Effects of Buoyancy and Rotation on the Polarity Reversal Frequency Of

Home , Geomagnetic reversal

GJI gji_4234 Dispatch: May 22, 2009 CE: RMQ Journal MSP No. No. of pages: 14 PE: Gregor 1 Geophys. J. Int. (2009) doi: 10.1111/j.1365-246X.2009.04234.x 2 3 4 5 Effects of buoyancy and rotation on the polarity reversal frequency 6 7 of gravitationally driven numerical dynamos 8 9 10 Peter Driscoll and Peter Olson 11 Department of Earth and Planetary Sciences, Johns Hopkins University, MD, USA. E-mail: [email protected] 12 13 14 Accepted 2009 April 28. Received 2009 April 28; in original form 2008 December 12 15 16 17 SUMMARY 18 We present the results of 50 simulations of the geodynamo using a gravitational dynamo model 19 20 driven by compositional convection in an electrically conducting 3-D fluid shell. By varying 21 the relative strengths of buoyancy and rotation these simulations span a range of dynamo 22 behaviour from strongly dipolar, non-reversing to multipolar and frequently reversing. The 23 polarity reversal frequency is increased with increasing Rayleigh number and Ekman number. 24 Model behaviour also varies in terms of dipolarity, variability of the dipole and core–mantle 25 boundary field spectra. A transition region is found where the models have dipolar fields and 26 moderate (Earth-like) reversal frequencies of approximately 4 per Myr. Dynamo scaling laws 27 are used to compare model dipole intensities to the geodynamo. If the geodynamo lies in a 28 similar transition region then secular evolution of the buoyancy flux in the outer core due to GJI Geomagnetism, rock magnetism and palaeomagnetism 29 the growth of the inner core and the decrease in the Earth’s rotation rate due to tidal dissipation 30 can account for some of the secular changes in the reversal frequency and other geomagnetic 31 32 field properties over the past 100 Myr. 33 Key words: ???. 34Q1 35 36 37 1 INTRODUCTION 38 periods of no recorded reversals, the magnetic superchrons. Com- 39 Palaeomagnetic records show that the geomagnetic field has under- pilations of geomagnetic polarity that go back as far as 500 Ma 40 gone nearly 300 polarity reversals over the last 160 Myr (Constable (Pavlov & Gallet 2005) have revealed a long-period oscillation in 41 2003) and perhaps as many as a thousand polarity reversals over the reversal frequency. They also show a decrease over the last 83 Myr − 42 last 500 Myr (Pavlov & Gallet 2005). It is challenging to resolve from the current reversal frequency of R ≈ 4Myr 1 going back 43 the intensity and structure of the geomagnetic field during reversals to the most recent Cretaceous Normal superchron (CNS), where 44 because these events last on the order of a few thousand years, only there are no recorded reversals for a period of 40 Myr. Prior to the 45 a snapshot in most sedimentary records and too long to record in CNS there is a steady increase in reversal frequency going back in − 46 most volcanic events. However, the consensus view is that polar- time with a maximum of R ≈ 5Myr 1 around 170 Ma. The pattern 47 ity reversals occur when the geomagnetic dipole field intensity is repeats as this maximum is preceded by a steady decrease in the 48 anomalously low (Valet & Meynadier 1993). The palaeomagnetic reversal rate going back to the Kiaman superchron, which ended 49 record also shows a number of excursions, some of which may be around 260 Ma. Prior to the Kiaman superchron the record is very 50 failed reversals (Valet et al. 2008), when the dipole axis wanders incomplete, although evidence has been found for a third major 51 substantially from the pole but recovers to its original orientation. Moyero superchron around 450–500 Ma (Gallet & Pavlov 1996). 52 Generally a period of strong dipole intensity follows such an excur- This oscillation in reversal frequency over the past 500 Myr, with 53 sion or reversal. Overall the geomagnetic field shows no polarity a periodicity of approximately 200 Myr between superchrons, is 54 bias, that is there is no preference for dipole axis alignment parallel the primary evidence for an ultra-low frequency oscillation in the 55 or antiparallel to the Earth’s rotation axis (Merrill et al. 1996). It geodynamo. 56 is expected theoretically that there is no preference for one polar- The causes of geomagnetic polarity reversals are obscure, partly 57 ity over the other from the induction equation where every term is due to the sparsity of the palaeomagnetic record and the fact that the 58 linear in the magnetic field B, the Navier–Stokes equation where source of the geomagnetic field, inside the outer core, is inaccessi- 59 the only magnetic term is the Lorentz force which is quadratic in ble to direct magnetic measurement. Many mechanisms have been 60 B, and the homogeneous magnetic boundary conditions (Gubbins proposed to explain magnetic field reversals (e.g. Constable 2003; 61 & Zhang 1993). Glatzmaier & Coe 2007). One of the fundamental questions regard- 62 Although polarity reversals seem to occur chaotically throughout ing polarity transitions according to Merrill & McFadden (1999) 63 the palaeomagnetic record there is striking evidence of very long is whether reversals are initiated by processes intrinsic or extrin- 64 timescale trends in the mean reversal frequency, including long sic to the geodynamo. Extrinsic influences that have been consid-

C 2009 The Authors 1 1 2 P.Driscoll and P.Olson 2 3 ered include heterogeneities on the core–mantle boundary (CMB, The outer core contains approximately 6–10 per cent light ele- 4 Glatzmaier et al. 1999; Christensen & Olson 2003; Kutzner & Chris- ment (denoted here by Le) mixture by mass, the remaining mass 5 tensen 2004), the rate at which heat is extracted from the core into being iron and nickel (denoted here by Fe). The Le constituents 6 the mantle (McFadden & Merrill 1984; Courtillot & Besse 1987; are unknown but are most likely some combination of S, O and Si, 7 Courtillot & Olson 2007), and the effects of the conducting inner based on their density when mixed with Fe at high temperatures 8 core (Hollerbach & Jones 1995; Gubbins 1999; Wicht 2002). Pos- and pressures (Alfe et al. 2000). The radius of the inner core is 9 sible intrinsic reversal mechanisms include advection of reversed determined by the intersection of the geotherm with the melting 10 magnetic flux from the interior of the core to high latitudes by curve for an Fe–Le mixture. As the core cools and the tempera- 11 meridional upwellings (Sarson & Jones 1999; Wicht & Olson 2004; ture decreases, the intersection of the geotherm and the melting 12 Takahashi et al. 2007; Aubert et al. 2008), a switch from an equato- curve moves to larger radii, hence the inner core grows. The drop 13 rially antisymmetric to a symmetric magnetic field state (Nishikawa in density across the inner core boundary (ICB) of 4.5 per cent is 14 & Kusano 2008), and mixing of axisymmetric magnetic flux bun- greater than the predicted drop of 1.8 per cent by just melt alone 15 dles of opposite polarity (Olson et al. 2009). In addition to physical (Alfe et al. 2000) and can be explained by the way the inner core 16 causes, statistical models have been developed to account for the grows: it preferentially rejects Le during the crystallization process 17 reversal timescales (Hulot & Gallet 2003; Jonkers 2003; Ryan & providing an enrichment of Le on the outer core side of the ICB. 18 Sarson 2007), but do not address the long-timescale trends in re- The fluid enriched in Le, being positively buoyant compared to the 19 versal frequency. There are so many factors that could effect the average outer core Le concentration, is a source of convection and 20 geodynamo reversal mechanism that it is beyond of the scope of gravitational energy for the dynamo. There is also a contribution 21 this study to consider them all. Instead, we focus on the sensitivity due to latent heat release during crystallization of the inner core 22 of the dipole field reversal frequency to changes in buoyancy and (Fearn 1998) but is not included in the model. 23 rotation in one class of numerical dynamos. Our restriction to model only compositional convection in this 24 In this study we use a numerical magnetohydrodynamic (MHD) study is justified as an approximation because compositional con- 25 dynamo model to investigate how reversal frequency varies as a vection is a more efficient way to drive the dynamo (Gubbins et al. 26 function of the input parameters, specifically buoyancy flux and 2004) and because even if the CMB heat flow is subadiabatic com- 27 rotation rate. We compute dynamos with fixed input parameters positional convection can still sustain a dynamo (Nimmo 2007b). 28 and infer changes in the geodynamo reversal frequency from the An entropy balance in the core shows that the thermodynamic ef- 29 thermochemical evolution of the core through this region of pa- ficiency of individual sources of buoyancy are proportional to the 30 rameter space. Driscoll & Olson (2009) investigate dynamos with temperature at which they operate ∝ (1 − T o/T ), where T o is 31 evolving input parameters and refer to the type of evolution applied the temperature at the CMB (Braginsky & Roberts 1995; Labrosse 32 here as staged evolution because the dynamo models represent dis- & Macouin 2003; Nimmo 2007a). Therefore, sources of buoyancy 33 crete stages in the evolution of the core. The main goals of this that are spread throughout the core will have a lower efficiency than 34 paper are: (i) to systematically explore dynamo polarity reversals sources near the ICB. Roberts et al. (2001) estimates the efficiency 35 over a range of parameter space, keeping the diffusive parameters of a purely thermal dynamo with no inner core to be ≈ 3per 36 constant while varying the buoyancy and rotation, (ii) to compare cent, whereas the efficiency of a thermochemical dynamo is larger 37 dynamo behaviour with previous results using scaling laws and (iii) ≈ 18 per cent. Intuitively this makes sense because most of the 38 to compare the dynamo model reversal frequencies with that of the gravitational energy released at the ICB is available for magnetic 39 geodynamo. In Section 2, we introduce the governing equations of field generation, whereas thermal convection requires that a large 40 the dynamo model and the form of the relevant non-dimensional portion of the heat is conducted down the core adiabat. 41 control parameters. In Section 3, we present our dynamo models in The governing equations of the gravitational dynamo model in- 42 terms of dipole strength, dipole variability, reversal frequency, mag- clude the Boussinesq conservation of momentum, conservation of 43 netic field spectra at the CMB, and apply dynamo scaling laws. In mass and continuity of magnetic field, buoyancy transport, and mag- 44 Section 4 we use scaling laws to compare these dynamos to the geo- netic induction. They are, in dimensionless form (Olson 2007) 45 dynamo and discuss the implications of these results on the changes Du 46 in geodynamo reversal frequency over the last 100 Ma. Section 5 E −∇2u + 2zˆ × u +∇P Dt 47 contains a summary, conclusions and avenues for future work. ERa r 48 = χ + Pm−1(∇×B) × B (1) 49 Pr ro 50 2 DYNAMO MODEL 51 ∇·(u, B) = 0(2) 52 The precise thermal and chemical state of the core is highly uncer- Dχ 53 tain, however it is generally accepted that convection in the core is = Pr−1∇2χ − 1(3) 54 driven by a combination of both thermal and compositional sources Dt 55 of buoyancy. Dynamo models have demonstrated the ability to re- ∂B 56 produce a generally Earth-like magnetic field structure using a vari- =∇× × + −1∇2 , ∂ (u B) Pm B (4) 57 ety of sources of buoyancy and boundary conditions (Dormy et al. t 58 2000; Kutzner & Christensen 2000). We use a gravitationally driven where u is the fluid velocity, zˆ is the direction along the rotation 59 dynamo, in which the buoyancy is produced by compositional in- axis, P is a pressure perturbation, r is the radial coordinate, r o is 60 stead of thermal gradients. The large adiabatic thermal gradient in the radius of the outer boundary, χ is the perturbation in Le mass 61 the core requires that a substantial amount of heat is lost to the concentration, B is magnetic field and t is dimensionless time. The 62 mantle. This loss of heat from the core subsequently causes the radius ratio in the model is fixed at an approximate Earth ratio 63 crystallization and growth of the inner core which is responsible for r i /r o = 0.35. The velocity is scaled by ν/D,whereD is the shell 64 driving compositional convection in the outer core. thickness. The magnetic field B is scaled by Elsasser number units

