Surface self-organized irregular and periodic structures such as ripples, cones, and bubbles were frequently observed on rf-cavity surfaces subjected to vacuum breakdown development, on the first wall on the Tokamak chamber, on the glass surfaces irradiated by laser beams, on the satellite surfaces bombarded with micro-meteorites and they are subject of our detailed study via experiment and computer simulation. Scanning Electron Microscope (SEM) studies of rf cavity copper surface arc pits were conducted that show that the surface is covered with irregular and regular ripple structures with a characteristic length of 2 microns.  In addition, a variety of sharp edges, corners and cracks in the surface were observed.  This paper studies the field enhancements that would be present in these edges, corners and crack systems, and how they might function as field emitters by using atomistic and finite element simulations.  We describe numerical finite-element calculations of realistic geometries that produce high enhancement factors and are consistent with experimental data.  We find that a variety of structures can produce the expected values of enhancement factors, although with small surface areas, so that a number of sources must contribute. Atomistic mechanisms and a new plasma model of the high-gradient breakdown in rf cavities were developed that can explain most of the existing experimental data.