Gravitational dynamos and the low-frequency geomagnetic secular variation

Abstract

Self-sustaining numerical dynamos are used to infer the sources of low-frequency secular variation of the geomagnetic field. Gravitational dynamo models powered by compositional convection in an electrically conducting, rotating fluid shell exhibit several regimes of magnetic field behavior with an increasing Rayleigh number of the convection, including nearly steady dipoles, chaotic nonreversing dipoles, and chaotic reversing dipoles. The time average dipole strength and dipolarity of the magnetic field decrease, whereas the dipole variability, average dipole tilt angle, and frequency of polarity reversals increase with Rayleigh number. Chaotic gravitational dynamos have large-amplitude dipole secular variation with maximum power at frequencies corresponding to a few cycles per million years on Earth. Their external magnetic field structure, dipole statistics, low-frequency power spectra, and polarity reversal frequency are comparable to the geomagnetic field. The magnetic variability is driven by the Lorentz force and is characterized by an inverse correlation between dynamo magnetic and kinetic energy fluctuations. A constant energy dissipation theory accounts for this inverse energy correlation, which is shown to produce conditions favorable for dipole drift, polarity reversals, and excursions.

Fluctuations of the Earth's magnetic field with time scales longer than a few years are collectively referred to as geomagnetic secular variation. Secular variations induced by the time-variable dynamics in the Earth's core that maintain the geodynamo span an enormous frequency range, from about one cycle per decade to one cycle per million years and longer (1, 2). A variety of theories have been proposed for the relatively short-period secular variation, and some have been successful in explaining geomagnetic observations. For example, the decade-to-century fluctuations that dominate the historical secular variation have been interpreted in terms of frozen-flux transport of magnetic field by large scale flow (3, 4), magnetohydrodynamic oscillations (5–7), and turbulent cascade processes (8) in the fluid outer core. In contrast, there is no consensus on the origin of the low-frequency secular variations that dominate the archeomagnetic and paleomagnetic records. This low-frequency variability consists mostly of large-amplitude, broad-band dipole moment fluctuations with time scales ranging from a few thousand years to about one million years. Archeomagnetic variations define the high-frequency end of this range (9, 10), and a slow drift of the dipole moment amplitude defines the low-frequency end (11). Proposed explanations for these fluctuations include external forcing by orbital variations (12, 13), kinematic dynamo waves (14), and tide-induced instabilities (15), in addition to purely statistical models such as power-law noise (16).

Large-amplitude, low-frequency secular variation has implications for polarity reversals, excursions, and other extreme geomagnetic events (17, 18). Paleomagnetic measurements indicate a close relationship between polarity reversals and the phase of the low-frequency secular variation, with polarity reversals and excursions—transient, large-amplitude dipole tilt events—occurring at times when the dipole moment is particularly weak (11). The connection between low-frequency secular variation and polarity reversals is supported by recent estimates of the geomagnetic power spectrum (19), which show a nearly uniform distribution of variance over this frequency band.

Spontaneous polarity reversals are seen in numerical dynamos and laboratory fluid dynamos, although in most cases the reversal mechanism is not well understood. Polarity reversals have been reported in one laboratory dynamo experiment in liquid sodium (20). Numerical models have shown that the frequency of polarity reversals can be highly stochastic (21), and, in convection-driven dynamos, reversal frequency generally increases with the strength of the convective forcing (22, 23) and is sensitive to the choice of boundary conditions (24, 25). Dynamo models have also shed light on the kinematic behavior of the magnetic field during the brief transition periods when polarity changes occur (26, 27).

This paper focuses on the underlying causes of the intrinsic low-frequency magnetic variability in self-sustaining numerical dynamos driven by compositional convection, the so-called gravitational dynamo mechanism, which is thought to be the main power source of the geodynamo (28). In addition to the intrinsic low-frequency geomagnetic variation, there are also ultralow-frequency geomagnetic variations, evidenced by the slow modulation in the frequency of polarity reversals over Phanerozoic time and the existence of long polarity magnetic superchrons (29). These variations have characteristic time scales ranging from tens to hundreds of millions of years (17, 18) and are often attributed to changes in the dynamics of Earth's mantle affecting the long-term energy budget of the geodynamo (30). Although time variable mantle dynamics are not explicitly considered in this study, the results here suggest how ultralow-frequency geomagnetic variations might occur.

Gravitational Dynamo Equations and Parameters.

