Astron. Astrophys. 317, 140-163 (1997) 2. Simulations of convectionAll simulations were performed with a modified version of the explicit hydrodynamical code PROMETHEUS (Fryxell et al. 1989; Müller et al. 1991), which is a direct Eulerian implementation of the Piecewise Parabolic Method of Colella & Woodward (1984). 2.1. Convection inside the proto-neutron starThe simulations concentrating on the convective processes inside the proto-neutron star have been performed neglecting effects due to neutrinos completely. This is justified, since the main purpose of the simulations was to demonstrate the existence of the convective instability inside the proto-neutron star and to reveal the relevant length scales and timescales of the convective overturn. We followed these processes only over a period of about 30-50 ms and do not expect any crucial modification of the gas flow by neutrino-energy gain or loss. In particular, the convective velocities seen in our simulations are so high that convective mixing is probably faster than neutrino diffusion. Neutrino viscosity should also be negligible, because the Reynolds number of neutrino shear viscosity is relatively large for the considered densities - g/cm^{3}, temperatures -10 MeV, length scales cm and velocities cm/s in the convective region. Estimates based on the expressions given by van den Horn & van Weert (1981) yield for the dynamical shear viscosity of neutrinos in the diffusive regime and for the corresponding Reynolds number where is the temperature in 5 MeV, the density in g/cm^{3}, the fluid velocity in cm/s and the typical length scale in cm. All simulations were performed with an elaborate, vectorized equation of state, which contains contributions from neutrons, protons, alpha particles, and a representative heavy nucleus in nuclear statistical equilibrium. Electrons, treated as an arbitrarily degenerate and relativistic gas, positrons, and photons were taken into account. The initial model was a spherically symmetrical (non-rotating) post-bounce model of Hillebrandt (1987) which represented the approximately iron core of a star about 12 ms after core bounce. As this model was computed with a general relativistic correction to the gravitational potential, the model had to be relaxed before it was mapped onto the multi-dimensional grid. This was achieved by evolving the model with the 1D Lagrangian hydrodynamics code of Janka (unpublished) for another 17 ms. Then all waves created by the initial force imbalance had died out. In order to save a significant amount of computer time we used an inner boundary condition at a fixed finite radius km, which corresponds to an interior mass . Hence, we did not simulate the flow in the inner part of the proto-neutron star. We assumed hydrostatic equilibrium at the inner boundary and approximated the interior mass by an equivalent central point mass. The usage of such an inner boundary is well justified for sufficiently short times, because initially the inner part of the proton-neutron star is convectively stable (see Figs. 1 and 2). Of course, one must take care that the computational grid extends far enough into the stable region. Self-gravity of the matter in the computational domain was computed with the efficient algorithm of Müller & Steinmetz (1995), which solves Poisson's equation in integral form by an expansion into spherical harmonics. In the 3D simulations a spherically symmetrical potential was used, which results from the angular-averaged density distribution outside the inner boundary and the central point mass. As the overall matter distribution remains nearly spherically symmetrical during the evolution of the convective instability, this approximation should cause only small errors. Two-dimensional test calculations with and without a spherically symmetrical potential confirmed the smallness of the errors. The multi-dimensional simulations were started by mapping the one-dimensional post-bounce model onto the 2D (or 3D) grid and by perturbing the radial velocity on the whole grid with a random perturbation of amplitude. When reducing the perturbation to the results changed only little. 2.1.1. Two-dimensional simulationsThe two-dimensional simulations of convection inside the proto-neutron star were performed in spherical coordinates using a computational grid of 400 (non-equidistant) radial and 90 (equidistant) angular zones. Axial and equatorial symmetry was assumed. The radius of the inner boundary was set to 15 km (see above), while the outer edge of the grid was located at a radius of 2000 km. In Figs. 1 and 2 the and entropy distributions, respectively, are displayed at several moments of time during the evolution. In both figures the solid curve gives the initial distribution.
