The statistics of solutions to a family of one-dimensional random-force-driven advection-diffusion equations is studied using high resolution numerical simulations. The equation differs from the usual Burgers equation by the non-local form of the nonlinear interaction term mimicking the non-locality of the Navier–Stokes equation. It is shown that under an appropriate choice of random forcing the statistical properties of the solution (energy spectrum and scaling exponents of structure functions) coincide with those of Kolmogorov turbulence. Also, a generalization is proposed which allows intermittency effects to be modeled. ©1997 American Institute of Physics.