  In order for a double-detonation model to be viable for normal type Ia
supernovae, the adverse impact of helium-burning ash on early-time observables
has to be avoided, which requires that the helium envelope mass should be at
most 0.02 solar mass. Most of the previous studies introduced detonation by
artificial hot spots, and therefore the robustness of the spontaneous helium
detonation remains uncertain. In the present work, we conduct a self-consistent
hydrodynamic study on the spontaneous ignition of the helium envelope in the
context of the double-degenerate channel, by applying an idealized
one-dimensional model and a simplified 7 isotope reaction network. We explore a
wide range of the progenitor conditions, and demonstrate that the chance of
direct initiation of detonation is limited. Especially, the spontaneous
detonation requires the primary envelope mass of >~ 0.03 solar mass. Ignition
as deflagration is instead far more likely, which is feasible for the lower
envelope mass down to ~ 0.01 solar mass, which might lead to subsequent
detonation once the deflagration to detonation transition (DDT) would be
realized. High-resolution multi-dimensional simulations are required to further
investigate the DDT possibility, as well as accurately derive the threshold
between the spontaneous detonation and deflagration ignition regimes. Another
interesting finding is the effect of the composition; while mixing with the
core material enhances detonation as previously suggested, it rather narrows
the chance for deflagration due to the slower rate of 12C(alpha,gamma)16O
reaction at the lower temperature ~108K, with the caveat that we presently
neglect the proton-catalyzed reaction sequence of 12C(p,gamma)13O(alpha,p)16O.
