboundary (dark circles), secular variation (SV; light grey circles), secularacceleration (open circles), velocity at the top of the free stream (blacktriangles) and flow acceleration (open triangles) for the reference dynamomodel. The spectra are normalized by the total power in the respective field.Broken lines are for a variation proportional to n2(2n + 3), anchored at then = 1 term of SV and acceleration, respectively. The dash-dotted line is fora SV ∝ n(n + 1) anchored at n = 8.

eq. (23). Thus we can estimate the contributions to the low-degreeSA in a similar way, by replacing in eq. (31) either V (nv) by V ′(nv)or W (nv) by W ′(nv). Doing so we first find that the contributionfrom the second term on the right-hand side of eq. (26), whichdepends on the flow acceleration, is indeed larger than that of thefirst term. In the case of low-degree SA, the large-scale flow andflow acceleration (nv < 13) contribute only 19–22 per cent. Most ofthe low-degree SA is caused by the interaction of the accelerationof the small-scale flow with the magnetic field at a similar scale.

We estimated the relative contributions of large- and small-scaleflow to the low-degree SV and SA also for other models whosemagnetic Reynolds numbers are of order 1000. The results arebroadly comparable to those in the reference model, although inmost of these cases the contributions of the large-scale flow issomewhat larger, up to 75 per cent for SV and 35 per cent for SA. Inthese models the spectral maxima of U and U are shifted towardslower degree compared to the spectrum of the reference model.

The weak contribution of the large-scale flow to the SA, as com-pared to a rather strong contribution to the SV, allows us to ra-tionalize the finding that W ′(ns) starts to fall below the expecteddependence on n2

s (2ns + 3) for ns ≥ 3 (Fig. 10), whereas W ′ ′(ns)follows this dependence rather closely up to at least ns = 10. Accord-ing to eq. (30) the SV (SA) spectrum should have this functionaldependence on ns if the field change (acceleration) at ns is onlycaused by flow (flow acceleration) with degrees nv ≥ ns. This isnearly the case for SA, but not so for SV. If flow with nv < ns werethe only contributor, a weaker dependence proportional to ns(2ns +3) would result from eq. (30). Since contributions from flow at nv

< ns become less and less negligible for SV as ns increases, W ′(ns)has a weaker dependence on ns than in the asymptotic limit whereinduction is caused only by small-scale flow.

6 D I S C U S S I O N A N D C O N C LU S I O N S

In the analysis of the geomagnetic field models a mean secular varia-tion (SV) timescale τ o

SV = 438 yr is obtained. The individual valuesof the models do not differ by more than their uncertainties from thisvalue, see Table 1. The secular acceleration (SA) timescales of thevarious satellite epoch models are similar, only the value of the ob-servatory model C3FM2 deviates from the mean value τ o

SA = 13 yrby more than its uncertainty. A reason for that could be the differenttemporal resolution of the field models and the differences in thetemporal regularizations of the field models between C3FM2 andthe satellite epoch models. Contrary to the satellite epoch modelsthe SA timescale of C3FM2 shows a high temporal variability forlow spherical harmonic degrees and seems to fluctuate mildly athigher degree terms. While the near invariance of the higher degreeterms can be explained by the increased efficiency of the a prioriconstraints applied in the geomagnetic field modelling, the varianceof the lower degree may imply the presence of some global shorttimescale processes. Very likely, this variability is related to pro-cesses of SA generation, as a similar variability is not evident for theSV timescale. Moreover, the lack of this variability in the satelliteepoch models could suggest that the models are covering epochs ofdifferent SA regimes (for SV, a difference of regimes at differentepochs had been suggested by Voorhies 2004). We note that also thedynamo models exhibit strong variability of decadal averages of theSA timescale at the lowest harmonic degrees (Fig. 4) that decreasessomewhat towards higher degree.

At first glance, the close proximity of the derived geomagneticSA timescale to the length of the solar cycle might suggest that it isnot of internal origin at all. However, for periodic processes τ SA isnot equal to the period T , but rather to the inverse angular frequencyω−1 = T /(2π ), so that an external signal with the periodicity of thesolar cycle would have a timescale τ SA ≈ 2 yr.

