3D Simulations of Oxygen Shell Burning with and without Magnetic Fields
MMNRAS , 1–10 (2020) Preprint 5 January 2021 Compiled using MNRAS L A TEX style file v3.0
3D Simulations of Oxygen Shell Burning with and without Magnetic Fields
Vishnu Varma ★ and Bernhard Müller , † School of Physics and Astronomy, 10 College Walk, Monash University, Clayton, VIC 3800, Australia ARC Centre of Excellence for Gravitational Wave Discovery – OzGrav
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present a first 3D magnetohydrodynamic (MHD) simulation of convective oxygen and neon shell burning in a non-rotating18 𝑀 (cid:12) star shortly before core collapse to study the generation of magnetic fields in supernova progenitors. We also run a purelyhydrodynamic control simulation to gauge the impact of the magnetic fields on the convective flow and on convective boundarymixing. After about 17 convective turnover times, the magnetic field is approaching saturation levels in the oxygen shell withan average field strength of ∼ G, and does not reach kinetic equipartition. The field remains dominated by small to mediumscales, and the dipole field strength at the base of the oxygen shell is only 10 G. The angle-averaged diagonal components of theMaxwell stress tensor mirror those of the Reynolds stress tensor, but are about one order of magnitude smaller. The shear flowat the oxygen-neon shell interface creates relatively strong fields parallel to the convective boundary, which noticeably inhibitthe turbulent entrainment of neon into the oxygen shell. The reduced ingestion of neon lowers the nuclear energy generation ratein the oxygen shell and thereby slightly slows down the convective flow. Aside from this indirect effect, we find that magneticfields do not appreciably alter the flow inside the oxygen shell. We discuss the implications of our results for the subsequentcore-collapse supernova and stress the need for longer simulations, resolution studies, and an investigation of non-ideal effectsfor a better understanding of magnetic fields in supernova progenitors.
Key words: stars: massive – stars: magnetic fields – stars:interiors – MHD – convection – turbulence
Rigorous three-dimensional (3D) simulations of neutrino-drivencore-collapse supernovae have become highly successful in recentyears (e.g., Melson et al. 2015; Lentz et al. 2015; Takiwaki et al.2014; Burrows et al. 2019), and have made clear headway in explain-ing the properties of supernova explosions and the compact objectsborn in these events (Müller et al. 2017; Müller et al. 2019; Burrowset al. 2020; Powell & Müller 2020; Bollig et al. 2020; Müller 2020).The latest 3D models are able to reproduce a range of explosion en-ergies up to 10 erg (Bollig et al. 2020), and yield neutron star birthmasses, kicks, and spins largely compatible with the population ofobserved young pulsars (Müller et al. 2019; Burrows et al. 2020).A number of ingredients have contributed, or have the potentialto contribute, to make modern neutrino-driven explosion modelsmore robust. Various microphysical effects such as reduced neutrinoscattering opacities due to nucleon strangeness (Melson et al. 2015)and nucleon correlations at high densities (Horowitz et al. 2017;Bollig et al. 2017; Burrows et al. 2018), muonisation (Bollig et al.2017), and large effective nucleon masses (Yasin et al. 2020) can beconducive to neutrino-driven shock revival.In addition, a particularly important turning point has been theadvent of 3D progenitor models and the recognition that aspheric-ities seeded prior to collapse can precipitate “perturbation-aided”neutrino-driven explosions (Couch & Ott 2013; Couch et al. 2015; ★ E-mail: [email protected] † E-mail: [email protected]
Müller & Janka 2015; Müller 2016; Müller et al. 2017; Müller 2020;Bollig et al. 2020). In perturbation-aided explosions, the moderatelysubsonic solenoidal velocity perturbation in active convective shellsat the pre-collapse stage with Mach numbers of order ∼ . secular impact of 3D effects not capturedby spherically symmetric stellar evolution models based on mixing-length theory (Biermann 1932; Böhm-Vitense 1958). The details ofconvective boundary mixing by processes such as quasi-steady turbu-lent entrainment (Fernando 1991; Strang & Fernando 2001; Meakin © a r X i v : . [ a s t r o - ph . S R ] J a n Varma & Müller & Arnett 2007b) or violent shell mergers (Mocák et al. 2018; Ya-dav et al. 2020) have received particular attention for their potentialto alter the core structure of massive stars and hence affect the dy-namics and final nucleosynthesis yields of the subsequent supernovaexplosion.Three-dimensional simulations of convection during advancedburning stages have so far largely disregarded two important aspectsof real stars – rotation and magnetic fields. The effects of rotationhave been touched upon by the seminal work of Kuhlen et al. (2003),but more recent studies (Arnett & Meakin 2010; Chatzopoulos et al.2016) have been limited to axisymmetry (2D). Magnetohydrody-namic (MHD) simulations of convection, while common and maturein the context of the Sun (for a review see, e.g., Charbonneau 2014)have yet to be performed for advanced burning stages of massivestars.Simulations of magnetoconvection during the pre-supernova stage,both in rotating and non-rotating stars, are a big desideratum for sev-eral reasons. Even in slowly rotating massive stars, magnetic fieldsmay have a non-negligible impact on the dynamics of neutrino-drivenexplosions (Obergaulinger et al. 2014; Müller & Varma 2020), andalthough efficient field amplification processes operate in the super-nova core (Endeve et al. 2012; Müller & Varma 2020), it stands toreason that memory of the initial fields may not be lost entirely, es-pecially for strong fields in the progenitor and early explosions. Abetter understanding of the interplay between convection, rotation,and magnetic fields in supernova progenitors is even more criticalfor the magnetorotational explosion scenario (e.g., Burrows et al.2007; Winteler et al. 2012; Mösta et al. 2014; Mösta et al. 2018;Obergaulinger & Aloy 2020a,b; Kuroda et al. 2020), which proba-bly explains rare, unusually energetic “hypernovae” with energies ofup to ∼ erg. Again, even though the requisite