Turbulence in a three-dimensional deflagration model for Type Ia supernovae: I. Scaling properties

📝 Abstract

We analyze the statistical properties of the turbulent velocity field in the deflagration model for Type Ia supernovae. In particular, we consider the question of whether turbulence is isotropic and consistent with the Kolmogorov theory at small length scales. Using numerical data from a high-resolution simulation of a thermonuclear supernova explosion, spectra of the turbulence energy and velocity structure functions are computed. We show that the turbulent velocity field is isotropic at small length scales and follows a scaling law that is consistent with the Kolmogorov theory until most of the nuclear fuel is burned. At length scales greater than a certain characteristic scale, turbulence becomes anisotropic. Here, the radial velocity fluctuations follow the scaling law of the Rayleigh-Taylor instability, whereas the angular component still obeys Kolmogorov scaling. In the late phase of the explosion, this characteristic scale drops below the numerical resolution of the simulation. The analysis confirms that a subgrid-scale model for the unresolved turbulence energy is required for the consistent calculation of the flame speed in deflagration models of Type Ia supernovae, and that the assumption of isotropy on these scales is appropriate.

💡 Analysis

We analyze the statistical properties of the turbulent velocity field in the deflagration model for Type Ia supernovae. In particular, we consider the question of whether turbulence is isotropic and consistent with the Kolmogorov theory at small length scales. Using numerical data from a high-resolution simulation of a thermonuclear supernova explosion, spectra of the turbulence energy and velocity structure functions are computed. We show that the turbulent velocity field is isotropic at small length scales and follows a scaling law that is consistent with the Kolmogorov theory until most of the nuclear fuel is burned. At length scales greater than a certain characteristic scale, turbulence becomes anisotropic. Here, the radial velocity fluctuations follow the scaling law of the Rayleigh-Taylor instability, whereas the angular component still obeys Kolmogorov scaling. In the late phase of the explosion, this characteristic scale drops below the numerical resolution of the simulation. The analysis confirms that a subgrid-scale model for the unresolved turbulence energy is required for the consistent calculation of the flame speed in deflagration models of Type Ia supernovae, and that the assumption of isotropy on these scales is appropriate.

📄 Content

DRAFT VERSION JULY 30, 2021 Preprint typeset using LATEX style emulateapj v. 03/07/07 TURBULENCE IN A THREE-DIMENSIONAL DEFLAGRATION MODEL FOR TYPE IA SUPERNOVAE: I. SCALING PROPERTIES F. CIARALDI-SCHOOLMANN, W. SCHMIDT, J. C. NIEMEYER, Lehrstuhl für Astronomie und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany F. K. RÖPKE AND W. HILLEBRANDT Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany Draft version July 30, 2021 ABSTRACT We analyze the statistical properties of the turbulent velocity field in the deflagration model for Type Ia supernovae. In particular, we consider the question of whether turbulence is isotropic and consistent with the Kolmogorov theory at small length scales. Using numerical data from a high-resolution simulation of a thermonuclear supernova explosion (Röpke et al., 2007), spectra of the turbulence energy and velocity struc- ture functions are computed. We show that the turbulent velocity field is isotropic at small length scales and follows a scaling law that is consistent with the Kolmogorov theory until most of the nuclear fuel is burned. At length scales greater than a certain characteristic scale that agrees with the prediction of Niemeyer and Woosley (1997), turbulence becomes anisotropic. Here, the radial velocity fluctuations follow the scaling law of the Rayleigh-Taylor instability, whereas the angular component still obeys Kolmogorov scaling. In the late phase of the explosion, this characteristic scale drops below the numerical resolution of the simulation. The analysis confirms that a subgrid-scale model for the unresolved turbulence energy is required for the consistent calculation of the flame speed in deflagration models of Type Ia supernovae, and that the assumption of isotropy on these scales is appropriate. Subject headings: hydrodynamics — instabilities — methods: statistical — turbulence — Supernovae: general

1. INTRODUCTION The mechanism of thermonuclear explosions of white dwarf (WD) stars, giving rise to Type Ia supernovae (SNe Ia) is still not understood in full detail. For three reasons, a key to this problem is the understanding of turbulent thermonuclear combustion in the deflagration phase—the phase of subsonic flame propagation which commences the explosion process. First, it is required for the correct modeling of the flame prop- agation in the deflagration model of SNe Ia (see Hillebrandt & Niemeyer 2000, for a review of SN Ia explosion scenar- ios). Second, in alternative models, burning starts out in the deflagration mode, and this is an essential ingredient to the overall explosion process. Third, turbulence in the deflagra- tion phase sets the conditions for a possible deflagration-to- detonation transition (DDT) in the delayed detonation sce- nario (Röpke 2007; Woosley 2007). The necessary insight into the details of the turbulent combustion process, however, is hampered by the fact that full-star simulations of thermonu- clear supernova explosions cannot resolve the structure of the deflagration flame. At the large scales accessible to simula- tions, the flame propagation is determined by flame instabil- ities and turbulence. These effects significantly boost the ef- fective burning speed and, in the pure deflagration model of SNe Ia, lead to the flame acceleration required to explode the WD (Reinecke et al. 2002; Gamezo et al. 2003). In order to describe the flame propagation in such simulations, the in- teraction of the flame with turbulent velocity fluctuations has to be modeled. These modeling approaches yield an effective flame propagation speed on the numerically resolved scales— the so-called turbulent burning speed. The problem of calculating this turbulent flame propagation speed in three-dimensional simulations of thermonuclear su- pernova explosions1 has been the subject of a lively debate. One school of thought holds the view that the effective prop- agation speed in the flamelet regime would naturally be given by the velocity scale vRT(ℓ) ∝(geffℓ)1/2 associated with the Rayleigh-Taylor (RT) instability induced by buoyancy in the gravitational field geff for any length scale ℓ(Sharp 1984). As a subgrid scale model (ℓ= ∆, where ∆is the numerical cutoff length), this scaling relation is easily implemented and ap- pears to be motivated by the basic physics of thermonuclear combustion in Type Ia supernovae (Gamezo et al. 2003). In opposition to this view, Niemeyer & Hillebrandt (1995) and Niemeyer & Kerstein (1997) argued that inevitably turbulent velocity fluctuations v′(ℓ) are dominated by the turbulent cas- cade at length scales ℓsmall compared to the scale of en- ergy injection by the RT instability and, hence, should follow the Kolmogorov scaling v′(ℓ) ∝ℓ1/3. Niemeyer & Woosley (1997) estimated the transition length ℓK/RT between the RT- dominated length scales (the “large scales” ℓ≳ℓK/RT) and the regime of the turbulent cascade (the “small scales” ℓ≲ℓK/RT) to be of the order 10km. Since the cutoff

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