The conservation of momentum, mass and magnetic field continuity, buoyancy transport, and magnetic induction equations for convection and magnetic field generation in an electrically conducting fluid in a rotating, self-gravitating spherical shell can be written in dimensionless form as

where B, u, P, and χ are the magnetic induction, fluid velocity, pressure perturbation, and buoyancy variable, respectively, t is time, r is the radius vector, ε is the buoyancy source (all dimensionless), and E, Pr, Pm, and Ra are the Ekman, Prandtl, magnetic Prandtl, and Rayleigh numbers, respectively. The conservation of momentum (Eq. 1) is the Boussinesq form of the Navier–Stokes equation in a spherical (r, θ, φ) coordinate system rotating at angular velocity Ωẑ in which gravity increases linearly with radius. It includes the inertial and Coriolis accelerations plus the pressure, viscous, buoyancy, and Lorentz forces. Eqs. 1–4 have been nondimensionalized by using the shell thickness D = r_o–r_i as the length scale (r_o and r_i are the inner and outer radii) and the viscous diffusion time D²/ν as the time scale (ν is the kinematic viscosity). The fluid velocity is scaled by ν/D and the magnetic field is scaled by (ρ_o Ω/σ)^1/2, where ρ_o is outer core mean density and σ is its electrical conductivity. The radius ratio is fixed at r_i/r_o = 0.35, approximating the Earth's inner core boundary/core–mantle boundary radius ratio. Appropriate boundary conditions for the composition variable are χ = 1 at r = r_i, fixed light element concentration at the inner core boundary, and ∂χ/∂ r = 0 at r = r_o, zero light element flux at the core–mantle boundary. Mechanical and electrical conditions on both spherical boundaries are no-slip and electrically insulating.

Three of the four control parameters in Eqs. 1–4 have conventional definitions: E = ν/ΩD², Pr = ν/κ, where κ is the diffusivity of the buoyancy variable, and Pm = ν/η, where η = 1/μ_oσ is the magnetic diffusivity (μ_o is magnetic permeability). The fourth control parameter, Ra, controls the strength of the buoyancy forces driving the convection. In the gravitational dynamo mechanism, this parameter indicates the rate of chemical evolution of the core, as follows. As the core cools and the inner core solidifies, light elements (an uncertain mixture of sulfur, oxygen, silicon, carbon, etc., comprising ≈15% of the core by mass, here denoted by Le) are partitioned into the liquid phase near the inner-core boundary and dense elements (primarily a mixture of iron and nickel, here denoted by Fe) are partitioned into the solid phase and incorporated into the inner core. The increased concentration of light elements reduces the density of the liquid phase, making it positively buoyant in the outer core, and the resulting compositional convection provides the kinetic energy for the geodynamo (31, 32). Meanwhile the outer core becomes progressively less dense, owing to its increasing light element concentration.

Convection produced by this chemical segregation can be modeled numerically by using a buoyancy sink formulation (33). The light element concentration in the outer core is represented as a sum of a spatially uniform, slowly increasing part χ_o(t), plus a perturbation χ. Let χ̇_o denote the secular increase of χ_o, which is assumed to occur on such a long time scale that both χ_o and χ̇_o can be taken to be constant over the duration of a dynamo calculation. If κ represents the light element diffusivity in the outer core and if χ is scaled by D²χ̇_o/ν, then ε = −1 in Eq. 3. With this definition of χ, the Rayleigh number in Eq. 1 is

where Δρ = ρ_o − ρ_Le is the density difference between the outer core and the light element end-member mixture and g_o is gravity at the core–mantle boundary.

The dynamo model output consists of the induced magnetic field, fluid velocity, and light element concentration as a function of time. Volume average variables to be analyzed include the dimensionless rms internal magnetic field and fluid velocity plus their energy densities, denoted by B, u, E_k and E_m, respectively. In terms of the rms magnetic field and velocity, the dimensionless kinetic and magnetic energy density per unit mass are E_k = u²/2 and E_m = (PmE)⁻¹ B²/2, respectively. Surface average variables include the rms magnetic field on the core–mantle boundary and its dipole part, denoted by B_o and B_d, respectively. Time averages of these quantities are denoted by overbars, and deviations from time averages are denoted by primes. Two ratios are of particular significance for comparing dynamo model output to the Earth, the time average dipolarity of the magnetic field on the core–mantle boundary d = B_d/B_o and the dipole variability, the standard deviation of the dipole strength relative to its time average s = |B′_d|/B_d.

