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Electric field noise is a major source of motional heating in trapped ion quantum computation. While the influence of trap electrode geometries on electric field noise has been studied in patch potential and surface adsorbate models, only smooth surfaces are accounted for by current theory. The effects of roughness, a ubiquitous feature of surface electrodes, are poorly understood. We investigate its impact on electric field noise by deriving a rough-surface Green's function and evaluating its effects on adsorbate-surface binding energies. At cryogenic temperatures, heating rate contributions from adsorbates are predicted to exhibit an exponential sensitivity to local surface curvature, leading to either a large net enhancement or suppression over smooth surfaces. For typical experimental parameters, orders-of-magnitude variations in total heating rates can occur depending on the spatial distribution of absorbates. Through careful engineering of electrode surface profiles, our results suggests that heating rates can be tuned over orders of magnitudes., Comment: 12 pages, 5 figures