Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

Abstract

We investigate the transfer properties of energy and helicity fluctuations in fully developed homogeneous and isotropic turbulence by changing the nature of the nonlinear Navier-Stokes terms. We perform a surgery of all possible interactions, by keeping only those triads that have sign-definite helicity content. In order to do this, we apply an exact decomposition of the velocity field in a helical Fourier basis, as first proposed by Constantin & Majda (Commun. Math. Phys, vol. 115, 1988, p. 435) and exploited in great detail by Waleffe (Phys. Fluids A, vol. 4, 1992, p. 350), and we evolve the Navier-Stokes dynamics keeping only those velocity components carrying a well-defined (positive or negative) helicity. The resulting dynamics preserves translational and rotational symmetries but not mirror invariance. We give clear evidence that this three-dimensional homogeneous and isotropic chiral turbulence is characterized by a stationary inverse energy cascade with a spectrum E back (k) ∼ k −5/3 and by a direct helicity cascade with a spectrum E forw (k) ∼ k −7/3 . Our results are important to highlight the dynamics and statistics of those subsets of all possible Navier-Stokes interactions responsible for reversal events in the energy-flux properties, and demonstrate that the presence of an inverse energy cascade is not necessarily connected to a two-dimensionalization of the flow. We further comment on the possible relevance of such findings to flows of geophysical interest under rotations and in thin layers. Finally we propose other innovative numerical experiments that can be achieved by using a similar decimation of degrees of freedom.

Dimensional transition in rotating turbulence

“…Such mechanism is consistent with the previous findings reported in [39,40], which showed that an inverse energy cascade develops also in homogeneous isotropic 3D turbulent flow when mirror symmetry is broken such that helicity has a well-defined sign at all wave numbers.…”

Cascades of energy and helicity in axisymmetric turbulence

“…The present work is further timely since a number of works have been recently dedicated to the investigation of the dynamics of 3D turbulence in which only a part of the helical modes is retained [35,36], or in which the weight of the different helical contributions is artificially modified [39]. It was shown that in such a triadically modified flow, the energy cascades towards the large scales, and helicity towards the small ones.…”

“…However, helicity can always be finite locally, and an arbitrary velocity field can be written as the sum of helical modes [31], a decomposition used for instance to study the role of helicity using shell models [32][33][34]. Recent studies have considered Navier-Stokes turbulence where artificially only helical modes of a single polarity were retained [35,36]. In these flows, an inverse cascade occurs, even in the 3D isotropic case.…”

Note on the triad interactions of homogeneous turbulence

“…There is no doubt that triad truncation has intrinsic interest, just as do other truncations like standard Galerkin truncation, or truncations based on helical modes, as adopted by Cambon & Jacquin (1989), Waleffe (1992) and more recently Biferale, Musacchio & Toschi (2013). The special interest of triad truncation lies in the fact that evolution under this truncation can be explicitly calculated and compared with evolution under the Euler equations starting from the same initial conditions.…”