THREE-DIMENSIONAL SIMULATIONS OF RAYLEIGH–TAYLOR MIXING IN CORE-COLLAPSE SUPERNOVAE

Supernova (SN) 1987A has furnished astronomers with perhaps the best opportunity to study a core-collapse explosion to date. One of the most exciting discoveries to emerge from this event is the evidence for large-scale, extensive mixing in the SN ejecta. γ-ray lines emitted by the decay of ⁵⁶Co were detected half a year earlier than expected for a spherically symmetric explosion model (Matz et al. 1988). Modeling the light curve in 1D requires a large amount of ⁵⁶Ni to be mixed outward and H and He to be mixed inward (Utrobin 2004). The Bochum event, in which fine-structure Hα lines are observed 2 weeks after the explosion, is explained by the ejection of 10⁻³ M_☉ of ⁵⁶Ni moving with a velocity in excess of 4000 km s⁻¹ into the far hemisphere of 1987A (Hanuschik et al. 1988). The Rayleigh–Taylor (RT) instability was posited as a possible explanation for the large-scale mixing implied for this explosion. Many groups using both smoothed particle hydrodynamics and grid-based codes simulated the evolution of 87A-like progenitor models, first in two-dimensional (2D), then in three-dimensional (3D). It soon became apparent that more than just the RT instability is needed to explain the high ⁵⁶Ni velocities as well as the large degree of outward mixing of heavy elements and inward mixing of lighter elements. Simulations that posit modest initial perturbations are unable to replicate these features of the explosions. Kifonidis et al. (2006) find that by following the explosion from early times, and using large, low-order perturbations in the inner layers, they are able to replicate the high ⁵⁶Ni velocities and large amounts of mixing seen in the explosion. The standing accretion shock instability (SASI; Blondin et al. 2003; Blondin & Mezzacappa 2006; Scheck et al. 2006; Burrows et al. 2006, 2007; Marek & Janka 2009) provides a mechanism by which the inner parts of the ejecta could be deformed.

Most studies of the RT instability in core-collapse supernovae (SNe) use 1987A-like progenitor models. However, SNe in the high-redshift universe, as well as many modern day SNe which die as red, not blue, giants, explode with pre-SN structures which differ significantly from 1987A. This may greatly influence the evolution of the RT instability in their outer layers. Simulations of rotating zero-metallicity SNe (Ekström et al. 2008; Hirschi 2007) indicate that they may die as extended red, rather than compact blue, giants. Rotation mixes CNO elements to the base of the hydrogen-burning shell, where they catalyze the CNO cycle, greatly boosting the burning rate of H. This leads to the formation of a large convective envelope, effectively turning a compact blue star into a giant red one. Models indicate that the shell boost induced convection effectively mixes together the helium shell and the hydrogen envelope. The dense helium shell has been found to be important to the post-bounce evolution of 1987A-like progenitors (Hammer et al. 2009). Rotation has less of an effect on early stars with very low, but not zero, metallicity, such as the Z = 10⁻⁴ Z_☉ progenitor models studied in Joggerst et al. (2010) as well as the current work. These stars remain blue. Solar metallicity stars end their lives as red giants, but retain their helium shells. Red stars should evolve differently from 1987A-type models after the SN shock is launched. They are larger, so the shock takes longer to traverse the star, and the RT instabilities have more time to develop (Chevalier 1976). Previous simulations of core-collapse explosions in two dimensions (Arnett et al. 1989; Hachisu et al. 1990, 1992; Fryxell et al. 1991; Herant & Benz 1991, 1992; Mueller et al. 1991) have primarily focused on 87A-like progenitor models. Studies of RT induced mixing in three dimensions have exclusively focused on attempts to replicate observations of 1987A, and have found significant differences between simulations performed in two and three dimensions. In the most successful of these 2D attempts, Kifonidis et al. (2003, 2006) found that to reproduce the high ⁵⁶Ni velocities observed in 1987A large initial asymmetries of low mode order were necessary. In studies of mixing in both red and blue supergiants (Joggerst et al. 2009; Herant & Woosley 1994) mixing is found to be more vigorous in red stars than in blue stars. Joggerst et al. (2010) evolve 36 models spanning three rotation rates, three explosion energies, three masses, and two metallicities in two dimensions, finding that mixing is more vigorous and goes on longer in stars that die as red, as opposed to blue, giants, in higher energy explosions, and in less massive stars. The literature on 3D simulations is sparser. Kane et al. (2000) find that for a single-mode perturbation of 10% amplitude, RT fingers grow 30%–40% faster in 3D than in 2D. The most recent paper on the post-explosion hydrodynamics in 1987A (Hammer et al. 2009) uses a successful explosion model from Scheck (2007) to initialize a 3D simulation, along with several 2D simulations derived from meridional slices through the initial explosion model. They find that for this model, which is highly asymmetric in the inner regions, the RT instability grows faster and ballistically moving clumps of ⁵⁶Ni reach the hydrogen envelope in 3D. These clumps are effectively stopped by the dense He shell in the suite of 2D models initialized from slices through the Scheck (2007) model.

