Intermittency and non-universality of pair dispersion in isothermal compressible turbulence

Statistical properties of the pair dispersion of Lagrangian particles (tracers) in incompressible turbulent flows provide insights into transport and mixing. We explore the same in transonic to supersonic compressible turbulence of an isothermal ideal gas in two dimensions, driven by large-scale solenoidal and irrotational stirring forces, via direct numerical simulations. We find that the scaling exponents of the order-$p$ negative moments of the distribution of exit times – in particular, the doubling and halving times of pair separations – are nonlinear functions of $p$. Furthermore, the doubling and halving time statistics are different. The halving-time exponents are universal – they satisfy their multifractal model-based prediction, irrespective of the nature of the stirring. However, the doubling-time exponents are not. In the solenoidally-stirred flows, the doubling time exponents can be expressed solely in terms of the multifractal scaling exponents obtained from the structure functions of the solenoidal component of the velocity. Moreover, they depend strongly on the Mach number, Ma, as elongated patches of high vorticity emerge along shock fronts at high Ma. In contrast, in the irrotationally-stirred flows, the doubling-time exponents do not satisfy any known multifractal model-based relation, and are independent of Ma. Our findings are of potential relevance to astrophysical disks and molecular clouds wherein turbulent transport and mixing of gases often govern chemical kinetics and the rates of formation of stars and planetesimals.

💡 Research Summary

This paper investigates the statistics of pair dispersion for Lagrangian tracer particles in two‑dimensional, isothermal, compressible turbulence. The authors perform direct‑numerical simulations (DNS) of the Navier–Stokes equations for an ideal gas, forcing the flow at large scales either solenoidally (divergence‑free) or irrotationally (curl‑free). Four simulation sets are considered: solenoidal forcing at transonic (Ma≈1) and supersonic (Ma>1) Mach numbers (runs S1, S2) and irrotational forcing at the same Mach regimes (runs C1, C2). Each run uses a 4096² grid and an equal number of Lagrangian particles, allowing the authors to track the separation R(t) of particle pairs over time.

Because the traditional mean‑square separation ⟨R²(t)⟩ exhibits only a very limited inertial‑range scaling, the authors adopt an “exit‑time” approach. For a given reference separation R they define a doubling time τ_D(R) (the first instant when the pair distance exceeds 3R/2) and a halving time τ_H(R) (the first instant when the distance falls below 3R/4). They then study the negative moments ⟨τ_D^{-p}⟩∝R^{-χ_D^p} and ⟨τ_H^{-p}⟩∝R^{-χ_H^p} for orders p=1…6. In the multifractal framework of turbulence, one expects a bridge relation χ_{D,H}^p = p – ζ^p, where ζ^p are the scaling exponents of the p‑th order Eulerian velocity structure functions.

The key findings are:

1. Halving‑time exponents (χ_H^p) are universal. In all four runs χ_H^p follows the multifractal bridge relation p – ζ^p within statistical error, and shows no dependence on the forcing type or Mach number. This indicates that the contraction of particle separations is governed by one‑dimensional shock structures (∇·u < 0), which are present irrespective of the flow’s solenoidal or compressive content.

2. Doubling‑time exponents (χ_D^p) are non‑universal. For solenoidal forcing (S1, S2) the authors find χ_D^p = p – ζ_s^p, where ζ_s^p are the scaling exponents of the structure functions of the solenoidal velocity component u_s. Moreover, χ_D^p varies systematically with Mach number: higher Ma leads to stronger intermittency (more pronounced curvature of ζ_s^p) and thus larger deviations from the incompressible prediction. This suggests that the expansion of particle separations is dominated by solenoidal motions, particularly elongated high‑vorticity patches that appear near shock fronts at high Mach numbers.

   In contrast, for irrotational forcing (C1, C2) χ_D^p does not satisfy any known bridge relation and shows no systematic dependence on Mach number. The growth of separations in these runs appears to be influenced by regions of positive divergence (∇·u > 0) and by the compressive component of the velocity field, indicating a more complex interplay that is not captured by the standard multifractal model.

3. Multifractal spectra from dynamic exponents. Using Legendre transforms, the authors convert χ_H^p and χ_D^p into singularity spectra D_H(h) and D_D(h). D_H(h) collapses onto the usual multifractal spectrum and approaches unity as h→0 for all runs, confirming that shocks (one‑dimensional structures) dominate halving events. D_D(h) behaves differently: for solenoidal forcing at high Mach number it also approaches unity at h≈0, reflecting the emergence of quasi‑one‑dimensional vortex sheets; at lower Mach number D_D(h) remains above one, indicating more space‑filling structures. For irrotational forcing D_D(h) stays close to one across the range, consistent with compressive structures being the primary agents of separation growth.

4. Implications for turbulent mixing. The coexistence of two distinct exit‑time statistics, each with its own intermittency properties, shows that Richardson’s classical picture of pair dispersion (a single effective diffusivity) breaks down in compressible turbulence. The non‑universality of χ_D^p implies that mixing rates in astrophysical gases (e.g., molecular clouds, protoplanetary disks) cannot be described by a single scaling law; they depend on the nature of the forcing and on the Mach number.

In summary, the paper demonstrates that compressible turbulence exhibits dynamic multiscaling of particle pair dispersion, with halving times obeying universal multifractal predictions while doubling times display forcing‑ and Mach‑dependent deviations. These results highlight the need for theoretical frameworks that go beyond the conventional multifractal model to capture the combined effects of shocks, compressive motions, and solenoidal vortical structures on turbulent transport. The authors suggest that similar phenomena are likely to persist in three‑dimensional flows, with important consequences for astrophysical mixing and star/planetesimal formation processes.