A mass invariant in a compressible turbulent medium

The gist Predicting measurable statistical properties of density fluctuations in supersonic compressible turbulent flow involves confirming a mass invariant, Minv, which remains constant during the decaying phase of turbulence and can be used to relate non-Gaussian evolution to the variance of the logdensity field without free parameters.

The Invariant Minv

Minv is defined by Minv = E(ρ)Var ρ E(ρ) (l ρ c) 3 where ρ is the density field and l ρ c is the correlation length This invariant was derived by Pierre Dumond, Jaupart & Chabrier (2021) extended this invariant to non-isotropic flows in a time-evolving background and showed that it has the dimension of a mass The basic mathematical expressions of the invariant derived by [10] and by [11] are the same In Sec. III we present the results of our numerical test of the time invariance of Minv in a steady state background for two different equation of states We confirm that Minv remains constant during the decaying phase of turbulence The invariant is independent of the Mach number, we show that this invariant can be used to relate the non-gaussian evolution of the logdensity probability distribution function to its variance with no free parameters

Numerical Setup and Verification

The numerical setup uses the hydrodynamical code RAMSES [13] without adaptive mesh refinement (fixed cartersian grid) The turbulence is forced using the Ornstein-Uhlenbeck forcing on the acceleration [15–17] We carried out simulations at different large scale Mach numbers: M = 2.5, 3.5, 5, 7 and 10 to test the invariant under different conditions To test the temporal invariance of Minv suggested by Ref. [10] and Ref. [11], we first drive the turbulence for at least 2 turbulent crossing times Tcross before stopping the driving We study only a subsystem of the whole simulation domain, choosing a sub-box with a volume of 1/8 of the whole box In this sub-box, while the volumic mean density ¯ρ is not constant but exhibits small variations (of the order of 5-10%), the statistical mean density E(ρ) is constant The invariant that is tested is thus the one derived by Ref. [10] for which the statistical mean density E(ρ)(t) does not change with time and the flow is statistically isotropic, in contrast to the more general derivation of Ref. [11]

Modeling Correlation Length Evolution

We now present a model for predicting the evolution of the correlation length with the Mach number Starting with a first order assumption that the logdensity s = ln(ρ/E(ρ)) is a 3D gaussian random field This form reproduces particularly well the one measured in our numerical simulations as shown in Fig. 6 The model only holds when the turbulence is forced and fully developed We now compare the prediction of our model with the measurements of the correlation lengths made in the simulations after reaching the statistically stationary regime and before turbulence starts to decay Some discrepancies (around 1.5 - 2 σ) between the measurements and the model are observed at high Mach, especially for the logdensity correlation length

Modeling Non-Gaussianity Correction

To better predict the evolution of the mass invariant with Mach number, we use a log-Poisson model to describe the logdensity PDF This introduces a Mach-dependent parameter T to reproduce the negative skewness of the density PDF observed in compressible turbulence simulations The expression of the invariant then becomes Minv = E(ρ)L 3 inj π/2Cρ˜(0)× Z ∞ 0 Cρ˜(qLinj)˜q 2d˜q exp σ squared s 1 + 3T + 2T squared − 1 This non-gaussian correction concerns only the relation between the density and the logdensity variance, not the one between the density and logdensity correlation lengths Our model reproduces well at high Mach, while it significantly underestimates the value at low Mach

Mach Number Dependence

The invariant should remain constant for any Mach number in this regime However, by considering the limit M → 0 we can see that this reasoning is limited At very low Mach, the variance of the density field tends to 0, while the correlation length cannot go to infinity because it is limited by the injection length Once we reach the supersonic regime, the cross-correlation becomes negligible The accretion timescale τacc should be greater than the turbulent crossing time Tcross = Linj/p E(v 2) Minv should thus be constant soon after the time such that σ squared s = 1, i.e. in the supersonic regime, as observed in the simulation

Conclusion and Astrophysical Relevance

In this article, we have studied numerically the accuracy of the mass invariant introduced by Ref. [10] for compressible isotropic turbulence and extended by Ref. [11] We show that the predicted time invariance of the quantity Minv is well verified during the decaying phase of isotropic turbulence Even though we use the gaussian random field assumption which is a first order assumption to describe the statistical properties of the supersonic flow, the derived model reproduces well the evolution of the correlation length l ρ c with M Finally, we argue why Minv should not depend on the Mach number from the trans to supersonic regime (M ∼ 7) and show that this invariant can be a powerfull tool to predict the evolution of the logdensity PDF with the Mach number with no free parameters In the astrophysical context, gravity plays a major role in the formation of structures in gravo-turbulent molecular clouds The invariant may represent a fundamental quantity for the formation of structures in molecular clouds, as it would determine the characteristic mass of the most correlated structures induced by turbulence in a gravitational system This is quite a fundamental result for our understanding of compressible turbulence as it provides an exact way to infer the evolution of the main statistical properties of a compressible turbulent flow

