Reversals of the Geomagnetic Field: Constraint on Convection Intensity in the Earth’s Core

Abstract

Modern geodynamo models allow the generation of a magnetic field without reversals and with frequent reversals. The transition from one regime to another is associated with a relatively small change in the intensity of the generation sources. From this, it is usually concluded that the geodynamo system is located near such a transition, which, generally speaking, requires a more detailed justification. Such a transition leads to other changes in the behavior of the geomagnetic field, which are analyzed in this paper based on modern geodynamo models, in particular, the degree to which the dipole of the magnetic field is violated, the change in its strength, and the ratio of the decay time and growth of the dipole during the reversal.

APPENDIX

Let us consider the dynamo equations in a spherical layer ${{r}_{1}} \leqslant r \leqslant {{r}_{0}},$ where ($r,\theta ,\varphi$) is a spherical coordinate system, r₀ = 1, and r_i = 0.35. Entering the following units for speed V, time t, pressure P and magnetic field B, ν/d, d²/ν, ϱν²/d², and $\sqrt {2\Omega \varrho \nu \mu }$, where d = r₀ – r_i is the unit of length, ν is the coefficient of kinematic viscosity, $\varrho$ is the density of matter, and μ is the magnetic permeability, we write the system of dynamo equations in the form

$$\begin{gathered} \frac{{\partial {\mathbf{B}}}}{{\partial t}} = \nabla \times ({\mathbf{V}} \times {\mathbf{B}}) + {\text{P}}{{{\text{m}}}^{{ - 1}}}\Delta {\mathbf{B}},\,\,\,\,\nabla \cdot {\mathbf{V}} = 0,\,\,\,\,\nabla \cdot {\mathbf{B}} = 0, \\ \frac{{\partial {\mathbf{V}}}}{{\partial t}} + ({\mathbf{V}} \cdot \nabla ){\mathbf{V}} = - \nabla P - \frac{2}{{\text{E}}}{{{\mathbf{1}}}_{{\mathbf{z}}}} \times {\mathbf{V}} + \frac{{{\text{Ra}}}}{{{\text{Pr}}}}T{{{\mathbf{1}}}_{r}} \\ + \,\,\Delta {\mathbf{V}} + \frac{{\text{1}}}{{{\text{EPm}}}}\left( {\nabla \times {\mathbf{B}}} \right) \times {\mathbf{B}}, \\ \frac{{\partial T}}{{\partial t}} + ({\mathbf{V}} \cdot \nabla )\left( {T + {{T}_{0}}} \right) = {{\Pr }^{{ - 1}}}\Delta T. \\ \end{gathered} $$

(A.1)

The dimensionless Prandtl, Ekman, Rayleigh, and magnetic Prandtl numbers are given in the form ${\text{Pr}} = \frac{\nu }{\kappa },$ ${\text{E}} = \frac{\nu }{{2\Omega {{L}^{2}}}},$ ${\text{Ra}} = \frac{{\alpha {{g}_{o}}\delta T{{d}^{3}}}}{{\nu \kappa }}$ and ${\text{Pm}} = \frac{\nu }{\eta },$ where κ is the coefficient of molecular thermal conductivity, α is the coefficient of volumetric expansion, g_o is the acceleration of gravity, δT is the unit of temperature perturbation T relative to the “diffusion” (nonconvective) temperature distribution ${{T}_{0}} = \frac{{{{r}_{i}}(r - 1)}}{{r({{r}_{i}} - 1)}},$ and η is the coefficient of magnetic diffusion.

System (A.1) is closed by vacuum boundary conditions for the magnetic field at r₀, r_i and by zero boundary conditions for the velocity field and temperature perturbations. The work uses the pseudo-spectral, MPI-code Magic adapted for the Gentoo operating system. For expansions in 65 Chebyshev polynomials and 128 spherical functions, 16 cores were used on Intel (R) Xeon (R) CPU E5-2640 computers. The used code is an amazing example of how, thanks to the enormous efforts of German scientists (Wicht, 2002; Gastine and Wicht, 2012), the pioneering prototype code developed at Los Alamos by Harry Glatzmaier (Glatzmaier and Roberts, 1995), was made publicly available on GitHub.