For comparison with the geomagnetic secular variation, it is customary to express the dynamo fluid velocity in magnetic Reynolds number units rather than the ordinary Reynolds number units implied by the nondimensionalization in Eqs. 1–4. The conversion from u in Reynolds number units to u_η in magnetic Reynolds number units is given by u = Pm⁻¹ u_η. Another way to compare dynamo velocities uses the local Rossby number Ro_ℓ defined in ref. 34. Dynamo model and geomagnetic time are usually compared in units of the dipole free decay time. In terms of the scaling used in Eqs. 1–4, the dimensionless dipole free decay time in a sphere with radius r_o and uniform electrical conductivity is

and the corresponding frequency is f_d = 1/t_d. For the Earth's core, one dipole decay time corresponds to approximately 20 kyr.

Results

Fig. 1 summarizes the statistical behavior of gravitational dynamos with E = 6.5 × 10⁻³, Pr = 1, and Pm = 20 as a function of Rayleigh number Ra. Time averages and standard deviations of rms internal magnetic field strength, dipole field strength on the core–mantle boundary, rms fluid velocity, and dipolarity of the magnetic field on the core–mantle boundary are shown. Filled and open circles in Fig. 1 denote time average values of reversing and nonreversing dynamos, respectively. The squares denote results of nonmagnetic convection calculations at Ra = 0.6 × 10⁵ and 1 × 10⁵. Error bars denote one standard deviation, and the dashed line indicates the critical Rayleigh number for dynamo onset, Ra_crit ≃ 0.31 × 10⁵, for which the critical magnetic Reynolds number is u_η(Ra_crit) ≃ 40. Further details of these calculations are given in Methods and supporting information (SI) Figs. 8–11 and SI Table 1. The five cases with the lowest Ra values in Fig. 1 are balanced dynamos, with constant or nearly constant rms magnetic field and fluid velocity. The pattern of convection in these dynamos is invariant with time but propagates westward (retrograde) relative to the rotating coordinate system. Their magnetic fields are strongly dipolar near Ra_crit but become substantially less dipolar as Ra is increased. Above Ra ≃ 0.55 × 10⁵, the convection pattern becomes time-dependent and the balanced dynamos are replaced by more highly variable dynamos with chaotic time dependence and generally stronger magnetic fields. For the chaotic dynamos, the time average velocity increases monotonically with the Rayleigh number of the convection approximately as u_η ∝ Ra^7/8, whereas their time average rms magnetic field strength is nearly independent of Ra. The time average dipole strength decreases approximately as Ra⁻¹ in the chaotic regime, as does the time average dipolarity. Polarity reversals were recorded in all of the chaotic dynamos in Fig. 1, with Ra ≥ 0.68 × 10⁵, but not at lower Ra values. The distinction between reversing and nonreversing is somewhat arbitrary, however, because some dynamos in the later category might eventually reverse polarity if run for a sufficiently long time. For example, it is possible that reversals would occur in the Ra = 0.65 × 10⁵ case if it were continued longer, because its statistics differ only slightly from nearby reversing cases. The transition from nonreversing to reversing behavior coincides approximately with the peak dipolarity, with a small reduction in rms magnetic field strength and, more importantly, with a larger reduction in time average dipole field strength. The drop in the time average dipole field strength, together with its increased variability, means that weak dipole field states (dipole collapses) become more frequent as Ra increases. Because polarity reversals in these dynamos occur during dipole collapse events, reversals become increasingly frequent in the higher Ra cases. Some of the balanced dynamos in Fig. 1 also have rather weak time average dipole fields, but they lack dipole collapse events and are unlikely to reverse.

Fig. 1.

Gravitational dynamo statistics versus Rayleigh number with E = 6.5 × 10⁻³, Pr = 1, and Pm = 20. Time average values are denoted by circles, and standard deviations are denoted by error bars. Closed and open circles denote reversing and nonreversing dynamos, respectively. Squares denote nonmagnetic convection cases. The dashed line indicates critical Rayleigh number for dynamo onset. (Upper) rms internal magnetic field intensity (Left) and rms fluid velocity in magnetic Reynolds number units (Right). (Lower) rms dipole intensity on the core–mantle boundary (Left) and dipolarity on the core–mantle boundary (circles) and rms dipole tilt angle of nonreversing dynamos (asterisks) (Right).

