Profile of coherent vortices in two-dimensional turbulence

Abstract

We examine the velocity profile of coherent vortices appearing as a consequence of the inverse cascade of two-dimensional turbulence in a finite box in the case of static pumping. We demonstrate that in the passive regime the flat velocity profile is realized, as in the case of pumping short correlated in time. However, in the static case the energy flux to large scales is dependent on the system parameters. We demonstrate that it is proportional to f 4/3 where f is the characteristic force exciting turbulence.Turbulence is a chaotic state of fluid motions realized at large Reynolds numbers Re, see, e.g., [1]. In a number of cases turbulence appears to be effectively two-dimensional [2]. Already first theoretical works [3][4][5] devoted to two-dimensional turbulence reveal its principal difference from three-dimensional one. The difference is related to existence of two quadratic quantities (energy and enstrophy) conserved by two-dimensional Euler equation. That leads to two different cascades produced by the non-linear interaction: enstrophy is carried from the pumping scale to smaller scales (direct cascade) whereas energy is carried to larger scales (inverse cascade). The enstrophy is dissipated due to viscosity at scales smaller than the pumping length and the energy is dissipated due to bottom friction at scales larger than the pumping length.Statistical properties of the velocity fluctuations in the inverse cascade were investigated both experimentally [6] and numerically [7]. Results of the works are in a good agreement with the analytical theory developed for an unbound system [8]. Particularly, the normal Kolmogorov scaling is observed in the inverse cascade [2]. The normal scaling in the inverse cascade of twodimensional turbulence is in contrast with the anomalous scaling observed in three-dimensional turbulence [9]. Some theoretical arguments in favor of the normal scaling in the inverse cascade were presented in the work [10], where the feature was related to a leading role of the converging Lagrangian trajectories in the inverse cascade.In an unbound two-dimensional system the inverse cascade is terminated at a scale L α , which is determined by a balance between the energy flux ǫ (per unit mass) toward large scales and the bottom friction. Using the Kolmogorov estimate (ǫL α ) 1/3 for the velocity at the scale L α , one obtains from the balance L α = ǫ 1/2 α −3/2 , where α is the bottom friction coefficient. If the box size L is smaller than L α , then the energy carried by the inverse cascade to scales of the order of the box size L starts to accumulate there. The scenario is realized if the pumping scale l is much less than the box size L, l ≪ L, otherwise there is no space for the inverse cascade carrying the energy to the scale L.The accumulation of the energy at the scale L leads to appearing an intensive large-scale motion including big coherent vortices. The tendency towards the formation of the vortices was indicated already in the first works devoted to two-dimensional turbulence, both exp...

Dynamical symmetry approach to path integrals of quantum spin systems

“…This representation was found in [12][13][14][15][16][17] and more recently in [10,11]. The measure of integration changes correspondingly,…”

“…We call the procedure of going from the time-ordered exponential to the product of the ordinary exponentials a disentanglement transformation. Similar philosophy has been recently used in [10][11][12][13][14][15][16][17][18][19][20].…”

Structure of coherent vortices generated by the inverse cascade of two-dimensional turbulence in a finite box

“…The structure of the coherent vortices in the case of zero bottom friction, α = 0, was established in [19]. In this case just the viscosity determines the structure.…”

“…Here we utilized the equation (3), V 0 ∇Ω 0 = 0, the incompressibility condition and the boundary conditions. The relation (19) is correct both for the periodic setup and for the case of a box with zero velocity at the boundaries. It is a manifestation of the existence of a zero mode of the equation (3), that can be obtained by multiplying Ω 0 (or V 0 ) by a constant.…”

Velocity statistics inside coherent vortices generated by the inverse cascade of 2-D turbulence

“…then the expressions are strongly suppressed due to dissipation. It follows from the expressions (21,23) that the averages v 2 and u 2 are determined by the infrared integrals. Therefore…”