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Preparing for an Explosion: Hydrodynamic Instabilities and Turbulence in Presupernovae

Both observations and direct numerical simulations are discordant with predictions of conventional stellar evolution codes for the latest stages of a massive star's life prior to core collapse. We suggest that the problem lies in the treatment of turbulent convection in these codes, which ignores finite amplitude fluctuations in velocity and temperature, and their nonlinear interaction with nuclear burning. The hydrodynamic instabilities that may arise prompt us to discuss a number of far-reaching implications for the fates of massive stars. In particular, we explore connections to enhanced presupernova mass loss, unsteady nuclear burning and consequent eruptions, swelling of the stellar radius that may trigger violent interactions with a companion star, and potential modifications to the core structure that could dramatically impact calculations of the core-collapse mechanism itself. These modifications may be of fundamental importance to the interpretation of measured isotopic anomalies in meteorites, changing the predictions of both mixing and detailed nucleosynthesis, and of young core-collapse supernova remnants. They may also make possible the development of an early warning system for the detection of impending core collapse.


INTRODUCTION
A significant subset of supernovae (SNe) appear to have suffered heavy and episodic pre-SN mass loss (Smith et al. 2011a;Smith 2014). This has not been explained by standard stellar evolutionary models, nor have the effects of episodic mass loss been included in them (see Langer [2012] and Maeder & Meynet [2000]). In part this may be because stellar evolution calculations are one dimensional (1D), have large time steps which damp dynamic effects, and many do not even progress past carbon burning. Such an approach may be valid for lower masses, but at higher masses several physical processes conspire to cause problems: (1) luminosities near the Eddington limit, (2) adiabatic exponent γ approaching 4/3 (or less) due to formation of electron-positron pairs, (3) complex and unstable nuclear burning [quasi-equilibrium (QSE) and nuclear statistical equilibrium (NSE) during oxygen and silicon burning], and (4) accelerated evolution due to copious energy loss by neutrinos. All these effects can become important and problematic if the stars suffer hydrodynamic instability, and could lead to significant errors in models of pre-SN stars.
There is empirical evidence of pre-SN hydrodynamic instability in both evolved massive stars in the Milky Way and Local Group, and in the dense circumstellar material (CSM) that must exist around Type IIn SNe and related events (see §2 for details). In the Milky Way and other local galaxies, the massive stars which are most notoriously observed to experience violent episodic mass loss are the luminous blue variables (LBVs ;Conti 1984;Humphreys & Davidson 1994;Smith et al. 2004Smith et al. , 2011bClark et al. 2005). This class of objects unifies a number of different specific types of blue supergiant variable stars, not all of which are necessarily in identical evolutionary phases, and most of which have persisted without a SN explosion for centuries. Most relevant are the so-called giant eruptions of LBVs, exemplified by the 19th century Great Eruption of η Carinae (see below), although η Car is perhaps the most extreme example of the phenomenon.
Extragalactic Type IIn SNe, named for their narrow H emission lines, are unique among SNe because they require CSM that is massive enough to decelerate the fast SN ejecta, allowing radiation from slow, heated material to dominate the observed spectrum. In order for the shock interaction to occur simultaneously and veil the broad lines from the underlying SN photosphere, this CSM must reside very close to the star (within a few 10 15 cm) and its ejection by the star must therefore have been synchronized to occur within only a few years of core collapse. This synchronicity, fully appreciated only in the last decade or so, hints at some as-yet unidentified phenomenon associated with the final nuclear burning phases in the star's life. We explore this topic in more detail in this paper.
It has been proposed that eruptive LBV-like events are connected to SNe IIn (Smith et al. , 2010aGal-Yam et al. 2007), but this has been controversial. It is in direct conflict with stellar evolution models for massive stars, all of which currently assume that massive stars with M ZAMS ≥ 40 M ⊙ at near-Solar metallicity will shed their H envelopes via steady winds to make Wolf-Rayet (WR) stars before core collapse as SNe of Type Ibc. 6 With currently adopted mass-loss rates (see Smith 2014) models predict that the most massive stars shed all their H envelope 0.5-1 Myr before exploding. However, a direct LBV/SN connection was recently confirmed in the remarkable case of SN 2009ip, which was a very massive star (50-80 M ⊙ ) that was being studied in detail as an LBV with multiple eruptions (see Smith et al. 2010b;Foley et al. 2011). While under study, this same eruptive star was then seen to explode as a Type IIn SN in 2012 , providing the best observed case yet of unstable pre-SN evolution.
In this paper we take a closer look at the latest stages of pre-core-collapse evolution and discuss some intriguing possibilities for how these pre-SN events may occur. After summarizing the key observational evidence for episodic pre-SN mass loss in §2, we discuss the detailed treatment of convection in stellar interiors in §3, and we then discuss a number of implications for the resulting mass loss and stellar structure in late evolution in §4.

RELEVANT OBSERVATIONAL EVIDENCE
In the past decade, empirical evidence has made it very clear that massive stars can experience violent episodic mass-loss events in their late evolutionary phases. Eruptive mass loss may be more important than steady winds in the integrated mass shed by the star in its lifetime (Smith & Owocki 2006), but the current generation of stellar evolution models does not account for it. Our aim here is to provide only a brief overview of the different lines of observational evidence; a more thorough discussion of pre-SN mass loss will appear in a forthcoming review (Smith 2014). We wish to diagnose the physical mechanisms that lead to this observed mass loss, and to this end, we first summarize the most important aspects of the relevant empirical evidence, including the types of progenitor stars that undergo these pre-SN outbursts, as well as timescales, mass budget, and energy budget.

