A consequence of the zero fourth cumulant approximation

Recent investigations by Kraichnan (1961) and Ogura (1961) have raised doubts concerning the usefulness of the zero fourth cumulant approximation in turbulence dynamics. It appears extremely tedious to examine, by numerical computation, the consequences of this approximation on the turbulent energy spectrum although the appropriate equations have been established by Proudman & Reid (1954) and Tatsumi (1957). It has proved possible, however, to compute numerically the sequences of an analogous assumption when applied to an isotropic passive scalar in isotropic turbulence. The result of such computation, for specific initial conditions described herein, and for stationary turbulence, is that the scalar spectrum does develop negative values after a time approximately $2 \Lambda | {\overline {(u^2)}} ^{\frac {1}{2}}$, Where Λ is a length scale typical of the energy-containing components of both the turbulent and scalar spectra and $\overline {(u^2)}^{\frac {1}{2}}$ is the root mean square turbulent velocity.