strong magneticfields may be generated after collapse by amplification processes likethe magnetorotational insstability (Balbus & Hawley 1991; Akiyamaet al. 2003) or an 𝛼 - Ω dynamo in the proto-neutron star (Duncan &Thompson 1992; Thompson & Duncan 1993; Raynaud et al. 2020),a robust understanding of magnetic fields during late burning is in-dispensable for reliable hypernova models on several heads. For suf-ficiently strong seed fields in the progenitor, the initial field strengthsand geometry could have a significant impact on the development ofmagnetorotational explosions after collapse (Bugli et al. 2020). Fur-thermore, our understanding of evolutionary pathways towards hy-pernova explosions (Woosley & Heger 2006; Yoon & Langer 2005;Yoon et al. 2010; Cantiello et al. 2007; Aguilera-Dena et al. 2018,2020) is intimately connected with the effects of magnetic fields onangular momentum transport in stellar interiors (Spruit 2002; Hegeret al. 2005; Fuller et al. 2019; Takahashi & Langer 2020).Beyond the impact of magnetic fields on the pre-supernova evo-lution and the supernova itself, the interplay of convection, rotation,and magnetic fields is obviously relevant to the origin of neutronstar magnetic fields as well. It still remains to be explained whatshapes the distribution of magnetic fields among young pulsars, andwhy some neutron stars are born as magnetars with extraordinarilystrong dipole fields of up to 10 G (Olausen & Kaspi 2014; Tau-ris et al. 2015; Kaspi & Beloborodov 2017; Enoto et al. 2019). Arethese strong fields of fossil origin (Ferrario & Wickramasinghe 2005;Ferrario et al. 2009; Schneider et al. 2020) or generated by dynamoaction during or after the supernova (Duncan & Thompson 1992;Thompson & Duncan 1993)? Naturally, 3D MHD simulations of thelate burning stages cannot comprehensively answer all of these ques-tions. In order to connect to observable magnetic fields of neutronstars, an integrated approach is required that combines stellar evolu-tion over secular time scales, 3D stellar hydrodynamics, supernova modelling, and local simulations, and also addresses aspects like fieldburial (Viganò et al. 2013; Torres-Forné et al. 2016) and the long-time evolution of magnetic fields (Aguilera et al. 2008; De Grandiset al. 2020). However, 3D MHD simulations of convective burningcan already address meaningful questions despite the complexity ofthe overall problem.In this study, we present a first simulation of magnetoconvectionduring the final phase of oxygen shell burning using the ideal MHDapproximation. This simulation constitutes a first step beyond ef-fective 1D prescriptions in stellar evolution models to predict themagnetic field strength and geometry encountered in the inner shellsof massive stars at the pre-supernova stage. We also compare to acorresponding non-magnetic model of oxygen shell convection togauge the feedback of magnetic fields on the convective flow with aparticular view to two important issues. First, the efficiency of the“perturbation-aided” neutrino-driven mechanism depends criticallyon the magnitude of the convective velocities during shell burning,and it is important to determine whether magnetic fields can signif-icantly slow down convective motions as suggested by some recentsimulations of solar convection (Hotta et al. 2015). Second, magneticfields could quantitatively or qualitatively affect shell growth by tur-bulent entrainment, which has been consistently seen in all recent3D hydrodynamics simulations of late-stage convection in massivestars.Our paper is structured as follows. In Section 2, we describe thenumerical methods, progenitor model, and initial conditions used inour study and discuss the potential role of non-ideal effects . Theresults of the simulations are presented in Section 3. We first focuson the strength and geometry of the emerging magnetic field and thenanalyse the impact of magnetic fields on the flow, and in particularon entrainment at shell boundaries. We summarize our results anddiscuss their implications in Section 4.
We simulate oxygen and neon shell burning with and without mag-netic fields in a non-rotating 18 𝑀 (cid:12) solar-metallicity star calculatedusing the stellar evolution code Kepler (Weaver et al. 1978; Woosleyet al. 2002; Heger & Woosley 2010). The same progenitor model haspreviously been used in the shell convection simulation of Mülleret al. (2016). The structure of the stellar evolution model at the timeof mapping to 3D is illustrated in Figure 1. The model containstwo active convective shells with sufficiently short turnover times tomake time-explicit simulations feasible. The oxygen shell extendsfrom 1 . 𝑀 (cid:12) to 2 . 𝑀 (cid:12) in enclosed mass and from 3 ,
400 km to7 ,
900 km in radius, immediately followed further out by the neonshell out to 3 . 𝑀 (cid:12) in mass and 27 ,
000 km in radius.For our 3D simulations we employ the Newtonian magnetohy-drodynamic (MHD) version of the CoCoNuT code as describedin Müller & Varma (2020). The MHD equations are solved inspherical polar coordinates using the HLLC (Harten-Lax-van Leer-Contact) Riemann solver (Gurski 2004; Miyoshi & Kusano 2005).The divergence-free condition ∇ · B = 𝜓 to ˆ 𝜓 = 𝜓 / 𝑐 h , where the cleaning speed 𝑐 h is chosen to be the fast magnetosonic speed. The extended system ofMHD equations for the density 𝜌 , velocity v , magnetic field B , thetotal energy density 𝑒 , mass fractions 𝑋 𝑖 , and the rescaled Lagrange MNRAS000
000 km in radius.For our 3D simulations we employ the Newtonian magnetohy-drodynamic (MHD) version of the CoCoNuT code as describedin Müller & Varma (2020). The MHD equations are solved inspherical polar coordinates using the HLLC (Harten-Lax-van Leer-Contact) Riemann solver (Gurski 2004; Miyoshi & Kusano 2005).The divergence-free condition ∇ · B = 𝜓 to ˆ 𝜓 = 𝜓 / 𝑐 h , where the cleaning speed 𝑐 h is chosen to be the fast magnetosonic speed. The extended system ofMHD equations for the density 𝜌 , velocity v , magnetic field B , thetotal energy density 𝑒 , mass fractions 𝑋 𝑖 , and the rescaled Lagrange MNRAS000 , 1–10 (2020) imulations of Oxygen Shell Burning . . . . . . m a ss f r a c t i o n X i He C O Ne Mg Si S Ar m [ M (cid:12) ] ρ [ g c m − ] ρs s [ k b / nu c l o n ] Figure 1.
