The Joint Cascade of Energy and Helicity in Three-Dimensional Turbulence

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 wavenumbers, diverging in the limit of infinite Reynolds number. However, because of a tendency to restore