Excursions, Reversals, and Secular Variation: Different Expressions of a Common Mechanism?

Fluctuations in the geomagnetic field occur over a broad range of timescales. Short‐period fluctuations are called secular variation, whereas excursions and reversals are viewed as anomalous transient events. An open question is whether distinct mechanisms are required to account for these different forms of variability. Clues are sought in trends b of the axial dipole moment from six time‐dependent geomagnetic field models. Variability in b has a well‐defined dependence on the time interval (or window) for the trend. The variance of b reveals a simple relationship to trends during excursions and reversals. This connection hints at a link between reversals, excursions and secular variation. Stochastic models exhibit a similar behavior in response to random fluctuations in dipole generation. We find that excursions, reversals and secular variation can be distinguished on the basis of trend durations rather than differences in the underlying physical process. While this analysis does not rule out distinct physical mechanisms, the paleomagnetic observations suggest that such distinctions are not required.


Introduction
Intervals of stable geomagnetic polarity are disrupted by transient events known as excursions and reversals (e.g., Merrill & McElhinny, 1994).Excursions are commonly defined as angular departures of the virtual geomagnetic pole (VGP) by 45°from the time-average position (Merrill & McElhinney, 1983).Some excursions exhibit much larger departures and may even have brief reversals of the axial dipole before returning to the original polarity (Nowaczyk et al., 2012).The term reversal is usually restricted to events in which the dipole reverses polarity and remains in a stable reversed state.The duration of this stable state can vary and distinct terms (i.e., chrons, subchrons, cryptochrons) are used to categorize the observed intervals (Ogg, 2012).It is difficult to detect durations much less than a few tens of kyr in marine magnetic anomalies, so our direct knowledge of shorter durations (or chrons) is limited.
Early statistical predictions of chron lengths, based on a Poisson process (Cox, 1968), supported the existence of arbitrarily short chrons.Later proposals, based on a Gamma distribution with a shape parameter k > 1 (Naidu, 1971), suggested that short chrons were suppressed (e.g., Merrill & McElhinny, 1994).A shape parameter k > 1 may also arise from incomplete sampling of a Poisson process (McFadden, 1984), so a definitive interpretation of observed chrons is not yet established.A somewhat different view emerges from simulations of polarity reversals using a stochastic model (Buffett & Avery, 2019).Estimates of the shape parameter k for the Gamma distribution can be recovered from histograms of chron lengths in the simulations.Calculations show that the outcome for k depends on the time resolution of the time series.A simulation with a 20-to 30-kyr time resolution (comparable to the resolution of the marine magnetic record) yields a shape parameter k > 1.However, a higher resolution record yields k < 1, implying that short chrons occur with greater frequency than that expected for a Poisson process.In detail the high-resolution simulations reveal numerous short chrons in the vicinity of transition fields before the dipole settles into a stable new polarity (see Figure 1).These details are lost when the time resolution of the time series is too low.
The possibility of short chrons raises the question of how we distinguish reversals from excursions, especially when excursions can include brief intervals of reversed polarity.One option is to appeal to differences in the underlying physical mechanisms.For example, Gubbins (1999) suggests that excursions occur when a reversedpolarity magnetic field in the outer core is not sustained long enough to diffuse into the inner core.Evidence for distinct behavior might be sought in power spectra of temporal fluctuations in the geomagnetic field.Current estimates of the power spectra (Constable & Johnson, 2005) reveal little distinction at the relevant frequencies, which might be taken as evidence against a mechanistic difference between excursions and reversals.We take this suggestion one step further by exploring the possibility that transient events are part of a continuum of geomagnetic variation, extending from reversals and excursions to secular variation on historical timescales.One of the goals of the present study is to assess whether this interpretation is compatible with observations.A useful approach is based on trends in the axial dipole moment as the geomagnetic field enters a reversal or an excursion.We can assess whether the trends into reversals and excursions are similar in amplitude or persist over similar time intervals.We can also compare the trends during transient events to the trends observed during times of stable polarity.A connection between transient events and secular variation might suggest a common origin.We explore this possibility using six geomagnetic field models over a range of time scales.Models CALS10k2 and HFM.OL1.A1 (Constable et al., 2016) use paleomagnetic sedimentary records, together with archeomagnetic and lava flow data, to construct a low-degree spherical harmonic expansion of the geomagnetic field over the past 10 kyr.These models are used here to compute the variance of dipole trends over durations as long as 1 kyr.Models GGF100k (Panovska et al., 2021) and GGFSS70 (Panovska et al., 2018) use methods similar to those of the 10 kyr models, but extend the record of the geomagnetic field back in time.GGF100k uses a large global distribution of sediment cores over the interval 0-100 ka, whereas GGFSS70 uses a selective set of sedimentary records with higher sedimentation rates to cover the interval 15-70 ka.These models are used to quantify dipole trends into three well-established excursions.We also extend our description of the trend variance to timescales of several millennia.Models PADM2M (Ziegler et al., 2011) and SINT-2000(Valet et al., 2005) provide the longest records.These models describe fluctuations in the dipole moment over the past 2 million years using stacks of relative paleointensity in marine sediments, combined with estimates of absolute paleointensity from lava flows.Trends from these two models are important because they reveal distinctive changes in the trend variance when the duration exceeds the dipole decay time (see below).Transient trends for six reversals are computed from PADM2M; a second estimate for the Matuyama-Bruhnes reversal is computed from the GGFMB model (Mahgoub et al., 2023).
Stochastic models provide a quantitative framework for interpreting the observed dipole trends.They are also useful for assessing how measurement error, smoothing and incomplete sampling affect the observed trends.We begin in Section 2 by introducing the stochastic model and presenting a theoretical prediction for trend variance as a function of the trend duration w.These predictions are compared with the results from the six geomagnetic field models in Section 3. A large part of the variability in these field models is reproduced by the simple stochastic model.Dipole trends into excursions and reversals are computed from the geomagnetic field models in Section 4. Trends into reversals are typically twice the root-mean-square (rms) variation.In effect, reversals are associated with 2σ events.Excursions are associated with 2σ-3σ events over somewhat shorter timescales.There is a hint from GGFSS70 that trends increase at low dipole fields, which would motivate a modest change in the structure of the stochastic model.Overall, there is broad consistency between the trends into reversals and excursions compared with the statistics of trends during times of stable polarity.This agreement suggests a common physical mechanism for all forms of variability in the axial dipole.Interestingly, the recent historical trend is also a 2σ event over a much shorter time interval.Extending this historical trend for another 2 kyr to achieve a reversal would imply a 7σ event (Buffett et al., 2019).Such an unlikely event would be inconsistent with previous reversals in the paleomagnetic record.