This is the first paper to simulate the post-explosion hydrodynamics of models with a diversity of pre-SN structures in three dimensions. All models have the same mass (15 M_☉) and explosion energy (1.2 × 10⁵¹ erg at infinity), but two models explode as red giants and one as a more compact blue star. The red giant with zero metallicity effectively lacks a He shell because of convection arising from the hydrogen shell boost. The Z = 10⁻⁴ Z_☉ and Z = Z_☉ models both have helium shells, though the former is blue and the latter is red at the time of explosion. This diversity of models allows us to see if pre-SN structure has an impact on the subsequent evolution of nearly isotropic 2D as opposed to 3D models. These calculations set the stage for further study of asymmetric explosions in these diverse pre-SN models.

The numerical methods employed in this study are discussed in Section 2, and initial models and problem configuration are discussed in Section 3. Results are presented in Section 4, and are discussed and compared with previous 3D simulations in Section 5. We offer some conclusions in Section 6.

The multidimensional simulations described in this paper are performed with CASTRO (Almgren et al. 2010), an Eulerian adaptive mesh refinement (AMR) hydrodynamics code. Time integration of the hydrodynamics equations in CASTRO is based on an unsplit higher-order piecewise parabolic method; the additional features are described below. All simulations described here used either axisymmetric coordinates in 2D or Cartesian coordinates in 3D.

2.1. Equation of State

We followed 16 elements, from hydrogen through ⁵⁶Ni, so that our elemental abundances could later be used to compute spectra. The atomic weights and amounts of the elements are used to calculate the mean molecular weight of the gas required by the equation of state (EOS). The abundance of ⁵⁶Ni was followed separately in order to calculate the energy deposited by radioactive decay of ⁵⁶Ni and ⁵⁶Co.

The EOS in our simulations is as described in Joggerst et al. (2010). It assumed complete ionization and included contributions from both radiation and ideal gas pressure:

$$\begin{equation} P= f(\rho,T) \, \frac{1}{3} a T^4 + \frac{k_{B} T \rho }{m_p \mu } \end{equation} \tag{ 1 }$$

$$\begin{equation} E = f(\rho, T)\, \frac{a T^4}{\rho } + 1.5\frac{k_{B} T}{m_p \mu }, \end{equation} \tag{ 2 }$$

where P is the pressure, a is the radiation constant, k_B is Boltzmann's constant, T is the temperature, ρ is the density, m_p is the proton mass, μ is the mean molecular weight, and E is the energy.

The function f(ρ, T) is a measure of the contribution of radiation pressure to the EOS that accounts for the fact that radiation ceases to be trapped at some point. It is 1 in regions where radiation pressure is important, i.e., where gas is optically thick, and 0 in regions where radiation pressure is unimportant, i.e., where gas is optically thin, with a smooth transition in between. The function f(ρ, T) takes the form

$$f(\rho,T) = \left\lbrace \begin{array}{@{}ll} 0 & \mbox{if $\rho \ge 10^{-9}$ g cm$^{3}$} \\ \ms & \mbox{or $T \le T_{\rm neg}$} \\ \ms f(T)=e^{\frac{T_{\rm neg}-T}{T_{\rm neg}}} & \mbox{if $\rho < 10^{-9}$ g cm$^{3}$}\\ & \mbox{and $T > T_{\rm neg}$}\\ \ms \end{array}, \right.\vspace{0.1in}$$

where T_neg is the temperature at which contributions to the pressure from radiation are 100 times less than that contributed by the ideal gas pressure:

$$\begin{equation} T_{\rm neg}={\frac{3 k_b \rho }{100 m_p \mu a}}^{1/3}. \end{equation} \tag{ 3 }$$

Radiation pressure will begin to dominate the EOS above T_neg without some adjustment, even though this is unphysical, as radiation pressure is negligible in optically thin regions. Damping the radiation component of the EOS with the function f(ρ, T) in these regions provides a more physical solution.

