The Energy Flux Number and Three Types of Planetary Dynamo

Abstract

It is proposed that the existence and nature of a planetary dynamo can be characterized by a dimensionless number Φ ≡ F_eR/ϱλ^2Ω, called the energy flux number, where F_e is the energy flux available for dynamo generation, R is the core radius (or thickness of the dynamo generating region), ϱ is the fluid density, λ is the magnetic diffusivity and Ω is the angular velocity. For Φ ≲ 1, there is no dynamo. For 1 ≲ Φ ≲ 10^(2.5) there is an “energy-limited dynamo”, in which F_e is insufficient to enable the dynamo to reach the dynamically desirable state Λ ≡ B^2/8πϱλΩ ∼ 1, where B is a typical field amplitude (in Gauss). For 10^(2.5) ≲ Φ ≲ 10^5, there is a dynamically determined dynamo (Λ ∼ 1) in which the magnetic Reynolds number of turbulent eddies is small. For Φ ≳ 10^5, there is a turbulent dynamo. Probable planetary examples of these three dynamo states are Mercury (Φ ∼ 10^2-10^3), Earth (Φ ∼ 10^4) and Jupiter (Φ ∼ 10^(11)), respectively.