Comparisons between a reversing and a nonreversing chaotic dynamo are shown in Figs. 2–6. Figs. 2 and 3 show time series of rms velocity, rms internal magnetic field, core–mantle boundary rms dipole field, core–mantle boundary dipolarity, and dipole tilt angle at Ra = 1 × 10⁵ and Ra = 0.6 × 10⁵, in the reversing and nonreversing regimes, respectively. The ratio of the fluctuations to mean values for the rms fluid velocity and the rms magnetic field have comparable amplitudes in the two cases. However, the Ra = 1 × 10⁵ case has both a weaker time average dipole and larger dipole fluctuations, so there are occasional, short time intervals when its dipole field is very weak. As seen in Fig. 2, roughly one half of these dipole collapse events result in polarity reversals or dipole tilt excursions. In contrast, the Ra = 0.6 × 10⁵ case shown in Fig. 3 lacks these dipole collapse events and has not reversed its polarity.

Fig. 2.

Time series of reversing gravitational dynamo with Ra = 1.0 × 10⁵, E = 6.5 × 10⁻³, Pr = 1, and Pm = 20. From top to bottom, shown are graphs of rms fluid velocity in magnetic Reynolds number units, rms internal magnetic field intensity, rms dipole field intensity at the core–mantle boundary, dipolarity on the core–mantle boundary, and dipole tilt angle. Time axes are dipole free decay time units. Dashed horizontal lines denote time average values; dotted horizontal lines indicate standard deviations. The vertical dashed line denotes the time of Fig. 4 images.

Fig. 3.

Time series of nonreversing gravitational dynamo with Ra = 0.6 × 10⁵, E = 6.5 × 10⁻³, Pr = 1, and Pm = 20. From top to bottom, shown are graphs of rms fluid velocity in magnetic Reynolds number units, rms internal magnetic field intensity, rms dipole field intensity at the core–mantle boundary, dipolarity on the core–mantle boundary, and dipole tilt angle. Time axes are dipole free decay time units. Dashed horizontal lines denote time average values; dotted horizontal lines indicate standard deviations. The vertical dashed line denotes the time of Fig. 5 images.

Fig. 4.

Structure of the reversing gravitational dynamo at the time indicated in Fig. 2. Shown are contours of the following variables. (Upper) The radial component of the magnetic field on the core–mantle boundary (Left) and the radial component of the fluid velocity in magnetic Reynolds number units at radius r = 0.93r_o (Right). (Lower) Axial component of the magnetic field in the equatorial plane with velocity arrows superimposed (Left) and normalized light element concentration in the equatorial plane (Right).

Fig. 5.

Structure of the nonreversing gravitational dynamo at the time indicated in Fig. 3. Shown are contours of the following variables. (Upper) (Left) Radial component of the magnetic field on the core-mantle boundary. (Right) Radial component of the fluid velocity in magnetic Reynolds number units at radius r = 0.93r_o. (Lower) (Left) Axial component of the magnetic field in the equatorial plane with velocity arrows superimposed. (Right) Normalized light element concentration in the equatorial plane.

Fig. 6.

Power spectra of the reversing gravitational dynamo at Ra = 1 × 10⁵ shown in Fig. 2. Frequency axes are dipole decay frequency units f/f_d. (Upper) rms internal magnetic field intensity (Left) and rms dipole field intensity on the core–mantle boundary (Right). (Lower) rms dynamo velocity (black) compared with rms nonmagnetic convection velocity (gray) in magnetic Reynolds number units (Left) and rms axial dipole field intensity on the core–mantle boundary (Right). Dashed vertical lines denote velocity corner frequency; solid lines indicate various power-law slopes.

Likewise, about one half of the minima in the dipolarity time series in Fig. 2 are associated with polarity reversals or excursions, even though the events that produced the smallest dipolarity resulted in excursions rather than reversals. Although the dipole tilt angle in Fig. 3 shows occasional spikes, the dipolarity remains >0.5, indicating that a polarity reversal is extremely unlikely in this case. Figs. 2 and 3 also show that the velocity and magnetic field fluctuations have rather different frequency content, with high frequencies being relatively more energetic in the velocity time series. The spikes in the velocity time series correspond to the spin up or spin down of one or more of the vortices shown in Figs. 4Lower Right and Figs. 5 Lower Right, caused by the growth or decay of a buoyant upwelling. The velocity time series also contain low-frequency fluctuations that are out of phase (anticorrelate) with the low-frequency fluctuations that dominate the magnetic time series.

