  We consider a model of galaxy clusters in which the hot gas is in hydrostatic
equilibrium and maintains energy balance between radiative cooling and heating
by thermal conduction. We analyze the thermal stability of the gas using a
Lagrangian perturbation analysis. For thermal conductivity at the level of
20-40 % of Spitzer conductivity, consistent with previous estimates for cluster
gas, we find that the growth rate of the most unstable global radial mode is ~
6-9 times lower than the growth rate of local isobaric modes at the cluster
center in the absence of conduction. The growth time in typical clusters is ~
2-5 Gyr, which is comparable to the time since the last major merger episode,
when the gas was presumably well mixed. Thus, we suggest that thermal
instability is not dynamically significant in clusters, provided there is an
adequate level of thermal conduction. On the other hand, if the heating of the
gas is not the result of thermal conduction or any other diffusive process such
as turbulent mixing, then the thermal instability has a growth time under a Gyr
in the central regions of the cluster and is a serious threat to equilibrium.
We also analyze local nonradial modes and show that the Lagrangian technique
leads to the same dispersion relation as the Eulerian approach, provided that
clusters are initially in strict thermal equilibrium. Because cluster gas is
convectively stable, nonradial modes always have a smaller growth rate than
equivalent radial modes.
