We investigate in this paper the core-collapse supernova explosion mechanism in both one and two dimensions. With a radiation/hydrodynamic code based upon the PPM algorithm, we verify the usefulness of neutrino-driven overturn ("convection") between the shock and the neutrinosphere in igniting the supernova explosion. The two-dimensional simulation of the core of a 15 soalr mass star that we present here indicates that the breaking of spherical symmetry may be central to the explosion itself and that a multitude of bent and broken fingers is a common feature of the ejecta. As in one dimension, the explosion seems to be a mathematically critical phenomenon, evolving from a steady state to explosion after a critical mass accretion rate through the stalled shock has been reached. In the two-dimensional simulation the preexplosion convective phase lasted ～30 overturns (～100 ms) before exploding. The preexplosion steady state in two dimensions is similar to that achieved in one dimension, but in two dimensions, owing to the longer dwell time of matter in the overturning region, the average entropy achieved behind the stalled shock is larger. In addition, the entropy gradient in the convecting region is flatter. These effects, together with the dynamical pressure of the buoyant plumes, serve to increase the steady state shock radius (R_s) over its value in one dimension by 30%-100%. A large R_s enlarges the volume of the gain region, puts shocked matter lower in the gravitational potential well, and lowers the accretion ram pressure at the shock for a given M. The critical condition for explosion is thereby relaxed. Since the "escape" temperature (T_(esc)) decreases with radius faster than the actual matter temperature (T) behind the shock, a larger R_s puts a larger fraction of the shocked material above its local escape temperature. T > T_(esc) is the condition for a thermally driven corona to lift off a star. In one, two, or three dimensions, since supernovae are driven by neutrino heating, they are coronal phenomena, akin to winds, though initially bounded by an accretion ram. Neutrino radiation pressure is unimportant.
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