Non-linear saturation of g-modes in proto-neutron stars: Quieting the acoustic engine

According to Burrows et al.'s acoustic mechanism for core-collapse supernova explosions, the primary, l = 1, g-mode in the core of the proto-neutron star is excited to an energy of ~ 10⁵⁰ erg and damps by the emission of sound waves. Here we calculate the damping of the primary mode by the parametric instability, i.e. by non-linear, three-mode coupling between the low-order primary mode and pairs of high-order g-modes. We show that the primary mode is strongly coupled to highly resonant, neutrino damped pairs with n ≳ 10; such short wavelength interactions cannot be resolved in the simulations. We find that the parametric instability saturates the primary mode energy at ~ 10⁴⁸ erg, well below the energy needed to drive an explosion. We therefore conclude that acoustic power is unlikely to be energetically significant in core-collapse supernova explosions.