On the effective velocity created by a point vortex in two-dimensional hydrodynamics

We present statistics of velocity fluctuations in both the Lagrangian and Eulerian frame for weakly driven two-dimensional turbulence. We find that simultaneous inverse energy and enstrophy ranges present in the Lagrangian and Eulerian Fourier spectra are not directly echoed in real-space moments of velocity difference. The spectral ranges, however, do line up very well with ratios of the real-space moments {\em local} exponents, indicating that though the real-space moments ...

We present results on the connection between the vorticity equation and the shape of the single-point vorticity PDF. The statistical framework for these observations is cast in form of conditional averages. The numerical evaluation of these conditional averages provides insights into the intimate relation of dynamical effects like vortex stretching and vorticity diffusion and non-Gaussian vorticity statistics.

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator technics, we derive a Fokker-Planck equation describing the relaxation of a ``test'' vortex in a bath of ``field'' vortices at statistical equilibrium. The relaxation is due to the combined effect of a diffusion and a drift. The drift is shown to be responsible for the organization of point vortices at negative temperatures. A description that goes beyond the th...

It is well known that an inverse turbulent cascade in a finite ($2 \pi \times 2 \pi$) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length $l_f = 2\pi/k_f$ with $k_f$ ranging from $100$ to $12.5$. Previously, it was found that in the asy...

We study the 2D Navier--Stokes solution starting from an initial vorticity mildly concentrated near $N$ distinct points in the plane. We prove quantitative estimates on the propagation of concentration near a system of interacting point vortices introduced by Helmholtz and Kirchhoff. Our work extends the previous results in the literature in three ways: The initial vorticity is concentrated in a weak (Wasserstein) sense, it is merely $L^p$ integrable for some $p>2$, and the e...

Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3-D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a 2D point-vortex flow coupled to localized 3D perturbations (ergophages), such that ergophages can gain energy by altering vortex-vortex distances through an induced diverg...

Cellular suspensions such as dense bacterial flows exhibit a turbulence-like phase under certain conditions. We study this phenomenon of "active turbulence" statistically by using numerical tools. Following Wensink et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 14308 (2012)], we model active turbulence by means of a generalized Navier-Stokes equation. Two-point velocity statistics of active turbulence, both in the Eulerian and the Lagrangian frame, is explored. We characterize th...

We present a status report on a discrete approach to the the near-equilibrium statistical theory of three-dimensional turbulence, which generalizes earlier work by no longer requiring that the vorticity field be a union of discrete vortex filaments. The idea is to take a special limit of a dense lattice vortex system, in a way that brings out a connection between turbulence and critical phenomena. The approach produces statistics with basic features of turbulence, in particul...

Numerical and physical experiments on two-dimensional (2d) turbulence show that the differences of transverse components of velocity field are well described by a gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume existence of critical space-dimensionality $d=d_{c}$, at which the energy flux and all odd-order moments of velocity difference change sign and th...

The motion of assemblies of point vortices in a periodic parallelogram can be described by the complex position $z_j(t)$ whose time derivative is given by the sum of the complex velocities induced by other vortices and the solid rotation centered at $z_j$. A numerical simulation up to 100 vortices in a square periodic box is performed with various initial conditions, including single and double rows, uniform spacing, checkered pattern, and complete spatial randomness. Point p...