A simple toy model of the advective-acoustic instability I. Perturbative   approach
  Some general properties of the advective-acoustic instability are described and understood using a toy model which is simple enough to allow for analytical estimates of the eigenfrequencies. The essential ingredients of this model, in the unperturbed regime, are a stationary shock and a subsonic region of deceleration. For the sake of analytical simplicity, the 2D unperturbed flow is parallel and the deceleration is produced adiabatically by an external potential. The instability mechanism is determined unambiguously as the consequence of a cycle between advected and acoustic perturbations. The purely acoustic cycle, considered alone, is proven to be stable in this flow. Its contribution to the instability can be either constructive or destructive. A frequency cut-off is associated to the advection time through the region of deceleration. This cut-off frequency explains why the instability favours eigenmodes with a low frequency and a large horizontal wavelength. The relation between the instability occurring in this highly simplified toy model and the properties of SASI observed in the numerical simulations of stellar core-collapse is discussed. This simple set up is proposed as a benchmark test to evaluate the accuracy, in the linear regime, of numerical simulations involving this instability. We illustrate such benchmark simulations in a companion paper. 
T. Foglizzo