Inverse energy cascades in three-dimensional turbulence

Abstract

Three-dimensional (3D) turbulence is characterized by a dual forward cascade of both kinetic energy and helicity, a second inviscid flow invariant, from the integral scale of motion to the viscous dissipative scale. In helical flows, however, such as strongly rotating flows with broken mirror symmetry, an inverse energy cascade can be observed analogous to that of two-dimensional turbulence (2D) where a second positive-definite flow invariant, enstrophy, unlike helicity in 3D, effectively blocks the forward cascade of energy. In the spectral-helical decomposition of the Navier-Stokes equation it has previously been show that a subset of three-wave (triad) interactions conserve helicity in 3D in a fashion similar to enstrophy in 2D, thus leading to a 2D-like inverse energy cascade in 3D. In this work, we show both theoretically and numerically that an additional subset of interactions exist conserving a new pseudo-invariant in addition to energy and helicity, which contributes either to a forward or inverse energy cascade depending on the specific triad interaction geometry. Fully developed three-dimensional (3D) turbulence is char-acterised by a forward cascade of kinetic energy from the large integral scale of motion to the small Kolmogorov scale of viscous dissipation. In the large Reynolds number limit, η → 0, the production of enstrophy, the integral of the vor-ticity squared, by the stretching and bending term in the in-compressible Navier-Stokes equations (NSE) permits the vis-cous dissipation of energy at the Kolmogorov scale. In two-dimensional (2D) turbulence, the stretching and bending term is absent, and enstrophy is, in addition to energy, also an invis-cid invariant [1]. In this case, the dissipation of enstrophy prevents a simultaneous dissipation of energy at the Kolmogorov scale, effectively blocking the forward cascade of energy. The dual inviscid conservation of both quantities, E(k)dk and k 2 E(k)dk, the integrals over the spectral energy and enstro-phy densities, consequently implies a reversal of the energy cascade to larger scales (inverse cascade). In 3D turbulence, helicity, the integral of the scalar product of velocity and vor-ticity, is also an inviscid invariant [2]. Similarly to the enstro-phy spectrum, the helicity spectrum H(k) ∼ kE(k) dominates over the energy spectrum at small scales (large k), but unlike enstrophy, helicity is not sign definite. As a consequence, the increased dissipation of both signs of helicity compared to energy can be obtained without a net helicity production as long as the dissipation of both positive and negative helicities balance [3]. Inviscid conservation of helicity does therefore not prevent a forward cascade of energy [4]. In helical flows, however, such as strongly rotating flows with broken mirror symmetry, a simultaneous forward helic-ity cascade and inverse energy cascade can be observed [5]. In the spectral decomposition of the NSE, energy and helicity (and enstrophy in 2D) are conserved within each three-wave interaction (triad interaction). ...

Theory and simulation of real and ideal magnetohydrodynamic turbulence

“…Tile predicted value <EM> = 0.515834 differs from the time average in Table 3 by more than 10 standard deviations. Thus, there a small but si gnificant departure ill simulation results from the predictions of the absolute equilibrium ensemble theory [7,10,16,17]. Figure 2 indicates that EK IEM increase with decreasing PM, i.e., with increasing rl.…”

“…The initial formulation of the absolute equilibrium theory, developed in the 1970s [7,10,16,17] has been extended in several ways [30][31][32][33][34][35]. These extensions and their uses are described in the following sections.…”

“…Thus, we have an inverse cascade [ 10], in which energy flows to low k modes at the same time it is flowing to high k modes where it is lost. This can be quantified by plotting modal energy E(k) versus time, as is done in Figures 8 for E(k), k = 1; in Figure 9 for EM(k), k = 1; and in Figure 10, for E(k), k = 5.…”

“…Since D-b = V j = V-u = V-w = 0, we can define for 2-D MHD the following: (10) a) u= Vxtp=-kXVvf b) b= Vxa=-e,xVa.…”

“…In particular, we will consider both ideal (non-dissipative) and real (dissipative) 2-D MHD turbulence. In the ideal case, there is a statistical theory with which to compare numerical results [7,10,16,17,[30][31][32][33][34][35][36][37], and in the real cases we compare the distribution of energy with inertial decay laws [14,24] and with the distribution found in the ideal case. However, we do not add energy through spectral forcing to the real case, so that the dissipative simulations are autonomous dynamical systems [38] in which energy decays with time.…”