^{*}

Edited by: Angelo De Santis, INGV / G. D'Annunzio University, Italy

Reviewed by: Maria Luisa Osete, Complutense University of Madrid, Spain; Bejo Duka, University of Tirana, Albania; Philip Livermore, University of Leeds, UK

*Correspondence: Domenico G. Meduri

This article was submitted to Geomagnetism and Paleomagnetism, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Earth's axial dipole field changes in a complex fashion on many different time scales ranging from less than a year to tens of million years. Documenting, analysing, and replicating this intricate signal is a challenge for data acquisition, theoretical interpretation, and dynamo modeling alike. Here we explore whether axial dipole variations can be described by the superposition of a slow deterministic drift and fast stochastic fluctuations, i.e. by a Langevin-type system. The drift term describes the time averaged behavior of the axial dipole variations, whereas the stochastic part mimics complex flow interactions. The statistical behavior of the system is described by a Fokker-Planck equation which allows useful predictions, including the average rates of dipole reversals and excursions. We analyze several numerical dynamo simulations, most of which have been integrated particularly long in time, and also the palaeomagnetic model PADM2M which covers the past 2Myr. The results show that the Langevin description provides a viable statistical model of the axial dipole variations on time scales longer than about 1kyr. For example, the axial dipole probability distribution and the average reversal rate are successfully predicted. The exception is PADM2M where the stochastic model reversal rate seems too low. The dependence of the drift on the axial dipole moment reveals the nonlinear interactions that establish the dynamo balance. A separate analysis of inductive and diffusive magnetic effects in three dynamo simulations suggests that the classical quadratic quenching of induction predicted by mean-field theory seems at work.

Earth's internal magnetic field is produced in the liquid iron core in a complex highly nonlinear dynamo mechanism. Geomagnetic data provide a window into this process which can be exploited for clues on the internal dynamics. Information on shorter time scales of years to decades can rely on a combination of high quality satellite data and historical data. The respective models can describe the field up to spherical harmonic degree and order 14 beyond which the small scale crustal field starts to dominate (Jackson et al.,

A combination of the different data sources reveals that dipole variations have a broad temporal power spectrum shaped by the nonlinear interaction between processes acting on different time scales (Constable and Johnson,

Statistical analyses attempt to extract essential information from the intricate but limited geomagnetic signal. Constable and Parker (

Statistical approaches have also proven useful when dealing with full 3D numerical dynamo simulations where the wealth and complexity of information is hard to interpret. Wicht and Meduri (submitted), referred to as WM16 in the following, analyzed the probability distribution of axial and equatorial dipole moments in several different numerical dynamo models. With the exception of the largest Ekman number cases which seem the most remote from Earth conditions, the results obey a simple statistical systematic. Both equatorial dipole moment contributions have a Gaussian distribution with zero mean and a variance that is nearly independent of the Rayleigh number

WM16 suggest that the third Gaussian reflects a new weak dynamo state that facilitates reversals and excursions. The dynamo may enter this state once the dipole moment has decreased to at least 30% of its mean, a prerequisite for reversals and excursions that seems to agree with palaeomagnetic inferences (Channell et al.,

A more refined model that concentrates on the statistics of dipole moment variations was pioneered by Hoyng et al. (

ADM fluctuations can only be expected to be statistically uncorrelated on time scales longer than the typical time scale of convective motions. Hoyng et al. (_{c} of approximately 120yr for Earth obey the Langevin equation and simply use time resolutions τ > τ_{c} for analyzing the magnetic data. Drift and fluctuation functions are then related to the mean variation 〈_{i}) (with _{i + 1} = _{i} + τ) by

When a stochastic system starts at time

For example, _{m}, are separated by a hump around _{m} while the random fluctuations

_{0}(_{0} is the analytical solution of the stationary Fokker-Planck equation (Equation 27). The reversing dynamo model shown is E3R9. After Figure 2 in Schmitt et al. (

The stationary solution _{0}(_{0}(_{0}(_{x} as a measure for palaeosecular variation. The mean reversal waiting time 〈_{R}〉, i.e. the mean time between successive reversals, can be predicted from the mean first passage time of the potential well. Fokker-Planck theory also predicts a Poissonian occurrence of reversals and thus an Earth-like distribution of reversal waiting times (Schmitt et al.,

Hoyng et al. (_{0}(_{x} value of about 0.3〈

The dependence of drift and fluctuation on the ADM provides clues on the feedback between dynamo and Navier-Stokes equations. The interpretation is complicated by the nonlinear nature of the feedback mechanism but the related problems have been addressed by mean-field theory (Krause and Rädler, _{n} is a normalization factor. Brendel et al. (^{−1}. Surprisingly, however, the fluctuation

Kuipers et al. (^{−3} and three different Rayleigh numbers which included reversing cases covering much longer timespans than Sint-2000. At least two of the three analyzed cases show a drift

Buffett et al. (^{−1} that is similar to the value inferred by Brendel et al. (_{d} = 25kyr to τ_{d} = 56kyr estimated for Earth's fundamental dipole mode. The upper value is based on the higher electrical conductivity of _{d} could be explained by the fact that these additional modes have a more complex radial structure.

