We present results from the first self-consistent, multidimensional simulations of perturbed radio jets in a realistic, cluster cooling flow atmosphere. The supersonic, steady state cooling flow atmosphere used in this work is unique in that it is evolved numerically from an isothermal, equilibrium state. Jets of varying Mach number are passed through this atmosphere and their ultimate fate is found to depend strongly on Mach number. Low Mach number jets (Mj = 3) effectively stagnate due to the ram pressure of the cooling flow atmosphere while medium Mach number jets (Mj = 6-12) become unstable and disrupt in the cooling flow to form amorphous structures. High Mach number jets (Mj > 50) manage to avoid disruption and are able to propagate through the cooling flow. The disruption observed in the intermediate Mach number jets is due to the growth of Kelvin-Helmholtz (K-H) instabilities resulting from a small applied perturbation. Although the pressure and temperature gradients in the flow increase the growth rate of these instabilities, they do not induce a planar shock (Mach disk) which would cause the jet to become subsonic and turbulent. We show that our conceptual framework can be used to account for the observed properties of radio sources found in dominant cluster galaxies in cluster cooling flows.