Inverse Energy Cascade in Three Dimensional Isotropic Turbulence

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Inverse Energy Cascade in Three-Dimensional Isotropic Turbulence Luca Biferale,1 Stefano Musacchio,2 and Federico Toschi3 1Department of Physics & INFN, Universita Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 2CNRS, Laboratoire J. A. Dieudonne UMR 7351, Parc Valrose, 06108 Nice, France 3Department of Physics and Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands & CNR-IAC, Via dei Taurini 19, 00185 Rome, Italy (Received 2 November 2011; published 20 April 2012) We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature.

  • kmax ?

  • defined sign

  • positive helicity

  • energy cascade

  • ns equations

  • scale forcing

  • helicity

  • energy transfer

  • interaction between

  • truncated ns


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PRL108,164501 (2012)
P H Y S I C A LR E V I E WL E T T E R S
week ending 20 APRIL 2012
Inverse Energy Cascade in ThreeDimensional Isotropic Turbulence 1 23 Luca Biferale,Stefano Musacchio,and Federico Toschi 1 Department of Physics & INFN, Universita` Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 2 CNRS,LaboratoireJ.A.Dieudonn´eUMR7351,ParcValrose,06108Nice,France 3 Department of Physics and Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands & CNR-IAC, Via dei Taurini 19, 00185 Rome, Italy (Received 2 November 2011; published 20 April 2012) We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics. DOI:10.1103/PhysRevLett.108.164501PACS numbers: 47.27.i
Inviscid invariants of the Navier-Stokes (NS) equations are crucial in determining the direction of the turbulent energy transfer [1]. In some cases, as for fully isotropic and homogeneous turbulence in 2D, the presence of two posi-tively defined invariants (energy and enstrophy) does not allow a stationary transfer of both quantities, neither to small nor to large scales [2]. In the presence of two fluxes, they must necessarily flow in opposite directions [37] and this remains true even for turbulent systems in noninteger dimensions obtained by fractal Fourier decimation [8]. The fluid equations also possess two inviscid invariants in 3D: energy and helicity (i.e., the scalar product of velocity and vorticity). The inviscid conservation of helicity was dis-covered relatively recently [9,10]. At variance with energy, helicity is not positively defined. This allows for a simul-taneous small-scale transfer of energy and helicity, as confirmed by the results of two-point closures [1012] and direct numerical simulations [13,14]. Nevertheless, a reversal of the flux of energy has been observed in geo-physical flows subject to the Earth’s rotation [15,16] as well as in shallow fluid layers [1722]. In both cases, this phenomenon is accompanied by strong anisotropic effects and by a substantial two-dimensionalization of the flow, induced either by the rotation or by the effects of confine-ment. Moreover, rotations inject fluctuations into the hel-ical sector while a perfect two-dimensional flow has vanishingpointwisehelicity, vorticity always being or-thogonal to velocity. Here, we rationalize these findings, showing that inverse energy transfer is much broader than previously thought and is present in all flows in nature. In order to highlight this mechanism, we investigate in detail
0031-9007=12=108(16)=164501(5)
the transfer properties of NS equations in 3D homogeneous systems at changing the nature of the triadic nonlinear interactions. We show that an inverse energy cascade oc-curs also in 3D isotropic flow whenever parity invariance is broken and helicity acquires a well-defined sign at all wave numbers. The key new idea is to make a suitable surgery of the NS equations, such as to disentangletriad by triadthe properties of the nonlinear energy transfer. In particular, we show that the energy flux is always reversed by keeping only triadic interactions between sign-defined helical modes, preserving homogeneity and isotropy and breaking reflection invariance. The role played by helicity in the energy transfer mechanism of 3D flows has attracted a broad scientific interest (see, e.g., [14] and references therein). Dynamical systems have been developed to study in detail energy and helicity transfer at high Reynolds numbers [23,24]. Further, speculations connecting the ex-istence of intermittent bursts in the energy cascade induced by a ‘‘local’’ helicity blocking mechanism have been pro-posed [23]. Despite these important contributions, the understanding of the phenomenology of helicity remains ‘‘mysterious,’’ as summarized in the conclusion of a recent state-of-the-art numerical study [14]. Here, we present theoretical and numerical evidence of a new phenomenon induced by helicity conservation: a statistically stationary backwardenergy transfer can be sustained even in 3D fully isotropic turbulence. The starting point of our analysis is the well-known helical decomposition [12] of the velocity fieldvðxÞ, expanded in a Fourier series,uðkÞ: þ þ  uðkÞ ¼uðkÞhðkÞ þuðkÞhðkÞ;(1)
164501-1
2012 American Physical Society