Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection

Abstract

We investigate the utility of the convex hull of many Lagrangian tracers to analyze transport properties of turbulent flows with different anisotropy. In direct numerical simulations of statistically homogeneous and stationary Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD Boussinesq convection a comparison with Lagrangian pair dispersion shows that convex hull statistics capture the asymptotic dispersive behavior of a large group of passive tracer particles. Moreover, convex hull analysis provides additional information on the sub-ensemble of tracers that on average disperse most efficiently in the form of extreme value statistics and flow anisotropy via the geometric properties of the convex hulls. We use the convex hull surface geometry to examine the anisotropy that occurs in turbulent convection. Applying extreme value theory, we show that the maximal square extensions of convex hull vertices are well described by a classic extreme value distribution, the Gumbel distribution. During turbulent convection, intermittent convective plumes grow and accelerate the dispersion of Lagrangian tracers. Convex hull analysis yields information that supplements standard Lagrangian analysis of coherent turbulent structures and their influence on the global statistics of the flow.

A Model of Rotating Convection in Stellar and Planetary Interiors. I. Convective Penetration

“…This length scale may be interpreted as depth to which the strongest downflows travel before stopping. This behavior is at least roughly consistent with the model proposed here, and with the results of Pratt et al (2017b) and Pratt et al (2017a), although it is in contrast to the models proposed in Freytag et al (1996) and Herwig (2000). For the time being, the examinations in Julien et al (1996) and Brummell et al (2002) will have to suffice as evidence that the over-shooting is reduced with rotation rate, while the shape of the mixing region likely depend upon the model setup and equations solved in similar simulations.…”

“…A frequent assumption made of turbulent flows is to assume that the underlying distribution of a flow quantity is normal. However, it is shown through the simulations and analysis of Pratt et al (2017b) and Pratt et al (2017a) that the more rare events occurring in the tails of the actual distribution function, e.g. the extremal penetrative flows with a higher velocity and a greater entropy deficit than the average flow, have a larger impact than is expected in Gaussian turbulence models on the turbulent mixing coefficients.…”

“…Such a parameterization of mixing processes has been extensively examined (e.g., Freytag et al 1996;Herwig et al 1999;Denissenkov et al 2013;Viallet et al 2015;Lecoanet et al 2016). A complementary model has been established through an extreme-value statistical analysis of 3D penetrative convection simulations (Pratt et al 2017b). A frequent assumption made of turbulent flows is to assume that the underlying distribution of a flow quantity is normal.…”

Lagrangian Statistics for Dispersion in Magnetohydrodynamic Turbulence

“…Lagrangian statistics are known to be sensitive to extreme events in the fluctuating turbulent fields, for example, Boffetta and Sokolov (2002) and Yeung and Borgas (2004). In Pratt et al (2017), we developed an Figure 6. Energy spectra for the simulations in Table 2: (a) kinetic energy spectra and (b) magnetic energy spectra.…”

“…Here we revisit some of the fundamental results of those earlier studies, using new simulations at higher Reynolds number. We also expand on progress that has been made more recently to use Lagrangian statistics for anisotropic turbulence driven by convection and magnetoconvection (Pratt et al, 2017). We discuss further directions for future work.…”