2.2. Radioactive Decay of ⁵⁶Ni

Energy from the radioactive decay of ⁵⁶Ni to ⁵⁶Fe was deposited locally at each mesh point, in the same manner as described in Joggerst et al. (2010). The rate at which energy from the decay of ⁵⁶Ni to ⁵⁶Co was deposited in the grid is given by

$$\begin{equation} dE_{{\rm Ni}} = \lambda _{{\rm Ni}} X_{{\rm Ni}} e^{-\lambda _{{\rm Ni}}t}q(\mbox{Ni}). \end{equation} \tag{ 4 }$$

The decay rate of ⁵⁶Ni, λ_Ni is 1.315 × 10⁻⁶ s⁻¹, and the amount of energy released per gram of decaying ⁵⁶Ni is q(Ni), which we set to 2.96 × 10¹⁶ erg g⁻¹. X_Ni is the mass fraction of ⁵⁶Ni in the cell. The mass fraction of ⁵⁶Co at a given time can be expressed in terms of the mass fraction of initial ⁵⁶Ni by

$$\begin{equation} X_{{\rm Co}} = \frac{\lambda _{{\rm Ni}}}{\lambda _{{\rm Co}}-\lambda _{{\rm Ni}}} X_{{\rm Ni}} (e^{-\lambda _{{\rm Ni}}t}-e^{-\lambda _{{\rm Co}}t}), \end{equation} \tag{ 5 }$$

so that the energy deposition rate from ⁵⁶Co is

$$\begin{equation} dE_{\phi} = \frac{\lambda _{{\rm Ni}}}{\lambda _{{\rm Co}}-\lambda _{{\rm Ni}}} X_{{\rm Ni}} (e^{-\lambda _{{\rm Ni}} t}-e^{-\lambda _{{\rm Co}} t})) \lambda _{{\rm Co}} q(\mbox{Co}). \end{equation} \tag{ 6 }$$

We assumed a decay rate λ_Co = 1.042 × 10⁻⁷ s⁻¹ and an energy per gram of decaying ⁵⁶Co, q(Co), equal to 6.4 × 10¹⁶ erg g⁻¹.

2.3. Gravity

2.4. Adaptive Mesh Refinement

The simulations presented here were carried out in two distinct stages. First, each stellar model was evolved in one dimension using the code KEPLER to the point where the core became unstable to collapse. It was then artificially exploded by means of a piston located at the base of the oxygen shell with enough energy that the explosions had 1.2 × 10⁵¹ erg of energy at infinity. The models were evolved forward in one dimension for 20 s for models z15 and u15, and 100 s for s15, until all nuclear burning had ceased.

These profiles were then mapped onto a two-dimensional r–z or a three-dimensional Cartesian grid in CASTRO, and the calculation was evolved until the shock reached the edge of the simulation domain. The calculation was then stopped, the domain enlarged, and the calculation restarted within the larger domain (see Section 3.3). Data to fill the new regions of the enlarged domain were supplied from the same one-dimensional profile that was used to initialize the multidimensional calculation.

This process continued until the models had evolved out to radii where RT mixing ceased and the star was expanding essentially homologously. We computed three 15 M_☉ models, at zero, 10⁻⁴ Z_☉, and solar metallicities.

3.1. Progenitor Models

Three different SN models, representing three different metallicities, are presented in this paper. Model z15 has zero initial metallicity, and represents a Population III (Pop III) star. Model u15 has Z = 10⁻⁴ Z_☉, and model s15 has solar metallicity. All models are derived from rotating 15 M_☉ progenitors. The zero- and low-metallicity models presented in this paper are the same as presented in Joggerst et al. (2010) and Zhang et al. (2008). The solar metallicity model, model s15, is similar to the model presented in Woosley & Heger (2007). For moderate values of rotation, the amount of rotation was found to have little effect on its post-explosion evolution (Joggerst et al. 2010). There are, however, profound differences between rotating and non-rotating zero-metallicity SNe. Several studies (Ekström et al. 2008; Hirschi 2007; Joggerst et al. 2010) have found that a small amount of rotation in zero-metallicity massive stars dredges up enough carbon, nitrogen, and oxygen from the helium burning shell to catalyze a CNO cycle at the base of the hydrogen envelope. This leads to rapid burning and a shell boost that triggers convection, mixing most of the helium and hydrogen shells together, and "puffing up" the outer envelope of the star. This is apparent in Figure 1. Models with a small amount of metals (i.e., u15 in this paper) do not show this shell boost behavior and die as blue supergiants. Rotation also has little effect on the pre-SN structure of the solar metallicity model s15, which dies as a red giant.

