Hydrodynamics of core-collapse supernovae and their progenitors
LLiving Reviews in Computational Astrophysics manuscript No. (will be inserted by the editor)
Hydrodynamics of core-collapse supernovae and theirprogenitors
Bernhard M ¨uller
Received: : 26 November 2019 / Accepted: : 18 April 2020
Abstract
Multi-dimensional fluid flow plays a paramount role in the explosions ofmassive stars as core-collapse supernovae. In recent years, three-dimensional (3D)simulations of these phenomena have matured significantly. Considerable progresshas been made towards identifying the ingredients for shock revival by the neutrino-driven mechanism, and successful explosions have already been obtained in a numberof self-consistent 3D models. These advances also bring new challenges, however.Prompted by a need for increased physical realism and meaningful model validation,supernova theory is now moving towards a more integrated view that connects multi-dimensional phenomena in the late convective burning stages prior to collapse, theexplosion engine, and mixing instabilities in the supernova envelope. Here we reviewour current understasnding of multi-D fluid flow in core-collapse supernovae andtheir progenitors. We start by outlining specific challenges faced by hydrodynamicsimulations of core-collapse supernovae and of the late convective burning stages.We then discuss recent advances and open questions in theory and simulations.
Keywords
Supernovae · Massive stars · Hydrodynamics · Convection · Instabilities · Numerical methods
Contents a r X i v : . [ a s t r o - ph . S R ] J un Bernhard M¨uller2.1.1 Problem geometry and choice of grids . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Challenges of subsonic turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . 172.1.3 High-Mach number flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Treatment of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Hydrostatic balance and conservation properties . . . . . . . . . . . . . . . . . . . 242.2.2 Treatment of general relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.3 Poisson solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Reactive flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Late-stage convective burning in supernova progenitors . . . . . . . . . . . . . . . . . . . . . . 293.1 Interior flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Supernova progenitor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Convective boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Current and future issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Core collapse and shock revival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1 Structure of the accretion flow and runaway conditions in spherical symmetry . . . . . . . 474.2 Impact of multi-dimensional effects on shock revival . . . . . . . . . . . . . . . . . . . . 504.3 Neutrino-driven convection in the gain region . . . . . . . . . . . . . . . . . . . . . . . . 514.4 The standing accretion shock instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Perturbation-aided explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.6 Outlook: Rotation and magnetic fields in neutrino-driven explosions . . . . . . . . . . . . 614.7 Proto-neutron star convection and LESA instability . . . . . . . . . . . . . . . . . . . . . 635 The explosion phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 The early explosion phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Explosion energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Compact remnant properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4 Mixing instabilities in the envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
The death of massive stars is invariably spectacular. In the cores of these stars, nuclearfusion proceeds all the way to the iron group through a sequence of burning stages.At the end of the star’s life, nuclear energy generation has ceased in the degenerateFe core (i.e., the core has become “inert”), while nuclear burning continues in shellscomposed of progressively light nuclei further out. Once the core approaches its ef-fective Chandrasekhar mass and becomes sufficiently compact, electron captures onheavy nuclei and partial nuclear disintegration lead to collapse on a free-fall timescale, leaving behind a neutron star or black hole. In most cases, a small fraction ofthe potential energy liberated during collapse is transferred to the stellar envelope,which is expelled in a powerful explosion known as a core-collapse supernova , asfirst recognized by Baade and Zwicky (1934).How precisely the envelope is ejected has remained one of the foremost questionsin computational astrophysics ever since the first modeling attempts in the 1960s(Colgate et al. 1961; Colgate and White 1966). In this review we focus on the criticalrole of multi-dimensional (multi-D) fluid flow during the supernova explosion itselfand the final pre-collapse stages of their progenitors.For pedagogical reasons, it is preferable to commence our brief exposition ofmulti-D hydrodynamic effects with the supernova explosion mechanism rather thanto follow the sequence of events in nature, or historical chronology. ydrodynamics of core-collapse supernovae and their progenitors 3 erg of potential energy, which appears more than sufficient to account forthe typical inferred kinetic energies of observed core-collapse supernovae of order10 erg (see, e.g., Kasen and Woosley 2009; Pejcha and Prieto 2015).Transferring the requisite amount of energy from the young “proto-neutron star”(PNS) is not trivial, however. The simplest idea is that the energy is delivered whenthe collapsing core overshoots nuclear density and “bounces” due to the high in-compressibility of matter above nuclear saturation density, which launches a shockwave into the surrounding shells (Colgate et al. 1961). However, the shock wavestalls within milliseconds as nuclear dissociation of the shocked material and neu-trino losses drains its initial kinetic energy (e.g., Mazurek 1982; Burrows and Lat-timer 1985; Bethe 1990). The shock then turns into an accretion shock, whose radiusis essentially determined by the pre-shock ram pressure and the condition of hydro-static equilibrium between the shock and the PNS surface. It typically reaches a radiusbetween 100–200 km a few tens of milliseconds after bounce and then recedes again.Among the various ideas to “revive” the shock (for a more exhaustive overviewsee Mezzacappa 2005; Kotake et al. 2006; Janka 2012; Burrows 2013) the neutrino-driven mechanism is the most promising scenario and has been explored most com-prehensively since it was originally conceived – in a form rather different from themodern paradigm – by Colgate and White (1966). The modern version of this mech-anism is illustrated in Fig. 1b: A fraction of the neutrinos emitted from the PNS andthe cooling layer at its surface are reabsorbed further out in the “gain region”. If theneutrino heating is sufficiently strong, the increased thermal pressure drives the shockout, and the heating powers an outflow of matter in its wake.However, according to the most sophisticated spherically symmetric (1D) modelsusing Boltzmann neutrino transport to accurately model neutrino heating and cool-ing, this mechanism does not work in 1D (Liebend¨orfer et al. 2001; Rampp and Janka2000) except for the least massive supernova progenitors. For all other progenitors,it is crucial that multi-D effects support the neutrino heating. Convection occurs in thegain region because neutrino heating establishes a negative entropy gradient (Bethe1990), and was shown to be highly beneficial for obtaining neutrino-driven explo-sions by the first generation of multi-D models from the 1990s (Herant et al. 1992,1994; Burrows et al. 1995; Janka and M¨uller 1995, 1996). Another instability, thestanding-accretion shock instability (SASI; Blondin et al. 2003; Blondin and Mezza-cappa 2006; Foglizzo et al. 2006, 2007; Laming 2007), arises due to an advective-acoustic amplification cycle and gives rise to large-scale ( (cid:96) = ,
2) shock oscillations;it plays a similarly beneficial role in the neutrino-driven mechanism as convection(Scheck et al. 2008; M¨uller et al. 2012a). Rapid rotation could also modify the dy-namics in the supernova core and support the development of neutrino-driven explo-sions (Janka et al. 2016; Summa et al. 2018; Takiwaki et al. 2016). More precisely, the neutrino-driven mechanism works in 1D for progenitors with a steeply decliningdensity profile outside the core (M¨uller 2016) as in the case of electron-capture supernovae from super-AGB stars (Kitaura et al. 2006) or the least massive iron core progenitors (Melson et al. 2015b). Bernhard M¨uller
Fig. 1
Overview of the multi-D effects operating prior to and during a core-collapse supernova as dis-cussed in this review (below the dashed line) in the broader context of the evolution of a massive star. a) After millions of years of H and He burning, the star enters the neutrino-cooled burning stages (C,Ne, O, Si burning). These advanced core and shell burning stages tpyically proceed convectively becauseburning and neutrino cooling at the top and bottom of an active shell/core establish an unstable negativeentropy gradient. The interaction of the flow with convective boundaries can lead to mixing and transferenergy and angular momentum by wave excitation. Rotation may modify the flow dynamics. b) After thestar has formed a sufficiently massive iron core, the core undergoes gravitational collapse, and a youngproto-neutron star is formed. The shock wave launched by the “bounce” of the core quickly stalls, and islikely revived by neutrino heating in most cases. In the phase leading up to shock revival, neutrino heatingdrives convection in the heating or “gain” region, and the shock may execute large-scale oscillations duethe standing accretion shock instability (SASI). Rotation and the asymmetric infall of convective burningshells can modify the dynamics. There is also a convective region below the cooling layer at the proto-neutron star (PNS) surface. c) After the shock has been revived and sufficient energy has been pumpedinto the explosion by neutrino heating or some other mechanism, the shock propagates through the outershells on a time scale of hours to days. During this phase, the interaction of the (deformed) shock with shellinterfaces as well as reverse shock formation trigger mixing by the Rayleigh–Taylor (RT), the Richtmyer–Meshkov (RM) instability, and (as a secondary process) the Kelvin–Helmholtz (KH) instability. Once theshock breaks out through the stellar surface, the explosion becomes visible as an electromagnetic transient.Mixing instabilities continue to operate on longer time scales throughout the evolution of the supernovaremnant.
Evidently, multi-D effects are also at the heart of the most serious alternativeto neutrino-driven explosions, the magnetohydrodynamic (MHD) mechanism (e.g.,Akiyama et al. 2003; Burrows et al. 2007a; Winteler et al. 2012; M¨osta et al. 2014b),which likely explains unusually energetic “hypernovae”. But whether core-collapsesupernovae are driven by neutrinos or magnetic fields, it is pertinent to ask: Howimportant are the initial conditions for the multi-D flow dynamics that leads to shockrevival?1.2 The multi-D structure of supernova progenitorsFor pragmatic reasons, supernova models have long relied on 1D stellar evolutionmodels as input, or at best on “1.5D” rotating models using the shellular approx- ydrodynamics of core-collapse supernovae and their progenitors 5 imation (Zahn 1992). For non-rotating progenitors, spherical symmetry was eitherbroken by introducing perturbations in supernova simulations by hand, or due to gridperturbations. For rotating and magnetized progenitors, spherical symmetry is bro-ken naturally, but on the other hand the stellar evolution models do not provide thedetailed multi-D angular momentum distribution and magnetic field geometry, whichmust be specified by hand.In reality, even non-rotating progenitors are not spherically symmetric at the onsetof collapse. Outside the iron core, there are typically several active convective burningshells (Fig. 1a) that will collapse in the wake of the iron core within hundreds ofmilliseconds. It was realized in recent years that the infall of asymmetric shells canbe important for shock revival (Couch and Ott 2013; M¨uller 2015; Couch et al. 2015;M¨uller et al. 2017a).The multi-D structure of supernova progenitors is thus directly relevant for theneutrino-driven mechanism, but the potential ramifications of multi-D effects dur-ing the pre-collapse phase are in fact much broader: How do they affect mixing atconvective boundaries, and hence the evolution of the shell structure on secular timescales? How do they affect the angular momentum distribution and magnetic fieldsin supernova progenitors?1.3 Observational evidence for multi-D effects in core-collapse supernovaeObservations contain abundant clues about the multi-D nature of core-collapse su-pernova explosion. Large birth kicks of neutron stars (Hobbs et al. 2005; Faucher-Gigu`ere and Kaspi 2006; Ng and Romani 2007) and even black holes (Repetto et al.2012) cannot be explained by stellar dynamics alone and require asymmetries in thesupernova engine. There is also evidence for mixing processes during the explosionand large-scale asymmetries in the ejecta from the spectra and polarization signa-tures of many observed transients (e.g., Wang and Wheeler 2008; Patat 2017), andfrom young supernova remnants like Cas A (Grefenstette et al. 2014).The relation between the asymmetries in the progenitor and the supernova core,and the asymmetries in observed transients and gaseous remnants is not straightfor-ward, however. The observable symmetries are rather shaped by mixing processesthat operate as the shock propagates through the stellar envelope (Fig. 1c). Rayleigh–Taylor instability occurs behind the shock as it scoops up material and decelerates(Chevalier 1976; Bandiera 1984), and the interaction of a non-spherical shock withshell interfaces can give rise to the Richtmyer–Meshkov instability (Kifonidis et al.2006). The asymmetries imprinted during the first seconds of an explosion providethe seed for these late-time mixing instabilities, and 3D supernova modellling is nowmoving towards an integrated approach from the early to the late stages of the to bet-ter link the observations to the physics of the explosion mechanism (e.g., Hammeret al. 2010; Wongwathanarat et al. 2013; M¨uller et al. 2018; Chan et al. 2018, 2020),and, in future, even to the multi-D progenitor structure.
Bernhard M¨uller
Modeling the late stages of nuclear burning and the subsequent supernova explosioninvolves solving the familiar equations for mass, momentum, and energy conserva-tion with source terms that account for nuclear burning and the exchange of energyand momentum with neutrinos. Viscosity and thermal heat conduction mediated byphotons, electron/positrons, and ions can be neglected, and so we have (in the New-tonian limit and neglecting magnetic fields), ∂ ρ∂ t + ∇ · ( ρ v ) = , (1) ∂ ρ v ∂ t + ∇ · ( ρ v ⊗ v ) + ∇ P = − ρ ∇ Φ + Q m , (2) ∂ ρ ( ε + v / ) ∂ t + ∇ · (cid:2) ρ ( ε + v / ) v + P v (cid:3) = − ρ v · ∇ Φ + Q e + Q m · v , (3)in terms of the density ρ , the fluid velocity v , the pressure P , internal energy density ε , the gravitational potential Φ , and the neutrino energy and momentum source terms Q e and Q m . If we take ε to include nuclear rest-mass contributions, there is no source ydrodynamics of core-collapse supernovae and their progenitors 7 term for the nuclear energy generation rate; otherwise an additional source term ˙ Q nuc appears on the right-hand side (RHS) of Eq. (3). These equations are supplementedby conservation equations for the mass fractions X i of different nuclear species andthe electron fraction Y e (net number of electrons per baryon), ∂ ρ X i ∂ t + ∇ · ( ρ X i v ) = ˙ X i , burn , (4) ∂ ρ Y e ∂ t + ∇ · ( ρ Y e v ) = ˙ Q Y e , (5)where the source terms ˙ X i , burn and ˙ Q Y e account for nuclear reactions and the changeof the electron fraction by β processes.In the regime of sufficiently high optical depth, the effect of the neutrino sourceterms could alternatively be expressed by non-ideal terms for heat conduction, vis-cosity, and diffusion of lepton number (e.g., Imshennik and Nadezhin 1972; Bludmanand van Riper 1978; Goodwin and Pethick 1982; van den Horn and van Weert 1983,1984; Yudin and Nadyozhin 2008), but this approach would break down at low opticaldepth. The customary approach to Eqs. (1–5) is, therefore, to apply an operator-splitapproach and combine a solver for ideal hydrodynamics for the left-hand side (LHS)and the gravitational source terms with separate solvers for the source terms dueto neutrino interactions and nuclear reactions. Simulations of the Kelvin–Helmholtzcooling phase of the PNS over time scales of seconds form an exception; here onlythe PNS interior is of interest so that it is possible and useful to formulate the neutrinosource terms in the equilibrium diffusion approximation (Keil et al. 1996; Pons et al.1999).2.1 HydrodynamicsA variety of computational methods are employed to solve the equations of ideal hy-drodynamics in the context of supernova explosions or the late stellar burning stages.Nowadays, the vast majority of codes use Godunov-based high-resolution shock cap-turing (HRSC) schemes with higher-order reconstruction (see, e.g., LeVeque 1998b;Mart´ı and M¨uller 2015; Balsara 2017 for a thorough introduction). Examples includeimplementations of the piecewise parabolic method of Colella and Glaz (1985) orextensions thereof in the Newtonian hydroydnamics codes P ROMETHEUS (Fryxellet al. 1989, 1991; M¨uller et al. 1991), which has been integrated into various neutrinotransport solvers by the Garching group (Rampp and Janka 2002; Buras et al. 2006b;Scheck et al. 2006), its offshoot P
ROMPI (Meakin and Arnett 2007b), PPM
STAR (Woodward et al. 2019), VH1 (Blondin and Lufkin 1993; Hawley et al. 2012) asused within the C
HIMERA transport code (Bruenn et al. 2018), F
LASH (Fryxell et al.2000), C
ASTRO (Almgren et al. 2006), A
LCAR (Just et al. 2015), and F
ORNAX (Skin-ner et al. 2019). This approach is also used in most general relativistic (GR) hydro-dynamics codes for core-collapse supernovae like C O C O N U T (Dimmelmeier et al.2002; M¨uller et al. 2010), Z
ELMANI (Reisswig et al. 2013; Roberts et al. 2016), andGRH
YDRO (M¨osta et al. 2014a). Godunov-types scheme with piecewise-linear total-variation diminishing (TVD) reconstruction are still used in the F
ISH code (K¨appeli
Bernhard M¨uller et al. 2011) and in the relativistic F
UGRA code of (Kuroda et al. 2012, 2016b). The3DnSNe code of the Fukuoka group (e.g., Takiwaki et al. 2012), which is based onthe ZEUS code of Stone and Norman (1992), has also switched from an artificial vis-cosity scheme to a Godunov-based finite-volume approach with TVD reconstruction(Yoshida et al. 2019).Alternative strategies are less frequently employed. The V
ULCAN code (Livne1993; Livne et al. 2004) uses a staggered grid and von Neumann–Richtmyer artificialviscosity (Von Neumann and Richtmyer 1950). The SNSPH code (Fryer et al. 2006)uses smoothed-particle hydrodynamics (Gingold and Monaghan 1977; Lucy 1977;for modern reviews see Price 2012; Rosswog 2015). Although less widely used insupernova modeling today, the SPH approach has been utilized for some of the earlystudies of Rayleigh–Taylor mixing (Herant and Benz 1991) and convectively-drivenexplosions (Herant et al. 1992) in 2D and later for the first 3D supernova simula-tions with gray neutrino transport (Fryer and Warren 2002). Multi-dimensional mov-ing mesh schemes have been occasionally employed to simulate magnetorotationalsupernovae (Ardeljan et al. 2005) and reactive-convective flow in stellar interiors(Dearborn et al. 2006; Stancliffe et al. 2011). More recently “second-generation”moving-mesh codes based on Voronoi tessellation, such as T
ESS (Duffell and Mac-Fadyen 2011) and A
REPO (Springel 2010), have been developed and employed forsimulations of jet outflows (Duffell and MacFadyen 2013, 2015), and fallback su-pernovae (Chan et al. 2018). Spectral solvers for the equations of hydrodynamics,while popular for solar convection, have so far been applied only once for simulat-ing oxygen burning (Kuhlen et al. 2003), but never for the core-collapse supernovaproblem.Since Godunov-based finite-volume solvers are now most commonly employedfor simulating core-collapse supernovae and the late burning stages, we shall focuson the problem-specific challenges for this approach in this section.
The physical problem geometry in global simulations of core-collapse supernovaeand the late convective burning stages is characterized by approximate spherical sym-metry, and one frequently needs to deal with strong radial stratification and a largerange of radial scales. For example, during the pre-explosion and early explosionphase, the PNS develops a “density cliff” at its surface that is approximately in radia-tive equilibrium and can be approximated as an exponential isothermal atmospherewith a scale height H of H = kT R GMm b , (6)in terms of the PNS mass M , radius R , and surface temperature T , and the baryonmass m b . With typical values of M ∼ . M (cid:12) , R shrinking down to a final value of ∼
12 km, and a temperature of a few MeV, the scale height soon shrinks to a few100 m. Later on during the explosion, the scales of interest shift to the radius of theentire star, which is of order ∼ km for red supergiants. ydrodynamics of core-collapse supernovae and their progenitors 9 The spherical problem geometry and the multi-scale nature of the flow is a criticalelement in the choice of the numerical grid for “star-in-a-box” simulations. Cartesiangrids, various spherical grids, and, on occasion, unstructured grids have been used inthe field for global simulations and face different challenges.
Grid-induced perturbations.
Cartesian grids have the virtue of algorithmic simplicityand do not suffer from coordinate singularities, but also come with disadvantages asthey are not adapted to the approximate symmetry of the physical problem. The un-avoidable non-spherical perturbations from the grid make it impossible to reproducethe spherically symmetric limit in multi-D even for perfectly spherical initial condi-tions, or to study the growth of non-spherical perturbations in a fully controlled man-ner. The former deficiency is arguably an acceptable sacrifice, though it can limit thepossibilities for code testing and verification, but the latter can introduce visible arti-facts in simulations. For example, Cartesian codes sometimes produce non-vanishinggravitational wave signals from the bounce of non-rotating cores (Scheidegger et al.2010), and often show dominant (cid:96) = Handling the multi-scale problem.
Furthermore, a single Cartesian grid cannot eas-ily handle the multiple scales encountered in the supernova problem. Even with O ( ) zones, such a grid can at best cover the region inside 1000 km with accept-able resolution, but following the infall of matter for several 100 ms without bound-ary artifacts and the development of an explosion requires covering a region of atleast 10 ,
000 km. This problem is often dealt with by using adaptive mesh refinement(AMR; see, e.g., Berger and Colella 1989; Fryxell et al. 2000), which is usuallyimplemented as “fixed mesh refinment” for pre-defined nested cubic patches (e.g.,Schnetter et al. 2004). Other codes have opted to combine a single central Cartesianpatch or nested patches with a spherically symmetric region (Scheidegger et al. 2010)or multiple spherical polar patches (Ott et al. 2012) outside. For long-time simulationsof Rayleigh–Taylor mixing in the envelope, standard adaptive or pre-defined mesh re-finement may not be sufficiently efficient for covering the range of changing scalesand necessitate manual remapping to a coarser grids (“homographic expansion”) asthe simulation proceeds (Chen et al. 2013).In spherical coordinates, the multi-scale nature of the problem can be accommo-dated to a large degree by employing a non-uniform radial grid that transitions toroughly equal spacing in log r at large radii. Radial resolution can be added selec-tively in strongly stratified regions like the PNS surface (Buras et al. 2006b), or onecan use an adaptive moving radial grid (Liebend¨orfer et al. 2004; Bruenn et al. 2018).However, some care must be exercised in using non-uniform radial grids. Rapid vari-ations in the radial grid resolution ∆ r / r can produce artifacts such as artificial wavesand disturbances of hydrostatic equilibrium.It is also straightforward to implement a moving radial grid to adapt to changingresolution requirements or the bulk contraction/expansion of the region of interest; Of course, some problems can or need to be studied using simplified geometries (planar or cylindrical)or local simulations.0 Bernhard M¨uller see Winkler et al. (1984); M¨uller (1994) for an explanation of this technique. TheMPA/Monash group routinely apply such a moving radial grid in quasi-Lagrangianmode during the collapse phase (Rampp and Janka 2000), and, with a prescribedgrid function, in parameterized multi-D simulations of neutrino-driven explosions(Janka and M¨uller 1996; Scheck et al. 2006) and in simulations of convective burn-ing (M¨uller et al. 2016b). The Oak Ridge group uses a truly adaptive radial grid intheir supernova simulations with the C
HIMERA code (Bruenn et al. 2018). A movingradial mesh might also appear useful for following the expansion of the ejecta andthe formation of a strongly diluted central region in simulations of mixing instabili-ties in the envelope, but the definition of an appropriate grid function is non-trivial.Most simulations of mixing instabilities in spherical polar coordinates have there-fore relied on simply removing zones continuously from the evacuated region of theblast wave to increase the time step (Hammer et al. 2010) rather than implementinga moving radial mesh (M¨uller et al. 2018).Both fixed mesh refinement and spherical grids with non-uniform radial meshspacing only provide limited adaptability to the structure of the flow. Truly adaptivemesh refinement can provide superior resolution in cases where very tenuous, nonvolume-filling flow structures emerge. Mixing instabilities in the envelope are a primeexample for such a situation, and have often been studied using AMR in sphericalpolar coordinates (Kifonidis et al. 2000, 2003, 2006) in 2D and Cartesian coordinates(Chen et al. 2017).
The time step constraint in spherical polar coordinates.
