DIMENSIONAL DEPENDENCE OF THE HYDRODYNAMICS OF CORE-COLLAPSE SUPERNOVAE

While unable to capture the full scope of the differences between 2D and 3D flows, integral quantities offer the advantages of simplicity and widespread usage. In the context of the neutrino mechanism, the integrated heating and cooling in the post-shock flow are naturally important quantities. Figures 4, 5, and 6 show the total integrated heating, cooling, and net heating minus cooling rates, respectively, in the region between the neutrinosphere (where the optical depth to neutrinos is approximately unity) and shock. While there tends to be marginally more heating in the 3D models, they have significantly more cooling and, therefore, less net heating than their corresponding 2D models. If we focus on the gain region (i.e., only the region with net neutrino heating), then we see in Figure 7 that the 2D models have significantly more heating than their 3D counterparts. This arises, in part, because, prior to explosion, there is more mass in the gain region in 2D than in 3D, as shown in Figure 8. Finally, in spite of the higher heating rates in 2D, the average specific entropy in the gain region,

$$\begin{equation} \langle s \rangle _{\rm gain} = \frac{1}{M_{\rm gain}}\int _{\rm gain} \rho s dV, \end{equation} \tag{ 1 }$$

is larger in 3D than in 2D, as shown in Figure 9 and suggested by Nordhaus et al. (2010). This is consistent with the results shown in Hanke et al. (2012), who found somewhat larger average entropies in 3D than in 2D. We note, however, that a higher average entropy is not directly related to earlier explosions, as is clearly demonstrated by the 3D $L_{\nu _e}=2.1\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ model, which has the highest average entropy of any model shown but does not explode within the simulated time. In any case, since entropy depends on the integrated heating, not the heating rate, this suggests that perhaps some material is exposed longer to heating in 3D than in 2D, which we return to in Section 5.1.

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With the exception of the average entropies, the integral quantities just shown seem to indicate that 2D models might have more favorable conditions for explosion. Since our 3D models explode earlier, as shown in Figure 1, important differences between 2D and 3D must remain which these integral quantities have yet to elucidate.

Moving beyond integral quantities, we can look at radial profiles, averaged over solid angle, to distinguish the global structures of 2D and 3D models. In order to highlight the trends with luminosity and dimension, we compute two sets of profiles, one time-averaged from 200 ms to 300 ms post-bounce and another time-averaged from 450 ms to 550 ms post-bounce. The $L_{\nu _e}=2.3\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ models are excluded from the latter set of plots because they are already well into explosion. Time averaging is essential for the 2D profiles to minimize the large fluctuations that obscure their underlying quasi-steady structure.

Figure 10 shows the time- and spherically-averaged entropy profiles. Consistent with Figure 1, the entropy profiles clearly show that the shock radii are systematically larger in 3D than in 2D prior to explosion. The 3D models also have higher peak entropies than their corresponding 2D models, consistent with Figure 9. Interestingly, the peak entropies vary little over the luminosity range considered for models with a given number of dimensions, but there is a clear distinction between the 2D and 3D models. On the other hand, the entropy profiles between ∼50 km and ∼100 km are remarkably similar between all models.

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Figure 11 shows the profiles of radial velocity. As with the entropy profiles, the radial velocities in the post-shock region vary more with dimension than over the range of luminosities of models with a given number of dimensions. The magnitudes of the radial velocities in the post-shock flow are generally larger in 2D than in 3D. This remains true also when doing mass-weighted spherical averages, though the profiles are somewhat closer. The larger post-shock radial velocities in 2D are associated with lower densities and, therefore, lower heating rates for radii beyond ∼120–150 km as compared with 3D. This has the effect of lowering the temperatures, which suppresses cooling and moves the gain radius inward (relative to 3D) toward higher densities and neutrino fluxes. Figures 12 and 13 show the radial profiles of the net neutrino heating rate and bear out these arguments. In the end, the smaller gain radius in 2D leads to a larger integrated net heating rate in the gain region in 2D as shown in Figure 7 and a larger mass in the gain region as shown in Figure 8, even though the densities and net heating rates are larger in 3D at radii beyond ∼120 km and ∼150 km, respectively.

