Simple Model of the Evolution of Magnetic and Kinetic Energy of the Geodynamo

Abstract

The induction and momentum equations are simplified to a dynamical system for the kinetic and magnetic energies in Earth’s core. Stable stationary points of this system give a geomagnetic field of ~10 mT and the cosecant of the angle between the magnetic field vector and fluid velocity vector is on average about 500 at a known speed of ~1 mm/s and a generally accepted dynamo power of ~1 TW. With a generally known typical geomagnetic time on the order of 1000 years, harmonic secular variations on the order of several decades and rapid exponential changes on the order of several months, possibly associated with jerks, were obtained. All this agrees well with dynamo theory, paleomagnetic reconstructions, numerical modeling, and observations. A geomagnetic energy of ~10 mJ/kg is four orders of magnitude greater than the kinetic energy. Under conditions of such dominant magnetic energy, an analytical solution was obtained, which over time converges to stable stationary points. Apparently unlikely catastrophes with virtually zero magnetic energy near partially stable stationary points are discussed.