While spherical polar coor-dinates are well-adapted to the problem geometry, they also suffer from drawbacks.One of these drawbacks — among others that we discuss further below — is that theconverging cell geometry imposes stringent constraints on the time step near the gridaxis and the origin. The Courant–Friedrichs–Lewy limit (Courant et al. 1928) for thetime step ∆ t requires that ∆ t < r ∆ θ / ( | v | + c s ) in 2D and ∆ t < r sin θ ∆ ϕ / ( | v | + c s ) in 3D in terms of the grid spacing ∆ θ and ∆ ϕ in latitude and longitude, and the fluidvelocity v and sound speed c s . If ∆ θ = ∆ ϕ , this is worse than a Cartesian code withgrid spacing comparable to ∆ r by a factor ∆ θ (cid:28) ∆ θ (cid:28) ROMETHEUS -V ERTEX , C
HIMERA , C O C O N U T) sim-ulate the innermost region of the grid assuming spherical symmetry. The approxima-tion of spherical symmetry is well justified in the core, since the innermost region ofthe PNS is convectively stable during the first seconds after collapse and explosionuntil the late Kelvin–Helmholtz cooling phase. Even more savings can be achieved bytreating the PNS convection zone using mixing-length theory (M¨uller 2015), but thisis a more severe approximation that significantly affects the predicted gravitationalwave signals and certain features of the neutrino emission and nucleosynthesis. Con-cerns have also been voiced that the imposition of a spherical core region creates animmobile obstacle to the flow that leads to the violation of momentum conservation,which might have repercussions on neutron star kicks (Nordhaus et al. 2010a). Whileit is true that the PNS tends not to move in simulations with a spherical core region,Scheck et al. (2006) found (using a careful analysis based on hydro simulations in the ydrodynamics of core-collapse supernovae and their progenitors 11a) b) c)
Fig. 2
Alternative spherical grids that avoid the tight time step constraint at the axis of standard spher-ical polar grids: a) Grid with mesh coarsening in the ϕ -direction only. Only an octant of the entire gridis shown. b) Dendritic grid with coarsening in the θ - and ϕ -direction Image reproduced with permis-sion from (Skinner et al. 2019), copyright by AAS. c) Overset Yin-Yang grid (Kageyama and Sato 2004;Wongwathanarat et al. 2010a) with two overlapping spherical polar patches in yellow and cyan. accelerated frame comoving with the PNS) that the assumption of an immobile PNSdoes not gravely affect the dynamics in the supernova interior and the PNS kick inparticular.Even with a 1D treatment for the innermost grid zones, one is still left with a se-vere time-step constraint at the grid axis in 3D. A number of alternatives to sphericalpolar grids with uniform spacing in latitude θ and longitude ϕ can help to remedythis. The simplest workaround is to adopt uniform spacing in µ = cos θ instead of θ . In this case, one has sin θ = ( N θ − ) / / N θ ≈ √ N − / θ in the zones adjacentto the axis for N θ zones in latitude instead of sin θ ≈ N − θ /
2, so the time step limitscales as ∆ t ∝ √ N − / θ N − φ instead of ∆ t ∝ N − θ N − φ /
2, where N ϕ is the number ofzones in longitude. Alternatively, one can selectively increase the θ -grid spacing inthe zones close to the axis. However, the time step constraint at the axis is still morerestrictive than at the equator in this approach, and the aspect ratio of the grid cellsbecomes extreme near the pole, which can create problems with numerical stabilityand accuracy.One approach to fully cure the time step problem, which was first proposed forsimulations of compact objects by Cerd´a-Dur´an (2009), consists in abandoning thelogically Cartesian grid in r , θ , and ϕ and selectively coarsening the grid spacing in ϕ (and possibly θ ) near the axis (and optionally at small r ) as illustrated in Fig. 2. Sucha mesh coarsening scheme has been included in the C O C O N U T-FMT code (M¨uller2015) with coarsening in the ϕ -direction, and as a “dendritic grid” with coarsen-ing in the θ - and ϕ -direction in the F ORNAX code (Skinner et al. 2019) and the3DnSNe code (Nakamura et al. 2019). Mesh coarsening can be implemented follow-ing standard AMR practice by prolongating from the coarser grids to the finer gridsin the reconstruction step. Alternatively, one can continue using the hydro solver ona fine uniform grid in θ and ϕ , and average the solution over coarse “supercells”after each time step, followed by a conservative prolongation or “pre-reconstruction” step back onto the fine grid to ensure higher-order convergence. This has the ad-vantage of retaining the data layout and algorithmic structure of a spherical polarcode, but care is required to to ensure that the prolongation of the conserved vari-ables does not introduce non-monotonicities in the primitive variables, which limitsthe pre-reconstruction step to second-order accuracy in practice (M¨uller et al. 2019).A possible concern with mesh coarsening on standard spherical polar grids is that itmay favor the emergence of axis-aligned bipolar flow structures during the explosionphase in supernova simulations (M¨uller 2015; Nakamura et al. 2019). In practice,however, strong physical seed perturbations easily break any grid-induced alignmentof the flow with the axis. As the more simulations with mesh coarsening becomeavailable (M¨uller et al. 2019; Burrows et al. 2020), it does not appear that axis align-ment is a recurring problem.Filtering in Fourier space, which has long been used in the meteorology com-munity (Boyd 2001), provides another means of curing the restrictive time step con-straint near the axis and has been implemented in the C O C O N U T-FMT code (M¨ulleret al. 2019). This can also be implemented with minimal interventions in a solverfor spherical polar grids, and is attractive because the amount of smoothing that isapplied to the solution increases more gradually towards the axis than with meshcoarsening schemes. M¨uller et al. (2019) suggests that this eliminates the problemof axis-aligned flows. On the downside, simulations with Fourier filtering may occa-sionally encounter problems with the Gibbs phenomenon at the shock.More radical solutions to the axis problem include overset grids and non-orthogonalspherical grids. An overset Yin-Yang grid (Kageyama and Sato 2004) has been im-plemented in the P
ROMETHEUS code (Wongwathanarat et al. 2010a; Melson 2013)and used successfully for simulations of supernovae and convective burning. TheYin-Yang grid provides near-uniform resolution in all directions, solves the time stepproblem, and also eliminates the delicate problem of boundary conditions at the axisof a spherical polar grid. The added algorithmic complexity is limited to interpolationroutines that provide boundary conditions; since each patch is part of a spherical po-lar grid, no modifications of the hydro solver for non-orthogonal grids are required.As a downside, it is more complicated – but possible (Peng et al. 2006) – to imple-ment overset grids in a strictly conservative manner. In future, non-orthogonal gridsspherical grids (Ronchi et al. 1996; Calhoun et al. 2008; Wongwathanarat et al. 2016)may provide another solution that avoids the axis problem and ensures conservationin a straightforward manner, but applications are so far limited to other astrophysicalproblems (Koldoba et al. 2002; Fragile et al. 2009; Shiota et al. 2010).
Boundary conditions.
The definition of the outer boundary conditions for Cartesiangrids can be more delicate and less flexible than for spherical grids. Simulations ofsupernova shock revival and the late convective burning stages usually do not coverthe entire star for efficiency reasons, and sometimes it can even be desirable to excisean inner core region, e.g. the PNS interior in supernova simulations (e.g., Janka andM¨uller 1996; Scheck et al. 2008; Ugliano et al. 2012; Ertl et al. 2016; Sukhbold et al.2016) or the Fe/Si core in the O shell burning models of M¨uller et al. (2017a). Tominimize artifacts near the outer and (for annular domains) the inner boundary, thebest strategy is often to impose hydrostatic boundary conditions assuming constant ydrodynamics of core-collapse supernovae and their progenitors 13 entropy, so that the pressure P , density ρ , and radial velocity v r in the ghost cells areobtained as d P = (cid:90) ρ g d r , (7)d ρ = (cid:90) c − d P , (8) v r = . (9)in terms of the radial gravitational acceleration g and the sound speed c s (cf. Zingaleet al. 2002 for hydrostatic extrapolation in the plane-parallel case). This can be read-ily implemented for spherical grid, and the same is true for inflow, outflow, or wallboundary conditions for excised outer shells or an excised core.In Cartesian coordinates, however, defining boundary conditions as a function ofradius is at odds with the usual strategy of enforcing the boundary conditions by pop-ulating ghost zones along individual grid lines separately. For pragmatic reasons, oneoften enforces standard boundary conditions (reflecting/inflow/outflow) on the facesof the cubical domain instead (e.g., Couch et al. 2015), which is viable as long as thedomain boundaries are sufficiently distant from the region of interest. Alternatively,one can impose fixed boundary conditions inside the cubical domain, but outside asmaller spherical region of interest (e.g., Woodward et al. 2018). Outflow conditionson an interior boundary, e.g. for fallback onto a compact remnant, can also be imple-mented relatively easily (Joggerst et al. 2009).On the other hand, the boundary conditions at the axis and the origin require care-ful consideration in case of a spherical polar in order to minimize artifacts from thegrid singularities. Conventionally, one uses reflecting boundary conditions to popu-late the ghost zones before performing the reconstruction in the r - and θ - direction,i.e., one assumes odd parity for the velocity components v r and v θ respectively, andeven parity for scalar quantities and the transverse velocity components. This usuallyensures that v r and v θ do not blow up near the grid singularities, but in some casesstronger measures are required; e.g., one can enforce zero v r or v θ in the cell nextto the origin/grid axis, or switch to step function reconstruction in the first cell. Onemay also need to impose odd parity for v ϕ or for better stability, or reconstruct theangular velocity component ω ϕ = v ϕ / r instead of v ϕ .No hard-and-fast rules for such fixes at the axis and the origin can be given, exceptperhaps that one should also consider treating the geometric source terms in sphericalpolar coordinates differently (see below) before applying fixes to the boundary condi-tions that reduce the order of reconstruction, or before manually damping or zeroingvelocity components. In fact, the symmetry assumptions behind reflecting boundaryconditions (i.e., v r → v θ →
0) are actually too strong. Strictlyspeaking, one should only impose the condition that the
Cartesian velocity compo-nents v x , v y , and v z are continuous across the singularity for smooth flow. In principle,this can be accommodated during the reconstruction by populating the ghost zonesfor r < θ <
0, and θ > π with values from the corresponding grid lines across theorigin or the axis, bearing in mind any flip in direction of the basis vectors e r , e θ , and e ϕ across the coordinate singularity. For the reconstruction along the radial grid line with constant θ and ϕ , this comes down to defining v r ( r ) = (cid:26) v r ( r , θ , ϕ ) , r > − v r ( r , π − θ , ϕ + π ) , r < v θ ( r ) = (cid:26) v θ ( r , θ , ϕ ) , r > v θ ( r , π − θ , ϕ + π ) , r < v ϕ ( r ) = (cid:26) v ϕ ( r , θ , ϕ ) , r > − v ϕ ( r , π − θ , ϕ + π ) , r < , (12)and, similarly, for reconstruction in the θ -direction along a grid line with constant r and ϕ : v r ( θ ) = v r ( r , − θ , ϕ + π ) , θ < v r ( r , θ , ϕ ) , < θ < π v r ( r , π − θ , ϕ + π ) , π < θ (13) v θ ( θ ) = − v θ ( r , − θ , ϕ + π ) , θ < v θ ( r , θ , ϕ ) , < θ < π − v θ ( r , π − θ , ϕ + π ) , π < θ (14) v ϕ ( θ ) = − v ϕ ( r , − θ , ϕ + π ) , θ < v ϕ ( r , θ , ϕ ) , < θ < π − v ϕ ( r , π − θ , ϕ + π ) , π < θ . (15)This allows for non-zero values of v r and v θ at the origin to reflect that matter canflow across the origin and the axis. Such special polar boundary conditions have beenimplemented for 3D light-bulb simulations of SASI and convection with F LASH (Fern´andez 2015), and are also used in the F
ORNAX code (Skinner et al. 2019). Inpractice, however, reflecting boundary conditions do not appear to pose a major ob-stacle for flows across the axis or the origin if the diverging fictitious force termsare treated appropriately (see below). The reason is that reflecting boundaries merelyslightly degrade the accuracy of the first cell interfaces away from the origin and theaxis; the fact that velocity components at the coordinate singularity are (incorrectly)forced to zero on the cell interfaces at r = θ =
0, and θ = π does not matter muchbecause these interfaces have a vanishing surface area, so that the interface fluxes must vanish anyway. Geometric source terms.
Another obstacle in spherical polar coordinates is the oc-currence of fictitious force terms in the momentum equation. In terms of the density ρ and the orthonormal components v i and g i of the velocity and gravitational accel- Broadly speaking, light-bulb simulations manually fix the neutrino luminosities and spectral proper-ties, or compute them based on simple analytic considerations, and also use simplified neutrino sourceterms.ydrodynamics of core-collapse supernovae and their progenitors 15 eration, the equations read, ∂ ρ v r ∂ t + r ∂ r ρ v r ∂ r + r ∂ r ρ v r v θ ∂ r + r ∂ r ρ v r v ϕ ∂ r + ∂ P ∂ r = ρ g r + ρ v θ + v ϕ r , (16) ∂ ρ v θ ∂ t + r sin θ ∂ sin θ ρ v r v θ ∂ θ + r sin θ ∂ sin θ ρ v θ ∂ θ + r sin θ ∂ sin θ ρ v θ v ϕ ∂ θ + r ∂ P ∂ θ = ρ g θ + ρ cot θ v ϕ − v r v θ r , (17) ∂ ρ v ϕ ∂ t + r sin θ ∂ ρ v r v ϕ ∂ ϕ + r sin θ ∂ ρ v θ v ϕ ∂ ϕ + r sin θ ∂ ρ v ϕ ∂ ϕ + r sin θ ∂ P ∂ ϕ = ρ g ϕ − ρ v r v ϕ + v θ v ϕ cot θ r , (18)where the fictitious force terms are singular at the origin and at the axis. Oftenstraightforward time-explicit discretization is sufficient for these source terms, es-pecially in unsplit codes with Runge-Kutta time integration. In a dimensionally splitimplementation it can be advantageous to include a characteristic state correction inthe Riemann problem due to (some of the) fictitious force terms (Colella and Wood-ward 1984). Time-centering of the geometric source terms can also lead to minordifferences (Fern´andez 2015).When stability problems or pronounced axis artifacts are encountered, one canadopt a more radical solution and transport the Cartesian momentum density ρ v =( ρ v x , ρ v y , ρ v z ) while still using the components v α in the spherical polar basis asadvection velocities, so that the fictitious force terms disappear entirely, ∂ ρ v ∂ t + √ γ ∂ √ γρ v v α ∂ x α + ∇ P = ρ g , (19)where α ∈ { r , θ , ϕ } and γ = r sin θ is the determinant of the metric. This has beenimplemented in the C O C O N U T-FMT code as one of several options for the solutionof the momentum equation. Transforming back and forth between the spherical polarbasis for the reconstruction and solution of the Riemann problem and the Cartesiancomponents for the update of the conserved quantities might appear cumbersome,but in fact one need not explicitly transform to Cartesian components at all. Insteadone only needs to rotate vectorial quantities from the interface to the cell center whenupdating the momentum components in the spherical polar basis. For example, foruniform grid spacing ∆ θ in the θ -direction, the flux difference terms from the θ -interfaces j and j + ρ v r and ρ v θ in zone j + / (cid:18) ∂ ρ v r , j + / ∂ t (cid:19) θ = cos ∆ θ / [ ρ v r v θ ∆ A ] j + sin ∆ θ / [( ρ v θ + P ) ∆ A ] j − cos ∆ θ / [ ρ v r v θ ∆ A ] j + + sin ∆ θ / [( ρ v θ + P ) ∆ A ] j + , (20) (cid:18) ∂ ρ v θ , j + / ∂ t (cid:19) θ = cos ∆ θ / [( ρ v θ + P ) ∆ A ] j − sin ∆ θ / [ ρ v r v θ ∆ A ] j − cos ∆ θ / [( ρ v θ + P ) ∆ A ] j + − sin ∆ θ / [ ρ v r v θ ∆ A ] j + , (21) where ∆ A is the interface area and ∆ θ is the grid spacing in the θ -direction, whichis assumed to be uniform here. The term for ρ v ϕ is not modified at all. Apart fromeliminating the fictitious force terms in favor of flux flux terms, this alternative dis-cretization of the momentum advection and pressure terms also complies with theconservation of total momentum (although discretization of the gravitational sourceterm may still violate momentum conservation).The singularities at the origin and the pole constitute a more severe problem forrelativistic codes using free evolution schemes for the metric, where they can jeopar-dize the stability of the metric solver. We refer to Baumgarte et al. (2013, 2015) for arobust solution to this problem that has been implemented in their NADA code; theyemploy a reference metric formulation both for the field equations and the fluid equa-tions that factors out metric terms that become singular and use a partially implicitRunge–Kutta scheme to evolve the problematic terms.
Angular momentum conservation.
A somewhat related issue concerns the violationof angular momentum conservation in standard finite-volume codes (both with Eu-lerian grids and moving meshes). This is a concern especially for problems such asconvection in rotating stars and magnetorotational explosions where rotation plays amajor dynamical role and the evolution of the flow needs to be followed over longtime scales. It is also an issue for question such as pulsar spin-up by asymmetricaccretion, although a post-processing of the numerical angular momentum flux canhelp to obtain meaningful results even when there is a substantial violation of angularmomentum conservation (Wongwathanarat et al. 2013).The problem of angular momentum non-conservation can be solved, or at leastmitigated, using Discontinuous Galerkin methods (Despr´es and Labourasse 2015;Mocz et al. 2014; Schaal et al. 2015), which are not currently used in this field how-ever, and can be avoided entirely in SPH (Price 2012). For a given numerical scheme,increasing the resolution is usually the only solution to minimize the conservationerror, but in 3D spherical polar coordinates (and in 2D cylindrical coordinates), onecan still ensure exact conservation of the angular momentum component L z along thegrid axis by conservatively discretizing the conservation equation for ρ v ϕ r sin θ , ∂ ρ v ϕ r sin θ∂ t + r sin θ ∂ ρ v r v ϕ r sin θ∂ ϕ + r sin θ ∂ ρ v θ v ϕ r sin θ∂ ϕ + r sin θ ∂ ρ v ϕ r sin θ∂ ϕ + ∂ P ∂ ϕ = ρ g ϕ r sin θ , (22)instead of Eq. (18). Incidentally, this angular-momentum conserving formulationemerges automatically in GR hydrodynamics in spherical polar coordinates if onesolves for the covariant momentum density components as in the C O C O N U T code(Dimmelmeier et al. 2002). However, as a price for exact conservation of L z , oneoccasionally encounters very rapid rotational flow around the axis, and enforcingconservation of only one angular component may add to artificial flow anisotropiesdue to the spherical polar grid geometry. Moreover, this recipe cannot be used forYin-Yang-type overset spherical grids or for non-orthogonal spherical grids. If angu-lar momentum conservation is a concern, one can, however, resort to a compromise ydrodynamics of core-collapse supernovae and their progenitors 17 . . . . . . . m [ M (cid:12) ]0 . . . . . . . h j i [ c m s − ] initial profilestandard formulation after 522 salternative formulation after 522 s Fig. 3
Profiles of the mass-weighted, spherically-averaged specific angular momentum (cid:104) j (cid:105) in 3D simula-tions of convective oxygen shell burning in a rapidly rotating gamma-ray burst progenitor with an initialhelium core mass of 16 M (cid:12) from Woosley and Heger (2006). The red and blue curves show the angularmomentum distribution after a simulation time of 522s (about 15 convective turnovers) for the standardformulation (red) of the fictitious force terms in Eqs. (17,18) and for the alternative formulation (blue) inEqs. (23,24). The initial angular momentum profile is shown in black. The alternative formulation reducesthe violation of global angular momentum conservation from 20% to 7%. However, for the standard for-mulation the effects of angular momentum non-conservation are much bigger locally than suggested bythe global conservation error. The fictitious force terms proportional to v r lead to a considerable loss ofangular momentum near the reflecting boundaries, even though the angular momentum flux through theboundaries is exactly zero, and there is a spurious increase of angular momentum at the bottom of theconvective oxygen burning shell outside the mass coordinate m = . M (cid:12) . by conservatively discretizing the equations for ρ v θ r and ρ v ϕ r , ∂ ρ v θ r ∂ t + r sin θ ∂ sin θ ρ v r v θ r ∂ θ + r sin θ ∂ sin θ ρ v θ r ∂ θ + r sin θ ∂ sin θ ρ v θ v ϕ r ∂ θ + ∂ P ∂ θ = ρ g θ r + ρ cot θ v ϕ r (23) ∂ ρ v ϕ r ∂ t + r sin θ ∂ ρ v r v ϕ r ∂ ϕ + r sin θ ∂ ρ v θ v ϕ r ∂ ϕ + r sin θ ∂ ρ v ϕ r ∂ ϕ + θ ∂ P ∂ ϕ = ρ g ϕ r − ρ v θ v ϕ cot θ r , (24)which eliminates some of the fictitious force terms. This sometimes considerably im-proves angular momentum conservation for all angular momentum components andworks for any spherical grid. Figure 3 illustrates the difference between the standardform of the fictitious force term and the alternative form in Eqs. (23,24) for a simula-tion of oxygen shell convection in a rapidly rotating gamma-ray burst progenitor. In the final stages of massive stars one encounters a broad range of flow regimes.The convective Mach number, i.e., the ratio of the typical convective velocity to the sound speed, during the advanced convective burning stages is fairly low, rangingfrom ∼ − or less (Cristini et al. 2017) to ∼ . . –10 . During the supernova, one finds convective Mach num-bers from a few 10 − in PNS convection to ∼ . . The flow in unstable regions is typically highly turbulent. Inthe gain region, one obtains a nominal Reynolds number of order 10 based on theneutron viscosity (Abdikamalov et al. 2015), but non-ideal effects of neutrino vis-cosity and drag play a role in the environment of the PNS (Burrows and Lattimer1988; Guilet et al. 2015; Melson et al. 2020). In the outer regions of the PNS convec-tion zone neutrino viscosity keeps the Reynolds number as low as ∼
100 during somephases (Burrows and Lattimer 1988), and in the gain region drag effects are still solarge that the flow cannot be assumed to behave like ordinary high-Reynolds numberflow (Melson et al. 2020). Later on, as the shock propagates through the envelopeand mixing instabilities develop, neutrino drag becomes unimportant, and the flow isagain in the regime of very high Reynolds numbers.Both the vast range in Mach number and the turbulent nature of the flow presentchallenges for the accuracy and robustness of numerical simulations. While even rel-atively simple HRSC schemes can deal with Mach numbers of Ma (cid:38)
AESTRO code of Nonaka et al. 2010, or by a time-implicit discretization of thefull compressible Euler equations (Viallet et al. 2011; Miczek et al. 2015). However,few studies (Kuhlen et al. 2003; Michel 2019) have used such methods to deal withlow-Mach number flow in the late stages of convective burning massive stars as yet;they have been employed more widely to study, e.g., the progenitors of thermonuclearsupernovae (Zingale et al. 2011; Nonaka et al. 2012)Part of the reason is that advanced HRSC schemes remain accurate and com-petitive down to Mach numbers of 10 − and below depending on the reconstructionmethod and the Riemann solver. Although it is impossible to decide a priori whethera particular choice of methods is adequate for a given physical problem, or what itsresolution requirements are, it is useful to be aware of strengths and weaknesses ofdifferent schemes in the context of subsonic turbulent flow. Unfortunately, our discus-sion of these strengths and weaknesses must remain rather qualitative because veryfew studies in the field have compared the performance of different Riemann solversand reconstruction schemes in full-scale simulations and not only for idealized testproblems. ydrodynamics of core-collapse supernovae and their progenitors 19 Fig. 4
Snapshots of the entropy from 2D simulations of the 20 M (cid:12) progenitor of Woosley and Heger(2007) using the C O C O N U T-FMT code at post-bounce time of 137ms (top row), 163ms (middle row),and 226ms (bottom row). The left and rights halves of the panels in the left column show the results forthe HLLE and HLLC solver with standard PPM reconstruction. The left and rights halves of the panelsin the right column show the results for second-order reconstruction using the MC limiter and 6th-orderextremum-preserving PPM reconstruction using the HLLC solver.0 Bernhard M¨uller
Riemann solvers.
For supernova simulations with Godunov-based codes, a varietyof Riemann solvers are currently used. Newtonian codes typically opt either for a(nearly) exact solution of the Riemann problem following Colella and Glaz (1985) or for approximate solvers that at least take the full wave structure of the Riemannproblem into account such as the HLLC solver (Toro et al. 1994). On the other hand,the majority of relativistic simulations still resort to the HLLE solver (Einfeldt 1988)because of the added complexity of full-wave approximate Riemann solvers for GRhydrodynamics; exceptions include the C O C O N U T code which routinely uses therelativistic HLLC solver of Mignone and Bodo (2005), the C O C O N U T simulationsof Cerd´a-Dur´an et al. (2005) using the Marquina solver (Donat and Marquina 1996),and the convection simulations with the W
HISKEY
THC code (Radice et al. 2016),which uses a Roe-based flux-split scheme.The use of simpler solvers in GR simulations is a concern because one-waveschemes behave significantly worse than full-wave solvers in the subsonic regime.The problem of excessive acoustic noise from the discontinuities introduced by thereconstruction is exacerbated because solvers like HLLE essentially have these dis-continuities decay only into acoustic waves. This results in stronger numerical diffu-sion, but can also create artificial numerical noise because the diffusive terms in theHLLE or Rusanov flux can generate spurious pressure perturbations from isobaricconditions. While higher-order reconstruction can beat down the numerical diffusionfor smooth flows, a strong degradation in accuracy is unavoidable in turbulent flowwith structure on all scales.Few attempts have as yet been made to quantify the impact of the more diffusiveone-wave solvers in supernova simulations. In an idealized setup, the problem wasaddressed by Radice et al. (2015), who conducted simulations of stirred isotropic tur-bulence with solenoidal forcing with a turbulent Mach number of Ma ∼ .
35, twodifferent Riemann solvers (HLLE vs. HLLC), and different reconstruction methodsand grid resolutions. Even when using PPM reconstruction, they still found substan-tial differences between the HLLE and HLLC solver in the spectral properties ofthe turbulence with the HLLE runs requiring about 50% more zones to achieve anequivalent resolution of the turbulent cascade to HLLC. Although it is not easy to ex-trapolate from the idealized setup of Radice et al. (2015) to core-collapse supernovasimulations, one must clearly expect resolution requirements to depend sensitively onthe Riemann solver. This is also illustrated by 2D supernova simulations comparingthe HLLE and HLLC solver using the C O C O N U T-FMT code as shown in Fig. 4:Starting from the initial seed perturbations, the HLLC model shows a faster growthof large-scale SASI shock oscillations during its linear phase and earlier emergenceof parasitic instabilities (see Sect. 4.4 for the physics behind the SASI) due to thesmaller amount of numerical dissipation. The evolution of the shock differs signifi-cantly during the first 100 ms of SASI activity, although the models become similar interms of shock radius and shock asphericity later on. However, even then HLLC runconsistently shows a higher entropy contrast and higher non-radial velocities withinthe gain region. The solver of Colella and Glaz (1985) is not exact in the strict sense because it involves a locallinearization of the equation of state.ydrodynamics of core-collapse supernovae and their progenitors 21
Reconstruction methods.