LBVs
The LBVs are a rather diverse class, consisting of a wide range of irregular variable phenomena associated with evolved massive stars (Conti 1984;Humphreys & Davidson 1994;van Genderen 2001;Smith et al. 2004Smith et al. , 2011b, excluding those seen in cool supergiants. The most relevant variability attributed to LBVs is the socalled "giant eruption", in which stars are observed to dramatically increase their luminosity for months to years, accompanied by severe mass loss. The best studied example is the Galactic object η Carinae, providing us 6 This conflict with stellar evolution models persists even if we admit that the "LBV-like" pre-SN eruptions inferred for SNe IIn may not necessarily be from the same unstable stars that make up nearby LBVs (many of which have not exploded within centuries of a past giant LBV eruption). The conflict is mainly in the substantial mass of H left in the envelopes of these very massive stars at the time of death. with its historically observed light curve (Smith & Frew 2011;Frew 2004), as well as its complex ejecta that contain 10-20 M ⊙ and ∼10 50 ergs of kinetic energy (Smith et al. 2003). Aside from the less well-documented case of P Cygni's 1600 AD eruption, our only other examples of LBV-like giant eruptions are in nearby galaxies. The associated optical transients are often discovered in dedicated searches for SNe, and hence, they have sometimes been referred to more generally as "SN impostors", since their connection to true LBVs is not always clear. A number of these have been identified, with peak luminosities similar to η Car or less (Van Dyk & Matheson 2012;Smith et al. 2011b). Typical bulk expansion speeds in the ejecta are 100-1000 km s −1 (Smith et al. 2011b). LBVs are generally thought to be very massive stars, but their mass range is known to extend down to 25 M ⊙ (Smith et al. 2004) and some of the extragalacitc SN impostors appear to have relatively low-mass progenitors around 8 M ⊙ (e.g., Prieto et al. 2008;Thompson et al. 2009).
Although the giant eruptions themselves are rarely observed because they are infrequent and considerably fainter than SNe, a large number of LBVs and spectroscopically similar stars in the Milky Way and Magellanic Clouds are surrounded by massive shell nebulae, indicating previous eruptions with a range of ejecta masses from 1-20 M ⊙ (Clark et al. 2005;Smith & Owocki 2006;Wachter et al. 2010;Gvaramadze et al. 2010). Thus, LBV mass loss is inferred to be important in late evolution of massive stars.
Traditionally, LBVs have been cast as super-Eddington winds driven by an increase in the star's bolometric luminosity (Humphreys & Davidson 1994;Shaviv 2000;Owocki et al. 2004;Smith & Owocki 2006), but there is growing evidence that some of them are explosive hydrodynamic ejections (see Smith 2008. Of course, these two cases (long-lived super-Eddington winds or sudden hydrodynamic explosions) may both operate.
2.2. Type IIn supernovae and pre-SN mass loss As noted above, the narrow H lines in SNe IIn require very dense CSM ejected shortly before core collapse. As the SN blast wave expands into the CSM, the SN ejecta are decelerated and the slow CSM is heated (e.g., Chugai et al. 2004;Chugai & Danziger 1994). Through conservation of momentum, one can infer the mass of CSM required to decelerate the fast SN ejecta (Smith et al. 2010a). Special cases of this are the super-luminous SNe IIn (SN 2006gy, SN 2006tf, SN 2003ma, etc.) where the bolometric luminosity of the CSM interaction demands very large masses up to 20 M ⊙ ejected a few years 7 before core collapse (Smith & McCray 2007;Smith et al. , 2008Smith et al. , 2010aWoosley et al. 2007;Rest et al. 7 In a 25 M ⊙ star, the neutrino-cooled stage takes about 300 yr, most of which is spent in C burning, with 3 years in Ne burning, 1 year O burning, and 4 days Si burning (e.g., Arnett 1996). Woosley et al. (2002) give similar numbers: C burning is 520 yr (1100 yr), Ne burning is 0.9 yr (0.6 yr), O burning is 0.4 yr (0.9 yr), and Si burning is 0.7 d (2 d) for an initially 25 M ⊙ (75 M ⊙ ) star. Limongi & Chieffi (2012) find similar values. The dramatic reduction in time scales as burning advances is due to enhanced neutrino cooling at higher temperatures. Mixing algorithms affect the precise numbers; higher mass stars evolve faster, and have little C to burn. All of these estimates are inaccurate to the extent that they are sensitive to empirically determined mass-loss rates imposed in 2011). For the characteristic Type IIn spectrum and high luminosity from CSM interaction to occur immediately after explosion, the CSM must be very close (within few to several 10 15 cm) of the star; from the widths of narrow lines in spectra (typically 100-600 km s −1 ; Smith et al. 2008Smith et al. , 2010aKiewe et al. 2012) the mass ejection must have occurred just a few years before hand. This is a strong hint that something violent (i.e., hydrodynamic) may be happening to these stars during O and Ne burning, so that the star's outer layers are already affected by the impending core collapse.
2.3. Earlier mass loss Some SNe IIn (and even some other SN types) show indications of more distant CSM at radii of 10 3 AU (1.5 × 10 16 cm) or more, suggesting heavy mass loss that occurred over a longer timescale than just a few years or a decade (at 100 km/s it takes 150 years to reach a thousand AU). The evidence for this is that some SNe IIn remain luminous and show signatures of strong CSM/shock interaction for years as the shock continues to overtake more distant CSM ejected in centuries to millennia before the SN. Many SNe IIn show bright infrared (IR) echoes from distant dusty CSM heated by the SN luminosity, with massive shells again residing at radii of up to a few light years . Some non-IIn explosions show IR echoes and may have dense CSM shells as well; an illustrative example is SN 1987A, where the SN blast wave began crashing into a CSM ring located 0.2 pc from the star after a delay of 10 yr. Thus, both persistent CSM interaction and IR echoes suggest that heavy mass loss is not limited to only the few years preceding core collapse. On timescales of 100-1000 yr or more, this may indicate that enhanced episodic mass loss may be linked to C burning, and possibly even He burning. Binary interaction may play a role. These more distant (older) CSM shells are harder to detect, so we do not have a reliable census of what fraction of all SN progenitors experience this phenomenon.