The mean reversal waiting time inferred by Buffett et al. (

The situation is even worse for the dynamo simulations analyzed by Buffett et al. (^{−5} and magnetic Prandtl number

The scope of this paper is to apply the Langevin and Fokker-Planck formalisms to much longer dynamo simulations than previously analyzed. Models introduced by WM16 form the basis but are supplemented by runs in the parameter range explored by Buffett et al. (

We employ the MHD code MagIC (Wicht, ^{2}, the (modified) Rayleigh number _{o}Δ_{i}∕_{o}. Physical properties are the kinematic viscosity ν, system rotation rate Ω, outer boundary reference gravity _{o}, thermal diffusivity κ, magnetic diffusivity λ, outer boundary radius _{o}, and inner boundary radius _{i}. The shell thickness _{o} − _{i} serves as a reference length scale. The spherical shell, that represents the Earth's outer core, is bounded by an electrically insulating mantle at _{o} and a conducting inner core at _{i} which rotates according to viscous and Lorentz torques (Wicht,

Two forms of convective driving are used. Thermal bottom driving is modeled using a fixed codensity jump Δ_{o} and a fixed codensity at _{i}. Homogeneously distributed internal sinks compensate the “light elements” entering the system at the inner boundary. This setup requires a modified Rayleigh number where the prescribed sinks instead of the imposed codensity jump define the codensity scale.

Most of the models analyzed here have been introduced by WM16 and use an Earth-like aspect ratio of ^{−3} and a magnetic Prandtl number of ^{−4} cover the same fundamental regimes but could only be run for much shorter periods because of the larger numerical costs. The respective Earth-like reversing case at E = 3 × 10^{−4} covers only 20 reversals. We have added two additional models in the regime discussed by WM16 with a low Ekman number of E = 3 × 10^{−5}, a relatively low Rayleigh number, and the two magnetic Prandtl numbers of _{c} = _{c} the critical Rayleigh number for the onset of convection) and

_{c} |
_{ℓ} |
_{o} |
_{geo} |
||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

E3R5 | D | 10^{−3} |
10 | Temp. | 250 | 4.5 | 571 | 202 | 0.04 | 2.6 | 0.60 |

E3R7 | D | 400 | 7.2 | 242 | 350 | 0.09 | 2.0 | 0.44 | |||

E3R9 | R | 500 | 8.9 | 17547 | 436 | 0.12 | 1.5 | 0.31 | |||

E3R13 | M | 750 | 13.4 | 229 | 592 | 0.17 | 1.7 | 0.18 | |||

E4R53C | D | 3 × 10^{−4} |
3 | Chem. | 3000 | 53 | 46 | 264 | 0.09 | 0.34 | 0.66 |

E4R78C | D | 4500 | 78 | 188 | 340 | 0.11 | 0.25 | 0.59 | |||

E4R106C | R | 6000 | 106 | 363 | 408 | 0.13 | 0.14 | 0.40 | |||

E4R159C | M | 9000 | 159 | 83 | 497 | 0.27 | 0.15 | 0.27 | |||

E5R18Pm05 | D | 3 × 10^{−5} |
0.5 | Temp. | 1500 | 17.6 | 47 | 105 | 0.05 | 0.20 | 0.95 |

E5R18Pm1 | D | 1.0 | Temp. | 1500 | 17.6 | 17 | 183 | 0.04 | 0.92 | 0.92 | |

Earth | R | 10^{−15} |
10^{−6} |
Flux | − | ≫1 | 35 | 2000 | 0.09 | 0.79 |

_{c} are listed in columns 3, 4, 6, and 7, respectively. Column 5 details the driving mechanism: an imposed constant temperature contrast across the shell (Temp.) or chemical convection (Chem.). The Prandtl number Pr is unity in all cases. Column 8 shows the total integration time in dipole decay units. For Earth we list the 2Myr covered by the palaeomagnetic field model PADM2M. The magnetic Reynolds number Rm, the outer boundary Elsasser number Λ_{o}, the local Rossby number Ro_{ℓ}, and the relative outer boundary dipole strength D_{geo} are time averaged values. Earth's core values refer to molecular diffusivities. The Rayleigh number is hard to estimate for Earth but thought to be highly supercritical. The estimate of Rm is derived from secular variation data. D_{geo} is estimated from the geomagnetic field model gufm1 (Jackson et al., _{ℓ} is the value reported by Christensen and Aubert (

WM16 also explore compositionally driven cases at E = 10^{−3}. Since the dynamics turned out to be very similar to the thermally driven cases at the same Ekman number, we refrain from analyzing these models here. Shorter sections of models E3R5, E3R9, and E3R13 have been analyzed by Kuipers et al. (