Similar concerns (not restricted to the low-Mach numberregime) as for the simpler Riemann solvers can be raised about the order of the recon-struction scheme. There is certainly a clear divide between second-order piecewiselinear reconstruction and higher-order methods like the PPM, WENO (weighted es-sentially non-oscillatory Shu 1997), and higher-order monotonicity-preserving (MPSuresh and Huynh 1997) schemes. In their simulations of forced subsonic turbulence,Radice et al. (2015) found similar differences between second-order reconstructionusing the monotonized central (MC; van Leer 1977) limiter — one of the shapersecond-order limiters — and runs using PPM or WENO as between the HLLC andHLLE solver. Again, the lower accuracy of second-order schemes is often clearlyvisible in full supernova simulations, which is again illustrated in Fig. 4 for the samesetup as above. Similar to the HLLE run, the simulations using the MC limiter showsa delayed growth of the SASI and less small-scale structure.Comparing the more modern higher-order reconstruction methods is much moredifficult. For smooth problems like single-mode linear waves solutions, going beyondthe original 4th-order PPM method of Colella and Woodward (1984) to methods of5th order or higher can substantially reduce numerical dissipation; in the optimalcase, the dissipation decreases with the grid spacing ∆ x as ∆ x − q for a q -th ordermethod (Rembiasz et al. 2017). However, the higher-order scaling of the numericaldissipation cannot be generalised to turbulent flow, because the dissipation of theshortest realizable modes at the grid scale does not increase as a higher power of q .Based on similarity arguments, one can work out that the effective Reynolds numberof turbulent flow increases only as Re ∼ ∆ x − / (M¨uller 2016) and not as ∆ x q asone might hope. The reason behind this limitation is that increasing the order ofreconstruction does not increase the maximum wavenumber k max of modes that canbe represented on the grid, it merely limits numerical dissipation to a narrow band ofwavenumbers below k max .For the moderately subsonic turbulent flow in core-collapse supernovae duringthe accretion phase, higher-order reconstruction often does not bring any tangibleimprovements for this reason. Fig. 4 again shows this by comparing runs using stan-dard 4th-order PPM and the 6th-order extremum-preserving PPM method of Colellaand Sekora (2008). In both cases, the evolution of the shock is very similar, eventhough the phases of the SASI oscillations eventually falls out of sync. It is not ob-vious by visual inspection that the higher-order method allows smaller structures todevelop. Only upon deeper analysis can small differences between the two methodsbe found, for example the model with extremum-preserving reconstruction maintainsa measurably higher entropy contrast in the gain region and a slightly higher turbulentkinetic energy in the gain region.There are nonetheless situations where it is useful to adopt extremum-preservingmethods of very high order in global simulations of turbulent flow. First, such meth-ods open up the regime of low Mach numbers to explicit Godunv-based codes. Usingtheir A PSARA code, Wongwathanarat et al. (2016) were able to solve the Gresho vor-tex problem (Gresho and Chan 1990) with little dissipation down to a Mach numberof 10 − with the extremum-preserving PPM method of Colella and Sekora (2008),which is about two orders of magnitude better than for the MC limiter (Miczek et al.2015), and about one order of magnitude better than for standard PPM. Modern higher-order methods can also be crucial in certain simulations of mix-ing at convective boundaries and nucleosynthesis. In the case of convective boundarymixing, this has been stressed and investigated by Woodward et al. (2010, 2014),who achieve higher accuracy for the advection of mass fractions in their PPM
STAR code by evolving moments of the concentration variables within each cell (which issomewhat reminiscent of the Discontinuous Galerkin method). They found that thisPiecewise-Parabolic Boltzmann method only requires half the resolution of standardPPM to achieve the same accuracy (Woodward et al. 2010). Higher-order extremum-preserving methods may also prove particularly useful for minimizing the numericaldiffusion of mass fractions in models of Rayleigh–Taylor mixing during the super-nova explosion phase, but this is yet to be investigated.
Some of the considerations for subsonic flow carry over to the supersonic and transsonicflow encountered during the supernova explosion phase where mixing instabilitiesalso lead to turbulence, but there are also problems specific to the supersonic regime.
Sonic points.
It is well known that the original Roe solver produces spurious expan-sion shocks in transsonic rarefaction fans, which needs to be remedied by some formof entropy fix (Laney 1998; Toro 2009). While other full-wave solvers – like the ex-act solver and HLLC – never fail as spectacularly as Roe’s, they are still prone tomild instabilities at sonic points. Under adverse conditions, these instabilities can beamplified and turn into a serious numerical problem. In this case, it is advisable toswitch to a more dissipative solver such as HLLE in the vicinity of the sonic point. Insupernova simulations, this problem is sometimes encountered in the neutrino-drivenwind that develops once accretion onto the PNS has ceased. It can also occur prior toshock revival in the infall region and severely affect the infall downstream of the in-stability, especially when nuclear burning is included. In this case, the problem can beeasily overlooked or misidentified because it usually manifests itself as an unusuallystrong stationary burning front, which may seem perfectly physical at first glance.
Odd-even decoupling and the carbuncle phenomenon.
Full-wave solvers like the ex-act solver and HLLC are subject to an instability at shock fronts (Quirk 1994; Liou2000): For grid-aligned shocks, insufficient dissipation in the direction parallel to theshock can cause odd-even decoupling in the solution, which manifests itself in ar-tificial stripe-like patterns downstream of the shock. When the shock is only locallytangential to a grid line, this instability can give rise to protrusions, which is known asthe carbuncle phenomenon. In supernova simulations, odd-even decoupling was firstrecognized as a problem by Kifonidis et al. (2000), and since then the majority ofsupernova codes (e.g., P
ROMETHEUS , F
LASH , C O C O N U T, F
ORNAX ) have opted tohandle this problem by adaptively switching to the more dissipative HLLE solver atstrong shocks following the suggestion of Quirk (1994). The C
HIMERA code (Bruennet al. 2018) adopts the alternative approach of a local oscillation filter (Sutherlandet al. 2003), which has the advantage of not degrading the resolution in the directionperpendicular to the shock, but has the drawback of allowing the instability to grow to ydrodynamics of core-collapse supernovae and their progenitors 23
Fig. 5
Odd-even decoupling in a 2D core-collapse supernova simulation of a 20 M (cid:12) star with the C O -C O N U T that uses the HLLC solver everywhere instead of switching to HLLE at shocks. The left and rightpanels show the radial velocity in units of the speed of light and the entropy s in units of k b / nucleon about10ms after bounce. The characteristic radial streaks from odd-even decoupling are clearly visible behindthe shock. a minute level (which may be undetectable in practice) before smoothing is applied tothe solution. The carbuncle phenomenon can also occur in Richtmyer-type artificialviscosity schemes and be cured by modifying the artificial viscosity (Iwakami et al.2008). The carbuncle instability remains a subject of active research in computationalfluid dynamics, and a number of papers (e.g., Nishikawa and Kitamura 2008; Huanget al. 2011; Rodionov 2017; Simon and Mandal 2019) have attempted to constructRiemann solvers or artificial viscosity schemes that avoid the instability without sac-rificing accuracy away from shocks, and may eventually prove useful for supernovasimulations. Kinetically-dominated flow.
In HRSC codes that solve the total energy equation, oneobtains the mass-specific internal energy ε by subtracting the kinetic energy v / e . In high Mach-number flow, one has e (cid:29) ε and v / (cid:29) ε ,and hence subtracting these two large terms can introduce large errors in the internalenergy density and the pressure and sometimes leads to severe stability problems. Asimilar problem can occur in magnetically-dominated regions in MHD. Sometimesthe resulting stability problems can be remedied by evolving the internal energy equa-tion ∂ ρε∂ t + ∇ · ( ρε v + P v ) − v · ∇ P = − ρ v · ∇ Φ + Q e + Q m · v , (25)instead of Eq. (3) in regions of high Mach number or low plasma- β . However, indoing so one sacrifices strict energy conservation, and hence one should apply thisrecipe as parsimoniously as possible.2.2 Treatment of gravityConvective burning and core-collapse supernovae introduce specific challenges in thetreatment of gravity. In the subsonic flow regimes, one needs to be wary of introduc- ing undue artifical perturbations from hydrostatic equilibrium and take care to avoidsecular conservation errors. Moreover, in the core-collapse supernova problem, gen-eral relativistic effects become important in the vicinity of the PNS. For nearly hydrostatic flow, one has ∇ P ≈ − ρ ∇ Φ , but this near cancellation is not au-tomatically reflected in the numerical solution when using a Godunov-based scheme.Instead, the stationary numerical solution may be one with non-zero advection termsthat are exactly (but incorrectly) balanced by the gravitational source term (Green-berg and Leroux 1996; LeVeque 1998a). Schemes that avoid this pathology are called well-balanced . The proper cancellation between the pressure gradient and the gravi-tational source term is particularly delicate if those two terms are treated in operator-split steps. Different methods have been proposed to incoroprate well-balancing intoGodunov-based schemes. One approach is to use piecewise hydrostatic reconstruc-tion (e.g., Kastaun 2006; K¨appeli and Mishra 2016). A related technique suggested byLeVeque (1998a) introduces discrete jumps in the middle of cells to obtain modifiedinterface states for the Riemann problem and absorb the source terms altogether.In practice, these special techniques are not used widely in the field for two rea-sons. First, it is not trivial to general these schemes to achieve higher-order accuracy.Second, one already obtains a very well-balanced scheme by combing higher-orderreconstruction, an accurate Riemann solver, and unsplit time integration. For splitschemes, one can ensure a quite accurate cancellation of the pressure gradient andthe source term by including a characteristic state correction as described by Colellaand Woodward (1984) for the original PPM method.Nevertheless, the cancellation of the pressure gradient and the gravitational sourceterm in hydrostatic equilibrium is usually not perfect and typically leads to minuteodd-even noise in the velocity field that is almost undetectable by eye. Computingthe gravitational source term ρ v · ∇ Φ in the energy equation using such a noisy ve-locity field v can lead to an appreciable secular drift of the total energy. For example,spurious energy generation can stop proto-neutron star cooling on simulation timescales longer than a second (M¨uller 2009). This problem can be circumvented bydiscretizing the energy equation starting from the form (M¨uller et al. 2010) ∂ ρ ( ε + v / + Φ ) ∂ t ∇ · (cid:2) ρ v ( ε + v / + Φ ) + P v (cid:3) = ρ ∂ Φ∂ t . (26)This guarantees exact total energy conservation if the time derivative of the gravi-tational potential is zero. Under certain conditions, exact total energy conservationcan be achieved for a time-dependent self-gravitating configuration as well, and themethod can also be generalized to the relativistic case.In principle, one can also implement the gravitational source term (in the New-tonian approximation) in the momentum equation in a conservative form by writing ρ g as the divergence of a gravitational stress tensor (Shu 1992). Such a scheme hasbeen implemented by Livne et al. (2004) in the V ULCAN code. However, this proce-dure involves a more delicate modification of the equations than in case of the energysource term, because it essentially amounts to replacing ρ by the finite-difference ydrodynamics of core-collapse supernovae and their progenitors 25 representation of the Laplacian ( π G ) − ∆ Φ in the momentum source term. Unlessthe solution for the gravitational potential is extremely accurate, large accelerationerrors may thus arise. Moreover, this approach does not work for effective relativisticpotentials (see Sect. 2.2.2). For these reasons, the conservative form of the gravita-tional source term has not been used in practice in other codes. Even though the issueof momentum conservation is of relevance in the context of neutron star kicks, con-servation errors do not seem to affect supernova simulation results qualitatively inpractice, and post-processing techniques can be used to infer neutron star velocitiesfrom simulations with good accuracy (Scheck et al. 2006). In core-collapse supernova simulations, the relativistic compaction of the proto-neutronstar reaches GM / Rc = . . M and a somewhat ex-tended radius R of the warm PNS. Infall velocities of 0 . . c are encountered.Hence general relativistic (GR) and special relativistic effects are no longer negligi-ble, though the latter is more critical for the treatment of the neutrino transport thanfor the hydrodynamics. For very massive neutron stars, cases of black hole formation,or jet-driven explosions, relativistic effects can be more pronounced.A variety of approaches is used in supernova modelling to deal with relativistic ef-fects. Purely Newtonian models have now largely been superseded. Using Newtoniangravity results in unphysically large PNS radii, and, as a consequence, lower neutrinoluminosities and mean energies and worse heating heating conditions than in the rel-ativistic case, even though the stalled accretion shock radius is larger than in GR be-fore explosion (M¨uller et al. 2012b; Kuroda et al. 2012; Lentz et al. 2012; O’Connorand Couch 2018b). As an economical alternative, one can retain the framework ofNewtonian hydrodynamics but incoroprate relativistic corrections in the gravitationalpotential based on the TOV equation (Rampp and Janka 2002). This approach wassubsequently refined by Marek et al. (2005); M¨uller et al. (2008) to account for someinconsistencies between the use of Newtonian hydrodynamics and a potential basedon a relativstic stellar structure equation, but full consistency can never be achievedin the pseudo-Newtonian approach. In the multi-D case, the relativistic potential re-places the monopole of the Newtonian potential, while higher multipoles are leftunchanged. From a purist point of view, this pseudo-Newtonian approach is delicatebecause one sacrifices global conservation laws for energy and momentum (whichwould still hold in a more complicated form in an asymptotically flat space in fullGR). In practice, this is less critical; in PNS cooling simulations by H¨udepohl et al.(2010) the total emitted neutrino energy was found to agree with the neutron starbinding energy (computed from the correct TOV solution) to within 1% for the mod-ified TOV potential (Case A) of Marek et al. (2005).If the framework of Newtonian hydrodyanmics is abandoned, one may still opt foran approximate method to solve for the space-time metric as in the C O C O N U T code(Dimmelmeier et al. 2005; M¨uller et al. 2010; M¨uller and Janka 2015). Elliptic for-mulations such as CFC (conformal flatness conditions, Isenberg 1978) and xCFC (amodification of CFC for improved numerical stability; Cordero-Carri´on et al. 2009)can be cheaper and more stable than free-evolution schemes based on the 3+1 decom- position (for reviews of these techniques, see Baumgarte and Shapiro 2010; Lehnerand Pretorius 2014) and maximally constrained schemes (Bonazzola et al. 2004;Cordero-Carri´on et al. 2012). However, full GR supernova simulations without theCFC approximation and with multi-group transport have also become possible re-cently (Roberts et al. 2016; Ott et al. 2018; Kuroda et al. 2016b, 2018). AlthoughCFC remains an approximation, it is exact in spherical symmetry, and comparisonswith free-evolution schemes have shown excellent agreement in the context of rota-tional collapse have shown excellent agreement even for rapidly spinning progenitors(Ott et al. 2007).Comparisons of pseudo-Newtonian and GR simulations have demonstrated thatusing an effective potential is at least sufficient to reproduce the PNS contraction,the shock evolution, and the neutrino emission in GR very well (Liebend¨orfer et al.2005; M¨uller et al. 2010, 2012b). While M¨uller et al. (2012b) still found better heat-ing conditions in the GR case than with an effective potential in their 2D models,this comparison was not fully controlled in the sense that two different hydro solverswere used, and the effect was related to subtle differences in the PNS convection zone,which may well be related to factors other than the GR treatment (cf. Sect. 4.7). Fur-ther code comparisons are desirable to resolve this. The pseudo-Newtonian approach,does, however, systematically distort the eigenfrequencies of neutron star oscillationmodes (M¨uller et al. 2008). In particular, the frequency of the dominant f-/g-modeseen in the gravitational wave spectrum is shifted up by 15–20% compared to thecorrect relativistic value (M¨uller et al. 2013).
In the Newtonian approximation, the gravitational field is obtained by solving thePoisson equation ∆ Φ = π G ρ . (27)In constrained formulations of the Einstein equations like (x)CFC, one encountersnon-linear Poisson equations.In simulations of supernovae and the late convective burning stages, the densityfield usually only deviates modestly from spherical symmetry and is not exceedinglyclumpy (except in the case of mixing instabilities in the envelope during the explo-sion phase when self-gravity is less important to begin with). For this reason, theusual method of choice for solving the Poisson equation (even in Cartesian geome-try) is to use the multipole expansion of the Green’s function (M¨uller and Steinmetz1995). Typically, no more than 10–20 multipoles are needed for good accuracy, andvery often only the monopole component is retained. Other methods have been usedoccasionally, though, such as pseudospectral methods (Dimmelmeier et al. 2005) andfinite-difference solvers (e.g., Burrows et al. 2007b), and the FFT (Hockney 1965;Eastwood and Brownrigg 1979) is a viable option for Cartesian simulations.Although it yields accurate results at fairly cheap cost, some subtle issues canarise with the multipole expansion. When projecting the source density onto spher-ical harmonics Y (cid:96) m to obtain multipole components ˆ ρ (cid:96) m = (cid:82) Y ∗ (cid:96) m ρ d Ω , a naive step-function integration can lead to a self-potential error (Couch et al. 2013) and destroy ydrodynamics of core-collapse supernovae and their progenitors 27 convergence with increasing mulitpole number N (cid:96) . This can be avoided either byperforming the integrals over spherical harmonics Y ∗ (cid:96) m analytically (M¨uller 1994), orby using a staggered grid for the potential (Couch et al. 2013). The accuracy of thesolution can also be degraded if the central mass concentration moves away fromthe center of the grid, which can be cured by off-centering the multipole expansion(Couch et al. 2013). Problems with off-centred or clumpy mass distributions can becured completely if an exact solver is used. In Cartesian geometry, this can be accom-plished econmically using the FFT, and an exact solver for spherical polar grid usinga discrete eigenfunction expansion has recently been developed as well (M¨uller andChan 2019). On spherical multi-patch grids, the efficient parallelization and com-putation of integration weights requires some thought and has been addressed by(Almanst¨otter et al. 2018; Wongwathanarat 2019).2.3 Reactive flowNuclear burning is the principal driver of the flow for core and shell convection in thelate, neutrino-cooled evolutionary stages of supernova progenitors. In core-collapsesupernovae nuclear dissociation and recombination play a critical role for the dynam-ics and energetics, and one of the key observables, the mass of Ni, is determined bynuclear burning.Approaches to nuclear transmutations differ widely between simulation codes,and range from the assumption of nuclear statistical equilbrium (NSE) everywhere insome core-collapse supernova models to rather large reaction networks. Naturally,theappropriate level of sophistication depends on the regime and the observables of in-terest. The theory of nuclear reaction networks is too vast to cover in detail here, andwe can only touch a few salient points related to their integration into hydrodynamicscodes. For a more extensive coverage, we refer to textbooks and reviews on the sub-ject (Clayton 1968; Arnett 1996; M¨uller 1998; Timmes 1999; Hix and Meyer 2006;Iliadis 2007).
Burning regimes.
As stellar evolution proceeds towards collapse, the ratio of the nu-clear time scale to both the sound crossing time scale and convective time scale de-creases, and the nuclear reaction flow involves an increasing number of reactions. Theburning of C, Ne, and to some extent of O is dominated by an overseeable number ofmain reaction channels, and the relevant reaction rates are slow compared to the rele-vant hydrodynamical time scales. During oxygen burning, quasi-equilibrium clustersbegin to appear and eventually merge into one or two big clusters during Si burning(Bodansky et al. 1968; Woosley et al. 1973) that are linked by slow “bottleneck” re-actions. For sufficiently high temperatures, NSE is established and the compositiononly depends on density ρ , temperature T , and the electron fraction Y e and is giventhe Saha equation. At higher densities during core-collapse, the assumption of non-interacting nuclei break down, and a high-density equation of state is required (seeLattimer 2012; Oertel et al. 2017; Fischer et al. 2017, for recent reviews); this regimeis not of concern here because the flow can be treated as non-reactive. Simple approaches.
In core-collapse supernova simulations, one sometimes simplyassumes NSE everywhere, which amounts to an implicit release of energy at the startof a simulation. Although the Si and O shell will still collapse in the wake of the Fecore, this is somewhat problematic, especially for long-time simulations where theeffect on the infall is bound to be more pronounced. For mitigating potential artifactsfrom the inconsistency of the composition and equation of state with the underlyingstellar evolution model, it can be useful to initialise supernova simulations using thepressure rather than the temperature of progenitor model.A considerably better and very cheap approach, known as “flashing”, is to usea few key α -elements and non-symmetric iron group nuclei in addition to protons,neutrons, and α -particles and burn them instantly into their reaction products andeventually into NSE upon reaching certain threshold temperatures (Rampp and Janka2002). Such an approach can capture the energetics of explosive burning in the shockand the freeze-out from NSE in neutrino-driven outflows reasonably well, but onlygives indicative results on the composition of the ejected matter. The choice of theproper NSE threshold temperature T NSE (cid:38)
Reaction networks.
In 2D, Baz´an and Arnett (1997) already conducted simulations ofconvective burning with 123 species, but the use of large networks in 3D simulationsis still prohibitively expensive. Modern 3D simulations of convective burning with theP
ROMPI (e.g., Meakin and Arnett 2006; Moc´ak et al. 2018), P
ROMETHEUS (M¨ulleret al. 2016b; Yadav et al. 2020), F
LASH (Couch et al. 2015) and 3D N SN E (Yoshidaet al. 2019) codes have therefore only use networks of 19–25 species consisting of α -elements, light particles, and at most a few extra iron-group elements. In multi-Dsupernova simulations with neutrino transport the use of such networks is feasible(von Groote 2014; Bruenn et al. 2013, 2016; Wongwathanarat et al. 2017), thoughthey have not been used widely yet. It is critical that such reduced reaction networksappropriately account for side chains and the effective reaction flow between lightparticles (Weaver et al. 1978; Timmes et al. 2000). Their use is problematic for Siburning which requires networks of more than a hundred species to accurately capturethe quasi-equilibrium clusters and the effects of deleptonization (Weaver et al. 1978)and for freeze-out from NSE with considerable neutron excess. Larger networks orspecial methods for quasi-equilibrium (Weaver et al. 1978; Hix et al. 2007; Guidryet al. 2013) will be required for reliable multi-D simulations of convective Si burning. ydrodynamics of core-collapse supernovae and their progenitors 29 Coupling to the hydrodynamics.
Some numerical issues arise when a nuclear net-work is coupled to a Eulerian hydrodynamics solver, or even if the composition isjust tracked as a passive tracer. One such problem concerns the conservation of par-tial masses, which is guaranteed analytically by a conservation equation (4) for eachspecies i , ∂ ρ X i ∂ t + ∇ · ( ρ X i v ) = ˙ X i , burn . (28)This equation can be solved using standard, higher-order finite-volume techniques.However, the solution also has to obey the constraint ∑ X i = , (29)which is not fulfilled automatically by the numerical solution, unless flat reconstruc-tion for the mass fractions is employed. One could enforce this constraint by rescalingthe mass fractions to sum up to unity, but this would violate the conservation of par-tial masses. Plewa and M¨uller (1999) developed the Consistent Multi-fluid Advection(CMA) method as the standard treatment to ensure both minimal numerical diffusionof mass fractions and enforce conservation of partial masses. This method involves arescaling and coupling of the interpolated interface values of the various mass frac-tions. Plewa and M¨uller (1999) demonstrated that simple methods for the advectionof mass fractions can easily result in wrong yields by a factor of a few for someisotopes in supernova explosions.Another class of problems is related to advection errors and numerical diffusion,especially at contact discontinuities and shocks, which can lead to artificial detona-tions or an incorrect propagation of physical detonations (Colella et al. 1986; Fryxellet al. 1989; M¨uller 1998). To ensure that detonations propagate at the correct physi-cal velocity, nuclear burning should be switched off in shocked zones (M¨uller 1998). Due to the extreme temperature dependence of nuclear reaction rates, similar prob-lems can arise away from discontinuities due to advection errors that produce a smalllevel of noise in the temperature. Artificial detonations can easily develop in highlydegenerate regions and around sonic points. Eliminating such artifacts may requireappropriate switches for pathological zones or very high spatial resolution (e.g., Ki-taura et al. 2006).
