78 Physics of the Earth and Planetary Interiors, 36 (1984) 78—84 Elsevier Science Publishers By.. Amsterdam — Printed in The Netherlands Self-consistent dynamo models driven by hydromagnetic instabilities D.R. Fearn and M.R.E. Proctor Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver St.. Cambridge. CB3 9E W (Gt. Britain) (Received 20 October, 1983; accepted 12 May, 1984) Fearn, D.R. and Proctor, M.R.E., 1984. Self-consistent dynamo models driven by hydromagnetic instabilities. Phys. Earth Planet. Inter., 36: 78—84. The dynamics of the Earth’s core are dominated by a balance between Lorentz and Coriolis forces. Previous studies of possible (magnetostrophic) hydromagnetic instabilities in this regime have been confined to geophysically unrealistic flows and fields. In recent papers we have treated rather general fields and flows in a spherical geometry and in a computationally simple plane-layer model. These studies have highlighted the importance of differential rotation in determining the spatial structure of the instability. Here we have proceeded to use these results to construct a self-consistent dynamo model of the geomagnetic field. An iterative procedure is employed in which an a-effect is calculated from the form of the instability and is then used in a mean field dynamo model. The mean zonal field calculated there is then input back into the hydromagnetic stability problem and a new a-effect calculated. The whole procedure is repeated until the input and output zonal fields are the same to some tolerance. 1. Introduction westward drift velocity suggest that the toroidal field in the core may be much larger than the It is now generally agreed that the Earth’s mag- poloidal field (see Moffatt, 1978), and that the netic field is maintained against ohmic decay by former plays a significant role in determining the the dynamo process (see for example Moffatt, balance of forces (the so called magnetostrophic 1978). Although the kinematic aspects of the balance). mechanism are now reasonably well understood, The idea that a magnetostrophic balance ob- rather less progress has been made on the fully tains its given support from consideration of the magnetohydrodynamic dynamo. There, the veloci- instabilities which occur in a rotating conducting ties that maintain the magnetic field, instead of fluid in the presence of a magnetic field. Soward being prescribed (as in the kinematic problem), are (1979, 1983) has discussed how in weak-field hy- produced by convective instability, and the prob- dromagnetic dynamos, the relaxation of the geo- lem becomes highly nonlinear. Busse (1975, 1976) strophic constraint by the magnetic field leads to a has made some analytic progress by supposing greatly reduced instability threshold, and ulti- that differential rotation in the Earth’s core is mately to runaway field growth. This process con- weak, so that the poloidal component B ~ and the tinues until Lorentz and Coriolis forces are corn- toroidal component BT of the large scale magnetic parable in magnitude; i.e., until the Elsasser num- field are of the same order of magnitude. The ber primary large scale force balance is then between A = B 2 /2~ (1 the Coriolis force and the pressure gradient (the / geostrophic balance). However, estimates of the is of order unity. (Here B is a-measure of the field 0031-9201/84/803.00 © 1984 Elsevjer Science Publishers B.V.