Helicity cascades in fully developed isotropic turbulence

Based on total helicity conservation in inviscid incompressible flows, the existence of simultaneous energy and helicity cascades is envisaged.

Kolmogoroff law for the energy spectrum of isotropic turbulence in the inertial range: E(k)ro..; t213k-518, where E is the rate of transfer of energy.
In this note we shall be concerned with the dynamics of the turbulent flow itself when helicity is present. This fully nonlinear problem has not been given much attention (cf., however, Refs. 3 and 4). Since there is presently no satisfactory theory of fully developed turbulence starting from first principles, we shall examine the influence of helicity from a phenomeno logical viewpoint.
The classical phenomenological theory uses as a starting point the conservation of the total energy E= H v2 dx dy dz (1) for inviscid :flows and the assumption that the non linear terms in the N a vier-Stokes equations give rise to an energy transfer from the large eddies to the smaller ones through a local energy cascade. Under these assumptions, a well-known argument leads to the There is, however, another conserved quadratic quantity in inviscid incompressible three-dimensional flows, namely, the total helicity2 H=fv·curlvdxdydz.
(2) Indeed, assuming that the velocity field vanishes at infinity and using the identity J A·curlB dx dy dz= f B·curlA dx dy dz To the conserved helicity one may associate a helicity spectrum H(k), where H(k) dk is the contribu tion from wavenumbers between k and k+dk to the mean helicity per unit volume (v•curlv). H(k), unlike E(k), is a pseudoscalar and not definite positive; however, using the definite-positiveness of the spectral velocity autocorrelation tensor, it may be shown that The situation is somewhat reminiscent of two dimensional turbulence where both the mean energy and mean-square vorticity are conserved; for two dimensional turbulence this leads to two distinct possibilities: an energy cascade and an ens trophy cascade, the energy spectrum following, respectively, a k-513 and a k-3 law .5•6 It must be stressed, however, that in three dimensions energy can be present without helicity whereas in two dimensions energy and enstrophy are always present simultaneously.
In view of the conservation of helicity, it is interesting to look for the possibility of a helicity cascade in three-dimensional turbulence. If there is a helicity inertial range, where E(k) and H(k) depend only on k and the rate of transfer of helicity 71, the energy and helicity spectra must, for dimensional reasons, take the form7 (8) The possibility of helicity cascades corresponding to local helicity transfer will now be investigated more closely using the simple dynamical argument of Kraichnan.6 Let Il(k) and 2:(k) denote the total rate of energy and helicity transfer from all wavenumbers <k to all wavenumbers> k. Following Kraichnan, we assume that II ( k) [2: ( k) ] is proportional to the ratio of the total energy (helicity) ,... ., kE(k) [,... ., kH(k)] available in wavenumbers of the order of k to some effective distortion time r(k) of flow structures of scale k-1 due to the shearing action of all wavenumbers <;; k.
The distortion time, which is the same for energy and helicity transfer, is given by6 r(k)"" (�k p2E(p) dp r11 2 We therefore write It is easily checked that the dominant contribution to r(k) comes from P""k in agreement with the idea of localness of both energy and helicity transfer. Notice that solution (11) is compatible with inequality (7) only for large enough k.
However, a problem arises with a pure helicity cascade: it appears difficult to inject helicity into the fluid without at the same time injecting some energy. Possibly this difficulty can be overcome, as for two dimensional turbulence, by assuming that energy and helicity are fed into the fluid at a certain wavenumber k;; helicity then cascades toward large wavenumbers according to (8) while energy cascades toward small wavenumbers (inverse cascade) according to the usual Kolmogoroff law. In the energy inverse cascade range, H(k) is proportional to e2'3k-2i3 and 2:(k)-+0 when k�, so there is no helicity transfer.
It may be asked under what conditions solutions of case (b) if they exist, do arise. When the injection of energy and helicity occur at a fixed wavenumber k;, a possible answer is that case (b) solutions appear when the helicity injection rate exceeds the critical value which is required by the compatibility of (7) and (11) for k= k;, above which stationary solutions of case a cannot exist anymore.
Since all the considerations in this note are purely phenomenological and cannot give conclusive evidence for the existence of helicity cascades, especially of case (b), we intend to test them numerically on a model of turbulence which leads to closed equations for the evolution of E(k) and H(k). As for experimental evidence, we believe that atmospheric turbulence may be a good place to look, because the rotation of the earth combined with the gradients of turbulent inten sity produces helicity.
It is a pleasure for us to thank Dr.