In the Introduction, we already outlined the motivation for multi-D simulations ofsupernova progenitors in broad terms. On the most basic level, multi-D models areneeded to properly intialize supernova simulations and provide physically correctseed perturbations for the instabilities that develop after collapse and in the explo-sion phase. This does not, in fact, presuppose that 1D stellar evolution models incor-rectly predict the overall spherical structure of pre-supernova progenitors; in principle Note also the use of front-tracking methods for unresolvable burning fronts (Reinecke et al. 2002;Leung and Nomoto 2019), which are commonly used for modelling Type Ia supernovae and the O defla-gration in electron-capture supernova progenitors.0 Bernhard M¨uller such an initialization might involve nothing but adding some degrees of freedom to1D stellar evolution models without any noticeable change of the spherically aver-aged stratification. Historically, however, simulations of late-stage convection havefocused on deviations of the multi-D flow from the predictions of traditional mixing-length theory (MLT; Biermann 1932; B¨ohm-Vitense 1958; Weiss et al. 2004) and notevolved progenitor models up to core collapse, whereas the initialization problem hasonly been tackled recently by Couch et al. (2015); M¨uller et al. (2016b, 2019); Yadavet al. (2020). In this section, we therefore address the interior flow in convective re-gions and boundary effects first before specifically discussing multi-D pre-supernovamodels.3.1 Interior flowLet us first consider the flow within convectively unstable regions. In MLT as imple-mented in modern stellar evolution codes such as K
EPLER (Weaver et al. 1978; Hegerand Woosley 2010) and M
ESA (Paxton et al. 2011), the convective velocity v conv insuch regions is tied to the superadiabaticity of the density gradient as encoded by theBrunt-V¨ais¨al¨a ω BV frequency and the local pressure scale height Λ , v conv = α (cid:112) Λ δ ρ / ρ g = αΛ ω BV , (30)where α is a tuneable parameter of order unity, and the MLT density contrast δ ρ isobtained from the the difference between the actual and density gradient ∂ ρ / ∂ r andthe adiabatic density gradient ( ∂ ρ / ∂ P ) s ( ∂ P / ∂ r ) , δ ρ = Λ ρω g = Λ (cid:20) ∂ ρ∂ r − (cid:18) ∂ ρ∂ P (cid:19) s ∂ P ∂ r (cid:21) = Λ (cid:18) ∂ ρ∂ r − c ∂ P ∂ r (cid:19) . (31)Note that stellar evolution textbooks usually express the convective velocity and den-sity contrast in terms of the difference between the actual and adiabatic temperaturegradient (Clayton 1968; Weiss et al. 2004; Kippenhahn et al. 2012), but Eqs. (30) and(31) are fully equivalent formulations that often prove less cumbersome.Using Eq. (30) for the convective velocity, Eq. (31) for the MLT density contrast,and the temperature contrast δ T = ( ∂ T / ∂ ln ρ ) P ( δ ρ / ρ ) , we then obtain the convec-tive energy flux F conv (Kippenhahn et al. 2012; Weiss et al. 2004), F conv = α e ρ c P δ T v conv (32) = − αα e ρ c P (cid:18) ∂ T ∂ ln ρ (cid:19) P Λ ω g , where c P is the specific heat at constant pressure, and α e is another tunable non-dimensional parameter. Similarly, by estimating the composition contrast δ X i usingthe local gradient as δ X i = α X Λ ∂ X i / ∂ r , we obtain the partial mass flux for species iF X i = ρ v conv δ X i = α X Λ ρ v conv ∂ X i ∂ r , (33) ydrodynamics of core-collapse supernovae and their progenitors 31 . . . . . m [ M (cid:12) ]02468 ˙ q nu c , | ˙ q ν | [ e r gg − s − ] Si shell O shell ˙ q nuc | ˙ q ν | v conv v c o n v [ k m s − ] Fig. 6
Energy generation rate ˙ q nuc (black), neutrino cooling rate | ˙ q ν | (red), and convective velocity v conv (blue) in an 18 . M (cid:12) progenitor (1D model, discussed in Yadav et al. 2020 in the innermost shells about1hr before collapse. At this stage, balanced power still obtains. Nuclear energy generation dominates at thebottom of the shells, while neutrino cooling dominates in the outer layers. The integrated energy generationand cooling rate for the entire shell, which are given by the areas under the black and red curve, nearlybalance each other. where α X is again a dimensionless parameter. When comparing 1D stellar evolutionmodels to each other or to multi-D simulations, one must bear in mind that slightlydifferent normalization conventions for Eqs. (32) and (33) are in use. Regardless ofthese ambiguities, these coefficients are of order unity, for example the K EPLER codeuses αα e = / αα X = / α = α e = / α X = / v Λ = α δ ρ / ρ gv conv , (34)which we can interpret as a balance between the rate of buoyant energy generation( α δ ρ / ρ v conv ) and turbulent dissipation ( ε ∼ v / Λ ). Furthermore the work doneby bouyancy must ultimately be supplied by nuclear burning. Using thermodynamicrelations, we find that the potential energy Λ δ ρ / ρ g liberated by bubbles rising orsinking by one mixing length is of the order of the enthalpy contrast δ h of the bubbles,which roughly equals the integral of the nuclear energy generation rate ˙ q nuc over oneturnover time τ = Λ / v conv , Λ δ ρ / ρ g ∼ δ h ∼ ˙ q nuc Λ / v conv . (35)Together, Eqs. (34) and (35) lead to a scaling law v conv ∼ ( ˙ q nuc Λ ) / for the typicalvalue of v conv in a convective shell.In nature, balance between nuclear energy generation, buoyant energy genera-tion, and turbulent dissipation is usually established over a few turnover times. Onlonger time scales, active burning shells also adjust by expansion or contraction until the total nuclear energy generation rate and neutrino cooling rate balance each other(Woosley et al. 1972), with the nuclear burning dominating in the inner region andneutrino cooling dominating in the outer region of the shell (Fig. 6).Because of the extremely strong temperature sensitivity of the burning rates, thisstate of balanced power is difficult to maintain when setting up multi-D simulationsand will only be reestablished over a long, thermal time scale. In fact, the problemof thermal adjustment has not yet been rigorously analyzed for any multi-D modelyet, and insufficient simulation time for thermal adjustment is a concern that needsto be addressed in future. However, the problem of thermal adjustment is mitigatedduring the latest phases of shell convection prior to collapse: As the core and thesurrounding shells contract, the nuclear burning rates accelerate to a point whereneutrino cooling and shell expansion by P d V work can no longer re-establish thermalbalance on the contraction time scale of the core, and the state of balanced power isphysically broken. Two-dimensional simulations of convective burning.
The first attempts to simulatelate-stage convection in massive stars by Arnett (1994); Bazan and Arnett (1994,1998) targeted oxygen burning in a 20 M (cid:12) star in a 2D shellular domain with theP ROMETHEUS code using a small, 12-species reaction network and neutrino coolingby the pair process. Starting from a simulation on a small wedge of 18 ◦ in Arnett(1994), Bazan and Arnett (1994, 1998) subsequently considered broader wedges incylindrical symmetry of up to 135 ◦ with a resolution of up to 460 ×
128 grid cells,as well as cases with meridional symmetry on a 2D grid ( r , ϕ ) in radius and longi-tude. The simulations were invariably limited to short periods (up to 400 s in Bazanand Arnett 1998) and only a few convective turnover times. One simulation (Baz´anand Arnett 1997) also tackled Si burning in 2D with a large network of 123 nuclei.These first-generation 2D models invariably found violent convective motions withMach numbers of 0.1–0.2 and velocities about an order of magnitude above the MLTpredictions, which cannot be accounted for by the aforementioned ambiguities inthe definition of the dimensionless coefficients. The convective structures invariablytended to grow to the largest angular scale allowed by the chosen wedge geometry,and large density perturbations were found at the convective boundaries. Bazan andArnett (1998) also stressed the high temporal variability of the convective flow, go-ing so far as to question whether a steady state is ever established before collapse.Longer simulations of the same 20 M (cid:12) model over 1200 ms by the same group usingthe V ULCAN code of (Livne 1993) showed the emergence of a steady state, albeitquite different from the 1D stellar evolution model due to convective boundary mix-ing (Asida and Arnett 2000),To a large extent, the pronounced differences between these first-generation sim-ulations and MLT predictions stem from the assumption of 2D flow. In 2D turbu-lence, the energy cascade is artificially inverted and goes from small to large scales(Kraichnan 1967). As a result, the flow tends to organise itself into large vortices, anddissipation occurs primarily in boundary layers (Falkovich et al. 2017; Clercx andvan Heijst 2017). Note that this thermal time scale is more difficult to define than during early burning stages whereradiative diffusion is important.ydrodynamics of core-collapse supernovae and their progenitors 33
Three-dimensional simulations of convective burning.
Consequently, 3D simulationsof convective burning obtained considerably smaller convective velocities. The first3D, full 4 π solid angle models of O shell convection (along with models of core hy-drogen burning) were presented by Kuhlen et al. (2003) for a 25 M (cid:12) star with andwithout rotation. Their simulations used the anelastic pseudospectral code of Glatz-maier (1984) to follow convection for about 90 turnovers in the non-rotating case,and approximated the burning and neutrino cooling rates by power-law fits. Differ-ent from the earlier 2D models, they found convective velocities in good agreementwith the 1D MLT prediction in the underlying stellar evolution model, but the stillobserved the emergence of large-scale flow patterns.The use of simplified burning and neutrino loss rates, the anelastic approxima-tion, and an explicit turbulent diffusivity in Kuhlen et al. (2003) still posed a con-cern, which was subsequently addressed by a series of 2D and 3D simulations ofO and C burning (Meakin and Arnett 2006, 2007b,a; Arnett et al. 2009) in wedge-shaped domains using the compressible P ROMPI code and a larger reaction network(25 species) than in the first generation of 2D models. These simulations confirmedthe significantly less violent nature of 3D convection compared to 2D (Meakin andArnett 2006, 2007b), and established good agreement between elastic and anelasticsimulations on the convective velocities and fluctuations of thermodynamic quanti-ties in the interior of convective zones, though anelastic codes cannot model fluctua-tions at convective boundaries very well (Meakin and Arnett 2007a). They also foundbalance between buoyant driving and turbulent dissipation (which is essentially a re-statement of the basic assumption of MLT) and observed rough equipartiton betweenthe radial and non-radial contributions to the turbulent kinetic energy (Arnett et al.2009). Their models still revealed differences from MLT in detail, such as differentcorrelation lengths for velocity and temperature and a non-vanishing kinetic energyflux (Meakin and Arnett 2007b). Moreover, Meakin and Arnett (2010) suggested thatthe implicit identification of the pressure scale height with the dissipation length inMLT might lead to an underestimation of the convective velocities. More recent workby the same group has stressed the time variability of the convective flow (Arnett andMeakin 2011a,b) and criticized the MLT assumption of quasi-stationary convectivevelocities. Specifically Arnett and Meakin (2011b) pointed to strong fluctuations inthe turbulent kinetic energy in the 3D oxygen shell burning simulation in a 23 M (cid:12) star by Meakin and Arnett (2007b), which they attempted to motivate by recourse tothe Lorenz model for convection in the Boussinesq approximation. The connectionbetween the simulations of convective burning and the Lorenz model for a viscous-conductive convection problem remains rather opaque, however.More recent work on 3D convection by other groups has vindicated rather thanundermined MLT as an approximation for the interior of convective zones. M¨ulleret al. (2016b) conducted a 4 π -simulation of O burning in an 18 M (cid:12) star up to thepoint of collapse and found that convection reaches a quasi-stationary state after afew turnovers with only small fluctuations in the turbulent kinetic energy. In line withMLT and as in Arnett et al. (2009), the average convective velocity is well describedby a balance of turbulent dissipation and buoyant driving in their model, and is in turn related to the average nuclear energy generation rate ˙ q nuc as v Λ ≈ . q nuc , (36)and even the profiles of the radial component of the turbulent velocity perturbationare in good agreement with the corresponding 1D stellar evolution model. A similarscaling was reported by Jones et al. (2017) based on idealized high-resolution simu-lations of O burning with a simple EoS and parameterized nuclear source terms andby Cristini et al. (2017) based on simulations of C burning in planar 3D geometry,also with a parameterized (and artificially boosted) nuclear source term. Jones et al.(2017) verified this scaling over a wider range of convective luminosities and Machnumbers by applying different boost factors to the nuclear generation rate.Regarding the dominant scales of the convective flow, the recent global 3D shellburning simulations (Chatzopoulos et al. 2014; Couch et al. 2015; M¨uller et al. 2016b;Jones et al. 2016; Yadav et al. 2020) confirm the emergence of large eddies with lowangular wavenumber (cid:96) that stretch across the entire convective zone. M¨uller et al.(2016b) verified quantitatively that the peak of the turbulent energy spectrum in (cid:96) agrees well with the wavenumber of the first unstable convective mode at the criticalRayleigh number (Chandrasekhar 1961; Foglizzo et al. 2006), (cid:96) = π ( R + + R − ) ( R + − R − ) . (37)Further simulations that also explored thinner shells (M¨uller et al. 2019) show a shifttowards higher (cid:96) and corroborate this scaling as illustrated in Fig. 7. Beyond thisdominant wavenumber, the turbulence exhibits a Kolmogorov spectrum (Chatzopou-los et al. 2014; M¨uller et al. 2016b).Naturally, the modern 3D models still exhibit differences to MLT in detail evenwithin convective zones. For example, ω often changes sign in the outer parts of aconvective layer in 3D, indicating that the spherically-averaged stratification is nomi-nally stable (Moc´ak et al. 2009; M¨uller et al. 2016b). M¨uller et al. (2016b) also remarkthat the spherically-averaged mass fraction profiles tend to be flatter in 3D than in 1D,due to the usual asymmetric choice α X = α e / α X = α e . Arigorous approach to quantify the structure of the convective flow and the differencesbetween 3D and 1D models is available in the form of spherical Reynolds decompo-sition, which has been pursued systematically by Viallet et al. (2013); Moc´ak et al.(2014); Arnett et al. (2015). The mere form of the Reynolds-averaged equations forbulk (i.e., spherically-averaged) and fluctuating quantities dictates that such an anal-ysis invariably finds dozens of terms that are implicitly set to zero in MLT. Assessment.
How are we to evaluate these commonalities and differences between3D simulations and 1D stellar evolution flow? For most purposes, the question isnot whether effects are missing in MLT-based 1D models (since the very purpose ofan approximation like MLT is to retain only the leading effects), but whether thosemissing effects matter over secular time scales or have an impact during the supernovaexplosion. As we shall discuss in detail in Sect. 4.5, the presence of asymmetries ydrodynamics of core-collapse supernovae and their progenitors 35 angular wavenumber ‘ − − n o r m a li ze d t u r bu l e n t e n e r g y s p ec tr u m E ( ‘ ) M (cid:12) ,Z = 014 . M (cid:12) ,Z = Z (cid:12) . M (cid:12) ,Z = Z (cid:12) Fig. 7
Dependence of the dominant eddy scale on the shell geometry illustrated by slices through 3Dsupernova progenitor models with convective burning and their turbulent energy spectra. The 2D slicesshow the radial velocity at the onset of collapse in progenitors of 12 M (cid:12) with metallicity Z = . M (cid:12) with Z = Z (cid:12) (top left) , and 12 . M (cid:12) with Z = Z (cid:12) (bottom left) with active convective O shells.The bottom right panel shows turbulent energy spectra E ( (cid:96) ) computed from the radial velocity around thecenter of the convective zone. The dominant wavenumber expected from Eq. (37) is indicated at the top;note that there is an uncertainty because the outer boundaries of the convective zones are fuzzy. The dottedlines show the slope of a Kolmogorov spectrum. The plots for the 12 M (cid:12) and 12 . M (cid:12) models have beenadapted from M¨uller et al. (2019). Image reproduced with permission, copyright by the authors. in convective shells indeed matters during the supernova, but the fact is also thatMLT and linear perturbation theory appear to predict the relevant parameters – thevelocities and dominant scales of convective eddies – quite well. As far as the secularevolution of convective burning shells is concerned, there is little evidence that MLTdoes not adequately describe the flow within convective shells. There is typically goodagreement in critical parameters for the shell evolution like the total nuclear burningrate. Many effects that MLT captures inaccurately and matter critically in models ofconvective envelopes and stellar atmospheres – such as the precise deviation of thestratification from superadiabticity – are of minor importance for the bulk evolutionof massive stars during the late burning stages. For more tangible consequence ofmulti-D effects on secular time scales, we need to consider convective boundaries inSect. 3.3. Simulations to the presupernova stage.
Only a few models of convective burninghave yet been carried up to the point of core collapse (Couch et al. 2015; M¨uller2016; M¨uller et al. 2016b, 2019; Yadav et al. 2020; Yoshida et al. 2019) becauseof several obstacles. In order to accurately follow the composition changes and thedeleptonization in the Fe core and Si shell (i.e., in the NSE and QSE regime) that drivethe evolution towards collapse, reaction networks with well over a hundred nuclei arerequired (Weaver et al. 1978). This is feasible in principle, but yet impractical forwell-resolved 3D simulations up to collapse. Furthermore, the initial transient phaseand imperfect hydrostatic equilibrium after the mapping from 1D to multi-D mayartificially delay the collapse.Two different strategies have been employed to circumvent these problems. Intheir simulation of Si burning in a 15 M (cid:12) star for 160 s, Couch et al. (2015) used anextended 21-species α -network with some iron group nuclei added to model coredeleptonization (Paxton et al. 2011). In order to force the core to collapse, they in-creased the electron capture rate on Fe by a factor of 50, and their 3D model in factreaches collapse more than six times faster than the corresponding 1D stellar evolu-tion model. This approach is problematic because any modification of the contractiontime scale of the core also affects the burning in the outer shells (M¨uller et al. 2016b).Using the same 21-species network, Yoshida et al. (2019) managed to evolve a 3Dsimulation of a 25 M (cid:12) star and several 2D simulations of different progenitors for thelast ∼
100 s without modifying the deleptonization rate. This suggests that multi-Dmodels can be evolved somewhat self-consistently to collapse even though the shortsimulation time is a concern in this particular case, since it remains unclear to whatextent the results are affected by the initial transient.The 3D studies of O shell burning in various progenitors by M¨uller (2016); M¨ulleret al. (2016b, 2019); Yadav et al. (2020) have followed a different approach and cir-cumvented the problems of QSE and deleptonization by excising the major part ofthe Fe core and Si shell and replacing them with an inner boundary condition thatis contracted according to a mass shell trajectory from the corresponding 1D stellarevolution model. This approach can be justified for many progenitors, which have noactive convective Si shell, or only weak convection in the Si shell.
Evolution towards collapse.
The convective flow in the contracting burning shellsshortly before collapse exhibits few noteworthy differences to the burning in quasi-hydrostatic shells described in Sect. 3.1. The 3D simulations of the different groups(Couch et al. 2015; M¨uller et al. 2016b; Yoshida et al. 2019) all show the emergenceof modes with a dominant wavelength of the order of the shell width according toEq. (37), and as far as comparisons have been performed, the convective velocitiesremain in good agreement with MLT until shortly before collapse. It is noteworthy,however, that the convective velocities and Mach numbers tend to increase signif-icantly during the last minutes before collapse because the temperature at the baseof the inner shells, and hence the burning rate, increase as they contract in the wakeof the core. The convective velocities then freeze out shortly before collapse oncethe burning rate changes on a time scale shorter than the turnover time scale. This ydrodynamics of core-collapse supernovae and their progenitors 37
10 15 20 25 30 35 M ZAMS [ M (cid:12) ]10 ‘ Si shellO shellSi shell with Ma < .
10 15 20 25 30 35 M ZAMS [ M (cid:12) ]0 . . . . . . . . M a Si shellO shell
Fig. 8
Convective Mach number (left) and dominant angular wave number (right) in the Si shell (black)and O shell (red) predicted from 1D single-star evolution models from the study of Collins et al. (2018).Image reproduced with permission, copyright by the authors. freeze-out seems to be captured adequately by time-dependent MLT so that 1D stel-lar evolution models provide good estimates for the convective velocities at the onsetof collapse. Bigger differences between 1D and 3D progenitor models can occur incase of small buoyancy barriers between the O, Ne, and C shell, in which case 3Dmodels are more likely to undergo a shell merger (Yadav et al. 2020).The evolution of the convective shells during collapse will be discussed in Sect. 4.5.
Progenitor dependence.
Since 3D simulations indicate that convective velocities andeddy scales can be estimated fairly well from 1D stellar evolution models, one canalready roughly outline the progenitor dependence of convective shell properties asshown by Collins et al. (2018). Considering the active Si and O shell burning shells atthe onset of collapse in over 2000 progenitor models, they find a number of systematictrends (Fig. 8): The O shell typically has a higher convective Mach number (0.1–0.3)than the Si shell, where usually Ma < .
1, but there are islands around 16 M (cid:12) and19 M (cid:12) in ZAMS mass where the convective Mach number in the Si shell reachesabout 0.15 and is higher than in the O shell. The highest convective Mach numbersof up to 0.3 are reached in the O shell of low-mass progenitors with small cores as Oburns deeper in the gravitational potential at higher temperatures. The general trendtowards lower convective velocities in the O shell with higher progenitor and coremass is modified by variations in shell entropy and the residual O mass fraction atthe onset of collapse. Deviations from this general trend also come about becausethe various C, Ne, O, and Si shell burning episodes do not always occur in the sameorder, and because of shell mergers.The O shell is usually thicker and therefore allows large-scale modes with wavenumbers (cid:96) <
10 to dominate. Large-scale modes are more prevalent in progenitorsabove 16 M (cid:12) with their more massive O shells. The first, thick Si shell is no longeractive at collapse in most cases, and there is typically only a thin convective Si shell(if any) between the Fe core and O shell at collapse, which will dominated by small-scale motions.Collins et al. (2018) also find a high prevalence of late shell O-Ne shell mergersamong high-mass progenitors. In about 40% of their models between 16 M (cid:12) and26 M (cid:12) such a merger was initiated within the last minutes of collapse. Although some of these trends follow from robust structural features and trends inthe progenitor evolution, these findings will need to be examined with different stellarevolution codes and may be modified in detail, especially when better prescriptionsfor convective boundary mixing on secular time scales become available.3.3 Convective boundaries
Mixing by entrainment.
As one of the most conspicuous features in their first 2Dmodels of O shell burning, Bazan and Arnett (1994); Baz´an and Arnett (1997) notedthe mixing of considerable amounts of C from the overlying layer into the active burn-ing region. Although mixing across convective boundaries (sometimes indistinctlycalled “overshooting”) had already been a long-standing topic in stellar evolution bythen, these results were noteworthy because Bazan and Arnett (1994); Baz´an andArnett (1997) found much stronger convective boundary mixing (CBM) than com-patible with overshoot prescriptions in 1D stellar evolution models of massive stars.Second, they observed that the mixed material can burn vigorously and thereby inturn dramatically affect the convective flow, i.e., there is the possibility of a feed-back mechanism that cannot occur in the case of envelope convection or surfaceconvection. Meakin and Arnett (2006) investigated this problem further in a situa-tion with active and interacting O and C shells and observed strong excitation of p-and g-modes at convective-radiative boundaries, which, as they suggested, might alsocontribute to compositional mixing.Critical steps beyond a mere descriptive analysis of CBM during the late burn-ing stages were finally taken by Meakin and Arnett (2007a), who established i) thepresence of CBM also in 3D (albeit weaker than in 2D), ii) identified the dominantprocess as entrainment driven by shear (Kelvin–Helmholtz and Holmb¨oe) instabil-ities at the convective boundary, and iii) verified that the mass entrainment ˙ M rateobeys a power law that can be motivated theoretically and has been verified in labo-ratory experiments of shear-driven entrainment (Fernando 1991; Strang and Fernando2001): ˙ M = π A ρ r v conv Ri − n b , (38)Here, A and n are dimensionless constants and Ri b is the bulk Richardson number,which can be expressed in terms of the integral scale L of the turbulent flow and thebuoyancy jump ∆ b across the boundary,Ri B = ∆ b Lv . (39)The buoyancy jump can be obtained by integrating the square of the Brunt-V¨aisalaover the extent of the boundary layer from r to r , ∆ b = r (cid:90) r − ω d r . (40)In the case of a thin boundary layer, this reduces to ∆ b = g δ ρ / ρ , where δ ρ / ρ isthe density contrast across the convective interface. From their simulations, Meakin ydrodynamics of core-collapse supernovae and their progenitors 39 and Arnett (2007b) determined values of A = .
06 and n = .
05 for the power-lawcoefficients. Since the work expended to entrain material against buoyancy the forceof buoyancy must be supplied by a fraction of the convective energy flux (an argumentwhich was independently redeveloped by Spruit 2015), one expects a value of n = B .Several subsequent 3D simulations (M¨uller et al. 2016b; Jones et al. 2017) haveconfirmed a value of n ≈ A ≈ .
1, however, but this may simply be due toambiguities in the definition and measurement of the integral length scale L , whichM¨uller et al. (2016b) identify with the pressure scale height Λ , and of the convectivevelocity v conv that enters Eq. (39) for the bulk Richardson number. Jones et al. (2017)expressed the entrainment law slightly differently by a proportionality ˙ M ∝ ˙ Q nuc tothe total nuclear energy generation rate ˙ Q nuc , which is equivalent to Eq. (38) with n =
1. Their simulations are particularly noteworthy because they employed sufficientlyhigh resolution to establish the entrainment law up to very high Ri b . Although theydo not explicitly state values of Ri b , one can estimate that their models reach up toRi b = n = .
74. This different power-law slopehas yet to be accounted for, but it is important to note that despite the shallowerpower law, Cristini et al. (2017, 2019) generally find lower entrainment rates thanMeakin and Arnett (2007b) for the same value of Ri b with a much smaller value of A = .
05. At Ri b ≈
20, their entrainment rate is actually in very good agreement withM¨uller et al. (2016b), and in the region of Ri b = n ≈
1. Since Cristini et al. (2017, 2019) also explore amuch broader range in bulk Richardson number than the aforementioned studies, onepossible interpretation could be that i) the value of A was overestimated in Meakinand Arnett (2007b), and that ii) the low value of n = .
74 may be due to a flatteningof the entrainment law below Ri b = b >
200 because of numerical resolution effects.
Shell mergers.
Convective boundary mixing can take on a dramatic form when thebuoyancy jump between two shells is sufficiently small for the neighboring shellsto merge entirely within a few convective turnover times. Such shell mergers havelong been known to occur in 1D stellar evolution models, in particular between O,Ne, and C shells (e.g., Sukhbold and Woosley 2014; Collins et al. 2018). This isbecause balanced power leads to very similar entropies in the O, Ne, and C shells,and hence small buoyancy jumps between the shells. When nuclear energy generationand neutrino cooling finally fall out of balance due to shell contraction, the entropyof the inner (O or Ne) shell frequently increases and overtakes the outer shell(s), sothat such mergers are particularly prevalent shortly before collapse as pointed outby Collins et al. (2018), who estimated that 40% of stars between 16 M (cid:12) and 26 M (cid:12) collapse during an ongoing shell merger. Although such mergers occur in 1D models,they may occur more readily in 3D, and 3D simulations are also necessary to capturethe composition inhomogeneities and nucleosynthesis during the dynamical mergerphase. Shell mergers have indeed been seen in several recent 3D simulations. M¨uller(2016) pointed out the breakout of a thin O shell through an inert, non-convective Olayer into the active Ne burning zone in a 12 . M (cid:12) star in the last minute before col-lapse, which, however, did not lead to a complete shell merger. Moc´ak et al. (2018)found a merger between the O and Ne shell in a 23 M (cid:12) model, and noted that the run-away entrainment leads to a peculiar quasi-steady with two distinct burning zones forO (at the base) and Ne (further out) within the same convective shell. However, theirsimulation only covered five turnover times and showed the merger occurring duringthe initial transient phase. Yadav et al. (2020) simulated an O-Ne shell merger in an18 . M (cid:12) over 15 turnover times, and were able to follow the evolution from the pre-merger phase with a soft, but clearly defined shell boundary and slow steady-state en-trainment through the dynamical merger phase to a partially mixed post-merger stateat the onset of collapse.. They stressed the emergence of large-scale asymmetries inthe velocity field (with extreme velocities of up to 1700 km s − ) and the compositionduring the merger, although the compositional asymmetries are already washed outsomewhat at the point of collapse. Impact on nucleosynthesis.
