Shatter or not: role of temperature and metallicity in the evolution of thermal instability

We test how metallicity variation (a background gradient and fluctuations) affects the physics of local thermal instability using analytical calculations and idealized, high-resolution 1D hydrodynamic simulations. Although the cooling function ($\Lambda[T,Z]$) and the cooling time ($t_{\rm cool}$) depend on gas temperature and metallicity, we find that the growth rate of thermal instability is explicitly dependent only on the derivative of the cooling function relative to temperature ($\partial \ln \Lambda/\partial \ln T$) and not on the metallicity derivative ($\partial \ln \Lambda/ \partial \ln Z$). For most of $10^4~{\rm K} \lesssim T \lesssim 10^7~{\rm K}$, both the isobaric and isochoric modes (occurring at scales smaller and larger than the sonic length covered in a cooling time [$c_s t_{\rm cool}$], respectively) grow linearly, and at higher temperatures ($\gtrsim 10^7~{\rm K}$) the isochoric modes are stable. We show that even the nonlinear evolution depends on whether the isochoric modes are linearly stable or unstable. For the stable isochoric modes, we observe the growth of small-scale isobaric modes but this is distinct from the nonlinear fragmentation of a dense cooling region. For unstable isochoric perturbations we do not observe large density perturbations at small scales. While very small clouds ($\sim {\rm min}[c_st_{\rm cool}]$) form in the transient state of nonlinear evolution of the stable isochoric thermal instability, most of them merge eventually.