Stochastic Model for Dipole Trends
A remarkably simple stochastic model is capable of reproducing much of the variability in the geomagnetic field models.We describe the dipole moment, x(t), over time, t, using a stochastic differential equation (e.g., Oksendal, 2003) where v(x) is the deterministic (drift) term and D(x) is the diffusion term, which specifies the amplitude of the random component.Individual random increments dW t are defined in terms of Brownian motion.Numerical integration of Equation 1 relies on an evaluation of the random increment over a finite time Δt.Integrating dW t over Δt gives a random variable defined by a normal distribution with zero mean and variance Δt.Combining this random component with a first-order Euler model for the deterministic term produces the standard Euler-Maruyama method (Kloeden & Platen, 1992).
Two simplifications are introduced to describe small fluctuations about the time average 〈x〉.First, we approximate v(x) with a linear term where γ represents the relaxation rate toward 〈x〉.Second, we replace D(x) with a constant D. The resulting model defines the well-studied Ornstein-Uhlenbeck process (Risken, 1996).
An extension of the linear drift is needed to describe large deviations from the time average, including the possibility of polarity reversals.Following Buffett and Puranam (2017) we introduce a nonlinear drift where |x| denotes the absolute value x.The nonlinear drift in Equation 3 changes sign when x reverses, consistent with symmetries in Maxwell's equations.We also expect v nl (x) ≈ v l (x) when the fluctuations about the time average are small.The linear approximation, v l (x), is particularly useful in deriving a prediction for the variance of the dipole trend over a prescribed duration w (see below).
Parameter values for the stochastic model are estimated using a Bayesian feature-based inversion (Morzfeld & Buffett, 2019).Data in the inversion include PADM2M and SINT-2000, combined with CALS10k.2 and the average reversal rate for the past 10 million years (Ogg, 2012).The recovered parameter values and uncertainties are γ = 0.1 ± 0.0033 kyr 1 and D = 0.34 ± 0.0072 × 10 44 A 2 m 4 kyr 1 .The time average is 〈x〉 = 5.23 × 10 22 A m 2 and the predicted standard deviation of x(t) for the linear model is Comparison of the linear stochastic model with the geomagnetic field models in Section 3 shows good agreement with the trends computed from SINT-2000, PADM2M and CALS10k.2.This agreement is not entirely surprising because these geomagnetic field models were used to construct the stochastic model.On the other hand, no explicit information about the trend was used in the inversion for the stochastic model.
The linear drift term in Equation 2 has a simple physical interpretation.It can be viewed as a superposition of persistent ohmic decay, γx, and a time-averaged generation term γ 〈x〉.Here we assume that the magnitude of the time-averaged generation term is equal to the time-averaged decay.When x(t) exceeds 〈x〉 we expect dipole decay to bring the dipole moment back toward the time average.Conversely, when x(t) falls below 〈x〉 the steady generation term restores the dipole moment to the time average.This interpretation of v l (x) allows us to relate the relaxation rate γ = τ 1 d to the dipole decay time τ d .Random contributions in the stochastic model are interpreted as fluctuations in the generation term about the time average.It is also possible to relate the magnitude of the diffusion term D to the mean-squared amplitude of helical convective motion in the core (Buffett et al., 2022).In either case we relate D to the vigor of convective motion in the core.

