Emergence of skewed non-Gaussian distributions of velocity increments in isotropic turbulence

Abstract

Skewness and non-Gaussian behavior are essential features of the distribution of short-scale velocity increments in isotropic turbulent flows. Yet, although the skewness has been generally linked to time-reversal symmetry breaking and vortex stretching, the form of the asymmetric heavy tails remain elusive. Here we describe the emergence of both properties through an exactly solvable stochastic model with a scale hierarchy of energy transfer rates. From a statistical superposition of a local equilibrium distribution weighted by a background density, the increments distribution is given by a novel class of skewed heavy-tailed distributions, written as a generalization of the Meijer G-functions. Excellent agreement in the multiscale scenario is found with numerical data of systems with different sizes and Reynolds numbers. Remarkably, the single scale limit provides poor fits to the background density, highlighting the central role of the multiscale mechanism. Our framework can be also applied to describe the challenging emergence of skewed distributions in complex systems. arXiv:1812.03199v2 [physics.flu-dyn]

Explicit construction of joint multipoint statistics in complex systems

“…Future work should address the inclusion of skewness of the velocity increment distribution into the framework which would thus make it more truthful to the original turbulence problem (i.e., the 4/5-law derived from the Navier-Stokes equation). This can be done by the introduction of a mean in the Gaussian ansatz (4), which has already been discussed in the context of superstatistics [49,83]. Moreover, dissipative effects might by incorporated by embedding the Ornstein-Uhlenbeck process which effectively leads to the differentiability of the velocity field [84,85].…”

Turbulence Signatures in High‐Latitude Ionospheric Scintillation

“…This has led fluid dynamicists, e.g., to study the probability distribution of the velocity increment U ( r + δ ) − U ( r ) instead of solving the Navier‐Stokes equation (Benzi et al., 1993), a nonlinear partial differential equation with no closed form solutions known in the case of turbulence; the probability distribution is then found to be skewed. Skewness would vanish only if there were invariance under time reversal, but for a turbulent dissipative flow this is not the case (for a mathematical proof, see Lawrance, 1991; Sosa‐Correa et al., 2019). This result has profound consequences since one is often led to believe that the probability distributions ought to be normal or lognormal, which would imply a zero skewness, a result that is inconsistent with observations.…”

Turbulence signatures in high--latitude ionospheric scintillation

“…This has led fluid dynamicists, for example, to study the probability distribution of the velocity increment U (r + δ) − U (r) instead of solving the Navier-Stokes equation (Benzi et al, 1993), a nonlinear partial differential equation with no closed form solutions known in the case of turbulence; the probability distribution is then found to be skewed. Skewness would vanish only if there were invariance under time reversal, but for a turbulent dissipative flow this is not the case (for a mathematical proof, see (Lawrance, 1991;Sosa-Correa et al, 2019)). This result has profound consequences since one is often led to believe that the probability distributions ought to be normal or log-normal, which would imply a zero skewness, a result that is inconsistent with observations.…”