Simple stochastic model for geomagnetic excursions and reversals reproduces the temporal asymmetry of the axial dipole moment

Significance

The geodynamo, the process that produces and controls the evolution of the Earth’s magnetic field, is still not fully understood. Despite the success of current numerical geodynamo models in reproducing some geomagnetic observations, their high complexity and computational requirements hinder their use for certain studies. Here, we present a simple model based on random fluctuations that is able to reproduce the temporal asymmetry of the dipolar strength of the field observed during polarity reversals. This behavior arises due to the existence of two parts of the model that are energetically unbalanced, imposing a preferred direction of energy transfer, and we propose that the same mechanism would be the cause of the temporal asymmetry of the geodynamo.

Abstract

We present a simple model for the axial dipole moment (ADM) of the geomagnetic field based on a stochastic differential equation for two coupled particles in a biquadratic potential, subjected to Gaussian random perturbations. This model generates aperiodic reversals and excursions separated by stable polarity periods. The model reproduces the temporal asymmetry of geomagnetic reversals, with slower decaying rates before the reversal and faster growing rates after it. This temporal asymmetry is possible because our model is out of equilibrium. The existence of a thermal imbalance between the two particles sets a preferential sense for the energy flux and renders the process irreversible.

The magnetic field of the Earth is generated in the outer core of our planet by a complex process called geodynamo. It is based on the movement of the electrical conducting fluid of the outer core powered by convection and influenced by the Coriolis force associated with the Earth’s rotation, giving rise to complex and chaotic dynamics. Because it is not possible to obtain direct measurements of relevant parameters (internal magnetic field, fluid velocity, temperature, composition, etc.) of the outer core deep below the core–mantle boundary (CMB), the exact mechanisms that trigger a field reversal are still poorly known (1, 2).

The study of the Earth’s magnetic field can be addressed from different perspectives. On the one hand, the geodynamo state is constrained by available magnetic field values above the Earth’s surface obtained by direct measurements or from paleomagnetic proxies such as archeological materials, volcanic rocks, or sedimentary records. These data are used to develop geomagnetic and paleomagnetic reconstructions that describe the evolution of the magnetic field. On the other hand, numerical and experimental models can be used to reproduce the outer core dynamics. Experimental models have been developed by means of metallic fluids, such as gallium (3) or sodium (4), rotating within a spherical shell. A different kind of model is provided by numerical simulations. Magnetohydrodynamical models simulate the dynamics of the core fluid from fundamental laws of electromagnetism, hydrodynamics, and thermodynamics. Numerical simulations provide detailed and useful information for investigating the geodynamo, but their computational cost is very high, requiring simulations to be run under some unrealistic conditions. Nevertheless, successive works have been pushing the parameters toward more realistic values, and in recent years, some breakthrough studies have reached a realistic force balance (5–7). Alternatives to those complex models are conceptual numerical models that represent in a simple manner the most important ingredients of the system in order to understand the main processes and magnitudes that control the geodynamo. The double-disk dynamo proposed by Rikitake (8), which showed a chaotic behavior and underwent stochastic reversals, was a historic model of this kind. Since then, several simple models have been studied including both fully deterministic models (8, 9) and models containing stochastic terms (10–14).

A relevant feature of the geomagnetic field behavior during reversals is the asymmetric behavior of the axial dipole moment (ADM). That is, reversals are preceded by a slow decrease of the ADM; then, the directional change takes place, and it is followed by a faster ADM recovery. Although the first description of asymmetric ADM variations during reversals (15) was considered to be an artifact of viscous remanent magnetization (16, 17), the sawtooth shape has been amply confirmed in relative paleointensity sedimentary records, in lava flows sequences (18, 19), and to a lesser extent, by ¹⁰Be records of the Bruhnes–Matuyama reversal (20). The analysis of Valet et al. (21) of the Sint-2000 stack revealed that reversals are preceded by a slow ADM decay that takes about 60 to 80 ky and are followed by a fast recovery of the ADM in only 10 to 20 ky. A similar asymmetry between the growing and decreasing rates has also been observed during stable periods (22, 23), suggesting that the underlying mechanisms may be active also outside transitional events.

