  The growth of structure from scale-free initial conditions is one of the most
important tests of cosmological simulation methods, providing a realistically
complex problem in which numerical results can be compared to rigorous analytic
scaling laws. Previous studies of this problem have incorporated gravitational
dynamics and adiabatic gas dynamics, but radiative cooling, an essential
element of the physics of galaxy formation, normally violates the conditions
necessary for self-similar evolution. We show that for any specified value of
the initial power spectrum index n [where P(k)~k^n], there is a family of
power-law cooling functions that preserves self-similarity by ensuring that the
cooling time in an object of the characteristic mass M_* is a fixed fraction
t_C of the Hubble time. We perform hydrodynamic numerical simulations with an
Einstein-de Sitter cosmology, a baryon fraction of 5%, Gaussian initial
conditions, two different power spectrum indices, and four values of t_C for
each index, ranging from no cooling to strong cooling. We restrict the
numerical simulations to two dimensions in order to allow exploration of a wide
parameter space with adequate dynamic range. In all cases, the simulations are
remarkably successful at reproducing the analytically predicted scalings of the
mass function of dissipated objects and the gas temperature distributions and
cooled gas fractions in collapsed systems. While similar success with 3-D
simulations must still be demonstrated, our results have encouraging
implications for numerical studies of galaxy formation, indicating that
simulations with resolution comparable to that in many current studies can
accurately follow the collapse and dissipation of baryons into the dense, cold
systems where star formation is likely to occur.