Progenitor eruptions
There have now been two 8 clear direct detections of a precursor eruption, seen as a transient source just a few years before the SN. SN 2006jc was the first object clearly seen to have a brief outburst 2 years before a SN. The precursor event in 2004 had a peak luminosity similar to that of η Car (absolute magnitude of −14) 9 , and was fairly brief (only a few weeks; Pastorello et al. 2007). No spectra were obtained for the precursor transient source, but the SN explosion 2 years later had strong narrow emission lines of He, indicating dense CSM (Pastorello et al. 2007;Foley et al. 2007). 10 stellar evolution models, which do not adequately include binary effects (see Smith 2014 for a detailed discussion).
8 Actually, as our manuscript was in the review process, Fraser et al. (2013) reported the direct detection of another pre-SN outburst in archival data one year before the Type IIn-P event SN 2011ht. 9 With no bolometric correction this is a luminosity of ∼ 10 41 erg s −1 .
10 The Type Ibn spectrum of SN 2006jc is quite significant. Strong narrow He lines (and very weak H lines) indicate a Herich/H-poor CSM. This, in turn, implies that the progenitor probably had a relatively compact stellar radius. The fact that SN 2006jc A much more vivid case was SN 2009ip, mentioned earlier. It was initially discovered and studied in detail as an LBV-like outburst in 2009, again with a peak absolute magnitude near −14 and a spectrum similar to LBVs. A quiescent progenitor star was detected in archival Hubble Space Telescope (HST) data taken 10 yr earlier, which indicated a very massive star of 50-80 M ⊙ (Smith et al. 2010b;Foley et al. 2011). The object then experienced several similar eruptions over 3 yrs that looked like additional LBV eruptions (unlike SN 2006jc, detailed spectra of these progenitor outbursts were obtained), culminating in a final SN explosion in 2012 Smith et al. 2014). The SN light curve was doublepeaked, with initially faint bump (−15 mag) that had very broad emission lines (probably the SN ejecta photosphere), and it rose quickly 40 days later to a peak of −18 mag, when it looked like a normal SN IIn (caused by CSM interaction, as the SN crashed into the slow material ejected 1-3 years earlier; see Mauerhan et al. 2013;Smith et al. 2014). A number of detailed studies of the SN have now been published Prieto et al. 2013;Pastorello et al. 2013;Fraser et al. 2013;Smith et al. 2013Smith et al. , 2014Margutti et al. 2014;Ofek et al. 2013a). A related case is the Type IIn SN 2010mc (Ofek et al. 2013b), which had a double-peaked light curve that was nearly identical to SN 2009ip (Smith et al. , 2014, but did not have the same extensive pre-SN observations or a detection of precursor outbursts. While there was initially some debate about the nature of these objects, their late-time data revealed them to be true core-collapse events, and Smith et al. (2014) demonstrated that their double-peaked light curves were well explained as core-collapse SNe from blue supergiants, but with strong CSM interaction.
Another possible case of a detected pre-SN outburst is the historical object SN 1961V (Smith et al. 2011b;Kochanek et al. 2011); its pre-1961 photometry may have indicated a precursor outburst, but it was far less clearly delineated than in SN 2006jc and SN 2009ip. In any case, if we take the full class of SNe IIn, which represent 8-9% of all core-collapse SNe in large galaxies (Smith et al. 2011a), the phenomenon is far too common to be caused by the pulsational pair instability that operates in very massive stars (as discussed below).  (Smith et al. 2011c). In this last case, however, the SN has not yet faded, so the candidate progenitor might be a massive young cluster; if so it still suggests a massive star above 30 M ⊙ (Smith et al. 2011c). All exhibited the same type of pre-SN eruptions as seen in SNe IIn indicates that an extended stellar radius and encounters with a companion may not be the only explanation for precursor outbursts, and that something deeper is at work. Woosley et al. (2007) suggested that the precursor outburst of SN 2006jc might have been a pulsational pair instability eruption.
four cases suggest progenitor stars that are much more massive than the typical red supergiant progenitors of SNe II-P (Smartt 2009). (This does not, however, preclude the possibility that some lower-mass stars produce SNe IIn as well, since the most luminous LBV-like progenitors are the easiest to detect.) 2.6. Statistics From a volume-limited sample in a survey with controlled systematics, Type IIn SNe appear to make up roughly 8-9% of all core-collapse events (Smith et al. 2011a). To get a sense of what this might mean we note that it corresponds roughly to the ratio of the number of stars in the range 30-100 M ⊙ to the total number in 8-100 M ⊙ (Arnett 1996, Table 14.4). SNe IIn are the most diverse and poorly understood of all SN typesany SN type can appear as a Type IIn if it has dense CSM. Some SNe IIn appear to be Type Ia with CSM, and some appear to be low-energy electron capture SNe from 8-10 M ⊙ stars, but the majority appear to be from more massive LBV-like stars (see, e.g., Smith 2014 and references therein).
These statistics indicate that the precursor eruptions leading to SNe IIn are far too common for all of them to be attributable to the pulsational pair instability (Arnett 1996, §11.7;Woosley et al. 2007;Chatzopoulos & Wheeler 2012), which is expected to occur only in very rare, very massive stars with initial masses around 100 M ⊙ or more. Some other instability must operate in most SNe IIn.
The relative rarity of SNe IIn is also important: As we consider implications for the dominant physical mechanism that may lead to pre-SN eruptions, we must be mindful that the vast majority of SNe (∼90%) do not suffer violent precursor outbursts that are powerful enough to yield dense, observable CSM -and it is important to know why. Is the relevant instability only significant at higher initial mass and luminosity ( §3, 4.1, 4.2)? Does the fraction of SNe IIn reflect rare interactions in a binary system at the right separation (e.g. §4.3), or some other special circumstance? The process is rare, but not too rare. Do stars of lower initial mass (say 10-30 M ⊙ ) that make the majority of SNe also encounter similar instabilities before core collapse, but they are less powerful and do not cause hydrodynamic ejection in these stars? Is there a continuum in energy and mass ejection in pre-SN eruptions that extends below what can readily be detected in bright SNe IIn, so that some fraction of "normal" SNe may also suffer pre-SN instability? If the pre-SN activity is too weak to cause severe mass ejection, does it nevertheless have potential implications for the initial conditions for core collapse ( §4.4)? Although these questions are inspired by observations and get to the heart of the puzzle, observations alone do not yet yield definitive answers. Below we discuss these issues from a theoretical perspective.

IN LATE STAGES
In parallel to the observational progress outlined in §2, theoretical progress, in the form of direct numerical simulations of 2D and 3D turbulence in late burning stages of stars, has provided valuable insights into the physical nature of this evolution (Bazàn & Arnett 1994, 1998Asida & Arnett 2000;Meakin & Arnett 2006, 2007bArnett & Meakin 2011a;Viallet, et al. 2013). These simulations indicate that 1D models, as presently implemented, will not capture essential aspects of turbulent convection (especially its time-dependent and fluctuating nature) or its influence on stellar structure and evolution. This, in turn, may lead to inaccuracies in pre-SN stellar structure that are fundamentally important.
In what follows we will use some common terms in a precise way; for clarity we define them here.
• perturbation: a change in a variable which is small compared to its average value, so that linear perturbation theory is valid.
• fluctuation: a change in a variable which is comparble to its average value, so that nonlinear terms are important, and linear analysis is invalid.
• eruption: an event in which the rate of change of the average value of a variable (the "background") is comparable to that of fluctuations. Such events are often associated with mass loss because, by this definition, the energy of the eruption is comparable to the internal energy, and hence the binding energy of the star.
In the stellar context, eruptions often lead to mass loss. They may be especially violent and significant in stars near the Eddington limit with loosely bound envelopes.

An Example of O-burning in 3D
Significant progress has been made in understanding such turbulent, dynamic behavior theoretically, by careful analysis of well-resolved 3D numerical simulations. Meakin & Arnett (2007b) examined the burning of 16 O through 8 turnover times in a 23 M ⊙ star, a few days prior to core collapse (as estimated by 1D models). There was no other significant burning on the grid, and the initial state was early in oxygen shell burning. The initial evolution in this 3D simulation was a rapid readjustment in which a dynamically self-consistent set of pressure and velocity fluctuations grew from low level noise (turbulent kinetic energy increased from less than 10 42 to more than 10 47 ergs) to form a convective region. The lack of consistent turbulent fluctuations is a common characteristic of 1D models: they are not self-consistent regarding the dynamics of convection. Even well chosen guesses at initial velocities do not help because they must be properly phased relative to entropy fluctuations, information we do not have from 1D models. The natural state of the convection, which develops in another turnover time, is dynamic, and structured in a complex way in both space and time, as demanded by the fluid flow equations in the turbulent regime.
As the simulation in Meakin & Arnett (2007b) advanced, the convection proceeded with pulses of turbulent kinetic energy.
The burning was mild, almost constant, and the fluctuations in kinetic energy and luminosity instead came from the turbulent flow. Arnett & Meakin (2011b) show that this dynamic behavior is similar to that generated in the Lorenz (1963) model of a convective roll, well known in meteorology and dynamic systems theory. During this early, mild stage of oxygen burning, the fluctuations did not (yet) induce significant changes in the energy generation rate, although fluctuations in turbulent kinetic energy and in enthalpy flux give variations of more that 50%.