The magnetic Reynolds number
_{ℓ} are listed in columns nine and ten of Table _{ℓ} < 0.1 the dynamo is typically dipole dominated and remains stable, i.e. neither undergoes reversals nor excursions (models E3R5, E3R7, E4R53C, E4R78C, and the two cases at E = 3 × 10^{−5}). Multipolar dynamos which reverse more or less continuously are characterized by _{ℓ} > 0.1 (models E3R13 and E4R159C). Dynamo models with Earth-like rare reversals where the axial dipole still dominates on time average and the reversals last much shorter than the stable polarity epochs are found at the transition (Wicht and Tilgner,

Column eleven in Table _{o} is the total RMS field strength at the outer boundary, ρ the outer core density, and μ the magnetic permeability. Since the magnetic field is scaled with (μ^{1∕2} here, the non-dimensional magnetic field amplitude at the outer boundary is identical to _{geo} shown in the last column of Table _{o} when considering only spherical harmonic contributions up to degree and order 11.

The time throughout this paper is scaled in the core dipole decay time
_{d} = 56kyr for Earth when based on the electrical conductivity ^{4}kgm^{−3}, the magnetic permeability of vacuum for μ, and Ω = 7.29 × 10^{−5}sec^{−1}.

The Langevin equation

The evolution of the probability distribution ^{ − 1}〈^{ − 1}(〈[^{2}〉)^{1∕2} already mentioned in Section 1. This highlights that

An ambiguity in solving the above time averages over stochastic fluctuations in the limit τ → 0 motivates two different formalisms for connecting the FPE (14) with the Langevin equation (11), the so-called Itô and Stratonovich approximations (Risken and Frank,

Equation (17), often referred to as the Euler-Maruyama integration scheme for the Langevin equation (Maruyama,

Constructing the drift and fluctuation profiles for the Langevin equation (or FPE) relies on binning the variations δ_{i}. The width reflects a compromise between the number of data within each bin and the fact that variations over the bin should be small.

After the discrete variations δ_{j}(τ) have been calculated for each data point of the discrete time series _{j} = _{j}), mean and RMS are evaluated by averaging over the _{i} data in each bin:

As discussed in Section 1, the Langevin description can only be expected to hold once τ exceeds the convective overturn time τ_{c}. This is confirmed by the fact that, for example, _{c} (Buffett and Matsui, _{i} and _{i} on τ for models E3R9 and E5R18Pm05. The solid vertical line marks the convective overturn time in the respective models. In model E3R9, for example, a time resolution τ of at least five times τ_{c} ≈ 0.1τ_{d} guarantees that drift and fluctuation profiles become nearly independent of τ. Similar inferences also hold for the other cases explored here, though the worse statistics of the shorter runs complicates the choice of τ. For model E5R18Pm05, for example, variations in both profiles are particularly strong for the lowest and the highest ADM bins which include relatively few instances (Figure _{c} ≈ 0.04τ_{d} has been chosen for analyzing this model. Tests with the different models have shown that a larger number of entries per bin _{i} is desirable to provide reliable profiles. The bin width Δ

^{22}Am^{2} is 1.47, 0.35, and 1.06 for models E3R9, E5R18Pm05, and PADM2M, respectively. The solid and dotted vertical lines mark the convective overturn time and the selected time resolution, respectively. In the palaeomagnetic model PADM2M time has been rescaled using an Earth's core dipole decay time of τ_{d} = 56kyr.

The bottom panel of Figure

Since PADM2M covers only few reversals and excursions, instances of low VADM are rare and we could not constrain this particularly interesting region with sufficient confidence. Adding to this problem is the fact that local field in a palaeomagnetic record is interpreted as being caused by the virtual axial dipole moment. Contributions from the equatorial dipole and higher field harmonics can therefore make the VADM significantly larger than the ADM during reversals or excursions. Drift and fluctuation are obviously harder to constrain and the profiles remain relatively uncertain, in particular for VADM values

Once an appropriate binning scheme and time resolution τ have been selected, the required functions can be modeled by least-square-fitting power series expansions to the binned values, for example

The steady-state probability density function (PDF) _{0}(_{x} predicted by _{0} provide measures to judge the similarity, at least when the PDF is simple enough.

Knowing _{0} allows us to infer additional statistical properties of the system. For example, for a stochastic particle starting at _{e}〉 is also often referred to as the mean first passage time at _{e}〉 is nearly independent of _{e} is the exponential

At the escape time _{e}, the stochastic particle will be located close to the top of the drift potential at _{R}〉 = 2〈_{e}〉 which can be shown to hold analytically (Hoyng et al., _{e}) and to estimate the mean escape time 〈_{e}〉 to verify the theoretical predictions discussed above.

Table ^{−3} models. The polynomial expansions (Equations 25 and 26) permit to draw profiles for ADM values that are never reached in the simulations. This does not necessarily make sense in all cases but allows, for example, a tentative prediction of the reversal rate where no reversals have been observed. We will discuss this point further below.