The deep trough in the profile is the result of neutrino losses and deleptonization around the neutrinosphere, where neutrinos decouple from the stellar gas and where their mode of propagation changes from diffusion to free streaming. Since electron neutrinos are created when electrons and protons combine to neutrons, the gas close to the proto-neutron star "surface" quickly neutronizes and the concentration of electrons, , decreases. The entropy maxima at and are caused by a combination of shock propagation and post-bounce oscillations of the collapsed iron core. Figs. 1 and 2 clearly show that for the initial entropy gradient as well as the initial gradient are negative, i.e., the stratification is convectively unstable. From the inner boundary at out to and for the gradient is negative but the entropy gradient is positive. The mass layer in between ( ) is stable as both gradients are positive. Outside the convectively unstable region the entropy gradient remains negative up to , while the initial gradient becomes positive. Even further out both gradients are positive up to the local entropy maximum at , implying a convectively stable stratification. In a narrow adjacent mass layer ( ) which is located immediately behind the shock the entropy gradient is strongly negative while the lepton gradient, which is positive at the inner edge of the layer, quickly becomes zero. According to the stability criterion this is a convectively unstable situation. We point out that such unstable stratifications can also be found in other core collapse simulations, as e.g., in post-bounce models of Bruenn (1993). Note that although the different regions may be rather narrow in terms of the mass coordinate M(r) , they can nevertheless have a sizable radial extension of several ten kilometers. The evolution of the angular-averaged electron number fraction and of the angular-averaged entropy is shown for the two-dimensional simulation in Fig. 1 and Fig. 2, respectively. While the trough in the initial -profile is filled in, further inside the adjacent, initially stable plateau is more and more eroded due to convective undershooting. The corresponding effect can be recognized in the evolution of the angular-averaged entropy distribution, where the trough in the profile at is filled in, while the entropy maximum at is removed. Due to convective overshooting both profiles are also flattened out to which is well beyond the outer edge of the convectively unstable mass layer at . Figs. 1 and 2 clearly exhibit convective mixing also immediately behind the shock wave in the region where only the entropy gradient is unstable. Here some convective over- and undershooting penetrate into the stable layers above and below and influence the and profiles within about around the convecting shell. According to Figs. 1 and 2 the convective instability needs a growth time of about 10 ms before significant modifications of the entropy and lepton number distributions occur. It then takes another 10 to 20 ms to completely homogenize both the entropy and the distributions in the unstable layers and in the adjacent regions influenced by convective under- and overshooting. Note in this respect that the extent of undershooting at the inner edge of the innermost unstable region steadily increases with time and has almost reached the inner boundary of the computational grid at the end of the simulation. This may be taken as an indication that eventually the whole proto-neutron star might be involved in this process und thus neutrino transport by convective motions could be more important than diffusive transport in deleptonizing the proto-neutron star. However, only simulations which cover a longer interval of the evolution and which also consider the whole proto-neutron star can definitely confirm this possibility. The development of the convective instability in the mantle of the proto-neutron star is further illustrated in Figs. 3 to 7. The right upper panels in Figs. 3a and 4a show that first the inner unstable layer develops Rayleigh-Taylor fingers that penetrate inward and plumes that rise outward. This is best seen from the entropy plot (Fig. 4) which shows the two entropy maxima (in orange and red) and the low-entropy zones (in blue) inside, between, and outside these maxima. The position of the shock wave is at the inner edge of the outermost blue region. The yellow-red ring immediately behind the shock marks the outer unstable layer where the convective instability requires a longer timescale (about 15-20 ms) to grow into the nonlinear regime (lower two panels in Fig. 4). The Rayleigh-Taylor fingers are subject to Kelvin-Helmholtz instabilities that lead to the formation of mushroom-like caps and strong bending of the mushroom stems. As a consequence the initially narrow fingers begin to merge into successively larger blobs (upper and lower left panels in Figs. 3a and 4a). After another 5-10 ms the whole unstable region is involved in a convective overturn (lower right panels in Figs. 3 and 4). Finally, after about 25-30 ms, the inner region is completely mixed and homogenized.
The convective mixing releases gravitational binding energy by establishing a more compact stratification in the mantle of the proto-neutron star (see also Figs. 7 and 17). The liberated binding energy is used to temporarily push the shock out to about 400 km which is more than twice its initial radius. This is illustrated in Fig. 5 which shows the evolution of the entropy distribution between 20 and 32 ms after the beginning of the simulation. The outward propagation of the shock is clearly seen, as well as the mushrooms rising up from the outer convectively unstable layer. The shock expansion comes to a halt at -40 ms and subsequently a re-contraction sets in.