From the point of view of thermodynamics, a temporally asymmetric behavior is a typical signature of dissipative processes in a system out of (thermal) equilibrium. In the case of the geodynamo, dissipation occurs through heat transfer from the Earth’s core to the mantle via the release of latent heat in the inner core boundary, by convection through the outer core, and by resistive losses of internal electric currents. Indeed, there is a profound connection between energy dissipation and irreversibility: the entropy production of a system is a measure of the difficulty of statistically distinguishing forward vs. backward trajectories (24, 25). Following this idea, we have tried to reproduce the asymmetric behavior of reversals by considering the ADM as the sum of two axial magnetic moments from two interacting components of the geodynamo characterized by different effective temperatures, which measure the strength of random perturbations. This thermal unbalance establishes a definite direction of the energy flow within the system and thus, irreversibility. This model can be used to improve our knowledge about the geodynamo and shed light on the dynamics of underlying processes that originate the observed characteristics at the Earth’s surface. Conceptual models, despite their simplicity or maybe because of it, are usually helpful to identify general mechanisms and minimal ingredients for a relevant system behavior, which may be later implemented or analyzed in more sophisticated models. This model may be applied not only to the Earth’s magnetic field but also, to other planetary dynamos that work under different regimes.

Materials and Methods

The Model.

We propose a simple stochastic system that mimics the behavior of the ADM of the geomagnetic field. This system is formed by two coupled and overdamped Brownian particles moving in a double-well potential. The two particles are characterized by different friction parameters and are affected by random forces simulated by independent Gaussian white noises of different magnitude. Furthermore, the two particles are coupled by an attractive force proportional to their distance, which tends to maintain them close to each other.

The system dynamics is thus governed by the following stochastic differential equations:

$$\begin{array}{c}\dot{x_1}=\frac{1}{{\gamma }_1}\left[-V^{\text{'}}\left(x_1\right)+K\left(x_2-x_1\right)+\sqrt{2k_B{T}_1{\gamma }_1}\hspace{0.17em}{\zeta }_1\left(t\right)\right]\\ \dot{x_2}=\frac{1}{{\gamma }_2}\left[-V^{\text{'}}\left(x_2\right)+K\left(x_1-x_2\right)+\sqrt{2k_B{T}_2{\gamma }_2}\hspace{0.17em}{\zeta }_2\left(t\right)\right]\end{array}\},$$ [1]

where x is the position of the particle, γ is the friction coefficient, V′ is the derivative of the potential (V) with respect to the position x, K is a coefficient that expresses the strength of the attraction force (much like the elastic constant of a spring), k_B is the Boltzmann constant, T is the temperature of the particle, and ζ(t) is a random variable that follows a normal distribution of zero mean and unit SD. Note that the temperature controls the intensity of the fluctuations through the factor $\epsilon =\sqrt{2k_{B}T\gamma }$. Both ζ₁ and ζ₂ are time uncorrelated and independent, with

$$\langle {\zeta }_1\left(t\right)|{\zeta }_2\left(t\right)\rangle =0$$ [2]

$$\langle {\zeta }_i\left(t\right)|{\zeta }_i\left(t\text{'}\right)\rangle =\delta \left(t-t\text{'}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},$$ [3]

where δ is the Dirac impulse. The potential has been estimated from the distribution of virtual axial dipole moment (VADM) values (red histogram in Fig. 1A) calculated from the Sint-800 record (26) and forced to be symmetric. To do that, we have considered the limit when the spring constant K is very high, forcing both particles to move together like a single particle, and applied the equation for the equilibrium probability density

$$\rho \left(x\right)={\rho }_0·e^{\frac{-1}{k_{B}T}V\left(x\right)}$$ [4]

Fig. 1.

(A) Frequency of VADM values as recorded by Sint-800 (red histogram) in terms of the probability density function (PDF), potential for a single Brownian particle to obtain the previous frequencies as its equilibrium distribution (blue circles), and biquadratic fitting of the potential (blue curve). (B) Scheme of our model: both particles are affected by the same potential and interact with an attractive force proportional to distance.

of a Brownian particle immersed in a fluid and affected by a potential (27), where ρ₀ is a normalization constant (to set the integrated probability to unity), k_B is the Boltzman constant, T is the temperature of the fluid, and V(x) is the potential. In this equation, the potential V is normalized by the thermal energy (k_BT), meaning that higher thermal fluctuations can push the particle farther into zones of high potential. In our case, this temperature is a measure for the typical amplitude of energy fluctuations, rather than a true temperature. The potential is then obtained from the probability function using

$$V\left(x\right)=-k_{B}T\text{ln}\left(\rho \left(x\right)\right)+k_{B}T\text{ln}\left({\rho }_0\right).$$ [5]