_{c} |
_{m} |
_{V} |
_{H} |
_{L} |
_{*} |
_{*}) |
^{⋆}(〈| |
^{⋆}(0) |
|||
---|---|---|---|---|---|---|---|---|---|---|---|

E3R5 | 0.021 | 0.201 | 20.62 | 21.42 | 0.14 | 0.37 | 0.46 | 10.87 | 15.20 | 12.86 | 36.57 |

E3R7 | 0.012 | 0.198 | 11.79 | 11.74 | 0.24 | 0.54 | 0.64 | 6.38 | 6.40 | 11.39 | 10.33 |

E3R9 | 0.010 | 0.145 | 6.32 | 7.30 | 0.48 | 1.03 | 1.02 | 4.23 | 2.70 | 11.47 | 9.58 |

E3R13 | 0.007 | 0.150 | 0.38 | 0.50 | − | 0.94 | 1.08 | 0 | 0 | 13.16 | 12.73 |

E4R53C | 0.018 | 0.100 | 7.73 | 7.82 | 0.12 | 0.56 | 0.36 | 3.46 | 6.09 | 5.87 | 9.68 |

E4R78C | 0.011 | 0.100 | 5.40 | 5.87 | 0.26 | 0.76 | 0.82 | 2.93 | 2.29 | 6.41 | 5.36 |

E4R106C | 0.009 | 0.140 | 2.63 | 2.73 | 0.60 | 3.64 | 3.10 | 1.95 | 0.42 | 5.45 | 5.18 |

E4R159C | 0.009 | 0.300 | 0.09 | 0.03 | − | 1.66 | 1.84 | 0 | 0 | 4.31 | 4.27 |

E5R18Pm05 | 0.040 | 0.300 | 9.95 | 10.06 | 0.06 | 0.47 | 0.25 | 4.32 | 10.78 | 1.99 | 25.47 |

E5R18Pm1 | 0.016 | 0.200 | 15.03 | 15.05 | 0.05 | 0.34 | 0.13 | 6.33 | 30.23 | 3.60 | 58.20 |

PADM2M | 0.0021 | 0.107 | 5.32 | 5.74 | 0.28 | 0.42 | 0.28 | 2.41 | 5.48 | 8.43 | 13.32 |

_{c} and the time resolution τ selected for the analysis, respectively. Columns 4 to 6 give the unsigned ADM mean 〈|x|〉, mode x_{m}, and relative standard deviation _{V} (Equation 30) as a measure for secular variation. τ_{H} (column 7) and τ_{L} (column 8) are the time scales characterizing the slow drift at x = x_{m} and x = 0 respectively. Column 9 gives the ADM value x_{*} where the drift d is maximum. The maximum drift value is listed in column 10. The last two columns report the rescaled fluctuation f^{⋆} at x = 〈|x|〉 and x = 0. For the multipolar dynamo models E3R13 and E4R159C all the listed measures are based on the signed axial dipole moment x instead of |x|. In the palaeomagnetic model PADM2M time has been scaled assuming a core dipole decay time of τ_{d} = 56kyr. The convective overturn time τ_{c}, the time resolution τ, and the time scales τ_{H} and τ_{L} are given in dipole decay units. The listed axial dipole moment statistics are in units of 10^{22}Am^{2}, and the drift d and rescaled fluctuation f^{⋆} values are in units of 10^{22}Am^{2}∕τ_{d}.

^{⋆} (squares) as function of the axial dipole moment ^{−3}

The ADM value _{m} where the drift (or mean variation) _{m} is the mode of the ADM distribution (dotted vertical lines in Figure _{m} does not coincide with the unsigned axial dipole moment mean 〈|_{m} (_{m}) the slow negative (positive) drift drives the system back toward the mean. The fluctuation (or RMS variation) ^{⋆}, always larger in amplitude than the drift, is instrumental in driving the system to other values that fill the ADM probability distribution.

Buffett et al. (_{H} defined by 〈_{H}. We use the power series expansion (25) to yield:
_{H} increases with the Rayleigh number, ranging from 0.37τ_{d} for model E3R5 to about one τ_{d} for the reversing case E3R9.

In the non-reversing cases (top two panels of Figure _{L} which is, like τ_{H} in Equation (29), defined by the drift derivative but now evaluated at _{H} and also increases with _{L} is not directly supported by data for E3R5 and E3R7 where the ADM always remains relatively strong. The somewhat large value of τ_{L} ≈ τ_{d} for the reversing model E3R9 is consistent with the increased probability for low ADM values which WM16 explained with a distinct low dipole moment state.

As expected, both time scales τ_{L} and τ_{H} roughly coincide in the multipolar case E3R13 where the drift decreases monotonically (Figure

The variation of the ADM value _{*} where the drift reaches its maximum reveals further interesting insights. Table _{*}) decreases significantly with increasing Rayleigh number. This can be compared with the amplitude of the rescaled fluctuation ^{⋆} characterized by the values at ^{⋆}(〈|

In the particle-in-well analogy, _{*} represents the point where the drift potential _{*} and 〈|

The fluctuation ^{⋆} has a concave shape in the non-reversing case E3R5 (Figure ^{⋆} monotonically increases with ^{⋆}(〈|

The drift and fluctuation curves for the E = 3 × 10^{−4} models shown in Figure _{H} and τ_{L} are particularly large for the reversing case E4R106C and reach more than three dipole decay times. The particular parameter combination promotes a much slower drift than in all other cases.