As the convective velocities reach and even partially exceed the local sound speed ( cm/s) strong pressure waves and even weak shock waves are generated by the convective flow. This is illustrated in Fig. 6, where the divergence of the velocity field is shown for the same four snapshots of time as displayed in Fig. 5. Besides the propagation of the shock wave, one recognizes several strong, interacting sound waves "emitted" from the convective region and propagating outward behind the supernova shock front. The convective layer itself shows a turbulent wave activity which reflects the non-stationary character of the convective overturn. Moreover, we see from Fig. 6 that the shock wave is not perfectly spherically symmetrical and exhibits large-scale deformations caused by the interaction with rising convective elements. The overall, moderately prolate deformation of the shock front along the symmetry axis of the two-dimensional computational grid for ms is produced by a large convective element that rises rapidly near the symmetry axis (see Figs. 3, 4 and 5). We point out, however, that its outward velocity may in fact be overestimated because of the imposed axial symmetry which implies that convective elements near the symmetry axis are blob-like while those at the equator are torus-like. The convective activity inside the proto-neutron star causes significant inhomogeneities in the temperature and density stratifications as illustrated by Fig. 7. The inhomogeneities vary in time and space implying non-radial mass motions and a time-dependent mass quadrupole moment associated with the outer layers of the proto-neutron star. Both effects generate gravitational radiation with typical frequencies in the range of several hundred Hz to about one kHz (see Sect. 3). In the transition from the upper panels (at after the start of the simulation) to the lower panels (at ) in Fig. 7 the more compact stratification of the later state is clearly visible.
Although not included in the present simulation, we may speculate about another, possibly very important effect of convection, which has implications for the explosion mechanism. Convective mixing around and below the neutrinospheric region near the proto-neutron star surface may be accompanied by an increase of the neutrino luminosities during the early phase of the supernova explosion. Since the convection is dynamical and violent and the gas velocities are close to the local speed of sound, neutrinos could be transported out of the dense interior of the newly-formed neutron star much faster than by diffusion. The corresponding increase of the neutrino emission may provide an important aid to the neutrino-powered explosion mechanism. Moreover, the neutron star shrinks faster due to the enhanced cooling of its surface layers. Since, in addition, the supernova shock is driven further out as a consequence of the convective settling of the outer parts of the proto-neutron star, ideal conditions for efficient neutrino-heating in the postshock region are established. 2.1.2. Three-dimensional simulationsThe two three-dimensional calculations were done on a grid ( ) of and zones, respectively. The non-equidistant radial zones covered the region km, while the equidistant angular zones covered a cone of opening angle and , respectively. In both simulations the cone was centered at and and periodic boundary conditions were imposed in the angular directions. The development of the convective instability inside the proto-neutron star looks qualitatively quite similar when comparing the results of 2D and 3D simulations. The growth rates of the convective instability are practically identical, as can be seen from the time evolution of the minimum value of the angular-averaged electron number fraction in the convective region (Fig. 8). In comparison with the 2D simulations the amount of overshooting and undershooting is somewhat smaller in the two 3D simulations which explains the smaller final , compared to , by less mixing of high- matter from the layers above and below into the convective region (see also Fig. 16). The qualitative similarity of the 2D and 3D results is also confirmed by Figs. 9 and 10, which show snapshots of the distributions of electron number fraction (Fig. 9) and of the entropy (Fig. 10) in a meridional cut of the computational domain 14 ms after the start of the simulation. The features present in Figs. 9 and 10 are qualitatively very similar to those found in the 2D simulations at the same epoch (see upper left panel of Fig. 3 and Fig. 4, respectively). However, the finger-like or blob-like structures seen in the 2D runs are actually toroidal structures (because of the assumed axial symmetry), whereas the inhomogeneities in the 3D simulations are genuine three-dimensional structures with no geometrical restrictions imposed. Their development from small fingers or bubbles to mushroom-like features which subsequently merge is illustrated in Fig. 11, which shows three snapshots of the surface at times t=12.2 ms, 13.6 ms, and 16.6 ms, respectively.
