On the statistics of coherent vortices in incoherent environments

Abstract

Coherent vortices are important building blocks of complex flows on all scales, governing dynamics in various applications from geophysics over engineering to the coherent-structures’ paradigm of turbulence. As a commonly studied prototype of this configuration, we consider a large-scale, line vortex evolving in a background flow of small-scale, incoherent turbulence, which, nevertheless, naturally occurs in trailing vortices and tornadoes. The main unsteadiness of the large-scale vortex in this configuration is called meandering. Despite a common observation in experiments since the 1970s and a multitude of studies, its origin and physical mechanism remain puzzling as of this writing. Nevertheless, we do have considerable experimental evidence that, in the range of typically assumed parameters, vortex meandering shares four universal characteristics; namely, (i) fluctuations of the vortex-centre position obey a Gaussian distribution with (ii) monotonously growing standard deviation downstream. Besides, (iii) the fluctuation kinetic energy and enstrophy increase downstream, whereas the main contribution comes from a dipolar vorticity fluctuation pattern confined to the vortex core. This corresponds to the (iv) typical spectral signature of power spectra spanning all resolved scales, whereas the variance level increases monotonously towards the low frequencies. We present the first theoretical model to explain all four experimental cornerstones of vortex meandering in one theory. Starting from the definition of the vortex centre, we first show that the meandering motion is linearly proportional to the leading expansion coefficients of a Karhunen–Loève decomposition of the vorticity field. This has been conjectured before but has never been shown. The configuration strongly suggests scale separation between the response times of the vortex and the surrounding turbulence. On this assumption, we derive a Langevin equation for the slow, large scales driven by the fast, small scales, represented as a stochastic forcing. The particular phenomenon of vortex meandering therewith is found to belong to the large class of Gauss–Markov processes. That is, vortex meandering is an (abstract) Brownian motion. From this understanding, we infer immediately that the Gaussian statistics are a consequence of the central limit theorem and that the growing standard deviation follows from a competition between external forcing and intrinsic resistance. In the equilibrium limit, we derive the spectral signature of the large scales as a power law of the frequency, response time and forcing variance. Eventually, comparing our theoretical model with an experimental database gathered at the ONERA, we find good overall agreement.