A new method of operating an inertial electrostatic confinement (IEC) device is proposed, and its performance is evaluated. The scheme involves an oscillating thermal cloud of ions immersed in a bath of electrons that form a harmonic oscillator potential. The scheme is called the periodically oscillating plasma sphere, and it appears to solve many of the problems that may limit other IEC systems to low gain. A set of self-similar solutions to the ion fluid equations is presented, and plasma performance is evaluated. Results indicate that performance enhancement of gridded IEC systems such as the Los Alamos intense neutron source device is possible as well as high-performance operation for low-loss systems such as the Penning trap experiment. Finally, a conceptual idea for a massively modular Penning trap reactor is also presented.