Profiles of selected mass fractions 𝑋 𝑖 (top), density 𝜌 , and specificentropy 𝑠 in the 18 𝑀 (cid:12) Kepler stellar evolution model at the time of mappingto CoCoNuT. The boundaries of the simulated domain are indicated bydashed vertical lines. Note subtle differences to Figure 1 in Müller et al.(2016), which shows the same stellar evolution model at the onset of collapse. multiplier ˆ 𝜓 reads, 𝜕 𝑡 𝜌 + ∇ · 𝜌 v = , (1) 𝜕 𝑡 ( 𝜌 v ) + ∇ · (cid:18) 𝜌 vv − BB 𝜋 + 𝑃 t I (cid:19) = 𝜌 g − (∇ · B ) B 𝜋 , (2) 𝜕 𝑡 𝑒 + ∇ · (cid:20) ( 𝑒 + 𝑃 t ) u − B ( v · B ) 𝜋 (cid:21) = 𝜌 g · v + 𝜌 (cid:164) 𝜖 nuc , (3) 𝜕 𝑡 B + ∇ · ( vB − Bv ) + ∇ · ( 𝑐 h ˆ 𝜓 ) = 𝜕 𝑡 ˆ 𝜓 + 𝑐 h ∇ · B = − ˆ 𝜓 / 𝜏. (5) 𝜕 𝑡 ( 𝜌𝑋 𝑖 ) + ∇ · ( 𝜌𝑋 𝑖 v ) = 𝜌 (cid:164) 𝑋 𝑖 , (6)where g is the gravitational acceleration, 𝑃 t is the total pressure, I is the Kronecker tensor, 𝑐 h is the hyperbolic cleaning speed, 𝜏 is the damping time scale for divergence cleaning, and (cid:164) 𝜖 nuc and (cid:164) 𝑋 𝑖 are energy and mass fraction source terms from nuclear reactions.This system conserves the volume integral of a modified total energydensity 𝑒 , which also contains the cleaning field ˆ 𝜓 , 𝑒 = 𝜌 (cid:18) 𝜖 + 𝑣 (cid:19) + 𝐵 + ˆ 𝜓 𝜋 , (7)where 𝜖 is the mass-specific internal energy. Further details on theMHD implementation will be presented in a code comparison paper(Varma et al., in preparation).Viscosity and resistivity are not included explicitly in the idealMHD approximation, and only enter through the spatial reconstruc- tion and the computation of the Riemann fluxes. While this “implicitlarge-eddy simulation” (ILES) approach is well-established for hy-drodynamic turbulence (Grinstein et al. 2007), the magnetohydrody-namic case is more complicated because the behaviour in the relevantastrophysical regime of low (kinematic) viscosity 𝜈 and resistivity 𝜂 may still depend on the magnetic Prandtl number Pm = 𝜈 / 𝜂 . Differ-ent from the regime later encountered in the supernova core wherePm (cid:29)
1, oxygen shell burning is characterised by magnetic Prandtlnumbers slightly below unity (Pm ∼ . ∼ ∼ × − in liquid sodium experiments (Pétrélis et al. 2007;Monchaux et al. 2007), provided that both the hydrodynamic andmagnetic Reynolds number are sufficiently high. An ILES approachthat places the simulation into a “universal”, strongly magnetisedregime (Beresnyak 2019) thus appears plausible. Moreover, even ifthe ILES approach were only to provide upper limits for magneticfields and their effects on the flow, meaningful conclusions can stillbe drawn in the context of shell convection simulations.The nuclear source terms are calculated with the 19-species nu-clear reaction network of Weaver et al. (1978). Neutrino cooling isignored, since it becomes subdominant in the late pre-collapse phaseas the contraction of the shells speeds up nuclear burning.The simulations are conducted on a grid with 400 × × 𝑟 , colatitude 𝜃 , and longitude 𝜑 with an exponentialgrid in 𝑟 and uniform spacing in 𝜃 and 𝜑 . To reduce computationalcosts, we excise the non-convective inner core up to 3000 km andreplace the excised core with a point mass. The grid extends to aradius of 50 ,
000 km and includes a small part of the silicon shell, theentire convective oxygen and neon shells, the non-convective carbonshell, and parts of the helium shell.In order to investigate the impact of magnetic fields on late-stage oxygen shell convection, we run a purely hydrodynamic, non-magnetic simulation and an MHD simulation. In the MHD simu-lation, we impose a homogeneous magnetic field with 𝐵 𝑧 = Gparallel to the grid axis as initial conditions. We implement reflectingand periodic boundary conditions in 𝜃 and 𝜑 , respectively. For thehydrodynamic variables we use hydrostatic extrapolation at the innerand outer boundary, and impose strictly vanishing advective fluxesat the inner boundary. Different from Müller et al. (2016), we do not contract the inner boundary to follow the contraction and collapse ofthe core. The inner and outer boundary conditions for the magneticfields are less trivial. In simulations of magnetoconvection in the Sun,various choices such as vertical boundary conditions ( 𝐵 𝑥 = 𝐵 𝑦 = 𝐵 𝜃 = 𝐵 𝜑 = MNRAS , 1–10 (2020)
Varma & Müller M a g n e t i c F i e l d S t r e n g t h [ G ] Maximum B-fieldAverage B-field
Figure 2.