Variance of Dipole Trend
A trend in the dipole moment is determined by a least-squares fit of x(t) to a linear function where b is the trend, a specifies the fit at the start of the window w and e(t) is the deviation from a linear trend.
Values for b depend on the interval w chosen for the least-squares fit, so we expect the variance of b to be a function of w.The linear stochastic model is used to compute the variance of the trend because it is possible to derive an analytical solution for the deviation from the time average ϵ(t) = x(t) 〈x〉.The general solution for ϵ (t) is where ϵ(t 0 ) defines the initial condition at t = t 0 .Memory of the initial condition is lost when t ≫ t 0 .In this limit the time evolution of ϵ(t) is defined by a sequence of random increments dW s in time, where s denotes the integration variable.
The trend of ϵ(t) over a time window w can be written as (Kenney & Keeping, 1962) where Cov(t, ϵ) is the covariance between t and ϵ and Var(t) is the variance of t.It is convenient to define the trend over a time interval w/2 ≤ t ≤ w/2 because the mean of t vanishes.The variance of t can then be written as Geochemistry, Geophysics, Geosystems 10.1029/2024GC011604 BUFFETT where E(⋅) denotes the expected value.The covariance between t and ϵ(t) is computed using the known solution in Equation 6. Combining these results in Equation 7defines the slope.The variance of b is given by where we use condition E(b) = 0.
Nijsse et al. ( 2019) derived a closed-form expression for σ 2 b (w).An expanded derivation is included in Appendix A using the notation adopted in this study.Here we simply give the general form of the variance where f is a known function of γw (or, equivalently, w/τ d ).Predictions based on Equation 10 reproduce trends computed from numerical integration of Equation 1, although small deviations are found at the shortest w (see Figure 2).These differences arise when there are only two or three discrete estimates of x(t) in the least-squares fit for the slope.By comparison, the theory underlying Equation 10 is based on a continuous description of x(t), so there is better agreement when a large number of discrete values of x(t) are used in the fit for b.

Sensitivity of Trends to Measurement Error and Smoothing
Another use of the stochastic model is to quantify the influence of measurement error and smoothing on the calculation of trends from the geomagnetic field models.We numerically integrate the stochastic model in Equation 1 to produce a long 4-Myr realization of the dipole moment; trends are recovered directly from the time series at discrete values for w.Random error is added to the time series to characterize the influence of measurement error on x(t).We also apply a moving average to the time series to approximate the smoothing due to gradual acquisition of magnetization in sediments.Trends recovered from these modified time series show how complications connected with observations alter the expected trends.
Figure 2 shows the trends computed from three solutions of the stochastic model.The first is an unmodified integration of the stochastic model, while the second includes random measurement error with a standard deviation 0.5 × 10 22 A m 2 .The third solution is altered by applying a moving average with a width of 0.5 kyr.The unmodified solution reproduces the theory at all but the shortest windows (as noted above).By comparison, random error causes a more substantial increase in σ b (w) at short w.Similarly the influence of smoothing is largest at small w.In fact, the standard deviation of trends from the smoothed solution is nearly independent of w when the duration is short.A similar behavior is observed in the trends from the geomagnetic field models (see Section 3).
We can also address whether a short numerical solution (or short geomagnetic field model) is representative of the long-term field behavior.For example, we may find an unusually large excursion in a short solution (or record), even though the average recurrence time for this event is much longer than the duration of the record.Longer solutions capture a more representative number of large events compared with a shorter solution.Consequently, it is of interest to explore the role of sampling in assessing trends from the field models.Two numerical experiments are motivated by specific results from the geomagnetic field models.We start with a 55-kyr simulation to mimic the record from GGFSS70.We run 100 ensembles of a 55-kyr solution using the stochastic model described above.We find a spread of individual estimates for σ b (w) that progressively expands about the theoretical prediction as w increases.Individual ensemble members can account for numerical estimates of σ b (w) from GGFSS70 when w > 2 kyr.Similarly trends from a 10-kyr simulation increase substantially when w > 500 years.
In both examples the spread of trends around the average (and hence the theory) narrows at short w.We rely on these results to interpret the trends from the geomagnetic field models.