^{−4}

Finally, Figure ^{−5}. Since the Rayleigh number is relatively small and the models have only been run for a comparatively short time, they sample only a limited ADM range. Little more can be said than that the linear drift and concave fluctuation curves are very similar to those discussed above. The difference in magnetic Prandtl number in these two low Ekman number cases has virtually no impact.

^{−5}

Figure _{H} is about 0.42τ_{d} or 23kyr, in agreement with the value found by Buffett et al. (_{L} = 16kyr which is similar to the 20kyr suggested by Brendel et al. (

_{d} = 56kyr.

The stochastic model represented by the Langevin and Fokker-Planck equations should be validated against the original data, i.e. against the numerical simulation results and the palaeomagnetic model PADM2M. The validation presented here has two levels. In the first level we compare with integrations of the discrete Langevin equation (17). Hundreds of such realizations, each integrated as long as the original numerical dynamo model, provide a statistical ensemble that allows us to estimate the error for statistics based on the respective integration time. The ensemble mean, on the other hand, is equivalent to very long integrations that usually cannot be afforded in the full numerical dynamo simulation. The second level of validation consists in a comparison with the analytical predictions from the FPE discussed in Section 2.2. An agreement between the results from very long integrations of the Langevin equation and the FPE confirms that the latter indeed describes the statistical behavior of the former and that we have correctly linked the two differential equations.

Figure

_{d} while PADM2M covers 2Myr or about 36τ_{d}. The horizontal dashed lines mark the unsigned axial dipole moment mean values. The stochastic realizations have been started with the same initial (V)ADM values of the original time series.

The unsigned ADM mean for model E3R9 is 〈|^{22}Am^{2} with a standard deviation of ^{4} realizations integrated as long as the original numerical model (more than 17 thousand dipole decay times or about 950Myr for Earth) yields a mean of _{m}. As expected,

_{e} |
||||||
---|---|---|---|---|---|---|

E3R5 | 20.58±0.12 | 20.92±0.26 | 0.15±0.01 | 224 | 221±8 | − |

E3R7 | 11.75±0.22 | 11.64±0.45 | 0.25±0.02 | 183 | 195±6 | − |

E3R9 | 6.42±0.03 | 6.78±0.25 | 0.469±0.003 | 12.8 | 13.8±0.6 | 16.5 |

E3R13 | 0.002±0.298 | 0.009±0.794 | − | 0.16 | 0.18±0.02 | − |

E4R53C | 7.73±0.13 | 7.73±0.25 | 0.12±0.01 | 57328 | − | − |

E4R78C | 5.48±0.12 | 5.48±0.25 | 0.23±0.02 | 437 | 444±16 | − |

E4R106C | 2.69±0.12 | 2.77±0.60 | 0.57±0.03 | 5.7 | 5.9±0.4 | 6.5 |

E4R159C | 0.005±0.374 | 0.006±0.887 | − | 0.03 | 0.09±0.02 | − |

E5R18Pm05 | 9.94±0.08 | 9.92±0.10 | 0.06±0.01 | 44152 | − | − |

E5R18Pm1 | 15.03±0.13 | 15.03±0.13 | 0.05±0.01 | 506445 | − | − |

PADM2M | 5.18±0.26 | 5.41±0.39 | 0.29±0.04 | 19.9 | 25.5±1.2 | − |

_{m}, and relative standard deviation _{V} (Equation 30). Columns 5 and 6 give the mean escape time 〈T_{e}〉 predicted by the Fokker-Planck theory (Equation 28) and the ensemble average estimate, respectively. The last column shows the mean escape time calculated using the theoretical relation 〈T_{e}〉 = 〈T_{R}〉∕2. The mean reversal waiting times 〈T_{R}〉 for models E3R9 and E4R106C are taken from WM16. For the multipolar cases E3R13 and E4R159C the listed statistics are based on the signed axial dipole moment x instead of |x|. The ensemble averages of 〈|x|〉, x_{m}, and S_{V} are performed over 10^{4} realizations and the errors denote the 68% confidence intervals. The mean escape time 〈T_{e}〉 for the stochastic model is evaluated using at least 2 × 10^{3} repeated realizations and the respective ensemble average is performed over at least 3 × 10^{3} of such estimates. The error on 〈T_{e}〉 denotes the 95% confidence interval. The statistics relative to the ADM are expressed in units of 10^{22}Am^{2} and 〈T_{e}〉 is given in dipole decay units.

Figure _{0} (thick light gray curves) agree nicely with the numerical simulations (open squares) and perfectly overlap long numerical realizations of the respective stochastic model (black curves). The very long simulation run for E3R9 provides a tight constraint and leads to very good agreement of all PDFs. The much shorter integration time for E4R106C, however, is also the reason for the PDF asymmetry. This asymmetry should ultimately vanish for longer integrations as shown by the stochastic realization.