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Finally, Figure 14 shows the radial profiles of the transverse kinetic energy. From 200 ms and 300 ms and between ∼100 km and the shock, the 2D models have nearly twice as much transverse kinetic energy as the 3D models. Again, the profiles vary much more with dimension than between the different luminosity models of the same dimension. At this stage in the evolution, the turbulence in the post-shock flow is artificially vigorous in 2D axisymmetric models, consistent with multidimensional simulations of stellar convection (Meakin & Arnett 2007). At later times, the transverse kinetic energies in the 3D models have grown to be roughly comparable with the 2D models, though the $L_{\nu _e}=2.2\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ models still differ by ∼30%.

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The radial structures of 2D and 3D models show interesting systematic differences, though none are obviously implicated in causing the discrepant explosion times. For example, Figure 13 shows that beyond 150 km, the 3D models have higher net specific heating rates. At the same time, Figures 6 and 7 show that the 2D models have higher integrated heating rates. Which is more important in producing explosions is not a priori clear. The 2D models tend to have larger transverse kinetic energies, which has been claimed to be conducive to explosion (Hanke et al. 2012), but our results seem at odds with this claim. Finally, consistent with Figure 9, the entropy in the gain region is systematically larger in 3D than in 2D, suggesting that there may be interesting differences in the temporally integrated heating, as discussed below in Section 5.1.

During the stalled accretion shock phase, fluid elements advect through the gain region and eventually settle onto the proto-neutron star. In spherical symmetry, the time to advect through the gain region (the integrated dwell time) is short and shared by all of the fluid elements belonging to the same mass shell. In multiple dimensions, the aspherical shock structure and post-shock turbulence lead to a distribution of dwell times, with some fraction of the mass being exposed longer to net neutrino heating (Murphy & Burrows 2008). This effect can increase the neutrino heating efficiency and has naturally been suggested to be one key element of the multidimensional picture (Burrows et al. 1995; Murphy & Burrows 2008). Here, we focus on the $L_{\nu _e}=2.1\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ models since the dwell time distribution is difficult to interpret once fluid elements begin to participate in explosion. We consider two related, but distinct, definitions of the dwell time distribution. The first is the standard integrated dwell time distribution introduced by Murphy & Burrows (2008), which answers the question, for a particular shell of accreting matter, what is the distribution of total integrated times spent in the region of net neutrino heating? Next, we introduce the instantaneous dwell time distribution. At any particular instant, the instantaneous dwell time distribution tells us the distribution of times all of the parcels of matter in the gain region have thus far been exposed to net heating. We suggest that the instantaneous dwell time distribution is of more direct relevance in producing explosions.

We measure the dwell-time distributions by following the trajectories of Lagrangian tracer particles which move according to the time-dependent velocity field. We initialize the particles at a radius of 400 km at 250 ms⁵ post-bounce and begin integrating their dwell times once they pass through the shock. In 3D, we use approximately 2¹⁸ particles distributed quasi-uniformly on the sphere. In 2D, the fewer degrees of freedom means that there are fewer independent trajectories for fluid elements of a given mass shell, even at the same resolution. To combat this, we use 16 shells of approximately 2¹² particles injected over 2 ms for a total of 2¹⁶ particles. In 2D, the particles are distributed uniformly in angle and we weight each particle's contribution to the dwell time distribution by sin θ to account for the fact that each particle represents an angle-dependent annulus.

The resulting integrated dwell time distributions are shown in Figure 15. We find that there are two important effects. First, the mean integrated dwell time, 〈τ〉, is about 5% longer in 2D. In a steady state, the mass in the gain region is related to the mean integrated dwell time simply by $M_{\rm gain}=\dot{M} \langle \tau \rangle$ and so a longer mean dwell time in 2D is consistent with the larger gain mass seen previously in Figure 8. Second, the integrated dwell time distribution in 3D has a longer, shallower tail than the 2D distribution. Integrating over the distributions, we find that about 25% of the accreted material spends more time in the gain region in 3D than in 2D. While the converse is also true—about 75% of the accreted material has a longer dwell time in 2D—much of this material simply settles onto the proto-neutron star without having had much neutrino heating or participated in the turbulent flow in the gain region. By recording the peak entropy reached by each tracer particle, we have found that there is a strong correlation between dwell time and peak entropy. Thus, the more prominent tail at long dwell times translates into having relatively more material in 3D exposed to heating for long times and, consequently, reaching high entropies. Which effect is more important—a longer mean integrated dwell time (2D) or a more prominent tail at long dwell times (3D)—is not immediately clear.