Figs. 4 and 5 compare the internal structure of the reversing and nonreversing dynamos. The convection patterns are broadly similar in the two cases, in terms of their light element distributions, the patterns of radial velocity just below the core–mantle boundary, and the number of anticyclonic vortices in the equatorial plane. Even the radial magnetic field on the core–mantle boundary has the same general structure in the two figures, apart from polarity. The primary difference between the reversing and nonreversing dynamos is the structure of B_z, the axial magnetic field in the equatorial plane. For the nonreversing case shown in Fig. 5, the two anticyclonic vortices each contain a concentrated magnetic flux bundle with the same polarity as the dominant large-scale magnetic field. In contrast, the two anticyclonic vortices in the reversing case shown in Fig. 4 contain magnetic flux bundles with opposite polarities. Although one polarity or the other dominates the field on the core–mantle boundary (normal polarity field, i.e., negative B_z happens to dominate at the time shown in Fig. 4), the field usually has a mixed polarity in the equatorial plane. The polarities of the two equatorial plane flux bundles do not actually change sign during reversals. Instead, their relative strengths change (see SI Movies 1 and 2). Similarly, the diffuse background field in the equatorial plane, which also has a mixed polarity in Fig. 4, simply changes its relative proportions of positive and negative during a reversal. The reversing dynamos in Fig. 1 have this mixed polarity internal magnetic field most of the time and always during polarity changes. What appears as a polarity reversal of the external field in these dynamos corresponds to a change in the relative strength of competing, opposite polarity internal field structures. The dipole strength and the dipolarity are lower and the time variability is higher in the reversing dynamos because of this internal competition, which is largely absent in the nonreversing chaotic dynamos. Opposite polarity field structures are able to stably coexist for long periods of time in the chaotic dynamos by virtue of their high magnetic Reynolds number, which leads to magnetic field concentration in flux bundles localized within a few rather large convective vortices. The spacing of the vortices acts to preserve the mixed polarity by separating the flux bundles in longitude, inhibiting field line reconnection, diffusive merging, and other effects that tend to destroy field structures with opposing polarities.

Figs. 6 shows power spectra of rms internal magnetic field, rms fluid velocity, plus rms core–mantle boundary dipole and axial dipole fields for the reversing dynamo case. Frequency is scaled by the dipole decay frequency f_d, vertical broken lines mark the knees or corner frequencies in the velocity spectra, and the solid lines indicate fits to various frequency power laws. The power spectra of the other chaotic dynamos have features seen in Fig. 6, including broad, low-frequency maximum near f/f_d = 0.1 (corresponding to a frequency of ≈5 Ma⁻¹ for the Earth) and progressively more negative slopes at higher frequencies in all spectra. The velocity spectrum has a corner frequency that is particularly well defined in Fig. 6, where it is marked by a small spectral line at ≈f/f_d = 20 (corresponding to a frequency of ≈1 kyr⁻¹ for the Earth). Between the low-frequency maximum and the velocity corner frequency, the magnetic spectra vary as fⁿ. In the reversing dynamos, the rms internal field decreases as n = −2, whereas the rms dipole and axial dipole are better fit by n = −7/3. In the nonreversing dynamos, the spectral slopes are less uniform and generally steeper. Power-law exponents have been fit to the magnetic fields of all of the gravitational dynamos in Fig. 1, and the results are given in SI Table 1. The dipole field spectra of the chaotic dynamos have exponents that vary from n = −3 for the nonreversing cases to n = −7/3 or n = −2 for the most frequently reversing cases. All of the chaotic dynamos have spectra power law exponents near n = −2 for their internal magnetic fields. SI Figs. 12 and 13 show that the amplitude and shape of the dipole power spectra in Fig. 6 compare favorably with geomagnetic power spectra derived from deep-sea sediment records (19).

Effects of the Lorentz Force

The chaotic dynamos have smaller time average velocities than nonmagnetic convection with otherwise identical parameters. From Fig. 1, the ratio of dynamo to nonmagnetic convection velocity is 0.78 at Ra = 6 × 10⁴ and 0.81 at Ra = 1 × 10⁵. Because the only difference between the pairs of dynamo and nonmagnetic convection calculations is the presence or absence of the Lorentz force, the slightly smaller time average velocity in the dynamos shows that the Lorentz force reduces the time average velocity of the convection, as previous dynamo modeling studies (35) have found.

