Current sheets having a transverse scale near the electron inertial length c/&ohgr;p are shown to be unstable. Two general classes of modes are found to occur, the single and double tearing modes and a current shear driven current-convective interchange mode. The tearing modes are global modes that relax the equilibrium to a lower magnetic energy state. The current-convective interchange mode is localized near points at which k⋅B0=0. The interchange mode has features in common with the collisionless tearing modes but is a local rather than global instability. An extensive analysis of the linear stability is presented, and the results are confirmed from linear initial value and eigenvalue calculations. Nonlinear three-dimensional initial value simulations are performed to determine the effects of the mode on the equilibrium. The ΔE/ΔB spectrum of the field fluctuations is computed and compared to satellite observations. The results support the hypothesis that short-scale broadband extremely low-frequency (BBELF) waves associated with Alfv¿nic structures are possibly the result of current decay through the electromagnetic current-convective instability. ¿ 2001 American Geophysical Union