Dynamo Scaling Laws and Applications to the Planets

Abstract

Scaling laws for planetary dynamos relate the characteristic magnetic field strength, characteristic flow velocity and other properties to primary quantities such as core size, rotation rate, electrical conductivity and heat flux. Many different scaling laws have been proposed, often relying on the assumption of a balance of Coriolis force and Lorentz force in the dynamo. Their theoretical foundation is reviewed. The advent of direct numerical simulations of planetary dynamos and the ability to perform them for a sufficiently wide range of control parameters allows to test the scaling laws. The results support a magnetic field scaling that is not based on a force balance, but on the energy flux available to balance ohmic dissipation. In its simplest form, it predicts a field strength that is independent of rotation rate and electrical conductivity and proportional to the cubic root of the available energy flux. However, rotation rate controls whether the magnetic field is dipolar or multipolar. Scaling laws for velocity, heat transfer and ohmic dissipation are also discussed. The predictions of the energy-based scaling law agree well with the observed field strength of Earth and Jupiter, but for other planets they are more difficult to test or special pleading is required to explain their field strength. The scaling law also explains the very high field strength of rapidly rotating low-mass stars, which supports its rather general validity.

A magnetic field evolution scenario for brown dwarfs and giant planets

“…Christensen & Aubert (2006) derived a magnetic field scaling relation for planets based on dynamo simulations that cover a broad parameter range. Recently, Christensen et al (2009) generalized the scaling law and showed that its predictions agree with observations for a wide class of rapidly rotating objects, from Earth and Jupiter to low-mass main sequence (spectral types K and M) and T Tauri stars (see also Christensen 2010). Here, we revisit the question of magnetic field strength at brown dwarfs and giant planets and the estimate for the radio flux from extrasolar planets, using this scaling law, which we believe to be on more solid grounds both theoretically and observationally than previously suggested scaling laws.…”

“…Christensen (2010) summarized scaling laws for planetary magnetic fields that were proposed by different authors. Most of them assume a strong relation between field strength and rotation rate.…”

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…The columns in these models typically have widths that are large, in fact, close to the scale of the system, /L ∼ E 1/3 0.1, and have been argued to be an essential feature of Earth-like models (e.g. Christensen & Aubert 2006;Christensen 2010). The highly coherent axial flow structures are responsible for generating dipolar magnetic fields that are well aligned with the rotation axis.…”

“…In fact, typical E ∼ 10 −4 models cannot generate Earth-like magnetic fields without the presence of axially coherent columns (e.g. Sreenivasan & Jones 2006b;Christensen 2010;Miyagoshi et al 2010;Soderlund et al 2012Soderlund et al , 2013.…”

The Solar-Stellar Connection

“…The Elsasser number Λ = B 2 /4πρczηΩ 0 , withρcz mean density in the convective envelope, η magnetic diffusivity, Ω 0 stellar rotation rate, B a characteristic magnetic field of the CZ, is useful to discuss this balance of terms in the Navier-Stokes (N.V.) equation. Depending on the amplitude of this number and on the balance assumed in the Navier-Stokes equation, various scaling of the magnetic field amplitude can be expected (Fauve and Pétrélis 2007;Christensen 2010): Fig. 12 Magnetic wreaths yielding in turn steady (case D3; top left), irregular (case D3a; top right) and quasi cyclic (cases D3b & S3; bottom left and right) magnetic butterfly diagrams Nelson et al (2013).…”