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 3 2 3 (ρ /σ )1/2,whereρ is outer core mean density, is rotation o o 3 RESULTS 4 rate and σ is electrical conductivity. Compositional convection is 5 modelled using a homogeneous buoyancy sink term in (3) (Kutzner All 50 dynamo models have input parameters Pr = 1, Pm = 20, 6 & Christensen 2000; Olson 2007). The Le concentration is scaled and span a range of E from 2 × 10−3 to 10 × 10−3 and Ra from 1 × 7 χ 2/ν χ 4 4 by ˙ o D ,where˙o is the secular change of outer core mean light 10 to 4.5 × 10 . All dynamos are run for a minimum of 22t d 8 element concentration, assumed constant in this model. Although (440 kyr), with an average run time of 68 t d (1.3 Myr), and a max- 9 this term is not constant over very long timescales, it is a good imum of 845 t d (17 Myr). We calculate the critical values of Ra 10 approximation in runs that are several Myr long because we expect for the onset of convection and dynamo action at three values of E. −1 −3 −3 −3 11 the change to be smallχ ö ≈ 0.01 Gyr according to the secular For E = 2 × 10 ,6× 10 and 10 × 10 we find the onset of 12 4 4 4 evolution described in Section 4.2. convection at Racrit = 2.4 × 10 , 1.1 × 10 and 0.7 × 10 ,respec- 13 The dimensionless parameters used to scale these equations in- tively. For the same E values we find the onset of dynamo action at 14 4 4 4 clude the Ekman E, Rayleigh Ra, Prandlt Pr and magnetic Prandtl Radyn = 2.9 × 10 , 1.1 × 10 and 0.7 × 10 , respectively. We find 15 Pm numbers, defined as follows: an approximate dependence of the critical Ra values on E of 16 ν = −3/4 17 E = (5) Racrit 225E (11) 2 18 D −7/8 19 Radyn = 127E . (12) 20 βg D5χ˙ Ra = o o (6) A theoretical prediction for the onset of convection in a rotating 21 κν2 sphere gives a −4/3 exponential dependence of Ra on E,and 22 crit for the onset of dynamo action predicts a −1/2 exponential depen- 23 ν = dence on E (Jones 2007). The discrepancy between the theoretical 24 Pr (7) κ predictions and our results is probably due to the limited range 25 of our relatively large E number models. Plotted in Fig. 1 are the 26 ν models at the onset of convection (triangles), at the onset of dy- 27 Pm = , (8) η namo action (plus symbols), and the relations (11) and (12) (solid 28 lines). 29 β =−ρ−1∂ρ/∂χ where o is the compositional expansion coefficient, Table 1 gives a summary of the parameters from all 50 dynamo 30 ν κ η go is gravity at the outer boundary, and , and are the viscous, models, and Fig. 1 is a regime plot of Ra versus E. Fully developed 31 chemical and magnetic diffusivities in the outer core. The relevant chaotic strong field dynamos are shown as circles, and one oscillat- 32 timescales of the dynamo model are the viscous diffusion time ing dynamo (at Ra = 3.3 × 104 and E = 2 × 10−3)isshownas 33 2 t = D /ν and the dipole decay time t d . The dipole decay time is 34 2 related to the viscous diffusion time by t d /t = Pm(r o/π D) .For 35 our models with Pm = 20, t d /t ≈ 4.81. We can express model times 36 in terms of Earth years using t d ≈ 20 kyr. 37 Boundary conditions, initial conditions, and resolution are the 38 same for all cases. Boundary conditions are no slip and electrically 39 insulating at both boundaries, isochemical at the inner boundary, 40

41 χ = χi at r = ri , (9) 42 χ 43 where i is a constant, and no flux at the outer boundary, 44 ∂χ 45 = 0atr = r . (10) ∂r o 46 47 We use an initial buoyancy perturbation (l, m = 3) of magnitude 48 0.1 and initial dipole field of magnitude 5, both in non-dimensional 49 units. 50 Due to the large number of dynamo models and relatively long 51 simulation times it is necessary to use a relatively sparse grid and 52 large time steps. We use the same numerical grid for all the cal- 53 culations with N r = 25 radial, N θ = 48 latitudinal and N φ = 54 96 longitudinal grid points, and spherical harmonic truncation at 55 l max = 32. This restricts us to relatively large Ekman numbers, 56 ∼10−3. The advantages and disadvantages of large Ekman numbers 57 Figure 1. Regime plot for all simulations. Symbols are shaded by reversal in modelling the geodynamo are described by Wicht (2005), Chris- −1 58 tensen & Wicht (2007), Aubert et al. (2008) and Olson et al. (2009). frequency R (Myr ) and scaled in size by dipole fraction f d ,definedas B at the CMB divided by the volume averaged B . Critical Ra values for 59 The current threshold Ekman number in geodynamo modelling is d rms the onset of convection and dynamo action are shown as triangles and plus −6 −7 60 around 10 –10 (Kageyama et al. 2008; Takahashi et al. 2008) symbols, respectively, and solid lines according to (11) and (12), respectively. 61 and polarity reversals have been found in low E dynamos (Glatz- All chaotic strong field dynamos are circles. A contour of γ core = 0.07 is 62 maier & Coe 2007; Takahashi et al. 2007; Rotvig 2009). However, shown for reference (dashed line), calculated by evaluating the scaling law in 63 these models require too much spatial and temporal resolution to (15) at core values. Non-reversing dynamos with enough dipole variability 64 allow for a large number of long simulations. (s > 0.3) to reverse if run longer are open circles with an inner filled circle.

C 2009 The Authors, GJI 1 4 P.Driscoll and P.Olson 2

3 Table 1. Results of 50 chaotic dynamo models listed in order of lowest reversal frequency R to highest. t run is the length of each run in 4 dipole decay times, R is the average reversal rate, d is dipolarity, s is variability of the dipole, f d is the dipole fraction, E m and E k are γ 5 the volume averaged magnetic and kinetic energies, Lod is the modified Lorentz number and is defined in (15). 4 −3 −1 6 Ra (× 10 ) E (× 10 ) t run R (Myr ) dsfd E m E k Lod γ Regime 7 2.53 3.00 154 0.00 0.66 0.06 0.32 167 5 0.017 0.208 i 8 3.39 3.00 54 0.00 0.65 0.08 0.29 199 11 0.017 0.159 i 9 2.54 4.00 32 0.00 0.65 0.10 0.28 137 9 0.019 0.148 i 10 2.54 5.00 48 0.00 0.63 0.16 0.25 103 14 0.018 0.105 i 11 1.97 6.00 48 0.00 0.63 0.17 0.25 75 11 0.019 0.104 i 12 3.39 4.50 44 0.00 0.60 0.13 0.20 119 22 0.014 0.078 i 13 2.26 6.00 48 0.00 0.59 0.25 0.20 69 16 0.014 0.072 i 14 3.10 5.00 64 0.00 0.58 0.14 0.19 102 22 0.014 0.072 i 15 3.67 4.50 32 0.00 0.58 0.14 0.19 130 26 0.014 0.074 i 16 2.82 5.50 48 0.00 0.58 0.17 0.19 87 21 0.013 0.067 i 17 1.97 6.50 112 0.00 0.58 0.35 0.20 60 14 0.014 0.072 i 2.54 6.00 46 0.00 0.57 0.26 0.18 70 21 0.013 0.060 i 18 2.26 6.49 48 0.00 0.56 0.28 0.17 59 18 0.012 0.056 i 19 3.39 4.00 70 0.00 0.55 0.17 0.22 133 19 0.015 0.095 i 20 3.39 5.00 56 0.00 0.55 0.24 0.17 102 28 0.012 0.061 i 21 3.95 4.50 48 0.00 0.55 0.24 0.17 126 32 0.012 0.062 i 22 2.54 6.50 125 0.00 0.54 0.31 0.16 65 23 0.012 0.050 i 23 3.67 5.00 80 0.00 0.53 0.24 0.16 110 33 0.012 0.057 i 24 4.23 4.50 95 0.00 0.53 0.26 0.16 131 37 0.011 0.058 i 25 2.82 6.00 112 0.00 0.53 0.34 0.15 71 26 0.011 0.048 i 26 2.26 7.02 64 0.00 0.52 0.46 0.14 50 21 0.010 0.041 i 27 3.95 5.00 845 0.48 0.52 0.29 0.15 112 38 0.011 0.051 ii 3.10 5.50 61 0.85 0.54 0.35 0.16 79 29 0.011 0.050 ii 28 3.67 5.50 48 1.08 0.50 0.35 0.14 95 38 0.010 0.045 ii 29 4.23 5.00 77 1.34 0.49 0.44 0.13 113 44 0.010 0.045 ii 30 2.54 7.00 64 1.61 0.47 0.58 0.11 50 28 0.008 0.031 ii 31 2.82 6.49 64 1.61 0.48 0.67 0.12 51 32 0.007 0.031 ii 32 3.67 5.80 64 1.62 0.49 0.34 0.13 95 40 0.010 0.043 ii 33 3.95 5.50 63 1.64 0.48 0.39 0.13 104 42 0.010 0.044 ii 34 3.10 6.00 62 1.66 0.50 0.43 0.13 67 33 0.009 0.038 ii 35 3.39 5.50 52 1.98 0.49 0.34 0.14 86 34 0.010 0.046 ii 36 3.10 6.50 45 2.30 0.49 0.41 0.14 76 33 0.010 0.041 ii 37 3.39 6.00 56 2.76 0.48 0.41 0.13 82 37 0.010 0.039 ii 4.23 5.50 57 3.63 0.45 0.59 0.12 100 49 0.009 0.036 ii 38 3.95 5.70 78 4.64 0.47 0.39 0.13 107 43 0.010 0.042 ii 39 3.67 6.00 48 5.41 0.46 0.50 0.12 92 42 0.010 0.038 iii 40 3.10 7.00 44 5.83 0.39 0.64 0.09 58 38 0.007 0.024 iii 41 3.39 8.00 52 5.98 0.29 0.88 0.06 35 53 0.004 0.011 iii 42 3.67 10.00 31 6.54 0.24 0.80 0.04 28 73 0.003 0.007 iii 43 3.95 10.00 22 6.87 0.23 0.91 0.04 29 81 0.003 0.006 iii 44 3.95 6.00 59 6.96 0.41 0.58 0.10 90 47 0.008 0.031 iii 45 3.67 7.00 42 7.37 0.34 0.78 0.08 59 51 0.006 0.019 iii 46 3.67 6.50 46 7.86 0.39 0.68 0.09 70 47 0.007 0.024 iii 3.67 7.50 38 8.14 0.30 0.94 0.06 40 58 0.004 0.012 iii 47 2.26 10.00 38 8.14 0.32 0.82 0.06 16 33 0.003 0.009 iii 48 2.82 10.00 50 8.19 0.27 0.84 0.05 20 51 0.003 0.007 iii 49 3.39 6.50 40 9.03 0.37 0.73 0.08 58 42 0.006 0.021 iii 50 3.39 7.00 39 9.15 0.35 0.66 0.08 53 45 0.005 0.019 iii 51 3.39 7.50 38 9.46 0.31 0.90 0.06 40 49 0.004 0.013 iii 52 3.39 10.00 29 14.19 0.27 0.80 0.05 27 66 0.003 0.008 iii 53 54 55 56 a square. The symbol for each dynamo is scaled in size according 57 3.1 Dynamo regimes to its dipole fraction f d = B d /B rms,whereB d is the rms dipole 58 intensity on the CMB and B rms is the volume averaged rms field. Like geomagnetic reversals, there is no universally accepted method 59 Each symbol is shaded according to its polarity reversal frequency of defining a dynamo model reversal. According to Constable 60 R, which is defined as the average number of axial dipole reversals (2003), only in hindsight can it be certain that a geomagnetic reversal 61 per Myr (or 50 t d ), with open circles for no reversals, black for in- has taken place. Alternatively, Merrill et al. (1996) define a geo- 62 frequent reversals, and light grey for all reversal frequencies larger magnetic reversal as a globally observed 180◦ change in the dipole 63 than 12.5 Myr−1. field direction lasting more than a few thousand years, with the new 64