Dipole Trends in Geomagnetic Field Models
Six geomagnetic field models are used to assess the statistics of dipole trends over short (10 kyr), intermediate (100 kyr) and long (2,000 kyr) time intervals.Each field model is subdivided into a set of n non-overlapping time intervals of length w.The number of intervals depends on the choice of window size w relative to the length of the field model.A trend is recovered for each window interval by a least-squares fit of x(t) to the linear function in Equation 5. When the geomagnetic field model is represented in spherical harmonics we recover the axial dipole moment from the g 10 (t) gauss coefficient using where a = 6371.2km is the reference radius and μ is the magnetic permeability of free space.Estimates for b at a particular window length w are used to compute a sample standard deviation σ b (w).Varying the window size w over all allowable values characterizes the statistical behavior of the dipole trends.The largest w for any given field model is limited by the number of samples needed for reliable statistics (typically n > 10).Combining the results from the short, intermediate and long field models defines the dipole variability over several orders of magnitude in w.This description of the variability serves as the basis for assessing trends into reversals and excursions.
Figure 3a shows individual estimates for σ b (w) from SINT-2000 and PADM2M, compared with the theoretical estimate from the stochastic model (denoted as σb (w)).Both models agree reasonably well with the theoretical estimate at w > 30 kyr, but fall below σb (w) at shorter w.This deviation from the theory can be attributed to the influence of smoothing in the sedimentary record due to post-depositional remanent magnetization (Roberts & Winklhofer, 2004).A comparable change in the theoretical trends can be obtained by applying a moving average to the results of the stochastic model (see Figure 2).The initial departure of the paleomagnetic models from the theoretical prediction in Figure 3a can be quantitatively reproduced with a smoothing time of 4 kyr.
It is also of interest to quantify the maximum dipole trend for each w in the paleomagnetic models.In this case we recover the maximum absolute value of the trend, based on our initial partitioning of the record into nonoverlapping intervals (A larger trend may be possible with a more selective choice of the time interval).
Figure 3b shows the maximum trend approaching the 3σ b level for many w.Exceptions for SINT-2000 at w < 10 kyr may reflect the influence of measurement noise.We conclude that individual fluctuations in the trends can occasionally exceed the 3σ b level.A similar conclusion is drawn from the stochastic model.Maximum trends from a two-million year simulation can reach or occasionally exceed 3σ b .Estimates for the maximum trends help to establish the likelihood of steep trends into reversals and excursions.
Results for models GGFSS70 and GGF100k are shown in Figure 4a.While there is broad similarity in the overall dependence of σ b (w) on the window length w, there is an offset in the overall amplitude of these two models.
There may also be a hint in GGFSS70 that σ b (w) departs from the general slope of the theoretical prediction at a lower w, which would be compatible with the higher temporal resolution of the observations.Focusing on GGF100k we find that the initial departure of σ b (w) from theory at short w can be quantitatively explained with a smoothing time of 300-400 years.This range of smoothing time is considerably shorter than that inferred from PADM2M and SINT-2000.At larger w the GGF100k model is consistent with the theoretical prediction, whereas the variability of GGFSS70 is higher by a factor of roughly 1.5.We might attribute the higher variance in GGFSS70 to increased resolution of excursions.If these variations are representative of typical variability of geomagnetic field, we could achieve a comparable increase in the theory, σb (w), by increasing the diffusion term D (see Equation 10).This change would increase the amplitude of σb (w) without altering the overall shape.
The question here is whether the variability of the geomagnetic field over the interval 15-70 ka is representative of the long-term average.We can explore this question by running an ensemble of stochastic models over a 55-kyr time period.While the ensemble average is consistent with the underlying theory, individual realizations can deviate enough to account for the trends recovered from GGFSS70 when w ≥ 2 kyr.In other words, the trends from GGFSS70 at w ≥ 2 kyr could be compatible with the theory (without a change in D).This possibility would reconcile the trends of GGFSS70 with trends from PADM2M and SINT-2000, which are both consistent with the original theory.From this perspective we might argue that GGFSS70 captures a period of unusually frequent excursions.On the other hand, the question of sampling is not relevant for the differences between GGFSS70 and GGF100k because we compute trends from both models over a common time interval (15-70 ka).Trends at shorter windows are computed from models CALS10k.2 and HFM.OL1.A1.Model HFM.OL1.A1 has larger temporal variance in the dipole component of the field, whereas CALS10k.2 has more complex spatial structure, reflecting larger contributions from multipolar components.Figure 3b shows the consequences for the standard deviation of dipole trends.An offset in the estimates of σ b (w) for the two models is comparable to the relative offset between the intermediate-time models in Figure 3a, possibly for the same reason.Both HFM.OL1.A1 and GGFS70 have larger temporal variance and less spatial complexity than their counterparts CALS10k.2 and GGF100k.Models CALS10k.2 and GGF100k have trends that are broadly consistent with the theoretical estimate σ b (w) between 0.3 < w < 3.0 kyr, once we account for the influence of smoothing.Models GGFSS70 and HFM.OL1.A1 could be brought into close agreement with the theory if we allow an increase in the diffusion term D by a factor 1.5 2 (see Equation 10).

