Modelling Turbulent Deflagrations in Type Ia Supernovae

Abstract

We present an overview of the current state of multidimensional modelling of type Ia supernovae and an example for surprising consequences of impoving the physics of the model. In this case, an improved handling of subgrid scale turbulence gives rise to lower burning rates and a decreased global energy release. While this result is too preliminary to be interpreted quantitatively, it shows that much work remains to be done before the turbulent deflagration model can be declared to be understood and/or insufficient to explain normal type Ia supernovae.

Full text

Modelling Turbulent Deflagrations in Type Ia Supernovae
J.C. Niemeyera,W.Schmidt
aand C. Klingenbergb
aInstitut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg
Am Hubland, D-97074 W¨urzburg, Germany
bInstitut f¨ur Angewandte Mathematik und Statistik, Universit¨at W¨urzburg
Am Hubland, D-97074 W¨urzburg, Germany
We pres e nt an ove r v i ew o f the current s t a t e o f multid i m e n s i onal m o d e l l i n g o f type Ia
supernovae and an example for surprising consequences of impoving the physics of the
model. In this case, an improved handling of subgrid scale turbulence gives rise to lower
burning rates and a decreased global energy release. While this result is too preliminary
to be interpreted quantitatively, it shows that much work remains to be done before
the turbulent deflagration model can be declared to be understood and/or insufficient to
explain normal type Ia supernovae.
1. INTRODUCTION
Type Ia supernovae (SNe Ia) have received a lot of attention recently in their roles as
cosmological distance indicators and prime witnesses of the accelerated expansion of the
universe (see [1,2] for reviews). Apart from cosmology, there is an urgent need for reliable
SN Ia models in the context of nucleosynthesis of heavy elements and galactic chemical
evolution. Yet despite more than three decades of research we are still debating some
fundamental aspects of the physics of the stellar explosions that give rise to these bright
and powerful events.
As is widely known, the most successful model for a SN Ia is the thermonuclear explosion
of a CO-White Dwarf driven to criticallity by accretion in a binary system until it reaches
the Chandrasekhar mass (e.g., [3]). “Successful” means that in principle, as demonstrated
by spherically symmetric explosion models, the right amounts of 56Ni and intermediate
mass elements can be produced (with adequate velocities) to explain the light curves and
spectra of typical SNe Ia. Unfortunately, in these models much of the essential explosion
physics is hidden in the parametrization of the thermonuclear burning speed. In the case
of the initial subsonic deflagration phase, the speed is basically a free parameter whereas
for the proposed secondary phase of a delayed detonation [4,5], the speed is fixed by
hydrodynamics but the onset of the detonation is dialed in by hand. Furthermore, in the
latter case the non-spherical geometry of an off-center detonation cannot be captured in
1D-simulations.
Much progress has been made recently in the field of multidimensional (2 and 3D) sim-
ulations of SN Ia explosions [6–8]. Parameter studies with 3D simulations of the turbulent
Nuclear Physics A 758 (2005) 431c–438c
0375-9474/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.nuclphysa.2005.05.080

deflagration phase have been presented in [9] and work is beginning now to model delayed
detonations in multiple dimensions [10,11]. However, in spite of similar results the inter-
pretation of the current state of affairs differs somewhat among various groups. While the
generally low 56Ni-masses and ejecta velocities of the 3D deflagration models as well as
the residual unburnt C+O near the stellar core has been interpreted as clear evidence for
delayed detonations in normal SNe Ia by [10], we argue here that this assertion is prema-
ture based on the following arguments: i) the trends in highly resolved 3D deflagration
models point toward a resolution of the core-CO problem as a more realistic representa-
tion of turbulent entrainment is reached by increasing the effective numerical Reynolds
number (limited to O(103)incurrentmodels);ii)delayeddetonationsrequiresponta-
neous deflagration-detonation-transitions (DDT) whose occurence in unbounded flows is
not well understood [12] and must therefore be initiated by hand in current models; and
iii) even if they occur, it is unclear whether they can really cure all the shortcomings of
pure deflagration models [11].
In the remainder of this article, we will sketch some new developments regarding the
modelling techniques for turbulent thermonuclear flames. This work is still in progress
and is meant to serve as an example of how much work still needs to be done to conclude
that we really understand the deflagration phase.
2. The modelling of turbulent deflagrations
In the pure deflagration model for type Ia supernovae, subsoncially propagating flame
fronts consume carbon and oxygen and produce heavier elements by thermonuclear burn-
ing. From the microscopic point of view, the propagation speed sis determined by the
thermal conductivity of the fuel and is typically much less than the speed of sound [13].
This process is called a deflagration. In numerical simulations of supernova explosions,
however, it is impossible to explicitly resolve the burning process on the characteristic
length and time scale of the thermonuclear reactions. In addition, the interaction be-
tween flame propagation and fluid turbulence folds and wrinkles the flame fronts on all
length scales larger than a certain cut-off, which is called the Gibson scale (see [14] for a
discussion of the relevant combustion physics in SNe Ia).
Although the Gibson scale is usually much larger than the microscopic burning scale,
i. e., the flame width, not even the whole range of scales above the Gibson scale can be nu-
merically resolved. As a consequence two approximations are made. Firstly, flame fronts
are numerically represented as discontinuities. Secondly, an effective propagation speed
stis introduced which accounts for the effect of turbulence on length scales smaller than
the size ∆of the cells in the numerical discretisation of the hydrodynamical equations.
Representing the flame fronts as discontinuities is achieved by means of the level set
method in the Prometheus code. Details about the method and implementation can be
found in [15]. For the determination of the turbulent flame speed st,ontheotherhand,
amodelbasedonthekineticenergyassociatedwithunresolvedturbulenteddieswas
formulated [16]. The basic idea is that turbulent eddies of size smaller than the numerical
resolution ∆will further wrinkle the physical flame front in comparison to the numerically
computed front. Because the rate of energy generation by theromuclear burning grows
in proportion to the surface area of the flames, this would produce too little burning in
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c432c