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 5 2 3 polarity exhibiting some overall stability. Following Cande & Kent 4 (1992), we define a dipole polarity reversal as an event in which the 5 dipole axis crosses the equator at least once and spends at least one 6 dipole decay time t d in both the normal and reverse polarity, before 7 and after the reversal event, without crossing the equator. In this 8 definition, we attempt to count only those events that involve the 9 transition from one stable dipole orientation to another, and exclude 10 isolated short-term dipole tilt reversals or cryptochrons. However, 11 it is common for the dipole axis to cross the equator several times 12 during a single complex reversal event in these simulations. 13 In general, we find that reversal frequency is increased by sep- 14 arately increasing the Rayleigh number or the Ekman number, or 15 by increasing both simultaneously. In Fig. 1 there is a transition 16 zone where the dynamos go from a non-reversing to a reversing 17 regime, which we will refer to as the reversal regime boundary. 18 For dynamos near this boundary the reversal statistics are probably

19 not well resolved because of the long timescales between reversals. Figure 2. Local Rossby number Rol versus a buoyancy flux-based Rayleigh 20 As a consequence, if the non-reversing dynamos near the bound- number Ra Q for all dynamos. Shading corresponds to reversal frequency as 21 ary were run much longer they would likely reverse, perhaps very in Fig. 1. 22 infrequently, and the regime boundary would be pushed to smaller 23 Ra and E values. However, it is clear from Fig. 1 that there exists 24 a reversal regime boundary in Ra–E space, analogous to the vari- with R = 4Myr−1. For comparison, average reversal frequencies 25 ous kinds of dynamo regime boundaries in Zhang & Busse (1988), of the geodynamo for three time periods over the last 83 Myr are 26 Kutzner & Christensen (2002) and Simitev & Busse (2005). shown as dashed lines (from Cande & Kent 1995). Further compar- 27 Here it is convenient to introduce a hybrid parameter Ra Q that ison with the geodynamo is discussed in Section 4. 28 is better for scaling convection-driven dynamos. Ra Q is a buoy- Fig. 3(b) shows dipolarity d = |Bd |/|B| at the CMB, where the 29 ancy flux-based Rayleigh number defined by Christensen & Aubert overbar denotes a time average. The present-day core field dipolarity 30 (2006), modified here to account for the buoyancy flux of the grav- d = 0.64 (dashed) is near the average of the most dipolar models 31 itational dynamo (e.g. Loper 2007) where d appears to asymptote, while reversing models are found for 32 ∗ − d ≤ 0.5. Dipolarity decreases with large convective forcing because Ra = r E 3 Pr 1(Ra − Ra ), (13) 33 Q crit both the flow and magnetic induction become increasingly smaller ∗ 34 where r = r o/r i and Racrit is calculated from (11). This hybrid scale, which is confirmed by the positive correlation between Ra Q 35 parameter allows us to plot dynamo output variables versus a single and Rol (Fig. 2). Takahashi et al. (2008) find a similar trend, that 36 control parameter that is a function of both buoyancy flux and the length scale of the kinetic and magnetic energy decreases with −2 37 rotation rate. The typical value of Ra Q = 10 for these models is increasing Ra. 38 larger than those of the models considered by Christensen & Aubert Non-reversing thermal convection dynamos sometimes have even 39 (2006). higher dipolarities of 0.8– 0.9 but they exhibit a sharp transi- 40 The Rossby number Ro, a measure of the balance of interial and tion down to d ≈ 0.3 at the onset of reversals (Christensen & 41 Coriolis forces, can be used to characterize the influence of rotation Aubert 2006). However, with compositional convection we find a 42 on the dynamo flow. It has been shown for numerical dynamos that smaller and smoother decrease in dipolarity at the onset of reversals 43 a local Rossby number, Rol = Ro(l u /π)wherel u is the characteris- (Fig. 3b) consistent with the compositional convection dynamos of 44 tic length scale of the flow, is better for scaling dynamos at various Kutzner & Christensen (2002). This smooth transition in dipolar- 45 Ekman numbers (Christensen & Aubert 2006; Olson & Christensen ity can be attributed somewhat to the preference for more dipolar 46 2006). Fig. 2 confirms that for these dynamo models Rol is strongly fields when convection is driven by buoyancy at the inner boundary 47 correlated with Ra Q , with the exception of two outlying reversing (Kutzner & Christensen 2000). Convection in the core is thought 48 cases both at high Ekman number (E ≥ 7 × 10−3). Therefore, as the to be some combination of thermal and chemical convection so 49 buoyancy flux is increased the flow velocity increases and the char- the transition from strong to weak dipolarities may be relatively 50 acteristic length scale of the flow decreases. Although we choose smooth. 51 to plot dynamo properties versus Ra Q instead of Rol , because Ra Q Fig. 3(c) shows the normalized standard deviation of the dipole = | |/| | 52 contains only input parameters, they can be interchanged in much s Bd Bd at the CMB, where the prime denotes deviations 53 of the following discussion. from the time average. There is an increase in dipole variability with 54 Our main focus is describing the reversal behaviour of the dy- Ra Q , and a threshold value of s = 0.3 that separates the reversing 55 namo models in terms of dimensionless variables and determin- (s > 0.3) and non-reversing (s < 0.3) regimes. The dynamos above 56 ing whether simple reversal regime boundaries can be identified. s = 0.3 that do not reverse are marked in Fig. 1 with an inner black −2 57 Fig. 3(a) shows that reversals are found above Ra Q = 10 and circle to indicate that they contain enough variability to reverse, and −2 58 increase with Ra Q . At higher Ra Q (>3 × 10 ) there is increased may if run longer. The estimated geomagnetic variability of s = 59 scatter in reversal frequency, due to the fact that these dynamos 0.4 over the last 160 Myr is shown for reference (Constable 2003). 60 are less dipolar with frequent excursions that become increasingly The correlation between dipole variability and reversal frequency 61 difficult to distinguish from short reversals and are run for a shorter found here is also evident for the geodynamo, for example during 62 period of time. The model reversal frequencies span the range of the Oligocene (23–34 Ma) when the reversal frequency is nearly 63 geodynamo reversal frequencies over the last 83 Myr, including 3Myr−1 the dipole variability is high with s ≈ 0.5, and during the 64 superchron-like states with R = 0 and modern Earth-like behaviour CNS the variability is much lower with s ≈ 0.28 (Constable 2003).

