Figure 1: initial temperature and density profiles. The departure of our model's temperature from an isentropic profile (black dashed line) marks the outer edge of the convective region. The vertical gray lines represent the cutoff density and sponge position used by the algorithm.
The convective flow preceeding the ignition of a Type Ia supernova is highly subsonic. Traditional, fully compressible hydrodynamics methods use explicit timestepping, and have a timestep that is limited by the sound-crossing time across a single computational cell. For low Mach numbers, this is prohibitively small. Our approach to modeling this problem is to use the low Mach number hydrodynamics code, Maestro. In the low Mach number hydrodynamics approach, the compressible hydrodynamics equations are reformulated to analytically filter soundwaves from the system. The fluid state is decomposed into a 1-d radial base state (with the density, ρ0, in hydrostatic equilibrium with the base state pressure, p0) and a 3-d full state. The velocity field is instantaneously determined by an elliptic constraint,
which captures the compressbility effects due to reactions (through S), and the background stratification of the star (through the density-like quantity, β0). The result is an algorithm that allows for timesteps determine by the fluid velocity, instead of the sound speed. For very low Mach number flows, this timestep is ~1/M larger than those allowed by undo an explicit fully-compressible code.
The calculation is started by placing the white dwarf in our 3-D Cartesian domain with 3843 zones (5x108 cm on a side). The entire star is modeled. Figure 1 shows our initial model, generated by the 1-d stellar evolution code, Kepler. We begin at the point when the central temperature has risen to 6x108 K.
Our initial white dwarf convection calculations are described in the following paper:
Immediately at the start of the 3-d calculation, thermonuclear carbon reactions heat the interior of the star, driving convection throughout the convectively unstable region. After a short tranisent, a convective flow develops throughout the inner ~solar mass of the white dwarf. As the reactions proceed, the central temperature of the star increases, increasing the reactions, and further heating up the interior of the star. Hot plumes cool as they rise and expand, preventing ignition until the central temperature crosses ~8x108 K, and the reactions outpace the cooling via expansion. The convective flow have a dipole pattern, which rapidly changes direction. Overall, we model the last ~2 hours of the convection. Our calculation ends when the first flame ignites.
Figure 2: The increase in peak temperature as a function of time in the white dwarf convection calculation.
Figure 2 shows the peak temperature in the white dwarf as a function of time. The temperature increases steadily for about 2 hours, before the final ignition—the point where the temperature rapidly increases to over 109 K. In inset shows the temperature diring the last ~250 s, where large fluctuations are evident, produced by hotspots that failed to ignite.
Figure 3: Vorticity field showing developing convection in the white dwarf, shown at 0, 50, 100, 200, 400, 800, 1600, 3200, and 6400 s. Only a single slice through the 3-d simulation is shown.
Figure 3 shows the magnitude of the vorticity in a single slice through the center of the star. In the early frames, we see a distinct separation between the convectively unstable interior and the surrounding convectively stable material. The late-time breakdown of this distinction seen in the last frame is under investigation and may be an artifact of not evolving the background state.
Figure 4: Radial velocity countours (red = outward flowing; blue = inward flowing) shown at a. 800 s; b. 3200 s; c. 3420 s; d. 7131.79 s.
Figure 4 shows radial velocity contours. Outward moving fluid is red and inward flowing fluid is blue. We see a distinct dipole nature to the flow, which was first observed in the work of Kuhlen et al. (2006, ApJ, 640, 407). In our calculation we see the dipole change direction on a timescale consistent with the convective turnover time.
Figure 5: Radial velocity surfaces (red: 4 x 106 cm/s and 2 x 106 cm/s; blue: -4 x 106 cm/s and -2 x 106 cm/s) and nuclear energy generation rate (yellow: 3.2 x 1012 erg/g/s; yellow-orange: 1 x 1013 erg/g/s; light green: 3.2 x 1013 erg/g/s; dark green: 1 x 1014 erg/g/s) close to the point of ignition. Only the inner (1000 km)3 are shown.
Figure 5 shows the inner 1000 km of the white dwarf, with the radial velocity again colored red (outward flow) and blue (inward flow) and contours of nuclear energy generation (yellow to green) show the concentration of the energy generation toward the center of the star. This snapshot is about 1 s before ignition. Ignition occured about 20 km from the center of the star.
nuclear energy generation + radial velocity movie
Currently, we are running new models with an improved algorithm (higher-order advection, increased coupling between the full state and base state, and a time-dependent base state), better energetics, and rotation.
This research is supported by a DOE/Office of Nuclear Physics Outstanding Junior Investigator award, grant No. DE-FG02-06ER41448, to Stony Brook, and by by the SciDAC Program of the DOE Office of Mathematics, Information, and Computational Sciences under the U.S. Department of Energy under contract No. DE-AC02-05CH11231 to LBNL. Computer time for this calculation was provided through a DOE INCITE award at the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.