Whereas the Lorentz force tends to reduce the time average kinetic energy of the convection, it has just the opposite effect on its time variability. This can be seen by comparing the reversing dynamo and its corresponding nonmagnetic convection velocity spectra in Fig. 6. The reversing dynamo has a large-amplitude, broadband velocity spectrum with peak variance at very low frequencies. The low-frequency portion of the dynamo velocity spectrum in Fig. 6 has a similar shape as the three magnetic spectra. The higher-frequency portion of the dynamo velocity spectrum also includes the prominent corner or knee referred to above. None of these features are present in the velocity spectra of the corresponding nonmagnetic convection. Instead, the nonmagnetic velocity spectrum consists of a few high-frequency spectral lines above a low-intensity background, and there is far less total variance than in the dynamo case.

In the time domain, the low-frequency variations of dynamo kinetic and magnetic energies are almost exactly out of phase. Fig. 7, which shows low-pass filtered time series of kinetic energy density and internal magnetic energy density for the Ra = 1 × 10⁵ reversing dynamo, was obtained by applying a running average of length t_d to the unfiltered time series in Fig. 2. The correlation coefficient between low-frequency kinetic and magnetic energy variations in Fig. 7 is −0.96. The inverse phase relationship also is due to the Lorentz force. An increase in the internal magnetic field strength increases the Lorentz force on the fluid and reduces the convective velocity; conversely, a reduction in the internal magnetic field strength reduces the Lorentz force on the fluid and enhances the convective velocity. The tradeoff between kinetic and magnetic energy gives the chaotic dynamos additional freedom that is missing from nonmagnetic convection and offers an explanation for why the chaotic dynamos have much larger-amplitude velocity fluctuations although slightly smaller time average velocities compared with their nonmagnetic convection counterparts. Variability occurs primarily at low frequencies in the dynamos because their magnetic fields are dominated by large-scale components with long free decay times.

Fig. 7.

Low-pass-filtered time series of the reversing gravitational dynamo shown in Fig. 2. (Top) Kinetic energy density. (Middle) Internal magnetic energy density (solid) compared with the nonlinear constant dissipation model (dashed). (Bottom) Internal magnetic energy density (solid) compared with the linear constant dissipation model (dashed).

The tradeoff between kinetic and magnetic energy fluctuations in Fig. 7 can be modeled by assuming constant total energy dissipation, as implied by the zero frequency limit of Eqs. 1–4 (34). In dimensionless form, the average viscous dissipation per unit mass can be written in terms of the kinetic energy density as Φ_k = δk−2E_k, where δ_k is the viscous dissipation length scale of the flow. By using the same nondimensionalization, the average Ohmic dissipation per unit mass can be written as Φ_m = δm−2E_mE_k, where δ_m is the Ohmic dissipation length scale. Constant total dissipation then implies

An alternative model assumes the Ohmic dissipation varies linearly with the internal magnetic energy (36) and yields

As a check, Φ_k and Φ_m were calculated at selected times for the dynamo and nonmagnetic convection cases at Ra = 1 × 10⁵, and it was found that their total Φ_k + Φ_m varied only by a few percent. The variability in δ_k or δ_m was not directly checked by calculation; however, it would readily be apparent as changes in the scale of velocity or magnetic field structures with time, which are not observed in these calculations. The consistency of the flow structure and the (nearly) constant dissipation accounts for the small time variations in the nonmagnetic convection cases. For the dynamos, the kinetic and magnetic energies appear as a product in the first term in Eq. 7 and as a sum in Eq. 8, either of which allows the kinetic and magnetic energies to vary inversely while maintaining constant total dissipation and fixed dissipation length scales.

Eqs. 7 and 8 were used to predict the low-frequency dynamo magnetic energy variations in terms of the kinetic energy variations, with the results shown in Fig. 7. The dashed curves Fig. 7 Middle and Bottom are the predicted low-frequency variation of the internal magnetic energy E_m(t) obtained by substituting E_k(t) into the above equations and solving for the best-fitting constant values of δ_m and δ_k. Both models provide good fits in terms of phase and amplitude, although the nonlinear model Eq. 7 fit is slightly better (correlation coefficient 0.95) than the linear model Eq. 8 (correlation coefficient 0.89), particularly at times when the magnetic energy is high.

