Results are presented here from several two dimensional numerical calculations of events in Type II supernovae. A new 2-D hydrodynamics and neutrino transport code has been used to compute the effect on the supernova explosion mechanism of convection between the neutrinosphere and the shock. This convection is referred to as exterior convection to distinguish it from convection beneath the neutrinosphere. The model equations and initial and boundary conditions are presented along with the simulation results. The 2-D code was used to compute an exterior convective velocity to compare with the convective model of the Mayle and Wilson 1-D code. Results are presented from several runs with varying sizes of initial perturbation, as well as a case with no initial perturbation but including the effects of rotation. The M&W code does not produce an explosion using the 2-D convective velocity. Exterior convection enhances the outward propagation of the shock, but not enough to ensure a successful explosion. Analytic estimates of the growth rate of the neutron finger instability axe presented. It is shown that this instability can occur beneath the neutrinosphere of the proto-neutron star in a supernova explosion with a growth time of {approximately} 3 microseconds. The behavior of the …