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3.2. Initialization of Multidimensional Data

3.3. Enlarging the Domain

In each multidimensional simulation presented here, when the shock encountered the large region of increasing ρr³ at the base of the hydrogen envelope of the pre-SN star (or, in the case of model z15, the helium–hydrogen envelope) it slowed, giving rise to a reverse shock. The reverse shock then propagated inward in mass toward the center of the star, and left a reversed pressure gradient in its wake. This triggered the RT instability first between the helium shell and the hydrogen envelope, and then between the helium and oxygen layers of the SN. For model z15, this instability first arose between the oxygen layer and the helium–hydrogen envelope. The instability grew, mixing together layers of increasing atomic mass as the reverse shock propagated inward through the star. Eventually, the reverse shock passed by, and mixing stopped once the RT fingers had dissipated their momenta.

In each simulation, the domain was enlarged five times, bringing the final effective resolution of each simulation to 16, 384ⁿ, where n is the dimensionality of the simulation. For model u15, four levels of refinement were retained at the final stage of the simulation; for models s15 and z15, two levels of refinement were retained.

Mixing had effectively ceased by 4.0 × 10⁵ s for model s15, 3.8 × 10⁴ s for model u15, and 1.7 × 10⁵ s for model z15. Mixing had stopped by similar times in 2D and 3D simulations. The models were run to twice this time to make sure that no additional mixing would occur.

Shown in Figure 2 are snapshots at identical times for the two- and three-dimensional simulations of models (from top) s15, u15, and z15. The abundances of He, O, Si, and ⁵⁶Ni are shown in a logarithmic scale, starting clockwise from top left. These snapshots correspond to times when mixing had effectively stopped in the different models. On the left are snapshots of the 2D models; on right are snapshots showing slices in the YZ-plane at the X-axis. Upon first inspection, there appears to be very little difference between the two- and three-dimensional simulations of all three models. The width of the mixed region appears the same between the 2D and 3D models, although mixing in the 3D models appears to be more complete in this region than in the 2D models. The shape of the instability differs slightly between the 2D and 3D simulations. The mushroom shape of the instability is more clearly defined in 2D than in 3D, especially for model u15, where the instability had less time to grow and thus retained its original shape more clearly. In the 3D models, the RT instability appears more elongated, in line with what was found in the single mode study of Kane et al. (2000). The greatest difference between the 2D and 3D simulations arises in model s15, shown at the bottom of Figure 2, where there is actually less mixing of ⁵⁶Ni in 2D than in 3D.

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In all models, the RT fingers have grown to the point where they have begun to interact with one another. Models z15 and s15, in which the RT instability had the longest time to grow, show the greatest degree of interaction. This interaction transfers energy and momentum in the transverse directions to the blast wave, leading to a transition to turbulence in 3D, and to a chaotic regime in 2D (Remington et al. 2006). Miles et al. (2005) state that this transition to turbulence begins to happen when the RT fingers are about 5–6 times longer than their initial wavelength. This has clearly happened in all simulations. Models z15 and s15 appear strikingly similar to the fully turbulent simulations presented in Miles et al. (2005).

The differences between 2D and 3D can be compared more quantitatively by examining Figure 3. This figure displays the radial average of abundances for significant elements in both 2D and 3D simulations of the three models after all mixing has stopped. The times shown are the same times as in Figure 2. Figure 3 shows clearly that the width of the mixing region in the 2D and 3D models is nearly the same. ⁵⁶Ni is the exception, and is slightly more mixed in 2D than in 3D in the simulations of model s15. Previous simulations of the RT instability in which these fingers do not interact show ≈30% faster growth rate in 3D than in 2D, resulting in a wider mixed region in 3D. We also see a faster initial growth rate in 3D than in 2D, but once the RT fingers begin to interact with one another, the growth rate in the 3D models decreases and the final width of the mixed region is the same between 2D and 3D. This is in line with simulations of the RT instability in a laboratory context presented in Miles et al. (2005), in which the RT fingers interacted.

