Magnetohydrodynamic solutions are presented for the problem of time-dependent magnetospheric convection. They include regions of both taillike and dipolelike flux tubes and the interaction and flux transfer between them. The only external input is the rate of dayside merging. The problem is solved as a free boundary problem for both the high-latitude boundary between taillike and dipolelike tubes and the low-latitude boundary defined by plasma penetration. A constraint of no equatorward component to the convection is used to resolve the &bgr; problem in the region where the plasma sheet is on dipolar field lines. This constraint imposes a three-dimensional nightside topology and determines the boundaries and the flux transfer rate into the nightside dipolar region. It also leads to a major asymmetry in the rate at which the amount of taillike flux can increase or decrease and hence accounts for the existence of the substorm phases. The narrow-channel approximation, developed a number of years ago by the author, is used to obtain analytic solutions. Extensive comparison of the results of the theory with the observed aspects of convection and substorms is carried out. Values predicted by the model are in agreement with observations, and most major properties of the substorm are explained, including the existence of activity at two separated latitudes in the recovery phase.