_{0}(

The good agreement of the VADM distribution from PADM2M with the simulated distribution suggests that the stochastic model successfully reproduces the variability of the palaeomagnetic time series (Figure _{0} overlaps with the distribution from a long numerical realization as expected. Both distributions closely follow the original data PDF. Since PADM2M covers only the last 2Myr of palaeomagnetic variations, however, the confidence intervals on the estimates of the ADM mean and mode remain relatively large (see Table _{0} and the data PDF occurs for low dipole field intensities: The stochastic model predicts a small, but still non-zero probability whereas the data probability is vanishingly small. One reason for this discrepancy might be related to the few instances of low VADMs in the palaeomagnetic record. The left flank of the original VADM data distribution is only somewhat more populated than the right flank. While this slightly asymmetric profile might indicate the presence of a “weak dynamo state” as discussed by WM16, it is certainly much less pronounced than in model E3R9 or E4R106C. The latter model shows a particularly high probability for low field intensities, but choosing slightly lower Rayleigh numbers for the simulation should reduce this effect. The analytical steady-state distributions of non-reversing and multipolar dynamos show similarly good agreements with simulation data and stochastic model realizations but are not illustrated here.

The relative standard deviation
_{V} are reported in Table ^{−3} and E = 3 × 10^{−4} cases _{V} increases with the Rayleigh number ^{−5} show a less pronounced SV of the order of 5% consistent with their highly dipolar character and the relatively small Rayleigh number. In PADM2M the relative standard deviation is _{V} values in the original numerical simulation (cf. Table _{V} must be based on the signed ADM _{x} for the multipolar case E3R13 (E4R159C) is 3.50 × 10^{22}Am^{2} (2.03 × 10^{22}Am^{2}). The ensemble average over 10^{4} realizations yields a standard deviation ^{22}Am^{2} ((1.98±0.16) × 10^{22}Am^{2}) which well agrees with the numerical simulation.

The stochastic model allows us to calculate the mean escape time 〈_{e}〉, i.e. the average time required by _{R}〉 for models E3R9 and E4R106C provided by WM16, using the relation 〈_{e}〉 = 〈_{R}〉∕2 (see Section 2.2).

The reversing and multipolar models at E = 10^{−3} and E = 3 × 10^{−4} present ensemble averages of 〈_{e}〉 close to the respective analytical predictions. Though the escape time _{e} is not well constrained by the simulation data for stable dipolar dynamos, the fitted drift and fluctuation functions nevertheless yield a prediction. The analytical 〈_{e}〉 ranges from few hundred dipole decay times for the E = 10^{−3} cases to very large values for the highly dipolar models at E = 3 × 10^{−5}. Since the stochastic realizations showed too few or no reversals for models E4R53C, E5R18Pm05, and E5R18Pm1 we could not list the respective estimates in Table _{e}〉 values from the stochastic realizations match the analytical predictions for the stable dipolar models E4R78C and E3R5 in the limits of statistical errors. The ensemble average of 〈_{e}〉 is close to the predicted value for E3R7.

The average Earth's reversal rate is about 4Myr^{−1} when estimated from palaeomagnetic reversal chronologies for the last 20Myr or so (Biggin et al., ^{−1}. The Fokker-Planck analysis, however, predicts a much smaller rate of only 0.45Myr^{−1}. A possible explanation for the discrepancy is the poor coverage of low VADMs in the PADM2M data. Figure _{*}) or larger fluctuations at small

Fokker-Planck theory also predicts that the escape times are exponentially distributed with a mean value 〈_{e}〉 given by Equation (28) and listed in Table _{e}) for the reversing dynamo models and for PADM2M together with the predicted exponential distributions. The analytical predictions are in very good agreement with the numerical realizations. Once more, this demonstrates the validity of the Fokker-Planck description of the stochastic process.

_{e} obtained from 10^{4} numerical realizations of the stochastic model for the reversing dynamo models E3R9 and E4R106C, and for the palaeomagnetic model PADM2M_{e}〉 the mean escape times (28) listed in Table

The stochastic model allows us to predict more than just the mean reversal waiting times. For example, we can access the mean waiting time 〈_{W}〉 for the axial dipole moment to reach any given value _{m}. Since _{m} is the most likely ADM value, this is a relevant scenario. Small variations can be achieved by direct fluctuations and can therefore be rather fast. However, Figure _{m} once several fluctuations have to team up to lead to more substantial variations. The combined fluctuations have to overcome the slow drift and the large 〈_{W}〉 values prove that this is quite difficult. The waiting times are generally longer than a simple dipole decay and larger for _{m} than for _{m}. The palaeomagnetic model PADM2M generally shows the longest mean waiting times and they are longer in model E3R9 than in E4R106C. The values at _{e}〉 listed in Table

_{W}〉 as function of the axial dipole moment _{m} (see Table ^{4} realizations of the stochastic model started with _{m}.