C 2009 The Authors, GJI 1 6 P.Driscoll and P.Olson 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 −1 27 Figure 3. Summary of dynamo behaviour as a function of Ra Q . Shading corresponds to reversal frequency as in Fig. 1. (a) Reversal frequency R (Myr ) 28 versus Ra Q . Contours of average R for the geodynamo are shown (dashed lines) for three periods over the last 83 Myr (Cande & Kent 1995). (b) Dipolarity d = 29 at the CMB versus Ra Q . Dashed line shows the present-day core field dipolarity d 0.64. (c) Dipole variability s at the CMB versus Ra Q . Dashed line shows = 30 the estimated geomagnetic dipole variability of s 0.4 over the last 160 Myr (Constable 2003). (d) Lod versus Ra Q and scaling law (15) with contours of γ = 0.025, 0.05, 0.1 and 0.2 (solid lines). 31 32 33 In order to compare these results with models in different regions models. Dipolarity may also be used to characterize this transition 34 of parameter space and with the geodynamo we require scaling in compositional convection models but as noted above d exhibits 35 laws that relate the relevant physical parameters. For magnetic field a much sharper transition for thermally driven dynamos. 36 scaling we use the modified Lorentz number, a non-dimensional We can use Fig. 3 to categorize these dynamos into three regimes. = 37 dipole field strength defined by Olson & Christensen (2006) as, Regime (i) contains the non-reversing (R 0) dynamos charac- γ> 38 1/2 terized by strong dipole field intensity ( 0.05) and dipolarity E > < 39 Lod = Bd , (14) (d 0.5), and relatively little variability (s 0.3). Regime (ii) con- Pm < < −1 40 tains the reversing (0 R 5Myr ) dynamos where the dipole <γ < < < 41 where B d is in Elsasser number units. A scaling law relating mag- intensity (0.025 0.05) and dipolarity (0.4 d 0.5) are < < 42 netic field intensity Lod to buoyancy flux Ra Q can be found by moderately strong and the variability (0.3 s 0.6) is high enough 43 simple dimensional analysis (also see Christensen & Aubert 2006). for the occasional reversal. Regime (iii) contains the frequently re- > −1 γ< 44 Our goal is to write a relationship between the magnetic field B versing (R 5Myr ) dynamos where the dipole intensity ( < 45 and the buoyancy flux F to some power, assuming independence 0.025) and dipolarity (d 0.4) are weak, and highly variable (s > 46 of the diffusivities and rotation rate (assuming the rotation rate is 0.6). It is the reversing regime (ii) that contains the long stable 47 sufficiently fast). First, we rewrite B by balancing the advection polarity chrons with widely spaced reversals, similar to those of 48 and Lorentz force terms in (1) which gives an expression for the the geodynamo over the last 83 Myr. Three of the statistics R, s √ γ 49 Alfven velocity vA ∼ B/ ρμ0. The simplest scaling law relating and used to characterize these regimes may be quite general, and −1 2 −3 1/3 50 vA, in units of [m s ], to F, in units of [m s ], is vA ∼ (FD) . therefore useful in characterizing dynamo regimes in other regions 51 This implies a relation between the modified Lorentz and Rayleigh of parameter space and for different modes of convection. 52 numbers of 53 1/3 Lod = γ Ra , (15) 3.2 Magnetic field spectra 54 Q

55 where γ is a proportionality coefficient. Fig. 3(d) shows Lod versus Fig. 4 shows the time averaged magnetic field power spectra at 56 Ra Q compared to contours of (15) with γ = 0.025, 0.05, 0.1 and 0.2. the CMB for several models from each dynamo regime, and an 57 The reversal regime boundary between non-reversing and reversing average of the geomagnetic power spectrum over the years 1840– 58 dynamos identified in Fig. 1 is well described in Fig. 3(d) by the 2000 (Jackson et al. 2000). The model spectra are scaled from 59 contour of γ = 0.05, demonstrating the validity of (15) for these dimensionless units to nT by equating (15) and (14), solving for B d , 60 dynamos. The dipolar models have 0.05 <γ <0.2, which is con- and using γ from Table 1. Among the four non-reversing models 61 sistent with the non-reversing, dipolar dominant thermal convection in Fig. 4(a), all chosen from regime (i), they display a very similar 62 dynamos scaled by Olson & Christensen (2006). We infer that γ power spectra up to degree l = 7: they are dipolar dominant and 63 0.05 and s 0.3 characterize the transition from non-reversing to also odd degree dominant. The odd degree dominance reflects the 64 reversing dynamos for both thermal and compositional convection strength of the axial terms in the dipole family and is to be expected

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 7 2 3 between the l = 1 and 2 terms, which explains why they have a 4 weaker dipolarity, substantial dipole variability, and the occasional 5 dipole field polarity reversal. The models in Fig. 4(b) scale the 6 closest to the geomagnetic spectrum over all the degrees shown. 7 Power spectra of four frequently reversing dynamos in Fig. 4(c), 8 all chosen from regime (iii), display no obvious odd or even degree 9 dominance and are relatively flat out to l = 9 due to the increased 10 vigor of convection. When the dipole intensity is close in magnitude 11 to the quadrupole, as it is for these dynamos, the dipole field is 12 no longer dominant and the dipole tilt can fluctuate rapidly with 13 frequent excursions and reversals. The geomagnetic spectrum is 14 substantially more dipolar than these models, but is qualitatively 15 similar from l = 2 to 9 where these spectra are all relatively flat. 16 17 18 3.3 A reversing dynamo in detail 19

20 Time-series of B d , dipole tilt, kinetic energy E k and magnetic en- 21 ergy E m from a strongly dipolar, reversing dynamo with Ra = 22 3.95 × 104 and E = 5.5 × 10−3 are shown in Fig. 5. There is a 23 reversal in this model at t d = 55 that we examine at three times 24 labelled t1 before, t2 during and t3 after the reversal. The reversal 25 process begins with a substantial decrease in B d of about an or- 26 der of magnitude relative to the average, and B d remains low for 27 about one t d . While the dipole amplitude is weak its axis crosses 28 the equator three times before stabilizing in the reversed polarity at 29 time t3. During the reversal process, not only is the dipolar portion 30 of the field weak but the magnetic energy of the system reaches a 31 local minimum, while the kinetic energy reaches a local maximum. 32 Following t3 the reversed polarity dipole field and energies recover 33 in magnitude, approaching their time average values. This trade- 34 off between magnetic and kinetic energies is described in detail by 35 Olson (2007) as an effect of the Lorentz force. Although reversal 36 events differ in detail (see Fig. 7), many of them follow a similar 37 pattern: during stable polarity chrons the magnetic field is strong 38 and the Lorentz force decreases the average fluid velocity. During 39 dipole collapses the magnetic field is weak and the reduced Lorentz 40 force increases the average fluid velocity and kinetic energy, which 41 in turn promotes induction and allows the dynamo to recover. 42 Not every dipole collapse results in a reversal, or even a dipole 43 axis excursion. For example, there is a major dipole collapse event 44 in Fig. 5 near t d = 10 that produces a temporary minimum in B d 45 and magnetic energy, and a temporary maximum in kinetic energy, 46 but does not result in much change in the tilt of the dipole axis. The 47 flow during this time does not change the large-scale structure of 48 the field and the dipole tilt is relatively stable. However, an earlier 49 dipole collapse near t = 5 does produce a dipole tilt excursion that Figure 4. Time averaged magnetic power spectra at the CMB from (a) four d 50 non-reversing dynamos of regime (i), (b) four reversing dynamos of regime briefly crosses the equator, but is not followed by a full reversal. 51 (ii) and (c) four frequently reversing dynamos of regime (iii). Parameters Fig. 6 shows maps of B r at the CMB, and B z and z-component of ω 52 (Ra, E) are shown in legends and the average core field from 1840 to 2000 vorticity z in the equatorial plane at the times t1 before, t2 during 53 (Jackson et al. 2000) is shown for reference. and t3 after the reversal. Before the reversal B r is predominantly 54 negative in the northern hemisphere and positive in the southern 55 from the relatively simple flow in these dynamos. The factor of hemisphere, that is, normal dipole polarity. B z also shows a pre- 56 10–100 between the l = 1 and 2 terms indicates that these dynamos dominantly southward orientation in the equatorial plane before the 57 are strongly dipolar and stable, with very rare dipole collapse events reversal. During the reversal B r loses its normal dipole polarity 58 or dipole excursions. Compared to the geomagnetic spectrum these and B z in the equatorial plane is concentrated in two flux bundles 59 models display a similar scaled dipole intensity and dipolarity, but with opposing polarity. The reversal involves a mixing of these flux 60 a stronger odd degree dominance. bundles until eventually the positive flux bundle supercedes the neg- 61 The spectra of four dipolar reversing dynamos in Fig. 4(b), all ative one and establishes the new reversed polarity. Note that even 62 chosen from regime (ii), also display an odd degree dominance, but though the dominant magnetic field in the equatorial plane changes 63 have smaller scaled dipole intensity. In contrast to the regime (i) sign after the reversal, the gross direction and pattern of the flow 64 dynamos in Fig. 4(a), these dynamos have only a factor of 5–10 does not change, i.e. the sign of ωz for the two main equatorial

C 2009 The Authors, GJI 1 8 P.Driscoll and P.Olson 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 4 −3 Figure 5. Time-series of dipole tilt (degrees), B d , kinetic energy E k and magnetic energy E m for a reversing dynamo (Ra = 3.95 × 10 , E = 5.5 × 10 ). 35 Background shading corresponds to reversed polarity and white corresponds to normal polarity. Times t 1, t 2 and t3 mark before, during and after the reversal. 36 Time shown in dipole decay times t d . 37 38 vortices is consistently negative before, during, and after the rever- ders substantially from the rotation axis, resulting in a tilt excursion. 39 sal process. However, not all dipole collapses lead to reversals. This is because 40 In numerical dynamos it has been found that the stability of the when the dipole axis wanders near the equator during the collapse 41 dipole field is correlated with the overall equatorial symmetry of the dipole has a chance of recovering in a state of reversed or normal 42 the velocity field and with the antisymmetry of the magnetic field polarity. There may be identifiable precursors in the dipole collapse 43 (Coe & Glatzmaier 2006; Takahashi et al. 2007; Olson et al. 2009). phase that distinguish reversals from excursions (e.g. Valet et al. 44 For the reversal at the end of the time-series in Fig. 5, we compute 2008; Olson et al. 2009). 45 the volume averaged ratio of the kinetic energy of the equatorially In Fig. 7(a), we compare the intensity of the axial dipole |B a | for 46 symmetric velocities to the sum of the symmetric plus antisymmet- many of the regime (ii) dynamos during polarity reversals, shifted 47 ric velocities is 0.82 before, 0.75 during and 0.78 after the reversal in time to the first equator crossing of the dipole axis. For each case 48 event. The volume averaged ratio of the magnetic energy of the |B a | is generally symmetric about the time of the reversal, that is, the 49 equatorial antisymmetric magnetic field to the symmetric plus anti- behaviour of |B a | before and after the reversal is the same, although 50 symmetric magnetic field is 0.62 before, 0.45 during and 0.61 after there may be a short recovery phase around t d = 0.1 − 0.3 where 51 the reversal event. As expected, during the reversal when the dipole |B a | remains unchanged. The peak intensity occurs near t d =±0.6 52 is low the magnetic field is less equatorially antisymmetric. Also on either side of the reversal, and the initial amplitude is recovered 53 during the reversal, the flow is less structured and less equatorially in most cases. There are two exceptional cases where |B a | stays near 54 symmetric. zero after the reversal and there are brief secondary polarity reversals 55 around t d = 0.5. Although most of these reversals look grossly 56 symmetric they clearly are not identical. A wide range of axial dipole behaviour characterizes this set of reversals, qualitatively 57 3.4 Comparison of reversals 58 similar to those of Coe et al. (2000), and unlike the almost identical 59 A simplified description of the field reversal process can be con- series of reversals found in the VKS dynamo experiment (Berhanu 60 structed by considering the dipole field strength and its time vari- et al. 2007). 61 ability. We have shown that if the average dipole strength is not Valet et al. (2005) show a similar superposition of virtual ax- 62 too high and there is enough variability in time then occasionally ial dipole moment (VADM) variations during several geomagnetic 63 there will be a random and large decrease in the dipole, a dipole reversal events over the last 2 Myr. Among the 5 reversals they 64 collapse event. Often during dipole collapses the dipole axis wan- analyze there is variability between each sequence (i.e. they are not

