Dynamics of Energy Condensation in Two-Dimensional Turbulence

Abstract

We report a numerical study, supplemented by phenomenological explanations, of "energy condensation" in forced 2D turbulence in a biperiodic box. Condensation is a finite size effect which occurs after the standard inverse cascade reaches the size of the system. It leads to emergence of a coherent vortex dipole. We show that the time growth of the dipole is self-similar, and it contains most of the injected energy, thus resulting in an energy spectrum which is markedly steeper than the standard k −5/3 one. Once the coherent component is subtracted, however, the remaining fluctuations have a spectrum close to k −1 . The fluctuations decay slowly as the coherent part grows.PACS numbers: 47.27.E-,92.60.hk A big difference between 2D and 3D turbulence is the generation of large scale structures from small scale motions [1,2]. This occurs because, if pumped at intermediate scales, the 2D Navier-Stokes equations favor energy transfer to larger scales [3,4,5,6], a phenomenon known as an inverse cascade. Simulations [7,8] and experiments [9,10,11] show that large scale accumulation of energy is observed if conditions permit the energy to reach the system size. In this letter, we study the "condensate" emerging in the form of two coherent vortices in a biperiodic box in 2D. Let us begin by briefly reviewing the classical 2D turbulence theory of Kraichnan, Leith and Batchelor (KLB) [3,4,5]. The essential difference with 3D turbulence is the presence of a second inviscid invariant, in addition to energy, the enstrophy. Stirring the 2D flow leads to emergence of two cascades. Enstrophy cascades from the forcing scale, l, to smaller scales (direct cascade) while energy cascades from the forcing scale to larger scales (inverse cascade). Viscosity dissipates enstrophy at the Kolmogorov scale, η, which is much smaller than l when the Reynolds number is large. The energy cascade is blocked at a scale ζ, ζ ≫ l, by a frictional dissipation (usually due to friction between the fluid and substrate although other mechanisms can be imagined) after a transient in time quasi-stationary regime. Then a stationary KLB turbulence is established [3,4,5]. Applying Kolmogorov phenomenology (see e.g.[1]) KLB predicts an energy spectrum scaling as k −3 in the direct cascade, and as k −5/3 in the inverse cascade. Here k is the modulus of the wave-vector. The KLB spectra imply that velocity fluctuations at a scale r, δv r scale as ǫ 1/3 l −2/3 r and (ǫr) 1/3 in the direct and inverse cascade ranges respectively. KLB theory is confirmed by simulations [12,13] and experiments [2,14], where a sufficient range of scales was available to form the cascades. If the frictional dissipation is weak so that ζ exceeds the system size L then ultimately the "condensate" regime emerges [3,7] where the standard KLB does not apply.One of the primary motivations for studying 2D turbulence is that it is structurally and phenomenologically similar to quasi-geostrophic turbulence [15,16] which describes planetary atmospheres [17]. In addition, a recent resurgence i...

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