  In this paper (Shivamoggi et al.), we explore a variant for the simple model
based on a binomial multiplicative process of Meneveau and Sreenivasan that
mimics the multi-fractal nature of the energy dissipation field in the inertial
range of fully developed turbulence (FDT), and uses the generalized fractal
dimension (GFD) prescription of Hentschel and Proccacia, Halsey et al. However,
the presence of an even infinitesimal dissipation in the inertial range is
shown to lead to a singularity in the GFD $D_q$ (at $q=1$) of the energy
dissipation field and leads to a breakdown of the Meneveau-Sreenivasan binomial
multiplicative formulation for a dissipative inertial cascade. The purpose of
this paper is to demonstrate that this can be resolved by introducing a new
appropriate ansatz for the definition of the GFD $D_q$ to incorporate the
effect of a scale-invariant dissipation via a phenomenological dissipative
parameter $K$ $( 0 < K < 1)$. This dissipation parameter is also shown to cause
a steeper energy spectrum in the inertial range, as to be expected. This ansatz
is then generalized to incorporate a more symmetric dissipation via two
dissipative parameters $K_1,$ and $K_2$ $( 0 < K_1$ and $K_2 < 1)$