With multi-D simulations of the late-burning stages firmlyestablished, it is critical to identify observable fingerprints of additional convec-tive boundary mixing. The nucleosynthesis yields may provide one such fingerprint,which has already been discussed by several studies, even though one can only drawconclusions based on qualitative arguments and on 1D models with artificially en-hanced mixing so far.Davis et al. (2019) pointed out that the assumptions for convective boundary mix-ing can significantly affect the yields of various α -elements (C, O, Ne, Mg, Si), sim-ply as a consequence of the change in shell structure. However, entrainment and shellmergers may leave more specific abundance patterns. In their investigation of O-Cshell mergers in 1D Ritter et al. (2018) found significant overproduction of P, Cl,K, Sc, and possibly p-process isotopes, and argue that the occurrence of shell merg-ers may have important consequences for galactic chemical evolution (GCE). Morerecently, Cˆot´e et al. (2020) considered a Si-C shell merger, for which they find sig-nificant overproduction of V and Cr, which allows them to strongly constrain therate of such events based on observed Galactic stellar abundances. This is relatedto the long-standing realization that the ashes of hydrostatic silicon burning underneutron-rich conditions cannot be ejected in large quantities because of GCE con-straints (Woosley et al. 1973; Arnett 1996).Supernova spectroscopy may also help constrain additional convective boundarymixing and shell mergers may via their nucleosynthetic fingerprints. For example,the ejection of neutron-rich material from the silicon shell that is mixed out by en-trainment before the explosion would lead to a supersolar Ni/Fe ratio as observed insome supernovae (Jerkstrand et al. 2015). Mixing of minimal amounts of Ca into O-rich zones can also have significant repercussions since only a mass fraction of a few10 − in Ca is required for Ca to be the domnant coolant during the nebular phase andquench O line mission in a shell (Fransson and Chevalier 1989; Kozma and Fransson1998). This diagnostic potential of supernova spectroscopy for convective boundarymixing needs to be explored further in the future, but further (macroscopic) mixing ydrodynamics of core-collapse supernovae and their progenitors 41 during the explosion presents a major complication as it is not straightforward todisentangle the effect of mixing processes prior and during the explosion. Secular effect on stellar evolution.
Evaluating the observational consequences of theconvective boundary mixing seen in 3D models is also difficult because there is stillno rigorous method for treating these processes in 1D stellar evolution codes. A crudeestimate for the shell growth by entrainment can be formulated by noting that thework required to entrain material with density contrast δ ρ / ρ against buoyancy mustbe no larger than a fraction of the time-integrated convective energy flux (Spruit 2015;M¨uller 2016). During the late burning stage, the convective energy flux is set by thenuclear energy generation rate, and hence one can estimate that the entrained mass ∆ M entr over the lifetime of a shell with mass M shell and radius r is roughly GMr δ ρρ ∆ M entr ∼ AM shell ∆ Q , (41)where A is the dimensionless coefficient in Eq. (38) and ∆ Q is the average Q -valuefor a given burning stage (M¨uller 2016). Based on Eq. (41), M¨uller (2016) estimatesthat O shells could grow by tens of percent in mass by entrainment; for Si shells oneexpects a smaller effect, for C shells, the effect may be bigger.How one can go beyond such simple estimates by using improved recipes forconvective boundary mixing in stellar evolution codes is still an unresolved question.A common approach, based on the simulations of surface convection by Freytag et al.(1996), models entrainment as diffusive overshooting with an exponential decay ofthe MLT diffusion coefficient outside the convective boundary. The length scale λ OV for the exponential decay can be calibrated against 3D simulations. This approachhas been followed by Cˆot´e et al. (2020); Davis et al. (2019) and by many works onadditional convective boundary mixing in low-mass mass stars, but has several is-sues. Entrainment is a very different process than diffusive overshoot that operatesin a distinct physical regime (high P´eclet number), and hence one should not expectthat it can be described by the same formalism (Viallet et al. 2015). It is also un-clear why diffusive overshoot should be applied only for compositional mixing inthe entrainment regime. The common approach of expressing λ OV by the pressurescale height is also open to criticism because the relevant length scale should be setby the convective velocities and the buoyancy jump, so that one would rather expect λ OV ∝ v / ∆ b .Staritsin (2013) proposed an alternative approach of extending convectively mixedregions with time following the entrainment law (38), which better reflects the physicsof the entrainment process. However, this approach has not been applied to the lateneutrino-cooled burning stages yet (i.e., precisely where entrainment should operate).It also has some conceptual issues, for example, the entrainment law (38) obviouslybreaks down if there is convection on both sides of a shell interface. Yet another ap-proach for extra mixing in 1D models was followed by Young et al. (2005), whohandle mixing based on the local gradient Richardson number Ri for shear flows,which is estimated using an elliptic equation for the amplitudes of waves excited byconvective motions (Young and Arnett 2005). This approach is physically motivated, but is still awaiting (and worthy of) a more quantitative comparison with 3D simula-tions beyond the qualitative discussion in Young and Arnett (2005) Flame propagation in low-mass progenitors.
Around the minimum progenitor massfor supernova explosions, multi-D effects can have a more profound effect than merelychanging shell masses on a modest scale, and may decide about the final fate of thestar. This regime is best exemplified by the electron-capture supernova channel ofsuper-asymptotic giant branch (SAGB) stars (see Jones et al. 2013; Doherty et al.2017; Nomoto and Leung 2017; Leung and Nomoto 2019 for a broader overviewand a discussion of uncertainties). In this evolutionary channel, the star builds upa degenerate core composed primarily of O and Ne. If this core grows to 1 . M (cid:12) ,electron captures on Ne and Mg trigger an O deflagration. Depending on the inter-play of deleptonization (which decreases the degeneracy pressure) and the nuclearenergy release, the core either contracts, collapses to a neutron stars, and explodes asan electron capture supernova, or the core does not collapse and explodes as a weakthermonuclear supernova (Jones et al. 2016). Since the flame is turbulent, its propaga-tion needs to be modelled in multi-D, similar to deflagrations in Type Ia supernovae.Simulations of this problem have been conducted by Jones et al. (2016); Kirsebomet al. (2019) in 3D and (Leung and Nomoto 2019; Leung et al. 2020) in 2D. Efforts toimprove the nuclear input physics and explore the sensitivity to the ignition geometry,general relativity, and flame physics are ongoing (e.g., Kirsebom et al. 2019; Leunget al. 2020).For slightly more massive stars, one encounters similar situations with convectively-bounded flames after off-center ignition of O or Si (Woosley and Heger 2015). Again,multi-D effects may significantly affect the final evolutionary phase before collapsein this regime, but multi-D simulations of such supernova progenitors are yet to becarried out (but see Lecoanet et al. 2016 for idealized 3D simulations relevant to thisregime).3.4 Current and future issuesSignificant progress notwithstanding, multi-D simulations of the late stages of con-vective burning still face further challenges. For the evolution towards collapse, mod-els will eventually need to include a more sophisticated treatment of burning anddeleptonization in the QSE and NSE regime and forgo the current approach of ei-ther using small networks or excising the Fe/Si core. Perhaps an even greater concernabout the conclusions of current multi-D simulations lies with the time-scale prob-lem, however. No 3D simulations have yet been evolved sufficiently long to reach thestate of balanced power (or to reveal why it would not be reached). This may haverepercussions for turbulent entrainment, which ultimately taps the energy in turbulentmotions and hence cannot be completely disconnected from the energy budget withina shell.Moreover, although a few attempts have been made to simulate convection inrotating shells in 2D (Arnett and Meakin 2010; Chatzopoulos et al. 2016) and 3D ydrodynamics of core-collapse supernovae and their progenitors 43 (Kuhlen et al. 2003), multi-D simulations have yet to investigate the angular mo-mentum distribution and angular momentum transport during the late pre-supernovastages in a satisfactory manner. Three-dimensional simulations are even more crit-ical for this purpose than in the non-rotating case since many relevant phenomenasuch as (Rossby waves, Taylor-Proudman columns) cannot be modeled adequately in2D. Simulations also need to explore a larger parameter space since the convectivedynamics will depend on the Rossby number Ro ∼ v conv / ( Ω R ) (where Ω is the ro-tational velocity). Furthermore, the time scales become a bigger challenge becausemodels need to be run for several rotational periods T = πΩ − and several convec-tive turnover times τ conv (whichever is longer), which is problematic since rotationin pre-supernova models is likely slow (e.g., Heger et al. 2005) so that Ro (cid:29) T (cid:29) τ conv . On the bright side, multi-D simulations may reveal much more interest-ing differences to 1D stellar evolution models: Both Kuhlen et al. (2003) and Arnettand Meakin (2010) found pronounced differential rotation developing from a rigidlyrotating initial state, and Arnett and Meakin (2010) suggest that convective shellsmight adjust to a stratification with constant angular momentum as a function of ra-dius rather than to uniform rotation as assumed in stellar evolution models. However,more simulations and more rigorous analysis is required to investigate these claims.The problem of rotation can obviously not be solved without including magneticfields in the long run. It is well known (e.g., Shu 1992) that the criterion for the in-stability of rotating flow becomes less restrictive in the MHD case, and effects suchas the magnetorotational instability (MRI, Balbus and Hawley 1991) or a small-scaledynamo may enforce a more uniform rotation profile than hydrodynamic convectionalone. But the importance of magnetic fields is not confined to the case rotating pro-genitors. Prompted by helioseismic measurements that indicate smaller convectivevelocities in the deeper layer of the solar convection zone (Gizon and Birch 2012;Hanasoge et al. 2012), some simulations of magnetoconvection in the Sun found asuppression of convective velocities by up to 50% compared to hydrodynamic simu-lations due to strong magnetic fields from a small-scale dynamo that reach equipar-tition with the turbulent kinetic energy (Hotta et al. 2015). Magnetic fields can alsoinhibit or enhance mixing in shear layers (Br¨uggen and Hillebrandt 2001) and mayhence affect convective boundary mixing. Thus, there is still plenty of ground left toexplore for simulations of the late burning stages. In the Introduction, we already outlined a variety of multi-D effects than can play arole in reviving the stalled supernova shock as a subsidiary agent to neutrino heat-ing (i.e., neutrino-driven convection in the gain region, the SASI, and progenitor as-phericities), or also as the main drivers of the explosion (rotation and magnetic fields).Historically, a number of works have also considered convection in the PNS interioras a means for precipitating explosions by enhancing the neutrino emission fromthe PNS (Epstein 1979; Wilson and Mayle 1988; Burrows and Lattimer 1988; Wil-son and Mayle 1993), but these hopes were not substantiated in subsequent decades.Nonetheless, convection inside the PNS remains important for various aspects of the supernova problem such as the neutrino and gravitational wave signals and the nucle-osynthesis conditions in the innermost ejecta.Since each of these phenomena has proved rich and complex over the years, itis no longer possible to treat them adequately within a chronological narrative of thequest for the supernova explosion mechanism. Nevertheless, ascertaining the explo-sion mechanism by means of first-principle simulations remains the overriding con-cern in supernova theory, and it is therefore still useful to recapitulate the progressin supernova explosion modelling from the advent of the first 2D models with a sim-plified treatment of neutrino heating and cooling in the 1990s (Herant et al. 1992;Shimizu et al. 1993; Yamada et al. 1993; Janka et al. 1993; Herant et al. 1994; Bur-rows et al. 1995; Janka and M¨uller 1995, 1996). A more detailed analysis of theindividual hydrodynamic phenomena beyond this chronicle of simulations is thenprovided in Sects. 4.1 to 4.7.
Neutrino-driven explosions in 2D.
Although the 2D simulations of the early and mid-1990s had shown multi-D effects to be helpful for shock revival, these models did notutilize neutrino transport on par with the best available methods for 1D simulations atthe time. In a first attempt to better model neutrino heating and cooling in 2D by us-ing the pre-computed neutrino radiation field from a 1D simulation with multi-groupflux limited diffusion, Mezzacappa et al. (1998) were unable to reproduce the suc-cessful explosions found in earlier 2D models. This led to a resurgence of interest inaccurate methods for neutrino transport, culiminating in the development of Boltz-mann solvers for relativistic (Yamada et al. 1999; Liebend¨orfer et al. 2001, 2004) andpseudo-Newtonian simulations (Rampp and Janka 2000, 2002). The explosion prob-lem was then revisited in 2D using various types of multi-group neutrino transportfrom the mid-2000s onwards. Neutrino-driven explosions were obtained in many ofthese 2D simulations for a wide range of progenitors (Buras et al. 2006a; Marek andJanka 2009; M¨uller et al. 2012b,a, 2013; Janka 2012; Janka et al. 2012; Suwa et al.2010, 2013; Bruenn et al. 2013, 2016; Nakamura et al. 2015; Burrows et al. 2018; Panet al. 2018; O’Connor and Couch 2018b), though still with significant differences be-tween the various simulation codes.
Challenges and successes in 3D.
Following isolated earlier attempts at 3D modellingusing the “light-bulb” style models of the 1990s (Shimizu et al. 1993; Janka andM¨uller 1996), or gray flux-limited diffusion (Fryer and Warren 2002), the role of3D effects in the explosion mechanism was finally investigated vigorously in thelast decade, starting again with simple light-bulb models (Nordhaus et al. 2010b;Hanke et al. 2012; Couch 2013; Dolence et al. 2013). Except for spurious results inNordhaus et al. (2010b), these light-bulb models indicated a similar “explodability” in2D and 3D. However, subsequent 3D models with rigorous neutrino transport provedmore reluctant to explode; indeed the first 3D models of 11 . M (cid:12) , 20 M (cid:12) , and 27 M (cid:12) progenitors using multi-group, three-flavour neutrino transport did not explode at all(Hanke et al. 2013; Tamborra et al. 2014a).Even though various groups have now obtained explosions in 3D simulations,shock revival usually occurs later than in 2D, and often requires additional (andsometimes hypothetical) ingredients to improve the heating conditions or a specific ydrodynamics of core-collapse supernovae and their progenitors 45 progenitor structure. For low-mass single-star (Melson et al. 2015b; M¨uller 2016;Burrows et al. 2019) and binary (M¨uller et al. 2018) progenitors just above the iron-core formation limit, 3D simulations readily yield explosions since the steep drop ofthe density outside the iron core implies a rapid drop of the accretion rate onto theshock after bounce. For more massive stars the record is mixed. For standard, non-rotating progenitors in the range between 11 . M (cid:12) and 27 M (cid:12) and unmodified, state-of-the-art microphysics, no explosions were found in simulations using the V ERTEX code (Hanke et al. 2013; Tamborra et al. 2014a; Melson et al. 2015a; Summa et al.2018) and the F
LASH -M1 code (O’Connor and Couch 2018a). On the other hand, theOak Ridge group obtained an explosion for a 15 M (cid:12) star (Lentz et al. 2015) with theirC HIMERA code, and the Princeton group observed shock revival in eleven out of four-teen models between 9 M (cid:12) and 60 M (cid:12) (Vartanyan et al. 2019b; Burrows et al. 2019;Radice et al. 2019; Burrows et al. 2020) with the F ORNAX code. In both cases theaccuracy of the microphysics, the neutrino transport, and gravity treatment appearscomparable to V
ERTEX . Three-dimensional simulations using other codes (that areconstantly evolving!) are more difficult to compare as they involve simplificationsin the microphysics or transport compared to V
ERTEX , C
HIMERA , and F
ORNAX ,although some of them compensate for this by higher resolution in real space and en-ergy space and a better treatment of gravity. At any rate, results obtained with othercodes such as C O C O N U T-FMT, F
UGRA , Z
ELMANI , and 3D N SN E add to the pic-ture of simulations straddling the verge between successful shock revival (Takiwakiet al. 2012, 2014; M¨uller 2015; Roberts et al. 2016; Chan et al. 2018; Ott et al. 2018;Kuroda et al. 2018) and failure (M¨uller et al. 2017a; Kuroda et al. 2018) for standard,non-rotating progenitors and standard or simplified microphysics.These different results may simply indicate that the neutrino-driven mechanismoperates close to the critical threshold for explosion in nature. Observations of su-pernova progenitors indeed indicate that black hole formation occurs already at rel-atively low masses down to to ∼ M (cid:12) for single stars (Smartt et al. 2009; Smartt2015). Since the lack of robust explosions in 3D persists to even lower masses, andsince strongly delayed explosions in 3D may turn out too weak to be compatiblewith observations, several groups have explored new avenues towards more robustexplosions. Some of the proposed ideas invoke modifications or improvements to themicrophysics that ultimately lead to improved neutrino heating conditions, such asstrangeness corrections to the neutral-current scattering rate (Melson et al. 2015a)and muonization (Bollig et al. 2017). Other studies have explored purely hydrody-namic effects. Among these, Takiwaki et al. (2016); Janka et al. (2016); Summa et al.(2018) pointed out that rapid progenitor rotation could be conducive to shock revivaleven without invoking MHD effects. Another idea posits that including seed perturba-tions from the late convective burning stages can facilitate shock revival. First studiedby Couch and Ott (2013) and M¨uller and Janka (2015) using parametric initial con-ditions, this “perturbation-aided” mechanism has subsequently been explored furtherusing pre-collapse perturbations from 3D models of the late burning stages, initiallywith ambiguous results in the leakage simulations of Couch et al. (2015) and thenin a number of 3D simulations using multi-group neutrino transport (M¨uller 2016;M¨uller et al. 2017a, 2019), where it led to robust explosions over a wider mass rangefrom 11 . M (cid:12) to 18 M (cid:12) for single stars. Neutrino-driven explosion models have thus matured considerably in recent years,but it would be premature to declare the problem of shock revival solved. The dis-crepancies between the results of different groups have yet to be sorted out, and thereis still no “gold standard” among the simulations that combines the best neutrinotransport, the best microphysics, 3D progenitor models, and general relativity. More-over, phenomenological models of neutrino-driven explosions (Ugliano et al. 2012;Pejcha and Thompson 2015; Sukhbold et al. 2016; M¨uller et al. 2016a) suggest thata different mechanism is still needed to explain hypernova explosions with energiesabove 2 × erg. Magnetohydrodynamic simulations.
The mechanism(s) behind hypernovae likely relyon rapid rotation and strong magnetic fields (Akiyama et al. 2003; Woosley andBloom 2006), but the importance of magnetic fields may not end there. There may bea continuous transition from neutrino-driven explosions to MHD-driven explosions(Burrows et al. 2007a), and strong magnetic fields may also a role in non-rotatingprogenitors as an important driving agent or as a subsidiary to neutrino heating (Ober-gaulinger et al. 2014).Although the ideas of Akiyama et al. (2003) quickly triggered first 2D MHDcore-collapse supernova simulations (e.g., Yamada and Sawai 2004; Sawai et al.2005; Obergaulinger et al. 2006; Shibata et al. 2006), there is still only a smallcorpus of magnetorotational supernova explosion models, especially if we focus onmodels of the entire collapse, accretion, and early explosion phase using reasonablydetailed microphysics and disregard parameterized models of relativistic and non-relativistic jets and of collapsar disks. Burrows et al. (2007a) presented 2D simula-tions of magnetorotational explosions of a 15 M (cid:12) progenitor (later followed by MHDsimulations of accretion-induced by collapse in Dessart et al. 2007) with the New-tonian radiation-MHD code V ULCAN and demonstrated the ready emergence of jetspowered by strong hoop stresses for sufficiently strong initial fields. Burrows et al.(2007a) made the important point that these non-relativistic jets are a distinctly dif-ferent phenomenon from the relativistic jets seen in long gamma-ray bursts (GRBs),which may be formed several seconds after shock revival.Like most other subsequent simulations, these models relied on parameterizedinitial conditions with artificially strong magnetic fields to mimic the purported fastamplification of much weaker fields in the progenitor by the magnetorotational insta-bility (Balbus and Hawley 1991; Akiyama et al. 2003). They also imposed the pro-genitor rotation profile by hand. The 2D studies of Obergaulinger and Aloy (2017,2020); Bugli et al. (2020) have recently explored variations in the assumed initialfield strength and topology and the assumed rotation profiles more thoroughly. Whilethey find considerable variation in the outcome of their models, it is interesting tonote that in some instances Obergaulinger and Aloy (2020) even find magnetorota-tional explosions for the unmodified rotation profile and magnetic field strength oftwo of the 35 M (cid:12) progenitor models from Woosley and Heger (2006), although itis not perfectly clear what the precise geometry of the field in the stellar evolutionmodels ought to be.The imposition of axisymmetry is an even bigger concern in the case of mag-netorotational supernovae than for nuetrino-driven explosion models. Several 3D ydrodynamics of core-collapse supernovae and their progenitors 47 r [km]10 ρ [ g c m - ] , P [ e r g c m - ] , T [ K ] cooling heating ρ PT shockr -4 r -3 r -1 PNS coreconvectively unstable a t m o s ph e r e s [ k b / nu c l e on ] , Y e v r Y e s r [km] -0.25-0.20-0.15-0.10-0.050.00 v r / c cooling heating v r Y e s PNS core convectively unstable deceleration region shock
Fig. 9
Schematic 1D structure of the supernova core after the formation of the gain region, illustrated byprofiles of the density ρ , pressure P , temperature T (left), radial velocity v r , entropy s , and electron fraction Y e (right). The profiles are taken from a 1D radiation hydrodynamics simulation of the 20 M (cid:12) progenitorof Woosley and Heger (2007) at a post-bounce time of 200ms. See text for details. simulations of MHD-driven explosions have by now been performed, but amongthese only Obergaulinger and Aloy (2020) included multi-group neutrino transport,whereas the others (Winteler et al. 2012; M¨osta et al. 2014b, 2018) employed a leak-age scheme. The prospects for successful magnetorotational explosions in 3D are stillsomewhat unclear. M¨osta et al. (2018) reported the destruction of the emerging jetsby a kink instability, although the jet can apparently be stabilised if the poloidal fieldstrength is comparable to the toroidal field strength (M¨osta et al. 2018). Moreover,the explosion dynamics already depends sensitively on the assumed initial field ge-ometry; strong dipole fields appear to be required for the most powerful explosions(Bugli et al. 2020). Given the vast uncertainties concerning the initial rotation rates,field strengths, and field geometries in the supernova progenitors, considerably morework is necessary before the magnetorotational mechanism can be considered robusteven for a small sub-class of progenitors. We will therefore focus only on the hydro-dynamics of neutrino-driven explosions in the subsequent discussion.4.1 Structure of the accretion flow and runaway conditions in spherical symmetryBefore analyzing the role of multi-D phenomena in core-collapse supernovae in greaterdepth, it is expedient to discuss the structure of the supernova core that emerges oncethe gain region has formed a few tens of milliseconds after collapse in an idealized,spherically-symmetric picture as shown in Fig. 9. Our discussion closely follows theworks of Janka (2001); M¨uller and Janka (2015); M¨uller et al. (2016a) which may beconsulted for further details. Structure of the accretion flow.
At this stage, the PNS consists of an inner core ofabout 0 . M (cid:12) (depending on the equation of state) with low entropy, which is sur-rounded by an extended mantle of about 1 M (cid:12) that was heated to entropies of about6 k b / nucleon as the shock propagated through the outer part of the collapsed iron coreand most of the Si shell. The mantle extends out to the neutrinosphere at high sub-nuclear densities, where neutrinos on average undergo their last interaction beforeescape (for details see Kotake et al. 2006; Janka 2017; M¨uller 2019b; Mezzacappa R ν , the pres-sure P is dominated by non-relativistic baryons, and neutrino interactions are stillfrequent enough to act as “thermostat” and maintain a roughly isothermal stratifica-tion, resulting in an exponential density profile (Janka 2001): ρ = ρ ν exp (cid:20) − GMm n rk b T ν (cid:18) − R ν r (cid:19)(cid:21) , (42)where M is the PNS mass, R ν , T ν , and ρ ν are the neutrinosphere radius, tempera-ture, and density, and m n is the neutron mass. To maintain rough isothermality withthe neutrinosphere, the accreted matter must cool as it is advected through the atmo-sphere. Below a density of about 10 g cm − , the pressure is dominated by relativisticelectron-positron pairs and photons, and around this point neutrino heating starts dodominate over neutrino cooling at the gain radius R g . Since the cooling and heatingrate scale with T ∝ P / and L ν E ν / r in terms of the matter temperature T and theelectron-flavor neutrino luminosity L ν and mean energy E ν (appropriately averagedover electron neutrinos and antineutrinos), balance between heating and cooling de-fines an effective thermal boundary condition for the radiation-dominated gain regionfurther out, P / ∝ L ν E ν R . (43)Before shock revival, the stratification between the gain radius is roughly adiabaticout to the shock , resulting in power-law profiles ρ ∝ r − and T ∝ r − for the tem-perature and density. Ahead of the shock, the infalling material moves with a radialvelocity of | v r | ≈ (cid:112) GM / r (i.e., a large fraction of the free-fall velocity), and the den-sity is given in terms of the mass acrretion rate ˙ M as ρ = ˙ M / ( π r | v r | ) . In a quasi-stationary situation, the stalled accretion shock will adjust to a radius R sh such thatthe jump conditions are fulfilled and the post-shock pressure P sh and the pre-shockram pressure P ram = ρ v r are related by P sh = ββ − P ram = ββ − M π R (cid:114) GMR sh ∝ ˙ MR − / . (44)Here β is the compression ratio in the shock, which varies from β ≈
10 early on,which is slightly larger than the value of β = γ = / β ≈ R sh ∝ P / . Recognizing that theheating rate ˙ q heat ∝ L ν T ν r − and the cooling rate ˙ q cool ∝ T ∝ P / balance at the Properly speaking, the EoS transition radius between the baryon-dominated and the radiation-dominated regime and the gain radius are close, but the gain radius is slightly larger (Janka 2001). Formany purposes it is not critical to distinguish them. This is because neutrino heating does not change the entropy appreciably as material traverses thegain region as long as the heating conditions are far from critical. Furthermore, mixing reduces the entropygradient in 3D once convection or SASI have developed.ydrodynamics of core-collapse supernovae and their progenitors 49 gain radius, and using the adiabatic stratification in the gain region and the jump con-ditions, one can go further and derive that the shock radius scales as (Janka 2012;M¨uller and Janka 2015) R sh ∝ ( L ν T ν ) / R / ˙ M / M / , (45)in spherical symmetry in terms of L ν , T ν , R g , M and the mass accretion rate ˙ M , whichis related to the density profile of the progenitor (Woosley and Heger 2012; M¨ulleret al. 2016a). Conditions for shock revival.
