  We study the initiation of the detonation in the gravitationally confined
detonation (GCD) model of Type Ia supernovae (SNe Ia). Initiation of the
detonation occurs spontaneously in a region where the length scale of the
temperature gradient extending from a flow (in which carbon burning is already
occurring) into unburned fuel is commensurate to the range of critical length
scales which have been derived from 1D simulations that resolve the initiation
of a detonation. By increasing the maximum resolution in a truncated cone that
encompasses this region, beginning somewhat before initiation of the detonation
occurs, we successfully simulate in situ the first gradient-initiated
detonation in a whole-star simulation. The detonation emerges when a
compression wave overruns a pocket of fuel situated in a Kelvin-Helmholtz cusp
at the leading edge of the inwardly directed jet of burning carbon. The
compression wave pre-conditions the temperature in the fuel in such a way that
the Zel'dovich gradient mechanism can operate and a detonation ensues. We
explore the dependence of the length scale of the temperature gradient on
spatial resolution and discuss the implications for the robustness of this
detonation mechanism. We find that the time and the location at which
initiation of the detonation occurs varies with resolution. In particular,
initiation of a detonation had not yet occurred in our highest resolution
simulation by the time we ended the simulation because of the computational
demand it required. We suggest that the turbulent shear layer surrounding the
inwardly directed jet provides the most favorable physical conditions, and
therefore the most likely location, for initiation of a detonation in the GCD
model.