The second term on the right-hand side of Eq. 5 is irrelevant in terms of dynamics since it only defines a reference energy level. The potential is thus determined by ρ(x) except for a multiplicative factor (k_BT). This factor can be set to unity without loss of generality since other terms in Eq. 1 can be normalized by it. The obtained potential (blue circles in Fig. 1A) is then fitted to a biquadratic polynomial of the form

$$V\left(x\right)=a{x}^4+b{x}^2+c=0$$ [6]

in order to force symmetry on the double-well potential and to obtain a continuous and differentiable function. Fig. 1B shows a sketch of the model. Particle 1, referred to as the “hot particle,” is characterized by a smaller friction coefficient and stronger fluctuations, while particle 2, the “cold particle,” will have a higher friction coefficient and weaker fluctuations. The ADM value is calculated from the position of both particles using a weighted mean

$$ADM\left(t\right)=w·x_1\left(t\right)+\left(1-w\right)·x_2\left(t\right)$$ [7]

with 0 < w < 1. Realistic ADM behaviors are obtained only if the weight 1 − w of the cold particle is larger than that of the hot particle: that is, w < 0.5. For demonstration purposes, we have chosen w = 0.2. This value will be maintained through the text. Note that w has no influence on the system dynamics since the ADM series is calculated from the simulation output.

Four-States Analysis.

In order to shed light on the role of the temperature difference between the two particles, we have analyzed the system using transition rates between different effective or macroscopic states of the model. Examination of the solutions of Eq. 1 shows that the phase space of the model can be effectively divided into four macrostates depending on the sign of the position of each particle (Fig. 2 and Dataset S1). Normal and reversed stable states are represented by macrostates 1 and 3, respectively. The other two (macrostates 2 and 4) are metastable and can be seen as an excursional state that is required to be able to fully reverse.

Fig. 2.

Partition of the phase space of the model into four macrostates (labeled from 1 to 4) as a function of the sign of the positions x₁ (hot particle) and x₂ (cold particle). Gray arrows represent the possible transitions.

Following the classical expression for the Kramers problem (28), the transition rate from one macrostate to another one will depend on the minimum energy increase required for the transition (i.e., the energy barrier). For instance, in the case where one particle stays in the bottom of a potential well while the other jumps over the central barrier, the minimum energy increase would coincide with the height of the barrier plus the spring energy increase due to the separation of the two particles. This gives the transition probabilities

$$k_{12}=k_{34}=\frac{A}{{\gamma }_1}·e^{\frac{-1}{k_B{T}_1}\left(E_b+\frac{E_s}{2}\right)}$$ [8]

$$k_{14}=k_{32}=\frac{A}{{\gamma }_2}·e^{\frac{-1}{k_B{T}_2}\left(E_b+\frac{E_s}{2}\right)}$$ [9]

$$k_{23}=k_{41}=\frac{A}{{\gamma }_2}·e^{\frac{-1}{k_B{T}_2}\left(E_b-\frac{E_s}{2}\right)}$$ [10]

$$k_{21}=k_{43}=\frac{A}{{\gamma }_1}·e^{\frac{-1}{k_B{T}_1}\left(E_b-\frac{E_s}{2}\right)},$$ [11]

where E_b is the height of the central barrier (from the bottom of the wells), E_s is the energy of the spring when the particles are in the bottom of different wells, and A is a constant factor.

The sequence of macrostate transitions 1 → 2 → 3 → 4 → 1 implies that the hot particle, which can move more easily and contributes by a smaller fraction to the total ADM, is jumping the barrier first and “helping” then the cold one to complete the transition. We can compare the probabilities of transitions associated to that sense and its opposite:

$$\frac{k_{\mathrm{\circlearrowleft }}}{k_{\mathrm{\circlearrowright }}}=\frac{k_{12}{k}_{23}{k}_{34}{k}_{41}}{k_{14}{k}_{43}{k}_{32}{k}_{21}}=e^{-2\frac{E_s}{k_B{T}_1}+2\frac{E_s}{k_B{T}_2}}=e^{\frac{2E_s}{k_B}\left(\frac{1}{T_2}\hspace{0.17em}-\hspace{0.17em}\frac{1}{T_1}\right)}.$$ [12]

As shown in Eq. 12, the temporal asymmetry does not depend on the friction coefficients but only on the particle temperatures in the way that if both temperatures are equal, the system exhibits temporal symmetry, but otherwise, asymmetry arises. Since T₂ < T₁, the sense 1 → 2 → 3 → 4 → 1, where the hot particle reverses first, is favored.

Furthermore, Eq. 12 clearly shows that “detailed balance”—the balance between transitions in both senses—is broken when temperatures are different. Our result is thus consistent with the observation that dissipation, and hence, irreversibility, is the result of breaking detailed balance (29).