Evolution of the volume-averaged (solid) and maximum (dashed)magnetic field strength within the oxygen shell. fields in the ghost zones to their initial values for a homogeneousvertical magnetic field. We argue that due to the buffer regions atour radial boundaries, and the lack of rotational shear, our choice ofmagnetic boundary conditions should not have a significant impacton the dynamically relevant regions of the star.
Both the magnetised and non-magnetised model were run for over 12minutes of physical time, which corresponds to about 17 convectiveturnover times. Very soon after convection develops in the oxygenshell, the turbulent convective flow start to rapidly amplify the mag-netic fields in this region. To illustrate the growth of the magnetic fieldwe show the root-mean-square (RMS) average and maximum valueof the magnetic field in the oxygen shell in Figure 2.The magneticfield, which we initialised at 10 G, is amplified by over two ordersof magnitude to over 10 G on average within the shell due to con-vective and turbulent motions. The average magnetic field strengthin the shell is still increasing at the end of the simulation, but thegrowth rate has slowed down, likely indicating that the model is ap-proaching some level of magnetic field saturation. While we cannotwith certainty extrapolate the growth dynamics without simulatinglonger, it appears likely that RMS saturation field strength will settlesomewhere around ≈ × G.A closer look reveals that the magnetic field is not amplified ho-mogeneously throughout the convective region. The convective ed-dies push the magnetic field lines against the convective boundaries,where the fields are then more strongly amplified by shear flows. Thisis visualised in Figures 3a and 3b, where both magnetic pressure andmagnetic field strengths appear concentrated at the convective bound-ary. The maximum magnetic field strength shown in Figure 2 (dashedline) therefore essentially mirrors the field at the inner boundary ofthe oxygen shell. We observe a very quick rise in magnetic fieldstrength at the boundary at the beginning of the simulation between20 -
50 s once the convective flow is fully developed. The rate ofgrowth of the maximum magnetic field strength after this increasesat approximately the same rate as the average magnetic field strength. To characterise the geometric structure of the magnetic field, weshow a radial profile of the dipole of the radial magnetic field compo-nent 𝐵 𝑟 at the end of the simulation at ≈
725 s (Figure 4). The dipolemagnetic field in the convective regions is approximately an order ofmagnitude smaller than the RMS average radial magnetic field, asidefrom the innermost boundary of our grid, where the dipole compo-nent is comparable to the total radial magnetic field. This behaviourat the boundary is likely an artefact of our choice of homogeneousmagnetic fields at the inner boundary. In general, however, the mag-netic fields in the convective zones appear dominated by higher-ordermultipoles. Disregarding the dipole fields at the inner boundary, thedipole magnetic field of ≈ G or below lies in the upper rangeof observed dipole magnetic fields of white dwarfs (Ferrario et al.2015), which have often been taken as best estimates for the dipolefields in the cores of massive stars.To further illustrate the small-scale nature of the magnetic fieldwithin the oxygen shell, we show angular power spectra, ˆ 𝑀 ℓ of theradial field strength at different times as a function of the sphericalharmonics degree ℓ inside the oxygen shell at a radius of ≈ 𝑀 ℓ is computed as:ˆ 𝑀 ℓ = 𝜋 ℓ ∑︁ 𝑚 = − ℓ (cid:12)(cid:12)(cid:12)(cid:12)∫ 𝑌 ∗ ℓ𝑚 ( 𝜃, 𝜑 ) 𝐵 𝑟 d Ω (cid:12)(cid:12)(cid:12)(cid:12) . (8)Very early in the simulation, the spectrum shows a significant ℓ = ℓ . The break in the spectrum movestowards smaller wave numbers. and the peak of the spectrum shiftsto larger scales from ℓ ≈
12 to ℓ ≈
7. Simulations of field amplifi-cation by a small-scale dynamo in isotropic turbulence often exhibita Kazantsev spectrum with power-law index 𝑘 − / on large scales.Our spectra show a distinctly flatter slope below the spectral speak,indicating that field amplification is subtly different from the standardpicture of turbulent dynamo amplification.This is borne out by a closer look at the magnetic field distribu-tion within the convective region. Somewhat similar to our recentsimulation of field amplification by neutrino-driven convection incore-collapse supernovae (Müller & Varma 2020), field amplifica-tion does not happen homogeneously throughout the convective re-gion and appears to be predominantly driven by shear flows at theconvective boundaries. To illustrate this, we compare the spherically-averaged diagonal components of the kinetic (Reynolds) and mag-netic (Maxwell) stress tensors 𝑅 𝑖 𝑗 and 𝑀 𝑖 𝑗 in the MHD model at thefinal time-step of the simulation at ≈
725 s (Figure 6). 𝑅 𝑖 𝑗 and 𝑀 𝑖 𝑗 are computed as 𝑅 𝑖 𝑗 = (cid:104) 𝜌𝑣 𝑖 𝑣 𝑗 (cid:105) , (9) 𝑀 𝑖 𝑗 = 𝜋 (cid:104) 𝐵 𝑖 𝐵 𝑗 (cid:105) , (10)where angled brackets denote volume-weighted averages. The mag-netic fields clearly remain well below equipartition with the total Strictly speaking, the most rigorous way to extract the dipole component ofthe magnetic field would use a poloidal-toroidal decomposition B = ∇× [∇×( P r ) ] + ∇ × ( T r ) where the scalar functions P and T describe the poloidaland toroidal parts of the field, and consider all components 𝐵 𝑟 , 𝐵 𝜃 , and 𝐵 𝜑 arising from the ℓ = P . Since the poloidal-toroidal decomposition cannot be reduced to a straightforward projection ontovector spherical harmonics, this analysis is left to future papers. Note that no explicit decomposition of the velocity field into fluctuatingMNRAS000