The dynamo simulations show that SA timescales are weaklydependent on spherical harmonic degree at low degrees, up to ap-proximately 10 for models with magnetic Reynolds number of or-der 1000. This agrees with the finding for the latest versions ofgeomagnetic satellite models (Holme et al. 2011, and this work).The dynamo models also suggest that for degrees n >∼ 10 the SAtimescale decreases significantly and asymptotically approaches a1/n-dependence for large n. So far, the quality of the satellite modelsis not sufficient to confirm or reject this. Also, the satellite modelsdo not show the weak decrease of the acceleration timescale fordegrees up to 10 that is found in the dynamo models. The effectiveincrease of SA damping with harmonic degree by the temporal reg-ularization, as discussed in Section 2, could slightly lengthen theSA timescales in the field models at, say, n > 5 and compensatea small decrease with n of the actual values. Assuming that thetrue spectrum of SA times is nearly flat at low n, as found in thedynamo models, the flatness of the spectra in the field models ar-gues against the assumption that the acceleration timescales at lowdegree are seriously overestimated because of the regularization. Ifthis were the case, one would expect a substantial increase of theinferred acceleration timescales with n caused by the effective scaledependence of the damping.

A simple frozen flux scaling theory explains this dependence ofthe SA time on harmonic degree and suggests that the geomagneticSA at low harmonic degrees is caused by the acceleration of theflow near the core surface as described by Lesur et al. (2010b).Although a large part of the geomagnetic SV can be explained bya steady core flow (e.g. Voorhies 1986; Bloxham 1992), Wadding-ton et al. (1995) and Jackson (1997) found that an acceptable fit to

C© 2012 The Authors, GJI, 190, 243–254



magnetic observatory data require a time-dependent core flow. Froman inversion of Geomagnetic Reference Field models covering theepoch 1945–1980 in terms of time-dependent core flow, Voorhies(1995) obtained a flow acceleration of 0.18 km yr−2. Analysingthe CHAOS satellite model and allowing for more flow complex-ity, Olsen & Mandea (2008) find that the core flow changes morerapidly. For their preferred scheme of inverting SV data for the coreflow expanded up to spherical harmonic degree 15 they infer therms flow acceleration 〈u2

h〉1/2 to be 0.52 km yr−2 at epoch 2003.5and 0.57 km yr−2 for 2004.7 (Olsen & Mandea 2008, supplemen-tary information). In our reference model the rms flow accelerationis larger (approximately 1.9 km yr−2), but most of the power is inspherical harmonic degrees beyond 15 (cf. Fig. 10). The flow ac-celeration in degrees 1–15 has an rms value of 0.39 km yr−2, quitecomparable to the value inferred by Olsen & Mandea (2008).

Applying the results of a statistical analysis of magnetic induc-tion by near-surface flow to the magnetic field and velocity powerspectra of our dynamo models, we find that SV at low harmonicdegree is caused by the interaction of large-scale flow with large-scale magnetic field and the interaction of small-scale flow withsmall-scale field in roughly equal amounts. This concurs with thefinding of Rau et al. (2000) and Amit et al. (2007) that the frozenflux inversion of magnetic field and SV data from dynamo modelsallows one to recover most of the actual velocity field, but also thatthe quality of the inversion for the large-scale flow degrades whensmall scales of the magnetic data are filtered out. With respect toSA at low degree, our analysis suggests that it is mostly caused bythe interaction of magnetic field at small scales, hidden from obser-vation in the geomagnetic field, with the flow acceleration at smallscales. The degree to which these results apply to geomagnetic datadepends on how well the velocity, flow acceleration and magneticfield spectra in the models represent the spectra in the Earth’s core.Because the magnetic Reynolds number, the ratio of advection tomagnetic diffusion, matches the core value, the shape of the mag-netic spectrum at high degrees in the model may represent the corespectrum reasonably well, although it will depend also to some de-gree on the shape of the flow spectrum. Possible differences in theflow spectrum between the model and the core are more difficult toassess, because some parameters that probably influence the flowspectrum are vastly different. One difference that we note is that themagnetic power spectrum in the range n = 2–12 is slightly bluishin the reference model (Fig. 10), whereas in the recent geomagneticfield at the CMB it is slightly reddish (McLeod 1996). This andpossible differences in the flow and flow acceleration spectra mayreduce the contribution from the interaction of unresolved compo-nents to the large-scale geomagnetic SV and acceleration comparedto what we find in dynamo models.

