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 M(circle dot) 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 similar to 30 overturns (similar to 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 hi. 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. We find that large and small eddies coexist, both before and after explosion. In the unstable region before explosion, columnar downflows are quasi-periodically formed and break up. These plumes excite nonlinear internal g-modes that feed back onto the convection and cause the plumes to meander over the neutrinosphere. The radial neutrino flux fluctuates with angle and time in response to the anisotropic mass flux onto the neutrinospheres by as much as a factor of 3. The boiling motion of the unstable region interior to the shock is epitomized by neutrino-heated bubbles that rise and collide episodically with the shock, whose radius oscillates in angle and time by as much as 30%. The angle-averaged neutrino luminosities vary by as much as 60% and decrease by a factor of 2 right after the explosion in a characteristic way. The region interior to the neutrinosphere has weakly unstable lepton and entropy gradients that drive persistent convective motions after core bounce. However, the effects of this convection on the driving neutrino luminosities seem dwarfed by the effects of convective dredge-up and by the wildly varying accretion component. We see no evidence of a ''building'' or accumulation of energy before explosion, save in the kinetic energy, in response to the decaying accretion ram. In fact, the total energy in the overturning region decreases steadily before explosion. In addition, we have noted a nontrivial dependence on the neutrino transport algorithm. We would eschew terms such as ''robust'' when referring to the effect of convection on the outcome of collapse. Neutrino energy is pumped into the supernova during the shock's propagation through the inner many thousands of kilometers and not ''instantaneously.'' Curiously, just after the explosion is triggered, the matter that will eventually be ejected is still bound. In addition, for a given asymptotic explosion energy, the amount of mass that reaches explosive nucleosynthesis temperatures is less than heretofore assumed. This may help to solve the Ni-56 overproduction problem encountered in previous models of explosive nucleosynthesis. The high-speed fingers that emerge from our model core seem a natural explanation for the nickel bullets seen in SN 1987A and the shrapnel inferred in some supernova remnants. In addition, the vigorous convective motions interior to the shock can impart to the residue recoil velocities and spins. The magnitudes of the former might be within reach of the observed pulsar proper motions, but extensive new calculations remain to be done to verify this. Within 100 ms of the explosion, a strong, neutrino-driven wind is blowing outward from the proto-neutron star that clears the interior of mass and, while operative, does not allow fallback. At the base of the rising explosion plumes (in the early wind), a few high-entropy (similar to 60) clumps are ejected, whose subsequent evolution may prove to be of relevance to the r-process.