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 9 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 colour18 in19 print20 21 22 23 24 25 26 27 28 29 30 Figure 6. B at the CMB and B and ω in the equatorial plane from the reversing model shown in Fig. 5. The top row shows time t before, the middle row 31 r z z 1 time t during and the bottom row time t after the reversal. 32 2 3 33 34 The discrepancy between the model versus palaeomagnetic rever- 35 sal sequences may be due in part to the limited number of samples 36 available to construct the VADM sequences. The data is compiled 37 from a number of palaeomagnetic measurements of the local mag- 38 netic field at specific sites, which are then transformed to produce 39 a global field intensity (e.g. Constable & Korte 2006). The global 40 intensity is then referred to as only the ‘virtual’ axial dipole moment 41 because the data may include non-dipole components. Wicht (2005) 42 and Wicht et al. (2009), who compare the true dipole intensity and 43 tilt time-series to site specific observations of the same quantities in 44 a reversing numerical dynamo model, find that VADM sequences 45 during reversals can appear to be shifted forward or backward in 46 time depending on the site. Therefore, better global coverage of 47 the reversal sequences may be significant and, although the VADM 48 sequences during reversals are extremely difficult to obtain, anal- 49 ysis of more reversals and excursions going back further than 2 50 Ma are needed to confirm whether this is a robust feature. Rever- Figure 7. (a) Composite of the magnitude of axial dipole |B | during 51 a sal sequences from dynamo models with more realistic parameters reversals of regime (ii) dynamos, shifted in time to the first equator crossing. 52 should also be investigated. (b) Axial dipole histogram (pdf) for a reversal from a model with Ra = 53 3.3 × 104 and E = 5.5 × 10−3, thick black solid line in (a). (c) Axial dipole Another useful diagnostic for comparing reversals are histograms 54 histogram (pdf) for a reversal from a model with Ra = 4.2 × 104 and E = (pdfs) of B a , shown as insets Figs 7(b) and (c). An example of a − 55 5.0 × 10 3, thick grey dotted line in (a). simple reversal is shown in Fig. 7(b) where B a spends little time 56 near zero and the reversal occurs quickly. The stable polarities are 57 clearly identifiable in the bimodal pdf with peaks around B a = 58 identical), similar to the variability in Fig. 7(a). However, one sig- ±0.7 and nearly zero probability at B a = 0. A contrasting example, 59 nificant difference is that the VADM sequences are generally less shown in Fig. 7(c), is of a complex reversal where the pdf is tri- 60 symmetric about each polarity reversal and undergo a prolonged modal with a clear local probability peak near B a = 0. Peaks near 61 dipole collapse phase followed by a rapid recovery of the dipole B a = 0 are attributable to multipolar fields, and may correspond 62 intensity, whereas many of the reversals shown in Fig. 7(a) exhibit to the quasi-stable reversing dynamo state identified by Nishikawa 63 a precipitous dipole collapse followed by a similarly quick recovery & Kusano (2008) that is occupied briefly during a reversal where −2 64 phase. |B a | < 10 .

C 2009 The Authors, GJI 1 10 P.Driscoll and P.Olson 2 3 4 IMPLICATIONS FOR THE = 1 × 1034 and E = 3 × 10−15 for the present-day core. Applying 4 GEODYNAMO the scaling laws in (13) and (21) to the core, using Racrit = Ra/2, −12 −5 5 gives Ra Q = 5.5 × 10 and Lod = 1.3 × 10 . Substituting 6 4.1 Scaling the dynamo models to the geodynamo these into (15) we find γ core = 0.07, which places the present-day 7 It is computationally unfeasible to resolve dynamo models with geodynamo near the reversal regime boundary in Fig. 3(d) and 8 γ = realistic core parameters so we must rely on scaling laws to extrap- among the reversing dynamos in Table 1. A contour of core 0.07 9 olate the dynamics of the available parameter space to Earth-like is shown in Fig. 1 (dashed line), calculated from a planar fit to the 10 γ values. There is no unique recipe for choosing which parameters to model values. 11 scale or how to scale them, although a number of scaling laws have 12 been proposed (e.g. Jones 2000; Christensen & Aubert 2006; Olson 13 4.2 Geodynamo evolution & Christensen 2006). Even though the model parameters are often 14 far from realistic, many numerical dynamos have been found that What changes in the geodynamo reversal frequency are expected 15 are qualitatively Earth-like in their field structure, time variability, as a consequence of the secular evolution of the core, in particular 16 and occasional polarity reversals (Dormy et al. 2000; Glatzmaier the secular evolution of the buoyancy flux due to inner core growth 17 2002). and secular changes in the planetary rotation rate? An answer to 18 We have shown that our dipolar reversing models are consistent this question can be found in the context of this study if the geody- 19 with the scaling law in (15) for γ 0.05 (Fig. 3d). In order to apply namo evolves along a trajectory in parameter space that includes the 20 this law to the geodynamo we consider compositional buoyancy regime boundary between non-reversing, superchron-like dynamos 21 introduced at the inner core boundary due to the rejection of light and reversing dynamos. Here we assume the core resides in a transi- 22 elements from the inner core as it grows. For this mechanism, we tion region, for example near the reversal regime boundary in Fig. 1 23 rewrite Ra as defined by (6) in terms of the inner core radius growth that separates regime (i) from regime (ii). We can then estimate the 24 rate r˙ as follows. Conservation of mass of light elements in the evolution vector of the core by calculating the fractional change in 25 i outer core can be written as buoyancy flux Ra/Ra and rotation E/E over the past 100 Myr. 26 We choose to project back 100 Myr because by this time the 27 d 0 = (ρoVoχo) (16) inner core had likely grown to near its present size and we can avoid 28 dt any complications associated with the possible jump in buoyancy 29 = ρ χ + ρ ˙ χ + ρ χ production at the time of inner core nucleation. Also, at 100 Ma 30 oVo ˙o oVo o ˙oVo o (17) the geodynamo was in the middle of the CNS and since that time 31 assuming complete partitioning of light elements at the inner core the reversal frequency has increased rather steadily, allowing us to 32 boundary, where V o is the volume of the outer core. Using the determine if it is possible to explain the observed increase in reversal 33 equation of stateρ ˙o =−ρoβχ˙o and solving forχ ˙o in (17) gives frequency since the last superchron solely in terms of the secular 34 change in buoyancy flux and rotation rate. 35 χ V˙ χ˙ =− o o . (18) To calculate changes in Ra and E, we assume simple linear evo- 36 o − βχ (1 o) Vo lutionary behaviour (Driscoll & Olson 2009), 37 ˙ / 38 We can write Vo Vo in terms of the change in inner core radius r˙i , Ra(t) = Ra + Ra (22) = s 39 assuming that r˙o 0, as 40 V˙ r 2r˙ E(t) = E + E, (23) o =−3 i i . (19) s 41 3 − 3 Vo ro ri 42 where refers to the change of a quantity over an interval of time 43 Using (18) and (19) in (6) gives t, and the subscript s denotes the value at the time of the start of the model t . In evaluating the time derivatives of (20) and (5) 44 β χ 5 2 s 3 go o D ri r˙i , 45 Ra = . (20) we consider the only time dependent quantities to be ri r˙i ,and , 2 3 3 ∗ κν (1 − βχo) r − r = − = / 46 o i where D r o r i is used. Using r r o r i this gives, 47 To scale the dipole field strength we write Lod from (14) using (5) Ra 5 3 r r˙ = 2 − + i + i (24) 48 and (8) giving ∗ ∗ 3 Ra r − 1 (r ) − 1 ri r˙i 49 Bd ση 50 Lod = , (21) ρ E 2ri 51 D o = − , (25) E ro − ri 52 where B d is now in Tesla. 53 To estimate Ra Q and Lod for the core we assume an outer core where all time-dependent quantities in (24) and (25) are evaluated 54 light element concentration χ o = 0.1, and assuming this represents at the present-day. 55 a mixture of Fe with FeO as the light element constituents we have We use a power-law representation of the growth history of the − − 56 β = 1andκ = 10 8 m2 s 1 (Gubbins et al. 2004). We assume inner core, that is, a simplified analytical formula that gives r i as a 57 a magnetic diffusivity of η = 2m2 s−1 (Jones 2007). We use an function of time since nucleation, 58 inner core radius of r = 1221 km and an outer core radius of r i o r (t) = atb, (26) 59 = 3480 km giving a shell thickness D = 2259 km, rotation rate i −5 −1 −2 60 = 7.292 × 10 s ,andattheCMBgo = 10.68 m s and where a and b are suitably chosen constants. This approximates the 61 B d = 0.26 mT (IGRF-10 epoch 2005). For the present-day rate of inner core growth history predicted by thermodynamically based −1 62 inner core growth we use r˙i = 814 km Gyr (see Section 4.2). The evolution models (e.g. Labrosse et al. 2001; Nimmo et al. 2004). −6 2 −1 63 viscosity of the outer core is assumed to be ν = 1.26 × 10 m s Subject to the initial r i (t = 0) = 0andfinalr i (t = τ) = 1221 km 64 (Terasaki et al. 2006). Using these values in (20) and (5) give Ra conditions, where τ = 1 Gyr is the assumed age of the inner core,

