1. Study on the growth and time-variability of fluctuations in super-sonic flows: toward more realistic investigations on SASI K. Takahashi & S. Yamada (Waseda Univ.)

2. Introduction Standing Accretion Shock Instability (SASI) is one of the mechanisms for successful explosion of core- collapse supernovae. Although analytical works before have assumed the steady flows in front of the standing shock, the real flows must be fluctuated as pointed out by Arnett & Meakin (2011), who revealed the dynamic structure of progenitors. (Arnett & Meakin 2011) Fe core Si burning O burning

3. Lai & Goldreich (2000) insisted that the perturbations in super-sonic flows grow, which imply that the fluctuations in Si/O layer will grow and have the impact on standing shock. Actually, Couch & Ott (2013) showed numerically that a perturbation in upstream flows leads the successful explosion even for a progenitor that fails to explode without the fluctuation. The importance of the fluctuating nature of upstream flow on shock dynamics in CCSNe

4. Aim & Scope Investigating the impact of fluctuating upstream flows on shock dynamics in core-collapse supernovae (e.g. SASI), more analytically in detail. Consistently, we begin the study of the growth and time-variability of fluctuation in super-sonic flows. And then we will start to investigate the impact on shock dynamics, SASI. (i.e., the investigation of downstream of the standing shock, future work).

5. Method 1 st step: Linearization We linearize the conservation laws of fluid to describe the time- and space-evolution of the perturbed super-sonic accreting flows in front of the standing shock. Assumptions * Polytropic EOS * Neglecting self-gravity * Background flow: spherical, super-sonic Bondi accretion flow Example of Bondi accretion (density at sonic point)

6. 2 nd step: Solving Laplace transformed eqs. We solve the linearized conservation laws by Laplace transform with respect to time, which reduces the PDEs to ODEs with respect to radius. Laplace transform Thanks to this property, partial derivatives w. r. t. time vanish from PDEs. And one obtains ODEs w. r. t. radius that have a parameter, s. We solve the ODEs from a neighborhood of the sonic point to the shock radius by Runge-Kutta method. Def. Property initial distribution of f at t = 0 We skip the details. See the poster.

7. Results; Input parameters The input parameters of the background Bondi accretion: supposing super-sonically accreting Si layer in 15 progenitor of Woosley & Heger (2007). We investigated the growth and time-evolution of the perturbation given as step function near the sonic point: supposing some accreting layer whose physical quantities slightly deviate from the background. Perturbation Background

8. 3 Amplitude 3 -4 15 -30 -8 14 Time evolution at r = 300km

9. Summary We have focused on the fluctuated nature of super- sonic flow in core-collapse supernovae and its impact on dynamics of the shock. First we have started the analytical study of the upstream: the growth and time-variability of the perturbation, using the Laplace transform method. We have found the perturbations can grow sufficiently and they oscillate at shock radius.