Trends Into Excursions and Reversals
Excursions and reversals are preceded by a rapid decline in the dipole moment.When this decline occurs erratically in time the identification of a start time can be subjective.We could define the start of an event as the time when x(t) initially drops below 〈x〉.This definition has the advantage of being well defined, but we may encounter cases where the dipole moment is persistently above the time average prior to the event.Neglecting the initial decline prior to reaching the average value would affect our assessment of the probability of this trend.We emphasize trend probabilities by normalizing the absolute value of the trend with the expected standard deviation σb (w) at the relevant window.Statistical inferences depend on the appropriate probability distribution for b/ σb .
For the linear stochastic model we expect b/ σb to be distributed as a standard normal N(0, 1).Independent of these assumptions we focus on b/ σb (w) (rather than b alone) to account for the important roles of magnitude and duration in assessing the likelihood of trends.
To illustrate the approach we use models GGF100k and GGFSS70 near the Laschamp excursion at 41 ka (see Figure 5).The dipole field is stronger than the time average prior to the excursion, so we define a start time by choosing a local maximum with x(t) > 〈x〉.Starting points for the Laschamp event in GGF100k and GGFSS70 are labeled as A and B, respectively.The end points for this event coincide with minima in x(t) in the two models, which we label as A′ and B′.Trends between A A′ and B B′ are then computed by least squares, recovering values b = 1.2 × 10 22 A m 2 kyr 1 for A A′ and b = 2.0 × 10 22 A m 2 kyr 1 for B B′. Window lengths for A A′ and B B′ are 3.8 and 2.8 kyr, respectively, so the predicted standard deviation, σb (w), is 0.429 and 0.509 (in units 10 22 A m 2 kyr 1 ).The ratio |b|/ σb gives 2.8 for GGF100k and 3.9 for GGFSS70 (see Table 1).
A somewhat lower |b|/ σb is recovered for the Laschamp event in GGFSS70 if we confine our attention to values x (t) > 2.5 × 10 22 A m 2 .Excluding the low-field values might be justified if the recovery of the dipole component is less certain when the non-dipole components are relatively large.Point B″ in Figure 5 defines the time when GGFSS70 initially drops below our threshold.The least-squares trend for the interval B B″ is b = 1.2 × 10 22 A m 2 kyr 1 over a window of 2.3 kyr.The predicted standard deviation is σb (w) = 0.568 × 10 22 A m 2 kyr 1 , so the initial part of the trend into the Laschamp excursion in GGFSS70 gives |b|/ σb = 2.1.This revised estimate makes the initial decline much more likely than the steeper decline at low x, where |b|/ σb > 4. A steep trend in GGFSS70 at low x may reflect real physical processes in the core, which would require an increase in the amplitude of the diffusion term, D(x), at low x.Alternatively, the steep trend at low x may be a consequence of incomplete separation of the dipole and non-dipole components.
Other excursions in the intermediate field models are handled in a similar way.We include excursion events at 34 ka (Mono Lake), 41 ka (Laschamp) and 65 ka (Norwegian-Greenland Sea) from GGFSS70, in addition to a record of the Laschamp event from GGF100k.Reversals are treated differently because we lack sufficient detail at small x to permit a least-squares fit.Instead, we identify the start of a reversal (at time t = t 0 ) by choosing a local maximum with x(t) ≥ 〈x〉.Setting x = 0 at the time of the reversal, t = t 1 , gives a trend where the window length is w = t 1 t 0 .Reversal trends for six events are computed from PADM2M (group A), together with a second estimate for the Matuyama-Bruhnes reversal (group B) from model GFFMB (Mahgoub et al., 2023).
Geochemistry, Geophysics, Geosystems  Note.Trends are computed between the start and end times in ka.The width of the trend is the difference between the start t 0 and end t 1 .The dipole moment x(t) at t 0 and t 1 is given in units 10 22 A m 2 .The absolute value of the trend b and the theoretical standard deviation σb are reported in units 10 22 A m 2 kyr 1 .excursions and reversals to occur too often.We note that our assessment of σb is largely based conditions of stable polarity.The linear stochastic model (and hence σb ) is restricted by construction to a stable polarity.Similarly the nonlinear stochastic model is construct from paleomagnetic models that spend the majority of time in a stable polarity.Estimates recovered for γ and D are heavily weighted by these stable conditions.A reasonable interpretation of Figure 6 is that excursions and reversals are a consequence of unusual, but statistically plausible fluctuations in the dipole.There may be a hint in model GGFSS70 that steeper trends develop at small x, suggesting a change in the mechanism for fluctuations when the dipole is weak.Such a behavior would be consistent with higher values for |b|/ σb in GGFSS70, particularly when these trends include low values for x.Alternatively, we might assess the Laschamp and Norwegian-Greenland Sea excursions in GGFSS70 in terms of the higher trend variance in this field model.Model GGFSS70 yields estimates for σ b (w) that are roughly by a factor of 1.5× higher than the stochastic model.Excursions in GGFSS70 do not exceed 3σ b events when the standard deviation is evaluated using the same geomagnetic field model.
What should we expect for excursions and reversals in the stochastic model?We explore this question using the nonlinear model to enable polarity reversals.A theoretical expression for trends in the nonlinear model is not practical, so we run an ensemble of 400 models with an initial condition x(0) = 〈x〉.To be specific we define excursions by the condition x ≤ 2.5 × 10 22 A m 2 , whereas reversals occur when x ≤ 0, independent of whether the solution subsequently returns to the original polarity or stays in the reversed polarity.The start time is identified by the first local maximum above the average (i.e., x ≥ 〈x〉).Excursions are recovered from solutions filtered with a 300-year running average to simulate the smoothing in the intermediate field models.Reversals are assessed after applying a 4-kyr running average to represent the lower resolution of long-time field models.
Individual estimates for excursions and reversals cluster around lines with a slope of approximately 1 (see Figure 7).Most events in the ensemble have Δx set by our event definitions, so the trend is mainly determined by the duration or time window.In other words, we expect b to vary as w 1 , in agreement with the nominal 1 slope.
Excursions and reversals are associated with unusual trends because most values are well above σb and occasionally above 3 σb .The average trend and duration for each event type is represented by a green star in Figure 7.
Excursions occur with an average duration of 3.7 kyr and an average magnitude of |b|/ σb = 2.3.By comparison, reversals occur over an average time interval of 21.8 kyr with an amplitude |b|/ σb = 2.5.These values are not too different from values recovered from the geomagnetic field models, particularly when b and σ b are based on the same field model.Excursions in the field models have durations between 1.2 and 3.8 kyr, whereas reversals have durations between 18 and 45 kyr.Both types of events have observed trends |b|/ σb ≈ 2 to 3. Similarities in the ratios (|b|/ σb ≈ 2 to 3) also tell us there is not much difference in the probability of excursions and reversals.Instead we distinguish between excursions and reversals solely on the basis of the duration of the trends.While the probability of both events may be comparable (based on |b|/ σb ), the recurrence time of excursions and reversals can be very different due to differences in the durations.