Results

The stochastic differential equations (Eq. 1) have been integrated following a fourth-order Runge–Kutta scheme (the model code has been developed in MATLAB and is available in Dataset S2). Fig. 3 shows some results from a simulation with the following parameters: γ₁ = 1, γ₂ = 2, k_BT₁ = 2, k_BT₂ = 1, K = 0.1 (that is, ε₁ = 2, ε₂ = 1). We remind here that our goal is not the precise determination of model parameters but the study of the different behaviors that the model may exhibit. The ADM (Fig. 3B) oscillates around its equilibrium value with abrupt collapses and might also reverse sign depending on the detailed movement of the particles (Fig. 3A and Movie S1). We consider that the model has produced a reversal when both particles change sign and reach the bottom of the opposite well. After the reversals have been detected and thus, the polarity of the series is well defined at any time, excursions are defined as the periods of opposite sign of the position of a single particle that reaches the bottom of the opposite well when it does not develop into a reversal.

Fig. 3.

Simulation results and comparison with observations. (A) Position of hot (red) and cold (blue) particles and signed ADM (black). The threshold for considering a reversal or excursion (ADM value at the bottom of the potential well) is also represented (dotted lines). (B) Unsigned ADM for the same simulation fragment. Reversals (R; orange) and excursions (E; green) are highlighted. (C) Fragment of a simulated ADM series at a higher scale to compare with (D) observed series of VADM (Sint-2000) (21). (E) Behavior of the ADM in the proximities of a reversal: stack of several reversals of the model (color lines) and mean behavior (black line) with error band (±1 SD). Time is relative to the moment of the reversals (set at t = 0). (F) Same with observed VADM series from Sint-2000. Adapted from ref. 21.

Reversals and excursions are distributed aperiodically, in agreement with the behavior of the geomagnetic field (Fig. 3 C and D). Furthermore, the parameter values used to generate the solution shown in Fig. 3 give an excursions/reversals ratio (R_e/r) of about seven, which is of the order of magnitude of the number of excursions identified in the current chron (period between successive reversals) (30).

Concerning the ADM variations near reversals, we obtain a pattern of slow decrease and fast recovery similar to that observed by Valet et al. (21) (Fig. 3 E and F). In order to quantify the asymmetry of that pattern, we define an asymmetry factor as follows. First, during each simulation, we detect all the reversal times with the above definition of reversal. For each reversal, we take two independent time windows of 150 time units before and after it. Then, we compute the area under the ADM curve after subtracting a reference value identified with the ADM corresponding to the bottom of the potential well. The asymmetry factor is the ratio between the areas before and after the reversal, so that an asymmetry factor greater than one means that the previous ADM decrease is slower than the subsequent recovery. We do this for every reversal, and finally, we calculate the mean value.

In the case of the real ADM data, as apparent from Fig. 3F, the reference value for the mean ADM in consecutive chrons is different, and this has been taken into account in the asymmetry computation. Furthermore, due to the low number of available reversals, we use directly the mean behavior for the area computation to avoid a too noisy result (SI Appendix, Supplementary Text has details).

With this definition, the mean asymmetry factor of the simulations shown in Fig. 3 is 1.36 ± 0.66, in agreement with the estimation of 1.51 from the Sint-2000 series. We have to point out, however, that the value derived from the simulations depends on the chosen weight factor, w, in Eq. 7.

Simulations with different parameter sets have been generated in order to assess the effects of parameter choice on the excursion/reversal frequency and on the asymmetry parameters (graphic results can be found in Dataset S1, and numerical data are in Dataset S3). The only fixed parameter was γ₁ = 1. We can do that without loss of generality since varying this parameter is equivalent to rescaling the time (SI Appendix, Supplementary Text). γ₂ controls how easily the cold particle can move with respect to the hot one. Higher values of γ₂ hinder the jumps of the cold particle, leading to rarer reversals and a higher R_e/r. The temperatures T₁ and T₂ set the energy of the system, which controls the frequency of jumps over the central maximum. If the hot particle gets hotter, it raises R_e/r, while if it is the cold particle that gets hotter, reversals are more favored than excursions and R_e/r decreases. The strength of the interaction between the particles is established by K. High values of this parameter cause the particles to move together, which hinders the occurrence of excursions. Furthermore, higher K values tend to render the reversals more symmetric (SI Appendix, Fig. S2). Finally, although w is not a dynamic parameter and does not affect R_e/r, it affects the shape of the ADM pattern. If w < 0.5, the ADM drops during the excursions (and the first stage of reversals). If w = 0.5, the ADM would vanish, and if w > 0.5, the ADM would change its sign during excursions.

Discussion

Here, we have considered a very simple model that can reproduce one of the most elusive characteristics of the field that other Brownian motion-based models presented so far (14, 31) have not been able to reproduce: the asymmetry of the dipole moment variation during reversals.

