The cascade model of Desnyansky and Novikov is extended to include energy flow inward as well as outward from a givenk‐space shell. The stationary solutions for the energy spectrum are of the 1941 Kolmogoroff form, &egr;2/3k−5/3f(k/kd), but the scaling functionf(x) has a nonanalytic dependence on a parameterCof the model which represents the relative strength of inward and outward energy flow inkspace. ForC<1,f(0) is finite, and the energy dissipation rate &egr; is nonzero in the limit of zero viscosity. ForC≳1,f(x) behaves asx−&zgr;for smallxwith &zgr;=2 logC/log 2. This leads to an inertial range exponent for the energy spectrum of 5/3+&zgr;, and to an &egr; (&ngr;) which goes slowly to zero as the viscosity, &ngr;, goes to zero. The model may play the role of a mean field theory for strong turbulence in which fluctuations leading to intermittency are neglected. The parameterCwould then be a smooth function of the spatial dimensionality. A comparison with other recent work indicates that the crossover phenomenon whenC=1 occurs between two and three dimensions.