By adding a weak magnetic field to a spherically symmetric fluid configuration that caricatures a stalled shock in the post-bounce supernova environment, we explore the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields. The SASI develops upon perturbation of the initial condition, and the ensuing flow generates--{em in the absence of rotation}--dynamically significant magnetic fields ($sim 10^{15}$ G) on a time scale that is relevant for the explosion mechanism of core-collapse supernovae. We describe our model, present some recent results, and discuss their potential relevance for supernova models.