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 11 2 b 3 we find a coefficient of a = r i (τ)/τ and a best-fitting exponent of long time constants so that cases near the regime boundary need 4 b = 2/3. Evaluating (26) and its derivatives today at t = τ gives to be run a very long time in order to obtain reliable reversal −1 −2 5 r˙i (τ) = 814 km Gyr and rï (τ) =−271 km Gyr . Evaluating statistics. −1 6 (26) at t s = 0.9 Gyr gives ri (ts ) = 1138 km, r˙i (ts ) = 843 km Gyr Despite these difficulties, we propose two scenarios that can ex- −2 7 and rï (ts ) =−312 km Gyr . plain the seafloor palaeomagnetic record of geodynamo reversal 8 To estimate changes in Earth’s rotation rate over the past 100 frequency, going from a non-reversing (superchron) state to a re- 9 Myr we use the length of day measurements from Phanerozoic versing state. Scenario (i), depicted in Fig. 8(b), assumes that the 10 marine invertebrates compiled by Zharkov et al. (1996) and Zhenyu regime boundary is complex and contains some non-linear struc- 11 et al. (2007). The length of day has increased over geological time ture, and that the core has evolved regularly (monotonically), as 12 due to tidal friction, a consequence of the torque exerted by the described in Section 4.2. In this scenario the geodynamo can evolve 13 Moon and Sun on the Earth’s tidal bulge (Lambeck 1980). A linear fromasuperchronstatetoareversingstateandbacktoasuper- 14 fit to the data in Zharkov et al. (1996) (Table 8.1) gives an estimate of chron state, etc., simply based on the shape of the regime boundary 15 the length of day at t s (100 Ma) to be 23.72 hr, corresponding to s itself. Assuming that the geodynamo resides near such a regime 16 = 7.36 × 10−5 rad s−1. We adopt this empirically derived estimate boundary in parameter space, the evolution depicted by the vector 17 because it is within the range of theoretically derived estimates of in Fig. 8(a) suggests a possible explanation for the change in rever- −5 −1 18 tidal breaking of s = 7.30 × 10 rad s by Stacey (1992) and sal frequency over the last 100 Myr solely by the secular evolution −5 −1 19 s = 7.45 × 10 rad s by Lambeck (1980). of the core across an irregular regime boundary. Scenario (ii), de- 20 The fractional changes in Ra and E over the past 100 Myr, using picted in Fig. 8(c), assumes that the regime boundary is simple and 21 (24) and (25), are that the core has evolved in a complex way, with at least one of 22 the parameters, buoyancy rate and/or rotation, varying in time. For Ra 23 =−0.071 (27) example, this variable evolution could involve a constant decrease 24 Ra in rotation rate but a sinusoidal variation in buoyancy flux, which 25 E could be caused by ultralow frequency fluctuations in mantle heat 26 =+0.082. (28) flow (Nakagawa & Tackley 2004; Butler et al. 2005; Courtillot & E 27 Olson 2007) among other possibilities. 28 As expected, Ra tends to decrease with time because the growth An important constraint on the evolution of the geodynamo is 29 rate of the inner core has decreased over the past 100 Myr, and E that despite oscillating from a superchron to a reversing state it 30 tends to increase with time due to the decrease in rotation rate and has maintained a rather steady dipole intensity over the last sev- 31 increase in inner core radius. eral hundred million years (Tauxe & Yamazaki 2007). Therefore, 32 since we have no evidence that the geodynamo has shut down over 4.3 Interpretation of changes in the geodynamo reversal 33 that time we can infer that dynamo evolution over the last several frequency 34 hundred millions years must not include a trajectory that crosses 35 Consider the simple core evolution described in Section 4.2 as an beneath the boundary of dynamo onset. In particular, we can rule 36 evolutionary vector in Ra–E space. We apply this evolution to the out dynamo evolution that includes a large enough decrease in Ra, 37 region of parameter space in Fig. 1 that contains the transition that is, anomalously low buoyancy flux, that would shut down the 38 from non-reversing to reversing dynamos. In Fig. 8(a), a magnified dynamo. Note that shutting down the dynamo by evolving from a 39 portion of Fig. 1, the evolutionary vector has been scaled in the dipolar reversing state to the dynamo onset boundary would require 40 Ra direction by (27) and in the E direction by (28), and placed a large decrease in Ra only, because a decrease in E over time would 41 within the diagram so that the tail corresponds to a non-reversing, require either an increase in or a decrease in r i , both of which 42 superchron-like state (representing conditions 100 Ma), while the are unphysical. We can estimate what change in Ra would corre- 43 head corresponds to a reversing state (representing the present-day). spond to dynamo shut down by extrapolating from an Earth-like 44 The detailed shape of the reversal regime boundary is challeng- model to the dynamo onset boundary using the evolutionary model 45 ing to map out using numerical dynamos due to the fact that described above. Consider, for example, starting with a dipolar re- 46 these models are chaotic in time and are therefore sensitive to versing dynamo at Ra = 3.95 × 104 and E = 5 × 10−3, then eq. (12) 4 47 the initial conditions and small changes in parameter values. We predicts the onset of dynamo action at Radyn = 1.3 × 10 .There- 48 have also found that these simulations often contain extremely fore, in order to shut down this dynamo an instantaneous change of 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Figure 8. (a) Magnified segment from Fig. 1 with an evolutionary vector placed arbitrarily in a non-reversing state that projects the geodynamo forward from 64 100 Ma to the present-day reversing state. (b) Schematic of scenario (i), a complex regime boundary and regular core evolution. (c) Schematic of scenario (ii), a simple regime boundary and variable core evolution.

C 2009 The Authors, GJI 1 12 P.Driscoll and P.Olson 2 3 Ra/Ra =−0.675 is required, corresponding to an instantaneous of the core is responsible for the systematic change in geomagnetic −1 4 decrease in r˙i from 814 to 264 km Gyr . This is an upper limit reversal frequency. Secular change in the buoyancy flux, due to the 5 on the instantaneous change required to kill the dynamo because growth of the inner core, and changes in rotation rate, due to tidal 6 it only includes the rate of inner core growth, while the effects of dissipation, over the past 100 Myr are large enough to evolve the 7 thermal stratification are not included and would certainly act to kill geodynamo from a non-reversing state to a reversing one, provided 8 the dynamo earlier. Also, this calculation assumes that the dynamo the geodynamo lies close to such a regime boundary. 9 regimes in our region of parameter space can be extrapolated to A natural extension of this study would be to include secular 10 Earth-like parameter values, which is entirely uncertain. Neverthe- evolution in the model, which will introduce additional parameters 11 less, we can conclude that such a drastic change in core evolution such as the rate of the evolution of Ra and E. Driscoll & Olson (2009) 12 is not expected, and that the dynamo fluctuations we consider to employ a step-wise evolution of Ra and E similar to the evolution 13 explain the seafloor reversal record are not enough to accidentally described in (27) and (28), and find that dynamos with simultaneous 14 shut down the dynamo. evolution in Ra and E are statistically similar to non-evolving fixed 15 parameter dynamos. They also demonstrate that a non-reversing 16 dynamo can evolve into a reversing regime with a steady increase 17 in Ra. An issue not addressed here is how a time dependent inner 5 CONCLUSIONS 18 core size might affect dynamo evolution (e.g. Heimpel et al. 2005). 19 We have systematized the behaviour of gravitational dynamos Whether or not the model behaviour found in this study can be 20 driven by compositional convection in a relatively small portion generalized to more Earth-like dynamo models is a topic for future 21 of Ra–E space, finding dynamos that exhibit a wide range of re- study with more sophisticated models. 22 versal frequencies, dipolarities, and magnetic field structures at the 23 outer boundary. A transition zone, or regime boundary, is identified 24 in the Ra–E plane that separates non-reversing from reversing dipo- ACKNOWLEDGMENTS 25 lar dynamos. We find that a simple scaling law (15) connects the We gratefully acknowledge support from the Geophysics Program 26 present-day geodynamo to the numerical dynamos in the reversing, of the National Science Foundation through grant EAR-0604974. 27 dipolar regime. The numerical dynamo code used in this study is available at: 28 The existence of a reversal regime boundary in the Ra–E plane www.geodynamics.org. 29 is likely to be a robust feature of convectively driven dynamos. 30 However, due to the large number and typical range of dynamo 31 parameters, the details of the reversal regime boundary may vary 32 from one model to the next. Accordingly, a more critical ques- REFERENCES 33 tion is whether reversing dynamos remain dipolar. Kutzner & Alfe, D., Gillan, M. & Price, G., 2000. Constraints on the composition 34 Christensen (2002) found that for most types of convection the of the Earth’s core from ab initio calculations, Nature, 405(6783), 172– 35 boundary between non-reversing and reversing behaviour was close 175. 36 to the boundary between dipolar and multipolar dynamos, but in Aubert, J., Aurnou, J. & Wicht, J., 2008. The magnetic structure of 37 many cases, the multipolar transition occurs first. The models pre- convection-driven numerical dynamos, Geophys. J. Int., 172, 945– 38 sented in this paper have the distinct property that the reversal 956. Berhanu,M.et al., 2007. Magnetic field reversals in an experimental turbu- 39 transition occurs before the transition to multipolar dynamos, so lent dynamo, EPL, 77(5), 59 001. 40 there exists a regime with reversals separated by long, stable dipo- Braginsky, S. & Roberts, P., 1995. Equations governing convection in Earth’s Q2 41 lar polarity chrons. core and the geodynamo, Geophys. Astrophys. Fluid Dynam., 79(1), 1–97. 42 This behaviour may be due to the fortuitous combination of pa- Butler, S., Peltier, W. & Costin, S., 2005. Numerical models of the Earth’s 43 rameters used in this study: large magnetic Prandtl number, large thermal history: effects of inner-core solidification and core potassium, 44 Ekman number, plus compositional convection and an insulating Phys. Earth planet. Inter., 152(1–2), 22–42. 45 inner core. Some of these choices might have favourably biased our Cande, S. & Kent, D., 1992. A new geomagnetic polarity time scale 46 results. For example, it has been observed in numerical studies that for the late Cretaceous and Cenozoic, J. geophys. Res., 97(B10), 13– 47 including a conducting inner core provides some degree of mag- 917. 48 netic polarity stability (Hollerbach & Jones 1995), which could be Cande, S., Kent, D., 1995. Revised calibration of the geomagnetic polarity timescale for the late Cretaceous and Cenozoic, J. geophys. Res. (Solid 49 critical for the spacing of geomagnetic reversals (Gubbins 1999). Earth), 100(4), 6093–6095. 50 However, Wicht (2002) finds this stabilizing effect to be signifi- Christensen, U. & Aubert, J., 2006. Scaling properties of convection-driven 51 cant only in frequently reversing models, with typical chron lengths dynamos in rotating spherical shells and application to planetary magnetic 52 of 10 kyr or less. Also, an investigation of the effects of various fields, Geophys. J. Int., 166(1), 97–114. 53 styles of convection by Kutzner & Christensen (2000) showed that Christensen, U. & Olson, P., 2003. Secular variation in numerical geodynamo 54 dynamo models with buoyancy sources concentrated at the inner models with lateral variations of boundary heat flow, Phys. Earth planet. 55 boundary tend to produce more stable dipolar magnetic fields than Inter., 138(1), 39–54. 56 models with buoyancy spread throughout the core. In this sense our Christensen, U. & Wicht, J., 2007. Numerical Dynamo Simulations, 57 gravitational dynamo models may be biased towards more stable Chapter 8, Vol. 8: Core Dynamics of Treatise on Geophysics, Elsevier 58 dipolar fields. Science. Q3 Coe, R. & Glatzmaier, G., 2006. Symmetry and stability of the geomagnetic 59 For the geodynamo, it is likely that slowly evolving, non- field, Geophys. Res. Lett., 33. 60 homogeneous thermal and topographic conditions on the CMB in- Coe, R.S., Hongre, L. & Glatzmaier, G., 2000. An examination of simulated Q4 61 fluence geomagnetic reversal behaviour (Glatzmaier et al. 1999), geomagnetic reversals from a paleomagnetic perspective, Phil. Trans. R. 62 and it is also likely that some of the change in reversal behaviour Soc. Lond. A, 358, 1141–1170. 63 over time is purely stochastic (Ryan & Sarson 2007). However, our Constable, C. & Korte, M., 2006. Is Earth’s magnetic field reversing?, Earth 64 dynamos demonstrate that it is possible that the evolving mean state planet. Sci. Lett., 246(1–2), 1–16.