725 s (Figure 6). 𝑅 𝑖 𝑗 and 𝑀 𝑖 𝑗 are computed as 𝑅 𝑖 𝑗 = (cid:104) 𝜌𝑣 𝑖 𝑣 𝑗 (cid:105) , (9) 𝑀 𝑖 𝑗 = 𝜋 (cid:104) 𝐵 𝑖 𝐵 𝑗 (cid:105) , (10)where angled brackets denote volume-weighted averages. The mag-netic fields clearly remain well below equipartition with the total Strictly speaking, the most rigorous way to extract the dipole component ofthe magnetic field would use a poloidal-toroidal decomposition B = ∇× [∇×( P r ) ] + ∇ × ( T r ) where the scalar functions P and T describe the poloidaland toroidal parts of the field, and consider all components 𝐵 𝑟 , 𝐵 𝜃 , and 𝐵 𝜑 arising from the ℓ = P . Since the poloidal-toroidal decomposition cannot be reduced to a straightforward projection ontovector spherical harmonics, this analysis is left to future papers. Note that no explicit decomposition of the velocity field into fluctuatingMNRAS000 , 1–10 (2020) imulations of Oxygen Shell Burning (a) (b)(c) (d) / M a g n e t i c F i e l d S t r e n g t h [ G ] R a d i a l V e l o c i t y [ k m / s ] S ili c o n M a ss F r a c t i o n Figure 3.
Snapshots of the equatorial plane in the MHD simulation at a time of 500 s, showing the inner part of the domain from the inner boundary at a radiusof 3000 km out to 12 ,
000 km. The panels display a) the ratio of magnetic to thermal pressure (i.e., inverse plasma- 𝛽 ), b) the magnitude of the magnetic fieldstrength, c) the radial velocity and d) the silicon mass fraction. turbulent kinetic energy, and appear to converge to saturation lev-els about one order of magnitude below, although longer simulationswill be required to confirm this. The non-radial diagonal components 𝑅 𝜃 𝜃 + 𝑅 𝜑𝜑 and 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 are generally higher than the respectiveradial components 𝑅 𝑟𝑟 and 𝑀 𝑟𝑟 . Throughout most of the domain,the Maxwell stresses are considerably smaller than the Reynoldsstresses, but it is noteworthy that the profile of the non-radial com-ponents of 𝑀 𝑖 𝑗 runs largely parallel to those of 𝑅 𝑖 𝑗 , just with adifference of slightly more than an order of magnitude. Peaks of themagnetic stresses at the shell interfaces suggest that field amplifica-tion is driven by shear flow at the convective boundaries. Convectivemotions then transport the magnetic field into the interior of the con- components and a spherically averaged background state is required since thebackground state is hydrostatic. vective regions and also generate radial field components. There areno humps of 𝑀 𝑟𝑟 within the convective zones that corresponds to thehumps in 𝑅 𝑟𝑟 , which indicates that little amplification by convectiveupdrafts and downdrafts takes place within the convection region.At the outer boundaries of the oxygen and neon shell, we observerough equipartition between 𝑅 𝑟𝑟 and 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 . The fact that thegrowth of the field slows down once the model approaches 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 ≈ 𝑅 𝑟𝑟 suggests that this “partial equipartition” may determinethe saturation field strength, but the very different behaviour at theinner boundary with 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 (cid:29) 𝑅 𝑟𝑟 argues against this. Itis plausible, though, that the saturation field strength is (or ratherwill be) determined at the boundary. Linear stability analysis of themagnetised Kelvin-Helmholtz instability (e.g. Chandrasekhar 1961;Sen 1963; Fejer 1964; Miura & Pritchett 1982; Liu et al. 2018) showsthat shear instability, which is critical for efficiently generating small-scale fields, are suppressed by magnetic fields parallel to the direction MNRAS , 1–10 (2020)
Varma & Müller M ]10 M a g n e t i c F i e l d S t r e n g t h [ G ] B r , = 1 contributionB r , RMS average Figure 4.
The angle-averaged root-mean-square (RMS) value (black) andthe dipole component (red) of the radial magnetic field component 𝐵 𝑟 as afunction of mass coordinate at a time of 725 s. M [ G ] k k Figure 5.
Power ˆ 𝑀 ℓ in different multipoles ℓ of the radial field componentof the magnetic field in the oxygen shell at different times. Dotted lines showthe slopes of Kolmogorov ( 𝑘 − / ) and Kazantsev ( 𝑘 / ) spectra. The low-wavenumber part of the spectrum is always distinctly flatter than a Kazantsevspectrum; at intermediate ℓ , a Kolmogorov spectrum obtains with a breakaround ℓ =
30 to a steeper slope in the dissipation range. of the shear flow; hence the generation of non-radial fields at theboundaries may be self-limiting. A naive application of the principleof marginal stability would suggest saturation occurs once the Alfvénvelocity at the boundary equals the shear velocity jump across theshell interface, which would imply 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 ≈ 𝑅 𝜃 𝜃 + 𝑅 𝜑𝜑 . Oursimulation suggests that saturation probably occurs at significantlysmaller values, and a more quantitative analysis of the saturationmechanism will clearly be required in future to elucidate the relationbetween the shear velocity (and possibly the width of the shear layer)and the saturation field strength. M ]10 S t r e ss [ g c m s ] R rr R + RM rr M + M Figure 6.
