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It is impractical to create gridded numerical models of coastal circulation with sufficient resolution around small topographic features, such as submarine canyons, and still have the alongshore boundaries placed beyond the decay distance of coastal trapped waves. Two solutions to this problem are to make the alongshore boundaries either open or periodic. Numerical simulations were performed with upwelling and downwelling winds to compare the effects of these different choices for boundary conditions. Several open boundary formulations were tried and three are discussed in detail. The offshore boundary was specified as "no gradient'' for all variables with no serious effect. The "modified'' Orlanski radiation condition is used for all variables at the alongshore boundaries, except the vertically integrated flow that has the strongest effect on the model solution. An alongshore pressure gradient, opposing the wind, develops in the model if the modified Orlanski radiation condition is applied to the barotropic flow, causing slower currents near the surface and deep undercurrents away from the shelf. The other cases, which combined either a radiation or a relaxation boundary condition with a local solution of the barotropic equations on the boundary, were at least initially similar to the periodic case but with slower alongshore flow. The initial impact of these differences on the circulation within the canyon was small. The models with the open boundaries were more stable (did not develop strong flow meanders) than the cases with periodic conditions as initial transients are not trapped, and amplified, within the domain. Thus, open cases, especially with the upwelling winds, could run for extended times.