We describe the results of 3D numerical simulations of oxygen shell burning and hydrogen core burning in a 23 M stellar model. A detailed comparison is made to stellar mixing-length theory (MLT) for the shell-burning model. Simulations in 2D are significantly different from 3D, in terms of both flow morphology and velocity amplitude. Convective mixing regions are better predicted using a dynamic boundary condition based on the bulk Richardson number than by purely local, static criteria like Schwarzschild or Ledoux. MLT gives a good description of the velocity scale and temperature gradient for shell convection; however, there are other important effects that it does not capture, mostly related to the dynamical motion of the boundaries between convective and nonconvective regions. There is asymmetry between upflows and downflows, so the net kinetic energy flux is not zero. The motion of convective boundaries is a source of gravity waves; this is a necessary consequence of the deceleration of convective plumes. Convective ''overshooting'' is best described as an elastic response by the convective boundary, rather than ballistic penetration of the stable layers by turbulent eddies. The convective boundaries are rife with internal and interfacial wave motions, and a variety of instabilities arise that induce mixing through a process best described as turbulent entrainment. We find that the rate at which material entrainment proceeds at the boundaries is consistent with analogous laboratory experiments and simulation and observation of terrestrial atmospheric mixing. In particular, the normalized entrainment rate E ¼ u E / H is well described by a power-law dependence on the bulk Richardson number Ri B ¼ ÁbL/ 2 H for the conditions studied, 20 P Ri B P 420. We find E ¼ ARi Àn B , with best-fit values log A ¼ 0:027 AE 0:38 and n ¼ 1:05 AE 0:21. We discuss the applicability of these results to stellar evolution calculations.

We study the three-dimensional (3D) hydrodynamics of the post-core-bounce phase of the collapse of a 27 M star and pay special attention to the development of the standing accretion shock instability (SASI) and neutrino-driven convection. To this end, we perform 3D general-relativistic simulations with a three-species neutrino leakage scheme. The leakage scheme captures the essential aspects of neutrino cooling, heating, and lepton number exchange as predicted by radiation-hydrodynamics simulations. The 27 M progenitor was studied in 2D by Müller et al., who observed strong growth of the SASI while neutrino-driven convection was suppressed. In our 3D simulations, neutrino-driven convection grows from numerical perturbations imposed by our Cartesian grid. It becomes the dominant instability and leads to large-scale non-oscillatory deformations of the shock front. These will result in strongly aspherical explosions without the need for large-scale SASI shock oscillations. Low--mode SASI oscillations are present in our models, but saturate at small amplitudes that decrease with increasing neutrino heating and vigor of convection. Our results, in agreement with simpler 3D Newtonian simulations, suggest that once neutrino-driven convection is started, it is likely to become the dominant instability in 3D. Whether it is the primary instability after bounce will ultimately depend on the physical seed perturbations present in the cores of massive stars. The gravitational wave signal, which we extract and analyze for the first time from 3D general-relativistic models, will serve as an observational probe of the postbounce dynamics and, in combination with neutrinos, may allow us to determine the primary hydrodynamic instability.

We analyze stellar convection with the aid of 3D hydrodynamic simulations, introducing the turbulent cascade into our theoretical analysis. We devise closures of the Reynolds-decomposed mean field equations by simple physical modeling of the simulations (we relate temperature and density fluctuations via coefficients); the procedure (CABS, Convection Algorithm Based on Simulations) is terrestrially testable and is amenable to systematic improvement. We develop a turbulent kinetic energy equation which contains both nonlocal and time dependent terms, and is appropriate if the convective transit time is shorter than the evolutionary time scale. The interpretation of mixing-length theory (MLT) as generally used in astrophysics is incorrect; MLT forces the mixing length to be an imposed constant. Direct tests show that the damping associated with the flow is that suggested by Kolmogorov (ε K ≈ ρ(u ′ ) 3 rms /ℓ D , where ℓ D is the size of the largest eddy and (u ′ ) rms is the local rms turbulent velocity). This eddy size is approximately the depth of the convection zone ℓ CZ in our simulations, and corresponds in some respects to the mixing length of MLT. New terms involving the local heating due to turbulent dissipation should appear in the stellar evolutionary equations, and are not guaranteed to be negligible. The enthalpy flux (stellar "convective luminosity") is directly connected to the buoyant acceleration, and hence to the scale of convective velocity. MLT tends to systematically underestimate the velocity scale, which affects estimates of chromospheric and coronal heating, mass loss, and wave generation. Quantitative comparison with a variety of 3D simulations reveals a previously unrecognized consistency. Extension of this approach to deal with rotational shear and MHD is indicated. Examples of application to stellar evolution will be presented in subsequent papers in this series.