Vertical Discontinuities in Self-Affine Surfaces Lead to Multi-affinity

Many systems of both theoretical and applied interest display multi-affine scaling at small length scales. We demonstrate analytically and numerically that when vertical discontinuities are introduced into a self-affine surface, the surface becomes multi-affine. The discontinuities may correspond to surface overhangs or to an underlying stepped surface. Two surfaces are numerically examined with different spatial distributions of vertical discontinuities. The multi-affinity is shown to arise simply from the surface of vertical discontinuities, and the analytic scaling form at small length scales for the surface of discontinuities is derived and compared to numerical results.