Recurrence Times for Excursions and Reversals
A well-established framework exists to estimate average recurrence times (e.g., Gardiner, 1985).The calculation is often described as a mean first passage time.The idea is to start with a solution inside some interval (say [x 0 , x 1 ]) and compute the mean time for the solution to leave this interval.We adopt an initial condition x(0) = 〈x〉 and an interval [0, ∞] to define a reversal.By this definition the reversal occurs once x(t) reaches x = 0.An excursion is defined by the interval [x l , ∞], where x l defines the threshold for a large angular deviation in the VGP.In practice we could take x l from the geomagnetic field models.The mean time T(x) for the dipole moment to exit the prescribed interval is governed by Gardiner (1985) v where x refers to the initial condition; different T are computed for different initial conditions.The usual boundary conditions, T(x 0 ) = T(x 1 ) = 0, are called absorbing boundary conditions.We can interpret T as the average time for an ensemble of solutions to leave the interval.Absorbing boundary conditions correspond to removal of solutions once they reach the edges of the interval.In practice we can shift an absorbing boundary condition at x → ∞ to a large finite value for x 1 without substantially modifying the solution for T. Alternatively we can replace the absorbing condition with which is usually called a reflecting boundary condition because solutions are not removed at the upper edge of the interval.Equation 13is solved numerically on a finite grid with an absorbing boundary condition at the lower edge of the interval (x = x 0 ) and a reflecting boundary condition x = x 1 .The choice of boundary condition at x 1 has little influence on the solution once the finite value of x 1 is large enough.
A potential U(x) is commonly introduced to represent the drift term in Equation 13.We define the potential using where the prime refers to differentiation with respect to the argument.Adopting the nonlinear drift term from Equation 3 gives a double potential well (see Figure 8).Superimposed on the potential well in Figure 8 is the steady-state probability distribution which is computed under the assumption of constant D. Solutions congregate near one of the two potential wells.Occasionally a solution over the barrier ΔU at x = 0 and enters into the opposite polarity.From our nonlinear drift term we have for the height of the barrier.Kramers (1940) used this representation to derive an approximate expression for T in a double potential well.His solution relies on the assumption that ΔU/D is large.This condition is only marginally satisfied for the parameter values in the stochastic model, motivating our more general approach using a numerical solution of Equation 13.Kramers formula in Equation 18gives T = 249 kyr, corresponding to reversal rate r = T 1 of approximately 4 Myr 1 .By comparison, the numerical solution of Equation 13gives T = 147.8kyr when x(0) = 〈x〉.The corresponding reversal rate is r = 6.7 Myr 1 .Only half of the solutions that reach x = 0 are expected to proceed into the opposite polarity; the other half return to the original polarity.Reversals in the marine magnetic anomalies have a wellestablished reversed state, so the expected detection rate for reversals in the stochastic model is r = 6.7/ 2 = 3.4 Myr 1 .By comparison the observed reversal rate is r = 4.2 Myr 1 over the past 10 Myr, based on the chronology of Ogg (2012).A simple way to match the observed reversal rate with the stochastic model is to allow a modest 17% increase the diffusion term D. (This change is not sufficient to account for the higher σ b (w) in models GGFSS70 and HLM.OL1.A1).Predictions for the recurrence time of excursions depends on the choice of x l .When x l = 2.5 × 10 22 A m 2 and x(0) = 〈x〉, we predict a recurrence time of 38 kyr.A Mono Lake excursion with x l = 3.21 × 10 22 A m 2 should have an average recurrence time of 23 kyr.