Simulating A More Extreme Case
Of particular relevance are the subsequent results of Arnett & Meakin (2011a) in which it was shown that 2D simulations, of combined O and Si burning shells in a 23 M ⊙ star, evolved from a quasi-steady state into eruptive behavior. The eruptions are violent and probably cause extensive mass loss, but do not blast the core completely apart. In this more extreme case the fluctuations in temperature induced by turbulent flow do affect the energy generation rate, and allow coupling between the flow, the oxygen burning, and the silicon burning. This is a new and complex problem, and the timedependence of burning may be of fundamental importance to observed transient sources. The prediction of this eruptive behavior is surprising, being entirely different from that inferred from 1D stellar evolution codes, for which O and Si burning stages are assumed to be quasi-static. However, indications of this violent behavior have been seen in all hydrodynamic simulations of Si burning which could show it (Bazàn & Arnett 1998;Meakin 2006;Arnett & Meakin 2011b). Another example of violent, nondisruptive behavior is the pulsational electron-positron pair instability SN models mentioned in §2.6, but they are extremely rare because they will be limited to the massive cores arising from stars with extremely high initial mass. Unsteady nuclear burning may be spread across a wider range of initial mass.

The importance of fluctuations
The calculations reviewed above illustrate that there is an important limitation in the standard 1D approach to the late stages of stellar evolution: it ignores finite amplitude fluctuations which have nonlinear interactions with the nuclear burning. Here fluctuations differ from perturbations in that the nonlinear terms are not small enough to discard, so the assumption of quasi-static burning is invalid.
Linear stability theory (e.g., Murphy, Burrows & Heger 2004) will not capture this behavior because the driving and damping terms are nonlinear (they involve second-and third-order correlations between fluctuations; Arnett & Meakin 2011b ). The problem gets worse (the nonlinear effects get bigger) as neutrino emission accelerates the evolution of the star in the final moments approaching core collapse (Fowler & Hoyle 1964). Unno, et al. (1989) have stressed that the standard version of stability theory now used is missing a potentially important term: the dynamic effects of convection, which is on the same level as the terms giving rise to the excitation mechanisms for the standard instabilities (e.g., the ǫ, κ, and δ mechanisms, op. cit., p. 241). Arnett & Meakin (2011b) have derived a turbulence model as a possible replacement for mixing length theory (MLT), with the constraint that it reproduce the dynamic, fluctuating behavior seen in the 3D simulations (Meakin & Arnett 2007b). It includes two advances which were published in the western literature only after the original work of Erika Böhm-Vitense (Böhm-Vitense 1958): the Lorenz model (Lorenz 1963) and the Kolmagorov damping at small scales (Kolmogorov 1962); see Arnett & Meakin (2011b) for details. We will rederive the Böhm-Vitense equation of mixing length theory, upon which most 11 stellar evolutionary codes are based, and we will show that it artificially suppresses fluctuations. This illustrates, explicitly and analytically, that conventional stellar evolution ignores a major dynamical aspect of late stages of stellar evolution -the same stages during which the existence of eruptions and explosive mass loss now have firm observational evidence (see §2), as well as confirmation in the multi-dimensional fluid-dynamic simulations (Arnett & Meakin 2011a).

MLT and the Lorenz model
It has long been known that nothing like an MLT blob appears in multi-dimensional well-resolved simulations (e.g., Freytag, Ludwig, & Steffan (1996), Stein & Nordlund (1998), Bazàn & Arnett (1998)), yet MLT has proven so useful that it is still the algorithm of choice for most stellar evolution codes (see Langer 2012 for a review).
Perhaps there is some truth in the mathematics of the MLT equations, if not the physics of the blob picture. If so, then it should be possible to derive the same (or very similar) equations from a conceptual picture more consistent with the behavior actually seen in the simulations. We do this below.
The mathematical basis for MLT as used in stellar evolution is the Böhm-Vitense cubic equation (Vitense 1953;Böhm-Vitense 1958) which may be written as where x = (∇ − ∇ e ) and W = ∇ r − ∇ a (see Kippenhahn & Weigert (1990), §7.2 for this notation). This equation is constructed from three equations (their Eq.s 7.6, 7.14 and 7.15), which are Here u 2 0 = gβ T H P , and where H P is the pressure scale height, ℓ m the mixing length, β T the coeficient of thermal expansion, C P the specific heat at constant pressue, and the other symbols have their usual meaning. The expressions ∇, ∇ e , ∇ a , and ∇ r represent the dimensionless temperature gradients ((∂ ln T /∂ ln P ) i ), for the background, the convective element, the adiabat, and that temperature gradient required to carry all the luminosity, respectively.  (2011b)). The speed of flow around the loop is u, and T i , i = 1, 2, 3, 4 represents the amplitude of the potential temperature fluctuation relative to T 0 , at different key points in and around the loop.

The Vortex Model
MLT is not unique; we illustrate this with another theoretical model that can produce a cubic equation like that of Böhm-Vitense (Eq. 1). To construct an alternative, we begin with the Lorenz (1963) model of a convective roll (see Fig. 1), which has been shown to exhibit key properties of the simulation behavior as noted above (Arnett, Meakin, & Young 2009;Arnett & Meakin 2011b). This model includes chaotic behavior and finite fluctuations in velocity and luminosity. The dissipation is assumed to be that implied by the Kolmogorov cascade, which affects the effective Peclet number 12 for the flow (Arnett & Meakin 2011b). Transformation of variables for non-dimensionality leads to where σ = Γ/K is the Prandtl number 13 , Γ is the inverse of the effective viscous damping time, K = ν T (2/ℓ) 2 is the thermal relaxation time, ν T is the thermal diffusivity, t is the time variable, ℓ is the diameter of the convective roll, and u is the convective speed around the roll. The aspect ratio of the roll is a so that b = 4/(1 + a 2 ) deals with the excess in vertical over horizontal heat conduction. Lorenz took b = 8/3 and σ = 10. These values give behavior similar to that found in 3D simulations; see Arnett & Meakin (2011b) for further discussion. Fig. 1 illustrates the temperatures around the roll. The mean temperature at the midplane of the roll is T 0 ; the amplitude of the variation in background temperature in the vertical direction is T 1 . The fluctuating temperature 12 The Peclet number is the ratio of the advective transport rate to the diffusive transport rate. In a turbulent medium its definition requires some care because of the turbulent cascade.
13 The Prandtl number is the ratio of the momentum diffusivity to the thermal diffusivity. amplitude in the horizontal direction is T 3 ; in the vertical it is T 2 . It is convenient to define the temperature amplitude difference T 4 = T 1 − T 2 . These are "potential temperatures": variations relative to the adiabatic run of temperature.
We have where r = (gℓ/2C P T 1 )σ is the Rayleigh number 14 . The only functional change with respect to the original set of equations (Lorenz 1963) is the dissipation term in the dX/dτ equation, with σX → X|X|/2. This is a consequence of the fact that, for stellar dimensions, viscosity is effective at the end of the turbulent cascade, not at the integral scale of the roll. Eq.s 7, 8 and 9 define a "vortex" model for convection if we use the identifications made below for stellar variables. Numerical solution of these equations in the context of hydrodynamic stellar structure gives rise to chaotic behavior similar to that found by Lorenz (1963); see Arnett & Meakin (2011b).