The results shown in Figure

Changes in the axial dipole moment are either of inductive or diffusive origin. The ratio of magnetic field production to the Ohmic decay is often estimated by the magnetic Reynolds number

To explore the role of the two processes in the drift and fluctuation contributions discussed above, we have performed additional shorter runs for models E3R5, E3R9, and E3R13. During these simulations we stored the axial dipole contribution of magnetic field induction and diffusion within the spherical shell. No-slip boundary conditions force the flow and thus the induction to vanish at both boundaries. We therefore analyze the axial dipole variation just below the outer Ekman boundary layer which was identified by inspecting the radial kinetic energy profile.

The analysis of both contributions follows the scheme already outlined above. MagIC actually solves for inductive and diffusive changes of all spherical harmonics contributions to the radial magnetic field. Variations in the axial dipole contribution _{10} of the radial field are directly related to variations in the axial dipole moment. For accessing induction and diffusion separately we integrate both contributions to the degree ℓ = 1 and order _{10} bins of width Δ_{10}.

Since the dipole field within the dynamo region is not a potential field, it cannot be expressed in terms of a dipole moment to facilitate a direct comparison with the ADM analysis presented in the previous sections. We therefore simply use non-dimensional values for time and _{10}, that are given in units of τ_{d} and (μ^{1∕2} respectively, and compare relative variations and the general form of drift and fluctuation profiles.

The left panels in Figure _{c} the profiles are very similar to those for the ADM sequences of the respective models presented in Section 3.1. The absolute diffusive contribution (dashed curves) increases roughly proportional to _{10} (or _{10} = 0. The diffusive effects level off for small _{10} values in model E3R5 which also shows some other distinct properties that we will discuss below.

^{(I)} (solid lines with open symbols), negative diffusion −^{(D)} (dashed lines with gray filled symbols) and the total _{c} when the Langevin and Fokker-Planck formalisms can be expected to hold (circles). The right column shows the respective rescaled fluctuation profiles ^{⋆}. For E3R9 (E3R13) the black profiles have been amplified by a factor 3 (5). The solid (dashed) vertical line marks the axial dipole mean (mode). All values are given in dimensionless units.

The inductive contribution (solid curves in Figure _{10}, on τ, and also on _{c}, this constructive term generally grows with increasing _{10}. The increase is roughly linear for model E3R5 but slightly levels off for larger dipole field intensities in models E3R9 and E3R13. Inductive and diffusive profiles cross at a value slightly larger than the mean axial dipole intensity 〈_{10}〉, with diffusion dominating for larger and induction for smaller values. This is equivalent to the point where the total drift (dotted curves) vanishes and defines the preferred _{10} value and/or the minimum in the respective drift potential as discussed above. Solid and dashed vertical lines in the left panels of Figure _{10}〉 and the crossing point respectively.

When τ is increased beyond the overturn time, the ^{(I)} profiles (solid black curves in Figure _{10} but still crosses the negative diffusive contributions. The decrease is likely a sign of magnetic flow quenching. The dominance of diffusive effects beyond and inductive effects below the preferred _{10} value becomes much more pronounced once the faster fluctuations have been filtered out. The inductive profiles for E3R9 and E3R13 increase roughly linearly for small _{10} values and seem to show quenching effects for larger field strengths, for case E3R9 more so than for E3R13. In the multipolar case E3R13 the induction profile remains always smaller than the diffusive one which explains the negative total drift toward _{10} = 0.

Profiles of the fluctuation contributions ^{⋆(I)} and ^{⋆(D)} are shown in the right panels of Figure _{10} and, contrary to the drift, inductive effects clearly dominate. The exception is once more E3R5 at τ = 0.1, the only model where quenching seems to significantly impact the fluctuation. For larger _{10} values, ^{⋆(I)} decreases so significantly that diffusive effects start to dominate the rescaled fluctuation in this model. Note that the fluctuation amplitude, contrary to the drift, also strongly depends on τ.

We refrain from fitting analytical functions to the drift and fluctuation profiles and use the ratio
_{10} value. We can, however, calculate the time scales that characterize purely inductive or purely diffusive mean or RMS variations, keeping in mind that they may not reflect the true variations which are generally a combination of both effects. For example, the resulting time scales for the mean variations ^{⋆}.

Figure ^{⋆} (gray curves) are shorter than the respective scales based on the mean

_{V} for the cases with τ > τ_{c} depicted in Figure

The diffusive drift time scales (dashed black curves) remain close to one for dynamo E3R5 but decrease to roughly 0.4 for E3R9 and 0.35 for E3R13. This indicates that dipole contributions with a more complex internal radial structure than the fundamental decay mode contribute to the axial dipole, as suggested by Buffett et al. (_{10} values but can reach much larger values when _{10} increases. The shortest drift time scales are clearly associated to mean inductive variations which therefore dominate the _{10} dynamics. These time scale estimates somewhat differ from those in Section 3.1 which were based on the total drift and thus on the difference between inductive and diffusive profiles.