Discussion and Conclusions
Generation of the dipole field is often attributed to helical convection in the liquid core through a mechanism known as the α effect (Moffatt, 1978).Convective flow lifts and twists the magnetic field to produce electric currents that reinforce the dipole.We envision a process that involves a large collection of convective eddies, each making constructive or destructive contributions to the dipole field.Averaging these contributions in time defines a mean generation rate, which balances the average rate of ohmic decay.Fluctuations in field generation are represented in the stochastic model as a random component with amplitude D. These fluctuations produce time variations in the dipole trends over a broad range of timescales.Predictions for σ b (w) from the linear stochastic model are similar in shape to the trends computed from the geomagnetic field models.Quantitative agreement is found for the trends from CALS10k.2, GGF100k, PADM2M and SINT2000.The other models (HLM.OLF.A1 and GGFSS70) have trends shifted to a higher level of variability without much change in the shape of σ b (w).
Better agreement with these two models could be achieved in the stochastic model by increasing D by a factor of 1.5 2 (or 2.25).
An infrequent series of fluctuations produce excursions and reversals in the stochastic model without requiring any fundamental differences in the underlying mechanisms.Maximum trends from PADM2M and SINT2000 occasionally reach or exceed the 3σ b level, based on the standard deviation computed from the fluctuations in these models.Maximum trends from the other geomagnetic field models give comparable results, relative to their standard deviations.Individual trends for reversals in PADM2M and GGFMB have amplitude |b|/ σb ≈ 2, which indicates that these events are unusual, but not implausible given the maximum trends noted above.
Excursions occur more frequently because the change in x for this type of event is smaller.We have previously shown that excursions like the Mono Lake event should occur every 23 kyr on average.On the other hand, an excursion that briefly reverses polarity (like the Laschamp event in GGFSS70) is indistinguishable from a reversal in the context of the stochastic model.The rapid decline into the Laschamp excursion has a trend of |b|/ σ b = 2.7 when evaluated using the observed σ b in GGFSS70.Applying the same elevated σ b in our interpretation of reversals (by introducing a 2.25× increase in D) would lower the reversal trends from |b|/σ b ≈ 2 to |b|/σ b ≈ 1.3 (Recall that σ 2 b in Equation 10 is proportional to D, so the change in |b|/σ b is proportional to D 1/2 ).We might expect the lower value for reversal trends to cause more frequent reversals.Indeed a solution of Equation 13 with the revised D shortens the reversal recurrence time to values less than one half the observed recurrence interval.This discrepancy raises the question of whether the higher variance in models GGFSS70 and HLM.OL1.A1 can be reconciled with the observed behavior of reversals.
A resolution is possible with one additional change to the parameter values.In fact, we can reconcile the trends from HFM.OL1.A1, GGFSS70, as well as SINT-2000 and PADM2M by increasing γ by a factor of 2 (in addition to the 2.25× increase in D).This change keeps the reversal trends in the range |b|/σ b ≈ 2 and gives a reasonable estimate for the reversal frequency.However, this change also shortens the effective dipole decay time by a factor of two, implying a larger turbulent magnetic diffusion (Holdenried-Chernoff & Buffett, 2022).In other words, the higher variance for trends in HFM.OL1.A1 and GFFSS70 requires larger convective velocities in the core to match the other paleomagnetic constraints.Alternatively, we might retain the original parameter values for the stochastic model (to match CALS10k.2,GGF100k, the two long-time field models and the reversal rate) by attributing the steep trends in GGFSS70 at low x to uncertainties in the dipole component when the multipolar components are relatively large.A related argument might apply to the higher trend variance in HFM.OL1.A1; the more restricted spherical harmonic expansion may cause multipole fluctuations to project into the dipole component.Identifying the origin of the offset in the variability of dipole trends in the short-and intermediatetime geomagnetic field models would help to resolve these questions.
We conclude with a brief comment about the current historical trend.The normalized trend over the past 180 years is |b|/ σb = 2.08, which is roughly comparable to trends into a reversal.However, this correspondence in |b|/ σb does not imply that a reversal is likely in the near future.We reach a reversed state (x < 0) at the current decline rate, b = 4.54, by sustaining this trend for the next 2 kyr.Increasing w from 0.18 to 2 kyr decreases σb from 2.12 to 0.61.As a result, the trend required to produce a reversal in the next 2 kyr has an amplitude |b|/ σb = 7.4.Such a high value makes the prospect of an imminent reversal is very unlikely.Similar conclusions have been reach by several independent approaches (Brown et al., 2018;Gwirtz et al., 2021Gwirtz et al., , 2022;;Morzfeld et al., 2017).The question of the next reversal might be explored by adopting a more likely trend with a longer duration w.Setting |b|/ σb (w) = 3 and taking present-day dipole x = 7.6 × 10 22 A m 2 (Huder et al., 2020) gives w = 14 kyr for the duration needed to reach the reversed state.A somewhat lower |b|/ σb would be more probable, but it would also imply a longer duration to achieve a reversal.An independent calculation predicts a 2% probability of reversing in the next 20 kyrs (Buffett & Davis, 2018).All of these results are broadly consistent and make a strong case against an impending magnetic reversal.

