Universal stirring force in the theory of isotropic turbulence

We consider the Navier-Stokes (Euler) equation ($0\leftarrow \frac{1}{L}<k<\mathrm{}\infty$) with an arbitrary pumping acting in the region $k\simeq \frac{1}{L}\to 0$. It is shown that after being projected onto a smaller space $\frac{1}{L}<<k_0<k<\mathrm{}\infty$ this model is equivalent to the one with random stirring [P. C. Martin and C. DeDominicis, Suppl. Prog. Theor. Phys. 64, 108 (1978) C. DeDominicis and P. C. Martin, Phys. Rev. B 11, 4077 (1975) and V. Yakhot, Phys. Rev. A 23, 1486 (1981)], and yields the energy spectrum $E\left(k\right)=\frac{a{k}^{-\frac{5}{3}}}{\ln \mathrm{kL}}$.