Nearly all existing models of the geodynamo are axisymmetric.
Since the axisymmetric part of the geomagnetic
field cannot, according to Cowling's theorem, be self--maintained, the emf created
by the non-axisymmetric components of fluid flow and magnetic field
must be retained.
For simplicity, it is usually parameterized by an 
effect.
Since turbulent induction is unimportant (see § 2), the 
effect
is created by asymmetric waves/instabilities of global scale.
Recent work on axisymmetric geodynamo models is reported in § 5 below;
here we describe studies of waves/instabilities.
Viewed from the inertial frame, a contained rotating fluid is filled with
vortex lines parallel to the rotation axis that impart the ``elasticity''
that inertial waves require.
In a non-rotating electrically conducting fluid, the ``elasticity'' of the lines of
force of the prevailing magnetic field is responsible for Alfvén waves.
In a rapidly rotating conductor, the Alfvén and inertial waves are
replaced by `fast' and `slow' waves.
The former resemble the inertial waves and have a timescale of the order of a
day; the latter act on timescales of order 
, where 
s
is the angular
velocity of Earth, 
H m
is the magnetic
permeability, 
kg m
is the core density,
m is the radius of the core and 
is a characteristic
strength of the (zonal) field.
If 
mT, 
years, which is similar to
timescales observed in the secular variation.
Slow waves are sometimes called `MC waves', because the primary dynamical balance
is between Magnetic and Coriolis forces.
In some circumstances MC waves become MC instabilities; these are much studied
in the hope of deriving constraints on the structure and strength of the field in the core.
It has also been argued that MC instabilities play a significant role in
the polarity reversal mechanism.
Some MC instabilities are of short time scale, of the order of 10
years. These
are the so-called ideal instabilities, where ``ideal'' refers to the fact that,
unlike the so-called resistive instabilities, they do not rely on the finite
resisitivity of the fluid.
Ideal and resistive instabilities are the counterparts of similar instabilities that
arise in laboratory plasmas but, because of the importance of Coriolis forces in the
core, they evolve there on longer time scales (see above).
London [1992a, b] examined MC waves and instabilities, supposing
that the prevailing magnetic field, B, is zonal and proportional in strength to
distance, s, from the rotation axis, Oz.
Assuming a geostrophic dynamical balance of the kind used in atmospheric dynamics,
he developed a uniform approximation for waves that have a short wavelength in
the s-direction, i.e. away from the rotation axis.
He showed [ London, 1992b] that these waves propagate in a westward direction.
In a later work [ London, 1993], he generalized to other zonal fields;
see also London [1994].
Stable field configurations may become unstable when the fluid is top heavy, and there is much interest therefore in MAC waves and instabilities, where the added `A' stands for ` Archimedean' (i.e. buoyancy) forces. Simple examples of MC and MAC instabilities have been analyzed by Kuang and Roberts [1991, 1992], Lan, Kuang and Roberts [1993]. Fearn and Kuang [1994] and Kuang [1994] stress the importance of the conductivity of the boundaries on the instabilities.
Bergman and Madden [1993] studied core convection, paying particular
attention to the steady mean poloidal circulation in the core, for which they argued
equatorial upwelling would occur.
Such a circulation has a profound effect on the functioning of an
dynamo; see § 5.
Compositional buoyancy is important for driving core convection in the considerations
of Kuang et al. [1994], and Bergman et al. [1994].
The surface of the inner core is probably constitutionally supercooled, so that a
mushy layer exists there. If no magnetic field is present, chimneys form in such a
layer through which the light fluid, released during fractionation inside the layer,
is ejected into the FOC.
Bergman et al. investigate how this mechanism is affected by the prevailing
magnetic field.
For simplicity, many investigations of core dynamics and the geodynamo ignore the SIC, by assuming that the entire core is fluid. At first sight, this unrealism seems not too serious: the SIC is only 4% of the volume of the core and 5% of its mass. Nevertheless, the SIC may have a disproportionate effect on core flows and field generation. It has long been known [ Stewartson, 1966] that, because of the rapid rotation of Earth, differential rotation between inner core and mantle, in a non--magnetic core, exerts a profound influence on the dynamics of the FOC. Ruzmaikin [1993] and Hollerbach [1993] pointed out that the same is likely to be true in corresponding MHD situations. The dynamics of the FOC has a different character inside and outside the tangent cylinder (TC), that is the circular cylinder drawn around the rotation axis and tangent to the SIC at its equator. As Stewartson showed, the TC is itself surrounded by a thin ``shear layer'' in which the flows inside and outside the TC adjust themselves to one another. Hollerbach [1993] showed how an axisymmetric magnetic field alters the structure of this layer. Hollerbach and Proctor [1993] observed that the significance of the TC and its adjustment layer may be even greater for the asymmetric fields; see also Hollerbach [1994]. Glatzmaier and Olson [1993] studied non-magnetic convection in a rotating sphere and showed that the amplitude of the convective motions is greatest outside the TC; see also Cardin and Olson [1994a]. In contrast, for the corresponding MHD situation, where a zonal magnetic field was imposed, Olson and Glatzmaier [1993, 1994] and Glatzmaier and Olson [1994] found that the convection was strongest inside the TC, the Taylor columns outside that cylinder being suppressed by the Lorentz force; see also Jones et al. [1994], Cardin and Olson [1994b] and below.