So far, we have assumed a stationary accretion flow inthis picture. The problem of shock revival is, however, related to the breakdown of sta-tionary accretion solutions (Burrows and Goshy 1993; Janka 2001), or more strictlyspeaking, to the development of non-linearly unstable flow perturbations (Fern´andez2012). The transition to runaway shock expansion can be understood in terms ofa competition of time scales, namely the advection or residence time τ adv that theaccreted material spends in the gain region, and the time scale τ heat for unbindingthe material in the gain region by neutrino heating (Thompson 2000; Janka 2001;Thompson et al. 2005; Buras et al. 2006a; Murphy and Burrows 2008). These can becomputed in terms of the binding energy E g and mass M g of the gain region, the massaccretion rate ˙ M , and the volume-integrated heating rate ˙ Q ν as τ adv = M g ˙ M , (46) τ heat = | E g | ˙ Q ν . (47)Transition to runaway expansion is expected if τ adv / τ heat (cid:38)
1, which is borne out by1D light-bulb simulations with a pre-defined neutrino luminosity (Fern´andez 2012).Alternatively, the runaway condition can be expressed in terms of a critical luminosity L crit above which there are no stationary 1D accretion solutions (Burrows and Goshy1993). Janka (2012) and M¨uller and Janka (2015) have pointed out that these twodescriptions are essentially equivalent by converting the time-scale criterion into apower law for the critical value of the “heating functional” L = L ν E ν , L crit = ( L ν E ν ) crit ∝ ( M ˙ M ) / R − / . (48)Other largely equivalent ways to characterize the onset of a runaway instability (atleast in spherical symmetry) are the notion that the Bernoulli parameter reaches zerosomehwere in the gain region around shock revival (Burrows et al. 1995; Fern´andez2012), and the antesonic condition c > / GM / r (Pejcha and Thompson 2012),which is effectively a condition for the flow enthalpy just like the Bernoulli parameter(M¨uller 2016). Qualitative description.
This simple picture is useful for qualitatively understandinghow multi-D effects modify the spherically-averaged bulk structure of the post-shockflow and hence affect the conditions for shock revival. It is intuitive from Eq. (44)that increasing the post-shock pressure (e.g., by turbulent heat transfer), or addingturbulent or magnetic stresses will increase the shock radius and modify Eq. (45) forthe spherically symmetric case. This will then affect the time scales τ adv and τ heat and thereby modify the conditions for runaway shock expansion driven by neutrinos.Moreover, certain multi-D phenomena may also facilitate runaway shock expansionmore directly by dumping extra energy into the gain region, which may take the formof thermal energy, turbulent kinetic energy, or magnetic energy. This is, of course,only a coarse-grain interpretation of the effect of multi-D effects, which needs to bebased on a more careful analysis of the underlying hydrodynamic phenomena.The studies from the 1990s also outlined qualitative explanations for the ben-eficial role of multi-D effects. Herant et al. (1994) interpreted convection as of anopen-cycle heat engine that continuously pumps transfers energy from the gain radius(where neutrino heating is strongest) further out into the gain region, and Janka andM¨uller (1996) similarly stress the importance of more effective heat transfer from thegain region to the shock. Herant et al. (1994) argued that large-scale mixing motionsare also advantageous because they continue to channel fresh matter to the coolingregion during the explosion phase so that the neutrino heating is not quenched whenthe shock is revived. Finally, Burrows et al. (1995) pointed out what we now sub-sume under the notion of turbulent stresses: As the convective bubbles collide withthe shock surface with significant velocities (or in modern parlance provide “turbulentstresses”) and thereby deform and expand it. Modified critical luminosity.
Since then, the impact of multi-D effects has been ana-lyzed more quantitatively. Several studies (Buras et al. 2006a; Murphy and Burrows2008; Hanke et al. 2012) showed that the advection time scale τ adv is systematicallylarger in multi-D, while the onset of runaway shock expansion is still determined bythe criterion τ adv / τ heat . This suggests that the runaway is still powered by neutrinoheating just as in 1D, and that multi-D effects facilitate explosions facilitate shock re-vival by somewhat expanding the stationary shock, keeping a larger amount of massin the gain region, and thereby increasing the heating efficiency. To a lesser extent,mixing also reduces the binding energy of the gain region (M¨uller 2016), but thisappears to be of secondary importance for shock revival.Building on the 1D picture from Sect. 4.1, the increase of the quasi-stationaryshock radius can be understood as the consequence of additional “turbulence” Note that this refers to a comparison of multi-D and 1D models for a given set of parameters ofthe accretion flow ( L ν , E ν , M , ˙ M , and R g . When comparing multi-D and 1D models at the threshold toexplosion (with different L ν and E ν ), the heating efficiency can be lower in multi-D (Couch and Ott 2015),but this does not mean that there is a different runaway mechanism (M¨uller 2016). It is important to stress that “turbulence” is something of a convenient misnomer in this context andrefers to any deviation from quasi-stationary, spherically-symmetric flow. This should be carefully distin-guished from the usual notion of turbulence in high-Reynolds number flow, although the two concepts arefrequently conflated.ydrodynamics of core-collapse supernovae and their progenitors 51 terms that arise in a spherical Reynolds or Favre decomposition of the flow, whoseimportance can be gauged by the square of the turbulent Mach number Ma in thegain region (M¨uller et al. 2012a). Using light-bulb simulations, Murphy et al. (2013)demonstrated quantitatively that the inclusion of Reynolds stresses (“turbulent pres-sure”) largely accounts for the higher shock radius in multi-D models. The criticalrole of the turbulent pressure was confirmed by Couch and Ott (2015) using a leak-age scheme and by M¨uller and Janka (2015) with multi-group neutrino transport.The resulting effect on the critical luminosity can be estimated by including aturbulent pressure term in Eq. (44), which ultimately leads to (M¨uller and Janka2015) ( L ν E ν ) crit ∝ ( M ˙ M ) / R − / (cid:18) + (cid:19) − / = ( L ν E ν ) crit , (cid:18) + (cid:19) − / , (49)where the critical luminosity in 1D, ( L ν E ν ) crit , , is modified by a correction factorcontaining the turbulent Mach number in the gain region. Although based on a rathersimple analytic model, Eq. (49) describes shock revival in 2D (Summa et al. 2016)and 3D models (Janka et al. 2016) remarkably well. This suggests that the critical pa-rameter for increased explodability in multi-D is indeed the turbulent Mach number,although, as argued by Mabanta and Murphy (2018), the larger accretion shock ra-dius may not be due to turbulent pressure alone. Even if other effects such as turbulentheat transport, turbulent dissipation (Mabanta and Murphy 2018), and even turbulentviscosity (M¨uller 2019a) play a role, one expects a scaling law similar to Eq. (49)simply because any leading-order correction to the 1D jump condition (44) from aspherical Reynolds decomposition will scale with Ma , only with a slightly differentproportionality constant than in Eq. (49). The turbulent Mach number itself will bedetermined by the growth and saturation mechanisms of the non-radial instabilitiesin the gain region as discussed in the following sections.4.3 Neutrino-driven convection in the gain regionConvection in the gain region develops because neutrino heating establishes a nega-tive entropy gradient. In many respects, this “hot-bubble convection” resembles con-vection on top of a quasi-hydrostatic spherical background structure as familiar fromthe earlier phases of stellar evolution, but there are subtle differences because theinstability occurs in an accrretion flow. Condition for instability.
Under hydrostatic conditions, the Ledoux criterion for con-vective instability can be written as (Buras et al. 2006b), C L = ∂ ρ∂ r − c ∂ P ∂ r = (cid:18) ∂ ρ∂ s (cid:19) P , Y e ∂ s ∂ r + (cid:18) ∂ ρ∂ Y e (cid:19) P , s ∂ Y e ∂ r > , (50) An alternative approach to account for the effect of the turbulent pressure is to use an effective adia-batic index γ > / in terms of the gradients of density, pressure, entropy s and electron fraction Y e . Usinga local stability analysis for a displaced blob, one finds a growth rate Im ω BV , wherethe Brunt-V¨ais¨al¨a frequency ω BV is defined as ω = − gC L ρ . (51)In a stationary accretion flow, the radial derivatives can be expressed in terms ofthe time derivatives ˙ s and ˙ Y e and the advection velocity v r , C L = v r (cid:34)(cid:18) ∂ ρ∂ s (cid:19) P , Y e ˙ s + (cid:18) ∂ ρ∂ Y e (cid:19) P , s ˙ Y e (cid:35) > . (52)Once the gain region forms around 80–100 ms after bounce, the electron fractiongradient plays a minor role for stability, and the material in the gain region can bewell described as a radiation-dominated gas with P ∝ ( ρ s ) / so that ω = − g ˙ q e v r c . (53)This has an important consequence: Different from a hydrostatic background, thestability of a heated accretion flow (or outflow) depends on the sign of the advectionvelocity rather than on the profile of the heating function. Using the aforementionedscaling for the heating rate and assuming a linear velocity profile behind the shock,we obtain an estimate ω ∝ GMR (cid:104) L ν E ν (cid:105) R (cid:18) β R sh GM (cid:19) / (cid:18) R sh R g (cid:19) . (54)More importantly, however, advection can stabilize the flow against convectionbecause perturbations only have a finite time to grow as they cross the gain region aspointed out by Foglizzo et al. (2006), who demonstrated that instability is regulatedby a parameter χ , χ = r sh (cid:90) r g Im ω BV | v r | d r . (55)Instability should only occur for χ (cid:38)
3. This has indeed been confirmed in a numberof parameterized (Scheck et al. 2008; Fern´andez et al. 2014; Fern´andez 2015; Couchand O’Connor 2014) and self-consistent simulations (M¨uller et al. 2012a; Hanke et al.2013) in 2D and 3D.
Dominant eddy scale.
Similar to the situation in convective shell burning, the lengthscale of the most unstable linear mode is determined by the width of the gain regionaccording to Eq. (37) (Foglizzo et al. 2006). In 3D this remains the characteristiclength scale of convective eddies during the non-linear saturation stage (e.g., Hankeet al. 2013). Since the ratio of the shock and gain radius typically lies in the range R sh / R g = . (cid:96) ≈ ydrodynamics of core-collapse supernovae and their progenitors 53 the accretion phase (Hanke et al. 2013; Couch and O’Connor 2014). Around andafter shock revival, large-scale modes with (cid:96) = (cid:96) = (cid:96) = (cid:96) = Non-linear saturation.
The evolution towards shock revival typically proceeds oversufficiently long time scales for hot-bubble convection to reach a quasi-stationarystate. Using 2D light-bulb simulations, Murphy et al. (2013) first demonstrated thatthis quasi-stationary state closely mirrors the situation in stellar convection (cf. Sect. 3.1),i.e., neutrino heating, buoyant driving, and turbulent dissipation balance each other(see also Murphy and Meakin 2011), and as a result the convective luminosity scaleswith the neutrino heating rate. Alternatively, the quasi-stationary state can be char-acterized by the notion of marginal stability; the flow adjusts itself that such that the χ -parameter for the spherically averaged flow converges to (cid:104) χ (cid:105) ≈ δ v scale as δ v = ( ˙ q ν ( R sh − R g )) / , (56)in terms of the average mass-specific neutrino heating rate ˙ q ν in the gain region. In3D, the convective velocities are slightly smaller (M¨uller 2016), δ v = . [ ˙ q ν ( R sh − R g )] / . (57)The smaller proportionality constant in 3D can be motivated by the tendency of theforward turbulent cascade to create smaller structures, which decreases the dissipa-tion length and increases the dissipation rate of the flow. Quantitative effect on shock revival.
Based on these scaling laws for the convectivevelocity, M¨uller and Janka (2015) determined that convective motions should reacha characteristic squared turbulent Mach number Ma ∼ . .
45 around the time ofshock revival. Using this value in Eq. (49) for the modified critical luminosity, theypredict a reduction of the critical luminosity by 15–25% due to convection, whichis in the ballpark of the numerical results (Murphy and Burrows 2008; Hanke et al.2012; Couch 2013; Dolence et al. 2013; Fern´andez et al. 2014; Fern´andez 2015)One might also be tempted to use Eqs. (56) and (57) to explain the lower explod-ability of self-consistent 3D models compared to their 2D counterparts. The nature ofthe differences between 2D and 3D is more complicated, however, since the criticalluminosity for shock revival is roughly equal in 2D and 3D light-bulb simulations.Evidently, there are effects that partly compensate for the smaller convective veloci-ties in 3D in some situations: The forward turbulent cascade (Melson et al. 2015b) andthe different behavior of the Kelvin–Helmholtz instability in 3D (M¨uller 2015) affectthe interaction between updrafts and downdrafts and can result in reduced cooling in
3D (Melson et al. 2015b). Moreover, compatible with earlier studies of the Rayleigh–Taylor instability for planar geometry (Yabe et al. 1991; Hecht et al. 1995), Kazeroniet al. (2018) found a faster growth of convective plumes and more efficient mixing ina planer toy model of neutrino-driven convection. Along similar lines, Handy et al.(2014) appealed to the higher volume-to-surface ratio of convective plumes in 3D toexplain the reduced critical luminosity in 3D in their light-bulb simulations. It is plau-sible that these factors establish a similar critical luminosity threshold for shock re-vival in light-bulb simulations, but they do not explain the much more decisive effectof dimensionality in self-consistent simulations. One possible explanation lies in thefact that explosions in self-consistent models usually occur in a short non-stationaryphase with a rapidly decreasing mass accretion rate and neutrino luminosity aroundthe infall of the Si/O shell interface; under these conditions the more sluggish emer-gence of large-scale modes in 3D due to the forward cascade may delay or inhibitshock revival (Lentz et al. 2015). Moreover, the more rapid response of the mass ac-cretion rate to shock expansion in 3D (Melson et al. 2015b; M¨uller 2015) might behurtful around shock revival because this effect can reduce the accretion luminosityand hence undercut neutrino heating before a runaway situation can develop.
Resolution dependence and turbulence.
Because of the turbulent nature of neutrino-driven convection, the spectral properties of the flow and the the resolution depen-dence in simulations have received considerable attention in the literature. Most self-consistent models with multi-group neutrino transport can only afford a limited res-olution (about 1 . ◦ in angle and about 100 zones or less in the gain region) anddo not reach a fully developed turbulent state, with Handy et al. (2014) going sofar as to speak of “perturbed laminar flow” instead. Various authors (Abdikamalovet al. 2015; Radice et al. 2015, 2016) have argued that considerably higher resolutionis needed to obtain clean turbulence spectra with a developed inertial range and aKolmogorov spectrum and raised concerns that a pile-up of kinetic energy (“bottle-neck effect”) at small scales might affect the overall dynamics. However, the detailedspectral properties of the flow are usually not critical, and integral properties of theflow are more important for the impact of convection on shock revival. The resolu-tion dependence nonetheless remains a concern for the question of shock revival, assome 3D resolution studies (Hanke et al. 2012; Abdikamalov et al. 2015; Robertset al. 2016) found a trend towards decreasing explodability with increased resolution.Recent work by Melson et al. (2020) resolved most of these concerns in a resolutionstudy using light-bulb simulations. They demonstrated that the resolution dependencein Hanke et al. (2012) was a spurious effect connected to details in their light-bulbscheme, and instead found a trend towards increased explodability at higher resolu-tion. In rough agreement with Handy et al. (2014), Melson et al. (2020) found that theoverall flow dynamics converges at an angular resolution of about 1 ◦ , which is not farfrom what most self-consistent simulations can afford (but one still needs to bear inmind that the resolution requirements depend on the details of the numerical scheme,cf. Sect. 2.1.2). Melson et al. (2020) also pointed out that neutrino drag plays a non-negligible role in the gain region, so that merely increasing the resolution does notadd physical realism beyond numerical Reynolds numbers of a few hundred unlessneutrino drag is also included as a non-ideal effect. Melson et al. (2020) speculate ydrodynamics of core-collapse supernovae and their progenitors 55 Fig. 10
The advective-acoustic mechanism for the standing accretion shock instability. Upward propagat-ing acoustic waves (blue) generate vorticity perturbations (red) as they interact with the accretion shock(orange circle). The vorticity perturbations are advected downward with the accretion flow to the PNS sur-face where they generate acoustic waves due to advective-acoustic coupling in the steep density gradient.Instability for a given mode obtains if the product of the amplitude ratios Q sh and Q ∇ for between ingoingand outgoing waves at the shock and PNS surface satisfies Q sh Q ∇ >
1. Image reproduced with permissionfrom Guilet and Foglizzo (2012), copyright by the authors. that findings of decreased explodability with higher resolution in Cartesian 3D mod-els may be explained because the grid-induced seed asphericities are lower at higherresolution.4.4 The standing accretion shock instabilityUsing adiabatic
2D simulations of spherical accretion shocks, the seminal work ofBlondin et al. (2003) demonstrated that another instability, dubbed “SASI” (standingaccretion shock instability), can operate in the supernova core even without a con-vectively unstable gradient in the gain region. This instability takes the form of large-scale ( (cid:96) = (cid:96) =
2) oscillatory motions of the shock, and it was imme-diately realized that it can support shock revival in a similar manner as convection. Inearly 2D supernova simulations, the SASI was sometimes confused with convectionbecause the two phenomena share superficial similarities like high-entropy bubblesand low-entropy accretion downflows. However, the SASI is set apart from convec-tion by dipolar (and sometimes quadrupolar) flow. Foglizzo et al. (2006) pointed outthat for the typical ratio between the shock and gain radius in the pre-explosion phasethere are no unstable convective modes with (cid:96) = , (cid:96) ≈ χ < (cid:96) = (cid:96) = R sh / R g . Amplification mechanism.
The stability of accretion shocks had in fact already beenanalyzed earlier in the context of accretion onto compact objects using linear per-turbation theory (Houck and Chevalier 1992; Foglizzo 2001, 2002), which provided useful groundwork for identifying the physical mechanism behind the SASI and ex-plaining the (cid:96) = , T SASI that is set by the sum of the advective and acoustic crossing times τ adv and τ ac between the shock and the deceleration region (Foglizzo et al. 2007), T SASI = τ adv + τ ac = (cid:90) r sh r ∇ d r | v r | + (cid:90) r sh r ∇ d rc s − | v r | . (58)The advective time scale usually dominates, and neglecting a weak dependence onthe PNS mass, one can determine empirically that the period of the (cid:96) = T SASI =
19 ms (cid:18) R sh
100 km (cid:19) / ln (cid:18) R sh R PNS (cid:19) , (59)where R PNS is the PNS radius. SASI-induced fluctuations in the neutrino emission(Lund et al. 2010; Tamborra et al. 2013; M¨uller and Janka 2014; M¨uller et al. 2019)and gravitational waves (Kuroda et al. 2016a; Andresen et al. 2017; Kuroda et al.2018) could provide direct observational confirmation for the SASI if this frequencycan be identified in spectrograms of the neutrino or gravitational wave signal.The growth rate of the SASI is set both by the period T SASI and the quality factor Q of the amplification cycle (Foglizzo et al. 2006, 2007), ω SASI = ln | Q | T SASI , (60)where Q depends on the coupling between vortical and acoustic waves at the shockand in the deceleration region, and hence on the details of the density profile and thethermodynamic stratification. Nuclear dissociation and recombination also affect theSASI growth rate and saturation amplitude (Fern´andez and Thompson 2009b,a). Interplay of SASI, convection, and neutrino heating.
In reality, the SASI grows inan accretion flow with neutrino heating, and in 2D, it is not trivial at first glanceto distinguish SASI and convection in the non-linear phase where both instabilitieslead to a similar (cid:96) = ydrodynamics of core-collapse supernovae and their progenitors 57 (Scheck et al. 2008) or multi-group (M¨uller et al. 2012a) neutrino transport, or sim-pler light-bulb models (Fern´andez et al. 2014): Different from convection-dominatedmodels SASI-dominated models clearly show an oscillatory growth of the multipolecoefficients of the shock surface and coherent wave patterns in the post-shock cavityin the linear regime, and maintain a rather clear quasi-periodicity even in the non-linear regime. The distinction between the two different regimes tends to becomemore blurred around shock revival, when large-scale convective modes emerge andthe periodicity of the SASI oscillations is eventually broken.The criterion χ ≈ χ >
3. For χ <
3, high quality factors ln | Q | ∼ χ < χ -parameter is still lacking.Three-dimensional simulations with neutrino transport (Hanke et al. 2013; Tam-borra et al. 2014b; Kuroda et al. 2016a; M¨uller et al. 2017a; Ott et al. 2018; O’Connorand Couch 2018a) as well as simplified leakage and light-bulb models (Couch andO’Connor 2014; Fern´andez 2015) show an even cleaner distinction between the SASI-and convection-dominated regimes for several reasons. The convective eddies remainsmaller in the non-linear stage than in 2D because of the forward cascade, and with-out the constraint of axisymmetry, the convective flow is not prone to artificial os-cillatory sloshing motions. The SASI, on the other hand, exhibits a cleaner periodic-ity prior to shock revival in 3D, and can develop a spiral mode that is very distinctfrom convective flow (e.g., Blondin and Mezzacappa 2007; Fern´andez 2010; Hankeet al. 2013; see also Sect. 5.3 for possible implications on neutron star birth periods).Self-consistent models show that the post-shock flow can transition back and forthbetween the convection- and SASI-dominated regime as the accretion rate and PNSparameters, and hence the χ -parameter change (Hanke et al. 2013). Saturation mechanism.
Guilet et al. (2010) argued that parasitic Kelvin–Helmholtzand Rayleigh–Taylor instabilities are responsible for the non-linear saturation of theSASI, and showed that this mechanism can explain the saturation amplitudes inthe adiabatic simulations of Fern´andez and Thompson (2009b). Assuming that theKelvin–Helmholtz instability is the dominant parasitic mode in 3D, one can derive(M¨uller 2016) a scaling law for the turbulent velocity fluctuations δ v in the saturatedstate, δ v ∼ ω SASI ( R sh − R g ) ∼ ln | Q | ( R sh − R g ) τ adv ∼ ln Q |(cid:104) v r (cid:105)| , (61)which is in good agreement with self-consistent 3D simulations. Interestingly, thisscaling results in similar turbulent velocities as in the convection-dominated regimefor conditions typically encountered in supernova core (M¨uller 2016).The saturation of the SASI can also be understood as a self-adjustment to marginalstability (Fern´andez et al. 2014), which is a closely related concept. As the SASI grows in amplitude, the flow is driven towards (cid:104) χ (cid:105) ≈
3, but stays slightly below thiscritical value (Fern´andez et al. 2014).
Effect on shock revival.
The SASI provides similar beneficial effects as convection toincrease the shock radius and bring the accretion flow closer to a neutrino-driven run-away, i.e., it generates turbulent pressure, brings high-entropy bubbles to large radii,channels cold matter towards the PNS, and converts turbulent kinetic energy thermalenergy throughout the gain region by turbulent dissipation. Due to the different in-stability mechanism (which feeds on the energy of the accretion flow directly insteadof the neutrino energy deposition), and the different flow pattern (which affects therate of turbulent dissipation), the quantitative effect on shock revival can be differ-ent from convection. Using light-bulb simulations Fern´andez (2015) indeed founda significantly lower critical luminosity in the SASI-dominated regime than in theconvection-dominated regime and a lower critical luminosity in 3D by ∼
20% com-pared to 2D, which he ascribed to the ability of the spiral mode to store more kineticenergy than sloshing modes in 2D. An even bigger difference to convective models(albeit with a different and very idealized setup) was found by Cardall and Budi-ardja (2015). Self-consistent simulations, on the other hand, have not found higherexplodability in 3D in the SASI-dominated regime (Melson et al. 2015a). The rea-son for this discrepancy could, e.g., lie in the feedback of shock expansion on neu-trino heating, but is not fully understood at this stage. Cardall and Budiardja (2015)also observed considerably more stochastic variations in shock revival in the SASI-dominated regime in their idealized models (i.e., a smeared-out critical luminositythreshold), but it again remains to be seen whether this is borne out by self-consistent3D models, where the SASI oscillations tend to be of smaller amplitude and shorterperiod than in Cardall and Budiardja (2015).4.5 Perturbation-aided explosionsProgenitor asphericities from convective shell burning can aid shock revival by af-fecting both the growth and saturation of convection of the SASI. That a higher levelof seed perturbations leads to a faster growth of non-radial instabilities behind theshock and thereby fosters explosions (as in the early studies of Couch and Ott 2013;Couch et al. 2015) may be intuitive, but appears less important in practice. In self-consistent simulations, shock revival typically occurs only once convection or theSASI have already reached the stage of non-linear saturation, and it is rather thepermanent “forcing” by infalling perturbations that matters (M¨uller and Janka 2015;M¨uller et al. 2017a). In either case, it is useful to separately consider a) how the initialperturbations in the porgenitor are translated to perturbations ahead of the shock, andb) how the infalling perturbations interact with the shock and the post-shock flow.
