Observations show substantial variations of the intensity of tropical and/or summertime deep convection on land that are not explained by standard measures of convective instability. One feature that distinguishes land surfaces is their heterogeneity. The possible importance of this is investigated here by calculating the response of a nonrotating atmosphere to localized, transient surface heating using both the linearized equations of motion and a cloud-resolving configuration of the Weather Research and Forecasting (WRF) numerical model with moist physics, each in 2D. Both models predict that the depth of the resulting surface heat low near storm center will be greatest for a particular horizontal scale of heating. The linear model reveals that this is a resonant scale determined by the product of the environmental buoyancy frequency, characteristic heating time scale, and thickness of the thermal boundary layer, and the resonance occurs when the aspect ratio of the applied heating matches the ratio of vertical and horizontal wavenumbers demanded by the dispersion relation for buoyancy (gravity) waves. For realistic conditions, the resonant horizontal scale is roughly 50 km. The numerical model indicates that other measures of convective intensity, such as updraft speed and storm height, are largely controlled by the depth of the heat low, despite the presence of conditional instability and the vigorous growth of moist convective plumes. Predictions here agree with reported observations of storm severity over islands of different sizes. These findings may help explain why observed geographical variations in storm intensity defy parcel theory and indicate that phenomena often attributed to parcel entrainment may instead be due largely to storm-scale dynamical constraints.