Interchange turbulence in two dimensions is investigated in the scrape-off layer (SOL) of fusion devices, when driven by a constant core particle influx. Contrary to the standard gradient-driven approach, density is allowed to fluctuate around its average profile. Transverse transport exhibits some of the features of self-organized critical systems, namely inward and outward avalanches, together with a frequency spectrum decrease in 1/f and f(-2) at intermediate and high frequencies, respectively. An avalanche occurs when the local radial density gradient exceeds the critical one. A self-sustained particle flux then follows the large radial structures of the electric potential. As observed experimentally, the radial profile of density relative fluctuations decreases from the wall into the core plasma, while that of electric potential relative fluctuations peaks inside the SOL. Equilibrium density exhibits the experimental exponential decrease. An analytical expression of the SOL width Delta(SOL) is obtained, which maximizes the linear growth rate, when the poloidal modulation of electric potential equilibrium is taken into account. The parametric dependencies of Delta(SOL) are compared to experimental data. (C) 1998 American Institute of Physics. [S1070-664X(98)02312-X].