Initial state and infall phase.
Typically, the Si and O shell (and sometimes a Neshell) are the only active convective shells that can reach the shock at a sufficientlyearly post-bounce time to affect shock revival. As described in Sect. 3.2, these shellsare characterized by Mach numbers Ma prog ∼ . ydrodynamics of core-collapse supernovae and their progenitors 59 Fig. 11
Interaction of infalling perturbations with the shock and the post-shock flow, illustrated by snap-shots of the entropy (in units of k b / nucleon, left panel) and the absolute value of the non-radial velocity(in units of kms − , right panel) in the 12 . M (cid:12) model of M¨uller et al. (2019) at a post-bounce time of510ms. The left panel also shows the deformation of the isodensity surface with ρ = × gcm − (redcurve). Due to the infalling density perturbations, the pre-shock ram pressure is anisotropic and createsa protrusion of the shock. Additional energy is pumped into non-radial motions in the gain region bothbecause of substantial lateral velocity perturbations ahead of the shock and because of the oblique infall ofmaterial through the deformed shock. different shells and progenitors, and can have a wide range of dominant angular wavenumbers (cid:96) . Due to its subsonic nature, the flow is almost solenoidal with ∇ · ( ρ v ) ≈ δ ρ / ρ ∼ Ma are small within convective zones. Viewedas a superposition of linear waves, the convective flow consists mostly of vorticityand entropy waves with little contribution from acoustic waves.From analytic studies of perturbed Bondi accretion flows in the limit r → linearly with theconvective Mach number at the pre-collapse stage (M¨uller and Janka 2015; Abdika-malov and Foglizzo 2020), δ P / P ∼ δ ρ / ρ ∼ Ma prog . (62)According to simulations (M¨uller et al. 2017a) and analytic theory (Abdikamalov andFoglizzo 2020), this scaling is roughly independent of the wave number (cid:96) . Shock-turbulence interaction and forced shock deformation.
The infalling perturba-tions affect the shock and the post-shock flow in several ways (M¨uller and Janka If strong acoustic perturbations were present at the pre-collapse stage, these modes with higher (cid:96) would grow faster during the linear stage (Takahashi and Yamada 2014), but quickly undergo non-lineardamping (M¨uller and Janka 2015).0 Bernhard M¨uller prog and wave number (cid:96) in the progenitor. They predicted a reduction of the critical luminosity functional L crit = ( L ν E ν ) crit by δ L crit L crit ≈ .
47 Ma prog (cid:96) η heat η acc (63)in terms of the heating efficiency η heat and the accretion efficiency η heat = L / ( GM ˙ M / R g ) .However, the analysis of M¨uller et al. (2016b) did not account in detail for the inter-action of the infalling perturbations with the shock. This has been investigated usinglinear perturbation theory (Takahashi et al. 2016; Abdikamalov et al. 2016, 2018;Huete et al. 2018; Huete and Abdikamalov 2019). As a downside, this perturbativeapproach cannot easily capture the non-linear interaction of the injected perturba-tions with fully developed neutrino-driven convection and the SASI, but Huete et al.(2018) recently incorporated the effects of buoyancy downstream of the shock. Themore sophisticated treatment of Huete et al. (2018) predicts a similar effect size asEq. (63). Phenomenology of perturbation-aided explosions.
Whatever its theoretical justifica-tion, Eq. (63) successfully captures trends seen in 2D and 3D simulations of perturbation-aided explosions starting from parameterized initial conditions or 3D progenitor mod-els. Both high Mach numbers (cid:38) . (cid:96) (cid:46) M (cid:12) model of M¨uller et al. (2017a). In leakage-based models with high heatingefficiency early on (Couch and Ott 2013; Couch et al. 2015), the effect is smaller,especially if the pre-collapse asphericities are restricted to medium-scale modes as inoctant simulations (Couch et al. 2015). ydrodynamics of core-collapse supernovae and their progenitors 61 By now, there is a handful of exploding supernova models that use multi-groupneutrino transport and 3D progenitor models (M¨uller et al. 2017a, 2019). While thisis encouraging, more 3D simulations are needed to determine to what extent convec-tive seed perturbations generally contribute to robust explosions. At present, one cannonetheless extrapolate the effect size based on the properties of convective shellsin 1D stellar evolution models using Eq. (63). Analysing over 2000 supernova pro-genitors computed with the K
EPLER code Collins et al. (2018) predict a substantialreduction of the critical luminosity due to perturbation by 10% or more in the massrange between 15 M (cid:12) and 27 M (cid:12) , and in isolated low-mass progenitors. Below 15 M (cid:12) ,the expected reduction is usually 5% or less, which could still make the convectiveperturbations one of several important ingredients for robust explosions. In the vastmajority of progenitors, only asphericities from oxygen shell burning are expected tohave an important dynamic effect.4.6 Outlook: Rotation and magnetic fields in neutrino-driven explosionsEarlier on, we already briefly touched simulations of magnetorotational explosionscenarios and the uncertainties that still beset this mechanism. It is noteworthy that ro-tation and magnetic fields could also play a role within the neutrino-driven paradigm. Rotationally-supported explosions.
Since early attempts to study the impact of rota-tion on neutrino-driven explosions either employed a simplified neutrino treatment(e.g., Kotake et al. 2003; Fryer and Warren 2004; Nakamura et al. 2014; Iwakamiet al. 2014) or were restricted to 2D in the case of models with multi-group transport(Walder et al. 2005; Marek and Janka 2009; Suwa et al. 2010), more robust conclu-sions had to wait for 3D simulations with multi-group neutrino transport (Takiwakiet al. 2016; Janka et al. 2016; Summa et al. 2018). The 3D simulations indicate thatthe overall effect of rapid rotation is to support neutrino-driven explosions. Centrifu-gal support reduces the infall velocities and hence the average ram pressure at theshock (Walder et al. 2005; Janka et al. 2016; Summa et al. 2018). Moreover, 3D neu-trino hydrodynamics simulations of rotating models tend to develop a strong spiralSASI (Janka et al. 2016; Summa et al. 2018). This is in line with analytic theory (Ya-masaki and Foglizzo 2008) and idealized simulations (Iwakami et al. 2009; Blondinet al. 2017; Kazeroni et al. 2018), which demonstrated that rotation enhances thegrowth rate of the prograde spiral mode and stabilises the retrograde mode. For suf-ficiently rapid rotation, an even more violent spiral corotation instability can occur(Takiwaki et al. 2016). There is also a subdominant adverse effect, since lower neu-trino luminosities and mean energies at low latitudes close to the equatorial plane aredetrimental for shock revival (Walder et al. 2005; Marek and Janka 2009; Summaet al. 2018), which is particularly relevant since the explosion tends to be alignedwith the equatorial plane in the case of rapid rotation (Nakamura et al. 2014). Summaet al. (2018) found that the overall combination of these effects can be encapsulatedby a further modification of the critical luminosity, ( L ν E ν ) crit = ( L ν E ν ) crit , (cid:18) + (cid:19) − / (cid:115) − j GMR sh . (64) Fig. 12
Entropy s in k b / nucleon (left panel) and the logarithm log P B / P gas of the ratio between themagnetic pressure P B and the gas pressure P gas (right panel) in a 3D simulation (left half of panels) and a2D simulation (right half of panels) of the slowly rotating progenitor 15 M (cid:12) progenitor m15b6 of Hegeret al. (2005) with the C O C O N U T-FMT code. The initial field is assumed to be combination of a dipolarpoloidal field and a toroidal field. Outside convective zones, the field strength is taken from the progenitor,inside convective zones, the magnetic pressure is set to a fraction of 10 − of the thermal pressure. Thefigures shows meridional slices 140ms after bounce. Field amplification is driven by convection. Strongfields are generated in regions of strong shear, but these strong field are highly localized, and the totalmagnetic energy in the gain region remains much smaller than the turbulent kinetic energy and thermalenergy. Here j is the spherically-averaged angular momentum of the shell currently fallingthrough the shock. The last factor accounts for the reduced pre-shock velocities, andthe effect of stronger non-radial flow in the gain region is implicitly (but not predic-tively) accounted for in the turbulent Mach number. Magnetic fields without rotation.
Given the expected pre-collapse spin rates, rotationis unlikely to have a major impact in the vast majority of supernova explosions. Itis harder to exclude a significant role of magnetic fields a priori . Even if the pro-genitor core rotates slowly and does not have strong magnetic fields, convection andthe SASI might furnish some kind of turbulent dynamo process that could generatedynamically relevant fields in the gain region. There could also be other processes toprovide dynamically relevant magnetic fields, e.g., the accumulation of Alfv´en wavesat an Alfv´en surface (Guilet et al. 2011) or the injection of Alfv´en waves generatedin the PNS convection zone (Suzuki et al. 2008).The simulations available so far do not suggest that sufficiently high field strengthscan be reached by a small-scale turbulent dynamo. In idealized 2D and 3D simula-tions of Endeve et al. (2010, 2012), the SASI indeed drives a small-scale turbulentdynamo, and strong field amplification occurs locally up to equipartition and super-equipartition field strengths, especially when a strong spiral mode develops. On largerscales, the magnetic field energy remains well below equipartition, however, and doesnot become dynamically important. The total magnetic energy in the gain region re-mains one order of magnitude smaller than the turbulent kinetic energy, and the fielddoes not organize itself into large-scale structures. The situation is similar in the 2Dneutrino hydrodynamics simulations of non-rotating progenitors of Obergaulinger ydrodynamics of core-collapse supernovae and their progenitors 63
Fig. 13
LESA instability in a simulation of an 18 M (cid:12) star at time of 453ms after bounce, illustrated by2D slices showing the electron fraction Y e (left) and the radial velocity in units of kms − in the PNSconvection zone. Note that the Y e distribution in the PNS convection zone between radii of 10km and20km shows a clear dipolar asymmetry, whereas the radial velocity field is dominated by small-scalemodes superimposed over a much weaker dipole mode. Image repoduced with permission from Powelland M¨uller (2019), copyright by the authors. et al. (2014) for initial field strengths of up to 10 G, with even lower ratios betweenthe total magnetic and turbulent kinetic energy in the gain region. Only for initialfield strengths of ∼ G, which yields magnetar-strength fields after collapse, doObergaulinger et al. (2014) find that magnetic fields become dynamically importantand accelerate shock revival. If the fossil field hypothesis for magnetars is correct andthe fields of the most strongly magnetized main-sequence stars translate directly tosupernova progenitor and neutron star fields by flux conservation (Ferrario and Wick-ramasinghe 2006), such conditions for magnetically-aided explosions might be stillrealized in nature in a substantial fraction of core-collapse events.Ultimately, a thorough exploration of resolution effects and initial field config-urations in the convection- and SASI-dominated regime will be required in 3D toconfidently exclude a major role of magnetic field in weakly magnetized, slowly ro-tating progenitors. First tentative results from 3D MHD simulations with neutrinotransport (Fig. 12) suggest a picture of fibril flux concentrations with equipartitionfield strengths, and sub-equipartition fields in most of the volume, akin to the situa-tion in solar convection (e.g., Solanki et al. 2006).4.7 Proto-neutron star convection and LESA instability
Prompt convection.
Convective instability also develops inside the PNS. As alreadyrecognized in the late 1980s (Bethe et al. 1987; Bethe 1990), a first episode of“prompt convection” occurs within milliseconds after bounce around a mass coordi-nate of ∼ . M (cid:12) as the shock weakens and a negative entropy gradient is established.The negative entropy gradient is, however, quickly smoothed out, and the convectiveoverturn has no bearing on the explosion mechanism, although it can leave a promi-nent signal in gravitational waves (see reviews on the subject; Ott 2009; Kotake 2013;Kalogera et al. 2019). Proto-neutron star convection.
Convection inside the PNS is triggered again latteras neutrino cooling establishes unstable lepton number (Epstein 1979) and entropygradients (see profiles in Fig. 9). PNS convection was investigated extensively in the1980s and 1990s as a possible means of enhancing the neutrino emission from thePNS, which would boost the neutrino heating and thereby aid shock revival (e.g.,Burrows 1987; Burrows and Lattimer 1988; Wilson and Mayle 1988, 1993; Jankaand M¨uller 1995; Keil et al. 1996). In particular, Wilson and Mayle (1988, 1993) as-sumed that PNS convection operates as double-diffusive “neutron finger” instabilitythat significantly increases the neutrino luminosity.None of the modern studies of PNS convection since the mid-1990s (Keil et al.1996; Buras et al. 2006a; Dessart et al. 2006) found a sufficiently strong effect of PNSconvection on the neutrino emission for a significant impact on shock revival. PNSconvection indeed increases the heavy flavor neutrino luminosity by ∼
20% at post-bounce times of (cid:38)
150 ms, leaves the electron neutrino luminosity about equal, buttends to decrease the electron antineutrino luminosity, and reduces the mean energyof all neutrino flavors (Buras et al. 2006a). This can be explained by the effects ofPNS convection on the bulk structure of the PNS, namely a modest increase of thePNS radius and a higher electron fraction (due to mixing) close to the neutrinosphereof ν e and ¯ ν e (Buras et al. 2006a). Convective instability appears to be governed bythe usual Ledoux criterion and does not develop as a double-diffusive instability inthe simulations. LESA instability.
Although PNS convection does not have a decisive influence onshock revival, its indirect effect on the gain region is quantitatively important; effec-tively PNS convection changes the inner boundary condition for the flow in the gainregion. PNS convection also has an important impact on the neutrino signal from thePNS cooling phase (e.g., Roberts et al. 2012; Mirizzi et al. 2016), and may providea sizable contribution to the gravitational wave signal (Marek et al. 2009; Yakuninet al. 2010; M¨uller et al. 2013; Andresen et al. 2017; Morozova et al. 2018).Moreover, starting with (Tamborra et al. 2014a), the dynamics of PNS convectionhas proved more intricate upon closer inspection in recent years with potential reper-cussions on nucleosynthesis and gravitational wave emission. In their 3D simulations,Tamborra et al. (2014a) noted that a pronounced (cid:96) = Y e is sensitive to the differences in electronneutrino and antineutrino emission. ydrodynamics of core-collapse supernovae and their progenitors 65 Since then this phenomenon has been reproduced by many 3D simulations us-ing very different methods for neutrino transport (O’Connor and Couch 2018a; Jankaet al. 2016; Glas et al. 2019; Powell and M¨uller 2019; Vartanyan et al. 2019a), andeven in the 3D leakage models of Couch and O’Connor (2014) The dipolar Y e asym-metry has also been seen in the 2D Boltzmann simulations of Nagakura et al. (2019).This demonstrates that the LESA is a robust phenomenon; claims that it depends onthe details of neutrino transport (Nagakura et al. 2019) are not convincing.The nature of this instabiity is still not fully understood. Tamborra et al. (2014a)initially suggested a feedback cycle between (cid:96) = external feedback cycle between asymmetries in the accretion flowand asymmetries in the PNS convection zone. Glas et al. (2019) demonstrated thatLESA can be even more pronounced in exploding low-mass progenitor models withlow accretion rates, which suggests that the mechanism behind LESA works within the PNS convection zone.This, however, leaves the question why the flow within the PNS convection zonewould organise itself to generate a dipolar lepton fraction asymmetry. Some papershave, however, formulated first qualitative arguments to suggest that there is an in-ternal mechanism for a dipole asymmetry in the lepton fraction, and that LESA mayjust be a very peculiar manifestation of buoyancy-driven convection. What appearsto play a role is that the lepton fraction gradient becomes stabilizing against convec-tion in the middle of the PNS convection zone (Janka et al. 2016; Powell and M¨uller2019). Janka et al. (2016) suggested that this can give rise to a positive feedbackloop because a hemispheric lepton asymmetry will attenuate or enhance the stabiliz-ing effect in the different hemispheres, thereby leading to more vigorous convectionin one hemisphere, which in turn maintains the lepton asymmetry. On a differentnote, Glas et al. (2019) sought to explain the large-scale nature of the asymmetry byapplying the concept of a critical Rayleigh number for thermally-driven convection(Chandrasekhar 1961). However, one still needs to account for the fact that the typicalscales of the velocity and lepton number perturbations appear remarkably differentin the PNS convection zone. This was confirmed by the quantitative analysis of Pow-ell and M¨uller (2019), who found a very broad turbulent velocity spectrum peakingaround (cid:96) =
20, which conforms neither to Kolmogorov or Bolgiano–Obukhov scal-ing for stratified turbulence. Powell and M¨uller (2019) suggested that this could beexplained by the scale-dependent effective buoyancy experienced by eddies of dif-ferent sizes as they move across the partially stabilized central region of the PNSconvection zone. Powell and M¨uller (2019) also remarked that the non-linear state ofPNS convection is characterized by a balance between the convective and diffusivelepton number flux. All these aspects suggest that the LESA could be no more thana manifestation of PNS convection, but that PNS convection is in fact quite dissim-ilar from the high-P´eclet number convection as familiar from the gain region or thelate convective burning stages. A satisfactory explanation of the phenomenon likelyneeds to go beyond concepts from linear stability theory and the usual global balancearguments behind MLT, and will have to take into account scale-dependent forcingand dissipation, and the “double-radiative” nature of the instability.
Since the stabilising lepton number gradient in the middle of the PNS convectionzone figures prominently in these attempts to understand LESA, one might justifiablyask whether there is some role for double-diffusive instabilities in the PNS after all.Local stability analysis (Bruenn et al. 1995; Bruenn and Dineva 1996; Bruenn et al.2004) in fact suggests that double-diffusive instabilities (termed lepto-entropy fin-gers and lepto-entropy semiconvection) should occur in the PNS. But why were suchdouble-diffusive instabilities never identified in multi-D simulations so far? Furthercareful analysis and interpretation of the simulation results and theory is in orderto clarify this. One possible interpretation could be that the characteristic step-likelepton number profile established by LESA (Powell and M¨uller 2019) is actually amanifestation of layer formation in the subcritical regime as familiar from semicon-vection (Proctor 1981; Spruit 2013; Garaud 2018). However, the slow, global turnovermotions in LESA do not readily fit into this picture. One should also beware pre-mature conclusions because PNS convection is an inherently difficult regime for nu-merical simulations due to small convective Mach numbers of order ∼ .
01 and theimportance of diffusive effects. The potential issues go beyond the question of res-olution and unphysically high numrical Reynolds numbers (cf. 2.1.2), and there areconcrete reasons to investigate these in more depth. For example, although differ-ent codes agree qualitatively concerning the region of instability and the qualitativefeatures of the convective flow, substantial differences in the turbulent kinetic en-ergy density have been reported in a comparison between the A
LCAR and V
ERTEX codes in the PNS convection zone, even though the agreement between the codes isotherwise excellent (Just et al. 2018). While it is unlikely that the uncertainties inmodels PNS convection have any impact on the problem of shock revival, they needto be addressed to obtain a better understanding of LESA and reliable predictions ofgravitational wave signals and the nucleosynthesis conditions in the neutrino-heatedejecta.
Regardless of whether the explosion is driven by neutrinos or magnetic fields, thereis no abrupt transition to a quasi-spherical outflow after shock revival. In this section,we shall focus on the situation in neutrino-driven explosions, which has already beenquite thoroughly explored.5.1 The early explosion phaseIn typical neutrino-driven models, the multi-dimensional flow structure in the earlyexplosion phase appears qualitatively similar to the pre-explosion phase at first glance.Buoyancy-driven outflows and accretion downflows persist for hundreds of millisec-onds to seconds and allow for simultaneous mass accretion and ejection. Because That the LESA is also seen in models without lateral diffusion may not an obstacle for this interpre-tation. Lateral diffusion is essential to obtain semiconvective overstability, but layer formation can occurbelow the threshold for overstability (Proctor 1981; Spruit 2013).ydrodynamics of core-collapse supernovae and their progenitors 67 of the ongoing accretion, high neutrino luminosities and hence high heating ratescan be maintained to continually dump energy into the developing explosion. Asthe shock radius slowly increases, large-scale (cid:96) = (cid:96) = Even in electron-capture supernova progenitors, which explode even without the helpof multi-dimensional effects (Kitaura et al. 2006), there is a brief phase of convectiveoverturn after shock revival (Wanajo et al. 2011).More recent 3D explosion models using multi-group transport (Takiwaki et al.2014; Melson et al. 2015a; Lentz et al. 2015; M¨uller 2015; M¨uller et al. 2017a, 2019;Burrows et al. 2020) have confirmed this picture, but paved the way towards a morequantitative theory of the explosion phase. In massive progenitors, shock expansionis usually sufficiently slow for one or two dominant bubbles of neutrino-heated ejectato form (Fig. 14). Only at the low-mass end of the progenitor spectrum (Melson et al.2015b; Gessner and Janka 2018) do the convective structures freeze out so quicklythat the neutrino-heated ejecta are organized in medium-scale bubbles instead of aunipolar or bipolar structure.The detailed dynamics of the outflows and downflows proved to be significantlydifferent in 3D compared to 2D, and that only restricted insights on explosion andremnant properties and nucleosynthesis can be gained from the impressive corpus ofsuccessful 2D simulations with multi-group transport (Buras et al. 2006a; Marek andJanka 2009; M¨uller et al. 2012b,a, 2013; Janka 2012; Janka et al. 2012; Suwa et al.2010, 2013; Bruenn et al. 2013, 2016; Nakamura et al. 2015; Burrows et al. 2018;Pan et al. 2018; O’Connor and Couch 2018b). Except at the lowest masses, the 2Dsimulations are uniformly characterized by almost unabated accretion through fastdownflows that reach directly to the bottom of the gain region, by outflows that areoften weak and intermittent, and by a halting rise of explosion energies. Long-time2D simulations showed that this situation can persist out to more than 10 s (M¨uller2015), and as a result, implausibly high neutron star masses are reached. The halt-ing growth of explosion energies in 2D can partly be explained by the topology ofthe flow which lends itself to outflow constriction by equatorial downflows, but theprimary difference between 2D and 3D lies in the velocity of the downflows. Melsonet al. (2015b) already noticed that the downflows appear to subside more quickly intheir 3D model of a low-mass progenitor, which they ascribed to the forward tur-bulent cascade in 3D; this led to a slight enhancement of the explosion energy by10% in 3D compared to 2D. In more massive progenitors with stronger accretion af-ter shock revival stronger braking of the downflows in 3D compared to 2D is evenmore evident (M¨uller 2015). Instead of crashing into a secondary accretion shock at Critiques of the neutrino-driven mechanism have occasionally overlooked (Papish et al. 2015) andthen ultimately rebranded the simultaneous outflows and downflows as “jittering jets” (Soker 2019). Inthis latest instalment, the alternative jittering-jet scenario seems to have come down to little more than aquestion of unconventional terminology for well-established phenomena in neutrino-driven explosions.8 Bernhard M¨ullera)
400 km C15-3D 400 ms b)c) d)
Fig. 14
Volume renderings of the entropy in different 3D supernova simulations showing the emergence ofstable large-scale plumes around and after shock revival as a common phenomenon despite differences inresolution and in the neutrino transport treatment. The outer translucent surface is the shock, the structuresinside are neutrino-heated high-entropy bubbles: a) M (cid:12) model of Lentz et al. (2015) with a unipolarexplosion geometry at a post-bounce time of 400ms. b ) 3 M (cid:12) He star model of M¨uller et al. (2019) at1238ms, with two prominent plumes in the 7 o’clock and 11 o’clock directions and weaker shock expan-sion on the opposite side. c) M (cid:12) model of Burrows et al. (2020, Fig. 8) with a more dipolar explosiongeometry at 651ms. d ) 11 . M (cid:12) model of Nakamura et al. (2019, Fig. 6) at a time of 991ms. Images re-produced with permission from [a] Lentz et al. (2015), copyright by AAS; [c] from Burrows et al. (2020)and [d] from Nakamura et al. (2019), copyright by the authors. ∼
100 km at a sizable fraction of the free-fall velocity, the downflows are gently de-celerated, and secondary shocks rarely form. M¨uller (2015) ascribed this pathologyof the 2D models to the behavior of the Kelvin–Helmholtz instability between theoutflows and downflows, which is stabilized at high Mach numbers in 2D, but canalways grow in 3D (Gerwin 1968). Since the typical Mach number of the downflowsis higher during the explosion phase, the assumption of 2D symmetry becomes evenmore problematic than during the accretion phase. ydrodynamics of core-collapse supernovae and their progenitors 69
Estimators for the explosion energy.
Strictly speaking, the final demarcation betweenejected and accreted material cannot be determined before the explosion becomeskinetically dominated after shock breakout, and the same holds true for the final ex-plosion energy E exp . It is, however, customary and useful to consider the diagnosticexplosion energy E diag (often shortened to “diagnostic energy” or “explosion energy”when there is no ambiguity), which is defined as the total energy of the material thatis nominally unbound at any given instance (Buras et al. 2006a; M¨uller et al. 2012b;Bruenn et al. 2013). By definition the diagnostic energy will eventually asymptoteto E exp , but might do so only over considerably longer time scales than can be sim-ulated with neutrino transport. In particular, the diagnostic energy can in principledecrease as the shock sweeps up bound material from the outer shells. To account forthis, one can correct E diag for the binding energy of the material ahead of the shock(“overburden”) to obtain a more conservative estimate for E exp (Bruenn et al. 2016).In practice, E diag usually rises monotonically because energy continues to be pumpedinto the ejecta over seconds, but there are exceptions, most notably in cases of earlyblack hole formation (Chan et al. 2018). In most cases, one expects that E diag levelsout after a few seconds and then provides a good estimate for E exp . Explosion energies from self-consistent simulations.