C 2009 The Authors, GJI 1 Polarity reversal frequency of numerical dynamos 13 2 3 Constable, C.G., 2003. Geomagnetic reversals: rates, timescales, preferred McFadden, P. & Merrill, R., 1984. Lower mantle convection and geomag- 4 paths, statistical models, and simulations, in Earth’s Core and Lower netism, J. geophys. Res., 89, 3354–3362. 5Q5 Mantle, Chapter 4, pp. 77–99, Taylor & Francis. Merrill, R. & McFadden, P., 1999. Geomagnetic polarity transitions, Rev. 6 Courtillot, V.& Besse, J., 1987. Magnetic field reversals, polar wander, and Geophys., 37(2), 201–226. 7 core-mantle coupling, Science, 237(4819), 1140–1147. Merrill, R., McElhinny, M. & McFadden, P., 1996. The Magnetic Field of Courtillot, V. & Olson, P., 2007. Mantle plumes link magnetic superchrons the Earth: Paleomagnetism, the Core, and the Deep Mantle, Academic 8 to phanerozoic mass depletion events, Earth planet. Sci. Lett., 260(3–4), Press. 9 Q12 495–504. Nakagawa, T. & Tackley, P., 2004. Effects of thermo-chemical mantle con- 10 Dormy, E., Valet, J. & Courtillot, V., 2000. Numerical models of the geody- vection on the thermal evolution of the Earth’s core, Earth planet. Sci. 11 namo and observational constraints, Geochem. Geophys. Geosyst., 1(10). Lett., 220(1–2), 107–119. 12Q6 Driscoll, P. & Olson, P., 2009. Polarity reversals in geodynamo models with Nimmo, F.,2007a. Energetics of the core, in Treatise on Geophysics, Chapter 13 core evolution, Earth planet. Sci. Lett., doi:10.1016/j.epsl.2009.02.017. 2, Vol. 8: Core Dynamics, pp. 31–66, Elsevier Science. 14Q7 Fearn, D., 1998. Hydromagnetic flow in planetary cores, Rep. Prog. Phys., Nimmo, F., 2007b. Thermal and compositional evolution of the core, in Q13 15 61, 175–235. Treatise on Geophysics, Chapter 9, Vol. 9, pp. 217–241, Elsevier Science. Q14 16 Gallet, Y. & Pavlov, V., 1996. Magnetostratigraphy of the Moyero river Nimmo, F., Price, G., Brodholt, J. & Gubbins, D., 2004. The influence of 17 section (north-western Siberia): constraints on geomagnetic reversal potassium on core and geodynamo evolution, Geophys. J. Int., 156(2), frequency during the early Palaeozoic, Geophys. J. Int., 125(1), 95– 363–376. 18 105. Nishikawa, N. & Kusano, K., 2008. Simulation study of the symmetry- 19 Glatzmaier, G., Coe, R., Hongre, L. & Roberts, P., 1999. The role of the breaking instability and the dipole field reversal in a rotating spherical 20 Earth’s mantle in controlling the frequency of geomagnetic reversals, shell dynamo, Phys. Plasmas, 15, 082 903. 21 Nature, 401, 885–890. Olson, P., 2007. Gravitational dynamos and the low-frequency geomagnetic Q15 22 Glatzmaier, G.A., 2002. Geodynamo models—how realistic are they?, Ann. secular variation, Proc. Natl. Acad. Sci., 104(51), 20 159. 23 Rev. Earth Planet. Sci., 30, 237–257. Olson, P. & Christensen, U., 2006. Dipole moment scaling for convection- Q16 24 Glatzmaier, G.A. & Coe, R.S., 2007. Magnetic polarity reversals in the driven planetary dynamos, Earth planet. Sci. Lett., 250(3–4), 561– 25 core, in Treatise on Geophysics, Chapter 9, Vol. 8: Core Dynamics, 571. 26Q8 pp. 283–298, Elsevier Science. Olson, P., Driscoll, P. & Amit, H., 2009. Dipole collapse and reversal pre- 27 Gubbins, D., 1999. The distinction between geomagnetic excursions and cursors in a numerical dynamo, Phys. Earth planet. Inter., 173, 121– reversals, Geophys. J. Int., 137(1), 1–4. 140. 28 Gubbins, D. & Zhang, K., 1993. Symmetry properties of the dynamo equa- Pavlov, V.& Gallet, Y., 2005. A third superchron during the early Paleozoic, 29 tions for paleomagnetism and geomagnetism, Phys. Earth planet. Inter., Episodes, 28(2), 78–84. 30 75, 225–241. Roberts, P.H., Jones, C.A. & Calderwood, A.R., 2001. Energy fluxes and 31 Gubbins, D., Alfe, D., Masters, G., Price, G. & Gillan, M., 2004. Gross ohmic dissipation in the earth’s core, in Earth’s Core and Lower Mantle, 32 thermodynamics of two-component core convection, Geophys. J. Int., Chapter 5, pp. 100–129, Taylor & Francis. 33 157(3), 1407–1414. Rotvig, J., 2009. An investigation of reversing numerical dynamos driven Q17 34 Heimpel, M., Aurnou, J., Al-Shamali, F. & Gomez Perez, N., 2005. A nu- by either differential or volumetric heating, Phys. Earth planet. Inter., 35 merical study of dynamo action as a function of spherical shell geometry, doi:10.1016/j.pepi.2009.04.003, in press. Q18 36 Earth planet. Sci. Lett., 236(1–2), 542–557. Ryan, D. & Sarson, G., 2007. Are geomagnetic field reversals controlled by 37 Hollerbach, R. & Jones, C., 1995. On the magnetically stabilizing role of turbulence within the Earth’s core? Geophys. Res. Lett., 34. the Earth’s inner core, Phys. Earth planet. Inter., 87(3), 171–181. Sarson, G. & Jones, C., 1999. A convection driven geodynamo reversal Q19 38 Hulot, G. & Gallet, Y., 2003. Do superchrons occur without any palaeomag- model, Phys. Earth planet. Inter., 111(3). 39 netic warning? Earth planet. Sci. Lett., 210(1–2), 191–201. Simitev, R. & Busse, F., 2005. Prandtl-number dependence of convection- Q20 40 Jackson, A., Jonkers, A. & Walker, M.R., 2000. Four centuries of geomag- driven dynamos in rotating spherical fluid shells, J. Fluid Mech., 532, 41 netic secular variation from historical records, Philos. Trans. Roy. Soc. A: 365–388. 42 Math., Phys. Eng. Sci., 358(1768), 957–990. Stacey, F., 1992. Physics of the Earth, Brookfield Press, Brisbane, Australia. 43 Jones, C., 2000. Convection-driven geodynamo models, Philos. Trans. Math. Takahashi, F., Matsushima, M. & Honkura, Y., 2007. A numerical study on 44 Phys. Eng. Sci., 358(1768), 873–897. magnetic polarity transition in an MHD dynamo model, Earth, Planets 45 Jones, C.A., 2007. Thermal and compositional convection in the outer core, Space, 59(7), 665–673. 46 in Treatise on Geophysics, Vo l . 8 , pp. 131–185, Elsevier Science. Takahashi, F., Matsushima, M. & Honkura, Y., 2008. Scale variability in 47 Jonkers, A., 2003. Long-range dependence in the Cenozoic reversal record, convection-driven MHD dynamos at low Ekman number, Phys. Earth Phys. Earth planet. Inter., 135(4), 253–266. planet. Inter., 167, 168–178. 48 Kageyama, A., Miyagoshi, T. & Sato, T., 2008. Formation of current coils Tauxe, L. & Yamazaki, T., 2007. Paleointensities, in Treatise on Geo- 49 in geodynamo simulations, Nature, 454(7208), 1106. physics, Chapter 13, Vol. 5: Geomagnetism, Elsevier Science, pp. 509– 50Q9 Kutzner, C. & Christensen, U., 2000. Effects of driving mechanisms in 563. Q21 51 geodynamo models, Geophys. Res. Lett., 27(1), 29–32. Terasaki, H., Suzuki, A., Ohtani, E., Nishida, K., Sakamaki, T. & Funakoshi, 52 Kutzner, C. & Christensen, U., 2002. From stable dipolar towards reversing K., 2006. Effect of pressure on the viscosity of Fe-S and Fe-C liquids up 53 numerical dynamos, Phys. Earth planet. Inter., 131, 29–45. to 16 GPa, Geophys. Res. Lett., 33, 22 307. 54 Kutzner, C. & Christensen, U., 2004. Simulated geomagnetic reversals and Valet, J. & Meynadier, L., 1993. Geomagnetic field intensity and re- Q22 55 preferred virtual geomagnetic pole paths, Geophys. J. Int., 157(3), 1105– versals during the past four million years, Nature, 366(6452), 234– 56 1118. 238. 57 Labrosse, S. & Macouin, M., 2003. The inner core and the geodynamo, Valet, J., Meynadier, L. & Guyodo, Y., 2005. Geomagnetic field strength Comptes rendus-Geoscience,´ 335(1), 37–50. and reversal rate over the past 2 Million years, Nature, 435, 802–805. 58 Labrosse, S., Poirier, J. & Le Mouel,¨ J., 2001. The age of the inner core, Valet, J.P., Plenier, G. & Herrero-Bervera, E., 2008. Geomagnetic ex- 59 Earth planet. Sci. Lett., 190(3-4), 111–123. cursions reflect an aborted polarity state, Earth planet. Sci. Lett., 60 Lambeck, K., 1980. The Earth’sVariable Rotation: Geophysical Causes and doi:10.1016/j.epsl.2008.07.056. 61 Q23 Q10 Consequences, Cambridge University Press. Wicht, J., 2002. Inner-core conductivity in numerical dynamo simulations, 62Q11 Loper, D.E., 2007. Turbulence and small-scale dynamics of the core, in Phys. Earth planet. Inter., 132(4), 281–302. 63 Treatise on Geophysics, Chapter 5, Vol. 8: Core Dynamics, pp. 131–186, Wicht, J., 2005. Palaeomagnetic interpretation of dynamo simulations, Geo- 64 Elsevier Science. phys.J.Int.,162(2), 371–380.