Appendix B Estimates

To estimate the correlation length, several other methods have been suggested in the litterature The first one consists in considering the integral length [21] as a proxy of the correlation length To avoid the computation of the ACF of the density field, which can be numerically expensive in high resolution simulations, we can also use the following estimate of the integral length [11, 22] The ratio between the correlation length and the integral length can be as large as 2, which means that the quantity Minv can be over- or underestimated by almost an order of magnitude The consistency of the estimate of the correlation length based on the integral of the ACF until its first zero can also be checked by comparing it with another estimates of the correlation length based on ergodic theory The study of the properties of Minv calculated from the integral length is therefore useless The consistency between our 10243 and 20483 simulations for Mach 3.5 and Mach 10 confirms the robustness of our calculation

Appendix A Convergence

We have increased the resolution of the simulation from 5123 to 20483 for the simulation where turbulence is forced at Lbox/7 Doubling the number of cells from 10243 to 20483 changes the correlation length by less than 5% At high Mach (Mach 7 and 10) the logdensity variance shows a slightly larger variation, until 15% when the resolution is doubled This problem highlights the difficulty of this study, which requires both a very good spatial resolution to properly resolve small scale turbulent structures, and a turbulent box large enough compared to the injection scale of turbulence to have sufficient statistics We also made a comparison in our 10243 simulations between the correlation length of the density field computed from the integration of the ACF to its first zeros and that computed from the ergodic theory At Mach 3.5 we get l l ist0 c /Linj = 0.168 ± 0.027 and l ergo c /Linj = 0.166 At Mach 10 we get l l ist0 c /Linj = 0.126 ± 0.020 and l ergo c /Linj = 0.

1. Bold header: Non-Gaussian PDF Modeling for Density Prediction. The improved AI system can predict the negative skewness of the density PDF observed in compressible turbulence simulations by using a log-Poisson model with a Mach-dependent parameter T, improving upon the Gaussian assumption which is noted to be an inaccurate first-order assumption.

2. Bold header: Invariant-Based Flow Property Prediction. The system can predict the evolution of the logdensity PDF with σs and M and compare it with the existing parameterisations of this relation in the literature using the derived expression for Minv, which is shown to be a powerful tool to predict the evolution of the logdensity PDF with the Mach number with no free parameters.

3. Bold header: Mach Number Regime Classification. The AI can determine whether Minv should be constant during this process by checking if it is in the supersonic regime (M ∼ 7) or subsonic regime, as Minv should therefore remain constant during this process when the condition for invariance is met.

4. Bold header: Correlation Length Estimation via Non-Gaussian Correction. The system can accurately predict the evolution of the correlation length lρc with M by incorporating a non-gaussian correction to better capture the evolution of the density variance σ2ρ˜ with M, as this model better agrees with the results of the simulations.

5. Bold header: Sub-Box Statistical Estimation. The system can perform accurate statistical averaging by using a sub-box size small enough compared with the whole box to be unaffected by the boundary conditions and (ii) large enough compared with the correlation length to resolve it properly, as this ensures that the volumic mean density ¯ρ is not constant but exhibits small variations (of the order of 5-10%), but the statistical mean density E(ρ) is constant.

Predicting the measurable statistical properties of density fluctuations in a supersonic compressible turbulent flow is a major challenge in physics. In 1951, Chandrasekhar derived an invariant under the assumption of the statistical homogeneity and isotropy of the turbulent density field and stationarity of the background density. Recently, Jaupart & Chabrier (2021) extended this invariant to non-isotropic flows in a time-evolving background and showed that it has the dimension of a mass. This invariant M inv is defined by M inv = E(ρ) Var (ρ E(ρ))(l c ρ) cubed where ρ is the density field and l c ρ is the correlation length. In this article, we perform numerical simulations of homogeneous and isotropic compressible turbulence to test the validity of this invariant in a medium subject to isotropic decaying turbulence. We study several input configurations, namely different Mach numbers, injection lengths of turbulence and equations of state. We confirm that M inv remains constant during the decaying phase of turbulence. Furthermore, we develop a theoretical model of the density field statistics which predicts without any free parameter the evolution of the correlation length with the variance of the logdensity field beyond the assumption of the gaussian field for the logdensity. Noting that M inv is independent of the Mach number, we show that this invariant can be used to relate the non-gaussian evolution of the logdensity probability distribution function to its variance with no free parameters.