the numerical simulation. However, the numerical smoothing of the flame front can be
compensated by enhancing the propagation speed. The corresponding velocity scale is
given by the mean kinetic energy per unit mass of eddies smaller than ∆, which is denoted
by ksgs and is called the subgrid scale turbulence energy.Hence,thefollowingansatzfor
calculating the flame propagation speed is made:
st=max(s, !2ksgs).(1)
This expression implies that stequals more or less the microscopic propagation speed
sif there is no or only little turbulence. On the other hand, if ksgs !s,thenthe
flame dynamics is largely dominated by turbulence and the propgation speed becomes
asymptotically independent of s.Thecomputationofksgs,inturn,entailstheproblemof
subgrid scale modelling.
The kind of subgrid scale model we adopted for simulating turbulent burning in super-
novae is based on a dynamical equation for the budget of turbulence energy:
Dsgs
Dtksgs−1
ρ∇·"ρCκ∆eff!ksgs∇ksgs#=
Cν∆eff!ksgs|S∗|2−(2
3+Cλ)ksgsdsgs −C$
(ksgs)3/2
∆eff
.
(2)
The left hand side corresponds to the Lagrangian time derivative of ksgs corrected by a dif-
fusion term, which accounts for the subgrid scale transport of ksgs.Ontherighthandside,
there is a production term, a compression term and a dissipation term. Kinetic energy on
subgrid scales is produced by non-linear turbulent interactions with the resolved velocity
field. The rate of production is proportional to the turbulent viscosity νsgs =Cν∆eff!ksgs
and the rate of strain |S∗|2=2Sik Sik −2
3d2,whereSik =1
2(vi,k +vk,i)isthesymmetrized
velo city derivative and d=vi,i the divergence. The rate of dissipation is given by the
dimensional expression ksgs/τ$,withthedissipationtimescaleτ$=∆
eff/(C$!ksgs). In
order to solve the above equation for ksgs,itisnecessarytocomputetheso-calledclosure
parameters, in particular, Cν,C$,Cκand Cλ.Inthefollowing,wewillconcentrateonthe
parameter of production, Cν.
The simplest possibility is to assume a constant value of Cν.Forstatisticallysta-
tionary and isotropic turbulence Cν≈0.5isactuallyafairapproximation. However,
the Rayleigh-Taylor driven turbulence in a supernova explosion is actually transient and
rather inhomogenous. For this reason, corrections of Cνcorrsponding to the local proper-
ties of the flow are required for a sound subgrid scale model. So far, a simple ad hoc rule
(a so-called wall proximity function) has been used in our simulations of type Ia super-
novae [16]. A presumbly much better approach makes use of the self-similarity properties
of turbulence in the limit of small length scales. The underlying assumption is that,
regardless of the large-scale structure of a flow, turbulent eddies become asymptotically
scale-invariant towards smaller length scales. This is known as inertial subrange of length
scales. If the resolution in a numerical simulation is sufficient, then the smallest resolved
as well as the subgrid scales will be within the inertial subrange and exhibit scale invari-
ance. It can be shown that the local value of the parameter Cνcan be estimated from
structural properties of the resolved flow in this case [17]. Because Cνthen varies in space
and time, we speak of a localised subgrid scale model.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c 433c

Figure 1. Total energy and mass as functions of time in simulations of thermonuclear su-
pernova explosions using different subgrid scale models for the calculation of the turbulent
burning speed.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c434c

Figure 2. Contour sections of the subgrid scale turbulence velocity qsgs for the model with
wall proximity functions. The white lines correspond to the flame surface.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c 435c

Figure 3. Contour sections of the subgrid scale turbulence velocity qsgs for the localised
model. The white lines correspond to the flame surface.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c436c

