Discontinuous Transition from Direct to Inverse Cascade in Three-Dimensional Turbulence

Abstract

Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three-dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflections, and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helical Fourier modes. The transition from a forward to inverse cascade is shown to occur at a critical point in a discontinuous manner with diverging fluctuations close to criticality. Our work thus supports the observation that purely isotropic and three-dimensional flow configurations can support inverse energy transfer when interactions are altered and that inside all turbulent flows there is a competition among forward and backward transfer mechanisms which might lead to multiple energy-containing turbulent states.

Closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence

“…For k < k f , the helical flux is zero and the kinetic one Π E is negative, showing a stable inverse cascade of energy for scales larger than the forcing one. Note that the shape of Π E is qualitatively in agreement with the one of [15] (figure 3 therein, curve marked with squares). Here we show for the first time in a clean way that the inverse transfer has a non trivial wavy shape around the front.…”

“…They consequently recovered the second scenario of [3], namely an inverse cascade of kinetic energy and a forward cascade of helicity, showing that all three dimensional turbulent flows indeed possess a sub-set of Fourier interactions potentially able to sustain an inverse energy cascade. Such a "surgery" of the Navier-Stokes equations was thoroughly investigated in multiple subsequent works [14][15][16] so that the details are not recalled here. The main findings are that by forcing at small scales the decimated Navier-Stokes (dNS) equations where helicity is made sign-definite, say positive here, kinetic energy is transferred to smaller wavenumbers with E(k) ∼ ǫ 2/3 k −5/3 .…”

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.…”

Energy cascade and intermittency in helically decomposed Navier–Stokes equations

“…Earlier studies have shown that starting from an homochiral Navier-Stokes simulation and by adding modes with the opposite sign of helicity, leads to a transition from inverse to direct energy cascade [20,33,34]. The transition is different, depending on the protocol used to add heterochiral interactions [10,20]. Unfortunately, the only way to study all potentially different triadic families is to recover to a fully spectral code [35] with the consequential limitations in the computational applications.…”