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The integrated dwell time distribution does not tell us directly about the instantaneous state of material in the gain region, as might be expected to bear on a model's ability to transition to explosion. A more relevant measure is the distribution of instantaneous dwell times for all of the parcels of matter in the gain region. This distribution tells us, at any particular instant, how long parcels of matter in the gain region have thus far been exposed to net heating. In a steady state, the instantaneous dwell time distribution (F(τ)) is related to the standard time-integrated dwell time distribution (f(τ)) discussed above by

$$\begin{equation} F(\tau) = \mathcal {N} \int _{\tau }^{\infty } f(\tau ^\prime) d\tau ^\prime, \end{equation} \tag{ 2 }$$

where $\mathcal {N}$ is a normalization factor that assures F(τ) integrates to unity. For CCSNe, the turbulence is established at some time t₀, so F(τ) = 0 for τ > t − t₀, with the above definition of F(τ) being correct only asymptotically. Aside from the normalization, F(τ) is just one minus the cumulative time-integrated distribution function. Figure 16 shows the instantaneous dwell time distribution, computed according to Equation (2). Apart from the first ∼50–100 ms, the distributions are very nearly exponential (∝exp (− τ/τ₀)), with a time constant of 55 ms in 3D and 43 ms in 2D. By computing the mean of the distributions, we find that at late times, the average parcel of matter in the gain region has been exposed to net heating for about 30% longer in 3D than in 2D, a much larger difference than is found for the time-integrated dwell time distributions. When the time-dependence of F(τ) is accounted for by varying the upper bound of integration when taking the mean, the ratio of the means in 2D and 3D becomes time-dependent. Soon after t₀, the mean instantaneous dwell time in 2D is longer than in 3D by up to 10%, a direct result of the relatively broad peak in the 2D integrated dwell time distribution. By 75 ms after t₀, the instantaneous dwell time in 3D becomes longer than in 2D as the tail of the integrated dwell time distribution becomes increasingly relevant. Thereafter, the fractional difference between the 3D and 2D mean instantaneous dwell times grows approximately as 0.3{1 − exp [ − t/(100 ms)]}. Thus, soon after turbulence develops, a typical fluid element in the gain region has been exposed to net heating for longer in 3D than in 2D. We note that this is consistent with explosion times in 2D and 3D approaching one another as the delay to explosion becomes shorter, as we have found and as previously reported (Nordhaus et al. 2010; Hanke et al. 2012). This is also likely the source of the larger entropies in 3D shown in Figures 9 and 10 and in other works (Nordhaus et al. 2010; Hanke et al. 2012). Importantly, this may also be one of the main reasons for the earlier explosions in 3D.

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Perhaps the single most distinguishing characteristic between the 2D and 3D turbulent post-shock flows is found in their energy spectra. As shown by Kraichnan (1967) and later confirmed experimentally in various contexts (see, e.g., Boffetta & Ecke 2012), turbulent cascades are different in 2D and 3D. In 3D turbulence, energy is the only constant of the motion and this leads to a single turbulent cascade that transfers energy from some driving wavenumber k_d toward larger k (smaller scales). In 2D, both energy and squared vorticity are constants of the motion which lead to cascades of energy and enstrophy (proportional to the mean squared vorticity). The enstrophy cascade transports enstrophy in k-space from the driving wavenumber k_d toward larger k (smaller scales). This leads to a characteristic k⁻³ scaling of the velocity energy spectrum for k > k_d (Kraichnan 1967). The energy cascade, by contrast, transports energy from k_d to smaller k (larger scales) and leads to a k^−5/3 scaling of the velocity energy spectrum for k < k_d, in direct analogy with the Kolmogorov theory of turbulence. This is the inverse energy cascade of 2D turbulence, which tends to exaggerate motions on the largest scales of the flow. These turbulence theories were developed within highly idealized setups, assuming, for example, steady isotropic turbulence, and do not necessarily apply directly to the turbulence seen in the core-collapse context. Nevertheless, we find, as did Hanke et al. (2012), that the basic predictions of these theories—the predominance of energy at the largest scales in 2D, the excess energy at the largest scales in 2D relative to 3D, and the shallower slope of the velocity energy spectrum for k > k_d (and therefore more energy for large k) in 3D relative to 2D—are all confirmed in our simulations.