Comparison with the Geodynamo

Because of the low viscosity of the outer core fluid and the Earth's rapid rotation, there is little prospect of making a direct numerical simulation of the geodynamo. The individual control parameters in this study are not representative of their Earth values, which are approximately Ra ≃ 10³⁰, E ≃ 10⁻¹⁴, Pr ≃ 10⁻¹, and Pm ≃ 10⁻⁶, respectively. Even the largest numerical dynamo models (23) are compelled to use some unrealistic control parameters. This inherent limitation has led to a search for scaling laws that would allow extrapolation of the results of numerical dynamos to planetary conditions (36–38). For example, it is sometimes asserted (39) that planetary dynamos saturate (equilibrate) with rms magnetic fields corresponding to an Elsasser number Λ = σB²/ρ_oΩ ≃ 1. The Elsasser number is particularly attractive as a scaling parameter because it is independent of the dynamo energy source. Unfortunately, it has been shown that Λ varies with the Rayleigh number in convection-driven dynamos (34), and generally the field strength does not conform to constant Λ. Fig. 1 shows that, for the gravitational dynamos in this study, the rms internal field strength increases with Ra in the balanced dynamo regime, then saturates somewhere in the range B = 3.5–4 (corresponding approximately to Λ ≃ 12–16), possibly decreasing at higher Ra. This behavior provides some limited support for the constant Elsasser number assumption for the internal magnetic field, although it does not address possible Elsasser number dependence on the other control parameters. However, Fig. 1 also shows that B_d varies inversely with Ra in the chaotic regime, so the external dipole field strength in these models depends strongly on the dynamo energy source and clearly does not have constant Elsasser number.

Other parameters that have been proposed for dynamo model scaling include the magnetic Reynolds number and the Rossby number. The magnetic Reynolds number for the flow in the core that induces the decadal geomagnetic secular variation is approximately 300–500 (3–6), compared with 100–300 for the dynamos in Fig. 1, so the models scale reasonably well in terms of this parameter. Recent numerical (34) and theoretical studies (40) indicate that dynamo properties, such as dipole strength and the transition from dipole-dominant to multipolar states, depend on the local Rossby number Ro_ℓ. Although the Rossby number for large-scale flow in the core is very small, Ro_ℓ may not be small, because the characteristic wave number of convection in the Earth's core is predicted to be rather large (41). Olson and Christensen (38) estimate Ro_ℓ = 0.05–0.2 for the geodynamo, similar to Ro_ℓ = 0.014–0.111 for the gravitational dynamos in Fig. 1.

Independent of scaling considerations, the magnetic field structure of these gravitational dynamos compares favorably with the present-day geomagnetic field structure in several respects. The axial dipole field in the dynamo models is mostly a product of the two pairs of high-latitude flux bundles seen in Figs. 4 and 5. These structures are similar to the high-latitude flux bundles on the core–mantle boundary in the present-day geomagnetic field (3, 42), which are the major contributors to the geomagnetic axial dipole moment. One difference is that the flux bundles in Figs. 4 and 5 drift rapidly westward, whereas their geomagnetic counterparts more stationary, possibly because the later are linked to heterogeneity in the lower mantle. Another significant property that these dynamo models share with the paleomagnetic field is dipole dominance during stable polarity chrons. The dipole dominance of the external field is enhanced in the gravitational dynamos because the magnetic field generation is concentrated deep in the fluid, close to the inner core boundary, with the effect that the nondipole moments of the field are strongly attenuated with radius.

Time domain statistics of the gravitational dynamo models in the range Ro_ℓ = 0.05–0.07 compare favorably with time domain statistics of the geodynamo. For example, the reversing dynamo at Ra = 1 × 10⁵ has Ro_ℓ ≃ 0.06, a dipole variability s ≃ 0.37, and a time average dipolarity d ≃ 0.53. The model dipolarity is similar to the present day geomagnetic field on the core–mantle boundary, when truncated at spherical harmonic ᶩ_max = 14 has d ≃ 0.64, or, alternatively, d ≃ 0.54 if the core–mantle boundary field spectrum is extrapolated to large ᶩ_max (43). Another point of comparison is the dipole variability parameter, estimated to be near s ≃ 0.4 for the paleomagnetic field over the past 160 Ma (44), virtually the same as this dynamo model. In terms of polarity reversal rates, Fig. 2 contains five reversals and excursions in 120 dipole decay times, equivalent to two such events per million years on Earth. When averaged over the entire sea-floor magnetic record, the rate of geomagnetic reversals also is about two per million years (18, 29). The ultra-low frequency modulation of the reversal rate seen in the paleomagnetic record is not explicit in these dynamo models, but the abrupt transition from reversing to nonreversing states in Fig. 1 suggests a possible explanation. A relatively small decrease in the Rayleigh number of the core, produced by a decrease in the total heat flow at the core–mantle boundary, for example, could switch the geodynamo from reversing to nonreversing behavior.