Figure 1 .
Figure1.Simulation of axial dipole moment x(t) as a function of time t using a stochastic model.A polarity reversal at t = 130 kyr is followed by several sign changes before x(t) settles into a stable reversed polarity at t = 150 kyr.A brief reversal at t = 180 kyr would likely be interpreted as an excursion in the paleomagnetic record because the dipole returns to the initial (reversed) polarity.

Figure 2 .
Figure 2. Influence of random error and smoothing on dipole trends.Random error increases σ b (w) at short w, whereas smoothing decreases σ b (w) at short w.Small deviations of the theory (solid line) from the numerical solution occur when a small number of discrete values of x(t) are used in the least-squares fit for b (see text).

Figure 3 .
Figure 3. Dipole trends from paleomagnetic models PADM2M and SINT-2000 (in units 10 22 A m 2 kyr 1 ) (a) The standard deviation of the dipole trend, σ b (w), is compared with the theoretical estimate σb (solid line).Departures from the theory at small w are attributed to the effects of smoothing in the sedimentary record.(b) Maximum trends are compared with 2σ b and 3σ b theoretical predictions.Individual maxima can reach or exceed 3σ b .

Figure 4 .
Figure 4. Standard deviation of dipole trends from (a) the intermediate-and (b) the short-time geomagnetic field models (in units 10 22 A m 2 kyr 1 ).In both cases the standard deviation of the dipole trend, σ b (w), is compared with a theoretical estimate σb (solid line), and the 2 and 3 sigma levels (dashed lines).

Figure 6
Figure 6 shows the absolute values of event trends compared with the theoretical prediction, σb , together with the 2σ b and 3σ b levels.The recent historical trend is included for comparison.Most of the event trends lie between |b|/ σb ≈ 2 and 3. Reversals are typically associated with |b|/ σb ≈ 2. Trends for the Mono Lake and Laschamp excursion in the GGF100k are below 3σ b , whereas the trends for the Norwegian-Greenland Sea and the Laschamp excursion in GGFSS70 are close to 4σ b .Accounting for only the initial part of the trend into the Laschamp excursion in GGFSS70 reduces the trend to 2.1σ b (see arrow in Figure 6).Trends into excursions and reversals are unusual because they exceed σb (w).Typical values for |b|/ σb during excursions and reversals are between 2 and 3. Lower values for |b|/ σb would more frequently and might cause

Figure 5 .
Figure 5. Axial dipole moment x(t) from the intermediate field models near the Laschamp excursion at 41 ka.The field models are generally above 〈x〉 prior to the excursion, so we identify starting points for GGF100k and GGFSS70 when x(t) is above the time average (points A and B).The minimum dipole moment occurs at A′ and B′ in GGF100k and GGFSS70.An intermediate point B″ is identified in GGFSS70 when x < 2.5 × 10 22 A m 2 .

Figure 6 .
Figure 6.Dipole trends for observed excursions and reversals.Excursions include Mono Lake (34 ka), Laschamp (41 ka) and Norwegian-Greenland Sea (65 ka) from GGFSS70 (group B) and the Laschamp excursion from GGF100k (group A).Restricting the Laschamp trend to the initial part of the record reduces the trend and shortens the window (see text).Reversals are computed from PADM2M (group A) and GGFMB (group B).The historical trend is also shown for comparison.

Figure 7 .
Figure 7. Ensemble of dipole trends for excursions and reversals from the nonlinear stochastic model.The prediction, σb (w), is based on the linear model.Individual estimates for |b| scatter around lines with a slope of roughly 1 (see text).Estimates for |b|/ σb occasionally lie above three.The average slope and duration are indicated by green stars.

Figure 8 .
Figure 8.A double potential well U(x) and steady-state probability p s (x) for the stochastic model.The most probable state (i.e., max( p s )) is located near the base of the potential wells.Random fluctuations allow the dipole moment to climb over the potential barrier ΔU at x = 0 to produce a reversal.

Table 1
Dipole Trends Into Excursions and Reversals