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 @ F,R/eA2P, called the energy flux number, where F , is the energy flux available for dynamo generation, R is the core radius (or thickness of the dynamo generating region), e is the fluid density, A is the magnetic diffusivity and sd is the angular velocity. For @ 5 I, there is no dynamo. For I 5 @ 5 102.5 there is an "energy-limited dynamo", in which F , is insufficient to enable the dynamo to reach the dynamically desirable state A = B2/8neMZ -I, where B is a typical field amplitude (in Gauss).

Dynamo Scaling Laws and Applications to the Planets

“…For estimating the variation of magnetic field strength during the evolutionary history of a planet, assumed a linear dependence of B 2 on the energy flux. Stevenson (1984) suggested that in cases of a comparatively low energy flux, the available energy limits the magnetic field strength to values that correspond to an Elsasser number < 1, whereas at high flux the field saturates at ≈ 1. His prediction for the lowflux regime results in rule #4 in Table 1.…”

“…(3) The Coriolis forces could be balanced to a large degree by a pressure gradient force (nearly geostrophic flow) and only the residual must be balanced by electromagnetic or other forces. Furthermore, Stevenson (1983Stevenson ( , 1984 pointed out that the Elsasser number rule (as well as several other proposed scaling laws) ignore the energy requirement for maintaining a magnetic field against ohmic dissipation. He suggested that the Elsasser number rule is only applicable when sufficient energy is available.…”

“…Finally, the scheme used in Fig. 3d makes the non-dimensional available flux equivalent to the 'energy flux number' , proposed by Stevenson (1984). Stevenson suggested that a change of regime occurs around = 3000 (or 2/3 ≈ 200; note that 2/3 is plotted in Fig.…”

“…The primary aim of a dynamo scaling theory is to explain the field strength in terms of fundamental properties of the planet U.R. Christensen ( ) Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany e-mail: christensen@mps.mpg.de Busse (1976) 3 B 2 ∝ ρ σ −1 Stevenson (1979) Elsasser number rule 4 B 2 ∝ ρR 3 c q c σ Stevenson (1984) at low energy flux 5 B 2 ∝ ρ R 5/3 c q 1/3 c Curtis and Ness (1986, modified) mixing length theory 6 B 2 ∝ ρ 3/2 R c σ −1/2 Mizutani et al (1992) 7 B 2 ∝ ρ 2 R c Sano (1993) 8 Starchenko and Jones (2002) MAC balance 9 B 2 ∝ ρR 4/3 c q 2/3 c Christensen and Aubert (2006) energy flux scaling or its dynamo region. It is not immediately obvious which quantities play the key role, but candidates are the radius R c of the electrically conducting fluid core of the planet, the conductivity σ and density ρ of the core, the rotation rate and the convected energy flux q c in the core.…”

Non-linear marginal convection in a rotating magnetic system

“…Both meridional and azimuthal magnetic fields have strengths comparable with that of the field emerging at the Earth's surface with the consequence that A is small. Stevenson (1984) has argued that the energy available in some planets may only be sufficient to maintain a wcak dynamo but for the Earth A of order unity may be expected. Recent investigations by Fearn (1985), on the other hand, suggest that very large fields are unlikely as the configuration is prone to the field gradient instabilities discussed recently by Acheson (1983).…”