Pattern evolution in thermally bistable media

We investigate numerically the time evolution of two-dimensional interfaces between the stable phases in a thermally bistable medium under the assumption of a uniform pressure. A model equation, including a generic bistable cooling function and a spatial coupling diffusion term, is studied with different initial conditions and perturbative forcing terms. We find, similarly to the previously studied one-dimensional case, that without any forcing the medium exhibits an inverse cascade on the way to phase separation, with increasingly larger structures predominating, but ultimately the phase imposed on the boundary wins over. Subjecting the medium to short pressure variations (as if the system passes through the pressure wave) may alter this evolution only temporarily. The introduction of spatiotemporal forcing on small scales, however, may give rise to persisting complex patterns with "clouds" having fractal boundaries in an appropriate scale range. In this model fluid motions are allowed only in the perpendicular direction, and several physical processes are ignored. Thus, the direct application of the model to a particular astrophysical system may be premature. However, we demonstrate here that thermal instability and heat diffusion alone (together with some random spatial excitation) are able to produce multidimensional cloudy complex structures, not unlike those observed in some astrophysical settings, notably the interstellar medium.