The most natural basis to represent the matter fields in the quasi-spherical post-shock flow is the basis of real spherical harmonics. We decompose the arbitrary scalar quantity Q into spherical harmonics with time- and radially-dependent coefficients

$$\begin{equation} a_{lm}(t,r) = \oint Q(t,r,\theta ,\phi) Y_l^m(\theta ,\phi) d\Omega, \end{equation} \tag{ 3 }$$

where

$$\begin{equation} Y_l^m(\theta ,\phi) =\left\lbrace \begin{array}{@{}ll}\sqrt{2} N_l^m P_l^m(\cos \theta) \cos m\phi &m>0,\\ \ms \ms N_l^0 P_l^0(\cos \theta) &m=0,\\ \ms \ms \sqrt{2} N_l^{|m|} P_l^{|m|}(\cos \theta) \sin |m|\phi &m<0 \end{array}\right. \end{equation} \tag{ 4 }$$

and

$$\begin{equation} N_l^m = \sqrt{\frac{2l+1}{4\pi }\frac{(l-m)!}{(l+m)!}}. \end{equation} \tag{ 5 }$$

In 2D, axisymmetry implies that all coefficients with m ≠ 0 are identically zero. Here, we consider $Q=\lbrace \rho ,P,s,\sqrt{\rho }v_i\rbrace$, where v_i represents the spherical velocity components {v_r, v_θ} in 2D and {v_r, v_θ, v_ϕ} in 3D. We compute the discrete energy spectrum as a function of spherical harmonic degree l as

$$\begin{equation} E(l) = \sum _{m=-l}^l a_{lm}^2, \end{equation} \tag{ 6 }$$

where it should be understood that a_lm, and, therefore, E(l), depend on time and radius (Burrows et al. 2012).

Figure 17 shows the energy spectra of $\sqrt{\rho }v_\theta$ for the $L_{\nu _e}=2.1\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ models at a radius of 150 km and time-averaged between 450 and 500 ms after bounce. We also reproduce the results of Hanke et al. (2012) for comparison. While these curves change in detail for different quantities, over time, and at different radii, the qualitative trends and relation between 2D and 3D are quite robust. At l = 1, 2D has an order of magnitude more energy in all quantities $Q=\lbrace \rho ,P,s,\sqrt{\rho }v_i\rbrace$, consistent with the idea of an inverse energy cascade that pumps energy into the largest scales. The results of Hanke et al. (2012) are even more extreme, with a factor ≈50 more energy in the l = 1 mode in 2D relative to 3D. In real-space, this excess energy at the largest scales manifests as the characteristic "sloshing" always found in 2D simulations (including those with coupled neutrino transport). This sloshing is typically associated with the development of the SASI (Blondin et al. 2003; Scheck et al. 2008; Foglizzo et al. 2007), but the inverse energy cascade will always produce excess energy at l = 1 in 2D, even if the turbulence is driven by, for example, neutrino-driven convection. The SASI may be capable, in principle, of producing significant energy in l = 1, but disentangling its effects from the inevitable l = 1 energy associated with the inverse cascade would seem to require 3D simulations. Whether the low-l energy is a result of the SASI or not, all of the 3D supernova simulations presented in the literature thus far show muted or nonexistent sloshing (Iwakami et al. 2008; Hanke et al. 2012; Burrows et al. 2012), though there has yet to be a fully self-consistent 3D radiation hydrodynamic simulation. Therefore, the results seen thus far suggest that these violent sloshing motions are an artifact of assuming axisymmetry, not a feature that must be incorporated into 3D models as suggested by Hanke et al. (2012). Importantly, this artifact is not a small effect; most of the energy in the flow in 2D simulations is at low-l. This artifact, in fact, dominates the flow and has poorly understood consequences on other coupled aspects of the problem, including the neutrino transport.

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At intermediate l, the spectra are significantly steeper in 2D (∼l^−2.6) than in 3D (∼l⁻¹). That these power-laws differ from those naively expected from simple theoretical arguments should not be surprising given that the turbulence analyzed here is, among other things, not steady-state or isotropic. Qualitatively, however, the expectation of a steeper slope in 2D relative to 3D is confirmed and we find our results to be generally consistent with those of Hanke et al. (2012).⁶ The transition to these slopes occurs around l ∼ 10 in 2D and l ∼ 4 in 3D, which each reflect the corresponding characteristic driving scales for convection. The lower l driving in 3D is likely the result of the larger radial extent of the gain region owing to the larger shock radius (Chandrasekhar 1961). The 3D model has significantly more energy at small-scales than the 2D model. At l ≳ 40 (spatial scales ∼10 km at a radius of 150 km), we begin to see the effects of grid-scale (2 km at this radius) dissipation associated with numerical diffusion.