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The 2D and 3D simulations do appear somewhat different in Figure 3 in that the 3D lines are smoother than the 2D lines. Because interactions between RT fingers can transfer momentum and energy perpendicular to the direction of the blast wave, mixing proceeds to a greater extent in 3D than in 2D. This reflects the fact that the 3D simulations have become fully turbulent (in the case of z15 and s15), while the 2D simulations are chaotic, as 3D simulations are able to transfer momentum and energy in two dimensions transverse to the blast wave, while 2D simulations have only one transverse dimension at their disposal. There is likely a sampling effect at play here, as well. In 2D, there simply are not as many RT fingers as there are in 3D, and thus there is effectively a smaller sample of the instabilities that grow. In 3D, there are more RT structures, so the space the fingers can occupy is better sampled, leading to a smoother radialized distribution.

The velocities obtained when mixing has effectively ceased are virtually identical between the 2D and 3D simulations, as shown in Figure 4. These simulations are shown at different times: 4.0 × 10⁵ s, 3.8 × 10⁴ s, and 1.7 × 10⁵ s for models s15, u15, and z15, respectively, and so direct comparison of one model with another would be misleading, as the material will slow down as time goes on. It is clear from Figure 4 that the ultimate velocities after mixing has stopped and the RT instability has frozen out are essentially the same in the 2D and 3D simulations. This is different from what Hammer et al. (2009) found in their simulations with larger degrees of initial asymmetry. They saw a high-velocity tail for ⁵⁶Ni, O, and Si. We do not see a high-velocity tail here.

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The 3D renderings shown in Figure 5 give a clearer picture of the shape of the RT instabilities in our simulations than the slice through the 3D plane shown in Figure 2. We show renderings of the outermost surfaces of carbon, oxygen, silicon, and nickel. Model s15 is shown in the left, model u15 in the center, and model z15 in the right. These renderings were taken at different times, but after mixing had frozen out. Only in model z15 have bits of the oxygen shell broken off and begun to move through the H–He shell that surrounds the heavy-element core of this star. For no models have clumps of heavier elements penetrated shells of lighter elements, as happened in the asymmetric explosion modeled by Hammer et al. (2009). Even the broken-off clumps of oxygen visible in the right-hand panel of Figure 5 are encased in a layer of carbon. It is also apparent by examining this figure that mixing has progressed further in the simulations where the progenitor star was a red giant, model s15 and z15, than for model u15, whose progenitor died as a more compact blue star. The original shape of the instability can be seen in the carbon surface (top middle panel). In addition, the ⁵⁶Ni at the center of the model is less deformed in this model than in previous models. The most vigorous mixing has taken place in model z15, which lacked a dense helium shell and died as a red giant.

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Previous simulations of the RT instability in core-collapse SNe have found significant differences between similar calculations performed in two and three dimensions. Kane et al. (2000) examined the growth of RT fingers arising from a single mode perturbation, and found that the instability grew 30%–35% faster in 3D than in 2D in a 1987A-type progenitor model. This faster growth rate was not sufficient to replicate observations of 1987A, however. In the more recent paper of Hammer et al. (2009), the authors performed simulations of a 1987A-type model in two and three dimensions, starting with a more realistic initial model from Scheck (2007). Their initial model was in the first stages of the explosion, and showed large-scale asymmetry in the heavy element core. They found, again, that single perturbations grew around 30% faster in 3D than in 2D. The authors proposed that artificial drag forces arising from the 2D geometry slow down the growth of the instability in 2D simulations relative to 3D simulations, where these artificial drag forces are not present. This was also described in Kane et al. (2000), who explain that the rising bubbles have less effective matter to push out of the way in 3D than in 2D.

The Hammer et al. (2009) study also found that in 2D, rising bubbles of heavy elements became entrained in the dense He shell left behind by the shock. In 3D, however, these bubbles were able to burst through the He shell to reach the H envelope, replicating one of the key observations of 1987A.

Studies of the RT instability outside the context of SN explosions have also found that a single RT instability grows about 30% faster in 3D than in 2D (Anuchina et al. 2004). Previous simulations of the RT instability in a laboratory context (Miles et al. 2005) have found that the width of the mixed region is very similar between 3D and 2D, provided that the individual RT fingers interact. Initially, the RT fingers grow faster in 3D than in 2D. Once the fingers begin to interact, 3D simulations transition to turbulence more completely than 2D simulations. 3D simulations can transfer momentum in two dimensions perpendicular to the blast wave, while 2D simulations can transfer momentum in only one transverse direction. This results in more complete mixing within the mixed region in 3D than in 2D, reducing the Atwood number in 3D relative to 2D.