The radial (solid) and non-radial (dashed) diagonal components ofthe Reynolds stress tensor 𝑅 𝑖 𝑗 (black) and Maxwell tensor 𝑀 𝑖 𝑗 (red) for theMHD convection model at 725 s as a function of enclosed mass. E k i n [ e r g ] E r (MHD) E , (MHD) E r (Hydro) E , (Hydro) Figure 7.
Evolution of the total radial (solid) and non-radial (dashed) con-vective kinetic energy within the oxygen shell for the purely hydrodynamic(purple) and MHD (black) model, respectively.
The slowing growth of the magnetic field indicates that feedbackeffects on the flow should become important during the later phaseof the simulation. It is particularly interesting to consider the effectof the strong fields tangential to the oxygen-neon shell interfaceon convective boundary mixing, though we also consider potentialeffects on the flow in the interior of the convective regions.To this end, we compare the MHD model to a purely hydrodynamicsimulation of oxygen and neon shell convection. Figure 7 comparesthe total kinetic energy in convective motions in the oxygen shellbetween the models. The radial and non-radial components 𝐸 𝑟 and MNRAS000
The slowing growth of the magnetic field indicates that feedbackeffects on the flow should become important during the later phaseof the simulation. It is particularly interesting to consider the effectof the strong fields tangential to the oxygen-neon shell interfaceon convective boundary mixing, though we also consider potentialeffects on the flow in the interior of the convective regions.To this end, we compare the MHD model to a purely hydrodynamicsimulation of oxygen and neon shell convection. Figure 7 comparesthe total kinetic energy in convective motions in the oxygen shellbetween the models. The radial and non-radial components 𝐸 𝑟 and MNRAS000 , 1–10 (2020) imulations of Oxygen Shell Burning R a d i u s [ c m ] (a) HydroMHD 0.500.510.520.530.54 O - s h e ll m a ss [ M ] (b) M [ M / s ] (c) E n e r g y g e n e r a t i o n r a t e [ e r g / s ] (d) Figure 8.
Effects of entrainment on the oxygen shell for both the MHD model (black) and the equivalent purely hydrodynamic model (purple). The panels showa) the inner and outer oxygen shell radii, b) the total mass within the oxygen shell, c) the entrainment rate (cid:164) 𝑀 into the oxygen shell, and d) the volume-integratednuclear energy generation rate throughout the shell. 𝐸 𝜃,𝜑 of the kinetic energy are defined as 𝐸 𝑟 = ∫ 𝑟 − ≤ 𝑟 ≤ 𝑟 + 𝜌𝑣 𝑟 d 𝑉, (11) 𝐸 𝜃,𝜑 = ∫ 𝑟 − ≤ 𝑟 ≤ 𝑟 + 𝜌 ( 𝑣 𝜃 + 𝑣 𝜑 ) d 𝑉, (12)where 𝑟 − and 𝑟 + are the inner and outer radii of the oxygen shell. Wecompute the inner and outer shell radii 𝑟 − and 𝑟 + as the midpoints ofthe steep, entropy slope between the oxygen shell and the silicon andneon shells below and above. Due to shell growth by entrainment, 𝑟 − and 𝑟 + are time-dependent (Figure 8a).For both models most of the turbulent kinetic energy is in thenon-radial direction. This is different from the rough equipartition 𝐸 𝑟 ≈ 𝐸 𝜃,𝜑 seen in many simulations of buoyancy-driven convection(Arnett et al. 2009). There is, however, no firm physical principlethat dictates such equipartition; indeed a shell burning simulation ofthe same 18 𝑀 (cid:12) progenitor with the Prometheus code also showedsignificantly more kinetic energy in non-radial motions (Müller et al.2016). Ultimately, the high ratio 𝐸 𝜃,𝜑 / 𝐸 𝑟 merely indicates that thefully developed flow happens to predominantly select eddies withlarger extent in the horizontal than in the vertical direction. In the low-Mach number regime, the anelastic condition ∇ · ( 𝜌 v ) ≈ Discounting stochastic variations, the radial component 𝐸 𝑟 of theturbulent kinetic energy for both the hydrodynamic and MHD modelare similar until the final ≈
300 s, at which point they start to devi-ate more significantly. Clearer differences appear in the non-radialcomponent 𝐸 𝜃,𝜑 , with the hydro model showing irregular episodicbursts in kinetic energy which are not mirrored in the MHD model.This points to feedback of the magnetic field on the flow, whose na-ture will become more apparent as we analyse convective boundarymixing in the models.To this end, we first consider the evolution of the boundaries 𝑟 − and 𝑟 + of the oxygen shell and the total oxygen shell mass 𝑀 O . Forcomputing the oxygen shell mass, we take the (small) deviation of theboundaries from spherical symmetry into account more accuratelythan when computing 𝑟 − and 𝑟 + , and integrate the mass containedin cells within the entropy range 3 . - . 