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

We show that some statistical properties of forced two-dimensional turbulence have an important sensitivity to the form of large-scale dissipation which is required to damp the inverse cascade. We consider three models of large-scale dissipation: linear "Ekman" drag, non-linear quadratic drag, and scale selective hypo-drag that damps only low-wavenumber modes. In all cases, the statistically steady vorticity field is dominated by almost axisymmetric vortices, and the probabil...

Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solution in powers of 1/N in a proper thermodynamic limit $N\to +\infty$, and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order $O(1/N)$. This equation was obtained previously [P.H. ...

It has been observed empirically that two dimensional vortices tend to cluster forming a giant vortex. To account for this observation Onsager introduced a concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter we will show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium, but becomes trapped in a non-equilibrium stationary state. We will show that the vortex distribution in...

In this paper, we introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a 2D turbulent decaying fluid. The number of vortices decreases as $N\sim t^{-\xi}$, with $\xi\approx 1$ instead of the value $\xi=4/3$ predicted by a na\"{\i}ve kinetic theory. For short time, we find an effective exponent $\xi\approx 0.7$ consistent with previous simulations and experiments. We show that the mean square displaceme...

We present a statistical analysis of the two-point vorticity probability density of the vorticity field generated in the inverse cascade of stationary two-dimensional turbulence.

The dynamics of two dimensional (2D) vortex fluctuations are investigated through simulations of the 2D Coulomb gas model in which vortices are represented by soft disks with logarithmic interactions. The simulations trongly support a recent suggestion that 2D vortex fluctuations obey an intrinsic anomalous dynamics manifested in a long range 1/t-tail in the vortex correlations. A new non-linear IV-exponent a, which is different from the commonly used AHNS exponent, a_AHNS an...

An inverse turbulent cascade in a periodic square box produces a coherent system-sized vortex dipole. We study the statistics of its motion by carrying out direct numerical simulations performed for various bottom friction $\alpha$, pumping intensity $\varepsilon$, and fluid hyperviscosity $\nu$. In the main approximation, coherent vortices can be considered as point vortices, and within this model, they drift at the same dipole velocity, which is determined by their circulat...

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density $n\ll 1$. When the more efficient three-body vortex mergings are considered {(involving vortices of different signs)}, we show tha...

Starting from the Liouville equation, and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order 1/N where N>>1 is the number of point vortices in the system (we assume that their individual circulation scales like \gamma ~ 1/N). It gives the first correction, due to graininess and correlation effects, to the...

The velocity fluctuations for point vortex models are studied for the {\alpha}-turbulence equations, which are characterized by a fractional Laplacian relation between active scalar and the streamfunction. In particular, we focus on the local dynamics regime. The local dynamics differ from the well-studied case of 2D turbulence as it allows to consider the true thermodynamic limit. This limit is not defined for 2D turbulence. We show an analytical form of the probability dens...