Hanke et al. (2012) showed that their results, especially in 3D, were sensitive to resolution. The comparison of energy spectra in Figure 17 affords us the opportunity to directly compare the effective resolutions of our independent calculations by comparing the scales at which dissipation begins to set in. As noted above, our 3D results begin to deviate from the l⁻¹ power-law at l ≈ 40, while the results in Hanke et al. (2012) begin to deviate at l ≈ 25, confirming our expectation that our models have nearly double the effective resolution. Hanke et al. (2012) argue that 3D models become less prone to explosion as resolution is increased, yet our higher resolution 3D models explode earlier than the 2D models. We also note that we carried out a 2D $L_{\nu _e}=2.2\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ simulation with double the resolution of the models presented in this work and found that it exploded marginally later (∼10 ms) than the model discussed herein, both countering the results of Hanke et al. (2012) and suggesting that our runs (in 2D, at least) are reasonably converged. The source of these apparent discrepancies is unclear.

A number of quantities have been proposed in the literature that are meant to distinguish exploding from non-exploding models and, furthermore, to define in a systematic way the time at which a model transitions into the exploding phase. The most widely used condition is based on the idea of a critical ratio of the advection to heating timescales (Janka & Keil 1998; Thompson 2000; Thompson et al. 2003). There are numerous ways of defining these timescales. Here, we define the advection time as

$$\begin{equation} t_{\rm adv} = \int _{R_{s}}^{R_{\rm gain}} \frac{dr}{\langle v_r \rangle } , \end{equation} \tag{ 7 }$$

where 〈v_r〉 is the spherically averaged radial velocity. In multidimensional models, the shock radius R_s and the gain radius R_gain are not uniquely defined bounds.⁷ We appeal to the radial profiles shown previously and define the shock radius as the outermost zero in the radial velocity gradient and the gain radius as the first zero crossing in the net heating rate interior to the shock (always around ∼100 km). Furthermore, to minimize the large fluctuations in t_adv that appear in 2D due to transient and localized fluctuations in v_r, we time-average the velocity profiles over a ±15 ms window. We define the heating timescale as

$$\begin{equation} t_{\rm heat} = \frac{\int _{R_{\rm gain}}^{R_{s}} (\langle \rho \varepsilon \rangle - \langle \rho \rangle \varepsilon _0(\langle \rho \rangle ,\langle Y_e\rangle)) 4 \pi r^2 dr}{\int _{R_{\rm gain}}^{R_{s}} \langle \rho \rangle (\langle \mathcal {H}-\mathcal {C}\rangle) 4 \pi r^2 dr}, \end{equation} \tag{ 8 }$$

where angle brackets indicate solid-angle averaging, ε is the specific internal energy, ε₀ is the zero-point energy of the EOS, and $\mathcal {H}-\mathcal {C}$ is the net heating rate per unit mass. The results are shown in Figure 18. After an initial transient phase, the models settle on an approximately (model-dependent) constant value. Rather than there being a particular critical value for the ratio of advection to heating timescales, explosions seem to be robustly connected to rapid growth from these model-dependent quasi-steady values. Nonetheless, we define an explosion time in a systematic manner by measuring the time at which this ratio exceeds 0.5 without later dropping below this value.

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An alternative explosion condition (the "antesonic" condition) was suggested by Pejcha & Thompson (2012). Based on parameterized 1D steady-state models, they suggested that explosions occur when max(c²_s/v_esc²) reaches a critical value ≈0.2. Murphy & Meakin (2011) tested this condition with their parameterized 2D simulations and found it to be consistent with their results. Müller et al. (2012) on the other hand, using 2D radiation hydrodynamic simulations, assessed the validity of the antesonic condition and suggest that it may not be a robust indicator of explosion, but, in any case, the critical value should at least be somewhat larger (∼0.35). Like Müller et al. (2012), we find that a larger value for the critical condition is required and we adopt 0.3 as the critical ratio. Unfortunately, max(c²_s/v_esc²) is a noisy quantity, with brief spikes as large as ≈0.5 that then return well below the critical value. The smoothed curves, however, seem to reliably indicate when explosions set in, but these smoothed curves are necessarily produced ex post facto. In other words, the antesonic condition is not a reliable indicator of explosion given the instantaneous value of the ratio. The evolutions for the 2D and 3D models, smoothed for clarity, are shown in Figure 19. As above, we identify the time of explosion as the time when the smoothed version of max(c²_s/v_esc²) exceeds 0.3 without later dropping below this value.