𝑘 B / nucleon. As shown byFigure 8 (panels a and b), the oxygen shell in the non-magnetic modelgrows slightly, but perceptibly faster without magnetic fields than inthe MHD model, starting at a simulation time of about 300 s. Thepresence of relatively strong magnetic fields in the boundary layerapparently reduces entrainment in line with the inhibiting effect ofmagnetic fields parallel to the flow on shear instabilities discussed inSection 3.1.The entrainment rate (cid:164) 𝑀 = d 𝑀 O / d 𝑡 (Figure 8c) is only slightlyhigher in the purely hydrodynamic model most of the time, but theentrainment rate exhibits occasional spikes, which do not occur inthe MHD model. We note that these spikes in (cid:164) 𝑀 occur at aroundthe same times as the bursts in non-radial kinetic energy (Figure 7). MNRAS , 1–10 (2020)
Varma & Müller
It appears that the stabilisation of the boundary by magnetic fieldsmostly suppresses rarer, but more powerful entrainment events thatmix bigger lumps of material into the oxygen shell. A comparisonwith the shell burning simulations of Müller et al. (2016) providesconfidence that this effect is robust. Because Müller et al. (2016)contract the inner boundary condition, convection grows more vig-orous with time and their entrainment rates are higher, adding about0 . 𝑀 (cid:12) to the oxygen shell within 300 s. Their resolution test (Fig-ure 20 in Müller et al. 2016) only showed variations of 0 . 𝑀 (cid:12) in the entrained mass between different runs of the same progeni-tor model. In our simulations, the oxygen shell only grows by about0 . 𝑀 (cid:12) from 300 s to 700 s (i.e, when the magnetic field does notgrow substantially any more), yet we find a difference in shell growthof about 0 . 𝑀 (cid:12) between the magnetic and non-magnetic model. Ittherefore seems likely that there is indeed an appreciable systematiceffect of magnetic fields on entrainment.The different entrainment rate also explains long-term, time-averaged differences in convective kinetic energy between the twosimulations. The convective kinetic energy is determined by the totalnuclear energy generation rate, which is shown in Figure 8d. Bothmodels show an overall trend downwards over time in nuclear energygeneration, which can be ascribed to a slight thermal expansion of theshell and the depletion of fuel. After about 200 s, the purely hydrody-namics model exhibits a higher energy generation rate, which persistsuntil the end of the simulations and becomes more pronounced. Thehigher energy generation rate in the non-magnetic model is indeedmirrored by a stronger decrease in the neon mass on the grid (Fig-ure 9) of about the right amount to explain the average difference inenergy generation rate at late times. We note, however, that the strongepisodic entrainment events in the purely hydrodynamic model are not associated with an immediate increase in nuclear energy gen-eration rate. This is mainly because the energy release from thedissociation of the entrained neon is delayed and spread out in timeas the entrained material is diluted and eventually mixed down toregions of sufficiently high temperature.The overall effect of magnetic fields on the bulk flow is rathermodest, though. We do not see a similarly strong quenching of theconvective flow by magnetic fields as in recent MHD simulationsof the solar convection zone (Hotta et al. 2015), who reported areduction of convective velocities by up to 50% at the base of theconvection zone. Such an effect is not expected as long as the mag-netic fields stay well below kinetic equipartition in the interior ofconvective shells as in our MHD simulation. Longer simulations willbe required to confirm that magnetic fields during late shell burningstages indeed remain “weak” by comparison and do not appreciablyinfluence the bulk dynamics of convection. We investigated the amplification and saturation of magnetic fieldsduring convective oxygen shell burning and the backreaction of thefield on the convective flow by conducting a 3D MHD simulation anda purely hydrodynamic simulation of an 18 𝑀 (cid:12) progenitor shortly be-fore core collapse. The simulations were run for about 12 minutes ofphysical time (corresponding to about 17 convective turnover times),at which point field amplification has slowed down considerably,though a quasi-stationary state has not yet been fully established.The magnetic field in the oxygen shell is amplified to ∼ Gand dominated by small-to-medium-scale structures with angularwavenumber ℓ ∼
7. The dipole component is considerably smallerwith ∼ G near the inner boundary of the oxygen shell and less fur- M a ss [ M ] HydroMHD
Figure 9.