For the simplest case of two incompressible fluids under constant acceleration with an initial perturbation of wavelength λ and amplitude η, such that η ≪ λ, the growth rate of the instability is given by Rayleigh (1883):

$$\begin{equation} \frac{d\eta }{dt}=\left(\frac{2\pi A g}{\lambda }\right)^{1/2}t, \end{equation} \tag{ 7 }$$

where g is a constant acceleration and t is the time.

A, the Atwood number, is defined by

$$\begin{equation} A \equiv \frac{(\rho _{2}-\rho _{1})}{(\rho _{2}+\rho _{1})}, \end{equation} \tag{ 8 }$$

where ρ₁ and ρ₂ are the densities of the light and heavy fluids, respectively.

As the densities of the two fluids become comparable, the Atwood number becomes smaller and the growth rate decreases. The growth rate of the bubbles in 3D is thus reduced, neatly canceling out the reduced drag these same bubbles experience in 3D as opposed to 2D. The width of the mixed region remains essentially the same between 2D and 3D for cases where the RT fingers interact with one another.

It could be supposed that the reason the final state of the SN remnant appears the same in our 2D and 3D models is that the perturbations arising from the grid were somehow larger in 2D than in 3D, and this effect exactly canceled out the faster growth rates we expect to see in 3D. But this is not the case. Perturbations arising from the grid are larger in 3D than in 2D. These perturbations arise because we try to fabricate a spherical object with boxes, as required by the RZ and Cartesian geometries used in our simulations. The maximum amount one would expect to depart from a sphere (or circle) in space is equivalent to the longest distance across a single cell, which in 2D is $\sqrt{2}\Delta x$ and in 3D is $\sqrt{3}\Delta x$. To make certain, we performed the same set of calculations in 2D at twice the resolution. These simulations give the same results as our less refined simulations, indicating that our simulations are numerically converged for the problem presented here.

As in the simulations of Miles et al. (2005), Kane et al. (2000), and Hammer et al. (2009), we find that the RT instability initially grows faster in 3D than in 2D. As the RT fingers grow to the point where they begin to interact with one another, the simulations begin to transition to a turbulent (3D) or chaotic (2D) regime. In 3D, momentum is transferred from the fingers in two dimensions, as opposed to only one dimension in 2D. The transition to turbulence in 3D results in more complete mixing within the mixing region than the 2D simulation, reducing the Atwood number in 3D relative to 2D. This has the effect of reducing the growth rate of the instability in 3D relative to 2D. The reduced growth rate in 3D relative to 2D at late times compensates for the fact that the bubbles do indeed experience less drag in 3D than in 2D, and grow faster at early times. By the time mixing has ceased, the width of the mixed region is nearly identical in 2D and 3D in models where the RT fingers have time to interact with one another.

It can be seen from Equation (7) that instabilities seeded by long wavelength perturbations grow more slowly than those seeded by short wavelength perturbations. Obviously, the interior of an SN deviates from the ideal case: the acceleration is not constant, the fluids are compressible, and η soon becomes of the same order as λ. In order for long wavelength perturbations to dominate before mixing ceases there must be significant initial power at long wavelengths. Because there is less time for the instability to grow in blue stars, we expect to see more traces of large perturbations at fairly low wavenumbers in these stars than in red stars. The simulations of Hammer et al. (2009) have a significant amount of power at long wavelengths, so these perturbations grow fast enough to dominate before mixing is truncated.

The initial perturbations in our simulations are a great deal smaller than those examined in previous studies. The spectrum of these perturbations is also flatter and made up of higher modes. The perturbations are more random, though, it should be noted, not entirely so, and are largest around the 30° and 60° marks on the grid. This can be seen during the initial stages of the development of the instability, though not at later times, for model s15. The enhanced perturbations at these points are likely behind the placement of the clumps visible in the 3D rendering of model z15 in Figure 5.

Mixing had more time to develop in our two red giant stars, models z15 and s15, than it generally does in the blue giant 1987A-type progenitor models studied in Kane et al. (2000) and Hammer et al. (2009). Mixing froze out after 4 × 10⁵ s and 1.7 × 10⁵ s for the red models s15 and z15, respectively, over an order of magnitude later than in the simulation of Hammer et al. (2009), in which mixing froze out after 9 × 10³ s. In our blue u15 model, mixing froze out after 4 × 10⁴ s, which is still nearly five times longer than in the Hammer et al. (2009) model. The individual RT instabilities in all simulations grew for sufficient times that they interacted, though more interaction took place in the red models. This was not the case in the single-mode perturbation explored in Kane et al. (2000), or the few large perturbation modes in Hammer et al. (2009). Smaller wavelengths grow faster than longer wavelengths, so longer wavelength perturbations are less likely to become nonlinear. In these models, as well as the single instability investigated in Anuchina et al. (2004), the RT fingers did not interact with one another, ensuring that the relative drag on the fingers was the only significant difference between 2D and 3D. This accounts for the around 30% faster growth rates discovered in these simulations. The simplicity of the initial perturbation spectrum can delay the transition to turbulence, and hence make mixing appear more efficient in 3D than in 2D (Miles et al. 2005).

