The core is only one component of the coupled core--mantle system. Each component profoundly affects, and is affected by, the other. Strictly, the core cannot be considered in isolation from the mantle but, when it is, the mantle is replaced by a set of conditions on the CMB. The resulting theoretical simplification is enormous, but sometimes is an over-simplification. In particular, to suppose that the form, and the physical state, of the CMB are uniform in space and unvarying in time is simplistic. Larson and Olson [1991] argue that variations in the convective regime in the mantle, and in particular the changing configuration of mantle plumes, control the rate of geomagnetic field reversals. It has recently been realized that lateral variations in the temperature of the CMB will have a strong effect on core motions and therefore on core--mantle coupling; see Sun et al. [1994]. A new type of geodynamo is also possible; see § 5. In this section we shall ignore lateral variations on the CMB, apart from topography.
The assumption (§ 1) of an adiabatic well--mixed core becomes suspect
near the CMB, and several authors have argued that a layer of comparatively
light fluid exists adjacent to the CMB.
Braginsky [1993] has christened this ``the hidden ocean of the core'' and
has argued that waves propagating in this stable layer may be partially responsible for
the short period geomagnetic secular variation.
It may also strongly affect core--mantle coupling, particularly
topographic coupling.
Waves in a stratified layer at the top of the core have been studied by
Bergman [1993], who developed a theory of magnetic Rossby waves based
on a generalization of Laplace's tidal equation in which the Lorentz force is
included and the induction equation is added.
He also developed 
plane solutions analytically and showed that the magnetic
field can release equatorially trapped Rossby waves.
Love and Bloxham [1994a] have recently investigated a new idea which
may lead to the abandonment of magnetic core--mantle coupling in comparison with other,
and especially topographic coupling, mechanisms.
Their reductio ad absurdum argument is based on an inverse problem: assuming that
core--mantle coupling is electromagnetic, they seek the time-varying toroidal field,
, at the CMB that creates an electromagnetic torque that best fits the length of day data.
They make three demands which they find cannot be simultaneously met:
(a) 
does not exceed the upper limit provided by dynamo theory (see also
Levy and Pearce [1991] who argue that 
is less than 10mT,
and is probably less than 1mT),
(b) the poloidal electric currents which generate that toroidal field and which leak
into the mantle do not exceed bounds on the electric field inferred from measurements
at Earth's surface [ Lanzerotti et al., 1992, 1993, 1994],
(c) the ohmic dissipation in the mantle caused by those currents does not exceed
the heat flux from the Earth.
They conclude that magnetic stresses cannot be the main factor in core--mantle coupling.
Their treatment of flux diffusion in the analysis leading to their conclusion merits further study.
Love and Bloxham [1994b] have recently proposed a second application
of their idea.
Diffusion of flux plays an important role in the study of Braginsky
and Le Mouël [1993], who are particularly interested in the inductive effects
of high shears in a ``
layer'' at the top of the core.
Kuang and Bloxham [1993] analyze how magnetic field in the upper
core affects topographic core--mantle coupling.
They find that the field may change the strength of the coupling by
several orders of magnitude but, for parameters appropriate to the core, the stress is of
order 10
N m
, which is adequate to account for the decadal variations in
Earth's rotation.
Angular momentum exchange between core and mantle is also discussed by
Bloxham and Kuang [1994].
Malkus has long urged that precessional driving of the core is an
important and, possibly, the dominant source of energy for core motions and
the geodynamo; e.g. see Malkus [1994].
Interest in this idea has been revitalized by the discovery that flows with
elliptical streamlines, somewhat similar to flows driven by the
luni--solar precession, are unstable.
Malkus has provided experimental demonstrations of the instability in an
elliptically distorted cylinder of fluid.
Experiments have also been performed by Vanyo [1991],
Vanyo et al. [1992, 1994b] and Wilde and V
anyo [1994]; see also Vanyo et al. [1994a],
Vanyo and Lods [1994].
So far, all studies have been non-magnetic, but it is hoped that they will
provide stepping stones to the corresponding MHD situations.
The 
effect created by precessionally--driven flows has already been
estimated by Barenghi et al. [1994].
The effects of the SIC on the forced nutation of the Earth have been studied
theoretically, and the results have been compared with observational data
by Mathews et al. [1991a, b] and Herring
et al. [1991]. Cognate issues are analyzed by de Vries and Wahr [1991].
The exchange of the z-component of angular momentum between core and mantle is accomplished via geostrophic motions in the core; these are zonal flows that depend only on distance s from the rotation axis, Oz, and on time t. By analyzing the field extrapolated downwards to the CMB, Jault and Le Mouël [1988] estimated the geostrophic flow in the recent past and could therefore monitor the angular momentum of the core as a function of time. They showed that its variations are roughly equal but opposite to those of the angular momentum of the mantle over the same period, as determined by changes in the length of day; the net angular momentum of the core--mantle system as a whole is constant. Jackson et al. [1993] have developed this theme and have used their analysis of the core geostrophic flow to predict, with encouraging results, variations in the length of day.