The joint cascade of energy and helicity in three-dimensional turbulence

Abstract

Three-dimensional (3D) turbulence has both energy and helicity as inviscid constants of motion. In contrast to two-dimensional (2D) turbulence, where a second inviscid invariant—the enstrophy—blocks the energy cascade to small scales, in 3D there is a joint cascade of both energy and helicity simultaneously to small scales. It has long been recognized that the crucial difference between 2D and 3D is that enstrophy is a nonnegative quantity whereas the helicity can have either sign. The basic cancellation mechanism which permits a joint cascade of energy and helicity is illuminated by means of the helical decomposition of the velocity into positively and negatively polarized waves. This decomposition is employed in the present study both theoretically and also in a numerical simulation of homogeneous and isotropic 3D turbulence. It is shown that the transfer of energy to small scales produces a tremendous growth of helicity separately in the + and − helical modes at high wave numbers, diverging in the limit of infinite Reynolds number. However, because of a tendency to restore reflection invariance at small scales, the net helicity from both modes remains finite in that limit. Since energy and helicity are not separately conserved in the + and − modes, there are four “fluxlike” quantities for both invariants, which correspond to transfer either out of large scales or into small scales and either to + helical or to − helical modes. The helicity fluxes out of large scales in the separate + and − channels are not constant in wave number up to the Kolmogorov dissipation wave number kE but only up to a smaller wave number kH, recently identified by Ditlevsen and Giuliani [Phys. Fluids 13, 3508 (2001); Phys. Res. E 63, 036304 (2001)]. However, contrary to their argument, the net helicity flux is shown to be constant all the way up to the Kolmogorov wave number: there is no shorter inertial range for helicity cascade than for energy cascade. The transfer of energy and helicity between + and − modes, which permits the joint cascade, is shown to be due to two distinct physical processes, advection and vortex stretching.