In order to compare the 2D and 3D results in a more quantitative way we have performed a normal mode analysis by Fourier transforming the electron number fraction (which is a good tracer of the inhomogeneities) at different epochs of the evolution. The relative power of the turbulent motions at different wave numbers and at different radial positions inside the convectively unstable layer is shown for the 3D simulation with cone in Fig. 12 by contour plots at three moments of time. A corresponding contour plot for the 2D simulation is given in Fig. 13 at t = 13.9 ms which is very close to the time of the second snapshot in Fig. 12. In Figs. 12 and 13 the left ordinate gives the wave number of the normal mode normalized to one quadrant , while the right ordinate gives the corresponding wavelength (in degrees) of the mode. This means that the typical angular size of rising or falling lumps of matter is half this wavelength. Note that in the 3D case, we have averaged the distribution in -direction before performing the Fourier transformation. If we instead average over the distribution in -direction the power spectrum does not change significantly. Fig. 12 shows that the evolution of the power spectrum is characterized by small-scale features growing into large-scale ones in the whole unstable layer, an effect which reflects the successive merging of smaller Rayleigh-Taylor fingers into larger ones as well as the increasing homogenization of the convective layer by the mixing process. At t = 16.6 ms (third panel in Fig. 12) all power has become concentrated into structures which have a wavelength larger than about 10 degrees, while early on in the evolution (first panel of Fig. 12) a significant amount of power is contained in features with a wavelength of several degrees only. Comparing the second panel of Fig. 12 with Fig. 13 one recognizes that at the epoch of maximum convective activity the structures in the 3D simulation are only about half as large with respect to their angular extent than those in the 2D simulation. In particular, in the three-dimensional case the dominant modes have (angular) wavelengths of about - (i.e., clump sizes of about - ), while the corresponding values are around - (i.e., clump sizes of approximately - ) in the two-dimensional simulation.
The angular-integrated angular kinetic energy as a function of time also reveals quantitative differences between the two- and three-dimensional results (see Fig. 14). The angular kinetic energy is about a factor of 3-4 smaller in the 3D simulations than in the 2D one. Fig. 14 also shows that the angular kinetic energy in the 3D simulation performed within the sector is significantly reduced compared to the 3D simulation performed within the sector. This reduction of the angular kinetic energy is caused by the insufficient angular extent of the computational domain, which hampers the development of the convective instability, because the dominant structures have wavelengths of about -30 which are too large to be correctly modelled in a sector. Note that early on in the evolution ( ms), when the dominant structures still have a small angular size, the angular kinetic energy of both 3D simulations agrees very well.
Further quantitative differences are found when comparing the root mean squared angular velocity of the 2D and 3D simulations at a given epoch. Fig. 15 shows that in the three-dimensional simulations is roughly a factor of two smaller than in the two-dimensional simulation. Moreover, in 2D shows larger variations as a function of the enclosed mass than in the 3D case. This is simply caused by the fact that the 2D convective structures are twice as large and have 3-4 times more specific angular kinetic energy than the 3D ones. Hence, in the two-dimensional simulation the flow quantities are more inhomogeneous and show larger spatial fluctuations. Compared to the result of the 3D simulation with the sector, the root mean squared angular velocity obtained in the 3D simulation with the angular sector turns out to be smaller in the unstable region where initially both the lepton and the entropy gradient are negative. As discussed before, this is a numerical artifact of the insufficiently large angular grid in case of the simulation performed with the cone.
Because the root mean squared angular velocity and the angular size of the structures is smaller in the three-dimensional simulations than in the two-dimensional models, the genuine three-dimensional finger-like and blob-like structures have less momentum than their two-dimensional counterparts. This explains why in the 3D simulations the extent of undershooting at the inner edge of the convectively unstable layer is reduced compared to the undershooting found in the 2D simulation (see Fig. 16). Note that the "reference" distribution, which is displayed as the solid curve in Fig. 16 and which all multi-dimensional simulations are compared with, refers to the result of a one-dimensional simulation which leaves the initial distribution (as function of the enclosed mass) unchanged due to the disregard of neutrinos in the hydrodynamical simulation and the lack of convective mixing in the 1D case. In two spatial dimensions convection penetrates by means of undershooting about 1.2 pressure scale heights into the adjacent convectively stable layers. In three spatial dimensions the undershooting is reduced to about 0.8 pressure scale heights, i.e., the depth of the undershooting region is smaller by about 50% than in the axisymmetrical models. Because of the insufficiently large angular grid in the 3D simulation performed with the sector, the extent of undershooting is less strong by another 10 to 20% in this case (Fig. 16).