The Steady Vortex Model
To make contact with MLT, we take the steady state solutions of the vortex model; we will call this the "steady vortex" model. We then have three algebraic equations: as in Lorenz (1963) for his steady state case. In MLT X|X| → X 2 by definition. Converting back to dimensional variables (see Fig. 1 and Arnett & Meakin 2011b), the first equation becomes where the Kolmogorov damping length is ℓ d = α d ℓ and the diameter of the roll is ℓ. Unlike MLT, α d is not an adjustable parameter, but may be determined directly from simulations: for a shallow convection zone α d ≈ 0.8, while for a deep one (strong stratification), α d ≈ 0.4 (Viallet, et al. 2013). The T variables are temperature amplitudes and u is the velocity amplitude as defined by Arnett & Meakin (2011b). We use potential temperature (temperature relative to its adiabatic value) to include some effects of compressibility and stratification (we emphasize that this includes some effects, not all; see Viallet, et al. (2013)).
The second dimensionless equation becomes where To keep the algebra concise, we definē which is similar to the mixing length quantity U defined above, The third equation becomes In order to once again make contact with MLT variables, we equate finite differences of amplitudes to derivatives. The temperature field in the vortex model is two-dimensional (horizontal and vertical), not onedimensional (radial) as in MLT. The horizontal derivative in density (temperature) is needed to drive a buoyant torque, so we identify and Eq. 13 becomes which is to be compared to Eq. 2. With the appropriate choice of mixing length ℓ m they are identical. This implies In the Lorenz model, ℓ has a precise meaning: the diameter of the roll, whereas in MLT ℓ m is an adjustable parameter of order H P . Here u 0 appears as a natural velocity scale, where u 2 0 = gβ T H P . We further identify the vertical temperature gradient relative to the adiabatic gradient by and finally the imposed vertical gradient relative to adiabatic as Using Eqs. 19 and 22 with 14 gives which is to be compared to Eq. 3. The two factors have many components in common; equating them implies which relates the MLT mixing length to the diameter of the convective roll. This differs from Eq. 21, but not drastically. This contradiction stems from the fact that in MLT, one adopts the length of a blob mean-freepath as being the same as its size for radiative cooling (Arnett, Meakin, & Young 2010); this is related to the "geometric factor" problem.
Using Eqs. 19, 22, and 23 with 18 gives which is to be compared to Eq. 4. There are two differences in these two equations. First, Eq. 4 has a factor ∇ r − ∇ while Eq. 26 has an apparently different factor ∇ r − ∇ e . This equation arises from the condition that the total flux is the sum of convective and radiative fluxes (flux of turbulent kinetic energy is ignored in both cases). Both theoretical models require this to be true, but define the radiative flux differently in the convective zone. Outside the convective region, ∇ r = ∇ = ∇ e , so there is no difference. Inside the convective region, the Lorenz model takes the whole region to be the convective "element" (this resembles the simulations), so that the radiative flux is that of the "element". MLT assumes that the "element" is distinct from the background, and that the radiative flux is that of the background. This difference is large only in the superadiabatic region, where neither theory resembles simulations nor is self-consistent (see above). In most stars the thickness of the superadiabatic region is determined by the hydrogen recombination region, and is much less than a pressure scale height. Such details are ignored in both MLT and the vortex model, but they are prominent in simulations (Stein & Nordlund 1998;Nordlund, Stein, & Asplund 2009). The second difference is in the coefficients multiplying these factors, equating them gives Removing the common factors gives While the choices (Eq. 21, 25, and 28) for mixing length needed for equality are surprisingly similar, they are not identical due to the different conceptual foundations used. Note that Equations 21 and 25 are necessarily inconsistent if we make the reasonable choice of using the α d determined by simulations. At this level of precision MLT is inconsistent with hydrodynamics. The Lorenz model is mathematically precise for the single mode of flow which was chosen; simulations show that several low-order modes (∼ 5) dominate, so that a single mode picture is oversimplified; e.g., see Arnett & Meakin (2011b). The MLT derivation has a plethora of factors of 2 and of geometry which are not unique, and MLT has adjustable parameters. It is unlikely that this difference in factors (of order unity) is significant.
Both MLT and Lorenz convection assume symmetry between up-flows and down-flows; this is inconsistent with simulations, which due to up-down asymmetry, have non-zero acoustic (F P ) and kinetic energy (F K ) fluxes (Meakin & Arnett 2007b;Arnett, Meakin, & Young 2009;Viallet, et al. 2013). The asymmetry increases for increasing stratification.

Böhm-Vitense Equation Recovered
As we have seen, the steady vortex model produces three basic algebraic equations almost identical to those making up the Böhm-Vitense version of MLT. Combining Eq.s 20, 24, and 26 gives an equation strikingly akin to Eq. 1 (the Böhm-Vitense cubic equation), These differ in (1) U andŪ as discussed above, and (2) the lack of an x 2 term due to the different definition of radiative flux in a convective zone. The x 2 = ∇ − ∇ e term may be neglected in both the large x and small x limits, and has only a modest effect in the transition region. Eq.s 1 and 29 have the same asymptotic behavior and similar constant factors. Physically this transition between large and small x corresponds to that between the adiabatic gradient and the radiative gradient. The central regions of massive stars are never in this transition region; it occurs in the Sun (and other stars) near the surface, associated with the hydrogen ionization zone. On purely mathematical grounds, the steady vortex model must give comparable agreement with observations as MLT. Inside a stellar evolutionary code the two are almost identical, and any small differences may be removed by slight adjustments of assumptions involving flow patterns, or mixing length and geometric parameters, for example. The mathematical equivalence of MLT and the steady vortex model via the Böhm-Vitense equation should serve as a warning. The physical pictures used in the two models are different, and therefore the interpretation of MLT in terms of physical processes is not unique. The steady vortex picture may prove more useful for physical interpretation, to the extent that it has connections to actual solutions of the Navier-Stokes equations in the form of 3D numerical simulations.
Both MLT and the steady vortex picture lack fluctuations. For the steady vortex picture this is because use of the steady state solutions removes the terms that cause chaotic behavior (the Lorenz strange attractor). These terms do not exist in MLT.
Thus, Eq.s 10, 11, and 12 (the steady vortex model) have no chaotic behavior while Eq.s 7, 8, and 9 (the dynamic vortex model) do. This explicitly demonstrates a fundamental difference between MLT and the more general dynamic vortex model. It is equivalent to using a damping (to get the steady state solution), and ignoring the dynamic aspects of turbulence. This neglect has potentially important consequences for models of the late stages of stellar evolution, which we will discuss below. The dynamic vortex model, which is more directly based on the 3D fluid dynamics equations than MLT, allows a more direct way to generalize stellar evolution codes by including additional physics, and a better conceptual base to deal with the physics involved. It can be shown that use of an acceleration equation in a vortex picture provides a way to include dynamics, boundary layers, and the turbulent cascade into a simple convection model (Arnett, et al., in preparation). In addition, it provides a natural link to the Richardson-Kolmogorov cascade and gives a balance between driving and damping of turbulent flow.
Stellar evolutionary models are based on an implicit assumption of stability. The most common tool for exploring instability is linear perturbation analysis (Cox 1980;Unno, et al. 1989), which deals with convection poorly (although perhaps adequately for the relatively stable main-sequence phase). The driving term for turbulence is quadratic and the damping term is cubic; neither is linear, so linear analysis is blind to them. However, numerical hydrodynamics is not so blind, and such simulations clearly deviate from stellar evolution models in the latest stages of a massive star's life, for which turbulent convection plays a dominant role. In this paper we have precisely identified the culprit -the chaotic behavior implicit in a convective roll. Steady-state solutions become irrelevant when they are unstable; the Lorenz (1963) model of a convective roll, with its stange attractor (chaotic behavior), illustrates this. It appears that the 3D simulations have similar chaotic behavior which give rise to convective pulses (Meakin & Arnett 2007b) and perhaps eruptions (Arnett & Meakin 2011a).
Why not just simulate the whole star? The combination of the need for high resolution to capture the turbulence (high numerical Reynolds number is required) and sufficient volume to include both the core and active burning shells, results in a computational problem that strains present-day computer facilities. This is presently being attempted (Meakin & Arnett, in preparation), but it is appropriate now to examine the implications of this new perspective.