We started this section with discussing the magnetic Reynolds number as a measure for the ratio of diffusive to inductive time scales. Our analysis allows us to calculate the ratio of these time scales based on, for example, the mean axial dipole variations _{c} which reflect the long time scale dynamo balance. The respective magnetic Reynolds numbers _{10} and cross unity where induction and diffusion profiles meet. Maximum values at low _{10} only reach about two for models E3R5 and E3R9 but never exceed unity for E3R13. This is in strong contrast to the much larger values between

The decrease of inductive effects with increasing field strength offers interesting clues on the nonlinear feedback between flow and magnetic field that establishes the dynamo balance. Classical mean-field theory predicts a quadratic quenching of the so-called α-effect that parametrizes the production of axisymmetric field via the interaction of non-axisymmetric field and non-axisymmetric flow (Roberts, _{α} is a normalization factor. Following Brendel et al. (_{10} = 0.

Figure ^{(I)}, the results nevertheless indicate that the quadratic quenching predicted by mean-field theory may indeed apply. The fitting parameters are (α = 3.27, _{α} = 2.13) for ^{(I)} and (α = 2.40, _{α} = 3.00) for ^{⋆(I)} which confirm the much weaker quenching of field fluctuations. Note, however, that fluctuations are more strongly quenched in model E3R5.

^{(I)} (black) and RMS induction ^{⋆(I)} (gray) for the dynamo model E3R9 with τ > τ_{c} depicted in Figure

Our analysis has confirmed that a stochastic Langevin model can describe the axial dipole moment variations in numerical dynamo simulations and in a palaeomagnetic field model. Separating the dynamics into a slow drift and faster stochastic fluctuations requires to exclude time scales where correlations of flow features still matter. On time scales longer than about a millennium, however, the stochastic model offers a viable description of the axial dipole field dynamics and provides useful prediction of, for example, its probability distribution (Hoyng et al.,

Fokker-Planck theory also allows us to predict the mean reversal rate which reasonably well agrees with the rates in the two reversing dynamo simulations explored here. For Earth, however, the stochastic model based on the palaeomagnetic PADM2M data predicts a rate about 10 times lower than the 4Myr^{−1} suggested by the marine magnetic anomaly record for the last 20Myr (Biggin et al.,

Figure

^{⋆} for the reversing dynamo model E3R9 (solid and dashed curves respectively) and for the palaeomagnetic model PADM2M (open circles and squares respectively)

Experiments with synthetic drift and fluctuation profiles could help to reconstruct the axial dipole moment variations at small field intensities leading to more realistic predictions of the reversal rate. Another problem are the slow changes of the Earth's reversal rate over time, for example due to variations in the lower mantle structure, and normal statistical fluctuations may also play a role (Biggin et al.,

The secular variation, measured by the relative standard deviation _{V}, is smaller in model PADM2M than in the two reversing dynamo models studied here. One possible reason is the coarse time resolution of the palaeomagnetic model. For the simulations we chose a time resolution τ of 15 times the convective overturn time while this factor has to be increased to about 60 for PADM2M. Perhaps more significant is the fact that the palaeomagnetic model is based on sedimentary data which are known to average out faster field variations due to the long locking time. It is therefore likely that the palaeomagnetic model misses extreme variations or averages them out. Both effects potentially lead to narrower distributions which could be tested for the numerical models, an analysis we plan to conduct in the future. Alternatively, a slight decrease in Rayleigh number may not only reduce the reversal rate but also lead to leaner probability distributions.

Following the ideas of Hoyng et al. (_{m}.

Buffett et al. (_{m} from both sides, i.e. also for _{m}. Our analysis confirms the more classical view that diffusion always acts destructively while any drift toward stronger fields is an inductive effect. The idea by Buffett et al. (

The fluctuations (or RMS axial dipole variations) act on time scales at least a factor two faster. The much weaker quenching effects suggest that the fluctuations are too fast to establish a balance. Note, however, that the fluctuation time scale decreases with the temporal resolution τ and we have chosen relatively large τ values here. Fluctuation quenching is bound to become even smaller when τ is decreased further.

Is the idea of an additional weak field state in reversing dynamos, suggested by WM16, supported by the analysis presented here? As stated above, the modes _{m} of the unsigned ADM distribution are determined by the values where the drift crosses zero with a negative slope (or equivalently where the drift potential has a local minimum). A distinct weak dipole moment state would thus require the drift to assume a negative slope around _{m}. This suggests that Earth-like rare reversals are actually not interludes where the dynamo ventures into the multipolar state. Instead they are facilitated by the simple fact that the drift amplitude decreases toward _{m}, shallower negative drift slopes around _{m}, a late turning point _{*} of the drift slope, or a small positive drift slope around

We have only started to analyze quenching effects and a more in depth comparison with predictions from mean-field theory would be certainly interesting. An extension of the internal induction and diffusion analysis to deeper depths, other field harmonics, and more dynamo cases could clarify how widely the simple picture we paint here actually applies. Unfortunately, it remains very difficult to include low Ekman number cases with E < 3 × 10^{−5} since the respective simulations are too computationally demanding.

Both authors contributed equally to running and analyzing the numerical simulations. The publication charges are funded by the MPDL.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.