The effect of convective overturn and mixing on the density stratification and on the shock propagation is illustrated in Fig. 17, which shows the density and mass distributions at t = 32 ms for the 1D, 2D and 3D simulations. One notices that the result for the 3D case lies in between the 1D and 2D results and that in two spatial dimensions the structural effects are largest. As already mentioned above, along with the convective overturn of the matter potential energy is liberated in the multi-dimensional simulations, which drives a re-expansion of the stalled shock wave. Hence, the shock is located at larger radii in the 2D and 3D simulations than in the 1D model. Consequently, the mass distribution at small radii is more compact in the multi-dimensional cases, i.e., more mass is located inside a radius of about 45 km than in the spherically symmetrical model (Fig. 17). Compared to the 2D model these effects are reduced in the 3D simulation, because the undershooting is weaker. Therefore the mass region involved in the mixing is smaller and the proto-neutron star becomes less compact, which in turn leads to less potential energy release and to a less strong re-expansion of the shock wave, at t = 32 ms, being the shock radius.
2.2. Convection in the hot-bubble regionWe have performed a second set of simulations, which were aimed at studying the convective instability in the hot-bubble region and which included a simple, but nevertheless reasonably well justified treatment of neutrino effects in the stellar matter (for details, see Janka & Müller 1995a, 1996). In these simulations the inner part of the collapsed stellar core slightly inside the neutrinosphere was cut out and time-dependent neutrino fluxes were imposed at this inner boundary. Thereby, the neutrino emission from the central part of the proto-neutron star was mimiced. The radius of the inner boundary was either kept fixed, or was allowed to shrink with time to take into account the contraction of the cooling neutron star. Due to neutrino heating an unstable entropy gradient builds up between the shock and the neutrinosphere. Depending on the imposed neutrino flux at the inner boundary, this hot-bubble region becomes convective after about 50-80 ms. Thus, in our second set of simulations two convective regions are present: (i) the lepton and entropy unstable layer inside the proto-neutron star just below the neutrinosphere, and (ii) the entropy-unstable hot-bubble region (see Table 1). Although we located the inner boundary somewhat inside the neutrinosphere, our computational domain did only partially encompass the convectively unstable layer in the proto-neutron star. Putting the inner boundary deeper inside the star reduces the time step appreciably and thus makes the simulation significantly more expensive, and, in particular, would have made the long time (up to 1 s past core bounce) hot-bubble simulations discussed in Janka & Müller (1995a, 1996) impracticable. Because only part of the inner convection zone is included in the simulations the radial extent of the convective eddies and of the undershooting is constrained and suppressed by the imposed inner boundary condition. Convection inside the proto-neutron star is therefore weaker in these models than we would expect in simulations without an inner boundary at finite radius. Hence, the gravitational waves produced by anisotropies in these models are primarily caused by non-radial motions in the hot-bubble region (see Sect. 3). The initial model was the collapsed iron core of a star at a time about 25 ms after core bounce. By that time the prompt shock wave had transformed into a standing accretion shock at a radius of about 115 km, enclosing a mass of . The matter behind the shock has small negative velocities and settles onto the forming neutron star. The initial model was provided by Bruenn (private communication; see also Bruenn 1993). The two-dimensional simulations of convection in the hot-bubble region were performed in spherical coordinates. The computational grid consisted of 400 radial zones distributed non-equidistantly between the time-dependent inner boundary and the fixed outer edge of the grid at 17000 km, with 100 radial zones out to 290 km and 42 radial zones within the innermost 100 km. Initially the inner boundary was located at 31.7 km ( g/cm^{3} ) with an interior mass (constant in time) of . Over a period of about 0.5 s the radius of the inner boundary was reduced to a value of 15 km. Thereby, we have simulated the contraction of the proto-neutron star due to lepton and energy losses. The gravitational potential was computed in the same way as in the 2D simulations of convection inside the proto-neutron star, i.e., as the superposition of contributions from the spherically symmetrical point-mass potential associated with the mass inside the inner boundary and from the 2D matter distribution in the computational domain. In angular direction between 90 and 180 equidistant zones were used. In all simulations axial symmetry was assumed, and in some simulations, in addition, equatorial symmetry was imposed. The latter restriction was found, however, to influence the growth of large-scale ( ) angular modes negatively (for more details see Janka & Müller 1995a, 1996) © European Southern Observatory (ESO) 1997 |