For long enough wavelength perturbations, spherical geometry will ensure that the perturbations diverge, so they never interact and thus never enter a turbulent regime. This will quickly lead to significant departures from axisymmetry. There may exist a transition mode perturbation which would be the longest wavelength mode that could lead to turbulent interaction. The instabilities will diverge without interaction if the wavelength of an instability is greater than the height at which it would become nonlinear. This transition mode ought to be dependent on pre-SN structure: red supergiants have longer time for the RT instability to grow, and thus potentially a lower mode than blue giants.

The RT instability may "forget" its initial conditions at late times. This can happen though an inverse cascade, where shorter wavelength bubbles merge to form longer wavelength structures. Under idealized conditions, the flow may become self-similar. This transition to self-similarity likely happens during the early stages of RT growth in our simulations, accounting for the similar end appearance of our simulations despite increasing the resolution (and decreasing the initial perturbations). This inverse cascade can only lead to the instabilities becoming so big. A necessary condition for self-similarity is that

$$\begin{equation} \lambda \ll h \ll L, \end{equation} \tag{ 9 }$$

where h is the height of the instability and L is the size of the region in which the instability is growing. The merger of short wavelength instabilities to larger wavelength instabilities should be truncated long before λ becomes comparable to L, which would produce large-scale asymmetries. Large-scale departures from symmetry must arise from the SN explosion mechanism itself, as they cannot arise through an inverse cascade of the RT instability.

Recent observations suggest that asymmetry in core-collapse-type SN explosions is common (Wang & Wheeler 2008). For Type II SNe (SNeII), this asymmetry is characterized by a dominant polarization angle, which is most likely due to a directed, jet-like flow. Polarization data show that most SNe also exhibit composition-dependent, large-scale departures from axisymmetry. The degree of departure from axisymmetry may be less in SNeII-P; Chugai et al. (2005) and Chugai (2006) are able to match observations of one SNeII-P, SN2004dj, by using a bipolar distribution of ⁵⁶Ni. This fits with the emerging picture of collapse SN mechanisms, which are likely to be inherently multidimensional and aspherical (Janka et al. 2007). Multidimensional, inherently assymmetric effects seem to be extremely important for reviving the stalled shock and successfully exploding a massive star (Burrows et al. 2006; Scheck et al. 2006; Marek & Janka 2009; Nordhaus et al. 2010). How relevant, then, are spherical explosion models? This is the first time the post-explosion hydrodynamics of a wide range of SN progenitors has been studied in 3D. It is reasonable to model Pop III and Pop II core-collapse SNe in 3D first as spherical explosions as no observations of these primordial SNe have yet been made. This study will contextualize future studies of SN progenitors with asymmetric explosions.

For SNeII in which departures from asymmetry are small and dominated by an axisymmetric component, 2D simulations may do as well as 3D simulations in reproducing the axisymmetric features of the SN remnant. In these stars, RT fingers of all but the lowest mode order are expected to grow far enough that they begin to interact with one another. Simulations in 2D of such stars can be used to predict nucleosynthesis and fallback for such explosions with a fair degree of accuracy, as was done in Joggerst et al. (2010). 2D calculations may also be used to more accurately predict light curves of SNe for which departures from axisymmetry do not dominate, which would use far less computational resources than similar calculations performed in 3D.

We have performed two- and three-dimensional simulations of SN models with three different metallicities: model z15, with Z = 0 Z_☉, model u15, with Z = 10⁻⁴ Z_☉, and model s15, with Z = Z_☉. The initial perturbations are seeded by the grid. Calculations performed at twice the resolution in 2D indicate that our solutions are numerically converged. We find no significant difference in the width of the mixed region between our 3D and 2D simulations of the same SN. Previous studies of RT mixing in 1987A-type SN models found that mixing was more efficient and individual RT instabilities grew about 30% faster in 3D than in 2D. Kane et al. (2000) studied single-mode density perturbations with amplitudes of 10% of mode l = 10, and found that perturbations grew 30%–35% faster in 3D than in 2D. The more recent study of Hammer et al. (2009) examined the post-explosion hydrodynamics of a 1987A-type model with a fully 3D explosion model from Scheck (2007) with large-scale asymmetries at early times. They found that the instability grew about 30% faster in 3D than in 2D.

