  In this paper we investigate how convective instabilities influence heat
conduction in the intracluster medium (ICM) of cool-core galaxy clusters. The
ICM is a high-beta, weakly collisional plasma in which the transport of
momentum and heat is aligned with the magnetic field. The anisotropy of heat
conduction, in particular, gives rise to instabilities that can access energy
stored in a temperature gradient of either sign. We focus on the heat-flux
buoyancy-driven instability (HBI), which feeds on the outwardly increasing
temperature profile of cluster cool cores. Our aim is to elucidate how the
global structure of a cluster impacts on the growth and morphology of the
linear HBI modes when in the presence of Braginskii viscosity, and ultimately
on the ability of the HBI to thermally insulate cores. We employ an idealised
quasi-global model, the plane-parallel atmosphere, which captures the essential
physics -- e.g. the global radial profile of the cluster -- while letting the
problem remain analytically tractable. Our main result is that the dominant HBI
modes are localised to the the innermost (~<20%) regions of cool cores. It is
then probable that, in the nonlinear regime, appreciable field-line insulation
will be similarly localised. Thus, while radio-mode feedback appears necessary
in the central few tens of kpc, heat conduction may be capable of offsetting
radiative losses throughout most of a cool core over a significant fraction of
the Hubble time. Finally, our linear solutions provide a convenient numerical
test for the nonlinear codes that tackle the saturation of such convective
instabilities in the presence of anisotropic transport.