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A final, phenomenological, explosion condition is simply when the average shock radius exceeds 400 km without later receding below this value, as used in Nordhaus et al. (2010). In this case, explosion times can be read directly from the shock radius evolution curves in Figure 1.

Table 1 shows the explosion times and corresponding mass accretion rates as determined by the three conditions above for the $L_{\nu _e}=2.2\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ and $L_{\nu _e}=2.3\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ models. All three conditions give comparable numbers, but the antesonic condition tends to give the earliest indication of explosion. We note, however, that this conclusion may be somewhat sensitive to the particular critical values adopted. That the three conditions give comparable explosion times is not surprising; the ratio of advection to heating times, the ratio of sound to escape speed, and the average shock radius all grow as models transition into explosion. None of these conditions is able to robustly predict when or even if an explosion will occur before the explosion begins. This suggests that while these conditions may be indicative of explosion, they are by no means the full story.

Table 1. Explosion Times and Accretion Rates

$L_{\nu _e}$a	Dimension	Explosion Condition
		tadv/theat		max(c2s/vesc2)		〈Rs〉
		texp	$\dot{M}$	texp	$\dot{M}$	texp	$\dot{M}$
		(ms)	(M☉ s−1)	(ms)	(M☉ s−1)	(ms)	(M☉ s−1)
2.2	2D	1031	0.182	920	0.196	954	0.188
	3D	698	0.213	611	0.224	695	0.214
2.3	2D	571	0.231	501	0.241	557	0.233
	3D	412	0.269	414	0.268	434	0.262

Note. ^a10⁵² erg s⁻¹.

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With the results shown in Table 1 in hand, we can quantify the delay between the 2D explosions and the earlier 3D explosions. Taking the minimum difference found between the explosion conditions, we find that the $L_{\nu _e}=2.3\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ 3D model explodes at least 87 ms before the corresponding 2D model, while the $L_{\nu _e}=2.2\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ 3D model explodes at least 259 ms earlier than the corresponding 2D model. These delays may translate into appreciable differences in the explosion energy at infinity (Yamamoto et al. 2012), with earlier explosions (3D) being more energetic.

What conditions trigger explosions in 3D? The multidimensional nature of the hydrodynamics is generally agreed to be important in producing such conditions. Turbulence in the post-shock flow leads to a distribution of dwell times for accreting parcels of matter, increasing the mass in the gain region and, therefore, the efficiency of neutrino heating. In this way, turbulence plays a crucial, but secondary, role as an aid to neutrino heating. Murphy & Meakin (2011) argue that convection modifies the quasi-steady global structure of the flow by introducing turbulent fluxes of, for example, enthalpy and entropy. Their model focuses on how turbulent convection modifies averaged radial profiles, rather than on the effects of particular turbulent fluctuations, and was able to account for the differences in, for example, the radial profiles of entropy between 1D and 2D models. While these are important roles for the post-shock turbulent motions, here we suggest that the turbulent fluctuations themselves, driven by neutrino heating, may be instrumental in triggering explosions.

The dominant hydrodynamic instability, aside from the explosion itself, in our parameterized 3D models is neutrino-heating-driven convection (Burrows et al. 2012; Murphy et al. 2012). In the nonlinear phase, the post-shock turbulent flow involves a complex interaction between buoyantly rising, neutrino-heating-driven plumes, negatively buoyant accretion streams, turbulent entrainment resulting from secondary Kelvin–Helmholtz instabilities, and dissipative interactions between rising plumes and the bounding shock. The evolution of a buoyant "bubble" depends on parameters including the heating rate, bubble size, and background inflow velocity. Small bubbles tend to be shredded by turbulent entrainment, while large bubbles can rise all the way to the shock and locally push it outward. Our simulations suggest that some bubbles can continue rising, locally pushing the shock to ever larger radii until the global explosion is triggered. Figure 20 shows a sequence of volume renderings illustrating that large buoyant features form in the flow and push the shock to larger radii. Figure 21 shows the evolution of the shock surface where the same local growth of the shock radius can be clearly seen. Interestingly, the growth of these features occurs not on a dynamical time, but appears to proceed quasi-statically. The same basic picture holds in all of our 3D simulations, suggesting that this may be a generic feature of the transition to explosion.

