The breakup of discrete auroral arcs on closed field lines is modeled by numerical calculations of nonlinear magnetohydrodynamics in a cavity. The boundary conditions are shown to be crucially important in determining the available mechanisms for nonlinear evolution and breakup of the arc. Nonlinear evolution is inhibited if the sign of the electric field is reversed in a reflecting Alfv¿n wave. The weaker the shear time of the arc compared with the Alfv¿n wave travel time across the cavity, the more strongly the nonlinear evolution is inhibited. Because different transverse length scales have different reflection properties related to their ability to pass through the auroral acceleration region, arcs with small horizontal scales are subject to the Kelvin-Helmholtz shear flow instability while large horizontal scales are not. Parallels are drawn to the observed breakup phenomenology of small-scale and large-scale arcs. ¿ 1999 American Geophysical Union