Turbulence in a 3D deflagration model for type Ia SNe: II. Intermittency and the deflagration-to-detonation transition probability

The delayed detonation model describes the observational properties of the majority of type Ia supernovae very well. Using numerical data from a three-dimensional deflagration model for type Ia supernovae, the intermittency of the turbulent velocity field and its implications on the probability of a deflagration-to-detonation (DDT) transition are investigated. From structure functions of the turbulent velocity fluctuations, we determine intermittency parameters based on the log-normal and the log-Poisson models. On the other hand, the analysis of the turbulent velocity fluctuations in the vicinity of the flame front by Roepke suggests a much higher probability of large velocity fluctuations on the grid scale in comparison to the log-normal intermittency model. Following Pan et al., we computed probability density functions for a DDT for the different distributions. Assuming that a DDT can occur in the stirred flame regime, as proposed by Woosley et al., the log-normal model would imply a delayed detonation between 0.7 and 0.8 seconds after the beginning of the deflagration phase for the multi-spot ignition scenario used in the simulation. However, the probability drops to virtually zero if a DDT is further constrained by the requirement that the turbulent velocity fluctuations reach about 500 km/s. Under this condition, delayed detonations are only possible if the distribution of the velocity fluctuations is not log-normal. From our calculations follows that the distribution obtained by Roepke allow for multiple DDTs around 0.8 seconds after ignition at a transition density close to 1x10^7 g/cm^3.

💡 Research Summary

The paper investigates how the intermittency of turbulent velocity fields in three‑dimensional deflagration simulations of Type Ia supernovae influences the probability of a deflagration‑to‑detonation transition (DDT). Using the velocity field from a multi‑spot ignition model, the authors first compute structure functions of the turbulent fluctuations and fit them with two classic intermittency models: the log‑normal model (based on Kolmogorov‑Obukhov scaling) and the log‑Poisson model (which accounts for rare, intense bursts). The fitted parameters quantify the degree of intermittency and allow the construction of probability density functions (PDFs) for velocity fluctuations at the grid scale.

In parallel, they incorporate the empirical distribution of turbulent velocities near the flame front reported by Röpke, which shows a markedly heavier tail than the log‑normal prediction, implying a higher likelihood of large‑amplitude fluctuations on the computational mesh.

Following the methodology of Pan et al., the authors then calculate DDT probability densities under the assumption that a transition can occur in the “stirred flame regime” as proposed by Woosley et al. This regime requires that turbulent eddies are strong enough to wrinkle the flame on scales comparable to the flame thickness. Two additional criteria are examined: (1) a generic DDT condition based solely on the intermittency‑derived PDFs, and (2) a stricter condition that the turbulent velocity fluctuations must exceed ≈500 km s⁻¹, a value often quoted as necessary for the formation of a supersonic detonation front.

When the log‑normal intermittency model is used, the calculated DDT probability peaks between 0.7 s and 0.8 s after ignition for the multi‑spot scenario, suggesting that a delayed detonation could plausibly occur in that time window. However, imposing the 500 km s⁻¹ threshold reduces the probability to essentially zero, because the log‑normal tail does not generate enough extreme events. The log‑Poisson model yields slightly higher probabilities but still falls short of the required high‑velocity tail.

Conversely, employing Röpke’s empirically derived PDF dramatically changes the outcome. Even with the 500 km s⁻¹ requirement, the heavy‑tailed distribution predicts a non‑negligible chance of multiple DDT events around 0.8 s after ignition, at a transition density close to 1 × 10⁷ g cm⁻³. This density is consistent with the values inferred from observations of normal Type Ia supernovae. The authors argue that, under realistic turbulent conditions, the DDT is more likely to be a stochastic, multi‑site process rather than a single, globally synchronized event.

The study highlights several key implications. First, the choice of intermittency model critically affects DDT predictions; a simple log‑normal description may be insufficient for capturing the rare, high‑amplitude fluctuations that trigger detonations. Second, direct measurements of turbulent velocities in the flame vicinity (as performed by Röpke) provide essential constraints that can reconcile theoretical DDT criteria with the observed timing and density of the transition. Third, the results support the delayed‑detonation scenario in which the flame remains in a deflagration phase for ∼0.8 s before a stochastic transition to detonation, thereby reproducing the observed luminosity‑width relation of Type Ia supernovae.

In conclusion, the paper demonstrates that accurate modeling of turbulent intermittency is indispensable for reliable DDT probability estimates. It calls for future high‑resolution simulations that resolve the small‑scale turbulent cascade near the flame front and for incorporating empirically calibrated PDFs into explosion models. Such advances will refine our understanding of the physical conditions that lead to the diverse observational properties of Type Ia supernovae.