A reduced circuit description of the U-shaped potential structure of a discrete auroral arc, consisting of the flank transmission line plus parallel-electric-field region, is used to provide the boundary condition for 1-d simulations of the double-layer evolution. The model yields asymptotic scalings of the double-layer potential, as a function of an anomalous transport coefficient &agr; and of the perpendicular length scale &phgr;a of the arc. The arc potential ϕDL scales approximately linearly with &agr;, and for &agr; fixed ϕDL ~&phgr;az. Using parameters appropriate to the auroral-zone acceleration region, potentials of ϕDL~10 kV scale to projected ionospheric dimensions ~1 km, with power flows of the order of magnitude of substorm dissipation rates.