IMPLICATIONS
In order for models to provide a coherent picture of the observations, we need some process to be able to inject 10 48 to 10 50 ergs 15 in the final couple of years before core collapse (i.e. during Ne, O, and Si burning), and we may need something similar but with lower energy to be more prolonged (several hundred years) during C burning and possibly even He burning. While Ne burning is weak and Si burning complex, both C and O burning release about 5 × 10 17 ergs g −1 of fuel burned (or equivalently, 10 51 ergs per M ⊙ of fuel burned), so that a modest amount of nuclear burning could directly provide the energy needed. This is relevant to the issue of shell instabilities (Arnett & Meakin 2011a).
Must the energy be supplied directly, or could it be the result of some less than completely efficient process? For core collapse to occur, at least 1.5 × 10 51 ergs (1.5 bethes) must be released by nuclear burning to convert helium burning products into iron-peak nuclei 16 . This is much larger than the gravitational binding energy of the star, so that to avoid explosion and mass loss this energy must be radiated away.
Neutrinos can do the job, but vigorous convection is demanded. It has long been known (Fowler & Hoyle 1964;15 This range of energy comes from the most extreme events (η Car, SN 2006gy's precursor eruption, etc.) to the least extreme LBV-like giant eruptions (such as V12 in NGC 2403, plus a number of objects in between. These exhibit a range of ∼100 in ejected mass at similar outflow speeds (see Smith et al. 2011b). Note that individual eruptions with energy well below 10 48 ergs may exist, but could be difficult to detect in external galaxies. On the other hand, energy injection of 10 47 erg or less may be absorbed by the stellar envelope, and might not lead to the hydrodynamic ejection that gives rise to a brief optical transient anyway (e.g., Dessart et al. 2010). 16 A mass of 1.5 M ⊙ converted from C 12 and O 16 releases about 5 × 10 17 ergs g −1 in burning to Si 28 (Fowler & Hoyle 1964), or 5 × 10 17 × 3 × 10 33 = 1.5 × 10 51 ergs. Further burning could release slightly more energy, but is countered by neutrino loss from nuclear weak interactions (Arnett 1996). Arnett 1996) that, because of the more sensitive temperature dependence of thermonuclear heating in comparison to neutrino cooling, that a convection zone in thermal balance will have heating at the bottom and cooling at the top. This generates an entropy gradient that drives convective transport of heat upward. To prepare the core for collapse requires turbulent convection to transport 1.5 bethes of energy, a factor of 15 more than the largest value needed to explain the observed pre-collapse ejection of mass. Any inefficiency in the transport process can therefore contribute to the injection of energy into a star's outer envelope. Depending on how much energy is diverted, the results can range from modest to catastrophic. It would take less than 7% of 1.5 bethes to supply the largest eruption energy (10 50 ergs) required above.
Below we highlight six ways in which the dynamical instability associated with turbulent convection in late burning stages might profoundly influence the star's pre-SN evolution, the stellar structure at the moment of collapse, the observational interpretation of supernova remnants, and isotopic abundances in meteoritic pre-solar grains.

Enhanced mass loss
A characteristic of turbulence is the presence of chaotic fluctuations in luminosity and velocity. Convective velocities rise from ∼0.5 km s −1 in He burning to 100 km s −1 in O burning, and 300 km s −1 in Si burning (Meakin & Arnett 2007b;Arnett & Meakin 2011a). This is subsonic in the core (mach number ∼ 0.01 to 0.1) but not in the envelope. Simulations show vigorous wave production at convective boundaries (Meakin & Arnett 2007b). Waves generated by vigorous turbulence travel outward and dissipate (e.g., steepen into shocks); consequently, the envelope becomes an absorber for this energy. If the envelope is already near Eddington luminosity, it can only rid itself of this extra energy by dynamical expansion or mass loss. For red giant structure, this would happen on a pulsation (sound travel) time scale, and appear as an eruption. For more condensed structure (a star stripped of its H envelope, for example), it might be seen as a vigorous wind 17 . Quataert & Shiode (2012) have discussed this mechanism of enhanced pre-SN mass loss driven by waves propagating through the star's envelope as a result of furious convection during O and Ne burning. This type of mechanism has the potential to explain strong winds that may produce dense CSM in the few years leading up to core collapse, although it is unclear if it can account for the most violent explosive events that occur.

Unsteady or explosive burning
The episodic nature of the observed pre-SN mass loss suggests that, in addition to strong winds that are quasisteady state by definition, we should consider hydrodynamic instabilities in shell burning with energy injection by explosive/unsteady nuclear burning. Both would be in operation once neutrino cooling dominates over photon cooling, that is, in the months and centuries prior to core collapse, from C burning to Si burning. There should be a wide variety of behavior because both the instabilities and the wave transport will be different for different core-envelope structures and rotational rates, both of which also depend upon the history of any binary interaction.
To make a bomb requires an energy source and an ignition mechanism to release that energy. The 1.5 bethes which are available from burning the ashes of He burning to iron-peak nuclei is a more than ample source; only a small fraction is required. For an ignition mechanism we already have the instability found by Arnett & Meakin (2011a). That simulation was terminated because the matter was flowing off the limited grid, but already had violently excited the lowest order mode available to fluctuations, with every indication of continued growth. Mild shocks were already forming. Despite the severe challenge to computer resources, it is important to repeat these simulations in 3D, on a 4π steradian grid, extended to later times (Meakin, et al. , in preparation). It is also important that the hydrodynamic algorithms used be non-damping; although anelastic, low Machnumber and implicit methods may allow larger time steps (Viallet, et al. 2011;Almgren, et. al 2010), they must be validated to insure that they give negligable artificial damping 18 for this problem, and that they seamlessly transition from turbulent to explosive flow.
There may be other instabilities to be found; the published library of well-resolved simulations of pre-collapse evolution is still small. While oxygen shell burning seems quasi-stable over 8 turnovers (Meakin & Arnett 2007b), it is vigorous; the possibility of eruption after a longer time remains open. Both C and Ne burning are more difficult to simulate directly because of their slower burning (which implies that more time steps are needed to calculate them over significant evolutionary times). We do not yet know if they harbor nonlinear instabilites which would become evident over the time scales needed to consume C and Ne.

