[Paper Review] Intermittent distribution of tracers advected by a compressible random flow

Jérémie Bec, Krzysztof Gawędzki+1 more|ArXiv.org|Oct 13, 2003

Fluid Dynamics and Turbulent Flows1 references3 citations

TL;DR

This paper establishes a quantitative link between the multifractal scaling of tracer density in a compressible random flow and the large deviations of finite-time Lyapunov exponents (stretching rates). Using the compressible Kraichnan model, it derives analytical expressions for the scaling exponents ξₙ of mass moments, showing that intermittency arises from rare, strongly stretching flow realizations, with numerical simulations confirming the predicted scaling and saturation at critical order n_cr.

ABSTRACT

Multifractal properties of a tracer density passively advected by a compressible random velocity field are characterized. A relationship is established between the statistical properties of mass on the dynamical fractal attractor towards which the trajectories converge and large deviations of the stretching rates of the flow. In the framework of the compressible Kraichnan model, this result is illustrated by analytical calculations and confirmed by numerical simulations.

Motivation & Objective

• To characterize the multifractal properties of tracer density in a compressible random flow.
• To relate the scaling exponents ξₙ of tracer mass moments to the large deviations of flow stretching rates (finite-time Lyapunov exponents).
• To establish a connection between the annealed Hentschel-Procaccia spectrum HP_an(n) and the statistical properties of the dynamical attractor.
• To validate the theoretical predictions through numerical simulations of the Kraichnan model with long-time averaging.

Proposed method

• Uses the compressible Kraichnan model with Gaussian, white-in-time velocity fields to model passive tracer advection.
• Defines the tracer density ρ* as the invariant measure on the dynamical attractor toward which Lagrangian trajectories converge.
• Introduces the local mass m_r(x) and its scaling exponent h_r(x) to characterize small-scale mass distribution.
• Derives the annealed Hentschel-Procaccia spectrum HP_an(n) via the scaling ⟨m_r^n⟩ ∼ r^ξₙ, with ξₙ related to the Legendre transform of the rate function S(h).
• Relates ξₙ to the large deviations of stretching rates σ_i via the generating function of the joint distribution of eigenvalues of the deformation tensor.
• Performs numerical simulations with up to 10⁵ tracers and 10⁸ turnover times to compute moments and verify scaling exponents.

Experimental results

Research questions

• RQ1 How do the scaling exponents ξₙ of tracer mass moments relate to the large deviations of finite-time Lyapunov exponents in compressible flows?
• RQ2 What determines the critical order n_cr beyond which ξₙ saturates, and how does it depend on compressibility?
• RQ3 Can the multifractal spectrum f(h) of the tracer measure be derived from the statistics of stretching rates?
• RQ4 How do the moments ⟨m_r^n⟩ scale with r, and what does this imply for the intermittency and non-Gaussianity of the tracer distribution?

Key findings

• The scaling exponents ξₙ for the mass moments of the tracer distribution are determined by the large deviations of the stretching rates of the flow, with a precise analytical relation derived.
• For the compressible Kraichnan model, the exponents ξₙ exhibit a non-linear dependence on n, indicating strong intermittency and non-Gaussian tails in the distribution of m_r.
• The critical order n_cr, beyond which ξₙ saturates, is determined by the integrability of the zero-mode solution of the Fokker-Planck operator in the collinear limit, with n_cr = 1/(6℘) for ℘ < 1/6 and n_cr = 1/(6(1−℘)) for ℘ > 1/6.
• Numerical simulations with up to 10⁸ turnover times confirm the theoretical predictions, showing good agreement with the analytical scaling exponents and saturation at n_cr.
• For integer n, the moment ⟨m_r^n⟩ is linked to the stationary correlation function F_{n+1} of the tracer density, which is a zero mode of a second-order differential operator M_{n+1}^†, and its homogeneity degree gives ξₙ.
• The quenched and annealed Hentschel-Procaccia spectra differ, with HP_qu(n) ≥ HP_an(n), and the annealed spectrum is fully determined by the scaling ξₙ.