Numerical simulations of the supernova (SN) neutrino self-induced flavor
conversions, associated with the neutrino-neutrino interactions in the deepest
stellar regions, have been typically carried out assuming the "bulb-model". In
this approximation, neutrinos are taken to be emitted half-isotropically by a
common neutrinosphere. In the recent Ref. \cite{Mirizzi:2011tu} we have removed
this assumption by introducing flavor-dependent angular distributions for SN
neutrinos, as suggested by core-collapse simulations. We have found that in
this case a novel multi-angle instability in the self-induced flavor
transitions can arise. In this work we perform an extensive study of this
effect, carrying out a linearized flavor stability analysis for different SN
neutrino energy fluxes and angular distributions, in both normal and inverted
neutrino mass hierarchy. We confirm that spectra of different nu species which
cross in angular space (where F_{\nu_e}=F_{\nu_x} and
F_{\bar\nu_e}=F_{\bar\nu_x}) present a significant enhancement of the flavor
instability, and a shift of the onset of the flavor conversions at smaller
radii with respect to the case of an isotropic neutrino emission. We also
illustrate how a qualitative (and sometimes quantitative) understanding of the
dynamics of these systems follows from a stability analysis.Comment: (v2: revised version. 10 pages, 10 eps figures. References updated.
Figures imrproved. Matches the version published in PRD.