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We can try to understand the conditions for the runaway growth of bubbles (and the associated triggered explosions) by considering the highly simplified toy model presented in the Appendix. In this model, the evolution of a bubble's outer radius (R_b) is determined by the competition between neutrino driving power (L_ντ, L_ν is the neutrino driving luminosity and τ is the effective optical depth of the bubble) and accretion power ($\alpha G M \dot{M}/R_b$, α is a constant defined in the Appendix) associated with the ram pressure of material immediately behind the shock. The evolution follows

$$\begin{equation} \frac{dR_b}{dt} = \frac{\Omega _0}{4 \pi } \frac{R_b^2}{G M M_b} \left(L_\nu \tau - \alpha \frac{G M \dot{M}}{R_b}\right), \end{equation} \tag{ 9 }$$

which has the solution

$$\begin{equation} R_b(t) = \frac{\alpha G M \dot{M}}{L_\nu \tau + e^{\lambda t} (\alpha GM\dot{M}/R_0 - L_\nu \tau)}, \end{equation} \tag{ 10 }$$

where

$$\begin{equation} \lambda = \alpha \frac{\Omega _0}{4\pi } \frac{\dot{M}}{M_b}. \end{equation} \tag{ 11 }$$

Here, Ω₀ is the bubble's constant solid angle and M_b is the bubble's fixed mass. Qualitatively, R_b(t) has two types of behavior. When the neutrino driving term exceeds the ram pressure term, R_b(t) grows without bound, presumably triggering the global explosion (Thompson 2000). When the ram pressure term dominates, the bubble recedes, perhaps allowing another bubble to take its place.

This simple model makes two predictions. First, the model predicts a characteristic growth of the maximum shock radius, at least in the quasi-static growth phase. Figure 22 shows the maximum shock radii (R_max) from the $L_{\nu _e}=2.2\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ and $L_{\nu _e}=2.3\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}$ 3D models along with fitted model solutions. To compute the fits, we fix M = 1.6 M_☉, $\dot{M}=0.25\,M_\odot \,{\rm s}^{-1}$, and M_b/Ω₀ = 2.5 × 10³⁰ g (chosen so that 4πM_b/Ω₀ ≲ M_gain) and fit the model solution to the data between 250 ms post-bounce and the time when R_max = 1000 km, leaving R₀, τ, and α as free parameters. Both fits give τ ≈ 0.04 (about 20% smaller than the average τ for the gain region) and α ≈ 0.25. We note that α ≈ 0.25 gives an "effective drag coefficient" C_d ≈ 12.5, which seems quite large, but it is nontrivial to estimate reliably what value C_d should take in this context, especially considering the crude dimensional analysis on which the equations are based. Given the simplicity of the model, the agreement between the hydrodynamical simulation results and the model predictions is quite surprising and encouraging.

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Second, there is a critical luminosity for a given mass accretion rate and shock radius above which a bubble will runaway. If the stalled shock radius scales as L^β, then the critical luminosity for runaway bubble growth is proportional to $\dot{M}^{1/1+\beta }$. Empirically, β ∼ 3, and if we adopt the parameters used above we find

$$\begin{eqnarray} L_{\rm crit} & \approx & 2.2 \left(\frac{M}{1.6\,M_\odot }\right)^{1/4} \left(\frac{\dot{M}}{0.25\,M_\odot \,{\rm s}^{-1}}\right)^{1/4} \nonumber\\ && \times \left(\frac{\tau }{0.04}\right)^{-1/4}\times 10^{52}\,{\rm erg}\,{\rm s}^{-1}, \end{eqnarray} \tag{ 12 }$$

which seems consistent with the critical explosion curves from parameterized multidimensional models shown in other works (Murphy & Burrows 2008; Nordhaus et al. 2010; Hanke et al. 2012; Couch 2012). Inasmuch as runway bubble growth is associated with explosions, this may be viewed as an alternative, albeit crude, derivation of the critical luminosity curve of Burrows & Goshy (1993). This may suggest that the reduction in the critical luminosity in going from 1D to 2D and 3D models might arise, in part, from the emergence of bubbles and their runaway growth. The crudeness of the model precludes us from distinguishing between 2D and 3D, but we suggest that important differences may include the different geometries of the bubbles (rings in 2D versus quasi-spherical in 3D) and the efficiency of turbulent entrainment.