Triggering violent binary interactions
There is another avenue for extremely sudden and violent events to occur. Recent observations of mainsequence O-type stars have shown definitively that the majority of massive stars (roughly 2/3) are born in binary systems with a separation small enough that the two stars will interact before they die (Sana et al. 2012;Chini et al. 2012;Kiminki et al. 2012). The fate of this interaction depends on the initial separation: the closest systems will exchange mass or merge on the main sequence, while binary systems with wider separations will interact only when the more massive star evolves off the main sequence and expands to fill its Roche lobe as a supergiant.
If the mechanisms mentioned above (unsteady or explosive burning, waves generated by vigorous convection) are able to inject an amount of energy into the star's envelope that is not quite sufficient to completely unbind the envelope, the result may be a swelling of the star's hydrostatic radius (i.e. a large pulsation). In that case, dramatic events may ensue if the star is in a binary sys- tem with an orbital period ranging from 10s of days to a few yr (depending on eccentricity). If the SN progenitor increases its radius significantly, a companion star that previously had been too distant to interact may suddenly find itself to be the victim of mass transfer, a merger, or a violent collision if it is in an eccentric orbit.
Consider the case of η Car, where something similar is known to have occurred: η Car is a binary with a 5.5 yr orbital period and an eccentricity e=0.9-0.95. Smith & Frew (2011) showed that brief ∼100 day brightening events occurred at times of periastron, and Smith (2011) argued that a stellar collision must have occurred because the periastron separation was much smaller than the required photospheric radius at that time. Mauerhan et al. (2013) pointed out that the multiple brief peaks in the few years leading up to the 2012 explosion of SN 2009ip were reminiscent of η Car's events, and suggested that violent periastron encounters may play a role. The SN impostor SN 2000ch may be yet another example (Pastorello et al. 2010;Smith et al. 2011b). A related idea was suggested by Chevalier (2012), who also included the intriguing possibility of mergers with a neutron star companion leading to SNe IIn, although he found that such events are probably too rare to account for the observed frequency of SNe IIn.
The possibility of energy injection that leads to envelope inflation might act to enhance the frequency of any such merger/collision events, because the progenitor suddenly finds itself to have a larger radius and is more able to interact with more widely separated companion stars (here "suddenly" means a time comparable to the orbital period). More importantly, if the increase in stellar radius is a result of energy injected during C, Ne, O, or Si burning phases, it provides a physical reason to expect such merger or collision events to be synchronized to within only a few years 19 before core collapse.
While debate continues over details involved in these calculations (Janka 2012), a unifying characteristic is that all results are highly sensitive to the core structure of the model progenitor. For example, this progenitor structure determines the rate at which mass rains down onto the proto-neutron star after core collapse (see O'Conner & Ott 2013, and their discussion of "compactness"; Ugliano et al. 2012). Real SNe are observed to explode nonetheless. Thus, it seems prudent to ask: What if the fault lies not so much in the details of the method of computing neutrino transport, the geometry (2D vs. 3D), or other aspects of the explosion, but in the structure of the progenitor itself ?
The progenitor models most commonly used by the groups doing core collapse simulations are those of  The actual duration of these phases of C, Ne, O, and Si burning might be modified by their dynamic behavior.
What effect might a proper treatment of convection and its associated hydrodynamic instabilities have on the stellar structure of a SN progenitor? Observations dictate that at least in some stars, pre-SN instabilities inject energy into the star's envelope that leads to significantly enhanced mass loss or even explosive ejection. This is not predicted by evolution models (but is indicated by the hydrodynamic simulations), yet something must be providing substantial extra energy on short timescales before collapse. One may surmize that successful mass ejection prior to core collapse is most likely to occur in more massive stars, since they are closer to the Eddington luminosity and have more loosely bound envelopes. What would happen, then, in lower-mass progenitor stars if the same process occurs, but where the injected energy is insufficient to cause observable eruptive/explosive mass loss? As in §4.3, one might infer that the outer layers may swell and that binary interaction might be more likely, or alternatively, that the mass loss might be less vigorous and the consequences less easily observed.
We will not know what the detailed changes will be until realistic 3D simulations illustrate the strongly nonlinear behavior. Some general trends are clear, however. For example, both the enhanced mass loss and unsteady burning move matter out of the potential well, changing 20 the central condensation of the core and making the progenitors easier to explode by the neutrino-transport mechanism (O'Conner & Ott 2013;Ugliano et al. 2012). The "compactness" parameter of O 'Conner & Ott (2013) is essentially the gravitational potential at the edge of the core; in hydrostatic equilibrium it acts as a pivot point for the density profile, so that higher central density, and lower compactness, is accompanied by lower mantle density (see Fig. 10.4 and 10.7 in Arnett 1996). The mapping of initial (ZAMS) mass to mass of the core at collapse may also be altered. This is a complex problem because of unaddressed issues in 1D evolution relating to convective boundaries (Meakin & Arnett 2007b;Viallet, et al. 2013) as well as large uncertainties in the treatment of mass loss (Smith 2014).
Turbulent fluctuations in the collapsing cores will also break spherical symmetry. They will be a result of both instantaneous driving by current burning and historical flows during the approach to collapse. Estimates for the amplitude and the form of these fluctuations are available. In the most advanced published simulation (Arnett & Meakin 2011a) there seems to be a complex interaction between O and Si burning shells, both wildly turbulent, driving pulsational modes of the core and mantle. The fluctuations are a factor of 10 or more larger than perturbations used to test core collapse simulations, and are of global rather than local scale (see Fryer & Young 2007;Wongwathanarat, Janka, & Müller 2010 for simulations with lower amplitude fluctuations). Such fluctuations may lead to enhanced energy flow, and to increasing neutrino luminosity, so they may make explosion more likely. Since a preprint of our paper was posted, simulations have confirmed our basic suggestion; Couch & Ott (2014) have explored just one aspect of this problem, with encouraging results. Using the Arnett & Meakin (2011a) data, they imposed a modest nonradial velocity (momentum conserving), corresponding to an initial convective flow in the mantle of their collapsing core. At 100 milliseconds after core collapse, this material passed through the stalled shock, and by 200 milliseconds the core is exploding. This small modification of the velocity field was sufficient to change a dud into a successful explosion by allowing the revival of the shock, showing the critical importance of realistic initial models for core collapse.