In order to compare the performance of the different subgrid scale models, we ran
three-dimensional simulations of supernova explosions with a resolution of 2563grid cells,
using axisymmetric initial conditions. The time evolution of several integrated quantities
is shown in figure 1. The total energy and the subgrid scale turbulence energy are plotted
in the top panels, whereas the bottom panels show the rate of kinetic energy production
and the total mass of the main burning product, 56Ni. The graphs labeled with WPF are
obtained for the simple subgrid scale model with the wall proximity function. The other
graphs apply for the localised model with two different initial values of qsgs =!2ksgs.
Remarkably, the localised subgrid scale model predicts a reduction in the explosion en-
ergy. This comes rather unexpected, because previous studies based on the simulation of
turbulent deflagration in a box indicated more rapid burning with the localised model.
However, if one considers the distribution ofturbulenceenergyanexplanationbecomes
apparent. In figure 3, two dimensional contour sections of qsgs are plotted for various stages
of the explosion in the simulation with the localised model. Comparing this with figure 2
which shows the corresponding plots in the case of the model with the wall proximity
function, one can see that the localised model predicts considerably higher intermittency
of subgrid scale turbulence. Although the enhanced intermittency causes a more wrinkled
flame surface, the propagation of the flames is inhibited in regions of very low subgrid
scale turbulence energy. Thus, the overall energetics of the burning process is reduced.
On the other hand, our treatment of subgrid scale dissipation is not fully satisfactory yet,
and it is possible that an improved closure will significantly alter the trends found so far.
For th e t ime b eing, our ma j or conclu s i on is that the gl o b al stati s t ics of the sup ernova
simulations appears to be sensitive to the subgrid scale model in use.
3. OUTLOOK
The results presented in the previous section show a dependence of the global energy
release on the microscopic dissipation mechanism encoded in the subgrid scale model,
albeit a very weak one, contrary to expectations from dimensional analysis and previous
simulations of deflagrations in a box. They are very preliminary and need to be supple-
mented by extensive parameter and resolution studies. This work is currently in progress.
However, these results demonstrate that our understanding of the turbulent deflagration
phase in type Ia supernova explosions is still incomplete. This fact, together with guidance
from Occam’s Razor, leads us to conclude that declaring delayed detonations necessary
for successful explosions is premature. This does not mean that delayed detonation do
not occur in general – they may still be needed to explain some peculiar events with high
ejecta velocities [18] – but more work is needed to explore the range of explosion energies
and nucleosynthesis products of pure turbulent deflagration explosions.
4. ACKNOWLEDGEMENTS
The research of JCN and WS was supported by the Alfried Krupp Prize for Young
University Teachers of the Alfried Krupp von Bohlen und Halbach Foundation.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c 437c

REFERENCES
1. B. Leibundgut, Ann. Rev. Astron. Astrophys. 39 (2001) 67.
2. W. Hillebrandt and J. C. Niemeyer, Ann. Rev. Astron. Astrophys. 38 (2000) 191.
3. M. Livio, The progenitors of type Ia supernovae, in Type Ia Super n ovae : Theory
and Cosmology,editedbyJensC.NiemeyerandJamesW.Truran,Cambridge,2000,
Cambridge Univ. Press.
4. A. M. Khokhlov, Astron. Astrophys. 245 (1991) 114.
5. S. E. Woosley and T. A. Weaver, in Supernovae, Les Houches Session LIV,edited
by J. Audouze, S. Bludman, R. Mochovitch and J. Zinn-Justin, page 63, Amsterdam,
1994, Elsevier.
6. J. C. Niemeyer, M. Reinecke, C. Travaglio and W. Hillebrandt, Small Steps Toward
Realistic Explosion Models of Type Ia Supernovae, in From Twilight to Highlight:
The Physics of Supernovae, page 151, 2003.
7. V. N. Gamezo, A. M. Khokhlov, E. S. Oran, A. Y. Chtchelkanova and R. O. Rosenberg,
Science 299 (2003) 77.
8. W. Hillebrandt, M. Reinecke, W. Schmidt, F.K. R¨opke, C. Travaglio and J.C.
Niemeyer, Simulations of turbulent thermonuclear burning in type Ia supernovae,
in Analysis and Numerics of Conservation Laws,editedbyG.Warnecke,Springer,
2004.
9. F. K. R¨opke and W. Hillebrandt, Astron. Astrophys. 420 (2004) L1.
10. V. N. Gamezo, A. M. Khokhlov and E. S. Oran, Physical Review Letters 92 (2004)
211102.
11. I. Golombek and J. C. Niemeyer, (2004), in preparation.
12. J. C. Niemeyer, Astrophys. J. Lett. 523 (1999) L57.
13. F. X. Timmes and S. E. Woosley, Astrophys. J. 396 (1992) 649.
14. J. C. Niemeyer and S. E. Woosley, Astrophys. J. 475 (1997) 740.
15. M. Reinecke, W. Hillebrandt, J. C. Niemeyer, R. Klein and A. Gr¨obl, Astron. Astro-
phys. 347 (1999) 724.
16. J. C. Niemeyer and W. Hillebrandt, Astrophys. J. 452 (1995) 769.
17. W. Schmidt, , J. C. Niemeyer and W. Hillebrandt, (2004), in preparation.
18. D. Branch, Type Ia supernovae: Spectroscopic surprises, in 3-D Signatures of Stellar
Explosions, 2003.
J.C. Niemeyer et al. / Nuclear Physics A 758 (2005) 431c–438 c438c