Previous dynamo modeling studies have shown that reversals occur in connection with velocity fluctuations. Sarson and Jones (45) found that reversals follow fluctuations in meridional circulation associated with surges in buoyancy. Wicht and Olson (26) found reversals initiated by reverse magnetic flux transported in plume-like upwellings, as have Aubert et al. (27). Because none of these mechanisms produce reversals when the dipole field is strong, deciphering the reversal phenomenon requires an understanding of why the dipole sometimes collapses. The present study reveals that the Lorentz force drives low-frequency dipole moment change in convection-driven dynamos, but the mechanism through which this occurs remains unclear. It may involve subtle interactions between the fluctuating convection and the magnetic fields, possibly as follows. High-frequency variations in the convection, in the form of transient plumes and vortices, induce magnetic field variations on their same short time scales. Part of the energy in these fluctuations reinforces the dynamo, part draws energy from it, and energy also is lost by Ohmic and viscous dissipation. In a chaotic flow, the production of magnetic energy tends to be slightly out of balance with the dissipative effects, so these dynamos evolve on time scales that are very long compared with the characteristic time scales of their convective and diffusive components. The magnetic field strengthens while magnetic energy production exceeds dissipation, but eventually the increased Lorentz force that accompanies the strengthened magnetic field reduces the kinetic energy, the rate of magnetic field generation declines, and the magnetic field begins to decay. The dynamo continually drifts between high-strength and low-strength states, occasionally reversing polarity when it is weak.

Methods

The numerical dynamo model in this study (MAG, available at www.geodynamics.org), was originally developed by G. Glatzmaier and has been benchmarked and was used previously in systematic dynamo studies (35, 38). The fluid velocity and the magnetic field vectors are represented as sums of poloidal and toroidal scalars, ensuring that the continuity equations are satisfied identically. The momentum and induction equations are then decomposed into four scalar evolution equations in terms of these functions and are advanced simultaneously using explicit time steps on a spherical finite difference grid, along with the buoyancy equation.

The five scalar variables also are represented in terms of surface spherical harmonics up to degree and order ᶩ, and in Chebyshev polynomials in radius. At each time step, the nonlinear terms are evaluated on the spherical finite difference grid, and the linear terms are evaluated by using the spherical harmonics and Chebyshev polynomials. Additional details about the numerical method are given in ref. 46.

The most severe restrictions on numerical models stem from the fact that it is not practical to make calculations with parameter values that are realistic for the Earth's core with existing codes (47–49). To remedy this, we set Pr = 1 in all calculations and choose relatively large values of the Ekman number, E = 6.5 × 10⁻³, but a few cases with E = 3 × 10⁻⁴ are included for comparison purposes. These choices then necessitate Pm = 5–20 to maintain a self-sustaining dynamo.

The finite difference grid and the truncation level of the spherical harmonics were chosen to ensure that the spectral power of magnetic energy attenuates by a factor of 100 or more from its peak ᶩ value and that the diffusive layers at the inner core and core–mantle boundaries are well resolved. The high Ekman number cases with E ≥ 6.5 × 10⁻³ use N_r = 25 radial grid intervals with N_r − 2 Chebyshev polynomials and harmonic truncation ᶩ_max = 32. The lower Ekman number cases use N_r = 35 and ᶩ_max = 48.

The calculations were initialized with random buoyancy perturbations and an axial dipole magnetic field. The dynamos were run until their statistical fluctuations became stationary. Short run times sufficed for the dynamos with constant or nearly constant magnetic and kinetic energies. A few, very long runs were made for some of the most complex cases, up to 900 viscous time units. Model output from the first few dipole decay times was excluded from the analysis to eliminate contamination by the transient adjustment to the initial conditions. Nonmagnetic convection calculations were made with the same set of parameters as several of the self-sustaining dynamo cases. Power spectra were computed with Hanning window tapers, normalized to preserve the variance of the raw time series. SI Table 1 lists the control parameters and summarizes the results of all calculations used in this study.