Both the observed SV timescales and the observed SA timescalesof the geomagnetic field can be reproduced by geodynamo modelsif their magnetic Reynolds number is approximately 1000, whichreinforces that in the Earth’s core Rm ≈ 1000. We note that themodels have been selected by criteria that are completely unrelatedto the timescales and spectra of magnetic variability, yet they are ableto match the scales inferred from geomagnetic observations. WhenTmag and Rm are resolved into the constituting components, themagnetic diffusivity η disappears on the right-hand side of eq. (18)and only weak dependencies on η with exponents of 0.3 and 0.25remain in eqs (19) and (21), respectively. Therefore the observedSV and SA timescales constrain more the rms velocity U in the corethan the combination U /η in the magnetic Reynolds number. Theinferred value is U = 0.6 mm s−1 (18.5 km yr−1), which includesflow at small length scales. This is in line with many earlier estimates

(e.g. Finlay & Amit 2011) and is based mainly on the SV, althoughnow also corroborated by the observed SA times. A new resultbased on the SA time and our finding that τflow ≈ τ o

SA is that for theabove value of U a characteristic value for the acceleration of thecore flow of U ≈ 2 km yr−2 is obtained. As argued earlier, most ofthe flow acceleration likely occurs at harmonic degrees >12.

In this paper we only addressed the statistical properties of SAin dynamo models and found that they can be matched with val-ues derived from observations. A more ambitious goal, beyond thescope of this paper, would be to also match characteristic patternof SA, although this may be of limited significance if most of theacceleration is at unresolved scales.

If the observed 10 yr geomagnetic SA time would reflect mainlythe effects of magnetic waves and torsional oscillations in the core,our models should be unable to reproduce it. For example, in our ref-erence dynamo model the Alfven wave timescale D/UA is similar tothe advection time D/U , whereas in the core the former is probablymore than an order of magnitude shorter. Our finding that dynamomodels can simultaneously match the observed timescales of geo-magnetic SV and SA suggests that core dynamics at decadal andsubdecadal timescales is not dominated by fundamentally differentprocesses than it is at timescales of >∼100 yr. Rather, there seemsto be a continuum between slow and rapid core motions driven byconvection. This does not mean that convective turbulence could notdrive torsional waves at short timescales, however, there is probablyno resonant amplification of these motions which would lead toenhanced power in this frequency band. Olson et al. (2012) foundthat dipole fluctuations in dynamo models have smooth broad-band(temporal) frequency spectra without spectral spikes or bands ofenhanced energy at high frequencies. Since our models can repro-duce the observed geomagnetic SV and acceleration timescales, thisprobably also holds for the geomagnetic frequency spectrum.

A C K N OW L E D G M E N T S

URC thanks Richard Holme for motivating him to study SA timein dynamo models and for useful discussions. IW was funded bythe Deutsche Forschungsgemeinschaft through Schwerpunktprojekt1488.

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Buffett, B.A. & Christensen, U.R., 2007. Magnetic and viscous coupling atthe core-mantle boundary: inferences from observations of the Earth’snutation, Geophys. J. Int., 171, 145–152.

Christensen, U.R. & Tilgner, A., 2004. Power requirement of the geodynamofrom ohmic losses in numerical and laboratory dynamos, Nature, 429,169–171.

Christensen, U.R., Holzwarth, V. & Reiners, A., 2009. Energy flux deter-mines magnetic field strength of planets and stars, Nature, 457, 167–169.

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Timescales of geomagnetic secular acceleration in satellite field models and geodynamo models