The resonant absorption of Alfv¿n waves in the plasma sheet boundary layer (PSBL) provides a significant source of energy for the heating of plasma sheet particles. We show that the energy absorbed is a function of the central plasma sheet (CPS) temperature. The heating curve when coupled with convective transport yields an equation of state for the steady state plasma sheet whose solution has the form of a mathematical catastrophe. We have previously suggested that this catastrophe is associated with the abrupt increase of energy dissipation during the onset of the substorm expansion phase. In this paper our master equation is generalized to include convection velocity Vx self-consistent with pressure anisotropy, and to retain the dynamics describing the transition across the thermal catastrophe. The dynamic terms allow evaluation of the time scale for the catastrophe to occur. The evolution of the plasma sheet through the growth phase to onset is traced in the quasi-static limit, assuming that the system passes slowly through a succession of equilibrium states described by the stationary limit of the master equation. The state variable for the plasma sheet is the temperature T; the control variables are the evolving lobe field Bl and the incident power flux, represented by W~b2, where b is the amplitude of the driving waves at the surface of the PSBL. Other parameters are reduced to initial conditions and then scale self-similarly with the evolving Bl. For a physically reasonable range of W, catastrophe occurs as Bl increases above a critical value, and in a relatively narrow range of local time, nominally 2200<LT<2400. The power dissipated after onset by resonant absorption is of order 109 W/R2e. The quasi-static evolution does not predict the gradual recovery phase; we discuss a possible roˆle in recovery for the formation of a second neutral line. ¿ American Geophysical Union 1989