## Abstract

It is generally accepted that the magnetic fields of planets and stars are produced by the self-exciting dynamo process (first proposed by Larmor) and that observed near-alignments of magnetic dipole axes with rotation axes are due to the influence of Coriolis forces on underlying fluid motions. The detailed role of rotation in the generation of cosmical magnetic fields has yet to be elucidated but useful insight can be obtained from general considerations of the governing magnetohydrodynamic equations. A magnetic field B cannot be maintained or amplified by fluid motion u against the effects of ohmic decay unless (a) the magnetic Reynolds number $R\equiv UL\overline{\mu \sigma}$ is sufficiently large, and (b) the patterns of B and u are sufficiently complicated (where U is a characteristic flow speed, L a characteristic length and $\overline{\mu}$ and $\overline{\sigma}$ are typical values of the magnetic permeability and electrical conductivity respectively). Axisymmetric magnetic fields will always decay (a result that suggests that palaeo-magnetic and archaeomagnetic data might show evidence that departures from axial symmetry in the geomagnetic field are systematically less during the decay phase of a polarity `reversal' or `excursion' than during the recovery phase). Dynamo action is stimulated by Coriolis forces, which promote departures from axial symmetry in the pattern of u when B is weak, and is opposed by Lorentz forces, which increase in influence as B grows in strength. If equilibrium is attained when Coriolis and Lorentz forces are roughly equal in magnitude then the system becomes `magnetostrophic' and the strengths of the internal and external parts of the field, $B_{\text{i}}$ and $B_{\text{e}}$ respectively, satisfy $B_{\text{i}}\lesssim B_{\text{s}}R^{{\textstyle\frac{1}{2}}}$ and $B_{\text{e}}\lesssim B_{\text{s}}R^{-{\textstyle\frac{1}{2}}}$ if $B_{\text{s}}\equiv (\overline{\rho}(\Omega +UL^{-1})/\overline{\sigma})^{{\textstyle\frac{1}{2}}}\approx (\overline{\rho}\Omega /\overline{\sigma})^{{\textstyle\frac{1}{2}}}$, ($\overline{\rho}$ being the mean density of the fluid and $\Omega $ the angular speed of rotation). The slow and dispersive `magnetohydrodynamic inertial wave' with a frequency that depends on the square of the Alfven speed|B|/ $(\mu \overline{\rho})^{{\textstyle\frac{1}{2}}}$ and inversely on $\Omega $ exemplifies magnetostrophic flow. Such waves probably occur in the electrically conducting fluid interiors of planets and stars, where they play an important role in the generation of magnetic fields as well as in other processes, such as the topographic coupling between the Earth's liquid core and solid mantle.