  The role played by magnetic fields in the intracluster medium (ICM) of galaxy
clusters is complex. The weakly collisional nature of the ICM leads to thermal
conduction that is channelled along field lines. This anisotropic heat
conduction profoundly changes the stability of the ICM atmosphere, with
convective stabilities being driven by temperature gradients of either sign.
Here, we employ the Athena magnetohydrodynamic code to investigate the local
non-linear behavior of the heat-flux driven buoyancy instability (HBI),
relevant in the cores of cooling-core clusters where the temperature increases
with radius. We study a grid of 2-d simulations that span a large range of
initial magnetic field strengths and numerical resolutions. For very weak
initial fields, we recover the previously known result that the HBI wraps the
field in the horizontal direction thereby shutting off the heat flux. However,
we find that simulations which begin with intermediate initial field strengths
have a qualitatively different behavior, forming HBI-stable filaments that
resist field-line wrapping and enable sustained vertical conductive heat flux
at a level of 10--25% of the Spitzer value. While astrophysical conclusions
regarding the role of conduction in cooling cores require detailed global
models, our local study proves that systems dominated by HBI do not necessarily
quench the conductive heat flux.