C 2009 The Authors, GJI 1 14 P.Driscoll and P.Olson 2 3 Wicht, J. & Olson, P., 2004. A detailed study of the polarity reversal generation in a rotating spherical shell, Geophys. Astrophys. Fluid Dyn., 4 mechanism in a numerical dynamo model, Geochem. Geophys. Geosyst., 44(1–4), 33–53. Q245 5(3). Zharkov, V., Molodensky, S., Brzezinski, A., Groten, E. & Varga, P., 1996. 6 Wicht, J., Stellmach, S. & Harder, H., 2009. numerical models of The Earth and its Rotation: Low Frequency Geodynamics, Wichmann, 7 the geodynamo: from fundamental cartesian models to 3D simula- Heidelberg. tions of ﬁeld reversals, in Geomagnetic Field Variations, Springer, Zhenyu, Z., Yaoqi,Z. & Guosheng, J., 2007. The periodic growth increments 8 p. 107. of biological shells and the orbital parameters of Earth-Moon system, 9 Q25 Zhang, K. & Busse, F.,1988. Finite amplitude convection and magnetic ﬁeld Environ. Geol., 51, 1271–1277. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

C 2009 The Authors, GJI 1 2 3 Queries 4 Journal: GJI 5 Paper: gji_4234 6 7 8 Dear Author 9 During the copy-editing of your paper, the following queries arose. Please respond to these by marking up your proofs with 10 11 the necessary changes/additions. Please write your answers on the query sheet if there is insufﬁcient space on the page proofs. 12 Please write clearly and follow the conventions shown on the corrections sheet. If returning the proof by fax do not write too 13 close to the paper’s edge. Please remember that illegible mark-ups may delay publication. 14 15 16 Query Query Remarks 17 18 Reference 19 Q1 Author: As per journal style only keywords from 20 the keyword list are allowed. No other keywords are 21 22 allowed, so we have deleted the keywords. Please 23 choose up to six key words from the attached list. 24 25 Q2 Author: Please provide last page number for refer- 26 ence Berhanu et al. (2007). 27 28 Q3 Author: Please provide publisher location for refer- 29 ence Christensen & Wicht (2007). 30 31 Q4 Author: Please provide page range for reference Coe 32 & Glatzmaier (2006). 33 34 Q5 Au: Please provide publisher location and editor 35 names for reference Constable (2003). 36 37 Q6 Author: Please provide page range for reference 38 Dormy et al. (2000). 39 40 Q7 Author: Please update reference Driscoll & Olson 41 (2009). 42 43 Q8 Author: Please provide editors and publisher location 44 for reference Glatzmaier & Coe (2007). 45 46 Q9 Author: Please provide last page number for refer- 47 ence Kageyama et al. (2008). 48 49 Q10 Author: Please provide publisher location for refer- 50 ence Lambeck (1980). 51 52 Q11 Au: Please provide editors and publisher location for 53 reference Loper (2007). 54 55 Q12 Author: Please provide publisher location for refer- 56 ence Merrill et al. (1996). 57 58 59 60 61 62 63 64 Query Query Remarks Reference Q13 Author: Please provide editors and publisher location for reference Nimmo (2007a).

Q14 Author: Please provide editors and publisher location for reference Nimmo (2007b).

Q15 Author: Please provide page range for refernece Nishikawa &Kusano (2008).

Q16 Author: Please provide complete page range for refernece Olson (2007).

Q17 Author: Please provide editors and publisher location for reference Roberts et al. (2001).

Q18 Author: Please update reference Rotvig (2009).

Q19 Author: Please provide page range for reference Ryan & Sarson (2007).

Q20 Author: Please provide page range for reference Sar- son &Jones (1999).

Q21 Author: Please provide editors and publisher location for reference Tauxe & Yamazaki (2007).

Q22 Author: Please provide last page number for reference Terasaki et al. (2006).

Q23 Author: Please update reference Valet et al. (2008).

Q24 Author: Please provide page range for reference Wicht &Olson (2004).

Q25 Author: Please provide editor names and publisher location for reference Wicht et al. (2009). May 22, 2009 17:40 GeophysicalJournalInternational gji2521-KeywordsList

Key words

Authors are requested to choose key words from the list below to describe their work. The key words will be printed underneath the summary and are useful for readers and researchers. Key words should be separated by a semi-colon and listed in the order that they appear in this list. An article should contain no more than six key words.

GEOPHYSICAL METHODS Palaeomagnetic secular variation Palaeoseismology Time series analysis Palaeomagnetism applied to Earthquake interaction, forecasting, Image processing tectonics and prediction Neural networks, fuzzy logic Palaeomagnetism applied to Seismicity and tectonics Numerical solutions geologic processes Body waves Fourier analysis Rapid time variations Surface waves and free oscillations Wavelet transform Remagnetization Interface waves Instability analysis Reversals: process, time scale, Guided waves Inverse theory magnetostratigraphy Coda waves Numerical approximations Rock and mineral magnetism Seismic anisotropy and analysis Satellite magnetics Seismic attenuation Persistence, memory, Marine magnetics and Site effects correlations, clustering palaeomagnetics Seismic tomography Probabilistic forecasting Marine electromagnetics Volcano seismology Spatial analysis Computational seismology Downhole methods GENERAL SUBJECTS Theoretical seismology Tomography Geomorphology Statistical seismology Interferometry Geomechanics Wave scattering and diffraction Thermobarometry Glaciology Wave propagation Fractals and multifractals Hydrogeophysics Acoustic properties Non-linear differential equations Ionosphere/atmosphere interactions Early warning Probability distributions Ionosphere/magnetosphere interactions Rheology and friction of Self-organization Gas and hydrate systems fault zones Ocean drilling GEODESY and GRAVITY Hydrology TECTONOPHYSICS Satellite geodesy Ultra-high pressure metamorphism Planetary tectonics Reference systems Ultra-high temperature metamorphism Mid-ocean ridge processes Sea level change Tsunamis Transform faults Space geodetic surveys Thermochronology Subduction zone processes Seismic cycle Heat flow Intra-plate processes Transient deformation Hydrothermal systems Volcanic arc processes Gravity anomalies and Earth structure Mantle processes Back-arc basin processes Geopotential theory Core, outer core and inner core Cratons Time variable gravity Continental margins: convergent Earth rotation variations COMPOSITION and PHYSICAL Continental margins: divergent Global change from geodesy PROPERTIES Continental margins: transform Lunar and planetary geodesy and gravity Microstructures Continental neotectonics Radar interferometry Permeability and porosity Continental tectonics: compressional Plate motions Plasticity, diffusion, and creep Continental tectonics: extensional Tides and planetary waves Composition of the core Continental tectonics: strike-slip Acoustic-gravity waves Composition of the continental crust and transform Composition of the oceanic crust Sedimentary basin processes GEOMAGNETISM and Composition of the mantle Oceanic hotspots and ELECTROMAGNETISM Composition of the planets intraplate volcanism Electrical properties Creep and deformation Oceanic plateaus and Electromagnetic theory Defects microcontinents Magnetotelluric Elasticity and anelasticity Oceanic transform and fracture Non-linear electromagnetics Equations of state zone processes Archaeomagnetism High-pressure behaviour Submarine landslides Biogenic magnetic minerals Fracture and flow Submarine tectonics and volcanism Dynamo: theories and simulations Friction Tectonics and landscape evolution Environmental magnetism Fault zone rheology Tectonics and climatic interactions Geomagnetic excursions Phase transitions Dynamics and mechanics of faulting Geomagnetic induction Dynamics of lithosphere and mantle Magnetic anomalies: modelling SEISMOLOGY Dynamics: convection currents, and and interpretation Controlled source seismology mantle plumes Magnetic and electrical properties Earthquake dynamics Dynamics: gravity and tectonics Magnetic fabrics and anisotropy Earthquake ground motions Dynamics: seismotectonics Magnetic mineralogy and petrology Earthquake source observations Heat generation and transport Magnetostratigraphy Seismic monitoring and test-ban Impact phenomena Palaeointensity treaty verification Hotspots May 22, 2009 17:40 GeophysicalJournalInternational gji2521-KeywordsList

2 Key words

Large igneous provinces VOLCANOLOGY Calderas Lithospheric flexure Physics of magma and Experimental volcanism Obduction tectonics magma bodies Tephrochronology Neotectonics Magma chamber processes Remote sensing of volcanoes Diapir and diapirism Magma genesis and partial melting Volcano monitoring Folds and folding Pluton emplacement Volcanic hazards and risks Fractures and faults Effusive volcanism Kinematics of crustal and mantle Mud volcanism GEOGRAPHIC LOCATION deformation Subaqueous volcanism Africa High strain deformation zones Explosive volcanism Antarctica Crustal structure Volcaniclastic deposits Arctic region Mechanics, theory, and modelling Volcano/climate interactions Asia Rheology: crust and lithosphere Atmospheric effects (volcano) Atlantic Ocean Rheology: mantle Volcanic gases Australia Lava rheology and morphology Europe PLANETS Magma migration and fragmentation Indian Ocean Planetary interiors Eruption mechanisms and North America Planetary volcanism flow emplacement Pacific Ocean Physics and chemistry of magma bodies South America

C 2009 RAS, GJI