Mixing in Presupernovae: Meteoritic Grains
The unsteady burning and its resulting eruptions may significantly change the mixing history of the progenitor. Convective boundaries are not sacrosanct; there is mixing across the boundaries even in quasistatic phases (Meakin & Arnett 2007b;Moćak, et al. 2008Moćak, et al. , 2009. Eruptions add to this, and turbulent mixing is faster and different from the "diffusive convection" algorithm currently in use. Isotopes (nuclei with the same proton number Z but different neutron number N) are good records of thermonuclear history because they are resistent to changes caused by chemistry (different Z). The development of precise microprobes and the discovery of grains in meteorites whose crystalization predated the solar system has given a new window into nucleosynthesis. The changed picture of presupernova evolution sketched above should have profound implications for the interpretation of isotopic anomalies in pre-solar grains in meteorites (for example, see Pignatari, et al. 2013;Travaglio, et al. 1999;Lugaro et al. 2001). Not only do we have possible changes in the bulk character of the initial models and the explosion mechanism to consider, but also new processes. First, turbulent fluctuations involve not only thermodynamic fields (e.g., temperature or density) and velocity, but also composition. Each turbulent burning zone will introduce a "cosmic scatter" in the detailed abundances; see Fig. 4 of Meakin & Arnett (2007b). Second, advective mixing often proceeds by "plumes", which allow matter from one level (and characteristic temperature and composition) to punch through other layers to finally mix in yet another level (with different composition), without homogenizing intermediate levels. This "layer-jumping" may be more likely as the advection becomes more and more violent. There seem to be unresolved issues in isotopic anomaly interpretation that this might help. Third, these two new processes would occur before the explosion shock from core collapse, which ejects the mantle from the core. Thermonuclear processing during this explosion would therefore be altered. 4.6. Mixing before, during, and after the explosion: Young SNRs Most of the interpretation of the observed asymmetries and mixing in young SN remnants has proceeded with the assumption that the explosion itself drove the asymmetries, rather than them being already developed prior to explosion. For core collapse this presents a problem. To form a collapsed core implies a radial compression factor of 100 or so. Asymmetries grow on compression, but decrease on expansion (this is the fundamental problem of inertial confinement fusion, Lindl 1998). Core asymmetries will tend to be made smaller (more spherical) on expansion by a large factor as compared to pre-existing asymmetries in the mantle. In addition, the mantle asymmetries are likely to be increased by local explosive burning.
Young core-collapse SN remnants may contain evidence for asymmetries directly related to turbulent fluctuations in pre-collapse phases. Tantalizing new maps of abundances in Cas A are now available (Milisavljevic & Fesen 2013;Isensee et al. 2012) which beg for detailed comparison with multidimensional simulations (Ellinger, et al. 2013). Pre-existing density fluctuations in the mantle seem indicated (Isensee et al. 2012). Such fluctuations will be caused by the type of turbulent convection in the progenitor that we described above. Advective turnover occurs on a timescale slightly longer than the explosive time scale, so that mixing will be partial, and will tend to preserve (in a distorted way) any layering in abundance. The more thoroughly burned matter will have higher entropy, and punch through overlying layers in some regions. Any 56 Ni produced will decay and heat the plasma, giving a characteristic modification (Ni bubbles) of the explosive outflow (Milisavljevic & Fesen 2013).

CONCLUSIONS
Both observations of pre-SN mass loss and direct numerical simulations of convection are discordant with predictions of conventional stellar evolution codes for the late stages of massive star evolution. In this paper, we have discussed the observed nature of the most dramatic discrepancies (pre-SN eruptions), and we have proposed that the problem in 1D stellar evolution codes lies in the treatment of turbulent convection. In particular, 1D codes ignore finite amplitude fluctuations in velocity and temperature, and their nonlinear interaction with nuclear burning. For most of a star's life this is probably a reasonable approximation (except perhaps near convective boundaries) but in the latest phases of evolution such fluctuations can become catastrophic in massive stars. The fluctuations are not allowed to occur in conventional 1D codes that impose steady state behavior, so their associated pre-SN eruptions are not predicted.
In order to derive MLT from a more general hydrodynamic formulation, the following assumptions are needed: (1) the damping is consistent with the Kolmogorov cascade, (2) the large scale dynamics could be approximated by a Lorenz convective roll, and (3) a steady-state solution was appropriate. While the first two are consistent with the 3D simulations and with experimental work on turbulent flows, the last is certainly not. Use of MLT therefore underestimates dynamic be-havior by artificially damping fluctuations in velocity and temperature.
This may be adequate (or not) through H and He core burning, but the traditional treatment of convection becomes increasingly unrealistic in the late stages of stellar evolution, which are accelerated by neutrino losses. This stage of cooling coincides with an evolutionary phase for which we have strong observational evidence of vigorous eruptive mass loss. In contrast, convection for photoncooled stages (H and He burning) is less strongly modified, although treatments of boundary layers, composition gradients, and discontinuities remain questionable.
Carbon burning is an interesting transitional stage; it is the first stage to be cooled primarily by neutrino emission, and the vigor of carbon burning depends upon the abundance of 12 C resulting from the previous helium burning (Arnett 1972b). The 12 C abundance also depends on the algorithm used for convective mixing (Arnett 1972a). The precise reaction rates remain elusive; see, e.g., Gai (2013); Brune & Sayre (2013). For small values of 12 C (massive cores), carbon will burn nonconvectively (and feebly), followed by weak neon burning (carbon burning produces 20 Ne, the fuel for neon burning). For large values of 12 C, carbon burning will be vigorous, lasting ∼ 3 × 10 11 seconds (10 4 years), to be compared with oxygen burning which lasts for ∼ 9 × 10 8 seconds (30 years) (Arnett 1974). Carbon burning occurs at a lower temperature, hence has a lower rate of neutrino cooling to balance than oxygen burning, and can thus last longer. These values are quoted for helium cores, which are more relevant to SNIbc events. While oxygen burning may be directly simulated in 3D, turbulent carbon burning, over time scales long enough for carbon exhaustion, remains beyond present computational capability. The behavior of massive stellar cores during carbon burning rests on 1D models and speculation at present, but observations suggest there may be enhanced episodic mass loss during this stage (see section 2.3).
We conclude that the use of MLT and "diffusive convection" are likely to be a dominant cause of the substantial discrepancies between 1D stellar models and actual observed pre-supernova stars. Realistic inclusion of turbulence in stellar codes is a frontier topic, as is the inclusion of rotation, binary interaction, and eruptive mass ejection, with which it interacts. Strong empirical evidence for eruptions and violent binary interaction occuring in the few years before core collapse suggests a link to instability from turbulent convection in the latest burning phases. Both interpretation of isotopic anomalies in pre-solar grains in meteorites, and initial models for gravitational collapse should be strongly influenced by the likely alteration of progenitor structure.
Finally, we note that if the surface layers of a SN progenitor star are indeed modified prior to the impending explosion in a predictable way, it now becomes conceivable to design an early warning system for core collapse.
The key would be to recognize the observational signature resulting from energy deposition caused by the type of mechanisms discussed herein. They could manifest as a sudden swelling of the star, sudden onset of violent binary interaction, or eruptive mass loss; less extreme cases may in principle be recognized by their asteroseismological signatures. Some of these eruptive effects have been observed in stars that have not yet experienced core collapse (such as nearby LBV stars that experienced eruptions in past centuries or millenia), so the observational patterns of pre-SN outbursts need to be studied in more detail. Indentification of pre-SN stars would be useful for detection of their neutrinos, gravitational waves, and early stages of nearby SNe and core collapse. Present and future transient surveys should be examined with this in mind.
This work was supported in part by NSF Award 1107445 at the University of Arizona. One of us (DA) wishes to thank the Aspen Center for Physics (ACP) and the Kavli Institute for Theoretical Physics (KITP) for their hospitality and support. We wish to thank Casey Meakin, Maxime Viallet, Christian Ott, Jeremiah Murphy, and Bill Paxton for helpful discussions.