If we consider only one particle within the potential (Movie S2 and SI Appendix, Fig. S3), we would obtain a model similar to those analyzed in other works (14, 31), which do not show any trace of asymmetry. Models with several identical particles, such as the “domino” model of Mori et al. (11), made of a ring of N interacting Brownian dipoles also exhibit a symmetric behavior since there are no differences between elements that can provide a preferential flux of energy or moment. Finally, more complex models that are not based on a Langevin description of random perturbations, as in Eq. 1, that are able to generate asymmetric patterns (9, 10, 32) include two or more different elements. Our model incorporates in the simplest manner two interacting and nonequivalent elements in the form of two particles with different temperatures.

The diffusion term, which is proportional to the amplitude of the random noise, does not depend on the ADM value, to which it acts as an additive noise term. This consideration has been found to be quite accurate for the geodynamo from the analysis of both VADM series and magnetohydrodynamical models (14, 31, 33).

For certain model parameter combinations, the excursional state appears to be a precursor stage of reversals, in which one particle jumps over the potential barrier, while the other is still in the original potential well (states 2 and 4 in Fig. 2). A reversal eventually occurs when the other particle jumps as well (states 1 and 3), helped by the attractive force between the two. The excursional and stable states are characterized by different mean ADM values. The existence of the excursional state has an effect on the overall ADM distribution, which shows a slight secondary maximum at about 3·10²² Am² for some parameter values (Dataset S1, p. 14). It is interesting to compare this feature with the results obtained by Heller et al. (34, 35), who found a bimodal distribution of virtual dipole moment values when they analyzed the last 320 My. Wicht and Meduri (36) also suggest the existence of two different states of low and high ADM, indicating that the low-ADM state is required for reversals and “grand excursions” to take place.

From a geophysical point of view, it is worth asking what each of the elements (particles) considered in our model represents. We have to remind here that the model temperatures are just parameters that control the intensity of the fluctuations; therefore, we cannot look for real “hot” and “cold” regions of the outer core. The following scenarios are proposed.

It has been suggested that the part of the outer core that falls within the tangent cylinder (TC; the imaginary cylinder parallel to the rotation axis and tangent to the inner core) is characterized by a strong and steady vortex quite isolated from the rest of the outer core (37), much like the polar vortices in the atmosphere, although other authors argued that the dynamics inside the TC is more complex (38). The inner core is completely contained inside the TC region so that it can directly interact only with the outer core part located inside the TC. The external magnetic field has been found to be closely related to the outer core dynamics within the TC (39). Furthermore, the inner core can only change its magnetic field by diffusion, so it would produce no fluctuation by itself, contributing to the “inertia” of that part of the system. In this case, the cold particle of our model might be associated with the TC interior (including the inner core) with lower temperature (fluctuations) and a higher friction (resistance to changes), while the hot particle would be the part of the outer core outside the TC. This interpretation is also supported by the differences in space and time spectra of the magnetic energy inside and outside the TC. It has been found that the inner region exhibits higher-energy fluctuations at lower frequencies with respect to the region outside the TC (6), implying that the TC region is more stable. This scenario is also compatible with the hypothesis of Hollerbach and Jones (40, 41) that the inner core acts to stabilize the magnetic field polarity and thus, opposes the reversal process.

Other possible scenarios might be related to temperature differences within the outer core, imposed by the boundary conditions at the CMB or at the inner core boundary. Some studies have pointed out that thermal and/or compositional variations in the CMB can regulate the geomagnetic field reversals (42). Accordingly, the helical cells would have different fluctuation levels if they are connected to a CMB region of greater or lesser heat flow. In this case, the cold particle would account for the zone where the convection is stronger, leading to higher field values and a more stable structure (less fluctuating).

It cannot be excluded that the difference in fluctuation levels may be related to the hemispheric asymmetry of the inner core. It has been found that there exists a zone of predominant solidification with the associated release of latent heat and light elements that increments the buoyancy and enhances the convection, while in the opposite hemisphere, there is little solidification or even melting, which generates a stratified layer that hinders the convection (43, 44).

Finally, the two particles in our model might represent different circulation patterns or field structures in the core, rather than discrete physical regions. In this scenario, the cold particle would be related to the largest structures, which have a major influence on the overall state, while the hot particle would represent the small patterns, which can more easily fluctuate.

In all the above cases, temporal asymmetry arises from the existence of irreversibilities in the system, which can be formulated in terms of general thermodynamic arguments. Any model displaying the sawtooth feature of the ADM evolution during reversals needs to incorporate a directed current or flow between two or more interacting parts, which arises from an out of equilibrium constraint.

Data Availability

All study data are included in the article and/or supporting information.