Unfortunately, E diag has not lev-elled off in most of the available self-consistent 3D explosion models, though thegrowth of the explosion energy has already slowed down significantly in some long-time simulations using the C O C O N U T-FMT code (M¨uller et al. 2017a, 2018, 2019).Even in 2D, only some of the models of the Oakridge group appear to have ap-proached their final explosion energy (Bruenn et al. 2016).This means that no final verdict on the fidelity of the simulations can be pro-nounced based on a comparison with observationally inferred explosion energies.The models of M¨uller et al. (2017a, 2018, 2019) and Bruenn et al. (2016), whoseexplosion energies are admittedly on the high side among modern simulations, havedemonstrated that neutrino-driven explosions can reach energies of up to 8 × erg.Similarly, plausible nickel masses of several 0 . M (cid:12) appear within reach, althoughno firm statements can be made for C O C O N U T-FMT models due to uncertaintiesin the Y e of the ejecta from the approximative transport treatment, and due to thehighly simplified treatment of nucleon recombination. Explosion energies beyond10 erg may simply be a matter of longer simulations, different progenitor models,and slightly improved physics; and there may be no conflict with the distribution ofobservationally inferred explosion energies (Kasen and Woosley 2009; Pejcha andPrieto 2015; M¨uller et al. 2017b) of Type IIP supernovae. First attempts to extrap-olate the non-converged explosion energies from simulations and compare them toobservations using a rigorous statistical framework (Murphy et al. 2019) indicate thatthe predicted values are still somewhat too low, but Murphy et al. (2019) also pointout that conclusions are premature due to biases and uncertainties in the comparison. We avoid the term “mass cut”, which is commonly used for describing artificial 1D explosion models.The boundary of the ejecta region is not a sphere, and does not correspond to a unique mass shell underrealistic conditions.0 Bernhard M¨uller
Growth of the explosion energy.
Even at this stage, the simulations already elucidatehow the energy of neutrino-driven explosions is determined. Upon closer inspection,the energy budget of the ejecta is quite complicated and includes contributions fromthe injection of neutrino-heated material from below, form nucleon recombinationand nuclear burning, from the accumulation of bound material by the shock, andfrom turbulent mixing with the downflows (for a broader discussion, see Marek andJanka 2009; M¨uller 2015; Bruenn et al. 2016). Nonetheless, a few key findings haveemerged. The most critical determinant for the growth of E diag is the mass outflowrate ˙ M out of neutrino-heated material. Neutrino heating only marginally unbinds thematerial, and the net contribution to E diag comes from the energy ε rec released bynucleon recombination, which occurs at a radius of about 300 km. To first order, theresulting growth rate of the diagnostics energy is (Scheck et al. 2006; Melson et al.2015b; M¨uller 2015), ˙ E diag ≈ ˙ M out ε rec . (65)In principle, 8 . ε rec furtherto about 5–6 MeV / nucleon.The mass outflow rate is roughly determined by the volume-integrated neutrinoheating rate and the energy required to lift the material out of the gravitational po-tential. One can argue that the relevant energy scale is the binding energy at the gainradius | e gain | , so that ˙ M out = η out ˙ Q ν | e gain | . (66)with some efficiency parameter η out that accounts for the fact that only part of theneutrino-heated matter is ejected. Initially, one finds η out < η out >
1, and helps compensate for the declining heating ratesas the supply of fresh material to the gain radius slowly subsides.The situation in 2D models is somewhat different (M¨uller 2015). Here, the ejectedmaterial comes from close to the gain radius, and the outflow efficiency η out is lowerthan in 3D. Although the lack of turbulent mixing results in a higher asymptoticspecific total energy and entropy in 2D, the net effect is a slower growth of the explo-sion energy than in 3D. Moreover, the higher entropies in 2D will result in reducedrecombination to the iron group and hence lower nickel masses. Despite these short-comings, 2D simulations remain of some use because they already allow extensiveparameters studies of explodability and explosion and remnant properties (Nakamuraet al. 2015). ydrodynamics of core-collapse supernovae and their progenitors 71 Accretion rates and remnant masses.
The forward cascade and the stronger Kelvin–Helmholtz instabilities between the outflows and downflows in 3D imply that theaccretion rate onto the PNS drops more quickly than in 2D (M¨uller 2015). As a result,some self-consistent 3D simulations have been able to determine firm numbers forfinal neutron star masses (Melson et al. 2015b; M¨uller et al. 2019; Burrows et al.2019, 2020), barring the possibility of late-time fallback. The predicted neutron starmasses appear roughly compatible with the range of observed values ( ¨Ozel and Freire2016; Antoniadis et al. 2016), but as with all other explosion and remnant properties,a robust statistical comparison is not yet possible.
Neutron star kicks.
Observations show that most neutron stars receive a considerable“kick” velocity at birth. The kick velocity is typically a few hundred km s − , but thereis a broad distribution ranging from very low kicks up to more than 1000 km s − (e.g.,Hobbs et al. 2005; Faucher-Gigu`ere and Kaspi 2006; Ng and Romani 2007). Thelarge-scale ejecta asymmetries that emerge during the explosion provide a possibleexplanation for this phenomenon (for an overview including other mechanisms suchas aniostropic neutrino emission, see Lai et al. 2001; Janka 2017).The 2D simulations of the 1990s could not yet naturally obtain the full range ofobserved kick velocities by a hydrodynamic mechanism (Janka and M¨uller 1994),unless unrealistically large seed asymmetries in the progenitor were invoked (Bur-rows and Hayes 1996). A plausible range of kicks was first obtained in parameterized2D simulations by Scheck et al. (2004, 2006), thanks to more slowly developing ex-plosions that allowed the (cid:96) = (cid:96) = ) rather than from pressure force and hydrodyanmic momentum fluxes onto thePNS; anisotropic neutrino emission was found to play only a minor role. Subsequentsimulations have not fundamentally changed this analysis. Although various studiesshowed that the momentum flux onto the PNS can be comparable to the gravitationalforce onto the PNS (Nordhaus et al. 2010a, 2012; M¨uller et al. 2017a), this doesnot invalidate the tugboat mechanism. Effectively, the contribution of each parcel ofaccreted material to the PNS momentum via the hydrodynamic flux and the gravita-tional tug almost cancel, and the net acceleration of the PNS is due to the gravitationalpull of the material that is actually ejected.Three-dimensional simulations have not altered this picture substantially. Eventhough 2D simulations tend to obtain higher kicks, values of several hundred km s − were already obtained in the parameterized 3D simulations of Wongwathanarat et al.(2010b, 2013). Recently, M¨uller et al. (2017a, 2019) performed sufficiently long 3Dsimulations with multi-group neutrino transport to extrapolate the final kick veloci-ties, which fall nicely within the observed range of up to 1000 km s − . The term “tugboat mechanism” was in fact suggested later by Jeremiah Murphy and introduced intothe literature in Wongwathanarat et al. (2013).2 Bernhard M¨uller
Based on the physics of the kick mechanism, various authors have posited a cor-relation between the kick and the ejecta mass (Bray and Eldridge 2016) or, usingmore refined arguments, on the explosion energy (Janka 2017; Vigna-G´omez et al.2018). Tentative support for a loose correlation comes from the small kicks obtainedin simulations of low energy, ultra-stripped supernovae (Suwa et al. 2015; M¨ulleret al. 2017a) and electron-capture supernovae (Gessner and Janka 2018), and frommore recent 3D simulations over a larger range of progenitor masses (M¨uller et al.2019).
Neutron star spins.
If the downflows hit the PNS surface with a finite impact pa-rameter, they also impart angular momentum onto the PNS. While this was realizedalready by Spruit and Phinney (1998), 3D simulations are needed for quantitative pre-dictions of PNS spin-up by asymmetric accretion. The predicted spin-up in 3D mod-els of non-rotating varies. Parameterized simulations (Wongwathanarat et al. 2010b;Rantsiou et al. 2011; Wongwathanarat et al. 2013) tend to find longer neutron starspin periods of hundreds of milliseconds to seconds (but extending down to 100 msin Wongwathanarat et al. 2013). Recent 3D simulations using multi-group trans-port (M¨uller et al. 2017a, 2019) obtain spin periods between 20 ms and 2 . Role of the spiral SASI mode.
In the first idealized simulations of the spiral mode ofthe SASI, Blondin and Mezzacappa (2007) noted a significant flux of angular mo-mentum into the “neutron star” (modeled by an inner boundary condition) that wouldlead to rapid neutron star rotation in the opposite direction to the SASI flow withangular frequencies of the order of 100 rad s − even in the case of non-rotating pro-genitors. This idea has been explored further in several numerical (Hanke et al. 2013;Kazeroni et al. 2016, 2017) and analytical (Guilet and Fern´andez 2014) studies. Thepotential for spin-up of non-rotating progenitors may be more modest than initiallythought; both numerical and analytical results suggest that the angular momentumimparted onto the PNS is only a few 10 erg s, corresponding to spin periods of hun-dreds of milliseconds (Hanke et al. 2013; Guilet and Fern´andez 2014). Moreover,part of the angular momentum contained in the spiral mode may be accreted aftershock revival, negating the separation of angular momentum previously achieved bythe SASI. Spin-up and spin-down by SASI in the case of rotating progenitors still ydrodynamics of core-collapse supernovae and their progenitors 73a) v r [ k m s - ] r [km]10 -7 -6 -5 -4 -3 -2 -1 ρ [ g c m - ] , P [ e r g c m - ] ρ Pv r d P /d r >0shockd ρ /d r <0 reverse shock(from C/He interface) RT-unstable region b) Fig. 15 a)
Emergence of Rayleigh–Taylor instability during the propagation of the shock through theenvelope, illustrated by spherically averaged profiles of density ρ , pressure P , and radial velocity v r froma 2D long-time simulation of a 9 . M (cid:12) star based on the explosion model of M¨uller et al. (2013). At thisstage (140s after the onset of the explosion), the shock has reached the He shell, and a reverse shock hasformed. Behind the forward shock, the pressure gradient is positive and decelerates the expansion of theejecta. Rayleigh–Taylor (RT) instability grows in a region with d ρ / d r < b) Mass fraction isocontours in a 3D model of mixing in SN 1987A.Note that while the biggest Ni-rich Rayleigh–Taylor clumps are seeded by large-scale asymmetries fromthe early explosion phase, these develop into the finger-like structures characteristic of the Rayleigh–Taylorinstability, and there is also considerable growth of small-scale plumes. Image reproduced with permissionfrom Wongwathanarat et al. (2015), copyright by ESO. merits further investigation; the idealized simulations of Kazeroni et al. (2017) sug-gest different regimes of random spin-up and spin-down for slow progenitor rotation,systematic spin-down for intermediate rotation, and weaker spin-down for high ro-tation rates in the regime of the corotation instability. The possibility of magneticfield amplification due to the induced shear in the PNS surface region in the case ofsignificant spin-up or spin-down by the SASI also needs to be explored.5.4 Mixing instabilities in the envelope
Structure of the flow in the later explosion phase.
As the propagating shock scoopsup matter and as the explosion energy levels off, the structure of the post-shock re-gion changes (Fig. 15a). Early on, the post-shock expansion velocities are subsonicand the outflows are accelerated by a positive pressure gradient, but eventually thepost-shock flow enters a Sedov-like regime where a positive pressure gradient is es-tablished and matter is decelerated behind the shock (Chevalier 1976). Generally,the shock velocity v sh also decreases as the mass M ej of the shock ejecta grows; itroughly scales as v sh ∝ ( E exp / M ej ) / . The shock can, however, transiently acceleratewhen it encounters density gradients steeper than ρ ∝ r − at shell interfaces (Sedov Note that deceleration of the post-shock matter and deceleration of the shock do not always coincide,though they are closely related phenomena. One can have ˙ v r < v sh > v sh ≈ . (cid:18) E exp M ej (cid:19) / (cid:18) M ej ρ r (cid:19) . . (67)The post-shock pressure and density profiles adjust to variations in shock velocityto establish something of a “quasi-hydrostatic” stratification behind the shock withan effective gravity that is directed outward . However, once the post-shock velocitiesbecome supersonic, the post-shock pressure profile can no longer globablly adjust tochanging shock velocities, and reverse shocks are formed. A first reverse shock formstypically forms at a few 1000 km as the developing neutrino-heated wind crashes intomore slowly moving ejecta. Later on, further reverse shocks emerge after the shockencounters various shell interfaces. Their strength depends on the density jump atthe shell interface. In hydrogen-rich progenitors, the reverse shock from the H/Heinterface is particularly strong (especially in red supergiant) and therefore sometimesreferred to simply as the reverse shock. Rayleigh–Taylor instability.
The non-monotonic variations in v sh result in mon-monotonicpost-shock entropy and density profiles, and some layers become Rayleigh–Taylorunstable (Chevalier 1976; M¨uller et al. 1991; Fryxell et al. 1991). In the relevanthighly compressible regime, the growth rate for the Rayleigh–Taylor instability froma local stability analysis is given by (Bandiera 1984; Benz and Thielemann 1990;M¨uller et al. 1991) ω RT = c s Γ (cid:115) ∂ ln P ∂ r (cid:18) ∂ ln P ∂ r − Γ ∂ ln ρ∂ r (cid:19) = (cid:115) g eff (cid:18) Γ ∂ ln P ∂ r − ∂ ln ρ∂ r (cid:19) , (68)where g eff = ρ − ∂ P / ∂ r is the effective gravity. The second form elucidates that sta-bility is determined by the sub- or superadiabaticity of the density gradient as forbuoyancy-driven convection. In the relevant radiation-dominated regime, composi-tion gradients have a minor impact on stability, and the entropy gradient is the de-ciding factor. However, since the effective gravity points outwards, positive entropygradients are now unstable. Such positive entropy gradients arise when the shockaccelerates at shell interfaces. One should note that the actual growth rate of pertur-bations depends on their length scale (Zhou 2017), and Eq. (68) roughly applies tothe fastest growing modes with a wavelength comparable to the width of the unstableregion. Since the unstable regions tend to be narrow, Rayleigh–Taylor mixing tendsto produce smaller, clumpy structures, but large-scale asymmetries can also growconsiderably for sufficiently strong seed perturbations.Intriguingly, it has been suggested that the overall effect of Rayleigh–Taylor mix-ing can roughly be captured in 1D by an appropriate turbulence model (Duffell 2016;Paxton et al. 2018). The key idea here is to incorporate the proper growth rate, sat-uration behavior, and a velocity-dependent mixing length (Duffell 2016). While afirst comparison of this model with 3D results from Wongwathanarat et al. (2015)proved encouraging (Paxton et al. 2018), a few caveats remain. A detailed analy-sis of Rayleigh–Taylor mixing in a 3D model of a stripped-envelope progenitor by ydrodynamics of core-collapse supernovae and their progenitors 75 M¨uller et al. (2017a) unearthed some basis for a phenomenological 1D descriptionof Rayleigh–Taylor mixing (most notably buoyancy-drag balance in the non-linearstage) and suggested some improvements to the model of Paxton et al. (2018), butcast doubt on the use of local gradient to estimate the density and composition con-trasts of the Rayleigh–Taylor plumes. In particular, the Rayleigh–Taylor instabilitysometimes produces partial inversions of the initial composition profiles, which can-not be modeled by diffusive mixing in 1D.In addition to the Rayleigh–Taylor instability, the Richtmyer–Meshkov instabil-ity (see Richtmyer 1960; Zhou 2017, for details of the instability mechanism) can de-velop because the shock is generally aspherical and hits the shell interfaces obliquely.The literature on mixing instabilities in supernovae is extensive, and we can only pro-vide a very condensed summary of extant numerical studies. We will exclusively fo-cus on the optically thick phase of the explosion and not consider the remnant phase.
Simulations of mixing in SN 1987A.
After a few earlier numerical experiments, sig-nificant interest in mixing instabilities was prompted by observations of SN 1987Athat pointed to strong early mixing of nickel (see Arnett et al. 1989b; McCray 1993,and references therein). Two-dimensional simulations of mixing instabilities (Arnettet al. 1989a; M¨uller et al. 1991; Fryxell et al. 1991; Hachisu et al. 1990; Benz andThielemann 1990; Herant and Benz 1992) in the wake of SN 1987A took a first steptowards explaining the observed mixing. The typical picture revealed by these modelsis that of a strong Rayleigh–Taylor instability at the H/He-interface with linear growthfactors of thousands (M¨uller et al. 1991), Some models (M¨uller et al. 1991; Fryxellet al. 1991) also showed a second strongly unstable region at the He/C-interface thateventually merges with the mixed region further outside. Mixing was dominated bymany small-scale plumes in these first-generation simulations. However, the maxi-mum velocities of nickel plumes still fell short by about a factor of two compared tothe observed velocities of up to ∼ − .Many subsequent studies have investigated stronger, large-scale initial seed per-turbations as a possible explanations for the strong mixing in SN 1987A and otherobserved Type II supernovae. Such seed perturbations are naturally expected in theneutrino-driven paradigm from the SASI and low- (cid:96) convective modes, and in magne-torotational explosions with veritable jets. Most simulations have explored the effectof large-scale seed perturbations by specifying them ad hoc (e.g., Nagataki et al.1998; Nagataki 2000; Hungerford et al. 2003; Couch et al. 2009; Ono et al. 2013;Mao et al. 2015; Ellinger et al. 2012). One must therefore be cautious in drawingconclusions on the role of “jets” in explaining the observed mixing. In fact, simula-tions with artificially injected jets rather serve to rule out kinetically-dominated jets inType IIP supernovae based on early spectropolarimetry, though thermally-dominatedjets are not excluded in principle (Couch et al. 2009).In their 2D AMR simulations, Kifonidis et al. (2000, 2003, 2006) followed amore consistent approach by starting from light-bulb simulations of neutrino-drivenexplosions. While the seed asymmetries from the early explosion phase initially led tohigh nickel clump velocities in the Type IIP model of Kifonidis et al. (2000, 2003), thefinal velocities were still too small because the clumps were caught behind the reverseshock and underwent fast deceleration by supersonic drag after crossing it. Kifonidis et al. (2000) found that this can be avoided with a more slowly developing and moreaspherical explosion. In this case, they found less clump deceleration because theclumps make it beyond the H/He interface before the reverse shock develops, andalso found strong downward mixing of hydrogen with the help of the Richtmyer–Meshkov instability.A very convincing picture of mixing in SN 1987A has emerged since the adventof 3D simulations. Simulations of single-mode perturbations by Kane et al. (2000)already suggested a faster growth of the Rayleigh–Taylor instability in 3D. A first3D simulation of mixing in SN 1987A based on a 3D explosion model using grayneutrino transport was conducted by Hammer et al. (2010), who were able to obtainrealistic mixing of nickel, hydrogen, and other elements even without the need forstrong initial shock deformation and a strong Richtmyer–Meshkov instability. Ham-mer et al. (2010) explain the reduced deceleration of plumes as a result of a morefavorable volume-to-surface ratio of the clumps in 3D compared to 2D, where theclumps are actually toroidal. Stronger mixing in 3D was also confirmed in a study notspecifically focused on SN 1987A (Joggerst et al. 2010b). The attainable nickel clumpvelocities are, however, quite sensitive to the progenitor structure (Wongwathanaratet al. 2015). Interestingly, the origin of the fastest and biggest clumps in Hammeret al. (2010) and in some of the other subsequent 3D simulations could be tracedback to the most prominent convective bubbles that formed around shock revival, i.e.the late-time instabilities still contain traces of the early asymmetries imprinted bythe neutrino-driven engine. As a next step towards model validation, synthetic lightcurves based on the 3D models of Wongwathanarat et al. (2015) were computed byUtrobin et al. (2015), and the results are encouraging. While the fit to the observedlight curve is still not perfect, the discrepancies likely indicate uncertainties in theprogenitor structure and the precise initial conditions after shock revival and not aproblem of the neutrino-driven explosion scenario for SN 1987A. Stripped-envelope supernovae.
Mixing instabilities are also highly relevant in thecontext of stripped-envelope supernovae. Due to the lack of a H envelope (or a smallmass of the H envelope in the case of Type IIb events), the early asymmetries arenot shredded as strongly by Rayleigh–Taylor mixing as in Type IIP supernovae, sothat spectroscopy and spectropolarimetry offer a more direct glimpse on global asym-metries generated by the engine (see Wang and Wheeler 2008 for observational di-agnostics of asymmetries). Moreover, the presence or absence of He lines in Ib/csupernovae is sensitive to the mixing of nickel (Dessart et al. 2012, 2015; Hachingeret al. 2012), and so is the detailed shape of the light curve (Shigeyama and Nomoto1990; Yoon et al. 2019).Two-dimensional simulations of Rayleigh–Taylor mixing in Ib/c supernovae werefirst conducted by Hachisu et al. (1991, 1994). These simulations were based on he-lium star models, which are a viable approximation for progenitors that lost their en-velope due to Case B/Case C mass transfer, but used artificially triggered explosions.Hachisu et al. (1991, 1994) found indications of stronger mixing in less massive he-lium stars. Baron (1992) interpreted this as pointing towards an association of Ib andIc supernovae with low- and high-mass helium stars, respectively. Kifonidis et al.(2000, 2003) triggered the explosion somewhat more realistically using a light-bulb ydrodynamics of core-collapse supernovae and their progenitors 77 scheme, but constructed their Ib supernova progenitor by artificially removing thehydrogen envelope at collapse (implying an inconsistent envelope structure); theirfinding on the mixing of nickel were qualitatively similar to Hachisu et al. (1991,1994).Wongwathanarat et al. (2017) took an ambitious step towards comparing stripped-envelope models with observations by performing a 3D simulation of a neutrino-driven explosion that matches the global asymmetries in the distribution of Ti and Ni and the neutron star kick in Cas A to an astonishing degree. The required pro-genitor for a Type IIb (i.e., partially stripped) supernova was again constructed bymanually removing part of the envelope at collapse, but in terms of simulation fi-delity this is likely less of an issue than the fact that the neutrino-driven explosionwas still tuned to match the desired explosion energy.A first 3D simulation of mixing in an ultra-stripped progenitor starting from aself-consistent explosion model was conducted by M¨uller et al. (2018) with a view toobservations of fast and faint Ic supernovae (Drout et al. 2013; De et al. 2018). Themodel showed mixing of substantial amounts of nickel in a few narrow dense plumesout to about half way through the He envelope. These findings are, however, difficultto extrapolate to other, less extreme stripped-envelope supernova progenitors. A morethorough exploration of mixing in Ib/c supernovae and a quantitative comparison of3D models of mixing instabilities with the spectropolarimtery of observed Ib/c eventsis called for.
Mixing and fallback.
Mixing instabilities have also been studied as a possible ex-planation of abundances in extremely metal-poor stars. Umeda and Nomoto (2003)suggested that the high [C/Fe] in some of these stars can be explained by invoking acombination of Rayleigh–Taylor mixing and fallback in the supernovae that suppos-edly contributed to their initial composition. Joggerst et al. (2009) conducted 2Dsimulations of this scenario using the F
LASH code. Their simulations indeed showedenhanced fallback in low- and zero-metallicity progenitors, and hence a possiblemechanism for low iron-group yields in metal-poor environments, but Rayleigh–Taylor mixing was not sufficient for ejecting the required amount of iron-group andintermediate-mass elements to match observed abundances. In a follow-up studythat surveyed a broader range of rotating progenitors with two different metallicities( Z = Z = − Z (cid:12) ) using the C ASTRO code, Joggerst et al. (2010a) were ableto find better matches of the supernova yields to abundance patterns from ultra-metalpoor stars. A similar study was conducted by Chen et al. (2017) to explain the abun-dances in the most iron-poor star to date (SMSS J031300.36-670839.3, Keller et al.2014) by fallback in an explosion with modest energy. However, all of these simu-lations were restricted to 2D and imposed seed perturbations by hand in sphericallysymmetric models. A first 3D simulation of a fallback supernova from collapse toshock revival by the neutrino-driven mechanism, through black hole formation, andon to shock breakout was performed by Chan et al. (2018). In their model, fallbackproceeds in a qualitatively different manner than in previous studies; the iron-group Jet-driven explosions could provide an alternative mechanism to explain the observed abundance pat-terns (e.g., Maeda and Nomoto 2003; Nomoto et al. 2006).8 Bernhard M¨uller material is accreted early, the post-shock flow involves global asymmetries during thefirst tens of seconds (which could potentially generate substantial black hole kicksand spins), but no mixing instabilities occur later on. While the work of Chan et al.(2018) has demonstrated the feasibility of a forward-modelling approach to fallbacksupernovae, they explored only a single progenitor, and a broader investigation isnecessary to understand the phenomenology of fallback in three dimensions.
Acknowledgements
I wish to thank S. Bruenn, A. Burrows, C. Collins, J. Guilet, T. Foglizzo, H.-Th. Janka,K. Kotake, E. Lentz, A. Mezzacappa, E. M¨uller, J. Powell, A. Skinner, T. Takiwaki, and A. Wongwatha-narat for their kind permission to reproduce figures from their papers. I am grateful to A. Heger for pro-viding 1D progenitor models at various evolutionary stages. This work was supported by the AustralianResearch Council (ARC) through Future Fellowship FT160100035. The author acknowledges supportfrom the ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav) as associate investigator.This research was undertaken with the assistance of resources obtained via NCMAS and ASTAC from theNational Computational Infrastructure (NCI), which is supported by the Australian Government and wassupported by resources provided by the Pawsey Supercomputing Centre with funding from the AustralianGovernment and the Government of Western Australia.
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