Total mass of neon in the entire computational domain for thepurely hydrodynamic simulation (purple) and the MHD simulation (black). ther outside. The profiles of the radial and non-radial diagonal com-ponents 𝑀 𝑟𝑟 and 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 of the Maxwell stress tensor mirrorthe corresponding components 𝑅 𝑟𝑟 and 𝑅 𝜃 𝜃 + 𝑅 𝜑𝜑 of the Reynoldsstress tensor, but remain about an order of magnitude smaller, i.e.,kinetic equipartition is not reached. However, 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 can ap-proach or exceed the radial component 𝑅 𝑟𝑟 at the convective bound-aries. The saturation mechanism for field amplification needs to bestudied in more detail, but we speculate that saturation is mediatedby the inhibiting effect of non-radial magnetic fields on shear insta-bilities at shell boundaries, which appear to be the primary driver offield amplification.We find that magnetic fields do not have an appreciable effecton the interior flow inside the oxygen shell, but observe a moderatereduction of turbulent entrainment at the oxygen-neon shell bound-ary in the presence of magnetic fields. Magnetic fields appear tosuppress stronger episodic entrainment events, although they do notquench entrainment completely. Through the reduced entrainmentrate, magnetic fields also indirectly affect the dynamics inside theconvective region slightly because they reduce the energy releasethrough the dissociation of ingested neon, which results in slightlysmaller convective velocities in the MHD model.Our findings have important implications for core-collapse su-pernova modelling. We predict initial fields in the oxygen shell ofnon-rotating progenitors that are significantly stronger than assumedin our recent simulation of a neutrino-driven explosion aided bydynamo-generated magnetic fields (Müller & Varma 2020). With rel-atively strong seed-fields, there is likely less of a delay until magneticfields can contribute the additional “boost” to neutrino heating andpurely hydrodynamic instabilities seen in Müller & Varma (2020).This should further contribute to the robustness of the neutrino-driven mechanism for non-rotating and slowly-rotating massive stars.Our simulations also suggest that the perturbation-aided mechanism(Couch & Ott 2013; Müller & Janka 2015) will not be substantiallyaffected by the inclusion of magnetic fields. Since magnetic fields donot become strong enough to substantially alter the bulk flow insidethe convective region, the convective velocities and eddy scales asthe key parameters for perturbation-aided explosions remain largelyunchanged.The implications of our results for neutron star magnetic fieldsare more difficult to evaluate since the observable fields will, to a MNRAS000
Total mass of neon in the entire computational domain for thepurely hydrodynamic simulation (purple) and the MHD simulation (black). ther outside. The profiles of the radial and non-radial diagonal com-ponents 𝑀 𝑟𝑟 and 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 of the Maxwell stress tensor mirrorthe corresponding components 𝑅 𝑟𝑟 and 𝑅 𝜃 𝜃 + 𝑅 𝜑𝜑 of the Reynoldsstress tensor, but remain about an order of magnitude smaller, i.e.,kinetic equipartition is not reached. However, 𝑀 𝜃 𝜃 + 𝑀 𝜑𝜑 can ap-proach or exceed the radial component 𝑅 𝑟𝑟 at the convective bound-aries. The saturation mechanism for field amplification needs to bestudied in more detail, but we speculate that saturation is mediatedby the inhibiting effect of non-radial magnetic fields on shear insta-bilities at shell boundaries, which appear to be the primary driver offield amplification.We find that magnetic fields do not have an appreciable effecton the interior flow inside the oxygen shell, but observe a moderatereduction of turbulent entrainment at the oxygen-neon shell bound-ary in the presence of magnetic fields. Magnetic fields appear tosuppress stronger episodic entrainment events, although they do notquench entrainment completely. Through the reduced entrainmentrate, magnetic fields also indirectly affect the dynamics inside theconvective region slightly because they reduce the energy releasethrough the dissociation of ingested neon, which results in slightlysmaller convective velocities in the MHD model.Our findings have important implications for core-collapse su-pernova modelling. We predict initial fields in the oxygen shell ofnon-rotating progenitors that are significantly stronger than assumedin our recent simulation of a neutrino-driven explosion aided bydynamo-generated magnetic fields (Müller & Varma 2020). With rel-atively strong seed-fields, there is likely less of a delay until magneticfields can contribute the additional “boost” to neutrino heating andpurely hydrodynamic instabilities seen in Müller & Varma (2020).This should further contribute to the robustness of the neutrino-driven mechanism for non-rotating and slowly-rotating massive stars.Our simulations also suggest that the perturbation-aided mechanism(Couch & Ott 2013; Müller & Janka 2015) will not be substantiallyaffected by the inclusion of magnetic fields. Since magnetic fields donot become strong enough to substantially alter the bulk flow insidethe convective region, the convective velocities and eddy scales asthe key parameters for perturbation-aided explosions remain largelyunchanged.The implications of our results for neutron star magnetic fieldsare more difficult to evaluate since the observable fields will, to a MNRAS000 , 1–10 (2020) imulations of Oxygen Shell Burning large degree, be set by processes during and after the supernova andcannot be simply extrapolated from the progenitor stage by magneticflux conservation. That said, dipole fields of order 10 G at the baseof the oxygen shell – which is likely to end up as the neutron starsurface region – are not in overt conflict with dipole fields of order10 G in many young pulsars inside supernova remnants (Enotoet al. 2019). However, considering the relatively strong small-scalefields of ∼ G with peak values over 10 G, it may prove diffi-cult to produce neutron stars without strong small-scale fields at thesurface. There is a clear need for an integrated approach towards theevolution of magnetic fields from the progenitor phase through thesupernova and into the compact remnant phase in order to fully graspthe implications of the current simulations.Evidently, further follow-up studies are also needed on the finalevolutionary phases of supernova progenitors. Longer simulationsand resolution studies will be required to better address issues likethe saturation field strength, the saturation mechanism, and the im-pact of magnetic fields on turbulent entrainment. The critical issue ofnon-ideal effects and the behaviour of turbulent magnetoconvectionfor magnetic Prandtl numbers slightly smaller than one at very highReynolds numbers deserves particular consideration. Some findingsof our “optimistic” approach based on the ideal MHD approximationshould, however, prove robust, such as the modest effect of magneticfields on the convective bulk flow and hence the reliability of purelyhydrodynamic models (Couch et al. 2015; Müller et al. 2016; Ya-dav et al. 2020; Müller 2020; Fields & Couch 2020) and even 1Dmixing-length theory (Collins et al. 2018) to predict pre-collapseperturbations in supernova progenitors. Future 3D simulations willalso have to address rotation and its interplay with convection andmagnetic fields.
ACKNOWLEDGEMENTS
We thank A. Heger for fruitful conversations. BM was supported byARC Future Fellowship FT160100035. We acknowledge computertime allocations from NCMAS (project fh6) and ASTAC. This re-search was undertaken with the assistance of resources and servicesfrom the National Computational Infrastructure (NCI), which is sup-ported by the Australian Government. It was supported by resourcesprovided by the Pawsey Supercomputing Centre with funding fromthe Australian Government and the Government of Western Australia.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the authors, subject to considerations of intellectual property law.
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