While, we also find a faster initial growth rate in 3D than in 2D in our simulations, the growth rate in 3D declines once the simulations transition to turbulence, and the final width of the mixed region is the same in 2D and 3D. Because the RT instability in our model is seeded by perturbations arising from the grid, our perturbations have a lower amplitude, and a flatter spectrum shifted toward higher modes than either the Kane et al. (2000) or Hammer et al. (2009) studies. In our progenitor models, the RT instability has more time to develop and grow than in previous simulations. With a growth time on the order of 10⁵ s, instabilities in models s15 and z15 grow for more than 10 times longer than the simulations of Hammer et al. (2009), in which mixing froze out at around 9 × 10³ s. In model u15, which died as a blue supergiant, mixing ceases after 4 × 10⁴ s, which is a shorter time than the red giant models in this survey but is still nearly five times longer than the simulations of Hammer et al. (2009).

In our progenitor models and with perturbations seeded by the grid, the RT instability has enough time to grow that the fingers of the instability begin to interact with one another. This interaction transfers momentum and energy transverse to the shock, allowing the 3D simulations to mix more fully than 2D simulations. The more efficient transfer of momentum and energy in 3D renders the 3D simulations fully turbulent in the mixed region. This higher degree of mixing in 3D than in 2D reduces the Atwood number in the 3D simulations relative to the Atwood number in the 2D simulations. A lower Atwood number leads to a reduction in the bubble growth rate in 3D, which compensates for the lower effective drag in 3D. The result is that the width of the mixed region is virtually identical between simulations performed in 2D and those performed in 3D. 3D simulations have transitioned to full turbulence and are more completely mixed than 2D simulations. This was found in a laboratory context in the simulations of Miles et al. (2005). For simulations in which interactions between the RT fingers do not occur, either because only one finger is modeled (Anuchina et al. 2004) or the RT fingers grow for a short enough time that they never interact (Kane et al. 2000), the greater effective drag experienced in 2D versus 3D will continue to dominate the simulation, ensuring that the width of the mixed region will be larger in 3D than in 2D. For simulations in which the RT instability has enough time to grow that the fingers begin to interact with one another, the width of the mixed region will be the same in 2D and 3D simulations.

Instabilities arising from long wavelength perturbations take longer to grow than those arising from short wavelength perturbations. Large amplitude perturbations of long wavelengths are necessary for such instabilities to grow substantially. These large-scale fingers may diverge before they can transition to turbulence, and thus may give rise to departures from axisymmetry. The large amplitude, long wavelength perturbations present in the more realistic explosion model of Hammer et al. (2009) ensure that such fingers do not interact, which results in the significant differences between their 2D and 3D simulations. RT theory predicts that an inverse cascade, in which shorter wavelength instabilities merge to form longer wavelength instabilities, should be truncated long before the wavelength λ becomes comparable to the size of the region in which the instability is growing. Large-scale departures from symmetry do not arise through an inverse cascade in our 3D simulations; they must have their origins in the explosion itself.

It is likely that many, if not most, core-collapse SN explosions are highly asymmetric. Observations not just of 1987A but of other SN, as well the "kicks" observed in pulsars, indicate that asymmetry is a near-universal phenomenon amongst the explosions of massive stars. From the theoretical side, it seems increasingly likely that viable SN explosions, like the SASI mechanism, are inherently asymmetric. In SNe that die as red giants, the RT instability will have enough time to develop that individual fingers will interact with one another. If the asymmetries in such explosions are dominated by axisymmetric components, 2D models might well be good predictors of the width of the mixed region and the axisymmetric features of the remnant. Future work should investigate asymmetric explosions in a wide variety of progenitor models.

Work at UCSC and LBL was supported in part by the SciDAC Program under contract DE-FC02-06ER41438. Work at LANL was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. S.E.W. was supported by the NSF and NASA as well as SciDAC National Science Foundation (AST 0909129) and the NASA Theory Program (NNX09AK36G). The simulations were performed on the open cluster Coyote at Los Alamos National Laboratory. Additional computing resources were provided on the Pleiades computer at UCSC under NSF Major Research Instrumentation award number AST-0521566. C.C.J. thanks Andrew Aspden for assistance with parallelizing a data reduction routine. Figure 5 was created using VisIt.