The time evolution of the strength of the Earth's virtual axial dipole moment (VADM) is analyzed by relating it to the Fokker-Planck equation, which describes a random walk with VADM-dependent drift and diffusion coefficients. We demonstrate first that our method is able to retrieve the correct shape of the drift and diffusion coefficients from a time series generated by a test model. Analysis of the Sint-2000 data shows that the geomagnetic dipole mode has a linear growth time of 13 to 33 kyr, and that the nonlinear quenching of the growth rate follows a quadratic function of the type [1-(x/x0)^2]. On theoretical grounds, the diffusive motion of the VADM is expected to be driven by multiplicative noise, and the corresponding diffusion coefficient to scale quadratically with dipole strength. However, analysis of the Sint-2000 VADM data reveals a diffusion which depends only very weakly on the dipole strength. This may indicate that the magnetic field quenches the amplitude of the turbulent velocity in the Earth's outer core.
Deep Dive into An analysis of the fluctuations of the geomagnetic dipole.
The time evolution of the strength of the Earth’s virtual axial dipole moment (VADM) is analyzed by relating it to the Fokker-Planck equation, which describes a random walk with VADM-dependent drift and diffusion coefficients. We demonstrate first that our method is able to retrieve the correct shape of the drift and diffusion coefficients from a time series generated by a test model. Analysis of the Sint-2000 data shows that the geomagnetic dipole mode has a linear growth time of 13 to 33 kyr, and that the nonlinear quenching of the growth rate follows a quadratic function of the type [1-(x/x0)^2]. On theoretical grounds, the diffusive motion of the VADM is expected to be driven by multiplicative noise, and the corresponding diffusion coefficient to scale quadratically with dipole strength. However, analysis of the Sint-2000 VADM data reveals a diffusion which depends only very weakly on the dipole strength. This may indicate that the magnetic field quenches the amplitude of the turbu
The strength of the geomagnetic dipole moment shows a considerable time variability, about 25% r.m.s. of the mean, over the course of thousands of years. Occasionally, the variability is so large that the sign of the dipole moment changes. These reversals happen roughly once per (2 -3) × 10 5 yr (Merrill et al., 1996). The geomagnetic field is the result of inductive processes in the Earth's liquid metallic outer core. Helical convection amplifies the magnetic field and balances resistive decay. Several groups have confirmed this idea with the help of numerical simulations (Glatzmaier and Roberts, 1995;Kuang and Bloxham, 1997;Christensen et al., 1999). A suitable measure of the geomagnetic dipole is the Virtual Axial Dipole Moment (VADM), of which several records have been published, e.g. by Guyodo and Valet (1999) and Valet et al. (2005). Since the dipole moment is the result of many processes taking place in the convecting metallic outer core that interact with each other in a complicated way, it makes sense to try to describe the time evolution of the VADM over long time scales as a stochastic process.
Before entering into details we recall that statistical modelling of the geomagnetic field has a long history. Constable and Parker (1988) were the first to give a complete characterization of the statistical properties of the geomagnetic field in terms of its spherical harmonic expansion coefficients. The distribution of the axial dipole was found to be symmetric and bi-modal, consisting of two gaussians shifted to the peak position of the two polarity states. They also showed that the expansion coefficients of the non-dipole field may, after appropriate scaling, be regarded as statistically independent samples of one single normal distribution with zero mean. This GGP (giant gaussian process) approach as it is now generally referred to, permitted computation of the average of any field-related quantity. Hulot and Le Mouël (1994) have extended the GGP approach by considering the evolution of the statistical properties with time, and Bouligand et al. (2005) have tested the GGP modelling technique on hydromagnetic geodynamo simulations.
Returning to the time evolution of geomagnetic dipole as a stochastic process, consider a stochastic equation of the type ẋ = v(x) + F (x)L(t) .
(1)
The function v(x) has the dimension of a velocity and represents the effective growth rate of x, sometimes called the drift velocity. The fluctuations are embodied in the term F (x)L(t) and they induce an additional diffusive motion of x.1 Here L(t) is a stationary random function with zero mean and a short correlation time τ c :
A short correlation time means that the duration τ c of the memory of L(t) is much shorter than all other time scales in the process. Under these circumstances the autocorrelation function of L(t) behaves as a δ-function of time. The probability distribution ρ(x, t) of x(t) determined by Eq. ( 1) obeys the Fokker-Planck equation (Van Kampen, 1992;Gardiner, 1990):
Here t is time, and v is again the effective growth rate of x. The diffusion coefficient is equal to The Fokker-Planck equation is a simple and versatile tool for modelling the dynamics of a stochastic process. That is to say, the statistical properties of a wide variety of different stochastic processes can be described by the Fokker-Planck equation (3). Hoyng et al. (2002) have shown that for theoretically plausible functions v(x) and D(x) the amplitude distribution of the Sint-800 data (Guyodo and Valet, 1999) is very well predicted by Eq. ( 3).
The purpose of this paper is twofold. We investigate whether the Sint-2000 VADM time series (Valet et al., 2005) can indeed be described by a Fokker-Planck equation (3). Secondly, we derive the dependence of the effective growth rate v and the diffusion coefficient D on the magnitude x of the VADM without making any prior assumption on the functional form. In doing so we are able to measure the linear growth rate of the dipole mode and its nonlinear quenching from the data. Likewise, the diffusion coefficient D(x) provides information on the convective flows in the outer core. This marks the difference between our approach and that of the GGP: we do not stop at giving a statistical desciption of the multipole coefficients of the geomagnetic field, but we extract information immediately related to the physics of the geomagnetic dipole.
After a brief discussion of the Sint-2000 data in Section 2, we develop in Section 3 a technique for extracting the functions v(x) and D(x) from a time series. Next, in Section 4, we validate the method with the help of an artificial VADM time series generated by a simple model to see how well we can retrieve the v(x) and D(x) that were used to generate the series. In Section 5 we apply the method the Sint-2000VADM data (Valet et al., 2005) and we discuss the implications of our findings for the geodynamo. A summary and our conclusions appear in Section 6.
The
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
